Eigenproblem driven triangular fuzzy analytic hierarchy process

Abstract

The eigenproblem plays a crucial role on checking acceptability of decision-makers’ pairwise comparison results and deriving priorities from preference relation matrices in the analytic hierarchy process (AHP). This paper analyzes two recent fuzzy eigenvector methods and illustrates their deficiencies. Two frameworks of additively normalized triangular fuzzy priorities (ANTFPs) are introduced to characterize a cluster of equivalent fuzzy priority vectors. Based on the approximation relationship between triangular fuzzy multiplicative preference relation matrices and their most appropriate ANTFPs, three eigenproblems with positive real matrices are established and an eigenvector based linear program is developed to obtain support interval based ANTFPs from fuzzy multiplicative preference relation matrices. The largest eigenvalue weighting consistency index and consistency ratio are defined and an acceptability checking method is then proposed by both examining acceptable consistency of fuzzy multiplicative preference relation matrices and acceptable vagueness of eigenvector based ANTFPs. Afterwards, an eigenproblem driven triangular fuzzy AHP is devised in detail. A numerical illustration including four fuzzy multiplicative preference relation matrices is provided and a comparative study is carried out to demonstrate the superiority and effectiveness of the presented models. Meanwhile, a multi-criteria graduate job selection problem is used to show the application of the proposed triangular fuzzy AHP.