Consistency stability intervals for a judgement in AHP decision support systems

Abstract

This paper focuses on the need to evaluate the consistency of human judgements in decision support systems (DSS). In this sense, and in the context of one of the most applied multicriteria methodologies, the analytic hierarchy process (AHP), we obtain the consistency stability interval (CSI) associated with each judgement, that is to say, the interval in which this can oscillate without exceeding a value of the consistency measure fixed in advance. To calculate these CSIs, we consider the row geometric mean method as the priorisation procedure, the geometric consistency index as the consistency measure and a local situation with one criterion. The proposed procedure has been implemented in a module that is easily adaptable to any Decision Support System based on AHP.

Introduction

The increasing complexity of decisional problems, especially those arising in the social field, is requiring the use of more flexible and open approaches which, employing software tools (decision support systems, DSS) as technological support, provide us with a more realistic and effective resolution of these problems than that offered by the traditional approach in decision making (Moreno-Jiménez et al., 1999, Moreno-Jiménez et al., 2001; DeTombe, 2001).

This traditional approach, based on utility theory (Fishburn, 1970), assumes the existence of a unique truth, independent of personal values. This neutrality of values, as well as many other restrictive hypotheses required by the substantive rationality paradigm (traditional approach), have become increasingly questioned, and even invalidated, from a practical point of view, especially in the field of cognitive psychology (Tversky and Kahneman, 1981; Prelec, 1998) and participatory policy analysis (DeTombe, 2001).

By contrast, current tendencies in complex decision making propose the integration of the objective with the subjective, the tangible with the intangible and the rational with the emotional (Moreno-Jiménez, 1998; Brans, 2000).

The contribution of the actors involved in the decision making process (Moreno-Jiménez, 1989; Roy, 1993) and, consequently, the importance given to the human factor in the new methodological approaches, suggests, independent of the school followed in the resolution process, that the consistency of the actors be evaluated when eliciting their judgements.

In this paper, and for the analytic hierarchy process (AHP) (Saaty, 1980) – one of the most useful methods for multicriteria decision making and the operational support of the cognitive constructivism considered in the multicriteria procedural rationality (MCPR) paradigm (Moreno-Jiménez et al., 1999; Moreno-Jiménez et al., 2001)––we derive the consistency stability interval (CSI) associated with each judgement. That is to say, we obtain the range of values in which this judgement can oscillate without the consistency measure exceeding a threshold fixed in advance.

In order to obtain the CSI for a judgement, we carry out an inverse sensitivity analysis of the consistence, taking into account a unique criterion, using the row geometric mean method (RGMM) as the priorisation procedure, and the geometric consistency index (GCI) as its associated consistency measure. The extension of these ideas to the case of an alternative, or to the whole matrix in AHP, in a manner analogous to the procedure presented in Aguarón and Moreno-Jiménez (2000a), will be the subject of a separate paper. Obviously, and despite the fact that we have obtained these intervals for a particular multicriteria technique (AHP), the underlying ideas could and should be extended to any other multicriteria technique employed in the resolution of complex problems.

Under the cognitive approach considered in the MCPR paradigm, these CSI increase our knowledge of the decision making process, complement the information provided by the priority stability intervals (Aguarón and Moreno-Jiménez, 2000a), and allow for the determination of the range of values in which a judgement can oscillate, with an acceptable consistency, without affecting a “property” previously considered for the alternatives (the best, the ranking,…).

The knowledge discovery provided by the CSI is especially relevant for group decision making (Aczel and Alsina, 1986; Saaty, 1989). In this case, the design of consensus paths between the actors and, more generally, the search for consensus among the participants involved in the negotiation process, requires that the acceptable consistency of the group (judgements) in the final decision be guaranteed.

The contribution of the actors in the decision making process, the development of a methodological approach which allows for this contribution to be considered, and, logically, the development of a technology which permits the incorporation of the actors’ knowledge into the DSS (Aguarón et al., 1996a), represents the framework in which this paper is based. In this sense, we have also implemented the algorithmic procedure to obtain the CSI in a module which has been integrated in PRIOR (Aguarón et al., 1996b), one of the available AHP-DSS. This program, just like other AHP-software, for example Expert Choice (Expert Choice, 1988) and HIPRE (Hämäläinen and Lauri, 1993), provides the multicriteria priorisation and selection of a discrete set of alternatives and, moreover, also offers the possibility of obtaining some kind of knowledge discovery.

The rest of the paper is organised as follows. Section 2 is dedicated to presenting the background of the proposed approach, specifically the aspects relative to MCPR, the Analytic Hierarchy Process, the consistency and the stability intervals. Section 3 presents the procedure and the theoretical results that allow us to obtain the CSIs for a judgement in AHP. Section 4 includes the description of the software module developed to implement the previous procedure, and its application to a particular case (3×3 matrix). Finally, Section 5 highlights the most important conclusions of the paper.