Keywords: consistent matrices, analytic hierarchy process, Perron eigenvalue, decision-making, water distribution systems, leaks, optimization.

Consistent matrices emerge in a natural way, for example, in the so-called analytic hierarchy process (AHP) [1], a well-established multiple attribute decision method. The AHP provides a useful way to establish relative scales among goods or commodities. These scales may be derived by making pairwise comparisons using numerical judgments from an absolute scale of numbers. In practice, nevertheless, such absolute numbers are not available. Instead, pairwise comparisons are replaced by judgements given by an expert or a panel of experts on a certain problem. Due to uncertainty and subjectivity, consistency is not frequently achieved. For a non-consistent matrix, but with entries that are assumed to be small perturbations of the unknown values, the Perron eigenvector still provides important information that allows decision makers to deal with complex decisions that involve certain dependence and feedback. Since judgements may lack a minimum level of consistency, some mechanism to improve consistency based on those inconsistent judgements is desirable. In this paper we prove a number of interesting properties verified by consistent matrices. Of special interest are the spectral properties and a characterization of a reciprocal, consistent matrix in terms of rank unity, which provides a basis for a consistency optimization procedure. This procedure gives a solution based on the minimization of the distance between two matrices in terms of the Frobenius norm. The main result states that a reduced number of decision variables, a number that turns out to be equal to the order of the matrix, are enough to achieve consistency. As a consequence, the process may be accomplished with no computational burden at all. These results are applied to a decision-making process in the management of a water supply company, aiming at minimizing water losses, considering a number of criteria, and involving two alternative policies regarding leakage control.

1

T.L. Saaty, "Relative Measurement and Its Generalization in Decision Making. Why Pairwise Comparisons are Central in Mathematics for the Measurement of Intangible Factors. The Analytic Hierarchy/Network Process", Revista de la Real Academia de Ciencias Serie A Matemáticas, 102(2), 251-318, 2008. doi:10.1007/BF03191825

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