Ranking of Independent and Dependent Fuzzy Numbers and Intransitivity in Fuzzy MCDA

Ranking of Fuzzy Numbers (FNs) is a key stage within Fuzzy Multi-Criteria Decision Analysis (FMCDA). However, the influence of FNs dependence on their ranking, including ranking alternatives within FMCDA, has not been studied yet. In this paper, for studying such an influence, the widely-used defuzzification based fuzzy ranking methods, Centroid Index and Integral of Means, along with their modifications, which are pairwise comparison defuzzification ranking methods, are explored. The authors argue that classical defuzzification ranking methods are intended to deal with independent FNs, whereas their modifications may be used for ranking of dependent FNs.It is proved, pairwise comparison Yuan's and defuzzification Integral of Means ranking methods are equivalent when ordering independent and can differ when ordering dependent FNs; at the same time, Yuan's and modified Integral of Means ranking methods are equivalent when ordering both independent and dependent FNs. Intransitivity of the two modified ranking methods when ordering dependent FNs as well as intransitivity of alternatives in FMCDA for Fuzzy Multi-Attribute Value Theory (FMAVT) as an example is proved. The distinctions in ranking of dependent FNs by all ranking methods under consideration are explored through ordering alternatives within FMAVT. For this, a real life case study is considered, and the distinctions in ordering alternatives by classical and modified ranking methods are demonstrated. Statistical analysis of distinctions in ordering alternatives by FMAVT with different ranking methods is implemented with the use of Monte Carlo simulation. The significance of distinctions for the choice and ranking multi-criteria problems as well as for justification of utilizing ranking methods under consideration in FMCDA are discussed.