The various mechanisms that represent the know-how of decision-makers are exposed to a common weakness, namely, a lack of consistency. To overcome this weakness within AHP (analytic hierarchy process), we propose a framework that enables balancing consistency and expert judgment. We specifically focus on a linearization process for streamlining the trade-off between expert reliability and synthetic consistency. An algorithm is developed that can be readily integrated in a suitable DSS (decision support system). This algorithm follows an iterative feedback process that achieves an acceptable level of consistency while complying to some degree with expert preferences. Finally, an application of the framework to a water management decision-making problem is presented.
One of the best established and most modern models of decision-making is AHP (analytic hierarchy process) [1], [2] and [3]. In AHP, the input format for decision-makers to express their preferences derives from pair-wise comparisons among various elements. Comparisons can be determined by using, for instance [4], a scale of integers 1–9 to represent opinions ranging from ‘equal importance’ to ‘extreme importance’ [5] (intermediate decimal values are sometimes useful). Homogeneous and reciprocal judgment yields an n×nn×n matrix AA with aii=1aii=1 and View the MathML sourceaij=1/aji,i,j=1,…,n. This last property is called reciprocity and AA is said to be a reciprocal matrix. The aim is to assign to each of nn elements, EiEi, priority values View the MathML sourcewi,i=1,…,n, that reflect the emitted judgments. If judgments are consistent, the relations between the judgments aijaij and the values wiwi turn out to be View the MathML sourceaij=wi/wj,i,j=1,…,n, and it is said that AA is a consistent matrix. This is equivalent to aijajk=aikaijajk=aik for View the MathML sourcei,j,k=1,…,n [6]. As stated by [7] and [2], the leading eigenvalue and the principal (Perron) eigenvector of a comparison matrix provides information to deal with complex decisions, the normalized Perron eigenvector giving the sought priority vector. In the general case, however, AA is not consistent. The hypothesis that the estimates of these values are small perturbations of the ‘right’ values guarantees a small perturbation of the eigenvalues (see, e.g., [8]). Now, the problem to solve is the eigenvalue problem View the MathML sourceAw=λmaxw, where View the MathML sourceλmax is the unique largest eigenvalue of AA that gives the Perron eigenvector as an estimate of the priority vector.
As a measurement of inconsistency, Saaty [5] proposed using the consistency index View the MathML sourceCI=(λmax−n)/(n−1) and the consistency ratio View the MathML sourceCR=CI/RI, where View the MathML sourceRI is the so-called average consistency index [5]. If View the MathML sourceCR<0.1, the estimate is accepted; otherwise, a new comparison matrix is solicited until View the MathML sourceCR<0.1. To overcome inconsistency in AHP while still taking into account expert know-how, the authors propose a model to balance the latter with the former. Our model incorporates an extended version of the linearization procedure described in [9], and integrates it along with AHP to produce optimal comparison matrices.
In this paper, by extending a linearization process already published by the authors [9], and describing an efficient implementation for the calculations, an algorithm is developed that follows an iterative feedback process and achieves an acceptable level of consistency while complying to some degree with expert preferences.
Our ultimate objective was to devise a method and then integrate it into a suitable DSS tool. This method would help practitioners build comparison matrices that both rely on their judgment and are efficient and reliable in deriving priorities.
An application to a real decision-making problem in water management has been presented. The study enhances the relevance of the economic aspects, showing the leading role in the decision played by planning development implementation and costs. The interesting aspect regarding the application of AHP is indeed the inclusion of social and environmental costs in decision-making.
