## 4 Model of Preferences and Judgments of a DM with a Non-Compensatory Aggregation of Preferences
### Making judgments of Satisfaction and Dissatisfaction  
Let \(x^{\star}\) be the best compromise solution obtained by an individual DM by exploiting the interval outranking relation of Section 4.1. **Assumption 4 (capacity to set the limiting boundary between classes). **_Let \(C_{\text{sat}}\) and \(C_{\text{dis}}\) denote classes \"satisfactory\" and \"unsatisfactory\", respectively. Consider \(\beta>0.5\) (\(\beta^{-}>0.5\)) and \(\lambda\) > 0.5 (\(\lambda^{-}>0.5\)). Taking \(x^{\star}\) as reference, the DM is able to set a boundary B= \(\{b_{1},\ldots\ b_{n}\}\) between \(C_{\text{sat}}\) and \(C_{\text{dis}}\) fulfilling:_  
1. \(f_{\text{f}}(b_{k})\) _are real numbers,_ \(j=1,\ldots\ N\) _and_ \(k=1,\ldots\ n\)_;_
2. _Each_ \(b_{k}\) (\(k=1,\ldots\ n\)) _belongs to_ \(C_{\text{sat}}\)_;_
3. _For all_ \(b_{k}\) _belonging to B, we have_ \(x^{\star}\text{P}_{r}(\beta,\lambda)b_{k}\)_;_
4. _There is no pair (_\(b_{i}\)_,_ \(b_{k}\)_) such that_ \(b_{i}\text{P}_{r}(\beta,\lambda)b_{k}\)_._  
Let us back to Definition 2, which establishes relations between solutions and the boundary. Replace \(\beta\) with the interval number \(\beta\). Assumption 4 guarantees the conditions to apply INTERCLASS-nB (see Section 2.4) to making judgments of satisfaction and dissatisfaction. From Definition 2, it follows that \(xS(\beta,\lambda)B\Rightarrow not\ (BP_{r}(\beta,\lambda)x)\) and \(BP_{r}(\beta,\lambda)x\Rightarrow not\ (xS(\beta,\lambda)B)\). If a solution \(x\) outranks the limiting boundary \(B,x\) is assigned to the best class by the \"pessimistic\" procedure (Definition 3); additionally, since \(xS(\beta,\lambda)B\Rightarrow not\ (BP_{r}(\beta,\lambda)x)\), \(x\) is also assigned to \(C_{\text{sat}}\) by the \"optimistic\" procedure (Definition 4) contrarily, if the boundary is preferred to \(x\), since \(BP_{r}(\beta,\lambda)x\Rightarrow not\ (xS(\beta,\lambda)B)\), \(x\) will be assigned to the worst class by both assignment rules. If \(not\ (xS(\beta,\lambda)B\) and \(not\ (BP_{r}(\beta,\lambda)x)\) are both fulfilled, then \(x\) is assigned to \(C_{\text{dis}}\) by the \"pessimistic\" rule and to \(C_{\text{sat}}\) by the \"optimistic\" procedure. In such a case, the DM may be doubtful about the class \(x\) should be assigned to. **Assumption 5 (compatibility with INTERCLASS-nB). **_Suppose that \(x\) and its pre-image satisfy the constraints imposed by the \(i\)-th DM. We assume that the DM is willing to accept the assignment suggested by INTERCLASS-nB._  
As a consequence of Assumption 5, even if \(xS(\beta,\lambda)B\) or \(not\ (BP_{r}(\beta,\lambda)x)\), the nonfulfillment of the constraints vetoes a satisfactory assignment. Based on this assumption, the definitions of what is a satisfactory (non-satisfactory) solution for a DM (who is compatible with the interval outranking model) are straightforward. **Definition 11. **_Suppose that an individual DM is compatible with the outranking model of Section 4._ The DM is said to be satisfied with a solution x iff the following conditions are fulfilled:  
1. \(xS(\beta,\lambda)B\)__
2. \(x\) _and its pre-image satisfy the constraints imposed by the DM._  
**Definition 12. **_Under the same premise of the above definition, the DM is said to be dissatisfied with a solution x iff at least one of the following conditions is fulfilled:\( \begin{array}{ll}\text{(A)}&BP_{r}(\mathbf{\beta},\mathbf{\lambda})x\\ \text{(B)}&x\text{ and/or its pre-image do not satisfy the constraints imposed by the DM.}\end{array} \)  
**Definition 13**. : _The DM is neither satisfied nor dissatisfied with a solution \(x\) if the following conditions are all fulfilled:_  
1. _not (_\(xS(\mathbf{\beta},\mathbf{\lambda})B\)_)_
2. _not (_\(BP_{r}(\mathbf{\beta},\mathbf{\lambda})x\)_)_
3. \(x\) _and its pre-image satisfy the constraints imposed by the DM._  
It is obvious that the above definitions follow from the INTERCLASS-nB assignment rules and Assumption 5. Suppose that the individual DM is satisfied with a solution \(x\).