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Highlights • The results of aggregation by weighted arithmetic mean are normalization-dependent. • Weighted arithmetic mean aggregation fails to reflect preference information well. • Serious problems caused by the arithmetic mean aggregation shown on examples. • Theorems claiming the superiority of geometric mean aggregation in AHP presented. • Weighted geometric mean recommended for aggregation of local priorities in AHP. Abstract The main focus of this paper is the aggregation of local priorities into global priorities in the Analytic Hierarchy Process (AHP) method. We study two most frequently used aggregation approaches - the weighted arithmetic and weighted geometric means - and identify their strengths and weaknesses. We investigate the focus of the aggregation, the assumptions made on the way, and the effect of different normalizations of local priorities on the resulting global priorities and their ratios. We clearly show the superiority of the weighted geometric mean aggregation over the weighted arithmetic mean aggregation in AHP for the purpose of deriving global priorities of alternatives. We also contribute to the literature on rank reversal in AHP. In particular, we show that a change of the normalization condition for the local priorities of alternatives may result in different ranking when the weighted arithmetic mean aggregation is used for deriving global priorities of alternatives, and we demonstrate that the ranking obtained by the weighted geometric mean aggregation is not normalization dependent. Moreover, we prove that the ratios of global priorities of alternatives obtained by the weighted geometric mean aggregation are invariant under the normalization of local priorities of alternatives and weights of criteria. We also propose three alternative approaches to aggregating preference information contained in local pairwise comparison matrices of alternatives into a global consistent pairwise comparison matrix of alternatives and prove their equivalence. [ABSTRACT FROM AUTHOR]

Analytic Hierarchy process (AHP) is a powerful method belonging to the full aggregation family of multi-criteria decision-making methods based on pairwise comparisons of objects. Since the information about the problem is usually not complete in real decision-making problems, it is difficult to express precisely the preferences on pairs of compared objects. This problem has been handled in the literature by introducing fuzziness into AHP. However, neither AHP nor its fuzzy extensions can deal with sorting decision-making problems, which form a significant part of decision-making problems. This paper presents the FAHPSort method — a fuzzy extension of the AHPSort method, which is an adaptation of the AHP method for sorting decision-making problems. The FAHPSort method handles the vagueness in the meaning of linguistic terms expressing the intensity of preference of one object over another one. Key properties of the FAHPSort method are described in the paper, and the method is illustrated in a decision-making problem. [ABSTRACT FROM AUTHOR]

Proper formulas for obtaining the fuzzy maximal eigenvalue and the corresponding fuzzy maximal eigenvector of a fuzzy pairwise comparison matrix are proposed in this paper. First, the formulas for obtaining the fuzzy maximal eigenvalue of a fuzzy pairwise comparison matrix proposed by Csutora & Buckley (2001) and by Ishizaka (2014) are reviewed, and the flaws in their formulas regarding the violation of the reciprocity of pairwise comparisons are pointed out. New formulas for obtaining the fuzzy maximal eigenvalue preserving the reciprocity of pairwise comparisons are then proposed. After, a fuzzy extension of Saaty's Consistency Index and Consistency Ratio is introduced in order to verify an acceptable level of inconsistency of a fuzzy pairwise comparison matrix. Further, the methods for obtaining the fuzzy maximal eigenvector corresponding to the fuzzy maximal eigenvalue of a fuzzy pairwise comparison matrix proposed by Wang & Chin (2006) and by Ishizaka (2014) are reviewed. The flaws in Ishizaka's method are pointed out, and Wang & Chin's method is studied and modified in order to preserve the reciprocity of pairwise comparisons. The fuzzy maximal eigenvalues and the corresponding fuzzy maximal eigenvectors obtained by the new formulas are confronted with those obtained by methods proposed by Csutora & Buckley (2001), Wang & Chin (2006) and Ishizaka (2014), and three numerical examples are given for better illustration. [ABSTRACT FROM AUTHOR]

The aim of the paper is to highlight the necessity of applying the concept of constrained fuzzy arithmetic instead of the concept of standard fuzzy arithmetic in a fuzzy extension of Analytic Hierarchy Process (AHP). Emphasis is put on preserving the reciprocity of pairwise comparisons during the computations. For deriving fuzzy weights from a fuzzy pairwise comparison matrix, we consider a fuzzy extension of the geometric mean method and simplify the formulas proposed by Enea and Piazza (Fuzzy Optim Decis Mak 3:39-62, 2004). As for the computation of the overall fuzzy weights of alternatives, we reveal the inappropriateness of applying the concept of standard fuzzy arithmetic and propose the proper formulas where the interactions among the fuzzy weights are taken into account. The advantage of our approach is elimination of the false increase of uncertainty of the overall fuzzy weights. Finally, we advocate the validity of the proposed fuzzy extension of AHP; we show by an illustrative example that by neglecting the information about uncertainty of intensity of preferences we lose an important part of knowledge about the decision making problem which can cause the change in ordering of alternatives. [ABSTRACT FROM AUTHOR]

In the last 30 years, several distinguished researchers have proposed and discussed different fuzzy versions of well-known Saaty's Analytic Hierarchy Process (AHP). The paper recently published by K. Zhü 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 Zhü's claims by showing that the evidences and the reasonings in his paper are very poor and far from proving the fallacy of fuzzy AHP. [ABSTRACT FROM AUTHOR]