## 2 Problem Description and Model Setting
### Problem Description  
To enhance the description of the model, the symbols employed in this paper are explicitly defined in Table 1 for better comprehension. A project consists of a set of activities \(A\) with \(A=\left\{A_{1},A_{2},\ldots,A_{n_{j}}\right\}\), where each activity \(A_{j}\in A\) is processed without preemption. An activity cannot be executed until all of its predecessor activities are completed due to precedence constraints between some activities. The activities are accomplished by a set of renewable resources \(R\) with \(R=\left\{R_{1},R_{2},\ldots,R_{n_{r}}\right\}\). Each resource \(R_{r}\in R\) masters several skills with certain familiarity levels from the set of skills \(S\) with \(S=\left\{S_{1},S_{2},\ldots,S_{n_{k}}\right\}\). Each activity \(A_{j}\in A\) requires a certain unit of skill requirement rather than resources. Hence, the resource requirement of each activity is now transformed into the skill requirement. For each resource \(R_{r}\in R\), it can be allocated to an activity \(A_{j}\) only if it masters the required skill of \(A_{j}\). In addition, the unit salary \(c_{r}\) of each resource \(R_{r}\) is constant no matter which skill is performed. The execution of each activity requires a resource combination \(p_{j}\), which consists of several resources that match the skill demand of the activity. The resource combination may vary in terms of resource quantity or specific composition in each discrete time period \(T_{t}\in T\) over the duration of the activity. The schedule for the F-MSRCPSP is to determine a start time period and a resource profile, which is composed of resource combinations \begin{table}
\begin{tabular}{l l} \hline \hline
**Indices** & \\ \hline \(j\) & Index of the activity, where \(j=1,2,\cdots,n_{j}\). \\ \(r\) & Index of the resource, where \(r=1,2,\ldots,n_{r}\). \\ \(t\) & Index of the time period, where \(t=1,2,\ldots,T\). \\ \(k\) & Index of the skill where, \(k=1,2,\cdots,n_{k}\). \\ \(p\) & Index of the resource combination. \\ \hline \hline
**Parameters** & \\ \hline \(A\) & The set of activities. \\ \(R\) & The set of resources. \\ \(S\) & The set of skills. \\ \(n_{j}\) & Number of activities. \\ \(n_{r}\) & Number of resources. \\ \(n_{k}\) & Number of skills. \\ \(T\) & Discrete time horizon. \\ \(A_{j}\) & Activity \(j\). \\ \(R_{r}\) & Resource \(r\). \\ \(T_{t}\) & Time period \(t\). \\ \(S_{k}\) & Skill \(k\). \\ \(p_{jt}\) & Resource combination \(p\) of \(A_{j}\) at \(T_{t}\). \\ \(c_{r}\) & The fixed-unit salary of \(R_{r}\). \\ \(l_{jr}\) & Minimum working period of \(R_{r}\) in the execution process of \(A_{j}\). \\ \(f_{max}\) & Project make-span. \\ \(AK_{j}^{k}\) & The requirement of \(S_{k}\) for \(A_{j}\). \\ \(RK_{r}^{k}\) & The requirement of \(S_{k}\) that \(R_{r}\) supplies per time period. \\ \(s_{j}f_{j}\) & Start and finished period of \(A_{j}\). \\ \(\underline{q}_{j}\) & The lower and upper usage bounds allocated to \(A_{j}\) for \(S_{k}\). \\ \(l_{jr}\) & Working length of \(R_{r}\) in the execution cycle of \(A_{j}\). \\ \(\varphi_{kt}\) & The set of current leftover resources that masters \(R_{k}\) at \(T_{t}\). \\ \(w_{jt}\) & Completed requirement of \(A_{j}\) for \(S_{k}\) at \(T_{t}\). \\ \hline \hline
**Decision Variables** & \\ \hline \(D_{ij}\) & \(D_{ij}=1\), if \(A_{i}\) is the predecessor of \(A_{j}\); else \(D_{ij}=0\). \\ \(x_{rt}\) & Decision variable, \(x_{rt}=1\), if \(R_{r}\), use \(S_{k}\) at \(T_{t}\); else \(x_{rt}=0\). \\ \(y_{rt}\) & Decision variable, \(y_{rt}=1\), if \(R_{t}\) is assigned at \(T_{t}\); else \(y_{rt}=0\). \\ \(z_{jrt}\) & Decision variable, \(z_{jrt}=1\), if \(R_{r}\) is assigned to \(A_{j}\) at \(T_{t}\); else \(z_{jrt}=0\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The notations used in the optimization model for the F-MSRCPSP. at each time period. The duration of the activity is unpredictable and depends on the allocated resource combination per time period during its processing.