## 4 Model of Preferences and Judgments of a DM with a Non-Compensatory Aggregation of Preferences
### Making judgments of Satisfaction and Dissatisfaction
Let us define the outranking credibility index of \(x\) with respect to the boundary \(B\) as follows:  
\[\sigma(x,B,\mathbf{\lambda})=max\ \{\sigma(x,b_{k},\mathbf{\lambda}),k=1,\ldots\ card(B) \tag{17}\]  
The value of \(\sigma\) can be interpreted as a measure of how much the DM feels satisfaction with \(x\). The following proposition states several consistency properties of the assignments from Definitions 11-13. **Proposition 2 (consistency properties of assignments)**. : _Under Assumptions 4 and 5, suppose that \(x\) and its pre-image satisfy the constraints imposed by the DM. The following propositions are valid:_  
1. \(x\) _is assigned to a single element of the set of classes (satisfactory, unsatisfactory, neither satisfactory nor unsatisfactory)._
2. _The assignment suggested for_ \(x\) _is independent of the assignment of other solutions._
3. _The class to which_ \(x\) _is assigned by the_ \(i\)_-th DM is independent of the assignment made by any other DM._
4. _Let_ \(y\) _be a feasible solution. Given_ \(\mathbf{\lambda}\)_, if_ \(x\) _and_ \(y\) _have the same interval outranking credibility indices with respect to the limiting profiles, then they are assigned to the same element of the set of classes (satisfactory, unsatisfactory, neither satisfactory nor unsatisfactory)._
5. _If there is_ \(b_{k}\in B\) _fulfilling_ \(x=b_{k}\)_, then_ \(x\) _is assigned to the satisfactory class._
6. _If there is_ \(b_{k}\) _such that_ \(xI(\mathbf{\beta},\mathbf{\lambda})b_{k}\) _and there is no_ \(b_{i}\in B\) _fulfilling_ \(b_{i}P_{r}(\mathbf{\beta},\mathbf{\lambda})x\)_, then_ \(x\) _is assigned to the satisfactory class._
7. _If_ \(x=x^{*}\)_, then_ \(x\) _is assigned to the satisfactory class._
8. _Let_ \(y\) _be a feasible solution such that_ \(y\)__\(D(\mathbf{\alpha})x\) _(_\(\alpha\geq\mathbf{\beta}\)_). If_ \(x\) _is assigned to the satisfactory class, then_ \(y\) _is assigned to the same class._  
Proof. : Proposition 2(a): The proof follows from two facts: (i) \(x\) has to fulfill one of the three Definitions 11-13, and (ii) the fulfillment of one definition excludes fulfillment of another. Proposition 2(b): The proof is obvious from Definitions 11-13. Proposition 2(c): The proof is obvious from Definitions 11-13. Proposition 2(d): The proof is obvious from Definitions 11-13. Proposition 2(e): \(x=b_{k}\), Assumption 4.i, Remark 2.a, Definition 2.i and Assumption 4.iv \(\Rightarrow xS(\mathbf{\beta},\mathbf{\lambda})B\)\(x\) is feasible and \(xS(\mathbf{\beta},\mathbf{\lambda})B\)\(\Rightarrow x\) is satisfactory for the DM (Definition 11). Proposition 2(f): \(xI(\mathbf{\beta},\mathbf{\lambda})b_{k}\Rightarrow xS(\mathbf{\beta},\mathbf{\lambda})b_{k}\) (Definition 1.iii) \(xS(\mathbf{\beta},\mathbf{\lambda})b_{k}\) and there is no \(b_{i}\in B\) such that \(b_{i}P_{r}(\mathbf{\beta},\mathbf{\lambda})x\Rightarrow xS(\mathbf{\beta},\mathbf{\lambda})B\) (Definition 2.i) \(x\) is feasible and \(xS(\mathbf{\beta},\mathbf{\lambda})B\)\(\Rightarrow x\) is satisfactory for the DM (Definition 11). Proposition 2(g): The proof follows trivially from Assumption 4.iii, Definition 2.i and Definition 11. Proposition 2(h): \(x\) is assigned to \(C_{sat}\Rightarrow xS(\beta,\lambda)B\) from Definition 11\(\Rightarrow\exists\)\(b_{k}\in B\) such that \(xS(\beta,\lambda)b_{k}\) and there is no \(b_{i}\in B\) with \(b_{i}P_{r}(\beta,\lambda)x\). From Proposition 1.i and Remark 2.b, \(yD(\alpha)x\) (\(\alpha\geq\beta\)) and \(xS(\beta,\lambda)b_{k}\Rightarrow yS(\beta,\lambda)b_{k}\). There is no \(b_{i}\in B\) with \(b_{i}P_{r}(\beta,\lambda)x\) and \(y\)\(D(\alpha)x\) (\(\alpha\geq\beta\)) \(\Rightarrow\) There is no \(b_{i}\in B\) with \(b_{i}P_{r}(\beta,\lambda)y\) counter-reciprocal of Proposition 1.ii. Combining (A) and (B), we have \(yS(\beta,\lambda)B\) (Definition 2.i) \(\Rightarrow y\) is assigned to \(C_{sat}\) (Definition 11). The proof is finished.