# 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
#### 3.1.2 Mathematical Algorithm of the Solution of a Vector Problem with Equivalent Criteria  
To solve the vector problems of mathematical programming of Equations (1)-(4), the methods based on axioms of the normalization of criteria and the principle of the guaranteed result [12; 21] are offered. Methods follow from Axiom 1 and the principle of optimality (Definition 3). We will present this as a number of steps as follows:  
The method of the solution of the vector problem of Equations (1)-(4) with equivalent criteria is presented in the form of the sequence of steps [25]. Step 1. The problem of Equations (1)-(4) is solved separately for each criterion, i.e., for \(\forall k\in K_{1}\) is solved at the maximum, and for \(\forall k\in K_{2}\) is solved at a minimum. As a result of the decision, we obtain \(X^{*}_{k}\), an optimum point by the corresponding criterion, \(k=\overline{1,K};f^{*}_{k}=f_{k}(X^{*}_{k})\), the \(k\)th criterion size at this point, \(k=\overline{1,K}\). Step 2. We define the worst value of each criterion on \(S;f^{0}_{k},k=\overline{1,K}\). For the problem of Equations (1)-(4) for each criterion of \(k=\overline{1,K}\), a minimum is solved as: \(f^{0}_{k}=\min f_{k}(X)\), \(G(X)\leq B\), \(X\geq 0,\,k=\overline{1,K}\). In addition, for Equations (1)-(4) for each criterion, a maximum is solved as: \(f^{0}_{k}=\max f_{k}(X)\), \(G(X)\leq B\), \(X\geq 0,\,k=\overline{1,K}\). As a result of the decision, we obtain \(X^{0}_{k}=\{x_{j},j=\overline{1,N}\}\), an optimum point by the corresponding criterion, \(k=\overline{1,K};f^{0}_{k}=f_{k}(X^{0}_{k})\), the \(k\)th criterion size at the point, \(X^{0}_{k},\,k=\overline{1,K}\). Step 3. For the system analysis of a set of Pareto optimal points, for this purpose optimum points of \(X^{*}=\{X^{*}_{k},\,k=\overline{1,K}\}\), are defined as sizes of criterion functions of \(F(X^{*})\) and relative estimates \(\lambda(X^{*})\), \(\lambda_{k}(X)=\frac{f_{k}(X)-f^{0}_{k}}{f^{*}_{k}-f^{0}_{k}},\,\forall k\in K\):  
\[ \begin{array}{rcl}F(X^{*})&=&\{f_{q}(X^{*}_{k}),q\,\,=\,\,\overline{1,K},k\, \,=\,\,\overline{1,K}\}&=&\left|\begin{array}{l}f_{1}(X^{*}_{1}),\,\ldots,f_ {k}(X^{*}_{1}),\\ \ldots\\ f_{1}(X^{*}_{k}),\,\ldots,f_{k}(X^{*}_{k})\\ \lambda_{1}(X^{*}_{1}),\,\ldots,\,\lambda_{k}(X^{*}_{1}),\\ \ldots\\ \lambda_{1}(X^{*}_{k}),\,\ldots,\,\lambda_{k}(X^{*}_{k})\\ \end{array}\right|.\end{array}  \tag{12}\]As a whole, for the problem \(\forall k\in\mathrm{K}\) the relative assessment of \(\lambda_{k}(X)\), \(k=\overline{1,K}\) lies within \(0\leq\lambda_{k}(X)\)\(\leq 1\), \(\forall k\in\mathrm{K}\). Step 4. Creation of the \(\lambda\)-problem. Creation of the \(\lambda\)-problem is carried out in two stages: initially build the maximize problem of optimization with the normalized criteria, which at the second stage will be transformed into the standard problem of mathematical programming called the \(\lambda\)-problem. For the construction of the maximize problem of optimization we use Definition 2, relative level \(\forall X\in\mathsf{S}\), \(\lambda=\min\limits_{k\in K}\lambda_{k}(X)\). The bottom \(\lambda\) level is maximized on \(X\in S\), as a result, we obtain a maximize problem of optimization with the normalized criteria:  
\[\lambda^{0}\ =\ \max\limits_{x}\min\limits_{k}\lambda_{k}(X),G(X)\leq B,X\geq 0. \tag{13}\]  
At the second stage we transform the problem of Equation (13) into a standard problem of mathematical programming:  
\[\lambda^{0}=\max\lambda,\rightarrow\lambda^{0}=\max\lambda, \tag{14}\]  
\[\lambda-\lambda_{k}(X)\leq 0,k\ =\ \overline{1,K},\rightarrow\lambda-\frac{f _{k}(X)-f_{k}^{0}}{f_{k}^{*}-f_{k}^{0}},k\ =\ \overline{1,K}, \tag{15}\]  
\[G(X)\leq B,\,X\geq 0,\to G(X)\leq B,\,X\geq 0, \tag{16}\]  
where the vector of unknowns of \(X\) has the dimension \(N+1\): \(X=\{\lambda,x_{1},\ldots,x_{N}\}\). Step 5. Solution of the \(\lambda\)-problem. The \(\lambda\)-problem of Equations (14)-(16) is a standard problem of convex programming for which decision standard methods are used.