A STUDY ON INTUITIONISTIC FUZZY OPERATORS

IN DECISION MAKING

SYNOPSIS SUBMITTED TO MADURAI KAMARAJ UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

AWARD OF THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

Researcher

R. NAGALINGAM

(Registration No. P4948)

Research Supervisor

Dr. S. RAJARAM

Associate Professor and Head

PG and Research Department of Mathematics

Sri S. Ramasamy Naidu Memorial College

Sattur – 626203.

India

MADURAI KAMARAJ UNIVERSITY

(University with potential for Excellence)

MADURAI – 625 021

TAMIL NADU

INDIA

May 2020

SYNOPSIS

The thesis entitled “A Study on Intuitionistic Fuzzy operators in decision making”

embodies the work done by Mr. R. Nagalingam, Part-time Research Scholar,

Sri S. Ramasamy Naidu Memorial College, Sattur, Virudhunagar District, Tamilnadu

under the guidance of Dr. S. Rajaram, Head and Associate Professor of Mathematics,

Sri S. Ramasamy Naidu Memorial College, Sattur - 626203, Virudhunagar District,

Tamilnadu.

German mathematician George Cantor (1843-1918) introduced fundamental set theory

and it is necessary for the whole mathematics. Set theory is actually the language of

science, mathematics and logic. The concept of vagueness is a long time challenge for

mathematicians and it is a crucial issue in the area of artificial intelligence in computer

science. To overcome this situation the concept fuzzy set was introduced by American

mathematician Zadeh. L.A [45]. He successfully used the fuzzy set concept to handle

uncertainity in decision making. In fuzzy set concept, a membership function is defined to

assign each element of the reference system, a real value in the interval [0, 1]. The

membership value of an element is zero indicates that the element does not belong to the

class. The membership value of an element is one indicates that element belongs to that

class and other values between zero to one indicate the degree of membership to a class.

The main drawback of fuzzy set theory is the inconclusive property because the

exclusiveness of non-membership function and the ignorance for the possibility of

hesitation margin. To overcome the above drawback Atanassov K.T [3] carefully studied

these drawback and proposed a new concept namely intuitionistic fuzzy sets[IFSs].

Intuitionistic fuzzy set is more adjustable and reasonable in dealing with vagueness and

fuzzy information which has given deep attention from literature. Intuitionistic fuzzy sets

accommodate both membership function and non-membership function with hesitation

margin.

Decision making on IFSs

Decision is an option made between the alternative courses of action in uncertainty

situation. Decision making is the study of identifying and selecting suitable alternatives

based on the data values and preferences of the decision maker. Decision making is an

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important activity of management sector and it is a huge part of all process of

implementation. The theory that controls decision making is called decision theory.

The concept of Intuitionistic Fuzzy Set has proven very interesting in various application

problems. Intuitionistic Fuzzy Set concept is a feasible tool to study the problems in

decision making and it seems to be gifted in solving many real life situations, medical

diagnosis, reasoning, portfolio selections and mathematical problems in Engineering. It is

also used in robotics, agriculture, control systems, computers, economy and many

Engineering fields. IFSs have been applied in the field of multi-criteria decision making

problems. Initially fuzzy relations, different distance methods, similarity measures,

max-min and min-max rule are used to solve decision making problems.

Decision making by using IFS Operator

An operator is a special symbol performing specific operations. Many operators have

been defined over intuitionistic fuzzy sets. Different operations and operators were

proposed by many researchers. At the beginning, Min. operator for the intersection and

the Max. operator for the union were given in L.A. Zadeh’s fuzzy set theory. In [1],

Anton Antonov proposed symmetrical difference operator over IFSs. Anton Cholakov [2]

defined a new operation over IFSs. Atanassov. K. T initiated and defined various

operations over intuitionistic fuzzy sets. The operations , $ and # were

defined by Atanassov. K. T [5] on IFSs. At first Atanassov. K. T [4] defined the level

operators and and established relations for each α, β [0,1]. Later on

Atanassov. K. T [8] introduced extended level operators over intuitionistic fuzzy sets.

Beloslav Riecan and Atanassov. K. T [15, 16] defined the operators division by n and

“ n ” extraction operation over intuitionistic fuzzy sets. In [19], De. S. K et al. defined

different operations on IFSs. An operator which maps intuitionistic fuzzy sets into fuzzy

sets was proposed by Vassilev. P [39]. This type of modal operators and a series of

their extensions were described in the two books of Atanassov. K. T [3, 5]. Generally

the IFS operators are classified into three categories namely modal, topological and level

operators. Yilmaz. S and Cockhan Cuvalcioglu. G [44] introduced new type of level

operators over temporal intuitionistic fuzzy sets. Sheik Dhavudh. S and Srinivasan. R [38]

proposed some level operators on L-fuzzy sets and proved some of their properties. In [9],

Atanassov. K.T proposed some relations between intuitionistic fuzzy negations and

intuitionistic fuzzy level operators and Baloui Jamkhanesh. E and

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Nadarajah. A [14] defined four new level operators and for α, β [0,1] over

generalised intuitionistic fuzzy sets and established some of their properties. In [28],

Liu. Q et., al proposed new operators and derived some new results in IFSs. In [33],

Parvathi. R and Geetha. S. P defined some level operators namely max-min implication

operators and operators and over temporal intuitionistic fuzzy sets. In [43], Xu

defined operational laws of intuitionistic fuzzy information, including the intuitionistic

fuzzy averaging operators and intuitionistic fuzzy aggregation operators.

Many day to day life application problems are solved by using IFSs operators. Different

IFS operators are also useful in solving Multiple Attribute Group Decision making

problem. In the last two decades, so many authors have paid more attention in solving

application problems in the various fields like decision making, medical diagnosis and

market prediction etc.,by IFSs operators. In [18], Cökhan Cuvalcioglu and Esra Aykut

applied some intuitionistic fuzzy modal operators to agriculture. In [20], Evdokia

Sotirova et al. applied inter criteria analysis approach to health related quality of life.

In [21, 22], Ejegwa Paul Augustine used intuitionistic fuzzy sets in career determination,

medical diagnosis and pattern recognition. In [25], Eulalia Szmidt and Janusz Kacprzyk

used intuitionistic fuzzy sets in some medical applications. In [27], Evdokia

Sotirova et al. applied inter criteria decision making method to the ranking of Universities

in the United Kingdom. In [29], Lyubka Doukovska and Vassia Atanassova used inter

criteria analysis approach in radar detection threshold analysis. Pathinathan. T [34] et al.

discussed max-min composition algorithm for predicting the best quality of two

wheelers.

In this thesis, we introduce new type of intuitionistic fuzzy set operators and extension

operators. By using the proposed operators we discussed eight application problems

which will be useful to our day to day life.

Structure of the Thesis

The thesis consists of following :

Chapter 1 Preliminaries

Chapter 2 New Intuitionistic Fuzzy Operators , and extension of

Chapter 3 Second degree homogeneous intuitionistic fuzzy operators

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Chapter 4 New type of extension of the level operators and on intuitionistic

fuzzy sets

Chapter 5 Different types of level operators and Modal operators analogous to

necessity and possibility operators on intuitionistic fuzzy sets

Chapter 6 The IFS Operators on New Generalized Intuitionistic Fuzzy set type

NGIFS

and One Hundredth of Intuitionistic Fuzzy Sets

ORGANIZATION OF THE THESIS

This research work is based on intuitionistic fuzzy operators and its applications on

decision making.

Chapter 1 deals with concepts in intuitionistic fuzzy sets , some basic definitions related

to the topic, overview of the related literature and organization of the thesis.

Chapter 2 deals with new operators , A(m,n) and extension of A(m,n). Some

equalities and theorems connected with the proposed operators are proved. A decision

making problem is discussed by using the operator to select the future

course of education. Another application problem regarding the selection of suitable

college to join after the completion of higher secondary course is established by using the

proposed operator

.

In Chapter 3, we have proposed some second degree homogeneous intuitionistic fuzzy

operators. Some theorems and properties have been verified for the proposed operators.

The operator “ ” is used in application problem to select two suitable players in the

place of two injured players of Junior Indian Foot ball team.

In Chapter 4, we proposed seven new type extensions of the level operators and

for every α, β [0, 1]. The fourth, fifth and sixth type of extension are proposed by

using Arithmetic Mean (A.M), Geometric Mean (G.M) and Harmonic Mean (H.M).

Some theorems related to the proposed extension operators are proved. Application

problems related to the operators , (A) and

are discussed.

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In Chapter 5, Different types of ten level operators and two types of modal operators

which are analogous to the necessity and possibility operators have been proposed.

Relations and operations between intuitionistic fuzzy sets are applied in the proposed

level operators. Some theorems related between the analogous type of necessity and

possibility modal operators and the existing operators have been proved. Two application

problems have been discussed. One of them is to select a suitable bride to validate the

level operator and another is to fix best channels in our T.V by using the

operators ( A) ( A) and ( A) ( A).

In Chapter 6, we have introduced new type of intuitionistic fuzzy sets namely OHIFSs

and new generalized intuitionistic fuzzy set collection NGIFS

. Some theorems

related to set relations and set operators in NGIFS

and OHIFSs are proved.

Important operators like modal, necessary and sufficient operators are introduced and

relationship between them have been proved in NGIFS

. By using the usual

operator “ + ” and “ some results in IFSs are discussed in OHIFSs also. An application

problem namely winner of the four state assembly election in India is predicted by

using the NGIFS operator ( ) ( ) , i = 1, 2, 3 and another

application problem for the selection of medical treatment among Allopathy, Ayurvedic

and Homeopathy is discussed by using the operator .

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APPENDIX

LIST OF PUBLICATIONS

Most of the results in this thesis have been part of the following research papers

published in various journals.

i) New Intuitionistic Fuzzy Operator and an application, Advances in Fuzzy

Mathematics, Vol 12, Number 4, 2017, ISSN 0973 - 533X, pp 881- 895.

ii) Intuitionistic Fuzzy Operator and its extentions, International Journal of

Mathematical Combinatorics, the Proceedings of the International Conference on

Discrete Mathematics and its Applications, M.S.University, (ICDMA 2018) Nov 2018,

Special Issue 1, ISSN 1937-1055, pp 144-154.

iii) New Generalized Intuitionistic Fuzzy sets NGIFS

,

International Journal of Computer Science, the Proceedings of the International

Conference on Algebra and Discrete Mathematics (ICADM 2018), Jan 08-10,

Special Issue ISSN No : 2348-6600, pp 69-75.

iv) One Hundredth of Intuitionistic Fuzzy Sets, Journal of Computer and

Mathematical Sciences, Vol 10(3), pp 445-453, March 2019, ISSN 0976-5727 (Print),

ISSN 2319-8133 (online)

v) Advanced intuitionistic fuzzy operators and its properties, International Journal

of Management, IT and Engineering, ISSN No, 2249-0558, pp 210-221.

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Communicated

i) Second degree homogeneous intuitionistic fuzzy operators, Springer, Special

edition, the Proceedings of the International conference on Applications of Basic

Sciences (ICABS 2019), Bishop Heber College, Tiruchirappalli, Tamilnadu, India.

ii) New type of extension of the level operators and on intuitionistic

fuzzy sets, Journal of the Maharaja Sayajirao University of Baroda, ISSN NO

0025-0422, UGC Care Group D Journal.

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List of papers presented at Conference

i) New distance measures, similarity measures between two intuitionistic fuzzy sets

and its application to decision making, CSIR sponsored conference

ICAMM 2018 organized by PSG college of Technology, Coimbatore, Taminadu on

January 6, 2018.

ii) New Generalized Intuitionistic Fuzzy Sets NGIFS

,

International conference on Algebra and Discrete mathematics (ICADM 2018)

organized by Madurai Kamaraj University, Madurai, Tamilnadu, January 08-10, 2018

iii) Intuitionistic Fuzzy Operator and its extentions, International Conference

on Discrete Mathematics and its Applications organized by Manonmaniam Sundaranar

University, Tirunelveli, Tamilnadu, India during January 18-20, 2018 (ICDMA 2018).

iv) One Hundredth of Intuitionistic Fuzzy Sets, International Conference on Graph

theory and its Applications (ICGTA 19) organized by Sadakathullah Appa College,

Tirunelveli, Tamilnadu on February 27, 2019.

v) Different type of level operators on intuitionistic fuzzy sets, National Conference

on Progress in Mathematics towards Industrial Applications (PMTIA-2019) conducted

at SRM IST, Ramapuram, Chennai, Tamilnadu, India during 27th

& 28th

of

September, 2019

vi) Second degree homogeneous intuitionistic fuzzy operators ” UGC, DST - FIST

& DBT Sponsored International Conference on Applications of Basic

Sciences (ICABS 2019) held on November 19 - 21, 2019, Bishop Heber

College, Tiruchirappalli, Tamilnadu, India.

vii) Modal operators analogous to necessity and possibility operators on

intuitionistic fuzzy sets, International Conference on Mathematical Analysis and

Computing (ICMAC-2019), December 23 - 24, 2019 organized by the Department

of Mathematics, SSN College of Engineering, Chennai, Tamilnadu, India.

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