# 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
#### 3.2.1 Axioms and the Principle of Optimality of Vector Optimization with a Criterion Priority
Set by the decision-maker, component \(p_{k^{\prime}}^{q}\) from the point of view of the decision-maker, shows how many times a relative estimate of \(\lambda_{k}(X)\), \(k=\overline{1,K}\) is greater than other relative estimates of \(\lambda_{k}(X)\), \(k=\overline{1,K}\). The vector of \(p_{k}^{q}\), \(k=\overline{1,K}\) is the numerical expression of the priority of the \(q\)th criterion over other criteria of \(k=\overline{1,K}\):  
\[ \begin{array}{l}P^{q}(X)\ \ =\ \{p_{k^{\prime}}^{q},k\ =\ \overline{1,K}\},p_{k}^{q}\geq 1, \forall X\in S_{q}\subset S,k\ =\ \overline{1,K},\forall q\in K.\end{array}  \tag{18}\]  
The vector problem of Equations (1)-(4), in which the priority of any criteria is set, is called a vector problem with the set priority of criterion. The problem of a task of a vector of priorities arises when it is necessary to determine the point \(X^{0}\in S\) by the set vector of priorities. In the comparison of relative estimates with a priority of criterion of \(q\in\mathbf{K}\), as well as in a task with equivalent criteria, we define the additional numerical characteristic of \(\lambda\) which we call the level. **Definition 7**. : (About the lower level among all relative estimates with a criterion priority). The \(\lambda\) level is the lowest among all relative estimates with a priority of criterion of \(q\in\) such that:  
\[\lambda\leq p_{k}^{q}\lambda_{k}(X),k\ =\ \overline{1,K},q\in K,\forall X\in S _{q}\subset S; \tag{19}\]  
The lower level for the performance of the condition in Equation (19) is defined as:  
\[\lambda\ =\ \min_{k\in K}\!\!p_{k}^{q}\lambda_{k}(X),q\in K,\forall X\in S _{q}\subset S. \tag{20}\]  
Equations (19) and (20) are interconnected and serve as a further transition from the operation of the definition of the minimum to restrictions, and vice versa. In Section 3.1, we gave the definition of a Pareto optimal point \(X^{0}\in S\) with equivalent criteria. Considering this definition as an initial one, we will construct a number of the axioms dividing an admissible set of \(S\) into, first, a subset of Pareto optimal points \(S^{0}\), and, secondly, a subset of points \(S_{q}\subset S\), \(q\in\mathbf{K}\), with priority for the \(q\)th criterion. **Axiom 2. ** (About a subset of points, priority by criterion). In the vector problem of Equations (12)-(13), the subset of points \(S_{q}\subset S\) is called the area of priority of criterion of \(q\in K\) over other criteria, if \(\forall X\in S_{q}\ \forall k\in K\lambda_{q}(X)\geq\lambda_{k}(X)\), \(q\
eq k\). This definition extends to a set of Pareto optimal points \(S^{0}\) that is given by the following definition. **Axiom 2a. ** (About a subset of points, priority by criterion, on Pareto's great number in a vector problem). In a vector problem of mathematical programming the subset of points \(S^{o}_{q}\subset S^{0}\subset S\) is called the area of a priority of criterion of q\(\in\)K over other criteria, if \(\forall X\in S^{o}_{q}\ \forall k\in K\lambda_{q}(X)\geq\lambda_{k}(X)\), \(q\
eq k\). In the following we provide explanations. Axiom 2 and 2a allow the breaking of the vector problem in Equations (1)-(4) into an admissible set of points \(S\), including a subset of Pareto optimal points, \(S^{0}\subset S\), and subsets:  
One subset of points \(S^{\prime}\in S\) where criteria are equivalent, and a subset of points of \(S^{\prime}\) crossed with a subset of points \(S^{\circ}\), allocated to a subset of Pareto optimal points at equivalent criteria \(S^{00}=S^{\prime}\cap S^{0}\). As will be shown further, this consists of one point of \(X^{0}\in S\), i.e., \(X^{0}=S^{00}=S^{\prime}\cap S^{0}\), \(S^{\prime}\in S\), \(S^{0}\in S\).