Mathematical Modelling and Applications
Volume 1, Issue 2, December 2016, Pages: 36-45

Utility of Correlation Measures for Weighted Hesitant Fuzzy Sets in Medical Diagnosis Problems

B. Farhadinia

Department Mathematics, Quchan University of Advanced Technology, Iran

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To cite this article:

B. Farhadinia. Utility of Correlation Measures for Weighted Hesitant Fuzzy Sets in Medical Diagnosis Problems. Mathematical Modelling and Applications. Vol. 1, No. 2, 2016, pp. 36-45. doi: 10.11648/j.mma.20160102.12

Received: April 7, 2016; Accepted: April 13, 2016; Published: October 28, 2016


Abstract: Due to importance of correlation measure in data analysis, some researchers have shown great interest in the concept of correlation measure for extensions of fuzzy sets, in particular, for a new extension known as hesitant fuzzy set (HFS). Recently, an extension of HFS called the weighted hesitant fuzzy set (WHFS) has been developed by Zhang and Wu [1] to allow the membership of a given element is defined in terms of several possible values together with their importance weight. But, Zhang and Wu’s definition of WHFS gives rise to a number of disadvantages which violate the well-known axioms for mathematical operations. To circumvent this issue, we refine the definition of WHFS and then we put forward some correlation measures for WHFSs. Finally, we give a practical example to illustrate the application of proposed correlation measures for WHFSs in medical diagnosis.

Keywords: Weighted Hesitant Fuzzy Set, Correlation Measure, Medical Diagnosis Problem


1. Introduction

Correlation measure is one of the most broadly applied indices in a variety of fields, such as pattern recognition and fuzzy multiple attribute decision making problems. Recently, some researchers have shown great interest in the concept of correlation measure for extensions of fuzzy sets as well as that for fuzzy sets, and applied it to the field of fuzzy decision making. Murthy and Pal [2], and Chiang and Lin [3] studied the correlation between two fuzzy sets. Gerstenkorn and Manko [4], and Mitchell [5] derived the correlation coefficient of intuitionistic fuzzy sets from different viewpoints.

A new generalization of fuzzy set called hesitant fuzzy set (HFS) [6] 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. Later, a number of other extensions of the HFS have been developed such as dual hesitant fuzzy sets (DHFSs) [7], generalized hesitant fuzzy sets (G-HFSs) [8], hesitant fuzzy linguistic term sets (HFLTSs) [9] and higher order hesitant fuzzy sets (HOHFSs) [10].

However, HFS [6] 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 [1] introduced the concept of a weighted hesitant fuzzy set, denoted hereafter by (Z-WHFS). To clarify the necessity of introducing WHFS, consider a situation in which L experts are asked to evaluate the membership degree of the element x in the set experts provide  experts provide  experts provide  such that . Keeping in the mind that these L experts cannot persuade each other to change their opinions. In such a situation, the membership degree of the element x in the set  has m possible values …, associated respectively with the weights of  In this regard, the membership degree of the element x in the set should be represented by a weighted hesitant fuzzy element (WHFE)

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 [1] 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 [10]-[17]. Due to the potential applications of correlation measures in HFS theory, Chen et al. [18], and Xu and Xia [19] have further studied them for HFSs. Farhadinia [7] proposed an approach for deriving the correlation measures of dual HFSs, and then extended the approach to the dual interval-valued HFS (IVHFS) theory. 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 correlation measures for WHFSs. Section 3 is shown the application of correlation measures of WHFSs in medical diagnosis problems. This paper is concluded in Section 4.

2. Correlation Measures for Weighted Hesitant Fuzzy Sets

This section starts with the definition of hesitant fuzzy sets (HFSs) which were first introduced by Torra [6] as an extension of fuzzy sets.

Definition 2.1 [6] Let X be a reference set, a HFS A on X is defined in terms of a function  when applied to X returns a subset of [0,1], i.e.,

where is a set of some different values in [0, 1], representing the possible membership degrees of the element to A.

For convenience, we call  a hesitant fuzzy element (HFE) [20] and denoted briefly by .

Example 2.2 Let  be a reference set, andbe the HFEs of  to a set A, respectively. Then A can be considered as a HFS, i.e.,

From a mathematical point of view, a HFS A can be seen as a fuzzy set if there is only one element in  which indicates that fuzzy sets are a special type of HFSs. That is, the theory for HFSs can also be applied to fuzzy sets.

Assumption 2.3 (See e.g. [11, 20]) Notice that the number of values in different HFEs may be different. Suppose that l(h) stands for the number of values in the HFE h. Hereafter, the following assumptions are made: (A1) All the elements in each HFE h are arranged in increasing/decreasing order, and then  is referred to as the jth largest/smallest value in the HFE h. (A2) If, for two HFEs  then To have a correct comparison, the two HFEsand should have the same length l. If there are fewer elements in  than in  an extension of  should be considered optimistically/pessimistically by repeating its maximum/minimum element until it has the same length with .

Hereafter, we assume that all HFEs have the same length N, and let  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 [1] 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 2.4 [1] 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

(1)

Where  is a set of some different values in [0, 1], denoting all possible membership degrees of the element  to the set is the weight of  such that for any

Zhang and Wu [1] called a weighted hesitant fuzzy element (Z-WHFE). A Z-WHFE is conveniently denoted by

Zhang and Wu [1] defined for three Z-WHFEs and some operations as follows:

(2)

(3)

(4)

(5)

(6)

(7)

(8)

By taking the above mathematical operations into consideration, one can easily find that Zhang and Wu [1] were careless about their definition of operations because such definitions inherit some fundamental disadvantages: Disadvantage 1. Zhang and Wu’s union and intersection operations given by (3) and (4) are not idempotent, that is, for any Z-WHFE

(9)

(10)

where and  are real functions of  such that and for .

To more explanation, assume that a company wants to classify some different cars. It asks 10 experts to provide their evaluation information of a car with respect to the safety criterion. 6 experts express their evaluation information by the value "70 percent", and others by the value "80 percent". Keeping in mind that these 10 experts cannot persuade each other to change their opinions. In such a situation, their evaluation information can be described by a Z-WHFE as . If we apply Zhang and Wu’s union and intersection definitions given by (3) and (4) to  it results in

;

.

From  one finds that near 4 experts are confident with "70 percent" about the safety of a car, and near 6 experts are confident with "80 percent". But, as observed from definition of WHFE  the number of experts who are confident with "70 percent" and "80 percent" are respectively 6 and 10. Such a comparison of confidence level can be made for  where near 8 experts are confident with "70 percent" about the safety of a car, and near 2 experts are confident with "80 percent", meanwhile, these numbers of experts have been already mentioned as 6 and 10 in the WHFE .

Disadvantage 2. By applying Zhang and Wu’s addition and multiplication definitions given by (7) and (8) to any Z-WHFE  does not give a reasonable result, that is,

Where  and  are real functions of  such that and for .

Once again consider the Z-WHFE . Then,

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 2.5 Let X be the universe of discourse. A weighted hesitant fuzzy set (WHFS) on X is defined as

(11)

Where , 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  to the set  is the weight of  such that  for any .

It is interesting to note that if we take for any  then the WHFS is reduced to a typical HFS.

Hereafter, for the convenience of representation, we denote the WHFE  by

Assumption 2.6 Notice that the number of values in different WHFEs may be different. Suppose that stands for the number of values in . Hereafter, the following assumptions are made: (A1) All the first component of elements in each  are arranged in increasing order, and then  is referred to as the jth largest value in . (A2) If, for some , then .

To have a correct comparison, the two WHFEs  and  should have the same length  If there are fewer elements in  than in  an extension of  should be considered optimistically by repeating the maximum first component of elements associated with zero weight until it has the same length with  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

Definition 2.7 Let

be three WHFEs. Then, some operations on the WHFEsand  are defined as the following:

(12)

(13)

(14)

(15)

(16)

(17)

In the above formulas, for  referred to as the normalized weights, are determined in two steps: (i) We first calculate the weight of jth component of the binary operation  by simply adding the weights  and  for  (ii) After the whole components of  are to be obtained, their weights are considered again and then normalized. In this regard, the normalized weights of the above binary operations are defined as follows:

(19)

In the case that the associative binary operation is iterated on the finite set of WHFEs , that is, we are interested to obtain  the normalized weights are constructed as

Example 2.8 Suppose that

and are two given WHFEs. Bearing Assumption 2.6 in mind, should be first extended as . Then, one gets

Theorem 2.9 Let

 and  be three WHFEs. Then, all operations , ,,, given in Definition 2.7 are also WHFEs.

Proof. We only prove that  is also WHFE. Known by the definition of  from (13), i.e.,  we need to show that

By definition of the normalized weight  one can get

This completes the proof.

Theorem 2.10 Let

and  be three WHFEs. Then,

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

It is noteworthy to say that properties given by (27)-(28) show the superiority of WHFS definition proposed here over that of Z-WHFS suggested by Zhang and Wu [1].

2.1. Correlation Measures for WHFSs

By correlation analysis in both classical set theory and fuzzy set theory, we can examine the joint relationship of two sets with the aid of a measure of interdependency of the two sets. In the following, the axiomatic definitions of correlation measure for WHFSs are described as:

Definition 2.11 A real-valued function  is called a correlation measure for WHFSs, if for WHFSs , on X,  satisfies the following properties:

By assuming that

(29)

(30)

we define the following correlation measure formulas for any two WHFSs  and  as

(31)

(32)

(33)

(34)

 

It is noted that if WHFSs  and  are reduced to HFSs A and B, then the above correlation measures are reduced to Xu and Xia’s measures [19] and Chen et al.’s measures [18].

In order to equip the WHFS theory with further correlation measures, we presented two other correlation measures for WHFSs by extending Jaccard’s [21] and Dice’s [22] correlation measures defined on the vector space as follows:

(36)

(37)

Theorem 2.12 The measure functions WHFSi ( (i = 1,..., 7) given respectively by (31)-(37) are correlation measures for WHFSs  and

Proof. It is necessary to show that each measure function satisfies the requirements (1)-(3) listed in Definition 2.11. The proof of (1) and (3) for  WHFS1 given by (31) is straightforward and we prove only ( 1). The left-hand side inequality is obvious where  WHFS1 ,. To show the assertion of the right-hand side inequality, recall the Cauchy-Schwarz inequality that

where for i = 1,..., n.

In view of the above inequality relation and letting

(38)

(39)

one can easily verify that

 

which implies that  WHFS1 , Thus, WHFS1 , is a correlation measure for WHFSs  and .

The correlation proof ofWHFS2 , is much like that of  WHFS1 ,. The only difference is that instead of the Cauchy-Schwarz inequality one should consider the following inequality

Where  for i = 1,..., n are those set as (38) and (39).

The correlation proof of WHFS3 , and WHFS4 , are respectively much like that of WHFS1 , and WHFS2 ,

The correlation proof of WHFS5 , is straightforward.

The correlation proof ofWHFS6 , andWHFS7 , comes from the fact that

Where  for i = 1,..., n are those set as (38) and (39).

Proposition 2.1 Let WHFSi, (i = 1,..., 7) be the correlation measures for WHFSs and , given respectively by (31)-(37). Ifand the values of  in B are k times the values of in A such that for  and, then

WHFS1 ,WHFS3 ,   (40)

WHFS2 ,WHFS4 ,  (41)

WHFS5 ,       (42)

WHFS6 ,      (43)

WHFS7 ,   (44)

Proof. We only prove (42), and the others can be easily verified.

Suppose that  and the values of  in B are k times the values of  in A such that  for  and. By taking the definition of correlation measures WHFS5 , into account, we find for

 

 

 

Therefore,

Trivially, for k = 1 we find that . On the other hand, for

3. Medical Diagnosis with Weighted Hesitant Fuzzy Information

In this portion, we implement the following medical diagnosis problem to illustrate the efficiency of the correlation measures for WHFSs. Although, the problem of medical diagnosis can be similarly re-modeled with WIVHFS data, we do not consider here such a problem because of having the same solution procedure.

Example 3.1 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

 

Note that the characteristics of the diagnosis "Viral fever" with respect to the symptoms "Headache", "Cough",Stomach pain", "Chest pain", denoted respectively by WHFEs  (j = 2, 3, 4, 5), form the WHFS  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 theconsidered 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.

Table 1. Symptoms characteristic for the considered diagnoses.

Table 2. Symptoms characteristic for the considered patients.

As can be seen from Tables 1 and 2, all WHFEs are not in the same size. To circumvent this issue, we implement Assumption 2.6. In this regard, The WHFEs with fewer elements are extended optimistically by repeating the maximum first component of elements associated with zero weight until it has the same length with others. For

example, the WHFE

During the process of deriving a diagnosis for each patient, the degree of dependence between the rows of Tables 1 and 2 should be analyzed. For instance, the first rows of Tables 1 and 2 which are regarded as the following two WHFSs

are taken into account to determine the correlated degree of Al and viral fever.

In order to proceed, we apply the correlation measures WHFS1, WHFS4, and WHFS7 to determine the degree of dependence between diagnoses and patients. The results obtained by the use of these correlation measures are shown in Tables 3-5, respectively.

Table 3. Values of _WHFS1 for each patient to the considered set of possible diagnoses.

Viral fever Malaria Typhoid Stomach problem Chest Problem
Al 0.7984 0.5376 0.5998 0.6467 0.7906
Bob 0.7261 0.7581 0.8549 0.7881 0.7781
Joe 0.803 0.6568 0.5697 0.6706 0.7356
Ted 0.8513 0.5778 0.8723 0.6798 0.8433

Table 4. Values of _WHFS4 for each patient to the considered set of possible diagnoses.

  Viral fever Malaria Typhoid Stomach problem Chest Problem
Al 0.7591 0.4221 0.5378 0.4653 0.7334
Bob 0.6551 0.6939 0.8179 0.6610 0.6192
Joe 0.7858 0.5542 0.5490 0.5185 0.6350
Ted 0.7961 0.5102 0.8650 0.5500 0.6956

Table 5. Values of _WHFS7 for each patient to the considered set of possible diagnoses.

  Viral fever Malaria Typhoid Stomach problem Chest Problem
Al 0.7974 0.5223 0.5963 0.6132 0.7884
Bob 0.7223 0.7551 0.8541 0.7761 0.7582
Joe 0.8028 0.6474 0.5694 0.6490 0.7277
Ted 0.8494 0.5733 0.8722 0.6649 0.8279

By comparing the results listed in Table 3, we observe that AL and Joe suffer from "Viral fever", Bob and Ted from "Typhoid".

Tables 3-5 present the same result for all correlation measures.

4. Conclusion

Weighted hesitant fuzzy set (WHFS) is a new extension of hesitant fuzzy set (HFS) where the membership degree of an element to a given set is expressed by several possible values together with their importance weight. In this contribution, we modified and emended a fault of WHFS definition proposed by Zhang and Wu [1] so as the modified definition of WHFS is acceptable in accordance with the well-known axioms for mathematical operations. Because of importance of correlation measure in data analysis, we then developed a series of correlation measures for WHFSs and employed them to solve the weighted hesitant fuzzy multi-attribute group decision making (MAGDM) vproblems. 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.

Acknowledgements

The author would like to express his sincere thanks to the editor-in-chief and referees for their helpful suggestions which improved the presentation of the paper. This work was supported by Quchan University of Advanced Technology under grant number 94/7628.


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