A Memetic Algorithm for Multiskill Resource-Constrained...
Transcript of A Memetic Algorithm for Multiskill Resource-Constrained...
Research ArticleA Memetic Algorithm for Multiskill Resource-ConstrainedProject Scheduling Problem under Linear Deterioration
Huafeng Dai 12 andWenming Cheng12
1School of Mechanical Southwest Jiaotong University 610031 Chengdu China2Technology and Equipment of Rail Transit Operation andMaintenance Key Laboratory of Sichuan Province 610031 Chengdu China
Correspondence should be addressed to Huafeng Dai annmyswjtueducn
Received 7 May 2019 Accepted 20 June 2019 Published 4 July 2019
Academic Editor Giovanni Berselli
Copyright copy 2019 Huafeng Dai and Wenming Cheng This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
This paper proposes a general variable neighborhood search-based memetic algorithm (GVNS-MA) for solving the multiskillresource-constrained project scheduling problem under linear deterioration Integrating a solution recombination operator and alocal optimization procedure the proposed GVNS-MA is assessed on two sets of instances and achieves highly competitive resultsOne set of benchmark instances is commonly used in the literature where the capability of the proposed algorithm to find highquality solutions is demonstrated compared with the state-of-the-art algorithms in the literatureThe other set revises the formerthrough incorporating the linear deterioration effect Two key components of the proposed algorithm are investigated to confirmtheir critical role to the success of the proposed method
1 Introduction
Scheduling is a form of decision-making that plays an essen-tial role in manufacturing and service industries A functionthat assigns tasks to resources to complete the project canbe formulated as an informal definition of the schedulingproblem However such simplification is not enough to coverall real-world applications with a variety of complex environ-ments Take the task duration for example the processingtime of any task is fixed and constant in classical schedulingtheory and system [1] while a hypothesis that the actualduration of the task is a linearly nondecreasing function ofits starting time is put forward by Gupta and Gupta [2] incontrast to the cases found in traditional literature Fromthen on many approaches concerning various schedulingtypes accompanying with the phenomenon denominated asdeterioration effect [3] constantly spring up
Scheduling with deterioration is very common Forinstance in traditional manufacturing industries like thefurniture some tasks need to be processed by a carpenterand the carpenter may lower his machining speed graduallydue to fatigue In this case the later a task is handled the
longer the time it needs to complete For another examplein steel rolling mills ingots need to be heated to requiredtemperature before rolling Heating time relies on the ingotsrsquocurrent temperature which depends upon the time it hasbeen waiting to be heated It is because the ingot cools downconsequently requiring more heating time in the period ofwaiting Similar cases often occur inmanufacturing financialmanagement steel production medicine treatment and soon Scheduling deteriorating jobs was first considered byBrowne and Yechiali [4] who assumed that task process-ing times are nondecreasing start time dependent linearfunctions Since then researchers have devoted considerableefforts to this area and lots of remarkable papers have beenpublished Alidaee and Womer [5] as well as Cheng et al[6]made the comprehensive overviews of existing schedulingproblems with various deteriorating mechanisms
In addition to changes in the definition of duration timein terms of research hotpot in the field of scheduling projectscheduling problem (PSP) provided a set of precedence-constrained tasks to be scheduled aiming at minimizinga given objective Furthermore tasks need to compete forscarce resources additionally in the resource-constrained
HindawiMathematical Problems in EngineeringVolume 2019 Article ID 9459375 16 pageshttpsdoiorg10115520199459375
2 Mathematical Problems in Engineering
project scheduling problem (RCPSP) which make it possiblethrough a better adaption to apply in production planningproject management or manufacturing etc Ultimately moreadaptable multiskill resource-constrained project schedulingproblem (MS-RCPSP) gives each resource a set of capabilitiesand each resource can bid for being assigned tasks
Scheduling with deterioration and MS-RCPSP are tworesearch hotspots of scheduling area and these two subfieldsare not absolutely separate In terms of most tasks of actualMS-RCPSP deterioration effects existed for the reason offatigued (humans ormachines)However there is no researchconcentrated on the integration of MS-RCPSP and deteriora-tion
In this paper we pick the deterioration mechanism aslinear deterioration As for the researched MS-RCPSP withlinear deterioration which is dubbed MS-RCPSPLD thedifferences compared to the scheduling problems with lineardeterioration in the reported literature can be explainedhere Foremost the resource or the machine in the usuallyresearched scheduling problem with deterioration mostly isrestricted to be single-mode which means that it owns onlyone capability and can execute just a kind of task
Moreover as theRCPSP is proven to beNP-hard [7] thereis no optimal solution that could be computed in polynomialtime And this paper also demonstrates that the MS-RCPSPand the MS-RCPSPLD are NP-hard because they are moregeneral problems compared with the single-mode RCPSPHence methods that find feasible solutions which maynot achieve global optimal but can obtained in acceptabletime are built by researchers In such cases soft computingmethods are used mostly heuristics and metaheuristics
Within the metaheuristics group of methodologies itis noticed that memetic algorithm (MA) as a quite simpleapproach gives promising results in the fields of both com-puter science and operational research [8] MA representsone of the recent growing areas of research in evolutionarycomputation and is introduced by Moscato [9] inspired byboth Darwinian principles of natural evolution and Dawkinsrsquonotion of a meme With MA the traits of Universal Darwin-ism are more appropriately captured in particular dealingwith areas of evolutionary algorithms that marry otherdeterministic refinement techniques for solving optimizationproblems [10] As a general framework MA provides thesearch with desirable trade-off between intensification anddiversification through the combined use of a crossoveroperator to generate new promising solutions and a localoptimization procedure to locally improve the generatedsolutions [11 12]
In next stages to make MA more efficient it needs tobe supported by other metaheuristics eg general variableneighborhood search (GVNS) proposed by Hansen et al[13] GVNS is a simple and effective metaheuristic forcombinatorial optimization yield through systematic changeof neighborhoods within a refinement local search It canavoid becoming mired in local optima and develop the searchdirection to an excellent solution ultimately by means ofexploring new regions throughout the whole solution spaceThe GVNS has attracted attentions and shows remarkableperformance compared to other basic variable neighborhood
search (VNS) variants Up to now various optimization prob-lems have taken the GVNS to pursue better solutions suchas the single-machine total weighted tardiness problem withsequence dependent setup times [14] the one-commoditypickup-and-delivery traveling salesmanproblem [15] and thevehicle routing problem [16]
As far as we know there is no published work on solvingthe MS-RCPSP by the GVNS-MA By natural extension itmakes sense to solve the MS-RCPSPLD with the GVNS-MAThemain goal of this paper is to get an insight to the problemintegrating the MS-RCPSP with linear deterioration HowGVNS can be effectively hybridized with MA to solve MS-RCPSP and MS-RCPSPLD is examined in the GVNS-MAThe proposed approach is demonstrated to show its effective-ness in comparison to other state-of-the-art methods
The remaining part of this paper is organized as followsIn Section 2 a short summary of existing publications ispresented Section 3 gives the description of the problemunder consideration In Section 4 the proposedGVNS-MA iselaborated Section 5 shows performed experiments and theirresults before concluding the paper in Section 6 finally
2 Related Work
21 Deterioration Effect There are many practical situationsthat any delay or waiting in starting time of a task may causeto increase its processing time and two main deteriorationmechanisms are discussed
Besides usual linear deterioration mechanism discussedin this paper situations in which taskrsquos processing times arerepresented by step functions characterized by a sharp changein processing time at the deadline points [17ndash22] also arefairly common Sundararaghvan and Kunnathur [23] firstlyconsidered the corresponding single-machine schedulingproblem for minimizing the sum of the weighted completiontimes while Mosheiov [17] provided some simple heuristicsfor these NP-hard problems tominimize themakespan NextJeng and Lin [24] introduced a branch and bound algorithmfor the single-machine problem with the same goal Theproblem was extended by Cheng et al [19] to the case withparallel machines where a variable neighborhood searchalgorithm (VNS) was proposed
As for the linear deterioration mechanism Mosheiov[25] considered the case that processing times increase ata common rate and job weights are proportional to theirnormal processing time He demonstrated that the optimalschedule is Λ-shaped if the optimal objective is to minimizethe total weighted completion time on a single machineMoreover simple linear deterioration where jobs have a fixedjob-dependent growth rate but no basic processing time wasfurther considered [17] Ji and Cheng [26] discussed corre-sponding methods refer to the parallel-machine schedulingand gave a fully polynomial-time approximation schemewhereas a single-machine scheduling problem with lineardeterioration was studied by Jafari and Moslehi [27] Wuet al [28] Wang and Wang [29] with respective goals ofminimizing the number of tardy jobs and earliness penaltiesas well as total weighted completion time More studies and
Mathematical Problems in Engineering 3
discussions about linear deteriorating scheduling problemswere shown in Bachman and Janiak [30] Oron [31] Wangand Wang [29] Wu et al [28]
22 MS-RCPSP For the sake of its practicality RCPSP hasreceived more and more attentions [32] U and Kusiak [33]Babiceanu et al [34] and Coban [35] deal with the RCPSPin a dynamic real time way To obtain particular schedulesat some point metaheuristics are the most used techniquesfor solving RCPSP including Tabu search [36 37] SimulatedAnnealing [38 39] or Genetic Algorithm [40] SwarmIntelligence metaheuristics [41] are also effective for solvingrelated issues For example Particle Swarm Optimization[42] Bee Colony Optimization [43] or the popular AntColony Optimization [44 45] Although classical RCPSP isdeeply investigated and numerous methods could be easilycompared using PSPLIB instances it is not same for MS-RCPSP There are not too many researches paying attentionto MS-RCPSP
Myszkowski et al [46] defined new problem MS-RCPSPand benchmark to compare the effectiveness of examinedapproaches Next a hybrid ant colony optimization approach(HAC) [47] which links classical heuristic priority and theworst solutions stored by ants to update pheromone valuewas proposed To research this problem more thoroughlya greedy randomized adaptive search procedure (GRASP)[48] a hybrid differential evolution and greedy algorithm[49] and a Co-Evolutionary algorithm [50] are developed toimprove the quality of solutions Before this period Alanzi etal [51] proposed a lower bound using a linear programmingscheme for the RCPSP to solve the new extended MS-RCPSPmodelwhile Santos and Tereso [52] developed a filtered beamas a bonus for early completion into account Zheng et al [53]presented a teaching-learning-based optimization algorithmwith a task-resource list-based encoding scheme combiningthe task list and the resource list and a left-shift decodingscheme where the balance between global exploration andlocal exploitation to achieve satisfactory performances wasmainly stressed Dai et al [54] did the related work on theintegration of MS-RCPSP and proposed step structures aswell as two mutation operators
3 Problem Statement
To define two considered MS-RCPSP and MS-RCPSPLD aswell asmake them clear the problem description and amixedinteger programming model can be described as followsLet 119866(119881 119864) be a task-on-node network consisting of a set119881 of nodes denoting task 119894 (119894 isin 119881) and a set 119864 of edgesrepresenting the precedence relationships between a pair oftasks (119894 119895) (119894 119895) isin 119864 Specifically each pair (119894 119895) isin 119864 meansthe task 119894 precedes 119895 In other words task 119895 cannot start untiltask 119894 finished The duration of each task 119894 in MS-RCPSP is aprior known and constant while the value of duration inMS-RCPSPLD turning out to hinge on the basic processing time119886119894 and deteriorating rate ℎ119894 can be calculated by the function119889119894 = 119886119894 + 119878119894 lowastℎ119894 where 119878119894 shall be the earliest available time ofassigned resource 119896 119894 isin 119881119896 To perform any task 119894 the specific
skill 119902119894 also is required To make the 119899 tasks completed aset of 119870 of 119898 renewable resources which unassigned taskscan compete for when resources are idle will be providedResources differ inmastered skills and let119876119896 denote the skillscovered by resource 119896 Naturally a subset 119881119896 of 119881 also isavailable to incorporate all tasks that can be processed byresource 119896 (119902119894 isin 119876119896 119894 isin 119881119896) In same way a subset ofresources 119870119894 including all resources which can be utilizedto handle task 119894 is obtained Moreover each resource canperform at most one task at a time and a task can be executedby at most one resource simultaneously The objective is tominimize the makespan (the completion time of all tasks)
Based on the above descriptionwe formulate the problemas a 0-1 integer programming model Firstly the binaryvariables 119909119895119896 receive value 1 if task 119895 is assigned to resource119896 and 0 otherwise As for the binary variables 119910119894119895 are set to 1if task 119894 is scheduled to precede task 119895 0 otherwiseThen MS-RCPSP and MS-RCPSPLD can be formulated by objectivefunction (1) subject to constraints (2) to (9)
119872119894119899119894119898119894119911119890 119862119898119886119909 (1)
subject to 119865119894 = 119878119894 + 119889119894 forall119894 isin 119881 (2)
sum119896isin119870119895
119909119895119896 = 1 forall119895 isin 119881 (3)
119909119895119896 = 0 forall119895 notin 119870119895 (4)
119904119895 minus 119904119894 ge 119889119894 forall (119894 119895) isin 119864 (5)
119910119894119895 + 119910119895119894 ge 119909119894119896 + 119909119895119896 minus 1
forall (119894 119895) isin 119864 119896 isin 119870(6)
119878119895 ge 119878119894 + 119889119894 minus 119872(1 minus 119910119894119895)
forall119894 = 119895 119894 119895 isin 119881(7)
119862119898119886119909 ge 119865119894 forall119894 isin 119881 (8)
119909119895119896 119910119894119895 isin 0 1
119904119895 ge 0
forall119894 isin 119881 119895 isin 119881(9)
In the above mode objective function (1) denotes theoptimal direction of MS-RCPSP and MS-RCPSPLD Equa-tion (2) defines the finishing time of any task Constraint (3)restricts exactly one resource to a specific task among all avail-able resources Constraint (4) respects the skill constraintsbetween resources and tasks Constraint (5) highlights theprecedence relationship between a pair of nodes (119894 119895) inother words task 119895 can start only after 119894 is finished Constraint(6) shows logical relationships between the assignment vari-ables and sequencing variables That is when resource 119896 issimultaneously assigned to task 119894 and task 119895 the sequencebetween two tasksmust be determined either 119894 precedes 119895 or 119895precedes 119894 Constraint (7) is the big-119872 formulation to enforce
4 Mathematical Problems in Engineering
the relationship between the sequencing variable and thecontinuous starting time variable Constraint (8) calculatesthe makespan of project while Constraint (9) regulars thedomains of the variables
4 GVNS-Based Memetic Algorithm
In this section we describe in detail the general solutionmethodology and the supporting procedures in the followingsubsections
41 Search Space and Evaluation Function For a givenproblem the GVNS-MA searches a space Ω composed of allpossible assignments respecting skill constraints in any orderof tasks including both legal and illegal configurations Thesize of the search space Ω is bounded by 119874(119898119899 lowast 2119899)
Based on preceding notations and depictions to evaluatethe quality of a candidate solution 119904 isin Ω we adopt anevaluation function which is defined as 119891(119904) induced by 119904
119891 (119904) = 119898119886119909119894119898119894119911119890 119865119894 119894 isin 119881 + 119872 lowast sum(119894119895)isin119864
120575 (119894 119895) (10)
120575 (119894 119895) =
1 (119894 119895) isin 119864 119878119895 lt 1198651198940 otherwise
forall119894 isin 119881 119895 isin 119881 (11)
where 119872 is a large positive constant such that 119872 997888rarr infin as119899 997888rarr infin The first part of (10) represents the completiontime of the last unfinished task and the last part is anaugmented penalty function wheresum(119894119895)isin119864 120575(119894 119895) denotes thedegree violation to precedence relationships If 120575(119894 119895) =0 holds for any (119894 119895) isin 119864 it demonstrates the solutionis feasible corresponding to a legal configuration and itsevaluation function will all depend on the first part equalto the completion time of the schedule When a probleminstance admits no solution able to satisfy 120575(119894 119895) = 0 forall(119894 119895) isin119864 the search space of GVNS-MA is empty and no feasiblesolution can be found Given two solutions 1199041015840 and 11990410158401015840 1199041015840is better than 11990410158401015840 if 119891(1199041015840) le 119891(11990410158401015840) This statement impliesan assumption that a better solution has fewer precedenceconstraint violations
42Methodology and General Procedure Let119875 denote a pop-ulation of 119901 candidate configurations Let 119904119887 119904119908 represent thebest solution attainable so far and the worst solution in 119875 (interms of the evaluation function in Section 41) respectivelyLet119875119886119894119903119878119890119905 be a set of solution pairs (119904119894 119904119895) initially composedof all possible pairs in 119875 Next the proposed GVNS-MA canbe described as depicted in Algorithm 1
GVNS-MA first builds an initial population 119875 including119901 candidate configurations by the procedures in Section 43Then the algorithmenters into awhile loopwhich constitutesthe main part of the GVNS-MA On each new generationthe subsequent operations are executed In the first place aconfiguration pair (119904119894 119904119895) is taken at random and deleted fromPairSet Next GVNS-MA builds with a crossover operator(see Section 44) a new configuration 119904119888 (the offspring)After that the offspring is used as a starting point to further
(1) Input Problem instance I the size of population(119901) the depth of GVNS 120572
(2) Output the best configuration found during thesearch
(3) 119875 = 1199041 119904119901 larr997888 Population initialization(119868) lowastSec-tion 43lowast
(4) 119875119886119894119903119878119890119905 larr997888 (119904119894 119904119895) 1 le 119894 lt 119895 le 119901(5) while 119875119886119894119903119878119890119905 = do(6) Isolate a solution pair (119904119894 119904119895) isin 119875119886119894119903119878119890119905 ran-
domly(7) 119875119886119894119903119878119890119905 larr997888 119875119886119894119903119878119890119905 (119904119894 119904119895)(8) 119904119888 larr997888 crossOver(119904119894 119904119895) lowastSection 44lowast(9) 119904 larr997888 GVNS Operator(119904119888 120572) lowastSection 45lowast(10) if 119891(119904) le 119891(119904119908) then(11) UpdatePopulation(119904 119875 119875119886119894119903119878119890119905 119891(119904)) lowastSec-
tion 46lowast(12) end if(13) end while(14) Return 119904119887 larr997888 arg min 119891(119904119894) 119894 = 1 119901
Algorithm 1 Main sketch of the proposed GVNS-MA
improve by the GVNS operator (see Section 45) Finally ifthe improved offspring 119904 is better than the worst solution 119904119908in 119875 it is used to update the population 119875 and PairSet Thedetailed update operations are described in Section 46 Thewhile loop continues until PairSet becomes empty At the endof the while loop the algorithm terminates and returns thebest configuration 119904119887 found during the search Note that thedepth of GVNS 120572 represents a maximum number betweentwo iterationswithout improvement regarded as the stoppingcriterion of the GVNS operator
43 Population Initialization In order to build the initialpopulation (in Algorithm 2) the construction operator togenerate a new solution is executed 3 times 119901 times From thescratch a new configuration is constructed as follows Theoperator starts from assigning each task with a randomresource satisfying skill constraint Subsequently all tasksassigned to same resource are sequenced randomly at the endof previous task-to-resource phase Then for each generatedsolution the GVNS operator with the evaluation function119891(119904) (see Section 41) is used to optimize it to a local optimumand the obtained 119901 best configurations are selected to formthe initial population The detailed procedures are describedin Algorithm 2
44 The Crossover Operator Within a memetic algorithmthe crossover operator is another essential ingredient whosemain goal is to bring the search process to new promisingsearch regions to diversify the search In this paper theoffspring of two parent configurations 1199041 = (1198601 1198791198761) 1199042 =(1198602 1198791198762) shows as 119904119900 = (119860119900 119879119876119900) To inherit the advantagesof parent solutions the tasks assigned to same resource in 11990411199042 are given priority to keep the task-to-resource assignmentunchanged As for the remain unassigned tasks and 1198791198760
Mathematical Problems in Engineering 5
(1) Input The set 119881 = V1 V2 V119899 of 119899 tasks the set 119870 =1198961 1198962 119896119898 of 119898 renewable resources skill constraintsprecedence relationships and the size of population (119901)
(2) Output The best 119901 configurations(3) for pop = 1 3119901 do(4) Set RA = 119881(5) while 119877119860 = do(6) Choose V119894 randomly from 119877119860 remove V119894 from 119877119860
randomly isolate 119896119895 isin 119870119894mdash the set covering all re-sources can perform V119894 and record this assignmentas 119860(V119894) = 119895
(7) end while(8) for 119895 = 1 119898 do(9) Generate a random task sequence 119879119876119895 for candidate
tasks assigned to 119896119895(10) end for(11) Calculate the objective value 119891(119860TQ) with the evalu-
ation function 119891(119904) (see Section 41)(12) 119904119887119901119900119901 larr997888 GVNS operator(119891(119860TQ) 120572) lowastSection 45lowast(13) end for(14) Sort the 3119901 configurations in an ascending sort of evalua-
tion function values and return the first 119901 solutions as theinitial population
Algorithm 2 Population initialization
they will be determined by same methods in Section 43Analogously we apply the GVNS operator (see Section 45)to the offspring to finally gain the candidate for furtherpopulation updatingThe principle of this operator is detailedin the procedure crossover (Algorithm 3)
45 GVNS Operator This section discusses the local opti-mization phase of GVNS MA a key part of memetic algo-rithm Its function ensures an intensified search to locatehigh quality local optima from any starting point Here wedesign a generable variable neighborhood search (GVNS)heuristic as the local refinement procedure which showsgood performance compared to other variable neighborhoodsearch variants in terms of local search capability
Given three neighborhood structures 1198731 11987321198733 and aninitial solution (1199040) our GVNS operator does the refinementas follows Attention the quality of any solution is evaluatedas depicted in Section 41 To start with the sequential orderSO which determines the applying sequence of these neigh-borhood structures is at random generated For exampleassuming that the sequence SO equals (3 1 2) the searchbegins from1198733 and ends at1198731 at the given iteration For eachneighborhood structure a new local optima 11990410158401015840 is obtainedby applying the corresponding local search procedures to theincumbent solution 1199041015840 set at 1199040 at the beginning of GVNSprocedure If 11990410158401015840 is better than 1199041015840 1199041015840 is updated with thenew solution 11990410158401015840 accepted as a descent to continue the localsearch for current neighborhood otherwise the search turnsto the next neighborhood structure in SO One iterationterminates until the last neighborhood structure in SO isexplored and then the search goes on with the next iteration
until the stopping criteria is met ie the best solution 1199041015840has not improved for 120572 consecutive iterations The generalsketch of GVNS operator is described in Algorithm 4 andthe neighborhoods employed as well as the technique tocalculate objective value rapidly are depicted in the followingsubsections
451 Move and Neighborhood Three neighborhoods119873119896 (119896 = 1 3) are adopted in GVNS MA The neighbor-hood 1198731 is defined by the swap move operator which swapstwo tasks processed by same resource and keeps previoustask-to-resource assignment unchanged As such given asolution 119904 the swap neighborhood 1198731(119904) of 119904 is composedof all possible configurations that can be applied with theswap move to 119904 The neighborhood 1198732 is designed on thebase of reversion which reverses all the tasks incorporatedinto two designated random tasks of one resource As forthe neighborhood 1198733(119904) it is designed by the alter moveoperator which alters assigned resource from the originalto another resource equipped with demanded skill for oneselected randomly task with a random position in the tasksequence of given resource To efficiently assess the qualityof any neighborhood solution we devise a rapid evaluationtechnique for neighborhood solutions which is committedgreatly to the computational efficiency of the GVNS-MA
452 Rapid Evaluation Mechanism Our rapid evaluationtechnique to neighborhood solutions realizes through effec-tively calculating the move value (Δ119891) which identifies thechange in the evaluation function 119891 (see Section 41) ofeach possible move applicable to the incumbent solution
6 Mathematical Problems in Engineering
(1) Input Two parent solutions 1199041 = (1198601 1198791198761) 1199042 = (11986021198791198762) Problem instance I
(2) Output The offspring 119904119900 = (119860119900 119879119876119900)(3) Set RA = 119881 contains 119899 tasks remaining to be assigned
resource to and 119899119906119898 represents the size of 119877119860(4) for 119903 = 1 119899 do(5) if 1198601(V119903) == 1198602(V119903) then(6) 119860119900(V119903) = 1198601(V119903)(7) RA=RAV119903(8) end if(9) end for(10) for 119903 = 1 num do(11) Assign randomly resource 119896119895 isin 119870119903 to V119903(12) end for(13)Obtain 119879119876119900 and refine the offspring 119904119900 = (119860119900 119879119876119900) in the
same way as Algorithm 2 lowastSection 43lowast
Algorithm 3 Procedure crossover
(1) Input Initial solution 1199040 a set of neighborhood structures119873119896 (119896 = 1 3) 120572
(2) Output The current best solution 119904119887 found during GVNSprocess
(3) Calculate the objective value 119891(1199040) according to the eval-uation function in Section 41
(4) 1199041015840 larr997888 1199040 lowast1199041015840 is the current solution lowast(5) 119904119887 larr997888 1199041015840 lowast119904119887 is the best solution found so farlowast(6) 119889 = 0 lowast119889 counts the consecutive iterations where 1199041015840 is not
improved lowast(7) repeat(8) Generate a random sequence (SO) to apply three
neighborhood structures(9) Apply the relevant mechanism (Section 451) in pre-
determined order specified by SO update 1199041015840 if a betterconfiguration is attained
(10) if 119891(1199041015840) lt 119891(119904119887) then(11) 119904119887 larr997888 1199041015840(12) reset the counter 119889 = 0(13) else(14) 119889 = 119889 + 1(15) end if(16) until 119889 == 120572(17) return 119904119887
Algorithm 4 GVNS operator
119904 It functions in the reduction of computational cost toevaluate any attainable neighborhood solution inspired bythe situation where the starting and finishing times of mosttasks will not be changed when neighborhood solutions aregenerated
For 1198731 and 1198732 generated by swap and reversion movethe set of tasks with changed starting and ending times onlyincorporate the elements ranking next to the isolated firsttask in the sequence of given resourceWithout considerationof deterioration only the elements located in the positionbetween two picked tasks are influenced As for 1198733 achieved
by altermove all the tasks lined up behind the designated taskin the sequence of its initial assigned resource and the newdistributed one are included Attention for three moves werecalculate the relevant parameters of above-mentioned tasksand the rest are ignored In addition the impact that the newsolutions is defying the precedence relationships to varyingdegrees will be respected in terms of 119872 lowast sum(119894119895)isin119864 120575(119894 119895) inevaluation function 119891 (see Section 41)
46 Updating population and PairSet As illustrated in Algo-rithm 1 the population119875 and thePairSet are updatedwhen an
Mathematical Problems in Engineering 7
Table 1 Settings of important parameters
Parameters Section Description Values119901 Section 42 population size for GVNS-MA 10120572 Section 45 depth of GVNS 50 000119872 Section 41 penalty value for a violation 1000 1 times 107
to precedence constraint
excellent offspring is obtained through the crossover operatorand improved further by GVNS operator First of all if itis better than the worst solution 119904119908 in 119875 for any improvedoffspring solution 119904 the worst configuration 119904119908 is replaced bythe offspring solution 119904 When the population is updated thePairSet should be updated accordingly all pairs containing 119904119908solution are deleted from setPariSet and all pairs generated bycombining 119904 solution with others in 119875 are incorporated intoPairSet
5 Computational Experiments and Results
This section plans to assess the proposed method GVNS-MA through having comparisons with the state-of-the-artmethods in the literature For the lack of known benchmarkdata for handingMS-RCPSPLDwe firstly apply the proposedGVNS-MA to solve the MS-RCPSP on exist benchmarkinstances in favor of argument its effectiveness Then onthis basis the GVNS-MA will be examined on the modifiedproblem set
51 Benchmark Instances For the purpose of assessingGVNS-MA fully and comprehensively computational exper-iments will be conducted on two sets of instances where thefirst set is composed of 30 benchmark instances irrespec-tive of deterioration and its available in Myszkowski et al[46] which are artificially created in a base of real worldobtained from the Volvo IT Department in Wroclaw Thefull information of each instance including tasks durationsresource capabilities or precedence between tasks has beengiven As for the other set it consists of 45 instances generatedwith some modifications on first set to consider the lineardeterioration The detail will be described in Section 54
52 Parameter Settings and Experiment Protocol OurGVNS-MA was programmed in MATLAB R2015b and all thereported computational experiments presented below wereexecuted on a personal computer equipped with an IntelCore i3 processor (310 GHz CPU and 2GB RAM) in theenvironment ofWindows 7 OS To eliminate the randomnessas much as possible twenty replications for each instance arecarried out
Table 1 shows the descriptions and settings of the param-eters adopted in GVNS-MA determined by preliminaryexperiments Our memetic algorithm rests upon only threeparameters the population size 119901 the depth of generalvariable neighborhood search 120572 and the price for a violationto precedence constraint 119872 For 119901 and 120572 we follow Lai andHao [55] and set 119901 = 10 120572 = 50000 while the parameter 119872
is set at 1000 for the first experimental group and 1 times 107 forthe second
53 Experimental Results without Deterioration Our firstexperimental group aims to evaluate the performance ofour GVNS-MA on the set of 30 known instances with atmost 200 tasks and 40 renewable resources Without regardto deterioration it means that the GVNS-MA will set thedeteriorating rates of all tasks at 0 when it deals with therelevant computations Table 2 records the computationalresults solved by the GVNS-MA with the goal of durationoptimization aswell as the results achieved by other referencealgorithms in the literature
Notice that the instance name (columns 1) contains itsfull description Take the instance named 100-10-26-15 asexample the number 100 represents the number of tasksincluded and 10 denotes the quantity of renewable resourcesprovided As for the number 26 and 15 it illustrates theamount of precedence relationships and the number ofdifferent introduced skills Column 2 of Table 2 indicates theprevious minimum objective values (119891119901119903119890119887) in the literaturewhich are compiled from the best solutions yield by tworecent and best performing algorithms namely GRASP[48] and DEGR [49] Columns 3 to 4 give the best resultsobtained by DEGR and GRASP The corresponding resultsof the GVNS-MA are given in columns 5 to 7 includingthe minimum objective value (119891119887119890119904119905) over 20 independentruns the average objective value (119891119886V119892) and the averagecomputing time in seconds (Time(s)) to reach 119891119887119890119904119905 The rowBest indicates a total number of instances where the specificmethod achieves optimal among three algorithms The bestone is indicated in italic In addition to verify whetherthere exists an essential difference between the best resultsof GVNS-MA and other reference algorithms the relativepercentage deviation (RPD) is defined by the equation
119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905
119891119901119903119890119887times 100 (12)
where a positive value of 119877119875119863 means an improvement ofresult achieved by GVNS-MA while the negative numberrepresents a worse solution
Table 2 discloses that the outcomes from our GVNS-MAare noteworthy compared to the state-of-the-art results inthe literature GVNS-MA improves the previous best knownresults for 19 instances and matches for 7 cases Comparedwith the 8 out of 30 cases solved by DEGR and 6 bestsolutions achieved by GRASP these data clearly indicatethe superiority of GVNS-MA compared to the previousexcellent methods Additionally it can be observed that
8 Mathematical Problems in Engineering
Table 2 Comparison of the GVNS-MA with other algorithms on known MS-RCPSP dataset [48] Best results are indicated in italic
instances 119891119901119903119890119887 DEGR GRASP GVNS-MA 119877119875119863()119891119887119890119904119905 119891119886V119892 119879119894119898119890(119904)
100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050
the improvement achieved by GVNS-MA is up to 939for instance 100-20-65-15 accompanying that the average119877119875119863() equals to 050
54 Experimental Results with Linear Deterioration Theprevious comparisons and discussions in Section 53 demon-strate the advantages of GVNS-MA in solving the relatedissues of MS-RCPSP In this section the aforementioneddataset with some modifications is used to assess the capabil-ity of GVNS-MA to solve the MS-RCPSPLDThe proceduresof generating the testing instances and analysis of the resultsare described below To make the benchmark instances meetthe considered linear deterioration precisely the deteriora-tion rate (119889119903119894) for task 119894 (119894 isin 119881) is generated randomly fromthree intervals (0 05] [05 1] and (0 1] similar to Cheng et
al [19] to shed light on the influence of the different valuerange of deterioration rate on its effectiveness
Since the extra included deterioration rate we providetwo additional heuristics for the initial population generationof GVNS-MA Both the two methods affect the phase ofgenerating 119879119876119895 determining the sequence of tasks assignedto resource 119895 119895 isin 119870 The first heuristic considers thesequence in descending order of deterioration rate (ℎ) whilethe other rests upon an ascending order of ratio (119886ℎ)of the basic processing time and deterioration rate Themethods adopting the former and latter heuristic to popula-tion initialization are dubbedGVNS-119872119860ℎ and GVNS-119872119860119886ℎrespectively
Here instanceswith 100 tasks fromMyszkowski et al [46]are isolated to attain the researched objects which fit with theunique nature of the MS-RCPSPLDmore To account for the
Mathematical Problems in Engineering 9
three intervals from which the deterioration rate is drawn3 extended cases are needed to solve for each instance Forconvenience these instances are denoted by adding a suffixfor identification to different intervals For example 100-10-26-15 1 represents the original case 100-10-26-15 is modifiedby adding the deterioration rates produced in (0 05] to thedurations of tasks In total there are 45(3 times 15) instancesrandomly generated
Due to zero known results in literature for same datasetthe improved tabu search (ITS) proposed by Dai et al [54]who discussed the MS-RCPSP under step deterioration anda path relinking algorithm (PR) [55] based on the populationpath relinking framework are programmed as referencealgorithms
Table 3 reports the computational results achieved by theITS PR GVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ on theset of 45 benchmark instances 119891119887119890119904119905 denotes the minimumobjective value and 119891119886V119892 is computed as the average objectivevalue of 20 runs
First Table 3 discloses that the solutions obtained byGVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ are better thanthe ITS and PR for any instance from the perspectiveof both quality of schedule and runtime To some extentthese results demonstrated the differences between lineardeterioration and step deterioration and the superiority ofmemetic algorithm framework Second these three methodsdiffering in the sort order of tasks in initialize phase behavesimilarly where GVNS-MA obtains the best 15 out of 45instances 17 for GVNS-MAℎ and 14 for GVNS119886ℎ in termsof 119891119887119890119904119905 Specifically GVNS-MA and GVNS-MAℎ attain theoptimal simultaneously for the instance 100-5-48-9-1 Froma view point of 119891119886V119892 and run time three methods alsohave a balanced performance Third as far as three differentintervals to generate deterioration rate are concerned thephenomenon did not happen that the relevant algorithmsdisplay strikingly different behavior In other words theperformance of the proposed algorithm is not sensitive to thesetting of deterioration rate
55 Analysis and Discussions In this section we study twoessential ingredients of the proposed GVNS-MA to getan insight to its performance One is the rapid evaluationmechanism the other is the role of the memetic framework
551 Importance of Rapid Evaluation Mechanism GVNS-MA with rapid evaluation mechanism only calculates therelevant parameters of some particular tasks rather thanall when the procedure computes the objective value ofa neighborhood solution To highlight the key role ofthe rapid evaluation mechanism two sets of comparisonexperiments are carried out on generated dataset with twoalgorithms GVNS-MA and GVNS-MA0 including sameingredients with GVNS-MA except for the computation ofobjective value When GVNS-MA0 figures up the value of aneighborhood solution it computes all relevant parametersagain
Table 4 records the experimental results carried out on thedataset [46] without consideration of deterioration whereas
Table 5 shows the comparisons of GVNS-MA and GVNS-MA0 about the set of 15 instances generated in Section 54on account of the indiscrimination in three intervals Col-umn 2 and 5 record the best attained by two algorithmsColumn 3 and 6 indicate the minimum time cost to a finalfeasible schedule with one run of procedure Note that thebest objective value cannot be guaranteed as the output ofshortest runtime As for the parameters in column 4 and 7they represent the mean runtime Finally two parameters119863119864119881119904ℎ119900119903119905119890119904119905 and 119863119864119881119886V119892 are used to disclose the runtimedeviation of two methods defined by equations
119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051
times 100 (13)
and
119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922
119879119886V1198921times 100 (14)
respectively The positive value of 119863119864119881119904ℎ119900119903119905119890119904119905() and119863119864119881119886V119892() means that GVNS-MA0 has better performanceand negative value tells GVNS-MA is prior to GVNS-MA0in terms of time cost And the rows Better and Worserespectively show the number of instances for which thecorresponding results of the associated algorithm are betterand worse than the other
The results summarized in Table 4 disclose that theGVNS-MA has an overwhelming advantage over GVNS-MA0 in terms of the computation time to solve MS-RCPSPleaving out the deterioration effect Indeed the shortestruntime 119879119904ℎ1199001199031199051198901199041199051 of the GVNS-MA method is better thanthe shortest runtime 119879119904ℎ1199001199031199051198901199041199052 of GVNS-MA0 for 30 out of30 representative instances and the average runtime 119879119886V1198921 isbetter for 28 out of 30 instances Meanwhile the average valueof 119863119864119881119904ℎ119900119903119905119890119904119905() equals -1379 accompanying with a highof -1880 percent in 119863119864119881119886V119892()
However focusing on Table 5 the results of twoapproaches are neck and neck and GVNS-MA lost its earlysuperiority in MS-RCPSP In terms of shortest runtimeGVNS-MA successes for 7 out of 15 tested instances whileGVNS-MA0 reaches optimal for the remain As for averageruntime GVNS-MA performs better for 9 out of 15 examplesand GVNS-MA0 achieves reversion in others 6 instancesWith these data it will be hard to judge the true benefits ofone approach versus the other
To figure out the reason of this phenomenon we shouldcome back to the inner rationale of rapid evaluation mecha-nism When GVNS-MA computes the completion time of aneighborhood solution it only recalculates the tasksrsquo relatedparameters influenced by the particular move InMS-RCPSPa move including swap reverse and alter will affect justa small number of tasks But for MS-RCPSPLD instancesany move can cumulatively effect on a large proportion oftasks because of the existing deterioration Consequently theruntime saved in computing some unchanged parametersmay not make up for the time spent on isolating the changedtasks
10 Mathematical Problems in Engineering
Table3Summaryandcomparis
onon
thes
etof
45newgeneratedinsta
nces
with
119899=10
0of
GVN
S-MA
GVN
S-119872
119860 ℎG
VNS-
119872119860 119886ℎand
theT
Sheuristic[54]
andPR
[55]B
estresultsare
indicatedin
italic
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-10-26-15
165572
6598
13899
963646
6492
66871
63274
6392
33092
661967
6379
730698
62632
63629
29422
100-10-26-15
260
1396
618416
28607
608971
625436
79283
587145
613934
37496
580557
615474
23879
572794
615156
34826
100-10-26-15
3172856
184175
31236
1646
96
175513
59776
160283
16936
22225
166221
17124
17541
163898
172228
3461
100-10-47-91
7008
72053
40304
6974
471232
10018
69445
70098
20666
67645
6915
731499
69673
7039
36314
100-10-47-92
700352
739287
39112
703439
721152
70563
672221
715138
2417
9672304
6959
20614
693553
702369
25114
100-10-47-93
188206
195206
4816
818358
190863
93463
174162
184389
30008
177323
183958
19383
173708
183368
41432
100-10-48-15
165868
6914
13991
567093
6890
88879
164
851
65526
3491
663253
66595
22686
6479
86610
226662
100-10-48-15
2638747
659809
42043
621357
658747
9116
161867
651609
4116
1590432
625462
27097
618507
6346
3233655
100-10-48-15
3170999
174232
29659
171864
178061
73395
159497
166887
23395
161483
166366
3216
9163501
170273
27056
100-10-64-91
70527
7572
53770
371049
7692
773024
6872
27092
931287
6884
71215
26024
67894
69682
2775
6100-10-64-92
705806
744775
39415
7040
32738141
6279
4663729
711114
3013
968421
719153
22806
696727
736833
3991
8100-10-64-93
193844
206206
3115
32006
42
2044
03
108918
189324
19445
35415
180842
200842
3104
193844
199996
31871
100-10-64-15
170119
71497
34626
71075
7279
683543
69346
70022
39097
6691
69112
21249
70042
7079
617116
100-10-64-15
2690964
744775
3899
8672578
72823
61684
6606
14685085
22491
62823
682957
3178
4626331
666418
2376
100-10-64-15
3193844
218506
31871
194247
204235
71318
1806
92
1860
08
25336
187149
192556
45438
185585
190957
25296
100-20-22-15
119089
19355
29631
1936
19739
5517
618883
19118
2222
18572
1898
21438
1878
18965
2474
2100-20-22-15
24615
847731
36854
46032
46714
6792
645371
46008
28455
45218
46627
2892
645833
4633
2312
4100-20-22-15
326328
2798
128528
27212
2697
36972
92597
26419
2812
25563
25918
17531
25575
26254
21352
100-20-46-15
126243
2673
33424
26214
26631
60354
25862
26329
4119
925511
25896
1215
325476
26006
17466
100-20-46-15
260
647
66244
34408
59367
63596
71878
5796
16165
17216
56542
5942
26696
5465
59285
22313
100-20-46-15
332421
3490
937523
33539
3417
977634
31695
32909
16433
31651
33263
29116
32841
33349
2791
100-20-47-91
19007
1975
929027
1916
319685
5776
318864
1912
28311
1874
719269
2732
18455
18892
30731
100-20-47-92
50591
53481
37119
47839
51484
102173
4803
49319
3549
46278
4876
435418
44966
47893
24241
100-20-47-93
30802
31827
4572
630631
31365
6995
829437
30352
57846
27712
29458
29737
28399
2879
927559
100-20-65-15
19013
492446
26413
89865
90543
6795
788801
90095
17505
86826
89518
15052
87305
89338
2843
100-20-65-15
2253899
2606
7628718
250126
253267
5876
3244762
250316
4099
7243449
24751
17659
242353
244944
2573
5100-20-65-15
3110
567
1115886
27491
109874
113685
53418
105215
109224
2513
8108595
111243
1591
1104829
106776
1412
6100-20-65-91
19113
1978
341248
1895
719548
79561
18369
18898
30561
18697
1914
61896
18694
19265
3097
9100-20-65-92
46242
4776
865469
4593
447443
58112
4495
746229
1768
44719
45985
22324
45593
46591
30606
100-20-65-93
28018
28776
4894
327512
2810
868532
2719
2772
124279
26587
27691
3101
26455
27845
2473
100-5-22-151
500988
514285
2897
8510285
505098
84526
486586
499318
43529
494364
500988
18444
498567
510285
22335
100-5-22-152
1193440
1227080
4319
4119
2300
1210150
61537
1118610
1199248
24322
1184940
1219514
20664
1138490
1226780
26656
100-5-22-153
663947
700818
5691
2661553
792627
7997
660
6322
652627
10639
592509
6279304
2619
8643856
6597526
17267
100-5-46
-151
675739
749763
3679
7652793
701457
137862
63503
679663
5075
8636903
687941
29888
649164
671982
36202
100-5-46
-152
1404
680
1453430
58595
1443529
1470970
108595
1298430
1370870
26564
1315940
1360
830
23536
1332210
1399014
2417
6100-5-46
-153
795664
870313
50687
7977713
863567
6891
1740178
796512
81792
4755866
7786062
16809
752568
80128
1876
3100-5-48-9
1563787
586832
31214
5666
27
78386
84807
55206
560407
2415
55206
563664
25018
554299
56657
2495
2100-5-48-9
21284230
1315720
3614
51289811
1364
650
71286
1222350
1285020
2795
8116
3860
1259332
2812
21233010
1268572
34659
100-5-48-9
3684299
692453
45335
66785
717279
29393
8657279
67622
4561
6604
26
670917
41286
64822
667213
828446
Mathematical Problems in Engineering 11
Table3Con
tinued
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-5-64
-151
581916
627374
55661
576589
618167
102415
5544
26
579234
3516
7546543
590352
29678
56942
580175
56619
100-5-64
-152
120944
01261510
31296
1190830
1257643
123533
1131450
1183014
40595
1118430
1151812
46305
1047250
1158556
3337
100-5-64
-153
642267
709602
3498
8634651
688889
89403
612857
6657858
3798
5626771
660614
44591
1624753
674996
261623
100-5-64
-91
550231
577524
37365
5544
81
566747
98693
528177
5404
36
3393
530748
543895
48396
515984
536747
2993
2100-5-64
-92
1214340
1271610
40553
1183479
1236750
11992
711140
60115
5584
41523
11640
101201126
40287
1121650
1159384
27286
100-5-64
-93
610356
648765
36395
6151502
632323
87448
604586
6210378
31363
594514
6171502
30543
595191
623223
2774
9119861
119890119904119905
00
00
1514
1716
1415
119879119900119905119886
11989745
4545
4545
4545
4545
45
12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
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2 Mathematical Problems in Engineering
project scheduling problem (RCPSP) which make it possiblethrough a better adaption to apply in production planningproject management or manufacturing etc Ultimately moreadaptable multiskill resource-constrained project schedulingproblem (MS-RCPSP) gives each resource a set of capabilitiesand each resource can bid for being assigned tasks
Scheduling with deterioration and MS-RCPSP are tworesearch hotspots of scheduling area and these two subfieldsare not absolutely separate In terms of most tasks of actualMS-RCPSP deterioration effects existed for the reason offatigued (humans ormachines)However there is no researchconcentrated on the integration of MS-RCPSP and deteriora-tion
In this paper we pick the deterioration mechanism aslinear deterioration As for the researched MS-RCPSP withlinear deterioration which is dubbed MS-RCPSPLD thedifferences compared to the scheduling problems with lineardeterioration in the reported literature can be explainedhere Foremost the resource or the machine in the usuallyresearched scheduling problem with deterioration mostly isrestricted to be single-mode which means that it owns onlyone capability and can execute just a kind of task
Moreover as theRCPSP is proven to beNP-hard [7] thereis no optimal solution that could be computed in polynomialtime And this paper also demonstrates that the MS-RCPSPand the MS-RCPSPLD are NP-hard because they are moregeneral problems compared with the single-mode RCPSPHence methods that find feasible solutions which maynot achieve global optimal but can obtained in acceptabletime are built by researchers In such cases soft computingmethods are used mostly heuristics and metaheuristics
Within the metaheuristics group of methodologies itis noticed that memetic algorithm (MA) as a quite simpleapproach gives promising results in the fields of both com-puter science and operational research [8] MA representsone of the recent growing areas of research in evolutionarycomputation and is introduced by Moscato [9] inspired byboth Darwinian principles of natural evolution and Dawkinsrsquonotion of a meme With MA the traits of Universal Darwin-ism are more appropriately captured in particular dealingwith areas of evolutionary algorithms that marry otherdeterministic refinement techniques for solving optimizationproblems [10] As a general framework MA provides thesearch with desirable trade-off between intensification anddiversification through the combined use of a crossoveroperator to generate new promising solutions and a localoptimization procedure to locally improve the generatedsolutions [11 12]
In next stages to make MA more efficient it needs tobe supported by other metaheuristics eg general variableneighborhood search (GVNS) proposed by Hansen et al[13] GVNS is a simple and effective metaheuristic forcombinatorial optimization yield through systematic changeof neighborhoods within a refinement local search It canavoid becoming mired in local optima and develop the searchdirection to an excellent solution ultimately by means ofexploring new regions throughout the whole solution spaceThe GVNS has attracted attentions and shows remarkableperformance compared to other basic variable neighborhood
search (VNS) variants Up to now various optimization prob-lems have taken the GVNS to pursue better solutions suchas the single-machine total weighted tardiness problem withsequence dependent setup times [14] the one-commoditypickup-and-delivery traveling salesmanproblem [15] and thevehicle routing problem [16]
As far as we know there is no published work on solvingthe MS-RCPSP by the GVNS-MA By natural extension itmakes sense to solve the MS-RCPSPLD with the GVNS-MAThemain goal of this paper is to get an insight to the problemintegrating the MS-RCPSP with linear deterioration HowGVNS can be effectively hybridized with MA to solve MS-RCPSP and MS-RCPSPLD is examined in the GVNS-MAThe proposed approach is demonstrated to show its effective-ness in comparison to other state-of-the-art methods
The remaining part of this paper is organized as followsIn Section 2 a short summary of existing publications ispresented Section 3 gives the description of the problemunder consideration In Section 4 the proposedGVNS-MA iselaborated Section 5 shows performed experiments and theirresults before concluding the paper in Section 6 finally
2 Related Work
21 Deterioration Effect There are many practical situationsthat any delay or waiting in starting time of a task may causeto increase its processing time and two main deteriorationmechanisms are discussed
Besides usual linear deterioration mechanism discussedin this paper situations in which taskrsquos processing times arerepresented by step functions characterized by a sharp changein processing time at the deadline points [17ndash22] also arefairly common Sundararaghvan and Kunnathur [23] firstlyconsidered the corresponding single-machine schedulingproblem for minimizing the sum of the weighted completiontimes while Mosheiov [17] provided some simple heuristicsfor these NP-hard problems tominimize themakespan NextJeng and Lin [24] introduced a branch and bound algorithmfor the single-machine problem with the same goal Theproblem was extended by Cheng et al [19] to the case withparallel machines where a variable neighborhood searchalgorithm (VNS) was proposed
As for the linear deterioration mechanism Mosheiov[25] considered the case that processing times increase ata common rate and job weights are proportional to theirnormal processing time He demonstrated that the optimalschedule is Λ-shaped if the optimal objective is to minimizethe total weighted completion time on a single machineMoreover simple linear deterioration where jobs have a fixedjob-dependent growth rate but no basic processing time wasfurther considered [17] Ji and Cheng [26] discussed corre-sponding methods refer to the parallel-machine schedulingand gave a fully polynomial-time approximation schemewhereas a single-machine scheduling problem with lineardeterioration was studied by Jafari and Moslehi [27] Wuet al [28] Wang and Wang [29] with respective goals ofminimizing the number of tardy jobs and earliness penaltiesas well as total weighted completion time More studies and
Mathematical Problems in Engineering 3
discussions about linear deteriorating scheduling problemswere shown in Bachman and Janiak [30] Oron [31] Wangand Wang [29] Wu et al [28]
22 MS-RCPSP For the sake of its practicality RCPSP hasreceived more and more attentions [32] U and Kusiak [33]Babiceanu et al [34] and Coban [35] deal with the RCPSPin a dynamic real time way To obtain particular schedulesat some point metaheuristics are the most used techniquesfor solving RCPSP including Tabu search [36 37] SimulatedAnnealing [38 39] or Genetic Algorithm [40] SwarmIntelligence metaheuristics [41] are also effective for solvingrelated issues For example Particle Swarm Optimization[42] Bee Colony Optimization [43] or the popular AntColony Optimization [44 45] Although classical RCPSP isdeeply investigated and numerous methods could be easilycompared using PSPLIB instances it is not same for MS-RCPSP There are not too many researches paying attentionto MS-RCPSP
Myszkowski et al [46] defined new problem MS-RCPSPand benchmark to compare the effectiveness of examinedapproaches Next a hybrid ant colony optimization approach(HAC) [47] which links classical heuristic priority and theworst solutions stored by ants to update pheromone valuewas proposed To research this problem more thoroughlya greedy randomized adaptive search procedure (GRASP)[48] a hybrid differential evolution and greedy algorithm[49] and a Co-Evolutionary algorithm [50] are developed toimprove the quality of solutions Before this period Alanzi etal [51] proposed a lower bound using a linear programmingscheme for the RCPSP to solve the new extended MS-RCPSPmodelwhile Santos and Tereso [52] developed a filtered beamas a bonus for early completion into account Zheng et al [53]presented a teaching-learning-based optimization algorithmwith a task-resource list-based encoding scheme combiningthe task list and the resource list and a left-shift decodingscheme where the balance between global exploration andlocal exploitation to achieve satisfactory performances wasmainly stressed Dai et al [54] did the related work on theintegration of MS-RCPSP and proposed step structures aswell as two mutation operators
3 Problem Statement
To define two considered MS-RCPSP and MS-RCPSPLD aswell asmake them clear the problem description and amixedinteger programming model can be described as followsLet 119866(119881 119864) be a task-on-node network consisting of a set119881 of nodes denoting task 119894 (119894 isin 119881) and a set 119864 of edgesrepresenting the precedence relationships between a pair oftasks (119894 119895) (119894 119895) isin 119864 Specifically each pair (119894 119895) isin 119864 meansthe task 119894 precedes 119895 In other words task 119895 cannot start untiltask 119894 finished The duration of each task 119894 in MS-RCPSP is aprior known and constant while the value of duration inMS-RCPSPLD turning out to hinge on the basic processing time119886119894 and deteriorating rate ℎ119894 can be calculated by the function119889119894 = 119886119894 + 119878119894 lowastℎ119894 where 119878119894 shall be the earliest available time ofassigned resource 119896 119894 isin 119881119896 To perform any task 119894 the specific
skill 119902119894 also is required To make the 119899 tasks completed aset of 119870 of 119898 renewable resources which unassigned taskscan compete for when resources are idle will be providedResources differ inmastered skills and let119876119896 denote the skillscovered by resource 119896 Naturally a subset 119881119896 of 119881 also isavailable to incorporate all tasks that can be processed byresource 119896 (119902119894 isin 119876119896 119894 isin 119881119896) In same way a subset ofresources 119870119894 including all resources which can be utilizedto handle task 119894 is obtained Moreover each resource canperform at most one task at a time and a task can be executedby at most one resource simultaneously The objective is tominimize the makespan (the completion time of all tasks)
Based on the above descriptionwe formulate the problemas a 0-1 integer programming model Firstly the binaryvariables 119909119895119896 receive value 1 if task 119895 is assigned to resource119896 and 0 otherwise As for the binary variables 119910119894119895 are set to 1if task 119894 is scheduled to precede task 119895 0 otherwiseThen MS-RCPSP and MS-RCPSPLD can be formulated by objectivefunction (1) subject to constraints (2) to (9)
119872119894119899119894119898119894119911119890 119862119898119886119909 (1)
subject to 119865119894 = 119878119894 + 119889119894 forall119894 isin 119881 (2)
sum119896isin119870119895
119909119895119896 = 1 forall119895 isin 119881 (3)
119909119895119896 = 0 forall119895 notin 119870119895 (4)
119904119895 minus 119904119894 ge 119889119894 forall (119894 119895) isin 119864 (5)
119910119894119895 + 119910119895119894 ge 119909119894119896 + 119909119895119896 minus 1
forall (119894 119895) isin 119864 119896 isin 119870(6)
119878119895 ge 119878119894 + 119889119894 minus 119872(1 minus 119910119894119895)
forall119894 = 119895 119894 119895 isin 119881(7)
119862119898119886119909 ge 119865119894 forall119894 isin 119881 (8)
119909119895119896 119910119894119895 isin 0 1
119904119895 ge 0
forall119894 isin 119881 119895 isin 119881(9)
In the above mode objective function (1) denotes theoptimal direction of MS-RCPSP and MS-RCPSPLD Equa-tion (2) defines the finishing time of any task Constraint (3)restricts exactly one resource to a specific task among all avail-able resources Constraint (4) respects the skill constraintsbetween resources and tasks Constraint (5) highlights theprecedence relationship between a pair of nodes (119894 119895) inother words task 119895 can start only after 119894 is finished Constraint(6) shows logical relationships between the assignment vari-ables and sequencing variables That is when resource 119896 issimultaneously assigned to task 119894 and task 119895 the sequencebetween two tasksmust be determined either 119894 precedes 119895 or 119895precedes 119894 Constraint (7) is the big-119872 formulation to enforce
4 Mathematical Problems in Engineering
the relationship between the sequencing variable and thecontinuous starting time variable Constraint (8) calculatesthe makespan of project while Constraint (9) regulars thedomains of the variables
4 GVNS-Based Memetic Algorithm
In this section we describe in detail the general solutionmethodology and the supporting procedures in the followingsubsections
41 Search Space and Evaluation Function For a givenproblem the GVNS-MA searches a space Ω composed of allpossible assignments respecting skill constraints in any orderof tasks including both legal and illegal configurations Thesize of the search space Ω is bounded by 119874(119898119899 lowast 2119899)
Based on preceding notations and depictions to evaluatethe quality of a candidate solution 119904 isin Ω we adopt anevaluation function which is defined as 119891(119904) induced by 119904
119891 (119904) = 119898119886119909119894119898119894119911119890 119865119894 119894 isin 119881 + 119872 lowast sum(119894119895)isin119864
120575 (119894 119895) (10)
120575 (119894 119895) =
1 (119894 119895) isin 119864 119878119895 lt 1198651198940 otherwise
forall119894 isin 119881 119895 isin 119881 (11)
where 119872 is a large positive constant such that 119872 997888rarr infin as119899 997888rarr infin The first part of (10) represents the completiontime of the last unfinished task and the last part is anaugmented penalty function wheresum(119894119895)isin119864 120575(119894 119895) denotes thedegree violation to precedence relationships If 120575(119894 119895) =0 holds for any (119894 119895) isin 119864 it demonstrates the solutionis feasible corresponding to a legal configuration and itsevaluation function will all depend on the first part equalto the completion time of the schedule When a probleminstance admits no solution able to satisfy 120575(119894 119895) = 0 forall(119894 119895) isin119864 the search space of GVNS-MA is empty and no feasiblesolution can be found Given two solutions 1199041015840 and 11990410158401015840 1199041015840is better than 11990410158401015840 if 119891(1199041015840) le 119891(11990410158401015840) This statement impliesan assumption that a better solution has fewer precedenceconstraint violations
42Methodology and General Procedure Let119875 denote a pop-ulation of 119901 candidate configurations Let 119904119887 119904119908 represent thebest solution attainable so far and the worst solution in 119875 (interms of the evaluation function in Section 41) respectivelyLet119875119886119894119903119878119890119905 be a set of solution pairs (119904119894 119904119895) initially composedof all possible pairs in 119875 Next the proposed GVNS-MA canbe described as depicted in Algorithm 1
GVNS-MA first builds an initial population 119875 including119901 candidate configurations by the procedures in Section 43Then the algorithmenters into awhile loopwhich constitutesthe main part of the GVNS-MA On each new generationthe subsequent operations are executed In the first place aconfiguration pair (119904119894 119904119895) is taken at random and deleted fromPairSet Next GVNS-MA builds with a crossover operator(see Section 44) a new configuration 119904119888 (the offspring)After that the offspring is used as a starting point to further
(1) Input Problem instance I the size of population(119901) the depth of GVNS 120572
(2) Output the best configuration found during thesearch
(3) 119875 = 1199041 119904119901 larr997888 Population initialization(119868) lowastSec-tion 43lowast
(4) 119875119886119894119903119878119890119905 larr997888 (119904119894 119904119895) 1 le 119894 lt 119895 le 119901(5) while 119875119886119894119903119878119890119905 = do(6) Isolate a solution pair (119904119894 119904119895) isin 119875119886119894119903119878119890119905 ran-
domly(7) 119875119886119894119903119878119890119905 larr997888 119875119886119894119903119878119890119905 (119904119894 119904119895)(8) 119904119888 larr997888 crossOver(119904119894 119904119895) lowastSection 44lowast(9) 119904 larr997888 GVNS Operator(119904119888 120572) lowastSection 45lowast(10) if 119891(119904) le 119891(119904119908) then(11) UpdatePopulation(119904 119875 119875119886119894119903119878119890119905 119891(119904)) lowastSec-
tion 46lowast(12) end if(13) end while(14) Return 119904119887 larr997888 arg min 119891(119904119894) 119894 = 1 119901
Algorithm 1 Main sketch of the proposed GVNS-MA
improve by the GVNS operator (see Section 45) Finally ifthe improved offspring 119904 is better than the worst solution 119904119908in 119875 it is used to update the population 119875 and PairSet Thedetailed update operations are described in Section 46 Thewhile loop continues until PairSet becomes empty At the endof the while loop the algorithm terminates and returns thebest configuration 119904119887 found during the search Note that thedepth of GVNS 120572 represents a maximum number betweentwo iterationswithout improvement regarded as the stoppingcriterion of the GVNS operator
43 Population Initialization In order to build the initialpopulation (in Algorithm 2) the construction operator togenerate a new solution is executed 3 times 119901 times From thescratch a new configuration is constructed as follows Theoperator starts from assigning each task with a randomresource satisfying skill constraint Subsequently all tasksassigned to same resource are sequenced randomly at the endof previous task-to-resource phase Then for each generatedsolution the GVNS operator with the evaluation function119891(119904) (see Section 41) is used to optimize it to a local optimumand the obtained 119901 best configurations are selected to formthe initial population The detailed procedures are describedin Algorithm 2
44 The Crossover Operator Within a memetic algorithmthe crossover operator is another essential ingredient whosemain goal is to bring the search process to new promisingsearch regions to diversify the search In this paper theoffspring of two parent configurations 1199041 = (1198601 1198791198761) 1199042 =(1198602 1198791198762) shows as 119904119900 = (119860119900 119879119876119900) To inherit the advantagesof parent solutions the tasks assigned to same resource in 11990411199042 are given priority to keep the task-to-resource assignmentunchanged As for the remain unassigned tasks and 1198791198760
Mathematical Problems in Engineering 5
(1) Input The set 119881 = V1 V2 V119899 of 119899 tasks the set 119870 =1198961 1198962 119896119898 of 119898 renewable resources skill constraintsprecedence relationships and the size of population (119901)
(2) Output The best 119901 configurations(3) for pop = 1 3119901 do(4) Set RA = 119881(5) while 119877119860 = do(6) Choose V119894 randomly from 119877119860 remove V119894 from 119877119860
randomly isolate 119896119895 isin 119870119894mdash the set covering all re-sources can perform V119894 and record this assignmentas 119860(V119894) = 119895
(7) end while(8) for 119895 = 1 119898 do(9) Generate a random task sequence 119879119876119895 for candidate
tasks assigned to 119896119895(10) end for(11) Calculate the objective value 119891(119860TQ) with the evalu-
ation function 119891(119904) (see Section 41)(12) 119904119887119901119900119901 larr997888 GVNS operator(119891(119860TQ) 120572) lowastSection 45lowast(13) end for(14) Sort the 3119901 configurations in an ascending sort of evalua-
tion function values and return the first 119901 solutions as theinitial population
Algorithm 2 Population initialization
they will be determined by same methods in Section 43Analogously we apply the GVNS operator (see Section 45)to the offspring to finally gain the candidate for furtherpopulation updatingThe principle of this operator is detailedin the procedure crossover (Algorithm 3)
45 GVNS Operator This section discusses the local opti-mization phase of GVNS MA a key part of memetic algo-rithm Its function ensures an intensified search to locatehigh quality local optima from any starting point Here wedesign a generable variable neighborhood search (GVNS)heuristic as the local refinement procedure which showsgood performance compared to other variable neighborhoodsearch variants in terms of local search capability
Given three neighborhood structures 1198731 11987321198733 and aninitial solution (1199040) our GVNS operator does the refinementas follows Attention the quality of any solution is evaluatedas depicted in Section 41 To start with the sequential orderSO which determines the applying sequence of these neigh-borhood structures is at random generated For exampleassuming that the sequence SO equals (3 1 2) the searchbegins from1198733 and ends at1198731 at the given iteration For eachneighborhood structure a new local optima 11990410158401015840 is obtainedby applying the corresponding local search procedures to theincumbent solution 1199041015840 set at 1199040 at the beginning of GVNSprocedure If 11990410158401015840 is better than 1199041015840 1199041015840 is updated with thenew solution 11990410158401015840 accepted as a descent to continue the localsearch for current neighborhood otherwise the search turnsto the next neighborhood structure in SO One iterationterminates until the last neighborhood structure in SO isexplored and then the search goes on with the next iteration
until the stopping criteria is met ie the best solution 1199041015840has not improved for 120572 consecutive iterations The generalsketch of GVNS operator is described in Algorithm 4 andthe neighborhoods employed as well as the technique tocalculate objective value rapidly are depicted in the followingsubsections
451 Move and Neighborhood Three neighborhoods119873119896 (119896 = 1 3) are adopted in GVNS MA The neighbor-hood 1198731 is defined by the swap move operator which swapstwo tasks processed by same resource and keeps previoustask-to-resource assignment unchanged As such given asolution 119904 the swap neighborhood 1198731(119904) of 119904 is composedof all possible configurations that can be applied with theswap move to 119904 The neighborhood 1198732 is designed on thebase of reversion which reverses all the tasks incorporatedinto two designated random tasks of one resource As forthe neighborhood 1198733(119904) it is designed by the alter moveoperator which alters assigned resource from the originalto another resource equipped with demanded skill for oneselected randomly task with a random position in the tasksequence of given resource To efficiently assess the qualityof any neighborhood solution we devise a rapid evaluationtechnique for neighborhood solutions which is committedgreatly to the computational efficiency of the GVNS-MA
452 Rapid Evaluation Mechanism Our rapid evaluationtechnique to neighborhood solutions realizes through effec-tively calculating the move value (Δ119891) which identifies thechange in the evaluation function 119891 (see Section 41) ofeach possible move applicable to the incumbent solution
6 Mathematical Problems in Engineering
(1) Input Two parent solutions 1199041 = (1198601 1198791198761) 1199042 = (11986021198791198762) Problem instance I
(2) Output The offspring 119904119900 = (119860119900 119879119876119900)(3) Set RA = 119881 contains 119899 tasks remaining to be assigned
resource to and 119899119906119898 represents the size of 119877119860(4) for 119903 = 1 119899 do(5) if 1198601(V119903) == 1198602(V119903) then(6) 119860119900(V119903) = 1198601(V119903)(7) RA=RAV119903(8) end if(9) end for(10) for 119903 = 1 num do(11) Assign randomly resource 119896119895 isin 119870119903 to V119903(12) end for(13)Obtain 119879119876119900 and refine the offspring 119904119900 = (119860119900 119879119876119900) in the
same way as Algorithm 2 lowastSection 43lowast
Algorithm 3 Procedure crossover
(1) Input Initial solution 1199040 a set of neighborhood structures119873119896 (119896 = 1 3) 120572
(2) Output The current best solution 119904119887 found during GVNSprocess
(3) Calculate the objective value 119891(1199040) according to the eval-uation function in Section 41
(4) 1199041015840 larr997888 1199040 lowast1199041015840 is the current solution lowast(5) 119904119887 larr997888 1199041015840 lowast119904119887 is the best solution found so farlowast(6) 119889 = 0 lowast119889 counts the consecutive iterations where 1199041015840 is not
improved lowast(7) repeat(8) Generate a random sequence (SO) to apply three
neighborhood structures(9) Apply the relevant mechanism (Section 451) in pre-
determined order specified by SO update 1199041015840 if a betterconfiguration is attained
(10) if 119891(1199041015840) lt 119891(119904119887) then(11) 119904119887 larr997888 1199041015840(12) reset the counter 119889 = 0(13) else(14) 119889 = 119889 + 1(15) end if(16) until 119889 == 120572(17) return 119904119887
Algorithm 4 GVNS operator
119904 It functions in the reduction of computational cost toevaluate any attainable neighborhood solution inspired bythe situation where the starting and finishing times of mosttasks will not be changed when neighborhood solutions aregenerated
For 1198731 and 1198732 generated by swap and reversion movethe set of tasks with changed starting and ending times onlyincorporate the elements ranking next to the isolated firsttask in the sequence of given resourceWithout considerationof deterioration only the elements located in the positionbetween two picked tasks are influenced As for 1198733 achieved
by altermove all the tasks lined up behind the designated taskin the sequence of its initial assigned resource and the newdistributed one are included Attention for three moves werecalculate the relevant parameters of above-mentioned tasksand the rest are ignored In addition the impact that the newsolutions is defying the precedence relationships to varyingdegrees will be respected in terms of 119872 lowast sum(119894119895)isin119864 120575(119894 119895) inevaluation function 119891 (see Section 41)
46 Updating population and PairSet As illustrated in Algo-rithm 1 the population119875 and thePairSet are updatedwhen an
Mathematical Problems in Engineering 7
Table 1 Settings of important parameters
Parameters Section Description Values119901 Section 42 population size for GVNS-MA 10120572 Section 45 depth of GVNS 50 000119872 Section 41 penalty value for a violation 1000 1 times 107
to precedence constraint
excellent offspring is obtained through the crossover operatorand improved further by GVNS operator First of all if itis better than the worst solution 119904119908 in 119875 for any improvedoffspring solution 119904 the worst configuration 119904119908 is replaced bythe offspring solution 119904 When the population is updated thePairSet should be updated accordingly all pairs containing 119904119908solution are deleted from setPariSet and all pairs generated bycombining 119904 solution with others in 119875 are incorporated intoPairSet
5 Computational Experiments and Results
This section plans to assess the proposed method GVNS-MA through having comparisons with the state-of-the-artmethods in the literature For the lack of known benchmarkdata for handingMS-RCPSPLDwe firstly apply the proposedGVNS-MA to solve the MS-RCPSP on exist benchmarkinstances in favor of argument its effectiveness Then onthis basis the GVNS-MA will be examined on the modifiedproblem set
51 Benchmark Instances For the purpose of assessingGVNS-MA fully and comprehensively computational exper-iments will be conducted on two sets of instances where thefirst set is composed of 30 benchmark instances irrespec-tive of deterioration and its available in Myszkowski et al[46] which are artificially created in a base of real worldobtained from the Volvo IT Department in Wroclaw Thefull information of each instance including tasks durationsresource capabilities or precedence between tasks has beengiven As for the other set it consists of 45 instances generatedwith some modifications on first set to consider the lineardeterioration The detail will be described in Section 54
52 Parameter Settings and Experiment Protocol OurGVNS-MA was programmed in MATLAB R2015b and all thereported computational experiments presented below wereexecuted on a personal computer equipped with an IntelCore i3 processor (310 GHz CPU and 2GB RAM) in theenvironment ofWindows 7 OS To eliminate the randomnessas much as possible twenty replications for each instance arecarried out
Table 1 shows the descriptions and settings of the param-eters adopted in GVNS-MA determined by preliminaryexperiments Our memetic algorithm rests upon only threeparameters the population size 119901 the depth of generalvariable neighborhood search 120572 and the price for a violationto precedence constraint 119872 For 119901 and 120572 we follow Lai andHao [55] and set 119901 = 10 120572 = 50000 while the parameter 119872
is set at 1000 for the first experimental group and 1 times 107 forthe second
53 Experimental Results without Deterioration Our firstexperimental group aims to evaluate the performance ofour GVNS-MA on the set of 30 known instances with atmost 200 tasks and 40 renewable resources Without regardto deterioration it means that the GVNS-MA will set thedeteriorating rates of all tasks at 0 when it deals with therelevant computations Table 2 records the computationalresults solved by the GVNS-MA with the goal of durationoptimization aswell as the results achieved by other referencealgorithms in the literature
Notice that the instance name (columns 1) contains itsfull description Take the instance named 100-10-26-15 asexample the number 100 represents the number of tasksincluded and 10 denotes the quantity of renewable resourcesprovided As for the number 26 and 15 it illustrates theamount of precedence relationships and the number ofdifferent introduced skills Column 2 of Table 2 indicates theprevious minimum objective values (119891119901119903119890119887) in the literaturewhich are compiled from the best solutions yield by tworecent and best performing algorithms namely GRASP[48] and DEGR [49] Columns 3 to 4 give the best resultsobtained by DEGR and GRASP The corresponding resultsof the GVNS-MA are given in columns 5 to 7 includingthe minimum objective value (119891119887119890119904119905) over 20 independentruns the average objective value (119891119886V119892) and the averagecomputing time in seconds (Time(s)) to reach 119891119887119890119904119905 The rowBest indicates a total number of instances where the specificmethod achieves optimal among three algorithms The bestone is indicated in italic In addition to verify whetherthere exists an essential difference between the best resultsof GVNS-MA and other reference algorithms the relativepercentage deviation (RPD) is defined by the equation
119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905
119891119901119903119890119887times 100 (12)
where a positive value of 119877119875119863 means an improvement ofresult achieved by GVNS-MA while the negative numberrepresents a worse solution
Table 2 discloses that the outcomes from our GVNS-MAare noteworthy compared to the state-of-the-art results inthe literature GVNS-MA improves the previous best knownresults for 19 instances and matches for 7 cases Comparedwith the 8 out of 30 cases solved by DEGR and 6 bestsolutions achieved by GRASP these data clearly indicatethe superiority of GVNS-MA compared to the previousexcellent methods Additionally it can be observed that
8 Mathematical Problems in Engineering
Table 2 Comparison of the GVNS-MA with other algorithms on known MS-RCPSP dataset [48] Best results are indicated in italic
instances 119891119901119903119890119887 DEGR GRASP GVNS-MA 119877119875119863()119891119887119890119904119905 119891119886V119892 119879119894119898119890(119904)
100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050
the improvement achieved by GVNS-MA is up to 939for instance 100-20-65-15 accompanying that the average119877119875119863() equals to 050
54 Experimental Results with Linear Deterioration Theprevious comparisons and discussions in Section 53 demon-strate the advantages of GVNS-MA in solving the relatedissues of MS-RCPSP In this section the aforementioneddataset with some modifications is used to assess the capabil-ity of GVNS-MA to solve the MS-RCPSPLDThe proceduresof generating the testing instances and analysis of the resultsare described below To make the benchmark instances meetthe considered linear deterioration precisely the deteriora-tion rate (119889119903119894) for task 119894 (119894 isin 119881) is generated randomly fromthree intervals (0 05] [05 1] and (0 1] similar to Cheng et
al [19] to shed light on the influence of the different valuerange of deterioration rate on its effectiveness
Since the extra included deterioration rate we providetwo additional heuristics for the initial population generationof GVNS-MA Both the two methods affect the phase ofgenerating 119879119876119895 determining the sequence of tasks assignedto resource 119895 119895 isin 119870 The first heuristic considers thesequence in descending order of deterioration rate (ℎ) whilethe other rests upon an ascending order of ratio (119886ℎ)of the basic processing time and deterioration rate Themethods adopting the former and latter heuristic to popula-tion initialization are dubbedGVNS-119872119860ℎ and GVNS-119872119860119886ℎrespectively
Here instanceswith 100 tasks fromMyszkowski et al [46]are isolated to attain the researched objects which fit with theunique nature of the MS-RCPSPLDmore To account for the
Mathematical Problems in Engineering 9
three intervals from which the deterioration rate is drawn3 extended cases are needed to solve for each instance Forconvenience these instances are denoted by adding a suffixfor identification to different intervals For example 100-10-26-15 1 represents the original case 100-10-26-15 is modifiedby adding the deterioration rates produced in (0 05] to thedurations of tasks In total there are 45(3 times 15) instancesrandomly generated
Due to zero known results in literature for same datasetthe improved tabu search (ITS) proposed by Dai et al [54]who discussed the MS-RCPSP under step deterioration anda path relinking algorithm (PR) [55] based on the populationpath relinking framework are programmed as referencealgorithms
Table 3 reports the computational results achieved by theITS PR GVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ on theset of 45 benchmark instances 119891119887119890119904119905 denotes the minimumobjective value and 119891119886V119892 is computed as the average objectivevalue of 20 runs
First Table 3 discloses that the solutions obtained byGVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ are better thanthe ITS and PR for any instance from the perspectiveof both quality of schedule and runtime To some extentthese results demonstrated the differences between lineardeterioration and step deterioration and the superiority ofmemetic algorithm framework Second these three methodsdiffering in the sort order of tasks in initialize phase behavesimilarly where GVNS-MA obtains the best 15 out of 45instances 17 for GVNS-MAℎ and 14 for GVNS119886ℎ in termsof 119891119887119890119904119905 Specifically GVNS-MA and GVNS-MAℎ attain theoptimal simultaneously for the instance 100-5-48-9-1 Froma view point of 119891119886V119892 and run time three methods alsohave a balanced performance Third as far as three differentintervals to generate deterioration rate are concerned thephenomenon did not happen that the relevant algorithmsdisplay strikingly different behavior In other words theperformance of the proposed algorithm is not sensitive to thesetting of deterioration rate
55 Analysis and Discussions In this section we study twoessential ingredients of the proposed GVNS-MA to getan insight to its performance One is the rapid evaluationmechanism the other is the role of the memetic framework
551 Importance of Rapid Evaluation Mechanism GVNS-MA with rapid evaluation mechanism only calculates therelevant parameters of some particular tasks rather thanall when the procedure computes the objective value ofa neighborhood solution To highlight the key role ofthe rapid evaluation mechanism two sets of comparisonexperiments are carried out on generated dataset with twoalgorithms GVNS-MA and GVNS-MA0 including sameingredients with GVNS-MA except for the computation ofobjective value When GVNS-MA0 figures up the value of aneighborhood solution it computes all relevant parametersagain
Table 4 records the experimental results carried out on thedataset [46] without consideration of deterioration whereas
Table 5 shows the comparisons of GVNS-MA and GVNS-MA0 about the set of 15 instances generated in Section 54on account of the indiscrimination in three intervals Col-umn 2 and 5 record the best attained by two algorithmsColumn 3 and 6 indicate the minimum time cost to a finalfeasible schedule with one run of procedure Note that thebest objective value cannot be guaranteed as the output ofshortest runtime As for the parameters in column 4 and 7they represent the mean runtime Finally two parameters119863119864119881119904ℎ119900119903119905119890119904119905 and 119863119864119881119886V119892 are used to disclose the runtimedeviation of two methods defined by equations
119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051
times 100 (13)
and
119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922
119879119886V1198921times 100 (14)
respectively The positive value of 119863119864119881119904ℎ119900119903119905119890119904119905() and119863119864119881119886V119892() means that GVNS-MA0 has better performanceand negative value tells GVNS-MA is prior to GVNS-MA0in terms of time cost And the rows Better and Worserespectively show the number of instances for which thecorresponding results of the associated algorithm are betterand worse than the other
The results summarized in Table 4 disclose that theGVNS-MA has an overwhelming advantage over GVNS-MA0 in terms of the computation time to solve MS-RCPSPleaving out the deterioration effect Indeed the shortestruntime 119879119904ℎ1199001199031199051198901199041199051 of the GVNS-MA method is better thanthe shortest runtime 119879119904ℎ1199001199031199051198901199041199052 of GVNS-MA0 for 30 out of30 representative instances and the average runtime 119879119886V1198921 isbetter for 28 out of 30 instances Meanwhile the average valueof 119863119864119881119904ℎ119900119903119905119890119904119905() equals -1379 accompanying with a highof -1880 percent in 119863119864119881119886V119892()
However focusing on Table 5 the results of twoapproaches are neck and neck and GVNS-MA lost its earlysuperiority in MS-RCPSP In terms of shortest runtimeGVNS-MA successes for 7 out of 15 tested instances whileGVNS-MA0 reaches optimal for the remain As for averageruntime GVNS-MA performs better for 9 out of 15 examplesand GVNS-MA0 achieves reversion in others 6 instancesWith these data it will be hard to judge the true benefits ofone approach versus the other
To figure out the reason of this phenomenon we shouldcome back to the inner rationale of rapid evaluation mecha-nism When GVNS-MA computes the completion time of aneighborhood solution it only recalculates the tasksrsquo relatedparameters influenced by the particular move InMS-RCPSPa move including swap reverse and alter will affect justa small number of tasks But for MS-RCPSPLD instancesany move can cumulatively effect on a large proportion oftasks because of the existing deterioration Consequently theruntime saved in computing some unchanged parametersmay not make up for the time spent on isolating the changedtasks
10 Mathematical Problems in Engineering
Table3Summaryandcomparis
onon
thes
etof
45newgeneratedinsta
nces
with
119899=10
0of
GVN
S-MA
GVN
S-119872
119860 ℎG
VNS-
119872119860 119886ℎand
theT
Sheuristic[54]
andPR
[55]B
estresultsare
indicatedin
italic
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-10-26-15
165572
6598
13899
963646
6492
66871
63274
6392
33092
661967
6379
730698
62632
63629
29422
100-10-26-15
260
1396
618416
28607
608971
625436
79283
587145
613934
37496
580557
615474
23879
572794
615156
34826
100-10-26-15
3172856
184175
31236
1646
96
175513
59776
160283
16936
22225
166221
17124
17541
163898
172228
3461
100-10-47-91
7008
72053
40304
6974
471232
10018
69445
70098
20666
67645
6915
731499
69673
7039
36314
100-10-47-92
700352
739287
39112
703439
721152
70563
672221
715138
2417
9672304
6959
20614
693553
702369
25114
100-10-47-93
188206
195206
4816
818358
190863
93463
174162
184389
30008
177323
183958
19383
173708
183368
41432
100-10-48-15
165868
6914
13991
567093
6890
88879
164
851
65526
3491
663253
66595
22686
6479
86610
226662
100-10-48-15
2638747
659809
42043
621357
658747
9116
161867
651609
4116
1590432
625462
27097
618507
6346
3233655
100-10-48-15
3170999
174232
29659
171864
178061
73395
159497
166887
23395
161483
166366
3216
9163501
170273
27056
100-10-64-91
70527
7572
53770
371049
7692
773024
6872
27092
931287
6884
71215
26024
67894
69682
2775
6100-10-64-92
705806
744775
39415
7040
32738141
6279
4663729
711114
3013
968421
719153
22806
696727
736833
3991
8100-10-64-93
193844
206206
3115
32006
42
2044
03
108918
189324
19445
35415
180842
200842
3104
193844
199996
31871
100-10-64-15
170119
71497
34626
71075
7279
683543
69346
70022
39097
6691
69112
21249
70042
7079
617116
100-10-64-15
2690964
744775
3899
8672578
72823
61684
6606
14685085
22491
62823
682957
3178
4626331
666418
2376
100-10-64-15
3193844
218506
31871
194247
204235
71318
1806
92
1860
08
25336
187149
192556
45438
185585
190957
25296
100-20-22-15
119089
19355
29631
1936
19739
5517
618883
19118
2222
18572
1898
21438
1878
18965
2474
2100-20-22-15
24615
847731
36854
46032
46714
6792
645371
46008
28455
45218
46627
2892
645833
4633
2312
4100-20-22-15
326328
2798
128528
27212
2697
36972
92597
26419
2812
25563
25918
17531
25575
26254
21352
100-20-46-15
126243
2673
33424
26214
26631
60354
25862
26329
4119
925511
25896
1215
325476
26006
17466
100-20-46-15
260
647
66244
34408
59367
63596
71878
5796
16165
17216
56542
5942
26696
5465
59285
22313
100-20-46-15
332421
3490
937523
33539
3417
977634
31695
32909
16433
31651
33263
29116
32841
33349
2791
100-20-47-91
19007
1975
929027
1916
319685
5776
318864
1912
28311
1874
719269
2732
18455
18892
30731
100-20-47-92
50591
53481
37119
47839
51484
102173
4803
49319
3549
46278
4876
435418
44966
47893
24241
100-20-47-93
30802
31827
4572
630631
31365
6995
829437
30352
57846
27712
29458
29737
28399
2879
927559
100-20-65-15
19013
492446
26413
89865
90543
6795
788801
90095
17505
86826
89518
15052
87305
89338
2843
100-20-65-15
2253899
2606
7628718
250126
253267
5876
3244762
250316
4099
7243449
24751
17659
242353
244944
2573
5100-20-65-15
3110
567
1115886
27491
109874
113685
53418
105215
109224
2513
8108595
111243
1591
1104829
106776
1412
6100-20-65-91
19113
1978
341248
1895
719548
79561
18369
18898
30561
18697
1914
61896
18694
19265
3097
9100-20-65-92
46242
4776
865469
4593
447443
58112
4495
746229
1768
44719
45985
22324
45593
46591
30606
100-20-65-93
28018
28776
4894
327512
2810
868532
2719
2772
124279
26587
27691
3101
26455
27845
2473
100-5-22-151
500988
514285
2897
8510285
505098
84526
486586
499318
43529
494364
500988
18444
498567
510285
22335
100-5-22-152
1193440
1227080
4319
4119
2300
1210150
61537
1118610
1199248
24322
1184940
1219514
20664
1138490
1226780
26656
100-5-22-153
663947
700818
5691
2661553
792627
7997
660
6322
652627
10639
592509
6279304
2619
8643856
6597526
17267
100-5-46
-151
675739
749763
3679
7652793
701457
137862
63503
679663
5075
8636903
687941
29888
649164
671982
36202
100-5-46
-152
1404
680
1453430
58595
1443529
1470970
108595
1298430
1370870
26564
1315940
1360
830
23536
1332210
1399014
2417
6100-5-46
-153
795664
870313
50687
7977713
863567
6891
1740178
796512
81792
4755866
7786062
16809
752568
80128
1876
3100-5-48-9
1563787
586832
31214
5666
27
78386
84807
55206
560407
2415
55206
563664
25018
554299
56657
2495
2100-5-48-9
21284230
1315720
3614
51289811
1364
650
71286
1222350
1285020
2795
8116
3860
1259332
2812
21233010
1268572
34659
100-5-48-9
3684299
692453
45335
66785
717279
29393
8657279
67622
4561
6604
26
670917
41286
64822
667213
828446
Mathematical Problems in Engineering 11
Table3Con
tinued
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-5-64
-151
581916
627374
55661
576589
618167
102415
5544
26
579234
3516
7546543
590352
29678
56942
580175
56619
100-5-64
-152
120944
01261510
31296
1190830
1257643
123533
1131450
1183014
40595
1118430
1151812
46305
1047250
1158556
3337
100-5-64
-153
642267
709602
3498
8634651
688889
89403
612857
6657858
3798
5626771
660614
44591
1624753
674996
261623
100-5-64
-91
550231
577524
37365
5544
81
566747
98693
528177
5404
36
3393
530748
543895
48396
515984
536747
2993
2100-5-64
-92
1214340
1271610
40553
1183479
1236750
11992
711140
60115
5584
41523
11640
101201126
40287
1121650
1159384
27286
100-5-64
-93
610356
648765
36395
6151502
632323
87448
604586
6210378
31363
594514
6171502
30543
595191
623223
2774
9119861
119890119904119905
00
00
1514
1716
1415
119879119900119905119886
11989745
4545
4545
4545
4545
45
12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
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Mathematical Problems in Engineering 3
discussions about linear deteriorating scheduling problemswere shown in Bachman and Janiak [30] Oron [31] Wangand Wang [29] Wu et al [28]
22 MS-RCPSP For the sake of its practicality RCPSP hasreceived more and more attentions [32] U and Kusiak [33]Babiceanu et al [34] and Coban [35] deal with the RCPSPin a dynamic real time way To obtain particular schedulesat some point metaheuristics are the most used techniquesfor solving RCPSP including Tabu search [36 37] SimulatedAnnealing [38 39] or Genetic Algorithm [40] SwarmIntelligence metaheuristics [41] are also effective for solvingrelated issues For example Particle Swarm Optimization[42] Bee Colony Optimization [43] or the popular AntColony Optimization [44 45] Although classical RCPSP isdeeply investigated and numerous methods could be easilycompared using PSPLIB instances it is not same for MS-RCPSP There are not too many researches paying attentionto MS-RCPSP
Myszkowski et al [46] defined new problem MS-RCPSPand benchmark to compare the effectiveness of examinedapproaches Next a hybrid ant colony optimization approach(HAC) [47] which links classical heuristic priority and theworst solutions stored by ants to update pheromone valuewas proposed To research this problem more thoroughlya greedy randomized adaptive search procedure (GRASP)[48] a hybrid differential evolution and greedy algorithm[49] and a Co-Evolutionary algorithm [50] are developed toimprove the quality of solutions Before this period Alanzi etal [51] proposed a lower bound using a linear programmingscheme for the RCPSP to solve the new extended MS-RCPSPmodelwhile Santos and Tereso [52] developed a filtered beamas a bonus for early completion into account Zheng et al [53]presented a teaching-learning-based optimization algorithmwith a task-resource list-based encoding scheme combiningthe task list and the resource list and a left-shift decodingscheme where the balance between global exploration andlocal exploitation to achieve satisfactory performances wasmainly stressed Dai et al [54] did the related work on theintegration of MS-RCPSP and proposed step structures aswell as two mutation operators
3 Problem Statement
To define two considered MS-RCPSP and MS-RCPSPLD aswell asmake them clear the problem description and amixedinteger programming model can be described as followsLet 119866(119881 119864) be a task-on-node network consisting of a set119881 of nodes denoting task 119894 (119894 isin 119881) and a set 119864 of edgesrepresenting the precedence relationships between a pair oftasks (119894 119895) (119894 119895) isin 119864 Specifically each pair (119894 119895) isin 119864 meansthe task 119894 precedes 119895 In other words task 119895 cannot start untiltask 119894 finished The duration of each task 119894 in MS-RCPSP is aprior known and constant while the value of duration inMS-RCPSPLD turning out to hinge on the basic processing time119886119894 and deteriorating rate ℎ119894 can be calculated by the function119889119894 = 119886119894 + 119878119894 lowastℎ119894 where 119878119894 shall be the earliest available time ofassigned resource 119896 119894 isin 119881119896 To perform any task 119894 the specific
skill 119902119894 also is required To make the 119899 tasks completed aset of 119870 of 119898 renewable resources which unassigned taskscan compete for when resources are idle will be providedResources differ inmastered skills and let119876119896 denote the skillscovered by resource 119896 Naturally a subset 119881119896 of 119881 also isavailable to incorporate all tasks that can be processed byresource 119896 (119902119894 isin 119876119896 119894 isin 119881119896) In same way a subset ofresources 119870119894 including all resources which can be utilizedto handle task 119894 is obtained Moreover each resource canperform at most one task at a time and a task can be executedby at most one resource simultaneously The objective is tominimize the makespan (the completion time of all tasks)
Based on the above descriptionwe formulate the problemas a 0-1 integer programming model Firstly the binaryvariables 119909119895119896 receive value 1 if task 119895 is assigned to resource119896 and 0 otherwise As for the binary variables 119910119894119895 are set to 1if task 119894 is scheduled to precede task 119895 0 otherwiseThen MS-RCPSP and MS-RCPSPLD can be formulated by objectivefunction (1) subject to constraints (2) to (9)
119872119894119899119894119898119894119911119890 119862119898119886119909 (1)
subject to 119865119894 = 119878119894 + 119889119894 forall119894 isin 119881 (2)
sum119896isin119870119895
119909119895119896 = 1 forall119895 isin 119881 (3)
119909119895119896 = 0 forall119895 notin 119870119895 (4)
119904119895 minus 119904119894 ge 119889119894 forall (119894 119895) isin 119864 (5)
119910119894119895 + 119910119895119894 ge 119909119894119896 + 119909119895119896 minus 1
forall (119894 119895) isin 119864 119896 isin 119870(6)
119878119895 ge 119878119894 + 119889119894 minus 119872(1 minus 119910119894119895)
forall119894 = 119895 119894 119895 isin 119881(7)
119862119898119886119909 ge 119865119894 forall119894 isin 119881 (8)
119909119895119896 119910119894119895 isin 0 1
119904119895 ge 0
forall119894 isin 119881 119895 isin 119881(9)
In the above mode objective function (1) denotes theoptimal direction of MS-RCPSP and MS-RCPSPLD Equa-tion (2) defines the finishing time of any task Constraint (3)restricts exactly one resource to a specific task among all avail-able resources Constraint (4) respects the skill constraintsbetween resources and tasks Constraint (5) highlights theprecedence relationship between a pair of nodes (119894 119895) inother words task 119895 can start only after 119894 is finished Constraint(6) shows logical relationships between the assignment vari-ables and sequencing variables That is when resource 119896 issimultaneously assigned to task 119894 and task 119895 the sequencebetween two tasksmust be determined either 119894 precedes 119895 or 119895precedes 119894 Constraint (7) is the big-119872 formulation to enforce
4 Mathematical Problems in Engineering
the relationship between the sequencing variable and thecontinuous starting time variable Constraint (8) calculatesthe makespan of project while Constraint (9) regulars thedomains of the variables
4 GVNS-Based Memetic Algorithm
In this section we describe in detail the general solutionmethodology and the supporting procedures in the followingsubsections
41 Search Space and Evaluation Function For a givenproblem the GVNS-MA searches a space Ω composed of allpossible assignments respecting skill constraints in any orderof tasks including both legal and illegal configurations Thesize of the search space Ω is bounded by 119874(119898119899 lowast 2119899)
Based on preceding notations and depictions to evaluatethe quality of a candidate solution 119904 isin Ω we adopt anevaluation function which is defined as 119891(119904) induced by 119904
119891 (119904) = 119898119886119909119894119898119894119911119890 119865119894 119894 isin 119881 + 119872 lowast sum(119894119895)isin119864
120575 (119894 119895) (10)
120575 (119894 119895) =
1 (119894 119895) isin 119864 119878119895 lt 1198651198940 otherwise
forall119894 isin 119881 119895 isin 119881 (11)
where 119872 is a large positive constant such that 119872 997888rarr infin as119899 997888rarr infin The first part of (10) represents the completiontime of the last unfinished task and the last part is anaugmented penalty function wheresum(119894119895)isin119864 120575(119894 119895) denotes thedegree violation to precedence relationships If 120575(119894 119895) =0 holds for any (119894 119895) isin 119864 it demonstrates the solutionis feasible corresponding to a legal configuration and itsevaluation function will all depend on the first part equalto the completion time of the schedule When a probleminstance admits no solution able to satisfy 120575(119894 119895) = 0 forall(119894 119895) isin119864 the search space of GVNS-MA is empty and no feasiblesolution can be found Given two solutions 1199041015840 and 11990410158401015840 1199041015840is better than 11990410158401015840 if 119891(1199041015840) le 119891(11990410158401015840) This statement impliesan assumption that a better solution has fewer precedenceconstraint violations
42Methodology and General Procedure Let119875 denote a pop-ulation of 119901 candidate configurations Let 119904119887 119904119908 represent thebest solution attainable so far and the worst solution in 119875 (interms of the evaluation function in Section 41) respectivelyLet119875119886119894119903119878119890119905 be a set of solution pairs (119904119894 119904119895) initially composedof all possible pairs in 119875 Next the proposed GVNS-MA canbe described as depicted in Algorithm 1
GVNS-MA first builds an initial population 119875 including119901 candidate configurations by the procedures in Section 43Then the algorithmenters into awhile loopwhich constitutesthe main part of the GVNS-MA On each new generationthe subsequent operations are executed In the first place aconfiguration pair (119904119894 119904119895) is taken at random and deleted fromPairSet Next GVNS-MA builds with a crossover operator(see Section 44) a new configuration 119904119888 (the offspring)After that the offspring is used as a starting point to further
(1) Input Problem instance I the size of population(119901) the depth of GVNS 120572
(2) Output the best configuration found during thesearch
(3) 119875 = 1199041 119904119901 larr997888 Population initialization(119868) lowastSec-tion 43lowast
(4) 119875119886119894119903119878119890119905 larr997888 (119904119894 119904119895) 1 le 119894 lt 119895 le 119901(5) while 119875119886119894119903119878119890119905 = do(6) Isolate a solution pair (119904119894 119904119895) isin 119875119886119894119903119878119890119905 ran-
domly(7) 119875119886119894119903119878119890119905 larr997888 119875119886119894119903119878119890119905 (119904119894 119904119895)(8) 119904119888 larr997888 crossOver(119904119894 119904119895) lowastSection 44lowast(9) 119904 larr997888 GVNS Operator(119904119888 120572) lowastSection 45lowast(10) if 119891(119904) le 119891(119904119908) then(11) UpdatePopulation(119904 119875 119875119886119894119903119878119890119905 119891(119904)) lowastSec-
tion 46lowast(12) end if(13) end while(14) Return 119904119887 larr997888 arg min 119891(119904119894) 119894 = 1 119901
Algorithm 1 Main sketch of the proposed GVNS-MA
improve by the GVNS operator (see Section 45) Finally ifthe improved offspring 119904 is better than the worst solution 119904119908in 119875 it is used to update the population 119875 and PairSet Thedetailed update operations are described in Section 46 Thewhile loop continues until PairSet becomes empty At the endof the while loop the algorithm terminates and returns thebest configuration 119904119887 found during the search Note that thedepth of GVNS 120572 represents a maximum number betweentwo iterationswithout improvement regarded as the stoppingcriterion of the GVNS operator
43 Population Initialization In order to build the initialpopulation (in Algorithm 2) the construction operator togenerate a new solution is executed 3 times 119901 times From thescratch a new configuration is constructed as follows Theoperator starts from assigning each task with a randomresource satisfying skill constraint Subsequently all tasksassigned to same resource are sequenced randomly at the endof previous task-to-resource phase Then for each generatedsolution the GVNS operator with the evaluation function119891(119904) (see Section 41) is used to optimize it to a local optimumand the obtained 119901 best configurations are selected to formthe initial population The detailed procedures are describedin Algorithm 2
44 The Crossover Operator Within a memetic algorithmthe crossover operator is another essential ingredient whosemain goal is to bring the search process to new promisingsearch regions to diversify the search In this paper theoffspring of two parent configurations 1199041 = (1198601 1198791198761) 1199042 =(1198602 1198791198762) shows as 119904119900 = (119860119900 119879119876119900) To inherit the advantagesof parent solutions the tasks assigned to same resource in 11990411199042 are given priority to keep the task-to-resource assignmentunchanged As for the remain unassigned tasks and 1198791198760
Mathematical Problems in Engineering 5
(1) Input The set 119881 = V1 V2 V119899 of 119899 tasks the set 119870 =1198961 1198962 119896119898 of 119898 renewable resources skill constraintsprecedence relationships and the size of population (119901)
(2) Output The best 119901 configurations(3) for pop = 1 3119901 do(4) Set RA = 119881(5) while 119877119860 = do(6) Choose V119894 randomly from 119877119860 remove V119894 from 119877119860
randomly isolate 119896119895 isin 119870119894mdash the set covering all re-sources can perform V119894 and record this assignmentas 119860(V119894) = 119895
(7) end while(8) for 119895 = 1 119898 do(9) Generate a random task sequence 119879119876119895 for candidate
tasks assigned to 119896119895(10) end for(11) Calculate the objective value 119891(119860TQ) with the evalu-
ation function 119891(119904) (see Section 41)(12) 119904119887119901119900119901 larr997888 GVNS operator(119891(119860TQ) 120572) lowastSection 45lowast(13) end for(14) Sort the 3119901 configurations in an ascending sort of evalua-
tion function values and return the first 119901 solutions as theinitial population
Algorithm 2 Population initialization
they will be determined by same methods in Section 43Analogously we apply the GVNS operator (see Section 45)to the offspring to finally gain the candidate for furtherpopulation updatingThe principle of this operator is detailedin the procedure crossover (Algorithm 3)
45 GVNS Operator This section discusses the local opti-mization phase of GVNS MA a key part of memetic algo-rithm Its function ensures an intensified search to locatehigh quality local optima from any starting point Here wedesign a generable variable neighborhood search (GVNS)heuristic as the local refinement procedure which showsgood performance compared to other variable neighborhoodsearch variants in terms of local search capability
Given three neighborhood structures 1198731 11987321198733 and aninitial solution (1199040) our GVNS operator does the refinementas follows Attention the quality of any solution is evaluatedas depicted in Section 41 To start with the sequential orderSO which determines the applying sequence of these neigh-borhood structures is at random generated For exampleassuming that the sequence SO equals (3 1 2) the searchbegins from1198733 and ends at1198731 at the given iteration For eachneighborhood structure a new local optima 11990410158401015840 is obtainedby applying the corresponding local search procedures to theincumbent solution 1199041015840 set at 1199040 at the beginning of GVNSprocedure If 11990410158401015840 is better than 1199041015840 1199041015840 is updated with thenew solution 11990410158401015840 accepted as a descent to continue the localsearch for current neighborhood otherwise the search turnsto the next neighborhood structure in SO One iterationterminates until the last neighborhood structure in SO isexplored and then the search goes on with the next iteration
until the stopping criteria is met ie the best solution 1199041015840has not improved for 120572 consecutive iterations The generalsketch of GVNS operator is described in Algorithm 4 andthe neighborhoods employed as well as the technique tocalculate objective value rapidly are depicted in the followingsubsections
451 Move and Neighborhood Three neighborhoods119873119896 (119896 = 1 3) are adopted in GVNS MA The neighbor-hood 1198731 is defined by the swap move operator which swapstwo tasks processed by same resource and keeps previoustask-to-resource assignment unchanged As such given asolution 119904 the swap neighborhood 1198731(119904) of 119904 is composedof all possible configurations that can be applied with theswap move to 119904 The neighborhood 1198732 is designed on thebase of reversion which reverses all the tasks incorporatedinto two designated random tasks of one resource As forthe neighborhood 1198733(119904) it is designed by the alter moveoperator which alters assigned resource from the originalto another resource equipped with demanded skill for oneselected randomly task with a random position in the tasksequence of given resource To efficiently assess the qualityof any neighborhood solution we devise a rapid evaluationtechnique for neighborhood solutions which is committedgreatly to the computational efficiency of the GVNS-MA
452 Rapid Evaluation Mechanism Our rapid evaluationtechnique to neighborhood solutions realizes through effec-tively calculating the move value (Δ119891) which identifies thechange in the evaluation function 119891 (see Section 41) ofeach possible move applicable to the incumbent solution
6 Mathematical Problems in Engineering
(1) Input Two parent solutions 1199041 = (1198601 1198791198761) 1199042 = (11986021198791198762) Problem instance I
(2) Output The offspring 119904119900 = (119860119900 119879119876119900)(3) Set RA = 119881 contains 119899 tasks remaining to be assigned
resource to and 119899119906119898 represents the size of 119877119860(4) for 119903 = 1 119899 do(5) if 1198601(V119903) == 1198602(V119903) then(6) 119860119900(V119903) = 1198601(V119903)(7) RA=RAV119903(8) end if(9) end for(10) for 119903 = 1 num do(11) Assign randomly resource 119896119895 isin 119870119903 to V119903(12) end for(13)Obtain 119879119876119900 and refine the offspring 119904119900 = (119860119900 119879119876119900) in the
same way as Algorithm 2 lowastSection 43lowast
Algorithm 3 Procedure crossover
(1) Input Initial solution 1199040 a set of neighborhood structures119873119896 (119896 = 1 3) 120572
(2) Output The current best solution 119904119887 found during GVNSprocess
(3) Calculate the objective value 119891(1199040) according to the eval-uation function in Section 41
(4) 1199041015840 larr997888 1199040 lowast1199041015840 is the current solution lowast(5) 119904119887 larr997888 1199041015840 lowast119904119887 is the best solution found so farlowast(6) 119889 = 0 lowast119889 counts the consecutive iterations where 1199041015840 is not
improved lowast(7) repeat(8) Generate a random sequence (SO) to apply three
neighborhood structures(9) Apply the relevant mechanism (Section 451) in pre-
determined order specified by SO update 1199041015840 if a betterconfiguration is attained
(10) if 119891(1199041015840) lt 119891(119904119887) then(11) 119904119887 larr997888 1199041015840(12) reset the counter 119889 = 0(13) else(14) 119889 = 119889 + 1(15) end if(16) until 119889 == 120572(17) return 119904119887
Algorithm 4 GVNS operator
119904 It functions in the reduction of computational cost toevaluate any attainable neighborhood solution inspired bythe situation where the starting and finishing times of mosttasks will not be changed when neighborhood solutions aregenerated
For 1198731 and 1198732 generated by swap and reversion movethe set of tasks with changed starting and ending times onlyincorporate the elements ranking next to the isolated firsttask in the sequence of given resourceWithout considerationof deterioration only the elements located in the positionbetween two picked tasks are influenced As for 1198733 achieved
by altermove all the tasks lined up behind the designated taskin the sequence of its initial assigned resource and the newdistributed one are included Attention for three moves werecalculate the relevant parameters of above-mentioned tasksand the rest are ignored In addition the impact that the newsolutions is defying the precedence relationships to varyingdegrees will be respected in terms of 119872 lowast sum(119894119895)isin119864 120575(119894 119895) inevaluation function 119891 (see Section 41)
46 Updating population and PairSet As illustrated in Algo-rithm 1 the population119875 and thePairSet are updatedwhen an
Mathematical Problems in Engineering 7
Table 1 Settings of important parameters
Parameters Section Description Values119901 Section 42 population size for GVNS-MA 10120572 Section 45 depth of GVNS 50 000119872 Section 41 penalty value for a violation 1000 1 times 107
to precedence constraint
excellent offspring is obtained through the crossover operatorand improved further by GVNS operator First of all if itis better than the worst solution 119904119908 in 119875 for any improvedoffspring solution 119904 the worst configuration 119904119908 is replaced bythe offspring solution 119904 When the population is updated thePairSet should be updated accordingly all pairs containing 119904119908solution are deleted from setPariSet and all pairs generated bycombining 119904 solution with others in 119875 are incorporated intoPairSet
5 Computational Experiments and Results
This section plans to assess the proposed method GVNS-MA through having comparisons with the state-of-the-artmethods in the literature For the lack of known benchmarkdata for handingMS-RCPSPLDwe firstly apply the proposedGVNS-MA to solve the MS-RCPSP on exist benchmarkinstances in favor of argument its effectiveness Then onthis basis the GVNS-MA will be examined on the modifiedproblem set
51 Benchmark Instances For the purpose of assessingGVNS-MA fully and comprehensively computational exper-iments will be conducted on two sets of instances where thefirst set is composed of 30 benchmark instances irrespec-tive of deterioration and its available in Myszkowski et al[46] which are artificially created in a base of real worldobtained from the Volvo IT Department in Wroclaw Thefull information of each instance including tasks durationsresource capabilities or precedence between tasks has beengiven As for the other set it consists of 45 instances generatedwith some modifications on first set to consider the lineardeterioration The detail will be described in Section 54
52 Parameter Settings and Experiment Protocol OurGVNS-MA was programmed in MATLAB R2015b and all thereported computational experiments presented below wereexecuted on a personal computer equipped with an IntelCore i3 processor (310 GHz CPU and 2GB RAM) in theenvironment ofWindows 7 OS To eliminate the randomnessas much as possible twenty replications for each instance arecarried out
Table 1 shows the descriptions and settings of the param-eters adopted in GVNS-MA determined by preliminaryexperiments Our memetic algorithm rests upon only threeparameters the population size 119901 the depth of generalvariable neighborhood search 120572 and the price for a violationto precedence constraint 119872 For 119901 and 120572 we follow Lai andHao [55] and set 119901 = 10 120572 = 50000 while the parameter 119872
is set at 1000 for the first experimental group and 1 times 107 forthe second
53 Experimental Results without Deterioration Our firstexperimental group aims to evaluate the performance ofour GVNS-MA on the set of 30 known instances with atmost 200 tasks and 40 renewable resources Without regardto deterioration it means that the GVNS-MA will set thedeteriorating rates of all tasks at 0 when it deals with therelevant computations Table 2 records the computationalresults solved by the GVNS-MA with the goal of durationoptimization aswell as the results achieved by other referencealgorithms in the literature
Notice that the instance name (columns 1) contains itsfull description Take the instance named 100-10-26-15 asexample the number 100 represents the number of tasksincluded and 10 denotes the quantity of renewable resourcesprovided As for the number 26 and 15 it illustrates theamount of precedence relationships and the number ofdifferent introduced skills Column 2 of Table 2 indicates theprevious minimum objective values (119891119901119903119890119887) in the literaturewhich are compiled from the best solutions yield by tworecent and best performing algorithms namely GRASP[48] and DEGR [49] Columns 3 to 4 give the best resultsobtained by DEGR and GRASP The corresponding resultsof the GVNS-MA are given in columns 5 to 7 includingthe minimum objective value (119891119887119890119904119905) over 20 independentruns the average objective value (119891119886V119892) and the averagecomputing time in seconds (Time(s)) to reach 119891119887119890119904119905 The rowBest indicates a total number of instances where the specificmethod achieves optimal among three algorithms The bestone is indicated in italic In addition to verify whetherthere exists an essential difference between the best resultsof GVNS-MA and other reference algorithms the relativepercentage deviation (RPD) is defined by the equation
119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905
119891119901119903119890119887times 100 (12)
where a positive value of 119877119875119863 means an improvement ofresult achieved by GVNS-MA while the negative numberrepresents a worse solution
Table 2 discloses that the outcomes from our GVNS-MAare noteworthy compared to the state-of-the-art results inthe literature GVNS-MA improves the previous best knownresults for 19 instances and matches for 7 cases Comparedwith the 8 out of 30 cases solved by DEGR and 6 bestsolutions achieved by GRASP these data clearly indicatethe superiority of GVNS-MA compared to the previousexcellent methods Additionally it can be observed that
8 Mathematical Problems in Engineering
Table 2 Comparison of the GVNS-MA with other algorithms on known MS-RCPSP dataset [48] Best results are indicated in italic
instances 119891119901119903119890119887 DEGR GRASP GVNS-MA 119877119875119863()119891119887119890119904119905 119891119886V119892 119879119894119898119890(119904)
100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050
the improvement achieved by GVNS-MA is up to 939for instance 100-20-65-15 accompanying that the average119877119875119863() equals to 050
54 Experimental Results with Linear Deterioration Theprevious comparisons and discussions in Section 53 demon-strate the advantages of GVNS-MA in solving the relatedissues of MS-RCPSP In this section the aforementioneddataset with some modifications is used to assess the capabil-ity of GVNS-MA to solve the MS-RCPSPLDThe proceduresof generating the testing instances and analysis of the resultsare described below To make the benchmark instances meetthe considered linear deterioration precisely the deteriora-tion rate (119889119903119894) for task 119894 (119894 isin 119881) is generated randomly fromthree intervals (0 05] [05 1] and (0 1] similar to Cheng et
al [19] to shed light on the influence of the different valuerange of deterioration rate on its effectiveness
Since the extra included deterioration rate we providetwo additional heuristics for the initial population generationof GVNS-MA Both the two methods affect the phase ofgenerating 119879119876119895 determining the sequence of tasks assignedto resource 119895 119895 isin 119870 The first heuristic considers thesequence in descending order of deterioration rate (ℎ) whilethe other rests upon an ascending order of ratio (119886ℎ)of the basic processing time and deterioration rate Themethods adopting the former and latter heuristic to popula-tion initialization are dubbedGVNS-119872119860ℎ and GVNS-119872119860119886ℎrespectively
Here instanceswith 100 tasks fromMyszkowski et al [46]are isolated to attain the researched objects which fit with theunique nature of the MS-RCPSPLDmore To account for the
Mathematical Problems in Engineering 9
three intervals from which the deterioration rate is drawn3 extended cases are needed to solve for each instance Forconvenience these instances are denoted by adding a suffixfor identification to different intervals For example 100-10-26-15 1 represents the original case 100-10-26-15 is modifiedby adding the deterioration rates produced in (0 05] to thedurations of tasks In total there are 45(3 times 15) instancesrandomly generated
Due to zero known results in literature for same datasetthe improved tabu search (ITS) proposed by Dai et al [54]who discussed the MS-RCPSP under step deterioration anda path relinking algorithm (PR) [55] based on the populationpath relinking framework are programmed as referencealgorithms
Table 3 reports the computational results achieved by theITS PR GVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ on theset of 45 benchmark instances 119891119887119890119904119905 denotes the minimumobjective value and 119891119886V119892 is computed as the average objectivevalue of 20 runs
First Table 3 discloses that the solutions obtained byGVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ are better thanthe ITS and PR for any instance from the perspectiveof both quality of schedule and runtime To some extentthese results demonstrated the differences between lineardeterioration and step deterioration and the superiority ofmemetic algorithm framework Second these three methodsdiffering in the sort order of tasks in initialize phase behavesimilarly where GVNS-MA obtains the best 15 out of 45instances 17 for GVNS-MAℎ and 14 for GVNS119886ℎ in termsof 119891119887119890119904119905 Specifically GVNS-MA and GVNS-MAℎ attain theoptimal simultaneously for the instance 100-5-48-9-1 Froma view point of 119891119886V119892 and run time three methods alsohave a balanced performance Third as far as three differentintervals to generate deterioration rate are concerned thephenomenon did not happen that the relevant algorithmsdisplay strikingly different behavior In other words theperformance of the proposed algorithm is not sensitive to thesetting of deterioration rate
55 Analysis and Discussions In this section we study twoessential ingredients of the proposed GVNS-MA to getan insight to its performance One is the rapid evaluationmechanism the other is the role of the memetic framework
551 Importance of Rapid Evaluation Mechanism GVNS-MA with rapid evaluation mechanism only calculates therelevant parameters of some particular tasks rather thanall when the procedure computes the objective value ofa neighborhood solution To highlight the key role ofthe rapid evaluation mechanism two sets of comparisonexperiments are carried out on generated dataset with twoalgorithms GVNS-MA and GVNS-MA0 including sameingredients with GVNS-MA except for the computation ofobjective value When GVNS-MA0 figures up the value of aneighborhood solution it computes all relevant parametersagain
Table 4 records the experimental results carried out on thedataset [46] without consideration of deterioration whereas
Table 5 shows the comparisons of GVNS-MA and GVNS-MA0 about the set of 15 instances generated in Section 54on account of the indiscrimination in three intervals Col-umn 2 and 5 record the best attained by two algorithmsColumn 3 and 6 indicate the minimum time cost to a finalfeasible schedule with one run of procedure Note that thebest objective value cannot be guaranteed as the output ofshortest runtime As for the parameters in column 4 and 7they represent the mean runtime Finally two parameters119863119864119881119904ℎ119900119903119905119890119904119905 and 119863119864119881119886V119892 are used to disclose the runtimedeviation of two methods defined by equations
119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051
times 100 (13)
and
119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922
119879119886V1198921times 100 (14)
respectively The positive value of 119863119864119881119904ℎ119900119903119905119890119904119905() and119863119864119881119886V119892() means that GVNS-MA0 has better performanceand negative value tells GVNS-MA is prior to GVNS-MA0in terms of time cost And the rows Better and Worserespectively show the number of instances for which thecorresponding results of the associated algorithm are betterand worse than the other
The results summarized in Table 4 disclose that theGVNS-MA has an overwhelming advantage over GVNS-MA0 in terms of the computation time to solve MS-RCPSPleaving out the deterioration effect Indeed the shortestruntime 119879119904ℎ1199001199031199051198901199041199051 of the GVNS-MA method is better thanthe shortest runtime 119879119904ℎ1199001199031199051198901199041199052 of GVNS-MA0 for 30 out of30 representative instances and the average runtime 119879119886V1198921 isbetter for 28 out of 30 instances Meanwhile the average valueof 119863119864119881119904ℎ119900119903119905119890119904119905() equals -1379 accompanying with a highof -1880 percent in 119863119864119881119886V119892()
However focusing on Table 5 the results of twoapproaches are neck and neck and GVNS-MA lost its earlysuperiority in MS-RCPSP In terms of shortest runtimeGVNS-MA successes for 7 out of 15 tested instances whileGVNS-MA0 reaches optimal for the remain As for averageruntime GVNS-MA performs better for 9 out of 15 examplesand GVNS-MA0 achieves reversion in others 6 instancesWith these data it will be hard to judge the true benefits ofone approach versus the other
To figure out the reason of this phenomenon we shouldcome back to the inner rationale of rapid evaluation mecha-nism When GVNS-MA computes the completion time of aneighborhood solution it only recalculates the tasksrsquo relatedparameters influenced by the particular move InMS-RCPSPa move including swap reverse and alter will affect justa small number of tasks But for MS-RCPSPLD instancesany move can cumulatively effect on a large proportion oftasks because of the existing deterioration Consequently theruntime saved in computing some unchanged parametersmay not make up for the time spent on isolating the changedtasks
10 Mathematical Problems in Engineering
Table3Summaryandcomparis
onon
thes
etof
45newgeneratedinsta
nces
with
119899=10
0of
GVN
S-MA
GVN
S-119872
119860 ℎG
VNS-
119872119860 119886ℎand
theT
Sheuristic[54]
andPR
[55]B
estresultsare
indicatedin
italic
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-10-26-15
165572
6598
13899
963646
6492
66871
63274
6392
33092
661967
6379
730698
62632
63629
29422
100-10-26-15
260
1396
618416
28607
608971
625436
79283
587145
613934
37496
580557
615474
23879
572794
615156
34826
100-10-26-15
3172856
184175
31236
1646
96
175513
59776
160283
16936
22225
166221
17124
17541
163898
172228
3461
100-10-47-91
7008
72053
40304
6974
471232
10018
69445
70098
20666
67645
6915
731499
69673
7039
36314
100-10-47-92
700352
739287
39112
703439
721152
70563
672221
715138
2417
9672304
6959
20614
693553
702369
25114
100-10-47-93
188206
195206
4816
818358
190863
93463
174162
184389
30008
177323
183958
19383
173708
183368
41432
100-10-48-15
165868
6914
13991
567093
6890
88879
164
851
65526
3491
663253
66595
22686
6479
86610
226662
100-10-48-15
2638747
659809
42043
621357
658747
9116
161867
651609
4116
1590432
625462
27097
618507
6346
3233655
100-10-48-15
3170999
174232
29659
171864
178061
73395
159497
166887
23395
161483
166366
3216
9163501
170273
27056
100-10-64-91
70527
7572
53770
371049
7692
773024
6872
27092
931287
6884
71215
26024
67894
69682
2775
6100-10-64-92
705806
744775
39415
7040
32738141
6279
4663729
711114
3013
968421
719153
22806
696727
736833
3991
8100-10-64-93
193844
206206
3115
32006
42
2044
03
108918
189324
19445
35415
180842
200842
3104
193844
199996
31871
100-10-64-15
170119
71497
34626
71075
7279
683543
69346
70022
39097
6691
69112
21249
70042
7079
617116
100-10-64-15
2690964
744775
3899
8672578
72823
61684
6606
14685085
22491
62823
682957
3178
4626331
666418
2376
100-10-64-15
3193844
218506
31871
194247
204235
71318
1806
92
1860
08
25336
187149
192556
45438
185585
190957
25296
100-20-22-15
119089
19355
29631
1936
19739
5517
618883
19118
2222
18572
1898
21438
1878
18965
2474
2100-20-22-15
24615
847731
36854
46032
46714
6792
645371
46008
28455
45218
46627
2892
645833
4633
2312
4100-20-22-15
326328
2798
128528
27212
2697
36972
92597
26419
2812
25563
25918
17531
25575
26254
21352
100-20-46-15
126243
2673
33424
26214
26631
60354
25862
26329
4119
925511
25896
1215
325476
26006
17466
100-20-46-15
260
647
66244
34408
59367
63596
71878
5796
16165
17216
56542
5942
26696
5465
59285
22313
100-20-46-15
332421
3490
937523
33539
3417
977634
31695
32909
16433
31651
33263
29116
32841
33349
2791
100-20-47-91
19007
1975
929027
1916
319685
5776
318864
1912
28311
1874
719269
2732
18455
18892
30731
100-20-47-92
50591
53481
37119
47839
51484
102173
4803
49319
3549
46278
4876
435418
44966
47893
24241
100-20-47-93
30802
31827
4572
630631
31365
6995
829437
30352
57846
27712
29458
29737
28399
2879
927559
100-20-65-15
19013
492446
26413
89865
90543
6795
788801
90095
17505
86826
89518
15052
87305
89338
2843
100-20-65-15
2253899
2606
7628718
250126
253267
5876
3244762
250316
4099
7243449
24751
17659
242353
244944
2573
5100-20-65-15
3110
567
1115886
27491
109874
113685
53418
105215
109224
2513
8108595
111243
1591
1104829
106776
1412
6100-20-65-91
19113
1978
341248
1895
719548
79561
18369
18898
30561
18697
1914
61896
18694
19265
3097
9100-20-65-92
46242
4776
865469
4593
447443
58112
4495
746229
1768
44719
45985
22324
45593
46591
30606
100-20-65-93
28018
28776
4894
327512
2810
868532
2719
2772
124279
26587
27691
3101
26455
27845
2473
100-5-22-151
500988
514285
2897
8510285
505098
84526
486586
499318
43529
494364
500988
18444
498567
510285
22335
100-5-22-152
1193440
1227080
4319
4119
2300
1210150
61537
1118610
1199248
24322
1184940
1219514
20664
1138490
1226780
26656
100-5-22-153
663947
700818
5691
2661553
792627
7997
660
6322
652627
10639
592509
6279304
2619
8643856
6597526
17267
100-5-46
-151
675739
749763
3679
7652793
701457
137862
63503
679663
5075
8636903
687941
29888
649164
671982
36202
100-5-46
-152
1404
680
1453430
58595
1443529
1470970
108595
1298430
1370870
26564
1315940
1360
830
23536
1332210
1399014
2417
6100-5-46
-153
795664
870313
50687
7977713
863567
6891
1740178
796512
81792
4755866
7786062
16809
752568
80128
1876
3100-5-48-9
1563787
586832
31214
5666
27
78386
84807
55206
560407
2415
55206
563664
25018
554299
56657
2495
2100-5-48-9
21284230
1315720
3614
51289811
1364
650
71286
1222350
1285020
2795
8116
3860
1259332
2812
21233010
1268572
34659
100-5-48-9
3684299
692453
45335
66785
717279
29393
8657279
67622
4561
6604
26
670917
41286
64822
667213
828446
Mathematical Problems in Engineering 11
Table3Con
tinued
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-5-64
-151
581916
627374
55661
576589
618167
102415
5544
26
579234
3516
7546543
590352
29678
56942
580175
56619
100-5-64
-152
120944
01261510
31296
1190830
1257643
123533
1131450
1183014
40595
1118430
1151812
46305
1047250
1158556
3337
100-5-64
-153
642267
709602
3498
8634651
688889
89403
612857
6657858
3798
5626771
660614
44591
1624753
674996
261623
100-5-64
-91
550231
577524
37365
5544
81
566747
98693
528177
5404
36
3393
530748
543895
48396
515984
536747
2993
2100-5-64
-92
1214340
1271610
40553
1183479
1236750
11992
711140
60115
5584
41523
11640
101201126
40287
1121650
1159384
27286
100-5-64
-93
610356
648765
36395
6151502
632323
87448
604586
6210378
31363
594514
6171502
30543
595191
623223
2774
9119861
119890119904119905
00
00
1514
1716
1415
119879119900119905119886
11989745
4545
4545
4545
4545
45
12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
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4 Mathematical Problems in Engineering
the relationship between the sequencing variable and thecontinuous starting time variable Constraint (8) calculatesthe makespan of project while Constraint (9) regulars thedomains of the variables
4 GVNS-Based Memetic Algorithm
In this section we describe in detail the general solutionmethodology and the supporting procedures in the followingsubsections
41 Search Space and Evaluation Function For a givenproblem the GVNS-MA searches a space Ω composed of allpossible assignments respecting skill constraints in any orderof tasks including both legal and illegal configurations Thesize of the search space Ω is bounded by 119874(119898119899 lowast 2119899)
Based on preceding notations and depictions to evaluatethe quality of a candidate solution 119904 isin Ω we adopt anevaluation function which is defined as 119891(119904) induced by 119904
119891 (119904) = 119898119886119909119894119898119894119911119890 119865119894 119894 isin 119881 + 119872 lowast sum(119894119895)isin119864
120575 (119894 119895) (10)
120575 (119894 119895) =
1 (119894 119895) isin 119864 119878119895 lt 1198651198940 otherwise
forall119894 isin 119881 119895 isin 119881 (11)
where 119872 is a large positive constant such that 119872 997888rarr infin as119899 997888rarr infin The first part of (10) represents the completiontime of the last unfinished task and the last part is anaugmented penalty function wheresum(119894119895)isin119864 120575(119894 119895) denotes thedegree violation to precedence relationships If 120575(119894 119895) =0 holds for any (119894 119895) isin 119864 it demonstrates the solutionis feasible corresponding to a legal configuration and itsevaluation function will all depend on the first part equalto the completion time of the schedule When a probleminstance admits no solution able to satisfy 120575(119894 119895) = 0 forall(119894 119895) isin119864 the search space of GVNS-MA is empty and no feasiblesolution can be found Given two solutions 1199041015840 and 11990410158401015840 1199041015840is better than 11990410158401015840 if 119891(1199041015840) le 119891(11990410158401015840) This statement impliesan assumption that a better solution has fewer precedenceconstraint violations
42Methodology and General Procedure Let119875 denote a pop-ulation of 119901 candidate configurations Let 119904119887 119904119908 represent thebest solution attainable so far and the worst solution in 119875 (interms of the evaluation function in Section 41) respectivelyLet119875119886119894119903119878119890119905 be a set of solution pairs (119904119894 119904119895) initially composedof all possible pairs in 119875 Next the proposed GVNS-MA canbe described as depicted in Algorithm 1
GVNS-MA first builds an initial population 119875 including119901 candidate configurations by the procedures in Section 43Then the algorithmenters into awhile loopwhich constitutesthe main part of the GVNS-MA On each new generationthe subsequent operations are executed In the first place aconfiguration pair (119904119894 119904119895) is taken at random and deleted fromPairSet Next GVNS-MA builds with a crossover operator(see Section 44) a new configuration 119904119888 (the offspring)After that the offspring is used as a starting point to further
(1) Input Problem instance I the size of population(119901) the depth of GVNS 120572
(2) Output the best configuration found during thesearch
(3) 119875 = 1199041 119904119901 larr997888 Population initialization(119868) lowastSec-tion 43lowast
(4) 119875119886119894119903119878119890119905 larr997888 (119904119894 119904119895) 1 le 119894 lt 119895 le 119901(5) while 119875119886119894119903119878119890119905 = do(6) Isolate a solution pair (119904119894 119904119895) isin 119875119886119894119903119878119890119905 ran-
domly(7) 119875119886119894119903119878119890119905 larr997888 119875119886119894119903119878119890119905 (119904119894 119904119895)(8) 119904119888 larr997888 crossOver(119904119894 119904119895) lowastSection 44lowast(9) 119904 larr997888 GVNS Operator(119904119888 120572) lowastSection 45lowast(10) if 119891(119904) le 119891(119904119908) then(11) UpdatePopulation(119904 119875 119875119886119894119903119878119890119905 119891(119904)) lowastSec-
tion 46lowast(12) end if(13) end while(14) Return 119904119887 larr997888 arg min 119891(119904119894) 119894 = 1 119901
Algorithm 1 Main sketch of the proposed GVNS-MA
improve by the GVNS operator (see Section 45) Finally ifthe improved offspring 119904 is better than the worst solution 119904119908in 119875 it is used to update the population 119875 and PairSet Thedetailed update operations are described in Section 46 Thewhile loop continues until PairSet becomes empty At the endof the while loop the algorithm terminates and returns thebest configuration 119904119887 found during the search Note that thedepth of GVNS 120572 represents a maximum number betweentwo iterationswithout improvement regarded as the stoppingcriterion of the GVNS operator
43 Population Initialization In order to build the initialpopulation (in Algorithm 2) the construction operator togenerate a new solution is executed 3 times 119901 times From thescratch a new configuration is constructed as follows Theoperator starts from assigning each task with a randomresource satisfying skill constraint Subsequently all tasksassigned to same resource are sequenced randomly at the endof previous task-to-resource phase Then for each generatedsolution the GVNS operator with the evaluation function119891(119904) (see Section 41) is used to optimize it to a local optimumand the obtained 119901 best configurations are selected to formthe initial population The detailed procedures are describedin Algorithm 2
44 The Crossover Operator Within a memetic algorithmthe crossover operator is another essential ingredient whosemain goal is to bring the search process to new promisingsearch regions to diversify the search In this paper theoffspring of two parent configurations 1199041 = (1198601 1198791198761) 1199042 =(1198602 1198791198762) shows as 119904119900 = (119860119900 119879119876119900) To inherit the advantagesof parent solutions the tasks assigned to same resource in 11990411199042 are given priority to keep the task-to-resource assignmentunchanged As for the remain unassigned tasks and 1198791198760
Mathematical Problems in Engineering 5
(1) Input The set 119881 = V1 V2 V119899 of 119899 tasks the set 119870 =1198961 1198962 119896119898 of 119898 renewable resources skill constraintsprecedence relationships and the size of population (119901)
(2) Output The best 119901 configurations(3) for pop = 1 3119901 do(4) Set RA = 119881(5) while 119877119860 = do(6) Choose V119894 randomly from 119877119860 remove V119894 from 119877119860
randomly isolate 119896119895 isin 119870119894mdash the set covering all re-sources can perform V119894 and record this assignmentas 119860(V119894) = 119895
(7) end while(8) for 119895 = 1 119898 do(9) Generate a random task sequence 119879119876119895 for candidate
tasks assigned to 119896119895(10) end for(11) Calculate the objective value 119891(119860TQ) with the evalu-
ation function 119891(119904) (see Section 41)(12) 119904119887119901119900119901 larr997888 GVNS operator(119891(119860TQ) 120572) lowastSection 45lowast(13) end for(14) Sort the 3119901 configurations in an ascending sort of evalua-
tion function values and return the first 119901 solutions as theinitial population
Algorithm 2 Population initialization
they will be determined by same methods in Section 43Analogously we apply the GVNS operator (see Section 45)to the offspring to finally gain the candidate for furtherpopulation updatingThe principle of this operator is detailedin the procedure crossover (Algorithm 3)
45 GVNS Operator This section discusses the local opti-mization phase of GVNS MA a key part of memetic algo-rithm Its function ensures an intensified search to locatehigh quality local optima from any starting point Here wedesign a generable variable neighborhood search (GVNS)heuristic as the local refinement procedure which showsgood performance compared to other variable neighborhoodsearch variants in terms of local search capability
Given three neighborhood structures 1198731 11987321198733 and aninitial solution (1199040) our GVNS operator does the refinementas follows Attention the quality of any solution is evaluatedas depicted in Section 41 To start with the sequential orderSO which determines the applying sequence of these neigh-borhood structures is at random generated For exampleassuming that the sequence SO equals (3 1 2) the searchbegins from1198733 and ends at1198731 at the given iteration For eachneighborhood structure a new local optima 11990410158401015840 is obtainedby applying the corresponding local search procedures to theincumbent solution 1199041015840 set at 1199040 at the beginning of GVNSprocedure If 11990410158401015840 is better than 1199041015840 1199041015840 is updated with thenew solution 11990410158401015840 accepted as a descent to continue the localsearch for current neighborhood otherwise the search turnsto the next neighborhood structure in SO One iterationterminates until the last neighborhood structure in SO isexplored and then the search goes on with the next iteration
until the stopping criteria is met ie the best solution 1199041015840has not improved for 120572 consecutive iterations The generalsketch of GVNS operator is described in Algorithm 4 andthe neighborhoods employed as well as the technique tocalculate objective value rapidly are depicted in the followingsubsections
451 Move and Neighborhood Three neighborhoods119873119896 (119896 = 1 3) are adopted in GVNS MA The neighbor-hood 1198731 is defined by the swap move operator which swapstwo tasks processed by same resource and keeps previoustask-to-resource assignment unchanged As such given asolution 119904 the swap neighborhood 1198731(119904) of 119904 is composedof all possible configurations that can be applied with theswap move to 119904 The neighborhood 1198732 is designed on thebase of reversion which reverses all the tasks incorporatedinto two designated random tasks of one resource As forthe neighborhood 1198733(119904) it is designed by the alter moveoperator which alters assigned resource from the originalto another resource equipped with demanded skill for oneselected randomly task with a random position in the tasksequence of given resource To efficiently assess the qualityof any neighborhood solution we devise a rapid evaluationtechnique for neighborhood solutions which is committedgreatly to the computational efficiency of the GVNS-MA
452 Rapid Evaluation Mechanism Our rapid evaluationtechnique to neighborhood solutions realizes through effec-tively calculating the move value (Δ119891) which identifies thechange in the evaluation function 119891 (see Section 41) ofeach possible move applicable to the incumbent solution
6 Mathematical Problems in Engineering
(1) Input Two parent solutions 1199041 = (1198601 1198791198761) 1199042 = (11986021198791198762) Problem instance I
(2) Output The offspring 119904119900 = (119860119900 119879119876119900)(3) Set RA = 119881 contains 119899 tasks remaining to be assigned
resource to and 119899119906119898 represents the size of 119877119860(4) for 119903 = 1 119899 do(5) if 1198601(V119903) == 1198602(V119903) then(6) 119860119900(V119903) = 1198601(V119903)(7) RA=RAV119903(8) end if(9) end for(10) for 119903 = 1 num do(11) Assign randomly resource 119896119895 isin 119870119903 to V119903(12) end for(13)Obtain 119879119876119900 and refine the offspring 119904119900 = (119860119900 119879119876119900) in the
same way as Algorithm 2 lowastSection 43lowast
Algorithm 3 Procedure crossover
(1) Input Initial solution 1199040 a set of neighborhood structures119873119896 (119896 = 1 3) 120572
(2) Output The current best solution 119904119887 found during GVNSprocess
(3) Calculate the objective value 119891(1199040) according to the eval-uation function in Section 41
(4) 1199041015840 larr997888 1199040 lowast1199041015840 is the current solution lowast(5) 119904119887 larr997888 1199041015840 lowast119904119887 is the best solution found so farlowast(6) 119889 = 0 lowast119889 counts the consecutive iterations where 1199041015840 is not
improved lowast(7) repeat(8) Generate a random sequence (SO) to apply three
neighborhood structures(9) Apply the relevant mechanism (Section 451) in pre-
determined order specified by SO update 1199041015840 if a betterconfiguration is attained
(10) if 119891(1199041015840) lt 119891(119904119887) then(11) 119904119887 larr997888 1199041015840(12) reset the counter 119889 = 0(13) else(14) 119889 = 119889 + 1(15) end if(16) until 119889 == 120572(17) return 119904119887
Algorithm 4 GVNS operator
119904 It functions in the reduction of computational cost toevaluate any attainable neighborhood solution inspired bythe situation where the starting and finishing times of mosttasks will not be changed when neighborhood solutions aregenerated
For 1198731 and 1198732 generated by swap and reversion movethe set of tasks with changed starting and ending times onlyincorporate the elements ranking next to the isolated firsttask in the sequence of given resourceWithout considerationof deterioration only the elements located in the positionbetween two picked tasks are influenced As for 1198733 achieved
by altermove all the tasks lined up behind the designated taskin the sequence of its initial assigned resource and the newdistributed one are included Attention for three moves werecalculate the relevant parameters of above-mentioned tasksand the rest are ignored In addition the impact that the newsolutions is defying the precedence relationships to varyingdegrees will be respected in terms of 119872 lowast sum(119894119895)isin119864 120575(119894 119895) inevaluation function 119891 (see Section 41)
46 Updating population and PairSet As illustrated in Algo-rithm 1 the population119875 and thePairSet are updatedwhen an
Mathematical Problems in Engineering 7
Table 1 Settings of important parameters
Parameters Section Description Values119901 Section 42 population size for GVNS-MA 10120572 Section 45 depth of GVNS 50 000119872 Section 41 penalty value for a violation 1000 1 times 107
to precedence constraint
excellent offspring is obtained through the crossover operatorand improved further by GVNS operator First of all if itis better than the worst solution 119904119908 in 119875 for any improvedoffspring solution 119904 the worst configuration 119904119908 is replaced bythe offspring solution 119904 When the population is updated thePairSet should be updated accordingly all pairs containing 119904119908solution are deleted from setPariSet and all pairs generated bycombining 119904 solution with others in 119875 are incorporated intoPairSet
5 Computational Experiments and Results
This section plans to assess the proposed method GVNS-MA through having comparisons with the state-of-the-artmethods in the literature For the lack of known benchmarkdata for handingMS-RCPSPLDwe firstly apply the proposedGVNS-MA to solve the MS-RCPSP on exist benchmarkinstances in favor of argument its effectiveness Then onthis basis the GVNS-MA will be examined on the modifiedproblem set
51 Benchmark Instances For the purpose of assessingGVNS-MA fully and comprehensively computational exper-iments will be conducted on two sets of instances where thefirst set is composed of 30 benchmark instances irrespec-tive of deterioration and its available in Myszkowski et al[46] which are artificially created in a base of real worldobtained from the Volvo IT Department in Wroclaw Thefull information of each instance including tasks durationsresource capabilities or precedence between tasks has beengiven As for the other set it consists of 45 instances generatedwith some modifications on first set to consider the lineardeterioration The detail will be described in Section 54
52 Parameter Settings and Experiment Protocol OurGVNS-MA was programmed in MATLAB R2015b and all thereported computational experiments presented below wereexecuted on a personal computer equipped with an IntelCore i3 processor (310 GHz CPU and 2GB RAM) in theenvironment ofWindows 7 OS To eliminate the randomnessas much as possible twenty replications for each instance arecarried out
Table 1 shows the descriptions and settings of the param-eters adopted in GVNS-MA determined by preliminaryexperiments Our memetic algorithm rests upon only threeparameters the population size 119901 the depth of generalvariable neighborhood search 120572 and the price for a violationto precedence constraint 119872 For 119901 and 120572 we follow Lai andHao [55] and set 119901 = 10 120572 = 50000 while the parameter 119872
is set at 1000 for the first experimental group and 1 times 107 forthe second
53 Experimental Results without Deterioration Our firstexperimental group aims to evaluate the performance ofour GVNS-MA on the set of 30 known instances with atmost 200 tasks and 40 renewable resources Without regardto deterioration it means that the GVNS-MA will set thedeteriorating rates of all tasks at 0 when it deals with therelevant computations Table 2 records the computationalresults solved by the GVNS-MA with the goal of durationoptimization aswell as the results achieved by other referencealgorithms in the literature
Notice that the instance name (columns 1) contains itsfull description Take the instance named 100-10-26-15 asexample the number 100 represents the number of tasksincluded and 10 denotes the quantity of renewable resourcesprovided As for the number 26 and 15 it illustrates theamount of precedence relationships and the number ofdifferent introduced skills Column 2 of Table 2 indicates theprevious minimum objective values (119891119901119903119890119887) in the literaturewhich are compiled from the best solutions yield by tworecent and best performing algorithms namely GRASP[48] and DEGR [49] Columns 3 to 4 give the best resultsobtained by DEGR and GRASP The corresponding resultsof the GVNS-MA are given in columns 5 to 7 includingthe minimum objective value (119891119887119890119904119905) over 20 independentruns the average objective value (119891119886V119892) and the averagecomputing time in seconds (Time(s)) to reach 119891119887119890119904119905 The rowBest indicates a total number of instances where the specificmethod achieves optimal among three algorithms The bestone is indicated in italic In addition to verify whetherthere exists an essential difference between the best resultsof GVNS-MA and other reference algorithms the relativepercentage deviation (RPD) is defined by the equation
119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905
119891119901119903119890119887times 100 (12)
where a positive value of 119877119875119863 means an improvement ofresult achieved by GVNS-MA while the negative numberrepresents a worse solution
Table 2 discloses that the outcomes from our GVNS-MAare noteworthy compared to the state-of-the-art results inthe literature GVNS-MA improves the previous best knownresults for 19 instances and matches for 7 cases Comparedwith the 8 out of 30 cases solved by DEGR and 6 bestsolutions achieved by GRASP these data clearly indicatethe superiority of GVNS-MA compared to the previousexcellent methods Additionally it can be observed that
8 Mathematical Problems in Engineering
Table 2 Comparison of the GVNS-MA with other algorithms on known MS-RCPSP dataset [48] Best results are indicated in italic
instances 119891119901119903119890119887 DEGR GRASP GVNS-MA 119877119875119863()119891119887119890119904119905 119891119886V119892 119879119894119898119890(119904)
100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050
the improvement achieved by GVNS-MA is up to 939for instance 100-20-65-15 accompanying that the average119877119875119863() equals to 050
54 Experimental Results with Linear Deterioration Theprevious comparisons and discussions in Section 53 demon-strate the advantages of GVNS-MA in solving the relatedissues of MS-RCPSP In this section the aforementioneddataset with some modifications is used to assess the capabil-ity of GVNS-MA to solve the MS-RCPSPLDThe proceduresof generating the testing instances and analysis of the resultsare described below To make the benchmark instances meetthe considered linear deterioration precisely the deteriora-tion rate (119889119903119894) for task 119894 (119894 isin 119881) is generated randomly fromthree intervals (0 05] [05 1] and (0 1] similar to Cheng et
al [19] to shed light on the influence of the different valuerange of deterioration rate on its effectiveness
Since the extra included deterioration rate we providetwo additional heuristics for the initial population generationof GVNS-MA Both the two methods affect the phase ofgenerating 119879119876119895 determining the sequence of tasks assignedto resource 119895 119895 isin 119870 The first heuristic considers thesequence in descending order of deterioration rate (ℎ) whilethe other rests upon an ascending order of ratio (119886ℎ)of the basic processing time and deterioration rate Themethods adopting the former and latter heuristic to popula-tion initialization are dubbedGVNS-119872119860ℎ and GVNS-119872119860119886ℎrespectively
Here instanceswith 100 tasks fromMyszkowski et al [46]are isolated to attain the researched objects which fit with theunique nature of the MS-RCPSPLDmore To account for the
Mathematical Problems in Engineering 9
three intervals from which the deterioration rate is drawn3 extended cases are needed to solve for each instance Forconvenience these instances are denoted by adding a suffixfor identification to different intervals For example 100-10-26-15 1 represents the original case 100-10-26-15 is modifiedby adding the deterioration rates produced in (0 05] to thedurations of tasks In total there are 45(3 times 15) instancesrandomly generated
Due to zero known results in literature for same datasetthe improved tabu search (ITS) proposed by Dai et al [54]who discussed the MS-RCPSP under step deterioration anda path relinking algorithm (PR) [55] based on the populationpath relinking framework are programmed as referencealgorithms
Table 3 reports the computational results achieved by theITS PR GVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ on theset of 45 benchmark instances 119891119887119890119904119905 denotes the minimumobjective value and 119891119886V119892 is computed as the average objectivevalue of 20 runs
First Table 3 discloses that the solutions obtained byGVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ are better thanthe ITS and PR for any instance from the perspectiveof both quality of schedule and runtime To some extentthese results demonstrated the differences between lineardeterioration and step deterioration and the superiority ofmemetic algorithm framework Second these three methodsdiffering in the sort order of tasks in initialize phase behavesimilarly where GVNS-MA obtains the best 15 out of 45instances 17 for GVNS-MAℎ and 14 for GVNS119886ℎ in termsof 119891119887119890119904119905 Specifically GVNS-MA and GVNS-MAℎ attain theoptimal simultaneously for the instance 100-5-48-9-1 Froma view point of 119891119886V119892 and run time three methods alsohave a balanced performance Third as far as three differentintervals to generate deterioration rate are concerned thephenomenon did not happen that the relevant algorithmsdisplay strikingly different behavior In other words theperformance of the proposed algorithm is not sensitive to thesetting of deterioration rate
55 Analysis and Discussions In this section we study twoessential ingredients of the proposed GVNS-MA to getan insight to its performance One is the rapid evaluationmechanism the other is the role of the memetic framework
551 Importance of Rapid Evaluation Mechanism GVNS-MA with rapid evaluation mechanism only calculates therelevant parameters of some particular tasks rather thanall when the procedure computes the objective value ofa neighborhood solution To highlight the key role ofthe rapid evaluation mechanism two sets of comparisonexperiments are carried out on generated dataset with twoalgorithms GVNS-MA and GVNS-MA0 including sameingredients with GVNS-MA except for the computation ofobjective value When GVNS-MA0 figures up the value of aneighborhood solution it computes all relevant parametersagain
Table 4 records the experimental results carried out on thedataset [46] without consideration of deterioration whereas
Table 5 shows the comparisons of GVNS-MA and GVNS-MA0 about the set of 15 instances generated in Section 54on account of the indiscrimination in three intervals Col-umn 2 and 5 record the best attained by two algorithmsColumn 3 and 6 indicate the minimum time cost to a finalfeasible schedule with one run of procedure Note that thebest objective value cannot be guaranteed as the output ofshortest runtime As for the parameters in column 4 and 7they represent the mean runtime Finally two parameters119863119864119881119904ℎ119900119903119905119890119904119905 and 119863119864119881119886V119892 are used to disclose the runtimedeviation of two methods defined by equations
119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051
times 100 (13)
and
119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922
119879119886V1198921times 100 (14)
respectively The positive value of 119863119864119881119904ℎ119900119903119905119890119904119905() and119863119864119881119886V119892() means that GVNS-MA0 has better performanceand negative value tells GVNS-MA is prior to GVNS-MA0in terms of time cost And the rows Better and Worserespectively show the number of instances for which thecorresponding results of the associated algorithm are betterand worse than the other
The results summarized in Table 4 disclose that theGVNS-MA has an overwhelming advantage over GVNS-MA0 in terms of the computation time to solve MS-RCPSPleaving out the deterioration effect Indeed the shortestruntime 119879119904ℎ1199001199031199051198901199041199051 of the GVNS-MA method is better thanthe shortest runtime 119879119904ℎ1199001199031199051198901199041199052 of GVNS-MA0 for 30 out of30 representative instances and the average runtime 119879119886V1198921 isbetter for 28 out of 30 instances Meanwhile the average valueof 119863119864119881119904ℎ119900119903119905119890119904119905() equals -1379 accompanying with a highof -1880 percent in 119863119864119881119886V119892()
However focusing on Table 5 the results of twoapproaches are neck and neck and GVNS-MA lost its earlysuperiority in MS-RCPSP In terms of shortest runtimeGVNS-MA successes for 7 out of 15 tested instances whileGVNS-MA0 reaches optimal for the remain As for averageruntime GVNS-MA performs better for 9 out of 15 examplesand GVNS-MA0 achieves reversion in others 6 instancesWith these data it will be hard to judge the true benefits ofone approach versus the other
To figure out the reason of this phenomenon we shouldcome back to the inner rationale of rapid evaluation mecha-nism When GVNS-MA computes the completion time of aneighborhood solution it only recalculates the tasksrsquo relatedparameters influenced by the particular move InMS-RCPSPa move including swap reverse and alter will affect justa small number of tasks But for MS-RCPSPLD instancesany move can cumulatively effect on a large proportion oftasks because of the existing deterioration Consequently theruntime saved in computing some unchanged parametersmay not make up for the time spent on isolating the changedtasks
10 Mathematical Problems in Engineering
Table3Summaryandcomparis
onon
thes
etof
45newgeneratedinsta
nces
with
119899=10
0of
GVN
S-MA
GVN
S-119872
119860 ℎG
VNS-
119872119860 119886ℎand
theT
Sheuristic[54]
andPR
[55]B
estresultsare
indicatedin
italic
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-10-26-15
165572
6598
13899
963646
6492
66871
63274
6392
33092
661967
6379
730698
62632
63629
29422
100-10-26-15
260
1396
618416
28607
608971
625436
79283
587145
613934
37496
580557
615474
23879
572794
615156
34826
100-10-26-15
3172856
184175
31236
1646
96
175513
59776
160283
16936
22225
166221
17124
17541
163898
172228
3461
100-10-47-91
7008
72053
40304
6974
471232
10018
69445
70098
20666
67645
6915
731499
69673
7039
36314
100-10-47-92
700352
739287
39112
703439
721152
70563
672221
715138
2417
9672304
6959
20614
693553
702369
25114
100-10-47-93
188206
195206
4816
818358
190863
93463
174162
184389
30008
177323
183958
19383
173708
183368
41432
100-10-48-15
165868
6914
13991
567093
6890
88879
164
851
65526
3491
663253
66595
22686
6479
86610
226662
100-10-48-15
2638747
659809
42043
621357
658747
9116
161867
651609
4116
1590432
625462
27097
618507
6346
3233655
100-10-48-15
3170999
174232
29659
171864
178061
73395
159497
166887
23395
161483
166366
3216
9163501
170273
27056
100-10-64-91
70527
7572
53770
371049
7692
773024
6872
27092
931287
6884
71215
26024
67894
69682
2775
6100-10-64-92
705806
744775
39415
7040
32738141
6279
4663729
711114
3013
968421
719153
22806
696727
736833
3991
8100-10-64-93
193844
206206
3115
32006
42
2044
03
108918
189324
19445
35415
180842
200842
3104
193844
199996
31871
100-10-64-15
170119
71497
34626
71075
7279
683543
69346
70022
39097
6691
69112
21249
70042
7079
617116
100-10-64-15
2690964
744775
3899
8672578
72823
61684
6606
14685085
22491
62823
682957
3178
4626331
666418
2376
100-10-64-15
3193844
218506
31871
194247
204235
71318
1806
92
1860
08
25336
187149
192556
45438
185585
190957
25296
100-20-22-15
119089
19355
29631
1936
19739
5517
618883
19118
2222
18572
1898
21438
1878
18965
2474
2100-20-22-15
24615
847731
36854
46032
46714
6792
645371
46008
28455
45218
46627
2892
645833
4633
2312
4100-20-22-15
326328
2798
128528
27212
2697
36972
92597
26419
2812
25563
25918
17531
25575
26254
21352
100-20-46-15
126243
2673
33424
26214
26631
60354
25862
26329
4119
925511
25896
1215
325476
26006
17466
100-20-46-15
260
647
66244
34408
59367
63596
71878
5796
16165
17216
56542
5942
26696
5465
59285
22313
100-20-46-15
332421
3490
937523
33539
3417
977634
31695
32909
16433
31651
33263
29116
32841
33349
2791
100-20-47-91
19007
1975
929027
1916
319685
5776
318864
1912
28311
1874
719269
2732
18455
18892
30731
100-20-47-92
50591
53481
37119
47839
51484
102173
4803
49319
3549
46278
4876
435418
44966
47893
24241
100-20-47-93
30802
31827
4572
630631
31365
6995
829437
30352
57846
27712
29458
29737
28399
2879
927559
100-20-65-15
19013
492446
26413
89865
90543
6795
788801
90095
17505
86826
89518
15052
87305
89338
2843
100-20-65-15
2253899
2606
7628718
250126
253267
5876
3244762
250316
4099
7243449
24751
17659
242353
244944
2573
5100-20-65-15
3110
567
1115886
27491
109874
113685
53418
105215
109224
2513
8108595
111243
1591
1104829
106776
1412
6100-20-65-91
19113
1978
341248
1895
719548
79561
18369
18898
30561
18697
1914
61896
18694
19265
3097
9100-20-65-92
46242
4776
865469
4593
447443
58112
4495
746229
1768
44719
45985
22324
45593
46591
30606
100-20-65-93
28018
28776
4894
327512
2810
868532
2719
2772
124279
26587
27691
3101
26455
27845
2473
100-5-22-151
500988
514285
2897
8510285
505098
84526
486586
499318
43529
494364
500988
18444
498567
510285
22335
100-5-22-152
1193440
1227080
4319
4119
2300
1210150
61537
1118610
1199248
24322
1184940
1219514
20664
1138490
1226780
26656
100-5-22-153
663947
700818
5691
2661553
792627
7997
660
6322
652627
10639
592509
6279304
2619
8643856
6597526
17267
100-5-46
-151
675739
749763
3679
7652793
701457
137862
63503
679663
5075
8636903
687941
29888
649164
671982
36202
100-5-46
-152
1404
680
1453430
58595
1443529
1470970
108595
1298430
1370870
26564
1315940
1360
830
23536
1332210
1399014
2417
6100-5-46
-153
795664
870313
50687
7977713
863567
6891
1740178
796512
81792
4755866
7786062
16809
752568
80128
1876
3100-5-48-9
1563787
586832
31214
5666
27
78386
84807
55206
560407
2415
55206
563664
25018
554299
56657
2495
2100-5-48-9
21284230
1315720
3614
51289811
1364
650
71286
1222350
1285020
2795
8116
3860
1259332
2812
21233010
1268572
34659
100-5-48-9
3684299
692453
45335
66785
717279
29393
8657279
67622
4561
6604
26
670917
41286
64822
667213
828446
Mathematical Problems in Engineering 11
Table3Con
tinued
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-5-64
-151
581916
627374
55661
576589
618167
102415
5544
26
579234
3516
7546543
590352
29678
56942
580175
56619
100-5-64
-152
120944
01261510
31296
1190830
1257643
123533
1131450
1183014
40595
1118430
1151812
46305
1047250
1158556
3337
100-5-64
-153
642267
709602
3498
8634651
688889
89403
612857
6657858
3798
5626771
660614
44591
1624753
674996
261623
100-5-64
-91
550231
577524
37365
5544
81
566747
98693
528177
5404
36
3393
530748
543895
48396
515984
536747
2993
2100-5-64
-92
1214340
1271610
40553
1183479
1236750
11992
711140
60115
5584
41523
11640
101201126
40287
1121650
1159384
27286
100-5-64
-93
610356
648765
36395
6151502
632323
87448
604586
6210378
31363
594514
6171502
30543
595191
623223
2774
9119861
119890119904119905
00
00
1514
1716
1415
119879119900119905119886
11989745
4545
4545
4545
4545
45
12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
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Mathematical Problems in Engineering 5
(1) Input The set 119881 = V1 V2 V119899 of 119899 tasks the set 119870 =1198961 1198962 119896119898 of 119898 renewable resources skill constraintsprecedence relationships and the size of population (119901)
(2) Output The best 119901 configurations(3) for pop = 1 3119901 do(4) Set RA = 119881(5) while 119877119860 = do(6) Choose V119894 randomly from 119877119860 remove V119894 from 119877119860
randomly isolate 119896119895 isin 119870119894mdash the set covering all re-sources can perform V119894 and record this assignmentas 119860(V119894) = 119895
(7) end while(8) for 119895 = 1 119898 do(9) Generate a random task sequence 119879119876119895 for candidate
tasks assigned to 119896119895(10) end for(11) Calculate the objective value 119891(119860TQ) with the evalu-
ation function 119891(119904) (see Section 41)(12) 119904119887119901119900119901 larr997888 GVNS operator(119891(119860TQ) 120572) lowastSection 45lowast(13) end for(14) Sort the 3119901 configurations in an ascending sort of evalua-
tion function values and return the first 119901 solutions as theinitial population
Algorithm 2 Population initialization
they will be determined by same methods in Section 43Analogously we apply the GVNS operator (see Section 45)to the offspring to finally gain the candidate for furtherpopulation updatingThe principle of this operator is detailedin the procedure crossover (Algorithm 3)
45 GVNS Operator This section discusses the local opti-mization phase of GVNS MA a key part of memetic algo-rithm Its function ensures an intensified search to locatehigh quality local optima from any starting point Here wedesign a generable variable neighborhood search (GVNS)heuristic as the local refinement procedure which showsgood performance compared to other variable neighborhoodsearch variants in terms of local search capability
Given three neighborhood structures 1198731 11987321198733 and aninitial solution (1199040) our GVNS operator does the refinementas follows Attention the quality of any solution is evaluatedas depicted in Section 41 To start with the sequential orderSO which determines the applying sequence of these neigh-borhood structures is at random generated For exampleassuming that the sequence SO equals (3 1 2) the searchbegins from1198733 and ends at1198731 at the given iteration For eachneighborhood structure a new local optima 11990410158401015840 is obtainedby applying the corresponding local search procedures to theincumbent solution 1199041015840 set at 1199040 at the beginning of GVNSprocedure If 11990410158401015840 is better than 1199041015840 1199041015840 is updated with thenew solution 11990410158401015840 accepted as a descent to continue the localsearch for current neighborhood otherwise the search turnsto the next neighborhood structure in SO One iterationterminates until the last neighborhood structure in SO isexplored and then the search goes on with the next iteration
until the stopping criteria is met ie the best solution 1199041015840has not improved for 120572 consecutive iterations The generalsketch of GVNS operator is described in Algorithm 4 andthe neighborhoods employed as well as the technique tocalculate objective value rapidly are depicted in the followingsubsections
451 Move and Neighborhood Three neighborhoods119873119896 (119896 = 1 3) are adopted in GVNS MA The neighbor-hood 1198731 is defined by the swap move operator which swapstwo tasks processed by same resource and keeps previoustask-to-resource assignment unchanged As such given asolution 119904 the swap neighborhood 1198731(119904) of 119904 is composedof all possible configurations that can be applied with theswap move to 119904 The neighborhood 1198732 is designed on thebase of reversion which reverses all the tasks incorporatedinto two designated random tasks of one resource As forthe neighborhood 1198733(119904) it is designed by the alter moveoperator which alters assigned resource from the originalto another resource equipped with demanded skill for oneselected randomly task with a random position in the tasksequence of given resource To efficiently assess the qualityof any neighborhood solution we devise a rapid evaluationtechnique for neighborhood solutions which is committedgreatly to the computational efficiency of the GVNS-MA
452 Rapid Evaluation Mechanism Our rapid evaluationtechnique to neighborhood solutions realizes through effec-tively calculating the move value (Δ119891) which identifies thechange in the evaluation function 119891 (see Section 41) ofeach possible move applicable to the incumbent solution
6 Mathematical Problems in Engineering
(1) Input Two parent solutions 1199041 = (1198601 1198791198761) 1199042 = (11986021198791198762) Problem instance I
(2) Output The offspring 119904119900 = (119860119900 119879119876119900)(3) Set RA = 119881 contains 119899 tasks remaining to be assigned
resource to and 119899119906119898 represents the size of 119877119860(4) for 119903 = 1 119899 do(5) if 1198601(V119903) == 1198602(V119903) then(6) 119860119900(V119903) = 1198601(V119903)(7) RA=RAV119903(8) end if(9) end for(10) for 119903 = 1 num do(11) Assign randomly resource 119896119895 isin 119870119903 to V119903(12) end for(13)Obtain 119879119876119900 and refine the offspring 119904119900 = (119860119900 119879119876119900) in the
same way as Algorithm 2 lowastSection 43lowast
Algorithm 3 Procedure crossover
(1) Input Initial solution 1199040 a set of neighborhood structures119873119896 (119896 = 1 3) 120572
(2) Output The current best solution 119904119887 found during GVNSprocess
(3) Calculate the objective value 119891(1199040) according to the eval-uation function in Section 41
(4) 1199041015840 larr997888 1199040 lowast1199041015840 is the current solution lowast(5) 119904119887 larr997888 1199041015840 lowast119904119887 is the best solution found so farlowast(6) 119889 = 0 lowast119889 counts the consecutive iterations where 1199041015840 is not
improved lowast(7) repeat(8) Generate a random sequence (SO) to apply three
neighborhood structures(9) Apply the relevant mechanism (Section 451) in pre-
determined order specified by SO update 1199041015840 if a betterconfiguration is attained
(10) if 119891(1199041015840) lt 119891(119904119887) then(11) 119904119887 larr997888 1199041015840(12) reset the counter 119889 = 0(13) else(14) 119889 = 119889 + 1(15) end if(16) until 119889 == 120572(17) return 119904119887
Algorithm 4 GVNS operator
119904 It functions in the reduction of computational cost toevaluate any attainable neighborhood solution inspired bythe situation where the starting and finishing times of mosttasks will not be changed when neighborhood solutions aregenerated
For 1198731 and 1198732 generated by swap and reversion movethe set of tasks with changed starting and ending times onlyincorporate the elements ranking next to the isolated firsttask in the sequence of given resourceWithout considerationof deterioration only the elements located in the positionbetween two picked tasks are influenced As for 1198733 achieved
by altermove all the tasks lined up behind the designated taskin the sequence of its initial assigned resource and the newdistributed one are included Attention for three moves werecalculate the relevant parameters of above-mentioned tasksand the rest are ignored In addition the impact that the newsolutions is defying the precedence relationships to varyingdegrees will be respected in terms of 119872 lowast sum(119894119895)isin119864 120575(119894 119895) inevaluation function 119891 (see Section 41)
46 Updating population and PairSet As illustrated in Algo-rithm 1 the population119875 and thePairSet are updatedwhen an
Mathematical Problems in Engineering 7
Table 1 Settings of important parameters
Parameters Section Description Values119901 Section 42 population size for GVNS-MA 10120572 Section 45 depth of GVNS 50 000119872 Section 41 penalty value for a violation 1000 1 times 107
to precedence constraint
excellent offspring is obtained through the crossover operatorand improved further by GVNS operator First of all if itis better than the worst solution 119904119908 in 119875 for any improvedoffspring solution 119904 the worst configuration 119904119908 is replaced bythe offspring solution 119904 When the population is updated thePairSet should be updated accordingly all pairs containing 119904119908solution are deleted from setPariSet and all pairs generated bycombining 119904 solution with others in 119875 are incorporated intoPairSet
5 Computational Experiments and Results
This section plans to assess the proposed method GVNS-MA through having comparisons with the state-of-the-artmethods in the literature For the lack of known benchmarkdata for handingMS-RCPSPLDwe firstly apply the proposedGVNS-MA to solve the MS-RCPSP on exist benchmarkinstances in favor of argument its effectiveness Then onthis basis the GVNS-MA will be examined on the modifiedproblem set
51 Benchmark Instances For the purpose of assessingGVNS-MA fully and comprehensively computational exper-iments will be conducted on two sets of instances where thefirst set is composed of 30 benchmark instances irrespec-tive of deterioration and its available in Myszkowski et al[46] which are artificially created in a base of real worldobtained from the Volvo IT Department in Wroclaw Thefull information of each instance including tasks durationsresource capabilities or precedence between tasks has beengiven As for the other set it consists of 45 instances generatedwith some modifications on first set to consider the lineardeterioration The detail will be described in Section 54
52 Parameter Settings and Experiment Protocol OurGVNS-MA was programmed in MATLAB R2015b and all thereported computational experiments presented below wereexecuted on a personal computer equipped with an IntelCore i3 processor (310 GHz CPU and 2GB RAM) in theenvironment ofWindows 7 OS To eliminate the randomnessas much as possible twenty replications for each instance arecarried out
Table 1 shows the descriptions and settings of the param-eters adopted in GVNS-MA determined by preliminaryexperiments Our memetic algorithm rests upon only threeparameters the population size 119901 the depth of generalvariable neighborhood search 120572 and the price for a violationto precedence constraint 119872 For 119901 and 120572 we follow Lai andHao [55] and set 119901 = 10 120572 = 50000 while the parameter 119872
is set at 1000 for the first experimental group and 1 times 107 forthe second
53 Experimental Results without Deterioration Our firstexperimental group aims to evaluate the performance ofour GVNS-MA on the set of 30 known instances with atmost 200 tasks and 40 renewable resources Without regardto deterioration it means that the GVNS-MA will set thedeteriorating rates of all tasks at 0 when it deals with therelevant computations Table 2 records the computationalresults solved by the GVNS-MA with the goal of durationoptimization aswell as the results achieved by other referencealgorithms in the literature
Notice that the instance name (columns 1) contains itsfull description Take the instance named 100-10-26-15 asexample the number 100 represents the number of tasksincluded and 10 denotes the quantity of renewable resourcesprovided As for the number 26 and 15 it illustrates theamount of precedence relationships and the number ofdifferent introduced skills Column 2 of Table 2 indicates theprevious minimum objective values (119891119901119903119890119887) in the literaturewhich are compiled from the best solutions yield by tworecent and best performing algorithms namely GRASP[48] and DEGR [49] Columns 3 to 4 give the best resultsobtained by DEGR and GRASP The corresponding resultsof the GVNS-MA are given in columns 5 to 7 includingthe minimum objective value (119891119887119890119904119905) over 20 independentruns the average objective value (119891119886V119892) and the averagecomputing time in seconds (Time(s)) to reach 119891119887119890119904119905 The rowBest indicates a total number of instances where the specificmethod achieves optimal among three algorithms The bestone is indicated in italic In addition to verify whetherthere exists an essential difference between the best resultsof GVNS-MA and other reference algorithms the relativepercentage deviation (RPD) is defined by the equation
119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905
119891119901119903119890119887times 100 (12)
where a positive value of 119877119875119863 means an improvement ofresult achieved by GVNS-MA while the negative numberrepresents a worse solution
Table 2 discloses that the outcomes from our GVNS-MAare noteworthy compared to the state-of-the-art results inthe literature GVNS-MA improves the previous best knownresults for 19 instances and matches for 7 cases Comparedwith the 8 out of 30 cases solved by DEGR and 6 bestsolutions achieved by GRASP these data clearly indicatethe superiority of GVNS-MA compared to the previousexcellent methods Additionally it can be observed that
8 Mathematical Problems in Engineering
Table 2 Comparison of the GVNS-MA with other algorithms on known MS-RCPSP dataset [48] Best results are indicated in italic
instances 119891119901119903119890119887 DEGR GRASP GVNS-MA 119877119875119863()119891119887119890119904119905 119891119886V119892 119879119894119898119890(119904)
100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050
the improvement achieved by GVNS-MA is up to 939for instance 100-20-65-15 accompanying that the average119877119875119863() equals to 050
54 Experimental Results with Linear Deterioration Theprevious comparisons and discussions in Section 53 demon-strate the advantages of GVNS-MA in solving the relatedissues of MS-RCPSP In this section the aforementioneddataset with some modifications is used to assess the capabil-ity of GVNS-MA to solve the MS-RCPSPLDThe proceduresof generating the testing instances and analysis of the resultsare described below To make the benchmark instances meetthe considered linear deterioration precisely the deteriora-tion rate (119889119903119894) for task 119894 (119894 isin 119881) is generated randomly fromthree intervals (0 05] [05 1] and (0 1] similar to Cheng et
al [19] to shed light on the influence of the different valuerange of deterioration rate on its effectiveness
Since the extra included deterioration rate we providetwo additional heuristics for the initial population generationof GVNS-MA Both the two methods affect the phase ofgenerating 119879119876119895 determining the sequence of tasks assignedto resource 119895 119895 isin 119870 The first heuristic considers thesequence in descending order of deterioration rate (ℎ) whilethe other rests upon an ascending order of ratio (119886ℎ)of the basic processing time and deterioration rate Themethods adopting the former and latter heuristic to popula-tion initialization are dubbedGVNS-119872119860ℎ and GVNS-119872119860119886ℎrespectively
Here instanceswith 100 tasks fromMyszkowski et al [46]are isolated to attain the researched objects which fit with theunique nature of the MS-RCPSPLDmore To account for the
Mathematical Problems in Engineering 9
three intervals from which the deterioration rate is drawn3 extended cases are needed to solve for each instance Forconvenience these instances are denoted by adding a suffixfor identification to different intervals For example 100-10-26-15 1 represents the original case 100-10-26-15 is modifiedby adding the deterioration rates produced in (0 05] to thedurations of tasks In total there are 45(3 times 15) instancesrandomly generated
Due to zero known results in literature for same datasetthe improved tabu search (ITS) proposed by Dai et al [54]who discussed the MS-RCPSP under step deterioration anda path relinking algorithm (PR) [55] based on the populationpath relinking framework are programmed as referencealgorithms
Table 3 reports the computational results achieved by theITS PR GVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ on theset of 45 benchmark instances 119891119887119890119904119905 denotes the minimumobjective value and 119891119886V119892 is computed as the average objectivevalue of 20 runs
First Table 3 discloses that the solutions obtained byGVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ are better thanthe ITS and PR for any instance from the perspectiveof both quality of schedule and runtime To some extentthese results demonstrated the differences between lineardeterioration and step deterioration and the superiority ofmemetic algorithm framework Second these three methodsdiffering in the sort order of tasks in initialize phase behavesimilarly where GVNS-MA obtains the best 15 out of 45instances 17 for GVNS-MAℎ and 14 for GVNS119886ℎ in termsof 119891119887119890119904119905 Specifically GVNS-MA and GVNS-MAℎ attain theoptimal simultaneously for the instance 100-5-48-9-1 Froma view point of 119891119886V119892 and run time three methods alsohave a balanced performance Third as far as three differentintervals to generate deterioration rate are concerned thephenomenon did not happen that the relevant algorithmsdisplay strikingly different behavior In other words theperformance of the proposed algorithm is not sensitive to thesetting of deterioration rate
55 Analysis and Discussions In this section we study twoessential ingredients of the proposed GVNS-MA to getan insight to its performance One is the rapid evaluationmechanism the other is the role of the memetic framework
551 Importance of Rapid Evaluation Mechanism GVNS-MA with rapid evaluation mechanism only calculates therelevant parameters of some particular tasks rather thanall when the procedure computes the objective value ofa neighborhood solution To highlight the key role ofthe rapid evaluation mechanism two sets of comparisonexperiments are carried out on generated dataset with twoalgorithms GVNS-MA and GVNS-MA0 including sameingredients with GVNS-MA except for the computation ofobjective value When GVNS-MA0 figures up the value of aneighborhood solution it computes all relevant parametersagain
Table 4 records the experimental results carried out on thedataset [46] without consideration of deterioration whereas
Table 5 shows the comparisons of GVNS-MA and GVNS-MA0 about the set of 15 instances generated in Section 54on account of the indiscrimination in three intervals Col-umn 2 and 5 record the best attained by two algorithmsColumn 3 and 6 indicate the minimum time cost to a finalfeasible schedule with one run of procedure Note that thebest objective value cannot be guaranteed as the output ofshortest runtime As for the parameters in column 4 and 7they represent the mean runtime Finally two parameters119863119864119881119904ℎ119900119903119905119890119904119905 and 119863119864119881119886V119892 are used to disclose the runtimedeviation of two methods defined by equations
119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051
times 100 (13)
and
119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922
119879119886V1198921times 100 (14)
respectively The positive value of 119863119864119881119904ℎ119900119903119905119890119904119905() and119863119864119881119886V119892() means that GVNS-MA0 has better performanceand negative value tells GVNS-MA is prior to GVNS-MA0in terms of time cost And the rows Better and Worserespectively show the number of instances for which thecorresponding results of the associated algorithm are betterand worse than the other
The results summarized in Table 4 disclose that theGVNS-MA has an overwhelming advantage over GVNS-MA0 in terms of the computation time to solve MS-RCPSPleaving out the deterioration effect Indeed the shortestruntime 119879119904ℎ1199001199031199051198901199041199051 of the GVNS-MA method is better thanthe shortest runtime 119879119904ℎ1199001199031199051198901199041199052 of GVNS-MA0 for 30 out of30 representative instances and the average runtime 119879119886V1198921 isbetter for 28 out of 30 instances Meanwhile the average valueof 119863119864119881119904ℎ119900119903119905119890119904119905() equals -1379 accompanying with a highof -1880 percent in 119863119864119881119886V119892()
However focusing on Table 5 the results of twoapproaches are neck and neck and GVNS-MA lost its earlysuperiority in MS-RCPSP In terms of shortest runtimeGVNS-MA successes for 7 out of 15 tested instances whileGVNS-MA0 reaches optimal for the remain As for averageruntime GVNS-MA performs better for 9 out of 15 examplesand GVNS-MA0 achieves reversion in others 6 instancesWith these data it will be hard to judge the true benefits ofone approach versus the other
To figure out the reason of this phenomenon we shouldcome back to the inner rationale of rapid evaluation mecha-nism When GVNS-MA computes the completion time of aneighborhood solution it only recalculates the tasksrsquo relatedparameters influenced by the particular move InMS-RCPSPa move including swap reverse and alter will affect justa small number of tasks But for MS-RCPSPLD instancesany move can cumulatively effect on a large proportion oftasks because of the existing deterioration Consequently theruntime saved in computing some unchanged parametersmay not make up for the time spent on isolating the changedtasks
10 Mathematical Problems in Engineering
Table3Summaryandcomparis
onon
thes
etof
45newgeneratedinsta
nces
with
119899=10
0of
GVN
S-MA
GVN
S-119872
119860 ℎG
VNS-
119872119860 119886ℎand
theT
Sheuristic[54]
andPR
[55]B
estresultsare
indicatedin
italic
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-10-26-15
165572
6598
13899
963646
6492
66871
63274
6392
33092
661967
6379
730698
62632
63629
29422
100-10-26-15
260
1396
618416
28607
608971
625436
79283
587145
613934
37496
580557
615474
23879
572794
615156
34826
100-10-26-15
3172856
184175
31236
1646
96
175513
59776
160283
16936
22225
166221
17124
17541
163898
172228
3461
100-10-47-91
7008
72053
40304
6974
471232
10018
69445
70098
20666
67645
6915
731499
69673
7039
36314
100-10-47-92
700352
739287
39112
703439
721152
70563
672221
715138
2417
9672304
6959
20614
693553
702369
25114
100-10-47-93
188206
195206
4816
818358
190863
93463
174162
184389
30008
177323
183958
19383
173708
183368
41432
100-10-48-15
165868
6914
13991
567093
6890
88879
164
851
65526
3491
663253
66595
22686
6479
86610
226662
100-10-48-15
2638747
659809
42043
621357
658747
9116
161867
651609
4116
1590432
625462
27097
618507
6346
3233655
100-10-48-15
3170999
174232
29659
171864
178061
73395
159497
166887
23395
161483
166366
3216
9163501
170273
27056
100-10-64-91
70527
7572
53770
371049
7692
773024
6872
27092
931287
6884
71215
26024
67894
69682
2775
6100-10-64-92
705806
744775
39415
7040
32738141
6279
4663729
711114
3013
968421
719153
22806
696727
736833
3991
8100-10-64-93
193844
206206
3115
32006
42
2044
03
108918
189324
19445
35415
180842
200842
3104
193844
199996
31871
100-10-64-15
170119
71497
34626
71075
7279
683543
69346
70022
39097
6691
69112
21249
70042
7079
617116
100-10-64-15
2690964
744775
3899
8672578
72823
61684
6606
14685085
22491
62823
682957
3178
4626331
666418
2376
100-10-64-15
3193844
218506
31871
194247
204235
71318
1806
92
1860
08
25336
187149
192556
45438
185585
190957
25296
100-20-22-15
119089
19355
29631
1936
19739
5517
618883
19118
2222
18572
1898
21438
1878
18965
2474
2100-20-22-15
24615
847731
36854
46032
46714
6792
645371
46008
28455
45218
46627
2892
645833
4633
2312
4100-20-22-15
326328
2798
128528
27212
2697
36972
92597
26419
2812
25563
25918
17531
25575
26254
21352
100-20-46-15
126243
2673
33424
26214
26631
60354
25862
26329
4119
925511
25896
1215
325476
26006
17466
100-20-46-15
260
647
66244
34408
59367
63596
71878
5796
16165
17216
56542
5942
26696
5465
59285
22313
100-20-46-15
332421
3490
937523
33539
3417
977634
31695
32909
16433
31651
33263
29116
32841
33349
2791
100-20-47-91
19007
1975
929027
1916
319685
5776
318864
1912
28311
1874
719269
2732
18455
18892
30731
100-20-47-92
50591
53481
37119
47839
51484
102173
4803
49319
3549
46278
4876
435418
44966
47893
24241
100-20-47-93
30802
31827
4572
630631
31365
6995
829437
30352
57846
27712
29458
29737
28399
2879
927559
100-20-65-15
19013
492446
26413
89865
90543
6795
788801
90095
17505
86826
89518
15052
87305
89338
2843
100-20-65-15
2253899
2606
7628718
250126
253267
5876
3244762
250316
4099
7243449
24751
17659
242353
244944
2573
5100-20-65-15
3110
567
1115886
27491
109874
113685
53418
105215
109224
2513
8108595
111243
1591
1104829
106776
1412
6100-20-65-91
19113
1978
341248
1895
719548
79561
18369
18898
30561
18697
1914
61896
18694
19265
3097
9100-20-65-92
46242
4776
865469
4593
447443
58112
4495
746229
1768
44719
45985
22324
45593
46591
30606
100-20-65-93
28018
28776
4894
327512
2810
868532
2719
2772
124279
26587
27691
3101
26455
27845
2473
100-5-22-151
500988
514285
2897
8510285
505098
84526
486586
499318
43529
494364
500988
18444
498567
510285
22335
100-5-22-152
1193440
1227080
4319
4119
2300
1210150
61537
1118610
1199248
24322
1184940
1219514
20664
1138490
1226780
26656
100-5-22-153
663947
700818
5691
2661553
792627
7997
660
6322
652627
10639
592509
6279304
2619
8643856
6597526
17267
100-5-46
-151
675739
749763
3679
7652793
701457
137862
63503
679663
5075
8636903
687941
29888
649164
671982
36202
100-5-46
-152
1404
680
1453430
58595
1443529
1470970
108595
1298430
1370870
26564
1315940
1360
830
23536
1332210
1399014
2417
6100-5-46
-153
795664
870313
50687
7977713
863567
6891
1740178
796512
81792
4755866
7786062
16809
752568
80128
1876
3100-5-48-9
1563787
586832
31214
5666
27
78386
84807
55206
560407
2415
55206
563664
25018
554299
56657
2495
2100-5-48-9
21284230
1315720
3614
51289811
1364
650
71286
1222350
1285020
2795
8116
3860
1259332
2812
21233010
1268572
34659
100-5-48-9
3684299
692453
45335
66785
717279
29393
8657279
67622
4561
6604
26
670917
41286
64822
667213
828446
Mathematical Problems in Engineering 11
Table3Con
tinued
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-5-64
-151
581916
627374
55661
576589
618167
102415
5544
26
579234
3516
7546543
590352
29678
56942
580175
56619
100-5-64
-152
120944
01261510
31296
1190830
1257643
123533
1131450
1183014
40595
1118430
1151812
46305
1047250
1158556
3337
100-5-64
-153
642267
709602
3498
8634651
688889
89403
612857
6657858
3798
5626771
660614
44591
1624753
674996
261623
100-5-64
-91
550231
577524
37365
5544
81
566747
98693
528177
5404
36
3393
530748
543895
48396
515984
536747
2993
2100-5-64
-92
1214340
1271610
40553
1183479
1236750
11992
711140
60115
5584
41523
11640
101201126
40287
1121650
1159384
27286
100-5-64
-93
610356
648765
36395
6151502
632323
87448
604586
6210378
31363
594514
6171502
30543
595191
623223
2774
9119861
119890119904119905
00
00
1514
1716
1415
119879119900119905119886
11989745
4545
4545
4545
4545
45
12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
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Submit your manuscripts atwwwhindawicom
6 Mathematical Problems in Engineering
(1) Input Two parent solutions 1199041 = (1198601 1198791198761) 1199042 = (11986021198791198762) Problem instance I
(2) Output The offspring 119904119900 = (119860119900 119879119876119900)(3) Set RA = 119881 contains 119899 tasks remaining to be assigned
resource to and 119899119906119898 represents the size of 119877119860(4) for 119903 = 1 119899 do(5) if 1198601(V119903) == 1198602(V119903) then(6) 119860119900(V119903) = 1198601(V119903)(7) RA=RAV119903(8) end if(9) end for(10) for 119903 = 1 num do(11) Assign randomly resource 119896119895 isin 119870119903 to V119903(12) end for(13)Obtain 119879119876119900 and refine the offspring 119904119900 = (119860119900 119879119876119900) in the
same way as Algorithm 2 lowastSection 43lowast
Algorithm 3 Procedure crossover
(1) Input Initial solution 1199040 a set of neighborhood structures119873119896 (119896 = 1 3) 120572
(2) Output The current best solution 119904119887 found during GVNSprocess
(3) Calculate the objective value 119891(1199040) according to the eval-uation function in Section 41
(4) 1199041015840 larr997888 1199040 lowast1199041015840 is the current solution lowast(5) 119904119887 larr997888 1199041015840 lowast119904119887 is the best solution found so farlowast(6) 119889 = 0 lowast119889 counts the consecutive iterations where 1199041015840 is not
improved lowast(7) repeat(8) Generate a random sequence (SO) to apply three
neighborhood structures(9) Apply the relevant mechanism (Section 451) in pre-
determined order specified by SO update 1199041015840 if a betterconfiguration is attained
(10) if 119891(1199041015840) lt 119891(119904119887) then(11) 119904119887 larr997888 1199041015840(12) reset the counter 119889 = 0(13) else(14) 119889 = 119889 + 1(15) end if(16) until 119889 == 120572(17) return 119904119887
Algorithm 4 GVNS operator
119904 It functions in the reduction of computational cost toevaluate any attainable neighborhood solution inspired bythe situation where the starting and finishing times of mosttasks will not be changed when neighborhood solutions aregenerated
For 1198731 and 1198732 generated by swap and reversion movethe set of tasks with changed starting and ending times onlyincorporate the elements ranking next to the isolated firsttask in the sequence of given resourceWithout considerationof deterioration only the elements located in the positionbetween two picked tasks are influenced As for 1198733 achieved
by altermove all the tasks lined up behind the designated taskin the sequence of its initial assigned resource and the newdistributed one are included Attention for three moves werecalculate the relevant parameters of above-mentioned tasksand the rest are ignored In addition the impact that the newsolutions is defying the precedence relationships to varyingdegrees will be respected in terms of 119872 lowast sum(119894119895)isin119864 120575(119894 119895) inevaluation function 119891 (see Section 41)
46 Updating population and PairSet As illustrated in Algo-rithm 1 the population119875 and thePairSet are updatedwhen an
Mathematical Problems in Engineering 7
Table 1 Settings of important parameters
Parameters Section Description Values119901 Section 42 population size for GVNS-MA 10120572 Section 45 depth of GVNS 50 000119872 Section 41 penalty value for a violation 1000 1 times 107
to precedence constraint
excellent offspring is obtained through the crossover operatorand improved further by GVNS operator First of all if itis better than the worst solution 119904119908 in 119875 for any improvedoffspring solution 119904 the worst configuration 119904119908 is replaced bythe offspring solution 119904 When the population is updated thePairSet should be updated accordingly all pairs containing 119904119908solution are deleted from setPariSet and all pairs generated bycombining 119904 solution with others in 119875 are incorporated intoPairSet
5 Computational Experiments and Results
This section plans to assess the proposed method GVNS-MA through having comparisons with the state-of-the-artmethods in the literature For the lack of known benchmarkdata for handingMS-RCPSPLDwe firstly apply the proposedGVNS-MA to solve the MS-RCPSP on exist benchmarkinstances in favor of argument its effectiveness Then onthis basis the GVNS-MA will be examined on the modifiedproblem set
51 Benchmark Instances For the purpose of assessingGVNS-MA fully and comprehensively computational exper-iments will be conducted on two sets of instances where thefirst set is composed of 30 benchmark instances irrespec-tive of deterioration and its available in Myszkowski et al[46] which are artificially created in a base of real worldobtained from the Volvo IT Department in Wroclaw Thefull information of each instance including tasks durationsresource capabilities or precedence between tasks has beengiven As for the other set it consists of 45 instances generatedwith some modifications on first set to consider the lineardeterioration The detail will be described in Section 54
52 Parameter Settings and Experiment Protocol OurGVNS-MA was programmed in MATLAB R2015b and all thereported computational experiments presented below wereexecuted on a personal computer equipped with an IntelCore i3 processor (310 GHz CPU and 2GB RAM) in theenvironment ofWindows 7 OS To eliminate the randomnessas much as possible twenty replications for each instance arecarried out
Table 1 shows the descriptions and settings of the param-eters adopted in GVNS-MA determined by preliminaryexperiments Our memetic algorithm rests upon only threeparameters the population size 119901 the depth of generalvariable neighborhood search 120572 and the price for a violationto precedence constraint 119872 For 119901 and 120572 we follow Lai andHao [55] and set 119901 = 10 120572 = 50000 while the parameter 119872
is set at 1000 for the first experimental group and 1 times 107 forthe second
53 Experimental Results without Deterioration Our firstexperimental group aims to evaluate the performance ofour GVNS-MA on the set of 30 known instances with atmost 200 tasks and 40 renewable resources Without regardto deterioration it means that the GVNS-MA will set thedeteriorating rates of all tasks at 0 when it deals with therelevant computations Table 2 records the computationalresults solved by the GVNS-MA with the goal of durationoptimization aswell as the results achieved by other referencealgorithms in the literature
Notice that the instance name (columns 1) contains itsfull description Take the instance named 100-10-26-15 asexample the number 100 represents the number of tasksincluded and 10 denotes the quantity of renewable resourcesprovided As for the number 26 and 15 it illustrates theamount of precedence relationships and the number ofdifferent introduced skills Column 2 of Table 2 indicates theprevious minimum objective values (119891119901119903119890119887) in the literaturewhich are compiled from the best solutions yield by tworecent and best performing algorithms namely GRASP[48] and DEGR [49] Columns 3 to 4 give the best resultsobtained by DEGR and GRASP The corresponding resultsof the GVNS-MA are given in columns 5 to 7 includingthe minimum objective value (119891119887119890119904119905) over 20 independentruns the average objective value (119891119886V119892) and the averagecomputing time in seconds (Time(s)) to reach 119891119887119890119904119905 The rowBest indicates a total number of instances where the specificmethod achieves optimal among three algorithms The bestone is indicated in italic In addition to verify whetherthere exists an essential difference between the best resultsof GVNS-MA and other reference algorithms the relativepercentage deviation (RPD) is defined by the equation
119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905
119891119901119903119890119887times 100 (12)
where a positive value of 119877119875119863 means an improvement ofresult achieved by GVNS-MA while the negative numberrepresents a worse solution
Table 2 discloses that the outcomes from our GVNS-MAare noteworthy compared to the state-of-the-art results inthe literature GVNS-MA improves the previous best knownresults for 19 instances and matches for 7 cases Comparedwith the 8 out of 30 cases solved by DEGR and 6 bestsolutions achieved by GRASP these data clearly indicatethe superiority of GVNS-MA compared to the previousexcellent methods Additionally it can be observed that
8 Mathematical Problems in Engineering
Table 2 Comparison of the GVNS-MA with other algorithms on known MS-RCPSP dataset [48] Best results are indicated in italic
instances 119891119901119903119890119887 DEGR GRASP GVNS-MA 119877119875119863()119891119887119890119904119905 119891119886V119892 119879119894119898119890(119904)
100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050
the improvement achieved by GVNS-MA is up to 939for instance 100-20-65-15 accompanying that the average119877119875119863() equals to 050
54 Experimental Results with Linear Deterioration Theprevious comparisons and discussions in Section 53 demon-strate the advantages of GVNS-MA in solving the relatedissues of MS-RCPSP In this section the aforementioneddataset with some modifications is used to assess the capabil-ity of GVNS-MA to solve the MS-RCPSPLDThe proceduresof generating the testing instances and analysis of the resultsare described below To make the benchmark instances meetthe considered linear deterioration precisely the deteriora-tion rate (119889119903119894) for task 119894 (119894 isin 119881) is generated randomly fromthree intervals (0 05] [05 1] and (0 1] similar to Cheng et
al [19] to shed light on the influence of the different valuerange of deterioration rate on its effectiveness
Since the extra included deterioration rate we providetwo additional heuristics for the initial population generationof GVNS-MA Both the two methods affect the phase ofgenerating 119879119876119895 determining the sequence of tasks assignedto resource 119895 119895 isin 119870 The first heuristic considers thesequence in descending order of deterioration rate (ℎ) whilethe other rests upon an ascending order of ratio (119886ℎ)of the basic processing time and deterioration rate Themethods adopting the former and latter heuristic to popula-tion initialization are dubbedGVNS-119872119860ℎ and GVNS-119872119860119886ℎrespectively
Here instanceswith 100 tasks fromMyszkowski et al [46]are isolated to attain the researched objects which fit with theunique nature of the MS-RCPSPLDmore To account for the
Mathematical Problems in Engineering 9
three intervals from which the deterioration rate is drawn3 extended cases are needed to solve for each instance Forconvenience these instances are denoted by adding a suffixfor identification to different intervals For example 100-10-26-15 1 represents the original case 100-10-26-15 is modifiedby adding the deterioration rates produced in (0 05] to thedurations of tasks In total there are 45(3 times 15) instancesrandomly generated
Due to zero known results in literature for same datasetthe improved tabu search (ITS) proposed by Dai et al [54]who discussed the MS-RCPSP under step deterioration anda path relinking algorithm (PR) [55] based on the populationpath relinking framework are programmed as referencealgorithms
Table 3 reports the computational results achieved by theITS PR GVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ on theset of 45 benchmark instances 119891119887119890119904119905 denotes the minimumobjective value and 119891119886V119892 is computed as the average objectivevalue of 20 runs
First Table 3 discloses that the solutions obtained byGVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ are better thanthe ITS and PR for any instance from the perspectiveof both quality of schedule and runtime To some extentthese results demonstrated the differences between lineardeterioration and step deterioration and the superiority ofmemetic algorithm framework Second these three methodsdiffering in the sort order of tasks in initialize phase behavesimilarly where GVNS-MA obtains the best 15 out of 45instances 17 for GVNS-MAℎ and 14 for GVNS119886ℎ in termsof 119891119887119890119904119905 Specifically GVNS-MA and GVNS-MAℎ attain theoptimal simultaneously for the instance 100-5-48-9-1 Froma view point of 119891119886V119892 and run time three methods alsohave a balanced performance Third as far as three differentintervals to generate deterioration rate are concerned thephenomenon did not happen that the relevant algorithmsdisplay strikingly different behavior In other words theperformance of the proposed algorithm is not sensitive to thesetting of deterioration rate
55 Analysis and Discussions In this section we study twoessential ingredients of the proposed GVNS-MA to getan insight to its performance One is the rapid evaluationmechanism the other is the role of the memetic framework
551 Importance of Rapid Evaluation Mechanism GVNS-MA with rapid evaluation mechanism only calculates therelevant parameters of some particular tasks rather thanall when the procedure computes the objective value ofa neighborhood solution To highlight the key role ofthe rapid evaluation mechanism two sets of comparisonexperiments are carried out on generated dataset with twoalgorithms GVNS-MA and GVNS-MA0 including sameingredients with GVNS-MA except for the computation ofobjective value When GVNS-MA0 figures up the value of aneighborhood solution it computes all relevant parametersagain
Table 4 records the experimental results carried out on thedataset [46] without consideration of deterioration whereas
Table 5 shows the comparisons of GVNS-MA and GVNS-MA0 about the set of 15 instances generated in Section 54on account of the indiscrimination in three intervals Col-umn 2 and 5 record the best attained by two algorithmsColumn 3 and 6 indicate the minimum time cost to a finalfeasible schedule with one run of procedure Note that thebest objective value cannot be guaranteed as the output ofshortest runtime As for the parameters in column 4 and 7they represent the mean runtime Finally two parameters119863119864119881119904ℎ119900119903119905119890119904119905 and 119863119864119881119886V119892 are used to disclose the runtimedeviation of two methods defined by equations
119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051
times 100 (13)
and
119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922
119879119886V1198921times 100 (14)
respectively The positive value of 119863119864119881119904ℎ119900119903119905119890119904119905() and119863119864119881119886V119892() means that GVNS-MA0 has better performanceand negative value tells GVNS-MA is prior to GVNS-MA0in terms of time cost And the rows Better and Worserespectively show the number of instances for which thecorresponding results of the associated algorithm are betterand worse than the other
The results summarized in Table 4 disclose that theGVNS-MA has an overwhelming advantage over GVNS-MA0 in terms of the computation time to solve MS-RCPSPleaving out the deterioration effect Indeed the shortestruntime 119879119904ℎ1199001199031199051198901199041199051 of the GVNS-MA method is better thanthe shortest runtime 119879119904ℎ1199001199031199051198901199041199052 of GVNS-MA0 for 30 out of30 representative instances and the average runtime 119879119886V1198921 isbetter for 28 out of 30 instances Meanwhile the average valueof 119863119864119881119904ℎ119900119903119905119890119904119905() equals -1379 accompanying with a highof -1880 percent in 119863119864119881119886V119892()
However focusing on Table 5 the results of twoapproaches are neck and neck and GVNS-MA lost its earlysuperiority in MS-RCPSP In terms of shortest runtimeGVNS-MA successes for 7 out of 15 tested instances whileGVNS-MA0 reaches optimal for the remain As for averageruntime GVNS-MA performs better for 9 out of 15 examplesand GVNS-MA0 achieves reversion in others 6 instancesWith these data it will be hard to judge the true benefits ofone approach versus the other
To figure out the reason of this phenomenon we shouldcome back to the inner rationale of rapid evaluation mecha-nism When GVNS-MA computes the completion time of aneighborhood solution it only recalculates the tasksrsquo relatedparameters influenced by the particular move InMS-RCPSPa move including swap reverse and alter will affect justa small number of tasks But for MS-RCPSPLD instancesany move can cumulatively effect on a large proportion oftasks because of the existing deterioration Consequently theruntime saved in computing some unchanged parametersmay not make up for the time spent on isolating the changedtasks
10 Mathematical Problems in Engineering
Table3Summaryandcomparis
onon
thes
etof
45newgeneratedinsta
nces
with
119899=10
0of
GVN
S-MA
GVN
S-119872
119860 ℎG
VNS-
119872119860 119886ℎand
theT
Sheuristic[54]
andPR
[55]B
estresultsare
indicatedin
italic
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-10-26-15
165572
6598
13899
963646
6492
66871
63274
6392
33092
661967
6379
730698
62632
63629
29422
100-10-26-15
260
1396
618416
28607
608971
625436
79283
587145
613934
37496
580557
615474
23879
572794
615156
34826
100-10-26-15
3172856
184175
31236
1646
96
175513
59776
160283
16936
22225
166221
17124
17541
163898
172228
3461
100-10-47-91
7008
72053
40304
6974
471232
10018
69445
70098
20666
67645
6915
731499
69673
7039
36314
100-10-47-92
700352
739287
39112
703439
721152
70563
672221
715138
2417
9672304
6959
20614
693553
702369
25114
100-10-47-93
188206
195206
4816
818358
190863
93463
174162
184389
30008
177323
183958
19383
173708
183368
41432
100-10-48-15
165868
6914
13991
567093
6890
88879
164
851
65526
3491
663253
66595
22686
6479
86610
226662
100-10-48-15
2638747
659809
42043
621357
658747
9116
161867
651609
4116
1590432
625462
27097
618507
6346
3233655
100-10-48-15
3170999
174232
29659
171864
178061
73395
159497
166887
23395
161483
166366
3216
9163501
170273
27056
100-10-64-91
70527
7572
53770
371049
7692
773024
6872
27092
931287
6884
71215
26024
67894
69682
2775
6100-10-64-92
705806
744775
39415
7040
32738141
6279
4663729
711114
3013
968421
719153
22806
696727
736833
3991
8100-10-64-93
193844
206206
3115
32006
42
2044
03
108918
189324
19445
35415
180842
200842
3104
193844
199996
31871
100-10-64-15
170119
71497
34626
71075
7279
683543
69346
70022
39097
6691
69112
21249
70042
7079
617116
100-10-64-15
2690964
744775
3899
8672578
72823
61684
6606
14685085
22491
62823
682957
3178
4626331
666418
2376
100-10-64-15
3193844
218506
31871
194247
204235
71318
1806
92
1860
08
25336
187149
192556
45438
185585
190957
25296
100-20-22-15
119089
19355
29631
1936
19739
5517
618883
19118
2222
18572
1898
21438
1878
18965
2474
2100-20-22-15
24615
847731
36854
46032
46714
6792
645371
46008
28455
45218
46627
2892
645833
4633
2312
4100-20-22-15
326328
2798
128528
27212
2697
36972
92597
26419
2812
25563
25918
17531
25575
26254
21352
100-20-46-15
126243
2673
33424
26214
26631
60354
25862
26329
4119
925511
25896
1215
325476
26006
17466
100-20-46-15
260
647
66244
34408
59367
63596
71878
5796
16165
17216
56542
5942
26696
5465
59285
22313
100-20-46-15
332421
3490
937523
33539
3417
977634
31695
32909
16433
31651
33263
29116
32841
33349
2791
100-20-47-91
19007
1975
929027
1916
319685
5776
318864
1912
28311
1874
719269
2732
18455
18892
30731
100-20-47-92
50591
53481
37119
47839
51484
102173
4803
49319
3549
46278
4876
435418
44966
47893
24241
100-20-47-93
30802
31827
4572
630631
31365
6995
829437
30352
57846
27712
29458
29737
28399
2879
927559
100-20-65-15
19013
492446
26413
89865
90543
6795
788801
90095
17505
86826
89518
15052
87305
89338
2843
100-20-65-15
2253899
2606
7628718
250126
253267
5876
3244762
250316
4099
7243449
24751
17659
242353
244944
2573
5100-20-65-15
3110
567
1115886
27491
109874
113685
53418
105215
109224
2513
8108595
111243
1591
1104829
106776
1412
6100-20-65-91
19113
1978
341248
1895
719548
79561
18369
18898
30561
18697
1914
61896
18694
19265
3097
9100-20-65-92
46242
4776
865469
4593
447443
58112
4495
746229
1768
44719
45985
22324
45593
46591
30606
100-20-65-93
28018
28776
4894
327512
2810
868532
2719
2772
124279
26587
27691
3101
26455
27845
2473
100-5-22-151
500988
514285
2897
8510285
505098
84526
486586
499318
43529
494364
500988
18444
498567
510285
22335
100-5-22-152
1193440
1227080
4319
4119
2300
1210150
61537
1118610
1199248
24322
1184940
1219514
20664
1138490
1226780
26656
100-5-22-153
663947
700818
5691
2661553
792627
7997
660
6322
652627
10639
592509
6279304
2619
8643856
6597526
17267
100-5-46
-151
675739
749763
3679
7652793
701457
137862
63503
679663
5075
8636903
687941
29888
649164
671982
36202
100-5-46
-152
1404
680
1453430
58595
1443529
1470970
108595
1298430
1370870
26564
1315940
1360
830
23536
1332210
1399014
2417
6100-5-46
-153
795664
870313
50687
7977713
863567
6891
1740178
796512
81792
4755866
7786062
16809
752568
80128
1876
3100-5-48-9
1563787
586832
31214
5666
27
78386
84807
55206
560407
2415
55206
563664
25018
554299
56657
2495
2100-5-48-9
21284230
1315720
3614
51289811
1364
650
71286
1222350
1285020
2795
8116
3860
1259332
2812
21233010
1268572
34659
100-5-48-9
3684299
692453
45335
66785
717279
29393
8657279
67622
4561
6604
26
670917
41286
64822
667213
828446
Mathematical Problems in Engineering 11
Table3Con
tinued
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-5-64
-151
581916
627374
55661
576589
618167
102415
5544
26
579234
3516
7546543
590352
29678
56942
580175
56619
100-5-64
-152
120944
01261510
31296
1190830
1257643
123533
1131450
1183014
40595
1118430
1151812
46305
1047250
1158556
3337
100-5-64
-153
642267
709602
3498
8634651
688889
89403
612857
6657858
3798
5626771
660614
44591
1624753
674996
261623
100-5-64
-91
550231
577524
37365
5544
81
566747
98693
528177
5404
36
3393
530748
543895
48396
515984
536747
2993
2100-5-64
-92
1214340
1271610
40553
1183479
1236750
11992
711140
60115
5584
41523
11640
101201126
40287
1121650
1159384
27286
100-5-64
-93
610356
648765
36395
6151502
632323
87448
604586
6210378
31363
594514
6171502
30543
595191
623223
2774
9119861
119890119904119905
00
00
1514
1716
1415
119879119900119905119886
11989745
4545
4545
4545
4545
45
12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
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Mathematical Problems in Engineering 7
Table 1 Settings of important parameters
Parameters Section Description Values119901 Section 42 population size for GVNS-MA 10120572 Section 45 depth of GVNS 50 000119872 Section 41 penalty value for a violation 1000 1 times 107
to precedence constraint
excellent offspring is obtained through the crossover operatorand improved further by GVNS operator First of all if itis better than the worst solution 119904119908 in 119875 for any improvedoffspring solution 119904 the worst configuration 119904119908 is replaced bythe offspring solution 119904 When the population is updated thePairSet should be updated accordingly all pairs containing 119904119908solution are deleted from setPariSet and all pairs generated bycombining 119904 solution with others in 119875 are incorporated intoPairSet
5 Computational Experiments and Results
This section plans to assess the proposed method GVNS-MA through having comparisons with the state-of-the-artmethods in the literature For the lack of known benchmarkdata for handingMS-RCPSPLDwe firstly apply the proposedGVNS-MA to solve the MS-RCPSP on exist benchmarkinstances in favor of argument its effectiveness Then onthis basis the GVNS-MA will be examined on the modifiedproblem set
51 Benchmark Instances For the purpose of assessingGVNS-MA fully and comprehensively computational exper-iments will be conducted on two sets of instances where thefirst set is composed of 30 benchmark instances irrespec-tive of deterioration and its available in Myszkowski et al[46] which are artificially created in a base of real worldobtained from the Volvo IT Department in Wroclaw Thefull information of each instance including tasks durationsresource capabilities or precedence between tasks has beengiven As for the other set it consists of 45 instances generatedwith some modifications on first set to consider the lineardeterioration The detail will be described in Section 54
52 Parameter Settings and Experiment Protocol OurGVNS-MA was programmed in MATLAB R2015b and all thereported computational experiments presented below wereexecuted on a personal computer equipped with an IntelCore i3 processor (310 GHz CPU and 2GB RAM) in theenvironment ofWindows 7 OS To eliminate the randomnessas much as possible twenty replications for each instance arecarried out
Table 1 shows the descriptions and settings of the param-eters adopted in GVNS-MA determined by preliminaryexperiments Our memetic algorithm rests upon only threeparameters the population size 119901 the depth of generalvariable neighborhood search 120572 and the price for a violationto precedence constraint 119872 For 119901 and 120572 we follow Lai andHao [55] and set 119901 = 10 120572 = 50000 while the parameter 119872
is set at 1000 for the first experimental group and 1 times 107 forthe second
53 Experimental Results without Deterioration Our firstexperimental group aims to evaluate the performance ofour GVNS-MA on the set of 30 known instances with atmost 200 tasks and 40 renewable resources Without regardto deterioration it means that the GVNS-MA will set thedeteriorating rates of all tasks at 0 when it deals with therelevant computations Table 2 records the computationalresults solved by the GVNS-MA with the goal of durationoptimization aswell as the results achieved by other referencealgorithms in the literature
Notice that the instance name (columns 1) contains itsfull description Take the instance named 100-10-26-15 asexample the number 100 represents the number of tasksincluded and 10 denotes the quantity of renewable resourcesprovided As for the number 26 and 15 it illustrates theamount of precedence relationships and the number ofdifferent introduced skills Column 2 of Table 2 indicates theprevious minimum objective values (119891119901119903119890119887) in the literaturewhich are compiled from the best solutions yield by tworecent and best performing algorithms namely GRASP[48] and DEGR [49] Columns 3 to 4 give the best resultsobtained by DEGR and GRASP The corresponding resultsof the GVNS-MA are given in columns 5 to 7 includingthe minimum objective value (119891119887119890119904119905) over 20 independentruns the average objective value (119891119886V119892) and the averagecomputing time in seconds (Time(s)) to reach 119891119887119890119904119905 The rowBest indicates a total number of instances where the specificmethod achieves optimal among three algorithms The bestone is indicated in italic In addition to verify whetherthere exists an essential difference between the best resultsof GVNS-MA and other reference algorithms the relativepercentage deviation (RPD) is defined by the equation
119877119875119863 () =119891119901119903119890119887 minus 119891119887119890119904119905
119891119901119903119890119887times 100 (12)
where a positive value of 119877119875119863 means an improvement ofresult achieved by GVNS-MA while the negative numberrepresents a worse solution
Table 2 discloses that the outcomes from our GVNS-MAare noteworthy compared to the state-of-the-art results inthe literature GVNS-MA improves the previous best knownresults for 19 instances and matches for 7 cases Comparedwith the 8 out of 30 cases solved by DEGR and 6 bestsolutions achieved by GRASP these data clearly indicatethe superiority of GVNS-MA compared to the previousexcellent methods Additionally it can be observed that
8 Mathematical Problems in Engineering
Table 2 Comparison of the GVNS-MA with other algorithms on known MS-RCPSP dataset [48] Best results are indicated in italic
instances 119891119901119903119890119887 DEGR GRASP GVNS-MA 119877119875119863()119891119887119890119904119905 119891119886V119892 119879119894119898119890(119904)
100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050
the improvement achieved by GVNS-MA is up to 939for instance 100-20-65-15 accompanying that the average119877119875119863() equals to 050
54 Experimental Results with Linear Deterioration Theprevious comparisons and discussions in Section 53 demon-strate the advantages of GVNS-MA in solving the relatedissues of MS-RCPSP In this section the aforementioneddataset with some modifications is used to assess the capabil-ity of GVNS-MA to solve the MS-RCPSPLDThe proceduresof generating the testing instances and analysis of the resultsare described below To make the benchmark instances meetthe considered linear deterioration precisely the deteriora-tion rate (119889119903119894) for task 119894 (119894 isin 119881) is generated randomly fromthree intervals (0 05] [05 1] and (0 1] similar to Cheng et
al [19] to shed light on the influence of the different valuerange of deterioration rate on its effectiveness
Since the extra included deterioration rate we providetwo additional heuristics for the initial population generationof GVNS-MA Both the two methods affect the phase ofgenerating 119879119876119895 determining the sequence of tasks assignedto resource 119895 119895 isin 119870 The first heuristic considers thesequence in descending order of deterioration rate (ℎ) whilethe other rests upon an ascending order of ratio (119886ℎ)of the basic processing time and deterioration rate Themethods adopting the former and latter heuristic to popula-tion initialization are dubbedGVNS-119872119860ℎ and GVNS-119872119860119886ℎrespectively
Here instanceswith 100 tasks fromMyszkowski et al [46]are isolated to attain the researched objects which fit with theunique nature of the MS-RCPSPLDmore To account for the
Mathematical Problems in Engineering 9
three intervals from which the deterioration rate is drawn3 extended cases are needed to solve for each instance Forconvenience these instances are denoted by adding a suffixfor identification to different intervals For example 100-10-26-15 1 represents the original case 100-10-26-15 is modifiedby adding the deterioration rates produced in (0 05] to thedurations of tasks In total there are 45(3 times 15) instancesrandomly generated
Due to zero known results in literature for same datasetthe improved tabu search (ITS) proposed by Dai et al [54]who discussed the MS-RCPSP under step deterioration anda path relinking algorithm (PR) [55] based on the populationpath relinking framework are programmed as referencealgorithms
Table 3 reports the computational results achieved by theITS PR GVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ on theset of 45 benchmark instances 119891119887119890119904119905 denotes the minimumobjective value and 119891119886V119892 is computed as the average objectivevalue of 20 runs
First Table 3 discloses that the solutions obtained byGVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ are better thanthe ITS and PR for any instance from the perspectiveof both quality of schedule and runtime To some extentthese results demonstrated the differences between lineardeterioration and step deterioration and the superiority ofmemetic algorithm framework Second these three methodsdiffering in the sort order of tasks in initialize phase behavesimilarly where GVNS-MA obtains the best 15 out of 45instances 17 for GVNS-MAℎ and 14 for GVNS119886ℎ in termsof 119891119887119890119904119905 Specifically GVNS-MA and GVNS-MAℎ attain theoptimal simultaneously for the instance 100-5-48-9-1 Froma view point of 119891119886V119892 and run time three methods alsohave a balanced performance Third as far as three differentintervals to generate deterioration rate are concerned thephenomenon did not happen that the relevant algorithmsdisplay strikingly different behavior In other words theperformance of the proposed algorithm is not sensitive to thesetting of deterioration rate
55 Analysis and Discussions In this section we study twoessential ingredients of the proposed GVNS-MA to getan insight to its performance One is the rapid evaluationmechanism the other is the role of the memetic framework
551 Importance of Rapid Evaluation Mechanism GVNS-MA with rapid evaluation mechanism only calculates therelevant parameters of some particular tasks rather thanall when the procedure computes the objective value ofa neighborhood solution To highlight the key role ofthe rapid evaluation mechanism two sets of comparisonexperiments are carried out on generated dataset with twoalgorithms GVNS-MA and GVNS-MA0 including sameingredients with GVNS-MA except for the computation ofobjective value When GVNS-MA0 figures up the value of aneighborhood solution it computes all relevant parametersagain
Table 4 records the experimental results carried out on thedataset [46] without consideration of deterioration whereas
Table 5 shows the comparisons of GVNS-MA and GVNS-MA0 about the set of 15 instances generated in Section 54on account of the indiscrimination in three intervals Col-umn 2 and 5 record the best attained by two algorithmsColumn 3 and 6 indicate the minimum time cost to a finalfeasible schedule with one run of procedure Note that thebest objective value cannot be guaranteed as the output ofshortest runtime As for the parameters in column 4 and 7they represent the mean runtime Finally two parameters119863119864119881119904ℎ119900119903119905119890119904119905 and 119863119864119881119886V119892 are used to disclose the runtimedeviation of two methods defined by equations
119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051
times 100 (13)
and
119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922
119879119886V1198921times 100 (14)
respectively The positive value of 119863119864119881119904ℎ119900119903119905119890119904119905() and119863119864119881119886V119892() means that GVNS-MA0 has better performanceand negative value tells GVNS-MA is prior to GVNS-MA0in terms of time cost And the rows Better and Worserespectively show the number of instances for which thecorresponding results of the associated algorithm are betterand worse than the other
The results summarized in Table 4 disclose that theGVNS-MA has an overwhelming advantage over GVNS-MA0 in terms of the computation time to solve MS-RCPSPleaving out the deterioration effect Indeed the shortestruntime 119879119904ℎ1199001199031199051198901199041199051 of the GVNS-MA method is better thanthe shortest runtime 119879119904ℎ1199001199031199051198901199041199052 of GVNS-MA0 for 30 out of30 representative instances and the average runtime 119879119886V1198921 isbetter for 28 out of 30 instances Meanwhile the average valueof 119863119864119881119904ℎ119900119903119905119890119904119905() equals -1379 accompanying with a highof -1880 percent in 119863119864119881119886V119892()
However focusing on Table 5 the results of twoapproaches are neck and neck and GVNS-MA lost its earlysuperiority in MS-RCPSP In terms of shortest runtimeGVNS-MA successes for 7 out of 15 tested instances whileGVNS-MA0 reaches optimal for the remain As for averageruntime GVNS-MA performs better for 9 out of 15 examplesand GVNS-MA0 achieves reversion in others 6 instancesWith these data it will be hard to judge the true benefits ofone approach versus the other
To figure out the reason of this phenomenon we shouldcome back to the inner rationale of rapid evaluation mecha-nism When GVNS-MA computes the completion time of aneighborhood solution it only recalculates the tasksrsquo relatedparameters influenced by the particular move InMS-RCPSPa move including swap reverse and alter will affect justa small number of tasks But for MS-RCPSPLD instancesany move can cumulatively effect on a large proportion oftasks because of the existing deterioration Consequently theruntime saved in computing some unchanged parametersmay not make up for the time spent on isolating the changedtasks
10 Mathematical Problems in Engineering
Table3Summaryandcomparis
onon
thes
etof
45newgeneratedinsta
nces
with
119899=10
0of
GVN
S-MA
GVN
S-119872
119860 ℎG
VNS-
119872119860 119886ℎand
theT
Sheuristic[54]
andPR
[55]B
estresultsare
indicatedin
italic
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-10-26-15
165572
6598
13899
963646
6492
66871
63274
6392
33092
661967
6379
730698
62632
63629
29422
100-10-26-15
260
1396
618416
28607
608971
625436
79283
587145
613934
37496
580557
615474
23879
572794
615156
34826
100-10-26-15
3172856
184175
31236
1646
96
175513
59776
160283
16936
22225
166221
17124
17541
163898
172228
3461
100-10-47-91
7008
72053
40304
6974
471232
10018
69445
70098
20666
67645
6915
731499
69673
7039
36314
100-10-47-92
700352
739287
39112
703439
721152
70563
672221
715138
2417
9672304
6959
20614
693553
702369
25114
100-10-47-93
188206
195206
4816
818358
190863
93463
174162
184389
30008
177323
183958
19383
173708
183368
41432
100-10-48-15
165868
6914
13991
567093
6890
88879
164
851
65526
3491
663253
66595
22686
6479
86610
226662
100-10-48-15
2638747
659809
42043
621357
658747
9116
161867
651609
4116
1590432
625462
27097
618507
6346
3233655
100-10-48-15
3170999
174232
29659
171864
178061
73395
159497
166887
23395
161483
166366
3216
9163501
170273
27056
100-10-64-91
70527
7572
53770
371049
7692
773024
6872
27092
931287
6884
71215
26024
67894
69682
2775
6100-10-64-92
705806
744775
39415
7040
32738141
6279
4663729
711114
3013
968421
719153
22806
696727
736833
3991
8100-10-64-93
193844
206206
3115
32006
42
2044
03
108918
189324
19445
35415
180842
200842
3104
193844
199996
31871
100-10-64-15
170119
71497
34626
71075
7279
683543
69346
70022
39097
6691
69112
21249
70042
7079
617116
100-10-64-15
2690964
744775
3899
8672578
72823
61684
6606
14685085
22491
62823
682957
3178
4626331
666418
2376
100-10-64-15
3193844
218506
31871
194247
204235
71318
1806
92
1860
08
25336
187149
192556
45438
185585
190957
25296
100-20-22-15
119089
19355
29631
1936
19739
5517
618883
19118
2222
18572
1898
21438
1878
18965
2474
2100-20-22-15
24615
847731
36854
46032
46714
6792
645371
46008
28455
45218
46627
2892
645833
4633
2312
4100-20-22-15
326328
2798
128528
27212
2697
36972
92597
26419
2812
25563
25918
17531
25575
26254
21352
100-20-46-15
126243
2673
33424
26214
26631
60354
25862
26329
4119
925511
25896
1215
325476
26006
17466
100-20-46-15
260
647
66244
34408
59367
63596
71878
5796
16165
17216
56542
5942
26696
5465
59285
22313
100-20-46-15
332421
3490
937523
33539
3417
977634
31695
32909
16433
31651
33263
29116
32841
33349
2791
100-20-47-91
19007
1975
929027
1916
319685
5776
318864
1912
28311
1874
719269
2732
18455
18892
30731
100-20-47-92
50591
53481
37119
47839
51484
102173
4803
49319
3549
46278
4876
435418
44966
47893
24241
100-20-47-93
30802
31827
4572
630631
31365
6995
829437
30352
57846
27712
29458
29737
28399
2879
927559
100-20-65-15
19013
492446
26413
89865
90543
6795
788801
90095
17505
86826
89518
15052
87305
89338
2843
100-20-65-15
2253899
2606
7628718
250126
253267
5876
3244762
250316
4099
7243449
24751
17659
242353
244944
2573
5100-20-65-15
3110
567
1115886
27491
109874
113685
53418
105215
109224
2513
8108595
111243
1591
1104829
106776
1412
6100-20-65-91
19113
1978
341248
1895
719548
79561
18369
18898
30561
18697
1914
61896
18694
19265
3097
9100-20-65-92
46242
4776
865469
4593
447443
58112
4495
746229
1768
44719
45985
22324
45593
46591
30606
100-20-65-93
28018
28776
4894
327512
2810
868532
2719
2772
124279
26587
27691
3101
26455
27845
2473
100-5-22-151
500988
514285
2897
8510285
505098
84526
486586
499318
43529
494364
500988
18444
498567
510285
22335
100-5-22-152
1193440
1227080
4319
4119
2300
1210150
61537
1118610
1199248
24322
1184940
1219514
20664
1138490
1226780
26656
100-5-22-153
663947
700818
5691
2661553
792627
7997
660
6322
652627
10639
592509
6279304
2619
8643856
6597526
17267
100-5-46
-151
675739
749763
3679
7652793
701457
137862
63503
679663
5075
8636903
687941
29888
649164
671982
36202
100-5-46
-152
1404
680
1453430
58595
1443529
1470970
108595
1298430
1370870
26564
1315940
1360
830
23536
1332210
1399014
2417
6100-5-46
-153
795664
870313
50687
7977713
863567
6891
1740178
796512
81792
4755866
7786062
16809
752568
80128
1876
3100-5-48-9
1563787
586832
31214
5666
27
78386
84807
55206
560407
2415
55206
563664
25018
554299
56657
2495
2100-5-48-9
21284230
1315720
3614
51289811
1364
650
71286
1222350
1285020
2795
8116
3860
1259332
2812
21233010
1268572
34659
100-5-48-9
3684299
692453
45335
66785
717279
29393
8657279
67622
4561
6604
26
670917
41286
64822
667213
828446
Mathematical Problems in Engineering 11
Table3Con
tinued
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-5-64
-151
581916
627374
55661
576589
618167
102415
5544
26
579234
3516
7546543
590352
29678
56942
580175
56619
100-5-64
-152
120944
01261510
31296
1190830
1257643
123533
1131450
1183014
40595
1118430
1151812
46305
1047250
1158556
3337
100-5-64
-153
642267
709602
3498
8634651
688889
89403
612857
6657858
3798
5626771
660614
44591
1624753
674996
261623
100-5-64
-91
550231
577524
37365
5544
81
566747
98693
528177
5404
36
3393
530748
543895
48396
515984
536747
2993
2100-5-64
-92
1214340
1271610
40553
1183479
1236750
11992
711140
60115
5584
41523
11640
101201126
40287
1121650
1159384
27286
100-5-64
-93
610356
648765
36395
6151502
632323
87448
604586
6210378
31363
594514
6171502
30543
595191
623223
2774
9119861
119890119904119905
00
00
1514
1716
1415
119879119900119905119886
11989745
4545
4545
4545
4545
45
12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
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8 Mathematical Problems in Engineering
Table 2 Comparison of the GVNS-MA with other algorithms on known MS-RCPSP dataset [48] Best results are indicated in italic
instances 119891119901119903119890119887 DEGR GRASP GVNS-MA 119877119875119863()119891119887119890119904119905 119891119886V119892 119879119894119898119890(119904)
100-10-26-15 236 236 250 237 2426 19178 -042100-10-47-9 256 256 263 253 2568 12490 117100-10-48-15 247 247 255 245 2509 17505 081100-10-64-9 250 250 254 247 2571 16536 120100-10-64-15 248 248 256 246 2506 17317 081100-20-22-15 134 134 134 133 1376 14953 075100-20-46-15 164 164 170 160 1632 13770 244100-20-47-9 138 138 180 132 1394 12870 435100-20-65-15 213 240 213 193 1980 10317 939100-20-65-9 134 134 134 134 1400 13893 000100-5-22-15 484 484 503 483 4840 13164 021100-5-46-15 529 529 552 528 5331 18948 019100-5-48-9 491 491 509 489 4905 13445 041100-5-64-15 483 483 501 480 4823 14627 062100-5-64-9 475 475 494 474 4752 16261 021200-10-128-15 462 462 491 479 4990 74632 -368200-10-50-15 488 488 522 488 5006 89529 000200-10-50-9 489 489 506 487 4932 79334 041200-10-84-9 517 517 526 509 5140 71920 155200-10-85-15 479 479 486 477 4818 56176 042200-20-145-15 245 245 262 252 2710 66008 -286200-20-54-15 270 270 304 291 3034 84746 -778200-20-55-9 257 262 257 257 2630 63997 000200-20-97-15 336 336 347 334 3382 72457 060200-20-97-9 253 253 253 253 2581 71620 000200-40-133-15 159 159 163 157 1650 77282 126200-40-45-15 164 164 164 159 1636 56558 305200-40-45-9 144 168 144 144 1520 62653 000200-40-90-9 145 160 145 145 1494 65424 000200-40-91-15 153 153 153 153 1576 62401 000119861119890119904119905 8 6 26119879119900119905119886119897 30 30 30119860V119890119903119886119892119890 050
the improvement achieved by GVNS-MA is up to 939for instance 100-20-65-15 accompanying that the average119877119875119863() equals to 050
54 Experimental Results with Linear Deterioration Theprevious comparisons and discussions in Section 53 demon-strate the advantages of GVNS-MA in solving the relatedissues of MS-RCPSP In this section the aforementioneddataset with some modifications is used to assess the capabil-ity of GVNS-MA to solve the MS-RCPSPLDThe proceduresof generating the testing instances and analysis of the resultsare described below To make the benchmark instances meetthe considered linear deterioration precisely the deteriora-tion rate (119889119903119894) for task 119894 (119894 isin 119881) is generated randomly fromthree intervals (0 05] [05 1] and (0 1] similar to Cheng et
al [19] to shed light on the influence of the different valuerange of deterioration rate on its effectiveness
Since the extra included deterioration rate we providetwo additional heuristics for the initial population generationof GVNS-MA Both the two methods affect the phase ofgenerating 119879119876119895 determining the sequence of tasks assignedto resource 119895 119895 isin 119870 The first heuristic considers thesequence in descending order of deterioration rate (ℎ) whilethe other rests upon an ascending order of ratio (119886ℎ)of the basic processing time and deterioration rate Themethods adopting the former and latter heuristic to popula-tion initialization are dubbedGVNS-119872119860ℎ and GVNS-119872119860119886ℎrespectively
Here instanceswith 100 tasks fromMyszkowski et al [46]are isolated to attain the researched objects which fit with theunique nature of the MS-RCPSPLDmore To account for the
Mathematical Problems in Engineering 9
three intervals from which the deterioration rate is drawn3 extended cases are needed to solve for each instance Forconvenience these instances are denoted by adding a suffixfor identification to different intervals For example 100-10-26-15 1 represents the original case 100-10-26-15 is modifiedby adding the deterioration rates produced in (0 05] to thedurations of tasks In total there are 45(3 times 15) instancesrandomly generated
Due to zero known results in literature for same datasetthe improved tabu search (ITS) proposed by Dai et al [54]who discussed the MS-RCPSP under step deterioration anda path relinking algorithm (PR) [55] based on the populationpath relinking framework are programmed as referencealgorithms
Table 3 reports the computational results achieved by theITS PR GVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ on theset of 45 benchmark instances 119891119887119890119904119905 denotes the minimumobjective value and 119891119886V119892 is computed as the average objectivevalue of 20 runs
First Table 3 discloses that the solutions obtained byGVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ are better thanthe ITS and PR for any instance from the perspectiveof both quality of schedule and runtime To some extentthese results demonstrated the differences between lineardeterioration and step deterioration and the superiority ofmemetic algorithm framework Second these three methodsdiffering in the sort order of tasks in initialize phase behavesimilarly where GVNS-MA obtains the best 15 out of 45instances 17 for GVNS-MAℎ and 14 for GVNS119886ℎ in termsof 119891119887119890119904119905 Specifically GVNS-MA and GVNS-MAℎ attain theoptimal simultaneously for the instance 100-5-48-9-1 Froma view point of 119891119886V119892 and run time three methods alsohave a balanced performance Third as far as three differentintervals to generate deterioration rate are concerned thephenomenon did not happen that the relevant algorithmsdisplay strikingly different behavior In other words theperformance of the proposed algorithm is not sensitive to thesetting of deterioration rate
55 Analysis and Discussions In this section we study twoessential ingredients of the proposed GVNS-MA to getan insight to its performance One is the rapid evaluationmechanism the other is the role of the memetic framework
551 Importance of Rapid Evaluation Mechanism GVNS-MA with rapid evaluation mechanism only calculates therelevant parameters of some particular tasks rather thanall when the procedure computes the objective value ofa neighborhood solution To highlight the key role ofthe rapid evaluation mechanism two sets of comparisonexperiments are carried out on generated dataset with twoalgorithms GVNS-MA and GVNS-MA0 including sameingredients with GVNS-MA except for the computation ofobjective value When GVNS-MA0 figures up the value of aneighborhood solution it computes all relevant parametersagain
Table 4 records the experimental results carried out on thedataset [46] without consideration of deterioration whereas
Table 5 shows the comparisons of GVNS-MA and GVNS-MA0 about the set of 15 instances generated in Section 54on account of the indiscrimination in three intervals Col-umn 2 and 5 record the best attained by two algorithmsColumn 3 and 6 indicate the minimum time cost to a finalfeasible schedule with one run of procedure Note that thebest objective value cannot be guaranteed as the output ofshortest runtime As for the parameters in column 4 and 7they represent the mean runtime Finally two parameters119863119864119881119904ℎ119900119903119905119890119904119905 and 119863119864119881119886V119892 are used to disclose the runtimedeviation of two methods defined by equations
119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051
times 100 (13)
and
119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922
119879119886V1198921times 100 (14)
respectively The positive value of 119863119864119881119904ℎ119900119903119905119890119904119905() and119863119864119881119886V119892() means that GVNS-MA0 has better performanceand negative value tells GVNS-MA is prior to GVNS-MA0in terms of time cost And the rows Better and Worserespectively show the number of instances for which thecorresponding results of the associated algorithm are betterand worse than the other
The results summarized in Table 4 disclose that theGVNS-MA has an overwhelming advantage over GVNS-MA0 in terms of the computation time to solve MS-RCPSPleaving out the deterioration effect Indeed the shortestruntime 119879119904ℎ1199001199031199051198901199041199051 of the GVNS-MA method is better thanthe shortest runtime 119879119904ℎ1199001199031199051198901199041199052 of GVNS-MA0 for 30 out of30 representative instances and the average runtime 119879119886V1198921 isbetter for 28 out of 30 instances Meanwhile the average valueof 119863119864119881119904ℎ119900119903119905119890119904119905() equals -1379 accompanying with a highof -1880 percent in 119863119864119881119886V119892()
However focusing on Table 5 the results of twoapproaches are neck and neck and GVNS-MA lost its earlysuperiority in MS-RCPSP In terms of shortest runtimeGVNS-MA successes for 7 out of 15 tested instances whileGVNS-MA0 reaches optimal for the remain As for averageruntime GVNS-MA performs better for 9 out of 15 examplesand GVNS-MA0 achieves reversion in others 6 instancesWith these data it will be hard to judge the true benefits ofone approach versus the other
To figure out the reason of this phenomenon we shouldcome back to the inner rationale of rapid evaluation mecha-nism When GVNS-MA computes the completion time of aneighborhood solution it only recalculates the tasksrsquo relatedparameters influenced by the particular move InMS-RCPSPa move including swap reverse and alter will affect justa small number of tasks But for MS-RCPSPLD instancesany move can cumulatively effect on a large proportion oftasks because of the existing deterioration Consequently theruntime saved in computing some unchanged parametersmay not make up for the time spent on isolating the changedtasks
10 Mathematical Problems in Engineering
Table3Summaryandcomparis
onon
thes
etof
45newgeneratedinsta
nces
with
119899=10
0of
GVN
S-MA
GVN
S-119872
119860 ℎG
VNS-
119872119860 119886ℎand
theT
Sheuristic[54]
andPR
[55]B
estresultsare
indicatedin
italic
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-10-26-15
165572
6598
13899
963646
6492
66871
63274
6392
33092
661967
6379
730698
62632
63629
29422
100-10-26-15
260
1396
618416
28607
608971
625436
79283
587145
613934
37496
580557
615474
23879
572794
615156
34826
100-10-26-15
3172856
184175
31236
1646
96
175513
59776
160283
16936
22225
166221
17124
17541
163898
172228
3461
100-10-47-91
7008
72053
40304
6974
471232
10018
69445
70098
20666
67645
6915
731499
69673
7039
36314
100-10-47-92
700352
739287
39112
703439
721152
70563
672221
715138
2417
9672304
6959
20614
693553
702369
25114
100-10-47-93
188206
195206
4816
818358
190863
93463
174162
184389
30008
177323
183958
19383
173708
183368
41432
100-10-48-15
165868
6914
13991
567093
6890
88879
164
851
65526
3491
663253
66595
22686
6479
86610
226662
100-10-48-15
2638747
659809
42043
621357
658747
9116
161867
651609
4116
1590432
625462
27097
618507
6346
3233655
100-10-48-15
3170999
174232
29659
171864
178061
73395
159497
166887
23395
161483
166366
3216
9163501
170273
27056
100-10-64-91
70527
7572
53770
371049
7692
773024
6872
27092
931287
6884
71215
26024
67894
69682
2775
6100-10-64-92
705806
744775
39415
7040
32738141
6279
4663729
711114
3013
968421
719153
22806
696727
736833
3991
8100-10-64-93
193844
206206
3115
32006
42
2044
03
108918
189324
19445
35415
180842
200842
3104
193844
199996
31871
100-10-64-15
170119
71497
34626
71075
7279
683543
69346
70022
39097
6691
69112
21249
70042
7079
617116
100-10-64-15
2690964
744775
3899
8672578
72823
61684
6606
14685085
22491
62823
682957
3178
4626331
666418
2376
100-10-64-15
3193844
218506
31871
194247
204235
71318
1806
92
1860
08
25336
187149
192556
45438
185585
190957
25296
100-20-22-15
119089
19355
29631
1936
19739
5517
618883
19118
2222
18572
1898
21438
1878
18965
2474
2100-20-22-15
24615
847731
36854
46032
46714
6792
645371
46008
28455
45218
46627
2892
645833
4633
2312
4100-20-22-15
326328
2798
128528
27212
2697
36972
92597
26419
2812
25563
25918
17531
25575
26254
21352
100-20-46-15
126243
2673
33424
26214
26631
60354
25862
26329
4119
925511
25896
1215
325476
26006
17466
100-20-46-15
260
647
66244
34408
59367
63596
71878
5796
16165
17216
56542
5942
26696
5465
59285
22313
100-20-46-15
332421
3490
937523
33539
3417
977634
31695
32909
16433
31651
33263
29116
32841
33349
2791
100-20-47-91
19007
1975
929027
1916
319685
5776
318864
1912
28311
1874
719269
2732
18455
18892
30731
100-20-47-92
50591
53481
37119
47839
51484
102173
4803
49319
3549
46278
4876
435418
44966
47893
24241
100-20-47-93
30802
31827
4572
630631
31365
6995
829437
30352
57846
27712
29458
29737
28399
2879
927559
100-20-65-15
19013
492446
26413
89865
90543
6795
788801
90095
17505
86826
89518
15052
87305
89338
2843
100-20-65-15
2253899
2606
7628718
250126
253267
5876
3244762
250316
4099
7243449
24751
17659
242353
244944
2573
5100-20-65-15
3110
567
1115886
27491
109874
113685
53418
105215
109224
2513
8108595
111243
1591
1104829
106776
1412
6100-20-65-91
19113
1978
341248
1895
719548
79561
18369
18898
30561
18697
1914
61896
18694
19265
3097
9100-20-65-92
46242
4776
865469
4593
447443
58112
4495
746229
1768
44719
45985
22324
45593
46591
30606
100-20-65-93
28018
28776
4894
327512
2810
868532
2719
2772
124279
26587
27691
3101
26455
27845
2473
100-5-22-151
500988
514285
2897
8510285
505098
84526
486586
499318
43529
494364
500988
18444
498567
510285
22335
100-5-22-152
1193440
1227080
4319
4119
2300
1210150
61537
1118610
1199248
24322
1184940
1219514
20664
1138490
1226780
26656
100-5-22-153
663947
700818
5691
2661553
792627
7997
660
6322
652627
10639
592509
6279304
2619
8643856
6597526
17267
100-5-46
-151
675739
749763
3679
7652793
701457
137862
63503
679663
5075
8636903
687941
29888
649164
671982
36202
100-5-46
-152
1404
680
1453430
58595
1443529
1470970
108595
1298430
1370870
26564
1315940
1360
830
23536
1332210
1399014
2417
6100-5-46
-153
795664
870313
50687
7977713
863567
6891
1740178
796512
81792
4755866
7786062
16809
752568
80128
1876
3100-5-48-9
1563787
586832
31214
5666
27
78386
84807
55206
560407
2415
55206
563664
25018
554299
56657
2495
2100-5-48-9
21284230
1315720
3614
51289811
1364
650
71286
1222350
1285020
2795
8116
3860
1259332
2812
21233010
1268572
34659
100-5-48-9
3684299
692453
45335
66785
717279
29393
8657279
67622
4561
6604
26
670917
41286
64822
667213
828446
Mathematical Problems in Engineering 11
Table3Con
tinued
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-5-64
-151
581916
627374
55661
576589
618167
102415
5544
26
579234
3516
7546543
590352
29678
56942
580175
56619
100-5-64
-152
120944
01261510
31296
1190830
1257643
123533
1131450
1183014
40595
1118430
1151812
46305
1047250
1158556
3337
100-5-64
-153
642267
709602
3498
8634651
688889
89403
612857
6657858
3798
5626771
660614
44591
1624753
674996
261623
100-5-64
-91
550231
577524
37365
5544
81
566747
98693
528177
5404
36
3393
530748
543895
48396
515984
536747
2993
2100-5-64
-92
1214340
1271610
40553
1183479
1236750
11992
711140
60115
5584
41523
11640
101201126
40287
1121650
1159384
27286
100-5-64
-93
610356
648765
36395
6151502
632323
87448
604586
6210378
31363
594514
6171502
30543
595191
623223
2774
9119861
119890119904119905
00
00
1514
1716
1415
119879119900119905119886
11989745
4545
4545
4545
4545
45
12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
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Mathematical Problems in Engineering 9
three intervals from which the deterioration rate is drawn3 extended cases are needed to solve for each instance Forconvenience these instances are denoted by adding a suffixfor identification to different intervals For example 100-10-26-15 1 represents the original case 100-10-26-15 is modifiedby adding the deterioration rates produced in (0 05] to thedurations of tasks In total there are 45(3 times 15) instancesrandomly generated
Due to zero known results in literature for same datasetthe improved tabu search (ITS) proposed by Dai et al [54]who discussed the MS-RCPSP under step deterioration anda path relinking algorithm (PR) [55] based on the populationpath relinking framework are programmed as referencealgorithms
Table 3 reports the computational results achieved by theITS PR GVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ on theset of 45 benchmark instances 119891119887119890119904119905 denotes the minimumobjective value and 119891119886V119892 is computed as the average objectivevalue of 20 runs
First Table 3 discloses that the solutions obtained byGVNS-MA GVNS-MAℎ and GVNS-MA119886ℎ are better thanthe ITS and PR for any instance from the perspectiveof both quality of schedule and runtime To some extentthese results demonstrated the differences between lineardeterioration and step deterioration and the superiority ofmemetic algorithm framework Second these three methodsdiffering in the sort order of tasks in initialize phase behavesimilarly where GVNS-MA obtains the best 15 out of 45instances 17 for GVNS-MAℎ and 14 for GVNS119886ℎ in termsof 119891119887119890119904119905 Specifically GVNS-MA and GVNS-MAℎ attain theoptimal simultaneously for the instance 100-5-48-9-1 Froma view point of 119891119886V119892 and run time three methods alsohave a balanced performance Third as far as three differentintervals to generate deterioration rate are concerned thephenomenon did not happen that the relevant algorithmsdisplay strikingly different behavior In other words theperformance of the proposed algorithm is not sensitive to thesetting of deterioration rate
55 Analysis and Discussions In this section we study twoessential ingredients of the proposed GVNS-MA to getan insight to its performance One is the rapid evaluationmechanism the other is the role of the memetic framework
551 Importance of Rapid Evaluation Mechanism GVNS-MA with rapid evaluation mechanism only calculates therelevant parameters of some particular tasks rather thanall when the procedure computes the objective value ofa neighborhood solution To highlight the key role ofthe rapid evaluation mechanism two sets of comparisonexperiments are carried out on generated dataset with twoalgorithms GVNS-MA and GVNS-MA0 including sameingredients with GVNS-MA except for the computation ofobjective value When GVNS-MA0 figures up the value of aneighborhood solution it computes all relevant parametersagain
Table 4 records the experimental results carried out on thedataset [46] without consideration of deterioration whereas
Table 5 shows the comparisons of GVNS-MA and GVNS-MA0 about the set of 15 instances generated in Section 54on account of the indiscrimination in three intervals Col-umn 2 and 5 record the best attained by two algorithmsColumn 3 and 6 indicate the minimum time cost to a finalfeasible schedule with one run of procedure Note that thebest objective value cannot be guaranteed as the output ofshortest runtime As for the parameters in column 4 and 7they represent the mean runtime Finally two parameters119863119864119881119904ℎ119900119903119905119890119904119905 and 119863119864119881119886V119892 are used to disclose the runtimedeviation of two methods defined by equations
119863119864119881119904ℎ119900119903119905119890119904119905 () = 119879119904ℎ1199001199031199051198901199041199051 minus 119879119904ℎ1199001199031199051198901199041199052119879119904ℎ1199001199031199051198901199041199051
times 100 (13)
and
119863119864119881119886V119892 () =119879119886V1198921 minus 119879119886V1198922
119879119886V1198921times 100 (14)
respectively The positive value of 119863119864119881119904ℎ119900119903119905119890119904119905() and119863119864119881119886V119892() means that GVNS-MA0 has better performanceand negative value tells GVNS-MA is prior to GVNS-MA0in terms of time cost And the rows Better and Worserespectively show the number of instances for which thecorresponding results of the associated algorithm are betterand worse than the other
The results summarized in Table 4 disclose that theGVNS-MA has an overwhelming advantage over GVNS-MA0 in terms of the computation time to solve MS-RCPSPleaving out the deterioration effect Indeed the shortestruntime 119879119904ℎ1199001199031199051198901199041199051 of the GVNS-MA method is better thanthe shortest runtime 119879119904ℎ1199001199031199051198901199041199052 of GVNS-MA0 for 30 out of30 representative instances and the average runtime 119879119886V1198921 isbetter for 28 out of 30 instances Meanwhile the average valueof 119863119864119881119904ℎ119900119903119905119890119904119905() equals -1379 accompanying with a highof -1880 percent in 119863119864119881119886V119892()
However focusing on Table 5 the results of twoapproaches are neck and neck and GVNS-MA lost its earlysuperiority in MS-RCPSP In terms of shortest runtimeGVNS-MA successes for 7 out of 15 tested instances whileGVNS-MA0 reaches optimal for the remain As for averageruntime GVNS-MA performs better for 9 out of 15 examplesand GVNS-MA0 achieves reversion in others 6 instancesWith these data it will be hard to judge the true benefits ofone approach versus the other
To figure out the reason of this phenomenon we shouldcome back to the inner rationale of rapid evaluation mecha-nism When GVNS-MA computes the completion time of aneighborhood solution it only recalculates the tasksrsquo relatedparameters influenced by the particular move InMS-RCPSPa move including swap reverse and alter will affect justa small number of tasks But for MS-RCPSPLD instancesany move can cumulatively effect on a large proportion oftasks because of the existing deterioration Consequently theruntime saved in computing some unchanged parametersmay not make up for the time spent on isolating the changedtasks
10 Mathematical Problems in Engineering
Table3Summaryandcomparis
onon
thes
etof
45newgeneratedinsta
nces
with
119899=10
0of
GVN
S-MA
GVN
S-119872
119860 ℎG
VNS-
119872119860 119886ℎand
theT
Sheuristic[54]
andPR
[55]B
estresultsare
indicatedin
italic
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-10-26-15
165572
6598
13899
963646
6492
66871
63274
6392
33092
661967
6379
730698
62632
63629
29422
100-10-26-15
260
1396
618416
28607
608971
625436
79283
587145
613934
37496
580557
615474
23879
572794
615156
34826
100-10-26-15
3172856
184175
31236
1646
96
175513
59776
160283
16936
22225
166221
17124
17541
163898
172228
3461
100-10-47-91
7008
72053
40304
6974
471232
10018
69445
70098
20666
67645
6915
731499
69673
7039
36314
100-10-47-92
700352
739287
39112
703439
721152
70563
672221
715138
2417
9672304
6959
20614
693553
702369
25114
100-10-47-93
188206
195206
4816
818358
190863
93463
174162
184389
30008
177323
183958
19383
173708
183368
41432
100-10-48-15
165868
6914
13991
567093
6890
88879
164
851
65526
3491
663253
66595
22686
6479
86610
226662
100-10-48-15
2638747
659809
42043
621357
658747
9116
161867
651609
4116
1590432
625462
27097
618507
6346
3233655
100-10-48-15
3170999
174232
29659
171864
178061
73395
159497
166887
23395
161483
166366
3216
9163501
170273
27056
100-10-64-91
70527
7572
53770
371049
7692
773024
6872
27092
931287
6884
71215
26024
67894
69682
2775
6100-10-64-92
705806
744775
39415
7040
32738141
6279
4663729
711114
3013
968421
719153
22806
696727
736833
3991
8100-10-64-93
193844
206206
3115
32006
42
2044
03
108918
189324
19445
35415
180842
200842
3104
193844
199996
31871
100-10-64-15
170119
71497
34626
71075
7279
683543
69346
70022
39097
6691
69112
21249
70042
7079
617116
100-10-64-15
2690964
744775
3899
8672578
72823
61684
6606
14685085
22491
62823
682957
3178
4626331
666418
2376
100-10-64-15
3193844
218506
31871
194247
204235
71318
1806
92
1860
08
25336
187149
192556
45438
185585
190957
25296
100-20-22-15
119089
19355
29631
1936
19739
5517
618883
19118
2222
18572
1898
21438
1878
18965
2474
2100-20-22-15
24615
847731
36854
46032
46714
6792
645371
46008
28455
45218
46627
2892
645833
4633
2312
4100-20-22-15
326328
2798
128528
27212
2697
36972
92597
26419
2812
25563
25918
17531
25575
26254
21352
100-20-46-15
126243
2673
33424
26214
26631
60354
25862
26329
4119
925511
25896
1215
325476
26006
17466
100-20-46-15
260
647
66244
34408
59367
63596
71878
5796
16165
17216
56542
5942
26696
5465
59285
22313
100-20-46-15
332421
3490
937523
33539
3417
977634
31695
32909
16433
31651
33263
29116
32841
33349
2791
100-20-47-91
19007
1975
929027
1916
319685
5776
318864
1912
28311
1874
719269
2732
18455
18892
30731
100-20-47-92
50591
53481
37119
47839
51484
102173
4803
49319
3549
46278
4876
435418
44966
47893
24241
100-20-47-93
30802
31827
4572
630631
31365
6995
829437
30352
57846
27712
29458
29737
28399
2879
927559
100-20-65-15
19013
492446
26413
89865
90543
6795
788801
90095
17505
86826
89518
15052
87305
89338
2843
100-20-65-15
2253899
2606
7628718
250126
253267
5876
3244762
250316
4099
7243449
24751
17659
242353
244944
2573
5100-20-65-15
3110
567
1115886
27491
109874
113685
53418
105215
109224
2513
8108595
111243
1591
1104829
106776
1412
6100-20-65-91
19113
1978
341248
1895
719548
79561
18369
18898
30561
18697
1914
61896
18694
19265
3097
9100-20-65-92
46242
4776
865469
4593
447443
58112
4495
746229
1768
44719
45985
22324
45593
46591
30606
100-20-65-93
28018
28776
4894
327512
2810
868532
2719
2772
124279
26587
27691
3101
26455
27845
2473
100-5-22-151
500988
514285
2897
8510285
505098
84526
486586
499318
43529
494364
500988
18444
498567
510285
22335
100-5-22-152
1193440
1227080
4319
4119
2300
1210150
61537
1118610
1199248
24322
1184940
1219514
20664
1138490
1226780
26656
100-5-22-153
663947
700818
5691
2661553
792627
7997
660
6322
652627
10639
592509
6279304
2619
8643856
6597526
17267
100-5-46
-151
675739
749763
3679
7652793
701457
137862
63503
679663
5075
8636903
687941
29888
649164
671982
36202
100-5-46
-152
1404
680
1453430
58595
1443529
1470970
108595
1298430
1370870
26564
1315940
1360
830
23536
1332210
1399014
2417
6100-5-46
-153
795664
870313
50687
7977713
863567
6891
1740178
796512
81792
4755866
7786062
16809
752568
80128
1876
3100-5-48-9
1563787
586832
31214
5666
27
78386
84807
55206
560407
2415
55206
563664
25018
554299
56657
2495
2100-5-48-9
21284230
1315720
3614
51289811
1364
650
71286
1222350
1285020
2795
8116
3860
1259332
2812
21233010
1268572
34659
100-5-48-9
3684299
692453
45335
66785
717279
29393
8657279
67622
4561
6604
26
670917
41286
64822
667213
828446
Mathematical Problems in Engineering 11
Table3Con
tinued
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-5-64
-151
581916
627374
55661
576589
618167
102415
5544
26
579234
3516
7546543
590352
29678
56942
580175
56619
100-5-64
-152
120944
01261510
31296
1190830
1257643
123533
1131450
1183014
40595
1118430
1151812
46305
1047250
1158556
3337
100-5-64
-153
642267
709602
3498
8634651
688889
89403
612857
6657858
3798
5626771
660614
44591
1624753
674996
261623
100-5-64
-91
550231
577524
37365
5544
81
566747
98693
528177
5404
36
3393
530748
543895
48396
515984
536747
2993
2100-5-64
-92
1214340
1271610
40553
1183479
1236750
11992
711140
60115
5584
41523
11640
101201126
40287
1121650
1159384
27286
100-5-64
-93
610356
648765
36395
6151502
632323
87448
604586
6210378
31363
594514
6171502
30543
595191
623223
2774
9119861
119890119904119905
00
00
1514
1716
1415
119879119900119905119886
11989745
4545
4545
4545
4545
45
12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
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Submit your manuscripts atwwwhindawicom
10 Mathematical Problems in Engineering
Table3Summaryandcomparis
onon
thes
etof
45newgeneratedinsta
nces
with
119899=10
0of
GVN
S-MA
GVN
S-119872
119860 ℎG
VNS-
119872119860 119886ℎand
theT
Sheuristic[54]
andPR
[55]B
estresultsare
indicatedin
italic
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-10-26-15
165572
6598
13899
963646
6492
66871
63274
6392
33092
661967
6379
730698
62632
63629
29422
100-10-26-15
260
1396
618416
28607
608971
625436
79283
587145
613934
37496
580557
615474
23879
572794
615156
34826
100-10-26-15
3172856
184175
31236
1646
96
175513
59776
160283
16936
22225
166221
17124
17541
163898
172228
3461
100-10-47-91
7008
72053
40304
6974
471232
10018
69445
70098
20666
67645
6915
731499
69673
7039
36314
100-10-47-92
700352
739287
39112
703439
721152
70563
672221
715138
2417
9672304
6959
20614
693553
702369
25114
100-10-47-93
188206
195206
4816
818358
190863
93463
174162
184389
30008
177323
183958
19383
173708
183368
41432
100-10-48-15
165868
6914
13991
567093
6890
88879
164
851
65526
3491
663253
66595
22686
6479
86610
226662
100-10-48-15
2638747
659809
42043
621357
658747
9116
161867
651609
4116
1590432
625462
27097
618507
6346
3233655
100-10-48-15
3170999
174232
29659
171864
178061
73395
159497
166887
23395
161483
166366
3216
9163501
170273
27056
100-10-64-91
70527
7572
53770
371049
7692
773024
6872
27092
931287
6884
71215
26024
67894
69682
2775
6100-10-64-92
705806
744775
39415
7040
32738141
6279
4663729
711114
3013
968421
719153
22806
696727
736833
3991
8100-10-64-93
193844
206206
3115
32006
42
2044
03
108918
189324
19445
35415
180842
200842
3104
193844
199996
31871
100-10-64-15
170119
71497
34626
71075
7279
683543
69346
70022
39097
6691
69112
21249
70042
7079
617116
100-10-64-15
2690964
744775
3899
8672578
72823
61684
6606
14685085
22491
62823
682957
3178
4626331
666418
2376
100-10-64-15
3193844
218506
31871
194247
204235
71318
1806
92
1860
08
25336
187149
192556
45438
185585
190957
25296
100-20-22-15
119089
19355
29631
1936
19739
5517
618883
19118
2222
18572
1898
21438
1878
18965
2474
2100-20-22-15
24615
847731
36854
46032
46714
6792
645371
46008
28455
45218
46627
2892
645833
4633
2312
4100-20-22-15
326328
2798
128528
27212
2697
36972
92597
26419
2812
25563
25918
17531
25575
26254
21352
100-20-46-15
126243
2673
33424
26214
26631
60354
25862
26329
4119
925511
25896
1215
325476
26006
17466
100-20-46-15
260
647
66244
34408
59367
63596
71878
5796
16165
17216
56542
5942
26696
5465
59285
22313
100-20-46-15
332421
3490
937523
33539
3417
977634
31695
32909
16433
31651
33263
29116
32841
33349
2791
100-20-47-91
19007
1975
929027
1916
319685
5776
318864
1912
28311
1874
719269
2732
18455
18892
30731
100-20-47-92
50591
53481
37119
47839
51484
102173
4803
49319
3549
46278
4876
435418
44966
47893
24241
100-20-47-93
30802
31827
4572
630631
31365
6995
829437
30352
57846
27712
29458
29737
28399
2879
927559
100-20-65-15
19013
492446
26413
89865
90543
6795
788801
90095
17505
86826
89518
15052
87305
89338
2843
100-20-65-15
2253899
2606
7628718
250126
253267
5876
3244762
250316
4099
7243449
24751
17659
242353
244944
2573
5100-20-65-15
3110
567
1115886
27491
109874
113685
53418
105215
109224
2513
8108595
111243
1591
1104829
106776
1412
6100-20-65-91
19113
1978
341248
1895
719548
79561
18369
18898
30561
18697
1914
61896
18694
19265
3097
9100-20-65-92
46242
4776
865469
4593
447443
58112
4495
746229
1768
44719
45985
22324
45593
46591
30606
100-20-65-93
28018
28776
4894
327512
2810
868532
2719
2772
124279
26587
27691
3101
26455
27845
2473
100-5-22-151
500988
514285
2897
8510285
505098
84526
486586
499318
43529
494364
500988
18444
498567
510285
22335
100-5-22-152
1193440
1227080
4319
4119
2300
1210150
61537
1118610
1199248
24322
1184940
1219514
20664
1138490
1226780
26656
100-5-22-153
663947
700818
5691
2661553
792627
7997
660
6322
652627
10639
592509
6279304
2619
8643856
6597526
17267
100-5-46
-151
675739
749763
3679
7652793
701457
137862
63503
679663
5075
8636903
687941
29888
649164
671982
36202
100-5-46
-152
1404
680
1453430
58595
1443529
1470970
108595
1298430
1370870
26564
1315940
1360
830
23536
1332210
1399014
2417
6100-5-46
-153
795664
870313
50687
7977713
863567
6891
1740178
796512
81792
4755866
7786062
16809
752568
80128
1876
3100-5-48-9
1563787
586832
31214
5666
27
78386
84807
55206
560407
2415
55206
563664
25018
554299
56657
2495
2100-5-48-9
21284230
1315720
3614
51289811
1364
650
71286
1222350
1285020
2795
8116
3860
1259332
2812
21233010
1268572
34659
100-5-48-9
3684299
692453
45335
66785
717279
29393
8657279
67622
4561
6604
26
670917
41286
64822
667213
828446
Mathematical Problems in Engineering 11
Table3Con
tinued
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-5-64
-151
581916
627374
55661
576589
618167
102415
5544
26
579234
3516
7546543
590352
29678
56942
580175
56619
100-5-64
-152
120944
01261510
31296
1190830
1257643
123533
1131450
1183014
40595
1118430
1151812
46305
1047250
1158556
3337
100-5-64
-153
642267
709602
3498
8634651
688889
89403
612857
6657858
3798
5626771
660614
44591
1624753
674996
261623
100-5-64
-91
550231
577524
37365
5544
81
566747
98693
528177
5404
36
3393
530748
543895
48396
515984
536747
2993
2100-5-64
-92
1214340
1271610
40553
1183479
1236750
11992
711140
60115
5584
41523
11640
101201126
40287
1121650
1159384
27286
100-5-64
-93
610356
648765
36395
6151502
632323
87448
604586
6210378
31363
594514
6171502
30543
595191
623223
2774
9119861
119890119904119905
00
00
1514
1716
1415
119879119900119905119886
11989745
4545
4545
4545
4545
45
12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
Hindawiwwwhindawicom Volume 2018
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AnalysisInternational Journal of
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Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 11
Table3Con
tinued
insta
nces
ITS
PRGVN
S-MA
GVN
S-119872
119860 ℎGVN
S-119872
119860 aℎ
119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)119891 119887119890119904119905
119891 119886V119892
119879119894119898119890
(119904)100-5-64
-151
581916
627374
55661
576589
618167
102415
5544
26
579234
3516
7546543
590352
29678
56942
580175
56619
100-5-64
-152
120944
01261510
31296
1190830
1257643
123533
1131450
1183014
40595
1118430
1151812
46305
1047250
1158556
3337
100-5-64
-153
642267
709602
3498
8634651
688889
89403
612857
6657858
3798
5626771
660614
44591
1624753
674996
261623
100-5-64
-91
550231
577524
37365
5544
81
566747
98693
528177
5404
36
3393
530748
543895
48396
515984
536747
2993
2100-5-64
-92
1214340
1271610
40553
1183479
1236750
11992
711140
60115
5584
41523
11640
101201126
40287
1121650
1159384
27286
100-5-64
-93
610356
648765
36395
6151502
632323
87448
604586
6210378
31363
594514
6171502
30543
595191
623223
2774
9119861
119890119904119905
00
00
1514
1716
1415
119879119900119905119886
11989745
4545
4545
4545
4545
45
12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
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12 Mathematical Problems in Engineering
Table 4 Summary and comparison of GVNS-MA and GVNS-MA0 on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 237 13458 19178 237 15541 24093 -1548 -2563100-10-47-9 253 10790 12490 253 11259 15526 -435 -2431100-10-48-15 245 14801 17505 245 17461 21594 -1797 -2336100-10-64-9 247 14261 16536 247 16258 21663 -1400 -3101100-10-64-15 246 10449 17317 246 12129 22218 -1608 -2830100-20-22-15 133 12946 14953 133 14547 17629 -1237 -1790100-20-46-15 160 10616 13770 160 12139 17566 -1435 -2757100-20-47-9 132 11482 12870 132 14251 17444 -2412 -3554100-20-65-15 193 9782 10317 193 11301 12922 -1553 -2525100-20-65-9 134 11229 13893 134 12616 15956 -1235 -1485100-5-22-15 483 10573 13164 483 12040 16790 -1387 -2754100-5-46-15 528 16097 18948 528 18298 22145 -1367 -1687100-5-48-9 489 11420 13445 489 13452 14637 -1779 -887100-5-64-15 480 12267 14627 480 16445 20254 -3406 -3847100-5-64-9 474 12463 16261 474 15410 20664 -2365 -2708200-10-128-15 479 65793 74632 479 80141 90258 -2186 -2094200-10-50-15 488 8355 89529 488 91344 98438 -933 -995200-10-50-9 487 74407 79334 487 82027 81820 -1024 -313200-10-84-9 509 66863 71920 509 72568 86285 -853 -1997200-10-85-15 477 50061 56176 477 56176 68883 -1222 -2262200-20-145-15 252 53414 66008 252 59008 71530 -1047 837200-20-54-15 291 74502 84746 291 78580 96701 -547 -1411200-20-55-9 257 57767 63997 257 62239 71394 -774 -1156200-20-97-15 334 59201 72457 334 66673 74914 -1262 -339200-20-97-9 253 56537 71620 253 68370 80950 -2093 1303200-40-133-15 157 68228 77282 157 72041 84472 -559 -930200-40-45-15 159 47609 56558 159 51130 61516 -740 -877200-40-45-9 144 57143 62653 144 61548 70062 -771 -1183200-40-90-9 145 61623 65424 145 68397 73817 ndash1099 -1283200-40-91-15 153 58802 62401 153 66487 76005 -1307 -2180119861119890119905119905119890119903 30 28 0 2119882119900119903119904119890 0 2 30 28119879119900119905119886119897 30 30 30 30119860V119890119903119886119892119890 -1379 -1880
These experimental results confirm that although therapid evaluation mechanism is not so critical for MS-RCPSPLD it is still quite useful to quickly solve MS-RCPSPinstances and constitutes a significant element of the pro-posed GVNS-MA
552 Influence of theMemetic Framework As shown in Lei etal [16] Mladenovicabcd [15] the GVNS approach has showngreat performance in a widespread academic application Soit is meaningful to research whether our GVNS-MA hasa significant advantage over the originally efficient GVNS
algorithm For this reason a comparative test between GVNSand GVNS-MA has been carried out For this experimentwe used the known dataset [46] with 20 times running foreach instance Same with GVNS-MA the stopping criteriaof GVNS is met when the maximum number betweentwo iterations without improvement reaches 5 times 104 Theexperimental results of two methods are recorded in Table 6where119863119864119881() = (119891minus119891119898119894119899)119891119898119894119899times100 and the other symbolshave same meanings as those of Table 2 As for the 119891 ad 119891119898119894119899in the equation they denote the objective value of the bestschedule solved by the particular algorithm and the best valueattainable until now respectivelyThe parameterDEV() can
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
Hindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
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Mathematical PhysicsAdvances in
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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
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Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 13
Table 5 Comparison of results of GVNS-MA and GVNS-MA0 on the set of 45 new generated instances in Section 54
instances GVNS-MA GVNS-MA0 119863119864119881119904ℎ119900119903119905119890119904119905() 119863119864119881119886V119892()119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199051 119879119886V1198921 119891119887119890119904119905 119879119904ℎ1199001199031199051198901199041199052 119879119886V1198922
100-10-26-15 1 63274 14608 27499 63508 16391 30999 -1221 -1273100-10-47-9 1 69445 20666 34951 69666 19736 36415 450 -419100-10-48-15 1 64851 19334 28655 64911 17483 24417 957 1479100-10-64-9 1 68722 22247 31241 67809 19171 34878 1383 -1164100-10-64-15 1 69346 19291 30949 67375 19047 29709 126 401100-20-22-15 1 18883 16171 25615 18722 14631 22534 952 1203100-20-46-15 1 25862 15494 24039 25660 14255 22886 800 480100-20-47-9 1 18864 22279 25872 18212 25205 29380 -1313 -1356100-20-65-15 1 88801 17261 20743 87085 22043 18768 -2770 952100-20-65-9 1 18369 23732 35453 18866 26417 33325 -1131 600100-5-22-15 1 486586 18444 27023 486602 22441 29576 -2167 -945100-5-46-15 1 63503 26437 45783 672042 27003 48858 -214 -672100-5-48-9 1 552060 15446 26423 552060 18928 31100 -2254 -1770100-5-64-15 1 554426 33839 41358 555552 32248 46102 470 -1147100-5-64-9 1 528177 22277 32346 528714 20070 35071 991 -842119861119890119905119905119890119903 7 9 8 6119882119900119903119904119890 8 6 7 9119879119900119905119886119897 15 15 15 15119860V119890119903119886119892119890 -329 -298
visually detect the gap between the current algorithm and thebest
Obviously Table 6 demonstrates that the GVNS-MAsignificantly outperforms the GVNS algorithm in generalFirst compared with the GVNS algorithm the GVNS-MAobtains better and worse results in terms of the minimumobjective value on 29 and 1 instances respectively Secondit can be seen that the obtained average Devs are 007 and583 respectively for the GVNS-MA and GVNS implyingthat there exists a huge difference between two methodsThird the runtimes of PR are obviously longer than GVNS-MA with worse solutions These outcomes indicate that thememetic part of the proposed GVNS-MA is very appropriatefor solving the related issue of MS-RCPSP
6 Conclusions
The proposed general variable neighborhood search-basedmemetic algorithm (GVNS-MA) for solving the MS-RCPSPand MS-RCPSPLD incorporates an effective neighborhoodsearch procedure and a random crossover operator whileapplying an original scheme for parent selection We testedthe proposed GVNS-MA on 30 benchmark instances com-monly used in the literature and 45 newly generatedinstances The computational results of the state-of-the-artalgorithms in the literature demonstrate that our algorithmis highly effective for solving MS-RCPSP Specifically itimproves or matches the previous best known results forall tested instances As for MS-RCPSPLD GVNS-MA has
a better performance than ITS for any instance in terms ofthe quality of solution and a considerable shorter runtimecompared to PR
The investigations of some essential ingredients of theproposed algorithm shed light on the behavior of the GVNS-MA First the rapid evaluation mechanism is particularlysuitable to solve MS-RCPSP instances Second the popu-lation evolution based memetic framework is significantlycontributed to the algorithmrsquos performance
Here we discussed the linear deterioration of the mul-tiskill tasks It would be interesting to investigate such ascheduling problem in other deterioration mechanisms tomeet various actual production conditions
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was partially supported by the National NaturalScience Foundation of China (no 51675450) and Sichuan
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
14 Mathematical Problems in Engineering
Table 6 Summary and comparison of GVNS-MA and GVNS on known MS-RCPSP dataset [46]
instances GVNS-MA GVNS1198911198871198901199041199051 119863119890V() 1198911198871198901199041199052 119863119890V()
100-10-26-15 237 000 260 970100-10-47-9 253 000 255 079100-10-48-15 245 000 255 408100-10-64-9 247 000 257 405100-10-64-15 246 000 248 081100-20-22-15 133 000 142 677100-20-46-15 160 000 161 063100-20-47-9 132 000 134 152100-20-65-15 193 000 205 622100-20-65-9 134 000 142 597100-5-22-15 483 000 486 062100-5-46-15 528 000 580 985100-5-48-9 489 000 493 082100-5-64-15 480 000 494 292100-5-64-9 474 000 477 063200-10-128-15 479 000 501 459200-10-50-15 488 000 516 574200-10-50-9 487 000 509 452200-10-84-9 509 221 498 000200-10-85-15 477 000 494 356200-20-145-15 252 000 285 1310200-20-54-15 291 000 303 412200-20-55-9 257 000 267 389200-20-97-15 334 000 347 389200-20-97-9 253 000 282 1146200-40-133-15 157 000 181 1529200-40-45-15 159 000 164 314200-40-45-9 144 000 176 2222200-40-90-9 145 000 173 1931200-40-91-15 153 000 160 458119861119890119905119905119890119903 29 1119864119902119906119886119897 0 0119882119900119903119904119890 1 29119879119900119905119886119897 30 30119860V119890119903119886119892119890 007 583
Science and Technology Program (nos 2019YFG0300 no2019YFG0285)
References
[1] M L Pinedo Scheduling Theory Algorithms and SystemsSpringer 2012
[2] J N D Gupta and S K Gupta ldquoSingle facility scheduling withnonlinear processing timesrdquo Computers amp Industrial Engineer-ing vol 14 no 4 pp 387ndash393 1988
[3] S Gawiejnowicz Time-Dependent Scheduling Springer 2008
[4] S Browne and U Yechiali ldquoScheduling deteriorating jobs on asingle processorrdquo Operations Research vol 38 no 3 pp 495ndash498 1990
[5] B Alidaee and N K Womer ldquoScheduling with time dependentprocessing times review and extensionsrdquo Journal of the Opera-tional Research Society vol 50 no 7 pp 711ndash720 1999
[6] T C E Cheng Q Ding and B M T Lin ldquoA concise surveyof scheduling with time-dependentprocessing timesrdquoEuropeanJournal of Operational Research vol 152 no 1 pp 1ndash13 2004
[7] J Blazewicz J K Lenstra and A H Rinnooy Kan ldquoSchedulingsubject to resource constraints classification and complexityrdquoDiscrete Applied Mathematics vol 5 no 1 pp 11ndash24 1983
[8] W E Hart J E Smith and N Krasnogor ldquoRecent advancesin memetic algorithmsrdquo Studies in Fuzziness amp Soft Computingvol 166 2005
[9] P Moscato ldquoOn evolution search optimization genetic algo-rithms and martial arts Towards memetic algorithmsrdquo CaltechConcurrent Computation Program 1989
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 15
[10] X Chen Y-S Ong M-H Lim and K C Tan ldquoA multi-facet survey on memetic computationrdquo IEEE Transactions onEvolutionary Computation vol 15 no 5 pp 591ndash607 2011
[11] J Layegh F Jolai and M S Amalnik ldquoA memetic algorithmfor minimizing the total weighted completion time on a singlemachine under step-deteriorationrdquo Advances in EngineeringSoftware vol 40 no 10 pp 1074ndash1077 2009
[12] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011
[13] PHansenNMladenovic J Brimberg and JAMPerez ldquoVari-able neighborhood searchrdquo European Journal of OperationalResearch vol 24 pp 593ndash595 1997
[14] G Kirlik and C Oguz ldquoA variable neighborhood search forminimizing total weighted tardiness with sequence dependentsetup times on a single machinerdquo Computers amp OperationsResearch vol 39 no 7 pp 1506ndash1520 2012
[15] N Mladenovicabcd ldquoA general variable neighborhood searchfor the one-commodity pickup-and-delivery travelling sales-man problemrdquo European Journal of Operational Research vol220 no 1 pp 270ndash285 2012
[16] H Lei G Laporte and B Guo ldquoA generalized variable neigh-borhood search heuristic for the capacitated vehicle routingproblem with stochastic service timesrdquo TOP vol 20 no 1 pp99ndash118 2012
[17] G Mosheiov ldquoScheduling jobs with step-deterioration Mini-mizing makespan on a single- and multi-machinerdquo Computersamp Industrial Engineering vol 28 no 4 pp 869ndash879 1995
[18] T E Cheng Q Ding M Y Kovalyov A Bachman andA Janiak ldquoScheduling jobs with piecewise linear decreasingprocessing timesrdquo Naval Research Logistics vol 50 no 6 pp531ndash554 2003
[19] W Cheng P Guo Z Zhang M Zeng and J Liang ldquoVariableneighborhood search for parallel machines scheduling problemwith step deteriorating jobsrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 928312 20 pages 2012
[20] P GuoW Cheng and Y Wang ldquoScheduling step-deterioratingjobs to minimise the total weighted tardiness on a singlemachinerdquo International Journal of Systems Science Operationsamp Logistics vol 0 pp 1ndash16 2015
[21] P Guo W Cheng and Y Wang ldquoA general variable neigh-borhood search for single-machine total tardiness schedulingproblem with step-deteriorating jobsrdquo Journal of Industrial andManagement Optimization vol 10 no 4 pp 1071ndash1090 2014
[22] C He C Wu and W Lee ldquoBranch-and-bound and weight-combination search algorithms for the total completion timeproblem with step-deteriorating jobsrdquo Journal of the Opera-tional Research Society vol 60 no 12 pp 1759ndash1766 2009
[23] P S Sundararaghavan and A S Kunnathur ldquoSingle machinescheduling with start time dependent processing times somesolvable casesrdquo European Journal of Operational Research vol78 no 3 pp 394ndash403 1994
[24] A A Jeng and B M Lin ldquoMakespan minimization in single-machine scheduling with step-deterioration of processingtimesrdquo Journal of the Operational Research Society vol 55 no3 pp 247ndash256 2004
[25] G Mosheiov ldquoV-shaped policies for scheduling deterioratingjobsrdquo Operations Research vol 39 no 6 pp 979ndash991 1991
[26] M Ji and T C E Cheng ldquoParallel-machine scheduling withsimple linear deterioration to minimize total completion timerdquoEuropean Journal of Operational Research vol 188 no 2 pp342ndash347 2008
[27] A Jafari and G Moslehi ldquoScheduling linear deteriorating jobsto minimize the number of tardy jobsrdquo Journal of GlobalOptimization vol 54 no 2 pp 389ndash404 2012
[28] C-C Wu W-C Lee and Y-R Shiau ldquoMinimizing the totalweighted completion time on a single machine under lineardeteriorationrdquoThe International Journal of Advanced Manufac-turing Technology vol 33 no 11-12 pp 1237ndash1243 2007
[29] D Wang and J-B Wang ldquoSingle-machine scheduling withsimple linear deterioration tominimize earliness penaltiesrdquoTheInternational Journal of Advanced Manufacturing Technologyvol 46 no 1ndash4 pp 285ndash290 2010
[30] A Bachman and A Janiak ldquoMinimizing maximum latenessunder linear deteriorationrdquo European Journal of OperationalResearch vol 126 no 3 pp 557ndash566 2000
[31] D Oron ldquoSingle machine scheduling with simple linear dete-rioration to minimize total absolute deviation of completiontimesrdquo Computers amp Operations Research vol 35 no 6 pp2071ndash2078 2008
[32] P Brucker A Drexl R Mohring K Neumann and E PeschldquoResource-constrained project scheduling notation classifica-tion models and methodsrdquo European Journal of OperationalResearch vol 112 no 1 pp 3ndash41 1999
[33] U Belhe and A Kusiak ldquoDynamic scheduling of design activ-ities with resource constraintsrdquo IEEE Transactions on SystemsMan and Cybernetics - Part A Systems and Humans vol 27 pp105ndash111 2002
[34] R F Babiceanu F F Chen and R H Sturges ldquoReal-timeholonic scheduling of material handling operations in adynamic manufacturing environmentrdquoRobotics and Computer-Integrated Manufacturing vol 21 no 4-5 pp 328ndash337 2005
[35] R Coban ldquoA context layered locally recurrent neural networkfor dynamic system identificationrdquo Engineering Applications ofArtificial Intelligence vol 26 no 1 pp 241ndash250 2013
[36] Z Akeshtech and F Mardukhi ldquoAn imperialist competitivealgorithm for resource constrained project scheduling withactivities flotationrdquo International Journal of Computer Scienceand Network Security vol 17 pp 125ndash134 2017
[37] J Poppenborg and S Knust ldquoA flow-based tabu search algo-rithm for the RCPSP with transfer timesrdquoOR Spectrum vol 38no 2 pp 305ndash334 2016
[38] A Laurent L Deroussi N Grangeon and S Norre ldquoA newextension of the RCPSP in a multi-site context Mathematicalmodel and metaheuristicsrdquo Computers amp Industrial Engineer-ing vol 112 pp 634ndash644 2017
[39] H Farughi A Amiri and F Abdi ldquoProject scheduling withsimultaneous optimization time net present value and projectflexibility for multimode activities with constrained renewableresourcesrdquo International Journal of Engineering Transactions BApplications vol 31 no 5 pp 780ndash791 2018
[40] M ETHumic D Sisejkovic R Coric and D Jakobovic ldquoEvolvingpriority rules for resource constrained project scheduling prob-lem with genetic programmingrdquo Future Generation ComputerSystems vol 86 pp 211ndash221 2018
[41] R Coban and C Burhanettin An Expert Trajectory Design forControl of Nuclear Research Reactors PergamonPress Inc 2009
[42] R Coban and I O Aksu ldquoNeuro-controller design by usingthe multifeedback layer neural network and the particle swarmoptimizationrdquo Tehnicki Vjesnik-Technicla Gazette vol 25 no 2pp 437ndash444 2018
[43] B Ata and R Coban ldquoArtificial bee colony algorithm basedlinear quadratic optimal controller design for a nonlinear
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
16 Mathematical Problems in Engineering
inverted pendulumrdquo International Journal of Intelligent Systemsamp Applications in Engineering vol 3 2015
[44] A Gonzalez-Pardo J Del Ser and D Camacho ldquoComparativestudy of pheromone control heuristics in ACO algorithms forsolving RCPSP problemsrdquo Applied Soft Computing vol 60 pp241ndash255 2017
[45] Q Chen K Lin and C C Wei ldquoApplication of acoalgorithm and different scheduling rules in multi-objectiveresource-constrained project scheduling problemmodificationand comparison with different scheduling rulesrdquo ComputerEngineering and Applications vol 53 pp 249ndash254 2017
[46] P B Myszkowski M E Skowronski and K Sikora ldquoA newbenchmark dataset forMulti-Skill resource-constrained projectscheduling problemrdquo in Proceedings of the Federated Conferenceon Computer Science and Information Systems (FedCSIS rsquo15) pp129ndash138 September 2015
[47] P B Myszkowski M E Skowronski Ł P Olech and K OslizłoldquoHybrid ant colony optimization in solvingmulti-skill resource-constrained project scheduling problemrdquo Soft Computing vol19 no 12 pp 3599ndash3619 2015
[48] P B Myszkowski and J J Siemienski ldquoGRASP applied tomultindashskill resourcendashconstrained project scheduling problemrdquoin Computational Collective Intelligence pp 402ndash411 2016
[49] P B Myszkowski Ł P Olech M Laszczyk and M ESkowronski ldquoHybrid differential evolution and greedy algo-rithm (DEGR) for solving multi-skill resource-constrainedproject scheduling problemrdquo Applied Soft Computing vol 62pp 1ndash14 2018
[50] P B Myszkowski M Laszczyk and D Kalinowski ldquoCo-evolutionary algorithm solving multi-skill resource-con-strained project scheduling problemrdquo in Proceedings of theFederated Conference on Computer Science and InformationSystems pp 75ndash82 2017
[51] F S Alanzi K Alzame andAAllahverdi ldquoWeightedmulti-skillresources project schedulingrdquoCommunications ampNetwork vol03 pp 1125ndash1130 2010
[52] M A Santos and A P Tereso ldquoOn the multi-mode multi-skillresource constrained project scheduling problem - a softwareapplicationrdquoAdvances in Intelligent and Soft Computing vol 96pp 239ndash248 2011
[53] H-Y Zheng L Wang and X-L Zheng ldquoTeachingndashlearning-based optimization algorithm for multi-skill resource con-strained project scheduling problemrdquo Soft Computing vol 21no 6 pp 1537ndash1548 2017
[54] H Dai W Cheng and P Guo ldquoAn improved tabu search formulti-skill resource-constrained project scheduling problemsunder step-deteriorationrdquo Arabian Journal for Science andEngineering vol 43 no 6 pp 3279ndash3290 2018
[55] X Lai and J K HaoATabu Search BasedMemetic Algorithm forThe Max-Mean Dispersion Problem Elsevier Science Ltd 2016
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Journal of
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Mathematical PhysicsAdvances in
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Hindawiwwwhindawicom Volume 2018
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International Journal of
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Operations ResearchAdvances in
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom