Pairwise Comparison Matrices: Some Issue on Consistency and a New Consistency Index

In multicriteria decision making, the pairwise comparisons are an useful starting point for determining a ranking on a set

X

 = {

x

1

,

x

2

,...,

x

n

} of alternatives or criteria; the pairwise comparison between

x

i

and

x

j

is quantified in a number

a

ij

expressing how much

x

i

is preferred to

x

j

and the quantitative preference relation is represented by means of the matrix

A

 = (

a

ij

). In literature the number

a

ij

can assume different meanings (for instance a ratio or a difference) and so several kind of pairwise comparison matrices are proposed. A condition of consistency for the matrix

A

 = (

a

ij

) is also considered; this condition, if satisfied, allows to determine a weighted ranking that perfectly represents the expressed preferences. The shape of the consistency condition depends on the meaning of the number

a

ij

. In order to unify the different approaches and remove some drawbacks, related for example to the fuzzy additive consistency, in a previous paper we have considered pairwise comparison matrices over an abelian linearly ordered group; in this context we have provided, for a pairwise comparison matrix, a general definition of consistency and a measure of closeness to consistency. With reference to the new general unifying context, in this paper we provide some issue on a consistent matrix and a new measure of consistency that is easier to compute; moreover we provide an algorithm to check the consistency of a pairwise comparison matrix and an algorithm to build consistent matrices.