Page - 86 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 68 6.3.VarianceAnalysis Varianceanalysis isapplied toevaluate thestatisticaldifferencebetweentheexperimental results and to observe the effect of the parameters on the quality of the results. It is used to determine factors thathaveasignificanteffectandtodiscover themost important factors. Theparametersof the problemareconsideredas factorsandtheirvaluesas levels.Weassumethat there isno interaction betweenfactors. TheF-Ratio is theratiobetweenFactorMeanSquareandMeanSquareresidue.AhighF-Ratio means that this factorsignificantlyaffects theresponse. Thep-valueshowsthestatistical significance of the factors: p-values that are less than0.05havea statistically significant effect on the response variable (RIBS)witha95%confidence level.Accordingto theF-Ratioandp-value, themost important factors in thecaseof twomachinesper stageare themutationoperatorandthecrossoveroperator variable.DFshowsthenumberofdegreesof freedom(Table10). Table10.Relative incrementof thebestsolution(RIBS)analysisofvariancefor twomachinesperstage. Source SumofSquares DF MeanSquare F-Ratio p-Value A:Crossover 1747 1 1747 9.755 0.00213 B:Mutation 11,967 1 11,967 66.833 9.58×10−14 C:Crossoverprobability 1519 1 1519 8.481 0.00411 D:Mutationprobability 654 1 654 3.652 0.05782 E:Population 1732 1 1732 9.673 0.00222 Residuals 27,934 156 179 In the case of fourmachinesper stage, themost important factors are themutationoperator andcrossoverprobability (Table11). Forsixmachinesperstage, themost important factorsare the mutationoperatorandmutationprobability (Table12). Table11.RIBSanalysisofvariance for fourmachinesperstage. Source SumofSquares DF MeanSquare F-Ratio p-Value A:Crossover 760 1 760 4.602 0.03349 B:Mutation 13,471 1 13,471 81.565 6.09×10−16 C:Crossoverprobability 1540 1 1540 31.223 1.00×10−7 D:Mutationprobability 5157 1 5157 9.326 0.00266 E:Population 3395 1 3395 20.557 1.15×10−5 Residuals 25,765 156 165 Table12.RIBSanalysisofvariance forsixmachinesperstage. Source SumofSquares DF MeanSquare F-Ratio p-Value A:Crossover 2587 1 2587 18.39 3.14×10−5 B:Mutation 17,785 1 17,785 126.47 2×10−16 C:Crossoverprobability 2886 1 2886 20.52 1.16×10−5 D:Mutationprobability 4840 1 4840 34.42 2.58×10−8 E:Population 2881 1 2881 20.49 1.18×10−5 Residuals 21,937 156 165 Figures 14–17 showmeans and 95%Least SignificantDifference (LSD) confidence intervals. Figure14showstheresultsobtainedformutationoperators for two, four,andsixmachinesperstage. WecanseethatoperatorDisplacement is thebestmutationamongthethreetestedoperators. Figure15 shows the results for the crossoveroperators,whereweobserve that thePMXoperator is thebest. Theresults for fourandsixstagesaresimilar. 86