A Hesitant Fuzzy Based Medical Diagnosis Problem

. In this paper, we refine the definition of weighted hesitant fuzzy set (WHFS), the concept that allows the membership of a given element is defined in terms of several possible values together with their importance weight, and then introduce some correlation measures for WHFSs. To illustrate the application of proposed correlation measures for WHFSs, we give a practical example in medical diagnosis


Introduction
A new generalization of fuzzy set called hesitant fuzzy set (HFS) (Torra (2010)) has received great attention in handling decision making problems where the decision makers have some hesitations among several possible memberships for an element to a set. However, HFS (Torra (2010)) has its inherent drawbacks, because it expresses the membership degrees of an element to a given set only by possible values without emphasizing on the importance of each possible value. In many practical decision making problems, the information provided by decision makers who are familiar with the area might often be described by the same preferences. In such situations, the value repeated several times is more important than that appeared only one time. Thus, the importance of possible membership degrees (i. e., their repetition rate) should be considered in improving the definition of HFS. To consider this fact, Zhang and Wu (2014) introduced the concept of a weighted hesitant fuzzy set, denoted hereafter by (Z-WHFS).
In this contribution, we will show that Zhang and Wu's definition of union, intersection, addition and multiplication operations for Z-WHFS have not been correctly set up. This motivates us to modify and emend a fault of WHFS definition proposed by Zhang and Wu (2014) so as not only the modified definition of WHFS is acceptable in accordance with the well-known axioms for mathematical operations, but also it allows that all information measures are to be defined reasonably as well as those defined for HFSs Farhadinia (2014a)-Farhadinia (2014e), and Farhadinia and Ban (2013). In this paper, we develop some correlation measures for WHFSs and then, the proposed correlation measures are applied to a medical diagnosis problem.
The present paper is organized as follows: Section 2 introduces some concepts related to WHFSs. Section 3 is presented a number of correlation measures of WHFSs. Section 4 shows the application of correlation measures in medical diagnosis problems. This paper is concluded in Section 5.

WHFS Conceptions
Definition 2.1. (Torra (2010)) Let X be a reference set, a HFS A on X is defined in terms of a function h (x) when applied to X returns a subset of [0, 1], i. e.
where h (x) is a set of some different values in [0, 1], representing the possible membership degrees of the element x ∈ X to A.
For convenience, we call h (x) a hesitant fuzzy element (HFE) (Xia and Xu (2011)) and denoted briefly by h . If, for two HFEs h , h , L(h ) ≠ l(h ), then l = max L(h ), l(h ) . To have a correct comparison, the two HFEsh andh should have the same length l. If there are fewer elements in h than in h , an extension of h should be considered optimistically/pessimistically by repeating its maximum/minimum element until it has the same length with h .
Hereafter, we assume that all HFEs have the same length N, and let h = ∪ h σ( ) throughout the paper. As can be seen from Definition 2.1, HFS expresses the membership degrees of an element to a given set only by several real numbers between 0 and 1 of equal importance, while in many real-world situations assigning exact values without importance weight to the membership degrees does not describe properly the imprecise or uncertain decision information. Thus, it seems to be difficult for the decision makers to rely on the present form of HFSs for expressing uncertainty of an element. To overcome the difficulty associated with the present form of HFSs, Zhang and Wu (2014) have attempted to introduce the concept of weighted hesitant fuzzy set (Z-WHFS) in which the membership degrees of an element to a given set can be expressed by several possible values together with their importance weight.
Definition 2.2. (Zhang and Wu (2014)) Let X be the universe of discourse. A Zhang and Wu's representation of weighted hesitant fuzzy set (Z-WHFS) on X is defined as where w (x) is a set of some different values in [0, 1], denoting all possible membership degrees of the element x ∈ X to the set w , w $γ ∈ )0,1, is the weight of γ such that∑ for any x ∈ X. Zhang and Zhang and Wu (2014) defined for three Z-WHFEs w =∪ % ∈/ 0 #γ , w % & , w =∪ % ∈/ 01 #γ , w % 1 & and w =∪ % ∈/ 02 #γ , w % 2 & some operations as follows: w ∩ w = ∪ γ ∈ w , γ ∈ w #min γ1, γ2 , w % 1 . w % 2 & . (4) By taking the above mathematical operations into consideration, one can easily find that Zhang and Wu (2014) were careless about their definition of operations because such definitions inherit some fundamental disadvantages (see, Farhadinia (2017) Here, in order to avoid the disadvantages arising from Zhang and Wu's definition of WHFS and mathematical operations on WHFSs, we redefine a weighted hesitant fuzzy set as follows.
Definition 2.3. Let X be the universe of discourse. A weighted hesitant fuzzy set (WHFS) on X is defined as Where w (x), referred to as the weighted hesitant fuzzy element (WHFE), is a set of some different values in [0, 1],denoting all possible membership degrees of the element x ∈ X to the set w , w I( ) (x) ∈ )0, 1, is the weight of for any x ∈ X.
It is interesting to note that if we take w I( ) (x) = ⋯ = w I(K L ) (x) = K L for any x ∈ X, then the WHFS w is reduced to a typical HFS.
Hereafter, for the convenience of representation, we denote the WHFE w (x) by w =∪ K L F〈h I( ) , w I( ) 〉J. To have a correct comparison, the two WHFEs w (x) and w (x) should have the same length L $ . If there are fewer elements in w (x) than in w (x), an extension of w (x) should be considered optimistically by repeating the maximum first component of elements associated with zero weight until it has the same length with w (x) This kind of extension is quite reasonable since the added element with zero weight is meant to be an element that does not really exist.
Throughout this paper, we assume that all WHFEs have the same length N, and let w =∪ 〈h σ( ) , w σ( ) 〉 .
By assuming that we define the following correlation measure formulas for any two WHFSs w and w _ as In order to equip the WHFS theory with further correlation measures, we presented two other correlation measures for WHFSs by extending Jaccard (1901)'s and Dice (1945)'s correlation measures defined on the vector space as follows:

WHF Information Used in Medical Diagnoses
In this portion, we implement the following medical diagnosis problem to illustrate the efficiency of the correlation measures for WHFSs.
Example 4.1. Consider the set of diagnoses D = {Viral fever, Malaria, Typhoid, Stomach problem, Chest problem}. The aim here is to assign a patient with the given values of the symptoms, S = {Temperature, Headache, Cough, Stomach pain, Chest pain} to one of the aforementioned diagnoses. Three medical experts El, (l = 1, 2, 3) are invited to provide their possible assessment of diagnoses with respect to symptoms. For each diagnosis with respect to each symptom, all of the medical experts provide anonymously their evaluated values. As an example, for the diagnosis "Viral fever" with respect to the symptom "Temperature", the evaluation value provided by medical experts E1 and E3 is 0.5; and E2's evaluation value is 0.7. In this regard, and noting that the weights of three medical experts are unknown, the evaluation of "Viral fever" with respect to "Temperature" can be represented by a WHFE as w (V iralfever, Temperature) = w = !〈0.5, 2 3 〉 , 〈0.7, 1 3 〉(. Note that the characteristics of the diagnosis "Viral fever" with respect to the symptoms "Headache", "Cough", "Stomach pain", "Chest pain", denoted respectively by WHFEs w , (j = 2, 3, 4, 5), form the WHFS w which is indicated in the first row of Table 1. The results evaluated for other diagnoses with respect to symptoms arecontained in a weighted hesitant fuzzy decision matrix, shown in Table 1.
Furthermore, suppose that the set of patients is P = {Al, Bob, Joe, Ted}, and the symptoms characteristic for the considered patients are evaluated and given by the three medical experts in the form of a weighted hesitant fuzzy matrix demonstrated in Table 2. Here, the main task is to seek a diagnosis for each patient. By comparing the results listed in Table 3, we observe that AL and Joe suffer from "Viral fever", Bob and Ted from "Typhoid".

Conclusion
In this contribution, we modified and emended a fault of WHFS definition so as it is acceptable in accordance with the well-known axioms for mathematical operations. We believe many future works can be developed by the use of the findings of this contribution which support the decision makers in making decisions effectively in WHFS-structured MAGDM problems.