Mediative Fuzzy Extension Technique and Its Consistent Measurement in the Decision Making of Medical Application

Department of Mathematics, C.C.S. University, Meerut-250004, India Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore 632014, Tamil Nadu, India Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484 887, India Department of Mathematics, MIT Campus, T.U, Janakpurdham, Nepal Department of Mathematics, Rajarshi Janak Campus, T.U, Janakpurdham 45600, Nepal


Introduction
e successful list of applications of fuzzy set theory [1] has been rising quickly. Since 1965, fuzzy set theory has played a vital part in the development of many areas of mathematics, engineering, and science. In the medical field, uncertainty exists in the process of disease diagnosis and medical knowledge concerned with the relationship between symptoms and disease. is relationship constitutes a huge amount of impreciseness and vagueness in diagnostic procedures. In medical diagnosis, the doctor generally depends upon the knowledge based on what he/she gathers from the patient's past history, physician examination, test reports, and medical images such as CT scan and X-rays. Each of these sources contains a large number of uncertainties. In order to face uncertainty, the physician knowledge is represented by fuzzy relationship [2] between symptoms and disease.
is fuzzy relation relates to the compositional fuzzy rule of inference in which each symptom is observed in the patient and to the set of various diseases. Adlassnig and Kolarz [3] elaborated this relational program to design a computerized system CADIAG-2 for medical diagnostic assistance. e approach of such a model is based on two types of relationships that presents tuck between symptoms and disease: occurrence and conformability relationship. Occurrence relation provided knowledge about the tendency of occurrence of symptoms when certain specific diseases are present, and the subsequent relation illustrates the power of symptoms toward validation of the existence of sickness. us, the CADIAG-2 system does not incorporate the relationship between symptoms and disease, but also between disease and symptoms themselves. Fordon and Bezdek [4] used another technique to model medical diagnosis by using fuzzy cluster analysis, in which the symptoms were designed with a membership function to represent each symptom category. Esogbue and Elder [5,6] have also illustrated some techniques to the medical diagnostic system. e fuzzy numbers are incomparable; for many fuzzy decision-making problems, many [7] methods have been suggested. e first method is based on the Hamming distance, and the second method is based on the α−cut method; furthermore, the supplementary method is based on the extension principle of fuzzy sets. As described earlier, the fuzzy theory takes interest in the description of the membership grade, and the non-membership degree is just complementary to the membership grade. However, the negation of the linguistic term does not satisfy the negation of the logical term in a valid mode. ese linguistic contradictions do not satisfy the logical contradiction. ere may be a few kinds of uncertainty present when selecting the membership degree. Membership function can be any function like triangular, trapezoidal, Gaussian, and exponential. Due to the uncertainty, nonmembership degree is less than or equal to the complement of the membership degree. Atanassov [8] recommended an intuitionistic fuzzy set to cover up the uncertainties in an appropriate way than the fuzzy sets. In the present work, the summation of the membership and non-membership grade is less or equal to "one," and hence the uncertainty or "unawareness" arises defining the membership grade. us, as compared to the fuzzy set theory, intuitionistic theory considers three types of uncertainties-membership grade, non-membership grade, and hesitation grade. For the studies of many decision-making problems [9], various comparison techniques of fuzzy numbers have been given in [10,11]. Jana et al. [12] used the Dombi operations to construct "Q-rung ortho pair fuzzy Dombi aggregation operators" and used the concept to build a mathematical model to solve the "multiattribute decision-making problems." e similarity measure technique has been widely used in many fields. Other study on similarity measure through interval-valued type-2 intuitionistic fuzzy set [13] has also been studied. is study shows the dual concept of some "semimetric distances" between various intuitionistic fuzzy sets.
However, due to the benefit of intuitionistic fuzzy set over the ordinary fuzzy set, intuitionistic fuzzy sets receive a little attention together with its corresponding measured techniques in [14,15]. e intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets have also been applied in the "dynamic hybrid multiattribute decision making (DHMADM)" [16], in which the contribution of dynamic weighted aggregation operators has also been described. Also, the Dombi operations were used to create some Pythagorean fuzzy Dombi aggregation operators [17] and build a mathematical model to solve the multiattribute decisionmaking problem in Pythagorean environment. Later, the concept of Pythagorean fuzzy multisets [18] has been used in medical diagnostic process.
Later in [19], the authors used the concept of mediative fuzzy logic in the management of contradictory information and showed the capability of mediative fuzzy logic over the intuitionistic and traditional fuzzy logics. ere are other applications of mediative fuzzy logic in medical field given in [9,20]. e basic objectives of the proposed approach are as follows: (1) ere are so many techniques that have been applied in the similarity measurement and ranking between two fuzzy sets or two intuitionistic fuzzy sets. (2) e well-defined extension principle will also extend over mediative fuzzy sets. (3) A compositional fuzzy relation between patients with symptoms and symptoms with disease has also been defined over meditative fuzzy sets. (4) We will define a priority contradictory function for the ranking purpose of two mediative fuzzy sets. e proposed tool will be easy to calculate and can easily be applicable in contradictory situations.
In the present work, we initiated with a comprehensive technique for a mediative fuzzy ranking method based on extension principle for mediative fuzzy set to construct a crisp order from the mediative fuzzy sets. In this paper, we are also giving an alternative approach for the better enhancement of medical diagnosis by using the utility of mediative fuzzy ranking analysis based on extension principle. e work in this present research article is divided into eight sections. In Section 2, we tried to explain a set of mathematical terms for easy communication with mediative fuzzy set and the extension principle for mediative fuzzy sets. Section 3 represents a mediative fuzzy relation in medical knowledge as shown in Figure 1. In Section 4 of the research paper, we formulated the algorithm. Section 5 and Section 5.1 introduce an intuitionistic fuzzy ranking method using the intuitionistic fuzzy extension and mediative fuzzy ranking method using the mediative fuzzy extension principle. Section 6 and Section 6.1 explain the applications of the proposed technique based on intuitionistic and mediative intuitionistic fuzzy extension in medical diagnosis. Section 7 presents the future aspect and real-life applications of the proposed technique. In Section 8, we tried to sketch the skeleton of the conclusion.

Basic Terminologies
In this section, we present some basic terminologies and the definitions associated with mediative fuzzy sets and the extension of mediative fuzzy sets.

Intuitionistic Fuzzy Set. Consider
, v A (x) ≤ 1called the membership and non-membership grades of

Intuitionistic Fuzzy
Number. An intuitionistic fuzzy setA is said to be an intuitionistic fuzzy number [21] if it satisfies the following axioms: (1)

Extension for Intuitionistic Fuzzy Sets. Let
. , r are intuitionistic fuzzy sets, and letY be another set. Let (2) en, the extension ∅ defines another intuitionistic fuzzy set in Y as where where ∅ − 1 (y) is the inverse image of y.

Mediative Fuzzy Set.
e mediative fuzzy set [9] is the set of such triplets over a given universal set which contains membership function, non-membership function, and index of uncertainty with contradictory function [9].
Let X be a universal set and M ⊆ X; then, a mediative fuzzy set Min X is defined as where is called the "contradictory factor" of Min X and with en, M is called mediative fuzzy set defined on X. μ M (x) and ] M (x) called the degree of membership and non-membership function respectively and is less then we know more about "x," whether x belongs or not belongs to X. Ifc M (x) + (ζ M (x)/2) has value large then we know less about "x," and if c M (x) + (ζ M (x)/2) � 0, then mediative fuzzy set coverts to traditional fuzzy set, and if μ M (x) and 1 − ] M (x) both are equals to 0 or 1, then it will convert into bi-valued logic.

2.5.
Extension . , r are the mediative fuzzy sets and let Y be another set. Let en, the extension ∅ defines another mediative fuzzy set B in Y as where where ∅ − 1 (y) is the inverse image of y.

Contradictory Function
Value. e value of contradictory function can be calculated by Medical experience R : Mediative fuzzy relation connecting patients P with disease D; R 1 : Mediative fuzzy relation on set of patients P with symptoms S; R 2 : Mediative fuzzy relation which represents the medical awareness to relate the symptoms S.

Mediative Fuzzy Relation in Medical Knowledge
For the purpose of our study, we have given a compositional rule for the intuitionistic fuzzy relationship to the diagnostic process as follows: R: mediative fuzzy relation connecting patients Pwith diseaseD. R 1 : mediative fuzzy relation on set of patients Pwith symptomsS. R 2 : mediative fuzzy relation that represents the medical awareness to relate the symptoms S to diseases and diseases D.

Algorithm
e algorithm is divided into two parts: part (A) presents the algorithm for intuitionistic fuzzy set which depends upon the membership and non-membership values and part (B) presents the algorithm for mediative fuzzy sets, which is based upon a single contradictory function value.

For Intuitionistic Fuzzy Sets.
Step 1: let us consider two intuitionistic fuzzy sets A(� A i ) and B(� B i ) and take their membership and non-membership values.
Step 2: calculate a "priority" function P on the ordered pair (A, B) based on their membership and nonmembership values.
Step 3: in step 3, we have the following conclusions:

For Mediative Fuzzy Sets.
Step 1: let us consider two mediative fuzzy sets A(� A i ) and B(� B i ) and find their contradictory values.
Step 2: find the "priority" contradictory function P on the ordered pair (A, B).
Step 3: after step 2, we have the following conclusions:

Ranking Method Based on Extension of
Intuitionistic Fuzzy Sets e intuitionistic fuzzy numbers are not comparable. In any intuitionistic fuzzy decision making, we need a comprehensive technique. We adopt an intuitionistic fuzzy ranking method based on extension principle to construct a crisp ordering from intuitionistic fuzzy numbers. In this section, we construct an intuitionistic fuzzy set P for intuitionistic fuzzy numbers called "priority." Let us consider two intuitionistic fuzzy sets A(� A i ) and B(� B i ) and then define a "priority" intuitionistic fuzzy set P on the ordered pair (A, B)as follows: If P(A) ≤ P(B)(for membership/M) and

Extension of Mediative Fuzzy
Sets. e mediative fuzzy sets are not comparable. In any mediative fuzzy decision making, we need a comprehensive technique. We adopt a mediative fuzzy ranking method based on extension principle to construct a crisp ordering from mediative fuzzy sets. In this section, we construct a mediative fuzzy set P for mediative fuzzy sets called "priority." Let us consider two mediative fuzzy sets A(� A i ) and B(� B i ) and then define a "priority" contradictory function P on the ordered pair (A, B)as follows: If P(A) ≤ P(B) ⇒A ≤ B where M stands for membership grade.

Applications Based on Extension of Intuitionistic Fuzzy Sets in Medical Diagnosis
Consider a patient p who has four symptoms represented by S � s 1 , s 2 , s 3 , s 4 and their severity level is defined by the intuitionistic fuzzy set: where I p (s i ) ∈ [0, 1], i � 1, 2, 3, 4 denotes the degree of membership and non-membership in the intuitionistic fuzzy set describing the patient p and is prescribed on the set S, which points out the level of severity of the symptom s i , i � 1, 2, 3, 4 for the patient. Now, our objective is to determine the proper diagnosis for this patient out of the three possible diseases D � d 1 , d 2 , d 3 . Each of these diseases is described by an intuitionistic fuzzy set, which gives the ordinary range of severity of respective symptom that may be familiar with the patient with the respective disease. So, the set of disease D is described by the following sets: For each j � 1, 2, 3, I d j denotes the intuitionistic fuzzy set where I d j (s i ) represents the membership and nonmembership degree of the symptom s i for the diseased j , j � 1, 2, 3. We further define an intuitionistic fuzzy relation R to represent the medical judgment that connects the set of symptoms S with their respective disease fromD that particularizes the efficiency of various symptoms s i for the diagnosis of the respective diseased j . Further, to the relevancy of respective symptom s i with respective disease d j defining an intuitionistic fuzzy relation R � R(s i , d j ) by weights matrix as; where the matrix R(s i , d j ) � S * D represents the weight of each symptom s i with their respective disease d j . For the diagnosis of the patient p, we use a clustering-based technique to find out that which diagnostic cluster as given in I d j , the patient is uttermost identical. Present technique is executed by enumerate a closeness measure between the symptoms s i and diseases d j To figure out this similarity, we use an intuitionistic fuzzy ranking method using intuitionistic fuzzy extension.  We may conclude that, according to the first priority, we have P(I p ) ≤ P(I d 1 ). When we apply the same method over d 2 , then the priority set gives P(I p ) ≥ P(I d 2 ) and ford 3 , we have P(I p ) ≥ P(I d 3 ). e most likely disease from this measured technique attains the maximum value. Results in the symptoms of patient p are uttermost similar to the disease d 1 .

Extension of Mediative Fuzzy Sets in Medical Diagnosis.
Consider a patient p who has four symptoms represented by S � s 1 , s 2 , s 3 , s 4 and their severity level is defined by the mediative fuzzy set: Mathematical Problems in Engineering where I p (s i ) ∈ [0, 1], i � 1, 2, 3, 4 denotes the degree of membership and non-membership in the mediative fuzzy set describing the patient p with some contradictory factor and is prescribed on the set S, which points out the level of severity of the symptom s i , i � 1, 2, 3, 4 for the patient. Now, our objective is to determine the proper diagnosis for this patient out of the three possible diseasesD � d 1 , d 2 , d 3 .
Each of these diseases is described by an intuitionistic fuzzy set, which gives the ordinary range of severity of respective symptom that may be familiar with the patient with the respective disease. So, the set of disease D is described by the following sets: For each j � 1, 2, 3, I d j denotes the mediative fuzzy set where I d j (s i )represents the membership and non-membership degree of the symptom s i for the disease d j , j � 1, 2, 3. We further define a mediative fuzzy relation Rto represent the medical judgment that connects the set of symptoms S with their respective disease from D that particularizes the efficiency of various symptoms s s i for the diagnosis of the respective disease d j . Further, to the relevancy of respective symptom s i with respective disease d j defining a mediative fuzzy relation R � R(s i , d j ) by weights matrix as; where the matrixR(s i , d j ) � S * D represents the weight of each symptom s i with their respective disease d j . For the diagnosis of the patient p, we use a clustering based technique to find out that which diagnostic cluster as given in I d j , the patient is uttermost identical. Present technique is executed by enumerate a closeness measure between the symptomss i and disease d j . To figure out this similarity, we use a mediative fuzzy ranking method using mediative fuzzy extension.

P I p � Sup
where M stands for membership value, P(I p ) � 0.

Future Aspect and Real-Life Applications
We extend the idea of mediative fuzzy extension over type-2 intuitionistic fuzzy sets. In this, the membership/nonmembership functions of type-1 mediative fuzzy sets are crisp and the two-dimensional can be taken as in type-1 intuitionistic fuzzy set. So, type-1 intuitionistic fuzzy sets are sufficient for dealing with the uncertainty of the present model of this study. On the one hand, type-2 intuitionistic fuzzy sets are capable of dealing with these uncertainties due to the three-dimensional nature of membership/nonmembership functions. So, mediative fuzzy set is not capable to instantly form the uncertainties present in any study. On the other hand, type-2 intuitionistic fuzzy sets have the capability to deal such type of uncertainties due to their membership/non-membership functions which are themselves fuzzy and three-dimensional. So, it gives the threedimensional degree that provides the added degree of freedom to handle such type of uncertainties. e arithmetic of type-2 intuitionistic fuzzy sets provided the clarity to deal the uncertainty, so everyone should believe in using the type-2 intuitionistic fuzzy extension principle which is the generalization of Zadeh's [22] fuzzy extension. We have given a type-2 intuitionistic fuzzy set characterized by [23] type-2 membership/non-membership function 0 ≤ μ A (x, u), ] A (x, u) ≤ 1: Let B be another set defined on Y and let ϕ: X × I x ⟶ Y × I y such that (y, u) � ϕ(x, u), (x, u) ∈ X × I x ; then, the extension ϕ defines a type-2 intuitionistic fuzzy set B in I y ⊆ [0, 1)as shown in Figure 2, where B � (y, u, μ B (y, u), ] B (y, u)): (y, u) � ϕ(x, u))}, ∀ x ∈ X. In decision making, type-2 intuitionistic fuzzy extension may play a vital role, and we may easily develop a comprehensive technique, by adopting a type-2 intuitionistic fuzzy ranking method based on extension principle to construct a crisp ordering for type-2 intuitionistic fuzzy sets. e type-2 intuitionistic fuzzy sets A and B can easily be compared by defining a priority function P on (A, B).

Conclusion
Since proper diagnosis is one of the most essential parts of decision making in the world of medical field, it is difficult to  handle a diagnostic process when there is a contradiction present in the experts' knowledge. So, in order to carry out the diagnosis process, we have used the extension of fuzzy logic in the form of mediative logic, to improve the impure and uncertain values of parameter present in the medical field. In decision-making problems, meditative fuzzy sets are very fruitful in finding the shape of the outputs of the problem. In the consecutive decision-making problems, an ordering technique is required. Mediative fuzzy sets provide a best ordering technique that can be used in consecutive decision-making problems. erefore, the whole work completed in this study enlightens the points as follows: (1) e concept of extension principle over mediative fuzzy set has been introduced; furthermore, a mediative fuzzy ranking-based technique is also described that provides a well ordering of the mediative fuzzy sets. (2) e given concept may also be used in decisionmaking for diagnosing the most likely same disease of this category. From the proposed measured technique, we can easily verify the patient symptoms most similar to a particular disease. (3) In the application section of the research work, for a patient p, we have taken a set of diseases D and a set of symptoms S whose levels of severity are described in the form of a mediative fuzzy set. On the basis of their priority value, we can easily find the most likely disease from the measured technique. is research paper also illustrates the mediative relation between symptom and disease on the basis of a mediative fuzzy ranking technique based on extension principle. (4) Furthermore, a new methodology for the interpretation of type-2 intuitionistic fuzzy set has also been discussed. Also, the given ranking method follows the popular existing extension principle based on a method that is described numerically. (5) One major advantage of mediative fuzzy logic over intuitionistic fuzzy logic is that, during the comparison process of mediative fuzzy sets, we need only one single value of the contradictory function as an output, which gives an easiest tool to compare two mediative fuzzy sets. In general, to do work with an intuitionistic fuzzy set, we need to consider both the values during the ranking process, i.e., membership degree and non-membership degree, which is quite difficult and time-consuming process. So, the proposed method presents the utility of mediative fuzzy logic over intuitionistic fuzzy logic. (6) A comparative study is also discussed in Table 1 between the existing approaches and the proposed approach, which shows the superiority of our proposed mediative fuzzy logic-based measurement approach.
On the basis of above, we may conclude that the present research paper describes the utility of mediative fuzzy logic in decision making. e uncertainties present in the membership degree and the non-membership degree of the mediative fuzzy set are described and their use in ranking technique has also been discussed. e given ranking method is effortlessly implemented and has a useful explanation over type-2 intuitionistic fuzzy sets.

Data Availability
We are sharing the source that includes the data that support the findings of our results and conclusion of the manuscript. e technique which we have given can be verified on the data which are available on the following hyperlinks: https:// www.mdpi.com/2073-8994/11/6/808 and https://www.icmr. gov.in/ (Micronutrient Profile of Indian Population by Indian Council of Medical Research).

Conflicts of Interest
e authors declare that they have no conflicts of interest.