Best-Worst Method (BWM) Overview

• BWM is a multi-criteria decision-making method that derives criterion weights from a reduced set of best-to-others and others-to-worst comparisons.
• It employs various optimization models—including linear, nonlinear, multiplicative, and taxicab-distance formulations—to balance consistency and computational efficiency.
• Extensions like fuzzy, interval-valued, and disaggregation-based variants enhance its applicability in complex decision problems by refining preference modeling.

Searching arXiv for recent and foundational BWM papers to ground the article. I’m checking arXiv for Best-Worst Method and related analytical, ordinal, and fuzzy variants. The Best-Worst Method (BWM) is a multi-criteria decision-making (MCDM) method for deriving criterion weights from a reduced set of pairwise comparisons. In its standard form, the decision-maker identifies the best criterion $c_b$ and the worst criterion $c_w$, then provides a best-to-other comparison vector and an other-to-worst comparison vector. This elicitation structure reduces the number of pairwise comparisons from $n(n-1)/2$ to $2n-3$, while preserving an explicit best/worst anchoring of the judgment process. Recent arXiv work has treated BWM not only as a practical weighting tool, but also as a family of optimization models with distinct analytical, consistency, and robustness properties, including linear, nonlinear, multiplicative, taxicab-distance, logarithmic-least-squares, fuzzy, and disaggregation-based variants (Csató, 2023, Ratandhara et al., 21 Jun 2025, Ratandhara et al., 8 Aug 2025).

1. Basic structure and elicitation logic

In the classical formulation, the criteria set is

$C=\{c_1,c_2,\dots,c_n\}.$

The decision-maker selects the best criterion $c_b$ and the worst criterion $c_w$, and provides two comparison systems:

• the best-to-other vector $A_b=(a_{b1},a_{b2},\dots,a_{bn})$,
• the other-to-worst vector $A_w=(a_{1w},a_{2w},\dots,a_{nw})^T$.

The intended weight vector $w_1,\dots,w_n$ should satisfy

$c_w$0

This best/worst anchoring is the defining feature of BWM. In the preference-disaggregation literature, the same logic has been described as a “consider-the-opposite” strategy: instead of requiring a full ranking or all pairwise comparisons, the expert compares all other items relative to two polar reference points, the best and the worst. The BWD extension argues that this reduces cognitive burden because the comparisons are structured around salient anchors rather than arbitrary pairwise judgments (Brunelli et al., 2024).

A standard implementation uses a $c_w$1 to $c_w$2 scale, with $c_w$3 meaning equality and $c_w$4 meaning extreme preference. This scale appears in the standard BWM setup, in BWD, and in application papers such as network selection in heterogeneous wireless networks (Mefgouda et al., 2 Apr 2025, Brunelli et al., 2024).

2. Core optimization models

The standard nonlinear BWM seeks weights by minimizing the maximum absolute deviation from the elicited ratios: $c_w$5 subject to

$c_w$6

$c_w$7

This formulation is central in recent analytical work on the nonlinear BWM, where the optimal value $c_w$8 is interpreted as a measure of fit between the weight vector and the elicited comparisons (Ratandhara et al., 8 Aug 2025).

A distinct linear BWM replaces ratio deviations by linear deviations: $c_w$9 subject to

$n(n-1)/2$0

for $n(n-1)/2$1, together with

$n(n-1)/2$2

The linear model is widely used because it yields a unique optimal weight vector, but recent analytical work emphasizes that it still required optimization software until a closed-form solution was derived (Ratandhara et al., 21 Jun 2025).

Other BWM variants preserve the best/worst elicitation structure but alter the loss function.

Model	Representative idea	Reported property
Nonlinear BWM	Minimize maximum ratio deviation	May have multiple optimal weight sets
Linear BWM	Minimize maximum linear deviation	Yields a unique optimal weight vector
Multiplicative BWM	Minimize maximum multiplicative inconsistency	Can have multiple optimal solutions
Taxicab BWM	Minimize total absolute deviation of all ratio errors	Can have finitely many or infinitely many optimal weight sets

The multiplicative BWM uses a multiplicative distance measure and minimizes the maximum multiplicative inconsistency. The taxicab-distance-based BWM instead minimizes the total absolute deviation of all ratio errors: $n(n-1)/2$3 Analytical studies of both models show that multiplicity of optimal solutions is structurally possible, rather than being a numerical artifact (Ratandhara et al., 2023, Ratandhara et al., 2024).

3. Pairwise-comparison representation, incomplete matrices, and ordinal violations

BWM can be represented through a reciprocal pairwise comparison matrix

$n(n-1)/2$4

with

$n(n-1)/2$5

In a BWM matrix, only comparisons involving the designated best and worst items are known: $n(n-1)/2$6 The resulting matrix is therefore incomplete. For such incomplete matrices, the logarithmic least squares method (LLSM) derives priorities by setting

$n(n-1)/2$7

constructing the Laplacian matrix $n(n-1)/2$8 of the comparison graph and the vector

$n(n-1)/2$9

and solving

$2n-3$0

Because the BWM comparison graph is connected, the LLSM solution is unique (Csató, 2023).

This matrix-based perspective makes precise a central pathology of BWM: the ordinal violation. A priority vector $2n-3$1 shows an ordinal violation if

$2n-3$2

In the BWM setting, this means either that the designated best item fails to receive the highest weight, or that the designated worst item fails to receive the lowest weight. The 2023 analysis of LLSM priorities for incomplete BWM matrices gives two sufficient conditions guaranteeing that ordinal violations do not occur. One theorem uses a dominance requirement based on a parameter $2n-3$3 together with a maximal-preference bound $2n-3$4; a second theorem adds the structural condition that the best-versus-worst judgment is at least as strong as any best-to-other or other-to-worst comparison, with maximal numerical preference bounded by

$2n-3$5

For the Saaty scale $2n-3$6, the second theorem yields a concrete corollary: if the best alternative is better than all others, the worst is worse than all others, and the best-versus-worst comparison is at least as large as any other comparison, then for

$2n-3$7

the LLSM priorities are guaranteed to have no ordinal violation (Csató, 2023).

4. Consistency theory and analytical frameworks

The basic BWM consistency condition is

$2n-3$8

where $2n-3$9. If this condition holds, the exact weights are uniquely determined by the ratio equations and the normalization

$C=\{c_1,c_2,\dots,c_n\}.$0

Recent analytical papers treat inconsistency as a geometric or algebraic deviation from this exact relation, and derive closed-form expressions for optimal weights, consistency indices, and consistency ratios.

For the linear BWM, the 2025 analytical framework partitions the inconsistency structure through

$C=\{c_1,c_2,\dots,c_n\}.$1

defines local inconsistency measures

$C=\{c_1,c_2,\dots,c_n\}.$2

and then derives a closed-form optimal solution. It also computes the consistency ratio

$C=\{c_1,c_2,\dots,c_n\}.$3

with explicit formulas for the linear-BWM consistency index. The same paper reports that the linear BWM can have limited responsiveness to data perturbations: different pairwise comparison systems may yield exactly the same weights and the same objective value. Its numerical examples report that four, nine, seventy-five, two, and thirty-six different pairwise comparison systems can produce the same solution in five separate cases (Ratandhara et al., 21 Jun 2025).

For the nonlinear BWM, the 2025 generalized analytical framework replaces direct optimization in weight space by optimization in comparison space through an optimally modified pairwise comparison system. It derives analytical formulas for optimal interval-weights, the best optimal weight set, the consistency index, and the consistency ratio, and shows that previously reported Saaty-scale CI values were not well-defined as true upper bounds in some cases. It also establishes several properties for the resulting CR, including normalization, zero iff consistency, permutation invariance with respect to criterion labels, non-increasing behavior under criterion elimination, and continuity in the preference values (Ratandhara et al., 8 Aug 2025).

The multiplicative and taxicab models extend the same analytical agenda. The multiplicative BWM derives an analytic form of optimal interval-weights, CI, and CR, and introduces a secondary objective function to select a unique best optimal weight set from the collection of all optimal sets (Ratandhara et al., 2023). The taxicab-distance-based model is analytically reduced to minimizing a univariate function $C=\{c_1,c_2,\dots,c_n\}.$4 over a finite candidate set $C=\{c_1,c_2,\dots,c_n\}.$5, and the analysis shows that the model may admit a unique optimum, finitely many optima, or infinitely many optima. This directly contradicts the earlier belief that taxicab BWM always yields a unique optimal weight set (Ratandhara et al., 2024).

5. Fuzzy, interval-valued, and disaggregation-based extensions

A major extension of BWM addresses imprecise judgments. The $C=\{c_1,c_2,\dots,c_n\}.$6-cut-interval fuzzy BWM begins from the observation that earlier fuzzy BWM models based on triangular fuzzy numbers used only the lower, modal, and upper values of a fuzzy judgment, leaving the rest of the fuzzy shape unused. To reduce this information loss, the proposed model represents each fuzzy number through its full family of $C=\{c_1,c_2,\dots,c_n\}.$7-cuts and seeks fuzzy weights whose entire $C=\{c_1,c_2,\dots,c_n\}.$8-cut structure fits the fuzzy comparisons. The general formulation imposes the ratio constraints for all $C=\{c_1,c_2,\dots,c_n\}.$9; because that yields infinitely many constraints, the method approximates the problem on a finite subset

$c_b$0

Its Degree of Approximation (DoA) is the mesh

$c_b$1

and the paper proves that the approximation error is bounded by this mesh. When multiple approximate optimal solutions exist, the set of attainable defuzzified weights for a criterion is an interval, and the method selects the center of that interval as the final crisp weight (Ratandhara et al., 2023).

A different line of development extends BWM from weight elicitation to preference disaggregation. Best-Worst Disaggregation (BWD) embeds the best/worst elicitation logic in an additive value-function model: $c_b$2 Instead of estimating only criterion weights, BWD uses best-to-others and others-to-worst judgments on a reference set of alternatives to fit piecewise linear attribute value functions and a global additive model. Its core optimization is a min-max model over two opposite comparison vectors, and it distinguishes compatibility of the fitted model from consistency of the elicited judgments. To quantify the latter, the paper defines an ordinal consistency ratio

$c_b$3

and a cardinal consistency ratio

$c_b$4

BWD also extends naturally to interval-valued judgments and introduces a combinatorial reference-set selection model to ensure coverage of criterion segments without Pareto dominance inside the selected reference set (Brunelli et al., 2024).

6. Applications, limitations, and adjacent best-worst paradigms

Recent applications show BWM serving as a weighting engine inside larger decision frameworks rather than as a stand-alone ranking procedure. In heterogeneous wireless networks, the BWM-GWO framework uses BWM to compute subjective weights encoding service-class preferences and Grey Wolf Optimization to derive objective weights aimed at reducing the rank reversal problem. The final weight vector is a convex linear combination,

$c_b$5

with the paper setting

$c_b$6

In that study, the criteria are cost per byte, security, data rate, packet delay, packet jitter, and packet loss rate, and the candidate radio access technologies are WiFi, WiMAX, LTE, and 5G. The reported result is that integrating BWM-GWO with MADM-based network selection reduces rank reversal occurrence by up to $c_b$7 while significantly improving user and service satisfaction compared to benchmark approaches (Mefgouda et al., 2 Apr 2025).

In sustainable-development-oriented agri-food supply chains, the analytical linear BWM was used to rank eighteen drivers across Industry 4.0, sustainability, and circular economy categories. The aggregated ranking identified $c_b$8 Resource efficiency as the most critical driver and $c_b$9 Artificial Intelligence as the least critical (Ratandhara et al., 21 Jun 2025). In logistics performance evaluation, BWD produced rankings aligned closely with expert preferences, with revised judgments becoming fully ordinally consistent and the interval-valued version achieving full compatibility at the cost of increased rank uncertainty (Brunelli et al., 2024).

Several limitations recur across the literature. The linear BWM may be weakly sensitive to some input variations (Ratandhara et al., 21 Jun 2025). The nonlinear, multiplicative, and taxicab formulations may all admit multiple optimal weight sets, which complicates interpretation unless a secondary objective or interval-weight analysis is used (Ratandhara et al., 8 Aug 2025, Ratandhara et al., 2023, Ratandhara et al., 2024). In the fuzzy setting, exact optimization of the entire $c_w$0-cut shape is difficult, which motivates finite-$c_w$1 approximation (Ratandhara et al., 2023). A practical implementation caveat also appears in the analytical linear-BWM paper, which reports a discrepancy between the analytically derived optimum and the output of a commonly used Excel Solver implementation in one numerical example (Ratandhara et al., 21 Jun 2025).

A persistent misconception concerns the term “best-worst” itself. In several neighboring literatures, it denotes something other than the MCDM weighting method. In best-worst choice theory, the object of study is a probability system $c_w$2 describing the probability that $c_w$3 is chosen as best and $c_w$4 as worst from an offered set $c_w$5. The relevant question there is random-utility rationalizability, characterized through nonnegativity of best-worst Block-Marschak polynomials, not criterion-weight elicitation (Colonius, 2020). A later technical note emphasizes that this random-utility literature is not the multi-criteria decision-making technique usually called the Best-Worst Method, even though both use “best” and “worst” language (Colonius, 2024).

Likewise, Best-Worst Scaling (BWS) in NLP and annotation studies is a comparative labeling scheme rather than an MCDM weighting algorithm. In those settings, annotators see 4-tuples of texts and choose the highest and lowest items with respect to a target property, then compute scores by

$c_w$6

This formulation has been used for automatic emotion-intensity annotation and historical biodiversity quantification with LLMs (Bagdon et al., 2024, Haider et al., 6 Feb 2025). The shared best/worst elicitation pattern suggests a family resemblance, but the mathematical goals differ: BWS produces interval-like item scores from comparative judgments, whereas BWM derives criterion weights for multi-criteria aggregation.