## 3 Discussion and Results Evaluation
The confidence interval of 95% for algorithms is also illustrated in Figure 5. According to the fact that intervals cross each other, we can statistically conclude that the obtained results from metaheuristic algorithms are equal to the obtained results from GAMS. Therefore, the proposed algorithms are capable of finding globally optimum solutions and are also an appropriate case for large-scale problems. Figure 4: S/N changes for NSGA-II factors. Figure 3: S/N changes for MOSA factors. After proving algorithm efficiency, large-scale problems are applied to evaluate algorithm performance in finding near-optimum solutions for large-scale problems. To analyze the results, we selected 32 samples which are placed in four categories regarding the number of suppliers, cross-docks, factories, and transportation vehicles. The first group includes problems with 3 suppliers, 4 cross-docks, 5 factories, and 7 transportation vehicles. The second group includes problems with 6 suppliers, 5 cross-docks, 7 factories, and 9 transportation vehicles. The third group includes problems with 8 suppliers, 7 cross-docks, 9 factories, and 11 transportation vehicles, and the fourth group includes problems with 10 suppliers, 9 cross-docks, 11 factories, and 13 transportation vehicles. Eight random samples are generated for each group. Then, the performance of the proposed algorithms is evaluated based on comparison criteria. In Figure 6, the proposed criteria are calculated and illustrated for all of the generated random problems. Additionally, Figure 6 represents algorithm efficiency in the SNS criterion. Since the less the SNS, the better it is, NSGA-II has the best performance among all the other algorithms. Furthermore, in smaller size problems, all of the algorithm's efficiencies are similar to each other. Figure 7 illustrates the proposed algorithm performance for the MID criterion. Approximately, in all of the sample problems, NSGA-II performs better and reaches high-quality solutions. Figure 8 shows algorithm efficiency for the Diversity criterion. NSGA-II still stands on the first stage for this criterion. Figure 5: Confidence interval of 95% for algorithms MID. Figure 6: Comparison of the proposed algorithm efficiency in SNS. Figure 9 illustrates the superiority of the proposed NSGA-II to MOSA and MOWFA when running speed is considered. According to this Figure 9, the MOSA algorithm has a similar performance to NSGA-II in small-size problems. However, when the size of the problem increases, the performance of NSGA-II is far better than the two other algorithms, especially MOWFA. In order to evaluate and compare more precisely, statistical analyses are used. At first, we normalize all the obtained results using relative percentage division (RPD). The RDP  
Figure 8: Comparison of the proposed algorithms efficiency in Diversity. Figure 7: Comparison of the proposed algorithm efficiency in MID. Table 5.0 Comparison of the proposed algorithms efficiency in NOEF. value shows the gap between each algorithm solution and the best-obtained solutions. We can calculate RPD as Equation (24):  
Table 5.1  
where i is the algorithm number and j is the example number. To determine algorithm efficiency, we use the confidence interval of 95%. The obtained results are illustrated in Table 5.2 The results show that in the SNS criterion, the NSGA-II algorithm performs significantly better than the two other algorithms. Although the MOSA and MOWFA algorithms are similar to each other in terms of results, the MOSA and NSGA-II algorithms are also similar in terms of MID. In this criterion, MOWFA has the worst performance. In the Diversity criterion, NSGA-II performs far better than MOFA and MOWFA. MOSA and MOFA are also similar to each other in terms of statistics. However, MOSA performs better than MOWFA. In the computational time criterion, MOWFA performs worse than the other two algorithms. In this criterion, NSGA-II has the best performance.