On Indeterminacy of Interval Multiplicative Pairwise Comparison Matrix

Abstract

The interval multiplicative pairwise comparison matrix (IMPCM) is widely used to model human judgments affected by uncertainty and/or ambiguity. To improve the quality of an IMPCM, consistency is not sufficient. The indeterminacy should also be within an acceptable threshold because a consistent IMPCM may be deemed unacceptable due to high indeterminacy. Regarding indeterminacy, two metrics have been proposed in the literature: the indeterminacy ratio and the indeterminacy index. The former is from a local view, and the latter is from a global view. We have proposed an acceptable IMPCM model, which guarantees that an inconsistent IMPCM can be transformed into a consistent IMPCM, and the maximal indeterminacy ratio can be reduced. However, there is still a research gap. That is, a concomitant question naturally arises: can the indeterminacy index be reduced as well? In this paper, we further prove that the indeterminacy index of an originally inconsistent IMPCM can be reduced under the proposed model. Three numerical examples are presented to illustrate the feasibility and superiority of the proposed model. We also flowcharted the proposed model from a pragmatic view such that we can judiciously reduce the indeterminacy index of the IMPCM to a certain satisfactory level. That is, by applying the proposed model once, the original inconsistent IMPCM can be transformed into a consistent IMPCM that will possess less indeterminacy than the original one has. Consequently, by successively applying the proposed model, we can reduce or even eventually eliminate the indeterminacy of the IMPCM. In other words, we can/may obtain an MPCM rather than an IMPCM. In addition to mathematical proofs, we present experimental results of computer simulations to corroborate our argument. In summary, this model is not only effective but also efficient because it only requires arithmetic operations without solving complex optimization problems.