Page - 89 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 68 25.98 26.18 26.38 26.58 16,900 17,100 17,300 17,500 17,700 ܧop (kW) Figure20.Pareto front forsixmachinesperstage. Tables 13–15 show the ordering of the front points regarding a combination of Eop, Cmax, andDesirability Index(DI) foreachPareto front.Wesee, forexample, that theparametercombination 156corresponds toahigherDI, followedby129,etc. Table13.Optimumparametercombinationselectionforbi-objectivegeneticalgorithm(GA)for two machinesperstage. No. Comb. Eop Cmax DI 1 156 6993.41 97.03 0.843 2 129 6968.70 97.15 0.816 3 128 7014.08 97.01 0.782 4 120 7046.87 96.94 0.677 Table14.Optimumparametercombinationselectionforbi-objectiveGAfor fourmachinesperstage. N Comb. Eop Cmax DI 1 156 12,381.55 44.38 0.926 2 129 12,370.82 44.45 0.906 3 147 12,442.75 44.32 0.878 4 128 12,362.17 44.67 0.775 5 120 12,355.79 44.74 0.730 Table15.Optimumparametercombinationselectionforbi-objectiveGAforsixmachinesperstage. N Comb. Eop Cmax DI 1 156 17,018.73 26.07 0.891 2 129 16,961.46 26.12 0.883 3 120 17,108.33 26.05 0.847 4 147 16,953.30 26.19 0.823 5 138 17,189.52 26.01 0.816 6 75 16,925.85 26.23 0.796 Table16showstheparametersof thesecombinations. ThecrossoverPMXhashighervaluesof DI.ThemutationoperatorDisplacement isused ineachof the topthreecombinations. Thecrossover probability ismaintainedat0.9 in the twomostsignificantcombinations. Themutationprobability remainsat thehighestwith0.2 inall three,asdoes thepopulationwith50 individuals. Themost influentialparameterscorrespondto thecombination156,whichhasgreaterDI in three cases (seeTables13–15). Ouranalysis showsthatcrossoverPMXandmutationoperatorDisplacementare thebestamong those tested. Thebestprobabilityofcrossover is0.9andthatofmutation is0.2. Thepopulationof50 individuals is statisticallymoresignificant. 89