Uncertainty index based consistency measurement and priority generation with interval probabilities in the analytic hierarchy process

• We capture inconsistency of multiplicative reciprocal comparison matrices (MRCMs).
• We construct an interval multiplicative reciprocal matrix from a MRCM.
• We introduce an uncertainty index to check acceptable consistency of MRCMs.
• We propose a relation between interval probabilities and the constructed matrix.
• We develop a two-stage procedure to generate interval probabilities from MRCMs.

The pairwise judgments provided by a decision-maker are often inconsistent in the analytic hierarchy process. Dealing with inconsistency among the given judgments by uncertainty and interval probabilities has received more attention in recent years. In this paper, an interval multiplicative reciprocal matrix is constructed to capture inconsistency and the decision-maker’s judgments from a multiplicative reciprocal comparison matrix (MRCM). A geometric mean based uncertainty index is defined to measure the uncertainty level of the constructed interval matrix, and used to check acceptable consistency of MRCMs. The paper devises a parameterization approximate relation between the normalized interval probabilities and the constructed interval matrix. A two-stage procedure consisting of two optimization models is developed to generate interval probabilities from MRCMs. The first stage minimizes the power-sum of absolute deviations between the logarithms of the sides of the approximate relation. The second stage aims to seek the most suitable interval probability vector among the optimal solutions derived from the previous stage such that inconsistency of the original MRCM is reflected by interval probabilities as much as possible. An interval probability based algorithm is further put forward for solving multi-criteria decision making problems with MRCMs. Two numerical examples and comparison analyses are presented to demonstrate the performance and validity of the proposed models.