Interval estimation of priorities in the AHP

Abstract

This paper extends and modifies the Analytic Hierarchy Process (AHP) and the Synthetic Hierarchy Method (SHM) of priority estimation to accommodate random data in the pairwise comparison matrices. It employs a Cauchy distribution to describe the pairwise comparison of alternatives in Saaty matrices, and shows how to modify these matrices in order to handle random data. The use of random data yields Saaty matrices that are not reciprocally symmetrical. Several variants of the AHP are then modified (i) to accommodate reciprocally asymmetric matrices, and (ii) to allow each priority estimate to be expressed on an interval of possible values, rather than as a single discrete point. The merits of interval estimation are illustrated by an example.

Introduction

The Analytic Hierarchy Process (AHP) has found widespread application in decision making problems involving multiple criteria in systems of many levels. The strongest features of the AHP are that it generates numerical priorities from the subjective knowledge expressed in the estimates of pairwise comparison matrices. In this study we treat the elements of the pairwise comparison matrices as realizations of random variables. This approach assumes that the evaluation of a priority ratio (or the evaluation of an object measured on an ordinary scale) may come from a range of possibilities, and the judge gives an evaluation that is his “most likely” or “best” estimate within the range of these possibilities. It is reasonable to expect that questions seeking the ranking of two alternatives can be answered precisely. However, questions such as “compare the motivation of managers in your organization”, or “state the relative importance of several information systems applications”, or “compare the performance of the professor of this course relative to those who taught you last year” do not have a precise answer. The answers to these questions may depend on personal taste and experience, specific (rather than general) knowledge and information, past experience, and more. Thus, interpreting such answers as realizations of random variables may be a better approximation of reality.

Moreover, it commonly happens that a judge will give different answers to the same question when it is posed at different times. Thus, another interpretation of our approach is that the judge knows the general structure of the distribution of the required evaluations, but has to estimate the parameters of this distribution. We assume that the judge is fully consistent in his evaluations. However, his estimates of the parameters may depend on events that happened just prior to the evaluation. For example, judging other similar or dissimilar problems, the construction of the questionnaire, the time of day (beginning of the day, after a long day of work), the setting of the evaluations, and the approach taken by the person who presents the questions; hence, his evaluation may change at different times or situations.

In a variety of recent studies, the ratios of priorities in the Saaty matrices were measured on an interval, or were assumed to be random variables. Among them are Saaty and Vargas (1987), Arbel, 1989, Arbel, 1991, Zahir (1991), Arbel and Vargas, 1992, Arbel and Vargas, 1993, Zhang and Yang (1992), Basak (1993), Salo (1993), Moreno-Jimenez and Vargas (1993), Xu and Wu (1993), Genest and Rivest (1994), Paulson and Zahir (1995), and MacKay et al. (1996). These studies, with the exception of MacKay et al. (1996), paid little attention to the precise type of distribution function that governs the underlying structure of the priority ratios. The study of MacKay et al. (1996) assumed that the priority ratios are truncated normal variates, which yields a nice interpretation of the estimation results, but entails a very complicated and inconvenient estimation procedure in practice.

The main purpose of this paper is to develop a practical and simple method for the estimation of priority ratios in the AHP approach, where pairwise comparisons are assumed to be realizations of random variables rather than being deterministic; that is, the judge gives his most likely or best estimate of the required evaluation within a range of possibilities. In addition, we show how to estimate the lower bound, the center and the upper bound of the range.

This study develops the structure of the Saaty matrices when the priority ratios are random variables. Specifically, each priority ratio is assumed to be a random variable with a Cauchy distribution (see Lipovetsky and Tishler, 1997 for the use of several other distribution functions). The analysis shows that the lower triangle elements of the Saaty matrix should be smaller than the reciprocals of the elements in the upper triangle. As a consequence, reciprocally nonsymmetric matrices must be used in order to correctly estimate the relevant priority vectors. This paper shows how to modify the conventional AHP and several of its variations to accommodate pairwise comparison matrices with random data.

The paper is structured in the following way. Section 2sets up the notation. Section 3analyzes the properties of the reciprocal elements of Saaty matrices when the priority ratios are described by the Cauchy distribution function. The use of Cauchy distribution for the initial pairwise comparison matrices is detailed in Section 4. Section 5introduces a modification of the AHP method for the estimation of the priority ratios by reciprocally nonsymmetric matrices. Section 6presents a numerical example which compares the evaluations of priorities using random data with those obtained using nonrandom data. Section 7summarizes the paper.