## 3 The MOFOA-HS for the Multi-Objective F-MSRCPSP
### Population Initialization
In addition, the dataset consists of 42 project instances, including six small sample instances with 10 and 15 activities, as well as 36 large sample instances with 78, 100, and 200 activities. Due to the lack of medium sample instances, we use intercept activities and preserve other parameters of certain large sample instances to generate them. The hardware and software platform parameters are listed as follows: Windows 10, AMD Ryzen 7 4800U CPU, 1.8 GHz, 16G RAM, and MATLAB R2016b. We select and modify 21 instances with various scales as experimental samples. In all the subsequent tests, each algorithm will run independently ten times. ### Parameter Setting

There are three key parameters in the MOFOA-HS, i.e., the number of individuals (NS), the number of searches per individual in the rejoining operator (\(S\)), and the learning rate of the experiences (\(c\)). To investigate the influence of these parameters on the MOFOA-HS, the design-of-experiment (DOE) method [33] is employed. Table 4 shows the four levels of each parameter, and the orthogonal array L\({}_{16}\)(4\({}^{3}\)) is selected. This is shown in \[I^{\prime}_{j}\in[1,\min_{j\in S_{j}}(I_{j^{\prime}})],\ A_{j}\in S2 \tag{16}\]1

For each parameter combination in Table 5, the MOFOA-HS is performed 10 times independently on the instance 50_10_15_13, which is a moderate-scale instance. \(PA_{i}(i=1,2,\dots,16)\) represents the obtained Pareto solutions, and the final solutions (FS) are the union of all sets \(PA_{i}\) with non-dominated sorting. Then, the relevance value (RV) of each parameter combination is calculated as \(RV(i)=|PA_{i}\cap FS|/FS|\). Thus, a larger RV means a better correlation. The relevance ranks of each parameter can be determined in \[I^{\prime}_{j}\in[1,\min_{j\in S_{j}}(I_{j^{\prime}})],\ A_{j}\in S2 \tag{16}\]2 Moreover, the changing trend of each factor parameter level for 50_10_15_13 can be depicted in \[I^{\prime}_{j}\in[1,\min_{j\in S_{j}}(I_{j^{\prime}})],\ A_{j}\in S2 \tag{16}\]3 \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Level** & **NS** & \(\mathbf{S}\) & \(\mathbf{c}\) \\ \hline
1 & 0.01356 & 0.04618 & 0.07614 \\
2 & 0.05792 & 0.05396 & 0.06813 \\
3 & 0.09137 & 0.06785 & 0.04809 \\
4 & 0.08592 & 0.08201 & 0.05764 \\ Delta & 0.07781 & 0.03584 & 0.02805 \\ Rank & 1 & 2 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The significant rank of each parameter for instance 50_10_15_13. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{4}{c}{**Factor Level**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** \\ \hline \(NS\) & 10 & 50 & 100 & 150 \\ \(S\) & 1 & 5 & 10 & 15 \\ \(c\) & 0.05 & 0.1 & 0.15 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Factor levels. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Experiment No.**} & \multicolumn{4}{c}{**Factor Level**} & \multicolumn{2}{c}{**RV (\%)**} \\ \cline{2-5}  & **1** & **2** & **3** & \\ \hline
1 & 1 & 1 & 1 & 0.011 \\
2 & 1 & 2 & 2 & 0.016 \\
3 & 1 & 3 & 3 & 0.000 \\
4 & 1 & 4 & 4 & 0.027 \\
5 & 2 & 1 & 2 & 0.045 \\
6 & 2 & 2 & 1 & 0.059 \\
7 & 2 & 3 & 4 & 0.050 \\
8 & 2 & 4 & 3 & 0.077 \\
9 & 3 & 1 & 3 & 0.067 \\
10 & 3 & 2 & 4 & 0.092 \\
11 & 3 & 3 & 1 & 0.111 \\
12 & 3 & 4 & 2 & 0.100 \\
13 & 4 & 1 & 4 & 0.062 \\
14 & 4 & 2 & 3 & 0.048 \\
15 & 4 & 3 & 2 & 0.110 \\
16 & 4 & 4 & 1 & 0.123 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Orthogonal array and RVs. From Figure 6, it can be seen that a greater \(NS\) and a smaller \(S\) are preferred, which implies that good results need strong exploration and moderate exploitation. As for \(c\), a value that is too large may cause premature convergence. According to the above analysis, the recommended parameter combination for the following numerical experiments is \(NS=100\), \(S=15\), and \(c=0.05\). ### Validation of the Mathematical Model

The software CPLEX 12.9.0 is used to verify the correctness of the F-MSRCPSP model. First, the nonlinear variables are linearized in the F-MSRCPSP model.