## 3 Characterization of GDM-MOPs Under-Study
Such a judgment is related to what the DMs consider as their maximum aspiration levels, which come from the best solution to Problem 11. Therefore, the DM should be able to determine such levels. The best solution to Problem 11 is ill-defined due to the conflicting nature of its objective functions. The following two definitions are useful to characterize a necessary condition to be the best compromise solution. **Definition 6** (a preferred solution by the i-th DM). : _A solution \(x\in O_{i}\) is preferred to \(y\in O_{i}\) by the i-th DM if the following statement is considered as true by them: \"\(x\) is at least as good as \(y\), but \(y\) is not at least as good as \(x\)\"._  
**Remark 1**. : _The interval preference relation from Definition 1 (\(xP_{r}(\beta,\lambda)y\)) and the dominance relation from Definition 5 (\(xD(\alpha)y\)) are good arguments in support to \"\(x\) is preferred to \(y\)\" for sufficiently high values of \(\beta\) and \(\alpha\)._  
**Definition 7** (necessary conditions to be the best compromise solution for the i-th DM). : \(x_{i}{}^{*}\in O_{i}\) _is the best compromise solution to Problem 11 for the i-th DM only if it fulfills two conditions:_  
1. _There is no_ \(y\) _in_ \(O_{i}\) _such that_ \(y\) _is preferred to_ \(x_{i}{}^{*}\) _by the i-th DM;_
2. _There are arguments to justify that the i-th DM considers_ \(x_{i}{}^{*}\) _as at least as good as many solutions that satisfy Condition_ \(A\)_._  
**Assumption 2** (capacity to identify the best compromise solution from the i-th DM's point of view). : _Let \(O_{i}\) be the image of the feasible region of Problem 11. Each individual DM is able to solve Problem 11, thus identifying their best compromise solution \(x_{i}{}^{*}\in O_{i}\)._  
The way in which the \(i\)-th DM can find their best compromise depends on the model of multi-criteria preferences; this should correspond to how the DM aggregates preferences on conflicting criteria. As stated in Section 2, there are three main paradigms: (**i**) the functional approach; (**ii**) the relational approach; and (**iii**) symbolic methods from Artificial Intelligence, mainly those based on rough sets theory. To the best of our knowledge, no symbolic multi-criteria decision method in the frame of interval numbers has been reported. The functional approach bases its prescriptions on a value or utility function, fulfilling axioms of full comparability and transitivity; these value or utility functions are compensatory models in which poor scores of some criteria can be compensated by good scores of other ones. There are a few papers that have used interval-based weighted sum functions as decision tools in project portfolio selection problems (e.g., [39; 40; 41]). To our knowledge, more complex forms of interval-based value functions have not been proposed to address multi-objective optimization problems; further, interval value functions have not been used in solving multi-criteria classification problems. The relational paradigm is generally focused on building and exploiting outranking relations; this approach can handle non-compensatory effects, veto situations, incomparability, and non-transitive preferences. In the frame of interval numbers, the method in ([20]) allows building credible interval outranking relations, and the INTERCLASS-nB method in ([21]) is able to suggest appropriate assignments in multi-criteria ordinal classification problem under imperfect information.