Relationship between the distance consensus and the consensus degree in comprehensive minimum cost consensus models: A polytope-based analysis

Abstract

Agreement in Group Decision-Making problems has recently been tackled through the use of Minimum Cost Consensus (MCC) models, which are associated with solving convex optimization problems. Such models minimize the cost of changing experts’ preferences towards reaching a mutual consensus, and establish that the distance between the modified individual preferences and the collective opinion must be bounded by the threshold ε>0. A recent MCC-based model, called the Comprehensive Minimum Cost Consensus (CMCC) model, adds another constraint related to a parameter γ∈[0,1] to the above constraint related to the parameter ε to enforce modified expert preferences in order to achieve a minimum level of agreement dictated by the consensus threshold 1−γ∈[0,1]. This paper attempts to analyze the relationship between the aforementioned constraints in the CMCC models from two different perspectives. The first is based on inequalities and allows simple bounds to be determined to relate the parameters ε and γ. The second one is based on Convex Polytope Theory and provides algorithms that compute more precise bounds to relate these parameters, and could also be applied to other similar optimization problems. Finally, several examples are provided to illustrate the proposal.