## 3 The MOFOA-HS for the Multi-Objective F-MSRCPSP
### Population Initialization
\begin{table}
\begin{tabular}{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
10\_3\_4\_3 & 0.4683 & 0.4683 & 0.4690 & 0.4733 & 0.4735 & 0.4776 & 0.4688 & 0.4714 \\
10\_5\_7\_5 & 0.2406 & 0.2497 & 0.5114 & 0.5309 & 0.3664 & 0.3963 & 0.2576 & 0.2728 \\
10\_7\_8\_6 & 0.2876 & 0.3088 & 0.2887 & 0.3123 & 0.2889 & 0.3099 & 0.3374 & 0.3451 \\
15\_3\_4\_3 & 0.3688 & 0.3776 & 0.4120 & 0.4229 & 0.4363 & 0.4448 & 0.4011 & 0.4116 \\
15\_6\_10\_9 & 0.1151 & 0.1247 & 0.1284 & 0.1379 & 0.1163 & 0.1310 & 0.1263 & 0.1416 \\
15\_9\_11\_9 & 0.2394 & 0.2429 & 0.2508 & 0.2644 & 0.2419 & 0.2648 & 0.2466 & 0.2635 \\
25\_4\_8

**Example 1**. : _The mean value of HV of MOFOA-HS is more than that of the other algorithms._

\[H_{0}:\theta_{mean}>\theta^{\prime}{}_{mean} \tag{25}\]

\[H_{1}:\theta_{mean}\leq\theta^{\prime}{}_{mean} \tag{26}\]

**Example 2**. : _The mean value of IGD of MOFOA-HS is less than that of the other algorithms._

\[H_{0}:\theta_{mean}<\theta^{\prime}{}_{mean} \tag{27}\]

\[H_{1}:\theta_{mean}\geq\theta^{\prime}{}_{mean} \tag{28}\]

Regarding the above example, \(H_{0}\) is the null hypothesis, and \(H_{1}\) is the alternative hypothesis; \(\theta\) and \(\theta^{\prime}\), respectively, represent the evaluation indicator values of MOFOA-HS and the other algorithms. All tests are carried out at the default \(\alpha\) = 0.05 significance level, and the test results for the examples are shown in Table 10, where \"Y\" and \"N\" mean, respectively, that the null hypothesis is accepted, and the null hypothesis fails to be accepted. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{5}{c}{**HV**} & \multicolumn{5}{c}{**IGD**} \\ \cline{3-8}  & & **NSGA-II** & **MOFOA** & **MOEA/D** & **NSGA-II** & **MOFOA** & **MOEA/D** \\ \hline
10\_3\_4\_3 & \(p\)-value & 0.31 & 0.29 & 0.01 & 0.75 & 0.79 & 1.00 \\  & Null hypothesis & Y & Y & N & Y & Y & Y \\
10\_5\_7\_5 & \(p\)-value & 0.96 & 1.00 & 0.95 & 0.14 & 0.93 & 1.00 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\
10\_7\_8\_6 & \(p\)-value & 0.48 & 0.06 & 0.01 & 0.66 & 0.94 & 1.00 \\  & Null hypothesis & Y & Y & N & Y & Y & Y \\
15\_3\_4\_3 & \(p\)-value & 0.72 & 0.62 & 0.78 & 0.77 & 0.81 & 0.61 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\  & \(p\)-value & 0.98 & 0.44 & 1.00 & 0.53 & 0.82 & 0.78 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\
15\_6\_10\_9 & Null hypothesis & Y & Y & Y & Y & Y \\
15\_9\_11\_9 & \(p\)-value & 0.87 & 0.70 & 0.47 & 0.78 & 0.91 & 0.92 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\
25\_4\_8\_9 & \(p\)-value & 0.41 & 0.69 & 0.67 & 0.83 & 0.95 & 0.85 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\
25\_5\_3\_15 & \(p\)-value & 0.93 & 1.00 & 0.95 & 0.91 & 1.00 & 0.84 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\
25\_5\_9\_9 & Null hypothesis & Y & Y & Y & Y & Y & Y \\
50\_4\_14\_9 & \(p\)-value & 0.79 & 0.88 & 0.61 & 0.57 & 1.00 & 0.86 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\
50\_5\_4\_14 & \(p\)-value & 0.97 & 0.00 & 0.20 & 0.20 & 1.00 & 0.78 \\  & Null hypothesis & Y & N & Y & Y & Y & Y \\
50\_10\_15\_13 & \(p\)-value & 0.81 & 0.80 & 0.99 & 0.61 & 1.00 & 0.44 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\
78\_5\_4\_15 & \(p\)-value & 0.29 & 0.76 & 0.15 & 0.89 & 0.93 & 0.91 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\
78\_5\_52\_9 & \(p\)-value & 1.00 & 0.27 & 1.00 & 0.17 & 1.00 & 0.29 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\
78\_10\_42\_15 & Null hypothesis & Y & Y & Y & Y & Y & Y \\
100\_5\_22\_14 & Null hypothesis & Y & Y & Y & Y & Y & Y \\
100\_10\_64\_9 & \(p\)-value & 0.15 & 0.15 & 0.08 & 0.96 & 0.97 & 0.96 \\  & Null hypothesis & Y & Y & Y & Y & Y & Y \\ \hline \hline \end{tabular}
\end{table} 
Table 10: \(t\)-test results at a = 0.05 significance level.