#### Abstract

The idea of composition relations on Fermatean fuzzy sets based on the maximum-extreme values approach has been investigated and applied in decision making problems. However, from the perspective of the measure of central tendency, this approach is not reliable because of the information loss occasioned by the use of extreme values. Based on this limitation, we introduce an enhanced Fermatean fuzzy composition relation with a better performance rating based on the maximum-average approach. An easy-to-follow algorithm based on this approach is presented with numerical computations. An application of Fermatean fuzzy composition relations is discussed in diagnostic analysis where diseases and patients are mirrored as Fermatean fuzzy pairs characterized with some related symptoms. To ascertain the veracity of the novel Fermatean fuzzy composition relation, a comparative analysis is presented to showcase the edge of this novel Fermatean fuzzy composition relation over the existing Fermatean fuzzy composition relation.

#### 1. Introduction

Diagnostic analysis of patients’ medical samples is a delicate assignment enmeshed with vagueness and hesitation. Many approaches have been posited to ameliorate this problem, like the introduction of fuzzy sets [1]. Though the fuzzy set seems to be promising in tackling uncertainties, it is unreliable because it considers the membership degree (MD) of the case under consideration without minding the possibility of hesitation. Sequel to this weakness, some generalized varieties of fuzzy sets have been put forward such as the intuitionistic fuzzy set (IFS) [2], the Pythagorean fuzzy set (PFS) [3, 4], and the Fermatean fuzzy set (FFS) [5, 6]. By including nonmembership degree (NMD) to of fuzzy set, the idea of IFSs was proposed and applied in numerous applicative areas. Boran and Akay [7] explored pattern recognition using a biparametric similarity measure. Some techniques of similarity measures and distance measures of IFSs have been used to handle pattern recognition problems [8–10]. In [11], a medical diagnosis was carried out based on composite relations. Similarly, in [12, 13], a diagnostic analysis was done based on a similarity measure approach.

The noticeable inadequacy of IFSs is that it only handles the scenario where the summation of membership degree (MD) and nonmembership degree (NMD) is not more than unity. Because of this drawback, intuitionistic fuzzy set of second type (IFSST) was proposed [3, 14], which is widely called Pythagorean fuzzy sets (PFSs) [4]. In PFS, the parameters and are characterized by such that . PFS has been applied to solve some hands-on problems such as medical diagnosis based on composite relation [15] and other sundry problems [16, 17]. A method for undertaking multi-attribute decision-making (MADM) under interval-valued Pythagorean fuzzy linguistic information was deliberated in [18]. A number of aggregation operators using Einstein t-conorm, Einstein operator, and Einstein t-norm under the Pythagorean fuzzy environment for decision-making were deliberated in [19, 20]. A new extension of the TOPSIS (technique for order preference by similarity to ideal solution) approach for multiple criteria decision-making (MCDM) with hesitant PFSs was discussed in [21]. Wang and Garg [22] developed some aggregation operators for PFS based on interactive Archimedean norm processes with application to MADM. In [23], a Choquet integral based interval type-2 trapezoidal fuzzy approach was applied in MCDM involving sustainable selection of supplier. A group decision-making approach was discussed in a dynamic feedback mechanism with an attitudinal consensus threshold for minimum adjustment cost [24].

In the same way as IFSs, the construct of PFSs is limited to handle a situation when and . In a quest to resolve this brainteaser, the intuitionistic fuzzy set of third type (IFSTT) also known as the Fermatean fuzzy set (FFS) was introduced [5, 6]. FFS has a broader scopes include , and with the ability to certainly handle indeterminate information in decision making. A number of operators on FFSs were elaborated in [25], and differential calculus of Fermatean fuzzy functions have been introduced [26]. A number of applications of FFSs in MCDM problems based on TOPSIS method, distance measures and certain weighted aggregated operators have been explored [6, 27–29]. Sari et al. [30] studied interval-valued Fermatean fuzzy sets and applied them to capital budgeting techniques. Jeevaraj [31] imposed ordering on interval-valued FFSs with application. A novel decision-making method based on Fermatean fuzzy WASPAS (weighted aggregated sum product assessment) for green construction supplier evaluation was discussed in [32]. In [33], some TOPSIS techniques via Fermatean fuzzy soft sets were discussed with application. Sahoo [34] presented certain score functions on FFSs with application to bride selection. Aydin [35] discussed a fuzzy MCDM method using Fermatean fuzzy theories, and Zhou et al. [36] applied the Fermatean fuzzy ELECTRE (Elimination Et Choix Traduisant la Realite) method to tackle multiple-criteria group decision making. Shahzadi et al. [37] discussed MADM via Fermatean fuzzy Hamacher interactive geometric operators.

The applications of FFSs based on TOPSIS and MCDM methods have been discussed in [33–37]. Some applications of FFSs in the selection of COVID-19 testing centres using aggregation operators, the SAW (simple additive weighting) approach, the VIKOR (VIekriterijumsko KOmpromisno Rangiranje) approach, and the ARAS (additive ratio assessment) approach were considered in [38, 39]. The concept of Fermatean fuzzy composition relations has been studied based on maximum-extreme values with application to diagnostic analysis using simulated data [40].

The idea of composition relation has been presented in intuitionistic fuzzy settings [11], Pythagorean fuzzy setting [41] and Fermatean fuzzy settings [40] based on the maximum-extreme values approach with applications. This approach, though presented in different frameworks, cannot be reliable because it makes use of only minimum and maximum values. This present work puts forward a new composition relation under the Fermatean fuzzy domain based on the maximum-average approach. To express the applicability of the new Fermatean fuzzy composition relation, a case of diagnostic analysis is considered via the approach where diseases and patients are viewed as Fermatean fuzzy pairs. More concepts related to this study have been studied in [42–45].

The specific objectives of the work are to (i) reiterate the max-min-max approach of composition relation [11, 40, 41] in a Fermatean fuzzy setting, (ii) present an enhanced Fermatean fuzzy composition relation based on the max-average approach, (iii) numerically demonstrate the max-min-max approach in conjunction with the new Fermatean fuzzy composition relation, (iv) decide patients’ medical status in a Fermatean fuzzy environment based on Fermatean fuzzy composition relations via max-min-max approach and maximum-average approach, respectively, and (v) present a comparative analysis to showcase the edge of the new Fermatean fuzzy composition relation over the approach in [11, 40, 41]. The summary of the paper follows: Section 2 presents the basis of FFSs and the existing Fermatean fuzzy composition relation [40], Section 3 discusses the new Fermatean fuzzy composition relation via the maximum-average approach, Section 4 dwells on diagnostic analysis of patients’ medical status where diseases and patients are presented as Fermatean fuzzy values, and Section 5 synopses the paper with recommendations for future work.

#### 2. Fermatean Fuzzy Sets

Some fundamentals of FFSs have been presented in [6, 28, 29, 40]. Let designates a fixed set for this work.

*Definition 1. *A FFS in is a generalized fuzzy set of the formwhere define MD and NMD of for . For a FFS in ,represents the FFS index or hesitation margin of .

In an IFS, , and . For PFS, , and . For the case of FFS, we have , and .

Now, some properties of FFSs are presented including equality, inclusion, complement, union, and intersection.

*Definition 2. *Suppose and in are FFSs, then(i)(ii) iff , (iii) iff , (iv) iff , (v)(vi)Now, we present a Fermatean fuzzy pairs (FFPs) thus

*Definition 3. *FFP is designated by such that where . A FFP evaluates the FFS for which the components ( and ) are interpreted as MD and NMD.

For simplicity sake, we write a FFS as .

##### 2.1. Fermatean Fuzzy Composite Relation

Composite relation has been established under IFS, PFS, and FFS [11, 40, 41] to enhance the applications of IFSs, PFSs, and FFSs in decision making. Suppose and are two sets. A Fermatean fuzzy relation (FFR) from to is a FFS in comprises of MD and NMD . A FFR from to is denoted by or .

*Definition 4. *Suppose and are FFRs in and , which can also be written as and . Then, the Fermatean fuzzy composite relation (FFCR) of is defined bywhere and .

Using Definition 4, the FFCR is computed byThe composite relation presented in [11, 40, 41] uses the extreme values, i.e., the maximum of the minimum of the membership degrees and the minimum of the maximum of the nonmembership degrees. The result from this approach is not reliable, judging from the knowledge of the measure of central tendency. Because of this limitation, we modify the technique in [11, 40, 41] based on the maximum-average approach.

#### 3. Enhanced Fermatean Fuzzy Composite Relation

This section presents a FFCR based on the maximum of the mean values of membership degrees, and the minimum of the mean values of nonmembership degrees to enhance better performance.

*Definition 5. *Let and be FFRs of and , which can also be written as and . Then, the FFCR of is defined bywherefor . Certainly,From Definition 5, the new FFCR is computed by

*Definition 6. *Suppose is a FFR in , then the inverse of denoted as in is defined by

*Definition 7. *Suppose and are FFRs in , then(i) iff and (ii) iff and (iii)(iv)(v),

Theorem 1. *If , and are FFRs in , then*(i)*(ii)**(iii)**(iv)**(v)**(vi)**(vii)**if and then *(viii)*if and then *(ix)*If then *(x)*If then *

*Proof. *(i) Assume , thenand similarly we haveTo prove (ii), we haveSimilarly, we haveThe proof of (iii) is similar to (ii). The proof of (iv) follows:Similarly, .

Now, we prove (v) as follows:Similarly, by using Definition 7 we haveThe proofs of (vi)–(x) are straightforward.

Theorem 2. *Suppose we have two FFRs in and in , respectively, then .*

*Proof. *Firstly, we show the result with respect to the membership degree. Then,Similarly, we have

Theorem 3. *Suppose is a FFR in , and is a FFR in and . Then,*(i)*(ii)*

*Proof. *(i)We show that the FFRs are commutative as follows: Similarly,(ii)Now, we proof the associativity as the FFRs as follows:for all .

On the contrary, we getfor all .

Theorem 4. *Let and be FFRs of , and and be FFRs of , then we have the following properties:*(i)* for every FFR in *(ii)* for every FFR in *(iii)* for every FFR in *(iv)* for every FFR in *(v)* if and are FFRs in *

*Proof. *(i) Assume . Then, and . Now, we haveSimilarly,The proofs of (ii)–(iv) are similar.

(v) If , then and . Then,Similarly,

Theorem 5. *Let be FFRs in , and be FFR in . Then, we have*(i)*(ii)*

*Proof. *By Theorem 1, we have and . Thus, and . Hence, we get , which proves (i). The proof of (ii) is similar.

Theorem 6. *Suppose are FFRs in , and is a FFR in . Then, we have*(i)*(ii)*

*Proof. *We first proof (i) as follows:Similarly, we obtainThe proof of (ii) is similar.

##### 3.1. Numerical Illustration of FFCRs

An example is given to show the supremacy of the new FFCR over the existing FFCR [11, 40, 41].

Given that and are FFSs in defined by

Using the existing FFCR , the minimum of the membership degrees between FFSs and for are implying that

Similarly, the maximum of the nonmembership degrees between FFSs and for are , implying that

Now, the FFCR between and is .

Using the new FFCR in Definition 5, the mean values of the membership degrees between FFSs and for are . Thus,

Again, the mean values of the nonmembership degrees between FFSs and for are . Thus,

Again, the mean values of the nonmembership degrees between FFSs and for are . Thus,

The new FFCR between FFSs and is .

From the results, it is certain that the new FFCR is better than the approach in [11, 40, 41] because the FFCR between and is greater for the new approach (i.e., while the existing approach yields 0.4162, the new approach yields 0.5698). This justifies the advantage of taking the mean values of the parameters of FFS over taking the extreme values.

#### 4. Fermatean Fuzzy Composite Relation in Determination of Patients’ Medical Status

This section discusses an application of FFCRs in diagnosis analysis of a patient’s medical status using a simulated database of disease diagnosis. For the sake of simulation, take as a set of symptoms, as a set of diseases, and as a set of patients. Then, we represent a medical knowledge in Fermatean fuzzy pairs based on a FFR from to indicated by to bespeak the grades of association and otherwise between and . In the Fermatean fuzzy medical diagnostic process, the symptoms of the diseases are determined, the medical knowledge of the patients based on Fermatean fuzzy values is formulated, and the diagnosis on the basis of the composition using the existing FFCR and the new FFCR are determined.

##### 4.1. New FFCR between Patients and Diseases

Suppose the medical condition of a patient is described in terms of a set of symptoms , then is taken to be assigned a diagnosis based on via a FFR from to designated as as simulated by medical knowledge in terms of degrees of association and otherwise.

We construct a FFR from to represented by as . Then, the FFCR of and (i.e., ) signifies the medical condition of the patients with regards to the ailments given by the MD and NMD in equation (36).for all the patients and ailments.

The result for which is the greatest determines the diagnosis of the patient . For easy computation of the FFCR, the algorithm which describes the step-to-step computational processes of the composite relation between the patients and the diseases is given as follows: Step 1: Institute a relation between and as FFPs Step 2: Institute a relation between and as FFPs Step 3: Find MD and NMD of between the patients and diseases with respect to the clinical symptoms Step 4: Calculate FFCR between the patients and diseases using the information from Step 3 Step 5: Decide the diagnosis on the basis of the relation for which the FFCR is maximum

The algorithm can be represented as a flowchart.

##### 4.2. Application Example

Assume patients visit a medical lab to ascertain their health conditions. After the vital signs of the patients were collected, the following symptoms , namely, high temperature, headache, stomach pain, cough, and chest pain were observed. From medical knowledge of the consultation, we simulate FFR , as shown in Table 1.

After the medical consultations guided by the vital signs, the patients are suspected to be infected by viral fever (*V*), malaria (*M*), typhoid fever (*T*), stomach problem (*S*), and heart problem (*H*). Similarly, FFR is given in Table 2. The simulated data in Tables 1 and 2 were used in [11] (S. K. De, R. Biswas, A. R. Roy (2001) An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets and Systems 117(2) 209–213) to demonstrate the application of IFSs in medical diagnosis. However, the data are extended to Fermatean fuzzy values in this work.

By applying our new approach, we get the parameters MD and NMD, as shown in Table 3.

By applying the approach of [11] in a Fermatean fuzzy setting, the parameters MD and NMD of FFCR are computed, as shown in Table 4.

After calculating the indexes of the FFPs, the results for FFCRs using the existing approach [11] are contained in the following matrix:

From the matrix, the following diagnosis are deduced:(i)Patient is diagnosed with malaria fever with a reasonable proportion of typhoid fever(ii)Patient is diagnosed with stomach problem(iii)Patient is diagnosed with malaria fever with a reasonable proportion of typhoid fever(iv)Patient is diagnosed with malaria fever

None of the patient is suffering from viral fever and heart problem. has negative relation with stomach problem and heart problem; has negative relation with viral fever, malaria fever, and heart problem; has negative relation with stomach problem and heart problem; and also has negative relation with stomach problem and heart problem. From the analysis, it is sensible the physician administers the same treatment to patients and because they have the same infection load of malaria fever and typhoid fever.

Similarly, the results using the new FFCR are contained in the following matrix:

From the matrix, we obtain the following diagnosis:(i)Patient is suffering from malaria fever with a reasonable proportion of viral fever and typhoid fever(ii)Patient is suffering from stomach problem only(iii)Patient is suffering from malaria fever with a reasonable proportion of typhoid fever and viral fever(iv)Patient is suffering from malaria fever with a reasonable proportion of viral fever

In order to show the edge of FFSs over IFSs and PFSs in terms of the ability to restrict uncertainties based on the new composition approach, we make use of the data in Tables 1–3 to compute the composite relation between each patients and diseases. By using the data as intuitionistic fuzzy data, we get the results in the matrix that follows:

From the results using intuitionistic fuzzy data, the following diagnoses are given: patient is suffering from the same disease as given by our approach; patient is suffering from stomach problem with a reasonable proportion of viral fever, which is different from the diagnosis of our approach (because of the inability of IFS to reasonably curb uncertainties); patient is suffering from the same diseases as given by our approach; patient is suffering from malaria fever with a reasonable proportion of viral fever, and equal proportion of typhoid fever, stomach problem, and heart problem (different from the diagnoses of our approach due to the inability of IFS to reasonably curb uncertainties). Though the values of the composite relation using intuitionistic fuzzy data are greater than our approach, it is certainly because of the inability of IFS to reasonably restrict the uncertainties in the process of diagnosis.

In addition, by using the data as Pythagorean fuzzy data, we get the following results: