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AdvancesinProductionEngineering&Management ISSN1854‐6250

Volume11|Number4|December2016|pp259–270 Journalhome:apem‐journal.org

http://dx.doi.org/10.14743/apem2016.4.225 Originalscientificpaper

  

A combined zone‐LP and simulated annealing algorithm for unequal‐area facility layout problem 

Xiao, Y.J.a,b, Zheng, Y.c, Zhang, L.M.d,*, Kuo, Y.H.e aJiangsu Key Laboratory of Modern Logistics, School of Marketing and Logistic Management, Nanjing University of Finance and Economics, Nanjing , China  bBusiness school, Nanjing University, Nanjing, China cCollege of Automobile and Traffic Engineering, Nanjing Forestry University, Nanjing, China dSchool of Management and Engineering, Nanjing University, Nanjing, China eStanley Ho Big Data Decision Analytics Research Centre, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong 

  

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

Facilitylayoutproblem(FLP)isoneofwell‐knownNP‐hardproblemsandhasbeendemonstratedtobeusefulinenhancingtheproductivityofmanufactur‐ingsystemsinpractice.Thispaperfocusesontheunequal‐areaFLP(UA‐FLP)whosegoalistolocatedepartmentswithdifferentareaswithinagivenfacilitysoastominimizethetotalmaterialhandlingcost.Anovelapproach,whichwecallacombinedzone‐linearprogramming(zone‐LP)andsimulatedannealingalgorithm, is developed for solving the UA‐FLP. The zone‐LP approach is alayoutconstructiontechniquefortheunequal‐areadepartmentsandconsistsoftwophases.Inthefirstphase,azoningalgorithmisimplementedtodeter‐mine the relative positions between the departments. In this algorithm, forthe sake of problem simplification and computational efficiency, each de‐partmentistreatedasarectanglewithanallowableaspectratioandtheareaofthefacilityisassumedtobeunbounded.Inthesecondphase,byusingtherelativepositionsobtained inthe firstphaseas input,a linearprogramming(LP) model is developed to identify the exact locations and dimensions ofdepartments within the facility with specified sizes while satisfying theirmaximumaspectratiorequirementandtheshapeconstraints.Wealsodesignasimulatedannealingalgorithmtoimprovetheplacingsequence.Finally,ourcomputational results suggest that ourproposed algorithm is efficient com‐paredwiththebestexistingapproachintheliterature.

©2016PEI,UniversityofMaribor.Allrightsreserved.

  Keywords:FacilitylayoutproblemUnequalareaZone‐LPapproachSimulatedannealing

*Correspondingauthor:zhanglm@nju.edu.cn(Zhang,L.M.)

Articlehistory:Received10August2016Revised9October2016Accepted15October2016 

  

1. Introduction  As the competition among the global marketplaces increases rapidly, the provision of high‐qualityproductsorservicesatlowcosttocustomersbecomesmoreandmorecrucialforcom‐panies to capture themassmarket. Facilities planning “determines how an activity’s tangiblefixedassetsbestsupportachievingtheactivityobjectives”[1]andthusplaysanimportantroleinthecostreductionandproductivitygaininindustrialfirms.

Facility planning consists of facilities location, facility system design, facility layout designandhandling systemdesign.As a crucial part of facilitiesplanning, facility layoutdesign is toarrange departments interactingwith each otherwithin a facility so as tominimize the totalmaterialhandlingcost.The layoutdesignofa facilitysignificantly impactsmanufacturingcostand theproductivity andefficiencyof the system.According to [1],materialhandling cost ac‐

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counts for20%to50%of thetotalmanufacturingcost;aneffectiveplanningof facilitiescanreduce such cost by at least 10% to 30%. It is essential to design an efficient layout beforemanufacturingsystemdesignsincefuturerearrangementofdepartmentscanresultinaconsid‐erableexpense.Forthesereasons,facilitylayoutproblem(FLP)hasbeenstudiedextensivelyforoverthreedecades[2‐10].

Therehavebeennumerousstudiesadoptingmathematicalprogrammingapproachestoob‐tain optimal solutions for FLP.Montreuil [11] proposed the firstmixed integer programming(MIP)modelforsolvingtheUA‐FLP.However,theirproposedMIPmodelcanbesolvedfortheproblemswithnomorethansixdepartments.Melleretal.[12]improvedtheformulationoftheMIPmodelofMontreuil[11]byintroducingatighteneddepartmentareaconstraintandseveralclassesofvalid inequalities.Bytheirenhancement, thenumberofdepartmentsof thesolvableproblemsincreasedtoeight.MotivatedbyMelleretal.[12],Sheralietal.[13]furtherimprovedtheaccuracyforapproximatingthedepartmentareasbyusingapolyhedralouterapproximationscheme.Moreover, Sherali et al. [13] investigated the effectivenessof the classes of valid ine‐qualitiesproposedbyMelleretal.[12]andreportedthattheMIPmodelwassolvedmosteffi‐cientlybyincorporatingonlytwoinequalitiesamongthemintotheformulation.Withthisfind‐ing,theproblemswithuptoninedepartmentscouldbeoptimallysolved.Recently,Melleretal.[14]developedanewmathematicalformulationforUA‐FLPbasedonasequence‐pairrepresen‐tation.Inthisformulation,thefeasibleregionoftheproblemwastightenedsuchthatthenum‐berofdepartmentsofthesolvableproblemshasbecomeeleven.

Exactsolutionmethodssuchasmathematicalprogrammingapproachesarenotapplicabletoobtainoptimalsolutionsforlarge‐scaleproblemssinceFLPisNP‐hard[2].Hence,heuristicap‐proachesbasedonMIPmodelshavebeendesignedtoconstructlayoutsolutionsofhighqualityefficiently [15‐17].Oneof thewell‐knownalgorithms for solvingFLP isbasedon theuseof aslicingtree[18,19],whichisabinarytreeusedtoformasubdivisionofarectanglebyrecursivecomputations.Intheslicingtree,eachinnernodecontainsaguillotinecutoperator(whichcutstheareahorizontallyorvertically),andeachleafnoderepresentsadepartment.Alayoutcanbegeneratedaccordingtoaslicingtreebyapplyingasetofguillotinecuts.LiuandMeller[20]de‐velopedasequence‐pairapproach,whereapairofsequencesisusedtodeterminetherelativelocationsbetweendepartments.Byemployingthoserelativelocations,areducedMIPmodelisutilizedtoconstruct the layoutsolutions.Bothofabovemethodareprimarilyproposed in theVLSI (very largeScale Integration)design. Similarly,BozerandWang [21]presentedagraph‐pairtechniquetofixtherelativelocationsbetweendepartmentsandthenalinearprogram(LP)issolvedtoobtainthelayoutsolutions.

Thispaperproposesanovelapproach,whichwecallacombinedzone‐LPandsimulatedan‐nealing(SA)algorithm,tosolvetheproblemefficiently.Thezone‐LPapproachisa layoutcon‐structiontechniquefortheUA‐FLPandconsistsoftwophases.Inthefirstphase,thezonealgo‐rithm is implemented to determine the relative positions between departments. In this algo‐rithm, for thesakeofproblemsimplificationandcomputationalefficiency,eachdepartment isconsideredtobearectanglewithanallowableaspectratioandtheareaofthefacilityisassumedtobeunbounded.Inthesecondphase,byusingtherelativepositionsobtainedinthefirstphaseasinput,anLPissolvedtodeterminetheexactlocationsanddimensionsofdepartmentswithinthefacilitywithspecifiedsizeswhilesatisfyingtheirmaximumaspectratiorequirementandtheshapeconstraints.Furthermore,wealsodevelopanSAalgorithmtoimprovethesolutionquali‐ty.Finally,weconductacomputationalstudytoexaminetheperformanceofourproposedalgo‐rithmandcompareitwithotherexistingsolutionapproachesintheliterature.

Ourpaper isorganizedas follows. In thenextsection,wewilldescribe theproblemofUA‐FLP.Section3willpresentourproposedsolutionmethodology.InSection4,weconductcompu‐tationalexperimentstostudytheperformanceofourproposedalgorithm.Section5concludesourpaper.

2. Problem description  

InUA‐FLP,theareasofthedepartmentsaregivenandtheirdimensionsarelimitedbyamaxi‐mumaspectratio(MAR),whichisdefinedasthelargestallowableratioofthelongestsidetothe

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shortestsideofthedepartment.Theaspectratioofadepartmentrangesfrom1/MARtoMAR.Furthermore,materialflowquantitiesandthelocationofinput/output(I/O)pointsaregiven.Intheproblem,theI/Opointsareassumedtobelocatedatthecentroidsofdepartments.Duringthemanufacturing process,materials are transferred between I/O points for each pair of de‐partments.Theobjectiveistominimizethetotaltraveldistance(TTD),whichisthesumproductofthedistancesbetweendepartmentsandtheirmaterialflowquantities.Forsimplicity,thedis‐tancebetweendepartments isdefinedas therectilineardistancebetween the inputpointandtheoutputpoint.

Table1providesanexampleofasettingofsixdepartmentswiththeirrequiredareas( )andmaximumaspectratio ( )and thematerial flowquantitiesbetweeneachpairof thedepartments.

Table1Anexampleofasettingofsixdepartmentswiththeirrequiredareas( )andmaximumaspectratio( )

andthematerialflowquantitiesbetweeneachpairofthedepartments

Materialflowquantities

1 2 3 4 5 61 — 4 2 5 7 1 16 42 — — 4 3 3 4 16 43 — — — 2 2 6 25 44 — — — — 7 4 36 45 — — — — — 3 36 46 — — — — — — 9 4

3. Solution methodology 

It iswell‐known thatUA‐FLP isNP‐hard.For this reason,wedevelopaheuristic algorithm tosolveUA‐FLPoflargesize.OnecomplicatingfactorofUA‐FLPistodeterminetherelativeposi‐tionsbetweenthedepartmentssubjecttotheconditionthattheirareasdonotoverlap.Tocir‐cumvent this difficulty, we propose a solution methodology that integrates the zone‐LP ap‐proachandanSAalgorithmtoobtaingood‐qualitysolutions.Thezone‐LPapproachisanewlyproposedlayoutconstructiontechniquebasedonanimprovedzonealgorithmandthesolutionforanLP.Whenaninitialsolutionisobtainedfromthepreviousstage,SAisemployedtosearchwithinthesolutionspaceandactstoavoidthesearchprocedurebeingtrappedatalocalopti‐mum.

3.1 Zone‐LP approach 

The zone‐LP approach consists of two phases. In the first phase, for the simplification of theproblemandmoreefficientcomputations,weconsidertheshapeofeachdepartmentasarec‐tanglewithanallowableaspectratio.Asanexample,asshowninFig.1,thesixdepartmentsinTable1aretreatedassquares.Giventheallowableaspectratiosofthedepartments,wedevelopanimprovedzonealgorithmtodeterminetheirrelativepositions.ThezoneconceptforUA‐FLPisfirstintroducedbyXiaoetal.[22]toreduceitscomputationalcomplexityandhasbeenshowntobecomputationallyefficient.Inthesecondphase,byusingtherelativepositionsobtainedinthefirstphaseasinput,wesolveanLPtoobtaintheexactlocationsofthedepartmentswithinthefacility,andtheirdimensionssubjecttotheirshapeconstraintsaresatisfied.

Fig.1Sixdepartmentstobelocatedwithinthefacilitywhereeachofthemisofafixedaspectratio

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3.2 Improved zone algorithm  

Theimprovedzonealgorithmlocatesdepartmentsonebyoneaccordingtoasequencesuchthatauniquelayoutisgenerated.Givenaninitialsequenceforlocatingthedepartmentswithinthefacility, say [1‐4‐2‐5‐3‐6], we begin the algorithm by placing the first department in the se‐quence(i.e.Department1)atthecenterofthefacility.Then,fourzonesthatarerespectivelyontheleft,above,ontheright,andbelowthisdepartmentcanbeidentified.Asanexample,thefourzonesareshadedinthegraphsofFig.2.Thenextstepistodeterminethezonethatthefollowingdepartment in the sequence (i.e.Department4) tobe located.We firstdetermine theoptimallocationsof thisdepartmentwithin the fourzonesbyusingamedianpoint(MP)method.Thedepartmentwillthenbelocatedatthebestzonethatresultsintheminimumtotaldistance.Oncethedepartmentislocatedwithinthefacility,thesetofzoneswillbeupdatedforthedetermina‐tionofthelocationofthenextdepartmentinthesequence.Thatprocedureisrepeateduntilalldepartmentsarelocated.Formoredetailaboutzoneupdatingprocedure,wereferthereadertoXiaoetal.[22].

Fig2Thefourzones(shadedingrey)forthepossiblelocationofthedepartmentinourexample

To generalize the expressionsof the zones, for 1, … , , let be the ‐thdepartment inthesequenceandlet , … , , … , bethesetofzonesthat canbelocated,wherejistheindexofzone.Notethat consistsofonlyonezonethatistheentirerectanglefacility.Foreachzoneof , isrepresentedby , , , ,where , and , arethecoordinatesofthebottomleftcornerandthetoprightcornerofthezone,respectively.Forstepofdeterminingthelocationofthedepartment,theMPmethodisappliedtoobtaintheoptimalpositionof withineachzone in . InthisMPmethod,an ideal locationof thecen‐troidof isfirstidentifiedinsuchawaythatthesumofrectilineardistancesweightedbytheflowquantitiesbetween andthedepartmentsthathavepreviouslybeenlocated,denotedbytheset ,isminimized.Theobjectivefunctionisformulatedasfollows:

Minimize D | | | | (1)

To determine the optimal location, ∗, y∗ , the objective function Eq. 1 is reformulated asEquations Eq. 2 and Eq. 3. Therefore, the values of ∗and ∗can be calculated separately. InEquations Eq. 2 and Eq. 3, and are piecewise linear and convex functions,whereall ( ),for ,arearrangedinincreasingorder,i.e. ’ ’ ⋯ | |’ ’ ’⋯ | |’ . The flow quantities associated with ( ), for , are also sorted by

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, , … , | | ( , , … , | | ).Todeterminethevaluesof∗and ∗,wemakeuseofProp‐

erty1.

Minimize | | | ′| | ′| ⋯ | | | |′ (2)

Minimize | | | ′| | ′| ⋯ | | | |′ (3)

Property 1. Suppose that ∗is the MP and ’satisfies the median condition, i.e.∑∑| | /2and∑| | ∑| | /2. Then ∗= ’. Similarly, if ∗MP and ’, satisfies the

MPcondition∑ ∑| | /2and∑| | ∑| | /2then, ∗= ’.ByProperty1,theoptimallocationof (i.e.theoptimalsolutionforformulationEq.1)can

bedeterminedbysearchingtheMPsuchthatbothofthetotalweightssummingoverthosebe‐foretheMPinthesequenceandsummeringoverthoseaftertheMPinthesequenceisnomorethanthetotalweightsummingoverallthedepartments.Inotherwords,theMPisthefirstpointwherethepartialsumofweights fromthe firstdepartment inthesequenceto itself isno lessthan half of the total weights of all departments. Let , be the partial sums of theweightsatpointsummingfromthefirstdepartmenttothe ‐thdepartmentinthesequencein ‐direction and ‐direction respectively, and be the total weights summing over all depart‐ments.With the sequence [1‐4‐2‐5‐3‐6] for locating thedepartmentswithin the facility in thepreviousexample,weillustrateinFig.3forhowtheaforementionedprocedurecandeterminetheMP.InFig.3,thelocationsoffourdepartmentshavepreviouslybeendetermined.The between ,i.e.Department3andthefourdepartmentsthathavebeenlocatedin ‐directionis:

2 – 5 2 – 0 4| – 0| 2| – 7|. In this example, 11, 1 2,2 2 2 4, 3 2 2 4 8 /2.Hence, theMP is ’ 0and ∗ ’

0 . Similarly, 2 – 0 2| – 0| 2 – 0 4 – 4 . And 1 2 , 2 2 2 4 , 3 2 2 2 6 /2 . Hence, the MP in ‐direction is ’ 0 and∗ ’ 0.Therefore,theoptimallocationofDepartment3is(0,0),whichismarkedbythedashedrectangleinFig.3.

Nevertheless,thisoptimallocationdeterminedforDepartment3,infact,isinfeasiblebecauseitoverlapswiththeotherdepartmentsasshownininFig.3.Inthiscase,wehavetodetermineanewlocationforDepartment3tocontinueourUA‐FLPprocedure.Tothisend,wedeterminethelocationineachzoneforDepartment3suchthatthelocationisclosesttothepreviouslydeter‐minedoptimal location inbothx‐directionandy‐direction.Therationale is that suchpositioncanminimizeTTDforeachzone.Forexample,thebestpositionsforDepartment3inthosecon‐sideredzonesareshowninFig.4(a)‐4(f),whicharerespectivelytheclosestlocationstotheop‐timallocationthatwaspreviouslydetermined.Thedepartmentistobelocatedatthebestposi‐tionthatattainstheminimum amongthosezones.Inthisexample,Department3hasthesame atthepositionsasshowninFig.4(b)andFig.4(d).Thus,thistieisbrokenbyran‐domlypickingoneof the twopositions for locatingDepartment3.The locationof the lastde‐partmentinthesequenceofthisexampleisalsodeterminedinthesameway.Fig.5showsthefinallayoutofthefacility.

Fig.3OptimallocationofDepartment3(representedbyadottedsquare)

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Fig.4Closestlocationineachzonetotheoptimallocation

Fig.5Layoutofthesixdepartmentsoffixedshapes

Todetermineif thedepartmentcanbelocatedattheintendedposition,wealsohavetoconsid‐erthedimensionofeachzone whichcanbemeasuredby

and ,

where and arerespectivelythewidthandtheheightofthezone.Let and bethewidthandtheheightofthedepartmenttobelocated,respectively.Ifazonesatisfiesthecondi‐tionsthat and ,itisfeasibletolocateDepartment inthiszone;otherwise,check if thesubsequentzonecanaccommodateDepartment .Theprocedureof themodifiedzonealgorithmissummarizedasfollows:

A combined zone‐LP and simulated annealing algorithm for unequal‐area facility layout problem 

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Step1. Set 1.Thecentroidof isplacedat(0,0)inthecoordinatesystem.Step2. Set 1.Createthezoneset, 1, … , ,byusingthezoneupdatingpro‐

cedure.Determinetheoptimallocation, ∗, ∗ ,for .Let ∗betheminimum amongthezonescontainedinthezoneset.Set 1, ∗ ∞.

Step3. If /2 ∗ – /2and /2 ∗ /2(i.e. withitscentroidlocatedattheoptimallocationcanfitentirelywithinzone ),gotoStep3.1;otherwise,gotoStep3.2.Step3.1 isoptimallylocatedat x∗, y∗ .GotoStep5.Step3.2 If ≥ and ≥ ,locate attheclosestpossiblepositionin tothe

optimal location. Calculate . If ∗, thenTTD∗ TTD ; else,∗keepsunchanged.

Step3.3 Set 1.If ,gotoStep3;otherwise,gotoStep4.Step4. isplacedinthezonewhose attains ∗,gotoStep5.Step5. If ,gotoStep2;otherwise,allthedepartmentshavealreadybeenlocatedwithinthe

facility.Terminatetheprocedure.

3.3 Linear programming (LP) 

Withfacilitylayoutdeterminedbytheimprovedzoningalgorithminthefirstphase,therelativepositionsofeachpairofdepartmentscanbeobtained.Byusingthisinformation,anLPmodelisformulatedtooptimizethedimensionsofdepartmentswithinaspecifiedfacilitywhilesatisfyingthephysicalconstraintsimposedontheirdimensions.Theadditionalnotationsusedinthisfor‐mulationarelistedasfollows:

Widthofthefacilityonthe floorplan Heightofthefacilityonthefloorplan Totalnumberofdepartmentstobelocated, Indicesusedtorepresentthedepartments, 1, … , and 1, … , . MaterialflowquantityfromDepartment toDepartment ( ). Arearequirementofdepartment Maximumaspectratioofdepartment, UpperandlowerboundofthelengthofDepartmentm inthex‐direction, UpperandlowerboundofthelengthofDepartmentm inthey‐direction, Indicators to denote if Departmentm is on the right/below Department n

respectively.More specifically,rmn =1 ifDepartmentm ison the leftofDe‐partmentn,and0otherwise;smn=1ifDepartmentmistobelowDepartmentn,and0otherwise

Thevalueofthetangentialsupportpointforthepolyhedralouterapproxima‐tiontotheareaconstraints, /2 /2

∆ Numberoftangentialsupportpoints,∆ 2 penaltyfortheviolationofthefloorboundarycondition,

,   HalfofwidthandheightofDepartment,   AmountofviolationofthefloorboundaryconditionforDepartment inthe

‐directionandthe ‐direction,respectively

OurLPmodeltodeterminethedimensionsofthedepartmentsisderivedfromtheMIPfor‐mulatedbySheralietal.[13]andpresentedbelow:

Minimize (4)

subjectto

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∀ (5)

∀ (6)

∀ (7)

∀ (8)

∀   (9)

∀   (10)

2 ∀   (11)

2 ∀   (12)

∀ , , : 1  (13)

∀ , , : 1  (14)

4 2 ∀ ,   (15)

x 2⁄Δ 1

2⁄ 2⁄ ∀ , 0, … , Δ 1  (16)

, , , , , 0 ∀   (17)

, 0 ∀   (18)

ObjectivefunctionEq.4consistsoftwoterms: betweendepartmentsandthepenaltyforviolationofthefloorboundarycondition.ConstraintsEq.5toEq.8determinethe betweendepartments.ConstraintsEq.9andEq.10measuretheviolationofthefloorboundarycondition(in ‐directionand ‐direction,respectively),ifitisinfeasibletolocateallthedepartmentswith‐inthefacility.ConstraintsEq.11andEq.12imposetheboundsonthewidthandheightforeachdepartment, where the upper bounds min , , min , , andlower bounds / , / . Constraints Eq. 13 and Eq. 14 ensure that thedepartmentsdonotoverlapwitheachother.ConstraintsEq.15isthepolyhedralouterapprox‐imationfortheareafunction, 4 ,withthenumberoftangentialsupportpointsequalto∆.Thevalueofthetangentialsupportpoints isgiveninEq.(16).

Asanexample,withtherelativepositionsofthedepartmentsobtainedbytheimprovedzonealgorithm(asshowninFig.5),thefinalfacilitylayoutforlocatingthesixdepartmentswiththeexactdimensionscannowbeobtainedbytheLP.TheirpositionsanddimensionsareshowninFig.6.

Fig.6Layoutsolutionofthesixdepartmentswiththeirdimensionsdetermined

   

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3.4 SA algorithm 

With the facility layoutsolutionobtainedby thezone‐LPapproach,we implementanSAalgo‐rithmtoimprovethequalityofthesolution.TheSAalgorithmthatweadoptisthesameastheprocedureproposedinXiaoetal.[22].Forthedetailofthealgorithm,wereferthereadertoXiaoetal.[22].

4. Computational experiments 

Inthissection,weconductcomputationalexperimentstoevaluatetheperformanceofourpro‐posed solutionmethodology forUA‐FLP – an integratedmethod of zone‐LP approach and SAalgorithm.Weuseasetofwidelytestedinstancesintheliteraturetoexaminetheefficiencyofthe solutionmethodology. There are five instances of different problem sizes included in thecomputationalexperiments:ProblemsO7(with7departments),O8(with8departments)andO9 (with9departments) fromMelleret al. [12] andTwo largest instancesSC30 (with30de‐partments)andSC35(with35departments)fromLiuandMeller[20].

Thecombinedzone‐LPandSAalgorithmisruntentimesforeachcomputationalinstance.Inthefirstphaseofzone‐LPalgorithm,thedimensionsofdepartmentsarefixedwiththeMARbe‐causealldepartments tend tohaveanarrowerrectangularshape togeta lower .Thepa‐rametercombinationused inSAwas 200°, 0.8,and 1000fromXiaoetal. [21].TheresultsonthecomputationalperformanceofourproposedapproachareshowninTable2.Wepresenttheaveragesandthestandarddeviationsofthecoststoexaminetherobustnessofourproposedmethod.Thecomputationalresultssuggestthattheproposedalgorithmappearstobequite robust.For the three small‐sizedproblems (O7,O8,O9), the standarddeviation isquitesmall,rangingfrom0to0.57(lessthan0.5%oftheaverage).Althoughthemagnitudeofthestandarddeviationbecomeslargeforthelarge‐sizedproblems(SC30,SC35),itsrelativeval‐ue isstill small (less than3%of theaverage).For thecomputationalefficiency,ourproposedsolutionmethodologycanobtainthefinalsolutionwithinareasonabletimeframe(lessthan1minuteforthesmall‐sizedproblemsandlessthan4minutesforthelarge‐sizedproblems).Giv‐enthatfacilitylayoutplanningisnotadailypractice(isusuallydeterminedforseveralyears),thiscomputationaltimeisnegligible,andourmethodissuitablefortheapplicationinpractice.

Table2ObjectivevaluesandthecomputingtimeobtainedbytheproposedalgorithmProblem Best Mean Worst Standarddeviation Averagetime(s)O7 89.25 89.25 89.25 0 33.13O8 185.00 185.30 186.00 0.46 36.93O9 185.00 185.45 186.5 0.57 41.80SC30 3441.57 3663.21 3792.71 103.50 180.27SC35 3347.94 3423.70 3555.89 69.06 225.94

Table3Objectivevalues,dimensionsofthefacilities,spaceutilizationsresultingfromthebestsolutionsfoundbyGRAPH[21]andourproposedalgorithm

ProblemGRAPH[21] Theproposedalgorithm

Diff.inTTD(%)TTD

Layoutdimen‐sion(WH)

Spaceutiliza‐tion(%)

TTDLayoutdimension

(WH)Spaceutilization

(%)O7 115.93 96.00 89.25 1213.5 68.52 23.01O8 239.00 96.00 185.00 1216.5 74.24 22.59O9 227.10 96.00 185.00 1516.5 63.03 18.54SC30 3601.20 1512 90.56 3441.57 12.7521.59 59.21 4.43SC35 3351.12 1615 80.00 3347.94 15.1424.7 51.34 0.09

Wealsocompareourproposedsolutionmethodologywithotherexistingalgorithms in theliteratureandusethemasbenchmarks.WefirstcomparethesolutionqualitiesobtainedbyourapproachandtheGRAPHalgorithm[21],whichattainsthebestknownsolutionsforthecompu‐tationalinstances.AsshowninTable3,ourproposedoutperformsGRAPH[21]intermsofTTDforallcomputationalinstances,butthespaceutilizationsresultingfromouralgorithmarelessthan those fromGRAPH. InourUA‐FLPs,departmentsarearrangedwithinanopen‐field floor

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(asshowninFig.4andFig.5)andthelossofconsiderationonspaceutilizationintheobjectivefunctionoftheproposedmodelmainlycausesthepoorperformanceonspaceutilization.There‐fore,amulti‐objectiveoptimizationconsideringminimizationofTTDandmaximizationofspaceutilizationmaybeourfutureresearchtopic.Forreference,thebestlayoutssolutionsobtainedbyourproposedapproachare shown inAppendix1.Theproposedalgorithmalsohas an ad‐vantageofashortercomputingtimeduetoitscapabilityofsearchingintherestrictedsolutionspaced.We conductanother study toexamine theefficiencyofourproposedmethodology. InadditiontoGRAPH[21],wealso includeSEQUENCE[20] for thecomparisonofcomputationalefficienciesofthedifferentapproaches.ComputationalresultsinTable4suggestthatouralgo‐rithmtakesasignificantlyshorter time tosolve theUA‐FLP.More importantly, thecomputingtimeappearstogrowalmost linearlyasthenumberofdepartments increases.Comparedwiththe approaches that appear to have an exponential computing time in the number of depart‐ments, our approach is apparentlymore suitable for problems of a larger size. The linear in‐creaseofcomputingtimemaybeattributedtotheproposedzone‐LPalgorithmwhichcancon‐structthelayoutsolutioneffectively.IneachiterationofSA,thenumberofzonesproducedforputtingdepartmentsbyMPmethodisO(N)accordingtothezonealgorithm.TheproposedLPmodelisjustusedtodeterminethedimensionsofdepartments.Ingeneral,thecomputingtimeisapproximatelyproportional tothenumberofgeneratedzones ineachSAiterationandthusgrowsalmost linearlyas thenumberofdepartments increases.This furtherdemonstrates thevalue of our proposed approach in advancing the computational performance for solvingUA‐FLP.

Table4AveragecomputingtimesoftheSEQUENCE[20],GRAPH[21],andourproposedalgorithm

InstanceComputingtime(s)

SEQUENCE [20] GRAPH [21] OurproposedalgorithmO7 1644.0 228.0 33.13O8 3056.0 390.0 36.93O9 3879.0 222.0 41.80SC30 7282.8 2442.0 180.27SC35 9590.4 4728.0 225.94

5. Conclusion and future research 

ThispaperdealswithUA‐FLPandproposesanovelapproach,whichwecallacombinedzone‐LPandSAalgorithm, for solving large‐sizedUA‐FLP.Thezone‐LPalgorithm isanew layout con‐structionmethodandconsistsoftwophases. Inthefirstphase,theshapesofdepartmentsarefixedtoarectanglewithanallowableaspectratio.Thosedepartmentsoffixedshapesarethenplacedsequentiallybyusinganimprovedzonealgorithm,whereanMPmethodisproposedtolocatingdepartments.Byusingrelativepositionsofthedepartmentsobtainedinthisfirstphase,anLP is formulated todetermine theexact locationsanddimensionsofdepartments.WealsoimplementanSAalgorithmtosearchthesequencesoflocatingthedepartmentswithinthefacili‐tyandtopreventthesearchprocedurefromgettingtrappedatalocaloptimum.Computationalexperiments indicate thatourproposedcombinedzone‐LPandSAhasareasonablygoodper‐formance, comparedwithexistingalgorithms in the literatureonwidely testedcomputationalinstances.

Wealsonotethattherecanbefuturedirectionsextendedfromthisresearch.Inreality,de‐partmentsofirregularshapesarecommonplace;afuturestudymayconsiderthefacilitylayoutproblemwiththeconsiderationofdepartmentswithirregularshapes.Moreover,theresultsofcomputing experiments suggest that our proposed solution methodology may generate solu‐tionsleadingtoalowspaceutilization.Forthiscase,wemayadoptamulti‐objectiveoptimiza‐tionapproachwhosegoalsare tominimize thematerialhandling cost and tomaximize spaceutilization.Thetrade‐offbetweenthesetwoperformancemeasureswouldalsobeaninterestingresearchtopicforinvestigationinthefuture.   

A combined zone-LP and simulated annealing algorithm for unequal-area facility layout problem

Acknowledgement The research has been supported by the National Natural Science Foundation of China (Grant No. 71501090, 71501093), the Natural Science Foundation of Jiangsu Higher Education Institutions of China (Grant No. 14KJD410001), the Natural Science Foundation of Jiangsu Province (Grant No. BK20150566), and the General Re-search Fund of Research Grants Council of Hong Kong (Grant No. 414313).

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Appendix 1 The best layouts obtained by our proposed algorithm for the experimental instances from Meller et al. [10] and Liu and Meller [19].

Layout with seven departments Layout with eight departments Layout with nine departments

Layout with 30 departments Layout with 35 departments

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