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solutions in B that are dominated by at least one solution in A . The formulation of C ( A , B ) is shown as follows. C ( A , B )= |{ b ∈ B | a ∈ A : a b }| | B | (28) The C ( A , B ) equal to one means that all solutions in B are dominated by some solutions in A . Meanwhile, C ( A , B ) equal to zero means there is no solution in B dominated by the solutions in A . When C ( A , B ) is greater than C ( B , A ) , then the quality of the solution A is better than that of B . The HV -metric of a non-dominated set can be represented by the size of portion of objective space that is dominated by those solutions collectively and bounded above by a reference point. The HV -metric can not only assess the proximity of the non-dominated set but also can assess the diversity of the non-dominated set. The HV -metric needs to be given a reference point, where we use the maximum value of the solutions obtained by all algorithms. When the HV -metric has a larger value, the approximate non-dominated solution set is better. Moreover, considering the randomness of the evolutionary algorithm, this study carries out the Wilcoxon signed-rank test to detect whether the results obtained by different algorithms are different significantly. The significance level is 0.05. A p -value of the Wilcoxon signed-rank test less than 0.05 indicates significant differences between the two algorithms. 4.3. Parameters Setting Table 7 summarizes the parameters of IMOEA/DTL, MOEA/D-MR, SPEA2, and NSGA-II. The population size and the maximum iterations are set as the same value to compare the performance of IMOEA/DTL, MOEA/D-MR, SPEA2, and NSGA-II. The parameters of NSGA-II, SPEA2, and MOEA/D-MR are set based on some literature and the trial and error method [31–35]. Moreover, considering the randomness of the evolutionary algorithm, three algorithms are run ten times for each scheduling problem, and the result is the average of ten times. Table 7. Parameters setting of MOEA/D-MR, Strength Pareto Evolutionary Algorithm 2 (SPEA2), Non-dominated Sorting Genetic Algorithm II (NSGA-II), and MOEA/DTL by Variable Neighborhood Search (IMOEA/DTL).
MOEA/D-MR
SPEA2
NSGA-II
IMOEA/DTL N : 100 Mt : 100 Size of EP: 100 T: 10 Ps: 0.7 Pm: 0.3 Pc: 0.7 M: 5 M1: 10
N : 100 Mt : 100 F: 0.5 CR: 0.2 H: 99 Tm: 10 Tr: 10
N : 100 Size of archive: 100 Mt : 100 Pc: 0.8 Pm: 0.2
N : 100 Mt : 100 Pc: 0.8 Pm: 0.2
M2: 5 M3: 3
4.4. Results and Discussion Table 8 describes the mean C -metric of the solutions obtained by IMOEA/DTL, MOEA/D-MR, SPEA2, and NSGA-II. W, X, Y, and Z represent the solutions obtained by MOEA/D-MR, NSGA-II, SPEA2, and IMOEA/DTL, respectively. From Table 8, it can be seen that C ( Z , W ) is larger than C ( W , Z ) , C ( Z , X ) is larger than C ( X , Z ) , and C ( Z , Y ) is larger than C ( Y , Z ) in 16 scheduling problems. The values of C ( Z , W ) , C ( Z , X ) and C ( Z , Y ) are close to one in the eight scheduling problems. Meanwhile, the values of C ( W , Z ) , C ( X , Z ) , and C ( Y , Z ) are almost equal to zero. The C -metric indicates that other
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