# Mathematical Apparatus of Optimal Decision-Making Based on Vector Optimization
## Conclusions
### Formation of Technical Specifications (Source Data) for the Numerical Simulation of the System
#### 8.2.3 Construction under the Conditions of Uncertainty  
Construction under the conditions of uncertainty involves the use of the qualitative and quantitative descriptions of the technical system obtained by the \"input-output\" principle shown in Table 6. Transformation of information (basic data of \(y_{2}(X)\), \(y_{3}(X)\)) to a functional type of \(f_{2}(X),f_{3}(X)\) is carried out by the use of mathematical methods (i.e., regression analysis).  
The basic data of Table 1 are created in the MATLAB system in the form of a matrix:  
\[I\ =\ |X,Y|\ =\ \{x_{i1}x_{i2}y_{i3}y_{i4},\,i\ =\ \overline{1,M}\}. \tag{96}\]  
For each experimental set function \(y_{k}\), \(k=2\), 3 regression was performed using the method of least squares \(\min\sum_{i\ =\ 1}^{M}\left(y_{i}-\overline{y_{i}}\right)^{2}\) in MATLAB. \(A_{k}\), a polynomial defining the interrelationship of the parameters \(X_{i}=\{x_{1i},x_{2i},x_{3i},x_{4i}|\) and functions \(\overline{y}_{ki}=f(X_{i}A_{k})\), \(k=2\), 3, is formed for this purpose.  
As a result of the calculations, we obtain the system of coefficients \(A_{k}=\{A_{0k},\,A_{1k},\,\ldots,\,A_{14k}\}\) which define the coefficients of the quadratic polynomial (function):  
\[\begin{array}{l}f_{k}(X,A)\ =\ A_{0k}+A_{1k}x_{1}+A_{2k}x_{2}+A_{3k}x_{3}+A_{4k}x_{4 }+A_{5k}x_{1}\ast x_{2}+A_{6k}x_{1}\ast x_{3}+A_{7k}x_{1}\ast x_{4}+A_{8k}x_{2 }\ast x_{3}\\ +A_{9k}x_{2}\ast x_{4}+A_{10k}x_{3}\ast x_{4}+A_{11k}x_{1}^{2}+A_{12k}x_{2}^{2} +A_{13k}x_{3}^{2}+A_{14k}x_{4}^{2},\ k\ =\ \ 2,\,3.\end{array} \tag{97}\]  
As a result of the calculations of the coefficients \(A_{k}\), \(k=2\), we obtain the \(f_{2}(X)\) function:  
\[\begin{array}{l}f_{2}(X)\ =\ 875.3+23.893\times x_{1}-30.866\times x_{2}-25. 858\times x_{3}-45\times x_{4}\\ -0.6984\times x_{1}\times x_{2}+0.4276\times x_{1}\times x_{3}+0.6793\times x_{1 }\times x_{4}\\ -0.1167\times x_{2}\times x_{3}+0.2969\times x_{2}\times x_{4}-0.0093\times x_{3 }\times x_{4}\\ +0.0362\times x_{1}^{2}+0.0331\times x_{2}^{2}+2.9158\times x_{3}^{2}+2.4052 \times x_{4}^{2}\end{array} \tag{98}\]As a result of the calculations of coefficients \(A_{k}\), \(k=3\), we obtain the \(f_{3}(X)\) function:  
\[\begin{array}{l}f_{3}(X)\ =\ 43.734+0.6598\times x_{1}+0.4493\times x_{2}-0.3094 \times x_{3}-\\ 1.8334\times x_{4}-0.01\times x_{1}\times x_{2}-0.0062\times x_{1}\times x_{3}+0. 0146\times x_{1}\times x_{4}\\ -0.013\times x_{2}\times x_{3}+0.0121\times x_{2}\times x_{4}-0.0004\times x_{3} \times x_{4}-0.0003\times x_{1}^{2}\\ -0.0002\times x_{2}^{2}+00.0254\times x_{3}^{2}+0.0939\times x_{4}^{2}\\ \text{restrictions }22\leq x_{1}\leq 88\text{, }0\leq x_{2}\leq 66\text{, }2.2\leq x_{3}\leq 8\text{, }2.2\leq x_{4}\leq 8\text{. }\end{array} \tag{98}\]  
The minimum and maximum values of experimental data \(y_{1}(X)\), \(y_{2}(X)\), \(y_{4}(X)\) are presented in the lower part of Table 1. The minimum and maximum values of the functions \(f_{1}(X),f_{2}(X),f_{4}(X)\) slightly differ from experimental data. For comparison, the settlement of these \(f_{4}(X)\) functions at the specified points of \(X\) presented in the right part of the eighth column of Table 7 are given. The index of correlation and coefficients of determination are presented in the lower lines of Table 7.  
Results of the regression analysis of Equations (97)-(99) are further used in the creation of the mathematical model of the technical system.