Decision Support
Incomplete pairwise comparison and consistency optimization

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Abstract

This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented.

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), letLijh=(rih+rhj-rij-0.5)2be 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 matrixR=0.50.50.50.81550.50.34230.50.50.65770.81550.50.34230.50.34230.50.86620.750.34230.18450.18450.13380.50.250.250.50.50.250.750.50.250.65770.65770.65770.750.750.5.The 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

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