On solving the Best-Worst Method in multi-criteria decision-making

Abstract

Decision-making often refers to ranking alternatives based on many involved criteria. Since the introduction of the Analytic Hierarchy Process (AHP) in 1980, pairwise comparisons of criteria have a long tradition in multi-criteria decision-making. One of the main concerns of the AHP refers to the inconsistency of decision makers in pairwise comparisons. Recently, the Best-Worst Method (BWM) was introduced to reduce the inconsistency by a concept that needs substantially less pairwise comparisons. The BWM includes solving a non-linear model (NLM) to derive the weights from the comparisons. A linear model (LM) was introduced in a follow-up to approximate the original NLM. This paper shows that the optimal weights of the proposed linear model (LM) may differ substantially from the optimal weights of the original NLM model. Moreover, this paper provides an MILP model approximation (MILM) which can be solved by standard optimization software and illustrates that its solution approximates the optimal weights of the original NLM model arbitrarily close. Since consistency in pairwise comparisons is usually not self-evident in practice, using approximation MILM to derive unique solutions of the original NLM, extends the applicability of the Best-Worst Method.