New additive consistency framework and utility derivation for interval fuzzy reciprocal preference relations

The interval fuzzy reciprocal preference relation (IFRPR) is one of commonly used frameworks characterizing decision-makers’ indeterminate preferences in multi-criteria decision making. Existing additive consistency frameworks and methods of generating additively consistent IFRPRs from interval fuzzy utility vectors as well as interval utility derivation methods often fail to obtain a satisfactory solution. To settle these challenging issues, this paper devises parametric transitivity equations to define additive consistency of IFRPRs. The parameter values are then determined and an equivalent definition is proposed for additively consistent IFRPRs. Important properties of consistent IFRPRs are developed and an additive consistency index is designed. A notion of equivalent interval fuzzy utility vectors is introduced and a novel framework is presented to normalize interval fuzzy utility vectors. Computational formulas are established for the generation between additively consistent IFRPRs and normalized interval fuzzy utility vectors. A goal programming model is built and converted into a linear program for deriving normalized interval fuzzy utility vectors from IFRPRs. Three illustrations including comparative analysis are provided to expose how the proposed models are utilized to identify whether an IFRPR is additively consistent and determine unknown preferences in an incomplete IFRPR as well as obtain normalized interval fuzzy utility vectors from IFRPRs.