In this paper, we investigate transportation problem in which supplies and demands are intuitionistic fuzzy numbers. Intuitionistic Fuzzy Vogel’s Approximation Method is proposed to find an initial basic feasible solution. Intuitionistic Fuzzy Modified Distribution Method is proposed to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. The solution procedure is illustrated with suitable numerical example.

1. International Journal of Mathematics Research.
ISSN 0976-5840 Volume 4, Number 4 (2012), pp. 411-420
© International Research Publication House
http://www.irphouse.com
The Transportation Problem in an Intuitionistic
Fuzzy Environment
R. Jahir Hussain1
and P. Senthil Kumar2
1
Associate Professor and 2
Research Scholar
PG and Research Department of Mathematics, Jamal Mohamed College,
Tiruchirappalli: 620 020 India
E-mail: hssn_jhr@yahoo.com, senthilsoft_5760@yahoo.com
Abstract
In this paper, we investigate transportation problem in which supplies and
demands are intuitionistic fuzzy numbers. Intuitionistic Fuzzy Vogel’s
Approximation Method is proposed to find an initial basic feasible solution.
Intuitionistic Fuzzy Modified Distribution Method is proposed to find the
optimal solution in terms of triangular intuitionistic fuzzy numbers. The
solution procedure is illustrated with suitable numerical example.
Keywords: Triangular intuitionistic fuzzy numbers, intuitionistic fuzzy
transportation problem, intuitionistic fuzzy vogel’s approximation method,
intuitionistic fuzzy modified distribution method, initial basic feasible
solution, optimal solution.
Introduction
The theory of fuzzy set introduced by Zadeh[8] in 1965 has achieved successful
applications in various fields. The concept of Intuitionistic Fuzzy Sets (IFSs)
proposed by Atanassov[1] in 1986 is found to be highly useful to deal with vagueness.
The major advantage of IFS over fuzzy set is that IFSs separate the degree of
membership (belongingness) and the degree of non membership (non belongingness)
of an element in the set .The concept of fuzzy mathematical programming was
introduced by Tanaka et al in 1947 the frame work of fuzzy decision of Bellman and
Zadeh[2].
In [4], Nagoor Gani et al presented a two stage cost minimizing fuzzy
transportation problem in which supplies and demands are trapezoidal fuzzy number.
In [7], Stephen Dinager et al investigated fuzzy transportation problem with the aid of
trapezoidal fuzzy numbers. In[6], Pandian.P and Natarajan.G presented a new

2. 412 R. Jahir Hussain and P. Senthil Kumar
algorithm for finding a fuzzy optimal solution for fuzzy transportation problem. In
[3], Ismail Mohideen .S and Senthil Kumar .P investigated a comparative study on
transportation problem in fuzzy environment.
In this paper , a new ranking procedure which can be found in [5] and is used to
obtain a basic feasible solution and optimal solution in an intuitionistic fuzzy
transportation problem[IFTP]. The paper is organized as follows: section 2 deals with
some terminology, section 3 provides the mathematical formulation of intuitionistic
fuzzy transportation problem, section 4 deals with solution procedure, section 5
consists of numerical example, finally conclusion is given.
Terminology
Definition 2.1: Let A be a classical set, be a function from A to [0,1]. A fuzzy
set with the membership function is defined by
, ; 0,1 .
Definition 2.2: Let X be denote a universe of discourse, then an intuitionistic fuzzy
set A in X is given by a set of ordered triples,
, , ;
Where , : 0,1 , are functions such that 0 1, .
For each x the membership represent the degree of membership and
the degree of non – membership of the element to respectively.
Definition 2.3: An Intuitionistic fuzzy subset A = {<x, µA(x), υA(x)> : x X } of the
real line R is called an intuitionistic fuzzy number (IFN) if the following holds:
i. There exist m R, µA(m) = 1 and υA(m) = 0, (m is called the mean value of
A).
ii. µA is a continuous mapping from R to the closed interval [0,1] and
, the relation 0 x x 1 holds.
The membership and non: membership function of A is of the following form
x
0 ∞
,
1
,
0 ∞
Where f1(x) and h1(x) are strictly increasing and decreasing function in
, and , respectively

3. The Transportation Problem in an Intuitionistic Fuzzy Environment 413
x
1 ∞
, ; 0 1
0
, ; 0 1
1 ∞
Here m is the mean value of A. α and β are called left and right spreads of
membership function x , respectively. α′ β′
represents left and right spreads
of non membership function x , respectively. Symbolically, the intuitionistic fuzzy
number is represented as AIFN =(m; , ; α′, β′
).
Definition 2.4: A Triangular Intuitionistic Fuzzy Number (ÃI
is an intuitionistic fuzzy
set in R with the following membership function x and non membership function
x : )
x
0
x
1
Where and x , x 0.5 for x
x . This TrIFN is denoted by
= , , , ,
Membership and non membership functions of TrIFN

4. 414 R. Jahir Hussain and P. Senthil Kumar
Ranking of Triangular intuitionistic Fuzzy Numbers
The Ranking of a triangular intuitionistic fuzzy number is completely defined by its
membership and non- membership as follows [5]:
Let ÃI
= (a,b,c) (e,b,f)
1
6
2 3
1
6
3 2
2
1
6
3 2
1
6
2 3
2
1
3
2
3
Rank (A) = (Sqrt ((xµ(A))2
+ (yµ(A))2
), Sqrt ((xυ(A))2
+ (yυ(A))2
))
Definition 2.5: Let and be two TrIFNs. The ranking of and by the R(.) on
E, the set of TrIFNs is defined as follows:
i. R( )>R( ) iff
ii. R( )<R( ) iff
iii. R( )=R( ) iff
Definition 2.6: The ordering and between any two TrIFNs and are defined
as follows
i. iff or and
ii. iff or
Definition 2.7: Let , 1,2, … , be a set of TrIFNs. If for all i,
then the TrIFN is the minimum of , 1,2, … , .
Definition 2.8: Let , 1,2, … , be a set of TrIFNs. If for all i,
then the TrIFN is the maximum of , 1,2, … , .
Arithmetic Operations
Addition: = , , , ,
Subtraction: ÃI
Θ B̃I
= , , , ,
Multiplication
A B l , l , l l , l , l
Where,

5. The Transportation Problem in an Intuitionistic Fuzzy Environment 415
min , , ,
= max { , , ,
min { , , , }
max { , , , }
Scalar multiplication
i. , , , , , 0
ii. , , , , , 0
Intuitionistic fuzzy transportation problem and its mathematical
formulation
Consider a transportation with m IF origins (rows) and n IF destinations (columns).
Let be the cost of transporting one unit of the product from ith
IF (Intuitionistic
Fuzzy) origin to jth
IF destination. , , , , be the quantity of
commodity available at IF origin i.
, , , , the quantity of commodity needed at intuitionistic
fuzzy destination j.
, , , , is the quantity transported from ith
IF origin to jth
IF destination, so as to minimize the IF transportation cost.
(IFTP) Minimize ∑ ∑
Subject to
, 1,2, … ,
, 1,2, … ,
0 , 1,2, … ,
1,2, … ,
Where m = the number of supply points
n = the number of demand points
, , , , is the number of units shipped from ith
IF origin
to jth
IF destination.
= the cost of shipping one unit from IF supply point i to IF demand point j.
, , , , is the intuitionistic fuzzy supply at supply point i and
, , , , is the IF demand at demand point j.

6. 416 R. Jahir Hussain and P. Senthil Kumar
The above IFTP can be stated in the below tabular form
Definition 3.1: Any set of intuitionistic fuzzy non negative allocations >(-
2δ,0,2δ)(-3δ,0,3δ) where δ is small positive number, which satisfies the row and
column sum is a IF feasible solution.
Definition 3.2: Any feasible solution is an intuitionistic fuzzy basic feasible solution
if the number of non negative allocations is at most (m+n-1) where m is the number of
rows and n is the number of columns in the transportation table.
Definition 3.3: Any intuitionist fuzzy feasible solution to a transportation problem
containing m origins and n destinations is said to be intuitionist fuzzy non degenerate,
if it contains exactly (m+n-1) occupied cells.
Definition 3.4: If an intuitionistic fuzzy basic feasible solution contains less than
(m+n-1) non negative allocations, it is said to be degenerate.
Solution of an intuitionistic fuzzy transportation problem
The solution of an IFTP can be solved two stages, namely initial solution and optimal
solution. Finding an initial solution of an IFTP there are numerous methods but
intuitionistic fuzzy vogel’s approximation method (IFVAM) is preferred over the
other methods, since the initial intuitionistic fuzzy basic feasible solution obtained by
this method is either optimal or very close to the optimal solution. We are going to
discuss IFVAM.
4.1 Intuitionistic fuzzy vogel’s approximation method
Step 1: Find the penalty cost, namely the difference between the smallest and next-
to-smallest costs for each row and display them to the right of the
corresponding row. If there are more than one least cost, the difference is

7. The Transportation Problem in an Intuitionistic Fuzzy Environment 417
zero. Similarly, compute the difference for each column.
Step 2: Among the penalties as found in step1, choose the maximum penalty. If this
maximum penalty is more than one, choose arbitrarily.
Step 3: In the selected row or column as by step2, allocate the maximum possible
amount to the cell with the least cost in the selected row or column i.e., =
min { , } by ranking procedure. If = , then delete the ith
row and
adjust the amount of IF demand. If = , then delete the jth
column and
adjust the amount of IF supply. If = = , then delete either ith
row or jth
column, but not both.
Step 4: Repeat step1 to step3 until all the intuitionistic fuzzy supply points are fully
used and all the intuitionistic fuzzy demand points are fully received.
4.2 Intuitionistic Fuzzy Modified Distribution Method
This proposed method is used for finding the optimal solution in an intuitionistic
fuzzy environment and the following step by step procedure is utilized to find out the
same.
1. Find out a set of numbers and for each row and column satisfying
for each occupied cell. To start with we assign an intuitionistic
fuzzy zero to any row or column having maximum number of allocations. If
this maximum number of allocation is more than one, select any one
arbitrarily.
2. For each empty (un occupied) cell, we find intuitionistic fuzzy sum and .
3. Find out for each empty cell the net evaluation value, = ,
this step gives the optimality conclusion.
i. If all 2δ, 0,2δ 3δ, 0,3δ the solution is IF optimal and a unique
solution exists.
ii. If 2δ, 0,2δ 3δ, 0,3δ then the solution is IF optimal, but an
alternate optimal solution exists.
iii. If at least one 2δ, 0,2δ 3δ, 0,3δ the solution is not IF optimal. In
this case we go to next step, to improve the total IF transportation cost.
4 Select the empty cell having the most negative value of from this cell we
draw a closed path drawing horizontal and vertical lines with corner cell
occupied. Assign sign + and – alternately and find the IF minimum allocation
from the cell having negative sign. This allocation having negative sign.
5 The above step yield a better solution by making one (or more) occupied cell
as empty and one empty cell as occupied. For this new set of intuitionistic
fuzzy basic feasible allocation repeat from the step1, till an intuitionistic fuzzy
optimal solution is desired.

8. 418 R. Jahir Hussain and P. Senthil Kumar
Numerical Example
Consider the 4× 4 IFTP
IFD1 IFD2 IFD3 IFD4 IF supply
IFO1 16 1 8 13 (2,4,5)(1,4,6)
IFO2 11 4 7 10 (4,6,8)(3,6,9)
IFO3 8 15 9 2 (3,7,12)(2,7,13)
IFO4 6 12 5 14 (8,10,13)(5,10,16)
IF demand(3,4,6)(1,4,8)(2,5,7)(1,5,8)(10,15,20)(8,15,22)(2,3,5)(1,3,6)
Since ∑ ∑ = (17, 27, 38) (11, 27, 44), the problem is balanced
IFTP. There exists an IF initial basic feasible solution.
IFD1 IFD2 IFD3 IFD4 IF supply
IFO1
(2,4,5) 1
(1,4,6)
(2,4,5)
(1,4,6)
IFO2
(-3,1,5) 4
(-5,1,7)
(-1,5,11) 7
(-4,5,14)
(4,6,8)
(3,6,9)
IFO3
(-2,4,10) 8
(-4,4,12)
(2,3,5) 2
(1,3,6)
(3,7,12)
(2,7,13)
IFO4
(-13,0,14) 6
(-21,0,22)
(-1,10,21) 5
(-6,10,26)
(8,10,13)
(5,10,16)
IF demand
(3,4,6)
(1,4,8)
(2,5,7)
(1,5,8)
(10,15,20)
(8,15,22)
(2,3,5)
(1,3,6)
Since the number of occupied cell having m+n-1 and are also independent, there
exist a non degenerate IF basic feasible solution. Therefore, the initial IF
transportation minimum cost is,
Min I
=(-112,131,381)(-233,131,502)
To find the optimal solution
Applying the intuitionistic fuzzy modified distribution method, we determine a set of
numbers , , , , , , , , , each nrow and column
such that
, , , , + , , , , for each occupied cell.
Since maximum number of allocations in row and column are same, so we give
intuitionistic fuzzy number , , , , (-2,0,2)(-3,0,3). The
remaining numbers can be obtained as given below.
= , , , , + , , , ,
, , , , =(6,8,10)(5,8,11)

9. The Transportation Problem in an Intuitionistic Fuzzy Environment 419
= , , , , + , , , ,
, , , , =(4,6,8)(3,6,9)
= , , , , + , , , ,
, , , , =(-6,-4,-2)(-7,-4,-1)
= , , , , + , , , ,
, , , , =(6,8,10)(5,8,11)
= , , , , + , , , ,
, , , , =(-3,-1,1)(-4,-1,2)
= , , , , + , , , ,
, , , , =(-8,-6,-4)(-9,-6,-3)
= , , , , + , , , ,
, , , , =(3,5,7)(2,5,8)
We find, for each empty cell of the sum , , , , and
, , , , . Next we find the net evaluation , , , ,
is given by
IFD1 IFD2 IFD3 IFD4 IF supply
IFO1
*(7,11,15)16
(5,11,17)
(2,4,5) 1
(1,4,6)
*(0,4,8) 8
(-2,4,10)
*(10,14,18) 13
(8,14,20)
(2,4,5)
(1,4,6)
IFO2
*(-1,3,7) 11
(-3,3,9)
(-3,1,5) 4
(-5,1,7)
(-1,5,11) 7
(-4,5,14)
*(4,8,12) 10
(2,8,14)
(4,6,8)
(3,6,9)
IFO3
(-2,4,10) 8
(-4,4,12)
*(7,11,15)15
(5,11,17)
*(-2,2,6) 9
(-4,2,8)
(2,3,5) 2
(1,3,6)
(3,7,12)
(2,7,13)
IFO4
(-13,0,14) 6
(-21,0,22)
*(6,10,14)12
(4,10,16)
(-1,10,21)5
(-6,10,26)
*(10,14,18) 14
(8,14,20)
(8,10,13)
(5,10,16)
IF demand
(3,4,6)
(1,4,8)
(2,5,7)
(1,5,8)
(10,15,20)
(8,15,22)
(2,3,5)
(1,3,6)
Where
= , , , , , = , , , ,
, , , , = -[ , , , , + , , , , ]
Since all , , , , >0 the solution is intuitionistic fuzzy optimal
and unique.The intuitionistic fuzzy optimal solution in terms of triangular
intuitionistic fuzzy numbers
=(2,4,5)(1,4,6), =(-3,1,5)(-5,1,7), =(-1,5,11)(-4,5,14), =(-2,4,10)(-
4,4,12), =(2,3,5)(1,3,6), =(-13,0,14)(-21,0,22), =(-1,10,21)(-6,10,26)

10. 420 R. Jahir Hussain and P. Senthil Kumar
Hence, the total intuitionistic fuzzy transportation minimum cost is
Min I
=(-112,131,381)(-233,131,502)
Conclusion
Mathematical formulation of intuitionistic fuzzy transportation problem and
procedure for finding an intuitionistic fuzzy optimal solution in two stages are
discussed with suitable numerical example. In the first stage, initial basic intuitionistic
fuzzy feasible solution using intuitionistic fuzzy vogel’s approximation method is
determined. In second stage, intuitionistic fuzzy optimal solution using intuitionistic
fuzzy modified distribution method is calculated. The new arithmetic operations of
triangular intuitionistic fuzzy numbers are employed to get the optimal solution in
terms of triangular intuitionistic fuzzy numbers. This method is a systematic
procedure, both easy to understand and to apply also; it can serve as an important tool
for the decision makers when they are handling various types of logistic problems
having intuitionistic fuzzy parameters.
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