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AdvancesinProductionEngineering&Management ISSN1854‐6250

Volume13|Number1|March2018|pp5–17 Journalhome:apem‐journal.org

https://doi.org/10.14743/apem2018.1.269 Originalscientificpaper

Comparison among four calibrated meta‐heuristic algorithms for solving a type‐2 fuzzy cell formation problem considering economic and environmental criteria  

Arghish, O.a, Tavakkoli‐Moghaddam, R.b,c,*, Shahandeh‐Nookabadi, A.d, Rezaeian, J.e

aDepartment of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran bSchool of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran cLCFC, Arts et Métiers Paris Tech, Metz, France dDepartment of Industrial and Systems Engineering, Isfahan University of Technology, Isfahan, Iran eDepartment of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran  

A B S T R A C T A R T I C L E   I N F O

In this paper, amathematicalmodel is proposed using economic and envi‐ronmental criteria for a type‐2 fuzzy (T2F) cell formation (CF) problememphasizing the effect of theman‐machine relationship aspect. ThismodelaimstoshowtheuseofthisaspectinCFtominimizethecostsofprocessing,materialmovement, energy loss, and tooling. For this purpose, a two‐stagedefuzzification procedure is used to convert the T2F variable into a crispvalue. Due to NP‐hardness of the model and problem, a genetic algorithm(GA)isusedtoderivetheappropriatesolutions.Furthermore,becausethereis no any existing benchmark to validate the performance of the proposedmodel, three tunedmeta‐heuristic algorithms,namely, differential evolution(DE),harmonysearch (HS)andparticle swarmoptimization (PSO),arepro‐posedandused.ThepresentresearchusestheTaguchimethodtoadjusttheparameters in the fourproposed algorithms. Furthermore, 15 examples areusedtovalidatethepresentedmodel.TheresultsshowthatPSOisthemostappropriatealgorithmforsolvingthemodel.

©2018PEI,UniversityofMaribor.Allrightsreserved.

  Keywords:Cellformation;Environmentalfactor;Geneticalgorithm;Particleswarmoptimization; Harmonysearch; Differentialevolution

*Correspondingauthor:tavakoli@ut.ac.ir(Tavakkoli‐Moghaddam,R.)

Articlehistory:Received29June2017Revised22January2018Accepted20February2018 

1. Introduction

Nowadays, the competitive environment throughout theworld has led the involvedmanufac‐turingindustriestosupplythehighestquality,affordableproductssuchthatrecentapproachesfocusmoreon theever‐growingmanufacturing costs including thoseassociatedwith location,energy, and transportation system. Group technology (GT) as one of the most efficientapproaches tries togrouppartsandmachines in termsof theirsimilarities inproductionpro‐cesses,functionalities,andgeometries[1].Cellularmanufacturing(CM)asanapplicationofGTin amanufacturing system is utilized to classify similar parts into families assigningdifferentmachinestocells[2].TheCMSdesigninvolvesfourprincipalstages,inwhicheachofthemcanbeconsideredasanindividualproblem,namelyCF,layout,schedulingandresourceassignment(RA) [3].As theCFproblemcomesupas the first stage in theCMSdesign, investigatorshaveattemptedtooptimallysolvetheproblem.

Forinstance,Majazi‐Delfard[4]introducedanon‐linearmodelforthedynamiccellformation(DCF)intermsofthequantityandlengthofintra‐andinter‐celltravels.Deljooetal.[5]provid‐

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6  Advances in Production Engineering & Management 13(1) 2018

edaGA‐basedsolutiontotheDCFproblemidentifyingerrorsinthemodelsrecommendedintheliterature, declining their helpful perspective and presenting a novel formulation for the DCFproblem.BagheriandBashiri[6]utilizedaLP‐metricapproachtoaproposedmodelcomprisingofCF,layoutandworkerassignmentcomponents.Xuetal.[7]providedabatalgorithm(BA)tothe dual flexible job‐shop scheduling problem considering the process sequence andmachineselection flexibility.Zupanetal. [8]presentedamethodbasedonacombinationofschmigallamodified triangularmethod, the schwerdfegercircular process, and a simulationmodelto thelayoutoptimizationofaproductioncellconsideringtheintensityofthematerialflow.Nieetal.[9]utilizedasimulationmodelforaToken‐orientedPetrinet‐basedflexiblemanufacturingcell.

Mahdavietal.[10]presentedatwo‐stageapproachtoaCFproblemconsideringintervalT2Finteractionalinterestsamongworkers.Inthefirststage,amulti‐depotmultipletravelingsales‐manproblemmodelisusedtoassignworkerstocells.Inthesecondstage,amathematicalmod‐elisappliedtoassignmachinestocells.Withrespecttothehistoricalperspective,sustainabilitywasconceptualizedinthelate1960’sandearly1970’sstressingtheenvironmentaleffectofin‐dustrial projects [11]. Sustainability was represented by Lozano [12] in three dimensions, inwhich themes ineconomic,environmental, andsocialaspects interact in the temporal respec‐tive.TheliteraturereviewalsorevealsthefactthatwhiletheCMSdesignhasreceivedconsider‐ableattention,socialandenvironmentalaspectshavebeenunderexploredwiththemajorityofthecriteriainvestigatedintheassociatedliteraturedevotedtoeconomicratherthansocialandenvironmentalissues[13].

Niakanetal. [14]proposedamathematicalmodelof theproblemcomprisingof twoobjec‐tivestoinvestigatethetrade‐offbetweenthetotalcostminimizationandsocialissuemaximiza‐tionattemptingtosolve theproblemusinganon‐dominatedsortinggeneticalgorithm(NSGA‐II).Inaddition,Niakanetal.[15]proposedamodelforthecostminimizationandthemachineenergy loss minimization incorporating a social constraint developing an NSGA‐II solve theproblem.Furthermore,Niakanetal.[16]introducedamodelpossessingtwoobjectivesfortheDCF problem minimizing production and worker costs and total production waste includingenergy, chemical material, rawmaterial, CO2 emissions. Proposing the social criteria as con‐straints, the researchers utilized theNSGA‐II andmulti‐objective simulated annealing (MOSA)algorithmstoprovideasolutionforthebi‐objectivemodel.

TheliteraturemostlyconsidersinputparametersintheCFtobedeterministic.However,inpractice,therearealargenumberofuncertainandimpreciseparametersinvolved.Asthequan‐tityofdatamaynotalwaysbesufficientfortheuncertainparametersprediction,fuzzylogicisutilizedasa robust instrument togauge thisuncertainty through themediumof thepersonalknowledge[17].Todescribetype‐2fuzziness,aT2Fvariablerepresentsamapextendingfromthefuzzypossibilityspacetotherealnumberspace[18].MillerandJohn[19] assertthatfurtheruncertaintydegreessuppliedthroughtheintervaltype‐2fuzzysets(IT‐2FS)logicmakesitpos‐sibletomoreoptimallyshowtheuncertaintyandvaguenessofresourceplanningmodels.Qinetal.[18]presentedthreecategoriesofcriticalvalues(CVs)foraregularfuzzyvariable(RFV)andthreereductionmethodsforaT2Fvariable.

Research conducted byKundu etal. [20] examines transportation problems using T2F pa‐rameterswhereinthefirstvaluesoftheT2FparametersweredefuzzifiedusingCV‐basedreduc‐tionmethodstotype‐1fuzzyvariables,andthecentroidmethodwasemployedfortotaldefuzz‐ification.

Thereviewoftherelatedliteraturerevealsthemostcommonly usedcriteriainCFtobetheassociatedcosts.Therefore,tobridgethisgap,amathematicalmodelwasdevelopedusingeco‐nomicandenvironmentalcriteriaandaT2FparameterforCFandRAproblemssimultaneously.Thenoveltyofthepresentresearchpartlyliesintheconsiderationofworker‐machinerelation‐shipforworkerallocationwheretheworkercanbeconsideredasasignificantindustrialsystemcomponent.Thepresentpaperconsistsoftheproblemstatementandmathematicalmodel(Sec‐tion 2), solutionmethodologies (Section 3), computational results (Section 4), and conclusion(Section5).

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2. Problem statement and definition of objective function

ThissectiondiscussesaT2Fmodel for theCFconsideringeconomic,environmentalandman‐machinerelationshipaspects.

2.1 Worker‐machine relationship  

Nowadays,numerousmachine toolsareeither totallyorpartiallyautomaticallyoperatedwiththeworkerbeingusually idle foraportionof thecycle.Thepotentialuseof this idle timecanenhanceworkerearningsandtheefficiencyofamanufacture.Theman‐machineprocesschartevidentlydepictstherespectiveidlemachinetimeandworkertimeareas,whichnormallyrep‐resent desirable locations to initiate effective improvements. Although this chart is generallyimplementedtospecifythenumberofmachinesassignedtoaworker,theapplicationofamath‐ematicalmodelcansubstantiallylessenthetimeneededtodoso.Suchidealsituationsaregen‐erallyreferredtoassynchronousservicingwiththenumberofmachinesassignedcomputedasshowninEq.1.

(1)

wheremrepresentsthenumberofmachinesbeingoperatedbyeachworker,sworkerservicingtimepermachine,rmachineworkingtimeandwwalkingtimebetweentwomachines,thenum‐ber of machinesmust be represented by a whole number; otherwise, we have the followingequation.

(2)

Ifthenumberofmachinesdoesnotrepresentawholenumber,theminimumtotalcostperpiececriteriacanbeusedfortheoptimumoperation.Thetotalcostperpiecefor and machinesaregiveninEqs.3and4.

C . . . ⁄ (3)

C . (4)

whereC1and are theworkerandmachinecosts, respectively.Thenumberofmachinesas‐signedtoworkersrepresentstheminimumtotalcostperpiece[21].

2.2 Environmental criteria 

Withthepassageoftime,humanshavedamagedtheenvironmentthroughthewasteproductionanduncontrolleduseofnaturalresourcestosatisfytheever‐increasingunprecedenteduselevel,whichwas farbeyond thenature’s capacity to restoreand/orregenerate itself.The fearofanuncertain future for theworldmade itnecessary tograsphow individuals,organizations, andgovernmentshavecooperatedtodiscoverapproachestopreventaglobalcollapse[22].Oneoftheobjective function terms is tominimize the total systemenergy losscost.TherelationshipbetweenmanandthemachineintheCFmodelleadstothefarbetteruseofbothworkersandmachinetime,andmoreoptimalbalanceintheworkcycle.Themanandmachinerelationshiptoolsdepicttheareas,inwhichmachineandworkeridletimestakeplace.Thus,theuseoftheseidle times can enhance worker earnings enhancing the production efficiency [21]. In the re‐searchers’proposedmodel,allthemachinesareconsideredasamulti‐functionaltask.Thema‐chines requirehighly skilledworkers thusprovidingworkerswith theopportunity to acquirenumerousskillsexpandingtheirpotential.

2.3 T2F set  

AT2F setwasproposed as an extension of an ordinary fuzzy set. LetΓ be the universe, Pos:A→ 0,1 beasetfunctionontheamplefieldAandPosisapossibilitycriterion.If(Γ, A, Pos)isapossibilityspace,thenanm‐aryregularfuzzyvectorξ=(ξ1,ξ2,...,ξm)isamapΓ→[0,1]mforanyt=(t1,t2,...,tm)∈[0,1]m,onehasthefollowingequation:

{γ∈Γ|ξ(γ)≤t}={γ∈Γ|ξ1(γ)≤t1,ξ2(γ)≤t2,…,ξm(γ)≤tm} ∈A (5)

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asm=1,ξisaRFV.Let :A→ 0,1 beasetfunctiononAsuchthat{ (A)| A}isanRFVsand isa fuzzypossibilitycriteria. Ifμ (1)=1, then isaregular fuzzypossibility

criteria.If(Γ, A, ) isafuzzypossibilityspace(FPS),thenanm‐aryT2Fvectorξ=(ξ1,ξ2,...,ξm)isamapΓ→�mforany , , . . . , ∈ ,asshowninEq.6.

∈ | ∈ | , , … , ∈A (6)

as 1, isaT2Fvariable[18].

Criticalvalue

Let , , , and , , beatrapezoidalandtriangularRFV,respectively.Then,wehavethefollowingitemsinTable1 [18].

Table1 ComponentofRFVstrapezoidal triangular

optimisticCVof 1⁄ 1⁄

pessimisticCVof 1⁄ 1⁄

TheCVof

21 2

12

12

12

1 212

21 2

12

1 212

Centroidmethod

Thecentroidmethodisthemostcommonlyusedmethodfortransformingthetype‐1fuzzyintocrispvalues.ThecentroidmethodcanbedefinedbytheEq.7forthediscretecase[23].

∗∑ .∑

(7)

Defuzzification

Inthepresentresearch,atwo‐stagedefuzzificationprocedureisemployedtotransformtheT2Fvariabletocrispvalue.Initially,theCVisutilizedforRFVsfortransformingtheT2Fintotype‐1fuzzy.Then,thecentroidmethodisemployedinordertotransformthetype‐1fuzzyintocrispvalues.

2.4 Assumptions 

For the CF considered in this paper, every single operation on each part classification can beexecuted onmulti‐functional and identicalmachines. Eachpart typedemand, each tool type’stoollife,maximumcellnumberandworker‐part‐machine‐tool‐workercombinationcompatibil‐ityaregiven.Theaveragequantityofenergywastedbyeachmachinetypeinaunitoftimeandenergyprice inaunitof time isalsoknownas is the totalservicingtimeofaworker foreachmachine. 

2.5 Notations and parameters 

Machinem working time for performing operation o on part pwith tool h byworkerg

1,ifmachinemisemployedtooperationoforpartpwithtoolhbyworkerg;and0,otherwise

Costrelatedtointer‐cellmovementforpartpinadistanceunit Costrelatedtointra‐cellmovementforpartpinadistanceunit

Demandforpartp Timecapacityformachinem

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Timecapacityforworkerg Upperboundofmachinesallowedincellk Averagedistanceamongcellskandk' Workerservicingtimepermachinemtoperformoperationoonpartpusingtool

hbyworkerg Walkingtimebetweenmachinem takentoprocessoperationo forpartpusing

toolhbyworkergtothenextmachine Operating cost onmachinem to process operationo on partp using toolh by

workerg 1,if ′;and0,if ′ Machine(m)costforatimeunit Worker(g)costforatimeunit Costoftoolh

Inter‐cellbatchsizemotionforpartp Intra‐cellbatchsizemotionforpartp Toollifeoftoolh

AveragequantityofenergywastedbyeachmachineminunittimeE Priceofenergyinunittime

2.6 Decision variables 

1,ifmachinemisusedforoperationoofpartpusingtoolhbyworkergincellk;and0,otherwise

Numberoftoolcopiesfortoolhonmachinem 1,ifoperationoofpartpisperformedonmachinemincellk;and0,otherwise

1,ifoperationoofpartpisperformedonmachinem;and0,otherwise1,ifmachinemisassignedtocellk;and0,otherwise

1 1,if(mu)machine isallocatedtoworkerg;and0,otherwise2 1,if(ml)machine isallocatedtoworkerg;and0,otherwise

Lowerwholequantityofmachinemallocatedtoworkerg Upperwholequantityofmachinemallocatedtoworkerg

C Totalcostperpiecefromonemachine( ) andworkergC Totalcostperpiecefromonemachine( u) andworkerg

Numberofmachinemassignedtoworkergincellk

2.7 Objective function 

Theobjectivefunctionoftheconsideredmodelistominimizethecostsofprocessing,materialmovement,energyloss,andtooling.

Min (8)

,  

1 ,  

2 1

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s. t. : . 1 , ∀ , (9)

, ∀ , , , , , (10)

, ∀ (11)

, ∀ , , , (12)

2 1 , ∀ , , (13)

. 1 , ∀ , (14)

. . . 2 , ∀ , (15)

. 1 , ∀ , (16)

. . . 2 , ∀ , (17)

., ∀ , , , , , (18)

. 1 , ∀ , , , , (19)

, ∀ , , , , (20)

. , ∀ , , , , (21)

1 , ∀ , (22)

2 , ∀ , (23)

1 2 1 , ∀ , (24)

. . , ∀ , (25)

, ∀ , , , (26)

, , 1 , 2 , , ∈ 0,1 (27)

, , , 0 , (28)

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Thefirstsentenceintheobjectivefunction(Eq.8)representsthetotalcostoftheprocess.Thesecondandthirdtermsrepresentthetotalmaterialtransportationcost.Thefourthtermisthecostofenergylossinthesystematatimeunit.Thefifthtermrepresentsthetotaltoolcost.Eq.9andEq.10expresstheoperation‐part‐machine‐tool‐workercombinations.Eq.11limitsthecellsize.Eq.12canbeusedtodefinethemachine‐cellcombination.Eq.13isusedtoensurethatthequantityofmachinemisassignedtoworkergincellk.Eq.14andEq.15expressthetimecapac‐ityoftheworker.Eq.16andEq.17expressthetimecapacityofthemachine.Eq.18ensuresthatthe upperwhole number ofmachinem is assigned toworkerg. Eq. 19 ensures that a lowerwholenumberofmachinemisassignedtoworkerg.Eq.20andEq.21expressthecostofapro‐ductionpercycle fromonemachine in the lowerandupperwholenumbersofmachinem as‐signedtoworkerg.Eq.22toEq.24guaranteethatthelowest ischosen.Eq.25representsthetool‐machinecombinations.Eq.26expressestheoperation‐part‐machine‐cellcombinations.Eq.27andEq.28canbeusedtodefinethetypeofvariables.

3. Used methodsInthepresentpaper,inordertosolvethepresentedCFmodel,aGAisimplementedandthreeDE,PSOandHSalgorithmsareemployedtotheobtainedaccreditoutcome.

3.1 Genetic algorithm 

Holland[24]wasthefirsttodeveloptheGA,whichrepresentedcodingtoachromosomeform.Subsequenttotheproductionofthefirstrandomchromosomes,evaluationofperformancewasundertaken using the fitness function. The remaining chromosomes and offspring produce ageneration through themedium of crossover andmutation. In the end, the elitism process isusedtoproducesolutions[25].TheGAusedfortheCFframeworkisasfollows.TwoelementsaregivenintheCFproblem.Thefirstelementrepresentstheassignmentofmachinestowork‐ersusing[Ma_Wo].Thismatrixisutilizedtodefinetheentiretyoftherelativeconstraints.Thesecondelement represents anoperation‐part‐machine‐cell‐tool‐worker [OP_Pa_N3],whereN3equals[Ma_Ce_To_Wo].Thesematricesareemployedtodefinetheentiretyoftherelativecon‐straints.

3.2 Differential evolution

TheDEalgorithmrepresentsarecentevolutionaryoptimizationtechniqueforcontinuousnon‐linear functions introduced by Noktehdan et al. [26] and Storn and Price [27]. The principalstagesinvolvedintheDEalgorithmaredefinedasfollows.Initially,arandompopulationgener‐ationisformedandtheobjectivefunctionisevaluated.Foreachindividualsolutioninthepopu‐lation,amutatedsolution isproducedasfollowsinEq.29.

(29)

whereFrepresentsascalar(F∈[0,1]),and , , representrandomly‐selectedsubjectsinthepopulationi( ).Thecrossoveroperationisemployedtoestablishatrialvector through shuffling the information incorporated in themutated vector and the currentsolutioninEq.30.

for and for (30)

whereCRrepresentsthecrossoverrate∈[0,1],whichneedstobespecifiedbytheuser,and representsarandomrealnumber∈[0,1]andjisthej‐thparameter.Acomparisonismadebe‐tweeneachtrialvector( )anditsparent( ),andthemoredesirableoneremainsinthepopu‐lationintheselectionstage[28].

3.3 Particle swarm optimization 

ThePSO algorithm represents a population‐based stochastic optimization algorithm extendedbyKennedy andEberhart [29] composedof a population (i.e., swarm) of candidate solutions,

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12  Advances in Production Engineering & Management 13(1) 2018

referred to as particles,which are in inwardmotion towards search spacewith a designatedvelocityinsearchofanoptimumsolution.Eachparticlemaintainsamemoryassistingitinpre‐servingthepathtakenbyitspreviousbestlocation.Theparticlepositionsareidentifiedasper‐sonalbestandglobalbest.Theprincipaliterativeprocesstoarriveatthesolutionisexecutedby:

(31)

(32)

Eq.31isusedtocalculatethei‐thparticlemovementinthek‐threplication,where , repre‐sentthevelocityandthecurrentlocationofthei‐thparticleinthek‐threplication,respectively.Eq.32isemployedtocalculatethelatestvelocityvectorofthei‐thparticleinthek‐thiteration,

representsthebestlocationofthei‐thparticle,wrepresentstheinertiafactorcontrollingthemagnitudeoftheoldvelocity, and representaccelerationconstants(i.e.,cognitiveandsocial)basedonthekindofsearch, localandglobal aredenoted and ,respec‐tively[30].

3.4 Harmony search 

Musicalperformancecanbedefinedasthequestforthelovelyharmonyamongallharmonies.Geem et al. [31] introduced an optimization algorithm on the basis of musical performance,knownasHS, searching for thebest solutionemanating fromtheobjective function.Thealgo‐rithm starts by playing a new harmony and comparing this harmonywith those in harmonymemory(HM),whichresultsintheimprovementintheharmonyqualityinastep‐by‐stepman‐ner.Subsequently, theHMupdatesandverifies thestopcriterion.Theentiretyof thedecisionvariables(notes)inHMtogetherwiththevaluesforthesenotesinthenewharmonyaredeter‐mined in the followingmanner: first, precise choiceof theHMdomain value. Second, randomselectionofthefullvaluedomainusingaselectionrateortheharmonymemoryconsideringrate(HMCR)betweenzeroandone.Third,selectionofidealidenticalvaluesfortheHMdomainwiththepitchadjustmentrate(PAR)betweenzeroandoneandafreedistancebandwidth(Bw)[32].

4. Results and discussion

To investigate and evaluate the performance of the fourmeta‐heuristic algorithms on the CF,some randomly‐selected numerical examples are produced. To solve the proposed model,MATLAB (R2016b) software isutilized toprovide the code thealgorithmson a laptophavingfiveIntelCorei5CPUand2GBRAM.TheTMisruninMINITABsoftwareversion17.3.1tocali‐bratetheparametersforasubsequentdataanalysis.

4.1 Defuzzification of T2F variable   

Theenergypricecoefficientinthefourthsentenceoftheobjectivefunctionrangesbetween4to8. ThecoefficientisrepresentedbythefollowingdiscreteT2Fvariable.

E3 with 3 0.1, 0.4, 0.74 with 4 0.9, 1, 15 with 5 0.1, 0.3, 0.4,0.6

E7 with 7 0.4, 0.5, 0.7,0.88 with 8 0.6, 0.8, 0.910 with 10 0.4, 0.6, 0.7

TosolvetheCFmodelunderconsiderationintheinitialstep,aCVreductionmethodisemployedtoconverttheenergypriceT2Fvariableinunittimetothecorrespondingtype‐1fuzzyvariable.Duringthesecondstep,acentroidmethod isperformedtoreducethetype‐1 fuzzyvariable tothecrispvalue.Theenergypricecrispvalueisobtainedusing 3.99,8.35).

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4.2 Generating random data 

15randomexamplesareproducedinvarioussizesthroughthegenerationofuniformlydistrib‐utedrandompointsforanumberofparametersgiven.Theattributesof15designedtestexam‐plesareshowninTable2.Also,Table3showsthecomponentsofthemodelinputparametersrequiredfor15probleminstances.

Table2 Attributeoftestexamples

Problem

Part

Operation

Machine

Cell

Tool

Worker

Problem

Part

Operation

Machine

Cell

Tool

Worker

Problem

Part

Operation

Machine

Cell

Tool

Worker

1 2 2 4 2 2 4 6 5 5 3 2 3 3 11 7 2 5 2 3 52 2 4 5 2 2 5 7 6 3 4 3 3 4 12 7 5 3 4 3 33 4 4 2 3 3 2 8 6 5 3 4 4 3 13 8 2 3 2 3 34 4 5 4 3 3 4 9 6 3 5 3 3 5 14 8 5 4 2 2 45 5 5 2 3 2 2 10 7 3 4 3 2 4 15 8 5 4 3 3 4

Table3DataidentifyingwithrandomtestproblemsParameter Amount Parameter Amount Parameter Amount Parameter Amount

100 500 E 3.99 8.35 , 55 9 20

5 10 10 30 0.02 0.01

10 15 50 75 0 1 2

0.5 0.7 50 80 2 3 1

4.3 Parameter calibration 

TheTMisusedtocalibratetheparametersintheGA,DE,PSOandHSalgorithms,asthevaluesofmeta‐heuristic algorithm parameters influence the solution quality. Nevertheless, the presentstudy,the“smallerisbetter”responseischosenasS/Nshouldbeminimized [25].ToperformtheTaguchiprocedure,theL^9designisemployedwiththevaluesandlevelsoftheGA,DE,PSOandHSalgorithmparametersoutlinedinTable4andthevaluesderivedaftermultipletestsontheexamplesoftheclassesusingthefrequentalgorithmruns.

Table4GA,DE,PSOandHSparameters,andlevelsAlgorithm Parameters (1) (2) (3) Algorithm Parameters (1) (2) (3)

POP 30 40 50 NOP 30 40 50GA Pc 0.5 0.6 0.7 PSO C1 1.5 2 2.5

Pm 0.01 0.05 0.1 C2 1.5 2 2.5NOG 100 200 300 NOG 100 150 200NOP 20 30 50 HMS 5 10 20

DE NOG 30 50 100 HS HMCR 0.9 0.95 0.99Pc 0.1 0.5 0.9 PAR 0.01 0.1 0.3‐ ‐ ‐ ‐ BW 0.1 0.5 0.9

4.4 Analysis of results and comparisons

TocomparetheresultsemanatingfromfouralgorithmsandelicitthebestmethodologytosolvetheCFhaving theT2Fvariable, eachof 15 examples, is solvedusingMATLAB (R2016b) soft‐ware. In this paper, theGA, PSO,HS andDE algorithms are used, inwhich the probability ofcrossover(Pc),thegenerationsnumber(NOG),theprobabilityofmutation(Pm)andthesizeofpopulation(POP)aretheGAparameters.Thegenerationsnumber(NOG),theaccelerationcoef‐ficients(C1,C2)andthesizeofpopulation(NOP)arethePSOparameters.Theharmonymemoryconsideringrate(HMCR),theBandwidth(BW),theharmonymemorysize(HMS)andthepitchadjusting rate (PAR) are the HS parameters. The size of population (NOP), the probability ofcrossover(Pc)andthegenerationsnumber(NOG)aretheDEparameters.

Tables5and6incorporatestheinputparameterandtheobjectivevaluesoffouralgorithmsineachexample,inwhichtheoptimalvaluesoftheparametersarederivedusingL^9designandtheTM.TocomparetheperformanceoftheGA,PSO,HS,andDEwithreferencetotheobjective

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function,anumberofapproachesareutilizedinthepresentresearch.Initially,theaverageandthestandarddeviationofeachofthe15exampleswereobtainedasshowninthelasttworowsinTable6.Theresultsfromaverageandstandarddeviationofthe15examplesshowthatPSOhasoutperformedGA,HS,andDE.ThegraphicalapproachdepictedinFig.1isalsoappliedtwicetocompare thealgorithmperformance in15producedexamples.Moreover, this figureshowsthatthePSOappearstorepresentabetterperformancethantheGA,HS,andDEintheobjectivefunctionintheentiretyoftheexamples.

Table5InputparametersofthePSO,FAandDEalgorithmsforthegeneratedproblemsHS PSO GA DE

Prob.No. HMS HMCR PAR BW NOP NOG POP NOG NOP NOG1 20 0.99 0.3 0.9 30 2 2 100 50 0.6 0.01 200 50 100 0.92 20 0.99 0.3 0.5 50 25 2 200 50 0.5 0.05 100 30 30 0.93 10 0.95 0.01 0.9 50 1.5 2 100 50 0.6 0.05 300 50 100 0.94 10 0.95 0.01 0.9 40 1.5 2.5 200 30 0.7 0.1 300 30 30 0.95 10 0.99 0.3 0.9 40 1.5 1.5 100 40 0.7 0.05 200 50 50 0.16 20 0.95 0.1 0.5 30 2 2 200 30 0.5 0.01 200 30 100 0.17 5 0.99 0.01 0.1 30 2 2.5 150 50 0.7 0.1 300 30 100 0.18 20 0.95 0.01 0.5 40 1.5 1.5 200 50 0.7 0.1 300 30 50 0.19 20 0.9 0.01 0.1 30 1.5 2.5 150 30 0.6 0.01 200 50 100 0.110 20 0.99 0.01 0.9 40 2 2.5 200 40 0.6 0.05 200 20 100 0.111 20 0.95 0.3 0.9 50 2 1.5 200 30 0.6 0.1 100 30 30 0.112 20 0.95 0.01 0.1 50 1.5 2 200 50 0.6 0.01 200 50 100 0.913 20 0.95 0.3 0.9 50 2 1.5 200 50 0.7 0.05 300 50 50 0.114 10 0.9 0.1 0.5 30 2 2 150 30 0.6 0.05 300 20 30 0.115  5 0.99 0.1 0.5 40 1.5 1.5 100 30 0.6 0.1 300 30 100 0.9

Table6ObjectivefunctionofthePSO,FAandDEalgorithmsforthegeneratedproblemsProblemNo. HS 

ObjectivefunctionPSO

ObjectivefunctionGA

ObjectivefunctionDE

Objectivefunction1 11579 11449 17874 125302 49371 34946 55498 537653 61232 59110 64833 634984 112440 89216 123982 1198705 134230 100420 143563 1391106 114140 101930 129390 1287657 99366 71208 101550 998758 127030 126750 148510 1441709 106320 105770 110289 10754010 112430 109290 115410 11398011 41477 37906 61187 4940512 215630 214010 265670 23172013 78911 70402 86832 8469514 221480 175750 240290 23873015  221080 220720 247720 225775

Average 113781 101925 127507 120895St.Dev 64446 61827 73328 68000

Fig.1Trendoftheobjectivefunctionvaluesofthegeneratedproblemsfortheproposedalgorithms

Comparison among four calibrated meta‐heuristic algorithms for solving a type‐2 fuzzy cell formation problem …

Advances in Production Engineering & Management 13(1) 2018  15

Table7AnalysisofvarianceresultstocomparethealgorithmsintermsofmeanobjectivefunctionvalueSource DF Adj.SS Adj.MS F‐Value P‐valueAlgorithms 3 5390788752 1796929584 0.40 0.754Error 56 2.51675E+11 4494189326Total 59 2.57065E+11

Fig.2Boxplotofthetotalcostvalues

Intheend,theone‐wayanalysisofvariance(ANOVA)methodisemployedtostatisticallyevalu‐atetheperformanceoftheGA,PSO,HS,andDE.ThisprocedurewasexecutedinMINITABsoft‐wareversion17.3.1.TheANOVAmethodoutputoutlinedinTable7demonstratesthatatacon‐fidence level of 95%, four algorithms reveal no significant differences in themean objectivefunction.Fig.2outlinestherespectiveperformanceofthefouralgorithms.

5. Conclusion

Inthispaper,aT2FCFproblemwasexaminedwitheconomicandenvironmentalcriteria.Intheproposedmodel,thecostsassociatedwithprocessing,materialmovement,energyloss,andtool‐ingwereminimized.Themostsalientbenefitsofthemathematicalmodelareasfollows:CFus‐ing economic and environmental criteria at desirable cost definedbyT2F andworker assign‐mentthroughtheman‐machinerelationshipaspect.TosolvethepresentedCFmodel,aGAwasutilized,inwhichthePSO,HS,andDEalgorithmswereemployedtoevaluatetheoutputsoftheproposedalgorithm.Anotherremarkableadvantageofthepresentresearchisthesolutionofthe15randomproblemsproducedas theoptimalratesof thealgorithmparametersusingTMforeachproblem.TheresultsemanatingfromthealgorithmsrevealthatthePSOalgorithmoutper‐formstheGA,DE,andHSalgorithmsintermsoftheobjectivefunctionon15randomproducedproblems.TheANOVAmethodwasalsoconductedtocomparetheperformanceoftheGA,PSO,HSandDEalgorithmsstatistically.Moreover,thetrendpatterndemonstratedthatPSOoutper‐formedGA,DE,andHSinthemajorityoftheproblems,inwhichastatisticallysignificantdiffer‐encewasnotobservedshowing thatvalid resultswerederivedusing thePSO. In theend, thefollowingrecommendationsforfurtherresearcharemade:

Theapplicationofthemodelcanbeextendedtoastochasticenvironment. Theproposedmodelcanbeconsideredwithreferencetoothercriteriaforsustainability. Theproposedmodelmaybeconsideredinthemulti‐periodplanninghorizon. TheResponseSurfaceMethodology(RSM)maybeimplementedtosettheparameters. Futureresearchcanconcentrateonothermeta‐heuristicalgorithms. ThemodelcanbeextendedtootherT2Fparameters.

DEGAPSOHS

300000

250000

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0

Algorithms

Tota

l Cos

t

Boxplot of Total Cost

Arghish, Tavakkoli-Moghaddam, Shahandeh-Nookabadi, Rezaeian

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