Luận án tiến sĩ toán học: Một số độ đo và biểu diễn mới của hệ mờ trực cảm và ứng dụng

VIETNAM NATIONAL UNIVERSITYVNU UNIVERSITY OF SCIENCE

ROAN THI NGAN

SOME NEW MEASURES AND REPRESENTATIONS OF

INTUITIONISTIC FUZZY SYSTEMS AND APPLICATIONS

PHD DISSERTATION IN MATHEMATICS

Hanoi - 2021

VIETNAM NATIONAL UNIVERSITYVNU UNIVERSITY OF SCIENCE

ROAN THI NGAN

SOME NEW MEASURES AND REPRESENTATIONS OF

INTUITIONISTIC FUZZY SYSTEMS AND APPLICATIONS

Major: Applied MathematicsCode: 9460112.01

This is to certify that to the best of my knowledge, the content of this thesisis my own work This thesis has not been submitted for any degree or other

purposes The experimental datasets in this thesis are clearly derived and

published in accordance with regulations The research results presented inthis thesis are honest and objective This thesis contains no material that ispreviously published, except that citations that have been clearly specified.

The co-authors totally agree for me to use the content of our publications for

the purpose of writing and reporting the dissertation at all levels.Here, if anything goes wrong, I assume full responsibility.

Hanoi, 2021

PhD Candidate

Roan Thi Ngan

First of all, I would like to express my sincere and profound gratitude to myresearch supervisors, Assoc Prof Dr Sc Bui Cong Cuong and Assoc Prof.Dr Le Hoang Son I am extremely grateful for the dedicated and valuablehelp that the supervisors have given me during the process of implementingthe thesis They have given me a lot of attention, guidance and valuable helpnot only in scientific research but also in life.

I would like to express my sincere thanks to the teachers, research group,and brothers in the Center for High-Performance Computing, University ofScience, the laboratory of the Department of Multimedia and Virtual Reality,VNU Information Technology Institute, and the Neuro-Fuzzy Systems withApplications seminar group for their.

I would like to express my sincere thanks to Prof Dr Sc Pham Ky Anh andthe other members in the Department of Computational and Applied Mathe-matics, Faculty of Mathematics, Mechanics and Informatics in particular andthe University of Science, VNU in general The comments after seminar re-ports, as well as the management of training, research environment, and fa-cilitation of the department and the university, help me a lot in completingthis thesis.

I would like to thank the project 911 of the Ministry of Education and ing and the European Union’s Erasmus Program for giving me the opportu-nity to develop my research I would like to express my gratitude to Assoc.Prof Dr Vu Van Manh and Assoc Prof Dr Juan-Miguel Martinez Rubio, fortheir support during the scholarship application.

Train-I would also like to sincerely thank colleagues in the Faculty of General ence in particular, Hanoi University of Natural Resources and Environmentin general for creating all favorable conditions.

Sci-Finally, this thesis will not be complete without the encouragement andsupport in all aspects of the family This thesis is to send to my family mem-bers, with all my deepest gratitude.

ii

LIST OF TABLES

1 INTRODUCTION & PRELIMINARY

11 PROBLEMS

1.2 LITERATUREREVIEW

1.21 Intuitionistic Fuzzy Measure

1.2.2 Intuitionistic Fuzzy Representation

1.2.3 Medical Diagnosis

13 MOTIIVAIIONS

14 OBJECTIVES

1.5 RESEARCH APPROACHEHS

1.5.1 Intuitionistic Fuzzy Measure

1.5.2 Intuitionistic Fuzzy Representation

1.8.3 Intuitionistic Fuzzy Relation and Similarity Measures 33

1.8.4 Complex Fuzzy Set and Complex Intuitionistic Fuzzy Set 351.8.5 Intuitionistic Fuzzy Systems 36

2.2.2 d6—Equalities for Intuitionistic Fuzzy Operations and EU ee 432.3 H-MAXDISTANCE 50

Re-2.3.1 H-max Distance Measure of Intuitionistic Fuzzy Sets 50

2.3.2 Distance Measure of IFSs with the Intuitionistic FuzzyT-normand T-conorm 59

2.4 EXPERIMENT ON UCI MEDICAL DATA 66

2.4.1 Medical Diagnosis based on the 6—Equality Measure 66

2.4.2 Medical Diagnosis based on the H-max Measure 74

2.5 EXPERIMENT ON DENTAL IMAGE DATA 80

2.5.1 Diagnosis Method based on the Modified H-max Measure 802.5.2 Experimental Environment and Datasets 84

2.5.3 PerformanceComparison 84

2.6 CONCLUDING REMARKS 86

NEW REPRESENTATIONS OF INTUITIONISTIC FUZZY SYSTEMSUNDER COMPLEX SET 8731 INTRODUCHON 87

3.2 IFS-C: INTUITIONISTIC FUZZY SYSTEMS BASED ON PLEXNUMBERS 88

COM-3.2.1 Polar Representation and a New Order Relation of IFSs 883.2.2 A New Distance Measure of IFS-C 93

3.3 REPRESENTING COMPLEX INTUITIONISTIC FUZZY SET BYQUATERNIONNDMBERS 95

3.3.1 A New Representation of Complex Intuitionistic FuzzySets based on Quaternion Numbers 3.3.2 Logic and Algebraic Operations

3.3.3 Quaternion Distance Measure 3.4 EXPERIMENT ON BENCHMARK MEDICAL DATASETS

3.4.1 PDM Decision-Making Model

3.4.2 Decision-Making Model based on Quaternion Distance

Measures 3.5 CONCLUDING REMARKSCONCLUSION

LIST OF PUBLICATIONS

109109

127129130145

APC Affinity Propagation Clustering

BCW Breast Cancer Wisconsin

CFC Complex Fuzzy ClassCFSs Complex Fuzzy Sets

CIFSs Complex Intuitionistic Fuzzy Sets

CIFSs-Q Complex Intuitionistic Fuzzy Sets-Quaternion

CM-SPA Cloud Model-Set Pair Analysis

C-ODM Cartesian-ODM

CTG Cardiotocography

DDS Dental Diagnosis System

Dermal Dermatology

DIHM Diagnosis from Image based on H-Max Measure

DIMHM Diagnosis from Image based on Modified H-Max MeasureDRD Diabetic Retinopathy Debrecen

EEI Edge-value and IntensityFIS Fuzzy Inference System

FKNN Fuzzy K-Nearest NeighborsFSs Fuzzy Sets

GCK Kruskal spanning treeGCP Prim spanning tree

GRA Gradient Feature

H-max Hamming-max

HS Haberman’s Survival

IFR(X x Y) The set of all IFRs on X x Y

IFRs Intuitionistic Fuzzy RelationsIFS(U) The set of all IFSs on U

IFSs Intuitionistic Fuzzy Sets

IFSs-C Intuitionistic Fuzzy Systems-ComplexILPD Indian Liver Patient

I-TSFIS Intuitionistic Time Series Fuzzy Inference System

LBP Local Patterns Binary FeatureLD Liver-Disorders

MAE Mean Absolute ErrorMSE Mean Squared ErrorNaN Not-a-Number

SPA Set Pair Analysis

SVM Support Vector Machinet-conorm triangle-conorm

LIST OF FIGURES

1.1 The change between IFSs 11

1.2 The order relation between two IFSs AandB 12

1.3 The histogram of the 1° attribute of the ILPD Data 25

1.4 The histogram of the 8" attribute of the ILPD Data 25

1.5 The dental X-ray images with the corresponding diseases 26

1.6 The white-black colorstrip[77] - 27

1.7 Basic structure of a fuzzy system[53] 37

21 Optimizing the thresholdvalue 69

2.2 The proposed model for medical diagnosis 69

2.3 Optimizing the disease threshold in proposed diagnosis model 752.4 The proposed model for medical diagnosis 75

2.5 Optimizing parameters 2 2 eee ee eee 832.6 ThemodelofDIMHM_ 83

3.1 Splitdomainson thesetl” eee 893.2 The order relation <œonL” co 943.3 Graphical representation of the products of unit quaternions asa 90°-rotation in 4D-space 96

3.4 Graphical representation of the complex degrees 97

3.5 The graphical representation of complex degrees in Polar form 1043.6 The order relation <,onQ* -204 1083.7 Optimizing the assessment threshold 112

3.8 Decision Making Model based on P-Distance Measure 112

3.9 MAEofmethodsonBCW_ 114

3.10 Optimizing the assessment threshold for the QDM model 115

3.11 A diagram oftheQDM model 115

3.12 An illustration of the functions Z;, B „t0, and | 117

3.13 The MAE results of the considered algorithms on ILPD Data 124

3.14 The MAE results of the considered algorithms on Diabetes Data 1243.15 The MAE results of the considered algorithms on HS Data 1243.16 The MAE results of the considered algorithms on Ecoli Data 1253.17 The MAE results of the considered algorithms on BCW Data 125

LIST OF TABLES

The descriptions of experimental datasets The values of dyam,de,dHay,dwxe SỈThe values of dpam,de,dHan, dwx dome 0 ee

Q, is intuitionistic fuzzy relation between the set of patients Pand the set of symptoms S with the data from the first group of

Q¿ is intuitionistic fuzzy relation between the set of patients Pand the set of symptoms S with the data from the second groupof decision makers 2.2 2 ee ee

Q3 is intuitionistic fuzzy relation between the set of patients Pand the set of symptoms S with the data from the third group

Symptoms characteristic for the patients considered

Symptoms characteristic for the diagnoses considered .

71

Diagnosed results for the proposed distance measure đr;„ given

methods on 3 considered datasets 113m records of a dataset encoded in the Cartesian form of quater-

nionnumbers 0 00 eee eee eee 116

Records on Viral Fever Iisease 119

The MAE values of the methods on the benchmark medical data.122Total time (sec) of the methods on the benchmark medical data 123

still some problems, including two major ones as follows.Problem 1: Intuitionistic fuzzy measure.

- Some measures have not yet been extended for intuitionistic fuzzy sets,

such as proximity measure of Pappis [80].

- Some distance measures do not satisfy the condition regarding the

inclu-sion relation between intuitionistic fuzzy sets on a finite space of points,that is, if A, B and C are three intuitionistic fuzzy sets on a finite space of

points X = {x1,x2, ,Xm} and A C BC C, thend (A,B) < d(A,C) andd(B,C) < d(A,C) For instance, for A, B and C being three intuitionisticfuzzy sets on X = {x}, where their membership degrees are 4 = 0.1,

vp = 0.1, and c = 0.3, and their non-membership degrees are v4 = 0.7,vg = 0.6, and vc = 0.6, and d is the Euclidean measure [109],

a(A,B) = ( ((fa — Ho)? + (va — vp)? + (xà — 7t8))) 1,

10

where 7r = 1 — #— 1, then đ(B,C) = 0.2 >d(A,C) + 0.17.

- Further, some existing measures give an equal rating that is not tight For

instance, for A, B and C being three intuitionistic fuzzy sets on X = {x},

d is the Hausdorff measure [40],

d(A,B) = max {|MA — p|,|UA — vel},

and va = 0.5, wp = 0.4, and c = 0.4, while vy = 0.5, vg = 0.4, andvc = 0.6, then d(A,B) = d(A,C) = 0.1 However, in this case d(A, B)should be smaller than d(A,C) because the change from A to C is more

soundness than that from A to B Specifically, from A to C, the bership degree (e.g the support degree in the election) decreases and atthe same time the non-membership degree (e.g the opposition degree inthe election) increases This is illustrated in Figure 1.1, where the greenpart includes the points that have the degrees of membership and non-membership being both less than those of the point A.

Figure 1.1: The change between IFSs

Problem 2: Intuitionistic fuzzy representation.

- To build a measure between objects, we firstly need to define the conceptof the order relation between them On the existing representations ofintuitionistic fuzzy sets in the literature, building a total order relation isof remarkable complication since various evaluation steps are employedalong the way, such as the order of Xu and Yager [131] must be based on

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the intermediate functions, or the score function,S=p-v,

and the accuracy function,

For instance, to compare two intuitionistic fuzzy sets A and B on X =

{x}, where pa = 0.5, vg = 0.1, wg = 0.6 and vg = 0.2, we need to

evaluate the score values, Sa = Sz = 0.4, and the accuracy values, Ha =0.6 < Hg = 0.8, and then A is smaller than B (see Figure 1.2).

Hp =0.8Hạ =0.6

O 0.5 0.6 1 #

Figure 1.2: The order relation between two IFSs A and B

As a result, it is difficult to analyze the properties of intuitionistic fuzzymeasures and logical operators This is the reason why partial orders arepreferred in the intuitionistic fuzzy system Recently, Tamir et al [113]have proposed a new representation of the intuitionistic fuzzy set basedon complex numbers However, Tamir et al still make use of the tra-ditional intuitionistic fuzzy order relation of Atanassov [10], which is anorder that is not total.

Further, if we consider the degrees of membership and non-membership

in two dimensions, the existing intuitionistic fuzzy set is difficult to cess multi-dimensional information of complex problems As it comes to

ac-elections, for example, the percentage of people who are most likely to

vote for a particular candidate is 70% ( = 0.7) while those vote against

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is 30% (v = 0.3) However, the percentage of people who actually vote

for the candidate is 55% while the percentage of people who actually vote

against is 20% Hence, information with regards to the decision making

behind vote requires consideration from differing dimensions such as thereliable membership degree, unreliable membership degree, reliable non-membership degree, and unreliable non-membership degree.

This thesis focuses on the solutions of these two problems of the istic fuzzy system, which are the measures and the representation of the intu-itionistic fuzzy sets Moreover, applications of the proposed methods for themedical diagnosis will be studied.

intuition-1.2 LITERATURE REVIEW

1.2.1 Intuitionistic Fuzzy Measure

The distance and similarity measure of IFSs which are two relative tions have attracted many researchers because of their potential, which pro-motes the development of the corresponding theory and their applications;therefore, they are widely used in pattern recognition, decision making, med-ical diagnosis and so on Basic criteria can be used to define a distance mea-sure or similarity measure of IFSs are the boundary, symmetry, coincidence,triangle inequality conditions and the condition regarding the inclusion rela-tion between IFSs.

representa Measurements do not take into account the condition of the inclusion relation:In 2000, Szmidt and Kacprzyk [109] proposed four distance measures of IFSsbased on the geometric interpretation Later, in 2004, Szmidt and Kacprzyk[112] modified their distance measures; Grzegorzewski [40] introduced dis-tances between IFSs based on the Hausdor metric In 2009, Xu and Yager [132]proposed an improved version of Szmidt and Kacprzyk’s work In 2012, Yang

and Chiclana [134], and Hatzimichailidis et al [44] proposed several distance

and similarity measures of IFSs.

- Measurements that take into account the condition of the inclusion relation:

In 2002, Dengfeng and Chuntian [31] introduced a similarity measure that

13

was also applied in pattern recognition to test the performance of the lation method In 2003, a similarity measure was proposed by Mitchell [70]to overcome the drawback of Dengfeng and Chuntian’s measure which may

calcu-give unreasonable results in some specific situations By considering some

unreasonable cases of the measure proposed by Dengfeng and Chuntian, in

2003, Liang and Shi [64] also presented several new similarity measures, andthis work further promoted the development of the similarity and distancemeasure of IFSs In 2004, Hung and Yang [52] proposed a distance measureaccording to the Hausdorff distance, and they also utilized the measure todevelop several similarity measures that can be effectively used in linguisticvariables Then, Hung and Yang [50], [51] gave some other similarity mea-sures for IFSs by following their previous works as well In 2005, a newdistance measure of IFSs was introduced by Wang and Xin [126] In 2009,Park et al [81] modified the inclusion relation between IFSs of Atanassov andproposed a new distance measure of IFSs In 2011, Ye [135] introduced a co-sine similarity measure that was also used in medical diagnosis and patternrecognition problems In 2013, Zhang and Yu [140] proposed two distancemeasures based on the lengths of lines and the areas of rectilinear figures,which can be regarded as a geometry-based method On the basis of trans-formation techniques, Chen and Chang [20] proposed a similarity measurefor IFSs Maheshwari and Srivastava [68] studied on divergence measuresthat satisfy the condition of Park’s inclusion relation [81] More discussion onmeasures of IFSs can be found in [75]- [82].

1.2.2 Intuitionistic Fuzzy Representation

Representation of sets is fundamental to an inference system As introducedabove, Zadeh’s FSs are represented through a membership function whilethat of Atanossov’s IFSs contains both the membership and non—membership

functions In 2002, Smarandache introduced the single-valued neutrosophic

sets [99], where an element z on a space Z is evaluated bya triplet including

the truth-membership, indeterminacy-membership, and falsity-membership

degrees which are limited on [0,1], such as the triplet (0.3, 0.7, 0.6) With such

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three parameters satisfying their sum in [0,1], the notion of a picture fuzzy

set was introduced by Cuong in 2013 [27], describing the degrees of tive membership, neutral membership, and negative membership, such as

posi-(0.3,0.02,0.6) Further, in 2012, Torra proposed the concept of the hesitant

fuzzy set [117] with the hesitant function h satisfying the condition when

applied to a reference set Z returns a subset of [0,1], for example, h(z) =

(0.3, 0.8] or h(z) = {0.3,0.1,0.5,0.7} Some research results and applications

of these sets can be found in [15], [93].

With the ability to capture compound features and convey multifaceted

information, complex numbers were proposed in expanding the fuzzy

the-ory to solve complicated problems In 2002, the complex fuzzy set (CFS) [88]was proposed by Ramot via the complex membership function in the form of

y(z) = p(z) e'“), where j = v—1, p and œ are both real-valued functions,

and the amplitude value p(z) € [0,1] For numerical example, 7(z) = 0.3e!%/,

where p(z) = 0.3 is the pure membership degree defined as in the conceptof the fuzzy set and the value w(z) = 1.6 indicates the considered specific

characteristic of z such as the periodicity Similarly, the complex intuitionisticfuzzy set (CIFS) was introduced by adding the non-membership function intothe form of a complex number [6].

Recently, since CIFS is restricted in the polar representation where only theamplitude terms are fuzzy functions that convey fuzzy information, Tamiret al [113] have proposed a new way to combine complex number in rep-resentation of intuitionistic fuzzy set, where the membership and the non-membership functions of an intuitionistic fuzzy set are respectively repre-sented as the real and imaginary parts of a complex function in the Cartesian

form One drawback of this formalism, however, the order relation is not a

totally order relation The operations defined are complicated and not readilyaccessible.

The existing representation of IFSs are just two-dimensional tion, which is a drawback of intuitionistic fuzzy sets when we need to solvemore complex problems in which information is sorted out and analyzed

representa-from different dimensions.

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1.2.3 Medical Diagnosis

As the world develops, issues regarding healthcare, economics, education,become increasingly complex In particular, world health is facing new dis-eases, new viruses causing fear and loss of human lives, global economic cri-sis, and many other consequences Given such a situation, decision-making

in general and medical diagnosis in particular urgently call for a need for lutions with high accuracy and efficiency.

so-As for decision-making, the approach to information must first and most be considered Fuzzy set, intuitionistic fuzzy set and some other typesof extended fuzzy sets were developed, such as type-2 fuzzy set, picturefuzzy set, neutrosophic set, etc In particular, intuitionistic fuzzy set is advan-tageous as its performance is relatively neat, the evaluation of information

fore-through the membership and non-membership degrees is relatively complete

and apparently whole in terms of sensory Moreover, the theory of

intuition-istic fuzzy set including intuitionintuition-istic fuzzy logic, measure, relation, system,

etc has relatively been completely constructed Therefore, further

devel-opment of applications of intuitionistic fuzzy sets and systems in

decision-making have been gaining wide attention.

Secondly, a decision-making method is to be developed, including an mation processing system including the evaluation process, handling rules,aggregating process, etc with the output as a decision Currently, there aremethods such as fuzzy inference systems (Mandani system, Takagi — Sugeno

infor-system, Tsukamoto system), intuitionistic fuzzy inference systems, machine

learning, and deep learning In medicine, diagnosis models are very diverse,

such as inference systems based on measurements of Grzegorzewski [40],

Hatzimichailidis et al [44], Park et al [81], Szmidt and Kacprzyk [109], [112],

Wang and Xin [126], Yang and Chiclana [134], etc The Sanchez’s inference

model [97] used the fuzzy relation to represent relationships between

patients-symptoms, symptoms-diseases, and patients-diseases De et al [29] extended

the Sanchez’s method with the theory of intuitionistic fuzzy sets.

Moreover, predicting dental diseases plays a significant role for treatmentof patients, especially in their early stage, as well as for studying the diseases

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in nature It is performed from examination of a dental X-ray image throughits structures namely bones, soft tissues, and teeth [75], [104], [105], [118]-[120] There are several machine learning methods which have been recentlyused in supporting dental diagnosis The fuzzy inference system (FIS) [78],for instance, is a common diagnosis model which uses fuzzy rules The fuzzyk-nearest neighbor method (FKNN) [19], was used in different problems ofhandling dental images A hybrid approach combining decision making,classification, and segmentation methods namely Dental Diagnosis System(DDS) [106] was introduced Some other methods include the Kruskal Span-ning Tree (GCK), the Prim Spanning Tree (GCP), and the Affinity PropagationClustering (APC) [120].

1.3 MOTIVATIONS

There are some problems of the prior researches after literature review:

1 Similar to the philosophy of Pappis [80] to build up the fuzzy proximitymeasure, that is the rule of inference called modus ponens, we believethe intuitionistic fuzzy measure proposed by this inspiration will be pre-served under the sup-min compositional rule of inference, which is anextension of the rule of modus ponens and is used in applications whileexecuting intuitionistic fuzzy algorithms.

2 A rigorous analysis of the axioms of an intuitionistic fuzzy inclusion lation is required, and this relation should be considered as one of thecriteria for constructing a distance measure.

re-3 In some cases, the existing distance measures give the values which are

not really convincing enough In the other words, the similarities

be-tween IFSs should be more rigorous Therefore, it is necessary to extendthe evaluation factors in establishing the measurements.

4 It is necessary to study the representation of IFSs that facilitate the plete ordering of intuitionistic fuzzy sets and at the same time, this orderrelation requires simplicity to set up logical operators.

com-17

5 The existing representations of intuitionistic fuzzy sets are just 2-D dimensional) representations These representations need to be expandedto approach complex problems in which information is analyzed from

(two-different dimensions.

6 In terms of applications, addressing the limitations of the intuitionistic

fuzzy system in medical diagnosis indicated above will improve

accu-racy of the diagnosis.

1.4 OBJECTIVES

The research objectives include the following issuses:

1 Intuitionistic fuzzy measures:

® Survey and analyse the existing fuzzy and intuitionistic fuzzy

¢ Study and propose new intuitionistic fuzzy measures that could

over-come the limitations of the previous measures.

2 Intuitionistic fuzzy representations:

® Survey and analyse the existing representations of intuitionistic fuzzy

3 Applications to medical diagnosis:

¢ Develop new intuitionistic fuzzy inference systems based on the

pro-posed measures and representations for decision-making in medicine.

® Do experiment on the benchmark medical data and compare with the

related methods.

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1.5 RESEARCH APPROACHES

1.5.1 Intuitionistic Fuzzy Measure

¢ Approach for Motivation 1:

Proximity measure was firstly discussed by Pappis [80] to demonstrate theimpractical significance of values of membership Let A and B be two fuzzy

sets on a universe U, and a(x) and g(x) representing their membership

functions, respectively A and B are said to be approximately equal if

sup|Ma(x) — ws(%)| < £,

where £ is a small nonnegative number and called the proximity measure.

Pappis believed that the max-min compositional rule of inference is preservedwith approximately equal fuzzy sets Another approach considered by Hongand Hwang [46], as a generalization of the work of Pappis [80], was mainly

based on the same philosophy of the max-min compositional rule of inference

that is preserved with respect to approximately equal fuzzy sets and imately equal fuzzy relation respectively.

approx-Cai [18] argued that both the Pappis et al approaches were limited to afixed value of ¢, ie they assumed that ¢ is constant and disregarded what“small nonnegative number” means However, different values of ¢ maymake different senses and the role of context is indeed important We alsonote that the notion of e-equality was introduced by Dubois and Prade [37].Two fuzzy sets A and B are said to be e-equality if

S(A,B) > e¢,

where S (A, B) is a similarity measure between A and B Evidently, there is aninherent relationship between proximity measure and e-equality, ie S (A,B)can be interpreted in terms of sup, |A (x) — pp (x)|.

In 2001, Cai [18] introduced 6-equalities of fuzzy set to overcome this

prob-lem in which two fuzzy sets are said to be ô-equal if they equal to a degree of

6 In other words, two fuzzy sets A and B are said to be ô-equality if

sup |Ha (x) — #p (x)| < 1-6,

19

where 0 < 6 < 1 As Cai explained in his paper, the advantage of using 1 — 6rather than e is that interpretation of ổ can comply with common sense Thatis, the greater the value of 6 is, the ‘more equal’ the two fuzzy sets are; andthey become ‘strictly equal’ when 6 = 1 The applications of ổ-equalitieshave important roles to fuzzy statistics and fuzzy reasoning Virant [124]

tested 5-equalities of fuzzy sets in synthesis of realtime fuzzy systems whileCai [18] used them for validating the robustness of fuzzy reasoning accom-panied with several reliability examples through 6-equalities Nonetheless,

there is no such notion in the context of the IFS set.

Inspired by the evaluation on the membership degrees in the Pappis’s imity measure and the Cai’s ổ—equality concept, the concept of ổ—equalitybetween two intuitionistic fuzzy sets will be formed based on the addition ofa similar evaluation on the non-membership degrees The sup-min compo-sitional rule of inference, when used with so defined ổ—equal intuitionisticfuzzy values, should give ổ—equal results.

prox-© Approach for Motivation 2:

As it comes to the existing limitations, the proposed distance measures anddivergence measures are not effective in some cases that requires the estab-

lishment of inclusion relation between IFSs [40], [68], [109], [112] Some thors modified the inclusion relation between IFSs [10] to obtain new distance

au-measures such as Park et al [81] Nonetheless, the modified inclusion tion between IFSs is not a suitable way to further mathematical logic reason-

rela-ing Further in [68], Maheshwari and Srivastava proved that their divergence

measure satisfies basic properties of a distance measure However, we noticethat the divergence measure proposed by Maheshwari and Srivastava [68] isa metric of IFSs based on the modified inclusion relation between IFSs of Parket al [81] instead of using the inclusion relation between IFSs of Atanassov asMaheshwari and Srivastava presented in [68] In other words, the inclusionrelation between IFSs is an important factor for building the distance measurebetween IFSs.

In some cases, the existing distance measures give the values which are

not really convincing enough For instance, the distance measure proposed

20

by Wang and Xin [126] gives out equal values while in reality the distinctionis clearly observed In another example, Szmidt & Kacprzyk [109] introducedthe Hamming distance measure between two intuitionistic fuzzy sets A and

B ona finite space of points X = {x1,X2, ,Xm} as follows

1 m

d(A,B) = 5 >- (Ha (xi) — mp (x¡)| + |tA (Xi) — vB

(3¡))-This measure will increase rigor when đ(A, B) = 0 if the cross-evaluation tween the degrees of membership and non-membership is considered, i.e.,adding consideration |4 — v4| and |p — vp| to the measurement formula.

be-In the other words, the previous distance measures have not thoroughlyevaluated intuitionistic fuzzy information because cross-evaluation betweenthe degrees of membership and non-membership has not been considered.

Hence, these measures were not really effective in the complex decision

mak-ing problems For example, medical diagnosis is not only based on currentsymptoms but also medical history of patients In this situation, if a distance

measure uses the cross-evaluation, it will be easy to evaluate all important

degrees between the membership degree and the non-membership degree ofa patient It is thus able to measure those degrees of a patient in the pasttime as well as the cross-time between the past and present Therefore, thisshows the difference and novelty of the proposal in terms of practical ap-plication Evaluating intuitionistic fuzzy medical information fully throughcross-evaluation will bring more information and accuracy of diagnosis for

Motivated by the above mentioned drawbacks, this research will develop a

new intuitionistic fuzzy distance measure based on the addition of the evaluation between the degrees of membership and non-membership andbased on checking the condition related to an appropriate ordering relation.

cross-1.5.2 Intuitionistic Fuzzy Representation

¢ Approach for Motivation 1:

Ramot et al [88] introduced the CFS - complex fuzzy sets in 2002 Theproposed formalism was based on the polar representation of complex num-

21

bers, where the amplitude is a fuzzy function and the phase is a general tion [88] CFSs are useful in solving complicated problems, such as multipleperiodic factor prediction problems [54] Alkouri et al [6], [8] used complexgrades of membership and non-membership to construct a generalization ofIFSs and CFSs called complex intuitionistic fuzzy set (CIFS) The initial ap-proach for CIFS used the Ramot CFS [88], where the functions of member-ship and non-membership employ the Ramot-based complex fuzzy sets Therange of the complex degree of membership is a unit disk in a complex plane.A decision-making model using the distance measure of CIFSs was presentedby Alkouri and Salleh [7], as an example of the theory.

func-Nevertheless, Ramot’s formalism and the derived CIFS [88], [6] are limitedto the polar representation, where the amplitude term is in the interval [0,1],that conveys indistinct information [115] To overcome this problem, Tamir etal [115] proposed a new concept of complex fuzzy class (CFC) via pure com-plex fuzzy membership degree; here the range of both the real and imaginarycomponents is the unit interval Furthermore, Tamir et al [114], [115] success-fully applied complex fuzzy classes to many problems in physics and stockmarkets In 2016, Mumtaz et al [5] extended the work presented in [114],

[115] to the new concept of complex intuitionistic fuzzy classes.

Recently, Tamir et al [113] have introduced the representation of istic fuzzy set via the complex number function in the Cartesian form

where and v are the functions of membership and non-membership, tively One drawback of this formalism, however, is that the order relation isnot a totally order relation Moreover, the operations defined are complicated

respec-and not readily accessible.

According to the research of Tamir et al [113], the complex number tion in the polar form

func-z= rel® = r (cos + j sin 8),

where r = |z| and Ø = arg (z) will provide new representation of

intuition-istic fuzzy sets based on two new evaluation functions, that is the modulus

22

function r and the argument function Ø.¢ Approach for Motivation 2:

As an extension of the complex number system, a quaternion number hasthe form of a + bi + cj + dk, where quaternion units i, j,k are the square roots

of —1, complying with the conditions /2 = j* = k* = ijk = —1, and a,b,c,d

are real values Although the quaternion multiplication has associative erty, it has not commutative property For examples, jk = —kj = i and

prop-# = (jk) = (jk) (ik) = (jk) (-ki) = —j()j = -7(-Dj = prop-# = -1

hence i* # j*k* Extending the research of Tamir et al [113], this research

de-velops an intuitionistic fuzzy representation based on quaternion numbers.Quaternion number representation can capture composite features and con-vey fuzzy information in four dimensions rather than in two, as in the com-plex number representation.

From the new representations, new order relationships and new distancemeasures will be formed.

1.6 CONTRIBUTIONS

The thesis proposes three main contents including1 Intuitionistic fuzzy measures:

¢ The 5—equality measure.

se The H-max distance measure.2 Intuitionistic fuzzy representations:

e A new representation of intuitionistic fuzzy systems based on

com-plex numbers and a new measure named P—distance.

e Anew representation of intuitionistic fuzzy systems based on nion numbers and a quaternion measure via this representation.

quater-3 Applications to Medical Diagnosis:

® A decision-making system based on the ổ—equality.

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¢ Two different medical diagnosis models from numeric data and age data based on the H-max measure.

im-® A complex intuitionistic fuzzy decision-making system based on theP—distance measure.

® A quaternion measure method for decision-making problems.

The results of the thesis were published in 5 papers [RP1]-[RP5].

1.7, MEDICAL DATA

Table 1.1: The descriptions of experimental datasets.

Dataset No elements No attributes No classes

ILPD 583 8 2LD 345 5 2PIDD 768 5 2

Diabetes 389 4 2Heart 270 4 2HS 306 3 2

Ecoli 336 5 2

DRD 1151 17 2Dermal 358 34 2

CTG 2126 20 2BCW 683 9 2

For testing the proposed diagnosis models in this thesis, the benchmark

medical data have been taken from UCI Machine Learning Repository [122]

such as Heart, ILPD Indian Liver Patient Dataset, PIDD (Pima Indians abetes Data Set), Liver-Disorders (LD), Haberman’s Survival Data Set (HS),Ecoli Data Set (Ecoli), Diabetic Retinopathy Debrecen Data Set (DRD), Der-matology Data Set (Dermal), Cardiotocography Data Set (CTG), and Breast

Di-Cancer Wisconsin (Original) Data Set (BCW), while the remaining benchmarkdataset Diabetes has been taken from Department of Biostatistics, VanderbiltUniversity [42] These data contain 2 classifications, their attribute character-

istics are real numbers, and where each object corresponds to a record Table

1.1 gives an overview of all those datasets Note that, the original Ecoli dataset

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has 8 classes named CP, IM, IMU, IML, IMS OM, OML, and PP Here, they aregrouped into two new classes labeled 1 including CP, IM, IMU, IML, and IMS;and 2 including OM, OML, and PP.

Figure 1.4: The histogram of the 8! attribute of the ILPD Data

Figures 1.3 and 1.4 show the histograms of values of 2 of the 8 attributesof the ILPD dataset In Figure 1.3, we can see the distribution of the values

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of the first attribute of the ILPD dataset Specifically, there are more than

500 elements (patients) with values of Total Bilirubin in the interval |0, 10]

(mg/dL), the Total Bilirubin of the remaining elements are valued as in the

interval (10,80) (mg/dL) Furthermore, the diagnostic model from images

will be tested on the image dataset taking from Hanoi Medical UniversityHospital, Vietnam It includes 56 dental X-ray images with 5 labels whichare Decay, Root fracture, Missing teeth, Resorption of periodontal bone, andIncluse teeth (see Figure 1.5).

Figure 1.5: The dental X-ray images with the corresponding diseases

1.8 PRELIMINARY

1.8.1 Fuzzy Set and Intuitionistic Fuzzy Set

In the classical set theory, the degree of membership of an element in a set canonly be equal to 0 or 1 In some sense, the colors of a particular object canbe associated only with black or white, while gray which can indicate uncer-tainty or ambiguity is largely ignored (see Figure 1.6).

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Figure 1.6: The white-black color strip [77]

In 1965, Zadeh introduced the concept of fuzzy sets [137] where the valuedomain of the degrees of membership is the interval [0,1] as in the followingdefinition.

Definition 1.1 [137] Let U be a space of points A fuzzy set S in U, denoted

by S € FS(U), is characterized by a membership function pgs in [0,1] as,

For example, the degree of membership of Hung in the fuzzy set “potentialcandidates" equals 0.8 The fuzzy logic and set theory was widely recognized

as a real revolution in applied maths for its array of applications in fields likereasoning [26], control theory [35], [62], decision making [65], signal process-

ing [69], [72], medical diagnosis [96], recommender systems [100],

compres-sion [101], dental segmentation [104], [105], geo-demographic analysis [129],

and other fields [56], [63], [107].

The human perception of the color of an object is presumably to have some

varying degree of hesitation which is void of in the notion of Zadel’s fuzzy

sets In 1983, Atanassov developed fuzzy sets by proposing the concept of

intuitionistic fuzzy sets [9] which had immediately attracted the interest of

researchers An IFS represents the state of elements through degrees of

mem-bership and non-memmem-bership.

Definition 1.2 [10] Let U be a space of points An intuitionistic fuzzy set S in

U, denoted by S € IFS(U), is characterized by a membership function /s and

a non-membership function vg with a range in [0,1] such that 0 < ws + 1s < 1.

27

S has representations as follows

S = {(u, ps (M),1s (M)) : u € US (1.2)

and 7rs (1) = 1— (fs (u) + 1s (w)) is called the hesitancy degree of u to S.

According to his definition, when the hesitation degree is nonzero thenthe sum of the membership degree and the non-membership degree is less

than 1 This condition truly reflects the human’s sense about a particularissue, for example, “like”, “dislike”, and “undecided” For example, the de-gree of membership and non-membership of Hung in the fuzzy set “potential

candidates" equal 0.8 and 0.1, respectively, i.e., the hesitation degree here is

1— (0.8 + 0.1) = 0.1 So far, there have been a series of publications about

the success of applying the intuitionistic fuzzy logic and set theory to real life

problems, such as in the fields of decision making [24], computational

intel-ligence [25], modeling imprecision [33], medical diagnosis [102], [103], [110],[111], and pattern recognition [125].

For later convenience when working on the objects of the intuitionistic

fuzzy set theory, we call L* to be the set of intuitionistic fuzzy values u =

(1„12), Ì.©.,

L* = {u = (,uạ) |uy, uz € [0,1], + uạ < 1} (1.3)

1.8.2 Intuitionistic Fuzzy Order and Operations

It can be said that all solutions of a problem stem from inference In orderto implement a logical reasoning system, it is necessary to format a group ofpremise reasoning rules For example,

¢ Event A: “Hung has an lelts score of 8.0 and a clean resume.

s® Rule R: “If a candidate has an English language proficiency certificate

above 7.5 or a Toefl test of above 96 and does not have a criminal record,

then he is recommended."

¢ Inference: A and R infer “Hung is chosen".

It can be seen that the order, negation, and basic links “and”, “or” and

“in-fer" are the “chain couplings" of an inference In terms of mathematical logic,

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they are the basic logical operators: negation, t-norm, t-conorm, and

¢ Inference: A and R infer “Hung is chosen".

In this case, the classical set-theoretic operators are replaced by the tended operations such as the fuzzy operations or the intuitionistic fuzzyoperations However, their characteristics are constant, for instance, the na-tures of order relations are reflexive, anti-symmetric and transitive properties.Some fuzzy and intuitionistic fuzzy operators are shown as follows.

ex-Definition 1.3 [57] A fuzzy negation n is a nonincreasing |0, 1| — [0,1] tion that satisfies n (0) = 1,n (1) = 0 A fuzzy negationn is called an involu-tion iff n satisfies n (n (x)) = x,Vx € [0,1].

func-For example: ng (x) = 1—x,Vx € [0,1] is an involutive fuzzy negation,

called standard fuzzy negation.

Definition 1.4 [57] A mapping í : [0, 1]* — [0,1] isa fuzzy t-norm if f satisfies

all of the following conditions:

1 t(x,1) = x,Vx € [0,1] (border condition).

2 t(x,y) = t(y,x),Vx,y € [0,1] (commutativity).

3 t(x,t(y,z)) = t(t(x, y),z),Vx,y,z € [0,1] (associativity).

4 t(x,y) < t(x’,y'),Vx,x',y,y' € [0,1] |x < x,y < y’ (monotonicity).

Definition 1.5 [57] A mapping s : [0,1]? — [0,1] is a fuzzy t-conorm if s

satisfies of all the following conditions:

s(x,0) = x,Vx € [0,1] (border condition).2 s(x,y) = s(y,x),Vx,y € [0,1].

3 s(x,s(y,z)) = s(s(x,y),z),Vx,y,z € [0,1].

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4 s(x,y) < s(x’, y'),Vx,x,y,y' € [0,1||x < x,ụ < 0.

Definition 1.6 [57] Let t be a fuzzy t-norm, then,

® tis Archimedean iff t is continuous and t(x,x) < x,Vx € (0,1).® t is nilpotent iff f is Archimedean and 3x, y € (0,1],t(x,y) = 0.® tis strict iff t is Archimedean and Vx,y € (0,1],f(x,) 4 0.

Definition 1.7 [57] Let s be a fuzzy t-conorm, then,

® sis Archimedean iff s is continuous and s(x,x) > x,Vx € (0,1).® sis nilpotent iff s is Archimedean and 3x, y € [0,1),s(x,y) = 1.® s is strict iff s is Archimedean and Vx,y € [0,1),s(x,y) # 1.

Proposition 1.1 [49] An Archimedean t-norm t is strict iff there is an order

isomorphism f : [0,1] — [0,1] (i.e., f is continuous, strictly increasing, ƒ (0) =

O and f (1) = 1) satisfying f (x,y) = f-1 (F(x) f

(y))-Proposition 1.2 [49] An Archimedean t-norm t is nilpotent iff there is an

or-der isomorphism f on [0,1] satisfying f (x,y) = f-! ((f (x) +f (y) -—1) V0).

In Proposition 1.1, 1.2, ƒ is called generator function of í.

Proposition 1.3 [49] An Archimedean t-conorm s is strict iff there is an order

isomorphism f on [0,1] satisfying s (x,y) = f~1 (f (x) + f (y) — f (x) f(y)).

Proposition 1.4 [49] An Archimedean t-conorm s is nilpotent iff there is an

order isomorphism ƒ on [0,1] satisfying s (x,y) = f-! ((f (x) + f(y)) A1).

In Proposition 1.3, 1.4, f is called cogenerator function of s.

Definition 1.8 [32] Let x,y € L*, then the order relation on L* is defined by

X<pHY © #1 <1, X2 => 12, (1.4)

The lattice (LÝ, <rx) is complete [32], where the unit is 1x = (1,0) Besides

that, Park et al [81] introduced the order relation <1, as follows

x <‡ U«© XI < 1,12 > Yn, My = 1— xị— #2 > My = 1—q Ta (16)

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Definition 1.9 [49] An intuitionistic fuzzy negation N is a nonincreasingL* + L* function satisfying (0x) = 1p+, N (1p*) = Ops If NV (NM (x)) =

x,Vx € L*, then N is involutive.

For example, Ns (x) = (x2,x1),Vx € L* called standard intuitionistic

fuzzy negation.

Definition 1.10 [32] An intuitionistic fuzzy triangular norm T isa L*? — L*

function satisfying all the following conditions for all x, y,w,z € LẺ,

4.7 (x,y) ,2) =7 (%,7 (W,z)):

Example 1.1 Some intuitionistic fuzzy triangular norms are given below For all

x,y € L* and ÀA € [0,1],

° Ti (x,y) = (Xi A yi, X2 V W2),

© 72 (x,y) = (X1/1,X2 + Y2 — X22),

® 73 (x,y) = (max (0,x1 + yi — 1),min (1, x2 + y2)),

© 74 (x,y) = (max (0,A (x1 + y1) —A + (1— A) x11/¡),mỉn (1,x2 + y2)),

® 75 (x,y) = (max (0, x1 + Vị — 1)„X2 + 2 — x2y2),

® 76 (x,y) = (max (0,A (41 + y1) —A + (1— A) 191) 2 + 2 — X22).

Definition 1.11 [32] An intuitionistic fuzzy triangular co-norm S is a L** >

L* function satisfying all the following conditions for all x, y,w,z € L*,$(0:,x) =x;

S (x,y) <1+S (w,z) whenever x<p+w and y<1+z;

S(x,y) = S(.*);

S(S (x,y) ,2) = § (x8 (y,2)).

Example 1.2 Some intuitionistic fuzzy triangular co-norms are presented below.

For all x,y € L* and A € [0,1],

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© ổi (x,y) = (XỊ VựI,32 A 12),

© So (x,y) = (XI +1 — X1Y1, X2Y2)

© S3 (x,y) = (min (1, x1 + y1),max (0,32 +y2—1)),

© Sy (x,y) = (min (1, x1 + 1⁄4) „max (0,A (xa + 2) — A + (1— A) x2y2)),

© S5 (x,y) = (XI + y1 — X1Y1, max (0,x2 + 2 — 1)),

© 6 (x,y) = (Xị + ị — X1y1,max (0,A (x2 + y2) — A+ (1 — A) x2y2)).

Definition 1.12 [32] An intuitionistic fuzzy implication 7 is a L*? + L*

func-tion satisfying I(Oz+,Oz*) = 1y*, I(0¡>, 11x) = 1x, I(1i>, 11x) = 1i», I{1r:,0rx)

= U¡:, and the two following monotonicity:

1 if x<p+y, then Z(x,z)>r*(y,z), Vz € L*,2 if y<p*z, then Z(x,y)<y*(x,z), Vx € L*.

For example, Z(x,y) = (xa V yi,%1 A Y2),Vx,y € L*.

Definition 1.13 [10], [76] Some set operations are defined as follows Let

A,B € IFS(X), then

ACB pa (x) S #pg (X), Va (x) 2 1p (x), Vx € X, (1.7)

ACB & pa (x) < tp (x), va (x) > vp (x), 74 (x) > 7p (x), Vx € X, (1.8)

ANB = {(x,min {Hạ (x) „ty (x) } „max {v4 (x) ,vp (x)})|x € X},

AUB = { (x,max {pi (x), pp (x)},min {vy (x) ,ve (x)})| x € X},

AS = { (x,va (x) fa (x))| x € X}.

Proposition 1.5 [32] Let VV be an involutive intuitionistic fuzzy negation,

then there exist an involutive fuzzy negation n satisfing

N(x) = (n(1 — x2),1 —n(x1)),Vx € L*.

Definition 1.14 [49] An intuitionistic fuzzy t-norm T is called t-representable

iff there exist a t-norm f and a t-conorm s on [0,1] satisfing

T (x,y) = (E(X1,1)„,s (Xz,2)),Vx, € LẺ.

Definition 1.15 [49] Let S be an intuitionistic fuzzy t-conorm S is called

t-representable iff there exist a t-norm í and a t-conorm s on |0, 1| satisfing

S (x,y) = (s (%1,Y1) ,# (X2,y2)),Vx,y € L*.

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Definition 1.16 [49] Let A,B € IFS(X) Let N(x) = (n(1— xa),1— m0(#1)),T = (f,s) and S = (s,t) be an involutive intuitionistic fuzzy negation, a t-

representable intuitionistic fuzzy t-norm, and a t-representable intuitionisticfuzzy t-norm, respectively Then,

AUsB = { (x, (MA (x) He (X))„† (va (x),vg (x)))|x © X}:

Definition 1.17 [49] Let V, 7 and S be an intuitionistic fuzzy negation, an

intuitionistic fuzzy t-norm, and a intuitionistic fuzzy t-norm, respectively If

X(S(x,w)) = TIN (2) NY), NT (ay) = SIN (2) NY), Vary € LY,

then 7 and S are called dual w.rt N and (7, M,®) is called a De Morgan

intuitionistic fuzzy triple.

Proposition 1.6 [32] Let (7, V, S) bea De Morgan intuitionistic fuzzy triple,where A is involutive, then 7 is t-representable iff S is t-representable.

1.8.3 Intuitionistic Fuzzy Relation and Similarity Measures

Definition 1.18 [16] Let X and Y be two spaces of points An intuitionistic

fuzzy relation (IFR) R between X and Y (R € IFR(X x Y)) is an

intuitionis-tic fuzzy set in X x Y,ie.,

R= {((x,Y¥) PR (XY) VR (x,J)):x€ X,u€ Y}, (1.9)

where [ip (x,y): X x Y — [0,1], vz (x,y) : X x Y — [0,1] satisfy the

0 < pr (x,y) + 2z (x,y) < LV (x,y) eXx Y (1.10)

Definition 1.19 [126] A mapping M, : L** — [0,1] is a similarity measure

on L* if it fulfils the axioms as follows:

33

1.0<M; (x,) < 1

2 If x = y, then M, (x,y) = 1;3 Mg (x,y) = Ms (y, x);

4 If x<p*y<zp+z, then M, (x,z) < Ms (x,y) and M, (x,z) < Ms (y,Z).

Definition 1.20 [126] Let A, B,C € IFS(X) A mapping d : IFS(X) x IFS(X) >

R is a distance measure of IFSs if it fulfils the axioms as follows:

i, d(A,B) >0

ii, d(A,B) =d(B,A).

11, đ(A,B) = Oiff A= B.

iv, If AC B CC thend(A,C) > d(A,B) andd(A,C) > d(B,C).

Definition 1.21 [18] Let A,B ¢ IF(U), then A and B are said to be ổ—equal

denoted by A = (6) B, if

sup |4 (x) — #g(x)| <1-6, 0<6<1 (1.11)

In this way, we say A and B construct 6—equality.

Now, let A,B € IFS(X = {#\,xa, ,x„ }) and 1, v;, and 71; are their

mem-bership, non-memmem-bership, and hesitancy function, respectively, i = 1;2 Someof the existing measures d = d(A, B) are listed as follows.

Definition 1.22 [109] Hamming distance measure

đHam = 5, dL Ha (Xi) — ta (xi) | + [1 (41) — v2 (X¡)|), (1.12)

1 m

diam = 2m 2 (lim (xi) — po (xi)| + [vi (xi) — 92(3¡)| + |7n(X¡) — Za(3¡)|), (113)

Definition 1.23 [109] Euclidean distance measure

dp = (= » (0n (xi) — Ha (xi))? + (tị (xi) = 12 (1;))Ÿ))ˆ⁄3, (1.14)

= ((ge1 (x2) — ta(xi))ˆ + (va (x2) — 92(xi))” + (m(xi) — 7a(3;))”))12,

Definition 1.24 [112] Distance measure of Szmidt & Kacprzyk (2004)

1 li (x) — ta (x2) | + [tị (xi) — 92 (xi) | + | (xi) — 7a (xi)|

(SK = m Ua Gai) = 92 (x0 3Ì (i) HC) mi (x)= a()| 2 (9

Definition 1.25 [40] Hausdorff distance measure

dHau = * }_ max {| (x1) — 2 (xi) |, [v1 (x4) — v2 (xi) |}, (1.17)

Definition 1.26 [126] Distance measure of Wang and Xin (2005)

dav = gy Dv (X) = a (+ la (2) — 1289|

+2max {| (xi) — Ma (xi) |, [tt (xi) — 92 (xi) |), (1.18)

Definition 1.27 [81] Distance measure of Park et al (2009)

1 m

dj = âm: (la (xi) — Ha (i)| + [ti (Xi) — v2 (3i)| + |7 (Xi) — 72 (3) |

+2max {| (Xi) — Ma (xi) |, [ti (Xi) — 92 (xi) |, |7 (1) — Z2 (xi)|}), 12)

Definition 1.28 1 Divergence measure of Maheshwari et al (2016)

đụs = —log;( Elvin 1) wa (xi) + Vi (8i) v2 (xi) + ym (xi) 72 (01).

1.8.4 Complex Fuzzy Set and Complex Intuitionistic Fuzzy Set

In 2002, Ramot et al [88] introduced the concept of complex fuzzy set, where

the membership function is the complex function in the polar form including

the fuzzy amplitude function and the general phase function.

Definition 1.29 [89] A complex fuzzy set S over a universe X, is formed by

S = {(,s(x)):x€ X}, (1.21)

where the complex-valued membership function, 7s, has the form ps.el ls,

here j = —1, the amplitude function, ps, satisfies ps : X — [0,1], and the

function /s is real-valued on X.

In 2012, Alkouri et al [6], [8] proposed the complex intuitionistic fuzzy

set based on the Ramot CFS [88] by accreting the function of complex-valued

35

Definition 1.30 [6] A complex intuitionistic fuzzy set S on a universe U is

characterized by the membership function jis (x) = rg (x) e/ “us(*) and the

non-membership +s (x) = ks (x) e MAN respectively, where i = /—1, each

of rs (x) and kg (x) belongs to [0,1] such that 0 < rg (x) + kg (x) < 1, alsoWy (x) and wy, (x) are real-valued.

Furthermore, Tamir et al [115] (2011) and Mumtaz et al [5] (2016) proposedthe complex fuzzy and complex intuitionistic fuzzy classes by combining therelation of an element and a set and the relation of a set and a class.

Definition 1.31 [115] A complex fuzzy class Ï over a universe X has the

representation as follows

T = {(E,x, wp (E,x)) : E € 2%,x € Ä}, (1.22)

where 2* is the power-set of X The pure complex membership degree is the

degree of membership of E in land the membership degree of x in E, which

is tr (E,x) = pr (E) + Jmị (x), where ty (E)„ wị (x) € [0,1].

Similarly, the concept of complex intuitionistic fuzzy class was introducedby adding to the concept of complex fuzzy class the degree of pure complexnon-membership [5] In 2016, complex numbers in the Cartesian form were

employed in the representations of IFSs in a simpler fashion.

Definition 1.32 (Tamir et al [113]) Let A be an IFS characterized by thecomplex number function Z = ji + jv, where ji, : X — [0,1] satisfying

ji+v € [0,1] are the functions of membership and non-membership,

respec-tively As a set of ordered pairs, the IFS A can be represented as:

A ={(3,z)|#e€ X,2 = ji(x) + jv (x)} (1.23)

Further, an IFS A is defined to be a subset of an IFS B, iff 2, < Zp, ie.

1.8.5 Intuitionistic Fuzzy Systems

Basically, a fuzzy system (fuzzy inference system) is described as Figure 1.7.

An intuitionistic fuzzy system (intuitionistic fuzzy inference system) is

sim-ilar to a fuzzy system, only changing the execution environment from the

fuzzy environment to intuitionistic fuzzy environment.

36