## 6 Numerical Experiments
### Results and Discussion  
To study the performance of the proposed algorithm, we use MATLAB 9.40 (R2018a) software to carry out the experiments. Based on the simulation results, the proposed PSACO has been compared with PSO and ACO in detail. The specific results from the PSO, ACO, PSACO are shown in following figures. Figures 7-10 compare the search performance between the original ACO and PSO algorithms and the hybrid algorithm. The curves shown in the pictures describe the search processes up to the 60 th iteration. We have two findings. Firstly, the three result curves of these algorithms have a similar tendency. The fitness value in the initial phase decreases sharply and searching efficiency reduces quickly in the final stages of the iteration. Secondly, the converging speed of PSACO is the fastest among the three algorithms. It is clearly reflected in Figure 10 when the three algorithms are shown in the same graph in which the black, blue and red curves represent PSO, ACO and PSACO respectively. The PSO's searcher fitness value decreases gradually and unstably at the initial stage until it reaches its optimal solution of 1.4844 at the 20 th iteration, revealing a defect that the algorithm tends to be trapped into local optima. The ACO's searching fitness value quickly decreases and becomes stable at the 4 th iteration with its optimal solution of 1.6588. This states that ACO has the worst result for lacking available feedback information. As for the hybrid algorithm, PSACO rapidly achieves the optimal stable result 1.4432 at the 10 th iteration. The curve of PSACO has the best solution value. It conquers the flaw of redundant iterations that PSO generates, especially where the problem is on a large scale. It utilizes the optimal solutions generated by PSO to update initial pheromone and the searching speed brought by ACO. Hence, Hence, compared with PSO and ACO, PSACO has lower time cost and better performance. Figure 6: Optimization of parameters of ACO. (**a**) ant number; (**b**) pheromone coefficient; (**c**) heuristic coefficient; (**d**) pheromone volatilization coefficient. We undertake an analysis of variance (ANOVA) for three algorithms to validate their objective function values statistically. For each algorithm, we conducted 60 runs. Table 2 shows the result. The test proved to be significant as the \(p\) value is \(0.00<0.05\), meaning that there are great performance disparities among the three algorithms. In order to further study the significant differences between the algorithms, we conducted a Tukey post hoc test which is displayed in Figure 11. The Tukey test shows that PSACO-PSO, ACO-PSO, ACO-PSACO vary differently. Among them PSACO-PSO has closer unanimity and PSACO-ACO and ACO-PSO have great differences. Figure 12 illustrates the boxplot graphs of three algorithms. It further proves that PSACO possess better statistical performance by measuring the dispersion of data used to distinguish it from PSO and ACO. As can be seen from the figure, PSACO is the lowest and also has a larger data spread. Figure 8: Iterative process diagrams of ACO. Figure 7: Iterative process diagrams of PSO. Figure 9: Iterative process diagrams of PSACO. Figure 11: Tukey test at 95% confidence interval (CI). Figure 12: BoxPlot graph for algorithm results. Figure 10: Comparison among PSO, ACO and PSACO. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **Degree of Freedom (DF)** & **Adj SS** & **Adj MS** & **F-Value** & _p-Value_ \\ \hline
**Factor** & 2 & 0.6324 & 0.316205 & 117.61 & 0.000 \\
**Error** & 147 & 0.4759 & 0.002689 & & \\
**Total** & 149 & 1.1083 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Analysis of variance (ANOVA) results for PSO, ACO and PSACO. The experimental results of the different algorithms are summarized in Table 3. For each algorithm, five best supplier combinations in supply chain network are listed.