Page - 79 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 68 andthreepopulations ispresented. Foreachcaseofmachinesperstage, thegoal is tofindthemost desirablesolutions thatprovide thebest resultsconsideringbothobjectives. 3.3.DesirabilityFunctions Getting solutions on thePareto front does not resolve themultiobjectiveproblem. There are severalmethodologies to incorporatepreferences in thesearchprocess [25]. These methodologies are responsible for giving guidance or recommendations concerning selectingafinal justifiablesolutionfromtheParetofront[26].OneofthesemethodologiesisDesirability Functions (DFs) thatare responsible formapping thevalueof eachobjective todesirability, that is, tovalues inaunitlessscale in thedomain[0,1]. Letusconsiderthattheobjective fi isZi⊆R; thenaDFisdefinedasanyfunction di :Zi→ [0, 1] that specifies thedesirabilityofdifferent regionsof thedomainZi forobjective fi [25]. The Desirability Index (DI) is the combination of the individual values of the DFs in one preferentialvalueintherange[0,1]. ThehigherthevaluesoftheDFs, thegreatertheDI.Theindividual inthefinal frontwiththehighestDIwillbethebestcandidate tobeselectedbythedecisionmaker[27]. Thealgorithmcalibrationofdesirability functions isapplied to thefinal solutionsof thePareto front to get thebestcombinationofparameters. 4.Model Many published works address heuristics for flow shops that are considered production environments [3,23,28–30]. We consider the following problem: A set J = {1, 2, 3, 4, 5} of jobs availableat time0mustbeprocessedonasetof6consecutivestagesS= {2, 3, 4, 5 , 6, 7}. of the production line, subject tominimizationof totalprocessingtimeCmax andenergyconsumptionof the machinesEop. Eachstage i∈SconsistsofasetMi={1, . . . , mi}ofparallelmachines,where |Mi|≥1. Thesets M2 and M5 have unrelatedmachines and the sets M3, M4, M6, and M7 have identicalmachines. Weconsider threedifferentcasesofmi={2, 4, 6}machines ineachstage. Each job j∈ Jbelongs toaparticulargroupGk. Jobs1,3,4,5∈ JbelongtoG1, job2∈ Jbelongs toG2. Each job isdividedintoasetTjof tj tasks,Tj= { 1, 2, . . . , tj } . Jobsandtasks in thesamegroup canbeprocessedsimultaneouslyonly instage2∈Saccordingto thecapacityof themachine. Instages 3, 4, 5, 6, 7∈S, the tasksmustbeprocessedbyexactlyonemachine ineachstage (Figure2). Each task t∈Tj consistsofasetofbatchesof10kgeach. The taskscanhavedifferentnumbersof batches. Thesetofall the tasks togetherofall the jobs isgivenbyT={1, 2, . . . , Q}. Eachmachinemustprocessanamountof“a”batchesat the timeofassigningatask. Let pil,qbe theprocessing timeof the taskq∈Tjonthemachine l∈Mi at stage i. LetSil,qpbe the setup(adjustment) timeof themachine l fromstage i toprocess the task p∈Tj afterprocessing taskq. Eachmachinehasaprocessbuffer for temporarilystoring thewaiting taskscalled“work inprogress”. Thesetuptimes instage2∈Sareconsideredtobesequencedependent; instages3, 4, 5, 6, 7∈S, theyaresequence independent. Usingthewell-knownthreefieldsnotationα |β |γ forschedulingproblemsintroducedin[31], theproblemisdenotedas follows: FH6, (RM(2),(PM(i))i=34, RM (5) ,(PM(i))i=67)|Ssd2,Ssi37 | { Cmax, Eop } . (2) Encoding Eachsolution(chromosome) isencodedasapermutationof integers. Each integerrepresentsa batch(10kg)ofagiven job. Theenumerationofeachbatch isgiven inarangespecifiedbytheorderof the jobsandthenumberofbatches. 79