Novel Approach Intuitionistic Fuzzy Transportation Problems by Ritesh kumar Dubey

1. Submitted by:-
Mr. RITESH KUMAR DUBEY
MATS UNIVERSITY INDIA
Presentation
On
Novel Approach Intuitionistic Fuzzy
Transportation Problems

2. INDEX
INTUITIONISTIC
FUZZY
TRANSPORTATION
PROBLEM
ABSTRACT
DEFINITION
INTRODUCTION
EXAMPLE
BASIC
OPERATIONS
CONCLUSION
REFERENCE
NOVEL
APPROACH

3. Abstract
Keyword: Intuitionistic Fuzzy Transportation
Problems
in real-life decisions usually we have to suffer through different states of
uncertainties. In this article, we formulate a transportation problem in which
costs, supplies and demands all are different types of real, fuzzy or
intuitionistic fuzzy numbers that s the data has different types of
uncertainties.

4. In several real life situations , there is need of shipping the product from
different origins (sources) to different destinations and the aim of the decision maker
is to find how much quantity of the product from which origin to which destination
should be shipped so that all the supply points are fully used and all the demand
points are fully received as well as total transportation cost is minimum for a
minimization problem or total transportation profit is maximum for a maximization
problem.
Since, in real life problems, there is always existence of impreciseness in the
parameters of transportation problem and in the literature Atanssov (1983), pointed
out that it is better to use intuitionistic fuzzy set as compared to fuzzy set Zadeh
(1965) to deal with impreciseness.
Introduction

5. The major advantage of Intuitionistic Fuzzy Set (IFS) over fuzzy set is
that Intuitionistic Fuzzy Sets (IFSs) separate the degree of membership
(belongingness) and the degree of non membership (non belongingness) of
an element in the set. In the history of Mathematics, Burillo et al. (1994)
proposed definition of intuitionistic fuzzy number and studied its properties.

6. Fuzzy sets can be thought of as an extension of classical sets. In a
classical set (or crisp set), the objects in the set are called elements or members
of the set.
An element x belonging to a set A is defined as x Є A an element that is
not a member in A is noted as x ɆA. A characteristic function or membership
function µA(x) is defined as an element in the universe U having a crisp value of
1 or 0. For every x Є U,
µA(x) = { 1 for x ɆA 0 for x Є A }
This can also be expressed as µA(x) Є {0 ,1}
Definition

7. In crisp sets the membership function takes a value of 1 or 0. For fuzzy sets, the
membership function takes values in the interval [0, 1].
The range between 0 and 1 is referred to as the membership grade or degree of
membership.
A fuzzy set A is defined as:
A = { (x, µA(x) ) x Є A, µA(x) Є [0, 1]}
Where µA(x) is a membership function belongs to the interval [0, 1].
Fuzzy set theory has equivalent operations to those of crisp set theory. It includes
functions such as equality, union and intersection etc. Membership functions can
be defined as the degree of the truthfulness of the proposition

8. Fuzzy Set : Let X be a universal set. Then a fuzzy set A in X is defined by:
A= {(x, µA (x) ) | x €X} where μ A: X → [0, 1].
Intuitionistic Fuzzy Sets: Let a set X be fixed. An IFS A in X is an object of
the following form :
A = {(x, µA ( X ) , VA(x ) ) x € X },
When Va(x) = 1 - |1a(x) for all x € X is ordinary fuzzy set. In addition, for
each IFS A in X, if
µA(x)= 1- µx - Vx Then µA(x) is called the degree of indeterminacy of x to
A, or called the degree of hesitancy of x to A. Especially, if µA(x) = 0, for all
x € X then the IFS, A is reduced to a fuzzy set.

9. The whole concept can be illustrated with this example. Let's talk about
people and "youthfulness” In this case the set S (the universe of
discourse) is the set of people.
A fuzzy subset YOUNG is also defined, which answers the question "To
what degree is person x young?" To each person in the universe of
discourse, we have to assign a degree of membership in the fuzzy subset
YOUNG. The easiest way to do this is with a membership function based
on the person's age.
young(x) = {1, if age(x) <= 20,
(30-age(x))/10, if 20 < age(x) <= 30,
0, if age(x) > 30 }
Example

10. A graph of this looks like:
Given this definition, here are some example values:
Person Age degree of youth
Johan 10 1.00
Edwin 21 0.90
Laiba 25 0.50
Arosha 26 0.40
Ahmad 28 0.20
Ravi 83 0.00
So using the definition, we'd say that the degree of truth of the statement
“Laiba is YOUNG" is 0.50.

11. Types of Membership Function
Zadeh has proposed a series of membership function like Triangular
Singleton, L function etc. Depending on the types of function different
fuzzy sets are obtained.
Triangular: Defined by its lower limit a, its upper limit b and the
modal value m, so that a<m<b. We call the value b-m margin when it is
equal to the value m-a.
Fig-Triangular Fuzzy Set

12. Singleton: It takes value 0 in all the universe of discourse except in
the point m, where it takes the value 1.It is the representation of a crisp
value.
Fig: Singleton Fuzzy Set

13. L function: This function is define by two parameters a and b in the
following way
Fig-L-fuzzy set

14. Trapezoid Function: Defined by its lower limit a and its upper limit d and the upper
and lower limit of its nucleus, b and c respectively
In general the trapezoid function adapts quite well to the definition of any
concept, with the advantage that it is easy to define, represent and simple
to calculate.

15. Some basic operation on Intuitionistic Fuzzy sets

16. Conclusion
TP having uncertainty as well as hesitation in prediction of the transportation cost
has been investigated. In the TP considered in this study, the values of
transportation costs are represented by intuitionistic fuzzy numbers and the values
of supply and demand of the products are represented by real numbers. Here, we
proposed an efficient computational solution approach for solving intuitionistic
fuzzy TP based on classical transportation algorithms. Here, we shall point out that
the IFTP studied in this paper is not in the form of a problem whose demands and
supplies are as intuitionistic fuzzy numbers too. Therefore, further research on
extending the proposed method to overcome these shortcomings is an interesting
stream of future research.

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B. A. Nagoor Gani, A. Edward Samuel, and D. Anuradha, Optimization of Air Lift Operation under Fuzzy Environment,
Applied Mathematical Sciences, Vol. 4, 2010, no. 55, 2723 – 2732.
C. K.T. Atanssov, Intuitionistic fuzzy sets, Fuzzy sets and Systems, Vol. 20, No.1, 87-96, 1986.
D. Kumar, P. S., & Hussain, R. J. (2014) A systematic approach for solving mixed intuitionistic fuzzy transportation
problems. International Journal of Pure and Applied Mathematics, 92,181–190.
E. P. Grzegorzewski, E. Mr´owka, Trapezoidal approximations of fuzzy numbers, fuzzy sets and systems, 153 (2005)
115-135.
Reference

18. THANK
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