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International Journal of Pure and Applied Mathematics
Volume 101 No. 5 2015, 687-692
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
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ijpam.eu
FINDING THE INITIAL BASIC FEASIBLE SOLUTION
OF A FUZZY TRANSPORTATION PROBLEM
BY A NEW METHOD
S. Narayanamoorthy1, S. Kalyani2
1,2Department of Applied Mathematics
Bharathiar University
Coimbatore, 641 046, INDIA
Abstract: Transportation model plays a vital role in logistics and supply
chain management for reducing cost and improving service. Many methods are
proposed previously to find the initial basic feasible solution like North-West
corner method, least cost method, row minima method column minima method,
Russell’s method, Vogel’s approximation method etc., In this paper the initial
basic feasible solution for a fuzzy transportation problem is obtained using a
new method. This method is illustrated by a numerical example. Further it
has been compared with the existing methods.
AMS Subject Classification: 90C08, 90C90, 90C29, 90C70, 90B06
Key Words: fuzzy transportation, Vogel’s approximation method
1. Introduction
The transportation problem refers to a special case of linear programming prob-
lem. The basic transportation problem was developed by Hitchcock [6]. In
Mathematics and Economics transportation theory is a name given to the study
Received: March 12, 2015 c
2015 Academic Publications, Ltd.
url: www.acadpubl.eu
688 S. Narayanamoorthy, S. Kalyani
of optimal transportation and allocation of resources. The problem was formal-
ized by French Mathematician Gaspard Monge in 1781. Tolstoi was one of
the first to study the transportation problem mathematically. Transportation
problem deals with the distribution of single commodity from various sources
of supply to various destinations of demand in such a manner that the total
transportation cost is minimized. In order to solve a transportation problem the
decision parameters such as availability, requirements and the unit transporta-
tion cost of the model must be fixed at crisp values. But in real life applications
supply, demand and unit transportation cost may be uncertain due the several
factors. These imprecise data may be represented by fuzzy numbers.
The idea of fuzzy set was introduced by Zadeh [10] in 1965. Bellman and Zadeh
[1] proposed the concept of decision making in fuzzy environment. After this
pioneering work many authors have studied fuzzy linear programming problem
techniques such as Fang. S.C [5], H.Rommelfanger [7] and H.Tanaka [8] etc.,
Chanas et al developed a method for solving fuzzy transportation problems
by applying the parametric programming technique using the Bellman - Zadeh
criterion. Chanas and Kuchta [4],[2] proposed a method for solving a fuzzy
transportation problem with Crisp objective function which provides only crisp
solution to the given problem. The fuzzy transportation problem can be solved
by fuzzy linear programming techniques [3]. But most of the existing tech-
nique provides the crisp solution of the fuzzy transportation problem. Ranking
method is used to change fuzzy numbers into crisp form. In this paper, an
algorithm is proposed for the new method and is compared with VAM using
numerical example.
2. Preliminaries
Definition 1. A Fuzzy set A is defined as the set of ordered pairs
(X, µA(x)) , where x is an element of the universe of discourse U and µA(x)
is the membership function that attributes to each X∈U a real number∈[0,1]
describing the degree to which X belongs to the set.
Definition 2. A crisp set is a special case of Fuzzy set, in which the
membership function takes only two values 0 and 1.
Definition 3. A fuzzy number ˜
A= (a, b, c, d) is said to be a trapezoidal
FINDING THE INITIAL BASIC FEASIBLE SOLUTION... 689
fuzzy number if its membership function is given by,
µ˜
A=
x−a
b−a, a ≤x≤b
1, b ≤x≤c
d−x
d−c, c ≤x≤d
0,otherwise
Definition 4. Let ˜a= (a1, a2, a3, a4) and ˜
b= (b1, b2, b3, b4) be two trape-
zoidal fuzzy numbers, the arithmetic operators on these numbers are as follows
Addition: ˜a+˜
b= (a1+b1, a2+b2, a3+b3, a4+b4)
Subtraction: ˜a−˜
b= (a1−b4, a2−b3, a3−b2, a4−b1)
Definition 5. We define a ranking a function which maps each fuzzy
number into the real line. F(R) represents the set of all trapezoidal fuzzy
numbers. If R be any ranking function, then R(˜a) = a1+a2+a3+a4
4.
3. Mathematical Formulation of Fuzzy Transportation Problem
A fuzzy Transportation Problem can be stated as
minz =Pm
i=1 Pn
j=1 cij xij
Subject to
n
X
j=1
xij =ai, i = 0,1,2, ..., m
m
X
i=1
xij =bj, j = 0,1,2, ..., n
xij ≥0, i = 0,1,2, ..., m and j= 0,1,2, ..., n in which the transportation cost
cij , supply aiand bjquantities are fuzzy quantities.
The necessary and sufficient condition for the fuzzy linear programming is
given as
Pm
i=1 ai=Pn
j=1 bj
690 S. Narayanamoorthy, S. Kalyani
4. Computational Procedure
The Computation procedure for Fuzzy Transportation Problem by new method:
To find the initial basic feasible solution of a fuzzy transportation problem
the algorithm is proposed:
Step 1 : Find the Fuzzy penalty cost ie. fuzzy difference between the
maximum fuzzy cost and minimum fuzzy cost in each row and column.
Step 2 : Choose a row or column with maximum penalty by ranking
method. If maximum penalty is more than one choose any one arbitrarily.
Step 3 : From the selected row or column choose a fuzzy minimum cost
and allocate as much as possible in that cell depending on supply and demand.
Step 4 : Delete the row or column which is fully exhausted. Repeat the
process till all the rim requirements are satisfied.
This process gives the initial basic feasible solution and we can optimize
using MODI method if it is non-degenerate.
5. Numerical Examples
Example 1: To illustrate the new method let us consider a fuzzy transportation
problem:
1 2 3 4 Supply
1 (1,2,3,4) (1,3,4,6) (9,11,12,14) (5,7,8,11) (1,6,7,12)
2 (0,1,2,4) (−1,0,1,2) (5,6,7,8) (0,1,2,3) (0,1,2,3)
3 (3,5,6,8) (5,8,9,12) (12,15,16,19) (7,9,10,12) (5,10,12,17)
Demand (5,7,8,10) (1,5,6,10) (1,3,4,6) (1,2,3,4)
The crisp transportation table is:
1 2 3 4 Supply
1 2.5 3.5 11.5 7.75 6.5
2 1.75 0.5 6.5 1.5 1.5
3 5.5 8.5 15.5 9.5 11
Demand 7.5 5.5 3.5 2.519
FINDING THE INITIAL BASIC FEASIBLE SOLUTION... 691
The solution table is:
1 2 3 4
1 2.5 3.5(5.5) 11.5(1) 7.75
2 1.75 0.5 6.5(1.5) 1.5
3 5.5(7.5) 8.5 15.5(1) 9.5(2.5)
Thus the crisp transportation cost by New Method is Rs.121.
Example 2: To illustrate the new method we choose another problem [9].
Now the Problem is solved by the new method:
1 2 3 4 Supply
1 (−4,0,4,16) (−4,0,4,16) (−4,0,4,16) (−2,0,2,8) (0,4,8,12)
2 (8,16,24,32) (8,14,18,24) (4,8,12,16) (2,6,10,14) (4,8,18,26)
3 (4,8,18,26) (0,12,16,20) (0,12,16,20) (8,14,18,24) (4,8,12,16)
Demand (2,6,10,14) (2,2,8,12) (2,6,10,14) (2,6,10,14) (8,20,38,54)
Thus the crisp transportation cost by New Method is Rs.267.2.
The New Method is compared with the Fuzzy VAM and the result is tabu-
lated below:
Exampl e F uzz yV AM N ewM ethod OptimumSol ution(MODI)
1 122.5121 121
2 272 267.2 267.2
6. Conclusion
A New method is proposed to find the initial basic feasible solution of the Fuzzy
Transportation Problem. From the comparision given above, the New Method
gives the better initial basic feasible solution when compared to Fuzzy VAM.
This method is easy to understand and to Compute.
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