7% of the demand. T is randomly generated in the range of 0 and 1. Combining ki, zi, and T we get the tth individual xt = (ki, zi, T). Create initial population randomly.Step 2 ��Calculate the objective function, that is, the total cost and the total shortage quantity of all items.Step Ku 0059436 3 ��Calculate the Pareto front and crowding distance of each individual.Step 4 ��Differential operations: while stopping criterion is not met, implement mutation and crossover for each individual. After that, the number of population is two times the original one.Step 5 ��Genetic operations: select the individuals according to the front rank and crowding distance. Then the number of population is the same as the original one.Step 6 ��When the number of iteration reaches a predefined maximum number, output the optimal results; otherwise, repeat Steps 2�C5.

5. Contrastive Example and Results Analysis 5.1. Basic Data of Numerical ExampleThe data come from Qu et al. [9]. Table 1 describes the items to be replenished and the center warehouse correspondingly. Tables Tables22 and and33 are the related parameters of items and distances between suppliers and warehouse, respectively.Table 1Supply relationship between items and suppliers.Table 2Parameters of items.Table 3Distances between suppliers and warehouse.In the following, two approaches named LP and MOEA are compared. The comparison contains two aspects: the Pareto solutions and some specific solutions obtained by each method. In the meanwhile, three algorithms used in each method are compared with each other. Table 4 reports related parameters of HDE, DE, and GA.

Table 4Parameters of the algorithms.For LP-based approach, we directly set F1max (T, ki, zi) = 10500, F1min (T, ki, zi) = 7500, F2max (T, ki, zi) = 120, and F2min (T, ki, zi) = 0 according to the advice of the decision makers. This approach is also widely used by other scholars (Wee et al. [18]).5.2. Comparisons for LP-Based and MOEA-Based SolutionsIn this section, the above numerical example is handled using LP and MOEA. For LP, the weight of each objective must be assigned firstly. In order to compare with MOEA, the objectives can be converted to single index by setting the total cost and total shortage quantity with the same weights for MOEA when the Pareto solutions are obtained. The best results for LP when w1 = 0.56 are presented in Table 5.

As to MOEA, the highest index after converting is shown in Table 6.Table 5Results for LP with HDE, DE, and GA (w1 = 0.56).Table 6Results for MOEA with HDE, DE, and GA (w1 = 0.56).Table 5 shows that HDE and DE are Batimastat better than GA for LP; Table 6 implies that HDE is better than GA and GA is better than DE for MOEA. In order to further verify the conclusion, we obtained for different w1, w1is set from 0.1 to 0.9 and the results are reported in Table 7.Table 7Results for LP with HDE, DE, and GA (w1 varies).