# Mathematical Apparatus of Optimal Decision-Making Based on Vector Optimization
## Conclusions
### Formation of Technical Specifications (Source Data) for the Numerical Simulation of the System  
We will consider a problem \"Numerical modeling of the system\" in which data on some set of functional characteristics (definiteness conditions), discrete values of characteristics (an uncertainty condition) and the restrictions imposed on the functioning of the technical system are known [6; 7; 8; 9; 10; 13; 15; 20; 22]. It is given that the system function is defined by four parameters \(X=\{x_{1},x_{2},x_{3},x_{4}\}\), a vector of (operated) variables. Basic data for the solution of the problem are the four characteristics (criterion) of:  
\(F(X)=\{f_{1}(X),f_{2}(X),f_{3}(X),f_{4}(X)\}\), whose size of assessment depends on a vector of \(X\). _The definiteness condition._ For the first and third characteristics of \(f_{1}(X)\) and \(f_{3}(X)\) functional dependence on parameters \(X\) is known (indexing of formulas within the individual section (methods)):  
\[ \begin{array}{l}f_{1}(X)\equiv 269.867\ -\ 1.8746\times x_{1}-\ 1.7469\times x_{2}+0.8939\times x_{3}\\ +1.0937\times x_{4}+0.0484\times x_{1}\times x_{2}-\ 0.0052\times x_{1}\times\ x_{3}-\\ 0.0141\times x_{1}\times\ x_{4}+\ 0.0037\times x_{2}\times\ x_{3}-0.0052\times x _{2}\times\ x_{4}-\\ 0.0002\ \times x_{3}\times\ x_{4}+\ 0.0119\times x_{1}^{2}+\ 0.0035\ \times x_{2}^{2}-\ 0.002\times x_{3}^{2}\\ -0.0042\times x_{4}^{2},\end{array}  \tag{84}\]  
\[ \begin{array}{l}f_{4}(X)\ =\ 19.253\ -\ 0.0081\ \times x_{1}-\ 0.7005\ \times x_{2}-\ 0.3605\ \times x_{3}\\ +\ 0.9769\ \times x_{4}+\ 0.0126\ \times x_{1}\times x_{2}+\ 0.0644\ \times x_{1}\times\ x_{3}-0\times x_{1}\times\ x_{4}\\ +\ 0.0396\ \times x_{2}\times\ x_{3}+\ 0.0002\ \times x_{2}\times\ x_{4}+\ 0.0004\ \times x_{3}\times\ x_{4}-\\ 0.0016\ \times x_{1}^{2}+\ 0.0027\ \times x_{2}^{2}+\ 0.0045\ \times x_{3}^{2}-\ 0.0235\ \times x_{4}^{2},\end{array}  \tag{85}\]  
restrictions: \(22\leq x_{1}\leq 88\), \(0\leq x_{2}\leq 66\), \(2.2\leq x_{3}\leq 8.8\), \(2.2\leq x_{4}\leq 8.8\)  
_The uncertainty condition._ For the second and fourth characteristic results of the experimental data, the sizes of parameters and corresponding characteristics are known. Numerical values of parameters \(X\) and characteristics of \(y_{2}(X)\) and \(y_{4}(X)\) are presented in Table 7. \begin{table}
\begin{tabular}{c c c c c c} \hline \(x_{1}\) & \(x_{2}\) & \(x_{3}\) & \(x_{4}\) & \(y_{2}(X)\)\(\rightarrow\)_min_ & \(y_{3}(X)\)\(\rightarrow\)_max_ \\ \hline  
\hline \end{tabular}
\end{table} 
Table 7: Numerical values of parameters and characteristics of the system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(x_{1}\) & \(x_{2}\) & \(x_{3}\) & \(x_{4}\) & \(y_{2}\)(X) \(\rightarrow\)_min_ & \(y_{3}\)(X) \(\rightarrow\)_max_ \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 7: _Cont._In the decision, from the assessment size of the first and third characteristic (criterion), it is possible to obtain: \(f_{1}(X)\)\(\rightarrow\)max\(f_{3}(X)\)\(\rightarrow\)max, and for the second and fourth characteristic: \(y_{2}(X)\)\(\rightarrow\)min \(y_{4}(X)\)\(\rightarrow\)min. Parameters \(X=\{x_{1},\,x_{2},\,x_{3},\,x_{4}\}\) change according to the following limits:  
\[x_{1}\in[22.\ 55.\ 88.],\,x_{2}\in[0.\ 33.\ 66.],\,x_{3}\in[2.2\ 5.5\ 8.8],\,x_{4}\in[2.2 \ 5.5\ 8.8].\]  
The requirements are as follows: to construct a model of the system in the form of a vector problem, to solve a vector problem with equivalent criteria, to choose a priority criterion, to establish the numerical value of the priority criterion, and to make the best decision (optimum) with a specified priority criterion. Note that using the MATLAB system, the author developed the software for the decision making of a vector problem of mathematical programming. The vector problem includes four variables (parameters of the technical system): \(X=\{x_{1},\,x_{2},\,x_{3},\,x_{4}\}\) and four criteria (characteristics) of \(F(X)=\{f_{1}(X),f_{2}(X),f_{3}(X),f_{4}(X)\}\).