Decision SupportIncomplete pairwise comparison and consistency optimization
Introduction
Pairwise comparison is a well established technique in decision making. In Saaty’s AHP [17], as an example, pairwise comparison matrices (PCM in the following) are used to derive the priorities for n alternatives by means of the so-called eigenvector method.
Nevertheless, in some cases we have to face a problem with missing judgements, thus obtaining an incomplete comparison matrix. This may happen, for instance, when the number of the alternatives, n, is large. In such cases it may be practically impossible, or at least unacceptable from the point of view of the decision maker, to perform all the n(n − 1)/2 required comparisons to complete the PCM. A trade-off between the completeness of the available information and the need to keep reasonably small the number of questions to be submitted to the decision maker is then unavoidable. Moreover, as it has been pointed out in [9], it can be convenient/necessary to skip some direct critical comparison between alternatives, even if the total number of alternatives is small. Some methods have been proposed in the literature to derive the priorities of n alternatives from an incomplete n × n PCM, [1], [3], [9], [10], [14], [19], [23], [24].
In this paper we define a measure of the inconsistency of a PCM using an index introduced in [6] and then we propose to calculate the missing elements of an incomplete PCM by maximizing the global consistency (i.e. minimizing the inconsistency) of the ‘completed’ matrix. The paper is organized as follows: in Section 2 we define the problem, introduce the necessary notations and briefly summarize some methods proposed in the literature to solve the problem of incomplete comparisons. In Section 3 we describe our method and we solve the one and two-dimensional cases (i.e. when one or two comparisons between the alternatives are missing); then we extend our results for the general case of p missing comparisons. Finally, we present some results obtained by applying our method on a case proposed in [18] by Saaty.
Section snippets
The problem formulation
Let Λ = {A1, A2, … , An} be a set of n alternatives and let the judgements of a decision maker be expressed by pairwise comparisons. If all pairs of alternatives (Ai, Aj) with i < j are considered, then it is necessary to perform n(n − 1)/2 comparisons. With this data, one can obtain the upper diagonal triangle of an n × n matrix. The remaining elements of the matrix are easily derived, as it is usually assumed that by comparing Ai with Aj, the comparison of Aj with Ai is automatically assigned. Clearly,
A new method for incomplete comparisons
Our method is based on an (in)consistency index introduced in [6], which directly refers to the definition (3) of consistency for a PCM.
As mentioned before, we assume that the preferences are expressed by an additive PCM R = [rij], rij ∈ [0, 1]. Following [6] and taking into account (3), letbe the inconsistency contribution associated with the triplet of alternatives {Ai, Aj, Ah}. This definition is meaningful, as the following lemma holds. Lemma 1 Lijh is invariant under permutations of
Numerical results
We propose some numerical experiences on a problem, concerning the choice of a job, which has been proposed and studied by Saaty in [18], page 85. The pairwise comparison matrix obtained by Saaty is transformed, by means of (5), into the equivalent additive matrixThe inconsistency index (10) of R is ρ = 3.78. Now, let us assume that the
Conclusions and final remarks
We think that our proposal is a natural way to solve the problem of missing data in pairwise comparison. We propose to complete the PCM coherently with the available judgements by directly referring to the definition (3) of consistency. From the computational point of view our method is rather simple: to calculate the optimal values, we only have to solve a nonsingular linear system. Our future research effort will be directed at comparing our method with other approaches by suitable numerical
References (25)
- et al.
Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations
Fuzzy Sets and Systems
(2001) SSB utility theory: An economic perspective
Mathematical Social Sciences
(1984)Alternative modes of questioning in the analytic hierarchy process
Mathematical Modelling
(1987)Incomplete pairwise comparisons in the analytic hierarchy process
Mathematical Modelling
(1987)- et al.
Multiperson decision-making based on multiplicative preference relations
European Journal of Operational Research
(2001) - et al.
Some issues on consistency of fuzzy preference relations
European Journal of Operational Research
(2004) A scaling method for priorities in hierarchical structures
Journal of Mathematical Psychology
(1977)Fuzzy preference orderings in group decision making
Fuzzy Sets and Systems
(1984)Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation
International Journal of Approximate Reasoning
(2004)A least deviation method to obtain a priority vector of a fuzzy preference relation
European Journal of Operational Research
(2005)
Consistency measures for pairwise comparison matrices
Journal of Multi-Criteria Decision Analysis
Cited by (217)
Integrating incomplete preference estimation and consistency control in consensus reaching
2024, Information FusionA lexicographically optimal completion for pairwise comparison matrices with missing entries
2024, European Journal of Operational ResearchConsensus reaching with minimum adjustment and consistency management in group decision making with intuitionistic multiplicative preference relations
2023, Expert Systems with ApplicationsNecessity of two normalities for the priority vectors of additively consistent fuzzy preference relations with application to group decision making
2023, International Journal of Approximate ReasoningA novel perspective on the inconsistency indices of reciprocal relations and pairwise comparison matrices
2023, Fuzzy Sets and Systems