A Generalized Analytical Framework for the Nonlinear Best-Worst Method

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.

💡 Research Summary

The paper addresses a fundamental limitation of the nonlinear Best‑Worst Method (BWM), namely the existence of multiple optimal weight vectors that are traditionally selected by solving a nonlinear optimization problem with commercial software. While an analytical approach proposed by Wu et al. (2022) can derive closed‑form interval weights and a secondary objective to pick a single “best” weight set, the authors demonstrate that this approach is only valid when the pairwise comparison values are expressed on the Saaty 1‑9 scale and when a single decision‑maker (DM) is involved. Consequently, the existing method fails for alternative scales, group decision‑making, and situations with multiple best or worst criteria, and the consistency index (CI) used in the conventional consistency ratio (CR) is ill‑defined.

To overcome these shortcomings, the authors develop a generalized analytical framework that works with any preference scale and any number of DMs. The core of the framework is a “optimal modification” of the pairwise comparison system (PCS). They introduce modified comparison values (\tilde a_{ij}) and a scalar (\eta) such that (|\tilde a_{ij}-a_{ij}|\le\eta) for all entries, while enforcing the consistency condition (\tilde a_{bi},\tilde a_{iw}=\tilde a_{bw}). By partitioning the set of criteria into three subsets (D₁, D₂, D₃) according to the sign of (a_{bi}a_{iw}-a_{bw}), they analytically solve two quadratic equations for each criterion to obtain the smallest positive root (\varepsilon_i). The optimal (\eta^*) is then selected as the maximum (\varepsilon_i) from D₁ or D₂, depending on a simple inequality, or as the larger of the two when both subsets are non‑empty. This yields a closed‑form expression for the optimal modified PCS, which in turn provides explicit formulas for the lower and upper bounds of each weight: \