Mixed-integer linear-fractional programming model and it’s linear analogue for reducing inconsistency of pairwise comparison matrices☆

Abstract

Pairwise comparison matrices are often used in multicriteria decision making (MCDM). The most critical part of this technique is the inconsistency, which emerges for logical reasons but can cause significant problems during the decision making process. Hence it is necessary to keep inconsistency below an acceptable threshold. In order to support the decision maker (DM) in making a rational decision, we must keep in mind the following: Our suggestion should be as close to the DM’s original result as possible, moreover it should have as low inconsistency as possible. We have studied various linear programming (LP) models that are used for reducing the inconsistency of pairwise comparison matrices (PCM) (Bozóki et al., 2011, 2015). These models, however aim at fulfilling only one of the previously mentioned two goals at a time. Therefore, the optimal solutions given may differ from each in the other respect, which is not taken as an objective but as a constraint in the model. So, they cannot be considered as equally good optimal solutions from a wider perspective. Based on our experiences concerning these models, we have developed a mixed-integer linear-fractional programming (MILFP) model that takes both mentioned goals as objectives by combining them into a linear-fractional objective function. We also provide the linear analogue (LA) of our MILFP model using an appropriate adaptation and combination of the Charnes-Cooper transformation and Glover’s linearization scheme.