Deriving a minimum distance-based collective preorder: a binary mathematical programming approach

Deriving the “closest” (minimal distance) collective judgment to all individual opinions is a complex aggregation problem that has been widely studied in group decision-making literature. However, most of the existing literature does not consider individual opinions expressed as partial preorders (i.e., a preference system which includes the incomparability relation). In this paper, we propose a method based on binary linear programming to derive a minimum distance-based collective preorder from individual preferences relational systems (p.r.s.). This method is threefold. First, each member determines a preorder (partial or total) over the set of alternatives. Second, an aggregation algorithm is proposed to derive at least one collective and not necessary transitive p.r.s. at minimum distance from all individual preorders. Third, a binary linear programming optimization will transform each non-transitive collective p.r.s. into a collective preorder (i.e. a transitive p.r.s.). The proposed method has three main advantages: (1) it deals with incomparability (partial preorders), (2) the relative importance of the members is explicitly considered and (3) the collective p.r.s. obtained after the aggregation step might be “exploited” according to different decision-making problematics (i.e. ranking, choice and sorting).