Ijcot v4p309

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

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(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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[3] H. Basirzadeh, R. Abbasi, A new approach for ranking fuzzy numbers based on α−cuts, JAMI, Journal of Appied Mathematics & Informatic,26(2008) 767-778.

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[6] H. W. Lu, C. B. Wang, An Index for ranking fuzzy numbers by belieffeature, Information and management science, 16(3) (2005) 57-70.

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