# 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\"  
As an \"object for optimal decision-making,\" we use a \"technical system\". The problem of the choice of optimum parameters of technical (engineering) systems according to functional characteristics arises during the study, analysis, and design of technical systems, and is connected with quality production. The problem includes the solution of the following tasks:  
* Creation of a mathematical model, which defines the inter-relation of each functional characteristic from parameters of the technical system, i.e., it is formed from the vector problem of mathematical programming;
* Choice of methods of the decision: we suggest using the methods based on normalization of criteria and the principle of the guaranteed result with equivalent criteria and with the set criterion priority;
* Development of the software which realizes these methods;
* The statement of a problem is executed according to [4; 6; 20; 22]. The technical system depends on \(N\), a set of design data: \(X=\{x_{1},\,x_{2},\,\ldots,\,x_{N}\}\), where \(N\) is the number of parameters, each of which lies in the set limits:  
\[x_{j}^{\min}\leq x_{j}\leq x,\,j\,\,=\,\,\overline{1,N},\,or\,X^{min}\leq X \leq X^{max}\]  
where \(x_{j}^{\min},\,x_{j}^{\max},\,\forall j\in N\) are the minimum and maximum limits of change of the vector of parameters of the technical system. The result of the functioning of the technical system is defined by a set \(K\) of technical characteristics of \(f_{k}(X)\), \(k=\overline{1,K}\) which functionally depend on design data \(X=\{x_{j},j=\overline{1,N}\}\), in total these represent a vector function: \(F(X)=\langle f_{1}(X)f_{2}(X)\,\ldots\,f_{K}\,(X)\rangle^{\mathrm{T}}\). The set of characteristics (criteria) is subdivided into two subsets \(K_{1}\) and \(K_{2}\): \(K=K_{1}\cup K_{2}\). \(K_{1}\) is a subset of technical characteristics, the numerical values of which are desired to be as high as possible: \(f_{k}(X)\rightarrow\)**max**, \(k=\overline{1,K_{1}}\). \(K_{2}\) is a subset of technical characteristics, the numerical values of which are desired to be as low as possible: \(f_{k}(X)\rightarrow\)**min**, \(k=\overline{K_{1}+1,K}\), \(K_{2}\equiv\overline{K_{1}+1,K}\). The mathematical model should consider, firstly, the purposes of the technical system which are represented by the characteristics of \(F(X)\), and, secondly, the \(X^{min}\leq X\leq X^{max}\) restrictions. The mathematical model of the technical system which solves in general a problem of the choice of the optimum design decision (a choice of optimum parameters) is presented in the form of a vector problem of mathematical programming. \[Opt\;F(X) = \{\mathbf{max}\,F_{1}(X)\ =\ \{\mathbf{max}\,f_{k}(X),k\ =\ \overline{1,K_{1}}\}, \tag{1}\] \[\mathbf{min}F_{2}(X) = \{\mathbf{min}\,f_{k}(X),k\ =\ \overline{1,K_{2}}\},\] (2) \[G(X)\leq 0,\] (3) \[x_{j}^{\min}\leq x_{j}\leq x_{j}^{\max},\,j\ =\ \overline{1,N}, \tag{4}\]  
where \(X\) is the vector of controlled variables (constructive parameters) in Equation (1), \(F(X)=\{f_{k}(X),\)\(k=\overline{1,K}\}\) represents the vector criterion for each component of the characteristics of the technical system in Equation (2), which functionally depends on the vector of variables \(X\), \(G(X)=(g_{1}(X)\,g_{2}(X)\,\ldots\,\,g_{M}(X))^{\mathrm{T}}\) represents a vector of the restrictions imposed on the functioning of the technical system, and \(M\) is a set of restrictions. Restrictions are defined in terms of technological, physical or similar processes, and can be presented by functional restrictions, for example, \(f_{k}^{\min}\!\leq\!f_{k}(X)\leq\!f_{k}^{\max}\), \(k=\overline{1,K}\).