Opinion-based optimal group formation

Most of classical decision making processes aim at selecting the "best" alternative or at ranking alternatives based on the opinions of decision makers. Often, such a process occurs among people (experts or decision makers) who are expected to achieve some shared consensus in ranking the alternatives. However, this is not likely to happen (especially for a large and heterogeneous collection of people) and decision makers tend to reveal groups characteristics derived from their different opinions. A major problem is that inconsistency in opinions arises as each expert has a limited knowledge, errors and misinterpretation of data can occur and thus it is not clear how groups can be identified to be internally consistent and non-conflicting. In this paper, we investigate the conditions under which experts can be split into different sub-groups that share coherent and consistent opinions but are mutually in conflict in the ordering of the alternatives. We face this problem by presenting a non-linear integer programming model where each decision maker specifies incomplete preferences on pairs of alternatives and the objective is to obtain groups having the least possible degree of inconsistency. From a theoretical standpoint, we show that the proposed problem is non-convex and NP-Hard. Moreover, we validate the proposed approach with respect to a case study related to the 2018 Italian political elections. Specifically, we analyze the opinions of 33 decision makers and we show that the proposed technique is able to identify sub-groups characterized by large internal consistency, i.e., the members of each sub-groups express similar judgements upon the different options, while such options are evaluated very differently by the different sub-groups. Interestingly, while dividing the decision makers in three sub-groups, we obtain group rankings that reflect the structure of the Italian political parties or coalitions at the time, i.e., left-wing, right-wing and populists, even if such kind of information has not been directly provided by the decision makers nor used within the proposed case study.