A Fuzzy Best-Worst Multi-Criteria Group Decision-Making Method

This paper proposed a novel fuzzy best-worst multi-criteria group decision-making method to solve the group decision-making (GDM) problem with multi-granular linguistic approach, which is an effective and promising technique to tackle this issue. In the proposed method, the selectable multi-granularity linguistic term sets (LTS) are firstly provided for experts to expressed their individual assessment information. Then, the improved fuzzy BWM is employed to calculate the weights of criteria with the form of fuzzy numbers. In current several studies using the BWM for group decision-making, only two unified best and worst criteria are given, which cannot reflect the evaluation of the best and worst criteria by different experts, resulting in the omission of information. Moreover, the difference between the best and worst criteria initially given and the experts’ ideas will cause the experts to be inaccurate in the comparison of each criterion. Therefore, in this paper, in order not to omit too much information, each expert will determine the best and the worst criteria. The evaluation information of each expert is integrated into two comparison vectors according to the transformation formula proposed in this paper. What’s more, an improved input-based consistency measurement is proposed, which can provide the DMs with a clear guideline on the revision of the inconsistent judgement(s). Finally, two examples are given to illustrate the effectiveness and applicability of the proposed method.


I. INTRODUCTION
People rank alternatives to express the importance, preference, or likelihood of a decision. As a very important branch of decision-making theory, the multi-criteria decisionmaking (MCDM) problems are gaining momentum in a wide range of real-world situations, such as engineering, economics, and management [1]- [5]. The essence of MCDM is the sorting of all the alternatives and then the selection of optimal one by employing certain approach and existing decision information with consideration of different criteria [6]. So, before alternatives are ranked, the decision criteria system and the importance (weight) of the criteria need to be given. After that, the performance of the alternatives with respect to the criteria can be constructed, which is called The associate editor coordinating the review of this manuscript and approving it for publication was Oussama Habachi . the performance matrix. Finally, different techniques can be applied to solve the problem.
To date, considerable MCDM techniques have been conducted to find efficient ranking decisions based on the preferences of the decision makers, such as TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) [7], [8], ANP (the Analytic Network Process), ELECTRE (Elimination and Choice Translation Reality) [9], [10], VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) [11], AHP (the Analytic Hierarchy Process) [12], and BWM (best-worst method) [13]. In particular, the AHP and the BWM are both pairwise comparison-based MCDM methods, but the BWM needs only 2n-3 pairwise comparisons and the AHP needs n(n-1)/2 pairwise comparisons, which makes the BWM a more data efficient method compared to AHP.
Despite these considerable techniques, due to the limited expertise, estimation inaccuracies and knowledge lack of decision makers, the application of fuzzy information to VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ reflect the decision information is an excellent way in many practical MCDM problems. Meanwhile, many fuzzy-based MCDM methods have been proposed and widely used in recent years, such as fuzzy TOPSIS [14], fuzzy ELEC-TRE [15], and fuzzy BWM [6], [16]. However, as the increasing complexity of the decisionmaking environment, many organizations have moved from a single decision maker to a group of experts to accomplish problems. In order to consider the evaluation preferences of different experts, different multi-granularity linguistic term sets are proposed. Multi-granular fuzzy linguistic modeling has been frequently used in group decision making field due to its capability of allowing each expert to express his/her preference using his/her own linguistic term set [17]- [20]. For example, Francisco et al. proposed the first fusion method to handle multi-granularity linguistic information with the use of fuzzy set theory [20]. Zhen Zhang and Chonghui Guo focused on dealing with multi-granularity uncertain linguistic group decision making problems with incomplete weight information. The uncertain linguistic evaluation information of each decision maker was transformed to trapezoidal fuzzy numbers, and then two optimization models were established to minimize the deviation between each decision maker's evaluation and the group's collective evaluation on each alternative [19]. Francisco and Luis expressed the linguistic information by means of 2-tuples, which are composed by a linguistic term and a numeric value assessed in [0.5,0.5) [21]. Moreover, Herrera and Martínez conducted linguistic hierarchies term sets, and applied 2-tuple linguistic representation model to unify multi-granularity information without loss of information to solve GDM problems with multi-granularlity linguistic information.
About the BWM in GDM, there are some recent developments [18], [22]- [26]. Reference [16] obtained the final set of criteria through discussion when selecting the best and worst criteria, but this is the result of the compromise of different experts, which will cause information loss in the stage of selecting the best and worst criteria. Furthermore, since experts need to re-accept a new set of best and worst criteria, it is easier to cause inconsistencies in the values obtained from subsequent comparisons between criteria. In addition, many researches conduct the consistency test after the final result (such as criteria weight and criteria performance) has been calculated. For inconsistent results, this will increase the unnecessary workload of the calculation.
In this study, we propose a fuzzy best-worst multi-criteria group decision-making method. The main contributions of this study include: (1) It introduces multiple experts into the fuzzy BWM method and allows them to use different Linguistic Term Set (LTS) to express the different assessments of the linguistic variables; (2) In the selection of the best and worst criteria and the subsequent comparison and scoring, the method proposed in this paper allows the experts to score according to their own judgments of the best and worst criteria, and finally uses mathematical formulas to carry out the full information conversion without information loss; (3) Several methods to integrate the opinions of experts and the advantages and disadvantages of each method are proposed, and the geometric average method is used to integrate the fuzzy set of the experts' language conversion, which presents the opinions of each expert completely and avoids the extreme value to influence the whole situation; and (4) a pre-emptive fuzzy consistency test method is constructed to test the consistency of expert opinions before mathematical model operation. This method can reduce the calculation amount compared with the traditional consistency test which needs to be modified and recalculated after the inconsistency occurs.
The remainder of this paper is organized as follows. In Section II, preliminaries about the multigranular fuzzy linguistic GDM problems and the consensus process are presented. The proposed model is described in detail in Section III. In section IV, the input-based consistency ratio is selected and extended so that the inconsistency of each expert can be corrected in time by pre-comparison. In Section V, two examples are given to illustrate the effectiveness and applicability of the proposed method, while in Section VI, we draw the conclusions.

A. TRIANGULAR FUZZY NUMBERS (TFN)
There are many researches on fuzzy set and fuzzy degree, such as triangular fuzzy number, trapezoidal fuzzy number and polygonal fuzzy number. This paper uses triangular fuzzy number and gives the sorting formula of triangular fuzzy number Definition 1: Given two TFNã 1 = (l 1 , m 1 , n 1 ),ã 2 = (l 2 , m 2 , n 2 ) and any positive real number λ. Let ⊕ and ⊗ denote the extended addition and multiplication operation defined by the extension principle, and two main operations ofã 1 andã 2 can be expressed as follows [27]: Definition 2: Let the graded mean integration representation (GMIR) R(ã) of a TFNã represent the ranking of triangular fuzzy number [28]- [30]. Letã i = (l i , m i , n i ), and the GMIR R(ã i ) of TFNã i can be calculated by

B. THE SIMPLIFICATION OF COMPARISON
Suppose there are n criteria for a research object, and the fuzzy pairwise comparisons on these n criteria can be performed based on the linguistic variables (terms) of decision makers [6]. Then, the fuzzy comparison matrix can be obtained as follows: whereã ij represents the relative fuzzy preference of criterion i to criterion j, which is a triangular fuzzy number;ã ij = (1, 1, 1) when i = j. And as the same as the Best-Worst method (BWM), the FBWM has proved that the number of fuzzy reference comparisons are just 2n − 3.
To explain the above 2n − 3 comparisons, it can divide the pairwise comparisons into two main categories, namely reference comparisons and secondary comparisons [13].
Definition 3: Comparisonã ij is defined as a reference comparison if i is the best element and/or j is the worst element.
Definition 4: Comparisonã ij is defined as a secondary comparison if i nor j are the best or the worst elements.
For FBWM, there are n − 2 'Best-to-Others' fuzzy comparisons, n − 2 'Others-to-Worst' fuzzy comparisons and one 'Best-to-Worst' fuzzy comparison. So, it can conclude that there are 2n − 3 reference comparisons, and the rest pairwise comparisons are secondary comparisons.
As discussed above, the secondary comparisons are executed based on the knowledge about the reference comparisons. Each secondary comparisonã ij appears in two relation chains, two members of which are reference comparisons: a Bi ⊗ã ij =ã Bj ,ã ij ⊗ã jW =ã iW , which also means that a ij =ã Bj /ã Bi ,ã ij =ã iW /ã jW (i = j). (1) The set is ordered: (2) There is a negation operator: Neg (s In this study, we assume that different experts use the linguistic term set with different granularities to establish the parameters of the triangular membership functions.

D. TRANSFORMING LINGUISTIC INFORMATION INTO FUZZY NUMBERS
In order to apply fuzzy BWM and its related operations to GDM problems, the linguistic terms are usually transformed to fuzzy numbers. The linguistic term s q i in set S q , q = 1, 2, . . . , h can be approximately expressed in the following triangular fuzzy number: where l q i = 1 + 8 * max((i − 1)/T q , 0), m q i = 1 + 8 * i/T q and n q i = 1 + 8 * min((i + 1)/T q , 1). The membership function of triangular fuzzy number is where µ˜s q i (x) is the degree to which the value x belongs tos q i . Some multi-granularity linguistic term sets and the rules of transformation are listed in Table 1, 2 and 3, respectively.

III. THE PROPOSED METHOD
Let E = {E 1 , E 2 , . . . , E l }(l ≥ 2) be the set of decision makers, C = {c 1 , c 2 , . . . , c n }(n ≥ 2) be a finite set of decision criteria, where E i (i ∈ {1, 2, . . . , l}) and c j (j ∈ {1, 2, . . . , n}) denote the ith expert and the jth criteria. We assume that the experts are identical in equal importance and use multi-granularity linguistic term sets S 1 , S 2 , . . . , S h to express their preference information, where S l denote the lth preestablished finite and totally ordered linguistic term set with T l + 1 cardinalities, i.e. S q = {s q 0 ,s q 1 ,s q 2 , . . . ,s q T q }.
Step 1: Define the decision criteria system. The decision criteria system consists of a set of decision criteria, which is very important to make a decision on alternatives. Suppose there are n decision criteria {c 1 , c 2 , . . . , c n }. Then we can denote that whereÃ q represent the qth decision maker's pairwise comparison matrix,ã q ij show the relative of criterion i to criterion j. AndÃ represents the vector of the ith index compared to other indexes. Furthermore, the pairwise comparison matrixÃ q is considered to be perfectly consistent if:ã Step 2: Determine the best and the worst criterion. Based on the built decision criteria system, decision makers are asked to determine the best and the worst criterion. The best criterion for qth decision maker is represented as c q B , and the worst criterion for qth decision-maker is labeled as c q W . After which we can identify the two best and worst criteria that experts choose the most, named c B m and c W m . Note that if all the experts choose different criteria for the best and worst, we can choose either of them as c B m and c W m . However, it should be noted that the two indicators here are not necessarily the final best and worst criteria, and the final result should be the one calculated by the programming equation later. When making a decision, the expert selects the best and worst criteria just like a set of comparison benchmarks, and compares other criteria with these two criteria to obtain the ranking of each criterion. However, when there are many experts making decisions, the best and worst criteria in the eyes of each expert are often different, so that multiple groups ofÃ q B andÃ q W are obtained. In order to finally integrateÃ q B andÃ q W of different experts, it is necessary to select a set of highly recognized best and worst criteria as the basis point in advance.
Step 3: Execute the fuzzy reference comparisons for the best criterion.
In this step, we define the preferences of the best criterion over the other criterion. The obtained fuzzy Best-to-Others vector for qth decision maker isÃ Bj is the fuzzy preference of the best criterion B over the criterion j(j = 1, 2, . . . , n) and its value is a fuzzy number. Of which,ã q BB = (1, 1, 1).
With this transformation, we can ensure consistency in each expert's decisions with minimal loss of information. When an expert selects a basis point and then compares a set of criteria based on that basis, the results tend to be consistent, and the comparison results can then be used to calculate the comparison values of the other two criteria. Because this set of numbers is based onÃ q B , it is the same as the expert's original definition of the standard. It is worth mentioning that this transformation preserves the opinion of each expert and does not result in the loss of information.
Step 4: Execute the fuzzy reference comparisons for the worst criterion.
In this step, we define the preferences of all criteria i over the worst criterion. The obtained fuzzy Others-to-Worst vector for qth decision maker is: , whereã q iW is the fuzzy preference of the criterion i(i = 1, 2, . . . , n) over the worst criterion W and its value is a fuzzy number. Note thatã q WW = (1, 1, 1). In the same way, noted thatã ij =ã iW /ã jW , we can define the qth decision-maker' Step 5: Incorporate all expert evaluations. Since this paper uses the input-based consistency ratio, we need to check the consistency of each expert's comparison vectors before we proceed to this step. The specific method will be given in the following section IV. After we get the evaluation of each expert, we can construct the pairwise comparison matrix of each expert. the qth decision maker's pairwise comparison matrix is shown below: The main purpose of aggregation is to produce appropriate results from the pairwise comparison matrix. There are many methods to synthesize the decisions of multiple experts. By learning from the integration method of AHP [31], the next part will give examples of several suitable methods and evaluate the merits and demerits of these methods.
To the best of our knowledge, the mean method is a simple and effective method, which usually includes geometric mean and arithmetic mean, emphasizing the average of all judgments. As a result, the method can more respect the opinions of experts with the same weight. Arithmetic mean can merge all theÃ q by the equationÃ = (1 q) ⊗ (Ã 1 ⊕Ã 2 ⊕ · · · ⊕Ã q ). For the details,ã B m j andã jW m are calculated as follows: . Simple as it is, but there should be no extreme value due to its sensitivity.
As for geometric mean, it is less affected by extreme value and more suitable to average values.ã B m j andã jW m are calculated as follows:

2) MAX-MIN METHOD
Compared to the mean methods using an average solution, the max-min method extends the aggregated value range by including the 'worst' and the 'best' judgements. Max and min, as two aggregation operators, choose the largest and smallest values respectively [30]. They decide the upper and lower bounds of the aggregated TFN.
The aggregated TFNã B m j = (l B m j , m B m j , n B m j ) by max-min with geometric mean is: And the aggregated TFNã B m j = (l B m j , m B m j , n B m j ) by max-min with arithmetic mean is:

3) METHOD BASED ON CONSENSUS DEGREE
The aggregation principle of this method borrows the idea of weighted arithmetic mean, but there are great differences in the derivation of weight coefficient. The key part of this approach is how to derive reasonable weights so that expert opinion can be integrated according to the weights. This method introduces a 'consensus coefficient' variable which is a compromise between the weight of expert and the difference of its opinion from the opinions of all the others, and multiplies it by a separate judgment instead of the weight of the expert in the weighted arithmetic average. The process is as follows: Step The degree is calculated as follows: Step b: Calculate the average degree of agreement A(DM g ) of expert DMg (g = 1, 2, . . . , n) with all the others.
Step c: Calculate the relative degree of agreement RA(DM g ) of expert DMg (g = 1, 2, . . . , n) Step d: Calculate the consensus degree coefficient C(DM g ) of expert DMg (g = 1, 2, . . . , n) C(DM g ) = y 1 y 1 + y 2 * ω DM g + y 2 y 1 + y 2 * RA(DM g ) (22) ω DM g is the weight of expert DMg. In this paper, each expert has the same weight; y 1 and y 2 are the weight of the importance of experts and the weight of the relative degree of agreement of experts.
Step e: Aggregate the fuzzy judgements.

VOLUME 9, 2021
The resultã Step 6: Determine the optimal fuzzy weights for decision makers (ω * 1 ,ω * 2 , . . . ,ω * n ). The optimal fuzzy weight for each criterion is the one where, for each fuzzy pairω B m /ω j andω j /ω W m , it should haveω B m /ω j =ã B m j andω j /ω W m =ã jW m [8]. To find the optimal weights of the criteria, it should minimize the maximum gaps between obtained weights and the performance of decision maker. Therefore, in order to satisfy these conditions for all j, the absolute gaps ω B m /ω j −ã B m j and ω j /ω W m −ã jW m for all j can be built. So, we can obtain the constrained optimization problem for determining the optimal fuzzy weights (ω * 1 ,ω * 2 , . . . ,ω * n ) as follows: ,ã jW = (l jW , m jW , n jW ). As discussed in [13], Eq.(24) may result in multiple optimal solutions. Then we can minimize the maximum among the set of ω B m −ã B m jωj , ω j −ã jW mω W . The constraint equation can be formulated as follows: What's more, Eq.(25) can be transferred to the following nonlinear programming problem: whereξ = (l ξ , m ξ , n ξ ). Considering l ξ ≤ m ξ ≤ n ξ , we supposeξ * = (k * , k * , k * ), k * ≤ l ξ . By solving Eq.(26), the optimal fuzzy weights (ω * 1 ,ω * 2 , . . . ,ω * n ) can be obtained.
So far, the main steps of the proposed method in this paper have been introduced, and the relevant process diagram is shown in Figure.1. With the aim of determining the fuzzy weight of criteria, we can execute the fuzzy comparison on relative criteria. Moreover, with the aim of determining the fuzzy weights of alternatives with respect to different criteria, the related alternatives should be fuzzily compared against each criterion. Finally, the fuzzy ranking scores of alternatives can be derived from the fuzzy weights of alternatives with respect to different criteria multiplied by the fuzzy weights of the corresponding criteria [31]. Compared with the best and worst criteria determined by a single leader in [16], this method reflects the fairness of group decision making. Compared with the multi-round discussion in [20], this method simplifies the number of iterations and introduces prior consistency comparison to achieve the same collective decision effect as the multi-round discussion.

IV. CONSISTENCY RATIO FOR FUZZY BWM IN GDM
When a decision-maker provides the pairwise comparisons in fuzzy BWM, it is important to check the acceptable inconsistency and ensure the rationality of the assessment.
To check how inconsistent a full set of pairwise comparisons may be, several consistency indices have been proposed. Liang et al. [32] proposed an input-based consistency measurement, which is simple to use and has several desirable properties. By using the simple calculation of the input-based consistency measurement, it is easy to provide a DM with immediate feedback. The consistency ratio can be obtained after the entire elicitation process has finished, which means that it provides a DM with a clear and immediate idea of his/her consistency level. Based on their research, this paper extends this method to the fuzzy environment.
Definition 5: The Input-based Consistency Ratio CR I is formulated as follows: where CR I is the global input-based consistency ratio for all criteria, CR I j represents the local consistency level associated with criterion c j . Besides we use CR IE n and CR IE n j to represent the global input-based consistency ratio and the local consistency level of the nth expert respectively. In this paper, we treat the triangular fuzzy numbers obtained by pairwise comparison accurately and convert them into crisp values. Defuzzification converts the fuzzy results produced by aggregation methods into crisp values. At the same time, according to Table 4 of consistency ratio threshold in [32], consistency evaluation of the results is given. In Table 4, let the Scales of the row dimension represents the estimated size of R(ã BW ). Because the size of R(ã BW ) may not be integer, and the row dimension data in the table are all integer, so it can approximate the integer value to obtain R(ã BW ) for convenience. It is worth noting that even though we can easily identify inconsistent judgments by using consistency measures, it is unrealistic to expect DM to achieve perfect consistency. However, this kind of prior consistency comparison can avoid the workload of recalculation caused by the possible inconsistencies in the last consistency test in [6], [17], [16], which is also one advantage of the method proposed in this paper.

V. CASE STUDIES
In this section, we give two examples for the different group decision-making problems to illustrate the method proposed in this paper. Note that, the mathematical models of the examples are solved by Lingo Version 17.0 software to obtain the optimal weights.

A. CASE 1
Imagine a situation where you want to buy a high costperformance car, but have no idea which car to choose, so ask three car experts to give you some ideas. At this point, the method mentioned in the paper can be applied.
By solving the problem, it can be obtained that 07) The optimal fuzzy weights of six criteria in this case is also shown in Figure.2.
Therefore, it can be seen that Quality (c 2 ) is the most important criterion in terms of supplier's willingness for supplier performance evaluation, the next important criteria are Safety (c 4 ), Price (c 1 ), and Comfort (c 3 ), and Style (c 5 ) is ranked as the least important criterion. When we defuzzy the obtained fuzzy numbers and turn them into R(ω j ) for sorting, through the result that R(ω 1 ,ω 2 ,ω 3 ,ω 4 ,ω 5 ) = (0.187, 0.397, 0.108, 0.259, 0.049), the same sorting results can be obtained. The dispersion degree of the fuzzy numbers also reflects the degree of inconsistency of experts' opinions to some extent.

B. CASE 2
In this case, the selection of the mobile phone is discussed, which was applied when the BWM approach was first proposed. The choice of mobile phone refers to six criteria, namely price (c 1 ), dimensions (c 2 ), weight (c 3 ), display (c 4 ), data inputs (c 5 ), and memory (c 6 ). In terms of personnel choices, five students were asked to rank six criteria to consider when buying a phone.

VI. CONCLUSION AND FURTHER RESEARCH
In this paper, a new model based on the fuzzy best-worst method in GDM environment is proposed to integrate DM opinions, which solves the consistency problem in fuzzy BWM and helps to provide immediate feedback on the consistency of pair comparison. At the same time, a number of experts are introduced to evaluate the opinions, and the transformation of expert opinions by using mathematical formulas is convenient for the integration of experts' opinions. Finally, the geometric average method is used for integration. The results show that the method is effective and suitable for group decision making problems.
In the examination of the degree of consistency of expert opinions, we refer to the input-based consistency ratio proposed in [32] and extend the input-based consistency ratio to the fuzzy environment. What's more, with the thresholds proposed in [32], the DM can decide whether to modify his/her previous assessment. This not only shows the ratio of DM violation ordinal degree of consistency, but also provides a convenient way to identify and correct the conflict.
There are several aspects of improvement for the proposed method in this paper, which are also the future research directions. Firstly, one of the defuzzification methods is used for improved input-based consistency ratio and constrained optimization equations. However, there are many other defuzzification methods that can be applied to the model, which can be a direction for future research. Secondly, the main purpose of aggregating the opinions of different experts is to produce appropriate results from pairwise comparison matrices. Each approach has its own strengths and weaknesses, and subsequent research can focus on the strengths and weaknesses of the different aggregation approaches. Thirdly, the proposed fuzzy best-worst multi-criteria group decision-making method in this paper can be employed in some real-world problems to further verify the effectiveness of the proposed method.