On reducing inconsistency of pairwise comparison matrices below an acceptance threshold. (English) Zbl 1339.91032

Summary: A recent work of the authors on the analysis of pairwise comparison matrices that can be made consistent by the modification of a few elements is continued and extended. Inconsistency indices are defined for indicating the overall quality of a pairwise comparison matrix. It is expected that serious contradictions in the matrix imply high inconsistency and vice versa. However, in the 35-year history of the applications of pairwise comparison matrices, only one of the indices, namely CR proposed by Saaty, has been associated to a general level of acceptance, by the well known ten percent rule. In the paper, we consider a wide class of inconsistency indices, including CR, CM proposed by W. W. Koczkodaj [Math. Comput. Modelling 18, No. 7, 79–84 (1993; Zbl 0804.92029)] and CI by J. I. Peláez and M. T. Lamata [Comput. Math. Appl. 46, No. 12, 1839–1845 (2003; Zbl 1121.91334)]. Assume that a threshold of acceptable inconsistency is given (for CR it can be 0.1). The aim is to find the minimal number of matrix elements, the appropriate modification of which makes the matrix acceptable. On the other hand, given the maximal number of modifiable matrix elements, the aim is to find the minimal level of inconsistency that can be achieved. In both cases the solution is derived from a nonlinear mixed-integer optimization problem. Results are applicable in decision support systems that allow real time interaction with the decision maker in order to review pairwise comparison matrices.


91B06 Decision theory
90C10 Integer programming
90C09 Boolean programming
90C25 Convex programming
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