Theory and MethodologyLocal stability intervals in the analytic hierarchy process
Introduction
The Analytic Hierarchy Process (AHP) (Saaty, 1977, Saaty, 1980, Saaty, 1994, Saaty, 1996) is a multicriteria decision making technique which allows us to resolve complex problems characterized by the existence of multiple actors, scenarios and criteria (tangible and intangible). This is accomplished through the construction of a ratio scale for the priorities associated with the alternatives of the problem, by means of hierarchical modelling and pairwise comparisons.
The application of any analytical procedure in the resolution of decision problems gives rise to two different stages: first, the solution, which is achieved at a specific moment in time, for various concrete values of the parameters of the model and on the basis of our existing knowledge of the problem; secondly, the exploitation of the model and its consequences. This second stage of the decision process, carried out under the traditional approaches by using a post-optimal analysis, is generating considerable interest throughout the multicriteria field, especially when the complexity and uncertainty of the problem limits the knowledge of the relevant aspects (see the discussion held in the Multicriteria list MCLIST at the beginning of 1998).
In such situations, where “the unknown is greater than the known”, we require a more flexible and realistic rationality than the merely substantive (Saaty, 1996, Brans, 1996, Moreno-Jiménez, 1996, Moreno-Jiménez, 1997, Moreno-Jiménez et al., 1997). Following this line, the current tendencies in multicriteria decision making are incorporating activities such as learning, justification, negotiation and the search for consensus between the actors involved in the decision process. In this context, our paper presents a tool (local stability intervals) which allows us to proceed further in this direction.
This tool, based on an inverse sensitivity analysis of the final ranking for the priorities of the alternatives, deals with the relationship between changes in the judgements and the rank reversal of the alternatives in two different situations (Roy, 1985): (1) the selection of the best alternative (P.α problem), and (2) the ranking of all alternatives (P.γ problem). In both situations, when the row geometric mean method (RGMM) is used to obtain the local priorities of the AHP for a problem with a single criterion1, we determine the local2 stability intervals and indexes for each judgement, for each alternative or element, and for the matrix associated with the criterion (reciprocal matrix of pairwise comparisons) which guarantees the upkeep of the best alternative and the complete ranking, respectively.
These values provide valuable information for the exploitation of the model, especially for the negotiation and monitoring processes. In practice, as with all sensitivity analyses of the models (Wolters and Marechal, 1995), a more extensive knowledge of the decision-making process is produced. In particular, control, learning and negotiation are improved, providing a very useful tool in the treatment of complex situations characterized by a high level of uncertainty and dynamism.
The paper is organised as follows. In Section 2 we briefly describe the basic methodological concepts required for the subsequent development of our tool. 3 Stability intervals for judgements, 4 Stability intervals for alternatives, 5 Stability intervals for a matrix consider the P.α and P.γ problems for three different situations, that is to say, the stability intervals for the judgements (Section 3), the stability intervals for the alternatives (Section 4) and the stability interval for the matrix (Section 5), respectively. Section 6 closes the paper with the most important conclusions.
Section snippets
Background
The AHP is a multicriteria decision-making technique that basically consists of four stages: (1) modelling; (2) valuation; (3) prioritization and (4) synthesis. In the first step (modelling), a hierarchy which represents the problem is constructed. In the most simple case, this has only three levels: the goal, or purpose, is placed in the top level (level 1); the relevant aspects of the problem (criteria) are included in the second level (level 2); and the alternatives being considered are
Stability intervals for judgements
Let A=(aij) with i,j=1,…,n be a reciprocal positive matrix corresponding to the paired comparisons of the n elements or alternatives Ai, i=1,…,n with respect to a single criterion, and let ωi be the local priorities of the alternatives obtained through the RGMM. In what follows, we suppose that the values of the priorities are not normalized (ωi=(∏aij)1/n), and, without loss of generality, that the elements are ranked as ω1⩾ω2⩾⋯⩾ωi⩾⋯⩾ωn−1⩾ωn. Lemma 1 If, in a paired comparison matrix A=(aij), the
Stability intervals for alternatives
If we wish to transfer the study carried out in Section 3 to an alternative Ar, it will be necessary to find a common stability interval for all the judgements in the corresponding row (arj, j=1,…,n with j≠r), within which we can vary the relative variations of each judgement without changing the rank of the best alternative (P.α problem) or the ranking of all the alternatives (P.γ problem). Lemma 2 If, in a paired comparison matrix A=(aij), the judgements in the rth row ar1,…,arn are converted into a′rs
Stability intervals for a matrix
In what follows, we will extend the idea of reciprocal stability intervals, considered in Section 4 for the alternatives, to the whole matrix in both situations (P.α and P.γ problems).
Conclusions
In this paper we have analyzed the relationship between changes in the judgements and the rank reversal of the alternatives for the AHP. Using the row geometric mean method to obtain the local priorities of the AHP for a problem with a single criterion, we have determined the local stability intervals and indexes for each judgement, for each alternative and for the matrix associated with the criterion which guarantee the maintenance of the best alternative (P.α problem) and the complete ranking
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