DSpace at VNU: Intuitionistic fuzzy recommender systems: An effective tool for medical diagnosis

Intuitionistic fuzzy recommender systems: An effective tool for medicaldiagnosis VNU University of Science, Vietnam National University, Viet Nam Article history: Received 12 May 2014 Re

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Intuitionistic fuzzy recommender systems: An effective tool for medical

diagnosis

VNU University of Science, Vietnam National University, Viet Nam

Article history:

Received 12 May 2014

Received in revised form 6 October 2014

Accepted 10 November 2014

Available online 20 November 2014

Keywords:

Accuracy

Fuzzy sets

Intuitionistic fuzzy collaborative ﬁltering

Intuitionistic fuzzy recommender systems

Medical diagnosis

a b s t r a c t

Medical diagnosis has been being considered as one of the important processes in clinical medicine that determines acquired diseases from some given symptoms Enhancing the accuracy of diagnosis is the centralized focuses of researchers involving the uses of computerized techniques such as intuitionistic fuzzy sets (IFS) and recommender systems (RS) Based upon the observation that medical data are often imprecise, incomplete and vague so that using the standalone IFS and RS methods may not improve the accuracy of diagnosis, in this paper we consider the integration of IFS and RS into the proposed method-ology and present a novel intuitionistic fuzzy recommender systems (IFRS) including: (i) new deﬁnitions

of single-criterion and multi-criteria IFRS; (ii) new deﬁnitions of intuitionistic fuzzy matrix (IFM) and intuitionistic fuzzy composition matrix (IFCM); (iii) proposing intuitionistic fuzzy similarity matrix (IFSM), intuitionistic fuzzy similarity degree (IFSD) and the formulas to predict values on the basis of IFSD; (iv) a novel intuitionistic fuzzy collaborative ﬁltering method so-called IFCF to predict the possible diseases Experimental results reveal that IFCF obtains better accuracy than the standalone methods of IFS such as De et al., Szmidt and Kacprzyk, Samuel and Balamurugan and RS, e.g Davis et al and Hassan and Syed

1 Introduction

In this section, we formulate the medical diagnosis problem and

give some illustrated examples in Section1.1 Section1.2describes

the relevant works using the intuitionistic fuzzy sets for the

med-ical diagnosis problem Section1.3summarizes the limitations of

those relevant works, and based on these facts the motivation

and ideas of the proposed approach are highlighted in Section1.4

Section 1.5 demonstrates our contributions in details, and their

novelty and signiﬁcance are discussed in Section 1.6 Lastly,

Section1.7elaborates the organization of the paper

1.1 The medical diagnosis problem

Medical diagnosis has been being considered as one of the most

important and necessary processes in clinical medicine that

determines acquired diseases of patients from given symptoms

According to Kononenko[20], diagnosis commonly relates to the

probability or risk of an individual developing a particular state

of health over a speciﬁc time, based on his or her clinical and

non-clinical proﬁle It is useful to minimize the risk of associated health complications such as osteoporosis, small bowel cancer and increased risk of other autoimmune diseases Mathematically, its deﬁnition is stated as follows

Deﬁnition 1 (Medical diagnosis) Given three lists: P = {P1, , Pn},

S = {S1, , Sm} and D = {D1, , Dk} where P is a list of patients, S a list of symptoms and D a list of diseases, respectively Three values

n, m, k 2 N+ are the numbers of patients, symptoms and diseases, respectively The relation between the patients and the symptoms is characterized by the set- RPS¼ fRPSðPi;SjÞj8i ¼ 1; ; n;8j ¼ 1; ; mg where RPS(Pi, Sj) shows the level that patient

Piacquires symptom Sj and is represented by either a numeric value or a (intuitionistic) fuzzy value depending on the domain of the problem Analogously, the relation between the symptoms and the diseases is expressed as RSD¼ fRSDðSi;DjÞj8i ¼ 1; ; m;8j ¼ 1; ; kg where RSD(Si, Dj) reﬂects the possibility that symptom Si would lead to disease Dj The medical diagnosis problem aims to determine the relation between the patients and the diseases described by the set- RPD¼ fRPDðPi;DjÞj8i ¼ 1; ; n;8j ¼ 1; ; kg where RPD(Pi, Dj) is either 0 or 1 showing that patient Piacquires disease Djor not The medical diagnosis problem can be shortly represented by the implication fRPS;RSDg ! RPD

http://dx.doi.org/10.1016/j.knosys.2014.11.012

⇑ Corresponding author at: 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam Tel.:

+84 904171284; fax: +84 0438623938.

E-mail addresses: sonlh@vnu.edu.vn , chinhson2002@gmail.com (L.H Son).

Contents lists available atScienceDirect

Knowledge-Based Systems

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / k n o s y s

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Example 1 Consider the dataset in [31] having four patients

namely P = {Ram, Mari, Sugu, Somu}, ﬁve symptoms S =

{Tempera-ture, Headache, Stomach-pain, Cough, Chest-pain} and ﬁve

dis-eases D = {Viral-Fever, Malaria, Typhoid, Stomach, Heart} The

relations between the patients – the symptoms and the symptoms

– the diseases are illustrated inTables 1 and 2, respectively

The relation between the patients and the diseases determined

by the medical diagnosis is illustrated inTable 3 Since the domain

of the problem is the intuitionistic fuzzy values, this relation is also

expressed in this form The most acquiring disease that the

patients suffer is expressed in Table 4, which is converted from

Table 3by a trivial defuzziﬁcation method considering the

maxi-mal membership degree of disease among all

Medical diagnosis is considered as an efﬁcient support tool for

clinicians to make the right therapeutical decisions especially in

the cases that medicine extends its predictive capacities using

genetic data[5] As being observed inTable 3, medical diagnosis

could assist the clinicians to enumerate the possible diseases of

patients accompanied with certain membership values Thus, it is

convenient for clinicians, who are experts in this ﬁeld, to quickly

diagnose and give proper medicated ﬁgures This fact clearly shows

the importance of medical diagnosis in medicine sciences

nowadays

1.2 The previous works Computerized techniques for medical diagnosis such as fuzzy set, genetic algorithms, neural networks, statistical tools and rec-ommender systems aiming to enhance the accuracy of diagnosis have been being introduced widely[20] Nonetheless, an impor-tant issue in medical diagnosis is that the relations between the patients – the symptoms (RPS) and the symptoms – the diseases (RSD) are often vague, imprecise and uncertain For instance, doc-tors could faced with patients who are likely to have personal problems and/or mental disorders so that the crucial patients’ signs and symptoms are missing, incomplete and vague even though the supports of patients’ medical histories and physical examination are provided within the diagnosis Even if information

of patients are clearly provided, how to give accurate evaluation to given symptoms/diseases is another challenge requiring well-trained, copious-experienced physicians These evidences raise the need of using fuzzy set or its extension to model and assist the techniques that improve the accuracy of diagnosis The deﬁni-tion of fuzzy set is stated below

Deﬁnition 2 A Fuzzy Set (FS) [49] in a non-empty set X is a function

l:X ! ½0; 1

wherel(x) is the membership degree of each element x 2 X A fuzzy set can be alternately deﬁned as,

An extension of FS that is widely applied to the medical progno-sis problem is Intuitionistic Fuzzy Set (IFS), which is deﬁned as follows

Deﬁnition 3 An Intuitionistic Fuzzy Set (IFS)[4]in a non-empty set

X is,

eA ¼ Dx;leAðxÞ;ceAðxÞE

jx 2 X

wherele

AðxÞ andceAðxÞ are the membership and non-membership degrees of each element x 2 X, respectively

leAðxÞ;ceAðxÞ 2 ½0; 1; 8x 2 X; ð4Þ

0 6leAðxÞ þceAðxÞ 6 1; 8x 2 X: ð5Þ

The intuitionistic fuzzy index of an element showing the non-deter-minacy is denoted as,

peAðxÞ ¼ 1 leAðxÞ þceAðxÞ; 8x 2 X: ð6Þ

Whenpe

AðxÞ ¼ 0 for "x 2 X, IFS returns to the FS set of Zadeh Some extensions of fuzzy sets are not appropriate for modeling uncertainty in the medical diagnosis such as the rough set[28], rough soft sets[11,12,16], intuitionistic fuzzy rough sets[50]and soft rough fuzzy sets & soft fuzzy rough sets[23] The limitations

of these sets, as pointed out by Yao[48], Rodriguez et al.[30], Jaf-arian and Rezvani[17]and many other authors lie to their intrinsic nature and how they are organized and operated such as (i) The positive and the boundary rules are considered in rough sets and their variants so that in cases of many concepts, the negative rules would be redundant; (ii) The modeling of linguistic information is limited due to the elicitation of single and simple terms that should encompass and express the information provided by the experts regarding the a linguistic variable; (iii) if exact membership degrees cannot be determined due to insufﬁcient information then

it is impossible to consider the uncertainty on the membership

Table 3

The relation between the patients and the diseases – R PD expressed by intuitionistic

fuzzy values.

P Viral_fever Malaria Typhoid Stomach Heart

Ram (0.4, 0.1) (0.7, 0.1) (0.6, 0.1) (0.2, 0.4) (0.2, 0.6)

Mari (0.3, 0.5) (0.2, 0.6) (0.4, 0.4) (0.6, 0.1) (0.1, 0.7)

Sugu (0.4, 0.1) (0.7, 0.1) (0.6, 0.1) (0.2, 0.4) (0.2, 0.5)

Somu (0.4, 0.1) (0.7, 0.1) (0.5, 0.3) (0.3, 0.4) (0.3, 0.4)

Table 4

The most acquiring diseases of patients.

Table 1

The relation between the patients and the symptoms – R PS

P Temperature Headache Stomach_pain Cough Chest_pain

Ram (0.8, 0.1) (0.6, 0.1) (0.2, 0.8) (0.6, 0.1) (0.1, 0.6)

Mari (0, 0.8) (0.4, 0.4) (0.6, 0.1) (0.1, 0.7) (0.1, 0.8)

Sugu (0.8, 0.1) (0.8, 0.1) (0, 0.6) (0.2, 0.7) (0, 0.5)

Somu (0.6, 0.1) (0.5, 0.4) (0.3, 0.4) (0.7, 0.2) (0.3, 0.4)

Table 2

The relation between the symptoms and the diseases – R SD

S Viral_fever Malaria Typhoid Stomach Heart

Temperature (0.4, 0) (0.7, 0) (0.3, 0.3) (0.1, 0.7) (0.1, 0.8)

Headache (0.3, 0.5) (0.2, 0.6) (0.6, 0.1) (0.2, 0.4) (0, 0.8)

Stomach_pain (0.1, 0.7) (0, 0.9) (0.2, 0.7) (0.8, 0) (0.2, 0.8)

Cough (0.4, 0.3) (0.7, 0) (0.2, 0.6) (0.2, 0.7) (0.2, 0.8)

Chest_pain (0.1, 0.7) (0.1, 0.8) (0.1, 0.9) (0.2, 0.7) (0.8, 0.1)

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function Thus, these types of fuzzy sets could not be used for the

application of medical diagnosis

The ﬁrst approach for the medical diagnosis problem was

drawn from the Sanchez’s notion of medical knowledge[32] Since

then several improvements of the Sanchez’s approach in

associa-tion with IFS and other advanced fuzzy sets have been introduced

De et al [9] fuzziﬁed the relations between the patients – the

symptoms and the symptoms – the diseases by intuitionistic fuzzy

memberships and derived the relation between the patients and

the diseases by means of intuitionistic fuzzy relations The

algo-rithm contains the following steps

1 Calculate the relation between the patients and the diseases

by intuitionistic fuzzy relations with the membership and

non-membership functions being expressed in Eqs.(7) and

(8), respectively

lPDðPi;DjÞ ¼ max

l¼1;m

minflPSðPi;SlÞ;lSDðSl;DjÞg

;

8i 2 f1; ; ng; 8j 2 f1; ; kg; ð7Þ

cPDðPi;DjÞ ¼ min

l¼1;m

maxfcPSðPi;SlÞ;cSDðSl;DjÞg

;

8i 2 f1; ; ng; 8j 2 f1; ; kg: ð8Þ

2 Perform the defuzziﬁcation through the SPD,

3 Determine the most acquiring diseases of patients based on

the maximal SPDand minimalpPD

Example 2 Consider the dataset in Example 1 The relation

between the patients and the diseases calculated by Eqs.(7) and

(8)is expressed inTable 5 The SPDmatrix is described inTable 6

Based upon this table, Ram, Sugu and Somu suffer from the Malaria

and Mari acquires Stomach the most

Samuel and Balamurugan[31]improved the method of De et al

[9]by a new technique named intuitionistic fuzzy max–min

com-position This method is analogous to that of De et al.[9]except

that Steps 2 & 3 are replaced by,

1 Compute WPD¼ ðlPD;1 cPDÞ

2 For each Pi ﬁnd maxjfminðlPDðPi;DjÞ; 1 cPDðPi;DjÞÞg and

conclude the most acquiring diseases

Example 3 Consider again the dataset in Example 1 The WPD

matrix is shown inTable 7 The reduction of WPDis presented in

Table 8 From this table, Ram, Sugu and Somu suffer from the

Malaria and Mari acquires Stomach the most

Another approach for the medical diagnosis is utilizing the

distance functions to calculate the relation between the patients

and the diseases from the relations between the patients – the

symptoms and the symptoms – the diseases as described in

[42–44,19,33] The general activities of these algorithms are,

1 Use the Hamming or Euclidean function to calculate the relation between the patients and the diseases as in Eqs.(10) and (11), respectively

RPD

ðPi;DjÞ ¼ 1

2m

Xm l¼1

lPSðPi;SlÞ lSDðSl;DjÞ

þjcPSðPi;SlÞ

cSDðSl;DjÞ

þ jpPSðPi;SlÞ pSDðSl;DjÞj

; ð10Þ

RPD

ðPi;DjÞ ¼ 1

2m

Xm l¼1

lPSðPi;SlÞ lSDðSl;DjÞ

þcPSðPi;SlÞ cSDðSl;DjÞ2

þðpPSðPi;SlÞ pSDðSl;DjÞÞ21=2

2 Conclude the possible diseases of patients based on the minimal distance criterion

Example 4 Use this method for the dataset inExample 1, we have the relations between the patients and the diseases by the Ham-ming (Table 9) or Euclidean function (Table 10) The most acquir-ing diseases of patients are highlighted in bold

Besides these approaches, some authors have extended them for special cases, e.g multi-criteria medical diagnosis and the mul-tiple time intervals modeling for the relation between the patients and the symptoms This requires the deployment on other advanced fuzzy sets such as the type-2 fuzzy sets[26], the inter-val-valued intuitionistic fuzzy sets[2], fuzzy soft set [25,47]and intuitionistic fuzzy soft set[1,21] The combination of these fuzzy sets with machine learning methods to handle the special cases such as the fuzzy-neural automatic system[27,24]and the

type-2 fuzzy genetic algorithm[45,14]was also investigated

Table 5

The relation between the patients and the diseases – R PD in the method of De et al [9]

expressed by intuitionistic fuzzy values.

Ram (0.4, 0.1) (0.7, 0.1) (0.6, 0.1) (0.2, 0.4) (0.2, 0.6)

Mari (0.3, 0.5) (0.2, 0.6) (0.4, 0.4) (0.6, 0.1) (0.2, 0.5)

Sugu (0.4, 0.1) (0.7, 0.1) (0.6, 0.1) (0.2, 0.4) (0.2, 0.5)

Somu (0.4, 0.1) (0.7, 0.1) (0.5, 0.3) (0.3, 0.4) (0.3, 0.4)

Table 6 The S PD matrix where bold values imply the most possible disease.

Table 7 The W PD matrix.

P Viral_fever Malaria Typhoid Stomach Heart Ram (0.4, 0.9) (0.7, 0.9) (0.6, 0.9) (0.2, 0.6) (0.2, 0.4) Mari (0.3, 0.5) (0.2, 0.4) (0.4, 0.6) (0.6, 0.9) (0.2, 0.5) Sugu (0.4, 0.9) (0.7, 0.9) (0.6, 0.9) (0.2, 0.6) (0.2, 0.5) Somu (0.4, 0.9) (0.7, 0.9) (0.5, 0.7) (0.3, 0.6) (0.3, 0.6)

Table 8 The reduction matrix where bold values imply the most possible disease.

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1.3 The limitations of the previous works

Considering the relevant works involving the usage of the IFS

set, we clearly recognize that IFS was used mainly for the

applica-tions of medical diagnosis among the advanced fuzzy sets

None-theless, these works have the following disadvantages

(a) The previous works calculate the relation between the

patients and the diseases (RPD) solely from those between

the patients – the symptoms (RPS) and the symptoms –

the diseases (RSD) In some practical cases where the relation

between the patients – the symptoms or the symptoms – the

diseases is missing, those works could not be performed

This fact is happened in reality since clinicians somehow

do not accurately express the values of membership and

non-membership degrees of symptoms to diseases or vive

versa;

(b) The information of previous diagnoses of patients could not

be utilized That is to say, a patient has had some records in

the patients-diseases databases (RPD) beforehand

Neverthe-less, the calculation of the next records of this patient is

made solely on the basis of both RPSand RSD Historic

diagno-ses of patients are not taken into account so that the

accu-racy of diagnosis may not be high as a result;

(c) The determination of the most acquiring disease is

depen-dent from the defuzziﬁcation method For instance, De

et al.[9]used SPDfor the defuzziﬁcation, Samuel and

Bala-murugan[31]relied on the reduction matrix from WPDand

Szmidt and Kacprzyk [42–44], Khatibi and Montazer[19]

and Shinoj and John[33]employed the distance functions

Independent determination from the defuzziﬁcation method

should be investigated for the stable performance of the

algorithm

(d) Mathematical properties of operations such as the fuzzy

implication in De et al.[9], Samuel and Balamurugan[31]

and the distance function in Szmidt and Kacprzyk[42–44],

Khatibi and Montazer[19]and Shinoj and John[33]were

not discussed in the equivalent articles Readers could not

know the theoretical bases of these operations and why they

were selected for the medical diagnosis problem

1.4 The motivation and ideas

From the disadvantages of the previous works, our idea in this

article is using the hybrid method between Recommender Systems

(RS) and the IFS set to handle them RS, which are a subclass of

decision support systems, can give users information about predic-tive ‘‘rating’’ or ‘‘preference’’ that they would like to assess an item; thus helping them to choose the appropriate item among numer-ous possibilities This kind of expert systems is now commonly popularized in numerous application fields such as books, docu-ments, images, movie, music, shopping and TV programs personal-ized systems The mathematical definition of RS is stated below Definition 4 (Recommender Systems – RS[29]) Suppose U is a set

of all users and I is the set of items in the system The utility function R is a mapping speciﬁed on U1 U and I1 I as follows

R : U1 I1! P

where R(u1, i1) is a non-negative integer or a real number within a certain range P is a set of available ratings in the system Thus, RS is the system that provides two basic functions below

(a) Prediction: determine Rðu;iÞ for any ðu;iÞ 2 ðU; IÞ n ðU1;I1Þ (b) Recommendation: choose i⁄

2 I satisfying i⁄

= arg maxi2IR(u, i) for all u 2 U

RS has been applied to the medical diagnosis problem Davis et al

[8]proposed CARE, a Collaborative Assessment and Recommenda-tion Engine, which relies only on a patient’s medical history in order

to predict future diseases risks and combines collaborative ﬁltering methods with clustering to predict each patient’s greatest disease risks based on their own medical history and that of similar patients

An iterative version of CARE so-called ICARE that incorporates ensemble concepts for improved performance was also introduced These systems required no specialized information and provided predictions for medical conditions of all kinds in a single run Hassan and Syed[13]employed a collaborative ﬁltering framework that assessed patient risk both by matching new cases to historical records and by matching patient demographics to adverse outcomes

so that it could achieve a higher predictive accuracy for both sudden cardiac death and recurrent myocardial infraction than popular classiﬁcation approaches such as logistic regression and support vector machines More works on the applications of RS could be ref-erenced in Duan et al.[10], Meisamshabanpoor and Mahdavi[22]

and our previous works in[7,38,40,39,41,34–37] Example 5 Consider the training dataset in Table 11 Taking a simple encoded method by multiplying the membership degree by

10 and adding the non-membership degree to it, we have a crisp training inTable 12

The method of Hassan and Syed[13]employed a collaborative ﬁltering including the traditional Pearson coefﬁcient to calculate the similarity between users and the k-nearest neighbor approxi-mation function to predict the blank values inTable 12 The results are shown inTable 13 If taking the maximal value among all for a given patient inTable 13then we can conclude that Ram, Sugu and Somu are suffered from Malaria and Mari acquires Stomach Analogously,Table 14shows the results of the method of Davis

Table 9

The relation between the patients and the diseases by the Hamming function where

bold values imply the most possible disease.

Table 10

The relation between the patients and the diseases by the Euclidean function where

bold values imply the most possible disease.

Table 11 The training dataset with ⁄ being the values to be predicted.

P Viral_fever Malaria Typhoid Stomach Heart Ram (0.4, 0.1) (0.7, 0.1) (0.6, 0.1) (0.2, 0.4) (0.2, 0.6) Mari (0.3, 0.5) (0.2, 0.6) (0.4, 0.4) (0.6, 0.1) (0.1, 0.7)

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et al.[8]where Ram is suffered from Malaria, Mari acquires

Stom-ach and Sugu and Somu have Typhoid

FromExample 5, we clearly recognize the following facts:

(a) RS could be applied to the medical diagnosis Yet in cases

that the relations are expressed by fuzzy memberships as

inTable 11, the accuracy of diagnosis in RS is dependent

on the encoded method In the other words, RS is effective

with the crisp dataset such asTable 12 but not the fuzzy

one, e.g.Table 11;

(b) The problem of the previous researches about the

depen-dence of the determination of the most acquiring disease

from the defuzziﬁcation method, e.g the maximal function

inExample 5still exists;

(c) RS works only if the training dataset is provided That is to

say, we must have the historic diagnoses of patients for

the prediction

From Sections1.3 and 1.4and illustrated examples, we clearly

recognize that the IFS and RS approaches have their own advantages

and disadvantages Thus, a combination of these approaches in order

to combine the advantages and eliminate the disadvantages could

handle the mentioned issues Scanning the literature, we realize that

some hybrid methods were also designed for the medical diagnosis

problem, to name but a few such as Davis et al.[8]combined

collab-orative ﬁltering methods with clustering; Kala et al.[18]integrated

genetic algorithms with modular neural network; Hosseini et al

[14]joined a type-2 fuzzy logic with genetic algorithm These

evi-dences show that the combination of groups of methods such as

between RS and IFS is a trendy approach for medical diagnosis

1.5 The contributions of this work

Based upon the observations, our contribution in this paper is a

novel intuitionistic fuzzy recommender system (IFRS) for medical

diagnosis consisting of the following components:

(a) The new deﬁnitions of single-criterion IFRS (SC-IFRS) and multi-criteria IFRS (MC-IFRS) that extend the deﬁnition of

RS (Definition 4) taking into account a feature of a user and a characteristic of an item expressed by intuitionistic linguistic labels (See Section2.1) These definitions are the basis for the deployment of similarity degrees used for the prediction of RPD(Pi, Dj) (Definition 1);

(b) The new deﬁnitions of intuitionistic fuzzy matrix (IFM), which is a representation of SC-IFRS and MC-IFRS in the matrix format and the intuitionistic fuzzy composition matrix (IFCM) of two IFMs with the intersection/union oper-ation Some interesting theorems and properties of IFM and IFCM are presented (See Section2.2);

(c) Some new similarity degrees of IFMs such as the intuitionis-tic fuzzy similarity matrix (IFSM) and the intuitionisintuitionis-tic fuzzy similarity degree (IFSD) The formulas to predict RPD(Pi, Dj)

on the basis of IFSD accompanied with an interesting theo-rem is proposed (See Section2.3);

(d) From the predicting formulas, a novel intuitionistic fuzzy collaborative ﬁltering method so-called IFCF is presented for the medical diagnosis problem (See Section2.4); (e) The validation of the IFCF method in comparison with the standalone methods of IFS such as De et al.[9], Szmidt and Kacprzyk [44], Samuel and Balamurugan[31] and RS, e.g Davis et al [8], Hassan and Syed [13] is made by both a numerical illustration on the dataset inExample 1and the experiments on benchmark medical diagnosis datasets from UCI Machine Learning Repository in terms of the accuracy of diagnosis (See Section3

1.6 The novelty and signiﬁcance of the proposed work According to the contributions in Section1.5and the limitations

of IFS and RS in Sections1.3 and 1.4, respectively, the novel and the significance of the proposed work are stressed as follows (a) The proposed work is different from the previous ones espe-cially the standalone IFS and RS methods Specifically, it employs the ideas of both the IFS set and RS in the defini-tions of SC-IFRS and MC-IFRS, which are the basis to develop some new terms and similarity degrees for the IFCF algo-rithm Furthermore, as being observed fromExample 1to

3, the determination of the relation between patients and diseases in the standalone IFS methods is performed by some operations such as the fuzzy implication in De et al

[9], Samuel and Balamurugan[31]and the distance function

in Szmidt and Kacprzyk[42–44] In the proposed work, this can be done through the intuitionistic fuzzy similarity degree (IFSD) in Section 2.3, which is developed based on SC-IFRS and MC-IFRS Comparing with the standalone RS methods such as Davis et al.[8]and Hassan and Syed[13], the similarity degree – IFSD in the proposed work is con-structed from the light of the IFS set but not by the Pearson coefﬁcient from the hard values such as inTable 12 Addi-tionally, the formulas to predict RPD(Pi, Dj) are also made according to the membership and non-membership func-tions but not by the hard values above These proofs demon-strate the novel of the proposed work;

(b) The proposed hybrid method could handle the issues of the standalone IFS and RS methods For instance, the limitations

of IFS relating to the missing relations and the historic diag-noses of patients stated in Section1.3(a) and (b) and the lim-itations of RS relating to the crisp and training datasets stated

in Section1.4(a) and (c) are solved by the integration of IFS and RS The deﬁciency of mathematical properties of opera-tions in Section1.3(d) is resolved by a number of interesting

Table 12

The crisp training dataset with ⁄ being the values to be predicted.

Table 13

The full dataset derived by the method of Hassan & Syed [13] where bold values imply

the most possible disease.

Table 14

The full dataset derived by the method of Davis et al [8] where bold values imply the

most possible disease.

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theorems and properties in Section2 Lastly, when predicting

RPD(Pi, Dj), users could ﬁnd a suitable defuzziﬁcation method

for the determination of the most acquiring disease;

(c) The proposal of this work is signiﬁcance in terms of both

theory and practice In the theoretical aspect, the proposed

work motivates researching on advanced algorithms of IFS

and RS especially the hybrid method between them to

enhance the accuracy of the algorithm Looking for details

in Section1.5, we recognize that the proposed method is

constructed on a well-deﬁned mathematical foundation,

which is not paid much attention in the previous researches

Thus, this guarantees the further deployment of other

advanced methods of both IFS and RS on such the

mathe-matical foundation In the practical side, the proposed work

contributes greatly to the medical diagnosis problem and

some extensions and variants of this method could be

quickly deployed for other socio-economic problems This

clearly afﬁrms the signiﬁcance of the proposed work

1.7 The organization of the paper

The rest of the paper is organized as follows Section2presents

the main contribution including the IFRS and its elements stated

in Section1.5 Section3validates the proposed approach through

a set of experiments involving benchmark medical diagnosis data

Section4draws the conclusions and delineates the future research

directions

2 Intuitionistic fuzzy recommender systems

In this section, we present the new deﬁnitions of

single-crite-rion IFRS (SC-IFRS) and multi-criteria IFRS (MC-IFRS) in Section2.1;

the new deﬁnitions of intuitionistic fuzzy matrix (IFM) and the

intuitionistic fuzzy composition matrix (IFCM) of two IFMs with

the intersection/union operation in Section2.2; the intuitionistic

fuzzy similarity matrix (IFSM), the intuitionistic fuzzy similarity

degree (IFSD) and the formulas to predict RPD(Pi, Dj) on the basis

of IFSD in Section2.3; a novel intuitionistic fuzzy collaborative

ﬁl-tering method so-called IFCF in Section2.4

2.1 The single-criterion and multi-criteria intuitionistic fuzzy

recommender systems

Recall P, S and D fromDeﬁnition 1being the sets of patients,

symptoms and diseases, respectively Each patient Pi("i 2 {1, ,

n}) (resp symptom Sj, "j 2 {1, , m}) is assumed to have some

features (resp characteristics) For the simplicity, we consider RS

including a feature of the patient and a characteristic of the

symptom denoted as X and Y, respectively X and Y both consist of

s intuitionistic linguistic labels Analogously, disease Di("i 2 {1,

., k}) also contains s intuitionistic linguistic labels A new

deﬁnition of RS under the lights of those of medical diagnosis

expressed by IFS and the traditional RS inDeﬁnition 1& 4

respec-tively is stated as follows

Deﬁnition 5 (Single-criterion Intuitionistic Fuzzy Recommender

Systems – SC-IFRS) The utility function R is a mapping speciﬁed

on (X, Y) as follows

R : X Y ! D

ðl1XðxÞ;c1XðxÞÞ;

ðl2XðxÞ;c2XðxÞÞ;u

ðlsXðxÞ;csXðxÞÞ

ðl1YðyÞ;c1YðyÞÞ;

ðlsYðyÞ;csYðyÞÞ

!

l1DðDÞ;c1DðDÞ

;

ðl2DðDÞ;c2DðDÞÞ;

ðlsDðDÞ;csDðDÞÞ

;

ð13Þ

where liX(x) 2 [0, 1] (resp ciX(x) 2 [0, 1]), "i 2 {1, , s} is the membership (resp non-membership) value of the patient to the linguistic label ith of feature X.ljY(y) 2 [0, 1] (resp.cjY(y) 2 [0, 1]),

"j 2 {1, , s} is the membership (resp non-membership) value of the symptom to the linguistic label jth of characteristicY Finally,

llD(D) 2 [0, 1] (resp.clD(D) 2 [0, 1]), "l 2 {1, , s} is the membership (resp non-membership) value of disease D to the linguistic label lth SC-IFRS provides two basic functions:

(a) Prediction: determine the values of ðllDðDÞ;clDðDÞÞ, "l 2 {1, , s};

(b) Recommendation: choose i⁄

2 [1, s] satisfying i

¼ arg maxi¼1;sfliDðDÞ þliDðDÞð1 liDðDÞ ciDðDÞÞg

Remark 1 (a) FromDefinition 5and Eq.(13), the medical diagnosis is rep-resented by the implication {Patient, Symtomp} ? Disease, which is identical to that ofDefinition 1 Thus, we clearly recognize that of SC-IFRS inDefinition 5is another represen-tation and an extension of medical diagnosis inDefinition 1

inspired by the ideas of RS inDefinition 4; (b) SC-IFRS inDefinition 5could be regarded as the extension of the traditional RS inDefinition 4in cases that

$i:liX(x) = 1 ^ciX(x) = 0; "j – i:ljX(x) = 0 ^cjX(x) = 1,

$i:liY(y) = 1 ^ciY(y) = 0; "j – i:ljY(y) = 0 ^cjY(y) = 1,

$i:liD(D) = 1 ^ciD(D) = 0; "j – i:ljD(D) = 0 ^cjD(D) = 1, then the mapping in(13)could be re-written as,

R : P S ! D

Now we extend SC-IFRS to handle the cases of multiple diseases

D = {D1, , Dk}

Deﬁnition 6 (Multi-criteria Intuitionistic Fuzzy Recommender Sys-tems – MC-IFRS) The utility function R is a mapping speciﬁed on (X, Y) below

R : X Y ! D1 Dk

ðl1XðxÞ;c1XðxÞÞ;

ðl2XðxÞ;c2XðxÞÞ; u

ðlsXðxÞ;csXðxÞÞ

ðlsYðyÞ;csYðyÞÞ

!

ðl1DðD1Þ;c1DðD1ÞÞ;

ðl2DðD1Þ;c2DðD1ÞÞ;

ðlsDðD1Þ;csDðD1ÞÞ

ðl1DðDkÞ;c1DðDkÞÞ;

ðl2DðDkÞ;c2DðDkÞÞ;

ðlsDðDkÞ;csDðDkÞÞ

: ð15Þ

MC-IFRS is the system that provides two basic functions below (a) Prediction: determine the values of ðllDðDiÞ;clDðDiÞÞ, "l 2 {1, , s}, "i 2 {1, , k};

(b) Recommendation: choose i⁄

2 [1, s] satisfying i

¼ arg maxi¼1;s Pk

j¼1wjliDðDjÞ þliDðDjÞð1 liDðDjÞ ciDðDjÞÞ

where wj2 [0, 1] is the weight of Djsatisfying the constraint:

Pk j¼1wj¼ 1

Trang 7

Example 6 In a medical diagnosis system, there are 4 patients

whose feature X is ‘‘Age’’ consisting of 3 linguistic labels {low,

med-ium, high} (s = 3) The symptom‘s characteristic Y is ‘‘Temperature’’

including 3 linguistic labels {cold, medium, hot} The diseases (D1,

D2) are {‘‘Flu’’, ‘‘Headache’’}, and both of them contain 3 linguistic

labels {Level 1, Level 2, Level 3} We would like to verify which ages

of users and types of temperature are likely to cause the diseases of

ﬂu and headache In this case we have a MC-IFRS system By using

the trapezoidal intuitionistic fuzzy number – TIFN ([3]) characterized

by a1;a2;a3;a4;a0

1;a0

4

with a0

16a16a26a36a46a0

4, the mem-bership (non-memmem-bership) functions of patients to the linguistic

label ith of feature X are:

llowðxÞ ¼

ð35 xÞ=15 20 < x 6 35

8

>

vlowðxÞ ¼

ðx 20Þ=15 20 < x 6 35

8

>

lmediumðxÞ ¼

0 x 6 20; x > 60

ðx 20Þ=15 20 < x 6 35

1 35 < x 6 45

ð60 xÞ=15 45 < x 6 60

8

>

vmediumðxÞ ¼

1 x 6 20; x > 60

ð35 xÞ=15 20 < x 6 35

0 35 < x 6 45

ðx 45Þ=15 45 < x 6 60

8

>

lhighðxÞ ¼

ðx 45Þ=15 45 < x 6 60

8

>

vhighðxÞ ¼

ð60 xÞ=15 45 < x 6 60

8

>

Based on Eqs (16)–(21), we calculate the information of

patients as follows

Alð25tÞ : highð0; 1Þ; mediumð0:33; 0:67Þ; lowð0:67; 0:33Þh i; ð22Þ

Bobð40tÞ : highð0; 1Þ; mediumð1; 0Þ; lowð0; 1Þh i; ð23Þ

Joeð45tÞ : highð0; 1Þ; mediumð1; 0Þ; lowð0; 1Þh i; ð24Þ

Tedð50tÞ : highð0:33; 0:67Þ; mediumð0:67; 0:33Þ; lowð0; 1Þh i: ð25Þ

Similarly, the membership (non-membership) functions of the

symptom to the linguistic label jth of characteristicY are:

lcoldðxÞ ¼

ð20 xÞ=15 5 < x 6 20

8

>

vcoldðxÞ ¼

ðx 5Þ=15 5 < x 6 20

1 x > 20

8

>

lmediumðxÞ ¼

0 x 6 5; x > 40

ðx 5Þ=15 5 < x 6 20

1 20 < x 6 35

ð40 xÞ=5 35 < x 6 40

8

>

<

>

:

vmediumðxÞ ¼

1 x 6 5; x > 40 ð20 xÞ=15 5 < x 6 20

0 20 < x 6 35

ðx 35Þ=5 35 < x 6 40

8

>

lhotðxÞ ¼

ðx 35Þ=5 35 < x 6 40

1 x > 40

8

>

vhotðxÞ ¼

ð40 xÞ=5 35 < x 6 40

0 x > 40

8

>

The information of symptom are shown as follows

ð4 CÞ : coldð1; 0Þ; mediumð0; 1Þ; hotð0; 1Þh i; ð32Þ ð16 CÞ : coldð0:267; 0:733Þ; mediumð0:733; 0:267Þ; hotð0; 1Þh i; ð33Þ ð39 CÞ : coldð0; 1Þ; mediumð0:2; 0:8Þ; hotð0:8; 0:2Þh i; ð34Þ ð25 CÞ : coldð0; 1Þ; mediumð1; 0Þ; hotð0; 1Þh i: ð35Þ

From Eqs (22)–(25), (32)–(35) we have a MC-IFRS described by

Table 15

InTable 15, the cells having question marks are needed to pre-dict the intuitionistic fuzzy values ðllDðDiÞ;clDðDiÞÞ;8l 2 f1; 2; 3g;

8i 2 f1; 2g

2.2 Intuitionistic fuzzy matrix and intuitionistic fuzzy composition matrix

Deﬁnition 7 An intuitionistic fuzzy matrix (IFM) Z in MC-IFRS is deﬁned as,

Z ¼

a11 a12 a1s

b21 b22 b2s

c31 c32 c3s

c41 c42 c4s

ct1 ct2 cts

0 B B B B B

@

1 C C C C C A

In Eq (36), t = k + 2 where k 2 N+ is the number of diseases in

Deﬁnition 6 The value s 2 N+ is the number of intuitionistic linguistic labels a1i, b2i, chi, " h 2 {3, , t}, "i 2 {1, , s} are the intuitionistic fuzzy values (IFV) consisting of the membership and non-membership values as in Deﬁnition 6 a1i¼ ðliXðxÞ;ciXðxÞÞ,

"i 2 {1, , s} represents for the IFV value of the patient to the linguistic label ith of featureX b2i¼ ðliYðyÞ;ciYðyÞÞ, " i 2 {1, , s} stands for the IFV value of the symptom to the linguistic label ith

of characteristic Y chi¼ ðliDðDh2Þ;ciDðDh2ÞÞ, "i 2 {1, , s},

"h 2 {3, , t} is the IFV value of the disease to the linguistic label ith Each line from the third one to the last in Eq (36)is related

to a given disease

Example 7 The ﬁrst line inTable 15describing the information of user Al (Age: 25) at the temperature 4C can be expressed by the IFM as follows

Z ¼

ð0:0; 1:0Þ ð0:33; 0:67Þ ð0:67; 0:33Þ ð1:0; 0:0Þ ð0:0; 1:0Þ ð0:0; 1:0Þ ð0:8; 0:1Þ ð0:2; 0:6Þ ð0:1; 0:9Þ ð0:1; 0:8Þ ð0:6; 0:35Þ ð0:3; 0:55Þ

0 B B

@

1 C C

Trang 8

Deﬁnition 8 Suppose that Z1and Z2are two IFM in MC-IFRS The

intuitionistic fuzzy composition matrix (IFCM) of Z1and Z2with the

intersection operation is,

að1Þ

11 að1Þ

12 að1Þ

1s

bð1Þ21 bð1Þ22 bð1Þ2s

cð1Þ31 cð1Þ32 cð1Þ3s

cð1Þ

41 cð1Þ

42 cð1Þ

4s

cð1Þ

t1 cð1Þ

t2 cð1Þ

ts

0

B

@

1 C C C C C C C A

að2Þ

11 að2Þ

12 að2Þ

1s

cð2Þ

41 cð2Þ

42 cð2Þ

4s

cð2Þ t1 cð2Þ t2 cð2Þ

ts

0 B B B B B B B

@

1 C C C C C C C A

¼

að12Þ11 að12Þ12 að12Þ1s

cð12Þ

31 cð12Þ

32 cð12Þ

3s

cð12Þt1 cð12Þt2 cð12Þts

0

B

@

1 C C C C C C C A

where

að12Þ1i ¼ að1Þ1i ^ að2Þ1i ¼ min lð1Þ

iXðxÞ;lð2Þ

iXðxÞ

;max cð1Þ

iXðxÞ;cð2Þ

iXðxÞ

;

bð12Þ2i ¼ bð1Þ2i ^ bð2Þ2i ¼ min lð1Þ

iYðyÞ;lð2Þ

iYðyÞ

;max cð1Þ

iYðyÞ;cð2Þ

iYðyÞ

;

cð12Þ

hi ¼ cð1Þhi ^ cð2Þhi

¼ min lð1Þ

iDðDh2Þ;lð2Þ

iDðDh2Þ

;max cð1Þ

iDðDh2Þ;cð2Þ

iDðDh2Þ

;

8i 2 f1; ; sg; 8h 2 f3; ; tg: ð41Þ

Example 8 Given 2 IFM below

Z1¼

0 B B

1 C

Z2¼

0 B B

1 C

The IFCM of Z1and Z2with the intersection operation is:

Z ¼ Z1 Z2¼

0 B B

1 C

C: ð44Þ

Deﬁnition 9 Suppose that Z1and Z2are two IFM in MC-IFRS The intuitionistic fuzzy composition matrix (IFCM) of Z1and Z2with the union operation is deﬁned as follows

að1Þ

11 að1Þ

12 að1Þ

1s

cð1Þ

41 cð1Þ

42 cð1Þ

4s

ts

0 B B B B B

@

1 C C C C C A

að2Þ

11 að2Þ

12 að2Þ

1s

cð2Þ

41 cð2Þ

42 cð2Þ

4s

ts

0 B B B B B

@

1 C C C C C A

¼

að12Þ11 að12Þ12 að12Þ1s

cð12Þ

31 cð12Þ

32 cð12Þ

3s

0 B B B B B

@

1 C C C C C A

Table 15

A MC-IFRS for medical diagnosis with ⁄ being the values to be predicted.

Alð25Þ :

highð0; 1Þ;

mediumð0:33; 0:67Þ;

lowð0:67; 0:33Þ

ð4 CÞ :

coldð1; 0Þ;

mediumð0; 1Þ;

hotð0; 1Þ

Level2(0.2, 0.6); Level2(0.6, 0.35); Level3(0.1, 0.9); Level3(0.3,0.55) Alð25Þ :

highð0; 1Þ;

lowð0:67; 0:33Þ

ð16 CÞ : coldð0:267; 0:733Þ;

mediumð0:733; 0:267Þ;

hotð0; 1Þ

Level2(0.6,0.2); Level2(0.2, 0.75); Level3(0.1,0.9); Level3(0.7,0.2) Bobð40Þ :

highð0; 1Þ;

mediumð1; 0Þ;

lowð0; 1Þ

ð39 CÞ :

coldð0; 1Þ;

mediumð0:2; 0:8Þ;

hotð0:8; 0:2Þ

Level2(0.1,0.8); Level2(0.1,0.9); Level3(0.0,0.95); Level3(0.0,0.9) Joeð45Þ :

highð0; 1Þ;

mediumð1; 0Þ;

lowð0; 1Þ

ð16 CÞ : coldð0:267; 0:733Þ;

mediumð0:733; 0:267Þ;

hotð0; 1Þ

Level2(0.2, 0.7); Level2(0.7, 0.3); Level3(1.0,0.0); Level3(0.1,0.85); Tedð50Þ :

highð0:33; 0:67Þ;

lowð0; 1Þ

ð25 CÞ : coldð0; 1Þ;

mediumð1; 0Þ;

hotð0; 1Þ

Level2(⁄, ⁄); Level2(⁄, ⁄); Level3(⁄, ⁄); Level3(⁄, ⁄);

Trang 9

að12Þ1i ¼ að1Þ1i _ að2Þ1i ¼ max lð1Þ

iXðxÞ;lð2Þ

iXðxÞ

;min cð1Þ

iXðxÞ;cð2Þ

iXðxÞ

;

bð12Þ2i ¼ bð1Þ2i _ bð2Þ2i ¼ max lð1Þ

iYðyÞ;lð2Þ

iYðyÞ

;min cð1Þ

iYðyÞ;cð2Þ

iYðyÞ

;

cð12Þ

hi ¼ cð1Þhi _ cð2Þhi

¼ max lð1Þ

iDðDh2Þ;lð2Þ

iDðDh2Þ

;min cð1Þ

iDðDh2Þ;cð2Þ

iDðDh2Þ

;

8i 2 f1; ; sg; 8h 2 f3; ; tg: ð48Þ

Example 9 Given 2 IFM inExample 8 The IFCM of Z1and Z2with

the union operation is:

Z ¼ Z1 Z2¼

ð0:0; 1:0Þ ð1:0; 0:0Þ ð0:67; 0:33Þ

ð1:0; 0:0Þ ð0:733; 0:267Þ ð0:0; 1:0Þ

ð0:8; 0:1Þ ð0:2; 0:6Þ ð1:0; 0:0Þ

ð0:1; 0:8Þ ð0:7; 0:3Þ ð0:3; 0:55Þ

0

B

1 C

C: ð49Þ

Theorem 1 The IFCM of Z1and Z2with the intersection (union)

oper-ation is an IFM

Proof 1 We prove the theorem with the intersection operation

only The theorem with the union operation is proven analogously

FromDeﬁnition 8, we know that

að12Þ1i ¼ að1Þ1i ^ að2Þ1i ¼ min lð1Þ

iXðxÞ;lð2Þ

iXðxÞ

;max cð1Þ

iXðxÞ;cð2Þ

iXðxÞ

;

8i 2 f1; ; sg;

bð12Þ2i ¼ bð1Þ2i ^ bð2Þ2i ¼ min lð1Þ

iYðyÞ;lð2Þ

iYðyÞ

;max cð1Þ

iYðyÞ;cð2Þ

iYðyÞ

;

8i 2 f1; ; sg;

cð12Þ

hi ¼ cð1Þhi ^ cð2Þhi

¼ min lð1Þ

iDðDh2Þ;lð2Þ

iDðDh2Þ

;max cð1Þ

iDðDh2Þ;cð2Þ

iDðDh2Þ

;

8i 2 f1; ; sg; 8h 2 f3; ; tg: ð52Þ

Since að1Þ1i and að2Þ1i are two IFV, the function of them – að12Þ1i is also an

IFV Similar conclusions are found for bð12Þ2i and cð12Þhi Thus, the IFCM

of Z1and Z2with the intersection operation is an IFM The proof is

complete h

Property 1 Given Z1, Z2and Z3being IFM The following properties

hold for these IFM

(a) Z1 Z2= Z2 Z1,

(b) (Z1 Z2) Z3= Z1 (Z2 Z3)

Proof 2 (a) Suppose that the IFCM of Z1and Z2is equipped with

the intersection operation Since að1Þ

1i and að2Þ 1i are two IFV, we obtain

að12Þ

1i ¼ að1Þ1i ^ að2Þ1i ¼ að2Þ1i ^ að1Þ1i ¼ að21Þ1i ; ð53Þ

bð12Þ

2i ¼ bð1Þ2i ^ bð2Þ2i ¼ bð2Þ2i ^ bð1Þ2i ¼ bð21Þ2i ; ð54Þ

cð12Þ

hi ¼ cð1Þhi ^ cð2Þhi ¼ cð2Þhi ^ cð1Þhi ¼ cð21Þhi : ð55Þ

It follows that,

Z1 Z2¼

að12Þ

11 að12Þ

12 að12Þ

1s

cð12Þ

31 cð12Þ

32 cð12Þ

3s

cð12Þ

41 cð12Þ

42 cð12Þ

4s

ts

0 B B B B B B

@

1 C C C C C C A

¼

að21Þ

11 að21Þ

12 að21Þ

1s

cð21Þ

31 cð21Þ

32 cð21Þ

3s

cð21Þ

41 cð21Þ

42 cð21Þ

4s

ts

0 B B B B B B

@

1 C C C C C C A

¼ Z2 Z1:

ð56Þ

The proof is analogously performed with the IFCM of Z1 and Z2

equipped with the union operation.(b) Suppose that the IFCM of

Z1and Z2is equipped with the intersection operation We have,

ðZ1 Z2Þ Z3¼

að123Þ

11 að123Þ

12 að123Þ

1s

bð123Þ21 bð123Þ22 bð123Þ2s

cð123Þ31 cð123Þ32 cð123Þ3s

cð123Þ

41 cð123Þ

42 cð123Þ

4s

0 B B B B B B

@

1 C C C C C C A

að123Þ1i ¼ a ð1Þ1i ^ að2Þ1i

^ að3Þ1i; 8i 2 f1; ; sg; ð58Þ

bð123Þ2i ¼ b ð1Þ2i ^ bð2Þ2i

^ bð3Þ2i; 8i 2 f1; ; sg; ð59Þ

cð123Þhi ¼ c ð1Þhi ^ cð2Þhi

^ cð3Þhi; 8i 2 f1; ; sg; 8h 2 f3; ; tg: ð60Þ

Because

að1Þ 1i ^ að2Þ1i

^ að3Þ1i ¼ að1Þ1i ^ a ð2Þ1i ^ að3Þ1i

bð1Þ2i ^ bð2Þ2i

^ bð3Þ2i ¼ bð1Þ2i ^ b ð2Þ2i ^ bð3Þ2i

cð1Þ

hi ^ cð2Þhi

^ cð3Þhi ¼ cð1Þhi ^ c ð2Þhi ^ cð3Þhi

It follows that

ðZ1 Z2Þ Z3¼ Z1 ðZ2 Z3Þ: ð64Þ

The proof is analogously performed with the IFCM of Z1 and Z2

equipped with the union operation h

2.3 The intuitionistic fuzzy similarity matrix and intuitionistic fuzzy similarity degree

Motivated by the ideas of Hung and Yang[15], we present the definition of intuitionistic fuzzy similarity matrix as follows Definition 10 Suppose that Z1and Z2are two IFM in MC-IFRS The intuitionistic fuzzy similarity matrix (IFSM) between Z1 and Z2 is defined as follows

eS ¼

eS11 eS12 eS1s

eS21 eS22 eS2s

eS31 eS32 eS3s

eS41 eS42 eS4s

eSt1 eSt2 eSts

0 B B B B B

@

1 C C C C C A

Trang 10

eS1i¼ 1

1 exp 1=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lð1Þ

iXðxÞ

q

lð2Þ

iXðxÞ q

þ

cð1Þ

iXðxÞ

q

cð2Þ

iXðxÞ q

eS2i¼ 1

1 exp 1=2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lð1Þ

iYðyÞ

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lð2Þ

iYðyÞ q

þ ffiffiffiffiffiffiffiffiffiffiffiffiffifficð1Þ

iYðyÞ

q

cð2Þ

iYðyÞ q

1 expð1Þ

Deﬁnition 11 Suppose that Z1and Z2are two IFM in MC-IFRS The

intuitionistic fuzzy similarity degree (IFSD) between Z1and Z2is

SIMðZ1;Z2Þ ¼aXs

i¼1

w1ieS1iþ bXs

i¼1

w2ieS2iþvX

t h¼3

Xs i¼1

whieShi; ð69Þ

where eS is the IFSM between Z1and Z2 W = (wij) ("i 2 {1, , t},

"j 2 {1, , s}) is the weight matrix of IFSM between Z1 and Z2

satisfying,

Xs

i¼1

w1i¼ 1; Xs

i¼1

w2i¼ 1; Xs

i¼1

whi¼ 1; 8h 2 f3; ; tg; ð70Þ

Remark 2 The formula of IFSD in Eq.(68)can be recognized as the

generalization of the hard user-based, item-based and the

rating-based similarity degrees in recommender systems [29] when

b=v= 0,a=v= 0 anda= b = 0, respectively

Deﬁnition 12 The formulas to predict the values of linguistic

labels of patient Pu("u 2 {1, , n}) to symptom Sj(" j 2 {1, ,

m}) according to diseases (D1, D2, , Dk) in MC-IFRS are:

lP u

iDðDjÞ ¼

Pn

v¼1SIMðPu;PvÞ lPv

iDðDjÞ

Pn

v¼1SIMðPu;PvÞ ;8i 2 f1; ; sg;

8j 2 f1; ; kg; 8u 2 f1; ; ng; ð72Þ

cP u

iDðDjÞ ¼

Pn

v¼1SIMðPu;PvÞ cPv

iDðDjÞ

Pn

v¼1SIMðPu;PvÞ ; 8i 2 f1; ; sg;

8j 2 f1; ; kg; 8u 2 f1; ; ng: ð73Þ

Theorem 2 The predictive IFM results inDeﬁnition 12are an IFV

Proof 3 We have the following fact

lP u

iDðDjÞ þcP u

iDðDjÞ ¼

Pn

iDðDjÞ þcPv

iDðDjÞ

Pn

v¼1SIMðPu;PvÞ ; ð74Þ

0 6lPv

iDðDjÞ þcPv

It is obvious that

lP u

iDðDjÞ þcP u

Since

SIMðPu;P1Þ lPv

iDðDjÞ þcPv

iDðDjÞ

6SIMðPu;P1Þ; ð78Þ SIMðPu;P2Þ lPv

iDðDjÞ þcPv

iDðDjÞ

6SIMðPu;P2Þ; ð79Þ

SIMðPu;PnÞ lPv

iDðDjÞ þcPv

iDðDjÞ

6SIMðPu;PnÞ: ð80Þ

It follows that

lP u

iDðDjÞ þcP u

iDðDjÞ ¼

Pn

iDðDjÞ þcPv

iDðDjÞ

Pn

v¼1SIMðPu;PvÞ 6

Pn

v¼1SIMðPu;PvÞ

Pn

v¼1SIMðPu;PvÞ¼ 1: ð81Þ

The proof is complete h 2.4 The intuitionistic fuzzy collaborative ﬁltering method

Fig 1)

3 Evaluation

In this section, we describe the experimental environment in Section3.1 The database for experiments is given in Section3.2

eShi¼ 1

1 exp 1=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lð1Þ

iDðDh2Þ

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lð2Þ

iDðDh2Þ q

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficð1Þ

iDðDh2Þ

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cð2Þ

iDðDh2Þ q

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