# 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
Assuming a linear nature of the change of criterion of \(f_{q}(X)\) in Equation (123) and according to a relative assessment of \(\lambda_{q}(X)\), using standard methods of linear approximation we calculate the proportionality coefficient between \(\lambda_{q}(X^{0})\), \(\lambda_{q}\), which we call \(\rho\):  
Figure 13:5  
Step 7. Calculation of the coordinates of priority criterion with the size \(f_{q}\). Assuming a linear nature of the change of a vector \(X^{q}=\{x_{1}\ x_{2}\}\), \(q=3\) we determine coordinates of a point of priority criterion with the size \(f_{q}\) with a relative assessment (95):  
Figure 13:6  
where \(X^{0}=\{X^{0}(1)=80.0\), \(X^{0}\ (2)=69.11\}\), \(X_{3}^{*}=\{X^{3^{3}}(1)=80.0\), \(X^{3^{3}}(2)=0.0\}\). As a result of the calculations, we obtain point coordinates: \(X^{q}=\{x_{1}=67.31\), \(x_{2}=21.27\}\). Step 8. Calculation of the main indicators of a point \(X_{q}\). For the obtained point \(X_{q}\), we calculate:  
* all criteria in physical units \(f_{k}(X^{q})=\{f_{k}(X^{q}),\ k=\overline{1,K}\}\): \(f(X^{q})=\{f_{1}(X^{q})=313.45,f_{2}(x^{q})=2575.7\), \(f_{3}(x^{q})=74.2,f_{4}(x^{q})=60.6\}\),
* all relative estimates of criteria \(\lambda^{q}=\{\lambda_{k^{\prime}}^{q},\ k=\overline{1,K}\}\), \(\lambda_{k}(X^{q})=\frac{f_{k}(X^{q})-f_{k}^{o}}{f_{k}^{*}-f_{k}^{o}},\ k= \overline{1,K_{1,K}}\), \(\lambda_{k}(X^{q})=\{\lambda_{1}(X^{q})=0.2405,\)\(\lambda_{2}(x^{q})=0.5102,\)\(\lambda_{3}(x^{q})=0.4825,\)\(\lambda_{4}(x^{q})=0.55861\overline{1,K}\),
* the minimum relative assessment: minLXq = min(LXq): minLXq = min(\(\lambda_{k}(X^{q})\)) = 0.2405,
* the vector of priorities \(P^{q}\
eq\{p_{k}^{q}=\frac{\lambda_{q}(X^{q})}{\lambda_{k}(X^{q})},k=\overline {1,K}\}\): \(P^{q}=(p_{1}^{3}=2.0061,p_{2}^{3}=0.9458,p_{3}^{2}=1.0\), \(p_{4}^{3}=0.8637\}\),
* the relative assessment taking into account a criterion priority: \(LX_{q}P_{q}=\lambda_{k}(X^{q}),*^{pq}=\{p_{1}^{3}\lambda_{1}(X^{q})\), \(p_{2}^{3}\lambda_{2}(X^{q})\), \(p_{3}^{3}\lambda_{3}(X^{q})\), \(p_{4}^{3}\lambda_{4}(X^{q})\}\)\(\lambda_{k}(X^{q})\). *^{pq}=\{0.7968\ 0.\ 7968\ 0.\ 7968\), \(7968\}\),
* the the minimum relative assessment taking into account a criterion priority: \(\lambda^{00}=\) min (\(p_{1}^{2}\lambda_{1}(X^{q})=0.4825,\)\(p_{2}^{2}\lambda_{2}(X^{q})=0.4825,\)\(p_{3}^{2}\lambda_{3}(X^{q})=0.4825,\)\(p_{4}^{2}\lambda_{4}(X^{q})\)) = 0.4825  
Any point from [PERSON]'s set \(X_{t}^{o}=\{\lambda_{t}^{o},X_{t}^{o}\}\)\(\in\)S\({}^{o}\) can be similarly calculated. Step 9. Analysis of results. The calculated size of criterion \(f_{q}(X_{t}^{o})\), \(\rho\)\(\in\)K is usually not equal to the set \(f_{q}\). The error of the choice of \(\Delta f_{q}=|f_{q}(X_{t}^{o})-f_{q}|=|74.2-80|=5.8\) is defined by an error of linear approximation, \(\Delta f_{q\%}=7.25\%\). In the course of modeling and simulation, as well as in the previous example, Section 7.5, the parametric restrictions of Equation (118) can be changed, i.e., some set of optimum decisions is obtained. We can choose a final version from, in our example, this set of optimum decisions: parameters of technical system \(X^{0}=\{x_{1}=52.9\), \(x_{2}=36.097\), \(x_{3}=8.8\), \(x_{4}=2.2\), \(\lambda^{0}=0.3179\), the parameters of the technical system at a given priority criterion \(q=2\): \(X^{q}=\{x_{1}=67.31\), \(x_{2}=21.27\}\). If the error \(\Delta f_{q}=|f_{q}(X^{00})-f_{q}|=|79.6-80|=0.4\), measured in physical units or as a percentage \(\Delta f_{q\%}=\Delta f_{q}/f_{q}\times 100=0.5\%\), is more than set \(\Delta f\), \(\Delta f_{q}>\Delta f\), \(t\), we pass to Step 2, else if \(\Delta f_{q}\leq\Delta f\), calculations come to an end. Geometric Interpretation of Results of the Decision in a Three-Dimensional Coordinate System in Physical Units  
In the course of modeling, the parametric restrictions of Equation (32) and functions can be changed, i.e., some set of optimum decisions is obtained.