Distances between intuitionistic fuzzy sets of second type with application to diagnostic medicine

: The concept of intuitionistic fuzzy sets of second type (IFSST) generalizes intuitionistic fuzzy sets (IFS) and thus, has many applications in decision making problems. The main feature of IFSST is that it is characterized by three parameters, namely: membership degree, non-membership degree and degree of indeterminacy in such a way that the sum of the square of each of the parameters is one. The purpose of this paper is to present the axiomatic deﬁnition of distance between IFSST, taking into account the three parameters that describe the sets and to investigate numerically, the validity of some distances between intuitionistic fuzzy sets introduced by E. Szmidt and J. Kacprzyk in IFSST environment. Finally, we explore the application of IFSST in diagnostic medicine by employing normalized Hamming distance of IFSST to calculate the distance between patients and diseases, because it provides a reliable distance with respect to other distances. Actually, by using the distance between patients and diseases (both in IFSST values), with recourse to the corresponding symptoms observe in the patients and of the diseases, we determine the illness of the paients. These distances are suggestible to be deployed in solving multicriteria decision making problems.


Introduction
Zadeh [24] introduced the idea of fuzzy set which has a membership function, µ, that assigns to each element of the universe of discourse, a number from the unit interval [0,1] to indicate the degree of belongingness to the set under consideration. Fuzzy set was generalized from classical sets theory by allowing intermediate situations between the whole and nothing. In a fuzzy set, a membership function which replaces the characteristic function in crisp sets, is defined to describe the degree of membership of an element to a class. Fuzzy set which is the extension of crisp set provides a means of representing and handling vagueness and imperfectly described knowledge. Notwithstanding, a fuzzy set cannot model vagueness precisely because there is no means to attribute reliability information to the membership degrees. The conspicuous presence of vagueness/imprecision in life necessitated researchers to develop some mathematical frameworks that can cope vagueness/imprecision more accurately than fuzzy sets [12].
There are cases where µ + ν ≥ 1 unlike in IFS (where µ + ν ≤ 1). This limitation in IFS naturally led to the generalization of IFS as intuitionistic fuzzy sets of second type (IFSST) proposed by Atanassov [3]. An identical set called Pythagorean fuzzy set has been studied in literature [23] without a recourse to IFSST. In fact, Pythagorean fuzzy set is the same as IFSST. We feel that the development of Pythagorean fuzzy set theory independent of IFSST does not augur well since IFSST was proposed in 1989 (twenty four years before Pythagorean fuzzy set).
IFSST is a reliable tool to deal with vagueness considering the membership grade, µ and the non-membership grade, ν satisfying the condition µ + ν ≥ 1. As a generalized set, IFSST has close relationship with IFS. This construct can be used to characterize uncertain information more sufficiently and accurately than IFS. Some operations on IFSST have been studied [14].
The notion of distances between IFS [15,16], which is very applicable in real-life situations, is the compelling motivation of this paper. In this paper, we extend the idea of distances between IFS [15,16] to IFSST and explore their application in medical diagnosis to determine the illness suffered by patients. We deploy the normalized Hamming distance for the application because it is the most reliable of the distance measures proposed in IFSST, after a reliability test is conducted. The paper is organized as follows: Section 2 provides some preliminaries on fuzzy sets, IFS as foundations to the idea of IFSST, while Section 3 covers the notion of IFSST and some distance measures between them with their numerical verifications. We present the application of IFSST to diagnostic medicine in Section 4. Finally, Section 5 concludes the paper and provides direction for future studies.

Preliminaries
We recall some basic notions of fuzzy sets and intuitionistic fuzzy sets. Definition 2.1 ([24]). Let X be a nonempty set. A fuzzy set A of X is characterized by a membership function µ A :

Fuzzy sets
Alternatively, a fuzzy set A of X is an object having the form where the function defines the degree of membership of the element, x ∈ X.
The closer the membership value µ A (x) to 1, the more x belongs to A, where the grades 1 and 0 represent full membership and full non-membership. In classical set theory, the membership of elements to a set is assessed in binary terms according to a bivalent condition; an element either belongs or does not belong to the set. Classical bivalent sets are called in fuzzy set theory crisp sets.
Let us consider two examples; (i) All employees of XYZ who are over 1.8 m in height.
(ii) All employees of XYZ who are tall.
The first example is a classical set with a universe (all XYZ employees) and a membership rule that divides the universe into members (those over 1.8 m) and non-members. The second example is a fuzzy set because some employees are definitely in the set and some are definitely not in the set, but some are borderline.
This distinction between the ins, the outs and the borderline is made more exact by the membership function, µ. If we return to our second example and let A represent the fuzzy set of all tall employees and x represent a member of the universe X (i.e. all employees), then µ A (x) would be µ A (x) = 1 if x is definitely tall or µ A (x) = 0 if x is definitely not tall or 0 < µ A (x) < 1 for borderline cases. Definition 2.2 ([1,2]). Let a nonempty set X be fixed. An IFS A of X is an object having the form

Intuitionistic fuzzy sets
where the functions define the degree of membership and the degree of non-membership, respectively, of the element x ∈ X to A, which is a subset of X, and for every x ∈ X, is the intuitionistic fuzzy set index or hesitation margin of x in X. The hesitation margin π A (x) is the degree of indeterminacy of the belonging of x ∈ X, to the set A and π A (x) ∈ [0, 1]. The hesitation margin is the function that expresses lack of knowledge of whether x ∈ X or x / ∈ X. Thus,

Intuitionistic fuzzy sets of second type
In this section, we succintly discuss some fundamentals of IFSST. 3,14]). Let X be a universal set. Then, an IFSST A of X is a set of ordered pairs defined by

Concept of intuitionistic fuzzy sets of second type
where the functions define the degree of membership and the degree of non-membership, respectively, of the element x ∈ X to A, which is a subset of X, and for every x ∈ X, We denote the set of all IFSST of X by IFSST(X).
This shows the validity of complementary law in IFSST.

Definition 3.7 ([3]
). Let A, B ∈ IFSST(X). Then, the following define union and intersection of A and B: Definition 3.8. Let A, B ∈ IFSST(X). Then, the sum of A and B is defined as and the product of A and B is defined as Remark 3.9. Let A, B, C ∈ IFSST(X). By Definitions 3.7 and 3.8, the following properties hold: Definition 3.10. Let A ∈ IFSST(X). Then, the level/ground set of A is defined by Certainly, A * is a subset of X.

Distances between intuitionistic fuzzy sets of second type
First and most, we review the concept of distance measure for IFS as presented by Szmidt and Kacprzyk [15,16]. For any two IFS A and B of X = {x 1 , ..., x n }, the following distance measures were proposed: All these distances have been tested to for reliability with the normalized Hamming distance as the most reliable one. Now, we give an analogous definition of distance measure for IFSST as follows: Definition 3.12. Let X be nonempty set and A, B, C ∈ IFSST(X). Then, the distance measure d between the IFSST is a function Let A and B be IFSST of X = {x 1 , ..., x n }. Then, we propose the following distance measures as extension of [15,16]: Proof. By Definition 3.12, the proof follows.

Numerical verification
We now verify whether these distance measures for IFSST satisfy the conditions in Definition 3.12. Recall that, every IFS is an IFSST but the converse is not true. Now, we evoke the following example from [16]. Clearly, A * = B * .
Applying the distances between IFSST, we obtain the following results:

Discussion
From Table 2, we observe that the distances between IFS (i.e., Hamming distance and Euclidean distance) [15,16] do not satisfy the conditions of distance measure for IFSST as seen in Defini- Notwithstanding, n−Hamming distance is the most reasonable/efficient of the distance measures discussed. Hence, we adopt n−Hamming distance for application to diagnostic medicine.

Application of IFSST to diagnostic medicine
In this section, we present an application of IFSST in diagnostic medicine. In a given pathology, suppose S is a set of symptoms, D is a set of diseases, and P is a set of patients. Using medical knowledge of diseases and symptoms, we know that some symptoms are perculiar to a particular disease than the other, that is, each disease has perculiar symptoms which reveal a degree of association, µ, and a degree of non-association, ν, between the symptoms and diseases. Now we discuss the concept of IFSST in diagnostic medicine. The methodology involves mainly three stages viz; (i) determination of symptoms (ii) formulation of medical knowledge based on IFSST values (iii) determination of diagnosis using n−Hamming distance between P and D.
Then, the distance, d between patient and disease with respect to symptoms is given as where s i ∈ S, and n is the number of symptoms. A patient is diagnosed to suffer from a particular disease if the distance, d between the disease and the patient is the least. To see the application of the method, let us make a hypothetical case.

Experimental example
Suppose there are four patients, P given as a set P = {Ene, Ehi, Ela, Ebo} to be examined for the set of diseases, D given as D = {viral fever, malaria fever, typhoid fever, stomach ulcer, chest problem}, which have the set of symptoms, S S = {temperature, headache, stomach pain, cough, chest pain}. From medical knowledge, After calculating the degree of indeterminatcy using π = 1 − [µ 2 + ν 2 ], we compute the distance between patients and diseases using Equation 1. The computational results are contained in Table 5 Table 5, the following diagnosis are determined based on the least value of the distances between P and D: Ene is diagnosed of malaria fever, Ehi is diagnosed of stomach ulcer, Ela is diagnosed of typhoid fever, Ebo is diagnosed of viral fever.

Conclusion
In this paper, we proposed an axiomatic definition of distance between intuitionistic fuzzy sets of second type. It was shown that distances between intuitionistic fuzzy sets of second type should be calculated by taking into account the three parameters that describe an intuitionistic fuzzy set of second type. Taking into account all the three parameters that describe intuitionistic fuzzy sets of second type when calculating distances ensure that the distances for intuitionistic fuzzy sets and intuitionistic fuzzy sets of second type can be easily compared. We verified the authenticity of the proposed distances between IFSST in comparison to Definition 3.12, and found that Hamming and Euclidean distances are not distance measures for IFSST. But, n−Hamming and n−Euclidean distances are appropriate distance measures for IFSST. To test the applicability of the proposed distance measures in real-life problems, a medical diagnostic problem was considered via n−Hamming distance, for a reliable output. The distances between IFSST studied in this work could be applied in decision making of real-life problems embedded with uncertainty.