AMathematicalModelandaSimulatedAnnealingAlgorithmfor...

Research ArticleA Mathematical Model and a Simulated Annealing Algorithm forBalancing Multi-manned Assembly Line Problem withSequence-Dependent Setup Time

Wucheng Yang 12 and Wenming Cheng12

1School of Mechanical Engineering Southwest Jiaotong University Chengdu 610000 China2Technology and Equipment of Rail Transit Operation and Maintenance Key Laboratory of Sichuan ProvinceChengdu 610031 China

Correspondence should be addressed to Wucheng Yang ywc20170317gmailcom

Received 19 September 2019 Revised 9 March 2020 Accepted 20 April 2020 Published 23 May 2020

Academic Editor Marzio Pennisi

Copyright copy 2020 Wucheng Yang and Wenming Cheng is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

Multi-manned assembly lines have been widely applied to the industrial production especially for large-sized products such ascars buses and trucks in which more than one operator in the same station simultaneously performs different tasks in parallelis study deals with a multi-manned assembly line balancing problem by simultaneously considering the forward and backwardsequence-dependent setup time (MALBPS) A mixed-integer programming is established to characterize the problem Besides asimulated annealing algorithm is also proposed to solve it To validate the performance of the proposed approaches a set ofbenchmark instances are tested and the lower bound of the proposed problem is also given e results demonstrated that theproposed algorithm is quite effective to solve the problem

1 Introduction

As flow-oriented production systems assembly lines havebeen widely used in the industrial production of highquantity standardized commodities since Ford developedsuch a line in 1913 With the usage and spread of assemblylines a combinatory problem named assembly line bal-ancing problem (ALBP) has aroused great interest of re-searchers [1ndash3] As the simplest problem of ALBP thesimple assembly line balancing problem (SALBP) is to assigntasks to an ordered sequence of stations such that theprecedence constraint and the cycle time constraint aresatisfied and one or more objectives are optimized In termsof the mathematical complexity the SALBP is strongly NP-hard since it can be subsumed as a special case of the binpacking problem which was being proved as a NP-hardproblem [4] us numerous researchers have been devotedto developing various approaches including exact algorithmheuristics and metaheuristics to solve the problem [5 6]

However when assembly lines are applied to producelarge-sized products such as cars trains or aircraft there isenough space to assign two or more workers to each stationthe SALBP is no longer suitable since it assumes that onlyone operator is allowed in one station In such conditionsthe multi-manned assembly line balancing problem(MALBP) is proposed to bridge the gap by resuming thatmore than one worker is allowed in one station as shown inFigure 1 Another very similar problems is two-sided as-sembly line balancing problem (TALBP) firstly addressed by[7] However the main difference between TALBP andMALBP is that only two workers can be in the same multi-manned workstation and there are preferred operation di-rections of tasks that restrict the assignments of them toworkstations [8] us the TALBP can be regarded as aspecial case of MALBP

As an extension of SALBP the MALBP is also NP-hardproblem and an increasing number of approaches have beendesigned to solve the problem To the best of authorsrsquo

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 8510253 16 pageshttpsdoiorg10115520208510253

knowledge Dimitriadis [9] is the first researcher to addressthe MALBP by proposing a two-level heuristic algorithm tostudy a real case from an automobile assembly plant Interms of the characteristics of the objective functions theMALBP can be divided into three versions MABLP-I tominimize the number of workersstations MALBP-II tominimize the cycle time and MALBP-C to minimize thetotal production cost

Regarding the MABLP-I Fattahi et al [10] firstly de-veloped a MIP model and an ant-colony-based algorithmand then Yilmaz and Yilmaz [11] corrected their mathe-matical model Kellegoz and Toklu [12] modified the branch-and-bound algorithm to minimize the length of the lineen the same author developed a priority-rule basedconstructive heuristic approach for the same problem [13]Roshani et al [14] designed a simulated annealing algorithmto solve the problem by simultaneously optimizing the lineefficiency the line length and the smoothness index Re-cently Kellegoz [15] modified the simulated annealing al-gorithm by performing on Gantt representations ofsolutions and a new MIP model requiring less number ofvariables and constraints was also developed Michels et al[16] proposed a bendersrsquo decomposition algorithm tominimize the number of workers as the primary objectiveFor other extension of MALBP-I Chen et al [17] addi-tionally considered the resource constraint Sahin andKellegoz [18] assumed that workers could walk betweendifferent stations Lopes et al [19] defined the flexible multi-manned assembly lines with flexible station frontiers inwhich multiple workers in the same station were allowed tostart operating as possible as they can Although theMALBP-II is a major practice for reconfiguration of theinstalled assembly lines research on the MALBP-II is muchless than that on the MALBP-I Roshani and Giglio [20]presented a mathematical model and two simulatedannealing-based algorithms for solving the MALBP-II onesolves it directly and another one solves it by repeatedlysolving the MALBP-I Besides the above mentioned twotime-oriented objectives the cost-oriented objective

gradually gets more attention since the production cost hasbeen the key to win in competitions among factories Re-garding the MALBP-C Roshani and Giglio [21] presented aMIP mathematical model to describe the problem Giglioet al [22] assumed workers were skilled and a MIP modelwas also built Sahin and Kellegoz [23] additionally con-sidered the resource constraint in the MALBP-C a MIPmodel and a particle swarm optimization algorithm wereboth developed to solve it

Furthermore despite it being certainly an argumentwhose setups are ubiquitous to the realistic work environ-ment and cannot be ignored for decision-making [24] mostresearchers still assumed setups as negligible in the litera-ture is phenomenon may be caused by the following (i)the influence of setups has been reduced to a large degree byusing some advanced manufacturing technologies such asflexible manufacturing system (ii) in some real cases thesetup time is ignored because it is too small (iii) researcherssimplified their problem by assuming the setup time as anassignment constraint (eg incompatible task) Howeverthere are still many environments where setup time issignificant especially in the case when a station is operated ator near full capacity e time required to perform a setupactivity is called as setup time Setup time can be classified assequence-dependent setup time and sequence-independentsetup time In assembly lines sequence-dependent setuptime occurs when the setup time of a task depends on whichtask was set up on the station prior to operate that task Withrespect to the sequence-independent setup time the se-quence-dependent setup time needs more consideration(such as scheduling tasks intra each station) Hence treatingthis kind of setup time separately from processing timeallows operations to be performed simultaneously and henceimproves performance Furthermore in multi-manned as-sembly lines setups usually cannot be ignored and should beconsidered more carefully with respect to the traditionalsingle-manned assembly lines e reason of this is that thelarge-sized products are often assembled in the former as-sembly lines us the setups tend to be large enough with

Station 2 Station 3Station 1

(a)


(b)

Figure 1 Configuration examples of a (a) simple assembly line and (b) multi-manned assembly line

2 Mathematical Problems in Engineering

respect to their task time such as the travel time of operatorscaused by its moving around the work piece (large-sized) toperform the assigned tasks [25]

Andres et al [26] made the first attempt to solve theassembly line balancing problem with setups (ALBPS)they also proposed a binary linear program model and agreedy randomized adaptive search procedure Since thenvarious metaheuristics were proposed for solving theproblem [27ndash29] en Scholl et al [30] extended theproblem by distinguishing the forward and backwardsetup time a MIP model and several heuristics were alsodeveloped For other extensions of ALBPS Nazarian et al[31] presented a MIP model for the multimodel ALBPSSahin and Kellegoz [32] defined the crossover setup time inu-shaped assembly lines Ozcan [33] defined the line-switching setup time in parallel assembly lines a binarylinear program model and a simulated annealing algo-rithm were proposed to solve it Akpinar et al [34] de-veloped a hybrid ant-colony optimization algorithm forsolving the mixed-model ALBPS then Akpinar andBaykasoglu [35 36] extended this problem by dis-tinguishing the forward and backward setup time(mALBPS) Ozcan and Toklu [37] firstly considered thetwo-sided assembly line balancing problem with setups(TALBPS) and developed a MIP model and a heuristicapproach to solve it Janardhanan et al [38] also extendedthe TALBPS to the robotic two-sided assembly lines byproposing aMIPmodel and a metaheuristic migrating birdoptimization algorithm Aghajani et al [39] extended theTALBPS to mixed-model two-sided robotic assemblylines and they proposed a MIP model and a simulatedannealing algorithm Furthermore Esmaeilbeigi et al [40]developed three formulations for the ALBPS and designedseveral possible improvements in the form of valid in-equalities and preprocessing approaches Akpinar et al[41] improved the model for the ALBPS and mALBPS andan exact procedure and introduced the benders decom-position algorithm

Although a great deal of research has been devoted tovarious approaches for solving MALBP according to ourbest knowledge no published paper on MALBP in the lit-erature has simultaneously considered forward and back-ward setup time before

In this study the MALBP is extended by considering thesequence-dependent setup time to minimize the number ofworkers as primary objective and minimize the number ofstations as secondary objective (MALBPS-I) A mixed-in-teger programming mathematical model is built to char-acterize the MALBPS and a metaheuristic algorithm basedon simulated annealing (SA) approach is also developed tosolve the problem

e rest of this article is organized as follows InSection 2 the problem to solve is formalized and the MIPmodel is also presented Besides an example of theproposed problem is given Section 3 is devoted to thedescription of the proposed SA algorithm In Section 4the design of experiment is presented and the results arediscussed e conclusion and future direction are givenin Section 5

2 The MALBPS-I

In this section the MALBPS-I is described in detail and theproblem assumptions are listed Finally before calculatingthe lower bound the proposed MIP formulation isdeveloped

21 Problem Definition A series of multi-manned stations(j 1 Smax) are utilized on the paced straight assemblylines to produce single model products A set of workersk (1 2 3 Wmax) are assigned to each multi-mannedstation A set of tasks i (1 2 3 n m) are being assignedto workers and stations to minimize the number of workersand the number of multi-manned stations without violatingthe cycle time constraint and the precedence constraint Asdepicted in Figure 2 worker 1 worker 2 and worker 3 areassigned to multi-manned station 1 worker 4 and worker 5are assigned to multi-manned station 2 Besides in such amulti-manned assembly line the sequence-dependent idletime may occur For example task h is delayed by its pre-decessor task i which is operated by different workers in thesame multi-manned station

In assembly lines setup time may occur in two ways theforward and backward setup time As we can see fromFigure 2 when a task i is immediately performed beforeanother task p operated by the same worker at the samemulti-manned station in the same cycle then a forwardsetup occurs for the same work piece to perform task p and aforward setup time fstip is added to the finish time of task pFurthermore when a task p is the last task operated by aworker and in the next cycle task i is the first task operatedby the same worker at the samemulti-manned station then abackward setup occurs for the next work piece to performtask i and a backward setup time bstpi is added to computethe global station time

Moreover the sequence-dependent idle time can be usedfor dealing with setup operations As depicted in Figure 2the sequence idle time Idle1 occurs in worker 1 us thefinish time of task p is calculated asftp fti + tp + max(Idel1 fstip) Besides the station idletime also can be used to deal with backward setup opera-tions As depicted in Figure 2 the station idle time Idle2occurs in worker 1 and a backward setup time bstpi alsooccurs if Idle2 ge bstpi then constraint (8) is satisfiedOtherwise the cycle time constraint is violated

22 Problem Assumptions e problem assumptions of theMALBPS-I are listed as below

Task time setup time and precedence diagram aredeterministic in nature and known in advanceAll stations are equally equipped and all workers areassumed having the same ability to perform any tasksMore than one worker is allowed to be assigned to eachstationForward and backward setup time may occur betweentwo adjacency tasks

Mathematical Problems in Engineering 3

e buffers or WIP are not allowed

23 Notations e notation is given in Table 1

24 e Mathematical Model e mathematical modelproposed by [10 11 37] are extended to develop a MIPmodel for the MALBPS-I in this study Task m is assumedas a virtue node with zero task time and it is a final node ofthe precedence graph thus the station which task m isassigned to is the final station e model is given asfollows

Min1113944jϵJ

1113944kisinK

wjk +1

Wmax times n + 11113944jisinJ

1113944kisinK

1113944sisinWS

j middot xmjks (1)

1113944jisinJ

1113944kisinK

1113944sisinWS

xijks 1 foralli isin I (2)

1113944iisinI

xijks le 1 forallj isin J k isin K s isinWS (3)

1113944iisinI

xijk(s+1) minus 1113944iisinI

xijks le 0

forallj isin J k isin K s isinWS and sltNmax

(4)

i

h

Idle1 Idle2p

Multi-manned station 1

Worker 1

Worker 2

m

Worker 3

Task time

Idle time

(a)

Forward setup time

Backward setup time

Assigned tasks

Worker 4

Assigned tasks

Worker 5


(b)

Figure 2 An example of multi-manned assembly lines with setup times


1113944gisinJ

1113944kisinK

1113944sisinWS

Nmax middot (g minus 1) + s( 1113857 middot xhgks

minus 1113944jisinJ

1113944kisinK

1113944sisinWS

Nmax middot (j minus 1) + s( 1113857 middot xijks

le 0foralli isin I h isin P(i)

(5)

fti minus fth + M middot 1 minus 1113944kisinK

1113944sisinWS

xhjks⎛⎝ ⎞⎠

+ M middot 1 minus 1113944kisinK

1113944sisinWS

xijks⎛⎝ ⎞⎠ge ti

foralli isin I h isin P(i) j isin J

(6)

ftp minus fti + M middot 1 minus 1113944kisinK

1113944

sltn

sgt 1xpjks

⎛⎝ ⎞⎠


1113944

sltn

sgt 0xijks

⎛⎝ ⎞⎠ + M middot 1 minus zipjk1113872 1113873

ge tp + fstipforalli isin I p isin Iandpne i j isin J k isin K

(7)

ftp minus fti + Ct + M middot 1 minus xpjk11113872 1113873 + M middot 1 minus ltijk1113872 1113873

+ M middot 1 minus zipjk1113872 1113873ge tp + bstip

foralli isin I p isin I j isin J k isin K

(8)

xijks + xhjk(s+1) le 1 + zihjk

foralli isin I h isin Iand hne iand h notin P(i) j isin J k isin K

(9)

xijks minus 1113944hisinIandhnemandhne iandh notin P(i)

xhjk(s+1) le ltijk

foralli isin I j isin J k isin K s isinWSand sltNmax

(10)

ltijk + xhjk1 le 1 + zihjk

forallh isin I i isin Iand i notin P(h) j isin J k isin K(11)

1113944iisinIandsisinWS

xijks minus Nmax middot wjk le 0 forallj isin J k isin K (12)

1113944kisinK

k middot wsjk minus 1113944kisinK

wjk 0 forallj isin J (13)

Table 1 Notations

Indicesi h p m A taskk A workerj A multimanned stationn e total number of tasksM A big numberCt e given cycle timefstip e forward setup time between task i and task pbstip e backward setup time between task i and task pWmax e maximum allowed number of workers in one stationSmax e upper bound of number of multimanned stationNmax e maximum allowed number of tasks for each workerP(i) Set of all immediate predecessors of task iPa(i) Set of all predecessors of task iS(i) Set of all immediate successors of task iSa(i) Set of all successors of task iWSk Set of tasks assigned to worker kθ A controlling parameter between 0 and 1ParametersI Set of tasks I 1 2 3 n mK Set of workers K 1 2 3 WmaxJ Set of stations J 1 2 3 SmaxWS Set of positions WS 1 2 3 Nmaxti e task time of task iDecision variablesxijks 1 if task i is assigned to the position s of station (j k) 0 otherwisefti e finish time of task iwjk 1 if at least one task is assigned to station (j k) 0 otherwiseWSjk 1 if k workers are used in station j 0 otherwiseIndicator variableszipjk 1 if task i is assigned to the immediately predecessor position of task p in station (j k) 0 otherwiseltijk 1 if task i is assigned to the last position of station (j k) 0 otherwise


wj(k+1) lewjk forallj isin J k isin Kand kltWmax (14)

1113944iisinIsisinWS

xijks minus wjk ge 0 forallj isin J k isin Kand kltWmax (15)

1113944kisinK

ws(j+1)k le 1113944kisinK

wsjk forallj isin Jand jlt Smax (16)

fti ge ti foralli isin I (17)

fti leCt foralli isin I (18)

xijks isin 0 1 foralli isin I j isin J k isin K s isinWS (19)

wjk isin 0 1 forallj isin J k isin K (20)

wsjk isin 0 1 forallj isin J k isin K (21)

zihjk isin 0 1 foralli isin Iforallj isin J k isin K h isin I (22)

ltijk isin 0 1 foralli isin Iforallj isin J k isin K (23)

Objective function (1) minimizes the number of workersas the primary objective and minimizes the number ofstations as the secondary objective Constraint (2) ensuresthat each task is assigned to one position s of one station(j k) Constraint (3) ensures that at most one task will beassigned to one position s of one station (j k) Constraint(4) ensures that the position will be opened in increasingorder Constraint (5) ensures that all precedence relationsamong tasks are satisfied Constraints (6)ndash(8) control thesequence-dependent finish time of tasks If task i and itsimmediate predecessor task h are assigned to the samestation j then constraint (6) becomes fti minus fth ge ti Con-straints (7) and (8) ensure when the forward or backwardsetup occurs then the forward or backward setup time mustbe considered When two tasks are assigned to the successiveposition in the same cycle of a station then constraint (7)becomes ftp minus fti + ge tp + fstip When two tasks areassigned to the successive position in the next cycle of astation then constraint (8) becomes ftp minus fti + Ct getp + bstip Constraint (9) ensures that if two tasks areassigned to two adjacent positions of station (j k) then zihjk

will be equal to one Constraint (10) ensures that if task i isthe last task of station (j k) then ltijk will be equal to oneConstraint (11) provides us to determine the backward setupbetween the last task and the first task of a station Constraint(12) ensures that if any task has been assigned to station(j k) then wjk will be equal to one Constraint (13) ensuresthat if k workers are assigned to station j then wsjk will beequal to one Constraint (14) observes the sequence ofworkersrsquo index in a multi-manned station Constraint (15)ensures that if no task is assigned to station (j k) then wjk

will be equal to zero Constraint (16) observes the sequenceof stationsrsquo index in lines Constraints (17) and (18) ensurethat the range of the finish time of task i is between itscompletion time and the cycle time Constraints (19)ndash(23)

are the internality constraints e lower bound of theproblem is given in Appendix A

25 An Example to Illustrate the MIP Model e Mertenproblem [42] with or without setup time are both solved op-timally by using the MIP model e setup time is generated inSection 4 and the detailed data is given in Table 2e cycle timeis set as seven and theWmax is set as three As shown in Figure 3when considering setups one more worker is needed with re-spect to the problem without setups

3 Proposed SA Algorithm for MALBPS-I

As an extension of MALBP the MALBPS is also strongly NP-hard problem us it is necessary to develop a heuristic ormetaheuristic-based algorithm to solve large-sized problem Inthis paper a simulated annealing (SA) approach is proposed forsolving the MALBPS-I Since the SA algorithm was introducedby Kirkpatrick et al [43] as an iterative random search tech-nique it has widely been used to solve various combinatorialoptimization problems including general assembly line bal-ancing problem [14 15 20 44 45] Basically the SA algorithmis a local search-based metaheuristic which derives its ac-ceptance mechanism from the annealing process to let thecurrent solution escape from local optima e detailed pro-cedure of the proposed SA algorithm is given below

31 Initial Solution Considering that the number of stationsis uncertain and it is essential to determine the task sequencein the proposed problem a priority-based coding method isadopted in which solutions are constructed according to apriority list (PL) of tasks en the initial solution is ran-domly generated as a sequence between 1 to n by a uniformdistribution (1 2 3 4 5 6 7) as shown in Figure 4 To obtaina feasible solution the assignable task with the lowest pri-ority value is being selected and then it is being assigned to aworker according to some given rules as Section 32 enthe process continues until all tasks are assigned

32 Building a Feasible Solution A feasible balancing so-lution is to determine how to assign works to stations andhow to assign tasks to workers without violating the pre-cedence constraint and the cycle time constraint e pro-cedure to build a feasible solution is given as Algorithm 1 inFigure 5 An example is also illustrated in Table 3 eprocedure to calculate the finish time (tFTl) of a task (i) isgiven as Algorithm 2 in Figure 6 e rules of accepting thetask assignment to current multi-manned station are definedas follows (if one of the following conditions (a or b) isfulfilled)

(a) e number of workers (L) working in currentstation equals to one

(b) A generated random number (0ltRlt 1) is not largerthan exp(minusδTc) where δ is the difference betweenthe mean idle time per worker in current station(Midle) and a predetermined upper bound of ac-ceptable idle time (UB)


Table 2 e detailed data of the example instance

Task (i) P(i) Task Forwardbackward sequence-dependent setup time1 2 3 4 5 6 7

1 sim 1 000019 010011 020023 001006 016023 021025 0150162 1 5 023024 000021 004015 011014 014024 017019 0040103 2 4 003012 021022 000022 022023 011014 008014 0060084 1 3 023023 004012 017018 000016 009010 018021 0120185 2 5 016021 018020 021023 018020 000009 003016 0170186 5 6 002005 014016 021022 017020 002008 000021 0080167 4 5 003011 015020 016019 004010 008016 016019 000017

5

3

Worker 3

Worker 2

Worker 5

Worker 4

MultimannedStation 2


7

7

7

7

0

0

0

04 61

Worker 1


702

7

Forward setup time

Backward setup time

Idle time

Task time

(a)

Forward setup time

Backward setup time

1Worker 1


702 4

70

5

3Worker 3

Worker 2 Worker 5

Worker 4



7

7

7

7

0

0

0

06

7Worker 6

Idle time

Task time

(b)

Figure 3e optimal task assignment to stations of multi-manned assembly lines without setup time (result (a)) and with setup time (result(b)) for Merten problem

1 2

4

3

5 6

7

1

65

4

5

5

3

PL sequence 1 2 4 5 3 6 7

1 2 3 4 5 6 7Task ID

Figure 4 An example of the coding method


Midle Ct times l minus 1113936iisinWSl

ti

l (24)

UB θ middotCt times THL minus 1113936iisinIti

THL (25)

THL 1113936iisinIti

Ct

1113890 1113891 (26)

δ Midle minus UB (27)

33 NeighborhoodGeneration Neighborhood structures aremain approaches to produce new solution in the SA algo-rithm by moving from a solution to its neighborhood oneFor the proposed SA algorithm the neighborhood structuresare designed the same as [14] swap and insert based op-erators An example is explained in Figure 7

34 Objective Function In this paper three factors are takeninto consideration including the line efficiency (LE) the linelength (LN) and smoothing index (SI) for the MALBPS-I

Moreover these criterions are combined together to build asingle objective function by using a minimum deviationmethod [14] Let LE0 LN0 and SI0 which are obtained froma initial solution be the least desirable objective value ofLE LN and SI e objective function of the proposedalgorithm is formulated as follows

LE 1113936iisinIti

Nw middot Ct

(28)

LN Ns (29)

SI 1113936jisinJ Ct minus 1113936iisinSL

ti1113872 11138732

Nw

(30)

Minimizef LEmax minus LE

LEmax minus LE0+

LNmin minus LN

LNmin minus LN0+

SImin minus SI

SImin minus SI0

(31)

where LEmax LNmin and SImin are respectively the mostdesirable objective value of LELN and SI For the problemof MALBPS-I the value of the LEmax is set as 100 LNmin isset as THLWmax and SImin is set as 0

Algorithm 1 Building a feasible solution for MABLPS-I

Input All parameters related to the problemsOutput The feasible solution(1)

(2)(3)(4)(5)(6)(7)

(8)(9)

(10)(11)(12)(13)(14)

(15)(16)(17)(18)(19)(20)(21)(22)

(23)

initialize Open a new station and set the current number of workers as L = Wmax Set allworkersrsquo load as WL = 0while all tasks have not been assigned do

Determin assignable tasks set as ALif AL is not empty then

while AL is not empety doSelect a task h from AL by using predetermined rulesCalculate finish time of task h of all available workers as tFTl as Algorithm 2Select the worker l with minimum value of tFTl and tFTl lt= Ctif l is empty then

AL = AL ndash hend if

end whileAssign the selected task to the selected worker and update all related parameters

else

if acceptance rules are satisfied thenAccept all assignment results to the current opened stationUpdate all related parameters

elseDelete all tasks have been assigned to current opened stationSet L = L ndash 1

end ifend if

end while

Figure 5 e procedure to build a feasible solution for MALBPS-I


Tabl

e3

Anexam

pleof

theprocedureto

build

afeasible

balancingsolutio

nfortheMALB

PS-I

Start

Step

1Step

2Step

3Step

4Step

5Step

6Ct

7

PL

[1245367]

N

s0

T 015

Ns

1

L2

AT

[1]

Select

task1

Rand

omly

select

aworker1in

station1

then

ft 1

t 1

1lt7

t 1+

bst

11119lt7

AT

[24]

Select

task2

Calculatethee

arlieststarttim

efor

allw

orkersasmin[11]

1thenrando

mly

select

aworker2in

station1

ft 2

f

t 1+

t 26lt7

t 2+

bst

22521lt7

AT

[345]

Select

task5

Calculate

theearliestfin

ishingtim

eforallw

orkers

asmin

[66]

6thenrando

mly

choo

seworker2

ft 5

f

t 2+

t 5+

fst

251114gt7

thencheck

station1

Midle

4

UB

102δ

298L

2

R0543ltEx

p(minusδT 0

081accept

theassig

nmentthenN

s2

Ns

2

L2

AT

[345]

Select

task5

Rand

omly

select

aworker1in

station2

then

ft 5

t 5

5lt7

t 5+

bst

55509lt7

AT

[346]

Select

task3

Calculatetheearlieststarttim

efora

llworkersas

min[50]

0thenselectw

orker2

ft 3

t 3

4t

3+

bst

33422lt7

AT

[46]

Select

task4


efora

llworkersas

min[54]

4thenselectworker2

ft 4

f

t 3+

t 4+

fst

34gt7

then

checkstation2

Midle

25

UB

102δ

148L

2

R0824ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s3

Ns

3

L2

AT

[46]

Select

task4

Rand

omlyselectaworker1

instation3

thenf

t 4

t 43lt7

t 4+

bst

44316lt7

AT

[67]

Select

task6


efora

llworkersas

min[30]

0thenselectw

orker2

ft 6

t 6

6lt7

t 6+

bst

66621lt7

AT

[7]

Select

task7


efora

llworkersas

min[36]

3thenselectw

orker1f

t 7

ft 4

+t 7

+f

st47gt7thencheck

station3

Midle

25

UB

102δ

148L

2

R0746ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s4

Ns

4

L2

AT

[7]

Select

task7

Rand

omly

select

aworker1in

station4

then

ft 7

t 7

5lt7

t 7+

bst

77517lt7

Alltasks

are

assig

ned


35 Description of Proposed SA Algorithm e proposed SAalgorithm starts with an initial solution (Section 31) in apredetermined temperature Tc (initialized as T0) and nav-igates around it through some predetermined neighborhood

structures (Section 33) to find better neighborhood solu-tion With the searching process of the algorithm thetemperature decreases during the iteration as Tc α middot Tcwhere α is the predetermined cooling rate In each

Algorithm 2 Calculated the tFTlInput All parameters related to the problemsOutput tFTl(1)(2)(3)(4)(5)(6)(7)(8)(9)

(10)(11)

(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

(29)(28)

(30)(31)

initialize Set the tFTl equals to the load of current worker l (WLl)while l lt= L do

Calculate the latest sequence-depedent start time of task h as LFThif WLl lt LFTh then

Determine the earliest start time of task h as Esth = LFTl Calculate the difference as d = LFTl mdash WLl

elseDetermine the earliest start time of task h as Esth = WLl Set the difference as d = 0

end ifif task p doesnrsquot exists then p and q is the lastfirst task has been assigned to thecurrent worker l respectively

if d gt 0 thenif fstph mdash d gt 0 then

fstph = fstph mdash delse

fstph = 0end if

end ifif th + Esth + fstph lt= Ct and Ct mdash ftq + fti mdash tq gt= bstiq then

Calculated tFTl = th + Esth + fstphelse

Set tFtl = 2 lowast Ctend if

elseif th + Esth lt= Ct and Ct mdash th gt= bsthh then

Calculate tFTl = th + Esthelse

Set tFTl = 2 lowast Ct

end ifend if

end while

Figure 6 e procedure to calculate the finish time of tasks

Generate p uniformly in [0 1]

Swap operator Insert operator

1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 5 4 6 7PL

1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 4 5 6 7PL

p gt 05 p lt = 05

Figure 7 An example of neighborhood structures for Merten problem


temperature the Tc remains the same for a period of time(TT the Marko chain length) and then it falls through thealgorithm the probability decreases and the result con-verges to the best solution e neighborhood solution willbe accepted as the probability exp(minusΔfTc) where Δf is thedifference of fitness between the current solution and itsneighbourhood solution e algorithm stops when the Tc islower than a predetermined value Tf en output the bestfound solution e procedure is given in Figure 8

4 Computational Results

In this section there are mainly two folds Firstly the testinstances are solved by using the MIP model e modelis solved by IBM ILOG CPLEX 1263 Secondly the testinstances in the literature are solved by the proposedSA algorithm which is coded by MATLAB2016a software All experiments are executed on PCwith Interreg Coretrade i3 34 GHz processor and 40 GB ofRAM

e test instances are a group of well-known Talbot dataset including 64 problems from [42] e forward setuptime is randomly generated as U[0 025 middot minforalliisinIti] forproblems with low-level setups and U[0 075 middot minforalliisinIti] forproblems with high-level setups e backward setup timeis randomly generated as U[0 115 middot 025 middot minforalliisinIti] forproblems with problems with low-level setups andU[0 115 middot 075 middot minforalliisinIti] for problems with high-levelsetups e number of maximum allowed workers is set astwo All the parameters of the proposed SA algorithm aredetermined through preliminary experiments by using theTaguchi method [46] To determine the values of param-eters including T0 Tf α andTT three test instances arerandomly selected as Mansoor (n 11 Ct 63)Kilbridge(n 45 Ct 92) and Arcus problem(n 111

Ct 5755) ree levels of T0(2000 1000 100) three levelsof Tf(0001 1 10) and three levels of α(099 095 09) areall tested e value of TT is fixed to n According to thenumber of levels and factors the Taguchimethod L9 is used forthe adjustment of the parameters Each test problem is solvedten times and a performance measureRPD is defined as

RPD fi minus fmin

fmintimes 100 (32)

where fmin and fi are the best solution obtained for a giveninstance and the objective value obtained for a trial re-spectively According to the results of the statistical analysisthe value ofT0 Tf andα are determined as 1000 0001 and099 respectively

41 e Result of the MIP Model e results of the MIPmodel for solving the MALBPS-I with different level setupsare given in Table 4 e model is coded by OPL languageand each test stops until the running time is larger than 7200seconds As we can observe from the table only very limitedsmall-sized (nle 20) test instances can be solved optimally in

the time limit e best found number of workersstations(NwNs) and the CPU time are also reported e bold textin the table shows that compared with the problem withoutsetups the same problem with setups need more stations orworkers and the higher the setups are the more stations orworkers are needed lowastmeans that not optimal results but thebest feasible results found by the model in time limit arereported

42eResult of the ProposedAlgorithm In this section the62 test instances are solved by using the proposed SA al-gorithm Each instance is solved ten times and the resultsare reported in Table 5 e Best and Mean mean the bestand mean found number of stationsworkers (Ns[Nw])respectively e LB is the optimal value of the ALBP-Ifrom [42] which can be regarded as the lower bound of thenumber of workers for the MALBP-I e BM is the op-timal result of number of stationsworkers (Ns[Nw]) forMALBP-I obtained by [10] e BHA is the best reportednumber of stationsworkers (Ns[Nw]) in the literature forthe MALBP-I without setups [14 15] e LB1 is the lowerbound of the number of workers given in this paper for theMALBPS-I

In order to evaluate the effectiveness of the proposedalgorithm the index G represents the gap between the bestfound number of worker and the lower bound of MALBPS-Iand can be calculated as G (Best( Nw)minus LB1)LB1 times 100Another index G1 represents the gap between the bestfound number of worker and the BHA( Nw) and can becalculated asG1 (Best( Nw) minus BHA(Nw))BHA(Nw) times 100 More-over Db represents the difference between the best andaverage found number of workers and Dc represents thevariance of number of workers found by the SA algorithm inten timesrsquo runs

e results in Table 5 show that the proposed SAalgorithm is able to find good solutions to MABLPS-I in areasonable time Besides a summary of computationalresults is given in Table 6 For all test instances the av-erage values of G from LB1 for problems with low-leveland high-level setups are 123 and 098 respectivelyese results show that the proposed algorithm producesvery close results to the lower bound Besides among 50test problems in Table 5 the proposed SA algorithmobtained 46 and 47 optimal solutions equal to the lowerbound for problems with low-level and high-level setupsrespectively In addition to this the average value of theLB1 and the average value of best found Nw for problemwith high-level setups are both larger than the value forproblems with low-level setups e results show that thesetups may increase the number of utilized workers andthe number of workers will be increased with the level ofsetups Furthermore the average value of Db and Dc forall test problems are 0007 and 0006 respectively whichshow that the proposed SA algorithm has a high rate ofconvergence and stability


Table 4 e results of the MIP model for problems with different level setups

Author n CtNo setup Low setups High setups

Ns[Nw] Time (s) Ns[Nw] Time (s) Ns[Nw] Time (s)

Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize

Generate initialsolution

Calculate thefitness value

Generate aneighborhood

solution

Acceptneighborhood

solution

Replace currentsolution

Terminationcondition satisfied

Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned

Calculate theobjective functionvalue and output it

Select a task

Enoughcapacity

Assign the task tocurrent worker

Accept theassignment of current

station

Accept all tasksassigned to current

station

Select a worker

Availableworker exists

Abandon all taskassigned to current

station

Yes

Yes

No

No

No

No

Yes

Yes

Adjust allowedworkersrsquo number

Building a feasible solution prodecure

Figure 8 e procedure of the proposed SA algorithm


Table 5 e computation results of the SA algorithm

Problem Cree different level setup times

Zero level Low level High levelLB BM BHA LB1 Best Mean G G1 Db Dc LB1 Best Mean G G1 Db Dc CPU (s)

Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)

14 8 sim 7[8] 8 7[9] 7[9] 1250 1250 0 0 8 7[9] 7[9] 1250 1250 0 0 539915 8 sim 7[8] 8 7[8] 7[8] 000 000 0 0 8 7[8] 7[8] 000 000 0 0 597721 5 sim 5[5] 6 4[6] 4[6] 000 2000 0 0 6 4[6] 4[6] 000 2000 0 0 248526 5 sim 4[5] 5 4[5] 4[5] 000 000 0 0 5 4[5] 4[5] 000 000 0 0 165435 3 sim 3[3] 4 3[4] 3[4] 000 3333 0 0 4 3[4] 3[4] 000 3333 0 0 61239 3 sim 2[3] 3 3[3] 3[3] 000 000 0 0 3 3[3] 3[3] 000 000 0 0 1356

Heskia (28)


Sawyer (30)


Kilbridge (45)

79 7 sim 5[7] 8 4[8] 4[8] 000 1429 0 0 8 5[8] 5[8] 000 1429 0 0 76482 6 sim 4[6] 7 4[7] 4[7] 000 1667 0 0 7 4[7] 4[7] 000 1667 0 0 099110 6 sim 3[6] 6 3[6] 3[6] 000 000 0 0 6 3[6] 3[6] 000 000 0 0 028138 4 sim 3[4] 5 3[5] 3[5] 000 2500 0 0 5 3[5] 3[5] 000 2500 0 0 038184 3 sim 2[3] 4 2[4] 2[4] 000 3333 0 0 4 2[4] 2[4] 000 3333 0 0 002

Tong (70)


Arcus (83)

5048 16 sim 10[16] 16 11[16] 112[16] 000 000 0 0 16 12[16] 12[16] 000 000 0 0 2785853 14 sim 10[14] 14 10[14] 10[14] 000 000 0 0 14 10[14] 107[14] 000 000 0 0 936842 12 sim 8[12] 12 9[12] 9[12] 000 000 0 0 12 9[12] 94[12] 000 000 0 0 1337571 11 sim 6[11] 11 8[11] 8[11] 000 000 0 0 11 8[11] 82[11] 000 000 0 0 468412 10 sim 6[10] 10 7[10] 7[10] 000 000 0 0 10 7[10] 79[10] 000 000 0 0 1648998 9 sim 6[9] 9 7[9] 7[9] 000 000 0 0 9 7[9] 7[9] 000 000 0 0 05410816 8 sim 5[8] 8 5[8] 59[8] 000 000 0 0 8 6[8] 6[8] 000 000 0 0 202

Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11

9 4[6] 396 4[7]lowast lt7200 4[7]lowast lt720010 4[5] 342 4[6]lowast lt7200 4[6]lowast lt720013 3[4] 214 3[4] 42718 3[4] 6119214 3[4] 402 3[4] 63121 3[4] 3691521 2[3] 231 2[3] 15786 2[3] 33264


5 Conclusion

In this paper the type-I problem of balancing multi-mannedassembly lines with sequence-dependent setup time is in-troduced and characterized A MIP and a simulatedannealing algorithm are proposed to solve the problem Todemonstrate the efficiency of SA algorithm and MIP modela computational experiment is conducted e results showthat the proposed approach is effective and successful for theproblem Furthermore according to our best knowledgethis paper is the first study on the multi-manned assemblyline balancing problem with forward and backward setuptime thus this paper gives a good starting point for futurestudies e following directions may be studied in future(1) more recent developed metaheuristics and exact algo-rithm should be developed (2) other extension and as-signment constraints such as u-shaped lines and zoningconstraints should be considered according to the need ofreal line (3) more effective procedure to build a feasiblesolution for the proposed should be developed (4) tighterlower bound should be given (5) dynamic job rotation toolscan be combined with the proposed problem [47] (6)multiple-user defined criteria should be considered and areal case should be studied [47]

Appendix

A The Procedure to Calculate the LowerBound of MALBPS-I

e optimal solution for ALBP-I from [42] is set as LBAndres Miralles and Pastor [26] designed a lower bound forALBPS-I then this method is being modified to develop thelower bound of MALBPS-I (LB1) and the detail of theprocedure is given below

Step 1 calculate LB1 as

LB1 max1113936iisinIti

CT LB1113888 1113889 (33)

If LB1 n then go to Step 5 otherwiseLBc LB1 and c 1

Step 2 calculate minimum forward setup times asSUFc e method is to add c lowest forward setuptimes between tasks using logical operator OR en

calculate the minimum backward setup times asSUBLBc

e method is to add LBc lowest backwardsetup times between tasks en add all possible setuptimes as SUc SUFc + SUBLBc

en calculate LBc as

LBc SUc + 1113936iisinIti

CT1113890 1113891 (34)

Step 3 q n minus LBc + 1

Step 4 if qle c thenLB1 LBc and go to Step 5otherwise go to Step 2Step 5 output LB1

Data Availability

e benchmark datasets that support the findings of thisstudy are previously reported by Scholl et al [30] and areavailable at httpsassembly-line-balancingdesalbpbenchmark-data-sets-1993 Other data used to supportthe findings of this study are available from the corre-sponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by the National Natural ScienceFoundation of China (Grants no51675450) and SichuanScience and Technology Program (Grant nos 2019YFG0300and 2019YFG0285)

References

[1] C Becker and A Scholl ldquoA survey on problems and methodsin generalized assembly line balancingrdquo European Journal ofOperational Research vol 168 no 3 pp 694ndash715 2006

[2] E Erel and S C Sarin ldquoA survey of the assembly line bal-ancing proceduresrdquo Production Planning amp Control vol 9no 5 pp 414ndash434 1998

[3] A Scholl and C Becker ldquoState-of-the-art exact and heuristicsolution procedures for simple assembly line balancingrdquoEuropean Journal of Operational Research vol 168 no 3pp 666ndash693 2006

[4] M R Garey and D S Johnson ldquoComputers and intractabilitya guide to the theory of NP-completenessrdquo 1990

Table 6 Summary of the computational resultsLevel of setups

Low HighAverage of G 123 098Average of G1 565 632Percentage of optimal solutions obtained 92 94Average of LB 822 822Average of BHA 822 822Average of LB1 840 844Average of best solutions 848 850Average of Db 0006 0008Average of Dc 0005 0007


[5] O Battaıa and A Dolgui ldquoA taxonomy of line balancingproblems and their solutionapproachesrdquo International Jour-nal of Production Economics vol 142 no 2 pp 259ndash2772013

[6] P Sivasankaran and P Shahabudeen ldquoLiterature review ofassembly line balancing problemsrdquo e International Journalof Advanced Manufacturing Technology vol 73 no 9ndash12pp 1665ndash1694 2014

[7] J J Bartholdi ldquoBalancing two-sided assembly lines a casestudyrdquo International Journal of Production Research vol 31no 10 pp 2447ndash2461 1993

[8] A Roshani P Fattahi A Roshani M Salehi and A RoshanildquoCost-oriented two-sided assembly line balancing problem asimulated annealing approachrdquo International Journal ofComputer Integrated Manufacturing vol 25 no 8 pp 689ndash715 2012

[9] S G Dimitriadis ldquoAssembly line balancing and groupworking a heuristic procedure for workersrsquo groups operatingon the same product and workstationrdquo Computers amp Oper-ations Research vol 33 no 9 pp 2757ndash2774 2006

[10] P Fattahi A Roshani and A Roshani ldquoA mathematicalmodel and ant colony algorithm for multi-manned assemblyline balancing problemrdquo e International Journal of Ad-vanced Manufacturing Technology vol 53 no 1ndash4 pp 363ndash378 2011

[11] H Yilmaz and M Yilmaz ldquoErratum to note to a mathe-matical model and ant colony algorithm for multi-mannedassembly line balancing problemrdquo e International Journalof Advanced Manufacturing Technology vol 89 no 5ndash8p 1941 2017

[12] T Kellegoz and B Toklu ldquoAn efficient branch and boundalgorithm for assembly line balancing problems with parallelmulti-manned workstationsrdquo Computers amp Operations Re-search vol 39 no 12 pp 3344ndash3360 2012

[13] T Kellegoz and B Toklu ldquoA priority rule-based constructiveheuristic and an improvement method for balancing assemblylines with parallel multi-manned workstationsrdquo InternationalJournal of Production Research vol 53 no 3 pp 736ndash7562015

[14] A Roshani A Roshani A Roshani M Salehi andA Esfandyari ldquoA simulated annealing algorithm for multi-manned assembly line balancing problemrdquo Journal ofManufacturing Systems vol 32 no 1 pp 238ndash247 2013

[15] T Kellegoz ldquoAssembly line balancing problems with multi-manned stations a new mathematical formulation and Ganttbased heuristic methodrdquo Annals of Operations Researchvol 253 no 1 pp 377ndash404 2017

[16] C Michels A Sato and L Magatatildeo ldquoA Bendersrsquo decom-position algorithm with combinatorial cuts for the multi-manned assembly line balancing problemrdquo European Journalof Operational Research vol 22 2019

[17] Y-Y Chen C-Y Cheng and J-Y Li ldquoResource-constrainedassembly line balancing problems with multi-mannedworkstationsrdquo Journal of Manufacturing Systems vol 48pp 107ndash119 2018

[18] M Sahin and T Kellegoz ldquoBalancing multi-manned assemblylines with walking workers problem definition mathematicalformulation and an electromagnetic field optimisation al-gorithmrdquo International Journal of Production Researchvol 19 2019

[19] T C Lopes G Vidal Pastre A S Michels and L MagatatildeoldquoFlexible multi-manned assembly line balancing problemmodel heuristic procedure and lower bounds for line lengthminimizationrdquo Omega vol 34 2019

[20] A Roshani and D Giglio ldquoSimulated annealing algorithmsfor the multi-manned assembly line balancing problemminimising cycle timerdquo International Journal of ProductionResearch vol 55 no 10 pp 2731ndash2751 2017

[21] A Roshani and D Giglio ldquoA mathematical programmingformulation for cost-oriented multi-manned assembly linebalancing problemrdquo IFAC-PapersOnLine vol 48 no 3pp 2293ndash2298 2015

[22] D Giglio M Paolucci A Roshani and F Tonelli ldquoMulti-manned assembly line balancing problem with skilledworkers a new mathematical formulationrdquo Spectrum vol 502017

[23] M Sahin and T Kellegoz ldquoA new mixed-integer linearprogramming formulation and particle swarm optimizationbased hybrid heuristic for the problem of resource investmentand balancing of the assembly line with multi-mannedworkstationsrdquo Computers amp Industrial Engineering vol 133pp 107ndash120 2019

[24] A Allahverdi ldquoe third comprehensive survey on sched-uling problems with setup timescostsrdquo European Journal ofOperational Research vol 246 no 2 pp 345ndash378 2015

[25] G Michalos ldquoDynamic job rotation for workload balancing inhuman based assembly systemsrdquo CIRP Journal ofManufacturing Science and Technology vol 2 no 3pp 153ndash160 2010

[26] C Andres C Miralles and R Pastor ldquoBalancing andscheduling tasks in assembly lines with sequence-dependentsetup timesrdquo European Journal of Operational Researchvol 187 no 3 pp 1212ndash1223 2008

[27] N Hamta S M T Ghomi F Jolai andM Akbarpour ShirazildquoA hybrid PSO algorithm for a multi-objective assembly linebalancing problem with flexible operation times sequence-dependent setup times and learning effectrdquo InternationalJournal of Production Economics vol 141 no 1 pp 99ndash1112013

[28] S A Seyed-Alagheband S M T F Ghomi and M ZandiehldquoA simulated annealing algorithm for balancing the assemblyline type II problem with sequence-dependent setup timesbetween tasksrdquo International Journal of Production Researchvol 49 no 3 pp 805ndash825 2011

[29] A Yolmeh and F Kianfar ldquoAn efficient hybrid genetic al-gorithm to solve assembly line balancing problem with se-quence-dependent setup timesrdquo Computers amp IndustrialEngineering vol 62 no 4 pp 936ndash945 2012

[30] A Scholl N Boysen and M Fliedner ldquoe assembly linebalancing and scheduling problem with sequence-dependentsetup times problem extension model formulation and ef-ficient heuristicsrdquo Spectrum vol 35 no 1 pp 291ndash320 2013

[31] E Nazarian J Ko and H Wang ldquoDesign of multi-productmanufacturing lines with the consideration of product changedependent inter-task times reduced changeover and machineflexibilityrdquo Journal of Manufacturing Systems vol 29 no 1pp 35ndash46 2010

[32] M Sahin and T Kellegoz ldquoIncreasing production rate inU-type assembly lines with sequence-dependent set-uptimesrdquo Engineering Optimization vol 49 no 8pp 1401ndash1419 2017

[33] U Ozcan ldquoBalancing and scheduling tasks in parallel as-sembly lines with sequence-dependent setup timesrdquo Inter-national Journal of Production Economics vol 213 pp 81ndash962019

[34] S Akpinar G M Bayhan and A Baykasoglu ldquoHybridizingant colony optimization via genetic algorithm for mixed-model assembly line balancing problem with sequence


dependent setup times between tasksrdquo Applied Soft Com-puting vol 13 no 1 pp 574ndash589 2013

[35] S Akpinar and A Baykasoglu ldquoModeling and solving mixed-model assembly line balancing problem with setups Part I amixed integer linear programming modelrdquo Journal ofManufacturing Systems vol 33 no 1 pp 177ndash187 2014

[36] S Akpinar and A Baykasoglu ldquoModeling and solving mixed-model assembly line balancing problem with setups Part II amultiple colony hybrid bees algorithmrdquo Journal ofManufacturing Systems vol 33 no 4 pp 445ndash461 2014

[37] U Ozcan and B Toklu ldquoBalancing two-sided assembly lineswith sequence-dependent setup timesrdquo International Journalof Production Research vol 48 no 18 pp 5363ndash5383 2010

[38] M N Janardhanan Z Li G Bocewicz Z Banaszak andP Nielsen ldquoMetaheuristic algorithms for balancing roboticassembly lines with sequence-dependent robot setup timesrdquoApplied Mathematical Modelling vol 65 pp 256ndash270 2019

[39] M Aghajani R Ghodsi and B Javadi ldquoBalancing of roboticmixed-model two-sided assembly line with robot setuptimesrdquo International Journal of Advanced ManufacturingTechnology vol 74 no 5ndash8 pp 1005ndash1016 2014

[40] R Esmaeilbeigi B Naderi and P Charkhgard ldquoNew for-mulations for the setup assembly line balancing and sched-uling problemrdquo Spectrum vol 38 no 2 pp 493ndash518 2016

[41] S Akpinar A Elmi and T Bektas ldquoCombinatorial Benderscuts for assembly line balancing problems with setupsrdquo Eu-ropean Journal of Operational Research vol 259 no 2pp 527ndash537 2017

[42] F B Talbot and J H Patterson ldquoAn integer programmingalgorithm with network cuts for solving the assembly linebalancing problemrdquo Management Science vol 30 no 1pp 85ndash99 1984

[43] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimizationby simulated annealingrdquo Science vol 220 no 4598pp 671ndash680 1983

[44] Y-Y Chen ldquoA hybrid algorithm for allocating tasks oper-ators and workstations in multi-manned assembly linesrdquoJournal of Manufacturing Systems vol 42 pp 196ndash209 2017

[45] A Roshani ldquoMixed-model multi-manned assembly linebalancing problem a mathematical model and a simulatedannealing approachrdquo International Journal of ProductionResearch vol 37 2017

[46] K-L Tsui ldquoAN overview OF taguchi method and newlydeveloped statistical methods for robust designrdquo IIE Trans-actions vol 24 no 5 pp 44ndash57 1992

[47] G Michalos A Fysikopoulos S Makris D Mourtzis andG Chryssolouris ldquoMulti criteria assembly line design andconfiguration - an automotive case studyrdquo CIRP Journal ofManufacturing Science and Technology vol 9 pp 69ndash87 2015


knowledge Dimitriadis [9] is the first researcher to addressthe MALBP by proposing a two-level heuristic algorithm tostudy a real case from an automobile assembly plant Interms of the characteristics of the objective functions theMALBP can be divided into three versions MABLP-I tominimize the number of workersstations MALBP-II tominimize the cycle time and MALBP-C to minimize thetotal production cost

Regarding the MABLP-I Fattahi et al [10] firstly de-veloped a MIP model and an ant-colony-based algorithmand then Yilmaz and Yilmaz [11] corrected their mathe-matical model Kellegoz and Toklu [12] modified the branch-and-bound algorithm to minimize the length of the lineen the same author developed a priority-rule basedconstructive heuristic approach for the same problem [13]Roshani et al [14] designed a simulated annealing algorithmto solve the problem by simultaneously optimizing the lineefficiency the line length and the smoothness index Re-cently Kellegoz [15] modified the simulated annealing al-gorithm by performing on Gantt representations ofsolutions and a new MIP model requiring less number ofvariables and constraints was also developed Michels et al[16] proposed a bendersrsquo decomposition algorithm tominimize the number of workers as the primary objectiveFor other extension of MALBP-I Chen et al [17] addi-tionally considered the resource constraint Sahin andKellegoz [18] assumed that workers could walk betweendifferent stations Lopes et al [19] defined the flexible multi-manned assembly lines with flexible station frontiers inwhich multiple workers in the same station were allowed tostart operating as possible as they can Although theMALBP-II is a major practice for reconfiguration of theinstalled assembly lines research on the MALBP-II is muchless than that on the MALBP-I Roshani and Giglio [20]presented a mathematical model and two simulatedannealing-based algorithms for solving the MALBP-II onesolves it directly and another one solves it by repeatedlysolving the MALBP-I Besides the above mentioned twotime-oriented objectives the cost-oriented objective

gradually gets more attention since the production cost hasbeen the key to win in competitions among factories Re-garding the MALBP-C Roshani and Giglio [21] presented aMIP mathematical model to describe the problem Giglioet al [22] assumed workers were skilled and a MIP modelwas also built Sahin and Kellegoz [23] additionally con-sidered the resource constraint in the MALBP-C a MIPmodel and a particle swarm optimization algorithm wereboth developed to solve it

Furthermore despite it being certainly an argumentwhose setups are ubiquitous to the realistic work environ-ment and cannot be ignored for decision-making [24] mostresearchers still assumed setups as negligible in the litera-ture is phenomenon may be caused by the following (i)the influence of setups has been reduced to a large degree byusing some advanced manufacturing technologies such asflexible manufacturing system (ii) in some real cases thesetup time is ignored because it is too small (iii) researcherssimplified their problem by assuming the setup time as anassignment constraint (eg incompatible task) Howeverthere are still many environments where setup time issignificant especially in the case when a station is operated ator near full capacity e time required to perform a setupactivity is called as setup time Setup time can be classified assequence-dependent setup time and sequence-independentsetup time In assembly lines sequence-dependent setuptime occurs when the setup time of a task depends on whichtask was set up on the station prior to operate that task Withrespect to the sequence-independent setup time the se-quence-dependent setup time needs more consideration(such as scheduling tasks intra each station) Hence treatingthis kind of setup time separately from processing timeallows operations to be performed simultaneously and henceimproves performance Furthermore in multi-manned as-sembly lines setups usually cannot be ignored and should beconsidered more carefully with respect to the traditionalsingle-manned assembly lines e reason of this is that thelarge-sized products are often assembled in the former as-sembly lines us the setups tend to be large enough with


(a)


(b)

Figure 1 Configuration examples of a (a) simple assembly line and (b) multi-manned assembly line







2 The MALBPS-I











Min1113944jϵJ

1113944kisinK

wjk +1


1113944kisinK

1113944sisinWS

j middot xmjks (1)

1113944jisinJ

1113944kisinK

1113944sisinWS


1113944iisinI


1113944iisinI


xijks le 0


(4)

i

h

Idle1 Idle2p


Worker 1

Worker 2

m

Worker 3

Task time

Idle time

(a)

Forward setup time

Backward setup time

Assigned tasks

Worker 4

Assigned tasks

Worker 5


(b)



1113944gisinJ

1113944kisinK

1113944sisinWS


minus 1113944jisinJ

1113944kisinK

1113944sisinWS



(5)


1113944sisinWS

xhjks⎛⎝ ⎞⎠


1113944sisinWS



(6)


1113944

sltn

sgt 1xpjks

⎛⎝ ⎞⎠


1113944

sltn

sgt 0xijks



(7)




(8)



(9)


xhjk(s+1) le ltijk


(10)





1113944kisinK



Table 1 Notations






1113944kisinK
























1 sim 1 000019 010011 020023 001006 016023 021025 0150162 1 5 023024 000021 004015 011014 014024 017019 0040103 2 4 003012 021022 000022 022023 011014 008014 0060084 1 3 023023 004012 017018 000016 009010 018021 0120185 2 5 016021 018020 021023 018020 000009 003016 0170186 5 6 002005 014016 021022 017020 002008 000021 0080167 4 5 003011 015020 016019 004010 008016 016019 000017

5

3

Worker 3

Worker 2

Worker 5

Worker 4



7

7

7

7

0

0

0

04 61

Worker 1


702

7

Forward setup time

Backward setup time

Idle time

Task time

(a)

Forward setup time

Backward setup time

1Worker 1


702 4

70

5

3Worker 3

Worker 2 Worker 5

Worker 4



7

7

7

7

0

0

0

06

7Worker 6

Idle time

Task time

(b)


1 2

4

3

5 6

7

1

65

4

5

5

3


1 2 3 4 5 6 7Task ID




ti

l (24)


THL (25)

THL 1113936iisinIti

Ct

1113890 1113891 (26)





LE 1113936iisinIti

Nw middot Ct

(28)

LN Ns (29)


ti1113872 11138732

Nw

(30)


LEmax minus LE0+

LNmin minus LN

LNmin minus LN0+

SImin minus SI

SImin minus SI0

(31)




(2)(3)(4)(5)(6)(7)

(8)(9)

(10)(11)(12)(13)(14)

(15)(16)(17)(18)(19)(20)(21)(22)

(23)






else



end ifend if

end while



Tabl

e3

Anexam

pleof

theprocedureto

build

afeasible

balancingsolutio

nfortheMALB

PS-I

Start

Step

1Step

2Step

3Step

4Step

5Step

6Ct

7

PL

[1245367]

N

s0

T 015

Ns

1

L2

AT

[1]

Select

task1

Rand

omly

select

aworker1in

station1

then

ft 1

t 1

1lt7

t 1+

bst

11119lt7

AT

[24]

Select

task2

Calculatethee

arlieststarttim

efor

allw

orkersasmin[11]

1thenrando

mly

select

aworker2in

station1

ft 2

f

t 1+

t 26lt7

t 2+

bst

22521lt7

AT

[345]

Select

task5

Calculate

theearliestfin

ishingtim

eforallw

orkers

asmin

[66]

6thenrando

mly

choo

seworker2

ft 5

f

t 2+

t 5+

fst

251114gt7

thencheck

station1

Midle

4

UB

102δ

298L

2

R0543ltEx

p(minusδT 0

081accept

theassig

nmentthenN

s2

Ns

2

L2

AT

[345]

Select

task5

Rand

omly

select

aworker1in

station2

then

ft 5

t 5

5lt7

t 5+

bst

55509lt7

AT

[346]

Select

task3


efora

llworkersas

min[50]

0thenselectw

orker2

ft 3

t 3

4t

3+

bst

33422lt7

AT

[46]

Select

task4


efora

llworkersas

min[54]

4thenselectworker2

ft 4

f

t 3+

t 4+

fst

34gt7

then

checkstation2

Midle

25

UB

102δ

148L

2

R0824ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s3

Ns

3

L2

AT

[46]

Select

task4

Rand

omlyselectaworker1

instation3

thenf

t 4

t 43lt7

t 4+

bst

44316lt7

AT

[67]

Select

task6


efora

llworkersas

min[30]

0thenselectw

orker2

ft 6

t 6

6lt7

t 6+

bst

66621lt7

AT

[7]

Select

task7


efora

llworkersas

min[36]

3thenselectw

orker1f

t 7

ft 4

+t 7

+f

st47gt7thencheck

station3

Midle

25

UB

102δ

148L

2

R0746ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s4

Ns

4

L2

AT

[7]

Select

task7

Rand

omly

select

aworker1in

station4

then

ft 7

t 7

5lt7

t 7+

bst

77517lt7

Alltasks

are

assig

ned





(10)(11)

(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

(29)(28)

(30)(31)








fstph = 0end if







end ifend if

end while




1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 5 4 6 7PL

1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 4 5 6 7PL

p gt 05 p lt = 05








RPD fi minus fmin

fmintimes 100 (32)











Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize




solution

Acceptneighborhood

solution



Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned


Select a task

Enoughcapacity



station


station

Select a worker



station

Yes

Yes

No

No

No

No

Yes

Yes








Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References



























































2 The MALBPS-I











Min1113944jϵJ

1113944kisinK

wjk +1


1113944kisinK

1113944sisinWS

j middot xmjks (1)

1113944jisinJ

1113944kisinK

1113944sisinWS


1113944iisinI


1113944iisinI


xijks le 0


(4)

i

h

Idle1 Idle2p


Worker 1

Worker 2

m

Worker 3

Task time

Idle time

(a)

Forward setup time

Backward setup time

Assigned tasks

Worker 4

Assigned tasks

Worker 5


(b)



1113944gisinJ

1113944kisinK

1113944sisinWS


minus 1113944jisinJ

1113944kisinK

1113944sisinWS



(5)


1113944sisinWS

xhjks⎛⎝ ⎞⎠


1113944sisinWS



(6)


1113944

sltn

sgt 1xpjks

⎛⎝ ⎞⎠


1113944

sltn

sgt 0xijks



(7)




(8)



(9)


xhjk(s+1) le ltijk


(10)





1113944kisinK



Table 1 Notations






1113944kisinK
























1 sim 1 000019 010011 020023 001006 016023 021025 0150162 1 5 023024 000021 004015 011014 014024 017019 0040103 2 4 003012 021022 000022 022023 011014 008014 0060084 1 3 023023 004012 017018 000016 009010 018021 0120185 2 5 016021 018020 021023 018020 000009 003016 0170186 5 6 002005 014016 021022 017020 002008 000021 0080167 4 5 003011 015020 016019 004010 008016 016019 000017

5

3

Worker 3

Worker 2

Worker 5

Worker 4



7

7

7

7

0

0

0

04 61

Worker 1


702

7

Forward setup time

Backward setup time

Idle time

Task time

(a)

Forward setup time

Backward setup time

1Worker 1


702 4

70

5

3Worker 3

Worker 2 Worker 5

Worker 4



7

7

7

7

0

0

0

06

7Worker 6

Idle time

Task time

(b)


1 2

4

3

5 6

7

1

65

4

5

5

3


1 2 3 4 5 6 7Task ID




ti

l (24)


THL (25)

THL 1113936iisinIti

Ct

1113890 1113891 (26)





LE 1113936iisinIti

Nw middot Ct

(28)

LN Ns (29)


ti1113872 11138732

Nw

(30)


LEmax minus LE0+

LNmin minus LN

LNmin minus LN0+

SImin minus SI

SImin minus SI0

(31)




(2)(3)(4)(5)(6)(7)

(8)(9)

(10)(11)(12)(13)(14)

(15)(16)(17)(18)(19)(20)(21)(22)

(23)






else



end ifend if

end while



Tabl

e3

Anexam

pleof

theprocedureto

build

afeasible

balancingsolutio

nfortheMALB

PS-I

Start

Step

1Step

2Step

3Step

4Step

5Step

6Ct

7

PL

[1245367]

N

s0

T 015

Ns

1

L2

AT

[1]

Select

task1

Rand

omly

select

aworker1in

station1

then

ft 1

t 1

1lt7

t 1+

bst

11119lt7

AT

[24]

Select

task2

Calculatethee

arlieststarttim

efor

allw

orkersasmin[11]

1thenrando

mly

select

aworker2in

station1

ft 2

f

t 1+

t 26lt7

t 2+

bst

22521lt7

AT

[345]

Select

task5

Calculate

theearliestfin

ishingtim

eforallw

orkers

asmin

[66]

6thenrando

mly

choo

seworker2

ft 5

f

t 2+

t 5+

fst

251114gt7

thencheck

station1

Midle

4

UB

102δ

298L

2

R0543ltEx

p(minusδT 0

081accept

theassig

nmentthenN

s2

Ns

2

L2

AT

[345]

Select

task5

Rand

omly

select

aworker1in

station2

then

ft 5

t 5

5lt7

t 5+

bst

55509lt7

AT

[346]

Select

task3


efora

llworkersas

min[50]

0thenselectw

orker2

ft 3

t 3

4t

3+

bst

33422lt7

AT

[46]

Select

task4


efora

llworkersas

min[54]

4thenselectworker2

ft 4

f

t 3+

t 4+

fst

34gt7

then

checkstation2

Midle

25

UB

102δ

148L

2

R0824ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s3

Ns

3

L2

AT

[46]

Select

task4

Rand

omlyselectaworker1

instation3

thenf

t 4

t 43lt7

t 4+

bst

44316lt7

AT

[67]

Select

task6


efora

llworkersas

min[30]

0thenselectw

orker2

ft 6

t 6

6lt7

t 6+

bst

66621lt7

AT

[7]

Select

task7


efora

llworkersas

min[36]

3thenselectw

orker1f

t 7

ft 4

+t 7

+f

st47gt7thencheck

station3

Midle

25

UB

102δ

148L

2

R0746ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s4

Ns

4

L2

AT

[7]

Select

task7

Rand

omly

select

aworker1in

station4

then

ft 7

t 7

5lt7

t 7+

bst

77517lt7

Alltasks

are

assig

ned





(10)(11)

(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

(29)(28)

(30)(31)








fstph = 0end if







end ifend if

end while




1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 5 4 6 7PL

1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 4 5 6 7PL

p gt 05 p lt = 05








RPD fi minus fmin

fmintimes 100 (32)











Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize




solution

Acceptneighborhood

solution



Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned


Select a task

Enoughcapacity



station


station

Select a worker



station

Yes

Yes

No

No

No

No

Yes

Yes








Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References

























































Min1113944jϵJ

1113944kisinK

wjk +1


1113944kisinK

1113944sisinWS

j middot xmjks (1)

1113944jisinJ

1113944kisinK

1113944sisinWS


1113944iisinI


1113944iisinI


xijks le 0


(4)

i

h

Idle1 Idle2p


Worker 1

Worker 2

m

Worker 3

Task time

Idle time

(a)

Forward setup time

Backward setup time

Assigned tasks

Worker 4

Assigned tasks

Worker 5


(b)



1113944gisinJ

1113944kisinK

1113944sisinWS


minus 1113944jisinJ

1113944kisinK

1113944sisinWS



(5)


1113944sisinWS

xhjks⎛⎝ ⎞⎠


1113944sisinWS



(6)


1113944

sltn

sgt 1xpjks

⎛⎝ ⎞⎠


1113944

sltn

sgt 0xijks



(7)




(8)



(9)


xhjk(s+1) le ltijk


(10)





1113944kisinK



Table 1 Notations






1113944kisinK
























1 sim 1 000019 010011 020023 001006 016023 021025 0150162 1 5 023024 000021 004015 011014 014024 017019 0040103 2 4 003012 021022 000022 022023 011014 008014 0060084 1 3 023023 004012 017018 000016 009010 018021 0120185 2 5 016021 018020 021023 018020 000009 003016 0170186 5 6 002005 014016 021022 017020 002008 000021 0080167 4 5 003011 015020 016019 004010 008016 016019 000017

5

3

Worker 3

Worker 2

Worker 5

Worker 4



7

7

7

7

0

0

0

04 61

Worker 1


702

7

Forward setup time

Backward setup time

Idle time

Task time

(a)

Forward setup time

Backward setup time

1Worker 1


702 4

70

5

3Worker 3

Worker 2 Worker 5

Worker 4



7

7

7

7

0

0

0

06

7Worker 6

Idle time

Task time

(b)


1 2

4

3

5 6

7

1

65

4

5

5

3


1 2 3 4 5 6 7Task ID




ti

l (24)


THL (25)

THL 1113936iisinIti

Ct

1113890 1113891 (26)





LE 1113936iisinIti

Nw middot Ct

(28)

LN Ns (29)


ti1113872 11138732

Nw

(30)


LEmax minus LE0+

LNmin minus LN

LNmin minus LN0+

SImin minus SI

SImin minus SI0

(31)




(2)(3)(4)(5)(6)(7)

(8)(9)

(10)(11)(12)(13)(14)

(15)(16)(17)(18)(19)(20)(21)(22)

(23)






else



end ifend if

end while



Tabl

e3

Anexam

pleof

theprocedureto

build

afeasible

balancingsolutio

nfortheMALB

PS-I

Start

Step

1Step

2Step

3Step

4Step

5Step

6Ct

7

PL

[1245367]

N

s0

T 015

Ns

1

L2

AT

[1]

Select

task1

Rand

omly

select

aworker1in

station1

then

ft 1

t 1

1lt7

t 1+

bst

11119lt7

AT

[24]

Select

task2

Calculatethee

arlieststarttim

efor

allw

orkersasmin[11]

1thenrando

mly

select

aworker2in

station1

ft 2

f

t 1+

t 26lt7

t 2+

bst

22521lt7

AT

[345]

Select

task5

Calculate

theearliestfin

ishingtim

eforallw

orkers

asmin

[66]

6thenrando

mly

choo

seworker2

ft 5

f

t 2+

t 5+

fst

251114gt7

thencheck

station1

Midle

4

UB

102δ

298L

2

R0543ltEx

p(minusδT 0

081accept

theassig

nmentthenN

s2

Ns

2

L2

AT

[345]

Select

task5

Rand

omly

select

aworker1in

station2

then

ft 5

t 5

5lt7

t 5+

bst

55509lt7

AT

[346]

Select

task3


efora

llworkersas

min[50]

0thenselectw

orker2

ft 3

t 3

4t

3+

bst

33422lt7

AT

[46]

Select

task4


efora

llworkersas

min[54]

4thenselectworker2

ft 4

f

t 3+

t 4+

fst

34gt7

then

checkstation2

Midle

25

UB

102δ

148L

2

R0824ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s3

Ns

3

L2

AT

[46]

Select

task4

Rand

omlyselectaworker1

instation3

thenf

t 4

t 43lt7

t 4+

bst

44316lt7

AT

[67]

Select

task6


efora

llworkersas

min[30]

0thenselectw

orker2

ft 6

t 6

6lt7

t 6+

bst

66621lt7

AT

[7]

Select

task7


efora

llworkersas

min[36]

3thenselectw

orker1f

t 7

ft 4

+t 7

+f

st47gt7thencheck

station3

Midle

25

UB

102δ

148L

2

R0746ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s4

Ns

4

L2

AT

[7]

Select

task7

Rand

omly

select

aworker1in

station4

then

ft 7

t 7

5lt7

t 7+

bst

77517lt7

Alltasks

are

assig

ned





(10)(11)

(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

(29)(28)

(30)(31)








fstph = 0end if







end ifend if

end while




1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 5 4 6 7PL

1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 4 5 6 7PL

p gt 05 p lt = 05








RPD fi minus fmin

fmintimes 100 (32)











Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize




solution

Acceptneighborhood

solution



Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned


Select a task

Enoughcapacity



station


station

Select a worker



station

Yes

Yes

No

No

No

No

Yes

Yes








Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References






















































1113944gisinJ

1113944kisinK

1113944sisinWS


minus 1113944jisinJ

1113944kisinK

1113944sisinWS



(5)


1113944sisinWS

xhjks⎛⎝ ⎞⎠


1113944sisinWS



(6)


1113944

sltn

sgt 1xpjks

⎛⎝ ⎞⎠


1113944

sltn

sgt 0xijks



(7)




(8)



(9)


xhjk(s+1) le ltijk


(10)





1113944kisinK



Table 1 Notations






1113944kisinK
























1 sim 1 000019 010011 020023 001006 016023 021025 0150162 1 5 023024 000021 004015 011014 014024 017019 0040103 2 4 003012 021022 000022 022023 011014 008014 0060084 1 3 023023 004012 017018 000016 009010 018021 0120185 2 5 016021 018020 021023 018020 000009 003016 0170186 5 6 002005 014016 021022 017020 002008 000021 0080167 4 5 003011 015020 016019 004010 008016 016019 000017

5

3

Worker 3

Worker 2

Worker 5

Worker 4



7

7

7

7

0

0

0

04 61

Worker 1


702

7

Forward setup time

Backward setup time

Idle time

Task time

(a)

Forward setup time

Backward setup time

1Worker 1


702 4

70

5

3Worker 3

Worker 2 Worker 5

Worker 4



7

7

7

7

0

0

0

06

7Worker 6

Idle time

Task time

(b)


1 2

4

3

5 6

7

1

65

4

5

5

3


1 2 3 4 5 6 7Task ID




ti

l (24)


THL (25)

THL 1113936iisinIti

Ct

1113890 1113891 (26)





LE 1113936iisinIti

Nw middot Ct

(28)

LN Ns (29)


ti1113872 11138732

Nw

(30)


LEmax minus LE0+

LNmin minus LN

LNmin minus LN0+

SImin minus SI

SImin minus SI0

(31)




(2)(3)(4)(5)(6)(7)

(8)(9)

(10)(11)(12)(13)(14)

(15)(16)(17)(18)(19)(20)(21)(22)

(23)






else



end ifend if

end while



Tabl

e3

Anexam

pleof

theprocedureto

build

afeasible

balancingsolutio

nfortheMALB

PS-I

Start

Step

1Step

2Step

3Step

4Step

5Step

6Ct

7

PL

[1245367]

N

s0

T 015

Ns

1

L2

AT

[1]

Select

task1

Rand

omly

select

aworker1in

station1

then

ft 1

t 1

1lt7

t 1+

bst

11119lt7

AT

[24]

Select

task2

Calculatethee

arlieststarttim

efor

allw

orkersasmin[11]

1thenrando

mly

select

aworker2in

station1

ft 2

f

t 1+

t 26lt7

t 2+

bst

22521lt7

AT

[345]

Select

task5

Calculate

theearliestfin

ishingtim

eforallw

orkers

asmin

[66]

6thenrando

mly

choo

seworker2

ft 5

f

t 2+

t 5+

fst

251114gt7

thencheck

station1

Midle

4

UB

102δ

298L

2

R0543ltEx

p(minusδT 0

081accept

theassig

nmentthenN

s2

Ns

2

L2

AT

[345]

Select

task5

Rand

omly

select

aworker1in

station2

then

ft 5

t 5

5lt7

t 5+

bst

55509lt7

AT

[346]

Select

task3


efora

llworkersas

min[50]

0thenselectw

orker2

ft 3

t 3

4t

3+

bst

33422lt7

AT

[46]

Select

task4


efora

llworkersas

min[54]

4thenselectworker2

ft 4

f

t 3+

t 4+

fst

34gt7

then

checkstation2

Midle

25

UB

102δ

148L

2

R0824ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s3

Ns

3

L2

AT

[46]

Select

task4

Rand

omlyselectaworker1

instation3

thenf

t 4

t 43lt7

t 4+

bst

44316lt7

AT

[67]

Select

task6


efora

llworkersas

min[30]

0thenselectw

orker2

ft 6

t 6

6lt7

t 6+

bst

66621lt7

AT

[7]

Select

task7


efora

llworkersas

min[36]

3thenselectw

orker1f

t 7

ft 4

+t 7

+f

st47gt7thencheck

station3

Midle

25

UB

102δ

148L

2

R0746ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s4

Ns

4

L2

AT

[7]

Select

task7

Rand

omly

select

aworker1in

station4

then

ft 7

t 7

5lt7

t 7+

bst

77517lt7

Alltasks

are

assig

ned





(10)(11)

(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

(29)(28)

(30)(31)








fstph = 0end if







end ifend if

end while




1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 5 4 6 7PL

1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 4 5 6 7PL

p gt 05 p lt = 05








RPD fi minus fmin

fmintimes 100 (32)











Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize




solution

Acceptneighborhood

solution



Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned


Select a task

Enoughcapacity



station


station

Select a worker



station

Yes

Yes

No

No

No

No

Yes

Yes








Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References

























































1113944kisinK
























1 sim 1 000019 010011 020023 001006 016023 021025 0150162 1 5 023024 000021 004015 011014 014024 017019 0040103 2 4 003012 021022 000022 022023 011014 008014 0060084 1 3 023023 004012 017018 000016 009010 018021 0120185 2 5 016021 018020 021023 018020 000009 003016 0170186 5 6 002005 014016 021022 017020 002008 000021 0080167 4 5 003011 015020 016019 004010 008016 016019 000017

5

3

Worker 3

Worker 2

Worker 5

Worker 4



7

7

7

7

0

0

0

04 61

Worker 1


702

7

Forward setup time

Backward setup time

Idle time

Task time

(a)

Forward setup time

Backward setup time

1Worker 1


702 4

70

5

3Worker 3

Worker 2 Worker 5

Worker 4



7

7

7

7

0

0

0

06

7Worker 6

Idle time

Task time

(b)


1 2

4

3

5 6

7

1

65

4

5

5

3


1 2 3 4 5 6 7Task ID




ti

l (24)


THL (25)

THL 1113936iisinIti

Ct

1113890 1113891 (26)





LE 1113936iisinIti

Nw middot Ct

(28)

LN Ns (29)


ti1113872 11138732

Nw

(30)


LEmax minus LE0+

LNmin minus LN

LNmin minus LN0+

SImin minus SI

SImin minus SI0

(31)




(2)(3)(4)(5)(6)(7)

(8)(9)

(10)(11)(12)(13)(14)

(15)(16)(17)(18)(19)(20)(21)(22)

(23)






else



end ifend if

end while



Tabl

e3

Anexam

pleof

theprocedureto

build

afeasible

balancingsolutio

nfortheMALB

PS-I

Start

Step

1Step

2Step

3Step

4Step

5Step

6Ct

7

PL

[1245367]

N

s0

T 015

Ns

1

L2

AT

[1]

Select

task1

Rand

omly

select

aworker1in

station1

then

ft 1

t 1

1lt7

t 1+

bst

11119lt7

AT

[24]

Select

task2

Calculatethee

arlieststarttim

efor

allw

orkersasmin[11]

1thenrando

mly

select

aworker2in

station1

ft 2

f

t 1+

t 26lt7

t 2+

bst

22521lt7

AT

[345]

Select

task5

Calculate

theearliestfin

ishingtim

eforallw

orkers

asmin

[66]

6thenrando

mly

choo

seworker2

ft 5

f

t 2+

t 5+

fst

251114gt7

thencheck

station1

Midle

4

UB

102δ

298L

2

R0543ltEx

p(minusδT 0

081accept

theassig

nmentthenN

s2

Ns

2

L2

AT

[345]

Select

task5

Rand

omly

select

aworker1in

station2

then

ft 5

t 5

5lt7

t 5+

bst

55509lt7

AT

[346]

Select

task3


efora

llworkersas

min[50]

0thenselectw

orker2

ft 3

t 3

4t

3+

bst

33422lt7

AT

[46]

Select

task4


efora

llworkersas

min[54]

4thenselectworker2

ft 4

f

t 3+

t 4+

fst

34gt7

then

checkstation2

Midle

25

UB

102δ

148L

2

R0824ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s3

Ns

3

L2

AT

[46]

Select

task4

Rand

omlyselectaworker1

instation3

thenf

t 4

t 43lt7

t 4+

bst

44316lt7

AT

[67]

Select

task6


efora

llworkersas

min[30]

0thenselectw

orker2

ft 6

t 6

6lt7

t 6+

bst

66621lt7

AT

[7]

Select

task7


efora

llworkersas

min[36]

3thenselectw

orker1f

t 7

ft 4

+t 7

+f

st47gt7thencheck

station3

Midle

25

UB

102δ

148L

2

R0746ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s4

Ns

4

L2

AT

[7]

Select

task7

Rand

omly

select

aworker1in

station4

then

ft 7

t 7

5lt7

t 7+

bst

77517lt7

Alltasks

are

assig

ned





(10)(11)

(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

(29)(28)

(30)(31)








fstph = 0end if







end ifend if

end while




1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 5 4 6 7PL

1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 4 5 6 7PL

p gt 05 p lt = 05








RPD fi minus fmin

fmintimes 100 (32)











Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize




solution

Acceptneighborhood

solution



Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned


Select a task

Enoughcapacity



station


station

Select a worker



station

Yes

Yes

No

No

No

No

Yes

Yes








Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References
























































1 sim 1 000019 010011 020023 001006 016023 021025 0150162 1 5 023024 000021 004015 011014 014024 017019 0040103 2 4 003012 021022 000022 022023 011014 008014 0060084 1 3 023023 004012 017018 000016 009010 018021 0120185 2 5 016021 018020 021023 018020 000009 003016 0170186 5 6 002005 014016 021022 017020 002008 000021 0080167 4 5 003011 015020 016019 004010 008016 016019 000017

5

3

Worker 3

Worker 2

Worker 5

Worker 4



7

7

7

7

0

0

0

04 61

Worker 1


702

7

Forward setup time

Backward setup time

Idle time

Task time

(a)

Forward setup time

Backward setup time

1Worker 1


702 4

70

5

3Worker 3

Worker 2 Worker 5

Worker 4



7

7

7

7

0

0

0

06

7Worker 6

Idle time

Task time

(b)


1 2

4

3

5 6

7

1

65

4

5

5

3


1 2 3 4 5 6 7Task ID




ti

l (24)


THL (25)

THL 1113936iisinIti

Ct

1113890 1113891 (26)





LE 1113936iisinIti

Nw middot Ct

(28)

LN Ns (29)


ti1113872 11138732

Nw

(30)


LEmax minus LE0+

LNmin minus LN

LNmin minus LN0+

SImin minus SI

SImin minus SI0

(31)




(2)(3)(4)(5)(6)(7)

(8)(9)

(10)(11)(12)(13)(14)

(15)(16)(17)(18)(19)(20)(21)(22)

(23)






else



end ifend if

end while



Tabl

e3

Anexam

pleof

theprocedureto

build

afeasible

balancingsolutio

nfortheMALB

PS-I

Start

Step

1Step

2Step

3Step

4Step

5Step

6Ct

7

PL

[1245367]

N

s0

T 015

Ns

1

L2

AT

[1]

Select

task1

Rand

omly

select

aworker1in

station1

then

ft 1

t 1

1lt7

t 1+

bst

11119lt7

AT

[24]

Select

task2

Calculatethee

arlieststarttim

efor

allw

orkersasmin[11]

1thenrando

mly

select

aworker2in

station1

ft 2

f

t 1+

t 26lt7

t 2+

bst

22521lt7

AT

[345]

Select

task5

Calculate

theearliestfin

ishingtim

eforallw

orkers

asmin

[66]

6thenrando

mly

choo

seworker2

ft 5

f

t 2+

t 5+

fst

251114gt7

thencheck

station1

Midle

4

UB

102δ

298L

2

R0543ltEx

p(minusδT 0

081accept

theassig

nmentthenN

s2

Ns

2

L2

AT

[345]

Select

task5

Rand

omly

select

aworker1in

station2

then

ft 5

t 5

5lt7

t 5+

bst

55509lt7

AT

[346]

Select

task3


efora

llworkersas

min[50]

0thenselectw

orker2

ft 3

t 3

4t

3+

bst

33422lt7

AT

[46]

Select

task4


efora

llworkersas

min[54]

4thenselectworker2

ft 4

f

t 3+

t 4+

fst

34gt7

then

checkstation2

Midle

25

UB

102δ

148L

2

R0824ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s3

Ns

3

L2

AT

[46]

Select

task4

Rand

omlyselectaworker1

instation3

thenf

t 4

t 43lt7

t 4+

bst

44316lt7

AT

[67]

Select

task6


efora

llworkersas

min[30]

0thenselectw

orker2

ft 6

t 6

6lt7

t 6+

bst

66621lt7

AT

[7]

Select

task7


efora

llworkersas

min[36]

3thenselectw

orker1f

t 7

ft 4

+t 7

+f

st47gt7thencheck

station3

Midle

25

UB

102δ

148L

2

R0746ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s4

Ns

4

L2

AT

[7]

Select

task7

Rand

omly

select

aworker1in

station4

then

ft 7

t 7

5lt7

t 7+

bst

77517lt7

Alltasks

are

assig

ned





(10)(11)

(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

(29)(28)

(30)(31)








fstph = 0end if







end ifend if

end while




1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 5 4 6 7PL

1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 4 5 6 7PL

p gt 05 p lt = 05








RPD fi minus fmin

fmintimes 100 (32)











Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize




solution

Acceptneighborhood

solution



Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned


Select a task

Enoughcapacity



station


station

Select a worker



station

Yes

Yes

No

No

No

No

Yes

Yes








Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References























































ti

l (24)


THL (25)

THL 1113936iisinIti

Ct

1113890 1113891 (26)





LE 1113936iisinIti

Nw middot Ct

(28)

LN Ns (29)


ti1113872 11138732

Nw

(30)


LEmax minus LE0+

LNmin minus LN

LNmin minus LN0+

SImin minus SI

SImin minus SI0

(31)




(2)(3)(4)(5)(6)(7)

(8)(9)

(10)(11)(12)(13)(14)

(15)(16)(17)(18)(19)(20)(21)(22)

(23)






else



end ifend if

end while



Tabl

e3

Anexam

pleof

theprocedureto

build

afeasible

balancingsolutio

nfortheMALB

PS-I

Start

Step

1Step

2Step

3Step

4Step

5Step

6Ct

7

PL

[1245367]

N

s0

T 015

Ns

1

L2

AT

[1]

Select

task1

Rand

omly

select

aworker1in

station1

then

ft 1

t 1

1lt7

t 1+

bst

11119lt7

AT

[24]

Select

task2

Calculatethee

arlieststarttim

efor

allw

orkersasmin[11]

1thenrando

mly

select

aworker2in

station1

ft 2

f

t 1+

t 26lt7

t 2+

bst

22521lt7

AT

[345]

Select

task5

Calculate

theearliestfin

ishingtim

eforallw

orkers

asmin

[66]

6thenrando

mly

choo

seworker2

ft 5

f

t 2+

t 5+

fst

251114gt7

thencheck

station1

Midle

4

UB

102δ

298L

2

R0543ltEx

p(minusδT 0

081accept

theassig

nmentthenN

s2

Ns

2

L2

AT

[345]

Select

task5

Rand

omly

select

aworker1in

station2

then

ft 5

t 5

5lt7

t 5+

bst

55509lt7

AT

[346]

Select

task3


efora

llworkersas

min[50]

0thenselectw

orker2

ft 3

t 3

4t

3+

bst

33422lt7

AT

[46]

Select

task4


efora

llworkersas

min[54]

4thenselectworker2

ft 4

f

t 3+

t 4+

fst

34gt7

then

checkstation2

Midle

25

UB

102δ

148L

2

R0824ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s3

Ns

3

L2

AT

[46]

Select

task4

Rand

omlyselectaworker1

instation3

thenf

t 4

t 43lt7

t 4+

bst

44316lt7

AT

[67]

Select

task6


efora

llworkersas

min[30]

0thenselectw

orker2

ft 6

t 6

6lt7

t 6+

bst

66621lt7

AT

[7]

Select

task7


efora

llworkersas

min[36]

3thenselectw

orker1f

t 7

ft 4

+t 7

+f

st47gt7thencheck

station3

Midle

25

UB

102δ

148L

2

R0746ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s4

Ns

4

L2

AT

[7]

Select

task7

Rand

omly

select

aworker1in

station4

then

ft 7

t 7

5lt7

t 7+

bst

77517lt7

Alltasks

are

assig

ned





(10)(11)

(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

(29)(28)

(30)(31)








fstph = 0end if







end ifend if

end while




1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 5 4 6 7PL

1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 4 5 6 7PL

p gt 05 p lt = 05








RPD fi minus fmin

fmintimes 100 (32)











Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize




solution

Acceptneighborhood

solution



Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned


Select a task

Enoughcapacity



station


station

Select a worker



station

Yes

Yes

No

No

No

No

Yes

Yes








Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References






















































Tabl

e3

Anexam

pleof

theprocedureto

build

afeasible

balancingsolutio

nfortheMALB

PS-I

Start

Step

1Step

2Step

3Step

4Step

5Step

6Ct

7

PL

[1245367]

N

s0

T 015

Ns

1

L2

AT

[1]

Select

task1

Rand

omly

select

aworker1in

station1

then

ft 1

t 1

1lt7

t 1+

bst

11119lt7

AT

[24]

Select

task2

Calculatethee

arlieststarttim

efor

allw

orkersasmin[11]

1thenrando

mly

select

aworker2in

station1

ft 2

f

t 1+

t 26lt7

t 2+

bst

22521lt7

AT

[345]

Select

task5

Calculate

theearliestfin

ishingtim

eforallw

orkers

asmin

[66]

6thenrando

mly

choo

seworker2

ft 5

f

t 2+

t 5+

fst

251114gt7

thencheck

station1

Midle

4

UB

102δ

298L

2

R0543ltEx

p(minusδT 0

081accept

theassig

nmentthenN

s2

Ns

2

L2

AT

[345]

Select

task5

Rand

omly

select

aworker1in

station2

then

ft 5

t 5

5lt7

t 5+

bst

55509lt7

AT

[346]

Select

task3


efora

llworkersas

min[50]

0thenselectw

orker2

ft 3

t 3

4t

3+

bst

33422lt7

AT

[46]

Select

task4


efora

llworkersas

min[54]

4thenselectworker2

ft 4

f

t 3+

t 4+

fst

34gt7

then

checkstation2

Midle

25

UB

102δ

148L

2

R0824ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s3

Ns

3

L2

AT

[46]

Select

task4

Rand

omlyselectaworker1

instation3

thenf

t 4

t 43lt7

t 4+

bst

44316lt7

AT

[67]

Select

task6


efora

llworkersas

min[30]

0thenselectw

orker2

ft 6

t 6

6lt7

t 6+

bst

66621lt7

AT

[7]

Select

task7


efora

llworkersas

min[36]

3thenselectw

orker1f

t 7

ft 4

+t 7

+f

st47gt7thencheck

station3

Midle

25

UB

102δ

148L

2

R0746ltEx

p(minusδT 0

091accept

theassig

nmentthenN

s4

Ns

4

L2

AT

[7]

Select

task7

Rand

omly

select

aworker1in

station4

then

ft 7

t 7

5lt7

t 7+

bst

77517lt7

Alltasks

are

assig

ned





(10)(11)

(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

(29)(28)

(30)(31)








fstph = 0end if







end ifend if

end while




1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 5 4 6 7PL

1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 4 5 6 7PL

p gt 05 p lt = 05








RPD fi minus fmin

fmintimes 100 (32)











Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize




solution

Acceptneighborhood

solution



Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned


Select a task

Enoughcapacity



station


station

Select a worker



station

Yes

Yes

No

No

No

No

Yes

Yes








Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References

























































(10)(11)

(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

(29)(28)

(30)(31)








fstph = 0end if







end ifend if

end while




1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 5 4 6 7PL

1 2 3 4 5 6 7Task ID

1 2 4 5 3 6 7PL

1 2 3 4 5 6 7Task ID

1 2 3 4 5 6 7PL

p gt 05 p lt = 05








RPD fi minus fmin

fmintimes 100 (32)











Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize




solution

Acceptneighborhood

solution



Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned


Select a task

Enoughcapacity



station


station

Select a worker



station

Yes

Yes

No

No

No

No

Yes

Yes








Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References



























































RPD fi minus fmin

fmintimes 100 (32)











Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize




solution

Acceptneighborhood

solution



Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned


Select a task

Enoughcapacity



station


station

Select a worker



station

Yes

Yes

No

No

No

No

Yes

Yes








Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References

























































Merten 7

7 3[5] lt1 3[6] 69 3[6] 758 3[5] 36 3[5] 36 3[5] 4610 3[3] 1 3[4] 56 3[4] 715 2[2] 1 2[3] 104 2[3] 5718 1[2] lt1 1[2] 51 1[2] 71

Bowman 8

20 4[5] lt1 4[5] 46 4[5] 3421 4[5] 3 4[5] 56 4[5] 4324 4[4] 1 4[4] 52 4[4] 4528 2[3] 1 2[4] 461 2[4] 3231 2[3] 1 2[3] 312 2[3] 233

Jaeschke 9

7 6[7] 5 6[8] 27 6[8] 448 5[6] 1 6[7] 86 6[7] 22710 4[4] 13 4[5] 1585 4[5] 115918 2[3] 1 2[3] 697 3[3] 1043

Start

Initialize




solution

Acceptneighborhood

solution



Yes

Yes

Output results

End

No

No

Problem initialize

Decode solution

All taskassigned


Select a task

Enoughcapacity



station


station

Select a worker



station

Yes

Yes

No

No

No

No

Yes

Yes








Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References

























































Jackson (11)

9 6 4[6] 4[6] 6 4[7] 4[7] 1667 1667 0 0 6 4[7] 4[7] 1667 1667 0 0 10310 5 4[5] 4[5] 5 4[6] 4[6] 2000 2000 0 0 5 4[6] 4[6] 2000 2000 0 0 15213 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 31614 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 16421 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 017

Mansoor (11)

54 4 3[4] 3[4] 4 3[4] 3[4] 000 000 0 0 4 3[4] 3[4] 000 000 0 0 05163 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 4 2[4] 2[4] 000 3333 0 0 10872 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 00281 3 2[3] 2[3] 3 2[3] 2[3] 000 000 0 0 3 2[3] 2[3] 000 000 0 0 002

Mitchell (21)


Heskia (28)


Sawyer (30)


Kilbridge (45)


Tong (70)


Arcus (83)


Arcus (111)


Table 4 Continued



Mansoor 11

54 3[4] 482 3[4] 33671 3[4] 4425463 2[3] 353 2[4]lowast lt7200 2[4] 4243672 2[3] 294 2[3] 11181 2[3] 1069581 2[3] 159 2[3] 7303 2[3] 10315

Jackson 11



5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References






















































5 Conclusion


Appendix





CT LB1113888 1113889 (33)





en calculate LBc as


CT1113890 1113891 (34)



Data Availability




Acknowledgments


References






















































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