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

470 Ann Oper Res (2013) 206:449–483Table 8 The ideal and nadir payoff values of Model (9)Ideal valuesProfit Cost TimeProfit 6.24E+8 229265.0 26.70Cost 2.48E+08 82362 14.00Time 1.14E+08 105242.0 4.000Nadir valuesProfit Cost TimeProfit 6.2E+7 105467.0 12.00Cost 5.18E+08 275440 25.50Time 4.58E+08 247483.0 47.30Table 9 The modified NSGA-II and EEC parametersInitializing the modified NSGA-II parametersInitializing the EEC parametersPop. no. 20 Project no. 12Archive size 35 Time period no. 10Maximum iteration no. 1000 Group no. 2Mutation rate Change linearly in [0.01, 0.05] Objective functions no. 3Cross rate Change linearly in [0.9, 0.7] Maximum cost allowed 247483Alpha 2 Maximum profit allowed 6.24E+8Beta 3 Minimum time allowed 4.0mine whether a candidate solution for a given method is non-dominated or not. The higherthis metric, the more the method has converged towards the real Pareto front.• Error ratio (ER): ER measures the non-convergence of the methods towards the realPareto front. The definition of the ER is given as:∑ Ni=1ER =e i(16)Nwhere N is the number of non-dominated solutions found, and{ 0 if the solution i belongs to the Pareto fronte i =1 otherwiseThe closer this metric is to 1, the less the solution has converged toward RS.• Generational distance (GD): This metric calculates the distance between RS and the solutionset. The definition of this metric is given as:∑ Ni=1GD =d iN(17)where, d i = min p∈PF {√ ∑mk=1 (zi k − zp k )2 } is the minimum Euclidean distance between solutioni and RS where |m| is the number of objective functions.