Minimum adjustment cost-based multi-stage goal programming models for consistency improving and consensus building with multiplicative reciprocal paired comparison matrices

Abstract

In group decision-making with multiplicative reciprocal paired comparison matrices (MRPCMs), existing research uses iterative procedures or optimisation models to improve consistency of individual assessments and build consensus. However, they often create numerous adjustments on original assessments and fail to achieve a comprehensive minimum adjustment cost. Furthermore, the adjustments in the resulting MRPCMs may be not within the predetermined continuous scale. To settle these issues, this article first introduces a logarithmic-distance-based consensus measurement framework. Four-stage sequential goal programming models are then developed to improve consistency of MRPCMs and build consensus with individual consistency control under a continuous or discrete scale. The first stage is to minimise the deviation between original assessments and adjusted ones. The second stage is to minimise the difference between the original priority information and the adjusted priority information. The third stage is to maximise the difference ratio between adjusted assessments and the neutral judgment characterised by ratio 1. The last stage is to minimise the number of modifications on original assessments. Afterwards, the article devises an interactive consistency improving procedure and an interactive group consensus building procedure. Three illustrative examples and comparisons with existing methods are offered to show the usability and efficiency of the developed models.