## 3 Method Description
### Theorems on the Election of a Vector Criterion  
Let \(\mathbf{F}^{+}(x)=[F_{1}(x),\ F_{2}(x),\ \dots,\ F_{n}(x),\ \dots,\ F_{N+1}(x)]\) and \(\beta_{n}\) be the confidence level corresponding to the \(F_{n}\) criterion for \(n=\overline{1,\ N+1}\), where the confidence levels \(\beta_{n,1},\ \beta_{n,2},\ \beta_{n,3}\) are defined for the \(\mathbf{F}^{+}\) criterion in the same way as \(\alpha_{n,1},\ \alpha_{n,2},\ \alpha_{n,3}\) for criterion \(\mathbf{F}\). Because, from the point of view of the modeled situation, it is important to know the results of introducing or removing a partial criterion, the following statements can be formulated and proved [2,4,5]:  
**Theorem 1**. : _(On the introduction of a partial criterion) [2,4,5]_  
_If \(\beta_{i,k}=\alpha_{i,k}\) for \(i=\overline{1,\ n;\ k=\overline{1,\ 3}}\),_  
_If \(\beta_{n+1,k}=\alpha_{N+1,k}\) for \(k=\overline{1,\ 3}\),_  
_If \(\beta_{j+1,k}=\alpha_{j,k}\) for \(j=\overline{n+1,\ N;\ k=\overline{1,\ 3}}\)_  
_this extended modification of the vector criterion is effective then and only then if:_  
\[\alpha_{N+1}>Q(F) \tag{13}\]  
**Proof of Theorem 1. **  
\[Q\Big{(}F^{+}\Big{)}-Q(F)=\frac{1}{3N+3}\sum_{l=1}^{N+1}\Bigl{(}\beta_{l,1}+\beta_ {l,2}+\beta_{l,3}\Bigr{)}-\frac{1}{3N}\sum_{l=1}^{N}\Bigl{(}\alpha_{l,1}+\alpha _{l,2}+\alpha_{l,3}\Bigr{)}\]  
Based on the assumptions of the theorem  
\[Q\Big{(}F^{+}\Big{)}-Q(F)=\frac{1}{N+1}(\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n }+\alpha_{N+1}+\alpha_{n+1}+\cdots+\alpha_{N})-\frac{1}{N}\sum_{l=1}^{N}\alpha _{l}\]  
Let \(C=\sum_{l=1}^{N}\alpha_{l}\),  
Then \((F^{+})-Q(F)=\frac{1}{N(N+1)}\bigl{(}NC+N\alpha_{N+1}-NC-C\bigr{)}=\frac{\alpha _{N+1}-Q(F)}{N+1}\). Because \(N+1>0\) then \(Q(F^{+})-Q(F)>0\) when \(\alpha_{N+1}>Q(F)\), which ends the proof. Conclusion: If \(\alpha_{N+1}>Q(F)\), then the partial criterion \(F_{N+1}\) can be included in the process of formulating the polyoptimization task, whereas in extreme cases, when \(\alpha_{N+1}=0\) or \(Q(F)=1\), it makes no sense to consider an additional partial criterion. **Theorem 2. ** (_On the removal of the partial criterion) [2,4,5]_  
_If_ \(\beta_{i,k}=\alpha_{i,k}\) _for_ \(i=\{1,\,2,\,\ldots,\,n-1,\,n+1,\,\ldots,\,N\}\)_;_ \(k=\overline{1,\,3}\)__  
_then reduced modification of the vector criterion is effective then and only then if:_  
\[\alpha_{N}<Q(F) \tag{14}\]  
**Proof of Theorem 2. **  
\[Q(F^{-})-Q(F)=\frac{1}{3N-3}\sum_{l=1}^{N-1}\Bigl{(}\beta_{l,1}+\beta_{l,2}+ \beta_{l,3}\Bigr{)}-\frac{1}{3N}\sum_{l=1}^{N}\Bigl{(}\alpha_{l,1}+\alpha_{l,2 }+\alpha_{l,3}\Bigr{)}\]  
Based on the assumptions of the theorem  
\[Q(F^{-})-Q(F)=\frac{1}{N-1}(\alpha_{1}+\alpha_{2}+\ldots+\alpha_{n-1}+\alpha _{n+1}+\ldots+\alpha_{N-1})-\frac{1}{N}\sum_{l=1}^{N}\alpha_{l}\]  
Let \(C=\sum_{l=1}^{N}\alpha_{l}\),  
Then \((F^{-})-Q(F)=\frac{1}{N(N-1)}\bigl{(}NC-N\alpha_{N+1}-NC-C\bigr{)}=\frac{ \alpha_{N-1}-Q(F)}{N-1}\). Because \(N-1>0\) then \(Q(F^{-})-Q(F)>0\) when \(\alpha_{N-1}<Q(F)\), which ends the proof. Conclusion: If \(\alpha_{N-1}<Q(F)\), the partial criterion \(F_{N-1}\) can be removed in the process of formulating the polyoptimization task. Therefore, it is advisable to withdraw partial criteria with the lowest confidence levels, and introduce criteria with the highest confidence level after verifying the modified \(F^{+,-}\) criteria using conditions of completeness, consistency, and nonredundancy. In practice, the addition or subtraction of a partial criterion causes the decision-maker to verify the value of some subjective confidence levels, while adding it lowers it, while reducing it increases it. Let \(\delta=(\delta_{1},\,\delta_{2},\,\ldots,\,\delta_{n},\,\ldots,\,\delta_{N})\) denote the vector of subjective confidence levels as a result of adding the \(F_{N+1}\) criterion. Therefore, the new, subjective confidence level is \(\alpha_{n,1}-\delta_{n}\). **Theorem 3.