## 2 Background
### Multi-Criteria Ordinal Classification Based on an Interval Outranking Approach  
There are more than one hundred multi-criteria decision-making (MCDM) methods. According to the model of the decision-maker's preferences, most of the methods can be grouped in one of three basic paradigms:  
* The functional paradigm, based on building functions that model the decision-maker's preferences (e.g., [51; 52]);
* The relational paradigm, based on building crisp or fuzzy binary relations (e.g., [53]);
* The symbolic paradigm, which is mainly related to artificial intelligence (e.g., [54]). Whatever the model of the DM's preferences, we can distinguish three MCDM fundamental problems: multi-criteria choosing, ranking, and sorting. Each MCDM method addresses one of these general problems. For choosing and ranking, multi-attribute utility theory (MAUT) ([51]) and the analytic hierarchy process (AHP) ([52]) are the most popular approaches. Although [PERSON] et al. in [55] proposed a variant of AHP for multi-criteria sorting (also called multi-criteria ordinal classification) problems, outranking and symbolic methods are more popular for this purpose. Perhaps the most known multi-criteria ordinal classification method is ELECTRE TRI-B ([56]), which was generalized to ELECTRE TRI-nB in ([57]). Such methods make use of an outranking relation, which means that for a pair of decision actions (\(x\),\(y\)), we should check the validity of the assertion \"action (alternative) \(x\) is at least as good as action (alternative) \(y\)\". Two effects (concordance and discordance) are combined to determine the degree of credibility of the outranking (cf. [53]). In ELECTRE TRI-B and nB, the boundary between adjacent classes is described by limiting actions. Unlike ELECTRE TRI-B, its extension allows characterizing the boundary between adjacent classes by several limiting profiles. Recently, ELECTRE TRI-nB was extended to handle imperfect information in terms of interval numbers ([21]). The so-called INTERCLASS-nB shares similar assignment rules and consistency properties with ELECTRE TRI-nB, but using the interval outranking approach proposed in ([20]). This handles imperfect information on criterion scores, weights, veto thresholds, and majority thresholds with interval numbers. Even missing criterion scores can be handled by the approach. According to [PERSON] et al. (2020), \"eliciting model preference parameters generally suffer from arbitrariness and imperfect knowledge (in particular, due to ill-determination). This occurs in a more acute way when there is a group of DMs, so that conflicting views are quite natural and frequent.\"  
The interval outranking approach and INTERCLASS-nB work with several interval preference relations:  
**Definition 1. **  
(i) \(xS(\beta,\lambda)y\Leftrightarrow\sigma(x,y,\lambda)\geq\beta\) (interval outranking);  
(ii) \(xP_{r}(\beta,\lambda)y\Leftrightarrow\sigma(x,y,\lambda)\geq\beta\) and \(\sigma(y,x,\lambda)<\beta\) (interval preference);  
(iii) \(xI(\beta,\lambda)y\Leftrightarrow\sigma(x,y,\lambda)\geq\beta\) and \(\sigma(y,x,\lambda)\geq\beta\) (interval indifference);  
where:  
\(\sigma\) is the credibility index of the interval outranking;  
\(\lambda\) is an interval number representing a majority threshold; \(\lambda>\) [0.5, 0.5] and \(\lambda^{min}\geq\) 0.5;  
\(\beta\) is a credibility threshold for establishing a credible crisp outranking relation; \(\beta>\) 0.5. The INTERCLASS-nB, adapted to the case of only two ordered classes, is presented below:  
Conditions on the limiting profiles: Let \(C_{1}\) and \(C_{2}\) be ordered classes (\(C_{2}\) is the most preferred). Set \(\beta>\) 0.5 and \(\Lambda>\) [0.5, 0.5] with \(\lambda^{min}\geq\) 0.5. The boundary between \(C_{1}\) and \(C_{2}\) is characterized by a set of limiting profiles, \(B=\{b_{j}\}\), such that:  
(i)  All \(b_{j}\) of \(B\) belongs to \(C_{2}\);  
(ii)  There is no pair (\(b_{j}\), \(b_{i}\)) such that \(b_{j}P_{r}(\beta,\lambda)b_{i}\).