## 3 Competency-Driven Staff Allocation
### Problem Formulation
The competency framework enables the completion of tasks in a process that satisfies the constraints above, in accordance with teacher allocation, \(X\), presented in Table 2. In a permissible teacher allocation. \(X\), teacher \(P_{1}\) carries out a task of course \(Z_{1}\), teacher \(P_{3}\) carries out a task of course \(Z_{2}\), and teacher \(P_{2}\) carries out two tasks of course \(Z_{3}\). Let us consider a situation in which teacher \(P_{2}\) is absent from the execution of tasks (teacher absence). In this situation, it is necessary to modify teacher allocation, \(X\) (Table 2), by substituting the teacher allotted to performing tasks of course \(Z_{3}\). As competency framework \(G\) (Table 1) shows, none of the remaining teachers has the competencies to complete tasks of course \(Z_{3}\). The absence of teacher \(P_{2}\) requires that two tasks of course \(Z_{3}\) should be delegated to other teachers. It is not possible to delegate both tasks to them because they lack competencies enabling the execution of course \(Z_{3}\). In this case, the following question should be considered: Does there exist a competency framework, \(G\), that guarantees a permissible allocation, \(X\), despite the absence of teacher \(P_{2}\)? This competency framework, \(G\), is presented in Table 3. Permissible teacher allocation, \(X\), where one task of course \(Z_{3}\) is allocated to teacher \(P_{1}\) and one to teacher \(P_{3}\) is presented in Table 4. In this situation, constraints (b) and (c) are satisfied (Table 4). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multirow{2}{*}{\(G\)} & \multicolumn{4}{c}{**Competencies Enabling the Execution of Courses, \(Z_{i}\)**} \\ \cline{3-5}  & & \(Z_{1}\) & \(Z_{2}\) & \(Z_{3}\) \\ \hline \multirow{3}{*}{\(P_{1}\)} & \(P_{1}\) & 1 & 1 & 0 \\  & \(P_{2}\) & 0 & 0 & 1 \\  & \(P_{3}\) & 1 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Competency framework, \(G\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multirow{2}{*}{\(X\)} & \multicolumn{4}{c}{**Number of Tasks of the Course**} \\ \cline{3-5}  & & \(Z_{1}\) & \(Z_{2}\) & \(Z_{3}\) \\ \hline \multirow{3}{*}{\(P_{1}\)} & \(P_{1}\) & 1 & 0 & 0 \\  & \(P_{2}\) & 0 & 0 & 2 \\ \cline{1-1}  & \(P_{3}\) & 0 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: A permissible teacher allocation, \(X\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multirow{2}{*}{\(G\)} & \multicolumn{4}{c}{**Competencies Enabling the Execution of Courses, \(Z_{i}\)**} \\ \cline{3-5}  & & \(Z_{1}\) & \(Z_{2}\) & \(Z_{3}\) \\ \hline \multirow{3}{*}{\(P_{1}\)} & \(P_{1}\) & 1 & 1 & 0 \\  & \(P_{2}\) & 0 & 0 & 1 \\  & \(P_{3}\) & 1 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Competency framework, \(G\). It is easy to note that the obtained competency framework, \(G\), presented in Table 3, ensures that tasks are completed despite the absence of teacher \(P_{2}\), which means the framework is robust to the type of disruption under consideration. In the example considered above, competency framework, \(G\) (Table 3), guarantees permissible teacher allocation, \(X\), regardless of which teacher is absent. In other words, the competency framework, \(G\), is fully robust to the absence of a single teacher. This situation is not always possible. Typically, the given competency framework will secure only certain variants of teacher absence. The number of scenarios for a given disruption for which a given competency framework guarantees an acceptable allocation of teachers to courses is determined by what is known as the robustness of a competency framework function.