## 4 Results
### Optimization of Project Duration and Team Composition  
As can be seen above, the preliminary estimation operates with these three main ingredients: project scope, team composition, and project duration. Given that the project scope is fixed (i.e., the NDE does not vary), the other two components are interdependent: changes in the team composition imply different project durations and vice versa: depending on the project duration, different team compositions are required. Manual selection of the best combination of these two ingredients requires time spent on calculations. To make this more efficient, a multiobjective optimization is proposed:  
\[F=L\sum_{m\in T}r^{m}\phi^{m} \rightarrow\min, \tag{16}\] \[L \rightarrow\min,\] (17) \[\Phi=\sum_{m\in T}\phi^{m} \rightarrow\min,\] (18) \[I=C-E \rightarrow\min, \tag{19}\]  
where \(T\) is the set of project team roles; \(m\in T\) is a particular role belonging to the team; \(r^{m}\in[0,1]\) is the normalized hourly rate of role \(m\in T\); \(W^{m}\) is the PWT of role \(m\in T\); \(\phi^{m}\) is the FTE of role \(m\in T\); \(L\) is the project duration; \(C\) is the NDC of the development team \(T_{D}\subset T\); \(E\) is the NDE; \(I\) is the development team idle time. It is worth noting that applying normalization to the cost-related variables brings the following benefits: (a) avoiding disclosure of commercially sensitive information; (b) avoiding too-big values of the objective function; (c) currency-independent calculations with further conversion of the normalized costs to a required currency. One of the main constraints to be satisfied is (14)--the chosen team composition and project duration have to allow the implementation of the project scope estimated as \(E\). Also, it is worth applying restrictions on the project duration:  
\[\left[L\right]_{\min}\leq L\leq\left[L\right]^{\max}. \tag{20}\]  
The other group of constraints is applicable to the project role FTEs:  
\[\left[\phi^{T^{*}}\right]_{\min}\leq\sum_{m\in T^{*}}\phi^{m}\leq\left[\phi^ {T^{*}}\right]^{\max}, \tag{21}\] \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Optimistic** & **Pessimistic** \\ \hline NDE, man-hours & 2007.6 & 3544.8 \\ DWTC & 3.8 * & 3.8 * \\ Development FTE & 7.5 & 7.5 \\ ND-FTE & 7.02 & 7.02 \\ Duration, hours & 1086.8 & 1918.9 \\ Working hours per month & 168 & 168 \\ Duration, months & 6.47 & 11.42 \\ Rounded duration, months & 6.5 & 11.5 \\ \hline \hline \end{tabular}  
* The DWTC value is based on the authors’ expert judgment.  
\end{table} 
Table 5: Duration estimation for the RTBPM-E project. where \(T^{*}\subseteq T\) is a subteam of project team \(T\). For example, a team has to include at least one middle software engineer: \(1\leq\sum\limits_{m\in T^{\text{mid}}}\phi^{m}\) (where \(T^{\text{mid}}\subseteq T\) is a subset of middle software engineers); or, the size of the whole team, \(T\), does not have to exceed 25 FTEs: \(\sum\limits_{m\in T}\phi^{m}\leq 25\). The interrelation of FTEs for different project roles is expressed with this type of constraint:  
\[\gamma_{1}\sum\limits_{m_{1}\in T_{1}}\phi^{m_{1}}\leq\gamma_{2}\sum\limits_{ m_{2}\in T_{2}}\phi^{m_{2}}, \tag{22}\]  
where \(T_{1}\subseteq T\) and \(T_{2}\subseteq T\) are a subteams of \(T\); \(\gamma_{1}>0\) and \(\gamma_{2}>0\) are constants. For example, 1 project manager cannot lead more than 15 team members: \(\sum\limits_{m_{1}\in T\setminus T^{\text{mid}}}\phi^{m_{1}}\leq 15\sum \limits_{m_{2}\in T^{\text{mid}}}\phi^{m_{2}}\) (where \(T^{\text{mgt}}\subseteq T\) is a subteam of project managers). Let us substitute real decision variables \(\phi^{m}\) with the corresponding integer variables \(f^{m}\):  
\[\phi^{m}=\mu^{m}f^{m},\;m\in T, \tag{23}\]  
where \(f^{m}\in\mathbb{N}_{0}=\{0,\,1,\,2,\,\ldots\}\) is a whole number of minimum FTEs for role \(m\in T\); \(\mu^{m}>0\) is a minimal possible step of FTE change for role \(m\in T\).