## 3 Model Formulation  
### Illustrative Example  
In order to introduce the issue of portfolio scheduling, taking into account the loss of competences of project team members, let us consider the following illustrative example shown in Figure 1. The project portfolio \(E\) consists of two projects, \(E_{1}\) and \(E_{2}\), with a known precedence between them. Each project contains several tasks. The \(E_{1}\) project requires the implementation of tasks \(Z_{1}\), \(Z_{2}\) and \(Z_{4}\), while the project \(E_{2}\) requires tasks \(Z_{1}\), \(Z_{2}\), \(Z_{3}\) and \(Z_{4}\). The completion time of each project is limited by the \(HE_{i}\) time horizon allocated to it. In the case under consideration, \(HE_{1}=4\) for the project \(E_{1}\), and \(HE_{2}=5\) for the project \(E_{2}\). The available team consists of three employees \(P=\{P_{1},P_{2},P_{3}\}\). The competences of the team members are listed in the \(G^{0}\) competence structure; see Table 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(G^{0}\) & \(P_{1}\) & \(P_{2}\) & \(P_{3}\) \\ \hline \(Z_{1}\) & 3 & 5 & 3 \\ \(Z_{2}\) & 4 & 4 & 4 \\ \(Z_{3}\) & 5 & 4 & 4 \\ \(Z_{4}\) & 4 & 4 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(G^{0}\) competence structure of team \(P\) employees performing tasks \(Z\). Figure 1: Project portfolio activity network \(E\). In general, the competence structure is defined as the matrix \(G^{n}=\left[g_{k,i}^{n}\right]_{k=1\ldots M;j=1\ldots Q}\)' whose elements \(g_{k,i}^{n}\) determine the level of competence of the \(k\)-th employee for the \(i\)-th task, in the \(n\)-th moment of time. In the example under consideration, \(g_{k,i}^{n}\in\{1,\ldots,5\}\). In the presented case, Table 1 means that before starting the implementation of projects (in the 0-th unit of time), the employee doing \(P_{1}\) has the competence to \(Z_{1}\) task at level 3 (\(g_{1,1}^{0}=3\)), to task \(Z_{2}\) at level 4 (\(g_{1,2}^{0}=4\)), to task \(Z_{3}\) at level 5 (\(g_{1,3}^{0}=5\)), etc. The adopted values of \(g_{k,i}^{n}\) determine the time of task execution according to the following principle:  
\[g_{k,i}^{n}=\left\{ \begin{array}{ll}5\text{ or }4&\text{ employee }P_{k}\text{ performs the task }Z_{i}\text{ in }t_{i}=1\text{ time unit}\\ 3\text{ or }2\text{ or }1&\text{ employee }P_{k}\text{ performs the task }Z_{i}\text{ in }t_{i}=\text{ }2\text{ time units}\end{array} \right.\]  
According to this principle, for the competence structure from Table 1, the times of task implementation are given in Table 2. It shows that, for example, an employee \(P_{1}\) performs a task \(Z_{1}\) during 2 units, and tasks \(Z_{2},\ Z_{3},\ Z_{4}\) during 1 unit, etc. The assignment of workers to tasks is represented in the form of a matrix \(X^{n}=\left[x_{k,i,j}^{n}\right]_{k=1\ldots K;i=1\ldots Q;j=1\ldots J}\), where: \(x_{k,i}^{n}\in\{0,1\}\), \(x_{k,i}^{n}=1\) when the employee \(P_{k}\) performs the task \(Z_{i}\) of the \(E_{j}\) project in \(n\)-th moment time; otherwise, \(x_{k,i}^{n}=0\). The following rules for assigning employees to the following tasks shall be adopted:  
* Only one employee can be assigned to one task at a time. * One employee can perform different tasks in the portfolio. * An employee can perform only one task per unit of time. An example of the assignment of team \(P\) employees to the tasks of set \(Z\) is presented in Table 3. This means that in the first moment of time, the employee \(P_{1}\) performs the task \(Z_{1}\), the employee \(P_{2}\) performs the task \(Z_{4}\) and the employee \(P_{3}\) performs the task \(Z_{2}\). According to the learning and forgetting effect, employees who do not provide a specific task lose the knowledge and skills related to it after some period of time. Curves that model this effect are the subject of many studies [9; 27], in which different shapes are given with different degrees of inclination. For simplicity, we assume that the competence level:  
* Increases by 1 after the task is completed by a specific employee.