Models for Multiple Attribute Decision Making with Some 2-Tuple Linguistic Pythagorean Fuzzy Hamy Mean Operators

: The Hamy mean (HM) operator, as a useful aggregation tool, can capture the correlation between multiple integration parameters, and the 2-tuple linguistic Pythagorean fuzzy numbers (2TLPFNs) are a special kind of Pythagorean fuzzy numbers (PFNs), which can easily describe the fuzziness in actual decision making by 2-tuple linguistic terms (2TLTs). In this paper, to consider both Hamy mean (HM) operator and 2TLPFNs, we combine the HM operator, weighted HM (WHM) operator, dual HM (DHM) operator, and dual WHM (DWHM) operator with 2TLPFNs to propose the 2-tuple linguistic Pythagorean fuzzy HM (2TLPFHM) operator, 2-tuple linguistic Pythagorean fuzzy WHM (2TLPFWHM) operator, 2-tuple linguistic Pythagorean fuzzy DHM (2TLPFDHM) operator and 2-tuple linguistic Pythagorean fuzzy DWHM (2TLPFDWHM) operator. Then some multiple attribute decision making (MADM) procedures are developed based on these operators. At last, an applicable example for green supplier selection is given.

Although, PFSs theory has been broadly applied to many domains, however, all the above approaches are unsuitable to depict the MD and NMD of an element to a set by 2TLSs.In order to overcome this issue, we develop the definition of 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs) based on the PFS [1,2] and 2-tuple linguistic terms (2TLTs) [68][69][70][71].And HM operator [72] and DHM operator [73] are famous aggregation operators which can depict interrelationships among any number of arguments assigned by a variable vector.Therefore, the HM and DHM operators can supply a robust and flexible mechanism to deal with the information fusion in MADM problems.Because 2-tuple linguistic Pythagorean fuzzy numbers (2TLPFNs) can easily describe the fuzzy information, and the HM and DHM operators can capture interrelationships among any number of arguments assigned by a variable vector, it is necessary to extend the HM and DHM operators to deal with the 2TLPFNs.
The aim of this paper is to combine the HM operator, weighted HM (WHM) operator, dual HM (DHM) operator, and dual WHM (DWHM) operator with 2TLPFNs to propose the 2-tuple linguistic Pythagorean fuzzy HM (2TLPFHM) operator, 2-tuple linguistic Pythagorean fuzzy WHM (2TLPFWHM) operator, 2-tuple linguistic Pythagorean fuzzy DHM (2TLPFDHM) operator and 2-tuple linguistic Pythagorean fuzzy DWHM (2TLPFDWHM) operator.Then some multiple attribute decision making (MADM) procedures are developed based on these operators.At last, an applicable example for green supplier selection is given.In order to do so, the rest of this paper is organized as follows.In Section 2, we develop the 2TLPFSs.In Section 3, we develop HM and DHM operators with 2TLPFNs.In Section 4, we present an example for green supplier selection.Conclusions are given in Section 5.

2TLTs
Definition 1. [68,69].Let S = {s i |i = 0, 1, . . ., t } be a linguistic term set with odd cardinality, where s i represents a possible value for a linguistic variable, and it should satisfy the following characteristics: (1) The set is ordered: s i > s j , if i > j; (2) Max operator: max s i , s j = s i , if s i ≥ s j ; (3) Min operator: min s i , s j = s i , if s i ≤ s j .For example, S can be defined as S = s 0 = extremely poor, s 1 = very poor, s 2 = poor, s 3 = medium, s 4 = good, s 5 = very good, s 6 = extremely good.
Herrera and Martinez [68,69] developed the 2-tuple fuzzy linguistic representation model based on the concept of symbolic translation.It is used for representing the linguistic assessment information by means of a 2-tuple (s i , ρ i ), where s i is a linguistic label for predefined linguistic term set S and ρ i is the value of symbolic translation, and ρ i ∈ [−0.5, 0.5).

PFSs
Let X be an ordinary fixed set, denoted by x.A Pythagorean fuzzy set (PFS) A in X is defined as following [1,2]: where the membership function u A (x) and the non-membership function v A (x) For a PFS {(x, u A (x), v A (x))|x ∈ X }, the ordered twofold components {u A (x), v A (x)}, are described as a Pythagorean fuzzy number (PFN), and each PFN can be expressed as

2TLPFHM Operator
In this section, we expand HM operator with 2TLPFNs and propose the 2-tuple linguistic Pythagorean fuzzy HM (2TLPFHM) operator.

Definition 7.
Let δ j = s ϕ j , φ j , s ϑ j , θ j (j = 1, 2, . . ., n) be a group of 2TLPFNs.The 2TLPFHM operator is: . ., n) be a group of 2TLPFNs.The fused result by 2TLPFHM operators is also a 2TLPFN where Proof.From the basic operations on 2TLPFN which are defined in Definition 5, we can get: Thus, x ⊗ j=1 δ i j Thereafter, Therefore, Hence, ( 7) is kept.
Then we need to give the proving process of that ( 7) is also a 2TLPFN.
From Property 1,
Then we shall prove that ( 23) is a 2TLPFN.We shall prove these two conditions.

Property 4.
(Monotonicity) Let δ a j = s ϕ a j , φ a j , s ϑ a j , θ a j (j = 1, 2, . . ., n) and , θ b j hold for al l j, then The proof is similar to 2TLPFWHM, it's omitted.

The 2TLPFDHM Operator
Wu et al. [73] proposed the DHM operator.Definition 9. [73].The DHM operator is shown as follows: where x is a parameter and x = 1, 2, . . ., n, i 1 , i 2 , . . ., i x are x integer values taken from the set {1, 2, . . . ,n} of k integer values, C x n denotes the binomial coefficient and C x n = n!
Then we shall prove that ( 39) is a 2TLPFN.We shall prove these two conditions.

Numerical Example
With the damage to the human environment and the earth's resources increasingly depleted, the traditional supply chain has been gradually not well adapted to the current needs of the times and social demand, thus, the concept of "green supply chain" was introduced.The construction of green supply chain has become the main challenges and trends to provide a green product and towards sustainable development of society for the current time, in which an important part of the core content and implementation of green supply chain is green supplier evaluation and selection, especially sustainable suppliers in line with environmental protection requirements.Since the selection of suppliers plays a decisive role in the green SCM, which directly determines the optimization and core competitiveness of the entire chain of the enterprise, therefore, how to efficiently identify a required supplier from a number of suppliers is a key issue in modern green supply chain management must be solved.Then we propose a example to select green suppliers with 2TLPFNs.There are five possible green suppliers in SCM A i (i = 1, 2, 3, 4, 5) to select.The experts group uses four attribute to assess the five green suppliers: 1 G 1 is the price factor; 2 G 2 is the delivery factor; 3 G 3 is the environmental factors; 4 G 4 is the product quality factor.The five green suppliers A i (i = 1, 2, 3, 4, 5) are to be assessed with 2TLPFNs (whose weighting vector ω = (0.26, 0.35, 0.21, 0.18), expert weighting vector ω = (0.25, 0.35, 0.40).), which are listed in Tables 1-3.

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Step 2. According to Table 4, we fuse all 2TLPFNs r ij by 2TLPFWHM (2TLPFDWHM) operator to derive the overall 2TLPFNs of the green suppliers A i .Let x = 3, then the fused results are listed in Table 5.
Table 5.The fused results of the green suppliers by 2TLPFWHM (2TLPFDWHM) operator.

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Step 3.According to Table 5 and the scores are shown in Table 6.

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Step 4. According to Table 6, the order is shown in Table 7 and the best green supplier is A 2 .
Table 7. Order of the green suppliers.

Influence of the Parameter on the Final Result
In order to depict the effects of parameters x in the 2TLPFWHM (2TLPFDWHM) operators, all the results are listed in Tables 8 and 9.
Table 10.Order of the green suppliers.

LPFWAA [74]
A 2 > A 4 > A 1 > A 5 > A 3 LPFWGA [74] A 2 > A 1 > A 4 > A 3 > A 5 From the above analysis, it can be seen that two methods have the same best green suppliers A 2 and two methods' ranking results are slightly different.This verifies that the 2TLPFWHM and 2TLPFDWHM operators we developed are reasonable and valid for MADM problems with 2TLPFNs.
In what follows, the comparisons of proposed approaches and the other methods with regard to some characteristics are shown in Tables 8-10.In light of Tables 8-10 some conclusions are summarized as follows: (1) The methods developed by Garg [74] aggregate the linguistic Pythagorean fuzzy information easily.The drawbacks of Garg's methods [74] are they assume that the input arguments are not correlated, that is, they fail to consider the relationships between the input arguments.Nevertheless, our developed operators can capture the correlations among all the input arguments, and fuse the 2TLPFNs more flexibly by the parameter vector.Therefore, our developed approaches are more general and flexible comparing with that proposed by Garg's methods [74].(2) Moreover, the methods developed by Garg [74] 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.Nevertheless, our developed operators have the ability that dynamic adjust to the parameter according to the decision maker's risk attitude.Thus, our method can overcome the drawbacks of the methods developed by Garg [74], because the 2TLPFWHM and 2TLPFDWHM operators operator can provides more flexible and robust in information fusion and make it more adequate to solve risk multiple attribute decision making in which the attributes are independent.

Table 4 .
The fused results by the 2TLPFNWA operator.

Table 6 .
The scores of the green suppliers.