Abstract
This paper studies the properties of an inconsistency index of a pairwise comparison matrix under the assumption that the index is defined as a norm-induced distance from the nearest consistent matrix. Under additive representation of preferences, it is proved that an inconsistency index defined in this way is a seminorm in the linear space of skew-symmetric matrices and several relevant properties hold. In particular, this linear space can be partitioned into equivalence classes, where each class is an affine subspace and all the matrices in the same class share a common value of the inconsistency index. The paper extends in a more general framework some results due, respectively, to Crawford and to Barzilai. It is also proved that norm-based inconsistency indices satisfy a set of six characterizing properties previously introduced, as well as an upper bound property for group preference aggregation.
Highlights
Pairwise comparison over a set of alternatives X = {x1, . . . , xn} is a well known and powerful method for preference elicitation in a decision problem
In the multiplicative approach (Saaty 1977), ai j is the relative preference of alternative xi over alternative x j, and it estimates the ratio between the weight wi of xi and the weight w j of x j, ai j
We showed that by defining an inconsistency index Id (A) by means of a distance induced by a norm, as in (12), it is possible to prove many relevant properties of this index
Summary
In Cavallo and D’Apuzzo (2009) and in some following papers by the same authors, a general framework for inconsistency evaluation is proposed, based on the algebraic structure of group. Our proposal can be considered as a generalization of the approach by Crawford and Williams (1985), where the logarithmic least square method (LLSM) is applied Their method corresponds to the Euclidean norm minimization after passing to the additive representation of preferences. This justifies, in our view, the numerous good properties of the LLSM and the related geometric mean solution. We assume the additive representation of preferences, and, we introduce an inconsistency index for a PCM defined as a norm-induced distance from the nearest consistent PCM.
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