q-Rung Orthopair Dual Hesitant Fuzzy Bonferron Mean Operators

Objectives/Methods: Taking into account the impreciseness and subjectiveness of decision makers (DMs) in complex decision-making situations, the assessment datum over alternatives given by DMs is consistently vague and uncertain. In meantime, to evaluate human’s hesitance, the q-rung orthopair dual hesitant fuzzy sets (q-RODHFSs) are defined which are more accurate for manipulation real MADM matters. To merge the datum in q-RODHFSs more precisely, in this research script, some Bonferroni mean (BM) operators in light of q-RODHFSs datum, which includes arbitrary number of being merged arguments, are developed and examined. Findings: Obviously, the novel defined operators can produce much accurate results than already existing methods. Additionally, some important measures of said BM operators are talked about and all the peculiar cases of them are studied which expresses that the BM operator is more dominant than others. Eventually, the MADM algorithm is furnished and the operators are utilized to choose the best alternative under q-rung orthopair dual hesitant fuzzy numbers (q-RODHFNs). Taking advantage of the novel operators and constructed algorithm, the developed operators are utilized in the MADM problems.

measures utilizing PFSs information for use in MADM problems. Wei and Wei (7) introduced a variety of cosine similarity measures for PFSs datum. Yet, practically, there may arise some relationships between more than one arguments, it is clear that previously studied collective operators are not authentic for such purpose. For the solution of such type of problems, the Bonferroni mean (BM) operator (8) as a reputed information collecting tool which have capability to acknowledge the interrelationship of the arguments, have been explored. Liang et al. (9) proposed some BM operators with PFSs information. Most likely, q-ROFS (10) are continuously expansive for the IFS, PFS and these two are its specific cases. Many researchers (11)(12)(13)(14)(15)(16)(17)(18)(19)(20) (21) developed a varriety of operators to aggregate the information presented q-ROFSs and its application in MADM. Taking advantage of the classical q-ROFSs, Liu and Liu (22) derived the definition of q-rung orthopair fuzzy linguistic sets (q-ROFLSs) and developed a few power BM aggregation operators for q-ROFLSs datum. Xu et al. (23) illustrated the concept of the q-rung otrhopair dual hesitant fuzzy set (q-RODHFS) and developed a few q-rung dual hesitant fuzzy HM operators for MADM.
Tang et al. (24) developed few Pythagorean fuzzy power aggregation operators and illustrated the idea of dual hesitant Pythagorean fuzzy sets (DHPFSs), as a combination of the PFSs and dual hesitant fuzzy sets (DHFSs) (25,26) (27) also developed some Hamacher aggregation operators utilizing DHPFSs. Jia et al. (28) developed a wide range of distance measures based on connection numbers of set pair analysis with dual hesitant fuzzy sets. Wang et al. (29) developed MM operators under DHPFSs datum. Apparently, there is no exploration led in light of BM operator to fuse q-RODHF information.
In past few years, numerous investigators studied the BM aggregation operators and their applications. The BM operations have the advantage of considering the relationship between the values being fused, thus the fused results are more reasonable and accurate. Clearly, DHq-ROFN is a meaningful tool to express evaluation information. BM operations are good to fuse evaluation information, so it's worth to develop some BM operators under dual hesitant q-rung orthopair fuzzy environments. The main novelty and contribution of our manuscript is developing some new BM operators to aggregate the dual hesitant q-rung orthopair fuzzy information. Evidently, these operators have the following advantages. (1) The DHq-ROFS can not only extend the scope of the assessment information to depict more fuzzy information, but also consider the human's hesitance, thus it is more useful and reasonable to derive decision-making results. (2) The BM operations can consider the relationship between fused arguments, obviously, BM operations are more suitable for handling practical MADM problems. Thus, it is of great significance to propose some new operators based on the dual hesitant q-rung orthopair fuzzy information and BM operations.
In the following text, we have developed a few BM aggregation operators to intertwine the q-RODHF datum. Furthermore, a portion of their alluring properties have additionally been considered and the unique instances of every operator is researched. At last, in light of these effective operators, a decision-making algorithm have been produced and a computative model is delineated to approve the methodology over some similar investigation with the current methodologies. To do as such, the rest of text is composed as pursues. Some basic knowledge about q-ROFSs, q-RODHFSs and BM have been reviewed in Section 2. In Section 3, we have talked about the BM and dual BM operators utilizing q-RODHFSs condition and then developed the q-rung orthopair dual hesitant fuzzy BM(q-RODHFBM) operator, the q-rung orthopair dual hesitant fuzzy weighted BM (q-RODHFWBM) operator, the q-rung orthopair dual hesitant fuzzy dual BM (q-RODHFDBM) operator and the q-rung orthopair dual hesitant fuzzy weighted dual BM (q-RODHFWDBM) operator. In Section 4, we will manufacture the MADM algorithm with q-RODHFNs. In Section 5, we will solve a numerical model for provider choice with q-RODHFNs and gave some similar investigation. Segment 6, finishes up the discussion with certain comments.

The q-RUNG ORTHOPAIR FUZZZY SET
The essential concepts and basic knowledge of q-rung orthopair fuzzy sets (q-ROFSs) (10) are quickly evaluated as pursues.
Definition 2.1. (10) Let χ be a universal set. A q-ROFS is an item owns the structure The level of indeterminacy is described as: Generally, written as o = (γ, η) a q-ROFN. Definition 2.2. (10) Let o = (γ, η) be a q-ROFN, the score and acuracy function has the form: to analyze the level of accuracy of the q-ROFN o = (γ, η) The bigger the value of T (o), the more the level of accuracy of the q-ROFN o is. Now we describe the comparison rule between two q-ROFNs as pursues: be score values of o 1 and o 2 , respectively, and let T (o 1 ) = γ q 1 + η q 1 and T (o 2 ) = γ q 2 + η q 2 be the accuracy degrees of o 1 and o 2 , respectively, then if Definition 2.5. (10) Let o 1 = (γ 1 , η 1 ), o 2 = (γ 2 , η 2 ) and o = (γ, η) be three q-ROFNs, and some basic operations on them are defined as follows:

The q-RUNG ORTHOPAIR DUAL HESITANT FUZZZY SET
In the light of q-ROFSs (10) and dual hesitant fuzzy sets (25) , (26) Xu et al. (23) introduced the idea and primary operations of the q-rung orthopair dual hesitant fuzzy sets (q-RODHFSs). Definition 2.6. (23) For any universal set χ, a q-rung orthopair dual hesitant fuzzy set (q-RODHFS) on χ is given as: telling the membership (favorable) degrees and non-membership (unfavorable) degrees of the element x ∈ χ to the set D respectively, with the criteria: is called a q-rung orthopair dual hesitant fuzzy number (q-RODHFN) simply written as d = (h, g), with the criteria: ρ ∈ h, κ ∈ g, 0 ≤ ρ, κ ≤ 1 and ∪ ρ∈h (max(ρ)) q + ∪ κ∈g (max(κ)) q ≤ 1. Moreover, the relationship among q-RODHFNs could be communicated as: , the score and accuracy functions are given as S(d) , where ♯h and ♯g are the numbers of the elements in h and g respectively, then, Let d i = (h i , g i )(i = 1, 2) be any two q-RODHFNs, we have these comparison rules: Definition 2.8. (23) Let d 1 = (h 1 , g 1 ), d 2 = (h 2 , g 2 ) and d = (h, g) be three q-RODHFNs, then, the basic working rules on the q-RODHFNs are defined as:

The q-RODHFBM OPERATOR
This segment stretches out BM and to fuse the q-RODHFNs, we will introduce the q-rung orthopair dual hesitant fuzzy Bonferroni mean (q-RODHFBM) operator, besides, some valuable properties of q-RODHFBM operator are talked about. Definition 2.10. Let d j = (h j , g j )( j = 1, 2, . . . , τ) be an assortment of q-RODHFNs. The q-rung orthopair dual hesitant fuzzy Bonferroni mean (q-RODHFBM) can be composed as: Theorem 1. Let d j = (h j , g j )( j = 1, 2, . . . , τ) be a list of q-RODHFNs. We can intertwine all the q-RODHFNs datum by utilizing the q-RODHFBM operator, the intertwined outcomes can be communicated in Eq.8, as pursues.
The By adjusting the estimations of parameter s,t and q, some unique instances of q-RODHFBM operator are discussed as pursues.
(1) For parameter q, there arise the accompanying exceptional cases Remark 1. When q = 1, the q-RODHFBM operator will turn to dual hesitant fuzzy BM (DHFBM) operator given as: https://www.indjst.org/ Remark 2. When q = 2, the q-RODHFBM operator will turn into dual hesitant Pythagorean fuzzy BM (DHPFBM) operator given as: (2) For parameter s and t, now we discuss these important cases.
Remark 3. When t → 0, then the q-RODHFBM will turn into the q-rung orthopair dual hesitant fuzzy arithmetic mean (q-RODHFAM) as shown below: Remark 4. If s = 2 and t → 0, then the q-RODHFBM will turn to the q-rung orthopair dual hesitant fuzzy square mean (q-RODHFSM) as shown below: Remark 5. If s = 1 and t → 0, then the q-RODHFBM will turn into the q-rung orthopair dual hesitant fuzzy geometric mean (q-RODHFGM) operator as shown below: https://www.indjst.org/ Remark 6. When s = t = 1, then the q-RODHFBM will turn into the q-rung orthopair dual hesitant fuzzy interrelated square mean (q-RODHFISM) operator as shown below

THE q-RODHFWBM OPERATOR
To get better results in MADM, it's good to take weighted attributes. In this segment we will introduce the q-rung orthopair dual hesitant fuzzy weighted Bonferroni mean (q-RODHFWBM) operator by this way.
Then we say q − RODHFW BM s,t τ the q-rung orthopair dual hesitant fuzzy weighted Bonferroni mean operator. Theorem 2. Let d j = (h j , g j )( j = 1, 2, . . . , τ) be an assortment of q-RODHFNs. The outcome value by using q-RODHFWBM operators is again a q-RODHFN, as shown below.
Proof. According to definition (2.8), we can obtain the following identities Hence, Eq.16 is preserved. Now, we must prove that Eq.16 is a q-RODHFN. For this we should prove these two criteria: Then, That means 0 ≤ ρ ≤ 1, on same lines, we can find 0 ≤ κ ≤ 1. Hence 1. is preserved. For (max(ρ)) q + (max(κ)) q ≤ 1, we have this expression .3}} be three q-RODHFNs, and let s = 1,t = 1 and q = 3 then using Eq.(16), we get for the membership (favorable) function ρ, the final outcomes are given as below.
Alike, we can find ρ 1) For parameter q, there exist following important cases Remark 7. When q = 1, the q-RODHFWBM operator will turn into dual hesitant fuzzy weighted BM (DHFWBM) operator as shown below: https://www.indjst.org/ Remark 8. When q = 2, the q-RODHFWBM operator will turn into dual hesitant Pythagorean fuzzy weighted BM (DHPFWBM) as shown below: 2) For parameter s and t, there exist some important cases.
Remark 9. When t → 0, the q-RODHFWBM will turn into the q-rung orthopair dual hesitant fuzzy weighted arithmetic mean (q-RODHFWAM) as shown below: Remark 10. If s = 2 and t → 0, the q-RODHFWBM will turn into the q-rung orthopair dual hesitant fuzzy weighted square mean (q-RODHFWSM) as shown below: https://www.indjst.org/ Remark 11. If s = 1 and t → 0, the q-RODHFWBM will turn into the q-rung orthopair dual hesitant fuzzy weighted geometric mean (q-RODHFWGM) operator as shown below: Remark 12. When s = 1 and t = 1, the q-RODHFWBM will turn into the q-rung orthopair dual hesitant fuzzy weighted interrelated square mean (q-RODHFWISM) operator as shown below:

The q-RODHFDBM OPERATOR
Now, we establish the dual BM (DBM) combining both the BM and dual operation.
Then we call DBM s,t the dual BM (DBM) operator. Now, we shall introduce the DBM operator for q-RODHFNs as follows. Definition 2.13. Let s,t > 0 and d j = (h j , g j )( j = 1, 2, . . . , τ) be a set of q-RODHFNs. If Then the name q-RODHFDBM s,t stands for the q-rung orthopair dual hesitant fuzzy dual Bonferroni mean operator.
Based on operations (1)-(4) of the q-RODHFNs stated in Section 2, we can drive the following result.
Proof. From definition (2.8), the proof follows easily. By adjusting the estimations of parameter s,t and q, some unique instances of q-RODHFBM operator are given as pursues.
(1) For parameter q, there arise some important cases Remark 13. When q = 1, the q-RODHFDBM operator will turn into dual hesitant fuzzy DBM (DHFDBM) operator as shown below: https://www.indjst.org/ Remark 14. When q = 2, the q-RODHFDBM operator will turn into dual hesitant Pythagorean fuzzy DBM (DHPFDBM) which can be presented in , (2) For parameter s and t, there exist these important cases.
Remark 15. If t → 0, then q-RODHFDBM will turn into the q-rung orthopair dual hesitant fuzzy dual arithmetic mean (q-RODHFDAM) operator as shown below: Remark 16. If s = 2 and t → 0, then the q-RODHFDBM will turn into the q-rung orthopair dual hesitant fuzzy dual square mean (q-RODHFDSM) as shown below: Remark 17. If s = 1 and t → 0, then the q-RODHFDBM will turn into the q-rung orthopair dual hesitant fuzzy dual geometric mean (q-RODHFDGM) operator as shown below: https://www.indjst.org/ Remark 18. If s = 1 and t = 1, then the q-RODHFDBM will turn into the q-rung orthopair dual hesitant fuzzy interrelated square mean (q-RODHFDISM) operator as shown below:

THE q-RODHFWDBM OPERATOR
In actual MADM, it's good to assign weights to each attribute. In this segment, we shall explore the q-rung orthopair dual hesitant fuzzy weighted dual Bonferroni mean (q-RODHFWDBM) operator as pursues.
Definition 2.14. Let s,t > 0 and a i (i = 1, 2, . . . , τ) be a set of nonnegative real numbers. If Then we call DBM s,t the dual BM (DBM) operator. Now, we will establish the DBM operator for q-RODHFNs as follows. Definition 2.15. Let s,t > 0 and d j = (h j , g j )( j = 1, 2, . . . , τ) be an assortment of q-RODHFNs. If Then q-RODHFWDBM s,t stands for the q-rung orthopair dual hesitant fuzzy weighted dual Bonferroni mean operator. Theorem 4. Let s,t > 0 and d j = (h j , g j )( j = 1, 2, . . . , τ) be an assortment of q-RODHFNs. The aggregated result after utilizing q-RODHFWDBM operators is again a q-RODHFN where as Eq.34, as shown here.
Therefore, Eq.36, as shown above. Hence, Eq.35 is preserved. Now to show that Eq.35 is a q-RODHFN. It should satisfy these two criteria as follows: By adjusting the estimations of parameter s,t and q, some unique instances of q-RODHFWBM operator are discussed as pursues.
(1) For some particular values of q, there exist following vital cases.
https://www.indjst.org/ Remark 19. When q = 1, the q-RODHFWDBM operator will turn into dual hesitant fuzzy weighted dual Bonferroni mean (DHFWDBM) operator as shown below: Remark 20. When q = 2, the q-RODHDFWDBM operator will turn into dual hesitant Pythagorean fuzzy weighted DBM (DHPFWDBM), as defined (2) For parameter s and t, there exist the following vital cases.
Remark 21. When t → 0, the q-RODHFWDBM will turn into the q-rung orthopair dual hesitant fuzzy weighted dual arithmetic mean (q-RODHFWDAM) operator as shown below: Remark 22. When s = 1 and t → 0, the q-RODHFWDBM will turn into the q-rung orthopair dual hesitant fuzzy weighted dual geometric mean (q-RODHFWDGM) operator as shown below: https://www.indjst.org/ Remark 23. When s = 2 and t → 0, the q-RODHFWDBM will turn into the q-rung orthopair dual hesitant fuzzy weighted dual square mean (q-RODHFWDSM) operator as shown below: Remark 24. When s = 1 and t = 1, the q-RODHFWDBM will turn into the q-rung orthopair dual hesitant fuzzy weighted dual interrelated square mean (q-RODHFWDISM) operator as shown below:

MODELS FOR MADM WITH q-RODHFNs
In the light of the q-RODHFWBM and q-RODHFWDBM operators, we shall furnish the model for MADM with q-RODHFNs. Let O = {O 1 , O 2 , . . . , O m } be a discrete set of alternatives, and K = {K 1 , K 2 , . . . , K τ } be collection of attributes, m×τ is the q-rung orthopair fuzzy decision matrix, where h i j set specify the level that the alternative O i satisfy the attribute K j given by the decision maker, g i j set specify the level that the alternative O i doesn't satisfy the attribute K j given by the decision maker, . . , m, j = 1, 2, . . . , τ. In the accompanying, we will utilize the q-RODHFWBM and q-RODHFWDBM operator to the MADM problems for q-RODHFNs.
Step 1 : We take advantage of q-RODHFNs of the matrix U , and utilize q-RODHFWBM operator to acquire d i (i = 1, 2, . . . , m) of the alternative O i .

Numerical Example
In this segment, we shall furnish an application to choose green providers in green inventory network the board (GINB) with q-RODHFNs. There are five possible green providers in GINB O i (i = 1, 2, 3, 4, 5) to decide. The specialists evaluate the five potential green providers with respect to the following attributes: 1. K 1 is the item quality factor; 2. K 2 is natural factors; 3. K 3 is conveyance factor; 4. K 4 is value factor. Five green providers O i (i = 1, 2, 3, 4, 5) are to be classified under q-RODHFNs with respect to four attributes with weight vector w = (0.4, 0.3, 0.1, 0.2) displayed in Table 1. Table 1. q-RODHFN decision matrixi( U) In the accompanying, we take the advantage of the operators developed for provider selection in provide network board with q-rung orthopair dual hesitant fuzzy numbers (q-RODHFNs) datum.
Step 1: We take advantage of the decision datum in matrix U , and the q-RODHFWBM operator to collect the collective preference values d i of the provider in green inventory network the board O i (i = 1, 2, 3, 4, 5). The collective preference values d i of the provider in green inventory network the board O i (i = 1, 2, 3, 4, 5) are listed below Step 2: Based on the q-RODHFWDBM operator, in order to select the most desirable supplier, we can develop an approach to multiple attribute decision making problems with q-rung orthopair dual hesitant fuzzy information, which can be described as following: Step 1 : Aggregate all q-rung orthopair dual hesitant fuzzy value d i j ( j = 1, 2, 3, 4) by using the dual hesitant q-rung orthopair fuzzy weighted DBM (q-RODHFWDBM) operator to derive the overall q-rung orthopair dual hesitant fuzzy values d i (i = 1, 2, 3, 4, 5) of the supplier A i . The overall performance values of all the supplier A 1 (here, we take q = 3,s=1,t=1) are given below, Step 2 : Calculate the scores s(A i )(i = 1, 2, 3, 4, 5) of the overall q-rung orthopair dual hesitant fuzzy values A i (i = 1, 2, 3, 4, 5) of the supplier A i : Step 3 : Rank all the suppliers in supply chain management A i (i = 1, 2, 3, 4, 5) in accordance with the scores s(A i )(i = 1, 2, 3, 4, 5) of the overall dual hesitant q-rung orthopair fuzzy values A i (i = 1, 2, 3, 4, 5) by using definition 2.15: 4 , and thus the most desirable supplier is A 2 . From the above analysis, it is easily seen that although the overall rating values of the alternatives are same by using two operators respectively.

Comparative analysis compared with existing magdm methods
To demonstrate the superiorities of the proposed method, we have compared our method with that (1) developed by Wang et al. 's (26) based on the dual hesitant fuzzy weighted averaging (DHFWA) operator, (2) presented by Tu et al. 's (27) , based on the dual hesitant fuzzy weighted Bonferroni mean (DHFWBM) operator, (3) putforwarded by Tang (24) , based on the dual hesitant Pythagorean fuzzy Heronian weighted averaging (DHPFHWA) operator, (4) proposed by, Xu et al. 's (23) based on the https://www.indjst.org/  (26) Tu et al. 's (27) dual hesitant Pythagorean fuzzy Heronian weighted averaging (DHPFHWA) operator. We utilized these methods to solve the above example, and the score functions and ranking results can be found in Table 2. First of all, Wang et al. 's (26) and Tu et al. 's (27) methods are based on DHFSs. Tang et al. 's (24) method is based on DHPFSs. As mentioned above, DHFS and DHPFS are two special cases of q-RDHFS. When q = 1, then q-RDHFS is reduced to DHFS, and when q = 2, q-RDHFS is reduced to DHPFS. Evidently, q-RDHFS is more general and can describe a greater information range and process more information in the process of MAGDM. For instance, if an attribute value provided by DMs is {{0.1, 0.2, 0.6, 0.7}, {0.1, 0.4, 0.5}}, then obviously, the pair {{0.1, 0.2, 0.6, 0.7}, {0.1, 0.4, 0.5}} is not valid for DHFSs and DHPFSs. Thus, our method is more general, powerful, and can process more information in MAGDM. Wang et al. 's (26) method is based on the simple weighted averaging operator. The drawback of this methods is that it does not consider the interrelationship between arguments. In other words, they assume all attributes are independent, which is not correct to some extent. In the abovementioned example, when choosing the most appropriate supplier, we need to consider not only the attribute values of each supplier but also the correlation between these attributes. Thus,Wang et al. 's (26) method is not suitable for dealing with this problem. As our method has the ability to capture variable correlations, it is more reasonable than Wang et al. 's method for addressing this problem. Xu et al. 's (23) is based on HM. Tu et al. 's (27) and our methods based on Bonferroni mean (BM). The prominent characteristic of BM and HM is that both can consider the interrelationship between arguments. Therefore, all the three can process the interrelationship among attribute values. However, Xu et al. 's (23) method and ours are better than Tu et al. 's (27) method. In addition, as Tu et al. 's (27) is a special case of our method (when q = 1), our method is more general, scientific, and applicable than Tu et al. 's (27) method.

Conclusion
In this article, we have examined the MADM problems under q-RODHFNs. we have utilized the BM operator and established some BM aggregation operators with q-RODHFNs. We have developed (q-RODHFBM) operator, (q-RODHFWBM) operator, DBM operator, (q-RODHFDBM) operator and (q-RODHFWDBM) operator. Also, the important merits of the examined operators are talked about. Furthermore, we have endorsed q-RODHFWBM and q-RODHFWDBM operators to construct decision-making steps to handle the q-rung orthopair dual hesitant fuzzy MADM problems. Finally, we take a solid example https://www.indjst.org/ for examining the green provider selection to exhibit our established model and to assert its efficiency and objectiveness. We have compared our results with q-RODHFWHM and q-RODHFWGHM operators, despite the fact that the results are minimal extraordinary and the ideal option is not changed. However, the q-RODHFWHM and q-RODHFWGHM operators just include the interrelationship of two arbitrary numbers but our introduced operators can include the interrelationship of any number arbitrary arguments, that indicates our established method is more decisive to handle the MADM problems. In the forthcoming, we will maintain our study about the MADM issues with the application and expansion of the presented operators to other realm.