## 3 Model Formulation
### Illustrative Example
* Decreases by 1 for every 2 units of time when an employee does not perform a task requiring this competence. In other words, the level of employees' competences changes during the implementation of subsequent projects. This means that the allocation \(X\) adopted to perform the tasks of portfolio \(E\) affects the structure of competences of the employees involved in its \begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{P_{1}}\) & \(\mathbf{P_{2}}\) & \(\mathbf{P_{3}}\) \\ \hline \(Z_{1}\) & 2 & 1 & 2 \\ \(Z_{2}\) & 1 & 1 & 1 \\ \(Z_{3}\) & 1 & 1 & 1 \\ \(Z_{4}\) & 1 & 1 & 1 \\ \hline \end{tabular}
\end{table} 
Table 2: Execution times of tasks of the \(Z\) set by employees of the \(P\) team. \begin{table}
\begin{tabular}{c c c c} \hline \(\mathbf{X^{1}}\) & \(\mathbf{P_{1}}\) & \(\mathbf{P_{2}}\) & \(\mathbf{P_{3}}\) \\ \hline \(Z_{1}\) & 1 & 0 & 0 \\ \(Z_{2}\) & 0 & 0 & 1 \\ \(Z_{3}\) & 0 & 0 & 0 \\ \(Z_{4}\) & 0 & 1 & 0 \\ \hline \end{tabular}
\end{table} 
Table 3: Assignment of \(X^{1}\) employees of team \(P\) to perform the tasks of the \(Z\) set. implementation. To this end, the concept of competence structure degree \(SG^{n}\) is introduced, which is the sum of competence levels \(g_{k,i}^{n}\) of the \(G^{n}\) structure. Consequently, for the \(G^{0}\) structure in Table 1, the competence structure degree follows \(SG^{0}=49\). Under the influence of the learning effect and forgetting, the degree of competence structure \(SG^{n}\) changes over time, i.e., this degree can:  
* Decrease to the minimum degree \(MINSG=K\times Q\times g_{MIN}=3\times 4\times 1=12\), where \(K\)--number of employees, \(Q\)--number of tasks and \(g_{MIN}\)--minimum level of competence. * Increase to the maximum degree \(MAXSG=K\times Q\times g_{MAX}=3\times 4\times 5=60\), where \(K\)--number of employees, \(Q\)--number of tasks and \(g_{MAX}\)--maximum level of competence. In the present case, \(MINSG=12\) and \(MAXSG=60\). It is arbitrarily assumed that after the completion of the project portfolio, the degree of competence structure should be \(SG\geq 25\) (i.e., around 25% of the maximum value). Assuming these assumptions, an answer to the following question is sought: Is there such a sequence of assignments \(X^{1}\), \(X^{2}\),..., \(X^{n}\) that guarantees the execution of the portfolio of projects \(E\) in the given horizons \(HE_{1}=4\) and \(HE_{2}=5\) while maintaining the degree of competence structure \(SG\geq 25\)? Figure 2 illustrates two examples of project delivery variants, which differ in the method of task assignment, without rotation and with rotation:  
* In variant (a), the rank \(SG^{5}=32\) was obtained, which corresponds to the situation where employees perform tasks without rotation, i.e., a specific task is carried out by only one employee; for example, a task \(Z_{1}\) performs only \(P_{2}\), a task \(Z_{2}\) performs only \(P_{1}\), etc. * In variant (b), the rank \(SG^{5}=34\) is obtained, which corresponds to the situation where employees perform tasks with rotation, i.e., a specific task is carried out by different employees; for example, a task \(Z_{1}\) performs \(P_{1}\) and \(P_{2}\), a task \(Z_{2}\) performs \(P_{2}\), and \(P_{3}\), etc. It follows that without rotation, a lower level of competence structure has been achieved, but this does not threaten the timely implementation of the project portfolio. The degree of competence structure \(SG^{n}\) proves to be crucial when unplanned events occur during the implementation of projects, hereinafter referred to as disruptions. For example, suppose that the disruption under consideration takes the form of an additional \(E_{3}\) project (network of activities as shown in Figure 3), which is expected to start when projects \(E_{1}\) and \(E_{2}\) are completed. An answer to the question is sought: For both early variants' (a) and (b) implementation of the project portfolio, is it possible to accept an additional project order \(E_{3}\), while maintaining the level of competence structure \(SG^{n}\geq 25\)?