# Mathematical Apparatus of Optimal Decision-Making Based on Vector Optimization
## Conclusions
### Formation of Technical Specifications (Source Data) for the Numerical Simulation of the System
#### 8.2.4 Construction of a Numerical Model of the System under Certainty and Uncertainty
Geometric Interpretation of Results of the Decision in a Three-Dimensional Coordinate System in Relative Units  
In an admissible set of points, \(\mathbf{S}\) formed by restrictions of Equation (32), 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 \(\mathbf{S}^{0}\)\(\subset\)\(\mathbf{S}\) in Figure 13. Coordinates of these points, and the characteristics of the technical system in relative units of \(\lambda_{1}(X)\), \(\lambda_{2}(X)\), \(\lambda_{3}(X)\), \(\lambda_{4}(X)\) are shown in Figure 14 in three-dimensional space, where the third axis \(\lambda\) is a relative assessment. **Discussion**. Looking at Figure 9, we can provide changes of all functions of \(\lambda_{1}(X)\), \(\lambda_{2}(X)\), \(\lambda_{3}(X)\), \(\lambda_{4}(X)\) in four-dimensional space. We consider, for example, an optimum point \(X_{3}^{*}\). The \(\lambda_{3}(X)\) function is created from the functions \(f_{3}(X)\) with variable coordinates \(\{x_{1}\), \(x_{2}\}\) and with constant coordinates \(\{x_{3}=8.8\), \(x_{4}=2.2\}\), taken from an optimum point \(\mathbf{X}^{0}\) (33). At point \(X_{3}^{*}\) the relative assessment of \(\lambda_{3}(X_{3}^{*})=0.83\) is shown in Figure 9 by a black point. However, we know that the relative assessment of \(\lambda_{3}(X_{3}^{*})\) obtained from the function \(f_{3}(X_{3}^{*})\) in the third step is equal to unity, which we designate as \(\lambda_{3}^{\Delta}(X_{31}^{*})=1\), and is shown in Figure 9 by a red point. The difference between \(\lambda_{3}^{\Delta}(X_{31}^{*})=1\) and \(\lambda_{3}(X_{3}^{*})=0.83\) is an error \(\Delta=0.17\) due to transitioning from four-dimensional (and generally \(N\)-dimensional) to two-dimensional space. The point \(X_{1}^{*}\) and appropriate relative estimates of \(\lambda_{1}(X_{1}^{*})\) and \(\lambda_{1}^{\Delta}(X_{1}^{*})\) are similarly shown. Thus, for the first time in domestic and foreign practice, the transition and its geometric illustration from an \(N\)-dimensional to a two-dimensional measurement of function is shown in vector problems of mathematical programming with the appropriate errors. The Solution of a Vector Problem of Mathematical Programming--Model of the System at the Given Priority of the Criterion  
The decision maker is usually the system designer. Step 1. We solve a vector problem with equivalent criteria. The algorithm of the decision is presented in Section 8.3. The numerical results of the solution of the vector problem are given above. Pareto's great number \(\mathsf{S}^{0}\subset\subset\mathsf{S}\) lies between optimum points \(\mathsf{X}_{1}^{*}\,\mathsf{X}^{0}\,\mathsf{X}_{3}^{*}\,\mathsf{X}^{0}\, \mathsf{X}_{4}^{*}\,\mathsf{X}^{0}\,\mathsf{X}_{2}^{*}\,\mathsf{X}^{0}\, \mathsf{X}_{1}^{*}\). We carry out the analysis of a great number of Pareto \(\mathsf{S}^{0}\subset\mathsf{S}\). For this purpose, we will connect auxiliary points: \(\mathsf{X}_{1}^{*}\mathsf{X}_{3}^{*}\mathsf{X}_{4}^{*}\mathsf{X}_{2}^{*}\, \mathsf{X}_{1}^{*}\) with a point \(\mathsf{X}^{0}\) which conditionally represents the center of a great number of Pareto. As a result, we obtain four subsets of points \(\mathsf{X}\in\mathsf{S}_{q}^{o}\subset\mathsf{S}^{0}\subset\mathsf{S}\), \(\mathsf{q}=\overline{1,4}\). The subset \(\mathsf{S}_{1}^{o}\subset\mathsf{S}^{0}\subset\mathsf{S}\) is characterized by the fact that the relative assessment \(\lambda_{1}\geq\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\), i.e., in the field \(\mathsf{S}_{1}^{o}\), the first criterion has priority over the others. Similarly, \(\mathsf{S}_{2}^{o}\), \(\mathsf{S}_{3}^{o}\), \(\mathsf{S}_{4}^{o}\) are the subsets of points where the second, third and fourth criterion has a priority over the others, respectively. We designate the set of Pareto optimal points \(\mathsf{S}^{0}=\mathsf{S}_{1}^{0}\cup\mathsf{S}_{2}^{0}\cup\mathsf{S}_{3}^{0} \cup\mathsf{S}_{4}^{o}\).