Application of Improved Best Worst Method (BWM) in Real-World Problems

: The Best Worst Method (BWM) represents a powerful tool for multi-criteria decision-making and deﬁning criteria weight coe ﬃ cients. However, while solving real-world problems, there are speciﬁc multi-criteria problems where several criteria exert the same inﬂuence on decision-making. In such situations, the traditional postulates of the BWM imply the deﬁning of one best criterion and one worst criterion from within a set of observed criteria. In this paper, an improvement of the traditional BWM that eliminates this problem is presented. The improved BWM (BWM-I) o ﬀ ers the possibility for decision-makers to express their preferences even in cases where there is more than one best and worst criterion. The development enables the following: (1) the BWM-I enables us to express experts’ preferences irrespective of the number of the best / worst criteria in a set of evaluation criteria; (2) the application of the BWM-I reduces the possibility of making a mistake while comparing pairs of criteria, which increases the reliability of the results; and (3) the BWM-I is characterized by its ﬂexibility, which is expressed through the possibility of the realistic processing of experts’ preferences irrespective of the number of the criteria that have the same signiﬁcance and the possibility of the transformation of the BWM-I into the traditional BWM (should there be a unique best / worst criterion). To present the applicability of the BWM-I, it was applied to deﬁning the weight coe ﬃ cients of the criteria in the ﬁeld of renewable energy and their ranking.


Introduction
In everyday life, we meet and analyze problems to find an optimal solution, i.e., the task of optimization. We meet them almost everywhere-in technical and economic systems, in the family, and elsewhere. The decision-making process and the choice of "the best" alternative is most frequently based on the analysis of more than one criterion and a series of limitations. When speaking about decision-making with the application of several criteria, decision-making may be referred to as multi-criteria decision-making (MCDM) [1,2]. The essence of the problem of MCDM is reduced to the ranking of an alternative from within the considered set by applying specific mathematical tools and/or logical preferences. Finally, a decision is made on the choice of the best alternative, taking into consideration different evaluation criteria. MCDM is an integral part of the contemporary science of decision-making and the science of management and systems engineering, which has broadly

Applications of BWM: A Literature Review
In order to calculate weights of evaluation criteria in an MCDM problem, some MCDM methods can be utilized, such as stepwise weight assessment ratio analysis (SWARA) [12], the analytic hierarchy process (AHP) [13][14][15], the analytic network process (ANP), the full consistency method (FUCOM) [16,17], criteria importance through intercriteria correlation (CRITIC) [18], Entropy [19], level-based weight assessment (LBWA) [20], and so on. As one of the latest weighting methods, BWM is based on pairwise comparisons to extract criteria weights. By only conducting 2n-3 comparisons, as mentioned before, the BWM overcomes the inconsistency problem encountered during pairwise comparisons.
During the past five years, the BWM has already been utilized in numerous real-world problems, such as energy, supply chain management, transportation, manufacturing, education, investment, performance evaluation, airline industry, communication, healthcare, banking, technology, and tourism. Moreover, there are numerous studies in which only the BWM method is used (singleton integration), as well as the papers employing this method together with other methods (multiple integrations).
Van de Kaa et al. [21] used the BWM to compare three communication factors and [22] applied the method to the evaluation of technical and performance criteria in supply chain management. Similarly, [23][24][25] studied the BWM to determine sustainable criteria weights in sustainable supply chain management. Both [26,27] applied the BWM to the selection of the mobile phone. In another study, the BWM was employed to evaluate cars [28]. Ghaffari [29] employed the method to evaluate the key success factors in the development of technological innovation. In addition, [30] applied the BWM in the development of a strategy for overcoming barriers to energy efficiency in buildings. This method is used by [31] to assess the factors influencing information-sharing arrangements. Furthermore, [24] employed the BWM to evaluate the research and development (R&D) performance of firms. Yadollahi et al. [32] applied the BWM in order to prioritize the factors of the service experience in the banking industry. Finally, [33] applied the method to the selection of the bioethanol facility location.
As mentioned above, the BWM has been combined with other robust techniques in order to obtain better results. For instance, fuzzy information and interval values were utilized to integrate with the method. To represent uncertainty in the BWM, [34,35] used fuzzy sets in manufacturing and performance evaluation, respectively. While [36] applied triangular fuzzy sets in performance evaluation, similarly, [37,38] employed the method with the variants of the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method in the supply chain management, the energy sector, and investment, respectively. Furthermore, researchers have integrated the Multicriteria Optimization and Compromise Solution (VIKOR) method with the BWM. For instance, [39][40][41] applied the BWM-VIKOR integration to supplier selection and the green performance of airports, respectively. In another study, [42] proposed a BWM-interval type-2 fuzzy TOPSIS framework for the selection of the most proper green supplier. In order to select a location for wind plants, [43] used the BWM and the MultiAtributive Ideal-Real Comparative Analysis (MAIRCA) integration. Moreover, [44] studied a rough BWM and Simple Aditive Weighting (SAW) approach to wagon selection. In order to assess firms' performance in product development, [45] applied the fuzzy BWM and the fuzzy Analytic Network Process (ANP) methodologies. Another study suggested the fuzzy BWM and the fuzzy COPRAS methodologies for the analysis of the key factors of sustainable architecture [46]. In order to assess and rank foreign companies, [47] proposed the BWM, ELimination Et Choice Translating REality (ELECTRE) III, and Preference Ranking Organization METHod for Enrichment of Evaluations (PROMETHEE) II multi-criteria models. Another study by [48] introduced the interval rough BWM-based Weighted Aggregated Sum Product ASsessment (WASPAS) and Multi-Attributive Border Approximation area Comparison (MABAC) models for the evaluation of third-party logistics providers. An integrated model including the BWM, TOPSIS, Gray Relational Analysis (GRA), and Weighted Sum Approach (WSA) was proposed for turning operations [49]. For web service selection, [50] employed the BWM, VIKOR, SAW, TOPSIS, and COmplex PRoportional ASsessment (COPRAS). Finally, [51] proposed the BWM-based MAIRCA multi-criteria methodology for neighborhood selection.
What is common to all these studies is that they apply the traditional algorithm of the BWM, which implies that one best criterion and one worst criterion are defined through a consensus. In the literature, there are numerous examples of studies implying the defining of criteria weight coefficients irrespective of whether there are one best or worst criterion, or several best or worst criteria [52][53][54][55]. In such studies, the algorithm of the traditional BWM would not be able to provide objective results, since it requires the adaptation of experts' preferences to one best/worst criterion. For that reason, the BWM-I that eliminates this problem and enables us to define criteria weights through a realistic perception of experts' preferences has been developed in this paper. The algorithm of the BWM-I is presented in the following section.

Improved Best Worst Method (BWM-I)
The BWM-I provides decision-makers with the possibility of choosing as many best/worst criteria as there are in the real decision-making problem. The determination of evaluation criteria weight coefficients by the application of the BWM-I implies the following steps: Step 1. Defining a set of evaluation criteria C = {c 1 , c 2 , . . . c n }, where n represents the total number of the criteria.
Step 2. Determining the best and the worst criteria, i.e., as many best and worst criteria as there are in the decision-making model. Simultaneously, m b and m w denote the number of the best and the worst criteria in the model, respectively.
Step 3. Determining the advantages of the best criterion/criteria from within the set C over the other criteria. A 9-degree numeric scale is used to determine the advantage(s). If the criteria C 1 and C 2 are marked as the best criterion, then an improved best-to-others vector (M-BO) is obtained by the application of expression (1), namely: where a Bn represents the advantage of the best criterion B over the criterion j, and m b represents the number of the best criteria in the model, whereas a BB = 1. It is clear that for m b = 1, expression (1) transforms into a classical best-to-others (BO) vector, as in the traditional BWM.
Step 4. Determining the advantages of all the criteria from within the set C over the worst criterion/criteria. In order to determine the advantage(s), as in Step 3, a 9-degree numeric scale is used. If we mark the criterion C n−1 and the criterion C n , i.e., m w = 2, as the worst criterion, then a modified others-to-worst vector (M-OW) is obtained by the application of expression (2), as follows: where a jW represents the advantage of the criterion j over the worst criterion W, m w represents the number of the worst criteria in the model, whereas a WW = 1. For m w = 1, expression (2) transforms into a classical OW vector, as in the traditional BWM.
Step 5. Calculating the optimal values of the weight coefficients of the criteria from within the set C, (w * 1 , w * 2 , . . . , w * n ). Since the BWM algorithm defining weight coefficients in the case when there is one or more than one best and/or worst criterion/criteria (i.e., m b ≥ 1 and m w ≥ 1) is considered here, the postulates for solving the optimization model must be defined.
The optimal values of weight coefficients are obtained once the condition stipulating that for each pair w B /w j and w j /w W , it is applicable that w B /w j = a Bj and w j /w W = a jW is met. Since we are considering the case where m b ≥ 1 and/or m w ≥ 1, it is necessary that the mentioned conditions should be revised, so there is the condition that w B /w j = m b a Bj and w j /w W = m w a jW , where the weight coefficients w B and w W represent the weights of the unique best and the unique worst criteria. The unique best and worst criteria (C B and C W ) represent all the criteria that are marked as the best and the worst criteria in the set C = {c 1 , c 2 , . . . c n }. In addition, since w B /w W = m b a BW /m w , we obtain It arises from the aforementioned factors that the weight coefficient of the unique best criterion (w B ) represents the sum of all the weight coefficients of the criteria that are marked as the best criteria in the set C = {c 1 , c 2 , . . . c n }, i.e., where w l represents the weight coefficients of all the criteria in the set C = {c 1 , c 2 , . . . c n } that are marked as the best criteria, whereas b represents the total number of the best criteria from the set C.
The unique worst criterion is defined similarly. The weight coefficient of the unique worst criterion (w W ) represents the sum of all weight coefficients of the criteria that are marked as the worst criteria in the set C = {c 1 , c 2 , . . . c n }, i.e., where w k represents the weight coefficients of all the criteria that are marked as the worst criteria in the set C = {c 1 , c 2 , . . . c n }, and v represents the total number of the worst criteria from within the set C. Since the optimal values of weight coefficients should meet the condition stipulating that the maximum absolute values of the differences should be The model presented in (5) is equivalent to the following model.
Should m b > 1 and/or m w > 1, then the total number of the criteria in the model is reduced (decreases) by the introduction of the unique best and the unique worst criteria. Then, we obtain a smaller number of comparisons, i.e., the total number of comparisons in the model is reduced It is clear that, should m b = m w = 1, the models (5) and (6) transform into the classical optimization BWM model [11]. Example 1. If a set of eight criteria C 1 , C 2 , . . . , C 8 is observed, in which there are two best and two worst criteria; if we know that the criteria C 1 = C 2 are marked as the best, then the unique best criterion (C B ) that represents both criteria in model (6) is introduced. If the criteria C 7 = C 8 are marked as the worst, then the unique worst criterion (C W ) represents the criteria C 7 and C 8 and in model (6). Then, the total number of the criteria in the model is reduced to six, since C 1 = C 2 = C B and C 7 = C 8 = C W . Thus, the total number of comparisons in pairs of criteria is reduced from 15 to 13.
Should m b > 1 and/or m w > 1, then, based on conditions (3) and (4), it follows that by solving model (6), the values of the weight coefficients of the best criterion and the worst criterion increased by the number of the best and the worst criteria are obtained. Therefore, after solving model (6), the obtained values of the weights w B and w W need to be divided by m b and m w in order to obtain the final values of the weight coefficients of the best and the worst criteria. For example, if m b = m w = 2, the final values of the best and the worst (w * B and w * W ) criteria obtained are w * The values of the weights of the remaining criteria remain unchanged, and they are taken from the solution to model (6).
In order to more easily understand the algorithm of the BWM-I, the following part is dedicated to solving a simple example including five criteria taken from a study by [28]; then, a complex model implying the defining of the weight coefficients of a total of the 28 criteria grouped into six clusters is considered in the case study (Section 3).

Example 2.
While buying a car, the buyer applies five criteria for the evaluation of the alternative (the car): Quality (C1), Price (C2), Comfort (C3), Safety (C4), and Style (C5). The buyer has the evaluated criteria per the algorithm of the traditional BWM, as shown in Table 1. Table 1. The best-to-others and others-to-worst pairwise comparison vectors.

Best-to-Others Vector Others-to-Worst Vector
Best: C2 and C4 Evaluation Worst: C5 Evaluation Based on the data accounted for in Table 1, it is possible to conclude that the buyer considers the criteria Price (C2) and Safety (C4) as the most significant, whereas the criterion Style (C5) is rated as the least significant.
The problem that appears here cannot be solved through the application of the traditional BWM, which requires the defining of the unique best and worst criteria. If we were to insist on the defining of the unique best criterion (as is required by the traditional BWM), then we would have to revise the BO vector to define a single best criterion. However, by doing so, we would exert an influence on the buyer's preferences, i.e., the buyer would not express his real preferences. Those revised preferences would further exert an influence on a non-objective choice of alternatives, which should be avoided. If the expert (in this case, the buyer) requires a high degree of rationality during the evaluation of the criteria, the multi-criteria decision-making methods also need to be used as support to such rational decision-making in order to meet that very same condition. Therefore, since it was impossible to apply the traditional BWM, the BWM-I was applied.
Based on the data from Table 1, we conclude that the number of the best criteria is m b = 2, whereas the number of the worst criteria is m w = 1. Based on that and expression (4), it is possible to define the model for the calculation of the optimal values of the weight coefficients of the BWM-I as follows: By solving the presented model, the values of the weights w B = 0.7088, w W = 0.0400, w * 1 = 0.1656, and w * 3 = 0.0856, as well as ξ = 0.140, are obtained. Based on condition (3), we obtain w * .0400 is obtained. So, the optimal values of the weight coefficients w j = (0.1656, 0.3544, 0.0856, 0.3544, 0.0400) T are obtained characterized by a high consistency ratio: Had the model of the traditional BWM [27] been applied to the presented example, optimization model (8) By solving model (8), the following vectors of the weight coefficients w j = (0.1638, 0.3505, 0.0847, 0.3616, 0.0396) T and ξ = 0.1401 are obtained. Based on the results obtained, we perceive that even though there is the defined condition that both best criteria (C2 and C4) are of the same significance, the values of the weight coefficients are different (w 2 w 4 ), i.e., w 2 = 0.3505 and w 4 = 0.3616. The different values of the weight coefficients of the criteria C2 and C4 are a consequence of undermining the condition of the transitivity of relations between criteria. This is confirmed by the value of the consistency ratio (CR), which is CR = 0.026, just as in model (7).
The shown example has demonstrated that the traditional BWM model can be applied to the determination of the weights of a larger number of the best/worst criteria, but only in the case when the consistency ratio is ideal, i.e., when CR = 0.00. However, we may realistically expect that more than one best/worst criterion and the value CR > 0 will appear in solving real-world problems, especially those with a greater number of criteria. In such cases, the BWM-I is inevitably applied. Given the fact that the BWM-I is capable of transforming itself into the traditional BWM (in the case when m b = m w = 1), its application is also logical for a future objective perception of and solving real-world multi-criteria problems.

Case Study: The Application of BWM-I
In this chapter, the application of the BWM-I in solving a renewable energy source evaluation problem implying the existence of a larger number of the best/worst criteria within the framework of the dimensions/criteria is presented. The most common criteria for a renewable energy source evaluation involve technical, environmental, social, risk, political, and economic aspects. Thus, we introduce a six-dimensional model in order to define the weights of the drivers for renewable energy sources, as shown in Figure 1, in which several criteria are considered for each dimension. The six dimensions are technical (C1), economic (C2), social (C3), environmental (C4), risk (C5), and political (C6); each dimension comprises three to six criteria. Moreover, the criteria for the evaluation of renewable energy sources were achieved by reviewing the existing literature [56][57][58][59][60][61][62][63][64]. Consequently, the evaluation comprised of six dimensions and 28 criteria. The criteria and their descriptions are listed in Table 2.

Main Criteria Sub-Criteria Code Definition References
Technical (C1) Efficiency C11 How technology is widespread at the regional, national, and international levels.
[ [57][58][59] Reliability C12 An energy system's ability to perform the required functions [56,58,60] Resource reserves C13 The availability of the energy source to generate energy [58] Technology maturity C14 The penetration of a specific technology in the energy mix at the regional, national, and international levels. [58,60] Safety of the system C15 The security of the workers and the local community

Main Criteria Sub-Criteria Code Definition References
Technical (C1) Efficiency C11 How technology is widespread at the regional, national, and international levels. [57][58][59] Reliability C12 An energy system's ability to perform the required functions [56,58,60] Resource reserves C13 The availability of the energy source to generate energy [58] Technology maturity C14 The penetration of a specific technology in the energy mix at the regional, national, and international levels. [58,60] Safety of the system C15 The security of the workers and the local community [56]

Main Criteria Sub-Criteria Code Definition References
Economic (C2)

Investment cost C21
All costs of products and services, except for the costs of labor or the cost of equipment maintenance [56,[58][59][60] Operation and maintenance cost C22 Operating the energy system adequately, as well as the costs related to the maintenance of the energy system [56,58] Return of investment C23 The time required to recover the investment [56,58] Energy cost C24 The cost of the energy-generating system [60,63] Operational life C25 The period during which the power plant can operate before being decommissioned [56] R&D cost C26 The expenses incurred for the R&D of technological innovations [65] Social (C3)

Social acceptance C31
The opinions of residents, local authorities, and other stakeholders on an energy project [56][57][58] Job creation C32 Jobs created per unit of the energy produced [57,58,61] Social benefits C33 The contribution of an energy system to the improvement and advancement of local society [56,58] Noise C34 The noise generated during the lifecycle under consideration [62] Visual impact C35 The aesthetics of the installations of the energy system [62] Environmental (C4) The dependency of countries on international legislations [57,58] Compatibility with the national energy policy

C62
The national energy policy related to renewable energy sources [58] Compatibility with the public policy C63 Voluntary agreements and general codes of conduct in line with national priorities [64] Government support C64 Approving and adapting to renewable energy sources. [64] After defining the set of the evaluation criteria, the following steps of the BWM-I (Steps 3 and 4) imply the formation of the M-BO and M-OW vectors of the dimensions/sub-criteria, as shown in Table 3.
Technical sub-criteria Economic sub-criteria Best: C21, C22 and C24 Preference Worst: C23 Preference Social sub-criteria Environmental sub-criteria Best: C43 and C45 Preference Worst: C41 and C44 Political sub-criteria Table 3 enables us to note that in some M-BO and M-OW vectors, there are several best and worst criteria. So, based on the M-BO and M-OW dimensions, we notice the existence of one best criterion (Environmental-C4), whereas there are two worst criteria (Risk-C5 and Political-C6). In the Economic Sub-Criteria group, there are three best criteria (Investment cost-C21, Operation and maintenance cost-C22, and Energy cost-C24) and one worst criterion (Return of investment-C23). In the Social Sub-Criteria group, there is one best criterion (Social acceptance-C31) and two worst criteria (Noise-C34 and Visual impact-C35). The Environmental Sub-Criteria group is characteristic, since it contains two best criteria (Impact on the environment and humans-C43 and Climate change-C45) and two worst criteria (GHG Emissions-C41 and Water use-C44). In the Political Sub-Criteria group, there are two best criteria (Compatibility with the national energy policy-C62 and Compatibility with the public policy-C63) and one worst criterion (Government support-C64). In the remaining sub-criteria groups (the Technical Sub-Criteria and the Risk Sub-Criteria), there are the unique best and worst criteria, for which reason the traditional postulate of the BWM is used to define the weight coefficients of these sub-criteria groups.
Based on the M-BO and M-OW vectors (Table 3), the optimization models for the calculation of the weight coefficients of the dimensions/sub-criteria were defined. A total of seven BWM-I models were defined, some of which are shown in the next part.
By solving the presented models, the optimal values of the weight coefficients of the dimensions/sub-criteria are obtained, as shown in Table 4. In Table 4, the global and local values of the weight coefficients of the criteria are presented. The global weights of the criteria were obtained by multiplying the weight coefficient of the dimension with the weight coefficients of the sub-criterion. By solving model (6), the values of ξ * , which are ξ * C1−C6 = 0.6277, ξ * C11−C15 = 0.2984, ξ * C21−C26 = 0.3542, ξ * C31−C35 = 0.3542, ξ * C41−C45 = 0.8939, ξ * C51−C53 = 0.2087, and ξ * C61−C64 = 0.2087 were obtained. The values of ξ * are used to determine the consistency ratio, as shown in Table 5. The analysis of the results of the BWM-I from Table 5 allows us to conclude that the values of the consistency ratio are satisfactory [27].
According to the findings shown in Table 4, the environmental dimension is determined to be the most crucial dimension, with the significance of 0.3972, only to be followed by the economic and technical dimensions, with the comparative weights of 0.2823 and 0.1674, respectively. According to Figure 1, in the pairwise comparison of the evaluation criteria, both "Impact on the environment and humans" and "Climate change" ranked as the priority factor from the environmental aspect, only to be followed by "Land use". Furthermore, the three criteria (Investment cost, Operation and maintenance cost, and Energy cost) ranked the first in the ranking related to the economic dimension. "Technology maturity" and "social acceptance" were the most important criteria in terms of technological and social dimensions, respectively. Overall, according to the global weights, the most important criteria were "Climate change" (0.1199), "Impact on the environment and humans" (0.1199), "Land use" (0.1084), and "Technology maturity" (0.0716), indicating that the Climate change, Impact on the environment and humans, Land use, and Technology maturity criteria represent the four most crucial evaluation criteria for the determination of a suitable renewable energy source.
In order to show the sensitivity analysis of the BWM-I model, in the next section, we simulated the changes in the input parameters of the BO and OW vectors. In each group of criteria, another best or worst criterion was added, while the values of the remaining criteria in BO and OW vectors remained unchanged.
In the Dimensions group, two best criteria were selected (C4 and C2), while the remaining values of the criteria remained unchanged. In the Technical Sub-Criteria group, two criteria-C12 and C11-were selected as the worst criteria. In the Economic Sub-Criteria group, in addition to the three best criteria, the two worst criteria were selected (C23 and C25). In the Social Sub-Criteria group, two best criteria, C31 and C32, were added to the input BO and OW vectors. In the Risk Sub-Criteria group, in addition to the best criterion C51 and criterion C52, it was selected as the best criterion. In the Political Sub-Criteria group, in addition to C64, criterion C61 was also chosen as the worst criterion. After the implementation of these changes, the results shown in Figure 2 were obtained. dimension. "Technology maturity" and "social acceptance" were the most important criteria in terms of technological and social dimensions, respectively. Overall, according to the global weights, the most important criteria were "Climate change" (0.1199), "Impact on the environment and humans" (0.1199), "Land use" (0.1084), and "Technology maturity" (0.0716), indicating that the Climate change, Impact on the environment and humans, Land use, and Technology maturity criteria represent the four most crucial evaluation criteria for the determination of a suitable renewable energy source.
In order to show the sensitivity analysis of the BWM-I model, in the next section, we simulated the changes in the input parameters of the BO and OW vectors. In each group of criteria, another best or worst criterion was added, while the values of the remaining criteria in BO and OW vectors remained unchanged.
In the Dimensions group, two best criteria were selected (C4 and C2), while the remaining values of the criteria remained unchanged. In the Technical Sub-Criteria group, two criteria-C12 and C11-were selected as the worst criteria. In the Economic Sub-Criteria group, in addition to the three best criteria, the two worst criteria were selected (C23 and C25). In the Social Sub-Criteria group, two best criteria, C31 and C32, were added to the input BO and OW vectors. In the Risk Sub-Criteria group, in addition to the best criterion C51 and criterion C52, it was selected as the best criterion. In the Political Sub-Criteria group, in addition to C64, criterion C61 was also chosen as the worst criterion. After the implementation of these changes, the results shown in Figure 2 were obtained.   By analyzing the results from Figure 2, we notice that the model is sensitive to changes in the number of best and worst criteria in the input data. Despite the changes in the input data, the degree of consistency of the considered models remained within acceptable limits. The authors believe that the presented analysis shows the stability and robustness of the modified BWM methodology.

Managerial Implications
Integrating some methods into decision-making methodologies will make a significant contribution to the particular body of knowledge. Furthermore, it is valuable that the existing methods are made more efficient by completing their deficiencies. In decision theory, MCDM methods are utilized to solve many real-world problems. Improvement and development of the functionless side of an existing approach is always appreciated to continuously improve this branch of operations research, because businesses, politicians, researchers, and industries need such arrangements to make more reliable decisions.
The aim of this paper is pertinent to the fact that the BWM method, which is one of the new approaches in the field of MCDM, is ineffective if there is more than one best/worst criterion. Thus, this work suggests a novel strategy to solve an MCDM problem via some specific modifications to the main structure of the traditional BWM method. As a result, decision-makers will be able to easily cope with the problem of more than one best/worst criteria often encountered in real-world problems. Furthermore, by making fewer pairwise comparisons (only 2n-5), they will not only have to deal with the problem of inconsistency but also save time. Therefore, it is as well believed that the present article will give a different point of view for future works.
The presented methodology eliminates deviations in expert preferences that occur as a consequence of adapting to the traditional BWM algorithm. The previous analysis showed apparent advantages, so it is expected that the proposed methodology will be accepted by the management when solving real-world problems. Most decision-makers readily accept tools that are logical and easy to understand. The BWM-I methodology can be included in the category of easy-to-understand decision-making tools. In particular, it is expected to be accepted and used by decision-makers who know the algorithm of the traditional BWM, as well as its advantages and disadvantages. In addition, the use of the BWM-I methodology as part of the set of tools that make up the decision support system will make it more acceptable to management structures. This tool will be acceptable for managers who require a more realistic view of the mutual relations between the criteria, as well as a realistic and rational view of expert preferences.
A few insights are extracted to increase the applicability of the proposed BWM-I methodology in real cases. Thereby, the implications are as follows: • By preferring the BWM-I model, authorities can make more accurate decisions.

•
Since the weight of each criterion is found according to the opinions of decision-makers, firms can improve their evaluation process through the BWM-I approach.

•
Firms can create a better competitive advantage over their business competitors by determining the best alternatives with the BWM-I model.
Knowing that the decision-making process is accompanied by greater or lesser uncertainties caused by a dynamic environment, such a system eliminates further adjustment and deviation of expert preferences. As a result of this feature, the demonstrated methodology can help companies establish a rational, systematic approach to evaluating the internal and external factors that affect their business. The flexibility of the methodology in terms of reducing the number of pairwise comparisons is also valuable. It is expected that the flexibility of the BWM-I methodology will enable its application in complex studies in which criteria and expert preferences differ and in which no consensus is required in expert preferences.

Conclusions
The BWM method represents a very powerful tool for multi-criteria decision-making and defining criteria weight coefficients. Generally viewed, while solving real-world problems, there are specific multi-criteria problems in which there are several criteria with the same influence on decision-making. The traditional postulate of the BWM implies that while defining priority vectors (BO and OW), one best criterion and one worst criterion should be chosen from within a set of the observed criteria. Then, the criteria are compared in pairs by defining the best-to-others (BO) and others-to-worst (OW) vectors. While defining the BO and OW vectors, the decision-maker may assign the same criteria preferences while comparing the BO and OW, which means that there may be several criteria with the same significance. However, the traditional BWM does not permit the defining of several best/worst criteria that will have the same significance, although it is frequently the case in real-world problems. As a result of that, by applying the traditional BWM, decision-makers are required to define one best/worst criterion should they believe that there are two or more best/worst criteria. In that way, the decision-maker's preferences are distorted to a certain extent, and no objective results are obtained. Should the small flexibility of the 9-degree scale be added to that as well, then the obtained values of criteria weights may significantly deviate from the preferences expressed by the decision-maker.
In this paper, the improvement of the traditional BWM is presented. The improved BWM (BWM-I) eliminates the shortcomings of the traditional BWM. It offers a possibility for decision-makers to express their preferences even in the cases when there is more than one best and worst criterion. The BWM-I was successfully tested on two examples in this paper. In the first example in Section 3, a case in which there are two best criteria is presented. The algorithm of the traditional BWM and the BWM-I was also applied to the same example. It was shown that the BWM-I has greater flexibility in expressing experts' preferences in relation to the traditional BWM. In the second example (Section 4), the BWM-I was applied to the defining of the weight coefficients of the criteria in the field of renewable energy and their ranking. In the presented example, all of the 28 criteria grouped into the six dimensions were subjected to evaluation. Through a combination of the seven models of the BWM-I, the advantages of the developed model and the possibilities of the objective processing of experts' preferences are demonstrated.
In comparison with the traditional BWM, the proposed BWM-I has several advantages according to the following: (1) Due to non-determinedness and imprecision in data, it is realistic that more than one best and/or worst criterion/criteria with the same significance may appear in experts' preferences. The BWM-I enables a realistic expression of experts' preferences irrespective of the number of the best/worst criteria in a set of evaluation criteria. (2) In case more than one best and worst criterion appear (m b > 1 and m w > 1) in the decision-making process, the application of the BWM-I reduces the number of comparisons from 2n-3 (in the traditional BWM) to 2n-5 (in the BWM-I). In that manner, the possibility of making a mistake while conducting a pairwise comparison of the criteria is also reduced, which further exerts an influence on the greater reliability of results.

Future Research
The proposed BWM-I represents a tool that is capable of being successfully integrated with other MCDM techniques. The development of the hybrid multi-criteria models for group decision-making that would be based on the integration of the BWM-I into other MCDM tools represents one of the future directions of its application. The second logical step for the future improvement of the BWM-I is its application in an uncertain environment, such as fuzzy, rough, grey, neutrosophic, and so on [67,68]. In the last few years, numerous linguistic approaches, such as the expansions of linguistic variables in a neutrosophic environment and the unbalanced linguistic approach, have been developed. The mentioned approaches have attracted considerable attention to the decision-making field through the possibility of applying linguistic variables in the decision-making process. Connecting these linguistic approaches with the BWM-I and research into the possibility of the linguistic modeling of preferences are interesting and promising topics in future research.
Author Contributions: Conceptualization, methodology, validation, D.P. and F.E.; writing-original draft preparation, review and editing, D.P., F.E., G.C. and M.A.A. All authors have read and agreed to the published version of the manuscript.