## 5 Calculation Results and Analysis
### Comparison of Liao's Benchmarks  
From the discussion in Section 5.1, the average completion time of the three algorithms of IDABC, HVNS and PSO for solving Carlier and Neron's benchmarks is smaller than that of GA, AIS, and B&B algorithms. Therefore, the algorithms IDABC, HVNS and PSO are selected as the comparison algorithms to solve the Liao problems in this section. In the source literature, the IDABC algorithm was coded in Visual C++ and run on an Intel Pentium 3.06 GHz PC with 2 GB RAM under the Windows 7 operating system, and each problem was tested 20 times and the termination criterion for all the tests was the computation time 100 s. The running environment of the HVNS algorithm in the literature [20] was Intel Core i5 3.3 GHz PC with 4 GB of memory. The algorithm PSO was used to solve the Liao problems, and the algorithm limited its run time to 200 s and each problem runs 20 times independently. The ISP-MCTS algorithm is used to solve the Liao problems in the computer running environment proposed in this paper, and the execution time for each problem is limited to 200 s.  
As shown in Table 7, the average completion time of the ISP-MCTS solution is smaller than that of the IDABC, HVNS and PSO, and the ISP-MCTS further updates the best solution of the three problems in Liao problems. The nonparametric test is proposed, and the null hypothesis H\({}_{0}\) assumes that \(\mu_{\text{SP-MCTS}}\geq\mu_{\text{IDABC}}\), and the alternative hypothesis H\({}_{1}\): \(\mu_{\text{SP-MCTS}}<\mu_{\text{IDABC}}\) with 0.05 significance level. Similarly, the hypotheses H\({}_{0}\): \(\mu_{\text{SP-MCTS}}\geq\mu_{\text{HVNS}}\), H\({}_{1}\): \(\mu_{\text{SP-MCTS}}<\mu_{\text{HVNS}}\). H\({}_{0}\): \(\mu_{\text{ISP-MCTS}}\geq\mu_{\text{PSO}}\), H\({}_{1}\): \(\mu_{\text{SP-MCTS}}<\mu_{\text{PSO}}\). As shown in Table 8, the ISP-MCTS algorithm performs significantly better than the PSO and AIS algorithms, and has certain advantages in terms of solution quality and stability compared with the HVNS algorithm. The six effective algorithms for solving HFSSP in recent years were selected for comparison with ISP-MCTS. The algorithms were: HVNS [20], IDABC [21], PSO [17], GA\({}_{\text{R}}\)[2], ISA [43] and artificial bee colony with permutation (ABC\({}_{\text{P}}\)) [21]. In the computer operating environment of this study, each algorithm solved 150 problems in Section 4.1. For the comparison algorithms, HVNS, DABC, and PSO solved the same problems as reported in the literature, whereas GA\({}_{\text{R}}\), ISA, and ABC\({}_{\text{P}}\) were used to solve complex HFSSP, and the ability to solve these algorithms was verified. The above algorithms  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multicolumn{3}{c}{**ISP-MCTS**} & \multicolumn{3}{c}{**IDABC**} & \multicolumn{3}{c}{**HVNS**} & \multicolumn{3}{c}{**PSO**} \\ \cline{2-13}  & **Ave.** & **Min. ** & **Std. ** & **CPU** & **Ave.** & **Min. ** & **Std. ** & **CPU** & **Ave.** & **Min. ** & **Std. ** & **CPU** & **Ave.** & **Min. ** & **Std. ** & **CPU** \\ \hline j30\(\times\)5e1 & 463.2 & 462 & 0.81 & 14.52 & 465.2 & 463 & 1.80 & 57 & 465.41 & 464 & 1.34 & 29.16 & 474.70 & 471 & 1.42 & 96.16 \\ j30\(\times\)5e  
directly select the optimal parameter combination of the source literature. All the algorithm runtimes were limited to the maximum CPU (t = \(\rho\) mn s) time, and \(\rho\) is: 10, 20, 30. All the above algorithms are programmed in MATLAB 2016a, and each of the 150 examples is run independently for 10 times, and the average of 10 running results is taken as the solution of each example. When \(\rho\) = 30, the best solution C\({}_{\text{min}}\) found by any algorithm is used to calculate the relative percentage increase (RPI). The results are shown in Tables 9-11.