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Consider a global company that wants to introduce a new product into several markets at the same time. Several product model alternatives with different features will need to be evaluated before the launch. The company assembles panels of experts in different countries to choose the best model alternative to be released in its markets simultaneously. Typically, multi-criteria decision-making (MCDM) methods are employed for solving these types of decision-making problems since MCDM seeks to identify and select an alternative from a set of alternatives using the preferences of experts in the field. Over the years, several methodologies for MCDM such as Analytic Hierarchy Process (AHP), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), ELimination Et Choix Traduisant la Realité (ELECTRE) and VlseKriterijuska Optimizacija I Komoromisno Resenje (VIKOR) have been developed and successfully applied in several domains. On the other hand, most widespread MCDM methods have also been criticized for several reasons in the literature (Shin et al., 2013). For example, the objective of the TOPSIS method is to identify a compromise solution that is closest to the Positive Ideal Solution and farthest from the Negative Ideal Solution. Then, a ranking index is determined based on these two distances; however, it does not take into account the relative importance or weights of these distances (Kuo, 2017). Furthermore, since TOPSIS employs Euclidean distances, it does not account for correlations which can lead to the influence of information overlap on the results (Xu et al., 2015)). AHP is one of the most popular techniques but also has some drawbacks; it does not take into account interactions between criteria (Mandic et al., 2015). Moreover, when the number of required comparisons increases, the experts tend to become more inconsistent in their choices (Mi et al., 2019). Rezaei (2015) proposed the Best-Worst Method (BWM) for determining the criteria weights which decreases the number of pairwise comparisons significantly and thus may lead to less expert inconsistency. Differently from AHP, the criteria comparisons are only made against the best (most satisfactory) and the worst (least satisfactory) criteria rather than among all possible combinations after determining the best and worst criteria. The best and the worst criteria are accepted as references, and decision-makers conduct only reference comparisons. Compared to the commonly known MCDM methods, BWM offers an optimization-based solution that reduces the number of pairwise comparisons, increases the reliability of weight coefficients, determines the consistency for criteria comparisons, and in the end, it increases the total consistency and ease of practical implications (Stević et al., 2018). Furthermore, when selecting a preferred alternative, researchers resort to other MCDM methods by considering the weights obtained with the BWM in the criterion comparison step. Moreover, recent BWM group decision-making studies such as Safarzadeh et al. (2018) have not allowed for different best and worst criteria by different experts. They entail getting the experts to agree on one best and one worst criterion before applying the BWM. The working principle of the original BWM is illustrated briefly in Fig. 1.

In group decision-making cases with multiple markets, the preferences of experts from different backgrounds can deviate significantly due to cultural and geographical differences. In this article, we extend the BWM framework to reconcile the preference differences among experts. The extension eliminates the necessity for agreeing on only one best and one worst criterion among the experts. Furthermore, we also propose to use a best-worst type approach in the evaluation of the alternatives when determining the scores of the alternatives. The methodology provides a common MCDM framework for both criteria evaluation and alternative selection to decision-makers. It is especially useful for decision-making processes that are conducted for different markets, and it helps global companies to evaluate the opinions of different experts from different countries within a BWM-based group decision-making tool. The proposed method is not limited to using it with different markets, but it can also be utilized for similar markets in which the assessments of the experts differ because of a volatile economy and unstable business environments.

Since its introduction, the BWM saw wide academic acceptance and is being compared to traditional methods like AHP (Mi et al., 2019). It is worth noting that there are other techniques with less number of pairwise comparisons than the BWM such as the Full Consistency Method (FUCOM) (Pamučar et al., 2018) or Level Based Weight Assessment (LBWA) (Žižović and Pamučar, 2019) that were developed in later years. Considering its popularity and its advantages over traditional methods such as AHP, we adopt the BWM in our solution process for the multi-market case and extend it accordingly for criteria weight calculations. We then provide BWM-based optimization models for alternative selection that can handle different preferred alternatives in separate markets. While other ranking techniques could also be adapted to handle the multi-market case we are considering here, we prefer to develop optimization models that are based on the original BWM so that both criteria weight determination and alternative ranking steps of the decision-making process are managed within a similar mathematical framework. Our models allow different experts to have different best and worst criteria or alternatives, and their differences are worked out within the mathematical models we develop.

We quantify the dissimilarity in the opinions of experts/markets by using a novel ratio named “Dissimilarity Ratio ($\textit{DR}$)”. $\textit{DR}$ enables users to check what the level of dissimilarity is, in other words, how the opinions of experts among different markets vary with regard to multiple criteria. $\textit{DR}$ also helps in determining the appropriateness of utilizing the extended BWM proposed here. High $\textit{DR}$ values indicate that using the extended BWM is more meaningful because the expert opinions are diverse. Therefore, before starting to use the extended BWM, $\textit{DR}$ values for all distinct market pairs should be computed. If the values are deemed to be not significantly different, then one could try to get the experts to agree on one best and one worst criterion. If some values are not close to zero, the models proposed here can be used. A summary of this new procedure is given in Fig. 2.

The rest of the study is organized as follows: Section 2 gives the relevant literature review regarding BWM and its applications. Section 3 outlines the details of the alternative selection methodology proposed in this research. Section 4 presents several scenario-based numerical examples to demonstrate the workings of the proposed mathematical models. Section 5 provides a sensitivity analysis of the models to the changes in the number of the significant input parameters followed by a conclusion.

The BWM is a recent addition to the MCDM arsenal that quickly became popular due to its efficiency and increased consistency in pairwise comparisons. Researchers continue to enrich the BWM literature with extensions and new applications. Some recent examples of theoretical contributions are Brunelli and Rezaei (2019) where the authors develop a multiplicative BWM, and Mohammadi and Rezaei (2019) where a probabilistic BWM is provided to handle group decision-making situations to aggregate different decision makers’ preferences to find the optimal criterion weights using a Bayesian hierarchical model. Rezaei et al. (2015) segment the suppliers by using the BWM, and then utilize the results for supplier development. Ahmad et al. (2017) propose a BWM-based framework for supply chain management applications in the oil and gas industry. The model takes into account many external factors such as economic and political stability. Ahmadi et al. (2017) develop a BWM-based framework to reveal the social sustainability of supply chains. van de Kaa et al. (2017) use BWM for the selection of biomass thermochemical conversion technology. A case study is conducted in the Netherlands. Ren et al. (2017) reveal the importance of criteria affecting sustainability assessment of the technologies for the treatment of urban sewage sludge by using the BWM. The TOPSIS method is preferred to complete the selection of the most satisfactory technology among the three alternatives. Guo and Zhao (2017) use a fuzzy BWM in order to handle linguistic terms (statements) of experts that make the process more imprecise. Aboutorab et al. (2018) combine the BWM with Z-numbers to handle uncertain information occurring during the decision-making process. The proposed model is applied to supplier development. Gupta (2018) proposes a model that integrates BWM with VIKOR, and utilizes it to rank the criteria of service quality for airlines and to select the best airline company. van de Kaa et al. (2018) rank the key success factors which have an impact on the substitution of standards with the help of BWM. Nawaz et al. (2018) utilize Markov Chain and BWM for cloud service selection which is a complex process because of the different satisfaction terms of decision makers. Omrani et al. (2018) develop a hybrid model using multi-response Taguchi-neural network-fuzzy best-worst method and TOPSIS to select the best power plant among the alternatives. Rezaei et al. (2018) prioritize the components of the logistics performance index by utilizing BWM. Salimi and Rezaei (2018) develop a model to evaluate R&D performance of 50 companies by using BWM. Shojaei et al. (2018) integrate Taguchi Loss Function, BWM, and VIKOR to evaluate the performance of airports. Kheybari et al. (2019) use BWM to choose the best location for bioethanol facilities in a sustainable way. Liao et al. (2019) propose a hesitant fuzzy BWM-based model to evaluate the success of hospitals, and conduct a comparative study to discuss the pros and cons of the proposed model. Malek and Desai (2019) focus on the barriers to sustainable production. They benefit from the BWM to rank the barriers in their importance. Hashemizadeh et al. (2020) propose a Geographic Information System-based BWM method for the site selection of a solar photovoltaic power plant. Ecer and Pamucar (2020) integrate fuzzy BWM with the traditional Combined Compromise Solution and use the proposed methodology for the selection of a sustainable supplier. Muravev and Mijic (2020) integrate BWM with the Multi-Attributive Border Approximation Area Comparison method to select the most suitable provider. Singh et al. (2021) utilize BWM to rank the enablers that help to apply environmental lean six sigma effectively. A case study is conducted in Indian Micro-Small and Medium Enterprises. In the study of Alidoosti et al. (2021) conversion technologies are measured from a socially sustainable perspective using BWM. Dwivedi et al. (2021) introduce a balanced scorecard model integrated with BWM in an insurance firm. This marks the first application in the insurance domain to evaluate performance across two distinct time periods. In a related study, Rahmati and Darestani (2022) adopt a BWM-TOPSIS hybrid model to determine the weights of criteria within the performance aspects of the balanced scorecard for insurance companies. Bayanati et al. (2022) present a methodology that integrates BWM and fuzzy VIKOR to assess and prioritize companies in the tire industry. The focus is on evaluating environmental risks associated with the industry’s sustainable supply chains. El Baz et al. (2022) explore the incorporation of sustainability factors in the implementation of Industry 4.0 technologies. They use BWM to prioritize sustainability drivers and externalities. In their research, Kheybari et al. (2023) propose a hybrid methodology that takes into account the significance of human health while making decisions related to the temporary locations of hospitals, using the BWM technique.

Recently, the focus of research changed from individual decision-making to group decision-making procedures due to the needs of concurrent engineering practices and multi-disciplinary studies. This shift makes MCDM applications more complex. As a result, finding solutions to group decision-making that can handle the complexity originating from the dissimilarities of expert opinions has become an attractive area for MCDM research. The BWM is also redesigned for group decision-making practices by some researchers. Mou et al. (2016) develop a more structured group decision-making method for the BWM in an uncertain environment with intuitionistic fuzzy multiplicative preferences. They firstly aggregate preferences by using a fuzzy multiplicative weighted geometric aggregation operator and they generate their own algorithm using max-min programming to rank the criteria. Guo and Zhao (2017) propose a fuzzy BWM based on a model that uses the graded mean integration representation. A non-linearly constrained model is established to find the fuzzy weights and select the alternatives. Hafezalkotob and Hafezalkotob (2017) develop a new group decision-making tool to overcome the subjectivity of the preferences by transforming the preferences into fuzzy numbers. The proposed model is helpful to aggregate the preferences of decision-makers from different levels in the organizational hierarchy. The preference degrees of decision makers and criteria are simultaneously handled as fuzzy numbers. Tabatabaei et al. (2019) introduced a novel approach for calculating the global weights of criteria and alternatives. Additionally, they proposed a new consistency ratio and a unified model capable of handling all formulations simultaneously. Haseli et al. (2021) presented a fresh group decision-making approach for the BWM, termed G-BWM, which facilitates the examination of experts’ inclinations in implementing democratic decision-making by leveraging the BWM framework. In the methodology proposed by Dehshiri et al. (2022), new programming approaches were devised to determine the criteria weights, reduce the number of constraints in the regular BWM methods, and combine the aggregation steps to elucidate the importance of criteria in the group decision-making process.

The majority of the previous studies evaluate the alternatives by utilizing other MCDM methods than the BWM such as AHP, TOPSIS, VIKOR, ELECTRE. Interestingly, although the research emphasizes that it is a favourable method for criterion-weighting, the BWM is not used in the alternative selection phase. Furthermore, existing BWM literature assumes that the experts will agree on one best and one worst criterion. However, this reconciliation may be very difficult to achieve for highly diverse markets, or for experts who hold very different opinions on a decision-making problem. Thus, this study makes the following important contributions:

Next, we give the details of our proposed methodology based on the BWM for alternative selection to close the above-mentioned gaps.

In this section, we explain the details of the proposed method. First, an outline is provided.

In this section, we report the results of several numerical experiments conducted to see how the proposed models behave under different scenarios. Computer runs were executed on an Intel i7-1165G7 CPU 2.8 GHz computer with 16 GB of RAM. Exact solutions to the reported problem instances were obtained with GAMS using the CPLEX solver.

For illustration purposes of our methodology, we modify examples in Rezaei (2016). We assume that a car company is going to introduce a new model into three different markets. Four designs with different features are under consideration for the new model. Buyers decide about their purchases based on five criteria: quality, price, comfort, safety, and style. We first give several examples for calculating the criterion weights under different scenarios followed by an example for the selection of the best alternative. In the first example, the markets are dissimilar and fully consistent with similar sales shares; in the second example, it is assumed that the sales shares are different, and finally, the first example is changed to demonstrate inconsistent markets. Note that criterion weights and alternative scores in the examples may not add up to exactly 1 due to rounding errors. Alternative selection example. We use the data for criterion preferences in Example 1. The assumed BW preferences for the four alternatives are given in Tables 20 and 21. In this example, the market shares are taken to be equal. Going through the score determination process using Model (3) the following alternative score matrix is obtained. The criterion weights are the weights in Table 9 using Model (1) with equal forecasted sales shares. The overall values of the four alternatives are then calculated with ${V_{i}}={\textstyle\sum _{j\in C}}{w_{j}}\cdot {v_{i,j}}$ as ${V_{1}}=0.233$, ${V_{2}}=0.232$, ${V_{3}}=0.193$, and ${V_{4}}=0.345$ which indicates that Alternative 4 is the best alternative.

Using Model (4) gives the alternatives’ final values as ${V_{1}}=0.218$, ${V_{2}}=0.239$, ${V_{3}}=0.191$, and ${V_{4}}=0.357$. Observe that while Alternative 4 is still the best alternative, the rankings of the alternatives have changed, and Alternative 2 is now better than Alternative 1. The alternative score matrix obtained from applying Model (4) is as follows.

Note that dissimilarity values and ratios among the markets concerning their alternative assessments can also be calculated using the dissimilarity formulae similar to the calculations in previous examples. Consistency ratios can also be found to see how consistent the experts were in their best-worst assessments of the alternatives. For the sake of brevity, we are not providing the dissimilarity calculations and consistency ratios.

In the previous section, we applied a scenario-based analysis to demonstrate the calculations needed in the proposed models. In this section, we conduct a sensitivity analysis for the models to check that they behave in a consistent manner with regard to changes to parameter values. In Models (3) and (4), the alternatives add yet another dimension for which there are endless scenarios; therefore, we restrict ourselves to reporting the impact of increasing the number of markets and criteria on Models (1) and (2). Since Models (3) and (4) are structurally similar to Models (1) and (2), they are also expected to behave similarly.

The presented numbers in Table 22 were obtained by randomly generating a best and worst (which was different than the generated best) criterion, and the corresponding best-to-others and others-to-worst comparisons on a 1–9 scale for 1,000 instances, and reporting the average objective function values (${z_{2}}$ for Model (1) and ξ for Model (2)) of those instances. The market shares were assumed to be equal. The lowest number of criteria was taken to be 3 to allow at least one other criterion besides the best and worst criteria. In Model (1), the ${z_{2}}$ values increase with the number of markets monotonically whereas they tend to increase with the number of criteria. With Model (2), ξ values behave the opposite way by decreasing.

Table 23 reports the number of times each criterion has the highest (indicated by h in the table) and lowest (indicated by l in the table) weights as calculated by both Model (1) and Model (2). We simulated 1,000 instances with 5 criteria and 3 markets with equal shares. The best and worst choices in each market are randomly determined based on given probabilities by making sure that the best and worst criteria are different from each other. We experimented with two different distributions. In the first scenario, each criterion was equally likely to be chosen whereas in the second case, different probabilities were assigned to the criteria as indicated in Table 23. When there were ties in the highest and lowest weight values, we increased the counts for each associated criterion. Hence, the sums of the counts are larger than 1,000. The distribution of weight rankings for both models more or less follows the probabilities assigned to the criteria. However, Model (2) has more cases compared to Model (1) where the criteria weights are tied, therefore, the counts increase with Model (2).

The sensitivity analysis results validate that the generated model outputs behave as expected.

In contemporary settings, decision-making processes have become increasingly intricate, inundating managers, researchers, and organizations with vast amounts of information in their pursuit of optimal choices. To address this complexity and enhance decision reliability, there has been a shift towards employing group-based methods, involving senior managers alongside multiple experts, rather than relying on individual decision-makers. In this paper, we provide a unified multiplicative MCDM methodology for alternative selection based on the BWM which allows for the experts to have different (dissimilar) best-worst choices and verified its working performance via numerical experiments and sensitivity analysis. Rather than asking the experts to find common ground among themselves before applying the mathematical models, the inclusion of the different best-worst choices directly in the models makes the selection process more efficient as it eliminates lengthy and possibly costly discussions for finding compromised preferences. This process enables the creation of a decision-making process without the requirement of assembling experts in one place. Furthermore, by using a BWM-type methodology both when determining the criterion weights and the alternative scores, we utilize the advantages of the BWM in the whole selection process and thus unify all of the stages in multi-criteria decision-making. A new assessment metric, the so-called dissimilarity ratio, quantifies the dissimilarity of the markets to see how meaningful it is to use the proposed methodology. By employing DR, one can ascertain the suitability of adopting the extended Best-Worst Method. Notably, higher DR values indicate greater significance in utilizing the extended BWM, signifying diverse expert opinions.

Overall, the suggested multiplicative models are novel additions to the quickly growing BWM-oriented research as a group decision-making tool. To extend the proposed methodology in the future, a fuzzy approach can be integrated to the model in order to deal with the vagueness of the linguistic terms in the evaluation of the criteria. Also, it would be interesting to compare the results with traditional MCDM techniques to discuss the advantages of the proposed methodology in terms of consistency and the time spent for running the applications. Moreover, the suggested methodology can be tested not only using numerical values with hypothetical data but also through its application in real-world scenarios in the future. Another future research avenue could be combining the extended BWM-based criterion weight calculations proposed here with other methodologies for ranking the alternatives such as some recent methodologies like CODAS (Keshavarz Ghorabaee et al., 2016), MABAC (Pamučar and Ćirović, 2015), MAIRCA (Pamučar et al., 2014) and RAFSI (Žižović et al., 2020) in the context of multiple markets.