Comparative Performance of Contradictory and Non-Contradictory Judgement Matrices in AHP Under Qualitative and Quantitative Metrics

Comparative Performance of Contradictory and Non-Contradictory Judgement Matrices in AHP Under Qualitative and Quantitative Metrics

Vishal Gupta (BITS Pilani, Pilani, India)
Copyright: © 2018 |Pages: 18
DOI: 10.4018/IJDSST.2018010102
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Abstract

Over the years, although AHP has proved its success in various diverse fields, many authors in the literature have also shown its shortcomings, often called as criticisms of AHP. One such criticism is allowing the consideration of contradictory judgement matrices. Such matrices violate the principle of ordinal transitivity and thus there does not exist any ranking of corresponding decision elements which satisfy all the judgements. In this paper, the results of our investigation towards measuring this criticism are further explored and discussed by comparing the quality of priority vector of contradictory judgement matrices and non-contradictory judgement matrices under Rank Reversals and the common frame work of “aggregated deviation”. The results further strengthen the notion of contradictory judgement matrices as a strong criticism of AHP for higher order judgement matrices and necessitate some proper avoidance (if not elimination) procedure for them.
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Introduction

Multi Attribute Decision Making (MADM) comprises of decision making techniques which address choice problems, i.e. selecting a best choice among many available choices (Köksalan et al., 2011; Yoon and Hwang, 1995). Analytic Hierarchy Process (AHP) is a widely used MADM technique in many diverse fields for selecting a best alternative among many alternatives (Satty, 2008; Satty and Forman, 2003). Since its inception, many authors have witnessed the superiority of AHP over other MADM techniques in different situations (Sgora et al., 2010; Gu et al., 2009; Stevens-Navarro and Wong, 2006; Salomon and Montevechi, 2001). Basically, AHP comprises of following five steps:

  • Step 1 - Problem structuring / developing a hierarchy: Modeling the problem is perhaps the most creative part of decision making because it can affect the outcome significantly. With respect to AHP, to structure the problem, the decision maker(s) and the facilitator sit together and decide upon hierarchical structure of the problem. In practice, there is no set procedure for this step; but a useful way to proceed for hierarchically structuring a decision is to come down from the goal as far as feasible, and in fact as far as one can, by decomposing it into most general and most easily controlled factors. Some suggestions for an elaborative design of a hierarchy are listed by Saaty (1994).

  • Step 2 - Pair-wise comparison matrix: Instead of directly allocating weights to the various criteria involved, AHP require comparing two criteria at a time. For it, each possible pair of criteria are compared, thus resulting in a pair wise comparison matrix; popularly called as judgement matrix (Saaty, 1994).

  • Step 3 - Estimating consistency and synthesizing judgements: AHP allows objective as well as subjective criteria. Subjective criteria are typically compared by human beings. Because many times human beings are inherently inconsistent (Satty, 2008), the entries in the judgement matrix reflect this inconsistency. Moreover, as the order of judgement matrix increases, the chance of judgement matrix being inconsistent also increases. One of the beauty and strength of AHP is that it allows certain degree of inconsistency to be present. For it the process of AHP also measures and quantifies the degree of inconsistency present in the judgement matrix in the form of finding Consistency Ratio (CR). As per the standard process (Saaty, 1994), if CR < 10%, the judgement matrix is considered for further processing; otherwise the decision maker has to reconsider the decisions.

  • Step 4 - Overall Priority Ranking: Once a consistent (or nearly consistent with CR<10%) judgement matrix is read; the corresponding decision elements can be ranked. There are various methods of getting this ranking, but Saaty argued that Principal Eigen Vector of the judgement matrix gives the best ranking of the decision elements (Saaty and Hu, 1998).

  • Step 5 - Aggregation: The last step is to synthesize the local priorities across all criteria in order to determine the global priorities. This step determines the global priorities of alternatives by synthesizing all the local priorities, thus ranking of all the available alternatives.

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