Stepping Up: Education Department Installs Biometric System to Ensure Based on Complex T-Spherical Hesitant Fuzzy Aczel-Alsina Aggregation Operators

The installation of biomatrix systems in educational departments are very awkward and complicated task because many employees are biased. The major influence of this manuscript is to evaluate the idea of a complex T-spherical hesitant fuzzy (CTSHF) set and its dominant and flexible operational laws. Furthermore, it is also very complicated to evaluate the idea of Aczel-Alsina aggregation operators based on CTSHF information, for this, first, we derive Aczel-Alsina operational laws under the consideration of the CTSHF values, and then we propose the CTSHF Aczel-Alsina weighted averaging (CTSHFAAWA) operator, CTSHF Aczel-Alsina ordered weighted averaging (CTSHFAAOWA) operator, CTSHF Aczel-Alsina weighted geometric (CTSHFAAWG) operator, CTSHF Aczel-Alsina ordered weighted geometric (CTSHFAAOWG) operator, and simplify their properties, called idempotency, monotonicity, and boundedness. Additionally, to find the major factor of the installation of the biometric systems in the education department based on the proposed operators, we demonstrate the procedure of multi-attribute decision-making (MADM) problems based on the derived operators. In the end, we compare the ranking values of the proposed techniques with the obtained results of prevailing methods to enhance the worth of the proposed operators.


I. INTRODUCTION
In ancient times we had the crisp function in this function we couldn't define the uncertainty later on Zadeh [1] produced the idea of a fuzzy set (FS) that can handle the problem of uncertainty in different situations.FSs are sets whose elements have degrees of membership, FS theory permits membership function valued in the interval [0,1].But in real life, it may not always be true that the degree of The associate editor coordinating the review of this manuscript and approving it for publication was Alba Amato .non-membership function is equal to one minus membership function because there may be some hesitant degree.Therefore, Atanassov [2], [3] proposed the intuitionistic FS (IFS).IFSs are sets whose elements have truth grades and falsity grades.We need to use IFS to represent the hesitation degree concerning both the truth grade and falsity grade of an element to a set.In real-life examples we use IFSs in many problems one of these is when we toss a coin we have two possible outcomes head or tail, it will give either head or tail at a time but it will not give both outcomes at the same time.This is a natural example of IFS.For handling such kinds of problems, the theory of Pythagorean fuzzy sets (PyFS) was derived by Yager [4].PyFS is the advanced shape of the IFSs and overcame their limitation because the structure of the IFS and PyFS are the same, but only changed their condition.In general, PyFS has a close relationship with IFS.In PyFS we can measure uncertainty more accurately and sufficiently than IFSs.Later on, Yager [5] presented the concept of q-rung orthopair fuzzy sets (q-ROFS) which can be considered an efficient method to describe the vagueness of multi-attribute decision-making problems.Further, Cuong [6] introduced the picture fuzzy set (PFS), which is a direct extension of FS and IFS by including the concept of positive, negative, and neutral membership degree of an element.But in many cases the theory of PFS has been neglected because of their limitation, therefore, the theory of spherical fuzzy set (SFS) and T-spherical fuzzy set (TSFS) was invented by Mahmoud et al. [7], which is a powerful concept with four membership functions i.e., membership, non-membership, refusal, and abstinence degree that can be deal with uncertain information as compared to existing FSs.Some applications of FS and their extensions are stated as follows, for instance, aggregation operators [8], [9], similarity measures [10], [11], decision-making techniques [12], [13], and different types of techniques [14], [15].
Complex FSs (CFSs) are generalized forms of classical fuzzy sets.They allow us to represent the uncertainty of more complex problems.Ramot et al. [16] introduced these sets in 2002.It can be defined by a mapping that is from a complex space to a unit interval [0,1].Therefore, we can say that it can associate a degree of membership between 0 and 1 with each element in a complex space.It has a lot of applications in different fields some of which are as follows; it is used in control systems, decision-making, artificial intelligence, etc.But specifically, it is used in situations where we observe that uncertainty is inherent and where we are facing complexity to define precise boundaries.A complex IFS (CIFSs) [17] can be defined by mapping from a complex space to a duplet of values, such as membership degree and non-membership.CIFS is a model to tackle some situations where we have a level of hesitancy that can be assigned to membership and non-membership values in a complex space.CIFSs aim to provide a more granular resolution of uncertainty than classical FSs.They find application in a variety of fields including decision-making, finding patterns, image processing, etc. where dealing with uncertainty is critical.Complex PyFS (CPyFS) [18] are the enlargement of PyFS.They give us a more clarified mechanism to demonstrate uncertainty and lack of accuracy in the decision-making process.The main intention of using CPyFS lies in their capability to tackle difficult circumstances where several sources of unsureness or incompatible facts and figures occur.By using CPyFS one can make more familiar and stalwart decisions by keeping in mind not only the truth grade and falsity grade but also the abstinence grade related to each alternative or substitute.Complex q-ROFSs (Cq-ROFSs) [19] are the advanced form of q-ROFS, they can be treated as well well-organized method to explain the vagueness of multi-attribute decision-making (MADM) problem.Akram et al. [20] introduced the concept of complex PFS (CPFS), it is a mathematical framework that elaborates the traditional FSs to show complex relationships and uncertainties most easily and efficiently.The major aim of CPFS is to provide a tool that can handle more difficult information in different applications such as decision-making, identifying patterns, and artificial intelligence.Ali et al. [21] introduced the concept of complex SFSs (CSFSs) which is the extended form of SFSs.The main aim of CSFSs is their capability to model complex, uncertain information most easily and efficiently as compared to classical FSs.It can deal with more crucial problems where inexactness and vagueness are at their peak.The complex TSFS (CTSFSs) [22] are the extension of TSFSs that can tackle more complex problems as compared to classical fuzzy sets.CTSFSs use four degrees i.e., membership, nonmembership, refusal, and abstinence degree that can deal with uncertain information as compared to predefined FSs.Some applications of CFS and their extensions are stated as follows, for instance, aggregation operators [23], similarity measures [24], decision-making techniques [25], and different types of techniques [26].
The idea of hesitant FS (HFS) was derived from Torra [27] in 2010.In HFSs an element can have more than one degree of FSs, it can help us to handle such situations where we have uncertainty more flexibly because it shows hesitancy when an element has a single membership degree.For example: If we are not sure about the grades of any object Whether an object has grades A, B, and C, it might have a membership degree of 0.9 to grade ''A'', 0.6 to grade ''B'', and 0.5 to grade ''C''.Furthermore, the complex HFSs (CHFSs) were exposed by Mahmood et al. [28].CHFs include more complex models to enhance HFS, which allows for representing both reluctance and granularity in degrees of membership.This means that it can be able to handle such situations where the degree of membership is not only arbitrary but can also vary in detail.
In CHFS theory we have truth grade in complex numbers and can be written in polar coordinates.For example: If we are not sure about the intensity of any object like if we are talking about the temperature, it can be cold, and how much cold and hot it is we can justify it in this complex situation of uncertainty we use CHFSs.Both HFSs and CHFSs are specially used in the decision-making process where we are highly unsure and where we have incomplete information, and we have to make decisions on this incomplete information.Application of HFSs & CHFSs is found in areas such as expert systems, identification of patterns, and decisionmaking.
Aczel Alsina t-norm and t-conorm were proposed by Aczel and Alsina [29] in 1982.It is an advanced form of classical logic AND operator, that can be usually used for connecting FSs.It is especially used in situations where need to model uncertainty, similarly to other t-norms, that can provide a facility to make logical decisions in such situations where uncertainty occurs.It can facilitate us to make changes in fuzzy logic statements and these statements can have a range between truth grade and falsity grade.In mathematics, it can be constructed for designing the theory of fuzzy sets and fuzzy logic.Some valuable applications are stated in the shape, for instance, Aczel-Alsina (AA) operators for IFSs [30], geometric AA operators for IFSs [31], AA operators for PyFSs [32], AA operators for q-ROFSs [33], AA operators for HFSs [34], AA operators for intuitionistic HFSs [35], AA operators for PFSs [36], geometric AA operators for PFSs [37], AA operators for CPFSs [38], AA operators for TSFSs [39], and AA operators for CTSFSs [40].After the above discussion, we analyzed that the theory of complex T-spherical hesitant fuzzy set is not proposed by anyone, therefore, keeping the advantages of the CTSHFSs, our major intention is listed below: 1) To evaluate the idea of the CTSHF set and their dominant and flexible operational laws.2) To derive the CTSHFAAWA operator, CTSHFAAOWA operator, CTSHFAAWG operator, and CTSHFAAOWG operator.3) To state some properties for the above-derived operators.4) To find the major factor of the installation of the biometric systems in the education department based on the proposed operators, we demonstrate the procedure of MADM problems based on the derived operators.5) To compare the ranking values of the proposed techniques with the obtained results of prevailing methods to enhance the worth of the proposed operators.This manuscript is arranged in the shape: In Section II, we stated the old idea of HFSs and CTSFSs.And their optional laws such as algebraic and Aczel-Alsina operational Laws for CTSFSs.In Section III, we evaluated the idea of the CTSHF set and their dominant and flexible operational laws.In Section IV, we derived the CTSHFAAWA operator, CTSHFAAOWA operator, CTSHFAAWG operator, and CTSHFAAOWG operator.We also stated some properties for the above-derived operators.In Section V, to find the major factor of the installation of the biometric systems in the education department based on the proposed operators, we demonstrated the procedure of MADM problems based on the derived operators.In Section VI, we compared the ranking values of the proposed techniques with the obtained results of prevailing methods to enhance the worth of the proposed operators.Some concluding remarks are listed in Section VII.

II. PRELIMINARIES
This section stated the old idea of HFSs and CTSFSs.And their optional laws such as algebraic and Aczel-Alsina operational Laws for CTSFSs.
Definition 1 [27]: Based on the universal set ψ uni , the main idea of HFS ηh is evaluated and computed in the form: Definition 2 [22]: Based on a universal set ψ uni , the main idea of CTSFS ηp is evaluated or computed in the form: Additionally, we further explained the value of truth grade ηp (x) = ηr (x) i2 ( ηi (x) ) and the falsity grade ηp (x) = ηr (x) i2 ( ηi (x) ) with a valuable characteristic: • C , Moreover, we described the main idea of refusal information such as: and the final version of the CTSF number (CTSFN) is derived by: ηj Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
‫ב‬s η1 For the above information, we exposed the following properties, such as:

III. CTSHFS
This section introduced the theory of CTSHFSs and their operational laws such as algebraic and Aczel-Alsina operational laws for any number of CTSHFSs.• C + sup ( j ηr (x)) • C + sup ( j ηr (x)) • C , sup ( ] M oreover, we described the main idea of refusal information such as: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
for the above data, we exposed the following properties, such as: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

IV. ACZEL-ALSINA AGGREGATION OPERATORS FOR CTSHFS
This section talked about the novel theory of the CTSHFAAWA operator, CTSHFAAOWA operator, CTSHFAAWG operator, and CTSHFAAOWG operator, and their properties such as idempotency, monotonicity, and boundness.Furthermore, the mathematical representation of the weight vector is stated by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Proof: Select the value of n = 2, we have as shown at the bottom of the next page.
Our target is correct for n = 2, moreover, we consider it for n = k, such as shown at the bottom of page 14.
Then, we evaluate it for n = k+1, such as shown at the bottom of page 15.
Hence, the main result has been proved successful for all possible values of n.Additionally, we examined the idea of idempotency, monotonicity, and boundness for the present techniques.
Proposition 1: Based on a universal set ψ uni and in the presence of CTSFNs   Proof: Straightforward.Additionally, we examined the idea of idempotency, monotonicity, and boundness for presented techniques.
Proposition 2: Based on a universal set ψ uni and in the presence of CTSFNs Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Then we prove that the resultant value of the above operator is again in the shape of CTSHFN, such as shown at the bottom of page 19.
Proof: Straightforward.Additionally, we examined the idea of idempotency, monotonicity, and boundness for presented techniques.
Proposition 3: Based on a universal set ψ uni and in the presence of CTSFNs Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Proof: Straightforward.Additionally, we examined the idea of idempotency, monotonicity, and boundness for presented techniques.

V. MADM TECHNIQUES FOR PROPOSED METHODS
The MADM technique is a famous and powerful method, which is used for collecting the finest decision from the group of alternatives.The MADM technique is one of the important parts of the decision-making method and most people have used it in distinct fields.In this section, we explain the MADM technique in the presence of examined techniques such as the CTSFAAWA operator and CTSHFAAWG operator to increase the worth of the presented theory.
To To calculate our problem, we also needed to calculate the matrix for the above alternatives and their attributes, where every item in the matrix will be included in the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.• C + sup ( j ηr (x)) • C + sup ( j ηr (x)) • C , sup ( . Further, we explain the main idea of refusal information such as: Finally for calculating some real-life problems, we find out the pattern of decision-making such as: Step 1: We arrange the information on CTSHFNs and try to calculate a decision matrix.
Step 2: We normalize the group of data in a decision matrix if needed, such as ηp , ηp , ηp for ηj c p = ηp , ηp , ηp for cost Step 3: We Aggregate the data in a normalized decision matrix if required, with the help of the CTSHFAAWA operator and CTSHFAAWG operator.
Step 4: We calculate the score value of the aggregate value.Step 5: We have to find out the ranking values based on score values and try to calculate the finest preference among the group of preferences.
Using the above methods of decision making we justified the proposed operators are superior and more powerful than the persuaded operators.

A. NUMERICAL EXAMPLE
In this example, we aim to illustrate the application of the installation of biometric systems for enhancing the quality of education., for this, we find the main factor that plays a valuable role in the installation of biometric systems in every education department, for this, we consider the following five factors, such as 1) Security Problems.
2) Attendance Tracking 3) Student Safety.4) Integrity with other Systems.5) Preventing unauthorized access.From the university administration, we have conducted these five factors which play an essential role in the implementation of the biometric systems in education departments.For this, we have used the weight vectors, such as (0.2, 0.3, 0.3, 0.2) T for the following four attributes, such as growth analysis, social impact, political impact, and environmental impact.Based on the above information, we are applying our proposed techniques to the below data.Finally for calculating some real-life problems, we find out the pattern of decision-making such as: Step 1: We arrange the information on CTSHFNs and try to calculate a decision matrix, see Table 1.
Step 2: We normalize the group of data in a decision matrix if needed, anyhow, the data in Table 1, do not need to be normalized.
Step 3: We Aggregate the data in a normalized decision matrix if required, with the help of the CTSHFAAWA operator and CTSHFAAWG operator, see Table 2.
Step 4: We calculate the score value of the aggregate value, see Table 3.
Step 5: We have to find out the ranking values based on score values and try to calculate the finest preference among the group of preferences, see Table 4.The best and finest optimal is ηA−2 p based on the above operators.Furthermore, we simplify the stability and influence of the parameters which are part of the proposed techniques, therefore, for different values of parameters, the ranking values are stated in Table 5.
The best and finest optimal is ηA−2 p based on the above operators, for some values of parameters, we different ranking results for geometric operators, where the best optimal is ηA−3 p for some values of parameters.

VI. COMPARATIVE ANALYSIS
In this section, we compare the derived operators with some existing techniques by including the data in Table 1.For showing the supremacy and validity of the proposed techniques, comparative analysis is the best way because without comparison any paper has no worth.For this, we try to arrange some existing operators and then try to compare the results with the proposed results to enhance the stability and rationality of the proposed operators.For this, we use the following techniques, such as Aczel-Alsina (AA) operators for IFSs [30], geometric AA operators for IFSs [31], AA operators for PyFSs [32], AA operators for q-ROFSs [33], AA operators for HFSs [34], AA operators for intuitionistic HFSs [35], AA operators for PFSs [36], geometric AA operators for PFSs [37], AA operators for CPFSs [38], AA operators for TSFSs [39], and AA operators for CTSFSs [40].Therefore, to use the data in Table 1, the comparative analysis is listed in Table 6.
22646 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

j = 1 , 2 ,
4 [40]: Based on a universal set ψ uni and in the presence , j = 1, 2, . . ., n, we have as shown at the bottom of page 5.Definition 5[22]: Based on a universal set ψ uni and in the presence . . ., n, we have

Definition 6 :
Based on a universal set ψ uni , the main idea of CTSHFS ηp is evaluated and computed in the form: ηp = x, ( ηp (x), ηp (x), ηp (x)) : x ∈ ψ uni Additionally, we further explained the value of truth grade ηp (x)

2 ,j = 1 , 2 , 7 :j = 1 , 2 , 8 :j = 1 , 2 , 9 :j
. . ., n} and the final version of the CTSHF number (. . ., n. Definition Based on a universal set ψ uni and in the presence of CTSHFNs ηj p = ηp , ηp , . . ., n. we have as shown at the bottom of page 7. Definition Based on a universal set ψ uni and in the presence of CPHFNs ηj . . ., n. we have as shown at the bottom of page 8. Definition Based on a universal set ψ uni and in the presence of CTSFNs ηj p = ηp , ηp , = 1, 2, . . ., n. we have

11 :jp 2 :j = 1 , 2 ,
Based on a universal set ψ uni and in the presence of CTSFNs ηj p = ηp , ηp , = 1, 2, . . ., n.Then we have examined the theory of the CTSHFAAOWA operator, such as CTSHFAAOWA η1 p , η2 p , . . ., ηn p Notice that o (j) ≤ o (j−1) .Theorem Based on a universal set ψ uni and in the presence of CTSFNs ηj . . ., n.Then we prove that the resultant value of the above operator is again in the shape of CTSHFN, such as shown at the bottom of page 18.

j = 1 , 2 ,p 4 :j
. . ., n.Then we have examined the theory of the CTSHFAAOWG operator, such as CTSHFAAOWG η1 p , η2 p , . . ., ηn p Notice that o (j) ≤ o (j−1) .Theorem Based on a universal set ψ uni and in the presence of CTSFNs ηj p = ηp , ηp , = 1, 2, . . ., n.Then we prove that the resultant value of the above operator is again in the shape of CTSHFN, such as shown at the bottom of page 20.

TABLE 5 .
CTSHF stability information for different values of parameters.

TABLE 6 .
Comparative analysis of the proposed and existing techniques.