Mutual Consensus and its Application in Minimum Cost Consensus Models

TL;DR

The paper addresses robust consensus in large-scale group decision making by introducing mutual consensus, a non-compensatory metric defined as $κ(x) = max_{i<j} |x_i - x_j|$, and integrating it into Minimum Cost Consensus (MCC) models to minimize adjustment costs while enforcing tight pairwise disagreement. It develops the Minimum Cost with Mutual Consensus (MCMC) model and proves that mutual consensus dominates classical measures, enabling a reduction of constraints via sorting and permutation, with a linearization path under symmetry. The work then connects mutual consensus to OWA-based MCC (OWA-MCC), showing both the potential linearization under symmetry and the inherent NP-hardness in general non-convex cases, and derives bounds linking MCC and OWA-MCC regions. To tackle practical instances, the authors introduce ApOWAMCC, an MCMC-based approximation algorithm that leverages δ_- and δ_+ bounds to efficiently approximate solutions for OWA-MCC, demonstrated through computational experiments on small and large GDM problems. Overall, the paper provides theoretical foundations for a robust, scalable consensus framework and a practical algorithm for approximating complex OWA-based MCC problems in large decision-making groups.

Abstract

This paper introduces the concept of {mutual consensus} as a novel non-compensatory consensus measure that accounts for the maximum disparity among opinions to ensure robust consensus evaluation. Incorporating this concept, several new Minimum Cost Consensus (MCC) models are proposed, and their properties are analyzed. To show their applicability, these mutual consensus-based MCC models are then considered in the context of the {OWA-MCC} model, which employs Ordered Weighted Averaging (OWA) operators for preference aggregation. Concretely, we include a linearized formulation under symmetry conditions as well as examples of the non-convexity of the feasible region in the general case. Finally, mutual consensus is utilized to obtain approximate solutions for the OWA-MCC model, demonstrating its practical effectiveness and advancing the theoretical and applied dimensions of consensus modeling in group decision-making.

Figures (2)

• Figure 1: Examples showing the non-convexity of the regions that determine the G-OWA-MCC model.
• Figure 2: Draft of the regions $\mathcal{R}_{\delta_-},\; \mathcal{R}_\varepsilon^{\Psi_\omega}$ and $\mathcal{R}_{\delta_+}$ for $\varepsilon=0.25$ and $\omega=(0.35,0.2,0.45)$.

Theorems & Definitions (36)

• Definition 1: MCC model benarieh2007zhang2011
• Definition 2: OWA operator yager1988
• Definition 3
• Definition 4: MCMC model
• Theorem 1
• proof
• Proposition 1
• proof
• Theorem 2
• proof