## 4 Calibration of the Proposed Algorithms
### Parameter Setting  
The Benchmark problems [42] and Liao problems [17] were introduced to verify the performance of the ISP-MCT algorithm. The scale of the problems ranged from 10 jobs of 5 stages to 30 jobs of 10 stages. In this section, the parameter combination (C,D) is optimized in three segments according to the search space magnitude. The order of magnitude is 0-10\({}^{5}\), 10\({}^{5}\)-10\({}^{6}\) and 10\({}^{6}\)-5 \(\times\) 10\({}^{6}\). The parameter combination (0.1, 32) focuses on the development of potential nodes during the search process. Parameter combination (1, 20,000) in the search process considers the exploration ability of unknown regions, and the combination of parameters (0.5, 10,000) in the search process takes into account the expansion and exploration capabilities. As shown in Table 3, j15c5d2 is selected for optimization analysis at each stage, and each parameter combination is operated independently for 20 times to obtain an average score. The depth is the distance between the deepest node and the root node, and the average depth is the average of 20 independent running depths.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **N-Expan and 4-Rule** & **NEH-LPT(\(\lambda\)) and 4-Rule** & **Random and 4-Rule** \\ \hline \(p\)-Value & 0.00 & 0.00 & 0.00 \\ Significant? (\(p<0.05\)) & Yes & Yes & Yes \\ \hline \hline \end{tabular}
\end{table}
Table 2: The hypothesis tests of 4-Rule with other heuristic schemes.  
Figure 2: Schematic of the HFSSP state evolution: (**a**) the relationship between the number of iterations and the node depth; (**b**) the number of callbacks and average depth of callbacks for each simulation policy.  
As shown in Table 3, in the order of magnitude of 10\({}^{5}\)-10\({}^{6}\), 10\({}^{6}\)-5 \(\times\) 10\({}^{6}\) within the specified time limit, the ISP-MCTS obtains the minimum average completion time and standard deviation, when the parameter combination is (0.1, 32). When applying the parameter combination (0.1, 32), the ISP-MCTS finds the minimum average completion time. The average depth of the search tree at the parameter combination (0.1, 32) is 57, then most of the nodes generated in the scheduling of the 15 jobs of 5 stages have been extended to the tree by the ISP-MCTS. Compared with the other two sets of parameters (0.5, 10,000) and (1, 20,000), the parameter combination (0.1, 32) expands the node deeper.  
In the order of magnitude 10\({}^{5}\)-10\({}^{6}\), 10\({}^{6}\)-5 \(\times\) 10\({}^{6}\) within the specified time limit, when the parameter combination is (0.5, 10,000), the ISP-MCTS finds that the average minimum completion time is optimal, and the algorithm stability is better than other combinations.  
With the increasing magnitude and search time, the exploratory parameter combination advantage becomes more and more obvious. Therefore, at the order of magnitude of 10\({}^{6}\)-5 \(\times\) 10\({}^{6}\), the exploration parameter (1, 20,000) was significantly better than the parameter combination (0.1, 32).  
It can be observed from the above that the order of magnitude is 0-10\({}^{5}\) within the specified time limit. When the parameter combination is (0.1, 32), the ISP-MCTS algorithm has the highest solution quality and the best stability. Therefore, when the order of magnitude is 10\({}^{5}\)-10\({}^{6}\) or 10\({}^{6}\)-5 \(\times\) 10\({}^{6}\), the algorithm selects the parameter combination (0.5, 10,000).