دستیابی به ثبات ماتریس در تحلیل سلسله مراتبی از طریق خطی سازی

Matrices used in the analytic hierarchy process (AHP) compile expert knowledge as pairwise comparisons among various criteria and alternatives in decision-making problems. Many items are usually considered in the same comparison process and so judgment is not completely consistent – and sometimes the level of consistency may be unacceptable. Different methods have been used in the literature to achieve consistency for an inconsistent matrix. In this paper we use a linearization technique that provides the closest consistent matrix to a given inconsistent matrix using orthogonal projection in a linear space. As a result, consistency can be achieved in a closed form. This is simpler and cheaper than for methods relying on optimisation, which are iterative by nature. We apply the process to a real-world decision-making problem in an important industrial context, namely, management of water supply systems regarding leakage policies – an aspect of water management to which great sums of money are devoted every year worldwide.

The analytic hierarchy process (AHP) [1] provides a useful method to establish relative scales that can be derived by making pairwise comparisons using numerical judgments from an absolute scale of numbers. This approach is essential, for example, when tangible and intangible factors need to be considered within the same pool. The various factors are arranged in a hierarchical or a network structure with the objective(s) at the top, followed by one or more layers of criteria, and finally, the alternatives at the bottom. The ability of alternatives to achieve the objective(s) is measured according to the criteria represented within the structure. To this end, the people involved in the process compare the criteria and the alternatives in pairs, make judgments, and compile the results into matrices (matrices of criteria or matrices of alternatives). Any two elements, for example criteria Ci and Cj are semantically compared. A value aij is proposed directly (numerically) or indirectly (verbally) that represents the judgment of the relative importance of the decision element Ci over Cj. Among the different approaches for developing such scales [2] the nine-point scale developed by Saaty [3] is one of the most popular. By using the Saaty scale, if the elements Ci and Cj are considered to be equally important, then aij = 1 (homogeneity). If Ci is preferred to Cj, then aij > 1, with an integer grade ranging from 2 to 9 that respectively corresponds to weak, moderate, …, until very strong, and extreme importance of Ci over Cj. Intermediate numerical (decimal) values in the scale may be used to model hesitation between two adjacent judgments [1], [4] and [5]. It is assumed that the reciprocal property aji = 1/aij always holds. Homogeneity also implies that aii = 1 for all i = 1, 2, … , n. In this way, a homogeneous and reciprocal n × n matrix of pairwise comparisons A is compiled. This approach is intended to embody expert know-how regarding a specific problem. Matrices such as A are positive matrices (matrices with only positive entries) that also exhibit homogeneity and reciprocity. There are different techniques to extract priority vectors from these comparison matrices [6], [7] and [8]. The eigenvector method, proposed by Saaty in his seminal paper [3] in 1977, stands out from the rest. Saaty proved that the Perron eigenvector of the comparison matrix provides the necessary information to deal with complex decisions that involve dependence and feedback - as analyzed in the context of, for example, benefits, opportunities, costs, and risks [9]. The required condition is that the matrix exhibits a minimum level of consistency. Consistency expresses the coherence that should (perhaps) exist between judgments about the elements of a set. Matrix consistency is defined as follows: a positive n × n matrix A is consistent if aijajk = aik, for i, j, k = 1, … , n. Although different measurements of inconsistency can be developed, in this paper we use the measurement proposed by Saaty [1] and [3]. We also use the intrinsic consistency threshold developed by Monsuur [10]. If consistency is unacceptable, it should be improved. Several alternatives, mostly based on various optimization techniques, have been proposed in the literature to help improve consistency, including [9], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] and [21]. For example, Saaty [11] proposes a method based on perturbation theory to find the most inconsistent judgment in the matrix. This action could be followed by the determination of the range of values to which that judgment can be changed and whereby the inconsistency could be improved – and then asking the judge to consider changing the judgment to a plausible value in that range. In the next section we develop a linearization technique that provides the closest consistent matrix to a given non-consistent matrix by using an orthogonal projection in a given linear space. Our method provides a closed form for achieving consistency, while methods relying on optimisation, which is non-linear for this problem, are iterative by nature. Section 3 presents an application to a real-world decision-making problem regarding leakage policies in water supply. Finally, the paper closes with conclusions.

AHP is a very well established technique for decision-making and enables the evaluation of complex multi-criteria problems through a hierarchical representation of the problem, including objectives, criteria, and alternatives. Although pairwise comparisons performed in AHP have been seen as an effective way for eliciting qualitative data, a major drawback is that judgments are rarely consistent when dealing with intangibles – no matter how hard one tries - unless they are forced in some artificial manner. In this paper, we show that when starting with an inconsistent matrix, consistency can be achieved in a direct (as opposed to iterative) and straightforward manner following the described process of linearization. For the studied problem, corresponding to the conclusions of a panel of experts in a water company in Spain and compiled after a comprehensive discussion in a workshop organized for the company’s personnel by the second author, we have shown that the alternative of active leakage control clearly outperforms the classical passive leakage control. The main factor influencing this fact comes from the consideration of costs or compensations for supply disruptions. It must be emphasized at this point that many water supply companies are liable for maintaining quality standards, and supply disruptions often result in numerous complaints from customers. Moreover, if disruptions become a major problem then political responsibilities may be felt, since many water companies are municipal or mixed private–public entities. These aspects represent a different kind of ‘toll’ that managers of water companies and politicians are very reluctant to pay. The obtained results have been applied to of a complex problem in engineering: the selection of a suitable policy to manage a water supply network and avoid water losses – a worrying and crucial issue in the management of a scarce resource. The results show that the inclusion of social and environmental costs clearly points in the direction of ALC as the best alternative in leakage control. In this specific case, the economic aspects are clearly left behind by a rise in other social and environmental factors – which are more subjective objectives. We must also note that legislation in many countries has been modified and new laws have been enforced to encourage cost recovery and environmental and social responsibility. Moreover, vast sums are invested around the world encouraging responsible consumption and raising awareness about the need to care for natural resources. For these kinds of decisions to be valid, they must be obtained from consistent matrices. Since, in general, consistency is poor, especially when many criteria are simultaneously considered, methods for improving consistency are necessary. Trial and error methods are clearly devised as very inefficient. Iterative techniques based on optimization may provide a good solution. Nevertheless, in this paper, consistency is achieved by using a closed formula. Of course, the obtained consistent matrix must be validated through suitable sensitivity analyses and feedback from the decision-maker.