Inferring Efficient Weights from Pairwise Comparison Matrices

Abstract

Several Multi-Criteria-Decision-Making methodologies assume the existence of weights associated with the different criteria, reflecting their relative importance.One of the most popular ways to infer such weights is the Analytic Hierarchy Process, which constructs first a matrix of pairwise comparisons, from which weights are derived following one out of many existing procedures, such as the eigenvector method or the least (logarithmic) squares. Since different procedures yield different results (weights) we pose the problem of describing the set of weights obtained by "sensible" methods: those which are efficient for the (vector-) optimization problem of simultaneous minimization of discrepancies.A characterization of the set of efficient solutions is given, which enables us to assert that the least-logarithmic-squares solution is always efficient, whereas the (widely used) eigenvector solution is not, in some cases, efficient, thus its use in practice may be questionable.

Efficiency Analysis of Simple Perturbed Pairwise Comparison Matrices

“…Blanquero et al [2] investigated several necessary and sufficient conditions on efficiency. One of these efficiency conditions is utilized in our paper, which applies a directed graph representation.…”

“…It is worth noting that efficiency follows also from the equivalence of the eigenvector method and the row geometric mean, also known as the optimal solution to the logarithmic least squares problem [9, Section 3.2]. Blanquero, Carrizosa and Conde [2,Corollary 7] proved that the weight vector calculated by the row geometric mean is efficient. …”

Inefficient weights from pairwise comparison matrices with arbitrarily small inconsistency

“…Our motivation is the paper of Blanquero, Carrizosa and Conde [1] discussing a general framework of (in)efficiency of a consistent approximation of a pairwise comparison matrix. Their remarkable example on page 282 is as follows.…”

“…Computational results in [1] are given with interval arithmetic, however, coordinates are now written truncated at 8 correct digits and we emphasize that the origin of the phenomenon in our focus is not a rounding error. The approximations X EM and X * coincide except for the third row and column, due to reciprocity, the latter is sufficient to be reported: The authors argue that X * is a better approximation of A than X EM because there exist three elements (and their reciprocals), which are closer to the corresponding elements of A while all the other approximations are the same.…”

Comparing Fundamentals of Additive and Multiplicative Aggregation in Ratio Scale Multi-Criteria Decision Making

“…Local preferences are assumed to be measured in ordinal scale in ranking models [6], in interval scale in multiple attribute utility models [7] and in ratio scale in AHP type models [8,9]. There are many methods proposed for deriving local preferences from pairwise comparison matrices [4,5,[13][14][15][16]. The local preference vectors x 1 ,x 2 ,…,x m and the criteria weight vector w are then aggregated into overall preference vector v=[v 1 ,v 2 ,…,v n ] by the additive aggregation [9,17] or the multiplicative aggregation [2].…”