# 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
As a result, the solutions of the vector problem of mathematical programming of Equations (71)-(73) with equivalent criteria and the \(\lambda\)-problem corresponding to Equations (74)-(75) are obtained:  
\(\mathbf{X}^{0}=\{X^{0},\lambda^{0}\}=\{X^{0}=\{x_{1}=33.027,\,x_{2}=69.54,\,x_{3}= 25.0,\,\lambda^{0}=0.4459\}\)] is an optimum point of the design data of the technical system. Point \(\mathbf{X}^{0}\) is presented in Figure 6. \(f_{k}(X^{0})\), \(k=\overline{1,K}\) represents sizes of criteria (characteristics of technical system):  
\[\{f_{1}(X^{0})=321.5,f_{2}(X^{0})=2901.7,f_{3}(X^{0})=370.2,f_{4}(X^{0})=19.1\}, \tag{77}\]  
and \(\lambda_{k}(X^{0})\), \(k=\overline{1,K}\) represents sizes of relative estimates:  
\[\{\lambda_{1}(X^{0})=0.4459,\,\lambda_{2}(X^{0})=0.4459,\,\lambda_{3}(X^{0})= 0.4459,\,\lambda_{4}(X^{0})=0.9609\}, \tag{78}\]  
\(\lambda^{0}=0.4459\) is the maximum lower level among all relative estimates measured in relative units: \(\lambda^{0}=min\) (\(\lambda_{1}(X^{0})\), \(\lambda_{2}(X^{0})\), \(\lambda_{3}(X^{0})\), \(\lambda_{4}(X^{0})\)) = 0.4459. A relative assessment, \(\lambda^{0}\), is called the guaranteed result in relative units, i.e., \(\lambda_{k}(X^{0})\). According to the characteristics of the technical system \(f_{k}(X^{0})\), it is impossible to improve, without worsening other characteristics. **Discussion**. We note that according to Theorem 1, at point \(\mathbf{X}^{0}\) criteria 1, 2, 3 are contradictory. This contradiction is defined by the equality of \(\lambda_{1}(X^{0})=\lambda_{2}(X^{0})=\lambda_{3}(X^{0})=\lambda^{0}=0.4459\), and for other criteria an inequality of \(\{\lambda_{4}(X^{0})=0.9609\}>\lambda^{0}\). Thus, Theorem 1 forms a basis for the determination of correctness of the solution of a vector problem. In a vector problem of mathematical programming, as a rule, for two criteria the equality holds: \(\lambda^{0}=\lambda_{q}(X^{0})=\lambda_{p}(X^{0})\), \(q\), \(p\in\mathbf{K}\), \(X\in S\), (in our example, three criteria), and for other criteria is defined as an inequality: \(\lambda^{0}\leq\lambda_{k}(X^{0})\) \"\(k\in\mathbf{K}\), \(q\
eq p\
eq k\). In an admissible set of points \(\mathbf{S}\) formed by the restrictions of Equation (76), the optimum points \(X_{1}\), \(X_{2}\), \(X_{3}\), \(X_{4}\) are united in a contour and presented as a set of Pareto optimal points, \(\mathbf{S}^{0}\subset\mathbf{S}\). For specification of the border of a great number of Pareto additional points are calculated: \(X_{12}^{o}\), \(X_{13}^{o}\), \(X_{42}^{o}\), \(X_{34}^{o}\) which lie between the corresponding criteria. For definition of a point \(X_{12}^{o}\), the vector problem was solved with two criteria of Equations (71), (72): \(\lambda_{1}\)X, \(\lambda_{2}\)X, (Equation (76)). Results of the decision are:  
\(X_{12}^{o}=[80.78\ 25.0\ 55.89]\), \(\lambda^{0}(X_{12}^{o})=0.9264\), F\({}_{12}=\{656.2\ 1426.0\ 101.7\ 142.7\}\),  
\(\text{L}_{12}=[\mathbf{0.9264}\ \mathbf{0.9264}\ 0.0261\ 0.4761]\).