Spherical q-linear Diophantine fuzzy aggregation information: Application in decision support systems

: The main goal of this article is to reveal a new generalized version of the q-linear Diophantine fuzzy set (q-LDFS) named spherical q-linear Diophantine fuzzy set (Sq-LDFS). The existing concepts of intuitionistic fuzzy set (IFS), q-rung orthopair fuzzy set (q-OFS), linear Diophantine fuzzy set (LDFS), and spherical fuzzy set have a wide range of applications in decision-making problems, but they all have strict limitations in terms of membership degree, non-membership degree, and uncertainty degree. We moot the article of the spherical q-linear Diophantine fuzzy set (Sq-LDFS) with control factors to alleviate these limitations. A Spherical q-linear Diophantine fuzzy number structure is independent of the selection of the membership grades because of its control parameters in three membership grades. An Sq-LDFS with a parameter estimation process can be extremely useful for modeling uncertainty in decision-making (DM). By using control factors, Sq-LDFS may classify a physical system. We highlight some of the downsides of q-LDFSs. By using algebraic norms, we o ﬀ er some novel operational laws for Sq-LDFSs. We also introduced the weighted average and weighted geometric aggregation operators and their fundamental laws and properties. Furthermore, we proposed the algorithms for a multicriteria decision-making approach with graphical representation. Moreover, a numerical illustration of using the proposed methodology for Sq-LDF data for emergency decision-making is presented. Finally, a comparative analysis is presented to examine the e ﬃ cacy of our proposed approach.


Introduction
Along with intricacies and ambiguity inherent in such situations, classical mathematics is not always effective when solving real-world problems. Zadeh [52] established the fuzzy set notion by assigning grades to possibilities with limit [0, 1]. Since Zadeh's approach to the fuzzy collection, fuzzy logic has been used to characterize imprecision, ambiguity and obscureness in a variety of fields [29,44,46]. Uncertainty-related problems arise frequently in DM, however they are difficult to predict and control on account of extensive modelling and regulating situation of these uncertainties.
The additional terms of non-membership degrees (NMD) to membership degrees (MD) with the premise that by adding MD and NMD restricted by unity, atanassov [2] gave notion about the thought of intuitionistic fuzzy set (IFS) as an extend form of FS. IF elements were used in a geometric description by attanassov [1]. Xu [47] represent weighted geometric notations for IFNs. Garg [20] utilized Einstein's t-norm operating rules for IFS. Apart from that, the researchers, create a new useful technique for collecting the OWA weights. By giving these reasons a low weight, the technique can reduce the impact of unjust arguments on the outcome of the decision. Xue [48] give their views about Measure-based granular uncertainty decision-making with intuitionistic fuzzy sets by using Choquet's integral, measure and representative payoffs. It is capable of handling problems in intuitionistic fuzzy environments. Khalil [31] look into two new distances: the absolute normalized Euclidean distance and the square hamming distance, both of which are used in decision-making as intuitionistic fuzzy sets.
The Pythagorean fuzzy set (PyFS) was introduced by Yager [49], which is an expansion of the IFS conception that satisfies the criteria that the summation of square of its MD and NMD is not exceeding by one. Farhadinia [17] proposed Pythagorean fuzzy decision technique to make decisions using similarity. Yager [50] added multiple aggregate operators (AOs) to the PyFS framework and Garg [23] updated PyFSs to include more integrated operational rules and associated aggregation operatives. Some Pythagorean fuzzy Dombi aggregation operators were constructed and explained in [33]. Garg in [21,22] utilized Einstein t-norm operational standards towards PyFNs. [36] created a couple of symmetric PyF AOs. Zeng [54] reported the information on probabilistic and ordered weighted averaging (OWA). Under the PyF framework, Garg [24] put forward various strategic DM ways to tackle MCDM problems with imminent probability. Deqing [35] suggest several distance measures that account for the four Pythagorean fuzzy set parameters for Pythagorean fuzzy sets and Pythagorean fuzzy numbers. Utilizing triangle conorms, Firozja [19] presents a novel similarity measure for Pythagorean fuzzy sets (S-norm). For more details, we refer to [13][14][15] A wide range of applications in various real-world sectors are concept based on IFSs and PyFSs. FSs, IFSs along with PyFSs ideas have a broad range of implementations in various physical-world sectors, but allow them hold their individual set of constraints that are appropriate to MD and NMD. To control these restrictions, Riaz [42] put forward the innovative invention of Linear Diophantine Fuzzy Set (LDFS) by adding control factors (CFs). The LDFS concept is most effectual and multifaceted than other models due to the inclusion of CFs. This collection expanded the space for MD and NMD by filling the cracks left by current structures and by adding CPs. LDFSs also gave two grades about information, the sum of which is not exceed by one and total of its factors such as the total of product of control factors with MD and NMD can not be surpass 1. Many researcher contribute to the LDFSs like: Iampan [16] introduced linear Diophantine fuzzy Einstein aggregation operators for multi-criteria decision-making problems for the established post-acute care (PAC) model network for the health restoration of patients with cerebrovascular disorders (CVDs). Decision making with linear Diophantine fuzzy relations and their algebraic properties was first described by Ayub [12]. Mahmood [37] presented linear Diophantine uncertain linguistic setting-based generalized hamacher aggregation operators and their applications in decision-making problems. Riaz [41] expressed his thoughts on a novel method for choosing third-party reverse logistics providers using linear Diophantine fuzzy prioritized aggregation operators. Farid [18] proposed using Einstein prioritized for linear Diophantine fuzzy aggregation operators to pick suppliers of sustainable thermal power equipment.
As a generalization of FS, IFS and PyFS, the concepts of spherical fuzzy sets (SFS) was introduced by Ashraf et al. [11]. The human thoughts are not ristricted to MD and NMD like [27,28,45]. So, Ashraf [9] presented the concept of spherical fuzzy set and its aggregation operators. The sphercal fuzzy set is an extention of picture fuzzy set containing three grades namely MD, ND (neutral degree) and NMD with limitation 0 ≤ (℘) + δ (℘) + F(℘) ≤ 1 but ashraf et al. presented the generalization set picture fuzzy set as spherical fuzzy set with constraint 0 ≤ 2 (℘) + δ 2 (℘) + F 2 (℘) ≤ 1. Also Ashraf et al. developed the spherical fuzzy Dombi aggregation operatives in [10]. The decision-making system based on cosine similarity was put forward by Rafiq et al. [40]. Barukab et al. [5] used Spherical fuzzy distance measurements to identify environmental influences on child development. In [6] Spherical fuzzy sets were utilized by Ashraf et al. to express the spherical fuzzy t-norms and t-conorms. Ashraf et al. [53] used application in the multi-attribute group decision making problem using symmetric sum based aggregation operators for spherical fuzzy information. In [25] Gündodu and Kahraman presented a novel technique for dealing with uncertainty in renewable energy utilising an analytic hierarchy process. The DM methodologies connected to child development effect environmental elements under SFSs were reported by Ashraf et al. [5]. Emergency decision model to deal with COVID19 under spherical fuzzy information in different ways is study in [8] and [7]. Jin et al. [30] used the logarithmic function to aggregate the uncertainty in decision-making issues and established the spherical fuzzy aggregation information. Gündogdu et al. [26] developed the QFD technique for spherical fuzzy information and explored how it may be used to linear delta robot technology. So, in some physicalworld situations the summation of the membership grades such as alternative meets to fulfil attribute given by DM is sometimes beat one, showing that SFS did not carry off his goal concerning to control factors, like 0 ≤ 2 (℘) + δ 2 (℘) + F 2 (℘) 1. For more detail decision making methods, we suggest fuzzy decision models [32][33][34]51].
Almagrabi [3] introduced the q-linear Diophantine fuzzy set (q-LDFS), an unique generalisation of the Pythagorean fuzzy set, q-rung orthopair fuzzy set, and linear Diophantine fuzzy set, and also highlighted its significant aspects. Furthermore, aggregation operators contribute significantly to the efficient aggregation of uncertainty in decision-making situations. The q-linear Diophantine fuzzy set cover the MD ( ) and NMD (F) with control factors (λ, µ) having restrictions 0 ≤ λ q +µ q F ≤ 1, ∀℘ ∈ U and 0 ≤ λ q + µ q ≤ 1. It gives us an open choice to select the MD and NMD values. The qlinear Diophantine fuzzy set Qiyas [39] used the q-rung linear Diophantine fuzzy to suggest some new distance and similarity measurements (q-ROLDF). The q-linear Diophantine fuzzy set only deals with MD and NMD but still there is a gap of ND neutral degree . We need to add new fuzzy numbers in fuzzy set theory in order to close this gap.
This research is driven by the desire to offer a novel hybrid structure of spherical q-linear Diophantine fuzzy set (Sq-LDFS) that includes both SFS and q-LDFS in order to evaluate the best option based on the record of attributes. This collection gives you a bird's-eye view of a variety of SFS generalized shapes. In addition, the research discusses certain aggregation operations for integrating spherical q-linear Diophantine fuzzy information in uncertain emergency condition. These operators are unique in that they can synthesis spherical q-linear Diophantine fuzzy information, further developing and enriching the idea of spherical q-linear Diophantine fuzzy aggregation operators. Furthermore, because the suggested aggregation operators are useful DM tools, they assist the development of multiple-criteria decision-making in the spherical fuzzy setting. This paper's contributions are mostly evident in the following areas: (1). The novel Sq-LDFS strategy with q > 1 is our initial target in filling this knowledge gap. We can solve the IF, PyF, q-OF, LDF and SFS structure (e.g. for 0.9 + 0.7 + 0.4 > 1), also 0.9 2 + 0.7 2 + 0.4 2 > 1 entrance of Sq-LDFS such that (0.9)(0.5) + (0.7)(0.4) + (0.4)(0.3) < 1, where the triplet (0.6, 0.7, 0.4) can be utilized for MD, neutral grade (NG) and NMD respectively. Because this proposed model is similar to familiar Diophantine equation (ax + by + cz = c) of number theory and the insertion of the qth degree of control factors it appears that Sq-LDFS is the best name for the established framework.
(2). The second objective is to implement the qth degree of control factors (CFs) capabilities in Sq-LDFS because qth factors cannot be handled by IFSs, PyFSs, q-ROFSs, LDFSs and SFSs. The designed system improves on current approaches and DM has complete freedom in selecting grades. By changing physical sense of connection this model also characterizes the problem. When setting q = 1, the respective assemblage is converted to SLDS. Furthermore, if we qth value increases, the Diophantine space extends giving boundary bounds a larger search space to transmit a broader range of fuzzy data. As a result, we may use Sq-LDFSs to describe a broader range of fuzzy data. By taking it in another way, we can keep adjusting the value of the factor q to decide the information expressive range, making Sq-LDFSs more idea and adaptable for unpredictable environments.
(3). Our third objective is to provide a clear connection between the present research and MADM problems. We derived decision support techniques to deal with multi-attribute difficulties in a parametric manner. Surprisingly both algorithms produce the same outcome.
This research contributions are follows as: (i). By merging the features of SFSs with q-LDFSs, we may offer some more sophisticated operational laws under spherical q-LDFSs based on algebraic t-norm and t-conorm.
(ii). Under spherical q-linear Diophantine fuzzy numbers, offer a collection of innovative aggregation operators using the defined algebraic t-norm and t-conorm. The significance of some fundamental features between the proposed aggregation operatives is also demonstrated.
(iii). To provide an unique MADM technique for solving decision making issues based on the proposed aggregation operatives.
(iv). A numerical demonstration as well as their complete evaluations, demonstrates the consistency and usefulness of the suggested method.
This work is organized as follows: Section 2 introduces the fundamental ideas of FS, IFS, PyFS, q-OFS, LDFS and SLDFSs. In Section 3, we explain the unique notion of Sq-LDFS and provide illustrations to demonstrate some Sq-LDFS procedures. The concept ofŚq-LDFSs is introduced in Section 4 for theŚq-LDFWA,Śq-LDFOWA operatives also provides the concept ofŚq-LDFSs for definingŚq-LDFWG,Śq-LDFOWG operatives, as well as distinct score for evaluatingŚq-LDFNs of different orders. The notion of MADM mathematical modelling is presented in Section 5 with the help of theŚq-LDFWA andŚq-LDFWG aggregation operators. In Section 6, we compare the proposed method to current methods in detail and examine the aggregated findings as the influence of score functions on the final selection. Section 7 outlines the conclusion of this project.

Primal concepts with some premises
In this section, we recall some significant and fundamental concepts of FSs, PyFSs, SFSs, LDFSs, q-LDFSs and SLDFSs. We also introduce some fundamental properties of the mentioned notations used in the study and briefly discuss the ideas and results employed in the rest of the work.

Definition 8.
[3] q-Linear Diophantine fuzzy set (q − LDFS ) q over a fixed set U depicted in the mathematical type as given:

Spherical q-linear Diophantine fuzzy set Ś q − LDFS
In this section, we initiate a novel notion of a spherical q-linear Diophantine fuzzy set (Sq-LDFS). In pure mathematics, there is a well-known linear Diophantine equation for three independent variables ax+by+cz = d, and the framework provided fits it. It's a little more challenging since the participation, abstention, and dissatisfaction categories in the picture fuzzy set, spherical fuzzy set, and SLDFS are confined in certain ways. To address these restrictions, we proposed the concept of Sq-LDFSs based on reference or control parameters. One significant component of this approach is that the decision-maker (DM) is not bound by grade membership (positive, neutral, or negative). This framework is frequently used to categories the problem by selecting several sorts of reference or control criteria. The Sq-LDFSs structure, its visual representation, and the use of diagrams to explain specific principles are all discussed.
Definition 9. A spherical q-linear Diophantine fuzzy set Ś q − LDFS over the non-empty fixed set U reflect the mathematical form as under: are MD, ND and N MD, also λ, µ, ω ∈ [0, 1] are CF. The mentioned degrees surely met the constraint These comparative factors may aid in the description or identification of pattern. We can arrange the pattern by changing the way in given factors.
In this part we have present the Spherical q-linear Diophantine fuzzy set which is the extension of q-linear Diophantine fuzzy set by extended the reference parameters and classify as: the summation of qth power of reference factors (RFs) by scalar multiplication with (MD) , (ND) and (N MD) . Our futuristic model of spherical q-linear Diophantine fuzzy set Ś q − LDFS is most flexible and having more efficacy than (q-LDFS) due to extended control factors (CF s) in spherical form. Our model strong correlation with (MADM) issues.
Now we present the definitions related to (Śq-LDFS) which are absolute spherical q-linear Diophantine fuzzy set and null spherical q-linear Diophantine fuzzy set.
Definition 10. The absolute spherical q-linear Diophantine fuzzy set is structured as: Definition 11. The null spherical q-linear Diophantine fuzzy set is the compliment of absolute spherical q-linear Diophantine fuzzy set is structured as: we are familiar with this, if (1). we set q = 1 in Definition 9,Śq-LDFS becomesŚLDFS, . we set q = 2 in Definition 9,Śq-LDFS becomesŚpherical quadratic DFS.
These are the major advantages of Sq-LDFS for distinct q values. It should be observed that as we increase the q values, the spherical Diophantine space stretches, giving the boundary parameters a larger search space to generate a larger spectrum of fuzzy data. Setting q = 1 gives Riaz's (ŚLDFS), while setting q = 2 makes Ashraf's (ŚFS) as displayed in Figure 1: Any IFS becomes LDFS, each LDFS becomes q-LDFS, also q-LDFS becomes Sq-LDFS by some additional terms as shown in Figure 2.
By combining 6 and 8 point of Definition 12, we gain We can retrieve it by using the right side of the equation.
Futhermore, apart from this Thus shown. In favour of (5) , the proof is given as 5) . , Thus proved.
In praise of equality (7), we have The proof of the remaining properties can be handled easily.

4.Śpherical q-linear Diophantine fuzzy weighted aggregation operator
This section presented the list of novel algebraic norm based aggregation information under spherical q-linear Diophantine fuzzy sets. Also the score function is introduced for ranking the Sq-LDFNs.
(3) . Now, we have to prove for n = + 1, then we havé Therefore, the equation is true for n = + 1. Hence proved.
(3) . Now, we have to prove for n = + 1, then we havé Therefore, the equation is true for n = + 1. Hence proved.

Mathematical modeling
This section consists of an algorithm based on the proposed list of novel aggregation operators under spherical q-linear Diophantine fuzzy (Sq-LDF) information to tackle the real world decision making problems. The set N g = N g 1 , N g 2 , N g 3 , ..., N g m contains the numbers of alternatives while ð = {ð 1 , ð 2 , ð 3 , ..., ð n } represents the numbers of attributes. Let = ( 1 , 2 , ..., n ) T be the weight vector met the criteria as ĥ > 0, ĥ ∈N ĥ = 1.
Consider the Sq-LDF decision matrix DM = (( s , δ s , F s ), (λ s , µ s , ω s )) (gm×n) , where s is the membership degree, δ s is the neutral degree, F s is the non-membership degree and λ s , µ s & ω s are the control factors for which the alternative N gm satisfies the (ð n ) attribute provided by the decisionmakers, such that s , F s , λ s , µ s ∈ [0, 1] as 0 ≤ λ q s s + µ s δ s + ω q s F s ≤ 1, (q = 1, 2, 3, ..., t) .
The algorithm contain the follows steps.
Step Step-2 Expert evaluation information required in standardŚq-LDF numbers: Prior to beginning the computations, the input data must be normalized in order to achieve the best result. It is therefore possible to standardized the Sq-LDF information.
Step-3 Evaluation of resultant weight vector as follows: Step-4 Using the proposed list of aggregation operators underŚq-LDFSs to compute the integrated (combined) aggregated value for individual attribute ð along with their weight vector.
Step-5 Utilizing the concept of graded functions, score function, quadratic function and expectation score functions to evaluate the scores for individual attribute ð from aggregated expert values.
Step-6 Individual rank the attribute based on the values of the score function, quadratic score function and expectation score functions.
Step-7 The attribute that get highest score has the highest ranking and must be selected for the final selection.
The flow chart of the algorithm is presented in Figure 3:  Step-2 The consider expert information is benefit type so we do not need to standardized the information.

Numerical illustration of the proposed algorithm
Step Step-4 Now we'll utilize the proposed aggregation operatorsŚq-LDFWA,Śq-LDFOWA, Sq-LDFGA and Sq-LDFOWG to compute the integrated (combine) Sq-LDF data computed in Table 4. Table 4. Sq-LDF aggregated information. Step-5 Now the scores are computed as follows in Tables 5-8.
Developed operators Expectation score Sq-LDFWA Step-7 The above tables shows that ℘ 5 is the best alternative (Due to the fact that the scoring functions for the data in Tables 5-8 for quadratic score function are not identical, hence they are sufficient to identify the optimal choice).

Comparative analysis and discussion
This section compare the proposedŚq-LDF aggregation operators with the existing methodology in the literature described in [42], demonstrating their ability to handle physical-world decision making problems under complex uncertainty. Because of the qth power, this notion is impressive in that it covers the valuation spaces of IFSs, SFSs, q-ROFSs, LDFSs and SLDFSs.
Cogency and integrity of the propose method: Our approach is adaptable and suited for all forms of input data. The proposed model is effective in dealing with uncertainty. This technique covers the areas of IFS, SFS, q-ROFS, LDFS and SLDFS with the addition of qth degree of CFs. By increasing the qth degree of factors, more membership, neutral and non-membership space is created, as well as the physical layout. We can utilize our strategy in a diversity of situations. we're using it in selection of required best university. The proposedŚq-LDFS may be simply modified to produce a variety of outputs.
Score function impact: We generalized and then implemented the previously described three types of score functions consists of three membership grades with their related control parameters, SF, QSF and ESF. Allow for a somewhat variable result because each SF has its own observation and ordering techniques. Tables 4-7 shows that the SF, ESF and QSF rankings differ slightly from one another. However, it's worth noting that the end result from both algorithms is practically identical for all scoring functions.
Aggregation versatility with various inputs and outputs: Due to qth power of control factors and the three membership grades increases grade space and can differ build on the scenarios in MADM approaches, this approach is significantly more versatile than others. It can also be utilized for a variety of input and output informations where the spherical and q-linear Diophantine fuzzy sets fail to fulfill their requirements.
Comparison of the suggested method to existing approaches and its superiority: Because Sq-LDFS handles qth simulations, it takes up a lot of space when compared to IFSs, SFSs, q-ROFSs, LDFSs and SLDFSs. [3] introduced q-LDFSs with additional qth degree, although q-LDFSs have some limitations and cannot handle the problem related to three grades. We expand the concept of q-LDFSs by merging it with SFS and proposed Sq-LDFSs to fill this knowledge gap. The SLDFS only serves for q = 1, but the strategy we propose serves for q ≥ 1.
The proposed approach and MADM difficulties are inextricably linked. Tables 11-13 shows the comparison of aggregation operation values based on the Sq-LDFWA, Sq-LDFOWA, Sq-LDFWGA and Sq-LDFOWGA and their graphical representation is in Figures 4 and 5. Table 11. Analysis of score using existing methods.  Table 12. Analysis of Quardratic score using existing methods.  Table 13. Analysis of Expectation score using existing methods.   We make comparison of SLDF weighted and geometric operators with Sq-LDF weighted and geometric operators. The Tables 11-13 shows the better results than the results of SLDFSs.
The fuzzy set theory, whose popularity has increased ever since Zadeh introduced it, is supported by the work in this paper. Specific academics expanded fuzzy set theory and the most significant of which are IFSs, PyFSs and SFSs. In order to address uncertainty in real-life problems are challenging to resolve utilizing fuzzy models. In 2019 the structure of Pythagorean fuzzy sets extended by [3] to initiate the concept of q-LDFS, in which they introduced the role of control factors, which hold the condition 0 ≤ λ q + µ q F ≤ 1, ∀℘ ∈ U and 0 ≤ λ q + µ q ≤ 1. However the total sum of degrees with scalar product with control factors provided by DM may be greater than one, i.e. λ q + µ q > 1, opposing the q-LDFS constraint. Consequently, q-LDFS, SFSs and SLDFSs are restricted to meet his target in terms of degrees, information are as follows in Table 14: Cann't fulfill the condition + F > 1 × q-ROFS Incapable to treat smaller"q" power in this state , q + F q > 1 for = 1 = F × SFS Incapable for this state, + δ + F > 1 for = δ = F = 1 ×

SLDFS
This collection take into account the of condition 0 ≤ λ + µδ + ωF ≤ 1, as well as the influence of control factors. SLDF operators are the sole existing method,which we compare to our suggested method.
× It limits the MADM and has an impact on the best decision. We offer the innovative idea of the Sq-LDFS, which is capable of dealing with these circumstances and eliminate these contradiction.

Conclusions
The manuscript briefly demonstrated how the Sq-LDFS framework extends all existing theories and provides a strong foundation with no limitations. The formal definition of Sq-LDFS was stated which is generalization of q-linear Diophantine fuzzy set by merging it with spherical fuzzy set to enhance the memberships space. Under the Sq-LDF context, set theoretical operations were introduced, and several aggregation operators were established. Some interesting properties of the proposed aggregation operators were explored. Furthermore, a MADM technique based on suggested aggregating operators and scoring functions was established. A case study was offered to demonstrate how the suggested strategy be used. As a limitation on our study, we only take into account five alternatives in order to demonstrate the veracity of the suggested strategy. The suggested technique works where the SFSs, q-linear Diophantine fuzzy set did not work. The SLDFS only works for q=1 but Sq-LDFS works for q ≥ 1. The methodology of the suggested technique may be converted into a computer program, allowing us to conduct our research for a small number of qualities and alternatives while using huge data and taking into account additional factors. Future research goals include investigating additional aggregating operators like Hamacher and Bonferroni, similarity and distance metrics, and extending the suggested operators to the Archimedean norm.