Scale transitivity in the AHP
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Scale transitivity in the AHP P Ji and R Jiang The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Transitivity is important in multicriteria decision-making. The analytic hierarchy process (AHP), as one of the widely used decision analysis tools, is criticized since it suffers from scale intransitivity. This paper first reviews and compares different scales from different aspects, then discusses the transitivity of AHP scales and derives a scale based on the transitivity, so it is naturally transitive. Besides, two approaches are provided to determine the scale parameter for the derived transitive scale. In order to deal with the transitivity problem, the AHP provides a consistency index for testing pairwise comparison consistency. So, finally, this paper proposes a consistency measure to reflect the judgmental inconsistency. Journal of the Operational Research Society (2003) 54, 896–905. doi:10.1057/palgrave.jors.2601557 Keywords: analytic hierarchy process; scale; transitivity; consistency
Introduction The analytic hierarchy process (AHP) is a useful decision analysis tool and has been widely used in many areas.1–5 Pairwise comparisons are fundamental in the use of the AHP.6 The values defined on a scale are used to represent the relative importance between two compared objects in terms of ratio. The scale to be used has a significant effect on the outcome’s consistency and accuracy. As such, the issue on scales or scaling systems has drawn extensively attentions, and several scales have been proposed in the literature.7–11 The AHP has received many critical debates because it does not have the transitivity but other decision analysis tools may have, like Multiple Attribute Utility Theory. This point has been discussed recently by Forman and Gass.2 In order to remedy this intransitivity, the AHP provides a consistency index for testing pairwise comparison consistency for a decision-maker to use.2 However, the consistency index is just an indicator, and it is not the solution to the transitivity. To solve the problem thoroughly, it is best for the AHP to provide a transitive scale. So, the aim of this paper is to derive a scale for the AHP based on the transitivity, and naturally, the scale is transitive. The paper is organized as follows. In the following section, we review the major existing scales in the AHP, and compare them from various aspects. The next section discusses the transitivity of a scale. In the section thereafter, we derive a scale based on the transitivity. The subsequent section discusses how to determine the scale parameter for
*Correspondence: P Ji, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. E-mail: [email protected]
the derived transitive scale. The penultimate section deals with the consistency measure associated with the proposed scale. Finally, we conclude.
Review of the existing scales Co
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