A Fuzzy Multidimensional Multiple-Choice Knapsack - Dr. Madjid ...

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Ann Oper Res (2013) 206:449–483 475Fig. 10The reference set• Diversification metric (DM): DM measures the spread of the solution set and is calculatedas follows:[∑ NDM = max ( ‖x i − y i ‖ )] 1 2(20)i=1where, ‖x i − y i ‖ is the Euclidean distance between the non-dominated solution x i and thenon-dominated solution y i . The aforementioned metrics as well as the CPU run time werecalculated for both methods in 10 different runs. The results are represented in Table 14and Table 15.As shown in Tables 14 and 15, the computation times for the modified NSGA-II methodis larger than the computation times for the EEC method. The modified NSGA-II hasmore non-dominated solutions in comparison with the EEC method. However, the modifiedNSGA-II had a smaller error ratio in all runs representing the lower non-convergenceof the NSGA-II method towards the RS. The modified NSGA-II method re-generated nondominatedsolutions which were considerably closer to the RS than the EEC method. Themodified NSGA-II method also re-generated non-dominated solutions with smaller averagevalues for SM. This shows that the solutions generated by the modified NSGA-II methodare more uniformly distributed throughout the RS in comparison with solutions generatedby the EEC method. The average DM value for the modified NSGA-II method is much morepromising in comparison with the average DM values for the EEC method. In other words,the modified NSGA-II method results in more scattered non-dominated solutions.Although the conclusion that the non-dominated solutions produced by the modifiedNSGA-II method dominate the solutions produced by the EEC method sounds logical, statisticalanalysis is used next to further test this conclusion.5.6 Statistical analysis and pruning the solutionsThe Kolmogorov-Smirnov test was performed for different metrics to determine whether theresults fit a normal distribution. The Kolmogorov-Smirnov test results for these metrics arepresented in Fig. 12.As shown in Fig. 12, there is not enough evidence to reject the null hypothesis of theKolmogorov-Smirnov test that states the populations of these metrics follow normal distributions.Hence, considering these findings, we performed a parametric statistical test tocheck whether there is a significant difference between the means for different metrics. Analysisof Variance (ANOVA) was used and the following hypotheses of one-way ANOVA wereformulated:
476 Ann Oper Res (2013) 206:449–483Fig. 11The comparison between EEC and RS and between modified NSGA-II and RSH 0 : the means of population metrics are all equalH 1 : the means of population metrics are not all equalThe ANOVA results are presented in Table 16.As shown in Fig. 16, there is not enough evidence to reject the null hypothesis. Therefore,the previous interpretations of Table 14 are statistically proven. Finally, we pruned the nondominatedsolutions generated by the modified NSGA-II method using the additive DEAModel (15). Model (15) produced 10 non-dominated solutions for the modified NSGA-II method presented in Table 17 where 4 of the 10 solutions (DMU 4 ,DMU 6 ,DMU 8 andDMU 10 ) were determined as efficient. This pruning procedure was intended to facilitate theportfolio selection process.
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A Fuzzy Multidimensional Multiple-Choice Knapsack - Dr. Madjid ...

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