## 3 Computational Results  
In this section, we confirm the effectiveness of the new priority rule and test the performance of the new heuristic and the effectiveness of the proposed method in solving PFSPs. The priority rules including PRO, PRLJP1, PRL, and the tie-breaking mechanism including TBO, TBFF [35], and TBLJP1 are taken as references. We combined different priority rules and tie-breaking mechanisms to obtain nine combination algorithms and tested them on the VRF benchmark [36]. All algorithms were coded in Plant Simulation 14.0 and ran on a CPU i3-7100 computer with 4.00G memory. As the time required to solve large-scale examples was much longer than we expected and the demands on the computer's performance were very high, we only calculated the first 37 instances of the VRF benchmark. This is also sufficient for justifying our conclusion, as it is unlikely that a practical production workshop would have more than 500 different products. To measure the solution quality of each algorithm, the relative percentage deviation (RPD) is employed as the performance measure:  
\[RPD=\frac{Cmax_{p}-UB_{p}}{UB_{p}}\times 100\% \tag{7}\]  
where \(Cmax_{p}\) represents the value obtained via heuristics on problem instance \(p\), and \(UB_{p}\) is the upper bound provided by the VRF benchmark. Table 1 shows the RPD values of nine algorithms on the VRF benchmark. As shown in Table 1, the average RPD values of the nine algorithms are 3.92, 3.76, 3.70, 3.80, 3.67, 3.61, 3.76, 3.69, and 3.56, respectively. The newly proposed algorithm NEHL (PRL with TBLJP1) outperforms all existing algorithms with an RPD value of 3.56, achieving the best on 10/37 problems of the VRF benchmark. The new priority rule PRL achieves the best on 18/37 problems compared to PRO and PRLJP1. Only comparing the NEHL and NEHLJP1 algorithms, the average RPD values of the two algorithms are 3.56 and 3.61, respectively, achieving the best results on 27/37 problems and 15/37 problems, respectively. Figure 7: Simulation interface in Plant Simulation software. As shown in Table 1, in the columns of TBO, the priority rules PRO and PRLJP1 and the new rule PRL achieve 3.92, 3.80, and 3.76, respectively. Similarly, the effectiveness of each priority rule can be determined by comparing each sub-column. By using the new PRL priority rule, the performance of TBO and TBLJP1 is significantly improved compared to the other priority rules, PRO and PRLJP1. This is a 9.18% improvement over the initial NEH algorithm [30] and a 1.40% improvement over the best current improved NEH algorithm [8]. To further compare the computational efficiency of nine algorithms, the average CPU time (ACT) is employed as a measure:  
\[ACT_{p}=\frac{\sum_{\forall j}CPU_{j,p}}{J} \tag{8}\]  
where \(CPU_{j,p}\) represents the computation time on instance \(j\) by heuristic \(p\), and \(J\) is the total number of instances. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & \multicolumn{3}{c}{**PR\({}_{\mathbf{O}}\)**} & \multicolumn{3}{c}{**PR\({}_{\mathbf{LJP1}}\)**} & \multicolumn{3}{c}{**PR\({}_{\mathbf{L}}\)**} \\ \cline{2-10}  & **TBO** & **TBFF** & **TBIJP1** & **TBO** & **TBFF** & **TBIJP1** & **TBO** & **TBFF** & **TBIJP1** \\ \hline  
& 1.86 & **1.67** & 2.14 & 1.87 & 1.75 \\ AVG & 3.92 & 3.76 & 3.7 & 3.8 & 3.67 & 3.61 & 3.76 & 3.69 & **3.56** \\ \hline \end{tabAs shown in Table 2, the computational efficiency of the nine algorithms in our solution framework is very slow, which differs greatly from the results in other literature using mathematical methods. The reason is that in the literature, the ACT time almost exclusively depends on the algorithm's complexity. However, in the discrete-event-simulation-based method presented in this paper, the total complexity is composed of the simulation model's complexity and the algorithm's complexity, while the simulation model complexity accounts for the majority.