A note on AHP group consistency for the row geometric mean priorization procedure
Introduction
The analytic hierarchy process (AHP), proposed by Thomas L. Saaty (1980), is one of the most commonly applied multicriteria decision making techniques. It combines tangible and intangible aspects in order to derive a ratio scale, the abstract scale of priorities, which is valid to make complex decisions. Two of the main characteristics of this approach are (i) the existence of an analytical measure to evaluate the inconsistency of the decision maker when eliciting the judgements and (ii) the possibilities that AHP offers in group decision making.
As regards group decision making (Saaty, 1980; Aczél and Saaty, 1983; Ramanathan and Ganesh, 1994; Forman and Peniwati, 1998), AHP considers two different approaches: the aggregation of individual judgements (AIJ) and the aggregation of individual priorities (AIP).
However, less attention has been given to the consistency in group decision making. Xu (2000), for a local context, proves that using the eigenvector method (EM) as the priorization procedure and the weighted geometric mean method (WGMM) as the aggregation procedure, if the individual decision makers have an acceptable inconsistency when eliciting the judgements, then so has the group. In what follows, and considering the same local context (unicriterion situation) and the same aggregation procedure (WGMM), we provide an analogous conclusion for the row geometric mean method (RGMM) and its associated (Aguarón and Moreno-Jiménez, 2003) consistency measure, the geometric consistency index (GCI).
This result guarantees that, when the RGMM is employed as the priorization procedure, the group inconsistency is at least as good as the worst individual inconsistency for both aggregation approaches (AIJ and AIP). Thus, to derive the group priorities in practical situations, we can use the simplest of the two aggregation approaches (usually the AIP), guaranteeing, as we prove in this paper, that it verifies the consistency requirement.
Section snippets
Background
Let A=(aij) be an n×n judgement matrix, and ω=(ω1,ω2,…,ωn) be its priority vector, where ωi>0, ∑iωi=1. The consistency in AHP is defined as the cardinal transitivity between judgements (Saaty, 1980), that is to say, aijajk=aik for all i, j, k. The consistency measures used for the two most extended priorization procedures in AHP, the EM and the RGMM are respectively the consistent index (CI) proposed by Saaty for the EM (Saaty, 1980), and the geometric consistency index (GCI) proposed by
Group consistency for the RGMM
Using the notation introduced in the last section, let e[k]ij=a[k]ijω[k]j/ω[k]i be the error of the k-th individual decision maker, k=1,…,m, when comparing alternatives i and j, i,j=1,…,n. Then, the geometric consistency index for the k-th decision maker can be expressed as
If the WGMM is employed as the aggregation approach, and the RGMM as that of the prioritization, then the group error for the judgement aij is given by the weighted
Numerical example
In order to see how these theoretical results work in practice, let us consider the example used in Xu (2000) for EM. This example consists of four alternatives (A, B, C and D) and four decision makers (I, II, III and IV), whose pairwise comparison matrices for the alternatives are shown below:
From these pairwise comparison matrices, we calculate, using the WGMM,
Conclusions
In this paper, we have proved a property in group decision making for the AHP, which guarantees that, using the RGMM priorization procedure, the geometric consistency index for the group judgements would be within an acceptable level of inconsistency, provided that the judgements from the different decision makers are also of an acceptable inconsistency.
This result, jointly with the priority and consistency stability intervals (Aguarón and Moreno-Jiménez, 2000; Aguarón et al., 2003), is being
Acknowledgements
This research has been partially supported by the “SIS-DECAP: Un Sistema Decisional para la Administración Pública” Research Project (ref: P072/99-E CONSI+D, Diputación General de Aragón, Spain). We also wish to thank Stephen Wilkins for helping with the English translation of the text.
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