Characteristics of positive reciprocal matrices in the analytic hierarchy process
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#2002 Operational Research Society Ltd. All rights reserved. 0160-5682/02 $15.00 www.palgrave-journals.com/jors
Characteristics of positive reciprocal matrices in the analytic hierarchy process SI Gass1* and SM Standard2 Robert H. Smith School of Business, University of Maryland, USA; and 2Department of Mathematics, University of Maryland, USA
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Positive reciprocal matrices (PRMs) are the basic elements used by the Analytic Hierarchy Process (AHP) for resolving an important class of multi-criteria decision problems. A PRM, A ¼ (aij), is square with all aij > 0 and aji ¼ 1=aij. We discuss characteristics of such matrices based on an analysis of both real-world and randomly generated sets. Journal of the Operational Research Society (2002) 53, 1385–1389. doi:10.1057=palgrave.jors.2601471 Keywords: analytic hierarchy process; decision analysis; pairwise comparison matrices
Introduction Given a decision problem in which possible alternative solutions are compared to each other with respect to multiple criteria, the Analytic Hierarchy Process (AHP) requires the decision maker (DM) to produce sets of pairwise comparison matrices. In general, the elements of each matrix represents the DM’s estimate on the importance of one criterion over another, or the preference of one alternative over another with respect to a criterion. Based on the axioms of the AHP,1,2 these elements, that is, the DM’s estimates, are positive with aji ¼ 1=aij, and 0 < aij 4 9 (1 to 9 scale). Such (n n) positive reciprocal matrices (PRMs) have a number of important properties with the key ones being: (1) the maximum eigenvalue lmax 5 n; and (2) the eigenvector associated with the maximum eigenvalue has all positive elements.1,2 We assume that the reader is familiar with how these facts are integrated within the AHP and the use of the fundamental 1 to 9 comparison scale. In this note we describe some results based on an analysis of real-world and randomly generated PRMs. Full details are given in work by Standard.3 Real-world PRMs We studied a set of 384 PRMs that were taken from realworld AHP analyses, as reported in the literature. We were first interested in the distribution of the numbers in the basic AHP comparison scale: 1 (equal), 2, 3 (moderate), 4, 5 (strong), 6, 7 (very strong), 8, 9 (extreme). The real-world use of these numbers, as given in the 384 PRMs, is shown in
*Correspondence: SI Gass, 4355 Van Munching Hall, University of Maryland, College Park, MD 20742-1815, USA. E-mail: [email protected]
Table 1. The row ‘Experimental probability’ shows the percentage of use of the numbers 1 to 9, with the 1s counted only for off-diagonal positions. The row ‘Probability without 1s’ contains similar frequencies, but here we did not count any 1s. The frequencies in Table 1 were unexpected. We had anticipated a greater use of the extreme comparison values of 7, 8, and 9. It would seem that for these real-world problems the DMs did not encounter very strong to extreme comparisons. That is, the criteria or the alternatives tended not to refl
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