Fuzzy extensions of PROMETHEE: Models of different complexity with different ranking methods and their comparison

Models of Fuzzy Multi-Criteria Decision Analysis (FMCDA) are based, as a rule, on different approaches to fuzzy extension of source MCDA methods. For this, simplified models are used to approximate the functions of fuzzy variables with propagation of parametric fuzzy numbers (FNs) through all calculations. In this paper, authors suggest a novel approach to fuzzy extension of MCDA methods, for PROMETHEE-I/II, through development of fuzzy PROMETHEE-I/II (FPOMETHEE-I/II) models of different complexity: in addition to simplified models, the standard fuzzy arithmetic (SFA), and transformation methods (TMs) are implemented for assessing functions of FNs corresponding to these models. For ranking of alternatives, two defuzzification based, and one pairwise comparison ranking methods are implemented within the developed models. Special attention is paid to analysis of the overestimation problem, which can occur when using SFA in the presence of dependent variables in corresponding expressions, and to “proper fuzzy extensions” of PROMETHEE-I/II (i.e., results of all functions of FNs within the model are in accordance with the extension principle) based on TMs and, for some models, on the SFA. One of the key goals of this contribution is comparison of the distinctions in ranking alternatives by different FPROMETHEE-II models. It is demonstrated by evaluating a large number of scenarios based on Monte Carlo simulation that the probability of distinction in ranking alternatives by “proper” and “approximated” FPROMETHEE-II models may be considered as significant for ranking multicriteria problems. Another goal of this paper is analysis of the correctness of FPROMETHEE-I/II models with respect to the basic MCDA axiom related to ranking of dominated and dominating alternatives. Authors demonstrate that the basic axiom can be violated, in the general case, by all developed FPROMETHEE-I/II models and suggest an approach to fix this problem.