# 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
As a result of the solution of the \(\lambda\)-problem, the following are obtained:  
\(\mathbf{X^{0}=\{\lambda^{0},\,X^{0}\}}\), an optimum point,  
\(f_{k}(X^{0})\), \(k=\overline{1,K}\), values of the criteria at this point,  
\(\lambda_{k}(X^{0})=\frac{f_{k}(X^{0})-f_{k}^{0}}{f_{k}^{*}-f_{k}^{0}}\), \(k=\overline{1,K}\), sizes of the relative estimates,  
\(\lambda^{0}\), the maximum relative estimates which represent the maximum bottom 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 \(\lambda^{0}\): \(\lambda_{k}(X^{0})\geq\lambda^{0}\), \(k=\overline{1,K}\) or \(\lambda^{0}\leq\lambda_{k}(X^{0})\), \(k=\overline{1,K}\), \(X^{0}\in S\), and according to Theorem 1 [12, 21], the point of \(X^{0}=\{\lambda^{0},\,x_{1},\ldots,x_{N}\}\) is Pareto optimal. 1.3 Implementation of the Decision using the Example of a Vector Problem of Linear Programming with Equivalent Criteria  
The use of the vector problem of Equations (1)-(4) for decision making is carried out in four stages: statement of the problem, construction of the mathematical model, software development for solving the vector problem, and, solution of the vector problem. These stages are carried out on the example of a model of an economic system presented by a vector problem of linear programming with equivalent criteria. Stage 1. Statement of the problem. As an economic system, a model of the production schedule of an enterprise is considered. It is given that the company, which produces heterogeneous products of four types, \(N=4\), uses resources of three types, \(M=3\), in production: labor (various specialties), material (different types of materials), power (equipment: welding, turning, etc.). The technological matrix of production is presented in Table 1. It also indicates the potential of the enterprise for each type of the resource \(b_{i}\), \(i=\overline{1,3}\), as well as income \(c_{j}^{1}\) and profit \(c_{j}^{2}\) from the sale of a unit of each type of product. It is required to make the production schedule of the enterprise, which includes indicators according to the nomenclature (by types of products) and on a volume basis, i.e., how many products of the corresponding type should be made by the enterprise so that income and profit can be realized as shown above. Construction and solution of the mathematical model follow. Stage 2. Construction of a mathematical model. As variables, we take the volume of products that the company produces: \(X=\{x_{1},\,\ldots,\,x_{N}\}\), \(N=4\). We express target orientation of the production schedule by means of a vector problem of linear programming (VPLP) which will take the form:  
\(\text{opt }F(X)=\{\max f_{1}(X)=(4.0x_{1}+5.0x_{2}+9.0x_{3}+11.0x_{4}),}\)  
\(\max f_{2}(X)=(2x_{1}+10x_{2}+6x_{3}+20x_{4})\)\(]\),  
with restrictions \(x_{1}+x_{2}+x_{3}+x_{4}\leq 15\),  
\(7x_{1}+5x_{2}+3x_{3}+2x_{4}\leq 120\),  
\(3x_{1}+5x_{2}+10x_{3}+15x_{4}\leq 100\), \(x_{1}\geq 0\), \(x_{2}\geq 0\), \(x_{3}\geq 0\), \(x_{4}\geq 0\). In this VPLP the following is formulated: it is required to find the non-negative solution of \(x_{1}\),..., \(x_{4}\) in the system of inequalities at which the \(f_{1}(X)\) and \(f_{2}(X)\) functions obtain maximum values. Stage 3. The software engineering of the solution of the VPLP. The \"Solution of a Vector Problem of Linear Programming\" program is presented in the annex to this section. Stage 4. Solution of a vector problem of linear programming. We show the solution of a problem of linear programming in the MATLAB system according to an algorithm of the solution of the VPLP on the basis of normalization of criteria and the principle of the guaranteed result. At first input data are prepared (the italicized font indicates the text of the program in the MATLAB system).