Taxicab distance based best-worst method for multi-criteria decision-making: An analytical approach

TL;DR

The paper addresses weight derivation in the Best-Worst Method (BWM) for multi-criteria decision-making by formulating a taxicab distance-based framework and deriving analytic solutions for optimal weights. It defines an optimal modification PCS, establishes a one-to-one mapping between modified PCS and weight sets, and expresses all optimal solutions via an auxiliary objective $\epsilon^*$, with $TD$ guiding optimality. A Mixed-Integer Linear Programming (MILP) approach computes a Consistency Index (CI) and a Consistency Ratio $CR=\frac{\epsilon^*}{CI}$, enabling precise consistency assessment without optimization software. Numerical examples reveal both finite and infinite families of optimal weight sets, and a comparison with nonlinear BWM shows the taxicab version can yield unique solutions while preserving localized inconsistencies. Overall, the framework advances analytical understanding of taxicab BWM, reduces computational dependence, and clarifies how inconsistencies influence weight determination in MCDM.

Abstract

The Best-Worst Method (BWM) is a well-known distance based multi-criteria decision-making method used for computing the weights of decision criteria. This article examines a taxicab distance based model of the BWM, with the objective of developing a framework for deriving the model's optimal weights by solving its associated optimization problem analytically. To achieve this, an optimal modification based optimization problem, equivalent to the original one, is first formulated. This reformulated problem is then solved analytically, and the optimal weight sets are derived from its solutions. Contrary to existing literature that asserts the uniqueness of optimal weight sets based on numerical examples, our findings reveal that, in some cases, the taxicab BWM leads to multiple optimal weight sets. A mixed-integer linear programming model is then employed to compute the consistency index. This framework provides a solid mathematical foundation that enhances understanding of the model. It also eliminates the requirement for optimization software, improving the model's precision and efficiency. Finally, the effectiveness of the proposed framework is demonstrated through numerical examples.

Figures (6)

• Figure 1: Flowchart of the analytical framework for the taxicab BWM
• Figure 2: Graph of $f$ in $[1,25]$ for Example 1
• Figure 3: Graph of $f$ in $[1,25]$ for Example 2
• Figure 4: Graph of $f$ in $[1,25]$ for Example 3
• Figure 5: Graph of $f$ in $[1,25]$ for Example 4

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Theorem 1
• Proposition 1
• proof
• Definition 3
• Definition 4
• Proposition 2
• proof
• Theorem 2