Generalized Hamacher Aggregation Operators Based on Linear Diophantine Uncertain Linguistic Setting and Their Applications in Decision-Making Problems

Hamacher aggregation operators are more flexible and more dominant to determine the interrelationships between any number of attributes. The goal of this manuscript is to elaborate on the principle of linear Diophantine uncertain linguistic sets and explored their useful Hamacher operational laws. The existing notions of intuitionistic uncertain linguistic sets, Pythagorean uncertain linguistic sets, and q-rung orthopair uncertain linguistic sets have certain applications in different fields. Unfortunately, these theories have their limitations related to the truth and falsity grades. To eradicate these limitations, the theory of linear Diophantine uncertain linguistic sets with the addition of reference parameters is massive flexible than the existing drawbacks. This notion removes the restrictions of prevailing methodologies, and the decision-maker can freely choose the grades without any limitations. This structure also categorizes the problem by changing the physical sense of reference parameters. Moreover, by using the investigated linear Diophantine uncertain linguistic information and Hamacher aggregation operators, we explored the linear Diophantine uncertain linguistic generalized Hamacher averaging operator and linear Diophantine uncertain linguistic generalized Hamacher hybrid averaging operator. Additionally, a multi-attribute decision-making (MADM) procedure is buildup based on the investigated operators under the linear Diophantine uncertain linguistic information. Certain numerical examples are illustrated by using initiated operators to determine the dominance and flexibility of explored operators. To find the consistency and supremacy of the presented operators, we compare the proposed work with certain prevailing operators and discussed their geometrical expressions to show that the introduced operators in this manuscript are extensively powerful and more useful than the prevailing drawbacks.


I. INTRODUCTION
The principle of the intuitionistic fuzzy set (IFS) was developed by Atanassov [1], by including the non-membership grade (NMG) v NMG as the generalization of the fuzzy set (FS) [2]. The theory of IFS has extensive effectiveness The associate editor coordinating the review of this manuscript and approving it for publication was Emre Koyuncu . and is more dominant as compared to FS, which deals with two-dimension information in the form of membership grade (MG) u MG and NMG v NMG at a time. The prominent characteristic of IFS is that the 0 ≤ u MG + v NMG ≤ 1. Certain scholars have used it in the environment of distinct fields, for instance, Beg and Rashid [3] utilized the principle of intuitionistic hesitant fuzzy sets. Moreover, Mahmood et al. [4] explored the improved intuitionistic hesitant fuzzy sets and discussed their application. Kumari and Mishra [5] initiated the parametric measures based on IFSs. Liu et al. [6] developed the variable-based hybrid approach by using the interval-valued IFSs. The similarity measures under the right-angle triangle based on IFSs were initiated by Garg and Rani [7]. Xue and Deng [8] explored the decision-making troubles based on measures under the granular uncertainty for IFSs. Thao [9] explored the entropy and diverges measures based on IFSs and discussed their application in supplier chain management. Jana and Pal [10] evaluated the bipolar intuitionistic fuzzy soft sets and their applications. Meng and He [11] developed the geometric interaction aggregation operators for IFSs.
The principle of IFS has achieved much attention from distinct scholars, but in certain situations, the principle of IFS has been neglected. For instance, if an individual provides information in the form of 0.6 for MG u MG and 0.5 NMG v NMG with a rule that is u MG +v NMG = 0.6+0.5 = 1.1 > 1, then the principle of IFS has cannot be working dominantly. For managing such complicated sorts of troubles, the flexible theory of Pythagorean FS (PFS) was initiated by Yager [12]. The prominent characteristic of PFS is that the 0 ≤ u 2 MG + v 2 NMG ≤ 1 that is 0.6 2 +0.5 2 = 0.36+0.25 = 0.66 < 1. Certain scholars have used it in the environment of distinct fields, for instance, Garg [13] initiated the linguistic PFSs, Wei and Wei [14] explored certain measures based on PFSs using cosine function, Xiao and Ding [15] proposed the diverges measures for PFSs, Ullah et al. [16] initiated certain distance measures for complex PFSs, Li and Lu [17] developed the similarity and distance measures for PFSs, Garg [18] investigated the improve the accuracy function based on intervalvalued PFSs, Yang and Hussain [19] explored the entropy measures based on PFSs.
Still, the principle of PFS has been neglected in certain situations, for instance, if an individual provides information in the form of 0.9 for MG u MG and 0.8 NMG v NMG with a rule that is u 2 MG +v 2 NMG = 0.9 2 +0.8 2 = 0.81+0.64 = 1.45 > 1, then the principle of PFS has cannot be working dominantly. For managing such complicated sorts of troubles, the flexible theory of q-rung orthopair FS (QROFS) was initiated by Yager [20]. The prominent characteristic of QROFS is that the 0 ≤ u q MG +v q NMG ≤ 1, q ≥ 1. The principle of IFS and PFS is the particle cases of the QROFS by using the value of q = 1 and q = 2. Certain scholars have used it in the environment of distinct fields, for instance, Ali [21] initiated another view of QROFSs, Liu and Wang [22] proposed aggregation operators for QROFSs, Peng and Liu [23] initiated the information measures based on QROFSs, Wang et al. [24] developed certain measures based on QROFSs using cosine function, Ali and Mahmood [25] investigated the Maclaurin symmetric mean operators based on QROFSs, Liu et al. [26] initiated the cosine similarity and distance measures for QROFSs, Liu and Wang [27] proposed Archimedean Bonferroni mean operators for QROFSs, Lin et al. [28] developed certain Heronian mean operators based on linguistic QROFSs, Khan et al. [29] initiated the knowledge measures by using QROFSs.
Still, the principle of QROFS has been neglected in certain situations, for instance, if an individual provides information in the form of 1 for MG u MG and 0.1 NMG v NMG with a rule that is u q MG +v q NMG = 1 q +0.1 q > 1 for any value of q, then the principle of QROFS has cannot be working dominantly. For managing such complicated sorts of troubles, the flexible theory of linear Diophantine FS (LDFS) was initiated by Riaz and Hashmi [30]. The prominent characteristic of LDFS is that the 0 ≤ α AMG u AMG ( )+β ANG v AMG ( ) ≤ 1. The principle of IFSs, PFS, and QROFS are the particle cases of the LDFS that is 0 ≤ 0.1 * 1+0 * 0.1 = 0.1 ≤ 1. Certain scholars have used it in the environment of distinct fields, for instance, Riaz et al. [31] developed the linear Diophantine fuzzy rough sets and their applications, Kamaci [32] explored the algebraic structure based on linear Diophantine fuzzy sets, Ayub et al. [33] explored the linear Diophantine fuzzy relation and their application in decision-making.
In numerous situations, the principle of FS has been failed, due to its structure, for instance, if an individual faces information in the shape of very good, good, normal, weak, very weak, then the principle of FS has cannot working dominantly. To manage such sorts of awkward situations, the principle linguistic variable (LV) was explored by Zadeh [34]. Moreover, Herrera and Martinez [35] modified the principle of LV is to explore the theory of a 2-tuple linguistic set (2-TLS), and Xu [36] investigated the uncertain LV (ULV). Liu et al. [37] initiated the Heronian mean operators for intuitionistic uncertain linguistic sets, Xu [38] developed the intuitionistic fuzzy aggregation operators, Xu and Yager [39] proposed certain geometric aggregation operators for IFSs. The principle of LDFS has been neglected in certain situations, for instance, if an individual provides information in the form of MG, NMG, and uncertain linguistic terms, then the principle of LDFS has cannot working dominantly. For managing such complicated sorts of troubles, the flexible theory of linear Diophantine uncertain linguistic set is initiated in this manuscript. Based on the above analysis, the main theme of the elaborated approaches in this study are discussed below: 1. To discuss the supremacy of the elaborated operators is also presented with the help of comparative analysis and geometrical expressions. The rest of this manuscript is summarized in the following ways: In section 2, we review the main idea of LDFS, ULS, and their important laws which are very useful for the investigated works. In section 3, we elaborated the principle of linear Diophantine uncertain linguistic sets and elaborated their useful Hamacher operational laws. In section 4, by using the elaborated linear Diophantine uncertain linguistic information and Hamacher aggregation operators, we explored the linear Diophantine uncertain linguistic generalized Hamacher averaging operator and linear Diophantine uncertain linguistic generalized Hamacher hybrid averaging operator are discovered. In section 5, a multi-attribute decision-making (MADM) procedure is buildup based on the investigated operators under the linear Diophantine uncertain linguistic information. Certain numerical examples are illustrated by using elaborated operators to determine the dominance and flexibility of explored operators. To find the consistency and supremacy of the elaborated operators, we compare the proposed work with certain prevailing operators and discussed their geometrical expressions to show that the elaborated operators in this manuscript are extensively powerful and more useful than the prevailing drawbacks. In section 6, we presented the conclusion of this study.

II. PRELIMINARIES
The flexible theory of linear Diophantine FS (LDFS) was initiated by Riaz and Hashmi [30]. The prominent characteristic of LDFS is that the 0 ≤ α AMG u AMG ( )+β ANG v AMG ( ) ≤ 1. The principle of IFSs, PFS, and QROFS are the particle cases of the LDFS. In this study, we review the main idea of LDFS, ULS, and their important laws which are very useful for the investigated works. Additionally, the symbol X is expressed the fixed set in all studies.
Definition 1 [30]: A LDFS A LD is elaborated by: Definition 2 [30]: For any two LDFNs (5), as shown at the bottom of the page.
Definition 3 [34]: A LTS is elaborated by: where must be odd and keep the resulting circumstances: 1. If > , then > ; 2. The pessimistic operator reg ( ) = with a condition + = −1; where a , b ∈ S, a and b are the lower and the upper limits, respectively. We call˜ the uncertain linguistic variable.

III. LINEAR DIOPHANTINE UNCERTAIN LINGUISTIC SETS
The principle of LDFS has been neglected in certain situations, for instance, if an individual provides information in the form of MG, NMG, and uncertain linguistic terms, then the principle of LDFS has cannot working dominantly. For managing such complicated sorts of troubles, the flexible theory of linear Diophantine uncertain linguistic set and their important laws are initiated in this manuscript.

IV. HAMACHER OPERATIONS AND THEIR OPERATORS FOR LINEAR DIOPHANTINE UNCERTAIN LINGUISTIC SETS
To determine the elaborated Hamacher laws for investigated LDULS, we revise the basics of Hamacher laws which are discussed below.
For ζ = 1, the Hamacher laws are converted for Algebraic laws such that T (x, y) = xy (18) S (x, y) = x+y−xy (19) For ζ = 2, the Hamacher laws are converted for Einstein laws such that S (x, y) = x+y 1+xy Based on the above analysis, we contract certain operational laws which are very important for the elaborated operators.
For any two LDULNs and ζ > 0, λ ≥ 0, then (22), as shown at the bottom of the page, (23) and (24), as shown at the bottom of the next page, and (25), as shown at the bottom of page 6. Theorem 2: For any two LDULNs Proof: The proof of Eq. (1) and Eq. (2) is straightforward. We only prove that Eq. (3), the Eq. (4) to Eq.
, as shown at the bottom of page 7.
By using the elaborated linear Diophantine uncertain linguistic information and Hamacher aggregation operators, we explored the linear Diophantine uncertain linguistic generalized Hamacher averaging operator and linear Diophantine uncertain linguistic generalized Hamacher hybrid averaging operator are discovered.

Definition 7: For any collection of LDULNs
ζ > 0, then linear Diophantine uncertain linguistic generalized Hamacher weighted averaging (LDULGHWA) operator is initiated by: Based on the above analysis, we discuss the monotonicity, idempotency, and boundedness to improve the quality of the proposed works.
Property 1: For any collection of LDULNs (A LD−1 , If the LDULGHWA operator lies between the max and min operators, then min

Definition 8: For any collection of LDULNs
ζ > 0, then linear Diophantine uncertain linguistic generalized Hamacher ordered weighted averaging (LDULGHOWA) operator is initiated by: Based on the above analysis, we discuss the monotonicity, idempotency, and boundedness to improve the quality of the proposed works.
Property 2: For any collection of LDULNs (A LD−1 ,

If LDULGHOWA operator lies between the max and min operators, then min
. Additionally, by using the value of parameters, we will discuss certain special cases of the elaborated operators.
1. For λ = 1, then the LDULGHWA operator is converted for HLDULWA operator, such that (28), as shown at the bottom of page 8. 2. For ζ = 1, then the HLDULWA operator is converted for LDULWA operator, such that (29), as shown at the bottom of page 8. 3. For ζ = 2, then the HLDULWA operator is converted for ELDULWA operator, such that . For λ → 0, then the LDULGHWA operator is converted for HLDULWGA operator, such that 5. For ζ = 1, then the HLDULWGA operator is converted for LDULWGA operator, such that (32), as shown at the bottom of the next page. 6. For ζ = 2, then the HLDULWGA operator is converted for ELDULWGA operator, such that (33), as shown at the bottom of the next page.
7. For λ = 1, then the LDULGHOWA operator is converted for HLDULOWA operator, such that (34), as shown at the bottom of the next page. 8. For ζ = 1, then the HLDULOWA operator is converted for LDULOWA operator, such that 9. For ζ = 2, then the HLDULOWA operator is converted for ELDULOWA operator, such that (36), as shown at the bottom of the next page. 10. For λ → 0, then the LDULGHOWA operator is converted for HLDULOWGA operator, such that (37), as shown at the bottom of page 10. 11. For ζ = 1, then the HLDULOWGA operator is converted for LDULOWGA operator, such that (38), as shown at the bottom of page 10. 12. For ζ = 2, then the HLDULOWGA operator is converted for ELDULOWGA operator, such that (39), as shown at the bottom of page 10.
. . , n, then (41), as shown at the bottom of the next page.
For n = 1, it is true. For n = k, we have LDULGHHWA (A LD−1 , A LD−2 , . . . , A LD−k ), as shown at the bottom of the next page.

V. MADM METHOD BASED ON PROPOSED OPERATORS
Certain scholars have explored different sorts of operators, measures, and methods by using certain prevailing ideas such as IFSs, PFSs, QROFSs, and LDFSs. Based on the above information, in this study, we goal to utilize the principle of elaborated operators by using the novel LDULSs. The impact of λ on the ranking results is investigated thoroughly.
One of the most important techniques in DM theory, the MADM is the best way to determine the best alternative from the family of alternatives, because of some finite attributes using the generalized Hamacher aggregation operators of LDULSs. The most favorable thing here is that the information is based on LDULNs which discuss the membership grades and non-membership grades are depended on the reference parameters. Let the collection of alternatives be A k (k is finite) and attributes be G j (j is finite) which form a decision matrix denoted by D k×j = (T ) k×j = where the terms in triplet denote the membership grades, abstinence, and non-membership grades of the information, where ω = (ω 1 , ω 2 , . . . , ω r ) T is the weight vector of A brief algorithm of the MADM process is illustrated in the following section.

A. ALGORITHM
The algorithm of MADM based on LDUL information and using investigated operators is proposed as follows: Step 1: In this step, we collect the information about alternatives given by the decision-makers. The decision-makers gave their opinion about alternates in the form of LDULNs which leads to the formation of the decision matrix.
Step 2: If there exist attributes of cost type, we normalize the decision matrix by taking the complement of each triplet in the matrix. If not, then we use the following investigated LDULGHWA operators to aggregate the data given in the decision matrix.
Step 3: In this step, we compute the expected values of the aggregated information using the below formula. For any LDULN then the expected value (EV) is elaborated by: For any LDULN then accuracy value (AV) is elaborated by: For any two LDULNs and Step 4: This step is based on the ranking of the alternatives. An illustrative example to see the viability of the proposed algorithm.

B. ILLUSTRATED EXAMPLE
An example of technology commercialization is adapted from [39] where the selection of the most favorable software    As shown above, we obtained the ranking results by using the proposed operator, which is discussed in the form of Table 4. The best option is A 3 . Further, we discuss the consistency of the elaborated operators by using different values of the parameters ζ and λ, which are discussed in the form of Table 5.
By using different values of the parameters ζ and λ, we obtained the same ranking results in the form of Table 5. We briefly explained the motivation of the proposed work. The concepts of intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PFSs), q-rung orthopair fuzzy sets (QROFSs), and linear Diophantine fuzzy sets have numerous applications in various fields of real life, but these theories have their limitations related to the membership and non-membership grades. To eradicate these restrictions, we introduce the novel concept of linear Diophantine uncertain linguistic set (LDULS) with the addition of reference parameters and uncertain linguistic terms. The proposed model of LDULS is more efficient and flexible rather than other approaches due to the use of reference parameters and ULVs. LDULS also categorizes the data in MADM problems by changing the physical sense of reference parameters and ULVs. This set covers the spaces of existing structures and enlarges the space for membership and non-membership grades with the help of reference parameters and ULVs. The motivation of the proposed model is given step by step in the whole manuscript. Now we discuss some important objectives of this paper.
1. The theory of LDULS is more generalized than IFSs, PFSs, QROFSs, LDFSs, and ULVs. 2. If we choose the information in the form of (0.5, 0.6), then by using the condition of IFSs that is the sum of both terms is limited to the unit interval, but 0.5+ 0.6 = 1.1 > 1, the theory of IFS has been failed for coping with such sorts of issues, the theory of LDULS is very comfortable to resolve the above issues. For this, we choose the reference parameters such as (0.1, 0.2), then by using the condition of LDULS is that 0.1 * 0.5+0.2 * 0.6 = 0.05+0.12 = 0.17 < 1.
We clarify that the IFS is the special case of the proposed LDULS. 3. If we choose the information in the form of (0.8, 0.9), then by using the condition of PFSs that is the sum of the square of both terms is limited to the unit interval, but 0.8 2 +0.9 2 = 0.64+0.81 = 1.45 > 1, the theory of PFS has been failed for coping with such sorts of issues, the theory of LDULS is very comfortable to resolve the above issues. For this, we choose the reference parameters such as (0.2, 0.2), then by using the condition of LDULS is that 0.2 * 0.8+0.2 * 0.9 = 0.16+0.18 = 0.34 < 1. We clarify that the theory of PFS is the special case of the proposed LDULS. 4. If we choose the information in the form of (0.1, 0.1), then by using the condition of QROFSs that is the sum of the q-powers of both terms is limited to the unit interval, but 1+1 = 2 > 1, the theory of QROFS has been failed for coping with such sorts of issues, the theory of LDULS is very comfortable to resolve the above issues. For this, we choose the reference parameters such as (0, 0.1), then by using the condition of LDULS is that 0.0 * 1+0.1 * 1 = 0+0.1 = 0.1 < 1. We clarify that the theory of QROFS is the special case of the proposed LDULS. VOLUME 9, 2021 5. If we choose the information in the form of ([s 1 , s 2 ] , (0.5, 0.3) , (0.5, 0.4)), then by using the condition of IFSs, PFSs, q-ROFSs, and LDFS have been failed, for coping with such sorts of issues, the theory of LDULS is a very proficient and reliable technique to resolve it. From the above analysis, the theory of IFSs, PFSs, QROFSs, and LDFSs is the special case of the proposed LDULS. In real-life problems, we come across many situations where we need to quantify the uncertainty existing in the data to make optimal decisions. To illustrate the significance of Linear Diophantine Uncertain Linguistic sets, we give an example. Suppose XYZ company decides to set up biometricbased attendance devices (BBADs) in all its offices spread all over the country. For this, the company consults an expert who gives the information regarding (i) models of BBADs, (ii) production dates of BBADs, and (ii) the Price of BBADs. The company wants to select the most optimal model of BBADs with its production date simultaneously. Here, the problem is three-dimensional, namely, the model of BBADs and the production date of BBADs. This type of problem cannot be modeled accurately using traditional fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets, q-rung orthopair fuzzy sets, and Linear Diophantine fuzzy set theory cannot tackle with triplet the dimensions simultaneously. The best way to represent all the information provided by the expert is by using Linear Diophantine uncertain linguistic set theory. The membership and non-membership terms in Linear Diophantine uncertain linguistic set may be used to give company's decision regarding the model of BBADs, the reference parameters may be used to represent the company's judgment in respect of production date of BBADs, and the uncertain linguistic terms may be used to represent company's judgment in respect of price level (ups and downs) of BBADs.

C. COMPARATIVE ANALYSIS
As shown above, we founded the ranking results by using the elaborated operator and applied them by interval-valued linear Diophantine uncertain linguistic types of information to find the effectiveness and proficiency of the discovered approaches. Additionally, to improve the quality of the proposed approach, we compare the discovered approaches with some existing approaches [30], [38], [37].
1. Riaz and Hashmi [30] elaborated the aggregation operators based on LDFSs. In [30], the authors combined the aggregation operators with LDFSs and determine the best optimal to show the dominance of the elaborated operators, but the theory proposed in [30] based on LDFS is the special case of the proposed operators based on LDULSs. Therefore, if we choose the proposed types of information, then the theory of Riaz and Hashmi [30] is not able to resolve it, because the elaborated approach is more general than the prevailing ideas in [30].

Aggregation operators based on intuitionistic uncertain
linguistic sets (IULSs) were elaborated by Xu [38], which is the mixture of the aggregation operators with IULSs. In [38], the authors combined the aggregation operators with IULSs and determine the best optimal to show the dominance of the elaborated operators, but the theory proposed in [38] based on IULS is the special case of the proposed operators based on -LDULSs. Therefore, if we choose the proposed types of information, then the theory of Xu [38] is not able to resolve it, because the elaborated approach is more general than the prevailing ideas in [38]. 3. Liu et al. [37] elaborated the Heronian mean (HM) operators based on IULSs. In [37], the authors combined the HM operators with IULSs and determine the best optimal to show the dominance of the elaborated operators, but the theory proposed in [37] based on IULS is the special case of the proposed operators based on LDULSs. Therefore, if we choose the proposed types of information, then the theory of Liu et al. [37] is not able to resolve it, because the elaborated approach is more general than the prevailing ideas in [37]. In the future, the principle of Hamacher aggregation operators for complex IFS [40], q-rung orthopair fuzzy graph [41], complex picture fuzzy sets [42], and m-polar fuzzy sets [43] to extend for the presented works.
From the above theories, we obtained the result that is the elaborated operator based on a new IV-LDULS is extensively useful and more dominant to manage awkward and inconsistent information in genuine issues.

VI. CONCLUSION
As the structure of Hamacher aggregation operators is powerful and massive suitable tool to cope with awkward and complicated information in realistic issues, so keeping in view the importance of Hamacher aggregation operators in the field of fuzzy logic and in the field of decision sciences many scholars have employed the theory of Hamacher aggregation operators in the environment of different fields [40]- [43]. In this manuscript the notion of Hamacher aggregation is applied in the environment of linear Diophantine uncertain linguistic sets and is applied in decision making problems. Based on the above analysis the main points of the presented work are discussed below: 1. The principle of linear Diophantine uncertain linguistic sets is explored with its useful Hamacher operational laws. 2. A MADM procedure is a buildup based on the investigated operators under the linear Diophantine uncertain linguistic information. 3. Certain numerical examples are illustrated by using initiated operators to determine the dominance and flexibility of explored operators. 4. The consistency and supremacy of the presented operators in the proposed work with certain prevailing operators is discussed. But there are some issues, when an intellectual gives information in the shape of ''yes'', ''abstinence'', and ''no'', then the principle of linear Diophantine uncertain linguistic set is enabled to manage with it. For managing with such sorts of issues, in future we will be exploring the ideas of picture linear Diophantine uncertain linguistic sets, spherical linear Diophantine uncertain linguistic sets, and T-spherical linear Diophantine uncertain linguistic sets. Further, we will extend the proposed operators for spherical fuzzy sets [44], complex spherical fuzzy sets [45], [46], and their modifications [47] to improve the quality of the research work. MUHAMMAD ASLAM received the Ph.D. degree from Quaid-i-Azam University Islamabad, Pakistan, in 2005. He is currently working as an Associate Professor with the Department of Mathematics, King Khalid University, Abha, Saudi Arabia. So far, he has published 115 research articles in well reputed peer-reviewed international journals. He has also produced six Ph.D. students and 33 M.S. students. His research interests include group graphs, coset diagrams, fuzzy sets, soft sets, rough sets, and decision making.
RONNASON CHINRAM was born in Ranong, Thailand, in 1975. He received the M.Sc. and Ph.D. degrees from Chulalongkorn University, Thailand. Since 1997, he has been with Prince of Songkla University, Thailand, where he is currently an Associate Professor in mathematics. Moreover, he is also the Head of the Division of Computational Science, and the Head of the Algebra and Applications Research Unit. He has more than 90 research publications in international well reputed journals. His research interests include semigroup theory, algebraic systems, fuzzy mathematics, and decision-making problems.