## 3 Method Description  
### Conditions for Selecting Subcriteria  
The set of \(N\) scalar partial criteria \(F_{1},\;F_{2},\;\ldots,\;F_{n},\;\ldots,\;F_{N}\) should meet three basic conditions: completeness, consistency, and nonredundancy [5]. The condition of completeness excludes the situation when the user chooses one of two different solutions, where these solutions have equal grades. However, the condition of completeness does not apply to the situation when the user chooses one of the two nondominated solutions in the sense of the dominance relationship. Let \(x,\;z\in X\), where \(X\) is a set of allowable states, and the vector criteria functions with \(n\) coordinates, which evaluate these states, are taken the form \(F(x)\) and \(F(z)\), then the condition of completeness can be formulated as follows:  
\[[(F(x)=F(z)\;)\;\Rightarrow\;the\;user\;do\;not\;prefers\;x\;rather\;then\;z] \tag{5}\]  
The consistency condition excludes the following situation: two different solutions \(x,\;z\in X\) with identical images \(F(x)=F(z)\) are modified. The first solution is modified by \"improving\" at least one of its partial assessments. Thus, \(\{F(x^{+}),\;F(x)\}\in R\), where \(R\) is the dominance relation of the polyoptimization task. On the other hand, the modification of the second solution consists of determining a solution with at least one \"worse\", in the sense of the dominance relationship, partial indicator. Therefore, \(\{F(z),\;F(z^{-})\}\in R\). It is not advisable for the user to choose the \(z^{-}\) solution instead of the \(x^{+}\) solution. This indicates an incorrect formulation of the criterion function. Formally, the criterion of consistency of a criterion function can be written in the form:  
\[\left[(F(x)=F(z))\wedge\left(F(x^{+}),\;F(x)\right)\in\mathbf{R}\wedge(F(z),\;F(z^ {-}))\in\mathbf{R}\;\Longrightarrow\;z\;then\;x^{+}\right]\;for\;x,\;x^{+},\;z,\; z^{-}\in\mathbf{X} \tag{6}\]  
The nonredundancy condition of partial criteria in the vector criterion is met when reducing one of the scalar criteria results in the remaining partial criteria not fulfilling the condition of completeness and consistency. Therefore, partial criteria cannot be reduced if they express user preferences. On the other hand, too many partial criteria cannot be taken into account because the nonredundancy condition will not be met. As a result of preliminary modeling, it is usually possible to determine some basic vector criteria (1). Multicriteria optimization problems often take into account restrictions that \"cut off\" the value of a criterion function:  
\[F_{n}(x)\leq r_{n}\;for\;n=1,\;N \tag{7}\]  
where \(r_{n}\) is the value of limitation of the partial criterion \(F_{n}(\cdot)\). Each vector criterion \(\mathbf{F}\) can be modified by removing or adding a partial criterion. Let the relationship below indicate the confidence level for the \(F_{n}\) criterion:  
\[\alpha_{n}=\frac{1}{3}(\alpha_{n,1}+\alpha_{n,2}+\alpha_{n,3}) \tag{8}\]  
where \(\alpha_{n,1},\ \alpha_{n,2},\ \alpha_{n,3}\in[0,1]\), \(\alpha_{n,1}\) is the subjective level of user confidence to the adequacy of the selection of the scalar partial \(F_{n}\) criterion, \(\alpha_{n,2}\) is relative accuracy of determining the \(F_{n}\) partial criterion, and \(\alpha_{n,3}\) is confidence level of exceeding the limit value \(r_{n}\). The total confidence level can be determined using the trust function \(Q:\mathbf{F}\rightarrow\mathbf{R}^{N}\):  
\[Q(F)=\frac{1}{N}\sum_{n=1}^{N}\alpha_{n}=\frac{1}{3N}\sum_{n=1}^{N}(\alpha_{n, 1}+\alpha_{n,2}+\alpha_{n,3}) \tag{9}\]  
where \(Q(F)\in[0,1]\). The extended modification of the \(\mathbf{F}^{+}\) vector criterion in relation to the base \(\mathbf{F}\) criterion is understood as a vector criterion created by adding the \(F_{N+1}\) partial criterion. Therefore,  
\[\mathbf{F}^{+}:\mathbf{X}\rightarrow\mathbf{R}^{N+1} \tag{10}\]  
where \(\mathbf{F}^{+}(x)=[F_{1}(x),\ \dots,\ F_{n}(x),\ F_{n+1}(x),\ \dots,\ F_{N}(x),\ F_{N+1}(x)]\)_for \(x\in\mathbf{X}\).