Taxicab Best-Worst Scaling (BWS)

• BWS is a quantitative method for comparative evaluation that infers criterion weights by minimizing taxicab distances between best and worst choices.
• The analytical framework reformulates the optimization problem through modified pairwise comparison systems, ensuring consistency and piecewise linear solutions.
• This method reveals conditions for unique, multiple, or infinite optimal weight sets, enhancing sensitivity analysis and reproducibility in multi-criteria decision-making.

Best-Worst Scaling (BWS) is a class of quantitative methodologies for comparative evaluation, rooted in the psychometric concepts of maximum-difference scaling and formalized as an optimization problem for inferring criterion weights or item scores from structured best/worst choices. BWS models are pervasive in multi-criteria decision-making (MCDM), applied annotation, and preference measurement in behavioral and computational sciences. One prominent strand is the distance-based BWM, where differences between implied and elicited pairwise preferences are minimized with respect to a norm. The taxicab distance based BWM, as analytically characterized in recent research, introduces new conceptual, mathematical, and computational advancements for the inference and uniqueness of optimal weights (Ratandhara et al., 26 Aug 2024).

1. Foundations of Taxicab Distance Based Best-Worst Method

The taxicab BWM is situated within distance-based approaches to BWM, where deviations between decision-maker (DM) input and implied weights are minimized in the L1 (taxicab) norm. The DM provides:

• A set of criteria $C = \{c_1, ..., c_n\}$.
• Indices $b$ (best) and $w$ (worst).
• Best-to-others: $A_b = (a_{b1}, a_{b2}, ..., a_{bn})$.
• Others-to-worst: $A_w = (a_{1w}, a_{2w}, ..., a_{nw})$.

The core analytical optimization problem is: $$\min_{w \in \mathbb{R}^n} \text{TD} = \sum_{i \in D} \left( \left| \frac{w_b}{w_i} - a_{bi} \right| + \left| \frac{w_i}{w_w} - a_{iw} \right| \right) + \left| \frac{w_b}{w_w} - a_{bw} \right|$$ subject to $\sum_j w_j = 1$, $w_j \ge 0$.

This convex, nonlinear problem differs from conventional Euclidean BWM, making existence, computability, and uniqueness of solutions non-trivial.

2. Analytical Reformulation and Mathematical Solution

A principal innovation is the reformulation from direct weight optimization to optimization over optimally modified consistent pairwise comparison systems (PCS). For each criterion $i \in D = C \setminus \{b,w\}$, the PCS is $(\tilde{a}_{bi}, \tilde{a}_{iw}, \tilde{a}_{bw})$, subject to the consistency condition: $\tilde{a}_{bi} \tilde{a}_{iw} = \tilde{a}_{bw}$ The modified minimization becomes: $$\min \sum_{i \in D} \left( |\tilde{a}_{bi} - a_{bi}| + |\tilde{a}_{iw} - a_{iw}| \right) + |\tilde{a}_{bw} - a_{bw}|$$ with domain restrictions $$\tilde{a}_{bi}, \tilde{a}_{iw}, \tilde{a}_{bw} \ge 0$$.

The weights $w_j$ follow from the consistent PCS: $$w_j^* = \frac{\tilde{a}_{jw}^*}{\sum_{i \in C} \tilde{a}_{iw}^*} \qquad \text{or} \qquad w_j^* = \frac{1}{\tilde{a}_{bj}^* \sum_{i \in C} 1/\tilde{a}_{bi}^*}$$

All possible minimally modified PCS (parameterized by a scalar $\tilde{a}_{bw}^*$ constrained to specific intervals) are enumerated. For each candidate, the total deviation is computed, and all global minimizers are collected as optimal weights.

The analytical solution involves:

• Piecewise definitions for minimal modifications, considering downside, upside, and consistent cases.
• Construction of piecewise affine objective functions $f(x)$ over $\tilde{a}_{bw}^*$, with explicit minimizer sets.
• Finite or continuous intervals of minimizers, leading to one, finitely many, or uncountably many (interval) solutions.

3. Uniqueness and Multiplicity of Optimal Weight Sets

Prior literature—guided by numerical experiments—often asserted the uniqueness of optimal weight sets in taxicab BWM [Amiri and Emamat, 2020]. The analytical treatment refutes this in general, demonstrating with both proof and explicit examples that:

• Unique solutions occur for specific input structures (e.g., perfectly compatible judgments).
• Multiple optimal weight sets (finite): There exist input configurations (Examples 2–4) with two distinct global minima, each yielding distinct weight sets.
• Infinitely many optimal weight sets: For certain input data, the minimal total deviation is achieved over a continuous interval of $\tilde{a}_{bw}^*$ values (Example 5), inducing a continuum of admissible weights.

This outcome has direct implications for result interpretation, sensitivity analysis, and method reproducibility in practical deployments.

4. Consistency Index, Ratio, and Analytical Computation

A foundational requirement is measurement of the internal consistency of DM inputs.

• The Consistency Index (CI) measures the degree to which the PCS can be rendered transitively consistent via minimal modifications. The analytic framework suggests that CI (and the derived Consistency Ratio (CR)) can be formulated as an explicit function of the (modified) input, paralleling approaches in nonlinear BWM.
• The proposed framework allows, in principle, for the construction of a mixed-integer linear programming (MILP) model for the exact (analytic) calculation of CI/CR. The precise MILP structure is noted as a research direction.

Advantages over numeric-only procedures include eliminating discretization error, ensuring all minimizers are found, and improving both reproducibility and interpretability.

5. Computational Advantages and Practical Implications

Key improvements are:

• No need for optimization software: The complete solution for any input system is found algebraically, which allows for easier auditing, transparency, reproducibility, and deployment in resource-limited contexts.
• Full characterization of the solution set: Whereas previous numerical approaches would yield a single solution (possibly missing other minima), the analytic framework ensures all valid minima (identifying all finitely or infinitely many unique weight sets) are produced.

In applied MCDM, these theoretical advances give practitioners:

• Enhanced control over sensitivity analyses; all admissible weights are characterized, so the impact of perturbing DM input or PCS structure is directly observable.
• Tools to rationalize multiple admissible results and robustify downstream analyses.

6. Illustrative Examples and Effectiveness

The analytical approach is validated with five canonical examples:

• Example 1: Single, unique weight set (full consistency).
• Examples 2–4: Multiple optimal weight sets—demonstrated graphically and analytically.
• Example 5: Infinitely many optimal weights—objective function $f(x)$ remains flat over a nontrivial interval.

For each, the approach details the explicit function $f(x)$, all global minima, and the associated PCS/weights. Visualizations confirm (non-)uniqueness, clarifying both method generality and diagnostic capacity.

Key Dimension	Taxicab BWM (Analytical Framework)
Optimization Type	Nonlinear, piecewise convex, L1 (taxicab) objective
Uniqueness	Not guaranteed; may be 1, finitely many, or infinite
Solution Method	Analytic reformulation: modification minimization over PCS
Consistency Measures	Analytically computable CI/CR, MILP formulation possible
Deployment	Software-free, explicit, reproducible

7. Impact on the Theory and Practice of BWM

This analytical framework establishes a new mathematical foundation for the taxicab BWM. By bridging consilience between the modified PCS and optimal weights, the framework generalizes the method's applicability and provides strong guarantees regarding result multiplicity and interpretability.

The findings clarify previous misconceptions about solution uniqueness and offer practical protocols for full-result auditing and transparent reporting. By facilitating analytical consistency diagnostics and permitting enumeration of all solutions, the method enhances both the methodological soundness of PCS-based MCDM and the reliability of BWM in decision support systems.

References to Core Mathematical Formulas

• Taxicab BWM objective function:

$$\min \text{ TD} = \sum_{i \in D} \left( \left| \frac{w_b}{w_i} - a_{bi} \right| + \left| \frac{w_i}{w_w} - a_{iw} \right| \right) + \left| \frac{w_b}{w_w} - a_{bw} \right|$$

• Final weights from optimally consistent PCS:

$$w_j = \frac{\tilde{a}_{jw}^*}{\sum_{i \in C} \tilde{a}_{iw}^*}$$

or

$$w_j = \frac{1}{\tilde{a}_{bj}^* \sum_{i \in C} 1/\tilde{a}_{bi}^*}$$

Conclusion

The taxicab distance based Best-Worst Method, as fully analytically specified, advances the theory of distance-based MCDM by enabling robust, closed-form, and exhaustive computation of criterion weights, consistency diagnostics, and sensitivity properties (Ratandhara et al., 26 Aug 2024). It resolves ambiguities about result uniqueness, providing practitioners with a complete view of admissible decision solutions and supporting advanced decision analytics without reliance on numerical solvers.