## 3 The MOFOA-HS for the Multi-Objective F-MSRCPSP
### Population Initialization
Then, the following four disjunctive cases apply:

\[I^{\prime}_{j}\in(\max_{f\in P_{j}}(I_{j^{\prime}}),J+1],\ A_{j}\in S1 \tag{15}\]

\[I^{\prime}_{j}\in[1,\min_{j\in S_{j}}(I_{j^{\prime}})],\ A_{j}\in S2 \tag{16}\]

\[I^{\prime}_{j}\in(\max_{f^{\prime}\in P_{j}}(I_{j^{\prime}}),\min_{j\in S_{j^{ \prime}}}(I_{j^{\prime}})],\ A_{j}\in S3 \tag{17}\]

\[I^{\prime}_{j}\in[1,J+1],\ A_{j}\in S4 \tag{18}\]

Figure 5 shows a project with four situations. \(\lambda\) and the local search of \(\lambda\) are shown, where the black arrows represent a selectable location rejoined in \(\lambda\) for four categories of activities. #### 3.5.2 Empirical Re-Arrangement Operator

The initialization of the activity sequence experience base is as follows:

\[ASB_{0}(j,j^{\prime})=\frac{\sum\limits_{n=1}^{NE}\left(I_{n,j^{\prime}}-I_{ n,j}\right)}{NE};\forall j,j^{\prime}\in A \tag{19}\]

where 0 represents the initialization status of \(ASB\), \(NE\) is the number of superior individuals in the Pareto set to provide the experience value about the activity sequence, and \(I_{n,j}\) is the position of \(A_{j}\) in the \(\lambda\) of the \(n-th\) individual. Figure 5: The rejoining operator. Record the relative position of one activity and the other activities in \(ASB\). In each generation, relative positions are updated with the experiences of the activity sequence provided by \(NE\) superior individuals among the population, which is as follows:

\[ASB_{g}(j,j^{\prime})=(1-c)ASB_{g-1}(j,j^{\prime})+\frac{c}{NE}\sum\limits_{n=1 }^{NE}\left(I_{n,j^{\prime}}-I_{n,j}\right) \tag{20}\]

where \(g\) represents the generation number, and \(c\) is the updating rate of the experience base. Each individual adjusts its activity sequence to obtain a new activity list \(\lambda^{\prime}\) under the guidance of \(ASB\). The step is as follows:

Step 1: Randomly generate two numbers \(1\leq p<q\leq n_{j}\). Step 2: Generate \(\lambda^{\prime}\) by inheriting the subsequence from \(\lambda\) as Equation (21). \[\pi^{\prime}(j)=\pi(j);\forall j\in[1,p-1]\cup\left[q+1,n_{j}\right],\pi^{ \prime}\in\lambda^{\prime},\pi\in\lambda \tag{21}\]

Step 3: Calculate the sum \(RP_{j}\) of the relative position value in the knowledge base of the activity \(\lambda^{\prime}\) to the non-sorted activities as Equation (22). \[RP_{j}=\sum\limits_{j^{\prime}}ASB\big{(}\pi(j),\pi(j^{\prime})\big{)};\forall j,j^{\prime}\in[p,q] \tag{22}\]

Step 4: Fill in the position from \(p\) to \(q\) in \(\lambda^{\prime}\) from small to large according to the \(RP_{j}\). #### 3.5.3 Empirical Re-Selection Operator

The experiential possibility of the resource pattern sub-knowledge base (\(RPB\)) is initialized as follows:

\[RPB_{0}=\frac{\sum\limits_{n=1}^{NE}\rho_{n}}{NE} \tag{23}\]

where \(\rho_{n}\) is the resource pattern list of the \(n-th\) individual in \(NE\) superior individuals, and \(RPB_{0}\) represents the probability of elite individuals selecting the GSS or the NGSS in the initial population phase. In each generation, update the experiences of the resource pattern provided by superior individuals among the population as follows:

\[I^{\prime}_{j}\in[1,\min_{j\in S_{j}}(I_{j^{\prime}})],\ A_{j}\in S2 \tag{16}\]0

## 4 Experiment Study

This paper chooses the intelligent multi-objective project scheduling environment (iMOPSE) benchmark dataset [21] for numerical tests, which is available at [[http://imopse.ii.pwr.wroc.pl/](http://imopse.ii.pwr.wroc.pl/)]([http://imopse.ii.pwr.wroc.pl/](http://imopse.ii.pwr.wroc.pl/)) (accessed on 8 January 2024). In order to adapt to the F-MSRCPSF problem, the following modifications are made to the definition of parameters in instances: the duration of the activity in instances is equal to \(AK_{j}^{k}\); the skill level of resources is equal to \(RK_{r}^{k}\); the \(\underline{q}_{jk}\) and \(\overline{q}_{jk}\) are randomly generated; and it is ensured that \(\overline{q}_{jk}\) will not exceed the number of resources with corresponding skills.