A goal programming approach to deriving interval weights in analytic form from interval Fuzzy preference relations based on multiplicative consistency

Abstract

This paper focuses on how to find an analytic solution of optimal interval weights from consistent interval fuzzy preference relations (IFPRs) and obtain approximate-solution-based interval weights in analytic form from inconsistent IFPRs. The paper first analyzes the popularly used interval weight additive normalization model and illustrates its drawbacks on the existence and uniqueness for characterizing ]0, 1[-valued interval weights obtained from IFPRs. By examining equivalency of ]0, 1[-valued interval weight vectors, a novel framework of multiplicatively normalized interval fuzzy weights (MNIFWs) is then proposed and used to define multiplicatively consistent IFPRs. The paper presents significant properties for multiplicatively consistent IFPRs and their associated MNIFWs. These properties are subsequently used to establish two goal programming (GP) models for obtaining optimal MNIFWs from consistent IFPRs. By the Lagrangian multiplier method, analytic solutions of the two GP models are found for consistent IFPRs. The paper further devises a two-step procedure for deriving approximate-solution-based MNIFWs in analytic form from inconsistent IFPRs. Two visualized computation formulas are developed to determine the left and right bounds of approximate-solution-based MNIFWs of any IFPR. The paper shows that this approximate solution is an optimal solution if an IFPR is multiplicatively consistent. Three numerical examples including three IFPRs and comparative analyses are offered to demonstrate rationality and validity of the developed model.