a b s t r a c tMatrices 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 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.

Abstract. Decision making (DM) processes are becoming increasingly complex. The reasons are manifold. DM usually involves many aspects; some are purely technical, while others are subjective and derived from social, political, and environmental factors, among others. As a result, they involve items that are not easily comparable under the same units of measurement. Problems are made even more complex by the fact that current governance processes tend to involve all the stakeholders in the DM process.In this paper we consider the AHP methodology (analytic hierarchy process), which is used to build consistent aggregate results from data provided by decision makers. As some of the actors involved may not be completely familiar with all the criteria under consideration, it is common that the body of opinion, expressed in terms of pairwise comparison, is incomplete. To overcome this weakness, we propose a framework that enables users to provide data on their preferences in a partial and/or incomplete way and at different times. This article is an advance towards a dynamic model of AHP. The authors have addressed the problem of adding a new criterion or deleting obsolete criteria. Here, we address the consistent completion of a reciprocal matrix as a mechanism to obtain a consistent body of opinion issued in an incomplete manner by a specific actor. This feature is incorporated into a process of linearization previously introduced by the authors, which is concisely presented. Finally, we provide an application for leakage control in a water supply company. The adoption of suitable control leakage policies in water supply is a problem of enormous interest in the water industry, particularly in urban hydraulics.

Decision making in engineering is becoming increasingly complex due to the large number of alternatives and multiple conflicting goals. Powerful decisionsupport expert systems powered by suitable software are increasingly necessary. In this paper, the multiple attribute decision method known as analytical hierarchy process (AHP), which uses pairwise comparisons with numerical judgments, is considered. Since judgments may lack a minimum level of consistency, mechanisms to improve consistency are necessary. A method to achieve consistency through optimisation is described in this paper. This method has the major advantage of depending on just n decision variables -the number of compared elements -and so is less computationally expensive than other optimisation methods, and can be easily implemented in virtually any existing computer environment. The proposed approach is exemplified by considering a simplified version of one of the most important problems faced by water supply managers, namely, the minimisation of water loss.

Abstract. AHP (analytic hierarchy process) is used to construct coherent aggregate results from preference data provided by decision makers. Pairwise comparison, used by AHP, shares a common weakness with other input formats used to represent user preferences, namely, that the input mode is static. In other words, users must provide all the preference data at the same time, and the criteria must be completely defined from the start. To overcome this weakness, we propose a framework that allows users to provide partial and/or incomplete preference data at multiple times. Since this is a complicated issue, we specifically focus on a particular aspect as a first attempt within this framework. For that reason, we re-examine a mechanism to achieve consistency in AHP, i.e. a linearization process, which provides consistency when adding a new element to the decision process or when withdrawing an obsolete criterion under the dynamic input mode assumption. An algorithm is developed to determine the new priority vector from the users' new input. Finally, we apply the new process to a problem of interest in the water field, specifically, the adoption of a suitable leak control policy in urban water supply.