## 4 Model of Preferences and Judgments of a DM with a Non-Compensatory Aggregation of Preferences
### The Interval Outranking Model  
The method requires the following assumption:  
**Assumption 3** (modeling arbitariness and ill-definition of parameters). : _The DM is willing to specify an outranking model in terms of weights, veto thresholds, a majority threshold, and a credibility threshold. In order to model imperfect information, those parameters can be specified as interval numbers. Therefore, the weights will be denoted as \(\mathbf{w}_{j}=\{w_{j}{}^{-},\,w_{j}{}^{+}\}\), the veto thresholds as \(\mathbf{v}_{j}=\{v_{j}{}^{-},\,v_{j}{}^{+}\}\), the majority threshold as \(\mathbf{\lambda}=[\lambda^{-},\,\lambda^{+}]\), and the credibility threshold as \(\mathbf{\beta}=[\beta^{-},\,\beta^{+}]\)._The credibility index \(\delta_{j}(a^{\prime},\,a)\) of the assertion \"\(a^{\prime}\) is at least as good as \(a\) regarding criterion \(f_{j}\)\" is  
\[\delta_{j}(a^{\prime},\,a)=P(f_{j}(a^{\prime})\geq f_{j}(a))\,j=1,\,\ldots,\,N \tag{12}\]  
where \(P\) is the possibility function of Equation (6). We say that criterion \(f_{j}\) is \(\gamma\)-concordant with the statement \"\(a^{\prime}\) is at least as good as \(a\) regarding criterion \(f_{j}\)\" (denoted \(a^{\prime}S_{j}a\)) if and only if \(a^{\prime}S_{j}a\) with a credibility index of at least \(\gamma\). The set of criteria \(f_{j}\) such that \(\delta_{j}(a^{\prime},\,a)\geq\gamma\) (\(\gamma=\min\{\delta_{j}\}\)) is called \(\gamma\)-possible concordance coalition with \"\(a^{\prime}\) is at least as good as \(a\)\" and is denoted by \(C(a^{\prime}S_{\gamma}a)\). \(\gamma\) is the credibility that all criteria in \(C(a^{\prime}S_{\gamma}a)\) agree that \"\(a^{\prime}\) outranks \(a^{\prime}\)\". All criteria not in \(C(a^{\prime}S_{\gamma}a)\) form the \(\gamma\)-possible discordance coalition, \(D(a^{\prime}S_{\gamma}a)\). For consistency reasons, the interval weights should satisfy the following constraints:  
\[\sum\limits_{j=1}^{N}w_{j}^{-}\leq 1,\] \[\sum\limits_{j=1}^{N}w_{j}^{+}\geq 1.\]  
The concordance index of \"\(a^{\prime}\) outranks \(a^{\prime}\)\" is an interval number denoted by \(c(a^{\prime},\,a)=[c^{-}(a^{\prime},\,a),c^{+}(a^{\prime},\,a)]\), and calculated as follows:  
\[c^{-}\big{(}a^{\prime},\,a\big{)}=\sum\limits_{fj\in C(a^{\prime}S_{\gamma}a )}w_{j}^{-} \tag{13}\]  
when  
\[\sum\limits_{fj\in C(a^{\prime}S_{\gamma}a)}w_{j}^{-}+\sum\limits_{fj\in D(a^ {\prime}S_{\gamma}a)}w_{j}^{+}\geq 1\]  
otherwise,  
\[c^{-}\big{(}a^{\prime},\,a\big{)}=1-\sum\limits_{fj\in D(a^{\prime}S_{\gamma }a)}w_{j}^{+} \tag{14}\]  
similarly,  
\[c^{+}\big{(}a^{\prime},\,a\big{)}=\sum\limits_{fj\in C(a^{\prime}S_{\gamma}a )}w_{j}^{+} \tag{15}\]  
when  
\[\sum\limits_{fj\in C(a^{\prime}S_{\gamma}a)}w_{j}^{+}+\sum\limits_{fj\in D(a^ {\prime}S_{\gamma}a)}w_{j}^{-}\leq 1\]  
otherwise,  
\[c^{+}\big{(}a^{\prime}\,a\big{)}=1-\sum\limits_{fj\in D(a^{\prime}S_{\gamma}a )}w_{j}^{-} \tag{16}\]  
Since the set of criteria in \(C(a^{\prime}S_{\gamma}a)\) is determined by \(\gamma\), the concordance index \(c\) depends on such a value. Such a dependence is denoted by \(c(a^{\prime},\,a,\gamma)\). Each possible concordance coalition \(C(a^{\prime}S_{\gamma}a)\) provides reasons in favor of \"\(a^{\prime}\) outranks \(a^{\prime}\)\", and each possible discordance coalition \(D(a^{\prime}S_{\gamma}a)\) provides reasons against it. In the following, we denote as \(d_{j}(a^{\prime},a)\) the credibility index of the statement \"\(a^{\prime}Sa\) is vetoed by \(g\)\". \(d_{j}(a^{\prime},a)\) is defined as \(P(f_{j}(a)\geq f_{j}(a^{\prime})+v_{j})\), where \(v_{j}\) is the interval veto threshold related to \(f_{j}\). The credibility of a possible outranking given a \(\gamma\)-possible concordance coalition and the corresponding discordance coalition is defined as follows:  
**Definition 8**. : _Let \(\Omega\) be the set {\(\delta_{j}>0,\,j=1,\,\ldots,\,N\)}.