## 3 The MOFOA-HS for the Multi-Objective F-MSRCPSP
### Population Initialization
Then, the \(\varepsilon\)-constraint method is adopted for multi-objective optimization of the transformed linear model. Since it is difficult for the CPLEX solver to reach the ideal solution in a reasonable amount of time in the larger scale case, the small instance 10\(3\)4_3 is employed to verify the correctness of the mathematical model in this paper. It can be seen that the optimal solution obtained by the F-MSRCPSP is the same as the exact solution solved by the CPLEX solver, as shown in Figure 7, which validates the effectiveness of the proposed model. ### Comparative Analysis

Next, we compare the MOFOA-HS to other algorithms including the non-dominated sorting genetic algorithm(NSGA-II) [34], the multi-objective fruit fly optimization algorithm (MOFOA) [10], and the multi-objective evolutionary algorithm based on decomposition (MOEA/D) [35]. Two evaluation indicators, Inverted Generational Distance (IGD) [36] and Hypervolume (HV) [37], are used to evaluate the performance of the algorithms. For fair comparisons, the NSGA-II, MOFOA, and MOEA/D also adopt the same solution representation, schedule generation scheme, and initialization mechanism as the MOFOA-HS. Based on the above parameter setting, comparative experiments are carried out by running 10 times independently in all the instances. The results of the comparative experiments are listed in Table 7, where the best results for each instance are marked in bold. Then, we use HV and IGD evaluation indicators to analyze the above obtained results. The results of the comparative experiment are determined in Tables 8 and 9. Figure 8 illustrates the boxplots of HV and IGD values among the MOFOA-HS, NSGAII, MOFOA, and MOEA/D in all instances. Figure 6: Level trend of the parameters. Figure 7: Comparisons of the obtained Pareto fronts. Figure 8: Boxplots of HV and IGD values for comparisons. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{2}{c}{**MOFOA-HS**} & \multicolumn{2}{c}{**NSGAII**} & \multicolumn{2}{c}{**MOFOA**} & \multicolumn{2}{c}{**MOEA/D**} \\ \cline{2-9}  & **Make-Span** & **Cost** & **Make-Span** & **Cost** & **Make-Span** & **Cost** & **Make-Span** & **Cost** \\ \hline



\hline \hline \end{tabular}
\end{table} 
Table 7: Computational results of four algorithms. It can be clearly seen that the computational results of MOFOA-HS are significantly better than those of the other algorithms. Specifically, MOFOA-HS has a larger HV value and a smaller IGD value in both the average and best situation. This indicates that MOFOA-HS has a better comprehensive performance than others to a great extent in all the instances. What is more, the performance of MOFOA-HS, NSGAII, MOFOA, and MOEA/D are analyzed with a \(t\)-test according to the above HV and IGD values. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{2}{c}{**MOFOA-HS**} & \multicolumn{2}{c}{**NSGAII**} & \multicolumn{2}{c}{**MOFOA**} & \multicolumn{2}{c}{**MOEA/D**} \\ \cline{2-9}  & **Best** & **Avg** & **Best** & **Avg** & **Best** & **Avg** & **Best** & **Avg** \\ \hline



\hline \hline \end{tabular}
\end{table} 
Table 8: HV results of four algorithms.