# Mathematical Apparatus of Optimal Decision-Making Based on Vector Optimization
## 7 Statement and Optimal Decision Making with Experimental Data in Problems with Three Parameters
### Construction of a Numerical Model of a System with Three Parameters in the Form of a Vector Problem
#### 7.2.2 Building a Model under the Conditions of Uncertainty  
Construction under the conditions of uncertainty entails the use of the qualitative and quantitative descriptions of the technical system obtained by the \"input-output\" principle in Table 5. Transformation of information (basic data \(y_{3}(X)\), \(y_{4}(X)\)) into functional types \(f_{3}(X),f_{4}(X)\) 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:  
\[I\ =\ |X,Y|\ =\ \{y_{i1}\ y_{i2},\ i\ =\ \overline{1,M}\}. \tag{67}\]  
For each experimental set function \(y_{k}\), \(k=\overline{1,2}\), regression using the method of least squares \(\min\!\sum_{i\ =\ 1}^{M}\,(y_{i}-\overline{y_{i}})^{2}\) in MATLAB is performed. \(A_{k}\), a polynomial defining the interrelationship of factors \(X_{i}=\{y_{1i},y_{2i}\}\) (67) and functions \(\overline{y}_{ki}=\!f(X_{i},A_{k})\), \(k=\overline{1,2}\) is constructed. As a result, we obtain a system of coefficients \(A_{k}=\{A_{0k},A_{1k},\ldots,A_{9k}\}\) which define the coefficients of a polynomial (function):  
\[ \begin{array}{l}\mbox{\it f}_{k}(X,A)\ =\ A_{0k}+A_{1k}x_{1}+A_{2k}x_{1}^{2}+A_{3 k}x_{2}+A_{4k}x_{2}^{2}+A_{5k}x_{3}+A_{6k}x_{3}^{2}+A_{7k}x_{1}*x_{2}+A_{8k}x_{1}* x_{3}+\\ A_{9k}x_{2}*x_{3},\ k\ =\ \overline{1,2}.\end{array}  \tag{68}\]  
As a result of the calculation of the coefficients \(A_{k}\), \(k=1\), we obtain the \(f_{1}(X)\) function:  
\[ \begin{array}{l}\mbox{\it f}_{1}(X)\ =\ 50.0\ +\ 11.55\ \times x_{1}+\ 3.55\ \times x_{2}+\ 1.0\ \times x_{3}\\ \mbox{\it+}\ 0.0144\ \times x_{1}\times x_{2}-0\ \times x_{1}\times x_{3}+0\ \times x_{2}\times x_{3}-\ 0.07\ \times x_{1}^{2}\\ \mbox{\it-}\ 0.07\ \times x_{2}^{2}-\ 0\ \times x_{3}^{2}.\end{array}  \tag{69}\]As a result of the calculations of the coefficients \(A_{k}\), \(k=\)2, we obtain the \(f_{2}(X)\) function:  
\[ \begin{array}{l}f_{2}(X)\ =\ -53.875\ +\ 0.7359\ \times x_{1}+51.3703\ \times x_{2}\\ +\ 0.3516\ \times x_{3}+\ 0.0072\ \times x_{1}\times x_{2}+\ 0.0519\ \times x_{1} \times x_{3}\\ +\ 0.0005\ \times x_{2}\times\ x_{3}-0.0066\times x_{1}^{2}-0.1454\ \times x_{2}^{2}+\ 0.0003\ \times x_{3}^{2} \end{array}  \tag{70}\]  
Parametric restrictions are similar to those of Equation (8). 2.3 Creation of a Mathematical Model of a Technical System under the Conditions of Definiteness and Uncertainty  
For the creation of a mathematical model of the technical system we used: the functions obtained from conditions of definiteness (Equations (64) and (65)) and uncertainty (Equations (69) and (70)), and parametric restrictions (Equation (66)). _Block 4._ We consider the functions of Equations (64), (65), (69) and (70) as the criteria defining the functioning of the technical system. A set of criteria \(K=4\) includes three criteria of \(f_{1}(X),f_{3}(X)\rightarrow\max\) and two of \(f_{2}(X),f_{4}(X)\rightarrow\min\). As a result, the model of the functioning of the technical system is presented as a vector problem of mathematical programming:  
\[ \begin{array}{l}\mathit{opt}\ F(X)\ =\ \{max\ F_{1}(X)\ =\ \{max\ f_{1}(X)\equiv 50.0\ +\ 11.55\ \times x_{1}}\\ +\ 3.55\ \times x_{2}+\ 1.0\ \times x_{3}+\ 0.0144\ \times x_{1}\times x_{2}-\ 0.0\ \times x_{1}\times x_{3}\\ +\ 0.0\ \times x_{2}\times x_{3}-0.07\times x_{1}^{2}-0.07\ \times x_{2}^{2}-\ 0.0\ \times x_{3}^{2},\\ max\ f_{3}(X)\ =\ 55.7188\ -\ 0.1187\ \times x_{1}+\ 0.1844\ \times x_{2}-\\ 0.0438\ \times x_{3}-\ 0.0002\ \times x_{1}\times x_{2}-\ 0.0023\ \times x_{1}\times x_{3}-\\ 0.0011\ \times x_{2}\times\ x_{3}+\ 0.0032\times x_{1}^{2}+0.0634\ \times x -\ 0\times x_{3}^{2}],\end{array}