International Journal of Scientific & Engineering Research, Volume 5, Issue 5, May-2014 76

ISSN 2229-5518

A Comparative Study of Optimization Methods for

Fuzzy Transportation Problems

Poonam Kumari

Department of Mathematics, Magadh Mahila College, Patna University

Email: poonamkumari1865@gmail.com

Abstract— Solution of a fuzzy transportation problem determines the transportation schedule that minimizes the total fuzzy transportation cost while satisfying the availability and requirement limits. A number of methods have been devised to solve a fuzzy transportation problem, but the solution obtained by some methods is not optimal. Obviously, the method which gives the optimal solution should be preferred and used in practice.

This work is intended to compare the performance of different methods available for solving a a fuzzy transportation problem and to find out the most appropriate one. For this purpose, a number of fuzzy transportation problems have been solved by different methods and the solution obtained has been tested for optimality. It has been found that the solution obtained by Russell;s Approximation Method is optimal for most of the fuzzy transportation problems.Therefore this method may be regarded as the the most appropriate method for solving a fuzzy transportation problem

Keywords— Fuzzy Transportation Problem, Trapezoidal Fuzzy Number, Ranking Function, Basic Feasible Solution, Optimal Solution

—————————— ——————————

1 INTRODUCTION

HE transportation problem is a special class of linear programming problems in which a commodity is to be transported from various sources of supply to various destinations of demand in such a way that the total transpor- tation cost is a minimum. In general, transportation problem is solved with the assumption that the decision parameters such as availability, requirement and the unit transportation cost are known exactly. But in real life applications, supply, demand and unit transportation cost may be uncertain due to several factors. These imprecise data may be represented by
fuzzy numbers.
The transportation problem, in which the transportation
costs, supply and demand quantities are represented in terms
of fuzzy numbers, is called a fuzzy transportation problem.
The objective of the fuzzy transportation problem is to deter-
mine the transportation schedule that minimizes the total
fuzzy transportation cost while satisfying the availability and
requirement limits. Most of the existing techniques provide
only crisp solution for fuzzy transportation problem. Chanas
etal [1] developed a method for solving fuzzy transportation
problems by applying the parametric programming tech- nique using the Bellman–Zadeh criterion [2]. Chanas and
In this paper, the fuzzy transportation problems using trapezoidal fuzzy numbers are discussed. The initial basic feasible solution of the same fuzzy transportation problem is obtained by different methods and then, U-V distribution method is used to find out the optimal solution for the total fuzzy transportation minimum cost.

2 OBJECTIVE

The aim of this paper is to compare the performance of differ- ent methods available for obtaining an initial basic feasible solution of the fuzzy transportation problem and to find out the most appropriate one.

3 PRELIMINARIES

3.1 Fuzzy Set

~

A fuzzy set A on a universal set X is a set of ordered pairs:
~ = µ ~

A

3.2 Fuzzy Number

~
Kuchta [3] proposed a method for solving a fuzzy transporta-
A fuzzy set A
defined on the set of real numbers R is said to
tion problem by converting the given problem to a bicriterial
transportation problem with crisp objective function which
provides only crisp solution to the given problem. Liu and
Kao [4] proposed a new method for the solution of the fuzzy
be a fuzzy number if its membership function µ ~ : R → [0,1] is
continuous and such that
 0 ∀x (− ∞, a]

transportation problem by using the Zadeh’s extension prin-
ciple. Using parametric approach, Nagoorgani and Abdul
f A

(x)

increasing on [a, b]
Razak [5] obtained a fuzzy solution for a two stage Fuzzy

µ ~ (x) = 

1 ∀x [b, c]
Transportation problem with trapezoidal fuzzy numbers. Omar et. alalso proposed a parametric approach for solving transportation problem under fuzziness. Pandian and Nata-

A

g A (x)


decreasing on [c, d]
rajan proposed a fuzzy zero point method to find the fuzzy optimal solution of fuzzy transportation problems.
 0 ∀x [d , ∞)

IJSER © 2014 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 5, Issue 5, May-2014 77

ISSN 2229-5518

where a, b, c ,d are real numbers, and the fuzzy number
~

TRANSPORTATION PROBLEM

denoted by A
ber.
= (a, b, c, d) is called a trapezoidal fuzzy num-
Mathematically, a fuzzy transportation problem can be stated as follows:

3.3 Trapezoidal Fuzzy Number

~ m n ~ ~
Minimize
~

Z = ∑∑Cij X ij

A fuzzy number A
= (a, b, c, d) is said to be trapezoidal if

i=1

j =1

f A and g A

are linear functions. The membership function
subject to the constraints
~
~ of a trapezoidal fuzzy number A = (a, b, c, d) is given by

A

n

j =1

m

~

X ij

~
a~ ,
~

i = 1,2,............m

 0 ,

x - a

b - a ,

x < a

a ≤ x b
where

X ij b j ,

i=1

X ij ≥ 0

j = 1, 2,............n

µ ~ (x) =  1 ,

A



d - x ,
d - c
 0 ,
b ≤ x c c x d

x > d

m = total number of sources
n = total number of destinations
a~ =fuzzy availability of the product at ith source
~
b j = fuzzy demand of the product at jth destination
~

3.4 Ranking Function

Let F(R) denote the set of all trapezoidal fuzzy numbers de- fined on the set of real numbers R. Then we define a ranking function as a function R : F(R)→R, which maps each fuzzy number into a real number.
Cij = fuzzy cost of transporting one unit of the product from ith source to jth destination
~ = fuzzy quantity of the product that should be transport- ed from ith source to jth destination
~ ~ a + b + c + d

If A = (a, b, c, d) , then R ( A) = .
The above fuzzy transportation problem is said to be bal-

4 ~ m ~ n ~

For any two trapezoidal fuzzy numbers
~

A1 = (a1 , b1 ,c1, d1 )

anced if ai = b j , otherwise it is called unbalanced.

and A 2 = (a2 , b2 , c2 , d 2 ) , we have

i =1

j =1

~ ~

A1 A 2

(ii) A1 A 2
~ ) ≤ R (~ )

R (A ) ≥ R (A2 )

5 METHODOLOGY

The solution of a fuzzy transportation problem involves two
~ ~ ~ ~

1

(iii) ~ ≈ ~ ⇔ R (~ ) = R (~ )

A1 A 2 A A

stages: Finding an initial basic feasible solution, and then,
finding the optimal solution from the initial basic feasible

1 2

solution. In this paper, the same approach has been used to

3.5 Arithmetic Operations on Trapezoidal Fuzzy

Numbers

solve a fuzzy transportation problem.
Some fuzzy transportation problems were selected from
Let

A1 = (a1 ,b1 ,c1, d1 ) and

A 2 = (a2 , b2 , c2 , d 2 ) be two trapezoi-

the literature and their initial basic feasible solution were ob- tained using different methods. Then, U-V distribution meth-
dal fuzzy numbers. Then we define
od was used to find out the optimal solution of the test prob-
~ ~
(i) A1 + A 2 = (a1 + a2 , b1 + b2 , c1+c2 , d1 + d 2 )
lems, and the results were compared with the true optimal
~ ~
(ii) A1 A 2 = (a1 d 2 , b1 c2 , c1 b2 , d1 a2 )
solution.
The brief summaries of three methods, namely North-

(iii) k~

(iv) ~ . ~

(ka1 , kb1 , kc1, kd1 ) if k > 0
= 
(kd1 , kc1 , kb1 , ka1 ) if k < 0
= (a', b', c', d ')
West Corner Rule, Vogel’s Approximation Method[6] and Russell’s Approximation Method [7], are given in this paper. The U-V distribution method to find out the optimal solution
is also described here.

A1 A 2

where

a' = min (a1a 2 , a1d 2,a 2 d 1, d1d 2 ) ,

c' = max (b1b 2 , b1c 2 ,b 2 c1, c1c 2 ) ,

b' = min (b1b 2 , b1c 2 ,b 2 c1, c1c 2 ),

d ' = max (a1a 2 , a1d 2,a 2 d 1, d1d 2 )

6 SOLUTION OF A FUZZY TRANSPORTATION PROBLEM

6.1 Finding an Initial Basic Feasible Solution

North-West Corner Rule

4 MATHEMATICAL FORMULATION OF A FUZZY

IJSER © 2014

The various steps of this method are as follows:
http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 5, Issue 5, May-2014 78

ISSN 2229-5518

Step 1. Select the north-west corner cell of the transportation table and allocate as much as possible so that either
Step 3. Subtract ~ from

a~ or

~

b j found in Step 3. Eliminate

the capacity of the first row is exhausted or the desti- nation required of the first column is satisfied, i.e.,
the row or column from the transportation table that
results in a zero supply or destination demand after
~
~ ~ ~

a~ b

are zero, other-

X 11 = min( a1 , b1 ).

~ ~
this subtraction. Stop if all i and j
wise go to Step 1.
Step 2. If

b1 > a1 , then move down vertically to the second

row and make the second allocation of magnitude

6.2 Test for Optimality

X 21 =

( a2 b1

X11 ) in the cell (2, 1). If b1 < a1 , then

U-V Distribution Method

~ min
~ , ~ - ~ ~ ~
move right horizontally to the second column and make the second allocation of magnitude
The various steps of this method are as follows:
Step 1. Assign a zero trapezoidal fuzzy number to any row
~ = min( a~ - ~
, ~ )
in the cell (1, 2). If
~ = a~ ,
or column having maximum number of allocations.

X12

1 X11 b2

b1 1

then we can make the second allocation of magni- tude 0 either in the cell (2, 1) or in the cell (1, 2).
Step 3. Repeat the procedure until all the demands are satis-
If the maximum number of allocations is more than
one, choose any one arbitrarily.
Step 2. For each basic cell, find out a set of numbers
~ ~ ~ ~ ~
fied.

Vogel’s Approximation Method

U i and V j satisfying U i + V j = Cij .

Step 3. For each non basic cell, find out the net evalution
The various steps of this method are as follows:
~ ~

U i V j

~

Cij .

Step 1. Find the fuzzy penalties, i.e., the fuzzy difference
between the smallest and the next smallest fuzzy
costs in each row and column.
~ ~ ~
Case 1. If U i + V j - Cij < 0 for all i, j, then the solution is
optimal and a unique solution exists.
Step 2. Select the row or column with the highest fuzzy pen-
Case 2. If
~ + ~ - ~

≤ 0 for all i, j, then the solution is

alty. If the highest penalties are more than one,

U i V j

Cij

choose any one arbitrarily.
optimal, but an alternate solution exists.
~ ~ ~
Step 3. In the selected row or column found in Step 2, find out the cell having the smallest fuzzy cost. Allocate to this cell as much as possible depending on the fuzzy availability and the fuzzy demands.
Case 3. If U i + V j - Cij > 0 for at least one i, j, then the so-
lution is not optimal. In this case, we go to the following step.
Step 4. Select the non basic cell having the largest positive
~ + ~ - ~
to enter the basis. Let the cell (r,
Step 4. Delete the row or column which is fully exhausted.
For the reduced fuzzy transportation table, again
value of U i

V j Cij

compute the fuzzy penalties, then go to Step 2 and repeat the procedure until all the demands are satis- fied.

Russell’s Approximation Method

The various steps of this method are as follows:
s) enter the basis. Allocate an unknown quantity, say
θ , to the cell (r, s). From this cell (r, s), draw a closed
path horizontally and vertically to the nearest basic cell with the restriction that the corner of the closed path must not lie in any non basic cell. Assign signs
+ and – alternately to the cells of the loop, starting
~ ~ ~ ~ ~
Step 1. Calculate the quantities Ui , V j and Cij Ui V j
i and j , where
for all
with a + sign for the entering cell.Then θ = mini-
mum of the allocations made in the cells having a
~ ~ negative sign. Add this value of θ to all cells having

U i = max {Cij } for i = 1, 2,……………….m

1≤ jn

V j = max {Cij } for j = 1, 2,……………….n

~ ~

im

1

~
+ sign and subtract the same from the cells having a
– sign. Then the allocation of one basic cell reduces to zero.This yields a better solution by making one (or
more) basic cell as non basic cell and one non basic
Step 2. Select the variables

X ij having the most negative

cell as basic cell.
~ ~ ~
value of Cij U i V j . If there are ties in the value of
Step 5. For the new set of fuzzy basic feasible solution ob-
~ − ~
− ~ , select ~
having the smallest unit cost
tained in Step 4, repeat the procedure until a fuzzy

Cij

~

U i V j ij

optimal solution is obtained.
~

Cij . If there are ties again in the value of

~

Cij , se-

7 EXAMPLE

lect X ij having the largest amount of remaining
source supply or destination demand.
~
For the fuzzy transportation problem given below, find the fuzzy quantity of the product transported from each source to
Set the activity level of

X ij equal to the smaller value

~
various destinations so that the total fuzzy transportation cost
between the source supply
~
demand b j .

ai and the destination

is minimum.

IJSER © 2014 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 5, Issue 5, May-2014 79

ISSN 2229-5518

TABLE 1

FUZZY TRANSPORTATION PROBLEM

We have
~ + ~ = ~
(0,0,0,0) + ~ = (8,16,24,32)⇒ ~ = (8,16,24,32)

U 2 V1

i

C21 V1 V1

~ + ~ = ~
(0,0,0,0) + ~ = (8,14,18,24)⇒ ~ = (8,14,18,24 )

U 2 V2

C22 V2 V2

~ + ~ = ~
(0,0,0,0) + ~ = (4,8,12,16)⇒ ~ = (4,8,12,16)

U 2 V3

C23 V3 V3

~ + ~ = ~
⇒ ~ + (8,16,24,32) = (- 4,0,4,16 )⇒ ~ =

U1 V1

C11 U1 U1

(- 36,-24,-12 ,8)

~ ~ ~ ~ ~
U 3 + V3 = C33 U 3 + (4,8,12,16) = (0,12,16,20)U 3 = (-16,0,8,16)

~ ~ ~ ~ ~

U 3 + V4 = C34 (-16,0,8,16) + V4 = (8,14,18,24 ) V4 = (- 8,6,18,40)

Now we compute the net evaluation ~ ~
- ~ for all the

Solution :

m n

~
non basic cells.
We have
Since

ai = b j , the problem is a balanced fuzzy

~ ~ ~

(- 44,-14,6,3 6 )

U1 + V2 C12 =

i=1

j =1

~ ~ ~
transportation problem.
U1 + V3 C13 = (- 48, - 20, 0, 28 )

TABLE 2

~ + ~ − ~
= (- 52, - 20, 6, 50 )

INITIAL BASIC FEASIBLE SOLUTION BY NORTH-W EST CORNER RULE

U1 V4

~ ~

C14

~

(- 22, - 4, 12, 38 )

U 2 + V4 C24 =

~

ai ~

+ ~ − ~
= (- 34, - 2, 24, 44 )

U 3 V1

C31

~ + ~ − ~
= (- 28, - 2, 14, 40 )

U 3 V2

C 32

~ ~ ~

(0,4,8,12)

Since U i + V j - Cij > 0 for some i and j , the solution is not
optimal

(4, 8,18,26)

TABLE 3

INITIAL BASIC FEASIBLE SOLUTION BY VOGELS APPROXIMATION METHOD

(4,8,12,16)

Dest. (j)→ Source (i)↓

D 1 D 2 D 3 D 4 ~

(8,20,38,54)

S1 (-4,0,4,16)

(0,4,8,12)

(-4,0,4,16) (-4,0,4,16) (-2,0,2,8) (0,4,8,12)

Since the number of basic cells are m+n-1=6, the solution is non degenerate fuzzy basic feasible solution.
The initial total fuzzy transportation cost is

S2 (8,16,24,32) (8,14,18,24) (4,8,12,16)

(-10, -

2,12,24)

(2,6,10,14)

(2,6,10,14)

(4, 8,18,26)

Minimum Z ( Z(1), Z(2), Z(3), Z(4))
= (-4,0,4,16) (0,4,8,12) + (8,16,24,32) (-10,-2,6,14)+ (8,14,18,24)

S3 (4, 8,18,26)

(-10,-

2,6,14)

(0,12,16,20)

(2, 2, 8, 12)

(0,12,16,20)

(-22, -6,12,

24)

(8,14,18,24) (4,8,12,16)

(2,2,8,12)+(4,8,12,16)(-22,-6,18,34)+(0,12,16,20) (-32,-12,16,36)
+ (8,14,18,24) (2,6,10,14)
~ (2,6,10,14) (2,2,8,12) (2,6,10,14) (2,6,10,14) (8,20,38,54)

b j

(-1328, -200, 972, 2528)

≈ 493

Test for Optimality

Since 2nd row has maximum number of allocations, we
Since the number of basic cells are m+n-1=6, the solution is non degenerate fuzzy basic feasible solution.
The initial total fuzzy transportation cost is
Minimum Z ( Z(1), Z(2), Z(3), Z(4))
= (-4,0,4,16) (0,4,8,12) + (4,8,12,16) (-10,-2,12,24) + (2, 6,10,14) (2,6,10,14)+(4,8,18,26)(-10,-2,6,14) +(0,12,16,20) (2,2,8,12)
take
~ = (0,0,0,0) . Now we compute ~
and ~
for all the
+(0,12,16,20) (-22,-6,12,24)
basic cells.

(-904, -96, 704, 1856)

IJSER © 2014 http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 5, Issue 5, May-2014 80

ISSN 2229-5518

390

Test for Optimality

Since 3rd row has maximum number of allocations, we take
The main virtue of the North West Corner Rule is that it is quick and easy. However, because it pays no attention to unit costs ( cij ), usually the solution obtained is far from optimal.
Vogel’s Approximation Method has been a popular crite-
~ = (0,0,0,0) . Now we compute
~ and
~ for all the basic
rion for many years since difference represents the minimum extra unit cost incurred by failing to make an allocation to the
cells.
We have
cell having the smallest unit cost in that row or column, this
criterion does take costs into account in an effective way.
Therefore, this method may give an optimal solution in some
~ + ~ = ~
(0,0,0,0) + ~ = (4, 8,18,26) ⇒ ~ = (4, 8,18,26)
cases. However, if the solution obtained by this method is not

U 3 V1

C31 V1 V1

~ + ~ = ~
(0,0,0,0) + ~ = (0,12,16,2 0) ⇒ ~ = (0,12,16,2 0)
optimal, we have to improve the UV method and continue

U 3 V2

C32 V2 V2

further till we get an optimal solution.
~ + ~ = ~
(0,0,0,0) + ~ = (0,12,16,2 0) ⇒ ~ = (0,12,16,2 0)
Russell’s Approximation Method is a much more recently

U 3 V3

C33 V3 V3

~ + ~ = ~
⇒ ~ + (0,12,16,20) = (4,8,12,16 )⇒ ~ =
proposed criterion that seems very promising. One distinct

U 2 V3

C23 U 2

U 2 advantage of Russell’s Approximation Method is that it is

(- 16,-8, 0 ,16)

patterned directly after part-1 of the iterative step for the
~ + ~ = ~
(-16,-8,-0 ,16) + ~ = (2,6,10,14)⇒ ~ =

U 2 V4

C24

V4 V4

transportation simplex method which somewhat simplifies

(-14,6,18,30)

the overall computer code.

~ ~

U1 V1

~ ~

C11 U1

+ (4,8,18,26) = (- 4,0,4,16 ) ~

(- 30,-18,-4,12)

~ ~ ~

9 CONCLUSION

Now we compute the net evaluation U i + V j - Cij for all the
non basic cells.
We have
For most of the fuzzy transportation problems, the initial
basic feasible solution obtained by Russell’s Approximation
Method is an optimal solution. For some fuzzy transportation
~ + ~ − ~
= (- 46,-10,12, 3 6 )
problems, the initial basic feasible solution obtained by Vo-

U1 V2

C12

gel’s Approximation Method is optimal, but it is not optimal
~ + ~ − ~
= (- 46, -10, 12, 36 )
for some other problems. On the other hand, the North-West

U1 V3

C13

~ + ~ − ~
= (- 52, -14, 14, 44)
Corner Rule usually gives the solution far from optimal.

U1 V4

C14

Therefore, Russell’s Approximation Method is preferred in
~ + ~ − ~
= (- 44, - 24, 2, 34 )
comparison to other methods for finding the initial basic fea-

U 2 V1

C21

~ + ~ − ~
= (- 40, -14, 2, 28 )
sible solution of a Fuzzy Transportation Problem.

U 2 V2

C22

~ + ~ − ~
= (- 38, -12, 4, 22 )

REFERENCES

U3 V4

C 34

~ ~ ~
Since U i + V j - Cij < 0 for all i and j, the solution is optimal.
For the given fuzzy transportation problem, the initial basic solution obtained by Russell’s Approximation Method is ex- actly the same as that obtained by Vogel’s Approximation Method and therefore Russell’s Approximation Method also gives a unique optimal solution for the given example.

8 RESULT & DISCUSSION

For the example given here, the initial basic feasible solution obtained by North West Corner Rule is not optimal, whereas the initial basic feasible solution obtained by Vogel’s Approx- imation Method and Russell’s Approximation Method is an optimal solution. For some fuzzy transportation problems, it has been found that the initial basic feasible solution obtained by Vogel’s Approximation Method is not optimal, but, the initial basic feasible solution obtained by Russell’s Approxi- mation Method is optimal. The initial basic feasible solution obtained by North West Corner Rule is far from optimal for most of the fuzzy transportation problems.

[1] Chanas S. , Kuchta D., “Fuzzy integer transporting problem”, Fuzzy

Sets and Systems 98(1998) pp. 291- 298.

[2] Bellman R Zadeh L.A, Decision making in a fuzzy environment, Man- agement Sci.17(B) (1970) 141-164.

[3] Chanas S. , Kuchta D., A concept of the optimal solution of the trans- portation problem with fuzzy cost coefficients, Fuzzy Sets and Sys- tems 82(1996), 299-305.

[4] Liu, S.T. and Kao, C . , Solving fuzzy transportation problems based on

Extension principle, European Journal of Operations Research, Vol.

153, 2004, 661-674.

[5] Nagoorgani.A and Abdul Razak.K, Two stage fuzzy transportation prob- lem, Journal of Physical Sciences, 10, 2006, 63-69.

[6] N.V. Reinfeld and W. R. Vogel, Mathematical Programming, Prentice- Hall, Englewood Cliffs, N. J., 1958.

[7] Edward J. Russell, “Extension of Dantzig’s, Algorithm for finding an Initial Near-Optimal Basis for the Transportation Problem”, Op- erations Research, 17: 187-191,1969.

IJSER © 2014 http://www.ijser.org