The geometric consistency index: Approximated thresholds

Abstract

Crawford and Williams [Journal of Mathematical Psychology 29 (1985) 387] suggested for the Row Geometric Mean Method (RGMM), one of the most extended AHP’s priorization procedure, a measure of the inconsistency based on stochastic properties of a subjacent model. In this paper, we formalize this inconsistency measure, hereafter called the Geometric Consistency Index (GCI), and provide the thresholds associated with it. These thresholds allow us an interpretation of the inconsistency tolerance level analogous to that proposed by Saaty [Multicriteria Decision Making: The Analytic Hierarchy Process, New York, 1980] for the Consistency Ratio (CR) used with the Right Eigenvector Method in Conventional-AHP.

Introduction

The Analytic Hierarchy Process (AHP) is a multicriteria decision making (MCDM) technique proposed by Saaty, 1977, Saaty, 1980, that integrates pairwise comparison ratios into a ratio scale. One advantage of this MCDM tool is that it allows us to measure the consistency of the decision maker when eliciting the judgements in a formal and elegant way.

Defining the consistency of a positive reciprocal pairwise comparison matrix, A=(a_ij), as the cardinal transitivity between the judgements, that is to say, a_ija_jk=a_ik, i,j,k=1,…,n, Saaty suggested that the inconsistency in Conventional-AHP, where the Right Eigenvector Method (EVM) is used as priorization procedure, can be captured by a single number (λ_max−n) which reflects the deviations of all a_ij from the estimated ratio of priorities ω_i/ω_j.

In this case, to provide a measure independent of the order of the matrix, n, Saaty proposed the use of the Consistency Ratio (CR). This is obtained by taking the ratio between λ_max−n to its expected value over a large number of positive reciprocal matrices of order n, whose entries are randomly chosen in the set of values {1/9,…,1,…,9}. For this consistency measure, he proposed a 10% threshold for the CR (5% and 8% for the 3 by 3 and 4 by 4 matrices, respectively) to accept the estimation of ω (Saaty, 1994). When the CR is greater than 10%, then, in order to improve the consistency, most inconsistency judgements, that is to say, those with a greater difference between a_ij and ω_i/ω_j, are usually modified and a new ω derived.

There are many other priorization procedures in the literature, but only a few of them present their corresponding indicators to evaluate the inconsistency. Furthermore, when these consistency indexes have been proposed (Crawford and Williams, 1985; Harker, 1987; Golden and Wang, 1989; Wedley, 1991; Takeda, 1993; Takeda and Yu, 1995; Monsuur, 1996; Escobar and Moreno-Jiménez, 1997; Aguarón, 1998), they lack a meaningful interpretation due to the absence of the corresponding thresholds. Obviously, if the priorization procedure is not the EVM, the Saaty approach to evaluate the consistency is not appropriate, by construction, and new consistency measures, related to the priorization procedure, are required.

Recently, and despite the strong defense of the EVM presented by the Saaty school (Saaty, 1990; Vargas, 1994, Vargas, 1997), the use of the Row Geometric Mean Method (RGMM), or Logarithmic Least Squares Method, as a priorization procedure in AHP has significantly increased (Ramanathan, 1997; Van den Honert, 1998; Levary and Wan, 1999) due fundamentally to its psychological (Gescheider, 1985; Lootsma, 1993; Barzilai and Lootsma, 1997; Brugha, 2000) and mathematical (Narasimhan, 1982; Jensen, 1984; Budescu, 1984; Barzilai, 1997; Aguarón and Moreno-Jiménez, 2000; Escobar and Moreno-Jiménez, 2000; Brugha, 2000) properties.

Crawford and Williams (1985) justified the RGMM by means of two different approaches: (1) the minimization of the log quadratic distance of errors (Logarithmic Least Squares Method); and (2) the maximum likelihood estimator of the priorities. The first is a deterministic approach and the second a stochastic one, where a multiplicative model for the perturbations has been supposed (a_ij=(ω_i/ω_j)π_ij, with π_ij independent and lognormal distributions with zero mean and constant variance π_ij∼Lognormal(0,σ)).

For this priorization procedure (RGMM), Crawford and Williams suggested that the estimator of the variance of the perturbations can be used as a measure of the consistency, where the lower the value, the better the consistency of the judgements. In what follows assuming the proposal of Crawford and Williams, and without entering into the analysis of the validity of the CR as a consistency measure in AHP, we calculate the thresholds that make this measure, called the Geometric Consistency Index (GCI), operative and that allow us to fix a tolerance level with an interpretation analogous to that considered for Saaty’s CR.

The paper has been structured as follows: Section 2 presents the two consistency measures considered in this paper (CR and GCI); Section 3 establishes a theoretical relation between the CR and the GCI, the validity of which is tested through a regression analysis; finally, Section 4 closes the paper with some comments about the GCI thresholds.