Cross-Trained Worker Assignment Problem in Cellular ...

Research ArticleCross-Trained Worker Assignment Problem in CellularManufacturing System Using Swarm Intelligence Metaheuristics

LangWu Fulin Cai Li Li and Xianghua Chu

College of Management Shenzhen University Shenzhen China

Correspondence should be addressed to Xianghua Chu xchuszueducn

Received 12 May 2018 Accepted 22 October 2018 Published 4 November 2018

Academic Editor Oliver Schutze

Copyright copy 2018 LangWu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Cross-trained worker assignment has become increasingly important for manufacturing efficiency and flexibility in cellularmanufacturing system because of the recent increase in labor cost Researchers mainly focused on assigning skilled workers totasks for favorable capacity or cost However few of them have recognized the need for skill level enhancement through cross-training to avoid excessive training especially for workload balance across multiple cells This study presents a new mathematicalprogramming model aimed at minimum training and maximum workload balance with economical labor utilization to addressthe worker assignment problem with a cross-training plan spanning multiple cells The model considers the trade-off betweentraining expenditure and workload balance to achieve a more flexible solution based on decision-makerrsquos preference Consideringthe computational complexity of the problem the classical swarm intelligence optimizers ie particle swarm optimization (PSO)and artificial bee colony (ABC) are implemented to search the problem landscape To improve the optimization performancea superior tracking ABC with an augmented information sharing strategy is designed to address the problem Ten benchmarkproblems are employed for numerical experiments The results indicate the efficiency and effectiveness of the proposed models aswell as the developed algorithms

1 Introduction

Following the recent emerging Industrial Revolution 40many manufacturers are engaged in finding new ways toincrease the productivity and flexibility of their manufactur-ing systems so they can cope with varied production envi-ronments such as multiproduct and small-batch productionOne of such solutions is to use cellular manufacturing sys-tem (CMS) which implements group technology to classifyfamilies of parts produced and allocate machine groups topart families CMS is a hybrid system that highlights thestrengths of job shop (flexibility in producing a wide varietyof products) and flow line (efficient flow and high productionrate) [1]

Previous studies on CMS have mainly focused on thecell formation problem which refers to the technology offorming appropriate part families and their correspondingmachine groups [2ndash4] However CMS includes not onlyparts and machines but also groups of tasks and workersOwing to the rapid increase in labor cost recently the

assignment of suitable workers to handle various tasks ineach manufacturing cell becomes an important factor in theimplementation of CMS

CMS comprises multiple manufacturing cells where eachcell includes multiple tasks in real-world manufacturingsituation Each worker is proficient in one or more skills atdifferent levels The standard task time is fixed generally Inpractice the actual task time varies due to the differencesof the skill levels of the workers Mutlu et al [5] noted thatskill types and skill levels of workers should be consideredduring worker assignment Cross-training can assist workersin obtaining more skills and enhance their skill levels Cross-trained workers are more flexible than specialized workerssince there is more opportunity to balance workload andrelieve overloaded stress among workers though there wouldbe significant additional costs to train multifunctional work-ers [6] The question is to what extent should the labor forcebe cross-trained And more precisely who should be cross-trained for which machine or task [7] This raises furtherquestions such as how to assign the cross-trained workers

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4302062 15 pageshttpsdoiorg10115520184302062

2 Mathematical Problems in Engineering

to various tasks so as to optimize some output measure[8] In previous studies the cross-training problem and theworker assignment problem were always handled separatelyHowever the associated trade-offs between the two problemshave not been answered yet In this study we aim to proposea framework to solve the worker assignment problem in CMSincorporating cross-training

A major challenge arising in this context concerns theprinciple of cross-trained worker assignment in CMS whereworker-task configuration among multiple cells is exploredbased on customer demand to seek workload balance ofassigned workers with minimum cross-training and laborcost Specifically two questions should be answered(1) Which worker should be assigned to which task in

multiple manufacturing cells(2)Which worker should be trained to reach which skill

level for the assigned task in multiple manufacturing cellsPrevious studies related to the above questions mainly

have two limitations (1) They only answered the firstquestion by assuming that the skill levels of workers arefixed while neglecting the need for skill level enhancementvia cross-training Skill level enhancement denotes that theassigned worker should be trained not only for higher effi-ciency in its existing skills but also for the acquisition of newskills And (2) they mainly paid attention to flow-line systemor a single manufacturing cell and ignored the complexinteraction among multiple cells during cross-trained workerassignment in CMS

To address this gap this paper presents a novel mathe-matical programming model that aims to minimize trainingcost workload imbalance and the number of assignedworkers based on customer demand The proposed modelnot only takes into account the problem of cross-trainedworker assignment for multiple cells but also determinesthe training requirements considering the trade-off betweentraining expenditure and workload balance in the followingscenarios(1) Training is recommended strongly for more exact

workload balance which results in more expenditure(2) Training is not recommended for less expenditure

which results in less exact workload balanceIn view of the complexity and computational burden

of the presented model we implement swarm intelligenceoptimizers for the global optimization Swarm intelligence(SI) algorithms motivated by collective behaviors in naturalsystem have achieved great successful applications in var-ious areas [9 10] As a result a number of swarm-basedmetaheuristics have been developed recently among whichparticle swarm optimization (PSO) [11] and artificial beecolony (ABC) [12] are two representative algorithms PSOwas proposed for global optimization by emulating the veloc-ity adjusting of fish schooling and bird flocking ABC wasdeveloped by simulating the information exchange throughbee dancing PSO and ABC have shown high efficiencyand effectiveness in solving real-world global optimizationproblems [9] In this study the global topology PSO (GPSO)local topology PSO (LPSO) and ABC are implementedto solve the proposed cross-trained worker assignment inCMS To improve the optimization performance a superior

tracking artificial bee colony (STABC) with an augmentedinformation sharing strategy is developed In STABC thereare two main contributions compared to the canonical ABC(1) instead of learning one dimension from neighbors inSTABC a bee can learn from others in all dimensions (2)instead of chasing the random individual in STABC eachbee either learns from its previous information or movetowards other superior bees Computational experiments andcomparisons are conducted to evaluate the efficiency of theswarm intelligence algorithms

Comparing with the previous research the contributionof this study lies in three aspects(1) A new model is presented to solve the worker assign-

ment problem formultiplemanufacturing cells incorporatingcross-training planning(2)The skill enhancement is integrated to avoid excessive

training and the trade-off between training expenditureand workload balance is combined to obtain more flexiblesolution in the model(3) Swarm intelligence optimizers are developed and

improved to solve the proposed mathematical model effi-ciently

Remainder of this paper is organized as follows Section 2reviews the literaturerelating to worker assignment Section 3presentsmodel for effective cross-trainedworker assignmentSection 4 describes the swarm intelligence metaheuristicsused and the algorithmic implementation for solving theproblem Section 5 describes experiments with computa-tional results to justify the proposed model and algorithmsSection 6 summarizes the conclusions and future researchdirections

2 Literature Review

Theapproaches regarding theworker assignment problemarebriefly reviewed in this sectionThe literature can be classifiedby considering two types ofmanufacturing systems flow-linemanufacturing system (FLMS) and cellular manufacturingsystem (CMS) The limitations in previous studies are sum-marized for the motivation of our work

21 Worker Assignment in Flow-Line Manufacturing SystemHopp et al [13] considered two cross-training strategiescherry picking and skill chain In zoned work sharing somemachinestasks on the line are shared between workers toseek capacity balance Inman et al [6] argued that cross-training should be used judiciously since it is costly and islimited by learning capacity and can confound the searchfor quality problems They presented a training strategycalled chaining in which workers are trained to perform asecond task and the assignments of task types to workersare linked in a chain Their research has shown that cross-training in chaining is a practical and effective strategy tocompensate for absenteeism on assembly lines Moreira etal [14] proposed simple heuristics for solving the assemblyline worker assignment and balancing problem Their ideawas to use task and worker priority rules to define whichworker and which set of tasks should be assigned to each

Mathematical Problems in Engineering 3

workstation by constructive heuristic framework Parvin etal [15] introduced a new canonical model of worker cross-training called a Fixed Task Zone Chain A new heuris-tic worker control policy was presented to design a zonestructure that can be balanced by assigning worker to workstation based on his or her skill set for maximum throughputMutlu et al [5] solved the assembly line worker assignmentand balancing problem when task times differ depending onoperator skills and concerns with the assignment of tasks andoperators to stations in order to minimize the cycle timeSaidi-Mehrabad et al [16] presented a novel integer linearprogramming model for dynamic manufacturing systems inthe presence of system configurations worker assignmentand production plan for each part type at each period Theobjective of worker assignment is to minimize training andsalary of worker costs Sungur and Yavuz [17] consideredqualification requirements and levels of workers to achieveassembly line balance and worker assignment problem Theysuggested that the workers should be ranked hierarchicallyaccording to their qualification requirements and levelsFrom the standpoint of the paper a higher qualified workerimplies a higher cost and lower process time

22 Worker Assignment in Cellular Manufacturing SystemThe proposed strategies of worker assignment in CMS canbe divided into two categories (1) simultaneous formation ofmanufacturing cells and worker assignment and (2) assign-ment of workers to cells after cell formation

221 Simultaneous Formation of Manufacturing Cells andWorker Assignment The strategy of forming manufacturingcells and worker assignment simultaneously was achievedfirst by Aryanezhad et al [18] The first part of the modelobjective function proposed in their study sought to mini-mize production cost intercell material handling cost andmachine costs in the planning horizon The second partinvolved human issues including hiring cost firing costtraining cost and salary into worker assignment in cellsHowever this study ignored efficiency of workersrsquo skills indifferent tasks Mahdavi et al [19] determined optimal cellconfigurations worker assignments and process plans usingan integer mathematical programming model The purposewas to minimize holding and backorder intercell materialhandling machine and reconfiguration and workers hiringfiring and salary costs Their model was solved by branch-and-bound (BampB)method using Lingo 80 softwareMahdaviet al [20] presented a fuzzy goal programming for groupingmachines parts and workers simultaneously and determin-ing production planning in dynamic virtual CMS Theirmodel aimed to minimize inventory holding and backordercosts as well as the number of exceptional elements in a cubicspace of machine-part-worker incidence matrix under con-straints of machine capacity worker capacity and customerdemand Bagheri and Bashiri [21] proposed a mathematicalmodel to simultaneously solve the cell formation operatorassignment and intercell layout problems Salary hiringfiring and training cost were considered while optimizing

worker assignment Their results indicated that consider-ation of the operator assignment problem has significantimpact on the overall system efficiency Niakan et al [22]proposed a new biobjective mathematical model to solvedynamic cell formation and skill-based worker assignmentproblem Environmental and social criteria were consideredin their research Due to the NP-hardness of the problemthey merged an efficient hybrid metaheuristic based on thenondominated sorting genetic algorithm with multiobjectivesimulated annealing Liu et al [1] built an integrated modelto solve the problems of machine grouping part schedulingworker assignment for minimizing material handling costsand the fixed and operating costs of machines and workersThey also developed a discrete bacteria foraging algorithmcombining with priority rule based parallel schedule gener-ation scheme for the intractable model Liu et al [23] builtanother integrated model to solve worker assignment andproduction planning problem again for CMS In the newmodel they assumed that all workers had the characteristicsof learning or forgetting

222 Assignment of Workers to Cells after Cell FormationMore previous studies focused on the problem of assigningworkers to tasks in cells after cell formation Askin andHuang[24] presented a mixed integer goal programming modelfor solving the worker assignment problem and coming upwith a training plan for technical and administrative skillsThemodel aimed to maximize team synergy between workerabilities and task requirements while minimizing trainingcost Greedy heuristic filtered beam search and simulatedannealing techniques were developed and tested to solve theproblem Norman et al [25] considered productivity outputquality and training costs in assigning workers to manufac-turing cells with the objective of maximizing the effectivenessof the organization Comparing with traditional workerassignments the model in their research did not include notonly technical skills but also human skills Ertay and Ruan[26] noted a data envelopment analysis to determine themost efficient number of operators and efficientmeasurementof labor assignment in CMS Fitzpatrick and Askin [27])developedmathematical models for forming effective humanteams by selecting suitable interpersonal construction andconsidering technical skill requirements from initial laborpools Meanwhile extensive cross-training policies were alsoattained to optimize team performance and a balancedplacement heuristic was proposed and evaluated Cesani andSteudel [28] simultaneously considered concepts of workloadsharing workload balancing and the presence of bottleneckoperations to classify labor strategies according to type ofmachine-operator assignments including dedicated sharedand combined assignment Simulation modeling was usedto test suitability of these concepts in an actual cell imple-mentation Suer and Tummaluri [7] developed mathematicalmodels to tackle three problems finding alternative cellconfigurations loading cells and finding crew sizes andassigning operators to operations They also proposed twoheuristic approaches for operator assignment McDonald etal [29] presented a mathematical model to solve the worker


assignment arising in a lean manufacturing cell The modelsought minimum cost of training inventory and poor qualityto determine skill training necessary for workers to meettasks and customer demand Their study also proposed theaddition of pay-for-skills reward in evaluating the costs andbenefits of cross-training However their study had onlyfocused on worker assignment for a single cell althoughmultiple cells exist in the CMS Murali et al [30] developedan artificial neural network model for assigning workforce invirtual cells under dual resource constrained context wherethe number of available workers is less than total numberof available machines Egilmez et al [31] developed a four-phased hierarchical methodology to address the stochasticskill-based manpower allocation problem In their studyworkerrsquos expected processing time and the standard deviationfor each operation were considered individually to optimizethe worker assignment to cells and maximize the productionrate of manufacturing system Table 1 presents a classificationof the previous studies

The aforementioned studies had pointed out that workerassignment plays an essential role in both flow-line andcellular manufacturing system To explore the problemof workload balance among assigned workers the cross-training strategy has been examined widely However mostof previous studies just determined which worker shouldbe trained for which task The question of an effectivestrategy of need for cross-training by prioritizing exact skilllevel of worker to avoid excessive training was ignoredThat is very few achievements specifically indicated howmany workers and in which skills they should be trained towhich level with minimum training and labor cost as wellas maximum workload balance Moreover the question oftrade-off between training expenditure andworkload balancecould lead to more flexible worker assignment and cross-training policies which did not attract significant interestin previous study Besides for the question of balancingworkload of assigned workers during cross-trained workerassignment most of previous studies paid attention to flow-line manufacturing system or a single manufacturing cellThe solution of this question formultiple manufacturing cellsshould receive more concern because it is more complex andpractical due to the interplay among cells

Considering the high complexity of the NP-hard prob-lem optimization software (LINGO CPLEX) were accompa-nied with long runtime consumption Consequently efficientmetaheuristics for the models have been attracting growingattention in recent years especially swarm intelligence meta-heuristics For example ABC was employed to address theuncovering community problem in complex networks [42]and a novel PSO algorithm was utilized for power systemoptimization [43] However few researchers used the swarmintelligence metaheuristics to solve the problem of cross-trained worker assignment for multiple cells

To fill the gaps in the literature this study aimsto build a new model to address the worker assign-ment problem among multiple manufacturing cells whileincorporating cross-training planning for workload balanceand develop solutions based on the swarm intelligencealgorithms

3 Proposed Model for Cross-TrainedWorker Assignment

This section elaborates the mathematical model for theproblem of cross-trained worker assignment in CMS Weassume that part families have already been formed andthe machines have been allocated to each part family Thetype of manufacturing cell is considered a divisional cellwhere tasks are divided among several workers in accordancewith their skill levels and task time Workers cooperate andshuttle among the workstations to complete one or severaldiscrete or successive tasks [44] Based on the feature ofcellular manufacturing system several part types with similaroperations form a part family which are processed by a groupof machines and workers in each manufacturing cell Thatmeans tasks for processing different part types in a cell aresimilar In this paper we consider skill as workerrsquos capabilityon assigned task including machine and hand operationsSome other operational assumptions and statements arestated as follows

(i) The initial skill level of each worker is given(ii)The processing time for each task for each part type is

known(iii) Each worker is assigned just to one cell(iv) Each task is assigned just to one worker(v) Machine breakdown and absenteeism are not consid-

ered(vi) All materials are delivered on time(vii) Training time is not considered(viii) The training cost of each worker for each task

depends on the task type and the skill level but not on the parttypes For instance part types 1 and 2 include task 1 which ishandled by worker 1 in a cell Training cost of worker 1 justdepends on task 1 but not on the part type

(ix) Each worker can handle all part types that arrive athis or her task stations

31 Model Notations Table 2 summarizes the index setsparameters and variables used in the proposed model

32 Mathematical Formulation The cross-trained workerassignment problem for multiple cells in CMS can be for-mulated as a nonlinear 0ndash1 integer programming model asfollows

119872119894119899 = 1205871 lowast119862

sum119888=1

119870

sum119896=1

119869

sum119895=1

119871

sum119897=1

(119908119897 minus119882119896119895) lowast 119879119862119895 lowast 119885119888119896119897119895 (1)

+1205872 lowast 120572 lowast119862

sum119888=1

119882119861119888 lowast sum119870119896=1119873119888119896sum119870119896=1119882119871119896 lowast 119873119888119896

(2)

+119862

sum119888=1

119870

sum119896=1

119873119888119896 lowast119872 (3)

subject to119870

sum119896=1

119910119888119896119895 = 1 for forall119888 119895 satisfying119875

sum119901=1

119879119888119901119895 gt 0 (4)


Table1Th

eclassificatio

nof

relatedliterature

Author

Manufacturin

gsystem

type

Num

bero

fcells

Skill

level

Cross-T

raining

Workloadbalance

Datan

ature

Solvingmetho

dHop

petal[13]

FLMS

radicradic

Stochastic

Heuris

ticPo

licies

Inman

etal[6]

FLMS

radicStochastic

Simulation

Moreira

etal[14]

FLMS

radicCertain

HGA

Mutluetal[5]

FLMS

radicCertain

IGA

Saidi-M

ehrabadetal[16]

FLMS

radicradic

Stochastic

LINGO

Sung

urandYavu

zetal[17]

FLMS

radicCertain

CPLE

XParvin

etal[15]

FLMS

radicradic

Stochastic

Heuris

ticPo

licies

Aryanezhadetal[18]

CMS

Multip

leradic

radicCertain

LINGO

Mahdavietal[19]

CMS

Multip

leCertain

LINGO

Mahdavietal[20]

CMS

Multip

leCertain

LINGO

Bagh

eriand

Bashiri

[32]

CMS

Multip

leradic

Stochastic

LINGO

Niakanetal[22]

CMS

Multip

leradic

radicStochastic

NSG

AII-

MOSA

Liuetal[1]

CMS

Multip

leStochastic

DBF

ALiuetal[23]

CMS

Multip

leradic

Certain

HBF

AAskin

andHuang

[24]

CMS

Multip

leradic

radicStochastic

Greedyheuristic

Norman

etal[25]

CMS

Sing

leradic

radicStochastic

CPLE

XErtayandRu

an[26]

CMS

Sing

leStochastic

DEA

FitzpatrickandAskin

[27]

CMS

Sing

leradic

Stochastic

Balanced

placem

enth

euris

ticCesaniand

Steudel[28]

CMS

Sing

leradic

Certain

Simulation

Suer

andTu

mmaluri[33]

CMS

Sing

leradic

Stochastic

Max

andMaxMin

heuristic

McD

onaldetal[29]

CMS

Sing

leradic

radicCertain

CPLE

XMuralietal[30]

CMS

Multip

leradic

radicCertain

ANN

Egilm

ezetal[31]

CMS

Multip

leradic

Certain

LINGO


Table 2 Model notations and definitions

Index sets Definitions119888 Index of cells (119888 = 1 2 119862)119901 Index of parts (119901 = 1 2 119875)119895 Index of tasks (119895 = 1 2 119869)119896 Index of workers (119896 = 1 2 119870)119897 Index of skill levels (119897 = 1 2 119871)Parameters Definitions119862 Number of cells119875 Number of parts119869 Number of tasks119870 Number of workers119871 Number of skill levels119879119862119895 Training cost of task 119895119879119888119901119895 Standard processing time of part type 119901 at task 119895 in cell 119888119889119888119901119895 Required throughput of part type 119901 at task 119895 in cell 119888 in one shift119882119896119895 Initial skill level weight of worker k at task 119895119908119897 Weight of skill level 119897119867 Available working time in one shift119881 Maximum permissible training cost120572 Penalty cost of workload imbalance119872 Large value1205871 1205872 Weight factorsVariables Definitions119885119888119896119897119895 1 if worker k with skill level l will be assigned to task j in cell c 0 otherwise119910119888119896119895 1 if worker k will be assigned to task j in cell c 0 otherwise119873119888119896 1 if worker k will be assigned into cell c 0 otherwise119882119871119896 Workload of worker k119882119861119888 Workload of bottleneck worker in cell c

119870

sum119896=1


sum119901=1

119879119888119901119895 = 0 (5)


sum119895=1

119910119888119896119895 ge 1 (6)


sum119895=1

119910119888119896119895 = 0 (7)

119862

sum119888=1

119873119888119896 le 1 forall119896 (8)

119871

sum119897=1

119885119888119896119897119895 = 119910119888119896119895 forall119888 119896 119895 (9)

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1

119885119888119896119897119895 lowast 119879119888119901119895 lowast 119889119888119901119895119908119897

le 119867 forall119888 119896 (10)

119882119871119896 =119862

sum119888=1

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1

119879119888119901119895 lowast 119885119888119896119897119895 lowast 119889119888119901119895119908119897 lowast 119867

forall119896 (11)

119882119861119888 = max119896isin(12119870)

(119882119871119896 lowast 119873119888119896) forall119888 (12)

119871

sum119897=1

119885119888119896119897119895 lowast 119908119897 ge 119910119888119896119895 lowast119882119896119895 forall119888 119896 119895 (13)

119862

sum119888=1

119870

sum119896=1

119869

sum119895=1

119871

sum119897=1

(119908119897 minus119882119896119895) lowast 119879119862119895 lowast 119885119888119896119897119895 le 119881 (14)

119910119888119896119895 119873119888119896 119885119888119896119897119895 = 0 119900119903 1 forall119888 119896 119897 119895 (15)

The first part of the above objective function is tominimize total skill training cost of the needed workers Themodel set skill level varies according to standard task timeA higher skill level refers to the worker needing less time tofinish the assigned task The cost increases when the workeris trained from a lower skill level to a higher one

The second part of the objective function is to minimizethe penalty cost of workload imbalance ie to maximizeworkload balance of assigned workers in each cell Thepenalty cost forces workload balance by reasonable workerassignment and cross-training 1205871 and 1205872 display the trade-off between training expenditure and workload balance Arelatively lower value of 1205871 and a higher value of 1205872 standfor the situation that training is recommended strongly formore exact workload balance but is accompanied with moreexpenditure Conversely a relatively higher value of 1205871 and a


lower value of 1205872 stand for the situation that training is notrecommended for less expenditure but is accompanied withless exact workload balance The decision-maker could setthe value of 1205871 and 1205872 according to the recommendation oftraining for skill level enhancement

The third part focuses on minimizing the number ofassigned workers Considering the current economic envi-ronment where labor cost is rapidly increasing this researchconsiders theminimumnumber of workers as the top priorityin the model

Constraints (4) and (5) ensure that all tasks should beallocated to the workers Constraints (6) and (7) guaranteethat one worker can handle more than one task Constraint(8) ensures that one worker could not be assigned to morethan one cell Constraint (9) guarantees that worker assignedto a task with one and only skill level Constraint (10) ensuresthat the customer demand is met Constraint (11) calculatesworkload of each assigned worker Constraint (12) definesworkload of the bottleneck worker in cell Constraint (13)ensures that the skill level of the assigned worker is not lessthan its initial level Constraint (14) ensures that the totalwork training cost does not exceed budget

4 Swarm Intelligence Metaheuristics forWorker Assignment

In this section two classical PSO variants ie GPSO andLPSO and ABC are introduced briefly Next we presentthe STABC with enhanced learning strategy to improvecomputational performanceThis is followed by the designedimplementations of the algorithms for the mathematicalmodel

41 Particle Swarm Optimization Particle swarm opti-mization (PSO) is a stochastic swarm-based metaheuristicinspired by the group behaviors of bird flocking and fishingschooling [11] In PSO a group of particles are employed tocollaboratively explore the problem landscape Each particlehas velocity and position information and moves towardsits historical best position in conjunction with populationbest position found so far Supposing that searching space isD-dimensional the m-th particlersquos velocity and position areupdated as follows

119901V119889119898 = 119901V119889119898 + 1199031 lowast 1199031198861198991198891198891119898 lowast (119901119887119890119904119905119889119898 minus 119883119889119898) + 1199032lowast 1199031198861198991198891198892119898 lowast (119892119887119890119904119905119889119898 minus 119883119889119898)

(16)

119883119889119898 = 119883119889119898 + 119901V119889119898 (17)

where 1199031 and 1199032 are two positive constriction coefficients1199031198861198991198891198891 and 1199031198861198991198891198892 are two random numbers between [0 1]119883119898 = (1198831119898 1198832119898 119883119863119898) is the position of particle m 119901V119898 =(119901V1119898 119901V2119898 119901V119863119898) is the velocity of particle m 119901119887119890119904119905119898 =(1199011198871198901199041199051119898 1199011198871198901199041199052119898 119901119887119890119904119905119863119898) is the best previous position forparticle m and 119892119887119890119904119905119898 = (1198921198871198901199041199051119898 1198921198871198901199041199052119898 119892119887119890119904119905119863119898) is thebest position discovered by the population so far

Global topology Local topology

Figure 1 Global and local topology structure

Many variants of PSO have been studied since theinception [45 46] A standard version is presented in [32]The standard PSO employs constricted factor 120594 as shown inthe following to guide population search

120594 = 210038161003816100381610038161003816100381610038162 minus 120593 minus radic1205932 minus 4120593

1003816100381610038161003816100381610038161003816 120593 = 1199031 + 1199032 (18)

where 1199031 and 1199032 are set to be constants to ensure the speed ofconvergence [45] With the constriction factor the originalvelocity equation is given as follows

119901V119889119898 = 120594 (119901V119889119898 + 1199031 lowast 1199031198861198991198891198891119898 lowast (119901119887119890119904119905119889119898 minus 119883119889119898) + 1199032lowast 1199031198861198991198891198892119898 lowast (119871119864119889119898 minus 119883119889119898))

(19)

where LEd represents the previous best information in pop-ulation It is GPSO when a neighbor could be anyone withinthe population otherwise it is LPSO The structures of thetwo patterns are shown in Figure 1The general procedures ofPSO are given as follows

(1) Initialize population size of particle swarm maxi-mum number of iterations and the constants (r1 andr2)

(2) Randomly generate a swarm of particles and initializethe velocity and position of each particle in the searchboundary

(3) Evaluate each particlersquos fitness value(4) Renew particlersquos personal best by comparing current

best information with its historical best information(5) Update the swarm best information using the best

neighborsrsquo information in population(6) Each particle updates its velocity by (19)(7) Each particle updates its position by (17)(8) Go to (3) if the stop criteria are not met otherwise

terminate the iteration

42 Artificial Bee Colony Algorithm Artificial bee colony(ABC) algorithm is inspired by the foraging behaviors in ahoney bee swarm for real-parameter optimization [12] InABC there are three types of bees employed bees onlooker


bees and scout bees The goal of the whole bees is tocollaboratively find the largest source of nectar The actuallocation of the food source represents a possible solution forthe optimization problems in ABC and the density of nectarcorresponds to a fitness value of solutionThenumber of foodsources represents the swarm size PS

In the iteration each employed bee explores the foodsource using the following

119887V119889119898 = 119909119889119898 + 120601119889119898 (119909119889119898 minus 119909119889119902) 119898 = 119902 (20)

where 119887V119889119898 denotes a candidate food source positionmm= 12 PS 120601119889119898 is a randomly produced value within [-1 1] 119909119889119898refers to the corresponding food source and 119909119889119902 is a randomlyselected food source

Each onlooker bee randomly selects its food sourcem for further refinement with a probability 119901119898 which isdetermined as follows

119901119898 =119891119894119905119899119890119904119904119898

sum119878119904=1 119891119894119905119899119890119904119904119904(21)

where119891119894119905119899119890119904119904119898 is the quality of them-th food source For theminimization problem the 119891119894119905119899119890119904119904119898 can be calculated usingthe following

119891119894119905119899119890119904119904119898 =

1(1 + 119891119898)

if 119891119898 ge 01 + 119886119887119904 (119891119898) if 119891119898 le 0

(22)

where 119891119898 is the m-th objective function value The value ofldquolimitrdquo to abandon the poor food source is suggested to be setto PSlowastD [47] The main steps of ABC are given below

(1) Initialize the number of food sources the limit andthe maximum number of iterations

(2) Randomly generate the positions of food sources inproblem landscape

(3) Evaluate the fitness value of food sources foundcurrently

(4) Each employed bee explores between a randomexem-plar and their own food source using (20)

(5) Apply greedy selection process to select the best PSpositions and record the updated food sources

(6) Calculate the probability of the food sources withwhich onlooker bees are going to follow by (21)

(7) Produce new solutionusing (20) for the onlooker beesdepending on 119901119894

(8) Conduct greedy selection and update food sourceinformation for onlooker bees

(9) Abandon the food sources with poor nectar and letscouts search for new food sources

(10) Terminate if the stop criteria are met otherwise go to(4)

43 Superior Tracking Artificial Bee Colony Algorithm Asa newly proposed algorithm while promising ABC stillsuffers from two weaknesses First the convergence speed isrelatively slow compared tomany existing swarm intelligencealgorithms Second the capability of exploring global optimais still underperforming on many multimodal problems [48]In the original search behavior of ABC only one dimensionof a bee is selected to learn and renew in each round ofinformation updating This search rule may result in the slowconvergence or inferior exploring capability [49] To addressthis issue the STABC is introduced to improve the searchcapability and convergence speed for global optimization

In STABC the employed bees and onlooker bees areupdated in accordance with the following equation

119887V119898 = 119909119898 + 119903119898 (119909119898 minus 119878119873119898) (23)

where 119887V119898 represents the food position of the m-th (m=1 2 PS) food source to be updated 119903119898 refers to arandomly generated D-dimensional vector and SN is the D-dimensional constructed exemplar for tracking in which theelements are either from itself or from the other superiorfood sources randomly chosen by roulette selection andtournament selection

The detailed procedure of constructing SNm is presentedas follows

(1) Initialize a given probability Pr to decide whichexemplar the current bee should chase itself or theother superior neighbors in the population

(2) For each dimension of 119878119873119889119898 (d = 1 toD) a valuewithin[0 1] is randomly generated(3) If the generated value is larger than Pr two food

positions are randomly chosen for comparison andthe d-th dimension of the better one will be learnedotherwise 119878119873119889119898 will be itself

(4) Repeat the above steps till all dimensions of SNm aredetermined

In STABC all elements of an individual will be likely toupdate at each iteration As a result each bee either exploitsthe local optima by learning from itself or explores the globallandscape by chasing the other individuals in the populationThis search strategy accelerates the convergence speeds ofthe population In addition it remains from ABC originalproperties that every individual could be learning exemplarsFigure 2 displays the flow chart of the STABC algorithm

44 Implementation of the SI-Based Optimizers for Cell For-mation In the proposed mathematical model the computa-tional complexity is dependent on the variable size of 119885119888119896119897119895which is equal to 119888 lowast 119896 lowast 119897 lowast 119895 The search dimension ofsolutionZcklj will be too large to optimize using the traditionalmathematical methods Our preliminary experiment indi-cates that it is extremely hard to obtain a suboptimum if thevariables are without proper treatment Meanwhile a feasiblesolution Z would be hard to locate due to the complexityand discontinuity of constraints To address these issues a SI-based metaheuristic is proposed to explore the global optima


Begin

Initialize food source positions Setting criteria and itc = 1

Fitness value calculation

Evaluate new solution

Greedy SelectionCalculate the probability

of the food source

Determine the chosenfood source by the

onlooker bees

Output

N

Y

itc gt criteria

itc = itc+1

Greedy SelectionRandomly generate a new

position for the abandonedfood sources


itc iteration counter

Trial successfully update record of each food source

Pr a threshold

randi an interger from a uniform discrete distribution

FS the position of the food sources

FV fitness value

Generate SNi i = xi + ri (xi - SNi)

xnewi = xi + i


xnewi = xi + i

Figure 2 Flowchart of the STABC algorithm

While implementing the swarm-based optimizers for cross-trained worker assignment problem three important issuesshould be considered encoding decoding and constrainshandling

An effective encoding method is developed in this studyThe original 0-1 planning problem is transformed into aninteger programming problem in which the targeting skilllevel and theworker are allocated to each task tomeet the pro-duction requirements For the population initialization them-th individual is represented as 119909119898 = [1199091198981 1199091198982 119909119898119895]j = 1 2 J where J denotes the value of tasks The range ofthe algorithmic representation 1199091198981 is [1 k+l]

In the decoding process the worker and skill levelindices need to be determined through the algorithmicrepresentation A cross-reference matrix is generated here forconverting individual 119909119898 to the indices of worker and skilllevel Details of procedures involved in the decoding processfor 119909119898 are shown in Figure 3

For the constraints handling process the basic penaltyfunctionmethod is adopted [50] Once the objective functionvalue of 119909119898 and the corresponding constraint violation havebeen obtained the penalty value is added to the objectivefunction value of 119909119898

Begin

Y

1 2 hellip K1 1 2 hellip K2 hellip hellip hellip helliphellip hellip hellip hellip hellipL hellip hellip hellip K+L N

Y

Y

N

c = 1 j = 1empty Zcklj

p = 1P

Cross-reference

Inputxmj

Outputindexvalue

kl

Z(cklj) = 1j = j+1

j gt J

c = c+1j =1

c gt C

Output Zcklj

any(Tcpj)gt0

Figure 3 Decoding procedure

5 Computational Experiment

To validate effectiveness and efficiency of the proposedmathematical model and optimizers computational ex-periments are performed in this section Ten problemsare randomly generated based on previous research(can be accessed at httpsdrivegooglecomopenid=0B18FVuMxAffXLW1mRzVLczlyem8) The experiments areimplemented using MATLAB R2012a and the desktop iswith Intel Core i5 CPU 320GHz and 8GB memory

51 Experiment Settings The cross-trained worker assign-ment problem is based on the formed manufacturing cellsin this research that is part and tasks types had beenfixed after cell formation in CMS Ten cell formation prob-lems are selected from previous literature (see Table 3) Asthe problems are generated with different sizes they arenot directly correlated to each other The different typesof problem size can comprehensively evaluate the scalableperformance of the proposed method on this mathematical


Table 3 Dimension of test problems

Problem No Number ofParts Tasks Cells Workers

P1 [34] 5 5 2 5P2 [35] 7 5 2 5P3 [36] 6 6 2 6P4 [37] 8 9 2 9P5 [38] 10 8 3 8P6 [36] 8 10 3 10P7 [39] 10 10 3 10P8 [35] 12 10 3 10P9 [40] 18 10 3 10P10 [41] 22 11 3 11

Table 4 Pattern of data generation

Parameters Values119879119862119895 lowastDU(500 1000)119879119888119901119895 DU(01 15)119889119888119901119895 DU(50 150)119882119896119895 Randomly generated in set (0 07 08 09 1 11)119908119897 w1=07 w2=08 w3=09 w4=10 w5=11lowastDU(m n) means discrete uniform distribution varying between m and nThe unit of time is minute

model The additional parameters are randomly generatedas shown in Table 4 To allow one-to-one correspondencebetween workers and tasks in extreme situation we assumedthat the number of available workers is equal to the numberof tasks in CMS The available working time of each workerin a shift is calculated as 8 hlowast60mlowast85=408m The trainingbudget in the factory is sufficient Penalty cost of workloadimbalance 120572 is assumed to be 100 With respect to the trade-off between training expenditure and workload balanceldquo1205871=1 1205872=10rdquo stands for the first situation which refers tothe pursuit of more exact workload balance although moretraining cost should be spent while ldquo1205871=10 1205872=1rdquo standsfor the second situation which refers to saving training costalthough workload balance may not be enforced very well

Themaximum number of iterations is set as 20000 whereall the candidate algorithms are found to converge fast Forthe standard PSO the constant120593=41 and the values120594=07298and r1=r2=205 are gathered (Bratton and Kennedy 2007)For the STABC the range of 119903119898 is empirically set as [0149] which could be a time-varying function as well [49]With respect to the threshold probability Pr of STABC themethod introduced in Liang et alrsquos research [51] is adoptedThe population size is set at 30 for all algorithms [47]

52 Experimental Results Each problem is run indepen-dently 20 times with the four algorithms (GPSO LPSO ABCand STABC) The CPU time for each run does not exceed90s In Table 5 119874119865119881119887 119874119865119881119908 and 119874119865119881119886 represent the bestworst and average objective function values (OFV) of theten problems after 20 runs under the two situations (1205871=1

0

25000

50000

75000

100000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

OFVb in the situation of 1=1 2=10

GPSOLPSO

ABCSTABC

Figure 4 119874119865119881119887 results for the case of 1205871=1 1205872=10

1205872=10) and (1205871=10 1205872=1)The best solutions of119874119865119881119887119874119865119881119908and 119874119865119881119886 found by the four algorithms are shown in boldBased on experimental results three measures to evaluate theeffectiveness of STABC algorithm are defined as follows(1) 119861119866lowast119878119879119860119861119862 denotes the gap of 119874119865119881119887 between STABC

and any other algorithms (GPSO LPSO or ABC) Forinstance 119861119866119866119875119878119874119878119879119860119861119862 means the rising percentage of 119874119865119881119887byGPSO over 119874119865119881119887 by STABC(2) 119882119866lowast119878119879119860119861119862 denotes the gap of 119874119865119881119908between STABC

and any other algorithms (GPSO LPSO or ABC) Forinstance119882119866119871119875119878119874119878119879119860119861119862 means the rising percentage of 119874119865119881119908 byLPSO over 119874119865119881119908 by STABC(3) 119860119866lowast119878119879119860119861119862 denotes the gap of 119874119865119881119886 between STABC

and any other algorithms (GPSO LPSO or ABC) Forinstance 119860119866119860119861119862119878119879119860119861119862 means the rising percentage of 119874119865119881119886 byABC over 119874119865119881119886 by STABC

With respect to the abovemeasures the presented STABCis employed as the basis to compare the performance of theinvolved algorithms A positive value indicates that STABCoutperforms the other algorithms on the optimized problem

Table 6 shows the comparison results on the ten problemsunder the two situations (1205871=1 1205872=10) and (1205871=10 1205872=1) Wesee thatmost of119861119866lowast119878119879119860119861119862119882119866lowast119878119879119860119861119862119860119866lowast119878119879119860119861119862 and all averagevalues are positive Thus better results with respect to119874119865119881119887119874119865119881119908 and 119874119865119881119886 can be attained by STABC compared toGPSO LPSO and ABC under both situations (1205871=1 1205872=10)and (1205871=10 1205872=1)

Figures 4 and 5 compare the 119874119865119881119887obtained by GPSOLPSO ABC and STABC (see Table 5)They demonstrate thatthe best solutions of STABC are better than the others Forthe relatively simple small-size problems such as problems1 2 and 3 most of the algorithms can easily identify thebest results With the increase of problem size STABCoutperforms the other algorithms on the relatively large-sizeproblems The improvement of STBACrsquos search capabilitycan be attributed to two properties (1) each individualrsquoslearning speed for exploration is improved by learning fromother bees in all dimensions and (2) the individualrsquos learningsources for exploitation are enriched by learning fromboth itshistorical information and the superior individualsThenovelstrategies can significantly enhance the global exploration


Table5Re

sults

from

GPS

OL

PSOA

BCand

STABC

GPS

OLP

SOABC

STABC

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

P11205871=1120587

2=10

33183

3330

733

212

33183

33344

33285

32852

42983

35125

3283

943122

34597

1205871=101205872

=141163

4195

841

590

41258

42753

42152

38955

48235

44105

3810

548837

42540

P21205871=1120587

2=10

3260

732

667

32638

32596

32714

3264

832259

32959

3237

032

259

33204

32499

1205871=101205872

=135260

35865

35293

35216

3586

035330

32602

40752

35266

3260

240

907

3339

2

P31205871=1120587

2=10

33497

33782

3360

733453

33769

33619

32772

33116

32885

3277

232

864

3280

21205871=101205872

=142745

44415

43450

42745

44344

43563

36861

41403

38321

3681

139

207

3742

4

P41205871=1120587

2=10

53681

5700

955124

53277

5604

554

768

63133

76480

67822

5249

364

168

60957

1205871=101205872

=166237

80183

75246

65879

81657

7606

455113

91142

79567

5251

577

852

6748

2

P51205871=1120587

2=10

54720

5564

455089

54771

55589

55163

53669

55151

54324

5360

754

088

5387

91205871=101205872

=159274

67264

63857

59871

69167

64942

52293

63298

57622

5153

657

031

5382

6

P61205871=1120587

2=10

55149

66403

60150

55108

66335

56981

4621

963596

54613

53296

5439

253

768

1205871=101205872

=170131

84437

79483

72363

86381

80558

55765

64943

61086

5166

461

076

5572

6

P71205871=1120587

2=10

65351

68255

66758

65730

69136

67249

64034

74700

65529

6346

464

682

6402

01205871=101205872

=182555

99389

92564

87731

103806

96119

66279

87778

76697

6302

674

477

6879

8

P81205871=1120587

2=10

75074

78381

7604

175630

7812

07644

673638

85692

80359

7363

783820

78977

1205871=101205872

=184676

97413

91203

85879

102599

93956

72901

90037

81322

7255

781

016

7790

5

P91205871=1120587

2=10

75750

78238

77138

76227

78917

77378

74097

85694

7666

773

342

7496

074

237

1205871=101205872

=197288

117201

105674

95973

120132

108403

81629

105846

89569

7117

692

650

7930

4

P10

1205871=1120587

2=10

86208

89887

88252

85905

98570

89542

85065

9544

887622

8388

186

995

8482

11205871=101205872

=194291

132599

121452

11040

4139528

123986

87300

121262

103559

8520

010

1088

9167

5


Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1

Figure 6 Training costs obtained by STABC

capability and local exploitation capability especially in large-size problems which are proved by the experimental results

The CPU time of all test problems by the four algorithmsfluctuates between 10s and 90s Although the advantage ofcomputational efficiency of STABC is not significant amongthe four algorithms it is efficient to obtain solution of theproblem In contrast the computational software Lingo isunable to obtain any feasible results in half an hour for mostof the experiments

In addition we compare the training cost and workloadbalance associated with the ten problems between the twoscenarios (1205871=11205872=10) and (1205871=101205872=1) see Figures 6 and 7All the training cost andworkload balance in scenario (1205871=101205872=1) is equal to or lower than that of scenario (1205871=1 1205872=10)This indicates that less training effort which is in the scenario(1205871=10 1205872=1) will lead to less balanced workload in allthe test problems though corresponding worker assignmentis reasonable It indicates that the cross-training strategyobtained by our proposed approach is effective while seekingworkload balance

6 Conclusion and Future Research

A novel mathematical programming model for solving theworker assignment problem in CMS while incorporatingcross-training is presented in this paper Then the swarmintelligence optimizers are introduced to obtain promisingsolution

800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1

Figure 7 Workload balance results obtained by STABC

The proposed model defines the objective function asminimum training cost optimizedworkload balance and theminimum number of assigned workers based on consideringcustomer demand available time task time limited trainingcost and initial skill level of worker This model not onlyassigns suitable workers with the desired skill levels to thevarious tasks in multiple cells but also determines necessityof skill enhancement by quantifying skill level needed toavoid excessive training As for the trade-off between trainingexpenditure and workload balance of workers decision-maker has more flexible selections on solution based onthe preference to cross-training or workload balance Theexperimental results demonstrate that the proposed model iseffective and feasible to produce interaction among cells forsolving worker assignment problem with reasonable cross-training policy

In addition the efficient swarm intelligence metaheuris-tics including GPSO LPSO and ABC are employed to solvethe CMS model To avoid premature convergence and lowexploitation STABCwith the enhanced information learningstrategy is proposed for the complex cross-trained workerassignment problem In this strategy an all-dimension-wiselearning mechanism is proposed to enable each individualto both exploit itself and track the promising individuals Tocompare the performances of the four algorithms we executeten computational experiments with respect to the severaldefinedmeasuresThe results reveal that STABC significantlyoutperforms the comparison algorithms in terms of thebest worst and average solutions of repeated experimentsWith increase in problem scale the performance differencesbetween STABC and PSO are more remarkable In additionthe convergence capability of the STABC is observed moreefficiently than the others over the experiments

In future research we will consider more useful andpractical issues to allocate cross-trained workers into tasks inCMS such as quality level of worker learning and forgettingcapability and work-sharing mechanism Additionally weintend to extend this problem from divisional cell to rotatingcell which is the other primary type of manufacturing cell

Data Availability

The original experiment data used to support the findingsof this study have been deposited in the repository ldquoTheNational Science Digital Libraryrdquo ([httpsnsdloercommons


orgcoursesexperiment-data-cross-trained-worker-assignment-problem-in-cellular-manufacturing-system-using-swarm-intelli-gence-meta-heuristicview]) The data are also enclosed asthe supplementary materials when we submitted themanuscript

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (Grant nos 71501132 71701079and 71371127) the Natural Science Foundation of GuangdongProvince (2016A030310067 and 2018A030310567) the ChinaPostdoctoral Science Foundation (2016M602528) and the2016 Tencent ldquoRhinoceros Birdsrdquo Scientific Research Foun-dation for Young Teachers of Shenzhen University

Supplementary Materials

The supplementary materials are the original data of 10experimental cases used to support the findings of this studywhich have been deposited in the repository ldquoThe NationalScienceDigital Libraryrdquo [httpsnsdloercommonsorgcours-esexperiment-data-cross-trained-worker-assignment-problem-in-cellular-manufacturing-system-using-swarm-intelligence-meta-heuristicview] Some types of the experiment datahave been employed from previous researches directlythe others have been randomly generated based onprevious research that had been explained in Section 5(Supplementary Materials)

References

[1] C Liu J Wang J Y-T Leung and K Li ldquoSolving cellformation and task scheduling in cellularmanufacturing systemby discrete bacteria foraging algorithmrdquo International Journal ofProduction Research vol 54 no 3 pp 923ndash944 2016

[2] G Papaioannou and J M Wilson ldquoThe evolution of cellformation problem methodologies based on recent studies(1997ndash2008) review and directions for future researchrdquo Euro-pean Journal of Operational Research vol 206 no 3 pp 509ndash521 2010

[3] H M Selim R G Askin and A J Vakharia ldquoCell formation ingroup technology review evaluation and directions for futureresearchrdquoComputers amp Industrial Engineering vol 34 no 1 pp3ndash20 1998

[4] Y Yin and K Yasuda ldquoSimilarity coefficientmethods applied tothe cell formation problem a taxonomy and reviewrdquo Interna-tional Journal of Production Economics vol 101 no 2 pp 329ndash352 2006

[5] O Mutlu O Polat and A A Supciller ldquoAn iterative geneticalgorithm for the assembly line worker assignment and balanc-ing problem of type-IIrdquo Computers amp Operations Research vol40 no 1 pp 418ndash426 2013

[6] R R Inman W C Jordan and D E Blumenfeld ldquoChainedcross-training of assembly line workersrdquo International Journalof Production Research vol 42 no 10 pp 1899ndash1910 2004

[7] J Slomp J A C Bokhorst and EMolleman ldquoCross-training ina cellularmanufacturing environmentrdquoComputers amp IndustrialEngineering vol 48 no 3 pp 609ndash624 2005

[8] S Sayin and S Karabati ldquoAssigning cross-trained workers todepartments A two-stage optimization model to maximizeutility and skill improvementrdquo European Journal of OperationalResearch vol 176 no 3 pp 1643ndash1658 2007

[9] X Chu T Wu J D Weir Y Shi B Niu and L LildquoLearningndashinteractionndashdiversification framework for swarmintelligence optimizers a unified perspectiverdquo Neural Comput-ing and Applications pp 1ndash21

[10] X Chu S X Xu F Cai J Chen andQQin ldquoAn efficient auctionmechanism for regional logistics synchronizationrdquo Journal ofIntelligent Manufacturing pp 1ndash17 2018

[11] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995

[12] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Technical report-tr06 Erciyes university engi-neering faculty computer engineering department 2005

[13] W J Hopp E Tekin and M P Van Oyen ldquoBenefits of skillchaining in serial production lines with cross-trained workersrdquoManagement Science vol 50 no 1 pp 83ndash98 2004

[14] M C O Moreira M Ritt A M Costa and A A ChavesldquoSimple heuristics for the assembly line worker assignment andbalancing problemrdquo Journal of Heuristics vol 18 no 3 pp 505ndash524 2012

[15] H Parvin M P VanOyen D G Pandelis D PWilliams and JLee ldquoFixed task zone chaining Worker coordination and zonedesign for inexpensive cross-training in serial CONWIP linesrdquoInstitute of Industrial Engineers (IIE) IIE Transactions vol 44no 10 pp 894ndash914 2012

[16] M Saidi-Mehrabad M M Paydar and A Aalaei ldquoProductionplanning and worker training in dynamic manufacturing sys-temsrdquo Journal of Manufacturing Systems vol 32 no 2 pp 308ndash314 2013

[17] B Sungur and Y Yavuz ldquoAssembly line balancing with hierar-chical worker assignmentrdquo Journal of Manufacturing Systemsvol 37 pp 290ndash298 2015

[18] M B Aryanezhad V Deljoo and S M J Mirzapour Al-E-Hashem ldquoDynamic cell formation and the worker assignmentproblem A new modelrdquoThe International Journal of AdvancedManufacturing Technology vol 41 no 3-4 pp 329ndash342 2009

[19] I Mahdavi A Aalaei M M Paydar and M SolimanpurldquoDesigning a mathematical model for dynamic cellular manu-facturing systems considering production planning and workerassignmentrdquo Computers amp Mathematics with Applications AnInternational Journal vol 60 no 4 pp 1014ndash1025 2010

[20] I Mahdavi A Aalaei M M Paydar and M SolimanpurldquoMulti-objective cell formation and production planning indynamic virtual cellular manufacturing systemsrdquo InternationalJournal of Production Research vol 49 no 21 pp 6517ndash65372011

[21] M Bagheri and M Bashiri ldquoA new mathematical modeltowards the integration of cell formation with operator assign-ment and inter-cell layout problems in a dynamic environmentrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 38 no 4 pp1237ndash1254 2014

[22] F Niakan A Baboli T Moyaux and V Botta-Genoulaz ldquoA bi-objective model in sustainable dynamic cell formation problem


with skill-based worker assignmentrdquo Journal of ManufacturingSystems vol 38 pp 46ndash62 2016

[23] C Liu J Wang and J Y-T Leung ldquoWorker assignment andproduction planning with learning and forgetting in manufac-turing cells by hybrid bacteria foraging algorithmrdquo Computersamp Industrial Engineering vol 96 pp 162ndash179 2016

[24] R G Askin and Y Huang ldquoForming effective worker teamsfor cellular manufacturingrdquo International Journal of ProductionResearch vol 39 no 11 pp 2431ndash2451 2001

[25] B A Norman W Tharmmaphornphilas K L Needy BBidanda and R C Warner ldquoWorker assignment in cellularmanufacturing considering technical and human skillsrdquo Inter-national Journal of Production Research vol 40 no 6 pp 1479ndash1492 2002

[26] T Ertay and D Ruan ldquoData envelopment analysis based deci-sion model for optimal operator allocation in CMSrdquo EuropeanJournal of Operational Research vol 164 no 3 pp 800ndash8102005

[27] E L Fitzpatrick and R G Askin ldquoForming effective workerteams with multi-functional skill requirementsrdquo Computers ampIndustrial Engineering vol 48 no 3 pp 593ndash608 2005

[28] HJ Cesani Steudel ldquoA study of labor assignment flexibilityin cellular manufacturing systemsrdquo Computers amp IndustrialEngineering vol 48 no 3 pp 571ndash591 2005

[29] T McDonald K P Ellis E M Van Aken and C PatrickKoelling ldquoDevelopment andapplication of aworker assignmentmodel to evaluate a lean manufacturing cellrdquo InternationalJournal of Production Research vol 47 no 9 pp 2427ndash24472009

[30] R Mural A Puri and G Prabhakaran ldquoArtificial neuralnetworks based predictive model for worker assignment intovirtual cellsrdquo International Journal of Engineering Science andTechnology vol 2 no 1 2010

[31] G Egilmez B Erenay and G A Suer ldquoStochastic skill-basedmanpower allocation in a cellular manufacturing systemrdquoJournal of Manufacturing Systems vol 33 no 4 pp 578ndash5882014

[32] D Bratton and J Kennedy ldquoDefining a standard for particleswarm optimizationrdquo in Proceedings of the IEEE Swarm Intelli-gence Symposium (SIS rsquo07) pp 120ndash127 Honolulu Hawaii USAApril 2007

[33] G A Suer andR R Tummaluri ldquoMulti-period operator assign-ment considering skills learning and forgetting in labour-intensive cellsrdquo International Journal of Production Researchvol 46 no 2 pp 469ndash493 2008

[34] Y Won and K C Lee ldquoGroup technology cell formationconsidering operation sequences and production volumesrdquoInternational Journal of Production Research vol 39 no 13 pp2755ndash2768 2001

[35] R Sudhakara Pandian and S S Mahapatra ldquoManufacturingcell formation with production data using neural networksrdquoComputers amp Industrial Engineering vol 56 no 4 pp 1340ndash1347 2009

[36] L Wu and S Suzuki ldquoCell formation design with improvedsimilarity coefficient method and decomposed mathematicalmodelrdquo The International Journal of Advanced ManufacturingTechnology vol 79 no 5-8 pp 1335ndash1352 2015

[37] F Alhourani ldquoCellular manufacturing system design consider-ing machines reliability and parts alternative process routingsrdquoInternational Journal of Production Research vol 54 no 3 pp846ndash863 2016

[38] W Hung M Yang and E S Lee ldquoCell formation using fuzzyrelational clustering algorithmrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 1776ndash1787 2011

[39] I Mahdavi B Javadi K Fallah-Alipour and J Slomp ldquoDesign-ing a new mathematical model for cellular manufacturingsystem based on cell utilizationrdquo Applied Mathematics andComputation vol 190 no 1 pp 662ndash670 2007

[40] G Prabhakaran T N Janakiraman and M SachithanandamldquoManufacturing data-based combined dissimilarity coefficientfor machine cell formationrdquo The International Journal ofAdvancedManufacturing Technology vol 19 no 12 pp 889ndash8972002

[41] W Hachicha F Masmoudi and M Haddar ldquoFormation ofmachine groups andpart families in cellularmanufacturing sys-tems using a correlation analysis approachrdquo The InternationalJournal of Advanced Manufacturing Technology vol 36 no 11-12 pp 1157ndash1169 2008

[42] Yuquan Guo Xiongfei Li Yufei Tang and Jun Li ldquoHeuristicArtificial Bee Colony Algorithm for Uncovering Communityin Complex NetworksrdquoMathematical Problems in Engineeringvol 2017 Article ID 4143638 12 pages 2017

[43] Hamza Yapıcı and Nurettin Cetinkaya ldquoAn Improved ParticleSwarmOptimization AlgorithmUsing Eagle Strategy for PowerLossMinimizationrdquoMathematical Problems in Engineering vol2017 Article ID 1063045 11 pages 2017

[44] K E Stecke Y Yin I Kaku and Y Murase ldquoSeru (cell)and reverse conversion part I definition self-evolution andtypologyrdquo inWorking paper Yamagata University 2008

[45] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[46] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002

[47] B Akay and D Karaboga ldquoParameter Tuning for the ArtificialBee Colony Algorithmrdquo in Computational Biology Journalvol 5796 of Lecture Notes in Computer Science pp 608ndash619Springer Berlin Heidelberg Berlin Heidelberg 2009

[48] D Karaboga B Gorkemli C Ozturk and N Karaboga ldquoAcomprehensive survey artificial bee colony (ABC) algorithmand applicationsrdquo Artificial Intelligence Review vol 42 pp 21ndash57 2014

[49] X Chu G Hu B Niu L Li and Z Chu ldquoAn superiortracking artificial bee colony for global optimization problemsrdquoin Proceedings of the 2016 IEEE Congress on EvolutionaryComputation CEC 2016 pp 2712ndash2717 Canada July 2016

[50] Z Michalewicz and M Schoenauer ldquoEvolutionary algorithmsfor constrained parameter optimization problemsrdquo Evolution-ary Computation vol 4 no 1 pp 1ndash32 1996

[51] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

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to various tasks so as to optimize some output measure[8] In previous studies the cross-training problem and theworker assignment problem were always handled separatelyHowever the associated trade-offs between the two problemshave not been answered yet In this study we aim to proposea framework to solve the worker assignment problem in CMSincorporating cross-training

A major challenge arising in this context concerns theprinciple of cross-trained worker assignment in CMS whereworker-task configuration among multiple cells is exploredbased on customer demand to seek workload balance ofassigned workers with minimum cross-training and laborcost Specifically two questions should be answered(1) Which worker should be assigned to which task in

multiple manufacturing cells(2)Which worker should be trained to reach which skill

level for the assigned task in multiple manufacturing cellsPrevious studies related to the above questions mainly

have two limitations (1) They only answered the firstquestion by assuming that the skill levels of workers arefixed while neglecting the need for skill level enhancementvia cross-training Skill level enhancement denotes that theassigned worker should be trained not only for higher effi-ciency in its existing skills but also for the acquisition of newskills And (2) they mainly paid attention to flow-line systemor a single manufacturing cell and ignored the complexinteraction among multiple cells during cross-trained workerassignment in CMS

To address this gap this paper presents a novel mathe-matical programming model that aims to minimize trainingcost workload imbalance and the number of assignedworkers based on customer demand The proposed modelnot only takes into account the problem of cross-trainedworker assignment for multiple cells but also determinesthe training requirements considering the trade-off betweentraining expenditure and workload balance in the followingscenarios(1) Training is recommended strongly for more exact

workload balance which results in more expenditure(2) Training is not recommended for less expenditure

which results in less exact workload balanceIn view of the complexity and computational burden

of the presented model we implement swarm intelligenceoptimizers for the global optimization Swarm intelligence(SI) algorithms motivated by collective behaviors in naturalsystem have achieved great successful applications in var-ious areas [9 10] As a result a number of swarm-basedmetaheuristics have been developed recently among whichparticle swarm optimization (PSO) [11] and artificial beecolony (ABC) [12] are two representative algorithms PSOwas proposed for global optimization by emulating the veloc-ity adjusting of fish schooling and bird flocking ABC wasdeveloped by simulating the information exchange throughbee dancing PSO and ABC have shown high efficiencyand effectiveness in solving real-world global optimizationproblems [9] In this study the global topology PSO (GPSO)local topology PSO (LPSO) and ABC are implementedto solve the proposed cross-trained worker assignment inCMS To improve the optimization performance a superior

tracking artificial bee colony (STABC) with an augmentedinformation sharing strategy is developed In STABC thereare two main contributions compared to the canonical ABC(1) instead of learning one dimension from neighbors inSTABC a bee can learn from others in all dimensions (2)instead of chasing the random individual in STABC eachbee either learns from its previous information or movetowards other superior bees Computational experiments andcomparisons are conducted to evaluate the efficiency of theswarm intelligence algorithms

Comparing with the previous research the contributionof this study lies in three aspects(1) A new model is presented to solve the worker assign-

ment problem formultiplemanufacturing cells incorporatingcross-training planning(2)The skill enhancement is integrated to avoid excessive

training and the trade-off between training expenditureand workload balance is combined to obtain more flexiblesolution in the model(3) Swarm intelligence optimizers are developed and

improved to solve the proposed mathematical model effi-ciently

Remainder of this paper is organized as follows Section 2reviews the literaturerelating to worker assignment Section 3presentsmodel for effective cross-trainedworker assignmentSection 4 describes the swarm intelligence metaheuristicsused and the algorithmic implementation for solving theproblem Section 5 describes experiments with computa-tional results to justify the proposed model and algorithmsSection 6 summarizes the conclusions and future researchdirections

2 Literature Review

Theapproaches regarding theworker assignment problemarebriefly reviewed in this sectionThe literature can be classifiedby considering two types ofmanufacturing systems flow-linemanufacturing system (FLMS) and cellular manufacturingsystem (CMS) The limitations in previous studies are sum-marized for the motivation of our work

21 Worker Assignment in Flow-Line Manufacturing SystemHopp et al [13] considered two cross-training strategiescherry picking and skill chain In zoned work sharing somemachinestasks on the line are shared between workers toseek capacity balance Inman et al [6] argued that cross-training should be used judiciously since it is costly and islimited by learning capacity and can confound the searchfor quality problems They presented a training strategycalled chaining in which workers are trained to perform asecond task and the assignments of task types to workersare linked in a chain Their research has shown that cross-training in chaining is a practical and effective strategy tocompensate for absenteeism on assembly lines Moreira etal [14] proposed simple heuristics for solving the assemblyline worker assignment and balancing problem Their ideawas to use task and worker priority rules to define whichworker and which set of tasks should be assigned to each





















119872119894119899 = 1205871 lowast119862

sum119888=1

119870

sum119896=1

119869

sum119895=1

119871

sum119897=1


+1205872 lowast 120572 lowast119862

sum119888=1


(2)

+119862

sum119888=1

119870

sum119896=1

119873119888119896 lowast119872 (3)

subject to119870

sum119896=1


sum119901=1

119879119888119901119895 gt 0 (4)


Table1Th

eclassificatio

nof

relatedliterature

Author

Manufacturin

gsystem

type

Num

bero

fcells

Skill

level

Cross-T

raining

Workloadbalance

Datan

ature

Solvingmetho

dHop

petal[13]

FLMS

radicradic

Stochastic

Heuris

ticPo

licies

Inman

etal[6]

FLMS

radicStochastic

Simulation

Moreira

etal[14]

FLMS

radicCertain

HGA

Mutluetal[5]

FLMS

radicCertain

IGA

Saidi-M

ehrabadetal[16]

FLMS

radicradic

Stochastic

LINGO

Sung

urandYavu

zetal[17]

FLMS

radicCertain

CPLE

XParvin

etal[15]

FLMS

radicradic

Stochastic

Heuris

ticPo

licies

Aryanezhadetal[18]

CMS

Multip

leradic

radicCertain

LINGO

Mahdavietal[19]

CMS

Multip

leCertain

LINGO

Mahdavietal[20]

CMS

Multip

leCertain

LINGO

Bagh

eriand

Bashiri

[32]

CMS

Multip

leradic

Stochastic

LINGO

Niakanetal[22]

CMS

Multip

leradic

radicStochastic

NSG

AII-

MOSA

Liuetal[1]

CMS

Multip

leStochastic

DBF

ALiuetal[23]

CMS

Multip

leradic

Certain

HBF

AAskin

andHuang

[24]

CMS

Multip

leradic

radicStochastic

Greedyheuristic

Norman

etal[25]

CMS

Sing

leradic

radicStochastic

CPLE

XErtayandRu

an[26]

CMS

Sing

leStochastic

DEA

FitzpatrickandAskin

[27]

CMS

Sing

leradic

Stochastic

Balanced

placem

enth

euris

ticCesaniand

Steudel[28]

CMS

Sing

leradic

Certain

Simulation

Suer

andTu

mmaluri[33]

CMS

Sing

leradic

Stochastic

Max

andMaxMin

heuristic

McD

onaldetal[29]

CMS

Sing

leradic

radicCertain

CPLE

XMuralietal[30]

CMS

Multip

leradic

radicCertain

ANN

Egilm

ezetal[31]

CMS

Multip

leradic

Certain

LINGO




119870

sum119896=1


sum119901=1

119879119888119901119895 = 0 (5)


sum119895=1

119910119888119896119895 ge 1 (6)


sum119895=1

119910119888119896119895 = 0 (7)

119862

sum119888=1

119873119888119896 le 1 forall119896 (8)

119871

sum119897=1

119885119888119896119897119895 = 119910119888119896119895 forall119888 119896 119895 (9)

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1

119885119888119896119897119895 lowast 119879119888119901119895 lowast 119889119888119901119895119908119897

le 119867 forall119888 119896 (10)

119882119871119896 =119862

sum119888=1

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1


forall119896 (11)

119882119861119888 = max119896isin(12119870)

(119882119871119896 lowast 119873119888119896) forall119888 (12)

119871

sum119897=1


119862

sum119888=1

119870

sum119896=1

119869

sum119895=1

119871

sum119897=1


119910119888119896119895 119873119888119896 119885119888119896119897119895 = 0 119900119903 1 forall119888 119896 119897 119895 (15)











(16)

119883119889119898 = 119883119889119898 + 119901V119889119898 (17)






1003816100381610038161003816100381610038161003816 120593 = 1199031 + 1199032 (18)



(19)












119887V119889119898 = 119909119889119898 + 120601119889119898 (119909119889119898 minus 119909119889119902) 119898 = 119902 (20)



119901119898 =119891119894119905119899119890119904119904119898

sum119878119904=1 119891119894119905119899119890119904119904119904(21)


119891119894119905119899119890119904119904119898 =

1(1 + 119891119898)

if 119891119898 ge 01 + 119886119887119904 (119891119898) if 119891119898 le 0

(22)














119887V119898 = 119909119898 + 119903119898 (119909119898 minus 119878119873119898) (23)










Begin





of the food source


onlooker bees

Output

N

Y

itc gt criteria

itc = itc+1






Pr a threshold



FV fitness value


xnewi = xi + i


xnewi = xi + i






Begin

Y


Y

Y

N


p = 1P

Cross-reference

Inputxmj

Outputindexvalue

kl

Z(cklj) = 1j = j+1

j gt J

c = c+1j =1

c gt C

Output Zcklj

any(Tcpj)gt0








P1 [34] 5 5 2 5P2 [35] 7 5 2 5P3 [36] 6 6 2 6P4 [37] 8 9 2 9P5 [38] 10 8 3 8P6 [36] 8 10 3 10P7 [39] 10 10 3 10P8 [35] 12 10 3 10P9 [40] 18 10 3 10P10 [41] 22 11 3 11






0

25000

50000

75000

100000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC










Table5Re

sults

from

GPS

OL

PSOA

BCand

STABC

GPS

OLP

SOABC

STABC

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

P11205871=1120587

2=10

33183

3330

733

212

33183

33344

33285

32852

42983

35125

3283

943122

34597

1205871=101205872

=141163

4195

841

590

41258

42753

42152

38955

48235

44105

3810

548837

42540

P21205871=1120587

2=10

3260

732

667

32638

32596

32714

3264

832259

32959

3237

032

259

33204

32499

1205871=101205872

=135260

35865

35293

35216

3586

035330

32602

40752

35266

3260

240

907

3339

2

P31205871=1120587

2=10

33497

33782

3360

733453

33769

33619

32772

33116

32885

3277

232

864

3280

21205871=101205872

=142745

44415

43450

42745

44344

43563

36861

41403

38321

3681

139

207

3742

4

P41205871=1120587

2=10

53681

5700

955124

53277

5604

554

768

63133

76480

67822

5249

364

168

60957

1205871=101205872

=166237

80183

75246

65879

81657

7606

455113

91142

79567

5251

577

852

6748

2

P51205871=1120587

2=10

54720

5564

455089

54771

55589

55163

53669

55151

54324

5360

754

088

5387

91205871=101205872

=159274

67264

63857

59871

69167

64942

52293

63298

57622

5153

657

031

5382

6

P61205871=1120587

2=10

55149

66403

60150

55108

66335

56981

4621

963596

54613

53296

5439

253

768

1205871=101205872

=170131

84437

79483

72363

86381

80558

55765

64943

61086

5166

461

076

5572

6

P71205871=1120587

2=10

65351

68255

66758

65730

69136

67249

64034

74700

65529

6346

464

682

6402

01205871=101205872

=182555

99389

92564

87731

103806

96119

66279

87778

76697

6302

674

477

6879

8

P81205871=1120587

2=10

75074

78381

7604

175630

7812

07644

673638

85692

80359

7363

783820

78977

1205871=101205872

=184676

97413

91203

85879

102599

93956

72901

90037

81322

7255

781

016

7790

5

P91205871=1120587

2=10

75750

78238

77138

76227

78917

77378

74097

85694

7666

773

342

7496

074

237

1205871=101205872

=197288

117201

105674

95973

120132

108403

81629

105846

89569

7117

692

650

7930

4

P10

1205871=1120587

2=10

86208

89887

88252

85905

98570

89542

85065

9544

887622

8388

186

995

8482

11205871=101205872

=194291

132599

121452

11040

4139528

123986

87300

121262

103559

8520

010

1088

9167

5


Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018




























119872119894119899 = 1205871 lowast119862

sum119888=1

119870

sum119896=1

119869

sum119895=1

119871

sum119897=1


+1205872 lowast 120572 lowast119862

sum119888=1


(2)

+119862

sum119888=1

119870

sum119896=1

119873119888119896 lowast119872 (3)

subject to119870

sum119896=1


sum119901=1

119879119888119901119895 gt 0 (4)


Table1Th

eclassificatio

nof

relatedliterature

Author

Manufacturin

gsystem

type

Num

bero

fcells

Skill

level

Cross-T

raining

Workloadbalance

Datan

ature

Solvingmetho

dHop

petal[13]

FLMS

radicradic

Stochastic

Heuris

ticPo

licies

Inman

etal[6]

FLMS

radicStochastic

Simulation

Moreira

etal[14]

FLMS

radicCertain

HGA

Mutluetal[5]

FLMS

radicCertain

IGA

Saidi-M

ehrabadetal[16]

FLMS

radicradic

Stochastic

LINGO

Sung

urandYavu

zetal[17]

FLMS

radicCertain

CPLE

XParvin

etal[15]

FLMS

radicradic

Stochastic

Heuris

ticPo

licies

Aryanezhadetal[18]

CMS

Multip

leradic

radicCertain

LINGO

Mahdavietal[19]

CMS

Multip

leCertain

LINGO

Mahdavietal[20]

CMS

Multip

leCertain

LINGO

Bagh

eriand

Bashiri

[32]

CMS

Multip

leradic

Stochastic

LINGO

Niakanetal[22]

CMS

Multip

leradic

radicStochastic

NSG

AII-

MOSA

Liuetal[1]

CMS

Multip

leStochastic

DBF

ALiuetal[23]

CMS

Multip

leradic

Certain

HBF

AAskin

andHuang

[24]

CMS

Multip

leradic

radicStochastic

Greedyheuristic

Norman

etal[25]

CMS

Sing

leradic

radicStochastic

CPLE

XErtayandRu

an[26]

CMS

Sing

leStochastic

DEA

FitzpatrickandAskin

[27]

CMS

Sing

leradic

Stochastic

Balanced

placem

enth

euris

ticCesaniand

Steudel[28]

CMS

Sing

leradic

Certain

Simulation

Suer

andTu

mmaluri[33]

CMS

Sing

leradic

Stochastic

Max

andMaxMin

heuristic

McD

onaldetal[29]

CMS

Sing

leradic

radicCertain

CPLE

XMuralietal[30]

CMS

Multip

leradic

radicCertain

ANN

Egilm

ezetal[31]

CMS

Multip

leradic

Certain

LINGO




119870

sum119896=1


sum119901=1

119879119888119901119895 = 0 (5)


sum119895=1

119910119888119896119895 ge 1 (6)


sum119895=1

119910119888119896119895 = 0 (7)

119862

sum119888=1

119873119888119896 le 1 forall119896 (8)

119871

sum119897=1

119885119888119896119897119895 = 119910119888119896119895 forall119888 119896 119895 (9)

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1

119885119888119896119897119895 lowast 119879119888119901119895 lowast 119889119888119901119895119908119897

le 119867 forall119888 119896 (10)

119882119871119896 =119862

sum119888=1

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1


forall119896 (11)

119882119861119888 = max119896isin(12119870)

(119882119871119896 lowast 119873119888119896) forall119888 (12)

119871

sum119897=1


119862

sum119888=1

119870

sum119896=1

119869

sum119895=1

119871

sum119897=1


119910119888119896119895 119873119888119896 119885119888119896119897119895 = 0 119900119903 1 forall119888 119896 119897 119895 (15)











(16)

119883119889119898 = 119883119889119898 + 119901V119889119898 (17)






1003816100381610038161003816100381610038161003816 120593 = 1199031 + 1199032 (18)



(19)












119887V119889119898 = 119909119889119898 + 120601119889119898 (119909119889119898 minus 119909119889119902) 119898 = 119902 (20)



119901119898 =119891119894119905119899119890119904119904119898

sum119878119904=1 119891119894119905119899119890119904119904119904(21)


119891119894119905119899119890119904119904119898 =

1(1 + 119891119898)

if 119891119898 ge 01 + 119886119887119904 (119891119898) if 119891119898 le 0

(22)














119887V119898 = 119909119898 + 119903119898 (119909119898 minus 119878119873119898) (23)










Begin





of the food source


onlooker bees

Output

N

Y

itc gt criteria

itc = itc+1






Pr a threshold



FV fitness value


xnewi = xi + i


xnewi = xi + i






Begin

Y


Y

Y

N


p = 1P

Cross-reference

Inputxmj

Outputindexvalue

kl

Z(cklj) = 1j = j+1

j gt J

c = c+1j =1

c gt C

Output Zcklj

any(Tcpj)gt0








P1 [34] 5 5 2 5P2 [35] 7 5 2 5P3 [36] 6 6 2 6P4 [37] 8 9 2 9P5 [38] 10 8 3 8P6 [36] 8 10 3 10P7 [39] 10 10 3 10P8 [35] 12 10 3 10P9 [40] 18 10 3 10P10 [41] 22 11 3 11






0

25000

50000

75000

100000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC










Table5Re

sults

from

GPS

OL

PSOA

BCand

STABC

GPS

OLP

SOABC

STABC

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

P11205871=1120587

2=10

33183

3330

733

212

33183

33344

33285

32852

42983

35125

3283

943122

34597

1205871=101205872

=141163

4195

841

590

41258

42753

42152

38955

48235

44105

3810

548837

42540

P21205871=1120587

2=10

3260

732

667

32638

32596

32714

3264

832259

32959

3237

032

259

33204

32499

1205871=101205872

=135260

35865

35293

35216

3586

035330

32602

40752

35266

3260

240

907

3339

2

P31205871=1120587

2=10

33497

33782

3360

733453

33769

33619

32772

33116

32885

3277

232

864

3280

21205871=101205872

=142745

44415

43450

42745

44344

43563

36861

41403

38321

3681

139

207

3742

4

P41205871=1120587

2=10

53681

5700

955124

53277

5604

554

768

63133

76480

67822

5249

364

168

60957

1205871=101205872

=166237

80183

75246

65879

81657

7606

455113

91142

79567

5251

577

852

6748

2

P51205871=1120587

2=10

54720

5564

455089

54771

55589

55163

53669

55151

54324

5360

754

088

5387

91205871=101205872

=159274

67264

63857

59871

69167

64942

52293

63298

57622

5153

657

031

5382

6

P61205871=1120587

2=10

55149

66403

60150

55108

66335

56981

4621

963596

54613

53296

5439

253

768

1205871=101205872

=170131

84437

79483

72363

86381

80558

55765

64943

61086

5166

461

076

5572

6

P71205871=1120587

2=10

65351

68255

66758

65730

69136

67249

64034

74700

65529

6346

464

682

6402

01205871=101205872

=182555

99389

92564

87731

103806

96119

66279

87778

76697

6302

674

477

6879

8

P81205871=1120587

2=10

75074

78381

7604

175630

7812

07644

673638

85692

80359

7363

783820

78977

1205871=101205872

=184676

97413

91203

85879

102599

93956

72901

90037

81322

7255

781

016

7790

5

P91205871=1120587

2=10

75750

78238

77138

76227

78917

77378

74097

85694

7666

773

342

7496

074

237

1205871=101205872

=197288

117201

105674

95973

120132

108403

81629

105846

89569

7117

692

650

7930

4

P10

1205871=1120587

2=10

86208

89887

88252

85905

98570

89542

85065

9544

887622

8388

186

995

8482

11205871=101205872

=194291

132599

121452

11040

4139528

123986

87300

121262

103559

8520

010

1088

9167

5


Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018






















119872119894119899 = 1205871 lowast119862

sum119888=1

119870

sum119896=1

119869

sum119895=1

119871

sum119897=1


+1205872 lowast 120572 lowast119862

sum119888=1


(2)

+119862

sum119888=1

119870

sum119896=1

119873119888119896 lowast119872 (3)

subject to119870

sum119896=1


sum119901=1

119879119888119901119895 gt 0 (4)


Table1Th

eclassificatio

nof

relatedliterature

Author

Manufacturin

gsystem

type

Num

bero

fcells

Skill

level

Cross-T

raining

Workloadbalance

Datan

ature

Solvingmetho

dHop

petal[13]

FLMS

radicradic

Stochastic

Heuris

ticPo

licies

Inman

etal[6]

FLMS

radicStochastic

Simulation

Moreira

etal[14]

FLMS

radicCertain

HGA

Mutluetal[5]

FLMS

radicCertain

IGA

Saidi-M

ehrabadetal[16]

FLMS

radicradic

Stochastic

LINGO

Sung

urandYavu

zetal[17]

FLMS

radicCertain

CPLE

XParvin

etal[15]

FLMS

radicradic

Stochastic

Heuris

ticPo

licies

Aryanezhadetal[18]

CMS

Multip

leradic

radicCertain

LINGO

Mahdavietal[19]

CMS

Multip

leCertain

LINGO

Mahdavietal[20]

CMS

Multip

leCertain

LINGO

Bagh

eriand

Bashiri

[32]

CMS

Multip

leradic

Stochastic

LINGO

Niakanetal[22]

CMS

Multip

leradic

radicStochastic

NSG

AII-

MOSA

Liuetal[1]

CMS

Multip

leStochastic

DBF

ALiuetal[23]

CMS

Multip

leradic

Certain

HBF

AAskin

andHuang

[24]

CMS

Multip

leradic

radicStochastic

Greedyheuristic

Norman

etal[25]

CMS

Sing

leradic

radicStochastic

CPLE

XErtayandRu

an[26]

CMS

Sing

leStochastic

DEA

FitzpatrickandAskin

[27]

CMS

Sing

leradic

Stochastic

Balanced

placem

enth

euris

ticCesaniand

Steudel[28]

CMS

Sing

leradic

Certain

Simulation

Suer

andTu

mmaluri[33]

CMS

Sing

leradic

Stochastic

Max

andMaxMin

heuristic

McD

onaldetal[29]

CMS

Sing

leradic

radicCertain

CPLE

XMuralietal[30]

CMS

Multip

leradic

radicCertain

ANN

Egilm

ezetal[31]

CMS

Multip

leradic

Certain

LINGO




119870

sum119896=1


sum119901=1

119879119888119901119895 = 0 (5)


sum119895=1

119910119888119896119895 ge 1 (6)


sum119895=1

119910119888119896119895 = 0 (7)

119862

sum119888=1

119873119888119896 le 1 forall119896 (8)

119871

sum119897=1

119885119888119896119897119895 = 119910119888119896119895 forall119888 119896 119895 (9)

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1

119885119888119896119897119895 lowast 119879119888119901119895 lowast 119889119888119901119895119908119897

le 119867 forall119888 119896 (10)

119882119871119896 =119862

sum119888=1

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1


forall119896 (11)

119882119861119888 = max119896isin(12119870)

(119882119871119896 lowast 119873119888119896) forall119888 (12)

119871

sum119897=1


119862

sum119888=1

119870

sum119896=1

119869

sum119895=1

119871

sum119897=1


119910119888119896119895 119873119888119896 119885119888119896119897119895 = 0 119900119903 1 forall119888 119896 119897 119895 (15)











(16)

119883119889119898 = 119883119889119898 + 119901V119889119898 (17)






1003816100381610038161003816100381610038161003816 120593 = 1199031 + 1199032 (18)



(19)












119887V119889119898 = 119909119889119898 + 120601119889119898 (119909119889119898 minus 119909119889119902) 119898 = 119902 (20)



119901119898 =119891119894119905119899119890119904119904119898

sum119878119904=1 119891119894119905119899119890119904119904119904(21)


119891119894119905119899119890119904119904119898 =

1(1 + 119891119898)

if 119891119898 ge 01 + 119886119887119904 (119891119898) if 119891119898 le 0

(22)














119887V119898 = 119909119898 + 119903119898 (119909119898 minus 119878119873119898) (23)










Begin





of the food source


onlooker bees

Output

N

Y

itc gt criteria

itc = itc+1






Pr a threshold



FV fitness value


xnewi = xi + i


xnewi = xi + i






Begin

Y


Y

Y

N


p = 1P

Cross-reference

Inputxmj

Outputindexvalue

kl

Z(cklj) = 1j = j+1

j gt J

c = c+1j =1

c gt C

Output Zcklj

any(Tcpj)gt0








P1 [34] 5 5 2 5P2 [35] 7 5 2 5P3 [36] 6 6 2 6P4 [37] 8 9 2 9P5 [38] 10 8 3 8P6 [36] 8 10 3 10P7 [39] 10 10 3 10P8 [35] 12 10 3 10P9 [40] 18 10 3 10P10 [41] 22 11 3 11






0

25000

50000

75000

100000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC










Table5Re

sults

from

GPS

OL

PSOA

BCand

STABC

GPS

OLP

SOABC

STABC

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

P11205871=1120587

2=10

33183

3330

733

212

33183

33344

33285

32852

42983

35125

3283

943122

34597

1205871=101205872

=141163

4195

841

590

41258

42753

42152

38955

48235

44105

3810

548837

42540

P21205871=1120587

2=10

3260

732

667

32638

32596

32714

3264

832259

32959

3237

032

259

33204

32499

1205871=101205872

=135260

35865

35293

35216

3586

035330

32602

40752

35266

3260

240

907

3339

2

P31205871=1120587

2=10

33497

33782

3360

733453

33769

33619

32772

33116

32885

3277

232

864

3280

21205871=101205872

=142745

44415

43450

42745

44344

43563

36861

41403

38321

3681

139

207

3742

4

P41205871=1120587

2=10

53681

5700

955124

53277

5604

554

768

63133

76480

67822

5249

364

168

60957

1205871=101205872

=166237

80183

75246

65879

81657

7606

455113

91142

79567

5251

577

852

6748

2

P51205871=1120587

2=10

54720

5564

455089

54771

55589

55163

53669

55151

54324

5360

754

088

5387

91205871=101205872

=159274

67264

63857

59871

69167

64942

52293

63298

57622

5153

657

031

5382

6

P61205871=1120587

2=10

55149

66403

60150

55108

66335

56981

4621

963596

54613

53296

5439

253

768

1205871=101205872

=170131

84437

79483

72363

86381

80558

55765

64943

61086

5166

461

076

5572

6

P71205871=1120587

2=10

65351

68255

66758

65730

69136

67249

64034

74700

65529

6346

464

682

6402

01205871=101205872

=182555

99389

92564

87731

103806

96119

66279

87778

76697

6302

674

477

6879

8

P81205871=1120587

2=10

75074

78381

7604

175630

7812

07644

673638

85692

80359

7363

783820

78977

1205871=101205872

=184676

97413

91203

85879

102599

93956

72901

90037

81322

7255

781

016

7790

5

P91205871=1120587

2=10

75750

78238

77138

76227

78917

77378

74097

85694

7666

773

342

7496

074

237

1205871=101205872

=197288

117201

105674

95973

120132

108403

81629

105846

89569

7117

692

650

7930

4

P10

1205871=1120587

2=10

86208

89887

88252

85905

98570

89542

85065

9544

887622

8388

186

995

8482

11205871=101205872

=194291

132599

121452

11040

4139528

123986

87300

121262

103559

8520

010

1088

9167

5


Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018









Table1Th

eclassificatio

nof

relatedliterature

Author

Manufacturin

gsystem

type

Num

bero

fcells

Skill

level

Cross-T

raining

Workloadbalance

Datan

ature

Solvingmetho

dHop

petal[13]

FLMS

radicradic

Stochastic

Heuris

ticPo

licies

Inman

etal[6]

FLMS

radicStochastic

Simulation

Moreira

etal[14]

FLMS

radicCertain

HGA

Mutluetal[5]

FLMS

radicCertain

IGA

Saidi-M

ehrabadetal[16]

FLMS

radicradic

Stochastic

LINGO

Sung

urandYavu

zetal[17]

FLMS

radicCertain

CPLE

XParvin

etal[15]

FLMS

radicradic

Stochastic

Heuris

ticPo

licies

Aryanezhadetal[18]

CMS

Multip

leradic

radicCertain

LINGO

Mahdavietal[19]

CMS

Multip

leCertain

LINGO

Mahdavietal[20]

CMS

Multip

leCertain

LINGO

Bagh

eriand

Bashiri

[32]

CMS

Multip

leradic

Stochastic

LINGO

Niakanetal[22]

CMS

Multip

leradic

radicStochastic

NSG

AII-

MOSA

Liuetal[1]

CMS

Multip

leStochastic

DBF

ALiuetal[23]

CMS

Multip

leradic

Certain

HBF

AAskin

andHuang

[24]

CMS

Multip

leradic

radicStochastic

Greedyheuristic

Norman

etal[25]

CMS

Sing

leradic

radicStochastic

CPLE

XErtayandRu

an[26]

CMS

Sing

leStochastic

DEA

FitzpatrickandAskin

[27]

CMS

Sing

leradic

Stochastic

Balanced

placem

enth

euris

ticCesaniand

Steudel[28]

CMS

Sing

leradic

Certain

Simulation

Suer

andTu

mmaluri[33]

CMS

Sing

leradic

Stochastic

Max

andMaxMin

heuristic

McD

onaldetal[29]

CMS

Sing

leradic

radicCertain

CPLE

XMuralietal[30]

CMS

Multip

leradic

radicCertain

ANN

Egilm

ezetal[31]

CMS

Multip

leradic

Certain

LINGO




119870

sum119896=1


sum119901=1

119879119888119901119895 = 0 (5)


sum119895=1

119910119888119896119895 ge 1 (6)


sum119895=1

119910119888119896119895 = 0 (7)

119862

sum119888=1

119873119888119896 le 1 forall119896 (8)

119871

sum119897=1

119885119888119896119897119895 = 119910119888119896119895 forall119888 119896 119895 (9)

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1

119885119888119896119897119895 lowast 119879119888119901119895 lowast 119889119888119901119895119908119897

le 119867 forall119888 119896 (10)

119882119871119896 =119862

sum119888=1

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1


forall119896 (11)

119882119861119888 = max119896isin(12119870)

(119882119871119896 lowast 119873119888119896) forall119888 (12)

119871

sum119897=1


119862

sum119888=1

119870

sum119896=1

119869

sum119895=1

119871

sum119897=1


119910119888119896119895 119873119888119896 119885119888119896119897119895 = 0 119900119903 1 forall119888 119896 119897 119895 (15)











(16)

119883119889119898 = 119883119889119898 + 119901V119889119898 (17)






1003816100381610038161003816100381610038161003816 120593 = 1199031 + 1199032 (18)



(19)












119887V119889119898 = 119909119889119898 + 120601119889119898 (119909119889119898 minus 119909119889119902) 119898 = 119902 (20)



119901119898 =119891119894119905119899119890119904119904119898

sum119878119904=1 119891119894119905119899119890119904119904119904(21)


119891119894119905119899119890119904119904119898 =

1(1 + 119891119898)

if 119891119898 ge 01 + 119886119887119904 (119891119898) if 119891119898 le 0

(22)














119887V119898 = 119909119898 + 119903119898 (119909119898 minus 119878119873119898) (23)










Begin





of the food source


onlooker bees

Output

N

Y

itc gt criteria

itc = itc+1






Pr a threshold



FV fitness value


xnewi = xi + i


xnewi = xi + i






Begin

Y


Y

Y

N


p = 1P

Cross-reference

Inputxmj

Outputindexvalue

kl

Z(cklj) = 1j = j+1

j gt J

c = c+1j =1

c gt C

Output Zcklj

any(Tcpj)gt0








P1 [34] 5 5 2 5P2 [35] 7 5 2 5P3 [36] 6 6 2 6P4 [37] 8 9 2 9P5 [38] 10 8 3 8P6 [36] 8 10 3 10P7 [39] 10 10 3 10P8 [35] 12 10 3 10P9 [40] 18 10 3 10P10 [41] 22 11 3 11






0

25000

50000

75000

100000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC










Table5Re

sults

from

GPS

OL

PSOA

BCand

STABC

GPS

OLP

SOABC

STABC

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

P11205871=1120587

2=10

33183

3330

733

212

33183

33344

33285

32852

42983

35125

3283

943122

34597

1205871=101205872

=141163

4195

841

590

41258

42753

42152

38955

48235

44105

3810

548837

42540

P21205871=1120587

2=10

3260

732

667

32638

32596

32714

3264

832259

32959

3237

032

259

33204

32499

1205871=101205872

=135260

35865

35293

35216

3586

035330

32602

40752

35266

3260

240

907

3339

2

P31205871=1120587

2=10

33497

33782

3360

733453

33769

33619

32772

33116

32885

3277

232

864

3280

21205871=101205872

=142745

44415

43450

42745

44344

43563

36861

41403

38321

3681

139

207

3742

4

P41205871=1120587

2=10

53681

5700

955124

53277

5604

554

768

63133

76480

67822

5249

364

168

60957

1205871=101205872

=166237

80183

75246

65879

81657

7606

455113

91142

79567

5251

577

852

6748

2

P51205871=1120587

2=10

54720

5564

455089

54771

55589

55163

53669

55151

54324

5360

754

088

5387

91205871=101205872

=159274

67264

63857

59871

69167

64942

52293

63298

57622

5153

657

031

5382

6

P61205871=1120587

2=10

55149

66403

60150

55108

66335

56981

4621

963596

54613

53296

5439

253

768

1205871=101205872

=170131

84437

79483

72363

86381

80558

55765

64943

61086

5166

461

076

5572

6

P71205871=1120587

2=10

65351

68255

66758

65730

69136

67249

64034

74700

65529

6346

464

682

6402

01205871=101205872

=182555

99389

92564

87731

103806

96119

66279

87778

76697

6302

674

477

6879

8

P81205871=1120587

2=10

75074

78381

7604

175630

7812

07644

673638

85692

80359

7363

783820

78977

1205871=101205872

=184676

97413

91203

85879

102599

93956

72901

90037

81322

7255

781

016

7790

5

P91205871=1120587

2=10

75750

78238

77138

76227

78917

77378

74097

85694

7666

773

342

7496

074

237

1205871=101205872

=197288

117201

105674

95973

120132

108403

81629

105846

89569

7117

692

650

7930

4

P10

1205871=1120587

2=10

86208

89887

88252

85905

98570

89542

85065

9544

887622

8388

186

995

8482

11205871=101205872

=194291

132599

121452

11040

4139528

123986

87300

121262

103559

8520

010

1088

9167

5


Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018











119870

sum119896=1


sum119901=1

119879119888119901119895 = 0 (5)


sum119895=1

119910119888119896119895 ge 1 (6)


sum119895=1

119910119888119896119895 = 0 (7)

119862

sum119888=1

119873119888119896 le 1 forall119896 (8)

119871

sum119897=1

119885119888119896119897119895 = 119910119888119896119895 forall119888 119896 119895 (9)

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1

119885119888119896119897119895 lowast 119879119888119901119895 lowast 119889119888119901119895119908119897

le 119867 forall119888 119896 (10)

119882119871119896 =119862

sum119888=1

119869

sum119895=1

119871

sum119897=1

119875

sum119901=1


forall119896 (11)

119882119861119888 = max119896isin(12119870)

(119882119871119896 lowast 119873119888119896) forall119888 (12)

119871

sum119897=1


119862

sum119888=1

119870

sum119896=1

119869

sum119895=1

119871

sum119897=1


119910119888119896119895 119873119888119896 119885119888119896119897119895 = 0 119900119903 1 forall119888 119896 119897 119895 (15)











(16)

119883119889119898 = 119883119889119898 + 119901V119889119898 (17)






1003816100381610038161003816100381610038161003816 120593 = 1199031 + 1199032 (18)



(19)












119887V119889119898 = 119909119889119898 + 120601119889119898 (119909119889119898 minus 119909119889119902) 119898 = 119902 (20)



119901119898 =119891119894119905119899119890119904119904119898

sum119878119904=1 119891119894119905119899119890119904119904119904(21)


119891119894119905119899119890119904119904119898 =

1(1 + 119891119898)

if 119891119898 ge 01 + 119886119887119904 (119891119898) if 119891119898 le 0

(22)














119887V119898 = 119909119898 + 119903119898 (119909119898 minus 119878119873119898) (23)










Begin





of the food source


onlooker bees

Output

N

Y

itc gt criteria

itc = itc+1






Pr a threshold



FV fitness value


xnewi = xi + i


xnewi = xi + i






Begin

Y


Y

Y

N


p = 1P

Cross-reference

Inputxmj

Outputindexvalue

kl

Z(cklj) = 1j = j+1

j gt J

c = c+1j =1

c gt C

Output Zcklj

any(Tcpj)gt0








P1 [34] 5 5 2 5P2 [35] 7 5 2 5P3 [36] 6 6 2 6P4 [37] 8 9 2 9P5 [38] 10 8 3 8P6 [36] 8 10 3 10P7 [39] 10 10 3 10P8 [35] 12 10 3 10P9 [40] 18 10 3 10P10 [41] 22 11 3 11






0

25000

50000

75000

100000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC










Table5Re

sults

from

GPS

OL

PSOA

BCand

STABC

GPS

OLP

SOABC

STABC

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

P11205871=1120587

2=10

33183

3330

733

212

33183

33344

33285

32852

42983

35125

3283

943122

34597

1205871=101205872

=141163

4195

841

590

41258

42753

42152

38955

48235

44105

3810

548837

42540

P21205871=1120587

2=10

3260

732

667

32638

32596

32714

3264

832259

32959

3237

032

259

33204

32499

1205871=101205872

=135260

35865

35293

35216

3586

035330

32602

40752

35266

3260

240

907

3339

2

P31205871=1120587

2=10

33497

33782

3360

733453

33769

33619

32772

33116

32885

3277

232

864

3280

21205871=101205872

=142745

44415

43450

42745

44344

43563

36861

41403

38321

3681

139

207

3742

4

P41205871=1120587

2=10

53681

5700

955124

53277

5604

554

768

63133

76480

67822

5249

364

168

60957

1205871=101205872

=166237

80183

75246

65879

81657

7606

455113

91142

79567

5251

577

852

6748

2

P51205871=1120587

2=10

54720

5564

455089

54771

55589

55163

53669

55151

54324

5360

754

088

5387

91205871=101205872

=159274

67264

63857

59871

69167

64942

52293

63298

57622

5153

657

031

5382

6

P61205871=1120587

2=10

55149

66403

60150

55108

66335

56981

4621

963596

54613

53296

5439

253

768

1205871=101205872

=170131

84437

79483

72363

86381

80558

55765

64943

61086

5166

461

076

5572

6

P71205871=1120587

2=10

65351

68255

66758

65730

69136

67249

64034

74700

65529

6346

464

682

6402

01205871=101205872

=182555

99389

92564

87731

103806

96119

66279

87778

76697

6302

674

477

6879

8

P81205871=1120587

2=10

75074

78381

7604

175630

7812

07644

673638

85692

80359

7363

783820

78977

1205871=101205872

=184676

97413

91203

85879

102599

93956

72901

90037

81322

7255

781

016

7790

5

P91205871=1120587

2=10

75750

78238

77138

76227

78917

77378

74097

85694

7666

773

342

7496

074

237

1205871=101205872

=197288

117201

105674

95973

120132

108403

81629

105846

89569

7117

692

650

7930

4

P10

1205871=1120587

2=10

86208

89887

88252

85905

98570

89542

85065

9544

887622

8388

186

995

8482

11205871=101205872

=194291

132599

121452

11040

4139528

123986

87300

121262

103559

8520

010

1088

9167

5


Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018
















(16)

119883119889119898 = 119883119889119898 + 119901V119889119898 (17)






1003816100381610038161003816100381610038161003816 120593 = 1199031 + 1199032 (18)



(19)












119887V119889119898 = 119909119889119898 + 120601119889119898 (119909119889119898 minus 119909119889119902) 119898 = 119902 (20)



119901119898 =119891119894119905119899119890119904119904119898

sum119878119904=1 119891119894119905119899119890119904119904119904(21)


119891119894119905119899119890119904119904119898 =

1(1 + 119891119898)

if 119891119898 ge 01 + 119886119887119904 (119891119898) if 119891119898 le 0

(22)














119887V119898 = 119909119898 + 119903119898 (119909119898 minus 119878119873119898) (23)










Begin





of the food source


onlooker bees

Output

N

Y

itc gt criteria

itc = itc+1






Pr a threshold



FV fitness value


xnewi = xi + i


xnewi = xi + i






Begin

Y


Y

Y

N


p = 1P

Cross-reference

Inputxmj

Outputindexvalue

kl

Z(cklj) = 1j = j+1

j gt J

c = c+1j =1

c gt C

Output Zcklj

any(Tcpj)gt0








P1 [34] 5 5 2 5P2 [35] 7 5 2 5P3 [36] 6 6 2 6P4 [37] 8 9 2 9P5 [38] 10 8 3 8P6 [36] 8 10 3 10P7 [39] 10 10 3 10P8 [35] 12 10 3 10P9 [40] 18 10 3 10P10 [41] 22 11 3 11






0

25000

50000

75000

100000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC










Table5Re

sults

from

GPS

OL

PSOA

BCand

STABC

GPS

OLP

SOABC

STABC

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

P11205871=1120587

2=10

33183

3330

733

212

33183

33344

33285

32852

42983

35125

3283

943122

34597

1205871=101205872

=141163

4195

841

590

41258

42753

42152

38955

48235

44105

3810

548837

42540

P21205871=1120587

2=10

3260

732

667

32638

32596

32714

3264

832259

32959

3237

032

259

33204

32499

1205871=101205872

=135260

35865

35293

35216

3586

035330

32602

40752

35266

3260

240

907

3339

2

P31205871=1120587

2=10

33497

33782

3360

733453

33769

33619

32772

33116

32885

3277

232

864

3280

21205871=101205872

=142745

44415

43450

42745

44344

43563

36861

41403

38321

3681

139

207

3742

4

P41205871=1120587

2=10

53681

5700

955124

53277

5604

554

768

63133

76480

67822

5249

364

168

60957

1205871=101205872

=166237

80183

75246

65879

81657

7606

455113

91142

79567

5251

577

852

6748

2

P51205871=1120587

2=10

54720

5564

455089

54771

55589

55163

53669

55151

54324

5360

754

088

5387

91205871=101205872

=159274

67264

63857

59871

69167

64942

52293

63298

57622

5153

657

031

5382

6

P61205871=1120587

2=10

55149

66403

60150

55108

66335

56981

4621

963596

54613

53296

5439

253

768

1205871=101205872

=170131

84437

79483

72363

86381

80558

55765

64943

61086

5166

461

076

5572

6

P71205871=1120587

2=10

65351

68255

66758

65730

69136

67249

64034

74700

65529

6346

464

682

6402

01205871=101205872

=182555

99389

92564

87731

103806

96119

66279

87778

76697

6302

674

477

6879

8

P81205871=1120587

2=10

75074

78381

7604

175630

7812

07644

673638

85692

80359

7363

783820

78977

1205871=101205872

=184676

97413

91203

85879

102599

93956

72901

90037

81322

7255

781

016

7790

5

P91205871=1120587

2=10

75750

78238

77138

76227

78917

77378

74097

85694

7666

773

342

7496

074

237

1205871=101205872

=197288

117201

105674

95973

120132

108403

81629

105846

89569

7117

692

650

7930

4

P10

1205871=1120587

2=10

86208

89887

88252

85905

98570

89542

85065

9544

887622

8388

186

995

8482

11205871=101205872

=194291

132599

121452

11040

4139528

123986

87300

121262

103559

8520

010

1088

9167

5


Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018











119887V119889119898 = 119909119889119898 + 120601119889119898 (119909119889119898 minus 119909119889119902) 119898 = 119902 (20)



119901119898 =119891119894119905119899119890119904119904119898

sum119878119904=1 119891119894119905119899119890119904119904119904(21)


119891119894119905119899119890119904119904119898 =

1(1 + 119891119898)

if 119891119898 ge 01 + 119886119887119904 (119891119898) if 119891119898 le 0

(22)














119887V119898 = 119909119898 + 119903119898 (119909119898 minus 119878119873119898) (23)










Begin





of the food source


onlooker bees

Output

N

Y

itc gt criteria

itc = itc+1






Pr a threshold



FV fitness value


xnewi = xi + i


xnewi = xi + i






Begin

Y


Y

Y

N


p = 1P

Cross-reference

Inputxmj

Outputindexvalue

kl

Z(cklj) = 1j = j+1

j gt J

c = c+1j =1

c gt C

Output Zcklj

any(Tcpj)gt0








P1 [34] 5 5 2 5P2 [35] 7 5 2 5P3 [36] 6 6 2 6P4 [37] 8 9 2 9P5 [38] 10 8 3 8P6 [36] 8 10 3 10P7 [39] 10 10 3 10P8 [35] 12 10 3 10P9 [40] 18 10 3 10P10 [41] 22 11 3 11






0

25000

50000

75000

100000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC










Table5Re

sults

from

GPS

OL

PSOA

BCand

STABC

GPS

OLP

SOABC

STABC

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

P11205871=1120587

2=10

33183

3330

733

212

33183

33344

33285

32852

42983

35125

3283

943122

34597

1205871=101205872

=141163

4195

841

590

41258

42753

42152

38955

48235

44105

3810

548837

42540

P21205871=1120587

2=10

3260

732

667

32638

32596

32714

3264

832259

32959

3237

032

259

33204

32499

1205871=101205872

=135260

35865

35293

35216

3586

035330

32602

40752

35266

3260

240

907

3339

2

P31205871=1120587

2=10

33497

33782

3360

733453

33769

33619

32772

33116

32885

3277

232

864

3280

21205871=101205872

=142745

44415

43450

42745

44344

43563

36861

41403

38321

3681

139

207

3742

4

P41205871=1120587

2=10

53681

5700

955124

53277

5604

554

768

63133

76480

67822

5249

364

168

60957

1205871=101205872

=166237

80183

75246

65879

81657

7606

455113

91142

79567

5251

577

852

6748

2

P51205871=1120587

2=10

54720

5564

455089

54771

55589

55163

53669

55151

54324

5360

754

088

5387

91205871=101205872

=159274

67264

63857

59871

69167

64942

52293

63298

57622

5153

657

031

5382

6

P61205871=1120587

2=10

55149

66403

60150

55108

66335

56981

4621

963596

54613

53296

5439

253

768

1205871=101205872

=170131

84437

79483

72363

86381

80558

55765

64943

61086

5166

461

076

5572

6

P71205871=1120587

2=10

65351

68255

66758

65730

69136

67249

64034

74700

65529

6346

464

682

6402

01205871=101205872

=182555

99389

92564

87731

103806

96119

66279

87778

76697

6302

674

477

6879

8

P81205871=1120587

2=10

75074

78381

7604

175630

7812

07644

673638

85692

80359

7363

783820

78977

1205871=101205872

=184676

97413

91203

85879

102599

93956

72901

90037

81322

7255

781

016

7790

5

P91205871=1120587

2=10

75750

78238

77138

76227

78917

77378

74097

85694

7666

773

342

7496

074

237

1205871=101205872

=197288

117201

105674

95973

120132

108403

81629

105846

89569

7117

692

650

7930

4

P10

1205871=1120587

2=10

86208

89887

88252

85905

98570

89542

85065

9544

887622

8388

186

995

8482

11205871=101205872

=194291

132599

121452

11040

4139528

123986

87300

121262

103559

8520

010

1088

9167

5


Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018









Begin





of the food source


onlooker bees

Output

N

Y

itc gt criteria

itc = itc+1






Pr a threshold



FV fitness value


xnewi = xi + i


xnewi = xi + i






Begin

Y


Y

Y

N


p = 1P

Cross-reference

Inputxmj

Outputindexvalue

kl

Z(cklj) = 1j = j+1

j gt J

c = c+1j =1

c gt C

Output Zcklj

any(Tcpj)gt0








P1 [34] 5 5 2 5P2 [35] 7 5 2 5P3 [36] 6 6 2 6P4 [37] 8 9 2 9P5 [38] 10 8 3 8P6 [36] 8 10 3 10P7 [39] 10 10 3 10P8 [35] 12 10 3 10P9 [40] 18 10 3 10P10 [41] 22 11 3 11






0

25000

50000

75000

100000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC










Table5Re

sults

from

GPS

OL

PSOA

BCand

STABC

GPS

OLP

SOABC

STABC

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

P11205871=1120587

2=10

33183

3330

733

212

33183

33344

33285

32852

42983

35125

3283

943122

34597

1205871=101205872

=141163

4195

841

590

41258

42753

42152

38955

48235

44105

3810

548837

42540

P21205871=1120587

2=10

3260

732

667

32638

32596

32714

3264

832259

32959

3237

032

259

33204

32499

1205871=101205872

=135260

35865

35293

35216

3586

035330

32602

40752

35266

3260

240

907

3339

2

P31205871=1120587

2=10

33497

33782

3360

733453

33769

33619

32772

33116

32885

3277

232

864

3280

21205871=101205872

=142745

44415

43450

42745

44344

43563

36861

41403

38321

3681

139

207

3742

4

P41205871=1120587

2=10

53681

5700

955124

53277

5604

554

768

63133

76480

67822

5249

364

168

60957

1205871=101205872

=166237

80183

75246

65879

81657

7606

455113

91142

79567

5251

577

852

6748

2

P51205871=1120587

2=10

54720

5564

455089

54771

55589

55163

53669

55151

54324

5360

754

088

5387

91205871=101205872

=159274

67264

63857

59871

69167

64942

52293

63298

57622

5153

657

031

5382

6

P61205871=1120587

2=10

55149

66403

60150

55108

66335

56981

4621

963596

54613

53296

5439

253

768

1205871=101205872

=170131

84437

79483

72363

86381

80558

55765

64943

61086

5166

461

076

5572

6

P71205871=1120587

2=10

65351

68255

66758

65730

69136

67249

64034

74700

65529

6346

464

682

6402

01205871=101205872

=182555

99389

92564

87731

103806

96119

66279

87778

76697

6302

674

477

6879

8

P81205871=1120587

2=10

75074

78381

7604

175630

7812

07644

673638

85692

80359

7363

783820

78977

1205871=101205872

=184676

97413

91203

85879

102599

93956

72901

90037

81322

7255

781

016

7790

5

P91205871=1120587

2=10

75750

78238

77138

76227

78917

77378

74097

85694

7666

773

342

7496

074

237

1205871=101205872

=197288

117201

105674

95973

120132

108403

81629

105846

89569

7117

692

650

7930

4

P10

1205871=1120587

2=10

86208

89887

88252

85905

98570

89542

85065

9544

887622

8388

186

995

8482

11205871=101205872

=194291

132599

121452

11040

4139528

123986

87300

121262

103559

8520

010

1088

9167

5


Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018











P1 [34] 5 5 2 5P2 [35] 7 5 2 5P3 [36] 6 6 2 6P4 [37] 8 9 2 9P5 [38] 10 8 3 8P6 [36] 8 10 3 10P7 [39] 10 10 3 10P8 [35] 12 10 3 10P9 [40] 18 10 3 10P10 [41] 22 11 3 11






0

25000

50000

75000

100000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC










Table5Re

sults

from

GPS

OL

PSOA

BCand

STABC

GPS

OLP

SOABC

STABC

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

P11205871=1120587

2=10

33183

3330

733

212

33183

33344

33285

32852

42983

35125

3283

943122

34597

1205871=101205872

=141163

4195

841

590

41258

42753

42152

38955

48235

44105

3810

548837

42540

P21205871=1120587

2=10

3260

732

667

32638

32596

32714

3264

832259

32959

3237

032

259

33204

32499

1205871=101205872

=135260

35865

35293

35216

3586

035330

32602

40752

35266

3260

240

907

3339

2

P31205871=1120587

2=10

33497

33782

3360

733453

33769

33619

32772

33116

32885

3277

232

864

3280

21205871=101205872

=142745

44415

43450

42745

44344

43563

36861

41403

38321

3681

139

207

3742

4

P41205871=1120587

2=10

53681

5700

955124

53277

5604

554

768

63133

76480

67822

5249

364

168

60957

1205871=101205872

=166237

80183

75246

65879

81657

7606

455113

91142

79567

5251

577

852

6748

2

P51205871=1120587

2=10

54720

5564

455089

54771

55589

55163

53669

55151

54324

5360

754

088

5387

91205871=101205872

=159274

67264

63857

59871

69167

64942

52293

63298

57622

5153

657

031

5382

6

P61205871=1120587

2=10

55149

66403

60150

55108

66335

56981

4621

963596

54613

53296

5439

253

768

1205871=101205872

=170131

84437

79483

72363

86381

80558

55765

64943

61086

5166

461

076

5572

6

P71205871=1120587

2=10

65351

68255

66758

65730

69136

67249

64034

74700

65529

6346

464

682

6402

01205871=101205872

=182555

99389

92564

87731

103806

96119

66279

87778

76697

6302

674

477

6879

8

P81205871=1120587

2=10

75074

78381

7604

175630

7812

07644

673638

85692

80359

7363

783820

78977

1205871=101205872

=184676

97413

91203

85879

102599

93956

72901

90037

81322

7255

781

016

7790

5

P91205871=1120587

2=10

75750

78238

77138

76227

78917

77378

74097

85694

7666

773

342

7496

074

237

1205871=101205872

=197288

117201

105674

95973

120132

108403

81629

105846

89569

7117

692

650

7930

4

P10

1205871=1120587

2=10

86208

89887

88252

85905

98570

89542

85065

9544

887622

8388

186

995

8482

11205871=101205872

=194291

132599

121452

11040

4139528

123986

87300

121262

103559

8520

010

1088

9167

5


Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018









Table5Re

sults

from

GPS

OL

PSOA

BCand

STABC

GPS

OLP

SOABC

STABC

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

119874119865119881119887

119874119865119881119908

119874119865119881119886

P11205871=1120587

2=10

33183

3330

733

212

33183

33344

33285

32852

42983

35125

3283

943122

34597

1205871=101205872

=141163

4195

841

590

41258

42753

42152

38955

48235

44105

3810

548837

42540

P21205871=1120587

2=10

3260

732

667

32638

32596

32714

3264

832259

32959

3237

032

259

33204

32499

1205871=101205872

=135260

35865

35293

35216

3586

035330

32602

40752

35266

3260

240

907

3339

2

P31205871=1120587

2=10

33497

33782

3360

733453

33769

33619

32772

33116

32885

3277

232

864

3280

21205871=101205872

=142745

44415

43450

42745

44344

43563

36861

41403

38321

3681

139

207

3742

4

P41205871=1120587

2=10

53681

5700

955124

53277

5604

554

768

63133

76480

67822

5249

364

168

60957

1205871=101205872

=166237

80183

75246

65879

81657

7606

455113

91142

79567

5251

577

852

6748

2

P51205871=1120587

2=10

54720

5564

455089

54771

55589

55163

53669

55151

54324

5360

754

088

5387

91205871=101205872

=159274

67264

63857

59871

69167

64942

52293

63298

57622

5153

657

031

5382

6

P61205871=1120587

2=10

55149

66403

60150

55108

66335

56981

4621

963596

54613

53296

5439

253

768

1205871=101205872

=170131

84437

79483

72363

86381

80558

55765

64943

61086

5166

461

076

5572

6

P71205871=1120587

2=10

65351

68255

66758

65730

69136

67249

64034

74700

65529

6346

464

682

6402

01205871=101205872

=182555

99389

92564

87731

103806

96119

66279

87778

76697

6302

674

477

6879

8

P81205871=1120587

2=10

75074

78381

7604

175630

7812

07644

673638

85692

80359

7363

783820

78977

1205871=101205872

=184676

97413

91203

85879

102599

93956

72901

90037

81322

7255

781

016

7790

5

P91205871=1120587

2=10

75750

78238

77138

76227

78917

77378

74097

85694

7666

773

342

7496

074

237

1205871=101205872

=197288

117201

105674

95973

120132

108403

81629

105846

89569

7117

692

650

7930

4

P10

1205871=1120587

2=10

86208

89887

88252

85905

98570

89542

85065

9544

887622

8388

186

995

8482

11205871=101205872

=194291

132599

121452

11040

4139528

123986

87300

121262

103559

8520

010

1088

9167

5


Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018









Table6Perfo

rmance

comparis

onam

ongGPS

OL

PSOA

BCand

STABC

119861119866119866119875119878119874

119878119879119860119861119862

119882119866119866119875119878119874

S119879119860119861119862

119860119866119866119875119878119874

119878119879119860119861119862

119861119866119871119875119878119874

119878119879119860119861119862

119882119866119871119875119878119874

119878119879119860119861119862

119860119866119871119875119878119874

119878119879119860119861119862

119861119866119860119861119862119878119879119860119861119862

119882119866119860119861119862119878119879119860119861119862

119860119866119860119861119862119878119879119860119861119862

P11205871=1120587

2=10

10

-228

-40

10

-227

-38

00

-03

15

1205871=101205872

=180

-141

-22

83

-125

-09

22

-12

37

P21205871=1120587

2=10

11

-16

04

10

-15

05

00

-07

-04

1205871=101205872

=182

-123

57

80

-123

58

00

-04

56

P31205871=1120587

2=10

22

28

25

21

28

25

00

08

03

1205871=101205872

=1161

133

161

161

131

164

01

56

24

P41205871=1120587

2=10

23

-112

-96

15

-127

-102

203

192

113

1205871=101205872

=1261

30

115

254

49

127

49

171

179

P51205871=1120587

2=10

21

29

22

22

28

24

01

20

08

1205871=101205872

=1150

179

186

162

213

207

15

110

71

P61205871=1120587

2=10

35

221

119

34

220

60

-133

169

16

1205871=101205872

=1357

382

426

401

414

446

79

63

96

P71205871=1120587

2=10

30

55

43

36

69

50

09

155

24

1205871=101205872

=1310

334

345

392

394

397

52

179

115

P81205871=1120587

2=10

20

-65

-37

27

-68

-32

00

22

17

1205871=101205872

=1167

202

171

184

266

206

05

111

44

P91205871=1120587

2=10

33

44

39

39

53

42

10

143

33

1205871=101205872

=1367

265

333

348

297

367

147

142

129

P10

1205871=1120587

2=10

28

33

40

24

133

56

14

97

33

1205871=101205872

=1107

312

325

296

380

352

25

200

130

Mean

114

78

11

113

0

99

120

2

59

15

7


0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018









0

20000

40000

60000

80000

100000

120000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10


GPSOLPSO

ABCSTABC


0300600900

120015001800

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Training Cost

1=1 2=101=10 2=1







800

850

900

950

1000

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

Workload Balance

1=1 2=101=10 2=1





Data Availability






Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018












Acknowledgments




References





























































Journal of












Journal of







Volume 2018






Volume 2018














































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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