1. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1416-1424
A Novel Approach for the fuzzy optimal solution of Fuzzy
Transportation Problem
M. Shanmugasundari *, K. Ganesan **
*Department of Mathematics, Faculty of Science Humanities,
SRM University, Kattankulathur, Chennai - 603203, India.
** Department of Mathematics, Faculty of Engineering and Technology,
SRM University, Kattankulathur, Chennai - 603203, India
ABSTRACT
In this paper we propose a new method Bellman and Zadeh [3] proposed the concept of
for the Fuzzy optimal solution to the decision making in Fuzzy environment. After this
transportation problem with Fuzzy parameters. pioneering work, several authors such as
We develop Fuzzy version of Vogels and MODI Shiang-Tai Liu and Chiang Kao[23], Chanas et al
algorithms for finding Fuzzy basic feasible and [4], Pandian et.al [19], Liu and Kao [17] etc
fuzzy optimal solution of fuzzy transportation proposed different methods for the solution of Fuzzy
problems without converting them to classical transportation problems. Chanas and Kuchta [4]
transportation problems. The proposed method proposed the concept of the optimal solution for the
is easy to understand and to apply for finding Transportation with Fuzzy coefficient expressed as
Fuzzy optimal solution of Fuzzy transportation Fuzzy numbers. Chanas, Kolodziejckzy, Machaj[5]
problem occurring in real world situation. To presented a Fuzzy linear programming model for
illustrate the proposed method, numerical solving Transportation problem. Liu and Kao [17]
examples are provided and the results obtained described a method to solve a Fuzzy Transportation
are discussed. problem based on extension principle. Lin
introduced a genetic algorithm to solve
Keywords - Fuzzy sets, Fuzzy numbers, Fuzzy Transportation with Fuzzy objective functions.
transportation problem, Fuzzy ranking, Fuzzy Nagoor Gani and Abdul Razak [12] obtained a
arithmetic fuzzy solution for a two stage cost minimizing fuzzy
transportation problem in which supplies and
I. INTRODUCTION demands are trapezoidal fuzzy numbers. A. Nagoor
The transportation problem is a special Gani, Edward Samuel and Anuradha [9] used
type of linear programming problem which deals Arshamkhan’s Algorithm to solve a Fuzzy
with the distribution of single product (raw or Transportation problem. Pandian and Natarajan [19]
finished) from various sources of supply to various proposed a Fuzzy zero point method for finding a
destination of demand in such a way that the total Fuzzy optimal solution for Fuzzy transportation
transportation cost is minimized. There are effective problem where all parameters are trapezoidal fuzzy
algorithms for solving the transportation problems numbers.
when all the decision parameters, i. e the supply In general, most of the existing techniques
available at each source, the demand required at provide only crisp solutions for the fuzzy
each destination as well as the unit transportation transportation problem. In this paper we propose a
costs are given in a precise way. But in real life, simple method, for the solution of fuzzy
there are many diverse situations due to uncertainty transportation problems without converting them in
in one or more decision parameters and hence they to classical transportation problems. The rest of this
may not be expressed in a precise way. This is due paper is organized as:
to measurement inaccuracy, lack of evidence, In section II, we recall the basic concepts
computational errors, high information cost, whether of Fuzzy numbers and related results. In section III,
conditions etc. Hence we cannot apply the we define Fuzzy transportation problem and prove
traditional classical methods to solve the the related theorems. In section IV, we propose
transportation problems successfully. Therefore the Fuzzy Version of Vogels Approximation Algorithm
use of Fuzzy transportation problems is more (FVAM) and fuzzy version of MODI method
appropriate to model and solve the real world (FMODI). In section V, numerical examples are
problems. A fuzzy transportation problem is a provided to illustrate the methods proposed in this
transportation problem in which the transportation paper for the fuzzy optimal solution of fuzzy
costs, supply and demand are fuzzy quantities. transportation problems without converting them to
classical transportation problems and the results
obtained are discussed.
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2. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1416-1424
II. PRELIMINARIES 2.3. Ranking of triangular fuzzy numbers
The aim of this section is to present some Several approaches for the ranking of fuzzy
notations, notions and results which are of useful in numbers have been proposed in the literature. An
our further consideration. efficient approach for comparing the fuzzy numbers
is by the use of a ranking function based on their
2.1 Fuzzy numbers graded means. That is, for every
A fuzzy set à defined on the set of real A   a1 , a 2 , a 3   F  R  , the ranking function

numbers R is said to be a fuzzy number if its
: F(R)  R by graded mean is defined as
membership function  A : R  0,1
   has the
  a  4a 2  a 3 
following characteristics  (A)   1 
(i) Ã is normal. It means that there exists an xR  6 
such that  A  x   1

For any two fuzzy triangular numbers
 is convex. It means that for every x , x  R, A   a1 , a 2 , a 3  and B   b1 , b 2 , b3  in F(R), we
 
(ii) A 1 2
have the following comparison
 A  x1  1    x 2   min  A  x1  ,  A  x 2  ,   0,1
  
(iii)  A is upper semi-continuous.    
(i). A  B if and only if  (A)  (B).


(iv) supp ( A ) is bounded in R.    
(ii). A  B if and only if  (A)  (B).
2.2. Triangular fuzzy numbers
    
(iii). A  B if and only if  (A)  (B).
A fuzzy number A in R is said to be a
triangular fuzzy number if its membership function
    
(iv). A  B  0 if and only if  (A)  (B)  0.
A : R   0,1 has the following characteristics.

 x  a1
 a1  x  a 2 A triangular fuzzy number A   a1 , a 2 , a 3  in F  R 

 a 2  a1 
 1 is said to be positive if  (A)  0 and denoted
 x  a2
A  x   
   
by A  0. Also if  (A)  0, then  
A  0 and if
 a3  x a2  x  a3
 a3  a2   
 (A)  0, then A  0. If  (A) 
 (B), then
  
 0
 otherwise the triangular numbers A and B are said to be
 
equivalent and is denoted by A  B.
We denote this triangular fuzzy number by
A   a1 , a 2 , a 3  . We use F(R) to denote the set of all
 2.4. Arithmetic operations on triangular fuzzy
triangular fuzzy numbers. Also if m  a 2 represents numbers
Ming Ma et al. [9] have proposed a new
the modal value or midpoint,   a 2  a1 represents fuzzy arithmetic based upon both location index and
the left spread and   a 3  a 2 represents the right fuzziness index functions. The location index
spread of the triangular fuzzy number number is taken in the ordinary arithmetic, whereas
A   a1 , a 2 , a 3  , then the triangular fuzzy number
 the fuzziness index functions are considered to
follow the lattice rule which is least upper bound in
A can be represented by the triplet A   m, ,   .
  the lattice L. That is for a, bL we define
That is A   a1 , a 2 , a 3    m, ,   .
 a  b  max a, b and a  b  min a, b .
For arbitrary triangular fuzzy numbers
  (a ,a ,a )   m ,  ,   , B  (b , b , b )
A 
1 2 3 1 1 1 1 2 3
  m 2 ,  2 , 2  and   , , ,  , the arithmetic
operations on the triangular fuzzy numbers are
defined by
A  B   m 1 , 1 , 1    m 2 ,  2 , 2 
 
 (m 1  m 2 , max{1 ,  2 }, max{1 , 2 }
 (m 1  m 2 , 1   2 , 1  2 )
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3. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and
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Vol. 3, Issue 1, January -February 2013, pp.1416-1424
In particular for any two triangular fuzzy numbers where c ij is the fuzzy unit transportation cost from ith

A  (a1 ,a 2 ,a 3 )   m 1 , 1 , 1  , B  (b1 , b2 , b3 )
 
source to the jth destination. The objective is to
  m 2 ,  2 , 2  , we define: minimize the total fuzzy transportation cost, in this
paper the fuzzy transportation problem is solved by
(i). Addition fuzzy version of Vogel’s and MODI method. This
A  B   a1 ,a 2 ,a 3    b1 ,b2 ,b3 
  fuzzy transportation problem is explicitly
  m1 , 1 , 1    m 2 ,  2 , 2 
represented by the following fuzzy transportation
table.

 m1  m 2 ,max  1 ,  2  ,max 1 , 2   Table 3.1 Fuzzy transportation table
(ii). Subtraction
A  B   a1 ,a2 ,a3    b1 , b2 , b3 
 
Destination
  m1 , 1 , 1    m 2 ,  2 , 2 

 m1  m 2 ,max 1 , 2  ,max 1 ,2   1 2 … n Supply
(iii).Multiplication 1 
c11 
c12 … 
c1n 
a1
AB   a1 ,a 2 ,a 3  b1 ,b2 ,b3 
Sources

2 
c 21 
c 22 …. 
c 2n 
a2
  m1 , 1 , 1  m 2 ,  2 , 2 
 
…
…
…
…
…
…
 m1m 2 ,max  1 ,  2  ,max 1 , 2 
(iv). Division m 
cm1 
cm2 …. 
cmn 
an
A  a1 ,a 2 ,a 3   m1
    
  ,max( 2 , 1 ),max(1 , 2 )  ,
 b ,b ,b   m
Demand b1 b2 bm
B 1 2 3  2 
3.2. Basic Theorems
if, B   b1 , b 2 , b 3  is non zero fuzzy number.

Theorem 3.1. (Existence of a fuzzy feasible
solution)
III. MAIN RESULTS The necessary and sufficient condition for the
Consider a fuzzy transportation with m existence of a fuzzy feasible solution to the fuzzy
sources and n destinations with triangular fuzzy transportation problem (3.1) is,
 m n
 b

numbers. Let a i  0 be the fuzzy availability at  ai 
 j (3.2)
  
source i and b , (b  0) be the fuzzy requirement i 1 j1
j j
  
at destinations j. Let cij (cij  0) be the unit (Total supply  Ttotal demand). .
fuzzy transportation cost from source i to destination

j. Let x ij denote the number of fuzzy units to be Proof: (Necessary condition)
Let there exist a fuzzy feasible solution to the fuzzy
transformed from source i to destination j. Now the transportation problem
problem is to find a feasible way of transporting the m n

Minimize Z    c x 
available amount at each source to satisfy the i 1 j1
ij ij
demand at each destination so that the total n
transportation cost is minimized. subject to  x ij  a i ,i  1, 2,3,..., m.
 
j1 (3.3)
m
3.1. Mathematical formulation of fuzzy  
 x ij  b j , j  1, 2,3,..., n.
transportation problem i 1
The mathematical model of fuzzy  
and x ij  0 for all i and j.
transportation problem is as follows n
From  x ij  a i , (i  1, 2,3,..., m), we have
 
m n j1

Minimize Z    cij x ij
  m n m
i 1 j1   x ij   a i
   3.4 
n i 1 j1 i 1
subject to  x ij  a i , i  1, 2,3,..., m
  m
j1  
Also from  x ij  b j , ( j  1, 2,3,..., n), we have
m i 1

 x ij  b j , j  1, 2,3,..., n
 (3.1) n m n
i 1  
  x ij   b j (3.5)
m n j1 i 1 j1

 a i   b j , i  1, 2,3,..., m;

i 1 j1 From equations (3.4) and (3.5), we have
 
j  1, 2,3,..., n and x ij  0 for all i and j.
m n

a  b
 i j
i 1 j1
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Vol. 3, Issue 1, January -February 2013, pp.1416-1424
(Sufficient condition) will be atleast 2m and 2n respectively. Suppose if
  
Since all a and b are positive, xij must be all the total number of basic variables is D, then
i j obviously D ≥ 2m , D ≥ 2n.
positive. Therefore equation (3.2) yields a feasible
solution. Case (i). If m < n, then m + n < n + n  m + n < 2n
 m + n < 2n < D m + n < D .
Theorem 3.2.
The dimension of the basis of a fuzzy
transportation are (m+n-1)(m+n-1).That a fuzzy Case (ii). If m  n, then m  n  m  n
transportation problem has only (m+n-1)  m  m  m  n  2m  m  n
independent structural constraints and its basic  m  n  2m  m  n  2m  D
feasible solution has only (m+n-1) positive
components.  m  n  D.
Proof: Consider a fuzzy transportation problem Case (iii). If m = n, then m + n = m + n
with m sources and n destinations,  m + m = m + n  2m = m + n
m n

Minimize Z    c x
   m + n = 2m  m + n = 2m < D
ij ij
i 1 j1
n
 m + n < D.
subject to  x ij  a i , i  1, 2,3,..., m.
  Thus in all cases D ≥ m + n. But the number of
j1
basic variables in fuzzy transportation problem is
m
 
 x ij  b j , j  1, 2,3,..., n. (m+n-1) which is a contradiction.
i 1
 
and x ij  0 for all i and j. Therefore there is atleast one equation,
either row or column having only one basic variable.
Let us assume that a fuzzy transportation has m
Let the rth equation have only one basic variable.
rows (supply constraint equations) and n columns
(demand constraint equations). Therefore there are Let x rs be the only basic variable in the rth row and

totally (m+n) constraint equations. s column ,then x rs  a r . Eliminate r th row from the
th
 
m n system of equation and substitute x rs  a r in sth
 
This is due to the condition that 
i 1
ai 
 b

j1
j  
column equation and replace b by b  b  a .
s s
 
s r
which is the last requirement constraint. Therefore After eliminating the r th row , the system has (m+n-
one of (m+n) constraints can always be derived 2) linearly independent constraints. Hence the
from the remaining (m+n-1). Thus there exists only number of basic variables is (m+n-2).
(m+n-1) independent constraints and its basic Repeat the process and conclude that in the
feasible solution has only (m+n-1) positive reduced system of equations there is an equation
components. which has only one basic variable. But if this is in sth
column equation, then it will have two basic
Theorem 3.3 variables.
The values of the fuzzy basic feasible solution are This concludes that in our original system

all differences between the partial sum of a i and of equations, there is an equation which has atleast
two basic variables. Continue the process to get the

the partial sum of b . j theorem.
m n
  
That is x ij    ri a i   s j b j , where ri ,s j are either
 
i 1 j1 IV. PROPOSED ALGORITHMS
1  1,1,1 or 0 =  0, 0, 0  .
 
4.1 Fuzzy Version of Vogels Approximation
Algorithm (FVAM)
Theorem 3.4
The fuzzy transportation problem has a triangular Step 1:
basis. From the Fuzzy Transportation table,
determine the penalty for each row or column. The
Proof: We note that every equation has a basic penalties are calculated for each row column by
variable, otherwise, the equation cannot be satisfied substituting the lowest cost element in that row or
   
for a i  0, b j  0 . Suppose every row and column column from the next cost element in the same row
equations has atleast two basic variables, since there or column. Write down the penalties below and
are m rows and n columns, the total number of basic aside of the rows and columns respectively of the
variables in row equations and column equations table.
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Step 2: Step 6:
Identify the column or row with largest On the closed path, identify the corners with – θ.
fuzzy penalty. In case of tie, break the tie arbitrarily. Select the smallest allocation among the corners
Select a cell with minimum fuzzy cost in the with – θ which indicate the number of units that can
selected column (or row, as the case may be) and be shifted to some other unoccupied cells. Add this
assign the maximum units possible by considering quantity to those corners marked with +θ and
the demand and supply position corresponding to subtract this quantity to those corners marked with –
the selected cell. θ on the closed path and cheek whether the number
of nonnegative allocations is (m+n-1) and repeat
Step 3:  
step 1 to step 7, till we reach  ij  0 for all i and j.
Delete the column/row for which the supply and
demand requirements are met.
4.3 Unbalanced Fuzzy Transportation Problem.
Step 4: Suppose the fuzzy transportation problem is
Continue steps 1 to 3 for the resulting fuzzy  m n 
transportation table until the supply and demand of unbalanced one, i. e , if 
 i 1
ai 
  
 
b j  , then

all sources and destinations have been met.  j1 
convert this into a balanced one by introducing a
4.2 Fuzzy Version of MODI Algorithm (FMODI) dummy source or dummy destination with zero
fuzzy unit transportation costs. Solve the resulting
Step 1: balanced fuzzy transportation problem using above
Given an initial fuzzy basic feasible solution of a said algorithms.
fuzzy transportation problem in the form of
allocated and unallocated cells of fuzzy V. NUMERICAL EXAMPLES
transportation table. Assign the auxiliary variables The following two numerical examples are
u i , i=1,2,3,…,m and v j , j=1,2,3,…,n for rows and
  taken from the paper “Simplex type algorithm for
 solving fuzzy transportation problem by Edward
columns respectively. Compute the values of ui and
Samuel et.al [9].
v j using the relationship cij  ui  v j for all i, j for
   
 
all occupied cells. Assume either ui or v j as zero Example 5.1
A company has two factories O1, O2 and
arbitrarily for the allocations in row/column. two retail stores D1, D2. The production quantities
per month at O1, O2 are (150, 201, 246) and (50, 99,
Step 2: 154) tons respectively. The demands per month for
For each unoccupied cell (i, j), compute the fuzzy D1 and D2 are (100,150,200) and (100,150,200) tons
opportunity cost ij using ij  cij  (ui  v j ).
   
respectively. The transportation cost per ton cij ,
 
i =1, 2; j=1, 2 are c11 =(15,19,29), c12 =(22,31,34),
Step 3:
 
c21 =(8,10,12) and c22 =(30,39,54).
(i) If all ij  0, then the current solution under

the test is optimal Solution: Transportation table of the given fuzzy
(ii) If at least one ij  0, then the current
 transportation problem is
Table 5.1 Fuzzy Transportation Problem
solution under the test is not optimal and proceeds to
the next step. D1 D2 Supply
O1 (15,19,29) (22,31,34) (150,201,246)
Step 4: O2 (8,10,12) (30,39,54) (50,99,154)
Select an unoccupied cell (i, j) with most negative
(100,150,200)
(100,150,200)
(200,300,400)
opportunity cost among all unoccupied cells.
Demand
Step 5:
Draw a closed path involving horizontal and vertical
lines for the unoccupied cells starting and ending at
the cell obtained in step 4 and having its other
corners at some allocated cells. Assign +θ and – θ To apply the proposed algorithms and the fuzzy
alternately starting with +θ for the selected arithmetic, let us express all the triangular fuzzy
unoccupied cells. numbers based upon both location index and
fuzziness index functions. That is in the form of
 m,  ,   given in table 5.2.
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Table 5.2 Balanced fuzzy transportation problem in The corresponding fuzzy optimal transportation cost
which all the triangular numbers are of the form is given by
 m,  ,   
m n
Minimize Z    c x
  ij ij
i 1 j1
D1 D2 Supply   6609,50  50r,55  55r  ,
where 0  r  1 can be suitably chosen by
(19, 4-4r, 10-10r)
the decision maker.
(150, 50-50r, 50-50r) (39, 9-9r, 15-15r) (31, 9-9r, 3-3r)
O1 (201, 51-51r, 45-45r) Case(i). When r=0, the fuzzy optimal transportation
cost in terms of the form (m, α, β) is (6609, 50, 55).
The corresponding fuzzy optimal transportation of
the form  a1 , a 2 , a 3  is (6559, 6609, 6664) and its
defuzzyfied transportation cost is 609.8.
(150, 50-50r, 50-50r) (10, 2-2r, 2-2r)
Case(ii). When r = 0.5, the fuzzy optimal
O2 (99, 49-49r, 55-55r) transportation cost in terms of the form (m, α, β) is
(6609, 25, 27.5). The corresponding fuzzy optimal
transportation cost of the form  a1 , a 2 , a 3  is (6584,
6609, 6636.5) and its defuzzyfied transportation cost
is 6609.4
Case(iii). When r=1, the fuzzy optimal
transportation cost in terms of the form (m, α, β) is
(6609,0,0). The corresponding fuzzy optimal
transportation cost of the form  a1 , a 2 , a 3  is
Demand
(0, 6609, 0) and its defuzzyfied transportation cost is
4406. Optimum fuzzy transportation cost is
(1150, 6609, 12882). Defuzzified fuzzy
By applying fuzzy version of Vogel’s transportation cost is 6745.
Approximation method (FVAM)), the initial fuzzy
basic feasible solution is given in table 5.3. Example 5.2
Find a fuzzy optimum solution for the unbalanced
Table 5.3 Initial fuzzy basic feasible solution fuzzy transportation problem given below.
D1 D2 Table 5.4. Unbalanced Fuzzy Transportation
(19, 4-4r, 10-10r) (31, 9-9r, 3-3r) Problem
O1
(51, 50-50r, 55-55r) (150, 50-50r, 50-50r)
B1 B2 Supply
(10, 2-2r, 2-2r)
O2 (39, 9-9r, 15-15r)
(99, 49-49r, 55-55r)
A1 (1,3,5) (3,5,13) (300,399,504)
The corresponding initial fuzzy transportation cost A2 (2,3,10) (3,4,11) (250,301,346)
is given by
IFTC  19, 4  4r, 10  10r  51, 50  50r, 55  55r  A3 (5,6,13) (1,3,5) (300,399,504)
  31, 9  9r, 3  3r 150, 50  50r, 50  50r 
 10, 2  2r, 2  2r  99, 49  49r, 55  55r 
(400,448,508)
(300,351,396)
  6609,50  50r,55  55r  .
Demand
By applying fuzzy version of MODI method
(FMODI), it can be seen that the current initial
fuzzy basic feasible solution is optimal.
Hence the fuzzy optimal solution in terms of
location index and fuzziness index is given by
Solution: Introduce a dummy destination B3 with
x11   51,50  50r,55  55r  ;
 (150,300,450) as its fuzzy demand and (0,0,0) as its
fuzzy transportation costs per unit. Hence the
x12  150,50  50r,50  50r  ;
 balanced fuzzy transportation with a dummy
x 21   99, 49  49r,55  55r  , where 0  r  1.
 destination becomes
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7. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and
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Vol. 3, Issue 1, January -February 2013, pp.1416-1424
Table 5.5 Balanced Fuzzy Transportation Problem By applying fuzzy version of Vogels Approximation
method, the initial fuzzy basic feasible solution is
B1 B2 B3 Supply given by
A1 (1,3,5) (3,5,13) (0,0,0) (300,399,504) Table 5.7 Initial fuzzy basic feasible solution
A2 (2,3,10) (3,4,11) (0,0,0) (250,301,346)
A3 (5,6,13) (1,3,5) (0,0,0) (300,399,504)
B1 B2 B3
(400,448,508)
(300,351,396)
(150,300,450)
(399,99-99r,105-105r)
Demand
(3,2-2r,2-2r)
(5,2-2r,8-8r)
(0,0,0)
A1
Now the balanced fuzzy transportation problem in
which all the triangular numbers are of the form
 m,  ,   is given in table 5.6.
(1,150-150r,150-50r)
Table 5.6 Balanced Fuzzy Transportation Problem
in which all the triangular numbers are of the form
(3,1-r,7-7r)
(4,1-r,7-7r)
 m,  ,  
(0,0,0)
A2
B1 B2 B3 Supply
(399,99-99r,105-105r)
(48,99-99r,105-105r)
(351,51-51r,45-45r)
(3,2-2r,2-2r)
(5,2-2r,8-8r)
(0,0,0)
(3,2-2r,2-2r)
(6,1-r,7-7r)
A1
(0,0,0)
A3
(301,51-51r,45-45r)
By applying fuzzy version of MODI method, it can
be seen that the current initial fuzzy basic feasible
(3,1-r,7-7r)
(4,1-r,7-7r)
(0,0,0)
solution is optimal. The fuzzy optimal solution in
A2 terms of location index and fuzziness index is given
by
x11   399, 99  99r, 105  105r  ;

x 21 
  49,150  150r, 150  150r  ;
(399,99-99r,105-105r)
x 23 
  252,150  150r, 150  150r  ;
(3,2-2r,2-2r)
(6,1-r,7-7r)
x 32 
  351,51  51r, 45  45r  and
(0,0,0)
A3
x 33 
  48,99  99r,105  105r  , where 0
 r  1.
The corresponding fuzzy optimal transportation cost
is given by
m n

Minimize Z    c x
  ij ij
i 1 j1
(300,150-150r,150-150r)
  2357,150  150r,150  150r 
(448,48-48r,60-60r)
(351,51-51r,45-45r)
where 0 ≤ r ≤ 1 can be suitably chosen by the
decision maker.
Demand
Case(i). When r = 0, the fuzzy optimal
transportation cost in terms of the form (m, α, β) is
(2397, 150,150). The corresponding fuzzy optimal
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8. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1416-1424

transportation cost of the form A  (a1 ,a 2 ,a 3 ) is [6]. A. Charnes, W. W. Cooper, The stepping
stone method for explaining linear
(2217, 2397, 2547). Its defuzzyfied transportation
programming calculation in transportation,
cost is 2392.
Management science, 1 (1954), 49-69.
Case(ii). When r = 0.5, the fuzzy optimal [7]. G. B. Dantzig, M. N. Thapa, Springer:
transportation cost in terms of the form (m, α, β) is L.P:2: Theory and Extensions, Princeton
(2397, 75, 75). The corresponding fuzzy optimal university Press New Jersey, 1963.

transportation cost of the form A  (a1 ,a 2 ,a 3 ) is [8]. D. Dubois and H. Prade, Fuzzy Sets and
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