The two original papers

Here are the two original papers which introduce the BWM, the first one presents the main philosophy and foundation of the method and a non-linear model, while the second one includes some properties of the original model as well as a linear model.

Rezaei, J. (2015). Best-worst multi-criteria decision-making method, Omega, 53, pp. 49-57.

In this paper, a new method, called best-worst method (BWM) is proposed to solve multi-criteria decision-making (MCDM) problems. In an MCDM problem, a number of alternatives are evaluated with respect to a number of criteria in order to select the best alternative(s). According to BWM, the best (e.g. most desirable, most important) and the worst (e.g. least desirable, least important) criteria are identified first by the decision-maker. Pairwise comparisons are then conducted between each of these two criteria (best and worst) and the other criteria. A minimax problem is then formulated and solved to determine the weights of different criteria. The weights of the alternatives with respect to different criteria are obtained using the same process. The final scores of the alternatives are derived by aggregating the weights from different sets of criteria and alternatives, based on which the best alternative is selected. A consistency ratio is proposed for the BWM to check the reliability of the comparisons. To illustrate the proposed method and evaluate its performance, we used some numerical examples and a real-word decision-making problem (mobile phone selection). For the purpose of comparison, we chose AHP (analytic hierarchy process), which is also a pairwise comparison-based method. Statistical results show that BWM performs significantly better than AHP with respect to the consistency ratio, and the other evaluation criteria: minimum violation, total deviation, and conformity. The salient features of the proposed method, compared to the existing MCDM methods, are: (1) it requires less comparison data; (2) it leads to more consistent comparisons, which means that it produces more reliable results.

Rezaei, J. (2016). Best-worst multi-criteria decision-making method: Some properties and a linear model, Omega, 64, pp. 126-130.

The Best Worst Method (BWM) is a multi-criteria decision-making method that uses two vectors of pairwise comparisons to determine the weights of criteria. First, the best (e.g. most desirable, most important), and the worst (e.g. least desirable, least important) criteria are identified by the decision-maker, after which the best criterion is compared to the other criteria, and the other criteria to the worst criterion. A nonlinear minimax model is then used to identify the weights such that the maximum absolute difference between the weight ratios and their corresponding comparisons is minimized. The minimax problem may result in multiple optimal solutions. Although, in some cases, decision-makers prefer to have multiple optimal solutions, in other cases they prefer to have a unique solution. The aim of this paper is twofold: firstly, we propose using interval analysis for the case of multiple optimal solutions, in which we show how the criteria can be weighed and ranked. Secondly, we propose a linear model for BWM, which is based on the same philosophy, but yields a unique solution.

Other major papers

Analytic framework for BWM

Wu, Q., Liu, X., Zhou, L., Qin, J., & Rezaei, J. (2024). An analytical framework for the best–worst method. Omega, 123, 102974.

Since the development of the best–worst method (BWM) in 2015, it has become a popular research focus in multi-criteria decision-making. The original optimization problem of the BWM is a nonlinear min–max model that can lead to multiple optimal solutions, while the linear model of the BWM produces a unique solution. The two models need to be solved by optimization software packages. In addition, although the linear model of the BWM can obtain a unique solution, it produces different feasible regions than the nonlinear model of the BWM, and it changes the objective function. This study aims to solve the nonlinear model of the BWM mathematically to obtain the analytical forms of the optimal solutions. First, we transform the original nonlinear model of BWM into an equivalent optimization model driven by the optimally modified comparison vectors. The equivalent BWM provides a solid basis for computing the analytical solutions. Second, for not-fully consistent pairwise comparison systems, we strictly prove that there is only one unique optimal solution with three criteria, and there might be multiple optimal solutions with more than three criteria. We further develop the analytical forms of these unique and multiple optimal solutions and the optimal interval weights. Third, we develop a secondary objective function to select a unique solution for the BWM. The secondary objective function retains all the characteristics of the original nonlinear model of the BWM, and we find the unique solution analytically. Finally, some numerical examples are examined, and a comparative analysis is performed to demonstrate the effectiveness of our analytical solution approach.

Consistency Thresholds of BWM

Liang, F., Brunelli, M., Rezaei, J. (2020). Consistency issues in the Best Worst Method: Measurements and thresholds, Omega, 96, 102175.

The Best-Worst Method (BWM) uses ratios of the relative importance of criteria in pairs based on the assessment done by decision-makers. When a decision-maker provides the pairwise comparisons in BWM, checking the acceptable inconsistency, to ensure the rationality of the assessments, is an important step. Although both the original and the extended versions of BWM have proposed several consistency measurements, there are some deficiencies, including: (i) the lack of a mechanism to provide immediate feedback to the decision-maker regarding the consistency of the pairwise comparisons being provided, (ii) the inability to consider the ordinal consistency into account, and (iii) the lack of consistency thresholds to determine the reliability of the results. To deal with these problems, this study starts by proposing a cardinal consistency measurement to provide immediate feedback, called the input-based consistency measurement, after which an ordinal consistency measurement is proposed to check the coherence of the order of the results (weights) against the order of the pairwise comparisons provided by the decision-maker. Finally, a method is proposed to balance cardinal consistency ratio under ordinal-consistent and ordinal-inconsistent conditions, to determine the thresholds for the proposed and the original consistency ratios.

Best-Worst Tradeoff Method (BWT)

Nonadditive Best Worst Method

Liang, Y., Ju, Y., Tu, Y., & Rezaei, J. (2022). Nonadditive best-worst method: Incorporating criteria interaction using the Choquet integral. Journal of the Operational Research Society, 1-12.

Mohammadi, M., & Rezaei, J. (2020). Bayesian Best-Worst Method: A Probabilistic Group Decision Making Model, Omega, 96, 102075.

The best-worst method (BWM) is a multi-criteria decision-making method which finds the optimal weights of a set of criteria based on the preferences of only one decision-maker (DM) (or evaluator). However, it cannot amalgamate the preferences of multiple decision-makers/evaluators in the so-called group decision-making problem. A typical way of aggregating the preferences of multiple DMs is to use the average operator, e.g., arithmetic or geometric mean. However, averages are sensitive to outliers and provide restricted information regarding the overall preferences of all DMs. In this paper, a Bayesian BWM is introduced to find the aggregated final weights of criteria for a group of DMs at once. To this end, the BWM framework is meaningfully viewed from a probabilistic angle, and a Bayesian hierarchical model is tailored to compute the weights in the presence of a group of DMs. We further introduce a new ranking scheme for decision criteria, called credal ranking, where a confidence level is assigned to measure the extent to which a group of DMs prefers one criterion over one another. A weighted directed graph visualizes the credal ranking based on which the interrelation of criteria and confidences are merely understood. The numerical example validates the results obtained by the Bayesian BWM while it yields much more information in comparison to that of the original BWM.

Multiplicative BWM

Brunelli, M., Rezaei, J. (2019). A multiplicative best-worst method for multi-criteria decision making, Operations Research Letters, 47(1), 12-15.

This communication examines the best–worst method for multi-criteria decision making from a more mathematical perspective. The central part of this manuscript is the introduction of a new metric into the framework of the best–worst method. This alternative metric does not change the original idea behind the best–worst method and yet it can be shown that it is not only mathematically more sound but also that it ultimately leads to an optimization problem which can be simply linearized and thus solved.

BWM Slides

In this presentation, you can find an explanation of the two models (non-linear BWM and linear BWM) with a numerical example.

BWM bibliographical database

You can also find the BWM bibliographical database here (the list is updated regularly).