## 2 Background
### Toward a Maximum Consensus  
According to [PERSON] and [PERSON] (2013), in GDM, reaching an acceptable agreement is not always possible. This comes from strong contradictions amongst important disjoint subsets of group members; for instance, there could be a clear opposition of a numerically significant minority that does not want to accept a preference coming from a not very strong majority of the group. Reaching a collective agreement is only possible when there is unanimity or an appreciable level of consensus. Among many ways to address and define consensus (e.g., [10; 14; 44; 45; 46; 47]), [PERSON] and [PERSON] (2013) consider a good consensus as associated with the fulfillment of two conditions:  
1. There is an important agreeing majority with a particular alternative (or solution);
2. There is no appreciable disagreeing minority.  
With Condition (B), [PERSON] and [PERSON] in [33] and [PERSON] et al. in [48] rejected \"majority dictatorship\", which neglects the importance of the intensity of disagreement and degrades fairness and equity concerning minorities.  
Let us consider a group with \(n_{g}\) members under the following general premise:  
Fundamental Premise: Each solution \(z\) in the decision variable space (with its corresponding objective values and feasibility) can be classified by the \(i\)-th DM into one and only one of the following classes: (i) \(z\) is a satisfactory solution; (ii) \(z\) is an unsatisfactory solution; (iii) the \(i\)-th DM is neither satisfied nor dissatisfied with \(z\).  
Point (iii) concerns situations in which a DM hesitates about the appropriate classification of a solution and confers to the above premise a very high generality.  
Let \(N_{sat}\) (resp. \(N_{dis}\)) denote the number of group members who are satisfied (resp. dissatisfied) by a particular solution \(z\). \(N_{sat}\) and \(N_{dis}\) are integer functions of \(z\). The ideal consensus (if possible) should correspond to a point \(z^{*}\) such that \(N_{sat}(z^{*})=n_{g}\) and \(N_{dis}(z^{*})=0\).  
Therefore, the ideal consensus is the ideal solution to the bi-objective optimization problem:  
\[\text{Maximize }N_{sat}(z)\text{, Minimize }N_{dis}(z) \tag{1}\]  
Unlike typical multi-objective optimization problems in which its ideal solution is not feasible, Problem 1 may reach the ideal point \(N_{sat}(z^{*})=n_{g}\) and \(N_{dis}(z^{*})=0\) as much as the preferences and beliefs of all the individual DMs become close enough through an effective consensus reaching process. Even when the ideal consensus is not possible, good agreements can be identified when \(N_{sat}\geq 2/3\)\(n_{g}\) (Condition A) and \(N_{dis}\approx 0\) or \(N_{dis}<<0.5\)\(n_{g}\) (Condition B).  
For a given closeness of preferences and beliefs of the individual DMs, the best consensus is a non-dominated solution to Problem 1. If this current consensus level is not high enough, a new CRP step should allow identifying better agreements.  
To make operational the process of solving Problem 1, we need a mapping between the decision variable space and the values \(N_{sat}\) and \(N_{dis}\). Therefore, we require a model that permits, for each DM and each \(z\) in the decision variable space, knowing to which class of satisfaction/dissatisfaction \(z\) is assigned to. Most of the remaining paper is devoted to discussing some preference models and assignment procedures to this end. Since in the objective space, \(z\) is described by multiple objective values, its assignment should be considered as a multi-criteria ordinal classification problem.