Research Article A Multiagent Evolutionary Algorithm for ...

Research ArticleA Multiagent Evolutionary Algorithm for theResource-Constrained Project Portfolio Selection andScheduling Problem

Yongyi Shou Wenwen Xiang Ying Li and Weijian Yao

School of Management Zhejiang University Hangzhou 310058 China

Correspondence should be addressed to Yongyi Shou yshouzjueducn

Received 5 November 2013 Revised 6 February 2014 Accepted 10 February 2014 Published 22 April 2014

Academic Editor Jyh-Hong Chou

Copyright copy 2014 Yongyi Shou et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A multiagent evolutionary algorithm is proposed to solve the resource-constrained project portfolio selection and schedulingproblem The proposed algorithm has a dual level structure In the upper level a set of agents make decisions to select appropriateproject portfolios Each agent selects its project portfolio independentlyThe neighborhood competition operator and self-learningoperator are designed to improve the agentrsquos energy that is the portfolio profit In the lower level the selected projects are scheduledsimultaneously and completion times are computed to estimate the expected portfolio profit A priority rule-based heuristic isused by each agent to solve the multiproject scheduling problem A set of instances were generated systematically from the widelyused Patterson set Computational experiments confirmed that the proposed evolutionary algorithm is effective for the resource-constrained project portfolio selection and scheduling problem

1 Introduction

Theproject portfolio selection problem (PPSP) together withits various extensions has been widely studied during thelast decade Given a set of project proposals and constraintsthe traditional PPSP is to select a subset of project proposalsto optimize the organizationrsquos performance objective [1]Mathematic models have been proposed in the literatureTheproject portfolio profit is regarded as the natural performanceobjective and utilized by most models for example thezero-one integer programming model [2] Since the PPSP isan NP-hard problem [3] metaheuristic algorithms such asevolutionary approaches are widely used [4]

Most studies on the PPSP generally dissever the inherentrelationship between portfolio selection and project schedul-ing The traditional PPSP is based on some assumptionsIt is assumed that an individual project has a fixed andunchangeable schedule [5] hence only the project selectiondecision is considered to impact the final portfolio profitHowever project scheduling tends to affect the portfoliofeasibility by adjusting the start and completion time ofits activities [6] Especially when resources are constrained

scheduling of project activities helps to better utilize thelimited resources and consequently to increase the portfo-lio profit [5] Inclusion of project activity scheduling as asubproblem of project portfolio selection helps improve theoverall organization performance even though it increasesthe complexity of decision making This combined prob-lem is termed as the resource-constrained project portfolioselection and scheduling problem (RCPPSSP)The RCPPSSPcan be described as a problem to select an optimal portfolioof projects and schedule their activities to maximize anorganizationrsquos stated objectives without exceeding availableresources or violating other constraints [7]

The RCPPSSP has attracted increasing attention in recentyears as a new research problem Owing to the dual levelstructure of the RCPPSSP most algorithms in the currentliterature are also composed of two parts In the upper leveldecisions are made to select project portfolios In the lowerlevel procedures of multiproject scheduling are adopted toimprove the performance of the selected portfolio Due tothe NP-hard nature of the project portfolio selection prob-lem researchers developed heuristics and metaheuristics toimprove the solution quality and computational efficiency

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 302684 9 pageshttpdxdoiorg1011552014302684

2 Mathematical Problems in Engineering

For example an implicit enumeration procedure was devel-oped for all possible project priority sequences with highprofit [7] An ant colony optimization (ACO) based on themax-min ant system [8] was proposed in which solutionswere encoded as walks of agents in a construction graphand transition probabilities were computed to determinethe probability of an arc of the graph being chosen bythe agents in the next iteration [8] An iterative multiunitcombinatorial auction algorithm [5] was also used to selectproject portfolios through a distributed bidding mechanismIn the lower level heuristics such as greedy heuristic [8]and priority rule-based heuristics [9] are widely adopted formultiproject scheduling

In recent years agent-based computation has been widelyapplied in distributed problem solving An agent is a self-contained problem solving entity [10] which exhibits theproperties of autonomy social ability responsiveness andproactiveness [11] In a multiagent optimization system(MAOS) [12] self-organization agents [13] interact to opti-mize their own problem solving with limited declarativeknowledge and simple procedural knowledge under ecolog-ical rationality [12] Specifically agents explore in parallelthrough three types of interactions namely cooperationcoordination and negotiation [14] Since interactions amongthe agents contribute to solution diversity and rapid conver-gence in some cases [15] it is recommended to embed theMAOS in evolutionary algorithms to improve the solutionquality [12 15ndash18] Recently multiagent evolutionary algo-rithms have been used for single project scheduling [19]

The objective of this paper is to develop a multiagentevolutionary algorithm for the RCPPSSP The master proce-dure in the upper decision level is designed by combiningthe neighborhood competition operator and self-learningoperator in the multiagent system A priority rule-basedheuristic is adopted for the subprocedure in the lower deci-sion level In Section 2 we present the resource-constrainedproject portfolio selection and scheduling problem and itsmathematical model Section 3 explains the multiagent evo-lutionary algorithm that we have developed for the RCPPSPComputational experiments and results are discussed inSection 4 Finally we conclude this paper in Section 5

2 Project Portfolio Selection and Scheduling

The objective of the resource-constrained project portfolioselection and scheduling problem is to maximize the projectportfolio profitThe problem can be recognized as a dual leveldecision problem [8]

The upper level is to select feasible project portfoliosunder resource constraints There are a set Ω of119873 candidateprojects from which we select the optimal portfolio A poolof119870 types of limited and renewable resources is available forall projects It is assumed that there is no other relationshipamong the projects besides resource competition

The lower level is to solve the multiproject schedulingproblem that is to determine the start (completion) timeof each activity without violating the precedence relationsor resource constraints [5] Given the resource constraints

scheduling project activities within a portfolio helps shortenthe project duration and increase the portfolio profit sincethe project profit is a decreasing function of the projectcompletion time [7]

Two sets of decision variables are designed in this paper119909119894is for project selection and 119888

119894119895119905for project activity schedul-

ing as shown in the following formulae

119909119894= 1 if project 119894 is included0 if project 119894 is excluded

119888119894119895119905= 1 if activity 119895 of project 119894 completes at time 1199050 otherwise

(1)

The notations used in this paper are listed in Notationsfor the RCPPSSP section

TheRCPPSSP can be formulated as a 0-1 integer program-ming model [5 7]

max119873

sum

119894=1

119871119865119894119869119894

sum

119905=119864119865119894119869119894

119887119894119905119888119894119869119894119905

(2)

st119871119865119894119869119894

sum

119905=119864119865119894119869119894

119888119894119895119905= 119909119894 forall119894 119895 (3)

119871119865119894119869119894

sum

119905=119864119865119894119869119894

119905119888119894119869119894119905le119879119894

forall119894 (4)

119871119865119894119895

sum

119905=119864119865119894119895

119905119888119894119895119905le

119871119865119894ℎ

sum

119905=119864119865119894ℎ

(119905 minus 119889119894ℎ) 119888119894ℎ119905 forall (119894 119895) forall (119894 ℎ) isin 119878119894119895 (5)

sum

(119894119895)isin119860119905

119909119894119903119894119895119896le 119877119896 forall119896 119905 (6)

119909119894isin 0 1 forall119894 (7)

119888119894119895119905isin 0 1 forall (119894 119895) forall119905 (8)

The objective (2) is to maximize the total profit of theselected project portfolio Constraint (3) ensures that allactivities of a selected project are completed It also enforcesthat every activity in unselected projects is not executedConstraint (4) is to guarantee the selected project is completebefore its deadline Constraint (5) describes the precedencerelations among activities which requires an activity tostart only after all its predecessors have been completedConstraint (6) ensures that in each time period the demandon any resource does not exceed its capacity Formulae (7)and (8) declare the decision variables

3 Multiagent Evolutionary Algorithm

To solve the RCPPSSP a multiagent evolutionary algorithm(MAEA) is proposed Corresponding to the dual level struc-ture of the RCPPSSP the MAEA has two levels as well

Mathematical Problems in Engineering 3

The master procedure for the upper level is to select projectportfolios and a priority rule-based heuristic [20] is designedas the subprocedure for the lower level to do multiprojectscheduling

31 Project Portfolio Selection The multiagent evolutionaryalgorithm is a combination of two theories multiagentsystems and evolutionary algorithms [17] Generally a mul-tiagent system [15] is composed of an environment a set ofobjects a set of agents a set of relations between objects(agents) and a set of operations During observation of theenvironment and interaction with other agents the fitnessvalue of an agent can be estimated and optimized on the basisof the possessed resources abilities and knowledge [17]

In this paper a multiagent evolutionary algorithm wasproposed to solve the project selection problemWe designeda multiagent system in which each agent selects projectportfolios according to its own preferences and environmentThe evolution of the agents is realized by the neighborhoodcompetition and self-learning operators In neighborhoodcompetition loser agents will be replaced by new generatedagents In this way the information and knowledge ofindividual agents will spread to the whole systemThewinneragents will conduct self-learning by applying its own knowl-edge for which a simple genetic algorithm is developed Sincemostmultiagent systems adopt the real-valued representationwhich is not appropriate for the project selection problem wedesigned an agent system based on a discrete representationand modified the operators correspondingly

311 Multiagent System The objective function (2) of theRCPPSSP can be simplified as the following formula

max119891 (x) x = (1199091 1199092 119909

119873) isin Θ (9)

where 119891(x) denotes the portfolio profit which is equal tosum119873

119894=1sum119871119865119894119869119894

119905=119864119865119894119869119894

119887119894119905119888119894119869119894119905

andΘ is an119873-dimensional search space ofthe project selection problemThe ldquoxrdquo in boldface represents avector which is a candidate solution in the search space Thecomponent ldquo119909

119894rdquo is a 0-1 variable which takes the value of 1

when project 119894 is selected or the value of 0 otherwiseAn agent for the RCPPSSP can be defined as follows

Definition 1 An agent denoted as a represents a candidatesolution x to the RCPPSSP The value of its energy is equalto its value of the objective function in (2)

a = x = (1199091 1199092 119909

119873) isin Θ

Energy (a) = 119891 (x) (10)

The agent a living in an environment makes decisionsautonomously to increase its energy as much as possible Torealize the local perceptivity of agents the environment isorganized as a lattice-like structure which can be defined asfollows [16]

Definition 2 All agents live in a lattice-like environmentdenoted as 119871 The size of 119871 is 119871 size times 119871 size 119871 size is determinedas 119871 size = 120588 times 119873 + 1 where 0 lt 120588 lt 1

(1 1) (1 2)

(2 1) (2 2)

(1 Lsize )

(2 Lsize )

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot



(L size 1) (L size 2) (L size Lsize )

Figure 1 Illustration of an agent lattice

Figure 1 illustrates an agent lattice In the agent lattice anagent denoted as a

119906V = (1198861 1198862 119886119873) is fixed on a latticepoint (119906 V) where 1 le 119906 V le 119871 size

312 Neighborhood Competition Operator In the agent lat-tice agents compete with their neighbors to gain moreresources so that their purposes that is objectives of theRCPPSSP can be achieved Reference [16] noted that theneighborhood competition operator facilitates informationdiffusion to the whole lattice To describe the neighborhoodcompetition operator we define the neighborhood as follows

Definition 3 All agents with a line or diagonal line con-necting to agent a constitute the neighborhood of agenta The competing neighborhood of a

119906V is denoted as 119873119862119906V

The perceptive range of an agentrsquos competing neighbor-hood 119877

119862determines the number of competing neighbors as

(2119877119862+ 1)2minus 1 where 0 le 119877

119862le (12)119871 size

For example when 119877119862is equal to 1 the number of agents

in the neighborhood119873119862119906V is 8

The basic rule for neighborhood competition operator isdefined as follows [16]

Rule 1 If the agent a119906V satisfies (11) it is a loser otherwise it

is a winner

Energy (a119906V) lt Energy (alowast

119906V) (11)

The winner survives in the agent lattice but the loserperishes and is replaced by a new agent a1015840

119906V generated fromthe local-best agent alowast

119906V which is defined as

alowast119906V = (119886

lowast

1 119886lowast

2 119886

lowast

119873) isin 119873

119862

119906V

Energy (alowast119906V) ge Energy (a) foralla isin 119873119862

119906V

(12)

Two alternative strategies [16] are adopted to generate thenew agent a1015840

119906V

Strategy 1 A set 119863 is composed of sequence numbers ofthe positions where the agent a

119906V takes different values from


0 1 1 0 0 0 1 01 1 0 0 1 0 1 10 0 1 1 0 1 0 0

1 1 1 0 1 0 1 1middot middot middot




Random

119834u

119834lowast

u

119834998400

u

Figure 2 Strategy 1 to generate a new agent

agent alowast119906V that is 119863 = 119894 | 1 le 119894 le 119873 119886119894 = 119886

lowast

119894 The new agent

a1015840119906V = (119909

1015840

1 1199091015840

2 119909

1015840

119873) is determined by formula (13) in which

1 le 119894 le 119873 and Random takes the values of 0 or 1 randomly

1198861015840

i = 119886lowast

119894 119894 notin 119863 or (119894 isin 119863 and Random = 0)

1 minus 119886lowast

119894 otherwise

(13)

One example of Strategy 1 is presented in Figure 2

Strategy 2 Mutation in the evolutionary algorithm is adoptedto transform agent alowast

119906V to a1015840

119906V which is represented in formula(14) In formula (14) 1 le 119894 le 119873 and Random is generatedrandomly in the interval of (01)

1198861015840

i =

119886lowast

119894 Random gt

1

119873


119894 otherwise

(14)

One example of Strategy 2 is presented in Figure 3In this paper a uniform random parameter 119863

119878isin (0 1) is

used to determine which strategy is to be applied to generatethe new agent Firstly we calculate the similarity between theloser agent a

119906V and the local-best agent alowast119906V with maximum

energy in the neighborhood by the following formula

|119863| =

119873

sum

119894=1

120578119894 (15)

where 120578119894= 1 119886119894=119886

lowast

119894

0 119886119894 = 119886lowast

119894

The higher the value of |119863| is the moresimilar a

119906V to alowast119906V is and the much lower the chance to get a

better solution through Strategy 1 is indicatedTherefore the rule to select an appropriate strategy is

designed as follows

Rule 2 When the value of |119863| satisfies (16) Strategy 1 isadopted to generate new agents Otherwise Strategy 2 isadopted

|119863|

119873lt 119863119878 (16)

313 Self-Learning Operator In order to survive in com-petition agents in the lattice may take actions to increasetheir energy by using their own knowledge [21] The self-learning operator [16 21] is designed to help agents achievethis purpose It is assumed that only winner agents have thechance to conduct self-learning

Definition 4 All agents with a line or diagonal line connect-ing to agent a constitute the neighborhood of agent aThe self-learning neighborhood of a

119906V is denoted as 119873119871119906V

1 1 0 0 1 0 1 1010 002 020 080 008 070 065 045

1 0 0 0 1 0 1 1




Random119834

lowast

u

119834998400

u


The perceptive range of an agentrsquos self-learning neighborhood119877119871determines the number of self-learning neighbors as

(2119877119871+ 1)2minus 1 where 0 le 119877

119871le (12)119871 size

The basic rule for the self-learning operator is defined asfollows [16 21]

Rule 3 If agent a119906V satisfies (17) it has the chance to execute

the self-learning operator

Energy (a119906V) ge Energy (a) foralla isin 119873119871

119906V (17)

A simple genetic algorithm (SGA) [22] is adopted torealize the self-learning of agent a

119906V in which the chromo-some takes the same representation as an agent The ℎthchromosome in the 119892th generation is denoted as follows

chromosome119892ℎ= x = (119909

1 1199092 119909

119873) isin Θ (18)

where 0 le 119892 le MaxGA and 1 le ℎ le PopSize MaxGAdenotes the maximum number of iterations and PopSizerepresents the population size The fitness function of achromosome is equal to the energy of its correspondingagent

In the self-learning process the initial population isgenerated as follows The first chromosome in the initialpopulation is equal to agent a

119906V and all other chromosomesare generated by the following formula

chromosome0ℎ= 1199101 1199102 119910

119873 2 le ℎ le PopSize

(19)

where 119910119894is determined by

119910119894= 1 minus 119909119894 Random le 119863

119878

119909119894 Random gt 119863

119878

(20)

Three types of operators are applied to generate the newpopulation in each generation including selection crossoverand mutation Binary tournament [23] is used for selectionThe elitist strategy [24] is employed to guarantee convergenceof the genetic algorithm One-point crossover is adopted Acrossover point is randomly selected and then the genes inthe two parent chromosomes after the crossover point areexchanged to generate two offspring chromosomes [25] Themutation locus is also selected randomlyThe selected gene isthen mutated by negation The probabilities of crossover andmutation are denoted as 119901

119888and 119901

119898respectively

314 Repair Mechanism In the case of resource constraintsand multiproject scheduling it is possible that some agents(solutions) are infeasible Especially the neighborhood com-petition operator and self-learning operator may lead to


infeasible agentsTherefore repair mechanisms are necessaryand have been proposed in the relevant literature [8]

In this paper the infeasible agents are repaired by remov-ing projects in sequence under a certain rule In an RCPPSSPthe profit of a candidate project depends on its completiontime which is unknown in advance so it is inconvenientto apply other rules except the random rule [8] Given aninfeasible portfolio a randomproject is selected and removedfrom the portfolio This process continues until feasibilityis achieved Once the repaired portfolio is feasible the newagent representing the repaired feasible portfolio is used toreplace the incumbent infeasible agent

32 Multiproject Scheduling A priority rule-based heuristicis applied to schedule the selected projects in the lowerdecision level Using the minimal slack (MINSLK) priorityrule [26] and the serial schedule generation scheme (SSGS)[27] a multiproject schedule is generated for the selectedportfolio In a multiproject environment the slack (SLK) ofan activity is computed as follows

SLK119894119895= LF119894119895minus EF119894119895 (21)

where LF and EF are the latest and earliest finish timesrespectively which are estimated by the critical path method(CPM)

Suppose a subsetΦ is selected in themaster procedure Inproject 119894 isin Φ there are 119869

119894activities which are denoted as (119894 119895)

The SSGS heuristic consists of sum119894isinΦ119869119894stages In each stage 119892

two disjoint activity sets are identified [27] The scheduledset 119878119892includes the activities which are already scheduled

and the decision set 119863119892contains the unscheduled activities

whose predecessors are all in the scheduled set 119878119892 According

to the MINSLK priority rule the activity with the minimumslack is selected from the decision set and scheduled as earlyas possible without violating the resource constraints Theactivity is then moved from the decision set to the scheduledset Algorithm 1 shows the pseudocode of the priority rule-based heuristic

To determine the feasibility of the multiproject schedulewe set a upper bound for each projectrsquos completion timedenoted as 119879

119894 Specifically for project 119894 in the selected

portfolio Φ if its completion time goes beyond its upperbound 119879

119894 the portfolioΦ is recognized as infeasible and shall

be repaired

4 Computational Experiments and Results

This section presents the experiment design and computa-tional analyses for investigating the performances of the pro-posed multiagent evolutionary algorithm for the RCPPSSPBased on the design of experiment (DOE) approach a setof instances was generated systematically Then parameterconfigurations of the algorithm were set up through testingof examples The proposed MAEA was then compared withother algorithms in the literature

41 Experiment Design An RCPPSSP instance consists ofa pool of candidate projects a set of profit profiles of all

Table 1 Experiment design

Profit decreasing rate () Resource tightness ()Cell 1 2 30Cell 2 2 60Cell 3 8 30Cell 4 8 60

projects and the resources available for all projects Threeproject pools with 10 projects in each pool and three otherpools with 20 projects in each pool were generated randomlyby 72 instances with three types of resources and differentnetworks selected from the widely used Patterson set [28]The above six project pools are denoted as PAT10 1 PAT10 2PAT10 3 PAT20 1 PAT20 2 and PAT20 3 respectively

It is assumed that a project achieves its base profit if itis complete at its critical path length (CPL) and the profitdecreases with a profit decreasing rate 120582

1as its completion

time increases Referring to [7] the base profit 119861119894and actual

profit 119887119894119905of project 119894 are calculated by formulae (22) and (23)

where 120590119894denotes the resource utilization coefficient which is

subject to a uniform distribution in the interval of [05 15]The upper bound of project completion time 119879

119894is based on

the critical path length by formula (24) and the relaxationrate 120582

2has the value of 04 in this paper Resource tightness

120596119896is introduced to estimate the capacity of resource 119896 from its

maximum resource demand119877max119896

in the critical pathmethod

119861119894= 120590119894

119870

sum

119896=1

119869119894

sum

119895=1

119903119894119895119896119889119894119895 (22)

119887119894119905= 119861119894(1 minus 120582

1(119905 minus CPL

119894)) (23)

119879119894= CPL

119894+min 10 120582

2CPL119894 (24)

119877119896= 120596119896119877max119896 (25)

Two levels of profit decreasing rate 1205821(2 or 8) and

resource tightness (30 or 60) were designed which formsfour experiment cells as shown in Table 1 This 22 experi-mental design crossed with our six project pools yielded 24instances to test the proposed algorithm

42 Parameter Configurations Parameters of the proposedMAEA were determined through experiments including themaximum generations of agents MaxMA the lattice size119871 size the coefficient 119863

119878 and the perceptive ranges 119877

119862and

119877119871 The parameters of the SGA for self-learning were also

assigned including the maximum generations MaxGA thepopulationsize PopSize the crossover probability 119901

119888 and the

mutation probability 119901119898

The number of generations and the lattice size weredetermined firstly With the increasing size of the agentlattice 119871 and more generations of evolution the multiagentalgorithm is more likely to find the optimal solution ina longer computation time Through testing on examplesMaxMA was set at 100 in this paper The suitable latticesize is proportional to the number of candidate projects We


Procedure of the priority rule-based heuristicBEGIN

INIT 1198780=

FOR119892 = 1TO sum119894isinΦ

119869119894

COMPUTE119863119892

SELECT (119894lowast 119895lowast) from119863119892lowast according to the MINSLK rule lowast

ASSIGN 119878119879119894lowast119895lowast lowast as early as possible lowast

119878119892= 119878119892minus1cup (119894lowast 119895lowast)

ENDFOREND

Algorithm 1 Pseudo-code of the priority rule-based heuristic

Table 2 Profit data for problem instances

Number Cell Project pool Problem size AlgorithmRanking MKP ranking MAEA

1 1 PAT10 1 10 707677 707677 7754632 1 PAT10 2 10 634367 634367 7128613 1 PAT10 3 10 770029 770029 8109664 1 PAT20 1 20 1351207 1351207 1498735 1 PAT20 2 20 1051092 1051092 11556386 1 PAT20 3 20 116729 116729 12520377 2 PAT10 1 10 69884 69884 7289518 2 PAT10 2 10 61999 61999 6391599 2 PAT10 3 10 770029 770029 79205210 2 PAT20 1 20 128297 128297 13703711 2 PAT20 2 20 1003322 1003322 107153812 2 PAT20 3 20 1158028 1158028 119224613 3 PAT10 1 10 43168 43168 48710514 3 PAT10 2 10 385056 385056 43004915 3 PAT10 3 10 432024 432024 53818816 3 PAT20 1 20 752215 755827 92885617 3 PAT20 2 20 632393 632393 70865818 3 PAT20 3 20 79029 79029 83833419 4 PAT10 1 10 394285 394285 44414520 4 PAT10 2 10 385056 385056 38505621 4 PAT10 3 10 432024 432024 46018422 4 PAT20 1 20 746419 748553 84723623 4 PAT20 2 20 617677 617677 62734424 4 PAT20 3 20 75448 75448 794527

Average 748685 748924 812070

estimated 119871 size = 120588 times 119873 + 1 by a coefficient 120588 = 04 in thispaper Since the designed 24 instances have 10 or 20 projectsthe lattice size 119871 size takes the values of 5 or 9 respectively

According to Definitions 3 and 4 when 119873 = 10 bothperceptive ranges119877

119862and119877

119871belong to the set 1 2 Similarly

when 119873 = 20 both 119877119862and 119877

119871belong to the set 1 2 3 4

Our testing showed that the proposed MAEA has a goodperformance when 119877

119862= 119877119871= 1

The coefficient 119863119878is used to select the strategy of

generating new agents to replace loser agents When 119863119878is

larger Strategy 1 is more likely to be applied which meansthe new agent will be more similar to the best agentin its neighborhood Consequently the multiagent algo-rithm may be easily trapped in a local optimum How-ever the stability of the algorithm will be affected when119863119878

is much smaller To tradeoff between the conver-gence and stability we set 119863

119878to 025 according to our

experimentsIn summary the parameters for the proposed multi-

agent evolutionary algorithm were determined as follows


Table 3 Average computation times (in seconds)

Ranking MKP-Ranking MAEACell 1 0018 0061 18155Cell 2 0019 0064 19060Cell 3 0025 0082 262155Cell 4 0026 0085 250638Average 0022 0073 137502

Table 4 Test statistics

Algorithm RankingmdashMKP Ranking RankingmdashMAEA MKP RankingmdashMAEA119885 minus1342a minus4197a minus4197a

Asymp Sig (2-tailed) 0180 0000 0000aBased on negative ranks

MaxMA = 100 120588 = 04 119877119862= 119877119871= 1 119863

119878= 025 MaxGA =

10 PopSize = 10 119901119888= 06 and 119901

119898= 02

43 Computational Results and Comparison For the above24 instances the profits achieved by the proposed MAEAare shown in Table 2 as well as the results of two otherbenchmarking algorithms in [29] namely Ranking andMKP ranking methods

In the Ranking method all candidate projects are rankedby a certain priority rule Projects are then scheduled one byone according to their priorities until the next project makesthe portfolio infeasible It was reported that the greedy ldquomaxprofitrdquo rule performs best [29] and hence was used in thispaper

The MKP ranking method also involves two stages [29]In the first stage the project selection problem is solved asa multidimensional 0-1 knapsack problem (MKP) and thenall selected projects are prioritized by the ldquomax profitrdquo ruleIn the second stage all projects selected in the first stageare scheduled sequentially In case a selected project cannotbe scheduled before its deadline due to resource constraintsor it has a negative profit the project is removed from theportfolio

The MAEA and benchmarking algorithms were imple-mented in C language on a PC with a CPU at 20GHz and2GB physical memory The average computation times areshown in Table 3 It is obvious that the profit decreasing ratehas a significant role in determining the computation time ofthe multiagent evolutionary algorithm If the profit decreasesfaster after the projectrsquos critical path length as in Cell 3 andCell 4 the algorithm takes a much longer time to search forportfolios with projects complete in time

The average profits of 24 instances achieved by the threealgorithms are 748685 748924 and 812070 respectively Itis observed that the MAEA has a higher average profit thanthe other two methods

To investigate the performance of the proposed MAEAthe Wilcoxon Signed Ranks Test was applied to analyzethe data in Table 2 Table 4 shows the paired comparisonoutcomes of statistical analysis The profits obtained by the

MAEA are significantly higher than the other two methodsat a significance level of 0001

5 Conclusions

In this paper the resource-constrained project portfolioselection and scheduling problem is formulated as a 0-1 integer programming model The problem has a duallevel structure Project scheduling in the lower level helpsincrease the portfolio profit by improving the resourceallocation among selected projects and rescheduling theiractivities A multiagent evolutionary algorithm is proposedto solve the RCPPSSP The algorithm adopts a dual levelstructure owing to the nature of the RCPPSSP In the upperlevel agents in an agent lattice are designed to search forfeasible portfolios automatically The neighborhood com-petition operator and self-learning operator are integratedto accelerate the evolution of agents In the lower leveleach agent adopts a priority rule-based heuristic to conductmultiproject scheduling to better utilize the scarce resourcesWe conducted an experiment to test the performance of theproposed algorithm A set of 24 instances were generatedfrom the Patterson set systematically Computational resultsshow that the proposed multiagent evolutionary algorithmhas an outstanding performance

Notations

Ω Set of candidate projects119894 Project index 119894 = 1 2 119873 where119873 denotes

the number of candidate projects119895 Activity index 119895 = 1 2 119869

119894 where 119869

119894is the

number of activities in project 119894119905 Time index 119905 = 0 1 119879 where 119879 is the upper

bound of project completion time119896 Resource index 119896 = 1 2 119870 where 119870 is the

number of resource typesΦ Set of selected projects Φ = 119894 | 119909

119894= 1 119894 isin Ω

(119894 119895) Activity 119895 in project 119894119889119894119895 Duration of activity (119894 119895)


119878119894119895 Set of immediate successors of activity (119894 119895)

119878119879119894119895 Start time of activity (119894 119895)

119862119879119894119895 Completion time of activity (119894 119895) 119862119879

119894119895= 119878119879119894119895+ 119889119894119895

119864119865119894119895 Earliest finish time of activity (119894 119895)

119871119865119894119895 Latest finish time of activity (119894 119895)

119860119905 Set of activities in process at time 119905

119877119896 Capacity of renewable resource 119896

119903119894119895119896 Quantity of resource 119896 required by activity (119894 119895)

for its execution119887119894119905 Profit of project 119894 when it is complete at time 119905

119879119894 Upper bound of the completion time of project 119894

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirconstructive comments The paper is supported by NationalNatural Science Foundation of China (Grant no 71072119)andZhejiangProvincialNatural Science Foundation ofChina(Grant no R7100297)

References

[1] F Ghasemzadeh and N P Archer ldquoProject portfolio selectionthrough decision supportrdquoDecision Support Systems vol 29 no1 pp 73ndash88 2000

[2] R L Schmidt ldquoA model for R amp D project selection with com-bined benefit outcome and resource interactionsrdquo IEEE Trans-actions on Engineering Management vol 40 no 4 pp 403ndash4101993

[3] K F DoernerW J Gutjahr R F Hartl C Strauss and C Stum-mer ldquoPareto ant colony optimization with ILP preprocessing inmultiobjective project portfolio selectionrdquo European Journal ofOperational Research vol 171 no 3 pp 830ndash841 2006

[4] A LMedaglia S BGraves and J L Ringuest ldquoAmultiobjectiveevolutionary approach for linearly constrained project selectionunder uncertaintyrdquo European Journal of Operational Researchvol 179 no 3 pp 869ndash894 2007

[5] Y-Y Shou and Y-L Huang ldquoCombinatorial auction algorithmfor project portfolio selection and scheduling to maximize thenet present valuerdquo Journal of Zhejiang University C vol 11 no7 pp 562ndash574 2010

[6] MA Coffin andBW Taylor III ldquoMultiple criteria RampDprojectselection and scheduling using fuzzy logicrdquo Computers andOperations Research vol 23 no 3 pp 207ndash220 1996

[7] J Chen and R G Askin ldquoProject selection scheduling andresource allocation with time dependent returnsrdquo EuropeanJournal of Operational Research vol 193 no 1 pp 23ndash34 2009

[8] W J Gutjahr S Katzensteiner P Reiter C Stummer and MDenk ldquoCompetence-driven project portfolio selection sched-uling and staff assignmentrdquo Central European Journal of Opera-tions Research vol 16 no 3 pp 281ndash306 2008

[9] T R Browning andA A Yassine ldquoResource-constrainedmulti-project scheduling priority rule performance revisitedrdquo Inter-national Journal of Production Economics vol 126 no 2 pp212ndash228 2010

[10] NR Jennings andMWooldridge ldquoApplying agent technologyrdquoApplied Artificial Intelligence vol 9 no 4 pp 357ndash369 1995

[11] M J Wooldridge and N R Jennings ldquoIntelligent agents theoryand practicerdquo Knowledge Engineering Review vol 10 no 2 pp115ndash152 1995

[12] X-F Xie and J Liu ldquoMultiagent optimization system for solvingthe traveling salesman problem (TSP)rdquo IEEE Transactions onSystems Man and Cybernetics B vol 39 no 2 pp 489ndash5022009

[13] G Di Marzo Serugendo M-P Gleizes and A KarageorgosldquoSelforganisation and emergence in MAS an overviewrdquo Infor-matica vol 30 no 1 pp 45ndash54 2006

[14] N R Jennings K Sycara and M Wooldridge ldquoA roadmap ofagent research and developmentrdquo Autonomous Agents andMulti-Agent Systems vol 1 no 1 pp 7ndash38 1998

[15] R Drezewski and L Siwik ldquoCo-evolutionary multi-agent sys-tem for portfolio optimizationrdquo Studies in Computational Intel-ligence vol 100 pp 271ndash299 2008

[16] W Zhong J LiuM Xue and L Jiao ldquoAmultiagent genetic algo-rithm for global numerical optimizationrdquo IEEE Transactions onSystems Man and Cybernetics B vol 34 no 2 pp 1128ndash11412004

[17] R Drezewski and L Siwik ldquoA review of agent-based co-evolu-tionary algorithms for multi-objective optimizationrdquo Adapta-tion Learning and Optimization vol 7 pp 177ndash209 2010

[18] J A Arauzo J Pajares and A Lopez-Paredes ldquoSimulating thedynamic scheduling of project portfoliosrdquo SimulationModellingPractice andTheory vol 18 no 10 pp 1428ndash1441 2010

[19] X Yuan C Xiao X Lv and J Liu ldquoA multi-agent genetic algo-rithm for resource constrained project scheduling problemsrdquoin Proceeding of Genetic and Evolutionary Computation Confer-ence pp 195ndash196 2013

[20] I S Kurtulus and S C Narula ldquoMulti-project scheduling anal-ysis of project performancerdquo IIE Transactions vol 17 no 1 pp58ndash66 1985

[21] W-C Zhong J Liu F Liu and L-C Jiao ldquoCombinatorial opti-mization using multi-agent evolutionary algorithmrdquo ChineseJournal of Computers vol 27 no 10 pp 1341ndash1353 2004

[22] M Srinivas and L M Patnaik ldquoGenetic algorithms a surveyrdquoComputer vol 27 no 6 pp 17ndash26 1994

[23] J Alcaraz and C Maroto ldquoA robust genetic algorithm forresource allocation in project schedulingrdquo Annals of OperationsResearch vol 102 no 1ndash4 pp 83ndash109 2001

[24] J F Goncalves J JMMendes andMG C Resende ldquoA geneticalgorithm for the resource constrained multi-project sched-uling problemrdquo European Journal of Operational Research vol189 no 3 pp 1171ndash1190 2008

[25] Y Shou andWWang ldquoRobust optimization-based genetic algo-rithm for project scheduling with stochastic activity durationsrdquoInformation vol 15 no 10 pp 4049ndash4064 2012

[26] EM Davies ldquoAn experimental investigation of resource alloca-tion in multiactivity projectsrdquo Operational Research Quarterlyvol 24 no 4 pp 587ndash591 1973

[27] R Kolisch ldquoSerial and parallel resource-constrained projectscheduling methods revisited theory and computationrdquo Euro-pean Journal of Operational Research vol 90 no 2 pp 320ndash3331996

[28] J H Patterson ldquoComparison of exact approaches for solvingthemultiple constrained resource project scheduling problemrdquoManagement Science vol 30 no 7 pp 854ndash867 1984


[29] J Chen Project selection scheduling and resource allocationfor engineering design groups [PhD dissertation] University ofArizona Tucson Ariz USA 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


For example an implicit enumeration procedure was devel-oped for all possible project priority sequences with highprofit [7] An ant colony optimization (ACO) based on themax-min ant system [8] was proposed in which solutionswere encoded as walks of agents in a construction graphand transition probabilities were computed to determinethe probability of an arc of the graph being chosen bythe agents in the next iteration [8] An iterative multiunitcombinatorial auction algorithm [5] was also used to selectproject portfolios through a distributed bidding mechanismIn the lower level heuristics such as greedy heuristic [8]and priority rule-based heuristics [9] are widely adopted formultiproject scheduling

In recent years agent-based computation has been widelyapplied in distributed problem solving An agent is a self-contained problem solving entity [10] which exhibits theproperties of autonomy social ability responsiveness andproactiveness [11] In a multiagent optimization system(MAOS) [12] self-organization agents [13] interact to opti-mize their own problem solving with limited declarativeknowledge and simple procedural knowledge under ecolog-ical rationality [12] Specifically agents explore in parallelthrough three types of interactions namely cooperationcoordination and negotiation [14] Since interactions amongthe agents contribute to solution diversity and rapid conver-gence in some cases [15] it is recommended to embed theMAOS in evolutionary algorithms to improve the solutionquality [12 15ndash18] Recently multiagent evolutionary algo-rithms have been used for single project scheduling [19]

The objective of this paper is to develop a multiagentevolutionary algorithm for the RCPPSSP The master proce-dure in the upper decision level is designed by combiningthe neighborhood competition operator and self-learningoperator in the multiagent system A priority rule-basedheuristic is adopted for the subprocedure in the lower deci-sion level In Section 2 we present the resource-constrainedproject portfolio selection and scheduling problem and itsmathematical model Section 3 explains the multiagent evo-lutionary algorithm that we have developed for the RCPPSPComputational experiments and results are discussed inSection 4 Finally we conclude this paper in Section 5

2 Project Portfolio Selection and Scheduling

The objective of the resource-constrained project portfolioselection and scheduling problem is to maximize the projectportfolio profitThe problem can be recognized as a dual leveldecision problem [8]

The upper level is to select feasible project portfoliosunder resource constraints There are a set Ω of119873 candidateprojects from which we select the optimal portfolio A poolof119870 types of limited and renewable resources is available forall projects It is assumed that there is no other relationshipamong the projects besides resource competition

The lower level is to solve the multiproject schedulingproblem that is to determine the start (completion) timeof each activity without violating the precedence relationsor resource constraints [5] Given the resource constraints

scheduling project activities within a portfolio helps shortenthe project duration and increase the portfolio profit sincethe project profit is a decreasing function of the projectcompletion time [7]

Two sets of decision variables are designed in this paper119909119894is for project selection and 119888

119894119895119905for project activity schedul-

ing as shown in the following formulae

119909119894= 1 if project 119894 is included0 if project 119894 is excluded

119888119894119895119905= 1 if activity 119895 of project 119894 completes at time 1199050 otherwise

(1)

The notations used in this paper are listed in Notationsfor the RCPPSSP section

TheRCPPSSP can be formulated as a 0-1 integer program-ming model [5 7]

max119873

sum

119894=1

119871119865119894119869119894

sum

119905=119864119865119894119869119894

119887119894119905119888119894119869119894119905

(2)

st119871119865119894119869119894

sum

119905=119864119865119894119869119894

119888119894119895119905= 119909119894 forall119894 119895 (3)

119871119865119894119869119894

sum

119905=119864119865119894119869119894

119905119888119894119869119894119905le119879119894

forall119894 (4)

119871119865119894119895

sum

119905=119864119865119894119895

119905119888119894119895119905le

119871119865119894ℎ

sum

119905=119864119865119894ℎ

(119905 minus 119889119894ℎ) 119888119894ℎ119905 forall (119894 119895) forall (119894 ℎ) isin 119878119894119895 (5)

sum

(119894119895)isin119860119905

119909119894119903119894119895119896le 119877119896 forall119896 119905 (6)

119909119894isin 0 1 forall119894 (7)

119888119894119895119905isin 0 1 forall (119894 119895) forall119905 (8)

The objective (2) is to maximize the total profit of theselected project portfolio Constraint (3) ensures that allactivities of a selected project are completed It also enforcesthat every activity in unselected projects is not executedConstraint (4) is to guarantee the selected project is completebefore its deadline Constraint (5) describes the precedencerelations among activities which requires an activity tostart only after all its predecessors have been completedConstraint (6) ensures that in each time period the demandon any resource does not exceed its capacity Formulae (7)and (8) declare the decision variables

3 Multiagent Evolutionary Algorithm

To solve the RCPPSSP a multiagent evolutionary algorithm(MAEA) is proposed Corresponding to the dual level struc-ture of the RCPPSSP the MAEA has two levels as well






max119891 (x) x = (1199091 1199092 119909

119873) isin Θ (9)


119894=1sum119871119865119894119869119894

119905=119864119865119894119869119894

119887119894119905119888119894119869119894119905





a = x = (1199091 1199092 119909

119873) isin Θ

Energy (a) = 119891 (x) (10)



(1 1) (1 2)

(2 1) (2 2)

(1 Lsize )

(2 Lsize )











119906V is denoted as 119873119862119906V



(2119877119862+ 1)2minus 1 where 0 le 119877

119862le (12)119871 size





is a winner


119906V) (11)




alowast119906V = (119886

lowast

1 119886lowast

2 119886

lowast

119873) isin 119873

119862

119906V


119906V

(12)


119906V




0 1 1 0 0 0 1 01 1 0 0 1 0 1 10 0 1 1 0 1 0 0





Random

119834u

119834lowast

u

119834998400

u



lowast


a1015840119906V = (119909

1015840

1 1199091015840

2 119909

1015840



1198861015840

i = 119886lowast



119894 otherwise

(13)



119906V to a1015840


1198861015840

i =

119886lowast

119894 Random gt

1

119873


119894 otherwise

(14)


119878isin (0 1) is




|119863| =

119873

sum

119894=1

120578119894 (15)

where 120578119894= 1 119886119894=119886

lowast

119894

0 119886119894 = 119886lowast

119894




designed as follows


|119863|

119873lt 119863119878 (16)



119906V is denoted as 119873119871119906V

1 1 0 0 1 0 1 1010 002 020 080 008 070 065 045

1 0 0 0 1 0 1 1




Random119834

lowast

u

119834998400

u



(2119877119871+ 1)2minus 1 where 0 le 119877

119871le (12)119871 size





119906V (17)




1 1199092 119909

119873) isin Θ (18)




chromosome0ℎ= 1199101 1199102 119910


(19)


119910119894= 1 minus 119909119894 Random le 119863

119878

119909119894 Random gt 119863

119878

(20)


119888and 119901

119898respectively






SLK119894119895= LF119894119895minus EF119894119895 (21)













be repaired








1as its completion





119894is based on






119861119894= 120590119894

119870

sum

119896=1

119869119894

sum

119895=1

119903119894119895119896119889119894119895 (22)

119887119894119905= 119861119894(1 minus 120582

1(119905 minus CPL

119894)) (23)

119879119894= CPL

119894+min 10 120582

2CPL119894 (24)

119877119896= 120596119896119877max119896 (25)





119862and



119888 and the





INIT 1198780=


119869119894

COMPUTE119863119892




ENDFOREND





Average 748685 748924 812070



119862and119877


when 119873 = 20 both 119877119862and 119877



119862= 119877119871= 1














MaxMA = 100 120588 = 04 119877119862= 119877119871= 1 119863

119878= 025 MaxGA =

10 PopSize = 10 119901119888= 06 and 119901

119898= 02








5 Conclusions


Notations



119894 where 119869

119894is the




119894= 1 119894 isin Ω






119894119895= 119878119879119894119895+ 119889119894119895










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














max119891 (x) x = (1199091 1199092 119909

119873) isin Θ (9)


119894=1sum119871119865119894119869119894

119905=119864119865119894119869119894

119887119894119905119888119894119869119894119905





a = x = (1199091 1199092 119909

119873) isin Θ

Energy (a) = 119891 (x) (10)



(1 1) (1 2)

(2 1) (2 2)

(1 Lsize )

(2 Lsize )











119906V is denoted as 119873119862119906V



(2119877119862+ 1)2minus 1 where 0 le 119877

119862le (12)119871 size





is a winner


119906V) (11)




alowast119906V = (119886

lowast

1 119886lowast

2 119886

lowast

119873) isin 119873

119862

119906V


119906V

(12)


119906V




0 1 1 0 0 0 1 01 1 0 0 1 0 1 10 0 1 1 0 1 0 0





Random

119834u

119834lowast

u

119834998400

u



lowast


a1015840119906V = (119909

1015840

1 1199091015840

2 119909

1015840



1198861015840

i = 119886lowast



119894 otherwise

(13)



119906V to a1015840


1198861015840

i =

119886lowast

119894 Random gt

1

119873


119894 otherwise

(14)


119878isin (0 1) is




|119863| =

119873

sum

119894=1

120578119894 (15)

where 120578119894= 1 119886119894=119886

lowast

119894

0 119886119894 = 119886lowast

119894




designed as follows


|119863|

119873lt 119863119878 (16)



119906V is denoted as 119873119871119906V

1 1 0 0 1 0 1 1010 002 020 080 008 070 065 045

1 0 0 0 1 0 1 1




Random119834

lowast

u

119834998400

u



(2119877119871+ 1)2minus 1 where 0 le 119877

119871le (12)119871 size





119906V (17)




1 1199092 119909

119873) isin Θ (18)




chromosome0ℎ= 1199101 1199102 119910


(19)


119910119894= 1 minus 119909119894 Random le 119863

119878

119909119894 Random gt 119863

119878

(20)


119888and 119901

119898respectively






SLK119894119895= LF119894119895minus EF119894119895 (21)













be repaired








1as its completion





119894is based on






119861119894= 120590119894

119870

sum

119896=1

119869119894

sum

119895=1

119903119894119895119896119889119894119895 (22)

119887119894119905= 119861119894(1 minus 120582

1(119905 minus CPL

119894)) (23)

119879119894= CPL

119894+min 10 120582

2CPL119894 (24)

119877119896= 120596119896119877max119896 (25)





119862and



119888 and the





INIT 1198780=


119869119894

COMPUTE119863119892




ENDFOREND





Average 748685 748924 812070



119862and119877


when 119873 = 20 both 119877119862and 119877



119862= 119877119871= 1














MaxMA = 100 120588 = 04 119877119862= 119877119871= 1 119863

119878= 025 MaxGA =

10 PopSize = 10 119901119888= 06 and 119901

119898= 02








5 Conclusions


Notations



119894 where 119869

119894is the




119894= 1 119894 isin Ω






119894119895= 119878119879119894119895+ 119889119894119895










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 1 0 0 0 1 01 1 0 0 1 0 1 10 0 1 1 0 1 0 0





Random

119834u

119834lowast

u

119834998400

u



lowast


a1015840119906V = (119909

1015840

1 1199091015840

2 119909

1015840



1198861015840

i = 119886lowast



119894 otherwise

(13)



119906V to a1015840


1198861015840

i =

119886lowast

119894 Random gt

1

119873


119894 otherwise

(14)


119878isin (0 1) is




|119863| =

119873

sum

119894=1

120578119894 (15)

where 120578119894= 1 119886119894=119886

lowast

119894

0 119886119894 = 119886lowast

119894




designed as follows


|119863|

119873lt 119863119878 (16)



119906V is denoted as 119873119871119906V

1 1 0 0 1 0 1 1010 002 020 080 008 070 065 045

1 0 0 0 1 0 1 1




Random119834

lowast

u

119834998400

u



(2119877119871+ 1)2minus 1 where 0 le 119877

119871le (12)119871 size





119906V (17)




1 1199092 119909

119873) isin Θ (18)




chromosome0ℎ= 1199101 1199102 119910


(19)


119910119894= 1 minus 119909119894 Random le 119863

119878

119909119894 Random gt 119863

119878

(20)


119888and 119901

119898respectively






SLK119894119895= LF119894119895minus EF119894119895 (21)













be repaired








1as its completion





119894is based on






119861119894= 120590119894

119870

sum

119896=1

119869119894

sum

119895=1

119903119894119895119896119889119894119895 (22)

119887119894119905= 119861119894(1 minus 120582

1(119905 minus CPL

119894)) (23)

119879119894= CPL

119894+min 10 120582

2CPL119894 (24)

119877119896= 120596119896119877max119896 (25)





119862and



119888 and the





INIT 1198780=


119869119894

COMPUTE119863119892




ENDFOREND





Average 748685 748924 812070



119862and119877


when 119873 = 20 both 119877119862and 119877



119862= 119877119871= 1














MaxMA = 100 120588 = 04 119877119862= 119877119871= 1 119863

119878= 025 MaxGA =

10 PopSize = 10 119901119888= 06 and 119901

119898= 02








5 Conclusions


Notations



119894 where 119869

119894is the




119894= 1 119894 isin Ω






119894119895= 119878119879119894119895+ 119889119894119895










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













SLK119894119895= LF119894119895minus EF119894119895 (21)













be repaired








1as its completion





119894is based on






119861119894= 120590119894

119870

sum

119896=1

119869119894

sum

119895=1

119903119894119895119896119889119894119895 (22)

119887119894119905= 119861119894(1 minus 120582

1(119905 minus CPL

119894)) (23)

119879119894= CPL

119894+min 10 120582

2CPL119894 (24)

119877119896= 120596119896119877max119896 (25)





119862and



119888 and the





INIT 1198780=


119869119894

COMPUTE119863119892




ENDFOREND





Average 748685 748924 812070



119862and119877


when 119873 = 20 both 119877119862and 119877



119862= 119877119871= 1














MaxMA = 100 120588 = 04 119877119862= 119877119871= 1 119863

119878= 025 MaxGA =

10 PopSize = 10 119901119888= 06 and 119901

119898= 02








5 Conclusions


Notations



119894 where 119869

119894is the




119894= 1 119894 isin Ω






119894119895= 119878119879119894119895+ 119889119894119895










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











INIT 1198780=


119869119894

COMPUTE119863119892




ENDFOREND





Average 748685 748924 812070



119862and119877


when 119873 = 20 both 119877119862and 119877



119862= 119877119871= 1














MaxMA = 100 120588 = 04 119877119862= 119877119871= 1 119863

119878= 025 MaxGA =

10 PopSize = 10 119901119888= 06 and 119901

119898= 02








5 Conclusions


Notations



119894 where 119869

119894is the




119894= 1 119894 isin Ω






119894119895= 119878119879119894119895+ 119889119894119895










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















MaxMA = 100 120588 = 04 119877119862= 119877119871= 1 119863

119878= 025 MaxGA =

10 PopSize = 10 119901119888= 06 and 119901

119898= 02








5 Conclusions


Notations



119894 where 119869

119894is the




119894= 1 119894 isin Ω






119894119895= 119878119879119894119895+ 119889119894119895










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119894119895= 119878119879119894119895+ 119889119894119895










Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Multiagent Evolutionary Algorithm for ...

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