Matrix representations of the inverse problem in the graph model for conflict resolution with fuzzy preference

Abstract

The inverse analysis of the graph model for conflict resolution (GMCR) determines all the preference relationships required by each decision-maker (DM) according to a given stability definition to ensure that the given state is stable. In the paper, a matrix representation approach for the inverse graph model with fuzzy preference relations is proposed to obtain all the preference relations for each DM making the stability result based on the fuzzy stability definition, including fuzzy Nash, fuzzy general metarationality, fuzzy symmetric metarationality, and fuzzy sequential stability definition. For the two-DM graph model, the fuzzy preference relation over states, unilateral movements, and fuzzy unilateral improvements are refreshed by using the matrix forms. Furthermore, for the multiple DMs model, the joint movements and joint improvement for a collection including at least two DM are represented by using matrix representation. Finally, the matrix relationship for specifying the required preference relation in the framework of inverse graph model with fuzzy preferences is derived. To illustrate the usefulness of the matrix representation of inverse graph model with fuzzy preference relations, this paper demonstrates it using a real-life conflict of the Elmira groundwater contamination conflict.