A new multi-objective particle swarm optimization method for solving ...

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K. Khalili-Damghani et al. / Reliability Engineering and System Safety 111 (2013) 58–75 69 Fig. 6. Comparison of the RS with different approaches in each dimension. (a) Reliability Objective, (a-1) Epsilon-Constraint vs. RS, (a-2) Modified TV-MOPSO vs. RS, (a-3) Modified NSGA-II vs. RS and (a-4) Proposed DSA-MOPSO vs. RS, (b) Cost Objective, (b-1) Epsilon-Constraint vs. RS, (b-2) Modified TV-MOPSO vs. RS, (b-3) Modified NSGA-II vs. RS and (b-4) Proposed DSA-MOPSO vs. RS and (c) Weight Objective, (c-1) Epsilon Constraint VS RS, (c-2) Modified TV-MOPSO VS RS, (c-3) Modified-NSGA-IIVS RS and (c-4) Proposed DSA-MOPSO VS RS.
70 K. Khalili-Damghani et al. / Reliability Engineering and System Safety 111 (2013) 58–75 5.3. Result of well-known benchmark case A more robust procedure is considered for the benchmark case. All algorithms are run 20 times to estimate the real Pareto front of benchmark case. This can acquire a proper infrastructure to compare their performances. The non-dominated solutions of each run formed regenerated Pareto fronts of each algorithm. The regenerated Pareto front of all procedures is presented in Fig. 3. It can be concluded from Fig. 3 that all procedures are capable to re-generate some parts of the Pareto front. Moreover, it can be included that the re-generated Pareto fronts of the DSAMOPSO and the NSGA-II methods have better diversity. The latter case should be analyzed using statistical measures. The performance metrics were selected and calculated for each algorithm. Fig. 4 represents the values of objective functions of different approaches. Section (a) of Fig. 4 illustrates the reliability objective. It can be concluded that the achieved reliability values of the DSAMOPSO method outperform the other approaches. Section (b) of Fig. 4 illustrates the cost objective. It can be concluded that the achieved cost values of the NSGA-II method are smaller than the other approaches. It is clear that there is a trade-off between reliability and cost objectives. As the reliability of the designed system goes higher its cost becomes higher too. For example, the proposed design of the DSAMOPSO method has the highest reliability and cost values and Table 12 Computational results of the accuracy metrics for the benchmark case MORAP. Run NNS ER GD DSA-MOPSO NSGA-II cTV-MOPSO AUGMECON DSA-MOPSO NSGA-II cTV-MOPSO AUGMECON DSA-MOPSO NSGA-II cTV-MOPSO AUGMECON 1 46 17 37 41 0.06 0.65 0.24 0.16 1.03 3.08 1.35 3.08 2 45 19 33 41 0.08 0.61 0.33 0.16 0.79 3.69 1.7 3.69 3 45 18 28 41 0.08 0.63 0.43 0.16 1.34 2.8 208 2.8 4 43 30 28 41 0.12 0.39 0.43 0.16 1.25 1.87 206 1.87 5 41 30 27 40 0.16 0.39 0.45 0.18 0.75 2.59 2.6 2.59 6 44 19 41 41 0.10 0.61 0.16 0.16 0.72 4.87 0.55 4.87 7 45 24 13 40 0.08 0.51 0.73 0.18 0.48 2.59 9.62 2.59 8 47 22 36 40 0.04 0.55 0.27 0.18 0.35 2.22 0.85 2.22 9 47 22 36 40 0.04 0.55 0.27 0.18 0.35 2.22 0.85 2.22 10 46 14 24 40 0.06 0.71 0.51 0.18 0.41 4.11 2.49 4.11 11 48 18 33 41 0.02 0.63 0.33 0.16 0.38 4.38 1.06 4.38 12 48 16 35 41 0.02 0.67 0.29 0.16 0.17 3.12 1.34 3.12 13 43 13 13 41 0.12 0.73 0.73 0.16 0.48 6.38 10.1 6.38 14 47 17 33 41 0.04 0.65 0.33 0.16 0.46 2.61 1.29 2.61 15 48 11 6 41 0.02 0.78 0.88 0.16 0.27 2.84 12.8 2.84 16 48 12 8 41 0.02 0.76 0.84 0.16 0.72 3.8 10.5 3.8 17 47 13 28 41 0.04 0.73 0.43 0.16 0.18 3.45 2.42 3.45 18 48 9 31 41 0.02 0.82 0.37 0.16 0.12 3.23 1.65 3.23 19 46 24 39 41 0.06 0.51 0.20 0.16 0.66 1.64 0.57 1.64 20 44 27 32 41 0.10 0.45 0.35 0.16 0.47 2.58 1.44 2.58 Ave. 45.8 18.8 28.1 40.8 0.07 0.62 0.43 0.17 0.57 3.2 23.9 3.2 Std. Dev. 2.02 6.08 10.3 0.44 0.04 0.12 0.21 0.01 0.34 1.12 62.8 1.12 Table 13 Computational results of the diversity metrics for the benchmark case of MORAP. Run SM DM DSA-MOPSO NSGA-II cTV-MOPSO AUGMECON DSA-MOPSO NSGA-II cTV-MOPSO AUGMECON 1 5.92 4.29 3.06 4.29 54.1 84.9 8944 62.3 2 2.84 5.36 3.12 5.36 54.1 86.2 8944 62.3 3 6.61 5.07 1425 5.07 54.2 86.2 10,000 62.3 4 4.58 4.77 1426 4.77 54.7 86.2 10,205 62.3 5 2.17 5.08 4.92 5.08 55.1 86.2 19,343 62.3 6 3.09 9.36 1.53 9.36 55.2 86.2 19,343 62.3 7 2.4 4.24 9.27 4.24 55.4 86.5 19,343 62.3 8 1.8 3.07 1.68 3.07 55.5 86.5 19,343 62.3 9 1.8 3.07 1.68 3.07 55.5 86.5 19,343 62.3 10 1.75 7.62 3.95 7.62 55.6 86.5 19,343 62.3 11 2.63 9.02 2.23 9.02 55.6 86.5 19,343 62.3 12 1.18 6.16 2.55 6.16 56 86.6 19,343 62.3 13 1.4 10.1 9.48 10.1 56 86.6 19,343 62.3 14 2.85 3.66 2.53 3.66 57.1 86.6 19,343 62.3 15 1.86 2.82 9.79 2.82 57.1 86.6 19,343 62.3 16 5.03 4.88 9.05 4.88 57.4 86.6 19,343 62.3 17 0.91 4.59 4.06 4.59 57.4 86.6 19,343 62.3 18 0.86 4.58 3.4 4.58 57.4 86.6 19,343 62.3 19 2.85 2.48 1.26 2.48 57.4 86.8 19,343 62.3 20 2.23 5.59 2.44 5.59 57.4 86.8 19,343 62.3 Ave. 2.74 5.29 146 5.29 55.9 86.4 17,379 62.3 Std. Dev. 1.61 2.18 438 2.18 1.19 0.4 4039 0
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