A Generalized Analytical Framework for the Nonlinear Best-Worst Method

TL;DR

This paper introduces a generalized analytical framework for the nonlinear best-worst method that accommodates any scale and multiple decision-makers, providing closed-form solutions and addressing limitations of existing approaches.

Contribution

It develops an analytical approach for the nonlinear best-worst method applicable to any scale and multiple decision-makers, including new formulas for consistency index and ratio.

Findings

Derived an analytical expression for optimal interval-weights.

Established formulas for consistency index and ratio.

Validated the approach with numerical examples and a real-world case study.

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.