Modern Approach in Pattern Recognition Using Circular Fermatean Fuzzy Similarity Measure for Decision Making with Practical Applications

Te circular Fermatean fuzzy (CFF) set is an advancement of the Fermatean fuzzy (FF) set and the interval-valued Fermatean fuzzy (IVFF) set which deals with uncertainty. Te CFF set is represented as a circle of radius ranging from 0 to � 2 √ with the center at the degree of association (DA) and degree of nonassociation (DNA). If multiple people are involved in making decisions, the CFF set, as an alternative to the FF and IVFF sets, can deal with ambiguity more efectively by encircling the decision values within a circle rather than taking an average. Using algorithms, a pattern can be observed computationally or visually. Machine learning algorithm utilizes pattern recognition as an instrument for identifying patterns and also similarity measure (SM) is a benefcial pattern recognition tool used to classify items, discover variations, and make future predictions for decision making. In this work, we introduce the CFF cosine and Dice similarity measures (CFFDMs and CFFSMs), and their properties are studied. Unlike traditional approaches of decision making, which emphasize a single number, the proposed CFFSMs observe the pattern over the circular region to help in dealing with uncertainty more efectively. We introduce an innovative decision-making method in the FF setting. Available bank loans and applicants’ eligibility levels are represented as CFF set using their FF criteria and are taken as loan patterns and customer eligibility patterns. Te loan is allocated to the applicant by measuring the CFFCSM and CFFDSM between the two patterns. Also, laptops are suggested to the customers by measuring the similarity between specifcation pattern and requirement pattern. Te correctness and consistency of the proposed models are ensured by comparison analysis and graphical simulations of the input and similarity CFFNs


Introduction
Pattern recognition is a developed but exciting feld that keeps growing swiftly.It provides a platform for developments in associated disciplines notably computer vision, image processing, gene expression analysis, protein structure prediction, medical image analysis, text classifcation, sentiment analysis, named entity recognition, speech recognition, gesture recognition, biometric identifcation, neural networks, and many more.
Te nearest neighbor algorithm, initiated in 1967, is the origin of pattern recognition.Statistical pattern recognition methods, machine learning algorithms, and deep learning techniques are some of the pattern recognition methods.
Relevant feature identifcation from the raw data is the frst stage in almost all pattern recognition tasks.Features that aid in recognizing between patterns could be statistical, structural, physical, or spectral attributes.Similarities in the data are then recognized and used to produce recommendations.Hagedoorn [1] used similarity matching for pattern recognition in his PhD thesis.Julian et al. [2] modifed Mitchell similarity and introduced new SM for pattern recognition.Liu [3] introduced new similarity measures between intuitionistic fuzzy sets and between elements.Yen et al. [4] applied the SM for pattern recognition in 2013.In the year 1980, Kono [5] identifed a pattern recognition system that can classify 22 kinds of patterns within 4 seconds in the minicomputer.
Te conventional methods of pattern recognition classify a collection of patterns into clusters based on similar characteristics.However, the majority of pattern recognition issues employ feature selection that is gathered at random, which leads to blending of pattern classes and ambiguity in object detection.
Zadeh [6] implemented fuzzy set (FS) theory in 1965, which assumes DA in [0, 1] instead of a characteristic function that maps to 0 or 1 in the crisp set.Fuzzy classifcation is a way of sorting elements into fuzzy sets with association functions identifed by the truth value of a fuzzy propositional function.Based on the exponential area of established intervals of membership functions, Dutta et al. [7] constructed a new Dice similarity measure for fuzzy numbers.Venugopal [8] used fuzzy indicator and Masoomi [9] applied neutrosophic indicator for performance analysis.In 1986, Pal et al. [10] and in 2006 Guo et al. [11] described fuzzy pattern recognition, which provides a powerful framework for modeling and analyzing complex patterns in real-world data.
Despite the fact that FSs are useful in instances of uncertainty and have no clear boundary, they fail to represent DNA.Both DA and DNA are important for the mathematical representation of real-life situations.For that need, Atanassov [12] coined the concept of an intuitionistic fuzzy (IF) set as an extension of Zadeh's FS theory in 1983.In the IF set environment, DA and DNA take the value in [0, 1], with their sum between 0 and 1. Chen [13] introduced a new SM between Atanassov's intuitionistic fuzzy sets.Liu et al. [14] examined the cosine SM between hybrid IF sets.To recognize the pattern of COVID-19, Saeed et al. [15] applied the IF tool.Ye [16][17][18] introduced IF and interval-valued IF cosine SM and the Dice SM based on the deducted IF sets which help in the feld of pattern recognition.Nwokoro [19] developed the IF approach for predicting maternal outcomes, Zhou et al. [20] applied the generalized IF similarity operator recognition principle, and Ejegwa [21] explored the IF correlation coefcient for classifcation process.Zhang [22] considered the IF score function for pattern recognition and medical diagnosis.
Tere have been drawbacks to the IF set, even though it has been utilized in various situations.Te IF set, in particular, lacks the capacity to cope with such information when a decision maker proposes it, but its total DA and DNA are greater than 1.In this connection, Yager et al. [23] recommended the Pythagorean fuzzy (PF) set theory in 2013 by modifying the IF set perspective as well.A PF set incorporates the feature that the aggregate value of the squares of DA and DNA lies between 0 and 1, thereby enhancing the acceptance range more than the IF set.Peng [24] used various parametric SMs, and Nguyen et al. [25] developed the exponential SM.Ejegwa et al. [26] applied the PF correlation coefcient for pattern recognition, decision making [27,28], disaster control, and diagnostic analysis [29].Zulqarnain [30][31][32][33][34][35][36] applied various forms of the FS Einstein aggregation operators, and Kuppusamy et al. [37] used a bipolar PF approach and its topological properties in decision making.
Te FF set developed by Senapati [38] has an extensively broad range of acceptance compared to the IF set and PF set by taking DA and DNA with the condition that the sum of cubes of DA and DNA lies between 0 and 1. Alahmadi et al. [39] used FF t-conorms for SM in pattern analysis.Deng et al. [40] applied FF distance measures.Khan et al. [41] derived a FF benchmark similarity using S-norm.Akram et al. [42,43] discussed transportation problem under FF environment.Ejegwa applied FF composite relation for pattern recognition [44].Also, he applied FF composite relation and FF correlation coefcient in diagnostic analysis [45,46].Onyeke [47] applied the modifed Senapati and Yager's FF distance and its application in students' course placement in tertiary institution.Kishorekumar et al. [48] and Revathy et al. [49] introduced interval-valued picture fuzzy geometric Bonferroni mean operators and FF normalized Bonferroni mean operator.Mishra et al. [50] applied t− norm and s− conorm and Wania et al. [51] used Fermatean Probabilistic Hesitant Fuzzy Set for MCDM.Sahoo [52] and Zhou et al. [53] applied the FF SM and FF ELECTRE methods in group decision making.Xu et al. [54] and Ashraf et al. [55] applied SMs for FF pattern recognition.Demir [56] utilized correlation coefcients for interval-valued Fermatean hesitant fuzzy sets for pattern recognition.Rahim et al. [57] examined improved cosine SM and distance measures-based TOPSIS method for cubic Fermatean fuzzy sets in 2023.
Te circular intuitionistic fuzzy (CIF) set, created by Atanassov [58] in 2020, is a profound enhancement to FS.Unlike an IF set, a CIF set provides each component as a circle.Tese are the sets in which each component of the universe has a value of DA and DNA as the center of a circle around them, and that circle has a radius of [0, ] such that the sum of DA and DNA within this circle is not more than 1.Also, in reference [59], he derived four distance measures of CIF set.Cakir et al. [60], Irem [61], and Alkan [62] applied the CIF set in decision making.Bolturk [63] discussed the diference between the IVIF set and the CIF set.As an extension of the CIF set, Olgun [64] introduced a circular Pythagorean fuzzy (CPF) set.Khan et al. [65] expanded the PF set as a circular and disc PF set.
Te circular Fermatean fuzzy (CFF) set introduced by Revathy et al. [66] is an amplifcation of the FF and IVFF sets.Te CFF set is a circle whose center is at (DA, DNA) and of radius at most with the sum of cubes of DA and DNA between 0 and 1. Te CFF set can address ambiguity in a better way when many diferent people are participating in 2 Journal of Mathematics decision making because it encircles various values within a circle instead of averaging them.We deal with highdimensional, more complicated data patterns in today's world.It might be difcult for traditional pattern recognition techniques to analyze these patterns and fnd appropriate knowledge efciently and successfully.Because of its wide range in the willingness of a decision maker, the CFF pattern recognition ofers a framework to optimize decision making by considering the most efcient paths through these patterns swiftly and promptly.Te decision makers' DA and DNA were addressed by specifc values in all of the previous sets, such as fuzzy, IF, PF, and FF.In our CFF set, the decision maker values are determined by the circles.Hence, the decision will be same if the decision makers DA and DNA lie in the circles.
Bank loans are advantageous way that people can utilize to achieve various kinds of intended goals.Loan allocation that is automated can speed up decisions for borrowers, minimize the human burden for lenders, and optimize the lending process.Lenders must, however, ensure their automated systems maintain to legal as well as moral norms, and borrowers need to review the details of any loan ofer thoroughly before accepting it.In this present work, we introduce CFF cosine and Dice similarity (CFFCSM and CFFDSM); also, their properties are investigated.Te CFF set utilizing the applicants' FF criteria represents the available bank loans and the applicants' eligibility level that are taken as loan patterns and customer eligibility patterns.Te applicant is allocated a loan based on the evaluation of the two patterns' CFFCSM and CFFDSM measurements.Additionally, customers are provided laptop suggestions based on how closely their expectations and need patterns coincide.
Te key factors that encouraged us to do the present study are as follows: (i) To deal with uncertainty, the FF sets are used in several felds, such as control system engineering, image processing, power engineering, industrial automation, robotics, consumer electronics, and optimization.If multiple decision makers are involved in the process of decision making, the average of their decision value is taken into account.(ii) Trough the CFF set environment, the decision values can be represented by circles which will assist to handle the uncertainty more efectively since the circles will express the decision makers' opinion in a region instead of a particular value.(iii) In the majority of decision-making aggregation methods, if one of the decision values has 0 in either DNA or DA, the aggregated value becomes zero.Tis can be avoided by using similarity measures instead of aggregation.(iv) Te CFF similarity measure will be a better tool for decision making than FF precisely because of its computational structure.
Te contribution of this present work is given below: (i) Te cosine and Dice similarity measures are defned in the CFF environment and applied for decision making in bank loan allocation and laptop suggestion using pattern recognition.
(ii) Set of FF decision values has been transformed into the CFFN and their graphical representation is demonstrated.Te consistency of the proposed method is represented by graphical representation which is the special feature of the work.(iii) Also, the comparative analysis with the existing methods and the statistical analysis with the help of SPSS software 27 are done for the proposed decision making.
Te article is organised as follows.Section 2 gives the preliminary.Section 3 describes the circular Fermatean fuzzy similarity measure.Section 4 explains the application of the CFFS in pattern recognition and analyzes and portrays the obtained result.Section 5 states the limitations, and fnally Section 6 ends up with the conclusion and future work.
Te nomenclature used in the current research is listed in Table 1.

Preliminaries
We now present some preliminary defnitions and notions necessary to apprehend our fndings.

􏽱
is the DI of x in F. Te components of the FF set are taken as the FF number (FFN) and it is represented by (1) (2)

Circular Fermatean Fuzzy Similarity Measure
In this section, we bring in CFF cosine similarity and CFF Dice similarity measures.

Application of Circular Fermatean Fuzzy Number in Pattern Recognition
In this section, we provide some applications to understand the pattern recognition model.In references [3,10,16], the authors have taken assumed data to validate their proposed model.In the same way, to check the efectiveness of our proposed model, we also have considered assumed data provided in Sections 4.1 and 4.2 for loan allocation and laptop selection.With our new algorithm, anyone can have expertise to suggest best out of available loans, products, electing leaders, etc., according to their eligibility and requirements.

Bank Loan Allocation Using Circular Fermatean Fuzzy
Similarity Measure.In the olden days, farmers used seeds and grains to borrow capital, with farm animals as repayment options.Te Arthasastra [67,68] written by Kautilya, a classical Indian book on the art of politics emanating from the 4th or 3rd century BC, indicates the existence of debtors, fnanciers, and interest rates.Since then, fnancing has evolved into a complicated fnancial procedure before shifting into a modern, simplifed method in the modern world.A short-term loan is a loan that can be received and reimbursed very quickly, typically within a few hours or even minutes after completing the application for the loan.Tis type of loan is mostly ofered by digital lenders, and the application procedure often happens quickly and online, with little documentation required.In this section, we entail how the pattern recognition model is used to allocate bank loans for customers.Te approaches in this procedure are described as follows: Step 1. Te expert team of the bank fxes the evaluation parameters of the specifc loan in terms of FFN including the no loan option.
Step 2. FFN evaluation criteria are converted into CCFN using Proposition 3 and it is taken as a CFF loan pattern.
Step 3. Te client's applications undergo scrutiny, and FFNs are given away to their parameters and expressed as CFFN, which is taken as CFF borrower patterns.
Step 4. Te similarity between the borrowers' criteria and evaluation parameters is measured using the CFFCSM and the CFFDSM.
Step 5. Possible loans for a customer are ranked according to the similarity values, from higher to lower.
Step 6. Te loan pattern that has the highest similarity value to the customer data pattern is allocated for the particular customer.
Te loan allocation framework is illustrated in the fowchart (Figure 2).
For allocation of loans to the customers, every bank has their own criteria.In this study, the personal loan (PL), jewelry loan (JL), mortgage loan (ML), and no loan (NL) are fxed with criteria, namely, regular income (RI), CIBIL score (CS), jewelry availability (JA), and asset value (AV).Te imprecise boundary makes it difcult to quantify the available loans and loan requirements by exact numbers.We consider the following loan pattern with assumed data in terms of DA and DNA using FFN for the criteria of the loans which are expressed in Table 2.
Figure 3 represents the decision makers' opinion on each criterion in the form of the circle with the mentioned colors.Te blue color shaded region represents the acceptance region.Te area of the circle outside the acceptance region is excluded in the further calculation procedure.
Lina, Louis, Iris, Denis, and Pierre's loan requirement application forms are scrutinized.Te regular in come, CIBIL score, jewellery availability and asset value are measured in terms of the FFN according to their DA and DNA.By using Proposition 3, the FFN is converted into CFFN.Te obtained CFFN is tabulated in Table 3.
For each person, their RI, CS, JA, and AV in terms of FFN are converted to CFFN.It takes the circular representation in the acceptance region and is taken as eligibility pattern corresponding to each criterion.Te visualization of CFFN is given in Figure 4.

6
Journal of Mathematics Bank requirement and customers' eligibility are represented as a circle instead of single numerical value.In such circumstances, measuring the similarity will give the efcient decision.Te CFFCSM and CFFDSM mentioned in (3) and ( 4) are utilized between the customers' data and available loans.Te particular loan that gets the highest similarity measure is allocated to the concerned customer.Te conclusions are tabulated in Table 4.
Te graph in Figure 5 exhibits the consolidated CFFCSM and CFFDSM of all fve applicants to the loan criteria.10, respectively, that are drawn in the 3D graphic calculator which confrms that our proposed method has efectively worked between customer criteria and loan requirements.

Bank Loan Allocation
Te CFFCSM and CFFDSM of Lina criteria pattern and the loan patterns are displayed in the graph of Figure 11.Since Lina's criteria values have a high similarity to those of no loan, she is denied for allocation of loan.From Figure 6, we can confrm that the shaded circular portion which is the eligibility pattern of Lina has high similarity of the circle that represents no loan pattern.
Personal loan needs considerable regular income and good CIBIL score as per the criterion requirement.Te graph in Figure 12 shows Louis has a high regular income and a sufcient CIBIL score, so he has been allocated a personal loan.Figure 7 shows that the eligibility pattern of Louis has high similarity with personal loan.
Iris does not have enough regular income and a good credit score, but she has a high value of jewelry and assets that she is allocated a jewelry loan as well as a mortgage loan.Also, from Figures 8 and 13, it is clear that eligibility criterion of Iris is similar to both jewelry loan and mortgage loan.
Denis has a regular income and a good CIBIL score, so he is allocated a personal loan.Te graph in Figure 14 shows the similarity values of all the fve loans.Also, it is evident from Figure 9 that Denis eligibility pattern has high similarity with personal loan.
Since Pierre has considerable jewelry compared to all other criteria, he is allocated a jewelry loan.Te graphs in   Te regression lines and the correlation coefcient between CFFCSM and CFFDSM were obtained using IBM SPSS software 27 for the fve borrowers and displayed in

Laptop Selection Using Circular
Fermatean Fuzzy Similarity Measure.In this section, we suggest a suitable laptop for the customer according to their requirement by using CFFCSM and CFFDSM like the procedure and fowchart (Figure 2) in Section 4. By taking into account the criteria of software adaptability, price, graphics quality, and security features like being free of bugs and hacking, we consider four types of laptops: gaming laptops (GL) for games; home and student laptops (HSL) for regular use; business laptops (BL) for commercial purposes; and MacBooks (MB) for high security.Te specifcations of all the laptops are tabulated in Table 5 in the form of FFN and CFFN.
Figure 17 illustrates the patterns of the available laptops.Te region enclosed between the circles and the curve x 3 + y 3 ≤ 1 in the yellow color shaded region represents the pattern.
Five persons' requirements under each criterion are tabulated in Table 6 in the form of FFN and CFFN.Using equations ( 3) and ( 4) , the CFFCSM and CFFDSM between the requirement criteria and the laptop pattern are calculated.Te laptop that has the highest similarity is suggested as in Table 7.
If the diference between the similarity measures of two laptops is very small, then both the laptops are suggested for the customer.Since the requirement of person 3 and person 5 has nearly equal similarity with business and gaming laptop, those two are suggested to them.
Figure 18 exhibits the consolidated CFFCSM and CFFDSM of all fve customers to the requirement criteria.

Comparison
Analysis.Circular Fermatean fuzzy sets, which capture data in circular form more efectively than existing fuzzy sets, provide a more robust framework for managing uncertainty in some felds depending on the particular requirements of the application and the type of          Journal of Mathematics means of CFFSM between the loan criteria and the applicants' eligibility criteria.Te graphical representation by 3D graph calculator for the similarity measurement is a new type of approach through our proposed circular Fermatean fuzzy set.It enhances the reliability of the application of circular Fermatean fuzzy set.Simulations of data analysis validate the potency of the suggested method.CFFCSM and CFFDSM ft is done through regression line using SPSS software 27 (see Figure 16).Te same procedure is applied to suggest the suitable laptop for the customers based on their required features.Our proposed formulas can be used to construct efcient algorithms which help to implement artifcial intelligence in banking sector as well as in purchase.

Journal of Mathematics
In future work, an efective CFF score function and CFF accuracy function will be set up for ranking addressed in limitation section.Also, their topological structure will be investigated in order to create the CFF topological space.
Various CFF aggregation operators, CFF distance measures, and CFF similarity measures will be defned and applied in machine learning and clustering to enhance their application in the felds of medical and image processing.Tus, with the help of programming languages, an algorithm for machine learning in artifcial intelligence will be developed by our proposed similarity measure which can be extended to the problem of electing the best leader through preelection survey from the public.

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Defining evaluation criteria of loan by expert's team Collection of applications from borrowers Assigning FFN for the loan criteria Assigning FFN for the assesment of the borrowers' criteria Converting FFN to CFFN loan pattern Converting FFN to CFFN borrowers pattern Similarity measure between loan and borrowers pattern Ranking of the similarity measure Suitable loan allocation

Figure 3 :
Figure 3: Graphical representation of loan requirement as CFFN.

Table 3 :
Figures 10 and 15 show the similarity values and visualization of all the fve loans which confrms that Pierre eligibility pattern has high similarity with jewelry loan.

Figure 5 :
Figure 5: Consolidated similarity measures of loan and customer pattern.

x 3 + y 3 Figure 8 :
Figure 8: Circular representation of Iris similarity measures for loans.

Figure 16 .
Figure 16.All fve customers' CFFCSM and CFFDSM are positively correlated by which we can ensure that our proposed similarity measures are consistent.

Figures 19 -
23 illustrate the suggestions of the laptop to the customers according to their requirement.Figures 19(a)-23(a) indicate the requirement pattern of the customer, and the shaded portion in Figures 19(b)-23(b) portrays the high similarity to the suggested laptop.All the fve persons are allocated with the laptop that has high similarity with their requirement.

Table 4 :
CFFCSM and CFFDSM between loans and customer criteria.

Table 6 :
Requirement of FFN and CFFN criteria.