VIKOR method for multiple criteria group decision making under 2-tuple linguistic neutrosophic environment

Abstract In this article, the VIKOR method is proposed to solve the multiple criteria group decision making (MCGDM) with 2-tuple linguistic neutrosophic numbers (2TLNNs). Firstly, the fundamental concepts, operation formulas and distance calculating method of 2TLNNs are introduced. Then some aggregation operators of 2TLNNs are reviewed. Thereafter, the original VIKOR method is extended to 2TLNNs and the calculating steps of VIKOR method with 2TLNNs are proposed. In the proposed method, it’s more reasonable and scientific for considering the conflicting criteria. Furthermore, the VIKOR are extended to interval-valued 2-tuple linguistic neutrosophic numbers (IV2TLNNs). Moreover, a numerical example for green supplier selection has been given to illustrate the new method and some comparisons are also conducted to further illustrate advantages of the new method.

For MADM problems, the way to express the assessment information is only one aspect, another vital aspect is selecting best alternative from a given alternative set. In previous document, some traditional decision making model had been applied to MADM problems, such as the EDAS model (Keshavarz Ghorabaee, Zavadskas, Olfat, & Turskis, 2015), the MABAC model (Pamucar & Cirovic, 2015), the COPRAS model (Roy, Sharma, Kar, Zavadskas, & Saparauskas, 2019), the TOPSIS model (Chen, 2000;Lai, Liu, & Hwang, 1994), The TODIM model (Gomes & Lima, 1979) and the GRA model (Li & Wei, 2014). As a powerful tool for handling MADM, The VIKOR (VIseKriterijumska Optimizacija I KOmpromisno Resenje) method (Opricovic & Tzeng, 2004), which owns precious merits of considering the compromise between group utility maximization and individual regret minimization, has been regards as a meaningful tool to apply in many decision making fields in past few years.
Comparing with these above mentioned methods, the VIKOR model has the advantage of taking the compromise between group utility maximization and individual regret minimization into consideration. Du and Liu (2011) extended the traditional VIKOR model to intuitionistic trapezoidal fuzzy environment. Park, Cho, and Kwun (2011) established the interval-valued intuitionistic fuzzy VIKOR model for MADM problems. Qin, Liu, and Pedrycz (2015) proposed an extension of VIKOR model based on interval type-2 fuzzy information. Based on extended hesitant fuzzy linguistic information, Ghadikolaei, Madhoushi, and Divsalar (2018) built new extended VIKOR model for MADM problems. Narayanamoorthy, Geetha, Rakkiyappan, and Joo (2019) developed an extended VIKOR model based on interval-valued intuitionistic hesitant fuzzy entropy for industrial robots selection. Yang, Pang, Shi, and Wang (2018) defined the linguistic hesitant intuitionistic VIKOR model for MADM. Wang, Zhang, Wang, and Li (2018) proposed the projection-based VIKOR model under picture fuzzy environment and applied it for the risk evaluation of construction project. Wu, Xu, Jiang, and Zhong (2019) presented the VIKOR model based on hesitant fuzzy linguistic term sets with possibility distributions.
According to above literature review, we can obtain that the 2-tuple linguistic neutrosophic set (2TLNS) can express the assessment information easily and reasonably, the VIKOR method can consider the conflicting criteria. Thus, to combine these two advantages, we shall propose some extended VIKOR models with 2TLNNs. The structure of our paper is organized as follows. Section 2 introduces the concepts, operation formulas, distance calculating method and some aggregation operators of 2TLNNs. Section 3 extends the original VIKOR method to 2TLNNs and introduce the calculating steps of VIKOR method with 2TLNNs. Section 4 extends the VIKOR method to IV2TLNNs and develops the calculating steps of VIKOR method with IV2TLNNs. Section 5 provides a numerical example for green supplier selection and introduces the comparison between our proposed methods with the existing method. Section 6 gives some summaries of our article.

2-tuple linguistic neutrosophic sets
Based on the concepts of 2-tuple linguistic fuzzy set (2TLS) and the fundamental theories of single valued neutrosophic set (SVNS), the 2-tuple linguistic neutrosophic sets (2TLNSs) which firstly defined by  can be depicted as follows.
Definition 1.  Let s 1 , s 2 , :::, s k be a linguistic term set. Any label s i shows a possible linguistic variable, and S ¼ fs 0 ¼ extremelypoor, s 1 ¼ verypoor, s 2 ¼ poor, s 3 ¼ medium, s 4 ¼ good, s 5 ¼ verygood, s 6 ¼ extremelygood:g, the 2TLNSs g can be depicted as: Where D À1 s a , / ð Þ, D À1 s b , u ð Þ and D À1 s v , c ð Þ 2 ½0, k represent the degree of the truth membership, the indeterminacy membership and the falsity membership which are expressed by 2TLNNs and satisfies the condition 0 D À1 s a , / ð Þþ Definition 2.  Assume there are three 2TLNNs Þ , s v , c ð Þ g, the operation laws of them can be defined: ; According to the Definition 2, it's clear that the operation laws have the following properties.

The aggregation operators of 2TLNNs
Definition 5. .

The VIKOR model for 2TLNNs MCGDM problems
Assume that fA 1 , A 2 , :::, A m g be a group of alternatives, fD 1 , D 2 , :::, D k g be a list of experts with weighting vector be fv 1 , v 2 , :::, v t g, and fG 1 , G 2 , :::, G n g be a list of criteria with weighting vector be fx 1 , x 2 , :::, x n g, thereby satisfyingx i 2 ½0, 1, v i 2 ½0, 1 and P n i¼1 x i ¼ 1, P t i¼1 v i ¼ 1: Construct the evaluation matrix g k ¼ ½g k ij mÂn , i ¼ 1, 2, :::, m, j ¼ 1, 2, :::, n, k ¼ 1, 2, :::, t, where k g means the estimate results of the alternative A i i ¼ 1, 2, :::, m ð Þbased on the criterion G j j ¼ 1, 2, :::, n ð Þby expert D k k ¼ 1, 2, :: :::, m, j ¼ 1, 2, :::, n, k ¼ 1, 2, :::, tÞ: Consider both the 2TLNNs theories and the traditional VIKOR model; we try to propose the VIKOR method with 2TLNNs to study MCGDM problems effectively. The method can be depicted as follows: Step 1. Construct the decision matrix g k ¼ ½g k ij mÂn , and utilize overall values of g k ¼ ½g k ij mÂn to g ¼ ½g ij mÂn by using equal (10) or (11); Step 2. Compute the positive ideal solution (PIS) A þ and the negative ideal solution (NIS)A À ; For benefit attribute For cost attribute Step 3. Based on the Equation (7) and the attribute weighting vector x j , we can calculate the values of s i and w i which express the average and the worst group scores of g i : where d is the normalized Hamming distance and 0 x j 1 means the weight of attributes which satisfies P n i¼1 x i ¼ 1: Step 4. Compute the values of Q i based on the results of s i and w i , the calculating formula is characteristic as follows.
where q means the coefficient of decision making strategic. q>0:5 depicts "the maximum group utility", q ¼ 0:5 depicts equality and q<0:5 depicts the minimum regret.
Step 5. To choose the best alternative by rank the values of Q i , the alternative with minimum value is the best choice.

The IV2TLNSs
To solve MCDM problems more effectively, we extend the 2TLNSs to interval-valued environment to propose the IV2TLNSs as follows.
Definition 6. Let s 1 , s 2 , :::, s k be a linguistic term set. Any label s i shows a possible linguistic variable, the IV2TLNSsg can be depicted as: where , k represent the degree of the truth membership, the indeterminacy membership and the falsity membership which are expressed by IV2TLNNs and satisfies the condi- an  IV2TLNN, the score and accuracy functions ofg can be expressed: For two IV2TLNNsg 1 andg 2 , based on the Definition 7, then Þ U g be two IV2TLNNs, then we can get the normalized Hamming distance: the Hamming distance d also has the following properties: ðP1Þ 0 dðg 1 ,g 2 Þ 1; ðP2Þ if dðg 1 ,g 2 Þ ¼ 0, theng 1 ¼g 2 ; ðP3Þ dðg 1 ,g 2 Þ ¼ dðg 2 ,g 1 Þ; ðP4Þ dðg 1 ,g 2 Þ þ dðg 2 ,g 3 Þ ! dðg 1 ,g 3 Þ: 4.2. The aggregation operators of IV2TLNNs Þ U g be a group of IV2TLNNs, then the IV2TLNNWA and IV2TLNNWG operators can be defined as follows.

Computing steps for MCGDM problems with IV2TLNNs
Assume that fA 1 , A 2 , :::A m g be a group of alternatives and fG 1 , G 2 , :::G n g be a list of criteria with weighting vector be fx 1 , x 2 , :::x n g, thereby satisfying x i 2 ½0, 1 and P n i¼1 x i ¼ 1: Construct the evaluation matrixg ¼ ½g ij mÂn , i ¼ 1, 2, :::, m, j ¼ 1, 2, :::, n Þ U g means the estimate results of the alternative A i i ¼ 1, 2, :::, m ð Þ based on the criterion G j j ¼ 1, 2, :::, n ð Þ : The calculating steps also can be depicted as follows: Step 1. Construct the decision matrixg ¼ ½g ij mÂn ; Step 2. Compute the positive ideal solution A þ and the negative ideal solution A À ; > > > > > > > = > > > > > > > ; ðj ¼ 1, 2, :::, nÞ (29) > > > > > > > = > > > > > > > ; ðj ¼ 1, 2, :::, nÞ For benefit attribute For cost attribute Step 3. Based on the Equation (26) and the attribute weighting vector x j , we can calculate the values ofs i andw i which express the average and the worst group scores ofg i : where d is the normalized Hamming distance and 0 x j 1 means the weight of attributes which satisfies P n i¼1 x i ¼ 1: Step 4. Compute the values of Q i based on the results ofs i andw i , the calculating formula is characteristic as follows.
where q means the coefficient of decision making strategic. q>0:5 depicts "the maximum group utility", q ¼ 0:5 depicts equality and q<0:5 depicts the minimum regret.
Step 5. To choose the best alternative by rank the values of Q i , the alternative with minimum value is the best choice.

Numerical for 2TLNNs MCGDM problems
After China's entering into the WTO, the economy has developed rapidly and it has held a high rate of economic growth. But the stamina of development is confronted with severe challenges: On the one hand, the international economic situation is continuously changing and many enterprises in China are limited by international green barriers; On the other hand, while enjoying great economic development achievements, people also realized that our country's environment and resources are becoming more and more serious. While China's economic development is growing at a high speed, the ecological environment and natural resources have been seriously injured and the contradiction between natural resource environment and social economic development has become increasingly obvious. Under the background of people's urgent need for environmental protection and healthy living, many enterprises in China are aware of the necessity and importance of green health and low carbon environmental protection for the survival and development of enterprises. Green suppliers selection is a classical MADM problem. In this chapter, we provide a numerical example to select best green suppliers selection by using VIKOR method with 2TLNNs. Assume that five possible green suppliers A i i ¼ 1, 2, 3, 4, 5 ð Þto be selected and four criteria to assess these green suppliers: ‹ G 1 is the product quality factor; › G 2 is environmental factors; fi G 3 is delivery factor; fl G 4 is price factor. The five possible green suppliers A i i ¼ 1, 2, 3, 4, 5 ð Þare to be evaluated with 2TLNNs with the four criteria by three experts (criteria weight x ¼ 0:32, 0:13, 0:35, 0:20 ð Þ , experts weight v ¼ 0:25, 0:35, 0:40 ð Þ :), which are given in Tables 1-3.
Step 1. Utilize overall values of g k ¼ g k ij h i mÂn to g ¼ g ij ½ mÂn by using 2TLNNWA operator, the aggregation results are listed in Table 4.    Step 2. Compute the values of A þ (PIS) and A À (NIS), for all attributes are benefit and based on the formula (16) and (17), we can obtain the (PIS) A þ and (NIS)A À as follows. Step 3. Based on the Equation (7)      Step 5. To choose the best alternative by rank the values of Q i , the ranking of Q i is Q 1 >Q 3 >Q 4 >Q 2 >Q 5 , and the best choice is g 1 : By altering the parameter q, we can derive the following results which listed in Table 5. From Table 5, we can easily find that the ordering of alternatives are same, which indicates our developed method has the robustness and can be applied to deal with practical decision making problems.

Comparative analyses
In this section, we compare our proposed VIKOR method under 2TLNNs with the 2TLNNWA and 2TLNNWG operators defined by . Based on the values of Table 4 and attributes weighting vector x ¼ 0:32, 0:13, 0:35, 0:20 ð Þ T , we can compute overall value g i by 2TLNNWA and 2TLNNWG operators.
We can get calculating results g i by 2TLNNWA operator:  Then, we calculate the alternative scores s g i ð Þ by score functions of 2TLNNs which are listed in Table 6.
The ranking of alternatives by 2TLNNWA and 2TLNNWG operators are listed in Table 7.
Compare the values of our proposed VIKOR method under 2TLNNs with 2TLNNWA and 2TLNNWG operators, the results are slightly different in ranking of alternatives and the best alternatives are same, VIKOR method with 2TLNNs can consider the conflicting attributes and can be more reasonable and scientific in the application of MCGDM problems.

Numerical case for MCDM problems with IV2TLNNs
In this chapter, if the evaluation values of five green suppliers are depicted by IV2TLNNs, then we can study the MCDM problems by using the VIKOR method with IV2TLNNs, the decision matrix are listed in Table 8 (attribute weighting vector x ¼ 0:4, 0:2, 0:1, 0:3 ð Þ T ).
Step 1. Construct the decision matrix (See Table 8) Step 2. Compute the values of A þ (PIS) and A À (NIS), for all attributes are benefit and based on the formula (18) and (19), we can obtain the (PIS) A þ and (NIS)A À as follows.   Step 3. Based on the Equation (11)  Step 4. Compute the values of Q i based on the results ofs i andw i , the calculating values are listed as follows. (Let q ¼ 0:4) Q 1 ¼ 0:0000, Q 2 ¼ 0:5024 , Q 3 ¼ 0:6274, Q 4 ¼ 0:4150, Q 5 ¼ 1:0000: Step 5. To choose the best alternative by rank the values of Q i , the ranking of Q i is Q 1 >Q 4 >Q 2 >Q 3 >Q 5 , and the best choice isg 1 : By altering the parameter q, we can derive the following results which listed in Table 9. From Table 9, we can easily find that the ordering of alternatives are same, which indicates our developed method has the robustness and can be applied to deal with practical decision making problems.

Comparative analyses
In this section, we compare our proposed the extend VIKOR method under IV2TLNNs with the IV2TLNNWA and IV2TLNNWG operators. Based on the values of Table 4 and attributes weighting vector x ¼ 0:4, 0:2, 0:1, 0:3 ð Þ T , we can utilize overallg ij by IV2TLNNWA and IV2TLNNWG operators.

Discussion
Based on above two numerical examples, we can easily find our proposed methods can express more fuzzy information and apply broadly situations in real MCGDM problems. Based on 2-tuple linguistic neutrosophic fuzzy set (2TLNS) and traditional VIKOR method, we develop the 2-tuple linguistic neutrosophic VIKOR method and the interval-valued 2-tuple linguistic neutrosophic VIKOR method; our research results can be more suitable for MCGDM problems than single-valued neutrosophic VIKOR method depicted in literature (Huang, Wei, & Wei, 2017). For the single-valued neutrosophic VIKOR method can't deal with MCGDM problems which the assessment results are depicted with 2TLNNs. Furthermore, in complicated decision-making environment, the decision maker's risk attitude is an important factor to think about. the VIKOR methods, which consider the compromise between group utility maximization and individual regret minimization, can make this come true by altering the parameters whereas other decision making ways such as the 2TLNNWA operator, the 2TLNNWG operator, the IV2TLNNWA operator and the IV2TLNNWG operator don't have the ability that dynamic adjust to the parameter according to the decision maker's risk attitude, so it is difficult to solve the risk multiple attribute decision making in real practice.

Disclosure statement
No potential conflict of interest was reported by the authors.