Einstein consistency of fuzzy preference relations

Abstract

Pairwise fuzzy preference matrices can be constructed using expert ratings. The number of pairwise preference values to be specified by the experts increases quadratically with the number of options. Consistency (transitivity) allows to reduce this quadratic complexity to linear complexity which makes this approach feasible also for large scale applications. Preference values are usually expected to be on a fixed finite interval. Additive preference is defined on such a finite interval. However, completing preference matrices using additive consistency may yield preferences outside this finite interval. Multiplicative preference is defined on an infinite interval and is therefore not suitable here. To overcome this problem we extend the concept of consistency beyond additive and multiplicative to arbitrary commutative, associative, and invertible operators. Infinitely many of such operators induce infinitely many types of consistency. As one example, we examine Einstein consistency, which is induced by the Einstein sum operator. Completing preference matrices using Einstein consistency always yields preferences inside the finite interval, which yields the first method that allows to construct large scale finite preference matrices using expert ratings. A case study with the real–world car preference data set indicates that Einstein consistency also yields more accurate preference estimates than additive or multiplicative consistency.