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

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Ann Oper Res (2013) 206:449–483 477Table 14The computational results of the accuracy metrics for the EEC and modified NSGA-II methodsRun NNS ER GDModifiedNSGA-IIEECModifiedNSGA-IIEECModifiedNSGA-IIEEC1 28 6 0.09677 0.8064516 967.74 4165.7182 29 6 0.06451 0.8064516 645.16 4165.7183 26 6 0.16129 0.8064516 1612.90 4165.7184 29 6 0.06451 0.8064516 645.16 4165.7185 31 6 0 0.8064516 0.00 4165.7186 23 6 0.25806 0.8064516 2580.65 4165.7187 23 5 0.25806 0.8387097 2580.65 4493.55328 28 5 0.09677 0.8387097 967.74 4493.55329 28 6 0.09677 0.8064516 967.74 4165.71810 31 6 0 0.8064516 0.00 4165.718Average 27.6 5.8 0.10967 0.8129032 1096.77 4231.285Standard deviation 2.83627 0.421637 0.09149 0.0136012 914.926 138.22749Table 15methodsThe computational results of the diversity metrics and time for the EEC and modified NSGA-IIRun SM DM CPU time (s)ModifiedNSGA-IIEECModifiedNSGA-IIEECModifiedNSGA-IIEEC1 3071.01 5056.481 2.2E+12 136797.72 376.453 96.1604092 2533.84 5056.481 2.2E+11 136797.72 383.766 90.6647323 3976.60 5056.481 2.2E+10 136797.72 378.672 91.9840354 2457.40 5056.481 2.2E+12 136797.72 364.828 92.1306275 0 5056.481 2.2E+12 136797.72 363.703 96.5602066 4465.15 5056.481 2.2E+11 136797.72 361.250 94.0117337 4465.15 4662.9979 2.2E+12 136798.03 361.250 95.4679848 3095.20 4662.9979 2.2E+13 136798.03 364.953 91.7528849 3095.20 5056.481 2.2E+13 136798.03 373.953 95.65128410 0 5056.481 2.2E+12 136798.03 379.359 93.429075Average 2715.95 4977.7844 2.2E+12 136797.85 370.81875 93.781297Standard deviation 1597.74 165.90704 1.0E+12 0.1579288 8.4836899 2.09983426 Conclusions and future research directionsIn the classic 0–1 MMKP, each object belongs to one and only one group. However, thisassumption is often not valid in the real-world applications as each object may belong tomultiple groups with different degrees of membership. We have applied fuzzy set theory tomodel this problem as a new fuzzy MMKP. An EEC and a modified NSGA-II method wereproposed and customized to solve the fuzzy MMKP.The proposed EEC and modified NSGA-II methods were applied to a real-world projectportfolio selection problem. Different sets of non-dominated solutions were generated us-
478 Ann Oper Res (2013) 206:449–483Fig. 12The Kolmogorov-Smirnov normality test results for the performance metricsing the proposed EEC and the modified NSGA-II methods. The performance of the twoproposed methods was compared with the results from the project portfolio selection problem.The accuracy and diversity metrics proposed by Yu and Gen (2010) wereusedinthiscase study to evaluate the performance of the EEC and the modified NSGA-II methods. Ac-
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A Fuzzy Multidimensional Multiple-Choice Knapsack - Dr. Madjid ...

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