An Extended EDAS Method with a Single-Valued Complex Neutrosophic Set and Its Application in Green Supplier Selection .

: The single-valued complex neutrosophic set is a useful tool for handling the data with uncertainty and periodicity. In this paper, a single-valued complex neutrosophic EDAS (evaluation based on distance from average slution) model has been established and applied in green supplier selection. Firstly, the deﬁnition of single-valued complex neutrosophic set and corresponding operational laws are brieﬂy introduced. Next, to fuse overall single-valued complex neutrosophic information, the SVCNEWA and SVCNEWG operators based on single-valued complex neutrosophic set, Einstein product and sum are proposed. Furthermore, the single-valued complex neutrosophic EDAS model has been established and all computing steps have been depicted in detail. Finally, a numerical example of green supplier selection and a comparison analysis have been given to illustrate the practicality and effectiveness of this new model.


Introduction
With the growth of the world economy, more and more companies are being founded. Some of the most significant competition among modern enterprises is in their supply chains, and a key of supply chain management is supplier selection. Suppliers play an important role in high quality products and customer satisfaction. A preeminent supplier can improve the competitiveness of the enterprise [1]. Meanwhile, enterprises should consider significant environmental issues, such as the green effect, and stress suffered from the government, associations and the public [2]. So, green supplier selection (GSS) has been proposed, which is a construct that can supervise supplier performance along with green technical standards [3], and green supply chain management (GSCM) [4] as become an emerging field whose aim is to find a balance between the economy and environment. Numerous researches and scholars have studied this popular topic over the past several years. Zhang et al. [5] established a nonlinear multi-objective optimization model to deal with GSS problem and used a Pareto genetic algorithm to solve the problem. Hosseini and Barker [6] developed some resilience-based supplier selection criteria and a Bayesian network to present an innovative decision method for GSS which can address the risk and uncertainty in decision making problem.
In practical GSCM, supply chain managers need to consider all suppliers with many conflicting attributes, and consider the trade-off to select the optimal supplier(s). Therefore, GSS is commonly regarded as a multi-attribute group decision making (MAGDM) problem. Up to now, there are many researchers have studied the issue. Based on the best-worst method (BWM) and Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) technique, Wu et al. [7] proposed a complex neutrosophic EDAS model is established and the computing steps are listed in Section 4. In Section 5, an example is given to illustrate the application of proposed method. In Section 6, a conclusion of this paper is given.

Single-Valued Neutrosophic Set
Definition 1. [22] Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function T A (x), an indeterminacy-membership function I A (x) and a falsity-membership function There is no restriction on the sum of T X , I x and Fx, so

Definition 2.
[23] Let X be a space of points (objects), with a generic element in X denoted by x. A single-valued neutrosophic set A in X is characterized by a truth-membership function T A (x), an indeterminacy-membership function I A (x) and a falsity-membership function Then, a single-valued neutrosophic set A can be denoted as

Single-Valued Complex Neutrosophic Set
Definition 3. [27] Let X be a space of points (objects), with a generic element in X denoted by x. A single-valued complex neutrosophic set S in X is characterized by a truth-membership function T S (x), an indeterminacy-membership function I S (x) and a falsity-membership function F S (x) that assigns a complex-valued membership grade to T S (x), I S (x) and F S (x) for all x ∈ X. The values of T S (x), I S (x), F S (x) and their sum may fall within the unit circle in the complex plane; the former is of the following form: For convenience a single-valued complex neutrosophic set S can be represented in set form as: and a single-valued neutrosophic number (SVCNN) can be denoted as S = T S , I S , F S which is a basic unit of single-valued complex neutrosophic set.
x ∈ X} be a SVCNS in X. Then, the complement of S is denoted as c (S) and specified by functions:

Definition 5. Let A and B be two SVCNSs which are defined by T
, respectively. Then, the operational rules of SVCNSs are defined as follows: (1) The sum of A and B, denoted as A ⊕ B, is defined as (2) The product of A and B, denoted as A ⊗ B, is defined as (3) The scalar multiplication of A is a single-valued complex neutrosophic set denoted as C = λA (λ > 0), defined as: The power of A is denoted as D = (A) λ (λ > 0), and defined as: λ Definition 6. Let X = {x 1 , x 2 , · · · , x n } be a universal of objects, A and B be two SVCNSs in X, and then the normalized Hamming distance between A and B is: Definition 7. Let A be a SVCNN; then, the score function S (A) of A is defined as: Definition 8. Let A be a SVCNN, and then the score function H (A) of A is defined as: Definition 9. let A 1 and A 2 be two SVCNNs, and S be the score function, H be the accuracy function.

The Einstein Operator with SVCNNs
The Einstein operator plays a significant role in as an aggregation operator. It consists of the Einstein product ⊗ E and the Einstein sum ⊕ E , where ⊗ E is a t-norm and ⊕ E is a t-conorm. They are defined as follows: [38] Γ In the following, we shall propose single-valued complex neutrosophic set operational rules based on the Einstein operator.

Definition 10. Let A and B be two SVCNSs which are defined by T
, respectively. Then, the Einstein operational rules of SVCNSs are defined as follows: (1) n) be a collection of SVCNNs, the single-valued, complex, neutrosophic, Einstein-weighted average (SVCNEWA) operator and single-valued, complex, neutrosophic, Einstein-weighted geometric (SVCNEWG) operator can be defined as follows: and where w i is the weight of A i with the condition w i ∈ [0, 1] and n ∑ i=1 w i = 1.
n) be a collection of SVCNNs, then the operation results by SVCNEWA and SVCNEWG operators are also a SVCNN where: Proof. In this part, we can prove the above theorem by mathematical induction.
(1) When n = 2, we have Therefore, when n = 2, the equation is true. (2) Assume that when n = k, Equation (10) is true. Then when n = k + 1, we have Thus, when n = k + 1, the equation is true.
So we can calculate SVCNEWA (A 1 , (w i A i ) for any n. In the same way, we can obtain the form of SVCNEWG operator shown in Theorem 1.

The EDAS Method with SVCNNs
In this section, a MAGDM approach by combining the proposed operators and EDAS method is presented.
Suppose there is a committee of r experts {E 1 , E 2 , · · · , E r } with the weight vector of experts Step 1: Construct the evaluation matrix of expert E d , and denote it as A d = a d ij m×n , i = 1, 2, · · · , m, j = 1, 2, · · · , n, d = 1, 2, · · · , r where a d ij is a SVCNN and represents the single-valued complex neutrosophic information of alternative A i versus attribute C j by expert E d .
Step 2: Normalize the evaluation matrix A d = a d ij m×n Step 3: According to the normalized decision making matrix A d = a d ij m×n and the weight vector v = {v 1 .v 2 , · · · , v r } of experts, we can fuse overall a d ij into a ij by using the SVCNEWA or SVCNEWG operator; then the aggregated decision making matrix A = a ij m×n with aggregated information is obtained and denoted as A = a ij m×n = T ij · e jω ij , I ij · e jψ ij , F ij · e jφ ij m×n .
Step 4: Compute the value of the average solution AV = AV j 1×n where Based on Definition 5, we can get So Step 5: Compute the positive distance from average PDA = P ij m×n and the negative distance from average NDA = N ij m×n , where s AV j and s a ij are score functions of AV j and a ij , respectively.
Step 6: Calculate the values of SP i and SN i which denote the weighted sums of PDA and NDA.
and w j is the weight of the j − th attribute.
Step 7: Normalize the values SP i and SN i to obtain NSP i and NSN i .
Step 9: Ranking the alternatives according to the values of AS i . The alternative with the highest AS i is the optimal one.

The Numerical Example for SVCNS MAGDM Problem
In this section, we provide a numerical example to select the best green supplier by using the proposed MAGDM approach.
Consider a small-sized trading service and transportation company who wants to seek a langfristig green supplier to purchase a new vehicle for its follow-up operation. The company will assess three potential suppliers A 1 , A 2 , A 3 . A managing committee E forms a group of three decision makers E 1 , E 2 , E 3 with different professional skills for the evaluation, and the decision makers' weight vector is v = {0.3, 0.2, 0.5}. During the selection process, there are five attributes to consider. They are: price/cost (C 1 ), quality (C 2 ), delivery (C 3 ), relationship closeness (C 4 ) and environmental management systems (C 5 ); and the corresponding weight vector is w = {0.2, 0.3, 0.25, 0.15, 0.1}. Meanwhile, the five attributes are benefit-type attributes. Three decision makers determined the suitability ratings of three potential suppliers versus the attributes by using the linguistic rating set S = {VL, L, F, G, VG} where The three suppliers are to be evaluated with SVCNNs which are listed in Tables 1-3.   Table 3. The evaluated values of decision maker E 3 .
To obtain the best green supplier, we utilize the proposed approach to evaluate the three suppliers.
Step 1: Aggregate the information ratings. According to the decision makers' weight vector v = {0.3, 0.2, 0.5}, we can obtain the aggregated decision matrix which is shown in Table 4 by using SVCNEWA operator .
Step 2: Compute the average solution AV = AV j 1×n . According to Equation (11), we can obtain:  Step 3: Calculate the score function of each evaluated value and average solution which is shown in Table 5. Table 5. The score values of a ij and AV j .
Step Step 6: Normalize the SP i and SN i .
Step 7: Calculate the appraisal score AS i .
Obviously, according to the values of AS i , we can rank all the alternatives as A 3 > A 2 > A 1 and A 3 is the best choice.

A Comparison Analysis
In this part, we make a simple comparative analysis. In Section 5.1 we aggregate the information ratings by using SVCNEWA operator, so in this part, we use another operator (SVCNEWG operator) and an EDAS method to obtain the best alternative(s). Meanwhile, we make a comparison through the results obtained by different aggregation operators. According to the steps in Section 4, we can obtain the values of appraisal score and the rankings of alternatives which are shown in Table 6. Table 6. The values of AS i and the rankings of alternatives.
From the results in Table 6, we can find that the values of appraisal score are slightly different, but the ranking of alternatives and the best alternative are the same, which indicate that the proposed method is practical and effective forthe MAGDM problem.

Conclusions
In modern enterprises, one of the most significant competitions is for green supply chain management, and green supplier selection is a vital factor of green supply chain management. So how to determine the optimal supplier plays an important role. In real life, green supplier selection can be regarded as a MAGDM problem. Therefore, in this paper, a single-valued complex neutrosophic EDAS model has been established and applied in green supplier selection. Meanwhile, considering that green supplier selection is a group decision making problem, two aggregation operators, namely, a single-valued, complex, neutrosophic, Einstein-weighted average operator and a single-valued, complex, neutrosophic, Einstein-weighted geometric operator have been proposed to fuse overall information ratings into a comprehensive value. Thus, in order to achieve these purposes, this paper firstly introduces the definition of the single-valued complex neutrosophic set and the corresponding operational laws. Next, to fuse overall single-valued complex neutrosophic information, some new aggregation operators of single-valued complex neutrosophic set based on Einstein product and sum, namely, SVCNEWA and SVCNEWA operators, have been proposed. Furthermore, the single-valued complex neutrosophic EDAS model has been established to solve MAGDM problem. Finally, a numerical example for green supplier selection and a comparison have been given. Although the aggregation operators are different in EDAS method, the rankings of alternatives and of the optimal one are the same, which illustrates the practicality and effectiveness of this new model. However, this model only considers the case of known or subjective decision makers' weight vector and attributes' weight vector; it does not discuss the unknown or objective weight vector. So in the future, some models need to be established to obtain a more objective weight vector and more reasonable evaluation results. Meanwhile, it is necessary to apply this single-valued complex neutrosophic EDAS method into different fields, such as venture capital, pattern recognition and comprehensive evaluation.

Conflicts of Interest:
The authors declare no conflict of interest.