## 5 An Example of Using the Proposed Method
### Simulation for Identifying the Desired Level of Costs
#### 5.2.2 Extension of Variable Domains  
This step of the procedure of solving the inverse form of the problem involves selecting variables whose domains can be extended. As a result of domain extension, the problem is solved again for new domains assigned to some variables, all other conditions being equal. The set of domains is as follows: \(D_{9}\) = [45], \(D_{10}\) = [800,..., 850], \(D_{11}\) = [50,..., 60], \(D_{12}\) = [20,..., 30], \(D_{13}\) = [10,..., 20], and \(D_{14}\) = [7]. The remaining constraints are the same as in the basic variant. An extension of domains for four variables causes an increase in the number of choice nodes in the explored search tree. Consequently, there is a need to use an effective technique to search the space of admissible solutions.  
Table 3 presents the results of searching for an admissible solution for different strategies of variable distribution. Different strategies of variable distribution in the constraint programming with exhaustive search (ES) are compared with regard to the number of nodes checked, depth, and the time needed to find solutions. The calculations were tested on an IntelCore (tm) i5-8300H 2.3-4 GHz, RAM 8 GB platform.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variant** & **Values of Variables** & \(V_{2}\) \\ \hline
1 & \(V_{9}\) = 45, \(V_{10}\) = 830, \(V_{11}\) = 55, \(V_{12}\) = 25, \(V_{13}\) = 15, \(V_{14}\) = 7 & 24.00 \\
2 & \(V_{9}\) = 45, \(V_{10}\) = 830, \(V_{11}\) = 54, \(V_{12}\) = 25, \(V_{13}\) = 15, \(V_{14}\) = 7 & 23.97 \\
3 & \(V_{9}\) = 45, \(V_{10}\) = 830, \(V_{11}\) = 56, \(V_{12}\) = 24, \(V_{13}\) = 15, \(V_{14}\) = 7 & 23.98 \\
4 & \(V_{9}\) = 45, \(V_{10}\) = 830, \(V_{11}\) = 55, \(V_{12}\) = 24, \(V_{13}\) = 15, \(V_{14}\) = 7 & 23.95 \\
5 & \(V_{9}\) = 45, \(V_{10}\) = 830, \(V_{11}\) = 54, \(V_{12}\) = 24, \(V_{13}\) = 15, \(V_{14}\) = 7 & 23.82 \\
6 & \(V_{9}\) = 45, \(V_{10}\) = 830, \(V_{11}\) = 56, \(V_{12}\) = 25, \(V_{13}\) = 14, \(V_{14}\) = 7 & 23.99 \\
7 & \(V_{9}\) = 45, \(V_{10}\) = 830, \(V_{11}\) = 55, \(V_{12}\) = 25, \(V_{13}\) = 14, \(V_{14}\) = 7 & 23.96 \\
8 & \(V_{9}\) = 45, \(V_{10}\) = 830, \(V_{11}\) = 54, \(V_{12}\) = 25, \(V_{13}\) = 14, \(V_{14}\) = 7 & 23.93 \\ \hline \hline \end{tabular}
\end{table}
Table 2: A set of possible solutions.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Distribution Strategy** & **Number of Nodes Checked** & **Depth** & **Time [s]** \\ \hline ES & 67,880 & 65 & 5.25 \\ CP Naïve & 32,852 & 48 & 1.41 \\ CP First-fail & 32,852 & 48 & 1.16 \\ CP Split & 32,852 & 48 & 1.07 \\ \hline \hline \end{tabular}
\end{table}
Table 3: A comparison of strategies for variable distribution.  
The results show that the application of the constraint programming reduces the computational time, which is especially important when there is a larger number of possible solutions. The user can obtain the entire set of solutions or one optimal solution. The extension of variable domains can generate many very similar solutions. The number of solutions can be reduced through considering only the minimum and maximum values of domains. Then, the decision maker can develop the scenario analysis towards the preferable direction to recognize all scenarios ensuring the desired cost. Constraint programming techniques allow to use strategies related to constraint propagation and variable distribution, significantly reducing the set of admissible solutions and the average computational time, improving in this way the interactive properties of a decision support system.