An analytic framework for the multiplicative best-worst method
Harshit Ratandhara, Mohit Kumar
TL;DR
The paper tackles the multiplicative Best-Worst Method (BWM) in MCDM, where non-uniqueness of optimal weight sets hinders interpretation. It introduces an analytic framework by formulating an equivalent optimally modified PCS model, deriving closed-form analytic solutions for optimal interval-weights and consistency metrics $CI$ and $CR$, and proving a one-to-one correspondence with optimal weight sets. A secondary objective is proposed to select a unique best optimal weight set, preserving the core multiplicative structure, validated through numerical examples and a real-world Sustainable Additive Manufacturing (SAM) driver-ranking case. The work yields an efficient, software-free method with immediate consistency feedback and offers pathways to extend the approach to other BWM variants.
Abstract
The Best-Worst Method (BWM) is a well-known Multi-Criteria Decision-Making (MCDM) method. This article deals with the multiplicative model of BWM. We first formulate an optimization model that is equivalent to the existing multiplicative model. This model provides a solid foundation for obtaining an analytic form of optimal interval-weights, Consistency Index (CI) and Consistency Ratio (CR). The proposed approach does not require any optimization software, which makes it easy to implement as well as time efficient. Also, the obtained analytical form of CR permits it to serve as an input-based consistency measure. After obtaining these analytic forms, a secondary objective function is introduced to select the best optimal weight set from the collection of all optimal weight sets. Finally, we discuss some numerical examples and a real-world application of the proposed approach in ranking the drivers of Sustainable Additive Manufacturing (SAM) to illustrate the proposed approach.