a study on intuitionistic fuzzy operators in …
TRANSCRIPT
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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