Comparative analysis of fuzzy multi-criteria decision making methods for selecting sustainable battery suppliers of battery swapping station

ABSTRACT Sustainable battery supplier (SBS) selection of battery-swapping station belongs to a complex multi-criteria decision-making (MCDM) problem due to the fact that included multiple and often conflicting criteria. To this end, this study presents a comparative analysis of selecting SBS using four fuzzy MCDM methods, and a case is executed to compare the ranking results obtained by the proposed four MCDM methods. From the weighting results of criteria, it is observed that economic criteria is the first priority in all evaluation criteria, followed by the technical, social, and environmental criteria in descending order. In terms of the priorities and recommendations of multiple SBSs for the battery swapping station, the comparative analysis demonstrates that the rankings of all alternatives obtained by all approaches are in high agreement, except that determined by fuzzy VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) slightly varies from the other three approaches, and none of these MCDM methods are deemed to be absolutely “perfect”. If possible, we recommend that more than one method should be applied to the same problem to provide a more comprehensive decision basis. If not possible, it is suggested to use the fuzzy VIKOR since it shows more superior potential in sustainable supplier decision analysis.


Introduction
With the continuously worsening environment and climate deriving from limited nonrenewable energy sources, the use of clean and renewable energy is of strategic significance to respond to this severe challenge (Zhang and Bai, 2017), and one of the main solutions to handle these issues is adopting electric vehicles (EVs) (Amiri et al., 2018).In recent years, many countries have devoted themselves to putting forward various controlling measures for energy utilization.Particularly, in order to promote the transformation to electrification, China, as one of the largest energy consumers, is vigorously encouraging the development of new energy vehicles (NEVs), namely EVs.However, one of the obstacles against the widespread use of the EVs is long charging time duration (Hannan et al., 2017).To address the problem of long charging time duration during the driving of electric vehicles, Chinese government has taken various measures to accelerate the construction of EV infrastructure (Lu et al., 2017).For instance, fast charging stations are constructed to shorten charging time duration, while they are exploring technological development, and still difficult to achieve widespread application.In addition to this, when potential peak periods happen, fast chargers cannot cope with the huge and uncertain power consumption (Hannan et al., 2017).
Battery swapping station (BSS) is a new concept to resolve this problem in which depleted EV batteries are substituted with a previously full-charged one in relatively less time.The batteries in the BSS are prepared for EVs in advance.When the EVs arrive at BSS, the battery's depleted energy power is directly replaced with fully charged batteries to continue driving.Since batteries are the core units of EVs, the prices of which occupy approximately 50% of the total operation cost.With the increasing global pressure for sustainable development (Gaziulusoy et al., 2015), lots of exhaust gases or air pollutants can be effectively reduced by using battery electric vehicles (BEVs).Therefore, it is very necessary to select sustainable battery suppliers (SBSs) for the BSS, which can not only enhance the environmental performance of the BSS and the satisfaction of customers, but also help the organizations or companies to increase the overall economic gains and environmental performance in the operation process of BSS (Hosseini and Sarder, 2019;Yang et al., 2014).
As discussed above, the contributions of this study aim to present multiple fuzzy multi-criteria decision-making (MCDM) methods to help decision-makers (DMs) select the most appropriate SBS.To begin with, a systematic sustainability evaluation criteria system of battery suppliers involving multiple criteria is established by collecting a large amount of literature and experts' opinions.It is worth noting that the criteria selected in this study are almost independent of each other.Then, Intuitionistic Fuzzy Sets (IFSs) are introduced to determine DMs' weights, and the linguistic variables combined with experts' knowledge and experience are intuitively used to obtain the weight of each criteria.Following this, the Triangular Fuzzy Sets (TFSs), as one of the most often used fuzzy numbers, are transformed into Interval Type-2 Fuzzy Sets (IT2FSs) to depict the sustainability performance of each battery supplier with respect to each criterion, which is observed to be more comprehensible and accurate.Finally, four fuzzy MCDM methods, including fuzzy Weighted Aggregates Sum Product Assessment (WASPAS), fuzzy VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR), fuzzy Similarity to Ideal Solution (TOPSIS), and fuzzy Gray Relational Analysis (GRA), are developed to identify differences and similarities in the ranking results obtained, and also to explore their advantages and disadvantages in selecting the optimal SBS under fuzzy environment.
The remainder of this study is organized below: Section 2 reviews the existing studies related to the MCDM methods and their applications to real-world supplier selection problems.The evaluation criteria system and their specific illustrations are summarized in Section 3. Section 4 presents relevant definitions and knowledge of IFSs and IT2FSs, along with the decision framework of the proposed fuzzy MCDM methods regarding SBS selection.Numerical results are studied in Section 5 to illustrate and verify the applicability and effectiveness of the proposed approaches.Finally, Section 6 executes sensitivity analysis and discussion, and a conclusion is drawn in Section 7.

Literature reviews
As the whole decision process of sustainable supplier selection requires objectives considering multiple criteria simultaneously in an uncertain environment, there have been various MCDM methods used for selecting the most suitable one among multiple sustainable alternatives (Puška and Stojanović, 2022;Song et al., 2023;Chai et al., 2023).An overview of the representative MCDM methods developed by previous studies in the field of sustainable supplier selection/management is summarized in Table 1.The existing MCDM methods in the abovementioned literature studies can be classified into qualitative, quantitative, and hybrid methods.Among them, hybrid methods are becoming increasingly popular due to the characteristics of correctness and flexibility.For instance, Govindan et al. (2013) incorporated the fuzzy concept into the general MCDM method for sustainable supplier selection, which brings about uncertain expressions being more realistic and reliable compared with crisp representations.
There has also been observed some limitations through various literature mentioned above.On the one hand, although lots of research scholars working on supplier selection problems have developed various MCDM methods for sustainable supplier selection, the majority of which had focused on the combination of traditional fuzzy set theory (FST) with various MCDM methods.There was less attention paid to the intuitionistic fuzzy set (IFS).IFS, as an extension of the ordinary fuzzy set, is deemed as a more ideal means for resolving realworld problems.Since compared to the common fuzzy set characterized by a membership degree, an IFS can be defined and represented by the three eigenvalues, including membership, non-membership and hesitation degrees, which can reveal the fuzzy nature of decision information more precisely.Particularly for decision makers (DMs), the definition and use of hesitation degree within IFS are beneficial.In the meantime, traditional fuzzy MCDM methods require to use the defuzzifying techniques to handle fuzzy numbers for implementing the comparison between fuzzy numbers.However, given that: (1) the DMs are usually dissatisfied with defining fuzzy linguistic variables via the fuzzy sets theory; (2) some counter-intuitive ranking results might be generated for similar fuzzy utilities; (3) the linguistic variable in the form of conventional fuzzy sets is not sufficiently clear (Grattan-Guinness, 1976).Fortunately, interval-valued fuzzy sets are proposed to describe imprecise/ vague information (Bigand and Colot, 2010), since a linguistic variable in the form of interval-valued fuzzy sets can present a more flexible and clearer in the decision-making process (Ashtiani et al., 2009).On this basis, to determine the distance between any two interval-valued fuzzy numbers, Vahdani et al. (2010) used the Hamming distance to resolve this problem, while it also has a drawback as it does not take into consideration negative values in any bounds of interval-valued fuzzy numbers (IVFNs) by conducting arithmetic operations (Ashtiani et al., 2009).On the other hand, the research field of the BSS is uncommon, especially SBS selection of BSS.To be specific, although there are numerous ongoing studies aimed at to enhance the utilization of EVs for sustainable development, the majority of research in this field focuses on mitigating range anxiety issues and improving the adaptability of EVs, and the studies on targeting the BSS aspect to promote EVs are relatively scarce.While the BSS enables the rapid exchange of discharged EV batteries with fully charged ones, presenting a solution that facilitates the promotion of EVs from a different perspective.Therefore, the studies on SBS selection in the overall operation of BSS should not be often neglected, where needs to be more relevant research (Koirala K, 2023).
In order to tackle these challenges, this study proposes multiple fuzzy MCDM methods based on IT2FSs to continue analyzing the selection problem of SBSs.To be more specific, the four fuzzy MCDM methods are developed to prioritize sustainable supplier alternatives for a battery swapping station, and they do not require special software and incorporate imprecise/vague information of each criterion regarding the alternatives in the decision-making process, which seem to be more superior than other MCDM techniques due to the simplicity in the computation process and lesser time in the calculation.Briefly, this study distinguishes itself from previous research as follows: (1) To the best of our knowledge, although some efforts on MCDM methods combined with IFS for sustainable supplier selection could be found in prior studies, in practice, it is almost scarce and even non-existent that the IFS was used to determine DMs' importance weights.The inclusion of the IFS contributes better to reducing the doubt in the decision-making process.
(2) The evaluation criteria and sub-criteria including technical aspect are selected from a comprehensive literature survey as well as DMs' opinions and preferences.This seems more precise in the real-world application since it realizes a balance between academic and practical perspectives.
(3) Due to the fact that the actual data are not readily available, the interval-valued fuzzy sets are introduced to express the imprecise decision information, which considers more flexibility, and ensures that the linguistic representation is clear enough.Meanwhile, the four fuzzy MCDM methods are developed to compare the ranking results obtained and to explore their advantages and disadvantages in selecting the optimal SBS under fuzzy environment.

Criteria system for sustainable supplier selection
According to the literature review above, the battery supplier selection of BSS is considered a typical multi-criteria decision-making problem in which selecting reasonable criteria plays an essential role in the whole decision process, directly affecting the accuracy of decision results.In past studies, most research on sustainability criteria have mainly focused on three aspects of the economic, environmental, and social (Alavi et al., 2021;Liu et al., 2019;Ortiz-Barrios et al., 2021;Tong et al., 2020;Mohammed et al., 2018;Mohammed et al., 2019;Lo et al., 2018).In other words, existing criteria related to the supplier selection activity were briefly divided into three main categories: economic, environmental, and social sustainability.Among them, economic sustainability indicates that sustainable suppliers should pay more attention to the long-term and stable return of investment on social capital; environmental sustainability indicates that suppliers should reduce the adverse impact of the resources on the surrounding environment and take into account improving the life quality of residents; and social sustainability indicates that suppliers should offer complete information disclosure to the public, provide employees' rights and pre-employment training for the underprivileged, and seek equity of stakeholders' rights.However, numerous existing sustainability evaluation criteria systems seldom considered another important aspect, namely technical sustainability, which leads to a lack of more comprehensive sustainability evaluation for battery suppliers.To our knowledge, the battery suppliers mainly provide battery products for BSS, so the technical problems of battery suppliers are very important.Moreover, the evaluation criteria for sustainable suppliers might be inconsistent in different areas, which brings an obstacle to the effective sustainability evaluation of such suppliers.To fill such a gap, this study establishes a more scientific and comprehensive criteria system for the sustainability evaluation of battery suppliers from the perspective of the four criteria, including economic sustainability, social sustainability, environmental sustainability, and technical sustainability.Meanwhile, the sub-criteria associated with each primary criteria are identified and selected by reviewing the related literature and recent study reports.Table 2 highlights a summary of the final evaluation criteria system for the battery supplier selection of BSS.It is worth noting that research and innovation ability, product diversity, battery quality and safety assurance ability, along with quality assurance period are considered into technical aspect for the first time.

Intuitionistic fuzzy set
IFS is practically considered an extension of traditional fuzzy set theory (FST), and it is an effective approach to cope with vagueness.
Definition 1. (Xia and Xu, 2011).Assume that A is an IFS belonging to a finite set X. Then the IFS A can be expressed as follows: where μ A x ð Þ : X !0; 1 ½ � represents the membership relationship function, and υ A x ð Þ : X !0; 1 ½ � is defined as the nonmembership relationship function, such that Definition 2. (Xu and Zhang, 2013).There is a third parameter of IFS called the hesitation degree.Assume that π A x ð Þ indicates the hesitation degree of x belonging to A, then the π A x ð Þ can be calculated below: It is apparent that for every x 2 X: It is worth noting that the smaller the value of π A x ð Þ, the more confident the information concerning x.On the contrary, the greater the value of π A x ð Þ, the more uncertain the information regarding x.

Interval type-2 fuzzy set
Type-1 fuzzy sets (T1FSs) were developed by Zadeh (1965), in which the membership degree of an element in a T1FS can be calculated as a real value between 0 and 1. Usually, a trapezoidal type-1 fuzzy number X ¼ x 1 ; x 2 ; Afterward, to describe uncertain information more effectively and precisely, IT2FSs were developed on the basis of T1FSs and characterized by primary and secondary membership (Hu et al., 2015).Some basic definitions related to IT2FSs are given by referring to Refs (Karnik and Mendel, 2001).
Definition 1. (Hu et al., 2015;Liang et al., 2019;Meniz, 2021) Criteria Sub-criteria Remarks References Economic (S1) Financial ability (C1) It refers to the overall status of suppliers' operating funds at a certain time, and some indexes, including cash ratio, asset-liability ratio and so on, can be used to measure the practical financial ability of their operating activities.(Khan et al., 2018;Luthra et al., 2017) Price of battery (C2) It refers to the actual price that BSS purchases from battery suppliers.In other words, the profits of BSS depend upon the purchase price of batteries to a large extent.(Mohammed et al., 2019;Liu et al., 2019;Chen et al., 2020;Khan et al., 2018;Luthra et al., 2017;Stević et al., 2020) Transportation cost (C3) It indicates that suppliers should have the ability to ship batteries at the lowest possible cost.The transportation cost is commonly greater than that of ordinary products due to requiring strict protection measures during the transportation process.(Luthra et al., 2017;Wang et al., 2021) Timely supply (C4) It mainly depends upon suppliers' lead time and rate of delivery in time.Suppliers should ensure the time interval from sending out a battery order to placing the BSS, and have the ability to offer batteries for BSS on time.(Mohammed et al., 2019;Chen et al., 2020;Khan et al., 2018;Luthra et al., 2017;Stević et al., 2020) Return policy (C5) It refers to the suppliers' service that if the purchased batteries exceed the actual demand of BSS, the superfluous batteries can be conducted buyback by the suppliers.(Wang et al., 2021) It indicates that suppliers' stakeholders should be familiar with their rights and strictly implement relevant responsibilities.(Chen et al., 2020;Luthra et al., 2017) Employees' rights (C7) It concentrates on satisfying the requirements related to employees, for instance, suppliers should promote and stimulate employees' career development, actively carry out safety and health education policies, and so on.(Mohammed et al., 2019;Liu et al., 2014;Luthra et al., 2017;Stević et al., 2020) Information disclosure (C8) It indicates that suppliers should provide detailed information to their customers and stakeholders during battery production, including safety certification, material used, carbon emissions, and so on.(Mohammed et al., 2018;Khan et al., 2018;Luthra et al., 2017;Stević et al., 2020) It refers to the consumption amount of raw materials, energy, power and water resource by suppliers in the process of battery production.(Khan et al., 2018;Wang et al., 2021) Pollutant discharge (C10) It refers to the pollution amount discharged into the surrounding environment or other facilities in the process of producing a battery containing hazardous material.(Mohammed et al., 2019;Liu et al., 2014;Khan et al., 2018;Luthra et al., 2017) Recycle system (C11) It refers to the suppliers' capability of collecting and recycling waste batteries.It is very necessary to construct a set of efficient recycling logistics systems to recover and dispose of harmful waste batteries.(Liu et al., 2015;Khan et al., 2018;Stević et al., 2020) Technical (S4)

Research and innovation ability (C12)
It refers to the suppliers' ability to continually update batteries with higher efficiency and performance.This ability can maintain strong technical advancement and avoid various potential risks caused by product supply.(Mohammed et al., 2019;Stević et al., 2020) Product diversity (C13) It refers to the suppliers' ability to produce different types of batteries to meet diverse BSS needs and service various EVs.(Wang et al., 2021) Battery quality and safety assurance ability (C14) It refers to the safety and quality levels of the batteries provided by suppliers, which can be measured from the perspectives of normal service life, low-temperature performance, charge and discharge efficiency, and so on.(Liu et al., 2019;Chen et al., 2020;Khan et al., 2018;Luthra et al., 2017;Stević et al., 2020) Quality assurance period (C15) It indicates that suppliers should make commitments that some quality problems can be handled for free in the process of normal battery use during a certain time.(Luthra et al., 2017;Wang et al., 2021) discourse represent the membership degree of the element x L i1 in the lower IT2TrFN X L i and the element x U i1 in the upper IT2TrFN X U i , respectively.The illustration of an IT2TrFN is displayed in Figure 1.
two IT2TrFNs, The distance between X1 and X2 , labeled as d X1 ; X2 À � is defined as: Definition 6.A linguistic variable represents a variable, the values of which are expressed by linguistic terms (Kacprzyk, 1986).Using linguistic terms, such as not important, important, very important, and so on, has been proved to be more intuitive and convenient to describe DMs' subjective judgments when facing with uncertain or imprecise information.

Decision framework of the proposed fuzzy MCDM methods
With the advantages of integrating fuzzy mathematics and ordinary MCDM methods, the fuzzy MCDM method is more superior used in this field.Hence, under uncertain environment, the four fuzzy MCDM methods, including fuzzy WASPAS, fuzzy VIKOR, fuzzy TOPSIS, and fuzzy GRA, are developed to evaluate the sustainability of battery suppliers quantitatively in this study.To put it another way, the decision framework of the proposed fuzzy MCDM methods integrates the IFSs, IT2FSs and four fuzzy MCDM methods in a systematic structure.Among them, the IFSs and IT2FSs are introduced to determine the DMs' weights and the criteria weights respectively, and the four fuzzy MCDM methods are used for evaluating and ranking SBSs.The research flow chart is depicted in Figure 2, which consists of two main sections: preparation section and decision-making section.The proposed approach can deal with complex decision problems involving lots of imprecise and vague decision information, as the IT2FSs present more flexibility when lacking of data.
A detailed procedure is presented in the following.

Data collection
In terms of the data collection of the selected criteria, a common practice is to invite several knowledgeable and experienced experts to express their opinions and preference regarding qualitative criteria (Chen et al., 2014).
Similarly in this study, the linguistic variables are used by experts to collect and handle uncertain information since they are much more approximate to humans' thinking and perception than precise numerical value.
Step 1. Weight and rate the selected criteria using IT2TrFNs determined by DMs' opinions.
This study employs a seven-point Likert-type scale to generate the "linguistic variables for the level of importance" and "linguistic variables for the ratings" Table S1 and Table S2 (see Supporting information section S1) respectively, which refer to Chen et al. (2022) and Hu et al. (2015).Here, it is required that DMs use the linguistic variables to assess the relative importance of various criteria and ratings of alternatives with respect to each criteria (Chen, 2000).Then, these linguistic variables are further transformed into the corresponding IT2TrFNs.Finally, the fuzzy importance weight and fuzzy ratings of each criterion are obtained directly.
Step 2. Determine the importance weights of DMs.The linguistic variables, as summarized in Table S3 (see Supporting information section S2), are firstly used to assess DMs' significance, which are then transformed into the corresponding intuitionistic fuzzy numbers.Suppose that is the intuitionistic fuzzy number for assessing the importance of the kth DM, the DM's weight can be defined as (Memari et al., 2019): Step 3. Create the aggregated interval type-2 trapezoidal fuzzy decision matrix and weight vector.
Suppose that the sustainability performances of alternatives are evaluated.If the decision-making group consists of K DMs, each DM gives lower and upper bound values for weight and rating matrix's elements of alternatives with respect to criteria with the help of IT2TrFNs.On this basis, the aggregated ratings of alternatives regarding each criterion are obtained using Eq. ( 15), and the aggregated criteria weights are calculated using Eq. ( 16).where w k denotes the weight of the kth DM, xk ij and ωk j are the rating and the weight of the jth criterion by the kth DM, and then xij and ωj are the average rating and importance weight with IT2TrFNs, respectively.

And xij
After that, an aggregated interval type-2 trapezoidal fuzzy decision matrix and weight vector under uncertain environment are established respectively below: where xij , "i; j denotes the interval type-2 trapezoidal fuzzy rating of alternative A i , i ¼ 1; 2;, regarding criterion C j , and ωj represents the fuzzy weight of criterion C j , j ¼ 1; 2; � � � n.

Decision-making
The deterministic MCDM methods, including WASPAS, VIKOR, TOPSIS, and GRA, are preferred when the information collected in the decision process are deterministic.However, in case that the decision problem contains lots of uncertainty, various fuzzy MCDM methods should be developed to cope with the imprecise information.This study focuses on multiple fuzzy MCDM methods involving the fuzzy WASPAS, fuzzy VIKOR, fuzzy TOPSIS and fuzzy GRA, and their treatments for the SBS selection have different procedures provided as follows.

Fuzzy WASPAS method.
WASPAS method, originally developed by Zavadskas et al. (2012), aims to combine the weighted sum method (WSM) with the weighted product method (WPM).The concise steps are presented in the following.
Steps 1-3.Refer to steps 1-3 of the data collection in Section 4.2.1.
Through collecting the performance rating xij of the ithalternative associated with the jth criterion, an aggregated interval type-2 trapezoidal fuzzy decision matrix for group decision-making is established below: Step 4. Compute the defuzzification weights of all criteria.To be more readily available, the linguistic variables used to assess the importance of various criteria are further transformed into the corresponding IT2TrFNs according to Table S1.On this basis, we suppose that 5 is the fuzzy weight of criterion C j with IT2TrFNs, where j ¼ 1; 2; � � � n, the value of its defuzzification ω j (i.e., the crisp weight of criterion C j ) is calculated by Eq. (20).
Step 5. Compute the relative importance P1 i based on WSM.
where ω j represents the weight of the jth criterion.Step 6.

Calculate the relative importance P2
i based on WPM.
Similarly, where ω j represents the weight of the jth criterion.
Step 7. Determine the total relative importance Pi of each alternative.
we calculate the defuzzification value of Pi for each alternative below: Step 9. Rank the alternatives and select the optimal one.The defuzzification value P i of the total relative importance is used to rank the alternatives.It is worth noting that the alternative with the greater P i value is more preferable.

Fuzzy VIKOR method. VIKOR was proposed by
Serafim Opeicovic in 1998, and it is one of the most common MCDM methods used for solving the decisionmaking problems.Particularly, for dealing with the complicated and conflicted multi-criteria decision problems, the method can give a compromise solution, namely an agreement formulated by mutual consent (Wu et al., 2016;Kumar et al., 2013).
Steps 1-3.Refer to steps 1-3 of the data collection in Section 4.2.1.
Step 5. Compute the defuzzification value of xij .
where x ij is the defuzzification value of xij .
Step 6. Determine the positive ideal solution (PIS) and the negative ideal solution (NIS) for each criterion.
For each sub-criterion, the defuzzification values of all IT2TrFNs in the interval type-2 trapezoidal fuzzy decision matrix are first calculated according to Eq. ( 25).After that, ranking the IT2TrFNs of the alternatives according to their defuzzification values, and we can identify and generate the PIS and NIS, labeled as x þ j and x À j (j ¼ 1; 2; � � � ; n), respectively.
Step 7. Calculate utility measure (S i ) and regret measure (R i ).
where S i is a utility measure, R i is a regret measure, andx ij is an element in decision matrix.x þ j and x À j is the criterion of the best and worst value.
Step 8. Compute the concordance index Q i value.
An alternative with the minimum Q i value is given as rank one, and the ranking of other options is determined according to the Q i value (Wu et al., 2016).
and the control parameter v refers to the strategic weight of the majority of criteria, namely maximum group utility.In this study, v ¼ 0:5.
Step 9. Determine the ranking of alternatives according to the Q i values.
If the alternative (A 0 ) with the best Q (minimum) value meets the following two conditions discussed simultaneously, it is recommended as a compromise solution.
Condition 1. "Acceptable advantage" Suppose that A 00 is the second optimal alternative according to the ranking list, Eq. ( 31) and Eq.(32) need to be met (Kumar et al., 2013).
where m is the number of alternatives.
Condition 2. "Acceptable stability" The best alternative A 0 also needs to be ranked first according to the S value and R value.
If one of the two conditions is not met, a compromise set of solutions is suggested as follows (Kumar et al., 2013): • Alternatives A 0 and A 00 only if Condition 2 is not satisfied.
To sum up, the best alternative is the alternative with a minimum Q i value.The rankings of the other alternatives are determined according to the Q i values in ascending order.The obtained ranking result is a compromise solution with consideration of a final ranking list of alternatives and an "advantage ratio" simultaneously.

Fuzzy TOPSIS method.
TOPSIS, as one of the most widespread MCDM methods, was initially developed by Hwang and Yoon (1981) and further extended by Chen and Hwang (1992).In this study, an extended fuzzy TOPSIS method based on IT2TrFNs is developed and two '"reference" points are introduced, including an interval type-2 trapezoidal fuzzy PIS (IT2TrF-PIS) and an interval type-2 trapezoidal fuzzy NIS (IT2TrF-NIS) (Yue, 2011).The fuzzy TOPSIS yields the best alternative via minimizing the distance to the IT2TrF-PIS and maximizing the distance to the IT2TrF-NIS.The ranking orders of alternatives are determined based on the comparison of Euclidean distances (Chen and Tsao, 2008).The fuzzy TOPSIS process is implemented below.
Step 5. Construct the weighted interval type-2 trapezoidal fuzzy decision matrix.
According to the above Eq.( 19) and Eq. ( 20), the weighted interval type-2 trapezoidal fuzzy decision matrix is obtained under fuzzy environment in the following: Step 6. Determine the IT2TrF-PIS and the IT2TrF-NIS.
After yielding the weighted interval type-2 trapezoidal fuzzy decision matrix with IT2TrFNs, the IT2TrF-PIS and IT2TrF-NIS of all alternatives under each sub-criterion, labeled as ẽþ j and ẽÀ j (j ¼ 1; 2; � � � ; n) respectively, are identified as follows (Chen and Lee, 2010): where B is related to benefit criteria, and C is related to cost criteria.
Following this, we obtain the positive and negative distance vectors for each alternative A i associated with each criterion.
Step 7. Compute the total distance d þ i and d À i .The total distance d þ i between each alternative A i and the IT2TrF-PIS vector E þ is computed as: where d ẽij ; ẽþ j � � denotes the crisp distance between two IT2TrFNs ẽij and ẽþ j .In a similar way, the total distance d À i between each alternative A i and the NIS vector E À is calculated as: where d ẽij ; ẽÀ j � � denotes the crisp distance between two IT2TrFNs ẽij and ẽÀ j .

Step 8. Calculate the relative closeness coefficient v i
The relative closeness coefficient v i of each alternative A i regarding the IT2TrF-PIS vector E þ is calculated as: Step 9. Sort the relative closeness coefficient v i values in a descending sequence.
The greater the v i value, the more the distance between A i and the IT2TrF-NIS vector E À , while the more similar between A i and the IT2TrF-PIS vector E þ .Therefore, the alternative with the maximum v i value should be selected as the optimal alternative

Fuzzy GRA method.
GRA method was originally proposed by Deng (1989) and has been widely used in many MCDM problems (Kuo et al., 2008).Similarly, this study introduces one extended fuzzy GRA method based on IT2TrFNs.Different from the fuzzy TOPSIS in Section 4.2.2.3 that employs Euclidean distances to calculate the distance between alternatives and the IT2TrF-PIS with the IT2TrF-NIS, the fuzzy GRA method uses gray relational grades to measure the closeness relationship between them, which is proposed for determining the ranking order of alternatives because of its advantages: (1) achieving multi-criteria sustainability evaluations of alternatives under uncertain conditions; (2) providing an integrated superiority index when lacking partial information.The procedure of the fuzzy GRA method is briefly presented in the following.
Step 6. Compute the gray relational coefficients.
According to the weighted interval type-2 trapezoidal fuzzy decision matrix, the gray relational coefficients r þ ij and r À ij between the ith alternative and the IT2TrF-PIS with the IT2TrF-NIS associated with the jth criterion are computed below: where and ρ is the distinguishing coefficient.In this study, the value of ρ is assigned to 0.5 (Shen et al., 2003).
Following this, the corresponding gray relational coefficient matrixes are yielded in the following: Step 7. Calculate the gray relational grade.
The gray relational grade is considered the average relational coefficient under multiple criteria.In terms of the ith alternative, the gray relational grades from a positive ideal solution and a negative ideal solution are given respectively below: Step 8. Determine the relative gray relational grade.
The relative gray relational grade of each alternative is determined below: Step 9. Rank and select the alternative with the highest G i The priority order of alternatives could be yielded after determining the relative gray relational grade G i of each alternative, and the higher the value of G i , the better the alternative A i .

Summary of the proposed fuzzy MCDM methods.
The above fuzzy MCDM methods aim to help DMs select the most suitable SBS based on multiple criteria under uncertain environment.Every method might have itself merits and demerits.It is difficult to claim that a particular method is superior to the others.The selection of an appropriate method is dependent upon the DMs' preferences to a large extent.The procedure of these MCDM methods has several common principles: (1) collecting the values of alternatives associated with each criterion; (2) multiplying the actual value by its corresponding weights; (3) ranking all alternatives in descending order.Table 3 briefly illustrates the comparisons of the four fuzzy MCDM methods.It is worthy noted that these three MCDM methods of fuzzy VIKOR, fuzzy TOPSIS, and fuzzy GRA aim to build an aggregating function measuring closeness to the reference point.In the decision process, the normalization process often needs to be conducted to eliminate the inconsistency of criterion units.Among them, the fuzzy VIKOR employs linear normalization, and the fuzzy TOPSIS and fuzzy GRA use vector normalization.

Numerical results
This section includes three subsections.The first subsection provides brief information regarding the alternatives considered.In the second subsection, the information about the decision-making process proposed in this study is illustrated in detail, and the comparison of results using different fuzzy MCDM methods is presented in the last subsection.

Problem description
To address the problem of battery swaps for EVs and promote the development of EV industry, H company prepared to build a BSS in Changsha City, Hunan province.A public procurement bidding activity for battery supplier selection of BSS was held in June 2022.According to the specific standards and requirements issued by the relevant department with the preliminary review of the bid in advance, a total of five potential EV battery suppliers were selected as the candidate bidding companies, labeled as A1, A2, A3, A4, and A5, which were located in Beijing, Shandong, Hubei, Henan and Anhui provinces, respectively.For the sake of confidentiality, the five suppliers' names were not disclosed in this study, and their details were summarized and listed in Table 4. Besides, in order to better select an appropriate supplier, four DMs (DM1, DM2, DM3, and DM4) with rich theoretical knowledge and practical experience were invited to form a decisionmaking group, and these DMs' profiles were presented in Table S4 (see Supporting information section S3).Next, the fuzzy MCDM methods, including fuzzy WASPAS, fuzzy VIKOR, fuzzy TOPSIS, and fuzzy GRA, were applied to this real case to evaluate and select the optimal sustainable alternative battery supplier quantitatively.

Decision-making process
To begin with, the criteria system regarding this decisionmaking problem is established in previous Section 3. Subsequently, the members of the decision-making group are invited to use linguistic variables to express their opinions and recognition with respect to the criteria.And an interval type-2 trapezoidal fuzzy matrix is naturally yielded, which seems to be more flexible and accurate than the traditional fuzzy decision matrix.Next, the decision procedure of the proposed approach is presented in the following.
Step 1: A decision-making group consisting of four DMs (DM1, DM2, DM3, and DM4) is generated to select the optimal SBS.Table 5 shows the four DMs' importance degrees on the decision-making group, and we present the detailed computational process of DMs' weights (see Supporting information section S4).The importance degree of each DM is considered based on the following three factors, including (i) relevant theoretical knowledge on sustainability performances of battery suppliers, (ii) practical experiences in battery supplier selection, and (iii) their organizational prestige and position.
Then, the four DMs use the linguistic variables for the importance weights to evaluate the relative importance of each sub-criteria, which is further converted into an interval type-2 trapezoidal fuzzy decision matrix to calculate the fuzzy weight of each sub-criteria.At last, the weighted average weights of the sub-criteria are yielded and summarized in Table S5 (see Supporting information section S5).
Step 2: In a similar manner, the DMs use the linguistic variables for rating alternative performance regarding each sub-criteria.The rating results of the five alternatives under all sub-criteria are provided in the form of linguistic variables (see Supporting information section S6).Then, they are further converted into IT2TrFNs to quantize the ratings of alternatives, constructing an aggregated interval type-2 trapezoidal fuzzy decision matrix, as shown in Table S6 (see Supporting information section S7).
Step 3: Based on Eq. ( 20) and the obtained weighted average weights of sub-criteria (see Table S5), the crisp-valued weights of which are obtained by using the process of defuzzifiness, as depicted in Figure 3 in an intuitive manner.Among them, the crisp-valued weight of each criterion is determined by calculating the sum of weights of the corresponding subcriterion related to the criterion.
As seen from Figure 3, the weights of the economic (S1), social (S2), environmental (S3), and technical (S4) aspects are 0.401, 0.170, 0.164, and 0.265, respectively.It is apparent that the economic aspect is the first priority in all evaluation criteria and is regarded as the most important for selecting the alternative SBSs for battery swapping stations, followed by the technical, social, and environmental aspects in descending order.Similarly, the most important considerations of the subcriteria are battery quality and safety assurance ability (C14, 0.089), followed by timely supply (C4, 0.085), research and innovation ability (C12, 0.084), pollutant discharge (C10, 0.078), return policy (C5, 0.075), employees' rights (C7, 0.071), and so on.
Step 4: The calculation results using the fuzzy WASPAS method.
According to the interval-valued fuzzy decision matrix in Table S6, the defuzzification values of the matrix's elements are obtained by using Eq. ( 25).Following this, we compute the relative importance P1 i , P2 i and Pi , along with the corresponding defuzzification values P i 1 ð Þ, P i 2 ð Þ and P i of each alternative.To ease of analysis, the results are presented in Figure 4 in an intuitive manner.
Step 5: The calculation results using fuzzy VIKOR method.
On the basis of the obtained interval type-2 trapezoidal fuzzy decision matrix, the defuzzification values of all IT2TrFNs are computed by Eq. ( 11).The calculation results of the five alternatives with respect to each sub-criteria are shown in Figure 5 in an intuitive way.
Regarding each sub-criterion, the PIS and the NIS of all alternatives, labeled as x þ j and x À j (j ¼ 1; 2; � � � ; n) respectively, are identified as follows: After that, we can calculate the ω j value, and the results are intuitively presented in Figure 6.For simplicity to express, we define At last, Figure 7 presents the S i , R i and Q i values of each alternative simultaneously.According to this figure, the A3 and A4 all have the highest and lowest values among the three types of values.Through checking Conditions 1 and 2 for alternatives, some conclusions are summarized as follows: (1) Acceptable advantage: according to Equation.(31) and Eq. ( 32), since condition 0.584 -0.000 ≥ 0.250 is met, the optimal alternative A 3 satisfies the requirement of acceptable advantage.(2) Acceptable stability: the alternative A 3 is considered the optimal one, and the S i and R i values of which also rank first.Thus, the alternative A 3 meets the requirement of acceptable stability.
To sum up, the final results of all alternatives satisfying conditions 1 and 2 are presented in Table 6.
Step 5: The calculation results using fuzzy TOPSIS method Based on the established weighted interval type-2 trapezoidal fuzzy decision matrix, we determine the IT2TrF-PIS and the IT2TrF-NIS for each criterion.Then, the crisp distance  10, which could be used to clearly illustrate the final ranking of each alternative.
Step 6: The calculation results using fuzzy GRA method To implement the fuzzy GRA method, we need to construct the weighted normalized decision matrix.On this basis, the gray relational coefficient r þ ij and r À ij between the ith alternative and the PIS with the NIS with respect to the jth criterion is calculated, respectively.The results of r þ ij and r À ij are shown in Figures 11 and 12. Furthermore, the values of the gray relational grade (i.e., G þ i and G À i ) and the relative gray relational grade (i.e., G i ) are available in Figure 13, which is more clear and easier to conduct the comparison of G þ i , G À i and G i values of each alternative.

Comparison of results using different fuzzy MCDM methods
Based on the previous decision-making process in Section 5.2, the values and ranking of all alternatives are obtained by using the four fuzzy MCDM methods (i.e., fuzzy WASPAS, fuzzy VIKOR, fuzzy TOPSIS, and fuzzy GRA), as shown in Table 7.
As seen from Table 7, it is obviously found that alternative A3 is the most suitable battery supplier, and the A4 is the lowestranking alternative in all methods.Meanwhile, the rankings of other alternatives obtained by the four fuzzy MCDM methods are similar to a great extent, except for Fuzzy VIKOR ranking alternative A5 as 2nd and alternative A3 as 3rd.To put it another way, the rankings of the alternatives determined by the fuzzy WASPAS, fuzzy TOPSIS, and fuzzy GRA are entirely the same, and are similar to that determined by the fuzzy VIKOR, namely three of the alternatives (i.e., 60%) in identical positions.To better make comparisons between the four MCDM methods, the corresponding values and rankings provided by these methods are depicted in Figure 14.

Sensitivity analysis in different scenarios
In this sub-section, we make several comparisons of the main criteria using the MCDM methods.On one hand, it is to        (1) Economic criteria In the multi-criteria battery supplier selection process, economic sustainability is regarded as a strategic criteria which has a significant effect on the sustainable development of battery swapping station.Next, by only taking into consideration the values of economic criteria in the fuzzy MCDM methods, the rankings of battery supplier selection are obtained by the four approaches and the detailed results are shown in Table S7 (see Supporting information section S8).For ease of comparison, the ranking results from these MCDM methods, namely fuzzy WASPAS, fuzzy VIKOR, fuzzy TOPSIS, and fuzzy GRA methods, are visualized in Figure 15.From Figure 15, it is obviously observed that the top-ranked alternatives are the same in all approaches, meaning that A3 is the most suitable battery supplier in terms of economic aspect.Moreover, the rankings of other alternatives obtained by these four approaches are entirely consistent, which further illustrates the performance of the proposed fuzzy MCDM methods.
(2) Social criteria Social criteria is always recognized as one of the most significant criteria in the process of sustainable suppliers.In a similar way, by only taking into account the values of social criteria in the fuzzy MCDM methods, the detailed results obtained by the four approaches are shown in Table S8 (see Supporting information section S8).For ease of comparison, the ranking results from these approaches are visualized in Figure 16.From Figure 16, it can be observed that the top-andbottom ranked alternatives are entirely the same among the proposed four approaches, meaning that A3 is the best battery supplier and A4 is the worst one in terms of social aspect.In addition, the rankings of other alternatives using these four approaches are completely in agreement, indicating the robustness of the fuzzy MCDM methods in social aspect.
(3) Environmental criteria Similarly, we use the fuzzy MCDM methods to only calculate the values of environmental criteria, and the obtained results and rankings are shown in Table S9 (see Supporting information section S8), which are presented in Figure 17 in an intuitive manner.As seen in Figure 17, the first-and-last ranked alternatives determined by all approaches are identical, namely the A3 is the best battery supplier and the A4 is the worst one regarding environmental aspect.Meanwhile, the rankings of other alternatives obtained by these four approaches also match perfectly, demonstrating that the solutions generated are reliable and stable using the fuzzy MCDM methods in environmental criteria.
(4) Technical criteria When only taking into consideration the values of technical criteria in the fuzzy MCDM methods, the calculation results of alternatives by the four approaches are summarized in Table S10 (see Supporting information section S8). Figure 18 displays the comparative results intuitively.From Figure 18, it is clearly concluded that the first-and-last ranked alternatives are consistent in all approaches, showing that the A3 and the A4 are the best and worst alternatives regarding technical aspect.Besides, there are slight differences among the intermediate rankings of the alternatives.Using the fuzzy VIKOR, a ranking    of the alternatives from A1 to A5 is yielded as 2-4-1-5-3.For the other fuzzy MCDM methods, the ranking result from A1 to A5 is 4-3-1-5-2.By comparing their results, two of the alternatives (40%) are in identical positions.This is mainly due to the fact that the fuzzy VIKOR employs linear means to eliminate the inconsistency of criterion units in the normalization process.
In summary, to explore the ranking of alternatives from the perspective of single criteria and demonstrate the efficiency of the proposed method, we conduct the sensitivity analysis of each criteria to compare the results of the four fuzzy MCDM methods.A total of four different scenarios are considered and generated with single criteria.The results reveal that the ranking of supplier A3 is the same as the ranking in terms of all criteria, which always ranks first in the decision and maintains its place in all scenarios, with the best sustainability according to DMs' opinion.On the contrary, supplier A4 ranks last in the four criteria and always maintains its ranking position in all scenarios.As a matter of fact, there are also some slight differences in rankings according to Table 7.With regard to technical criteria, when the fuzzy VIKOR methods are used to evaluate, the obtained ranking results are not exactly the same as those of other methods.Because the normalization process of the fuzzy VIKOR is quite different, the ranking results exhibit slight differences.The analysis above can not only help DMs to understand the principles and procedures of using different MCDM methods within the set of sustainability criteria but also examine the practicality of these approaches.

Correlation coefficient analysis
To further reveal the relationship among the ranking results obtained by using different MCDM methods, Spearman's correlation coefficient as an important index is introduced to demonstrate the connection between any two approaches and verify their validity of them.The correlation coefficients of the ranking results determined by different fuzzy MCDM methods are depicted in Table S11 (see Supporting information section S9), which are further visualized in Figure 19.From Figure 19, it is observed that the most of correlation coefficients of ranking results obtained by different MCMD methods are equal to 1.000.In practice, the correlation coefficient of two ranking results greater than 0.8 has meant a strong similarity between the results obtained by different methods.In addition, it is found that all correlation coefficients are greater than 0.9, especially the optimal battery supplier obtained by these four methods is the same.This indicates that there is a high similarity between the ranking results derived from the four fuzzy MCMD methods, further verifying the effectiveness and reliability of the proposed approaches for addressing the SBS selection problem under uncertain environment.

Managerial insights
To better improve the efficiency of battery supplier selection, some managerial insights are provided in the following when implementing this decision procedure.
(1) The criteria system is the foundation of battery supplier selection, which is associated with the collected sustainability performance and further investment preference.
The considered criteria are recommended to include not only classical sustainable criteria, such as economic, social, and environmental aspects, but also technical criteria representing technology advancement, such as research and innovation ability, product diversity, and so on.(2) The comparative analysis of the proposed four fuzzy MCDM methods highlights that the fuzzy WASPAS is relatively simpler and less time-consuming to use in comparison to the other three approaches.Chang et al.
(2001) also declaimed that the complexity of MCDM methods could prevent their widespread application in practice to a large extent.(3) The ranking results determined by all fuzzy MCDM methods presented high consistency amongst themselves.Although none of them is considerably superior to the others, the correlation coefficient analysis shows that it is suggested that fuzzy VIKOR would be preferred in case of only one method to be employed for the alternatives' ranking purpose.An important fact that makes the fuzzy VIKOR outclass other available approaches is that it can provide a compromise solution with a compromise ranking list of alternatives taken for comparison and an "advantage ratio".
For alternative suppliers, there are also some additional suggestions provided below: (1) From Table 7, it can be observed that battery supplier A3 is superior to the other four optional suppliers.And from Table S6, it can be seen that supplier A3 presents the best sustainability performance under sub-criteria C9 (resource consumption), C11 (recycle system) and C14 (battery quality and safety assurance ability) as well as the worst sustainability performance under subcriteria C8 (information disclosure) and C13 (product diversity).Therefore, it is recommended this supplier increase and improve information disclosure regarding battery products from the social perspective and pay high attention to diversified developments of battery products in terms of environmental aspect.(2) As seen from Table S6, battery suppliers A2 and A5 present perfect sustainability performances with respect to social and environmental aspects, but they have poor performances in economic and technical criteria, including research and innovation ability as well as product diversity of A2, and price of battery along with return policy of A5.Hence, it is advised that they should positively formulate related economic policies and develop advanced technologies.To be specific, supplier A2 needs to concentrate on the research and innovation ability with product diversity, and supplier A5 should update and adopt high-efficiency technology in reducing pollutant discharge.Particularly, supplier A5 presents moderate sustainability performances under most sub-criteria, the competitiveness of which needs to be improved comprehensively.In terms of battery suppliers A1 and A4, it is found that they present good overall sustainability performance in product diversity from Table S6.However, supplier A1 has the poorest sustainability competencies under the criterion of the price of the battery, resource consumption and quality assurance period.So, we suggest that supplier A1 save resource consumption and invest more in improving the quality of the battery.Similarly, the supplier A4 should pay more attention to transportation costs and resource consumption, and reduce pollutant discharge.

Conclusions
This research aims to select the best SBS which is capable of providing the needed batteries for the BSS.Considering different fuzzy MCDM methods might yield different results when applied to the same decision-making problem under uncertain environment, multiple fuzzy MCDM methods are developed for the sustainability assessment of battery suppliers from a comparative perspective.A case example of five alternative battery suppliers is studied to illustrate and test the application of the proposed four fuzzy MCDM methods, including fuzzy WASPAS, fuzzy VIKOR, fuzzy TOPSIS, and fuzzy GRA.The results demonstrate that the ranking of alternatives obtained by these approaches is in high agreement, except that determined by fuzzy VIKOR slightly varies from the other three approaches.Through identifying differences and similarities in the ranking results yielded by these methods and analyzing their advantages and disadvantages in selecting the optimal SBS, it is concluded that none of these MCDM methods is deemed to be absolutely "perfect."We recommend that, if possible, more than one method should be applied to the same problem to provide a more comprehensive decision basis.If not possible, it is suggested to use the fuzzy VIKOR since it shows more superior potential in sustainable supplier decision analysis.
Overall, compared with traditional MCDM methods, the main contributions of the proposed fuzzy methods are summarized below: (1) the IFS is introduced to determine DMs' weights, which is more flexible in resolving imprecise problems due to the fact that it extends and improves classical FST.It efficiently undertakes the ambiguity in available information and the essential fuzziness in human opinions and preferences by representing the degree of certainty in an interval form.(2) in this study, the performance ratings and weights of all criteria are expressed in forms of linguistics variables, which are characterized by using IT2FSs.This can more efficiently deal with the ambiguity existing in the available decision information along with the essential vagueness in human thinking, since IT2TrFNs can represent the imprecise decision information in a more flexible manner (Ashtiani et al., 2009).To sum up, through the above improvements of this study, the probability of occurring decision-making mistakes can be effectively reduced under fuzzy environment.
In the future work, since this study cannot consider the interrelated relationships among criteria when determining the criteria weights, the ANP and DEMATEL methods could be used to quantitatively reveal the interactional and influencing relationship among criteria in the system.Particularly, the DEMATEL method can visualize the complex relationship among criteria by using a causeeffect relationship diagram and more intuitively show the direct and indirect impact of the criteria.Moreover, the decision results yielded by the four fuzzy MCDM methods were greatly dependent upon the DMs' subjective opinions.Thus, more decision-makers in the process of SBS selection need to be interviewed to relieve this issue and enhance the accuracy of decision-making results in the future.To put it another way, how to attempt to use large-scale group decision-making methods to select the optimal SBSs is a subject worthy of investigation.

Figure 1 .
Figure 1.The illustration of an IT2TrFN.

Figure 2 .
Figure 2. The research flow chart of SBS selection.
ẽij and ẽþ j with ẽÀ j are calculated respectively.The results are shown in Figures 8 and 9 in an intuitive manner.After obtaining the crisp distance d ẽij ; ẽþ j � � and d ẽij ;

Figure 4 .
Figure 4. Results of the defuzzification values P i 1 ð Þ, P i 2 ð Þ and P i of the relative importance Pi 1 ð Þ, Pi 2 ð Þ and Pi .

Figure 5 .
Figure 5.The defuzzification decision matrix of alternatives.

Figure 6 .
Figure 6.The results of u ij for all alternatives.

Figure 7 .
Figure 7.The chart of S i , R i and Q i for each alternative.

Figure 10 .
Figure 10.Results of ̃ d þ i , d À i and v i .

Figure 13 .
Figure 13.Results of G þ i , G À i and G i .

Figure 15 .
Figure 15.The corresponding values and rankings of alternatives using different methods for economic criteria.

Figure 16 .
Figure 16.The corresponding values and rankings of alternatives using different methods for social criteria.

Figure 17 .
Figure17.The corresponding values and rankings of alternatives using different methods for environmental criteria.

Figure 18 .
Figure 18.The corresponding values and rankings of alternatives using different methods for technical criteria.

Table 1 .
The summary of the MCDM methods used for sustainable supplier selection/management.

Table 2 .
Criteria and sub-criteria for the SBS selection in sustainable supply chain.

Table 5 .
The importance of DMs and their weights.
Figure 3. Criteria and sub-criteria weights.

Table 6 .
S i R i and Q i values and rankings of alternatives.
(Guan et al., 2018)licability and effectiveness of the proposed approaches in solving real-world SBS problems; on another hand, it is to compare the sensitivity of the four MCDM methods.In what follows, the sensitivity analysis in different scenarios is carried out via the proposed four fuzzy MCDM methods.It should be noted that the practical SBS selection sometimes requires the ranking of alternatives from the perspective of single criteria and selecting the optimal alternative(Guan et al., 2018).

Table 7 .
Ranking results of alternatives in different methods.Figure 14.The corresponding values and rankings of alternatives using different methods.