## 5 Computational Experiments for Scalability Assessment  
In addition to qualitative experiments aimed at illustrating use cases of the proposed approach, quantitative experiments have also been carried out to assess the scale of solvable problems. For this purpose, an assessment of the problem-solving time of the analysis and synthesis was carried out for a different number: employees and tasks of projects that make up portfolio \(E\). In the experiments, it was assumed that portfolio \(E\) consists of two \(EP\) set projects (\(2/3\) of the number of tasks of portfolio \(E\)) and one additional \(EX\) project (\(1/3\) of portfolio \(E\) tasks). To implement portfolio \(E\), a team of employees with the following numbers is used: \(K=3,4,5,6\). For each experiment (computer parameters: Intel Core i7-M4800 MQ 2.7 GHz, 32 GB RAM), the expected levels of the competence structure \(SG^{*}\) were arbitrarily assumed.  
The conducted experiments allow for assessing the possibilities of searching for project portfolio implementation plans in situations of introducing new \(EX\) orders. The obtained results (see Table 6) demonstrate that the solution to analysis problems is always obtained in less than two seconds. The search for rotational plans (i.e., the synthesis problem) already requires significant computational outlays. Figure 10 shows that the time to solve the synthesis problem increases exponentially with the number of tasks of the project portfolio \(E\).  
The results of the experiments demonstrate that the proposed approach can be used to support decisions in the online mode (i.e., \(<\)10 min) in the synthesis of task rotation schedules for projects of no more than four employees and 27 tasks in portfolio \(E\).  
Figure 9: Solution to the synthesis problem in which the portfolio \(EP\) was executed according to the sequence of assignments \(X^{1}\), \(X^{2}\),..., \(X^{17}\) with task rotation but, however, with an additional \(EX\) order.  
Figure 10: Time to determine the solution to the synthesis problem: results of experiments are presented by solid lines while trend functions are represented by dashed lines.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \begin{tabular}{c} **Number of** \\ **Employees** \\ **\(K\)** \\ \end{tabular} & \begin{tabular}{c} **Number of Project** \\ **Portfolio \(E\) Tasks** \\ \end{tabular} & \begin{tabular}{c} **The Expected** \\ **Degree of** \\ **Competence** \\ **Structure \(SG^{\star}\)** \\ \end{tabular} & \
\begin{tabular}{c} **Time Spent on** \\ **Solving the** \\ **Analysis** \\ **Problem [s]** \\ \end{tabular} &
\begin{tabular}{c} **Time Spent on** \\ **Solving the** \\ **Synthesis** \\ **Problem [s]** \\ \end{tabular} \\ \hline \hline \multirow{4}{*}{3} & \multirow{2}{*}{15} & 45 & 1 & 16 \\ \cline{2-5}  & & 18 & 54 & 1 & 30 \\ \cline{2-5}  & & 21 & 63 & 1 & 57 \\ \cline{2-5}  & & 24 & 72 & 1 & 111 \\ \cline{2-5}  & & 27 & 81 & 1 & 240 \\ \cline{2-5}  & & 30 & 90 & 1 & 600 \\ \hline \multirow{4}{*}{4} & \multirow{2}{*}{15} & 15 & 60 & 1 & 18 \\ \cline{2-5}  & & 18 & 72 & 1 & 41 \\ \cline{2-5}  & & 21 & 84 & 1 & 88 \\ \cline{2-5}  & & 24 & 96 & 1 & 190 \\ \cline{2-5}  & & 27 & 108 & 1 & 360 \\ \cline{2-5}  & & 30 & 120 & 1 & 880 \\ \hline \multirow{4}{*}{5} & \multirow{2}{*}{15} & 15 & 60 & 1 & 55 \\ \cline{2-5}  & & 18 & 90 & 1 & 130 \\ \cline{2-5}  & & 21 & 105 & 1 & 280 \\ \cline{2-5}  & & 24 & 120 & 1 & 740 \\ \cline{2-5}  & & 27 & 135 & 2 & 1901 \\ \cline{2-5}  & & 30 & 150 & 2 & 3651 \\ \hline \multirow{4}{*}{6} & \multirow{2}{*}{15} & 15 & 75 & 1 & 240 \\ \cline{2-5}  & & 18 & 108 & 2 & 440 \\ \cline{1-1} \cline{2-5}  & & 21 & 126 & 2 & 1256 \\ \cline{1-1} \cline{2-5}  & & 24 & 144 & 2 & 2589 \\ \cline{1-1} \cline{2-5}  & & 27 & 162 & 2 & 6987 \\ \cline{1-1} \cline{2-5}  & & 30 & 45 & 2 & \textgreater{}7000 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Analysis and synthesis problem-solving times for different \(E\) project portfolios.