A Generalized Analytical Framework for the Nonlinear Best-Worst Method
Harshit M. Ratandhara, Mohit Kumar
TL;DR
This paper extends the nonlinear Best-Worst Method (BWM) by delivering a generalized analytical framework that is compatible with any preference scale and any number of decision-makers. It introduces interval-weights via optimally modified PCS and a secondary objective to select the best weight set, derives scale-invariant consistency indices and an input-based consistency ratio, and handles multiple best/worst criteria by enforcing equal weights. The approach is validated through six numerical examples and a real-world case ranking barriers to energy efficiency in buildings, demonstrating accurate analytic recovery of optimal weights and practical applicability. Overall, the framework eliminates reliance on optimization software, fixes inconsistencies in CI, and enhances decision-support for both individual and group MCDM problems.
Abstract
The nonlinear model of the best-worst method frequently produces multiple optimal weight sets, which are conventionally determined through optimization software. While an analytical approach exists that provides both a closed-form expression for the optimal interval-weights and a secondary objective function to determine the best optimal weight set, we demonstrate that this approach is only valid when preferences are quantified using the Saaty scale and only a single decision-maker is involved. To tackle this issue, we propose a framework compatible with any scale and any number of decision-makers. We first derive an analytical expression for optimal interval-weights and then select the best optimal weight set. After demonstrating that the values of consistency index for the Saaty scale in the existing literature are not well-defined, we derive a formula of consistency index. We also obtain an analytical expression for the consistency ratio, enabling its use as an input-based consistency indicator. Furthermore, we establish that when multiple best/worst criteria are present, weights may vary among best criteria and among the worst criteria. To address this limitation, we modify the original optimization model for weight computation in such instances, solve it analytically to obtain optimal interval-weights and then select the best optimal weight set using a secondary objective function. Finally, we demonstrate and validate the proposed approach using numerical examples and a real-world case study of ranking barriers to energy efficiency in buildings.