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

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Ann Oper Res (2013) 206:449–483 473Fig. 7The 3D-view of the non-dominated solutions for EEC and modified NSGA-IITable 13The upper-bound and the lower-bound values for the generated non-dominated solutionsEECProfit Cost TimeUpper bound 624000000 275440 28Lower bound 234000000 82065 4.7Modified NSGA-IIProfit Cost TimeUpper bound 550000000 208972 24.9Lower bound 397000000 150096 17.1Fig. 8The re-generated 3-D Pareto front for EEC and modified NSGA-II5.5 Diversity measures• Spacing metric (SM): SM measures the uniformity of the spread of the points in thesolution set. In order to calculate SM, ¯d, the mean value of all the d i , is calculated as
474 Ann Oper Res (2013) 206:449–483Fig. 9 The objective values ofEEC and modified NSGA-IIfollows:∑ Ni=1 ¯d = d i(18)NNext, SP, the standard deviation of the closest distances, is calculated as follows:√ ( ∑Ni=1SP =( )¯d − d i ) 2(19)N − 1
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A Fuzzy Multidimensional Multiple-Choice Knapsack - Dr. Madjid ...

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