Jana Krejčí

A Note on the Paper “Fuzzy Analytic Hierarchy Process: Fallacy of the Popular Methods”

In the last 30 years, several distinguished researchers have proposed and discussed different fuzzy versions of well-known Saatys Analytic Hierarchy Process (AHP). The paper recently published by K. Zhu in the European Journal of Operational Research heavily criticizes the fuzzy approaches to AHP, claiming the fallacy of all of them. Therefore, it seems to be necessary to clarify whether the criticisms are well-founded or not. The present paper aims to rebut Zhus claims by showing that the evidences and the reasonings in his paper are very poor and far from proving the fallacy of fuzzy AHP.

Fuzzified AHP in the evaluation of scientific monographs

Fuzzification of the analytic hierarchy process (AHP) is of great interest to researchers since it is a frequently used method for coping with complex decision making problems. There have been many attempts to fuzzify the AHP. We focus particularly on the construction of fuzzy pairwise comparison matrices and on obtaining fuzzy weights of objects from them subsequently. We review the fuzzification of the geometric mean method for obtaining fuzzy weights of objects from fuzzy pairwise comparison matrices. We illustrate here the usefulness of the fuzzified AHP on a real-life problem of the evaluation of quality of scientific monographs in university environment. The benefits of the presented evaluation methodology and its suitability for quality assessment of R&D results in general are discussed. When the task of quality assessment in R&D is considered, an important role is played by peer-review evaluation. Evaluations provided by experts in the peer-review process have a high level of subjectivity and can be expected in a linguistic form. New decision-support methods (or adaptations of classic methods) well suited to deal with such inputs, to capture the consistency of experts’ preferences and to restrict the subjectivity to an acceptable level are necessary. A new consistency condition is therefore defined here to be used for expertly defined fuzzy pairwise comparison matrices.

Additively reciprocal fuzzy pairwise comparison matrices and multiplicative fuzzy priorities

The transformations between multiplicatively and additively reciprocal fuzzy pairwise comparison matrices are dealt with, and formulas for obtaining multiplicative fuzzy priorities from additively reciprocal fuzzy pairwise comparison matrices are proposed in this paper. The formulas are based on the concept of constrained fuzzy arithmetic and preserve the additive reciprocity of pairwise comparisons. Further, the consistency issue is approached in the paper; the consistency check is employed directly in the formulas for obtaining multiplicative fuzzy priorities of objects both from multiplicatively and additively reciprocal fuzzy pairwise comparison matrices. Two definitions of consistency based on transitivity are employed in this paper—the traditional additive/multiplicative transitivity and the weak consistency. However, also several other transitivity conditions are discussed in this context. Finally, the proposed formulas for obtaining multiplicative fuzzy priorities from an additively reciprocal fuzzy pairwise comparison matrix and the formulas with employed consistency check are applied in an illustrative example. Triangular fuzzy numbers are used for the fuzzy extension in this paper. However, all the formulas can be modified easily to be applied on intervals, trapezoidal fuzzy numbers or any other fuzzy numbers given by

On multiplicative consistency of interval and fuzzy reciprocal preference relations

\alpha

Fuzzy Pairwise Comparison Matrices

α-cuts.

Incomplete Large-Dimensional Pairwise Comparison Matrices

Multiple-criteria decision making and evaluation problems dealing with a large number of objects are very demanding, particularly when the use of pairwise-comparison (PC) techniques is required. A major drawback arises when it is not possible to obtain all the PCs, due to time or cost limitations, or to split the given problem into smaller subproblems. In such cases, two tools are needed to find acceptable weights of objects: an efficient method for partially filling a pairwise-comparison matrix (PCM) and a suitable method for deriving weights from this incomplete PCM. This paper presents a novel interactive algorithm for large-dimensional problems guided by two main ideas: the sequential optimal choice of the PCs to be performed and the concept of weak consistency. The proposed solution significantly reduces the number of needed PCs by adding information implied by the weak consistency after the input of each PC (providing sets of feasible values for all missing PCs). Interval weights of objects are computed from the resulting incomplete weakly consistent PCM adapting the methodology for calculating fuzzy weights from fuzzy PCMs. The computed weight intervals, thus, cover all possible weakly consistent completions of the incomplete PCM. The algorithm works both with Saatys PCMs and fuzzy preference relations. The performance of the algorithm is illustrated by a numerical example and a real-life case study. The performed simulation demonstrates that the proposed algorithm is capable of reducing the number of PCs required in PCMs of dimension 15 and greater by more than 60% on average.

Discussion and Future Research

Abstract Extension of Saaty’s definition of consistency to interval and fuzzy reciprocal preference relations is studied in the paper. The extensions of the definition to interval and triangular reciprocal preference relations proposed by Wang (2005), Liu (2009), Liu et al. (2014) and Wang (2015a, 2015b) are reviewed and some shortcomings in the definitions are pointed out. Particularly, as was already shown by Wang (2015a, 2015b), the definitions of consistency proposed by Liu (2009) and Liu et al. (2014) are not invariant under permutation of compared objects. Wang’s (2015a, 2015b) definitions rectify this drawbacks. However, as is pointed out in this paper, Wang’s definitions of consistent interval and triangular reciprocal preference relations do not keep the reciprocity of pairwise comparisons, which is the substance of reciprocal preference relations. In this paper, definitions of consistent interval, triangular and trapezoidal reciprocal preference relations invariant under permutation of compared objects and preserving the reciprocity of pairwise comparisons are introduced. Useful tools for verifying the consistency are proposed and some properties of consistent interval and fuzzy reciprocal preference relations are derived. Furthermore, the new definition of consistency for interval reciprocal preference relations is compared with the definition of consistency proposed by Wang et al. (2005), and numerical examples are provided to illustrate the difference between the consistency definitions.

From Measurement to Decision with the Analytic Hierarchy Process: Propagation of Uncertainty to Decision Outcome

This chapter provides a critical review of well-known and in real-life multi-criteria decision making problems most often applied pairwise comparison methods. Three types of pairwise comparison matrices are studied in this chapter—multiplicative pairwise comparison matrices, additive pairwise comparison matrices with additive representation, and additive pairwise comparison matrices with multiplicative representation. The focus is put on the construction of pairwise comparison matrices, definitions of consistency, and methods for deriving priorities of objects from pairwise comparison matrices. Further, the transformations between the approaches for the three different pairwise comparison matrices are studied. The chapter pays a particular attention to two key properties of the pairwise comparison matrices and the related methods—reciprocity of the related pairwise comparisons and the invariance of the pairwise comparison methods under permutation of objects.