# 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 Equivalent Criteria  
#### 3.1.1 Axioms and the Principle of Optimality of Vector Optimization with Equivalent Criteria  
**Definition 1. ** (Definition of the relative assessment of criteria). In the vector problem of Equations (1)-(4), definitions are as follows: \(\lambda_{k}(X)=\frac{f_{k}(X)-f_{k}^{0}}{f_{k}^{*}-f_{k}^{0}}\), \(\forall k\in\mathbf{K}\) is the relative estimate of a point \(X\in S\)\(k\)th criterion, \(f_{k}(X)\) is the \(k\)th criterion at the point \(X\in S\), \(f_{k}^{*}\) is the value of the \(k\)th criterion at the point of optimum \(X_{k}^{*}\), obtained in the vector problem of Equations (1)-(4) of the individual \(k\)th criterion, \(f_{k}^{0}\) is the worst value of the \(k\)th criterion (anti-optimum) at the point \(X_{k}^{0}\) (superscript 0) on the admissible set \(S\) in Equations (1)-(4), at the task at \(max\) (3), (5), (6), the value of \(f_{k}^{0}\) is the lowest value of the \(k\)th criterion, \(f_{k}^{0}=\underset{X\in S}{\min}f_{k}(X)\ \forall k\in\mathbf{K}_{1}\) and the task \(\min f_{k}^{0}\) is the greatest,\(f_{k}^{0}=\underset{X\in S}{\max}f_{k}(X)\ \forall k\in K_{2}\). The relative estimate of \(\lambda_{k}(X)\), \(\forall k\in K\) is firstly measured in relative units and, secondly, the relative assessment of \(\lambda_{k}(X)\ \forall k\in K\) on the admissible set is changed from zero at a point of \(X_{k}^{0}\): \(\forall k\in K\underset{X\to X_{k}^{0}}{\max}\lambda_{k}(X)=0\), to the unit at the point of an optimum of \(X_{k}^{*}\forall k\in K\), \(\underset{X\to X_{k}^{*}}{\lim}\lambda_{k}(X)=1\), i.e., \(\forall k\in K\), \(0\leq\lambda_{k}(X)\leq 1\), \(X\in S\). This allows the comparison of the criteria, measured in relative units, by joint optimization. **Axiom 1**. : (About equality and equivalence of criteria at an admissible point of vector problems of mathematical programming)  
In vector problems of mathematical programming two criteria with the indexes \(k\in K\), \(q\in K\) shall be considered as equal at point \(X\in S\) if relative estimates of the \(k\)th and \(q\)th criterion are equal at this point, i.e., \(\lambda_{k}(X)=\lambda_{q}(X)\), \(k\), \(q\in K\). We will consider criteria equivalent in vector problems of mathematical programming at a point \(X\in S\) if, when comparing the numerical size of relative estimates of \(\lambda_{k}(X)\), \(k=\overline{1,K}\), for each criterion of \(f_{k}(X)\), \(k=\overline{1,K}\), and, respectively, relative estimates of \(\lambda_{k}(X)\), conditions are not imposed about the priorities of criteria. **Definition 2**. : (Definition of a minimum level among all relative estimates of criteria). The relative level \(\lambda\) in a vector problem represents the lower assessment of a point of \(X\in S\) among all relative estimates of \(\lambda_{k}(X)\), \(k=\overline{1,K}\):  
\[\forall X\in S,\ \lambda\leq\lambda_{k}(X),k\ =\ \overline{1,K}, \tag{5}\]  
The lower level for the performance of the condition of Equation (5) at an admissible point of \(X\in S\) is defined as:  
\[\forall X\in S,\ \lambda\ =\ \underset{k\in K}{\min}\lambda_{k}(X). \tag{6}\]  
Equations (5) and (6) are interconnected. They serve as a transition from Equation (6) of the definition of a minimum to the restrictions of Equation (5), and vice versa. The level \(\lambda\) allows the union of all criteria in a vector problem with one numerical characteristic of \(\lambda\), made over certain operations, thereby carrying out these operations over all criteria measured in relative units. The level \(\lambda\) functionally depends on the \(X\in S\) variable, by changing \(X\), we can change the lower level, \(\lambda\). From here we will formulate the rules of searching for the optimum decision. **Definition 3**. : (The principle of optimality with equivalent criteria).