# 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  
(Method of the decision in problems of vector optimization with a criterion priority) [25]. Step 1. We solve a vector problem with equivalent criteria. The algorithm of the decision is presented in Section 3.1.2. As a result of the decision we obtain:  
* optimum points by each criterion separately \(X^{*}_{k}\), \(k=\overline{1,K}\) and sizes of criterion functions in these points of \(f^{*}_{k}=f_{k}(X^{*}_{k})\), \(k=\overline{1,K}\), which represent the boundary of a set of Pareto optimal points,
* anti-optimum points by each criterion of \(X^{0}_{k}=\{x_{j},j=\overline{1,N}\}\) and the worst unchangeable part of each criterion of \(f^{0}_{k}=f_{k}(X^{0}_{k})\), \(k=\overline{1,K}\),
* \(X^{0}=\{\lambda^{0},X^{0}\}\), an optimum point, as a result of the solution of VPMP at equivalent criteria, i.e., the result of the solution of a maximine problem and the \(\lambda\)-problem constructed on its basis,
* \(\lambda^{0}\), the maximum relative assessment which is the maximum lower level for all relative estimates of \(\lambda_{k}(X^{0})\), or the guaranteed result in relative units, \(\lambda^{0}\) guarantees that all relative estimates of \(\lambda_{k}(X^{0})\) are equal to or greater than \(\lambda^{0}\): \[\lambda^{0}\leq\lambda_{k}\big{(}X^{0}\big{)},k\ =\ \overline{1,K},X^{0}\in S.\] (28)  
The person making the decision carries out the analysis of the results of the solution of the vector problem with equivalent criteria. If the received results satisfy the decision maker, then the process concludes, otherwise subsequent calculations are performed. In addition, we calculate:  
* in each point \(X^{*}_{k}\), \(k=\overline{1,K}\) we determine sizes of all criteria of \(q=\overline{1,K}\): \(\{f_{q}(X^{*}_{k})\), \(q=\overline{1,K}\}\), \(k=\overline{1,K}\), and relative estimates \(\lambda(X^{*})=\{\lambda_{q}(X^{*}_{k})\), \(q=\overline{1,K}\), \(k=\overline{1,K}\}\), \(\lambda_{k}(X)=\frac{f_{k}(X)-f^{0}_{k}}{f^{*}_{k}-f^{0}_{k}}\), \(\forall k\in K\): \[F(X^{*})\ =\ \left| \begin{array}{c}f_{1}(X^{*}_{1}),\ \ldots,f_{k}(X^{*}_{1}),\\ \ldots\\ f_{1}(X^{*}_{k}),\ \ldots,f_{k}(X^{*}_{k})\end{array} \right|\lambda(X^{*})\ =\ \left| \begin{array}{c} \lambda_{1}(X^{*}_{1}),\ \ldots,\ \lambda_{k}(X^{*}_{1}),\\ \ldots\\ \lambda_{1}(X^{*}_{k}),\ \ldots,\ \lambda_{k}(X^{*}_{k})\end{array} \right|\] (29)  
Matrices of criteria of \(F(X^{*})\) and relative estimates of \(\lambda(X^{*})\) show the sizes of each criterion of \(k=\overline{1,K}\) upon transition from one optimum point \(X^{*}_{k}\)\(k\)\(\in\)K to another \(X^{*}_{q}\), \(q\)\(\in\)K, i.e., on the border of a great number of Pareto. * at an optimum point at equivalent criteria \(X^{0}\) we calculate sizes of criteria and relative estimates: \[f_{k}\big{(}X^{0}\big{)},k\ =\ \overline{1,K};\lambda_{k}\big{(}X^{0}\big{)},k\ =\ \overline{1,K},\] (30) which satisfy the inequality of Equation (28). In other points \(X\)\(\in\)S\({}^{0}\), in relative units the criteria of \(\lambda=\underset{k\in K}{\min}\lambda_{k}(X)\) are always less than \(\lambda^{0}\), given the \(\lambda\)-problem of Equations (22) and (23). This information is also a basis for further study of the structure of a great number of Pareto. Step 2. Choice of priority criterion of \(q\in\) K.  
From theory (see Theorem 1) it is known that at an optimum point \(X^{0}\) there are always two most inconsistent criteria, \(q\in\) K and \(v\in\) K, for which in relative units an exact equality holds: \(\lambda^{0}=\lambda_{q}(X^{0})=\lambda_{p}(X^{0})\), \(q\), \(v\in\) K, \(X\in\) S. Others are subject to inequalities: \(\lambda^{0}\leq\lambda_{k}(X^{0})\)\(\forall k\in\) K, \(q\
eq v\
eq k\). As a rule, the criterion which the decision-maker would like to improve is part of this couple, and such a criterion is called a priority criterion, which we designate \(q\in\) K.  
Step 3. Numerical limits of the change of the size of a priority of criterion \(q\in\) K are defined.