# Cosine Similarity Measure between Hybrid Intuitionistic Fuzzy Sets and Its Application in Medical Diagnosis

Authors: Donghai Liu; Xiaohong Chen; Dan Peng
Journal: Computational and Mathematical Methods in Medicine
 (2018)

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## Abstract

In this paper, a cosine similarity measure between hybrid intuitionistic fuzzy sets is proposed. The aim of the paper is to investigate the cosine similarity measure with hybrid intuitionistic fuzzy information and apply it to medical diagnosis. Firstly, we construct the cosine similarity measure between hybrid intuitionistic fuzzy sets, and the relevant properties are also discussed. In order to obtain a reasonable evaluation in group decision, the weight of experts under different attributes is determined by the projection of individual decision information on the ideal decision information, where the ideal decision information is the average values of each expert’s evaluation. Furthermore, we propose a decision method for medical diagnosis based on the cosine similarity measure between hybrid intuitionistic fuzzy sets, and the patient can be diagnosed with the disease according to the values of proposed cosine similarity measure. Finally, an example is given to illustrate feasibility and effectiveness of the proposed cosine similarity measure, which is also compared with the existing similarity measures.


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## Body

## 1. Introduction

A similarity measure is an important tool for determining the degree of similarity between two objects in many fields, such as pattern recognition, medical diagnosis, and so on. Many similarity measures have been introduced [1–8]. Among them, some similarity measures of intuitionistic fuzzy sets (IFSs) have been proposed. For example, Li and Cheng [3] proposed a similarity measure between IFSs and applied it to pattern recognition. Huang and Yang [2] defined the similarity measure between IFSs based on the Hausdorff distance and used it to calculate the degree of similarity between IFSs. Nguuen [9] proposed a new knowledge-based similarity measure between IFSs and applied it to pattern recognition. However, due to the complexity and uncertainty of the decision-making environment, the membership degree and nonmembership degree of IFS need to be expressed by interval rather than the numerical value. Motivated by this, Atanassov and Gargov [10] introduced the concept of interval-valued intuitionistic fuzzy set (IVIFS), which is a generalization of IFS. Xu [11] proposed some distance and similarity measures between IVIFSs and applied them to pattern recognition.On the other hand, the cosine similarity measure based on Bhattacharyya distance was first proposed in Bhattacharyya [12]. Ye [7] proposed a cosine similarity measure for IFSs (CIFS) and applied it to pattern recognition. Furthermore, Ye [13] proposed the cosine similarity measure for IVIFSs (CIVIFS) and applied it to group decision-making problems. However, in the complex group decision-making problem, it is difficult to use a single value to express the alternative under all attributes. Because some attributes might be represented by IFSs, but other attributes are suitable to be represented by IVIFSs. At this time, the people should use hybrid intuitionistic fuzzy set to make a decision. However, the existing methods can not deal with the hybrid fuzzy information. As far as we know, no people studied the cosine similarity measure between hybrid IFSs. Motivated by this, we will introduce the cosine similarity measure with hybrid intuitionistic fuzzy information (CHIFS) in this paper. This generalization makes the CHIFS measure includes CIFS measure and CIVIFS measure as particular case.In addition, applying theCHIFS measure to group decision-making problems is very interesting. For example, Zhou and Wahab [14] use transmissibility incorporated with cosine similarity measure to investigate the structural damage detection. Furthermore, Zhou et al. [15] apply transmissibility function with distance measure to separate the intact patterns apart from the damaged pattern. In group decision-making problems, the weight of the experts under different attributes can be obtained by using the projection of individual decision information on the ideal decision information. Then, we aggregate all individual decisions into a collective one and apply the proposed cosine similarity measure between hybrid intuitionistic fuzzy sets to medical diagnosis.The rest of the paper is organized as follows. In Section2, we review the cosine similarity measure for IFSs and IVIFSs. In Section 3, we propose the CHIFS measure, some properties are also analyzed. In Section 4, we propose a decision method for medical diagnosis based on the cosine similarity measure between hybrid intuitionistic fuzzy sets. In Section 5, an example is given to illustrate the feasibility and effectiveness of the proposed CHIFS measure. Finally, the conclusion and further research are discussed in Section 6.


## 2. Preliminaries

Throughout this paper, letX=x1,x2,...,xn be a finite universal set. In this section, we briefly review the IFSs and IVIFSs, the cosine similarity measure between IFSs, and the cosine similarity measure between IVIFSs.


### 2.1. Intuitionistic Fuzzy Set

Definition 1. Let X be a fixed set, an intuitionistic fuzzy set (IFS) A in X is defined as:(1)A=xj,μAxj,νAxjxj∈X,where the functions μAxj and νAxj represent the membership degree and nonmembership degree of the element xj to the set A, respectively, such that 0≤μAxj+νAxj≤1∀xj∈X.The intuitionistic fuzzy indexπAxj=1−μAxj−νAxj, and we have 0≤πAxj≤1. For example, A=0.4,0.3 is an intuitionistic fuzzy number, and πA=0.3. The space of membership degree of IFS is shown in Figure 1.In particular, whenX has only one element, the IFS A=xj,μAxj,νAxjxj∈X is reduced to A=μAxj,νAxj, which we call it an intuitionistic fuzzy number (IFN).For any two IFSsA=xj,μAxj,νAxj and B=xj,μBxj,νBxj, the following operations are true [16]:(1)
A+B=μAxj+μBxj−μAxjμBxj,νAxjνBxj(2)
λA=1−1−μAxjλ,νAxjλ,λ>0(3)
A=BifμAxj=μBxj,νAxj=νBxjThe results of the operationsA+B and λA are still IFSs.Figure 1
The membership degree of IFS.


### 2.2. Interval-Valued Intuitionistic Fuzzy Set

Definition 2. Let X be a fixed set, an interval-valued intuitionistic fuzzy set A˜ is defined as follows:(2)A˜=xj,μA˜Lxj,μA˜Uxj,νA˜Lxj,νA˜Uxjxj∈X,where(3)μA˜Lxj≥0,νA˜Lxj≥0;0≤μA˜Uxj+νA˜Uxj≤1,∀xj∈X.The interval-valued intuitionistic fuzzy index is defined asπA˜xj=πA˜Lxj,πA˜Uxj, where(4)πA˜Lxj=1−μA˜Uxj−νA˜Uxj,πA˜Uxj=1−μA˜Lxj−νA˜Lxj.For example,A˜=0.3,0.4,0.1,0.4 is an interval-valued intuitionistic fuzzy number, the fuzzy index πA˜xj=0.2,0.6.Remark 1.
IfμA˜Lxj=μA˜Uxj,νA˜Lxj=νA˜Uxj, then the interval-valued intuitionistic fuzzy set is reduced to intuitionistic fuzzy set.
When the setX has only one element, the IVIFS A˜=xj,μA˜Lxj,μA˜Uxj,νA˜Lxj,νA˜Uxjxj∈X is reduced to A˜=μA˜Lxj,μA˜Uxj,νA˜Lxj,νA˜Uxj, which we call it an interval-valued intuitionistic fuzzy number(IVIFN).
LetA˜=xj,μA˜Lxj,μA˜Uxj,νA˜Lxj,νA˜Uxj and B˜=xj,μB˜Lxj,μB˜Uxj,νB˜Lxj,νB˜Uxj be two IVIFSs, the following operations are true [17]:(1)
A˜+B˜=μA˜Lxj+μB˜Lxj−μA˜LxjμB˜Lxj,μA˜Uxj+μB˜Uxj−μA˜UxjμB˜Uxj, νA˜LxjνB˜Lxj,νA˜UxjνB˜Uxj(2)
λA˜=1−1−μA˜Lxjλ,1−1−μA˜Uxjλ,νA˜Lxjλ,νA˜Uxjλ,λ>0(3)
A˜=B˜ if μA˜Lxj=μB˜Lxj,μA˜Uxj=μB˜Uxj,νA˜Lxj=νB˜Lxj and νA˜Uxj=νB˜Uxj


### 2.3. Cosine Similarity Measures for IFSs or IVIFSs

Definition 3 (Ye [7]). Let A=μAxj,νAxj and B=μBxj,νBxj be two IFSs in X, the cosine similarity measure between A and B is defined as follows:(5)CIFSA,B=1n∑i=1nμAxiμBxi+νAxiνBxiμA2xi+νA2xiμB2xi+νB2xi.The cosine similarity measure between two IFSsA and B satisfies the following properties:(1)
0≤CIFSA,B≤1(2)
CIFSA,B=CIFSB,A(3)
CIFSA,B=1 if A=BDefinition 4 (Ye [13]). Let A˜=μA˜Lxj,μA˜Uxj,νA˜Lxj,νA˜Uxj and B˜=μB˜Lxj,μB˜Uxj, νB˜Lxj,νB˜Uxj be two IVIFSs in X, the cosine similarity measure between two IVIFSs A˜ and B˜ is defined as follows:(6)CIVIFSA˜,B˜=1n∑i=1nμA˜LxiμB˜Lxi+μA˜UxiμB˜Uxi+νA˜LxiνB˜Lxi+νA˜UxiνB˜Uxi+πA˜LxiπB˜Lxi+πA˜UxiπB˜UxiμA˜L2xi+μA˜U2xi+νA˜L2xi+νA˜U2xi+πA˜L2xi+πA˜U2xi⋅H,where(7)H=μB˜L2xi+μB˜U2xi+νB˜L2xi+νB˜U2xi+πB˜L2xi+πB˜U2xi.The cosine similarity measure between two IVIFsA˜ and B˜ satisfies the following properties:(1)
0≤CIVIFSA˜,B˜≤1(2)
CIVIFSA˜,B˜=CIVIFSB˜,A˜(3)
CIVIFSA˜,B˜=1ifA˜=B˜


## 2.1. Intuitionistic Fuzzy Set

Definition 1. Let X be a fixed set, an intuitionistic fuzzy set (IFS) A in X is defined as:(1)A=xj,μAxj,νAxjxj∈X,where the functions μAxj and νAxj represent the membership degree and nonmembership degree of the element xj to the set A, respectively, such that 0≤μAxj+νAxj≤1∀xj∈X.The intuitionistic fuzzy indexπAxj=1−μAxj−νAxj, and we have 0≤πAxj≤1. For example, A=0.4,0.3 is an intuitionistic fuzzy number, and πA=0.3. The space of membership degree of IFS is shown in Figure 1.In particular, whenX has only one element, the IFS A=xj,μAxj,νAxjxj∈X is reduced to A=μAxj,νAxj, which we call it an intuitionistic fuzzy number (IFN).For any two IFSsA=xj,μAxj,νAxj and B=xj,μBxj,νBxj, the following operations are true [16]:(1)
A+B=μAxj+μBxj−μAxjμBxj,νAxjνBxj(2)
λA=1−1−μAxjλ,νAxjλ,λ>0(3)
A=BifμAxj=μBxj,νAxj=νBxjThe results of the operationsA+B and λA are still IFSs.Figure 1
The membership degree of IFS.


## 2.2. Interval-Valued Intuitionistic Fuzzy Set

Definition 2. Let X be a fixed set, an interval-valued intuitionistic fuzzy set A˜ is defined as follows:(2)A˜=xj,μA˜Lxj,μA˜Uxj,νA˜Lxj,νA˜Uxjxj∈X,where(3)μA˜Lxj≥0,νA˜Lxj≥0;0≤μA˜Uxj+νA˜Uxj≤1,∀xj∈X.The interval-valued intuitionistic fuzzy index is defined asπA˜xj=πA˜Lxj,πA˜Uxj, where(4)πA˜Lxj=1−μA˜Uxj−νA˜Uxj,πA˜Uxj=1−μA˜Lxj−νA˜Lxj.For example,A˜=0.3,0.4,0.1,0.4 is an interval-valued intuitionistic fuzzy number, the fuzzy index πA˜xj=0.2,0.6.Remark 1.
IfμA˜Lxj=μA˜Uxj,νA˜Lxj=νA˜Uxj, then the interval-valued intuitionistic fuzzy set is reduced to intuitionistic fuzzy set.
When the setX has only one element, the IVIFS A˜=xj,μA˜Lxj,μA˜Uxj,νA˜Lxj,νA˜Uxjxj∈X is reduced to A˜=μA˜Lxj,μA˜Uxj,νA˜Lxj,νA˜Uxj, which we call it an interval-valued intuitionistic fuzzy number(IVIFN).
LetA˜=xj,μA˜Lxj,μA˜Uxj,νA˜Lxj,νA˜Uxj and B˜=xj,μB˜Lxj,μB˜Uxj,νB˜Lxj,νB˜Uxj be two IVIFSs, the following operations are true [17]:(1)
A˜+B˜=μA˜Lxj+μB˜Lxj−μA˜LxjμB˜Lxj,μA˜Uxj+μB˜Uxj−μA˜UxjμB˜Uxj, νA˜LxjνB˜Lxj,νA˜UxjνB˜Uxj(2)
λA˜=1−1−μA˜Lxjλ,1−1−μA˜Uxjλ,νA˜Lxjλ,νA˜Uxjλ,λ>0(3)
A˜=B˜ if μA˜Lxj=μB˜Lxj,μA˜Uxj=μB˜Uxj,νA˜Lxj=νB˜Lxj and νA˜Uxj=νB˜Uxj


## 2.3. Cosine Similarity Measures for IFSs or IVIFSs

Definition 3 (Ye [7]). Let A=μAxj,νAxj and B=μBxj,νBxj be two IFSs in X, the cosine similarity measure between A and B is defined as follows:(5)CIFSA,B=1n∑i=1nμAxiμBxi+νAxiνBxiμA2xi+νA2xiμB2xi+νB2xi.The cosine similarity measure between two IFSsA and B satisfies the following properties:(1)
0≤CIFSA,B≤1(2)
CIFSA,B=CIFSB,A(3)
CIFSA,B=1 if A=BDefinition 4 (Ye [13]). Let A˜=μA˜Lxj,μA˜Uxj,νA˜Lxj,νA˜Uxj and B˜=μB˜Lxj,μB˜Uxj, νB˜Lxj,νB˜Uxj be two IVIFSs in X, the cosine similarity measure between two IVIFSs A˜ and B˜ is defined as follows:(6)CIVIFSA˜,B˜=1n∑i=1nμA˜LxiμB˜Lxi+μA˜UxiμB˜Uxi+νA˜LxiνB˜Lxi+νA˜UxiνB˜Uxi+πA˜LxiπB˜Lxi+πA˜UxiπB˜UxiμA˜L2xi+μA˜U2xi+νA˜L2xi+νA˜U2xi+πA˜L2xi+πA˜U2xi⋅H,where(7)H=μB˜L2xi+μB˜U2xi+νB˜L2xi+νB˜U2xi+πB˜L2xi+πB˜U2xi.The cosine similarity measure between two IVIFsA˜ and B˜ satisfies the following properties:(1)
0≤CIVIFSA˜,B˜≤1(2)
CIVIFSA˜,B˜=CIVIFSB˜,A˜(3)
CIVIFSA˜,B˜=1ifA˜=B˜


## 3. Cosine Similarity Measure with Hybrid Intuitionistic Fuzzy Information

In this section, we will propose the cosine similarity measure with hybrid intuitionistic fuzzy information (CHIFS) and some properties are also discussed.Definition 5. Let A be fuzzy set (FS) in X=x1,x2,...,xn, I and II be two subsets of the attribute set X, such that I∪II=X,I∩II=ϕ. If xj∈I, the value of fuzzy set A is characterized by IFSs, if xj∈II, the values of fuzzy set A is characterized by IVIFSs, then A is called hybrid intuitionistic fuzzy sets (HIFSs).Definition 6. Let A=xj,μAxj,νAxjxj∈X and B=xj,μBxj,νBxjxj∈X be two hybrid intuitionistic fuzzy sets, such that if the same attributes xj∈I, μAxj,νAxj, and μBxj, νBxj are IFSs, if the same attribute xj∈II, μALxj,μAUxj,νALxj,νAUxj, and μBLxj,μBUxj,νBLxj,νBUxj are IVIFSs, which we call A and B the same type hybrid intuitionistic fuzzy sets.Definition 7. Suppose A and B are the same type hybrid intuitionistic fuzzy sets, that is, if xj∈I, μAxj,νAxj, and μBxj,νBxj are IFSs, if the same attribute xj∈II, μALxj,μAUxj,νALxj,νAUxj, and μBLxj,μBUxj,νBLxj,νBUxj are IVIFSs, then the cosine similarity measure between hybrid intuitionistic fuzzy sets A and B is defined as follows:(8)CHIFSA,B=1n∑xi∈IμAxiμBxi+νAxiνBxi+πAxiπBxiμA2xi+νA2xi+πA2xiμB2xi+νB2xi+πB2xi+∑xi∈IIμALxiμBLxi+μAUxiμBUxi+νALxiνBLxi+νAUxiνBUxi+πALxiπBLxi+πAUxiπBUxiμAL2xi+μAU2xi+νAL2xi+νAU2xi+πAL2xi+πAU2xi⋅μBL2xi+μBU2xi+νBL2xi+νBU2xi+πBL2xi+πBU2xi.Remark 2.
IfI=ϕ, then CHIFS measure is reduced to CIVIFS measure.Remark 3.
IfII=ϕ, then CHIFS measure is reduced to CIFS measure.Theorem 1.
The cosine similarity measure between two hybrid intuitionistic fuzzy sets A and B satisfies the following properties:(1)
0≤CHIFSA,B≤1(2)
CHIFSA,B=CHIFSB,A(3)
CHIFSA,B=1 if A=BProof (1)
It is obvious that the property (1) is true according to the cosine value in0,1(2)
Because the multiplication of numbers satisfies the commutative law, if the positions ofA and B are exchanged in the computation of cosine measure, the result values will not change, so the property (3) is true.(3)
IfA=B,xi∈I, we have μAxi=μBxi and νAxi=νBxi.IfA=B,xi∈II, we have μALxi=μBLxi,μAUxi=μBUxi, νALxi=νBLxi, and νAUxi=νBUxi, then CHIFSA,B=1 is obvious obtained.


## 4. Multiple-Attribute Group Decision-Making with the Cosine Similarity Measure between Hybrid Intuitionstic Fuzzy Sets

In this section, we will apply theCHIFS measure between hybrid intuitionistic fuzzy sets to medical diagnosis. The CHIFS measure can be applied in many situations, such as pattern recognition, medical diagnosis, and so on. The main motivation for considering this model is that the representation of the decision information is very complex. We need several doctors correctly to evaluate the symptoms of the disease. The doctor usually provides his/her preferences for symptoms with IFSs or IVIFSs. Suppose that doctors are good at different diagnostic skills, we can obtain the weights of doctors based on the projection of individual decision on the ideal decision; then, all individual diagnosis decisions are aggregated into a collective one. At last, we apply the CHIFS measure to medical diagnosis.In a given pathology, suppose that a set of symptomsS=s1,s2,…,sn, a set of diagnoses A=A1,A2,…,Am and a set of medical experts E=e1,e2,…,et. Assume that a patient has all the symptoms, which can be represented by the hybrid intuition fuzzy set B˜, our aim is to diagnose what kind of diagnoses the patient B˜ belong to.In order to solve this problem, we first introduce some relevant concepts.Definition 8. Let A′=aijm×n=μAixj,νAixjm×n be a decision matrix, I and II be two subsets of the attribute set X=xjj=1,2,…,n, such that I∪II=X and I∩II=∅. If the attribute xj∈Ι, then the evaluation values aij are IFSs, if the attribute xj∈II, then the evaluation values aij are IVIFSs. In this case, A′ is called a hybrid intuitionistic fuzzy matrix.Definition 9. Let A′=A1,A2,…,AmT and B′=B1,B2,…,BmT be two hybrid intuitionistic fuzzy matrices, where Ai=ai1,ai2,…,ain and Bi=bi1,bi2,…,bini=1,2,…m, if they satisfy the following conditions:(1)
aij=μAixj,νAixj about xj in Ai if and only if bij=μBixj,νBixj about xj in Bi(2)
ai=μAiLxj,μAiUxj,νAiLxj,νAiUxj about xj in Ai if and only if bij=μBiLxj,μBiUxj,νBiLxj,νBiUxj about xj in Bi, then Ai and Bi are the same type vector, A′ and B′ are the same type hybrid intuitionistic fuzzy matrices.Now, suppose that a set of diagnosesA′=A1,A2,…,Am, where Ai is represented by IFS Ai=xj,μAixj,νAixj or IVIFS Ai=xj,μAiLxj,μAiUxj,νAiLxj,νAiUxj i=1,2,…,m. We should diagnose what kind of disease the patient B˜ belongs to. Furthermore, assume that the patient B˜ is represented by the same type intuitionistic fuzzy set as Ai. In the following, we will present the method for application of CHIFS measure to medical diagnosis, which involves the following steps:Step 1. Each medical expert provides his/her individual decision matrix about the relation between the diagnosis and the symptoms.Step 2. According to the expert’s diagnostic decision matrix Rk=rijkm×n, the ideal decision information should be close to the opinions of most doctors; then, we define the ideal relation R∗=rij∗m×n between the diagnosis Aii=1,2,…,m and the symptom sjj=1,2,…,n as follows:(9)(10)Ifj∈Ι,rij∗=μij∗,νij∗=1t∑k=1tμijk,1t∑k=1tνijk.(11)Ifj∈II,rij∗=μ˜ij∗,ν˜ij∗=1t∑k=1tμijLk,1t∑k=1tμijUk,1t∑k=1tνijLk,1t∑k=1tνijUk.Step 3. Medical experts may give unreasonable assessments when they encounter unfamiliar symptoms. So, it is not very reasonable to assume that each expert has equal weights. In order to obtain a reasonable evaluation, the weights of medical experts under different attributes are obtained by the projection of the individual evaluation on the ideal evaluation rij∗. The greater the weight of the expert is, the closer the evaluation value is to the ideal evaluation. The projection of each decision on the ideal decision is given by(12)ifj∈Ι,Prjrij∗rijk=μijkμij∗+νijkνij∗+πijkπij∗μij∗2+νij∗2+πij∗2,(13)ifj∈II,Prjrij∗rijk=μijLkμijL∗+μijUkμijU∗+νijLkνijL∗+νijUkνijU∗+πijLkπijL∗+πijUkπijU∗μijL∗2+μijU∗2+νijL∗2+νijU∗2+πijL∗2+πijU∗2.Then the weight of medical expert’s evaluation on different symptoms can be defined as(14)wijk=Prjrij∗rijk∑k=1tPrjrij∗rijk,k=1,2,…,t;i=1,2,…,m;j=1,2,…,n.Step 4. According to the recognition principle of maximum degree of cosine similarity measure, the process of diagnosis B˜ to Ak is derived by k=Max1≤i≤mCHIFSAi,B˜.


## 5. Numerical Example

In this section, the proposed cosine similarity measure between hybrid IFSs is applied in medical diagnosis to demonstrate its effectiveness.


### 5.1. Illustration of the Cosine Similarity Measures for Hybrid IFSs

Assume that a set of diagnosisA=A1viral  fever,A2typhoid,A3stomach  problem,A4chest  problem and a set of symptoms S=s1temperature,s2stomach  pain,s3cough,s4chest  pain. Suppose a patient has all the symptoms, which can be represented by the following hybrid intuitionistic fuzzy information (data obtained through a survey of doctors):(15)B˜=s1,0.5,0.4,s2,0.6,0.2,s3,0.5,0.6,0.2,0.3,s4,0.4,0.2.There are three medical experts evaluate each diagnosis with all the symptoms, which are represented by the hybrid IFSs, the results are shown in Tables1–3.Table 1
The relation between the diagnosis and the symptoms—expert 1.
s
1
s
2
s
3
s
4
A
1
0.5
,
0.4
0.5
,
0.3
0.4
,
0.6
,
0.1
,
0.3
0.4
,
0.4
A
2
0.7
,
0.3
0.7
,
0.2
0.3
,
0.5
,
0.4
,
0.5
0.6
,
0.2
A
3
0.8
,
0.1
0.6
,
0.4
0.6
,
0.7
,
0.2
,
0.3
0.6
,
0.3
A
4
0.7
,
0.2
0.5
,
0.2
0.5
,
0.7
,
0.1
,
0.2
0.5
,
0.3Table 2
The relation between the diagnosis and the symptoms—expert 2.
s
1
s
2
s
3
s
4
A
1
0.4
,
0.5
0.6
,
0.2
0.5
,
0.6
,
0.2
,
0.3
0.3
,
0.4
A
2
0.5
,
0.2
0.7
,
0.2
0.4
,
0.7
,
0.1
,
0.3
0.7
,
0.1
A
3
0.6
,
0.2
0.5
,
0.1
0.5
,
0.7
,
0.1
,
0.2
0.6
,
0.2
A
4
0.7
,
0.1
0.4
,
0.3
0.3
,
0.6
,
0.2
,
0.4
0.4
,
0.3Table 3
The relation between the diagnosis and the symptoms—expert 3.
s
1
s
2
s
3
s
4
A
1
0.5
,
0.3
0.6
,
0.2
0.4
,
0.6
,
0.2
,
0.3
0.5
,
0.4
A
2
0.7
,
0.2
0.4
,
0.4
0.5
,
0.7
,
0.1
,
0.3
0.6
,
0.3
A
3
0.6
,
0.3
0.7
,
0.3
0.6
,
0.8
,
0.1
,
0.2
0.7
,
0.2
A
4
0.5
,
0.2
0.5
,
0.3
0.3
,
0.6
,
0.1
,
0.4
0.6
,
0.1
According to step 2 in Section4, applying (10) and (11), respectively, the ideal relation between the diagnosis and the symptoms are shown in Table 4.Table 4
The ideal relation between the diagnosis and the symptoms.
s
1
s
2
s
3
s
4
A
1
∗
0.467
,
0.4
0.567
,
0.233
0.433
,
0.6
,
0.167
,
0.3
0.4
,
0.4
A
2
∗
0.633
,
0.233
0.6
,
0.267
0.4
,
0.633
,
0.2
,
0.367
0.633
,
0.2
A
3
∗
0.667
,
0.2
0.6
,
0.267
0.567
,
0.733
,
0.133
,
0.233
0.633
,
0.233
A
4
∗
0.633
,
0.167
0.467
,
0.267
0.367
,
0.633
,
0.133
,
0.333
0.5
,
0.233By step 3 in Section4, applying (12)–(14), we can calculate the weights of each medical expert for the diagnosis with respect to different symptoms, which are obtained in Tables 5–7.Table 5
The weights of the medical expert 1 forAi with respect to sj.
s
1
s
2
s
3
s
4
A
1
0.3427
0.3155
0.3391
0.3333
A
2
0.3614
0.3614
0.3052
0.3223
A
3
0.3761
0.3466
0.3293
0.3263
A
4
0.3458
0.3395
0.3314
0.3313Table 6
The weights of the medical expert 2 forAi with respect to sj.
s
1
s
2
s
3
s
4
A
1
0.3371
0.3423
0.3311
0.3148
A
2
0.2841
0.3614
0.3474
0.3531
A
3
0.3097
0.2821
0.3196
0.3193
A
4
0.3594
0.3210
0.3276
0.3106Table 7
The weights of the medical expert 3 forAi with respect to sj.
s
1
s
2
s
3
s
4
A
1
0.3202
0.3422
0.3298
0.3519
A
2
0.3545
0.2772
0.3474
0.3246
A
3
0.3142
0.3713
0.3511
0.3544
A
4
0.2948
0.3395
0.3410
0.3581
When the weight values of the experts are determined, the aggregated evaluating decision results provided by different experts are obtained in Table8.Table 8
The aggregated relation between the diagnosis and the symptom.
s
1
s
2
s
3
s
4
A
˜
1
0.4683
,
0.3933
0.5708
,
0.2273
0.4351
,
0.6
,
0.1581
,
0.3
0.4093
,
0.4
A
˜
2
0.5316
,
0.2316
0.6364
,
0.2424
0.4097
,
0.6494
,
0.1527
,
0.3506
0.6386
,
0.1786
A
˜
3
0.6918
,
0.175
0.6172
,
0.2431
0.5704
,
0.7398
,
0.1256
,
0.2286
0.6388
,
0.2283
A
˜
4
0.6513
,
0.1559
0.4699
,
0.2614
0.3739
,
0.6364
,
0.1255
,
0.3179
0.5115
,
0.2024From the previous formulaCHIFSA′,B, we can calculate the cosine similarity measure between A˜ii=1,2,3,4 and B˜ as follows:(16)CHIFSA˜1,B˜=0.9674,CHIFSA˜2,B˜=0.9477,CHIFSA˜3,B˜=0.9140,CHIFSA˜4,B˜=0.9511.We can conclude that the diagnosis of the patientB˜ is viral fever (A1).


### 5.2. Comparison Analysis

In this subsection, the existing cosine similarity measure is used to compare with the same numerical example. In the numerical example, the decision information is represented with hybrid IFS, we can transform it into a unified form. For example, the relation between the diagnosis and the symptoms under the attributes3 of experts is IVIFSs, and if we use the cosine similarity measure CIFS proposed by Ye [7] to calculate the numerical example, we should convert the corresponding IVIFSs to IFS according to the midpoints of IVIFSs. For example, 0.4,0.6,0.1,0.3 can be converted to (0.5). Then using the cosine similarity measure CIFS proposed by Ye [7], we can obtain the corresponding cosine similarity measure values: CIFSA1,B˜=0.9691, CIFSA2,B˜=0.9546, CIFSA3,B˜=0.9377, and CIFSA4,B˜=0.9586. That is to say, the diagnosis of the patient B˜ is still the viral fever A1. The proposed cosine similarity between hybrid IFS in this paper produces the same results as the existing methods. This means that the proposed method is feasible and effective, and it has some advantages in solving multiple criteria decision-making problems. On one hand, the method is more convenient to make decision for decision makers, who can express their preferences over the decision information by IFS or IVIFS simultaneously. On the other hand, because the information conversion will be lost in decision-making process, there are no information conversions between IFSs and IVIFSs in this model, the alternatives will be ranked directly based on the original decision information.


## 5.1. Illustration of the Cosine Similarity Measures for Hybrid IFSs

Assume that a set of diagnosisA=A1viral  fever,A2typhoid,A3stomach  problem,A4chest  problem and a set of symptoms S=s1temperature,s2stomach  pain,s3cough,s4chest  pain. Suppose a patient has all the symptoms, which can be represented by the following hybrid intuitionistic fuzzy information (data obtained through a survey of doctors):(15)B˜=s1,0.5,0.4,s2,0.6,0.2,s3,0.5,0.6,0.2,0.3,s4,0.4,0.2.There are three medical experts evaluate each diagnosis with all the symptoms, which are represented by the hybrid IFSs, the results are shown in Tables1–3.Table 1
The relation between the diagnosis and the symptoms—expert 1.
s
1
s
2
s
3
s
4
A
1
0.5
,
0.4
0.5
,
0.3
0.4
,
0.6
,
0.1
,
0.3
0.4
,
0.4
A
2
0.7
,
0.3
0.7
,
0.2
0.3
,
0.5
,
0.4
,
0.5
0.6
,
0.2
A
3
0.8
,
0.1
0.6
,
0.4
0.6
,
0.7
,
0.2
,
0.3
0.6
,
0.3
A
4
0.7
,
0.2
0.5
,
0.2
0.5
,
0.7
,
0.1
,
0.2
0.5
,
0.3Table 2
The relation between the diagnosis and the symptoms—expert 2.
s
1
s
2
s
3
s
4
A
1
0.4
,
0.5
0.6
,
0.2
0.5
,
0.6
,
0.2
,
0.3
0.3
,
0.4
A
2
0.5
,
0.2
0.7
,
0.2
0.4
,
0.7
,
0.1
,
0.3
0.7
,
0.1
A
3
0.6
,
0.2
0.5
,
0.1
0.5
,
0.7
,
0.1
,
0.2
0.6
,
0.2
A
4
0.7
,
0.1
0.4
,
0.3
0.3
,
0.6
,
0.2
,
0.4
0.4
,
0.3Table 3
The relation between the diagnosis and the symptoms—expert 3.
s
1
s
2
s
3
s
4
A
1
0.5
,
0.3
0.6
,
0.2
0.4
,
0.6
,
0.2
,
0.3
0.5
,
0.4
A
2
0.7
,
0.2
0.4
,
0.4
0.5
,
0.7
,
0.1
,
0.3
0.6
,
0.3
A
3
0.6
,
0.3
0.7
,
0.3
0.6
,
0.8
,
0.1
,
0.2
0.7
,
0.2
A
4
0.5
,
0.2
0.5
,
0.3
0.3
,
0.6
,
0.1
,
0.4
0.6
,
0.1
According to step 2 in Section4, applying (10) and (11), respectively, the ideal relation between the diagnosis and the symptoms are shown in Table 4.Table 4
The ideal relation between the diagnosis and the symptoms.
s
1
s
2
s
3
s
4
A
1
∗
0.467
,
0.4
0.567
,
0.233
0.433
,
0.6
,
0.167
,
0.3
0.4
,
0.4
A
2
∗
0.633
,
0.233
0.6
,
0.267
0.4
,
0.633
,
0.2
,
0.367
0.633
,
0.2
A
3
∗
0.667
,
0.2
0.6
,
0.267
0.567
,
0.733
,
0.133
,
0.233
0.633
,
0.233
A
4
∗
0.633
,
0.167
0.467
,
0.267
0.367
,
0.633
,
0.133
,
0.333
0.5
,
0.233By step 3 in Section4, applying (12)–(14), we can calculate the weights of each medical expert for the diagnosis with respect to different symptoms, which are obtained in Tables 5–7.Table 5
The weights of the medical expert 1 forAi with respect to sj.
s
1
s
2
s
3
s
4
A
1
0.3427
0.3155
0.3391
0.3333
A
2
0.3614
0.3614
0.3052
0.3223
A
3
0.3761
0.3466
0.3293
0.3263
A
4
0.3458
0.3395
0.3314
0.3313Table 6
The weights of the medical expert 2 forAi with respect to sj.
s
1
s
2
s
3
s
4
A
1
0.3371
0.3423
0.3311
0.3148
A
2
0.2841
0.3614
0.3474
0.3531
A
3
0.3097
0.2821
0.3196
0.3193
A
4
0.3594
0.3210
0.3276
0.3106Table 7
The weights of the medical expert 3 forAi with respect to sj.
s
1
s
2
s
3
s
4
A
1
0.3202
0.3422
0.3298
0.3519
A
2
0.3545
0.2772
0.3474
0.3246
A
3
0.3142
0.3713
0.3511
0.3544
A
4
0.2948
0.3395
0.3410
0.3581
When the weight values of the experts are determined, the aggregated evaluating decision results provided by different experts are obtained in Table8.Table 8
The aggregated relation between the diagnosis and the symptom.
s
1
s
2
s
3
s
4
A
˜
1
0.4683
,
0.3933
0.5708
,
0.2273
0.4351
,
0.6
,
0.1581
,
0.3
0.4093
,
0.4
A
˜
2
0.5316
,
0.2316
0.6364
,
0.2424
0.4097
,
0.6494
,
0.1527
,
0.3506
0.6386
,
0.1786
A
˜
3
0.6918
,
0.175
0.6172
,
0.2431
0.5704
,
0.7398
,
0.1256
,
0.2286
0.6388
,
0.2283
A
˜
4
0.6513
,
0.1559
0.4699
,
0.2614
0.3739
,
0.6364
,
0.1255
,
0.3179
0.5115
,
0.2024From the previous formulaCHIFSA′,B, we can calculate the cosine similarity measure between A˜ii=1,2,3,4 and B˜ as follows:(16)CHIFSA˜1,B˜=0.9674,CHIFSA˜2,B˜=0.9477,CHIFSA˜3,B˜=0.9140,CHIFSA˜4,B˜=0.9511.We can conclude that the diagnosis of the patientB˜ is viral fever (A1).


## 5.2. Comparison Analysis

In this subsection, the existing cosine similarity measure is used to compare with the same numerical example. In the numerical example, the decision information is represented with hybrid IFS, we can transform it into a unified form. For example, the relation between the diagnosis and the symptoms under the attributes3 of experts is IVIFSs, and if we use the cosine similarity measure CIFS proposed by Ye [7] to calculate the numerical example, we should convert the corresponding IVIFSs to IFS according to the midpoints of IVIFSs. For example, 0.4,0.6,0.1,0.3 can be converted to (0.5). Then using the cosine similarity measure CIFS proposed by Ye [7], we can obtain the corresponding cosine similarity measure values: CIFSA1,B˜=0.9691, CIFSA2,B˜=0.9546, CIFSA3,B˜=0.9377, and CIFSA4,B˜=0.9586. That is to say, the diagnosis of the patient B˜ is still the viral fever A1. The proposed cosine similarity between hybrid IFS in this paper produces the same results as the existing methods. This means that the proposed method is feasible and effective, and it has some advantages in solving multiple criteria decision-making problems. On one hand, the method is more convenient to make decision for decision makers, who can express their preferences over the decision information by IFS or IVIFS simultaneously. On the other hand, because the information conversion will be lost in decision-making process, there are no information conversions between IFSs and IVIFSs in this model, the alternatives will be ranked directly based on the original decision information.


## 6. Conclusion

The paper proposed the cosine similarity measure between hybrid intuitionistic fuzzy sets, and the proposed method would be quite good for some real-world applications, such as pattern recognition and medical diagnosis. Through the proposed cosine similarity measure, we can classify the patientB˜ in one of the diagnosis A1,A2,…Am. Finally, a numerical example illustrated the application and efficiency of the developed approach, which is also compared to the existing methods. In future research, we expect to develop further extensions of the CHIFS measure by adding the new characteristic, such as ordered weighted averaging operator, and we will also consider other applications of the proposed CHIFS measure.



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*Source: 3146873-2018-10-17.xml*