Page - 85 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 68 According to the selection of a number ofmachines per stage, the set ofmachines is chosen. Themachinesarenumberedfrom1to6. Byconsideringdifferentnumbersofmachinesperstage (2,4, or6),weconduct4860×3=14,580experiments. Ineachof the30runsofeachcombination, thebest individualapplyingdesirability function isobtained. Subsequentvaluesofeachobjective (Cmax and Eop) areusedtocalculateanaverageof30,obtainingasinglevalue foreachobjective. Table9.Parametersusedforcalibration. Factors Levels Population 20,30,50 Crossoveroperators OX,PMX Crossoverprobability 0.5,0.7,0.9 Mutationoperators Displacement,Exchange, Insertion Mutationprobability 0.05,0.1,0.2 Selection Binary tournament Stopcriterion 50 iterations Theperformanceof eachalgorithmis calculatedas the relative incrementof thebest solution (RIBS).TheRIBS iscalculatedwith the followingformula: RIBS= Heusol−Bestsol Bestsol ×100 (6) whereHeusol is thevalueof theobjective functionobtainedbythealgorithm,andBestsol is thebest valueobtainedduringtheexecutionofallpossiblecombinationsofparameters. AllexperimentswereperformedonaPCwithIntelCorei3CPUand4GBRAM.Theprogramming languageused to encode the algorithm isR. It provides advantages, including the scalability and libraries [38]. Thecalibrationof thealgorithmlastedapproximately15daysand9h. 6.2. ResidualAnalysis Toverify that thedataarenormalized, thesuppositionof thesuitabilityof themodelusingthe normality,homoscedasticity,andindependenceof residues isverified. Theresidualsarecalculated accordingto the followingformula [39]: ei= yi−yl, i=1, 2, 3, . . . , n (7) whereyi is theRIBSfor therun iandyi is theaverageof theexperiment. Figure3showsthegraphof thenormalprobabilityof residuals for6machinesper stage. Ascanbeseen (Figure13), thegraph complieswith the assumption of normality. The same results are obtained for 2 and 4machines perstage. Figure13.Graphof thenormalprobabilityofwaste forsixmachinesperstage. 85