International Journal of Computer & Organization Trends – Volume 4 – January 2014
An Efficient Algorithm to Obtain the Optimal Solution for Fuzzy Transportation Problems S.Krishna Prabha,S.Devi,S.Deepa Associate Professor,Department of Mathematics, PSNA CET-Dgl,Tamilnadu. Assistant Professor, Department of Mathematics, PSNA CET-Dgl,Tamilnadu. Lecturer,Department of civil Engineering, PSNA CET-Dgl,Tamilnadu Abstract: In
this
paper
we
solve
the
Fuzzy
Transportation problems by using a new algorithm namely EAVAM . We introduce an approach for solving a wide range of such
Keywords:
Optimization,
Transportation problem, ranking of fuzzy numbers, Fuzzy sets, Fuzzy transportation problem,
problems by using a method which applies it for ranking of the fuzzy numbers. Some of
1. Introduction
the quantities in a fuzzy transportation
Transportation
problem may be fuzzy or crisp quantities. In
applications in logistics and supply chain for
many fuzzy decision problems, the quantities
reducing the cost. When the cost coefficients
are represented in terms of fuzzy numbers.
and the supply and demand quantities are
Fuzzy numbers may be normal or abnormal,
known exactly, many algorithms have been
triangular or trapezoidal or any LR fuzzy
developed for solving the transportation
number. Thus, some fuzzy numbers are not
problem. But, in the real world, there are
directly comparable. First, we transform the
many cases that the cost coefficients and the
fuzzy quantities as the cost, coefficients,
supply and demand quantities are fuzzy
supply and demands, into crisp quantities by
quantities.
models
have
wide
using our method and then by using the
The transportation problem is one of the
EAVAM, we solve and obtain the solution of
oldest applications of linear programming
the problem. The new method is systematic
problems. The basic transportation problem
procedure, easy to apply and can be utilized
was originally developed by Hitchcock [3].
for all types of transportation problem
Efficient methods of solution derived from
whether maximize or minimize objective
the simplex algorithm were developed in
function.
1947, primarily by Dantzig [4] and then by
Finally we explain this method with a
Charnes and Cooper [1]. The transportation
numerical example.
problem can be modeled as a standard
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International Journal of Computer & Organization Trends – Volume 4 – January 2014 linear programming problem that can be
origin
solved by the simplex method. We can get
transportation problem can be represented
an initial basic feasible solution for the
as a single objective transportation problem
transportation problem by using the North-
or as a multi-objective transportation
West corner rule, Row Minima, Column
problem. Fuzzy transportation problem
Minima, Matrix Minima or the Vogel’s
(FTP)[6] is the problem of minimizing
Approximation Method. To get an optimal
fuzzy valued objective functions with fuzzy
solution for the transportation problem, we
source and fuzzy destination parameters.
use
(Modified
The balanced condition is both a necessary
Distribution Method).Charnes and Cooper
and sufficient condition for the existence of
[1] developed the Stepping Stone Method,
a feasible solution to the transportation
which provides an alternative way of
problem. Shan Huo Chen [15] introduced
determining the optimal solution The
the concept of function principle, which is
LINDO (Linear Interactive and Discrete
used to calculate the fuzzy transportation
Optimization)
cost.
the
MODI
method
package
handles
the
Oi
The
to
destination
Graded
Mean
Dj.
The
Integration
transportation problem in explicit equation
Representation Method, used to defuzzify
form and thus solves the problem as
the fuzzy transportation cost, was also
standard
introduced by Shan Huo Chen [14].
linear programming problem.
Consider m origins (or sources) Oi (i =1,....,m)and n destinations Dj (j = 1,...,n). At each origin Oi, let ai be the amount of a homogeneous product which we want to transport to n destinations Dj, in order to satisfy the demand for bj units of the product there. A penalty cij is associated with transport in a unit of the product from source i to destination j. The penalty could represent transportation cost, delivery time, quantity of goods delivered, under-used capacity, etc. A variable xij represents the unknown quantity to be transported from
ISSN: 2249-2593
In this problem we determine optimal shipping
patterns
between
origins
or
sources and destinations. Many problems which
have
nothing
to
do
with
transportation have this structure. In many fuzzy decision problems, the data are represented in terms of fuzzy numbers. In a fuzzy
transportation
problem,
all
parameters are fuzzy numbers. Fuzzy numbers may be normal or abnormal, triangular or trapezoidal. Thus, some fuzzy numbers are not directly comparable.
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International Journal of Computer & Organization Trends – Volume 4 – January 2014 Comparing between two or multi fuzzy
the unknown quantity to be transported
numbers and ranking such numbers is one
from origin Oi to destination Dj. However,
of the important subjects, and how to set
in the real world,
the rank of fuzzy numbers has been one of
problems are not single objective linear
the main problems. Several methods are
programming problems.
all transportation
introduced for ranking of fuzzy numbers. Here, we want to use a method which is introduced for ranking of fuzzy numbers, by Basirzadeh et al. [3]. Now, we want to apply
this
method
transportation parameters
for
problems,
can
be,
the where
triangular
The mathematical form of the above said problem is as follows Minimize Z = ∑
∑
fuzzy
∑
=
, i=1,2,…m.
all
∑
=
, j=1,2,…m.
fuzzy
(1)
≥ 0 , i= 1,2,..m, j=1,2,..m.
numbers. This method is very easy to
where cij is the cost of transportation of an
understand and to apply. At the end, the
unit from the ith source to the jth destination,
optimal solution of a problem can be
and the quantity xij is to be some positive
obtained in a fuzzy number or a crisp
integer or zero, which is to be transported
number.
from the ith origin to jth destination. An obvious necessary and sufficient condition
2. Terminology
for the linear programming problem given
Consider m origins (or sources) Oi (i =
in (1) to have a solution is that
1,...,m) and n destinations Dj (j = 1,...,n). At each origin Oi, let ai be the amount of a homogeneous product that we want to
∑
=∑
(2)
i.e. assume that total available is equal to the
transport to n destinations Dj, in order to
total required. If not true, a fictitious source
satisfy the demand for bj units of the
or destination can be added. It is has a
product there. A penalty cij is associated
feasible solution if and only if the condition
with transportation of a unit of the product from source i to destination j. The penalty could represent transportation cost, delivery time, quantity of goods delivered underused capacity, etc. A variable xij represents
ISSN: 2249-2593
(2) satisfied. Now, the problem is to determine xij , in such a way that the total transportation cost is minimum. Mathematically a fuzzy transportation problem can be stated as follows:
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International Journal of Computer & Organization Trends – Volume 4 – January 2014 Minimize Z = ∑
∑
∑
, i=1,2,…m.
=
∑
=
decision systems: 1. Contribution of the decision-maker in the (5) decision making process,
, j=1,2,…m.
≥ 0 , i= 1,2,..m, j=1,2,..m.
in which the transportation costs supply
Two factors play significant roles in fuzzy
(3)
and demand
,
quantities are
2. Simplicity of calculation. This paper attempts to propose a method for ranking and comparing fuzzy numbers to
fuzzy quantities. An obvious necessary and
account for the above-mentioned factors as
sufficient condition for the fuzzy linear
much as possible.
programming problem given in (2-3) to have a solution is that ∑
2.1 Definition
≈ ∑
is a Triangular-Fuzzy
A fuzzy number
(4)
number denoted (a1,a2,a3,) where a1,a2 and a3
A
considerable
number
of
methods
presented for fuzzy transportation problem.
( ) is given
and it’s membership function below
Some of them are based on ranking of the
fuzzy numbers. Some of the methods for
(x)
=
≤ ≤
≤
ranking of the fuzzy numbers for example,
0
≤
ℎ
have limitations, are difficult in calculation, or they are non-intuitive, which makes them inefficient
in
practical
applications,
Consider the following fuzzy transportation problem (FTP),
especially in the process of decision making. However, in some of these methods, as ones in which fuzzy numbers are compared
(FTP) Minimize: Z = ∑
∑
̃
Subject to
according to their centeroid point (see [2],
∑
≤
, i=1,2,…m.
[15], [16]), the decision maker does not play
∑
≥
, j=1,2,…m.
any role in the comparison between fuzzy
≥ 0 , i= 1,2,..m, j=1,2,..m.
numbers. Nevertheless, there are certain methods in which fuzzy numbers are compared in a parametric manner
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Where = ( a1,a2,a3), =( b1,b2, b3) and ̃ =(cij,cij,cij) representing the uncertain supply and demand for the transportation problems.
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International Journal of Computer & Organization Trends – Volume 4 – January 2014 (2). ⊖ = (a1,a2, a3) ⊖ (b1,b2,b3) = (a1+b1,a2-b2,a3+b3) (3). ⊗ = (a1,a2, a3) ⊗ (b1,b2,b3) = (s1,s2,s3)
(r)
where s1 = minimum{a1b1, a1b3, a3b1,a3b3}, = mn ̅(s2 r) = maximum{ a b , a b , a b ,a b }, s3 1 1 1 3 3 1 3 3
ω
3. A new Algorithm for solving fuzzy a b c Figure 1: an arbitrary Fuzzy number (0 ≤ ω ≤ 1)
transportation Problem Now, we introduce a new method for solving a fuzzy transportation problem
According to the above-mentioned
where the transportation cost, supply and
definition of a triangular fuzzy number,
demands are fuzzy numbers. The fuzzy
let = ( (r), ̅(r)), (0 ≤ r ≤ 1) be a fuzzy
numbers in this problem is triangular. The
number, then the value M( ), is
optimal solution for the fuzzy transportation
assigned to
problem can be obtained as a crisp or fuzzy
is calculated as follows:
form.
[2
( ) = ∫ { ( ) + +
+
̅( )} dr =
Step 1. Calculate the values M(.) for each fuzzy data, the transportation costs
]
supply which is very convenient for calculation.
and demand
̃
quantities which
are fuzzy quantities. Step 2. By replacing M( ̃
), M(
) and
M( ) which are crisp quantities instead of the ̃ ,
ω
and
quantities which are fuzzy
quantities, define a new crisp transportation problem. δ m β A triangular fuzzy number Definition let = (a1,a2, a3) and = (b1,b2,b3) be two triangular fuzzy numbers. Then (1). ⊕ = (a1,a2, a3) ⊕ (b1,b2,b3) = (a1+b1,a2+b2,a3+b3)
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Step 3. Solve the new crisp transportation problem, by usual method , and obtain the crisp optimal solution of the problem. Note Any solution of this transportation problem will contain exactly (m+n-1) basic
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International Journal of Computer & Organization Trends – Volume 4 – January 2014 feasible solutions. We note that the optimal
want to solve it by our method, and then we
solution Xij is to be some positive integer or
will compare the results.
zero, but the optimal solution Xij for a crisp
A company has two factories O1, O2 and two
transportation problem may be integer or
retail
non integer, because the RHS of the problem
quantities per month at O1, O2 are (150, 201,
are the measure of fuzzy numbers which are
246) and (50, 99, 154) tons respectively. The
real numbers. If you accept the crisp
demands per month for D1 and D2 are
solution then stop. The optimal solution is in
(100,150,200)
your hand. If you want fuzzy form of
respectively. The transportation cost per ton
solution, go to the next step.
̃
Step 4. Determine the locations of non zero
(22,31,34),
basic feasible solutions in transportation
(30,39,54).
tableau. The basis is rooted spanning tree,
Solution: Transportation table of the given
that is there must be at least one basic cell in
fuzzy transportation problem is
each row and in each column of the
Table 4.1 Fuzzy Transportation Problem
,
stores
D2.
and
̃ =
D1
The
production
(100,150,200)
i =1, 2; j=1, 2 are ̃
transportation tableau. Also, the basis, must be a tree, that is , the (m+n-1) basic cells
D1 ,
= (15,19,29), ̃ =
(8,10,12)
̃ =
and
D2
Supply
(15,19,29) (22,31,34) (150,201,
O1
should not contain a cycle. Therefore, there exist some rows and columns which have
tons
246) (8,10,12)
O2
(30,39,54) (50,99,15
only one basic cell. By starting from these
4)
cells, calculate the fuzzy basic solutions,
Dema
(100,150,
(100,150,
(200,300,
continue
nd
200)
200)
400)
until
obtain
(m+n-1)
basic
solutions.
According
to
the
above-mentioned
definition of a triangular fuzzy number,
4.Numerical example The following example may be helpful to
let
= ( (r), ̅(r)), (0 ≤ r ≤ 1) be a fuzzy
number, then the value M( ), is
clarify the proposed method: Example: Consider the following fuzzy
assigned to
is calculated as follows:
transportation problem .All the data in this
problem are triangular fuzzy numbers. We
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[2
( ) = ∫ { ( ) + +
+
http://www.ijcotjournal.org
̅( )}
dr
=
] Page 37
International Journal of Computer & Organization Trends – Volume 4 – January 2014 which is very convenient for calculation. Since
(
) =
(
D1
D2
y
), the given
problem is a balanced one.
20.
O1
Now, by using our method we change the
5
29. 49.2
5
fuzzy transportation problem into a crisp transportation problem. So, we have the following
reduced
fuzzy
Suppl
199.5
15 0
10
O2
transportation
40.5 100.5
100.
problem:
5 150
Deman D1
D2
Supply
O1
20.5
29.5
199.5
O2
10
40.5
100.5
Demand
150
150
300
d Table 4.3
As shown in the table 4.2, the result of of
the
fuzzy
numbers
O1
obtaining the measures which are not all integer. So, existing of a non integer value in
O2
a transportation problem follows this fact that the solution of the crisp transportation problem is not integer. We note that the solution in a usual transportation problem is integer, because its matrix is an unimodular matrix If we solve the new problem, we obtain the solutions as follows: X11 = 49.5, X12 = 150, X21 = 100.5
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300
Now, we can return to initial problem and obtain the fuzzy solution of the fuzzy transportation problem based on the data of table 4.3.
Table 4.2
defuzzification
150
D1
D2
Supply
(350,51,3
(100,150,
(150,201,
46)
200)
246)
(50,99,15
(50,99,15
4)
4)
Dema
(100,150,
(100,150,
(200,300,
nd
200)
200)
400)
Table 4.4 where the fuzzy optimal solution for the given fuzzy transportation problem is: = (350, 51,346) = (100,150,200) = (50, 99,154) The crisp value of the optimum fuzzy transportation cost for the given problem by 6444.5
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International Journal of Computer & Organization Trends – Volume 4 – January 2014 5. Conclusion In this paper, the transportation costs are considered as imprecise numbers described by triangular fuzzy numbers which are more realistic and general in nature. In this method, we had used a simple method to solve fuzzy transportation problem by using ranking of fuzzy numbers. This method can be used for all kinds of fuzzy transportation problem, instead of other existing methods. The new method is a systematic procedure, easy to apply and can be utilized for all types of transportation problem whether maximize or minimize objective function.
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knowledge-Based systems, 8(5) (2001) 573-592. [8] M. S. Bazarra, John J. Jarvis, Hanif D. Sherali, Linear programming and network flows, (2005). [9] M. Shanmugasundari, K. Ganesan , A Novel Approach for the fuzzy optimal solution of Fuzzy Transportation Problem, International Journal of Engineering Research and Applications (IJERA) Vol. 3, Issue 1, January February 2013, pp.1416-1424 [10] P. Grzegorzewski, E. Mr´owka, Trapezoidal approximations of fuzzy numbers,fuzzy sets and systems, 153 (2005) 115-135. [11] P. Pandian and G. Natarajan, A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems, Applied Mathematical Sciences, 4(2010), 79-90. [12] S. Abbasbandy, B. Asady, The nearest trapezoidal fuzzy number to a fuzzy quantity, Applied mathematics and computation, 159 (2004) 381-386.[11] [13] S. Abbasbandy, B. Asady, Ranking of fuzzy numbers by sign distance,Information science, 176 (2006) 2405-2416. [14] S. Chanas and D. Kuchta, A concept of the optimal solution of the transportation problem with fuzzy cost coefficients, , Fuzzy sets and Systems,82 (1996) 299-305. [15] S. Chanas, W. Kolodziejczyk and A. Machaj, A fuzzy approach to the transportation problem, Fuzzy Sets and Systems, 13 (1984) 211-221. [16] Shiang-Tai Liu and Chiang Kao, Solving fuzzy transportation problems based on extension principle, Journal of Physical Science, 10(2006), 63-69. [17] T. C. Chu, C. T. Tsao, Ranking fuzzy numbers with an area between the centroid point and original point, compute. math. Appl., 43 (2002) 111-117.
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