## 5 Model for a DM Whose Preferences Are Compatible with a Weighted-Sum Function
### Making judgments of Satisfaction and Dissatisfaction with the Functional Model  
We will present here an idea that is, to a certain extent, similar to the one on which INTERCLASS-nB and the proposal in Section 4.3 are inspired. If the DM is capable of setting a limiting boundary between the two classes, all feasible solutions preferred to the boundary should be assigned to the best class and vice versa. Formally: Let \(x^{\star}\) be the best compromise solution obtained by the individual DM following the procedure exposed in the previous section. **Assumption 7 (capacity to set the limiting boundary between classes)**. : _Let \(C_{\text{sat}}\) and \(C_{\text{dis}}\) denote classes \"satisfactory\" and \"unsatisfactory\", respectively. Consider a sufficiently high value of the credibility threshold \(\alpha\). Taking \(x^{\star}\) as reference, the DM is able to set a limiting boundary B= {b_1,... \(b_{n}\)} between \(C_{\text{sat}}\) and \(C_{\text{dis}}\) fulfilling:_  
1. _For all_ \(b_{k}\) _belonging to B, we have_ \(x^{\star}\) _is a-preferred to_ \(b_{k}\)_;_
2. _There is no pair (_\(b_{i}\)_,_ \(b_{k}\)_) in_ \(B\times B\) _such that_ \(b_{i}\) _is a-preferred to_ \(b_{k}\)_;_
3. _For all_ \(b_{k}\) _belonging to B, the DM hesitates about its assignment._  
**Definition 16 (\(\alpha\)-preference between a solution and the boundary)**. :
1. _[label=()]_
2. \(x\) _is_ \(\alpha\)_-preferred to B iff there is_ \(b_{k}\in B\) _such that_ \(x\) _is_ \(\alpha\)_-preferred to_ \(b_{k}\) _and there is no_ \(b_{i}\in B\) _such that_ \(b_{i}\) _is_ \(\alpha\)_-preferred to_ \(x\)_._
3. _The boundary B is_ \(\alpha\)_-preferred to_ \(x\) _iff there is_ \(b_{k}\in B\) _such that_ \(b_{k}\) _is_ \(\alpha\)_-preferred to_ \(x\) _and there is no_ \(b_{i}\in B\) _such that_ \(x\) _is_ \(\alpha\)_-preferred to_ \(b_{i}\)_._  
**Definition 17**. : _Consider \(x\in O\) and \(b_{k}\in B\). The credibility index of the statement \"\(x\) is preferred to \(b_{k}\)\" is defined as \(\sigma_{P}\) (\(x\), \(b_{k}\)) = Maximum \(\chi\) fulfilling at least one of the following conditions:_  
1. \(P(\textbf{U}(x)\geq\textbf{U}(b_{k}))=\chi\)__
2. \(xD(\chi)b_{k}\)__  
The credibility index \(\sigma_{P}\) (\(x\), \(B\)) of the statement \"\(x\) is preferred to the boundary \(B\)\" is defined as:  
\[\sigma_{P}\) (\(x\), \(B\)) = \(Max\) {\(\sigma_{P}\) (\(x\), \(b_{k}\))}, \(k=1,\,\ldots\,\,card\) (\(B\)) (19)\(\sigma_{P}\left(x,\,B\right)\) can be interpreted as the degree of credibility of the statement \"\(x\) is preferred to the boundary by the \(i\)-th DM\". **Definition 18**. : _Suppose that an individual DM is compatible with the weighted-sum function model and Assumptions 6 and 7. The DM is said to be satisfied with a solution \(x\) iff the following conditions are fulfilled:_  
* \(x\) is _\(\alpha\)-preferred_ to the Boundary \(B\)_;_
* \(x\) and its pre-image fulfill the constraints imposed by the DM. **Definition 19**. : _Suppose that an individual DM is compatible with the weighted-sum function model and Assumptions 6 and 7. The DM is said to be dissatisfied with a solution \(x\) if at least one of the following conditions is fulfilled:_  
* \(B\) is _\(\alpha\)-preferred_ to \(x\)_;_
* \(x\) and/or its pre-image do not fulfill the constraints imposed by the DM. **Definition 20**. : _The DM is neither satisfied nor dissatisfied with a feasible solution \(x\) iff the following conditions are held:_  
* \(x\) is not _\(\alpha\)-preferred_ to the Boundary \(B\)_;_
* \(B\) is not _\(\alpha\)-preferred_ to \(x\)_._  
**Proposition 3 (consistency properties of assignments)**. : _Suppose that \(x\) is feasible.