Abstract
The pairwise comparison (PC) matrix is often used to manifest human judgments, and it has been successfully applied in the analytic hierarchy process (AHP). As a PC matrix is formed by making paired reciprocal comparisons, symmetry is a striking characteristic of a PC matrix. It is this simple but powerful means of resolving multicriteria decision-making problems that is the basis of AHP; however, in practical applications, human judgments may be inconsistent. Although Saaty’s rule for the consistency test is commonly accepted, there is evidence that those so-called “acceptable” PC matrices may not be ordinally consistent, which is a necessary condition for a PC matrix to be accepted. We propose an ordinal consistency indicator called SDR (standard deviation of ranks), derive the upper bound of the SDR, suggest a threshold for a decision-maker to assess whether the ordinal consistency of a PC matrix is acceptable, and reveal a surprising fact that the degree of ordinal inconsistency of a small PC matrix may be more serious than a large one. We made a comparative analysis with some other indicators. Experimental results showed that the ordinal inconsistency measured by the SDR is invariant under heterogeneous judgment measurements with a varied spectrum of scales, and that the SDR is superior to the two compared indicators. Note that the SDR not only works for a multiplicative PC matrix but can also be used for additive and fuzzy PC matrices.
Highlights
Since Thurstone’s eminent article was published, the pairwise comparison matrix (PC matrix for short) has been a well-known method used to help decision-makers to manifest experts’ subjective judgments [1]
For a PC matrix A of order n, the upper bound of SDR[A] is n(n+1) 12
The upper bound of SDR[A] is a function of n, i.e., the size of a PC matrix A
Summary
Since Thurstone’s eminent article was published, the pairwise comparison matrix (PC matrix for short) has been a well-known method used to help decision-makers to manifest experts’ subjective judgments [1]. As the pairwise comparisons that we consider are formed by making paired reciprocal comparisons, symmetry is a striking characteristic of a PC matrix It is this simple, but powerful means of resolving multicriteria decision-making problems that is the basis of the AHP [14]; they are not necessarily consistent because this is a desirable property that is what we want to achieve, from the academic perspective and from the decision-maker’s perspective.
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