## 3 The MOFOA-HS for the Multi-Objective F-MSRCPSP
### Heuristic Strategies for Resource Combination Selection
#### 3.3.2 Non-Greedy Selection Strategy  
In order to achieve the goal of non-greedy selection, it is not necessary to start as soon as possible but at as low a cost as possible. We determine a fixed resource combination with the lowest cost and then give up scheduling if there are occupied resources. By analyzing  
Figure 4: The Gantt chart generated by the FSSGS.  
the characteristics of the F-MSRCPSP, we obtain a property that the total cost is minimized if and only if the resource combination with the lowest sum of unit salary for each activity is allocated.  
**Property 1.**: _Cost(\(\Pi\)) is the lowest cost, where \(\Pi\) expresses a solution if and only if \(\forall j,\Pi(j)=\arg\min\limits_{p_{j}\in\Omega_{j}}\ (\frac{AK_{j}^{k}}{\sum\limits_{r\in p_{j}} RK_{r}^{k}}.\sum\limits_{r\in p_{j}}c_{r})\)._  
**Proof.**  
**Sufficiency.**  
\[\begin{array}{ll}Cost(\Pi)&=\sum\limits_{j}C_{\Pi(j)}=\sum\limits_{j}\sum \limits_{t}\sum\limits_{r\in p_{j}}c_{r}\\ &=\sum\limits_{j}(d_{j}\cdot\sum\limits_{r\in p_{j}}c_{r})=\sum\limits_{j}( \frac{AK_{j}^{k}}{\sum\limits_{r\in p_{j}} RK_{r}^{k}}\cdot\sum\limits_{r\in p_{j}}c_{r})=\sum\limits_{j}\min\limits_{p_{j} \in\Omega_{j}}\ (\frac{AK_{j}^{k}}{\sum\limits_{r\in p_{j}} RK_{r}^{k}}\cdot\sum\limits_{r\in p_{j}}c_{r})\end{array} \tag{12}\]  
Therefore, _Cost_(\(\Pi\)) is the optimal cost.  
**Necessity.** By contradiction, suppose there exists a solution \(\Pi^{\prime}\), such that \(\mathit{Cost}(\Pi^{\prime})<\mathit{Cost}(\Pi)\).  
\[\begin{array}{ll}\mathit{Cost}(\Pi^{\prime})=\sum\limits_{j}\sum\limits_{t} \sum\limits_{r\in p^{\prime}_{j}}c_{r}=\sum\limits_{j}\frac{AK_{j}^{k}\cdot \sum\limits_{r\in p^{\prime}_{j}}c_{r}}{\sum\limits_{r\in p^{\prime}_{j}} RK_{r}^{k}}<\mathit{Cost}(\Pi)=\sum\limits_{j}\frac{AK_{j}^{k}}{\sum\limits_{r\in p _{j}} RK_{r}^{k}}\cdot\sum\limits_{r\in p_{j}}c_{r}\end{array} \tag{13}\]  
which means \(\exists\ j\), such that \(\frac{AK_{j}^{k}}{\sum\limits_{r\in p^{\prime}_{j}} RK_{r}^{k}}\cdot\sum\limits_{r\in p^{\prime}_{j}}c_{r}<\frac{AK_{j}^{k}}{\sum \limits_{r\in p_{j}} RK_{r}^{k}}\cdot\sum\limits_{r\in p_{j}}c_{r}\).  
This contradicts the condition that \(\forall j,\Pi(j)=\arg\min\limits_{p_{j}\in\Omega_{j}}\ (\frac{AK_{j}^{k}}{\sum\limits_{r\in p_{j}} RK_{r}^{k}}\cdot\sum\limits_{r\in p_{j}}c_{r})\) and completes the proof. \(\square\)  
Therefore, the NGSS means that the specific resource combination is given for each activity before execution, and it remains unchanged in each time period of the whole execution. In addition, all resources of the given resource combination need to be idle before they can be allocated to the activity. Once allocated to the activity, they will be occupied until the activity is completed. The principle of minimum cost according to the above is the only criterion for the selection of the resource combination that can reduce the cost. Flatten the Pareto front curve, and approach the optimal solution more closely.  
\[\underline{p}_{j}=\arg\min\limits_{p_{j}\in\Omega_{j}}\ (\frac{AK_{j}^{k}}{ \sum\limits_{r\in p_{j}} RK_{r}^{k}}\cdot\sum\limits_{r\in p_{j}}c_{r}) \tag{14}\]  
The generated minimum cost resource combination associated with each activity is unchanged during the execution of the algorithm and can be computed in advance.