A Decision Support System for Improving the Inconsistency in AHP

Introduction

The analytic hierarchy process (AHP), proposed by Thomas L. Saaty at the end of the 1970s, is a multi-criteria decision technique that has become one of the most commonly employed approaches to the resolution of complex problems (Subramanian and Ramanathan, 2012; Zyoud and Fuchs-Hanusch, 2017). Decision makers incorporate their preferences using pairwise comparisons and some degree of inconsistency is allowed when eliciting their judgements. Consistency is a particularly important issue as it is a requirement for the validity of the derived priority vector (Grzybowski, 2016).

Given a pairwise comparison matrix (PCM), with and , Saaty (1980) established that the matrix A is consistent if . This is a desirable property that reflects a certain rationality, logic, or formal coherence. There are many factors that may cause inconsistencies in the judgements elicitation process, such as (Aguarón et al, 2020): (1) the ambiguity and complexity of the problem; (ii) the knowledge of the actors in the matter under consideration; (iii) the affective aspects (mood, emotions, personality features, attitudes and motivations) that condition the behaviour of the actors; (iv) the level of attention (errors in the response) during the assessment process; and (v) the rationality of the procedure followed when incorporating preferences, especially when working with subjective aspects.

To measure the inconsistency different indicators have been proposed in the AHP literature. Two of the most widely used are the Consistency Ratio (CR) associated with the eigenvector (EV) prioritisation procedure and the Geometric Consistency Index (GCI) associated with the row geometric mean (RGM) prioritisation procedure. Other inconsistency measures for pairwise comparisons were proposed in the literature. Brunelli (2018) presents a survey of them as well as a study of their properties and relations. With regards to the improvement of inconsistency in AHP, different procedures have also been described in the literature. An overview of these approaches can be found in Khatwani and Kar (2017).

Aguarón et al. (2021) proposed, for the first time in the literature, a procedure for improving the inconsistency when the Row Geometric Mean (RGM) is used to derive the priorities and the Geometric Consistency Index (GCI) is employed to measure the inconsistency. This is a sequential procedure that, at each iteration, identifies the judgement that would improve the GCI faster and with greater intensity. In the proposed procedure the decision maker intervenes at the beginning indicating its permissibility threshold, that is, the maximum variation, in relative terms, that they would accept to modify the initial judgements. Limiting the variations of the judgements by the permissibility threshold guarantees that both the final judgements and the derived priority vector will be close to the initial values, as recommended by Saaty (2003).

The objective of the paper is to present a DSS that implements the Aguarón et al. (2021)’s procedure proposed for reducing the inconsistency in AHP by adapting it to be used interactively. The DSS also calculates the minimum permissibility necessary to achieve an allowable inconsistency level (below the required threshold). The value of this parameter (minimum permissibility) provides relevant information about the decision problem, in line with the cognitive multicriteria decision making paradigm (Moreno-Jiménez and Vargas, 2018), that can be used by the decision maker as a starting point to set their own permissibility.