Applied Mathematical Sciences, Vol. 9, 2015, no. 7, 337 - 343
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2015.410857
Max – Min Method to Solve Fuzzy
Transshipment Problem
P. Gayathri, K. Kannan and D. Sarala
Department of Mathematics
Sastra University, Thanjavur, India
Copyright © 2014 P. Gayathri, K. Kannan and D. Sarala. This is an open access article
distributed under the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper deals with fuzzy transshipment problem in which available commodity
frequently moves from one source to another source before reaching its actual
destination. Here Max – Min method is introduced to find initial feasible solution
for the large scale fuzzy transshipment problem.
Mathematics Subject Classification: 03E72
Keywords: Fuzzy Transportation Problem, Fuzzy Transshipment Problem, VAM
1 Introduction
Orden [1] has extended transportation problem to include the case when
transshipment is also allowed. Both the transportation problem and the
transshipment problem are also quite widely used for planning bulk distribution,
especially in the USA where the (road) distances travelled are large. In the
transshipment problem all the sources and destinations can function in any
direction. Usually, in the absence of transshipment, the transportation cost goes
higher. Hence transshipment is also very useful to reduce the transportation cost.
Garg and Prakash (1985) [2] studied time minimizing transshipment problem.
Afterwards, multi-location transshipment problem with capacitated production
and lost sales were studied by Ozdemir (2006) [3]. The concept of fuzzy numbers
and arithmetic operations with these numbers were first introduced and
investigated by Zadeh [4].In [5] Nagoor Gani et al.solving Transportation
Problem using Fuzzy Number and, Nagoor Gani et al [6,7]presented a two-stage
cost minimizing fuzzy transportation problem in which supplies and demands are
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P. Gayathri, K. Kannan and D. Sarala
trapezoidal fuzzy numbers. In [8], Stephen Dinagar et al investigated fuzzy
transportation problem with the aid of trapezoidal fuzzy numbers.
2 Preliminaries
2.1 Fuzzy Set: A fuzzy set A is defined by A   x,  A ( x)  : x  A,  A ( x) [0,1] .
In the pair  x,  A ( x)  , the first element x belong to the classical set A , the second
element  A ( x) belong to the interval [ 0 , 1 ], called Membership function.
2.2 Fuzzy number: A fuzzy set A on R must possess at least the following three
properties to qualify as a fuzzy number,
(i) A must be a normal fuzzy set ;
(ii)  A must be closed interval for every  [0,1]
(iii) The support of A, 0 A , must be bounded.
2.3 Triangular fuzzy number: It is a fuzzy number represented with three points
as follows: A = (a1, a2, a3). This representation is interpreted as membership
functions and holds the following conditions.
(i) a1 to a2 is increasing function
(ii) a2 to a3 is decreasing function
(iii) a1≤ a2 ≤ a3
2.4 Function principle operation of triangular fuzzy number
The following are the operations that can be performed on triangular fuzzy
numbers:
Let A = (a1, a2, a3) and B = (b1, b2, b3). Then
(i) Addition: A + B = (a1+b1, a2+b2, a3+b3)
(ii) Subtraction: A - B = (a1 - b3, a2 - b2, a3 - b1)
(iii) Multiplication: A x B = ( min (a1b1 , a1b3 , a3b1 , a3b3 ) , a2b2 , max ( a1b1 ,
a1b3 , a3b1 , a3b3 ) )
2.5 Graded mean integration method
The graded mean integration method is used to defuzzify the triangular fuzzy
number. The representation of triangular fuzzy number is A = (a1, a2, a3) and its
a  4a2  a3
defuzzified value is obtained by A = 1
6
2.6  - level set
The  - level set of the fuzzy number a and b is defined as the ordinary set
L (a, b ) for which the degree of their membership function exceeds the level
 [0,1].L (a, b)  {a, b  Rm | a (ai , b j )   , i  1, 2,....m, j  1, 2,....n}
Max – min method to solve fuzzy transshipment problem
339
2.7 Formulation of the fuzzy transshipment problem
The fuzzy transportation problem assumes that direct routes exist from each
source to each destination. However, there are situations in which units may be
shipped from one source to another or to other destinations before reaching their
destinations. This is called a fuzzy transshipment problem. The purpose of
transshipment, the distinction between a source and destination is dropped so that
a transportation problem with m sources and n destinations gives rise to a
transshipment problem with m + n sources and m + n destinations. The basic
feasible solution to such a problem will involve [(m + n) + (m + n) -1] or 2m + 2n
-1 basic variables and if we omit the variables appearing in the (m + n) diagonal
cells, we are left with m + n – 1 basic variables. Thus the fuzzy transshipment
problem may be written as:
m n m n
Maximize Z  

i 1 j 1, j  i
mn
cij xij Subject to

j 1, j  i
xij 
mn

j 1, j i
x ji  ai , i = 1, 2 , 3 , ……..
,m
m n

i 1,i  j
xij 
mn

i 1,i  j
x ji  b j , j = m + 1 , m + 2 , m + 3 , ………… , m + n
Where xij  0 , I , j = 1 , 2 , 3 , ……. , m + n , j  i
m
Where
m
 a  b
i 1
i
j 1
j
then the problem is balance otherwise unbalanced.
The above formulation is a fuzzy transshipment model, the transshipment model
is reduced to transportation form as:
m n m n
Minimize Z  

i 1 j 1, j  i
mn
cij xij
Subject to
x
j 1
ij
 ai  T ,
i = 1, 2 , 3 , ……..
,m
m n
m n
j 1
i 1
 xij  T , i = m + 1to m + n ,
mn
x
i 1
ij
x
ij
T ,
j = 1 to m
 b j  T , j = m + 1 to m + n where xij  0 , i , j = 1 to m + n , j  i
The above mathematical model represents a standard balanced transportation
problem with (m + n) origins and (m + n) destinations. T can be interpreted as a
buffer stock at each origin and destination. Since we assume that any amount o
goods can be transshipped at each point, T should be large enough to take care of
all transshipments. It is clear that the volume of goods transshipped at any point
cannot exceed the amount produced or received and hence we take
m
m
i 1
j 1
T   ai or  b j
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P. Gayathri, K. Kannan and D. Sarala
3 Max – min algorithm
Step 1: Balance the given transshipment problem if either ( total supply > total
demand ) or ( total supply < total demand ). Now the transshipment table in
fuzzy environment looks like fuzzy transportation table.
Step 2: In this fuzzy transportation table all the cost, supply and demand are in
fuzzy. Now convert all the fuzzy cost to α level cost.
Step 3: To apply Max – Min algorithm, subtract the difference of largest and
smallest cost elements of the above and below main diagonal elements from all
the elements of the cost matrix in the transportation table.
Step 4: Arrange all the costs of transportation table in ascending order. (If the
costs are repeated, Consider it once). Assign numbers 1, 2, 3, ……to these
arranged costs and form a new Transportation table with their corresponding
numbers for the costs.
Step 5: Since the supply & demand are in fuzzy, we use graded mean integration
method to find the max. Fuzzy number and min. fuzzy number.
Step 6: Identify a row (or column) for this max. Fuzzy number corresponding to
supply (or demand). Allocate the min. fuzzy number of supply (or demand)
corresponding to the min. cost of the identified row (or column)
Step 7: Subtract the allocated fuzzy number from the corresponding supply and
demand values. Delete a row (or column) with zero supply (or demand). Continue
this process until all rows and columns are satisfied.
Step 8: Compute the total fuzzy transportation cost for the feasible cost for the
feasible allocation using the original balanced fuzzy transshipment cost matrix.
4 Numerical example
Consider the following transshipment problem with two origins and three
destinations
O1
O2
O1 ( 0,0,0) (2,3,4)
O2 (3,4,5) (0,0,0)
D1
D2
D3
O1 (13,15,17) (7,9,11)
(6,8,10) (15,24,33)
O2
(5,7,9)
(7,9,11)
(6,8,10) (12,21,30)
(14,22,30) (10,18,26) (12,20,28)
O1
O2
D1 (14,16,18) (2,4,6)
D2
(5,7,9)
(4,5,6)
D3
(6,7,8)
(3,5,7)
(9,15,21) (9,15,21)
D1
D2
D3
D1 (0,0,0) (4,6,8) (3,5,7)
D2 (1,2,3) (0,0,0) (1,2,3)
D3 (1,2,3) (3,5,7) (0,0,0)
Max – min method to solve fuzzy transshipment problem
341
Solution: Now forming the transformed transportation problem
O1
O2
D1
D2
D3
O1
(0,0,0)
(2,3,4) (13,15,17) (7,9,11)
(6,8,10) (15,24,33)
O2
(3,4,5)
(0,0,0)
(5,7,9)
(7,9,11)
(6,8,10) (12,21,30)
D1 (14,16,18) (2,4,6)
(0,0,0)
(4,6,8)
(3,5,7)
(9,15,21)
D2
(5,7,9)
(4,5,6)
(1,2,3)
(0,0,0)
(1,2,3)
(9,15,21)
D3
(6,7,8)
(3,5,7)
(1,2,3)
(3,5,7)
(0,0,0)
(9,15,21)
(9,15,21) (9,15,21) (14,22,30) (10,18,26) (12,20,28)
Now applying the α-level to the fuzzy costs at α =0.5. Then for (0,0,0) = 0 ≤ x ≤ 0
we select 0, (2,3,4) = 2.5 ≤ x ≤ 3.5 we select 3,(13,15,17) =13.5 ≤ x ≤ 15.5 we
select15.Similarly for triangular fuzzy number at α =0.5, we select its middle
range.
So, the new transportation table is
O1
O2
D1
D2
D3
O1
0
3
15
9
8
(15,24,33)
O2
4
0
7
9
8
(12,21,30)
D1
16
4
0
6
5
(9,15,21)
D2
7
5
2
0
2
(9,15,21)
D3
7
5
2
5
0
(9,15,21)
(9,15,21) (9,15,21) (14,22,30) (10,18,26) (12,20,28)
To apply Max Min Algorithm, the newly transformed transportation table is
O1
O2
D1
D2
D3
O1
1
3
10
9
8
(15,24,33)
O2
4
1
7
9
8
(12,21,30)
D1
11
4
1
6
5
(9,15,21)
D2
7
5
2
1
2
(9,15,21)
D3
7
5
2
5
1
(9,15,21)
(9,15,21) (9,15,21) (14,22,30) (10,18,26) (12,20,28)
Now the proposed Max-Min algorithm is applied and shown below
O1
O2
D1
D2
D3
O1
1
3
10
9
8
(15,24,33)
( 9,5,21)
(-42,1,44) (-11,3,17) (-9,5,19) (-6,9,24)
(-25,4,33)
(-42,1,44)
O2
4
1
7
9
8
(12,21,13)
(9,15,21) (-9,6,4)
(-9,6,4)
D1
11
4
1
6
5
(9,15,21)
(9,15,21)
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P. Gayathri, K. Kannan and D. Sarala
D2
7
5
2
D3
7
5
2
1
(9,15,21)
5
2
(9,15,21)
1
(9,15,21)
(9,15,21)
(9,15,21) (9,15,21) (14,22,30) (10,18,26) (12,20,28)
(-7,7,21) (-11,3,17) (-9,5,19)
(-11,1,30)
Number of fuzzy units transported from origin to destinations O1 to D1 is (42,1,44) , O1 to D2 is (-11,3,17) , O1 to D3 is (-9,5,19) , O2 to D1 is (-9,6,4) and
their corresponding fuzzy cost are O1 to D1 is (13,15,17) , O1 to D2 is (7,9,11) ,
O1 to D3 is (6,8,10) , O2 to D1 is (5,7,9 ).
Therefore the initial basic feasible fuzzy transportation cost is
(-42,1,44)*(13,15,17)+(-11,3,17)*(7,9,11)+(-9,5,19)*(6,8,10)+(9,6,4)*(5,7,9)=108.5
For the above transshipment problem, the initial feasible solution by Vogel’s
Approximation Method is 129.3333. Hence the Max – Min algorithm gives better
result than the well known Vogel’s Approximation Method.
Conclusion
In this paper Max - Min method is introduced to find the initial feasible solution
to the Fuzzy transshipment problem. The main advantage of the proposed method
is to determine the feasible solution without finding penalties as in Vogel’s
Approximation method. Also this Max – Min method yields better solution for
the large scale fuzzy transshipment problem when compared with the solution of
Vogel’s Approximation method.
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Received: October 25, 2014; Published: January 3, 2015