# Mathematical Apparatus of Optimal Decision-Making Based on Vector Optimization
## 2 Statement of a Problem: Creation of the Mathematical Model \"Acceptance of an Optimal Solution\"
### Vector Optimization with a Criterion Priority
#### 3.2.2 Mathematical Method of the Solution of a Vector Problem with Criterion Priority
For priority criterion \(q\in\mathbf{K}\) from the matrix of Equation (29) we define the numerical limits of the change of the size of criterion:  
\[\bullet\quad\text{in physical units of }f_{q}\big{(}X^{0}\big{)}\,\leq\,f_{q}(X)\, \leq\,f_{q}\big{(}X^{*}_{q}\big{)},\,k\in\,\mathbf{K}, \tag{31}\]  
where \(f_{q}(X^{*}_{q})\) derives from the matrix of Equation (29) \(F(X^{*})\), all criteria showing sizes measured in physical units, \(f_{q}(X^{0})\) from Equation (30), and,  
\[\bullet\quad\text{in relative units of }\lambda_{q}\big{(}X^{0}\big{)}\leq\, \lambda_{q}(X)\,\leq\,\lambda_{q}\big{(}X^{*}_{q}\big{)},\,k\in\,\mathbf{K}, \tag{32}\]  
where \(\lambda_{q}(X^{*}_{q})\) derives from the matrix \(\lambda(X^{*})\), all criteria showing sizes measured in relative units (we note that \(\lambda_{q}(X^{*}_{q})=1\)), \(\lambda_{q}(X^{0})\) from Equation (29). As a rule, Equations (31) and (32) are given for the display of the analysis. Step 4. Choice of the size of priority criterion (decision-making). The person making the decision carries out the analysis of the results of calculations of Equation (42) and from the inequality of Equation (31) chooses the numerical size \(f_{q}\) of the criterion of \(q\in\mathbf{K}\):  
\[f_{q}\big{(}X^{0}\big{)}\,\leq\,f_{q}\,\leq\,f_{q}\big{(}X^{*}_{q}\big{)},\,q \in\,\mathbf{K}. \tag{33}\]  
For the chosen size of the criterion of \(f_{q}\) it is necessary to define a vector of unknown \(X^{0}\). For this purpose, we carry out the subsequent calculations. Step 5. Calculation of a relative assessment. For the chosen size of the priority criterion of \(f_{q}\) the relative assessment is calculated as:  
\[\lambda_{q}\,=\,\frac{f_{q}-f_{q}^{o}}{f_{q}^{*}-f_{q}^{o}}, \tag{34}\]  
which upon transition from point \(X^{0}\) to \(X^{*}_{q}\), according to Equation (32), lies in the limits: \(\lambda_{q}(X^{0})\leq\lambda_{q}\leq\lambda_{q}(X^{*}_{q})=1\). Step 6. Calculation of the coefficient of linear approximation. Assuming a linear nature of the change of criterion of \(f_{q}(X)\) in Equation (31) and according to the relative assessment of \(\lambda_{q}(X)\) in Equation (32), using standard methods of linear approximation we calculate the proportionality coefficient between \(\lambda_{q}(X^{\circ})\), \(\lambda_{q}\), which we call \(\rho\):  
\[\rho\,=\,\frac{\lambda_{q}-\lambda_{q}(X^{\circ})}{\lambda_{q}(X^{\circ}_{q})- \lambda_{q}(X^{\circ})},q\in K \tag{35}\]  
Step 7. Calculation of coordinates of priority criterion with the size \(f_{q}\). In accordance with Equation (33), the coordinates of the \(X^{q}\) priority criterion point lie within the following limits: \(X^{0}\leq X^{q}\leq X^{*}_{q}\), \(q\in\mathbf{K}\). Assuming a linear nature of change of the vector \(X^{q}=\{x^{q}_{1},\ldots,x^{q}_{N}\}\) we determine coordinates of a point of priority criterion with the size \(f_{q}\) with the relative assessment of Equation (32):  
\[ \begin{array}{l}X^{q}\,=\,\{x^{q}_{1}\,\,=\,\,x^{0}_{1}\,+\rho(x^{*}_{q}(1)\, -\,x^{0}_{1}),\\ \ldots\,,\\ x^{q}_{N}\,\,=\,\,x^{0}_{N}\,\,+\rho(x^{*}_{q}(N)\,-\,x^{0}_{N})\}.\end{array}  \tag{36}\]  
where \(X^{0}=\{x^{o}_{1},\ldots,x^{o}_{N}\}\), \(X^{*}_{q}=\{x^{*}_{q}(1),\ldots,x^{*}_{q}(N)\}\). 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 \(f_{k}(x^{q})=\{f_{k}(x^{q}),\,k=\overline{1,K}\}\),
* all relative estimates of criteria \(\lambda^{q}=\{\lambda^{q}_{k^{\prime}},\,k=\overline{1,K}\}\), \(\lambda_{k}(x^{q})=\frac{f_{k}(x^{q})-f_{k}^{o}}{f_{k}^{*}-f_{k}^{o}}\), \(k=\overline{1,K}\overline{1,K}\),* the vector of priorities \(P^{\text{I}}=\{p_{k}^{q}=\frac{\lambda_{q}(x^{q})}{\lambda_{k}(x^{q})}\}\), \(k=\overline{1,K}\}\),
* the maximum relative assessment \(\lambda^{qq}=\min\) (\(p_{k}^{q}\lambda_{k}(x^{q})\), \(k=\overline{1,K}\)). Any point from [PERSON]'s set \(X_{t}^{o}=\{X_{t}^{o},\,X_{t}^{o}\}\)\(\in\)\(S^{o}\) can be similarly calculated. Analysis of results.