رویکردی به تصمیم تحلیل سلسله مراتبی در یک زمینه پویا

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.

AHP (analytic hierarchy process) is a multicriteria decision process [23] that involves aggregating various comparisons to obtain a priority vector that is representative of coherent results. In other words, AHP generates consolidated priorities about a number of alternatives that represent the will, likes, or decisions revealed by the preference data provided by one or more actors, or groups of actors, involved in the decision-making process. Achieving consistency in AHP has become an important issue [11], [14], [18], [21] and [22] and different methods have been proposed [2], [3], [5], [6], [7], [9], [12], [15], [17], [24] and [29]. AHP uses a specific input format for decision makers to express their preferences regarding multiple criteria and alternatives, namely, pair-wise comparisons. This format may be not perfect — yet it expresses user preferences reasonably well in many practical situations. After all the preference data has been collected, an algorithm is applied to generate consistent consolidated results. However, a limitation of pair-wise comparisons is that before applying the decision model the experts must provide judgment data representing their preference with respect to all the elements involved. This kind of input is impossible in many practical situations. Consider, for example, the following two scenarios. Firstly, let us suppose that not all the elements for comparison are known or evident from the start. In leakage control, for example, only economic aspects have so far been widely considered. Nevertheless, environmental aspects have started to be considered as important, and even more recently, social elements have also started to play important roles in decision making on leakage control policies. A second scenario is when the consulted actors are unfamiliar with the effects of various items. As a result, it is difficult to collect complete preference information from decision makers at one time. It would be reasonable to allow decision makers to express their preferences at multiple times at their own convenience. In the meanwhile, partial results based on partial preference data could be generated from the data collected at multiple times — and this data could eventually be consolidated when the information is complete. To consider the above mentioned scenarios, the input mode of the traditional AHP needs to be extended from a static mode to a dynamic mode. The dynamic mode involves the dimension of time. In other words, it will not be compulsory for users to provide input preference data at one point in time. A user will be able to input his/her preferences at multiples times. The user only needs to express his/her preference each time for a subset of elements, rather than the complete set. A change from static to dynamic mode will probably have many repercussions in future studies. It is impossible to address all of these issues at this time. Therefore, in this study we initiate a new approach and focus on a specific sub-problem. In this paper, we restrict ourselves to the case where a new criterion is added to the pool of previously considered criteria. This case can obviously be extended to the case of adding more than one criterion. The withdrawal of an obsolete criterion is readily obtained as a corollary. The remainder of this paper is organized as follows. First, a short review of the linearization process [5] to achieve consistency is presented. In the methodology section, new results are presented that enable an efficient calculation of the new consistent matrix and its corresponding vector of priority after introducing a new criterion or withdrawing an obsolete criterion. Finally, the proposed methodology is applied to a real-world case in water leakage management, and conclusions are presented to close the paper.

In this work, we consider a new dimension in the traditional AHP methodology by considering that input may be either static or dynamic, depending on whether actors provide their preferences all at once or at multiple times. We think that this will open a window to various new variants of AHP methodology in the future. However, since it is not possible to address all potential related issues at once, in this paper we aim to solve a particular variant in which the introduction of a new criterion or the withdrawal of an obsolete criteria are allowed while avoiding the need for repeating all the calculations from scratch. This novel method is computationally inexpensive and is based on a linearization process previously introduced by the authors [5]. To check the performance of the algorithm, an experiment is performed that considers a decision-making problem in water supply regarding the adoption of either an active or a passive leakage control policy. Many future extensions regarding dynamic AHP may be devised. For example, in the second scenario considered in the introduction the consulted actors are unfamiliar with the effects of various items. In this case, collecting complete and quality preference information from decision makers at the same time cannot be expected. It is necessary to allow user preference data to be input at multiple times at their own convenience. As a result, the static input mode could be changed in such a way that partial results based on partial preference data could be generated from the data collected at multiple times, and new results could eventually be obtained when all the information is complete. Allowing for this dynamic input mode in traditional approaches would open the door to a large array of future research issues. Another issue worth exploring is the way a new criterion is added in a context of group decision making [8] and [10]. Various approaches can be devised. Individual judgments could be considered for the new criterion, then the geometric mean of the expert judgments obtained, and the process described in this paper applied to the resulting matrix. Another alternative could be the individual application of the process to the individual enlarged expert matrices and, finally, some type of voting system could be used to produce the final priority vector. Another possibility would compute interval bounds for the expert judgments regarding the new criterion and then perform the calculations using some interval arithmetic. In the case of many decision makers, various voting systems could be considered with different purposes, such as eliminating outliers, aggregating values, etc.