Fabric Selection Problem Based on Sine Hyperbolic Fractional Orthotriple Linear Diophantine Fuzzy Dombi Aggregation Operators

The paper aims are to impersonate some robust sine-hyperbolic operations laws to determine the group decision-making process under the fractional orthotriple linear Diophantine fuzzy set (FOLDFS) situation. The FOLDFS has a notable feature to trade with the dubious information with a broader membership representation space than the Spherical fuzzy set. Based on it, the present paper is classified into three phases. The first phase is to introduce new operational laws for FOLDFS using Dombi operational laws. The main idea behind the proposed operations is to incorporate the qualities of the sine hyperbolic function, namely periodicity and symmetric about the origin towards the decisions of the objects. Secondly, based on these laws, numerous operators (sine hyperbolic fractional orthotriple linear Diophantine fuzzy Dombi weighted averaging, sine hyperbolic fractional orthotriple linear Diophantine fuzzy Dombi ordered weighted averaging, sine hyperbolic fractional orthotriple linear Diophantine fuzzy Dombi hybrid averaging, sine hyperbolic fractional orthotriple linear Diophantine fuzzy Dombi weighted geometric, sine hyperbolic fractional orthotriple linear Diophantine fuzzy Dombi ordered weighted geometric, sine hyperbolic fractional orthotriple linear Diophantine fuzzy Dombi hybrid geometric) to aggregate the information are acquired along with their requisite properties and relations. Finally, an algorithm to interpret the multi-attribute group decision making problem is outlined based on the stated operators and manifest it with an illustrative example. A detailed comparative interpretation is achieved with some of the existing methods to reveal their influences.


I. INTRODUCTION
A important and crucial component of the decision making (DM) process used to extract best decision from a group of decisions is the multi-attribute group decision-making (MAGDM) technique. Information is required during the DM process in order to assess the offered decisions by specific The associate editor coordinating the review of this manuscript and approving it for publication was Giovanni Pau . classes, such as single or interval. While there is conventional knowledge available, it might be exceedingly difficult and demanding for specialists to make the right option in some invalidate situations. Zadeh [1] analyzed the fuzzy set (FS) utilized to handle ambiguous and imprecise data in order to manage it. Numerous applications have been made after their evaluation in a variety of fields, such as binary visualization of FSs, university-business corporations, susceptibility of self-organized systems, and more. Atanassov [2] further improved the intuitionistic fuzzy sets (IFSs). Hayat et al. [3] proposed new aggregation operators on group-based generalized intuitionistic fuzzy soft sets. After that, Yager introduced Pythagorean fuzzy set (PyFS), which is more expandable and has a wider range for expressing ambiguity, as an alternative to IFS, which is less extensible the MD plus NMD number being more than 1. Yager [4] devised the new Pythagorean fuzzy set (PyFS) logic as a result, which is a more generalized version of IFS. Pythagorean fuzzy set logic is characterized by the requirement that the value be 1 when squared with sum of its MD and NMD. The Pythagorean fuzzy number (PyFN) was subsequently developed by Zhang and Xu [5]. Zhang and Xu also suggested mathematical form of PyFS, Pythagorean fuzzy TOPSIS approach, and an order preference strategy comparable to the best outcome. Peng and Yang [6] developed a Pythagorean fuzzy maximum and minimum strategy to solve a MAGDM problem utilizing Pythagorean fuzzy information. Additionally, Peng and Yang recommended adding and subtracting PFNs methods. Reformat and Yager [7] used the PyFNs to control the jointly suggested system. Ren et al. [8] utilised the TODIM strategy to resolve the MADM issue. In [9], Garg proposed the newly developed generalised Pythagorean fuzzy aggregation operators (AOs). Deabes and Amin [10] proposed image reconstruction algorithm based on PSO-tuned fuzzy inference system for electrical capacitance tomography. Almotiri [11] developed an integrated fuzzy based computational mechanism for the selection of effective malicious traffic detection approach. Talpur et al. [12] defined a comprehensive review of deep neuro-fuzzy system architectures and their optimization methods. Alaoui et al. [13] defined a novel analysis of fuzzy physical models by generalized fractional fuzzy operators.
Despite this, Yager [14] created the idea of q-rung orthopair fuzzy sets (q-ROFSs), representing more decision setting, which is based on the total of qth powers of MD and NMD is less than or equal to 1 (ȃ q Ť (ȗ) + υ q Ť (ȗ) ≤ 1). PFSs are more IFS-specific as compared to q-ROFS. Liu and Wang [15], were created the q-ROF weighted geometric and averaging operators. Wei et al. [16] defined a few q-ROF Heronian mean (HM) AOs. Ali [17] developed two novel algorithms using q-ROFSs. Yager et al. [18] investigated plausibility and acceptance in the q-ROFS data as well as the notion of possibility and confidence. Yang and Pang provided new partitioned BM operators employing q-ROF data in their paper [19]. A strategy for green supplier selection (GSS) based on data with q-rung interval values and the q-ROFRH operator was put forth by Lei and Xu in their paper published in [20]. The point operators for q-ROFSs were supplied by Xing et al. [21]. Garg and Chen [22] presented q-ROFS aggregation operators and discussed a group DM problem. Ye and colleagues [23] investigated differential calculus using q-ROFSs. Peng et al. [24] found logarithmic and exponential operation principles for q-ROFNs. Qiyas et al. [25] studied a case study for hospital-based post-acute care-cerebrovascular disease using sine hyperbolic q-rung orthopair fuzzy Dombi aggregation operators. The q-ROFS can communicate more hazy information because it is plainly more broad-based than the IFS. Yang et al. [26] developed an aggregation and interaction aggregation soft operators on interval-valued q-rung orthopair fuzzy soft environment and application in automation company evaluation. Hayat et al. [27] defined new group-based generalized interval-valued q-ROF soft aggregation operators and their applications in sports decision-making problems.
In order to address this problem that picture fuzzy set (PFS) cannot address, Shahzaib et al. [29] initially presented the idea of Spherical fuzzy set (SFS). They then discovered numerous aggregation operations using Spherical fuzzy information. In contrast to PFSs, where all membership degrees must meet the condition ν Ť (ȗ) +ĉ Ť (ȗ) +ȇ Ť (ȗ) ≤ 1, SFSs demand ν 2 Ť (ȗ) +ĉ 2 Ť (ȗ) +ȇ 2 Ť (ȗ) ≤ 1. SFS is just a fuzzy set-added variant of PyFS. Huanhuan et al. [30] developed spherical linguistic FS, which integrates the ideas of IFS and SFS. Ashraf et al. [31], citation needed The authors described a few Spherical fuzzy aggregation techniques and looked at how they may be applied in decision-making using the Dombi approach. They also investigated the way that Spherical fuzzy t-norm and t-conorm are displayed in [32].
The PFS and SFS principles, which have major applicability in numerous areas of daily life, confine the MD, NuMD, and NMD. We developed a novel enlarged idea of fractional orthotriple fuzzy sets (FOFSs) to get over these restrictions (FOFS). Three membership gradesȃ Ť (ȗ) ∈ [0, 1] ,ĉ Ť (ȗ) ∈ [0, 1] andȇ Ť (ȗ) ∈ [0, 1] are included in the proposed FOFS framework. In a fixed setĤ , where f = a b , b ̸ = 0, a, b ∈ N, and for eachȗ ∈Ĥ ,ȃ f Ť (ȗ) +ĉ f Ť (ȗ) +ȇ f Ť (ȗ) ≤ 1, f ∈ Q + (set of positive rational numbers). We note that the fractional orthotriple fuzzy field broadens with increasing rung f, allowing observers to express their support for membership over a wider range. Be noted that we were able to obtain a FOFS that created more exact and correct rung fuzzy numbers in order to deal with ambiguity and false information. It is evident that PFS and SFS are the generic forms of FOFS, and by a = b (for PFS) and a = 2n, b = n for all n ∈ N for SFS, respectively, the corresponding set simplifies to PFS and SFS. Since FOFSs are more adaptive and better able to handle uncertain information, they reflect more complete fuzzy information.
To overcome these limitations, we created a novel expanded idea of fractional orthotriple fuzzy set (FOFS). Abosuliman et al. [34] initially presented the fractional orthotriple fuzzy set in order to widen the notion of the SFS. Naeem, et al. [35] created similarity matrix for FOFS and explained how they may be used in emergency scenarios including accidents. They did this by using cosine and cotangent functions. Qiyas et al. [36] proposed aggregation methods based on Banzhaf-Choquet-Copula for managing fractional orthotriple fuzzy data. Qiyas et al. [37] developed FOF rough Hamacher AOs and used them to the service quality of wireless network selection.
The research discussed above leads us to the conclusion that aggregation operators are essential in decision-making since they combine data from several cause into a single value. Through the use of FOLDFNs, DM is given additional latitude in how it might express judgement in situations requiring real-world decision-making. In this study, we established the notion of sine hyperbolic fractional orthotriple linear Diophantine fuzzy number (sinh-FOLDFN) to address such challenges using sine hyperbolic function and FOFNs. This is what spurs the present FOLDF research activity. The sine hyperbolic function is a significant function that also has the advantages of being amplitude and symmetric about the origin and meeting criteria of experts across the multiple time process. Based on the operations of Dombi t-norm and t-conorm to produce some more sophisticated operational laws for FOLDFNs is the main goal of the entire study. A list of geometric aggregate and averaging operators is given based on the indicated operating rules, along with a full explanation of the pertinent features. We offer an innovative application of alternative selection using suggested operators, to overcome issues with group DM. The effects of the parameters on how the alternatives rank are also extensively examined.
The contribution of this study are the following. 1) Firstly, we define a new set called sinh-FOLDFS and its basic properties.
3) Using Dombi operational laws, we propose a series of aggregation operators using sinh-FOLDFS information. 4) We define an algorithm using proposed aggregation operators.
6) The proposed method can be applied to real life DM problems to find the best optimal solution.
The following summarizes the remaining sections of the complete study: In Sec. II, we provide a brief explanation of some basic terminology that relate to FOLDFSs. First, we provide a definition for sine hyperbolic FOLDFNs in Sec. III along with a list of its features. Additionally, we derived sine hyperbolic operation laws for FOFSs using Dombi t-norm and t-conorm. In Sec. IV, we defined sine hyperbolic fractional orthotriple linear Diophantine fuzzy Dombi averaging operators. In Sec. V we defined sine hyperbolic fractional orthotriple linear Diophantine fuzzy Dombi geometric operators. On the basis of the given operators, we defined an algorithm. In Sec. VI, we solved an illustration of an alternate selection problem using defined operators. A comparison of several methods is presented in Sec. VII to support the applicability of the suggested approach. The end of the article was covered in Sec. VIII.

II. PRELIMINARIES
We discussed some key ideas regarding the domain of dis-courseĤ in this section that are crucial for FOFSs.

A. SINE HYPERBOLIC OPERATIONAL LAWS BASED ON FOLDFNs
We defined sine hyperbolic fractional orthotriple linear Diophantine fuzzy numbers (sinh-FOLDFNs) in this section and went through some of its fundamental characteristics. Following these, we used Dombi t-norm and conorm operation as well as a few fundamental features to develop the operation laws for sinh-FOLDFNs.

B. SINE HYPERBOLIC FRACTIONAL ORTHOTRIPLE LINEAR DIOPHANTINE FUZZY DOMBI OPERATION
In this section, we create sine hyperbolic FOLDF operational laws (sinh-FOLDFOLs) for FOLDFNs using the Dombi tnorm and t-conorm. We'll then develop some AOs for sinh-FOLDFNs after these.
Definition 2.13: Let FOLDFNs, where f ≥ 1 and λ > 0. Then, sine hyperbolic FOLDF operational laws (sinh-FOLDFOLs) are defined as; Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.

III. SINE HYPERBOLIC FRACTIONAL ORTHOTRIPLE LINEAR DIOPHANTINE FUZZY DOMBI AVERAGING OPERATORS
Using sinh-FOLDFOLs of FOLDFNs, the following geometric aggregation and weighted averaging operators are defined.

Definition 3.1:
Let . . , n) be a set of sine hyperbolic fractional orthotriple linear Diophantine fuzzy numbers (sinh-FOLDFNs). Then, sinh-FOLDFDWA operator is a mapping sinhŤ n → sinhŤ, such that; As a result, we are given an auxiliary theorem that describes the observed Dombi operations on sinh-FOFNs.
Thus, Equation (12) is correct for n = k +1. Hence, we prove that Equation (12) is hold for any n.
The following properties can be easily verified using the sinh-FOLDFDWA operator.
The following features can be easily illustrated by utilizing sinh-FOLDFDOWA. Theorem 3.6 (Idempotency): be a set of sinh-FOLDFNs and sinhŤ − = min(sinhŤ 1 , . . . , sinhŤ n ) and sinhŤ + = max(sinhŤ 1 , . . . , sinhŤ n ). Then, where the permutation of sinhŤ i (i = 1, . . . , n) is sinhŤ We discover that sinh-FOLDFDOWA operator weights are the accurate method of the organised placement of sinh-FOLDF values from Def. (III-B), and that sinh-FOLDFDWA operator weights are the effective method of computing sinh-FOLDF values from Def. (III-A). In the sinh-FOLDFDWA and sinh-FOFDOWA operators, weights show a number of connected components. Since these components are typically assumed to be the same, we define the sinh-FOLDFDHA operator to remove this type of limitation.
The below properties can be displayed very simply by utilising the sinh-FOLDFDWG operator.
The below properties can be displayed very simply by utilising the sinh-FOLDFDWG operator.

Theorem 4.3 (Boundedness):
The following theorem is proved by Dombi product operation on sinh-FOLDFNs.
The below properties are simply illustrated for sinh-FOLDFDOWG operator.
According to Definition (IV-B), sinh-FOLDFDWG operator weights are the efficient way to represent a sinh-FOLDF value, and sinh-FOLDFDOWG operator weights are the precise form of an organised location for sinh-FOLDF values. The sinh-FOLDFDWG and sinh-FOLDFDOWG operators use weights to express a number of interrelated factors. Since it is typical for these aspects to be the same, we add the sinh-FOLDFDHG operator to work around this restriction.
76898 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
When the operators are equal to ϖ = 1 n , . . . , 1 n , sinh-FOLDFDWG and sinh-FOLDFDOWG are regarded a special case of sinh-FOLDFDHG operator. Because of this, the sinh-FOLDFDHG operator is a generalised version of the sinh-FOLDFDWG and sinh-FOLDFDOWG operators that expresses the structured condition and degree of disagreement assertions.

V. APPROACH FOR MCGDM BASED ON sinh-FRACTIONAL ORTHOTRIPLE LINEAR DIOPHANTINE FUZZY DOMBI AGGREGATION OPERATORS
Let's say we have a DM problem with m possible solutions (ℜ 1 , . . . , ℜ m ) , n possible criteria and d experts (E 1 , . . . , E d ).
Let the expert and attribute weights, respectively, be,. Where ϖ, ε ∈ [0, 1] and n j=1 ϖ j , ε j = 1 are true. Each expert E k evaluates the offered options ℜ i in terms of FOLDFNs and rates them according to the attribute. The activities listed below are used.
Step 1: Collect the values for each alternative according to the sinh-FOLDFN in the form of decision matrix ℜ (k) .
Step 2a: Aggregate the different preferences ℜ Step 2b: Aggregate the different preferences ℜ Step 3a: Using sinh-FOLDFDOWA operator and weight vector ε, find cumulative values of alternative ℜ i .
Step 3b: Using sinh-FOLDFDOWG operator and weight vector ε, find cumulative values of alternative ℜ i .
Step 4: Using Equ. (7) to find the score values of ℜ i .
Step 5: According to the score value, rank all the alternatives.

VI. NUMERICAL EXAMPLE
The MCGDM problem is addressed in the following section using the sinh-FOLDF geometric operator and sinh-FOLDF averaging operator. The first scenario involves choosing the best textile from among five textile industries in order to produce the best fabric based on four parameters, with all the data being provided by sinh-FOLDFNs.

A. TEXTILE INDUSTRY SELECTION PROBLEM
Markets have become more open and competitive as a result of globalization, which has contributed to corporations changing their business practices. More and more businesses are focusing on internationalization in order to become less reliant on the home market and maximize profitability [40].
We used a scenario created by Akram et al. [28] as our model. A fashion designer makes an effort to choose the greatest textile sector to supply exquisite cloth. Assume that the three experts E 1 , E 2 and E 3 evaluated the five firms ℜ 1 , ℜ 2 , ℜ 3 , ℜ 4 and ℜ 5 based on four factors: durability (D), price (P), moisture absorption and heat conductivity (MH), and appearance and style (AS). The data are provided by the sinh-FOLDFNs listed in Tables 1, 2, and 3 and have the associated weights of ε = (0.4, 0.3, 0.2, 0.1) T for criteria and ϖ = (0.4, 0.3, 0.3) T for experts, respectively. The operators of sinh-FOFLDFDWA and sinh-FOFLDFDWG initially solved the matter. In this study, the same case is addressed using both the sinh-FOFLDFDWA and sinh-FOFLDFDWG operators. The following describes the precise computing procedure.
Step 1. The following Tables provided by experts: Step 2a. Using the sinh-FOLDFDWA operator and weight of experts are ϖ = (0.4, 0.3, 0.3) T and f = 3, ℘ = 1. Then, the new aggregated matrix is shown in Table 4.

VII. COMPARATIVE ANALYSIS
The comparison of sinh-FOLDFDWA and sinh-FOLDFDWG operators with Spherical fuzzy and fractional orthotriple fuzzy aggregation operators is established in this part. According to our findings, pre-existing aggregation operators in the context of SFSs and FOFSs are unable to handle data in the form of FOLDFNs.
A comparison study with a few other existing methods [29], [34], [35], [36], [37], [38], [41], [42], [44] was conducted in order to show the superiority of our specified sinh-FOLDF approach. Table 6 demonstrates that the alternative methods now in use are unable to solve the defined illustrated example of Section VI using FOLDF values. Although the procedure in [38], [41], [42], [44] includes a linear Diophantine fuzzy set of details, it cannot be used to solve the given model. From an examination of Table 6, it is clear that the methods now in use lack basic information and are unable to solve or rank the established case. As a result, the established approach is more effective and trustworthy than the other ways in use today.  Membership, neutral and non-membership grades are limited in spherical fuzzy sets and fractional orthotriple fuzzy sets, and parameterizations are not supported. This research gap is filled by sinh-FOLDFS. Sinh-FOLDFS has more room than SPSs and FOFSs. The role of reference parameters is introduced in sinh-FOFLDFS, and the limits on membership, neutral, and non-membership grades are removed. The suggested approach improves on previous methodologies, and the decision maker (DM) has complete freedom in selecting grades. The proposed model has a significant association with multi-attribute decision-making problems. The pair of reference parameters is crucial in this model. These assist us in expanding the space of membership and non-membership grades as well as providing parameterization in the model.

VIII. CONCLUSION
Sin hyperbolic fractional orthotriple linear Diophantian fuzzy models are more practical and valuable than Spherical fuzzy and fractional orthotriple fuzzy models because they provide more spaces between membership, neutral and nonmembership grades for conveying imprecise information. Dombi operators with general parameters have a high degree of freedom.
As Dombi operators have not yet been applied for PFSs, hence motivated from these operators, in this study, we have presented Dombi operations on sinh-FOLDFNs. We investigated averaging and geometric operators using Dombi operations to build specialized sinh-FOLDF aggregation operators, such as the sinh-FOLDFDWA, sinh-FOLDFDOWA, sinh-FOLDFDHA, sinh-FOLDFDWG, sinh-FOLDFDOWG, and sinh-FOLDFDHG operators. The many traits of the produced operators are being researched. The defined operators are used to find best alternative. We used operational laws using sine hyperbolic functions to deal the uncertinity in the data, which will guard against information loss throughout the investigation. The suggested strategy takes care of the DM issue as well. The study is based on the representation of the sinh-FOLDFOLs, and the resulting operators are generalizations of the existing operators from the given instances. The defined AOs are utilized for solving an MCGDM problem. Using a real-world situation, the effectiveness of the suggested operators is evaluated. Finally, a comparison study has been given to demonstrate how the suggested operators are advantageous. Thus, the suggested operators are more broad, consistent, and detailed in order to address DM concerns in the sinh-FOLDFS environment.
The proposed concept might eventually be expanded to include the Einstein, Hamacher, Frank, and Bipolar fuzzy sets.