Peide Liu, and Hui Gao

Abstract—In view of the environment competencies, selecting the optimal green supplier is one of the crucial issues for enterprises, and multi-criteria decision-making (MCDM)methodologies can more easily solve this green supplier selection(GSS) problem. In addition, prioritized aggregation (PA) operator can focus on the prioritization relationship over the criteria,Choquet integral (CI) operator can fully take account of the importance of criteria and the interactions among them, and Bonferroni mean (BM) operator can capture the interrelationships of criteria. However, most existing researches cannot simultaneously consider the interactions, interrelationships and prioritizations over the criteria, which are involved in the GSS process. Moreover, the interval type-2 fuzzy set (IT2FS) is a more effective tool to represent the fuzziness. Therefore, based on the advantages of PA, CI, BM and IT2FS, in this paper, the interval type-2 fuzzy prioritized Choquet normalized weighted BM operators with λ fuzzy measure and generalized prioritized measure are proposed, and some properties are discussed. Then,a novel MCDM approach for GSS based upon the presented operators is developed, and detailed decision steps are given.Finally, the applicability and practicability of the proposed methodology are demonstrated by its application in the sharedbike GSS and by comparisons with other methods. The advantages of the proposed method are that it can consider interactions, interrelationships and prioritizations over the criteria simultaneously.

I. INTRODUCTION

TO make further development, enterprises should strengthen their operational managements so as to satisfy the demand of customers and to survive in the fierce market competition [1], [2]. Another way, customers are also searching for lower cost product or service with greater quality [2]. Thus, the supplier selection is a crucial task for enterprises [3]–[6]. Once a supplier becomes a partner, the cooperative partnership between them will have very great influence on improving customer satisfaction and operations of the products or services [4]. For relieving the pressure of environmental protection, enterprises are forced to concern environmental competencies [7]. Accordingly, green supplier selection (GSS) is playing more and more vital role in the process of supplier selection, and now it has been a new and interest research topic which is attracting numerous researchers to pay attention to this issue.

On this regard, multi-criteria decision-making (MCDM)methods can select an optimal supplier from some potential green suppliers which are evaluated with regard of conflicting criteria. Nevertheless, in the real GSS environments, owing to ambiguity and uncertainty, the assessment values on various criteria, such as green research and development (R&D),green technology, green image, and green management system, are not able to be expressed effectively and fully by type-1 fuzzy set. In this case, as an efficient extension of type-1 fuzzy set, type-2 fuzzy set, initially developed by Zadeh [8], is an appropriate and available tool, and it can be applied to represent the fuzziness since it has obvious advantage in more precisely dealing with high-order uncertainties.

On the other hand, because the calculation amount of type-2 fuzzy set is rather huge, it is difficult to be applied to actual MCDM circumstances [9], [10]. However, as a special form of type-2 fuzzy set, interval type-2 fuzzy set (IT2FS) is a very effective tool in processing ambiguity and uncertainty [11],[12]. For years, IT2FSs have been positively employed to various real areas, such as GSS. Through the consistent efforts of researchers, the application of MCDM in the field of GSS has been gradually matured. For instance, Qinet al. [13]developed a GSS model in the context of IT2FSs based on prospect theory and extended TODIM (an acronym in Portuguese for interactive multi-criteria decision making)technique. Mousakhaniet al. [14] introduced a GSS method with IT2FS based on extended TOPSIS (technique for order preference by similarity to an ideal solution) method.Ghorabaeeet al. [15] developed an MCDM approach for dealing with GSS problem based on modified WASPAS(weighted aggregated sum product assessment) and entropy method. However, there is no research on extending certain aggregation operators into the context of IT2FSs for GSS.

Although the above interval type-2 fuzzy aggregation operators can process the interactions and interrelationships among conflicting criteria, they are based on a hypothesis that the conflicting criteria are with the identical priority level, so they are not available for processing the MCDM problems with prioritized criteria. For overcoming this shortcoming,Yager [21] originally developed a prioritized aggregation(PA) operator.

In real MCDM process, the prioritizations, interactions and interrelationships among criteria are universal. PA operator can reflect the priority over the criteria, CI operator can perfectly reflect the interactions among them, BM operator can capture the interrelationships of criteria, and the IT2FSs can express ambiguity and uncertainty. However, until now,there is no study on the interval type-2 fuzzy aggregation operators considering simultaneously the prioritizations,interactions and interrelationships among criteria.

Motivated by these ideas, in order to overcome above drawbacks, the purposes of this study are as follows:

1) The IT2FSs are applied to express the uncertainties of the GSS since they have obvious advantage of dealing with highorder uncertainties more precisely;

2) Interval type-2 fuzzy prioritized Choquet NWBM operator (IT2FPCNWBM) with λ fuzzy measure (FM) and generalized prioritized measure (GPM) by combining PA operator, CI operator and BM operator are proposed respectively, so as to consider interactions, interrelationships and prioritizations over the criteria simultaneously;

3) A novel GSS method based on the presented aggregation operators is developed;

4) A case of shared-bike GSS is conducted to validate the applicability and practicability of the proposed method, and a richer comparative analysis is utilized to explain the superiority and feasibility of the novel method.

The paper is organized as follows: Section II reviews fundamental theories of IT2FSs as well as several relevant aggregation operators. Section III briefly reviews some concepts and operational laws of IT2FS, fuzzy measure, and the basic definition of NWBM operator. Section IV proposes an IT2FPCNWBM operator with FM and GPM. Additionally,and their desirable properties are given. In Section V, a novel GSS method based on the presented aggregation operators is developed. A real case about shared-bike GSS is addressed in Section VI, and Section VII gives some conclusions.

II. LITERATURE REVIEW

Criteria determination and selection of MCDM method are the essential issues of GSS. Next, the current research situations of two issues are respectively analyzed and summarized.

A. Evaluation Criteria for GSS

A significant amount of research goes into developing the evaluation criteria of GSS. Generally, the evaluation criteria in GSS are determined based on three aspects known as economic, social and environment. Economic aspect aims to maximize the revenue flow while minimize the resource [22].For example, the universal evaluation criteria in economic aspect are cost/price [3], [4], [6], [22]–[24], quality [1],[25]–[27] and service [1], [4], [13], [14], [28], [29].Environmental aspect aims to reduce resource consumption and pollution. For example, the universal evaluation criteria in environment aspect are green R&D [4], [23], [25], [27], [28],[30], green technology [1], [3], [4], [23], [25], [27], [28], [30],green image [2], [12], [22], [26], [31], green management system [1], [4], [7], [26], [27], [30], [31]. Social aspect may consider some social issues, such as respect for policy and compliance to government [7], [12], [14], [27], [30], ethical issues and legal complaints [2], [13], [21], [32], [33].

Obviously, researchers develop different criteria systems for GSS in accordance with real situations. While nearly all of the literatures adopt the economic, environment and social indexes as GSS criteria, the sub-criteria utilized by each researcher are different. The GSS criteria are summarized and listed in Table I, which are accepted in recent literatures.

TABLE I EVALUATION CRITERIA AND ILLUSTRATION FOR GSS

B. MCDM Method for GSS

Most literatures have adopted the MCDM methods for GSS,which are from the following two domains:

1) Extended MCDM methods. For instances, Haeri and Rezaei [31] proposed GSS models based on the improved grey relational analysis and best-worst method. Dobos and Vorosmarty [1] presented a data envelopment analysis-based GSS method, where green factors served as the output variables of DEA model. Qinet al. [13] proposed an extended TODIM method for GSS in interval type-2 fuzzy environment. Rouyendeghet al. [24] developed a fuzzy TOPSIS method for GSS problem. Wuet al. [12] proposed an integrated approach for GSS based on the interval type-2 fuzzy best-worst and extended VIKOR methods.

2) MCDM methods with aggregation operators. For instances, Zhang [16] proposed an MCDM method based on GTIT2FWA operator and GTIT2FHA operator. Liuet al. [27]developed a novel GSS method by combining quality function deployment with partitioned BM operator in interval type-2 fuzzy environment. Wuet al. [22] developed a framework under interval type-2 fuzzy environment to select the optimal green supplier of electric vehicle charging facility.

Evidently, the MCDM methods with aggregation operators are more persuasive and more suitable than the traditional MCDM methods, because they can get the collective values of alternatives by aggregating all criteria values, and then rank them. Thus, it is significant to extend the MCDM methods with aggregation operators.

C. MCDM Method With Interval Type-2 Fuzzy Aggregation Operators

For considering the interaction relationships among conflicting criteria, Jinet al. [32] presented a method based on edge detection system by combining CI with IT2FSs. Nehi and Keikha [33] presented a hybrid technique based on CI operator and TOPSIS in the context of IT2FSs.

For considering the interrelationships between criteria,many researchers have extended the BM operator to fuzzy information, such as IT2FSs. Gonget al. [34] presented an MCDM method by the trapezoidal interval type-2 fuzzy geometric BM operator. Wanget al. [35] introduced two methods based on Frank bipolar neutrosophic choquet weighted BM operator and Frank bipolar neutrosophic choquet geometric BM operator, respectively. Because the weighted BM operator has no desirable property of reducibility, in order to overcome this drawback, Zhou and He[36] presented the NWBM operator. Nieet al. [37] introduced an MCDM method by Pythagorean fuzzy partitioned NWBM operator and the optimal Shaply FM. Yang and Li [38]proposed an MCDM approach based on single value neutrosophic NWBM operator.

For processing the MCDM problems with prioritized criteria. Numerous researchers extended PA operator to various fuzzy information, such as intuitionistic fuzzy set [39],neutrosophic set [40], and hesitant fuzzy set [32]. However,extended PA operator with IT2FSs is almost blank. Moreover,the above PA operators only solve the MCDM problems with rigorously ordered prioritization relationship (PR), i.e., each PI has only one criterion.

Although various extended MCDM methods for GSS are developed, a limitation still remains, i.e., the interactions,interrelationships and prioritizations over the criteria cannot be considered simultaneously. Evidently, this limitation is critical and should be handled urgently. Based on above reviews, the innovations of this paper include: 1) λ fuzzy measure ( λ-FM) and generalized prioritized measure (GPM)are utilized in the domain of GSS; 2) the T2FPCNWBM operator is proposed to deal with the problems with prioritizations, interrelationships and interactions at the same time; 3) a MCDM method with IT2FPCNWBM operator is developed to evaluate the green suppliers.

III. PRELIMINARIES

In the following subsection, some concepts and operational laws of IT2FS, fuzzy measure, and the basic definition of N WBM operator are briefly reviewed.

A. IT2FS

Definition 1 [41]:LetEbe the universe of discourse, a T2FSAcan be denoted as follows:

Fig. 1. A geometrical interpretation of an IT2FS [31].

B. Fuzzy Measure

C. NWBM Operator

IV. IT2FPCNWBM OPERATOR

In this section, we propose CI operator with λ-FM, CI operator with GPM, IT2FPCNWBM operator with λ-FM and IT2FPCNWBM operator with GPM. Then, we discuss the desirable properties of these aggregation operators.

A. CI operator With λ-FM and CI Operator With GPM

B. IT2FPCNWBM Operator With λ-FM and IT2FPCNWBM Operator With GPM

Finally,

Let

then

Finally,

Similarly, we can get

Proof:

Meanwhile,

and

The proof of Theorem 4 is similar to that of theorem 1, so it is omitted here.

Fig. 2. The flowchart of the proposed GSS method.

V. A GSS METHOD BASED ON IT2FPCNWBM-λ -FM OPERATOR AND IT2FPCNWBM-GPM OPERATOR

TABLE II LINGUISTIC TERMS AND THEIR CORRESPONDING IT2FSS [42], [45]

VI. APPLICATION CASE: GSS OF SHARED-BIKE

In this section, the effectiveness and applicability of the novel GSS method are validated by a real shared-bike procurement case. Then, by compared with some other available methodologies, the superiority of the novel GSS method is evidently displayed.

A. Case Description

Share economy is a new economic model that conforms to the green development concept. As the rapid development of the mobile Internet, the sharing economy has become popular all over the world. The trend of accelerating the development of the sharing economy in the future is irreversible and will become an important driving force for mankind to move from an industrial society to an information society. With the development of the sharing economy, there are a variety of shared products and services around people, such as the most common shared-bikes. The model combining bike travel and mobile internet technology has raised the efficiency of bike sharing and using, while the shared-bike mode has increased the efficiency of a bike from 5 minutes to 76 minutes, an increase of 16 times. The number of people who can be served has changed from 1 to at least 10, an increase of at least 10 times. By sharing, from the transfer of ownership to the sharing of usage rights, the useful efficiency of bike is improved, urban resource waste is reduced, more congestion is also reduced, more space is saved, and green low-carbon travel is promoted.

Now, there are many well-known shared-bike brands in the world, such as Mobike (China), LimeBike (USA), Getbike(India), Velobike (Russia), oBike (Singapore), Urbo (Ireland)and so on. In order to enhance the market competitiveness,numerous shared-bike companies begin to pay more and more attention to the purchases of shared-bike. Moreover, for demonstrating their social responsibility, the shared-bike companies encourage shared-bike suppliers to utilize green raw materials, green energy and green design. Obviously, this constitutes a typical MCDM problem, and our purpose is to utilize the developed novel GSS method to acquire the better shared-bike supplier for these companies.

B. The Application of Presented Method

TABLE III DECISION MATRIX GIVEN BY GOVERNMENTAL OFFICIAL G1

TABLE IV DECISION MATRIX GIVE N BY PROCUREMENT MANAGER G2

TABLE V DECISION MATRIX GIVEN BY GREEN PRODUCT DESIGNER G3

TABLE VI DECISION MATRIX GIVEN BY GREEN SUPPLY CHAIN PROFESSIONAL G4

C. Exploration of Influences of Parameters

1) The Influences of Prioritized Attitudinal Character T

For the different values of prioritized attitudinal characterT,the corresponding quantitiesθi(i=1,2,3) can be obtained by(27). Then, the ranking results can be gotten and shown in Fig 3.Then, we can know the optimal shared-bike supplier remainsχ2no matter what value the prioritized attitudinal characterTis. In [45], we know the prioritization degrees of criteria can be adjusted by parameterTwhich can represent the psychology of the decision-makers. Generally, the higher the value ofT, the more optimistic the DMs show. On the contrary, the lower the value ofT, the more pessimistic the DMs show. So, in real MCDM problems, based on the actual situations, the DMs can select some different values ofTto show their interests.

Fig. 3. d(ci(χ),) of χi (i=1,2,3,4) by the proposed method with parameter T.

2) The Influences of Parameterspandq

For different values ofpandq, the ranking results are illustrated in Figs. 4–7. From these figures, we can visually know that the ranking results are different owing to the different values ofpandq. That is to say, when thepandqare set different values, it means the different interrelationships between criteria can be captured, the ranking results are different. For instance, whenp= 2 andq= 5, the ranking result becomes χ2?χ4?χ1?χ3. Nevertheless, the optimal shared-bike supplier still remainsχ2.

Fig. 4. d(c1(χ),) of χ1 based on the novel GSS method.

Fig. 5. d(c2(χ),) of χ2 based on the novel GSS method.

Fig. 6. d(c3(χ),) of χ3 based on the novel GSS method.

Fig. 7. d(c4(χ),) of χ4 based on the novel GSS method.

Moreover, the smaller the values ofpandq, the smaller the values of the signed distance function. As a rule, the higher the values ofpandq, the more complex to compute, but it means that greater weight is given on the interrelationships between criteria. Different decision-makers have different attitudes toward risk. In [47], we know thepandqcan be applied to represent the risk preference of the decisionmakers. So, in real MCDM problems, based on the actual situations, the decision-makers can select some different values ofpandqto show their RP. When decision-maker is a risk seeker, smaller values can be set topandq. On the contrary, when decision-maker is a risk reverse, greater values can be set topandq. As a rule, we setp=q= 1, which is not only simple and effortless to calculate, but also reflects the interrelationships of the criteria.

D. Comparative Analysis

TABLE VII RANKING RESULTS OF DIFFERENT METHODS

TABLE VIII COMPARISONS OF CHARACTERISTICS FOR DIFFERENT METHODS

4) The presented method not only considers the psychology of decision-makers, but also considers the risk preference. The DMs can choose some different values ofT,pandqto show their individual personalities and risk preferences.

In short, the novel GSS method can simultaneously focus on the interactions, interrelationships and priority ranks over the criteria by combining PA operator and CI operator with BM operator. A comprehensive comparison of the characteristics for these methods is shown in Table VIII.

VII. CONCLUSION

The supplier selection is a crucial task for enterprise. GSS is playing more and more vital role in supplier selection. MCDM methods can enable enterprises to address this issue more easily. In this study, firstly, we respectively develop IT2FPCNWBM operator with FM and GPM by combining PA operator and CI operator with BM operator, such as IT2FPCNWBM- λ-FM operator and IT2FPCNWBM-GPM operator. Thereafter, a GSS method based on the developed operator is proposed. In the end, a real case of shared-bike GSS is conducted to illustrate the applicability and practicability of the novel GSS method, and a richer comparative analysis is utilized to explain the superiority and feasibility of the novel GSS method.

In future studies, some other aggregation operators, such as spherical, quadratic, harmonic, Hamy, Muirhead, Maclaurin symmetric mean, Heronian operator, can be applied to integrate the IT2FSs, and then to deal with the GSS problems.In addition, it is necessary to study the consensus reaching process of group decision making under fuzzy environment[48]. In the meantime, we can also utilize the developed novel method to process other real GSS problems, such as shared umbrella, shared power bank, shared logistics service, and so on.