Minimum adjustment consensus optimization models with fuzzy preference relations: The perspective of cardinal and ordinal consensus

In group decision-making (GDM), traditional consensus models have primarily focused on cardinal consensus. In reality, irrespective of whether the objective of GDM is to select the optimal alternative or to rank alternatives, it is imperative to establish a ranking that garners the utmost assent from all decision-makers (DMs). When preferences are articulated through fuzzy preference relations (FPRs), cardinal information emerges in numerical form, quantifying the degree of preference for alternatives, while ordinal relations are implicitly embedded within pairwise comparisons. To delve into both cardinal and ordinal consensus among DMs, this study introduces two consensus optimization models that strive to minimize adjustments to FPRs while fostering consensus in terms of preference intensity and ranking. To this end, we first propose two ordinal consensus measurement methods: one precisely discerns whether DMs have achieved consensus on the selection of the best alternative, while the other assesses the consistency of different preference rankings, taking into account the importance of positions. Based on these methods, two systems of inequalities are designed to explicitly govern both types of ordinal consensus. Subsequently, two consensus control rules are formulated, tailored to distinct objectives. These rules necessitate not only cardinal consensus among all DMs but also their alignment in terms of either the selection of the best alternative or the preference ranking. Ultimately, these rules are integrated as constraints into two mixed-integer programming models aimed at minimizing preference adjustments. The proposed models have been applied in a case study, confirming their practicality, with thorough comparative analyses demonstrating their effectiveness.