 Open Access
 Total Downloads : 918
 Authors : Utpal Kanti Das, Md. Ashraful Babu, Aminur Rahman Khan, Dr.Md. Sharif Uddin
 Paper ID : IJERTV3IS10120
 Volume & Issue : Volume 03, Issue 01 (January 2014)
 Published (First Online): 06012014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Advanced Vogel’s Approximation Method (AVAM): A New Approach to Determine Penalty Cost for Better Feasible Solution of Transportation Problem
Utpal Kanti Das Department of Computer Science & Engineering, IUBAT
International University of Business Agriculture and Technology, Uttara, Dhaka1230, Bangladesh,
Md. Ashraful Babu Department of Mathematics, IUBAT
International University of Business Agriculture and Technology, Uttara, Dhaka1230, Bangladesh,
Aminur Rahman Khan Department of Mathematics, Jahangirnagar
University, Savar, Dhaka1342, Bangladesh,
Dr.Md. Sharif Uddin Department of Mathematics, Jahangirnagar University, Savar, Dhaka1342, Bangladesh,
Abstract
In Operation Research, obtaining significant result for Transportation Problems is very important nowadays. Vogels Approximation Method (VAM) is the very efficient algorithm to solve the transportation problem for feasible solution which is nearer to optimal solution. In this paper we identified a computational error in VAM and approach a logical development of VAM algorithm. The main concept of VAM is to determine penalty cost which obtains from the difference of smallest and next to smallest cost in each row or column and make maximum allocation in lowest
of Linear Programming Problem (LPP) where Vogels Approximation Method (VAM) is known as efficient method to solve TP. The concept penalty cost (difference of smallest and next to smallest cost in row or column) makes this method more effective more than other methods such as North West Corner Rule (NWC), Least Cost Method (LCM) etc. But the way of determining penalty cost is not logical for some cases; we discuss this types of situation in [,] and developed an algorithm with new concept where difficulties are resolved in [,] and gives the feasible solution very close to optimal solution lower than VAM.
Consider a Transportation Problem with m sources
cost cell of that row or column which have largest
and n destinations where Cij is the unit transportation
penalty. The difficulty arises when smallest cost and next to smallest cost have same magnitude. In that case
cost from ith source to jth destination. Let
Si be the
we find a very logical concept to resolve this and developed a new algorithm Advanced Vogels Approximation Method (AVAM) to find a feasible
supply amount of ith source and D be the demand
j
amount of jth destination. We have to find the
solution of transportation problem which is very close
transported amount of commodity
xij
so that total
to optimal solution more than VAM.
Keyword: AVAM, VAM, Penalty Cost, Feasible Solution, Error Estimation, Transportation Problem (TP).

Introduction
Transportation problem is real life problem where commodities are transferred from factories to retail house so that total transportation cost should be minimized. In Operation research, TP is a special class
transportation cost will be minimized. The above problem in LPP model can be express as follows:
Minimize:
m n
Total cost Cij xij
i1 j 1
Subject to:
n
xij Si
m
j 1
xij Dj i1
xij 0
for i 1, 2,……m (Supply constrains) for j 1, 2,……n (Demand constrains)
i, j
Step3:

Identify the row and column with the largest penalty, breaking ties arbitrarily. Allocate as much as possible to the variable with the least cost in the selected row or column. Adjust the supply and demand and cross out the satisfied row or column. If a row and column are satisfies simultaneously, only one of them is crossed out and remaining row or column is
We have to convert this LPP in the following mathematical Model and applying transportation methods to find feasible solution.
assigned a zero supply or demand.

If two or more penalty costs have same largest magnitude, then select any one of them (or select most top row or extreme left column).
Source
Destination
Supply
D1
D2
D3
Dn
S1
C11
C12
C13
C1n
S1
S2
C21
C22
C23
C2n
S2
Sm
Cm1
Cm2
Cm3
Cmn
Sm
Demand
D1
D2
D3
Dn


Existing Algorithm: Vogels Approximation Method (VAM)
The Vogel Approximation method is an iterative procedure for computing a basic feasible solution of a transportation problem. This method is preferred over the two methods i.e. North West Corner Rule and Least cost Method. The algorithm of VAM is given below: Step1:

Identify the cells having minimum and next to minimum transportation cost in each row and write the difference (Penalty) along the side of the table against the corresponding row.

Identify the cells having minimum and next to minimum transportation cost in each column and write the difference (Penalty) along the side of the table against the corresponding column.
Step2: If minimum cost appear in two or more times in a row or column then select these same cost as a minimum and next to minimum cost and penalty will be zero.
Step4:
a. If exactly one row or one column with zero supply or demand remains uncrossed out, Stop.

If only one row or column with positive supply or demand remains uncrossed out, determine the basic variables in the row or column by the LeastCost Method.

If all uncrossed out rows or column have (remaining) zero supply or demand, determined the zero basic variables by the LeastCost Method. Stop.

Otherwise, go to Step1.
2.1. Finding Computational Error of Vogels Approximation Method (VAM)
The main concept of VAM algorithm is the penalty cost which is determined by the difference of smallest and next to smallest cost of each row and column where highest penalty indicate that one of the value of two minimum costs is too higher than another. For that case VAM select highest penalty cost and give allocation in lowest cost cell in that row or column for avoiding the probability of selecting higher cost in next iteration.
If there are two or more cells have same smallest magnitude then VAM selects these same smallest cost as minimum and next to minimum cost and penalty will be zero [2. Step2]. For that case penalty will be lowest in that row or column among all rows or columns so that there is no possibility to select that row or column for allocating commodities. If in that row or column have any higher cost other than same smallest costs then probability will be increased to select the higher cost in next iteration and total transportation cost may be increase.
In VAM algorithm, allocations are depends on penalty cost. For this above computational error to determine penalty in VAM, lowest cost may not be ensure in all teration so that total cost in feasible solution has a chance to be higher.



Proposed Algorithm for Advanced Vogels Approximation Method (AVAM)
Step4:

Select max(Pi , Pj ) . Set
xij :Amount of
In this section, we solved the computational error of VAM which discussed in above and proposed a new
commodity from ith
source
jth to destination;
algorithm named Advanced Vogels Approximation Method (AVAM). In our proposed algorithm (AVAM), when smallest cost appear in two or more
Select lowest cost of that row or column
which has largest penalty and allocate maximum possible amount
times in a row or column then penalty determined by
xij
i.e. min(Si , Dj ) . If the lowest cost
difference of two minimum cost taken one of them as a minimum and following smallest cost other than equal smallest costs as a next to minimum. As an example, if 3, 10, 3, 7, 9 are the costs of a row or column then select 3 as a smallest cost and select 7 as a next to smallest cost instead of 3 again and penalty will be 4. In that case penalty is not zero and if this penalty has the largest magnitude then probability of the chance of taking larger cost in next iteration will be decreased because of at least one more smallest cost remains. The algorithm of AVAM is given follows:
i
Step1: Set S : Supply amount of the ith source;
j
Set D : Demand amount of the jth
destination;
appear in two or more cells in that row or column then choose the extreme left or most top lowest cost cell.

If two or more penalty costs have same largest magnitude, then select any one of them (or select most top row or extreme left column).
Step5: Adjust the supply and demand and cross out the satisfied row or column. If row and column are satisfied simultaneously then crossed out one of them and set zero supply or demand in remaining row or column.
Step6:

If exactly one row or one column with zero supply or demand remains uncrossed out, Stop.
Set
Cij
:Unit transportation cost of
ith source

If only one row or column with positive
supply or demand remains uncrossed out,
to jth destination;
determine the basic variables in the row or
Check: if Si 0 and Dj 0, then Stop.
Step2:
column by the LeastCost Method.

If all uncrossed out rows or column have (remaining) zero supply or demand,

f
Si Dj or if
Si Dj
determined the zero basic variables by the
i j i j
then balance the transportation problem adding dummy demand or supply.

Set Cij : 0 for all dummy rows or columns.

Step3:

Identify the smallest and next to smallest cost of each row and column and calculate the difference between them which is called by penalty.
Set Pi :Row penalty and Set Pj : Column penalty.
Pi  Cih Cik  and Pj  Chj Ckj 

If smallest cost appear two or more times in a row or column then select one of them as a smallest and following smaller cost other than equal smallest costs as a next to smallest cost.

If there is no more cost other than equal
LeastCost Method. Stop.

Otherwise go to Step3.



Numerical Simulation
Consider some special types of transportation problems where smallest cost is appear in two or more in rows or columns and solve them using Advanced Vogels Approximation Method (AVAM) and compare these results with the solution of Vogels Approximation Method (VAM).
4.1. Example1
Source
Destination
Supply
D1
D2
D3
D4
S1
6
8
10
9
50
S2
5
8
11
5
75
S3
6
9
12
5
25
Demand
20
20
50
60
Consider a Mathematical Model of a transportation problem in given below:
smallest costs
i.e.
all costs are same then
select smallest and next to smallest as same and penalty will be zero.
Table1.1
Now solve this problem using AVAM and VAM respectively in below:
Solution of Example1 Using Advanced Vogels Approximation Method (AVAM):
Source
Destination
Supply
D1
D2
D3
D4
S1
6
8
0
10
50
9
50
S2
5
15
8
11
5
60
75
S3
6
5
9
20
12
5
25
Demand
20
20
50
60
Table1.2 Total Transportation Cost:
(0 8) (50 10) (15 5) (60 5) (5 6)
(20 9) 1085
Solution of Example1 Using Vogels Approximation Method (VAM):
Source
Destination
Supply
D1
D2
D3
D4
S1
6
20
8
10
30
9
50
S2
5
8
20
11
20
5
35
75
S3
6
9
12
5
25
25
Demand
20
20
50
60
Table1.3 Total Transportation Cost:
(206) (3010) (208) (2011) (355) (255) 1100
Optimal solution of Example1:
The optimal solution of Example1 determined by MODI is 1060.

Example2
Consider a Mathematical Model of a transportation problem in given below:
Source
Destination
Supply
D1
D2
D3
D4
D5
S1
4
4
9
8
13
100
S2
7
9
8
10
4
80
S3
9
3
7
10
6
70
S4
11
4
8
3
9
90
Demand
60
40
100
50
90
Table2.1
Now solve this problem using AVAM and VAM respectively in below:
td>
4
40
Source
Destination
Supply
D1
D2
D3
D4
D5
S1
4
60
9
8
13
100
S2
7
9
8
10
4
80
80
S3
9
3
7
60
10
6
10
70
S4
11
4
0
8
40
3
50
9
90
Demand
60
40
100
50
90
Solution of Example2 Using Advanced Vogels Approximation Method (AVAM):
Table2.2 Total Transportation Cost:
(60 4) (40 4) (80 4) (60 7) (10 6)
(0 4) (40 8) (50 3) 1670
Solution of Example2 Using Vogels Approximation Method (VAM):
Source
Destination
Supply
D1
D2
D3
D4
D5
S1
4
60
4
0
9
40
8
13
100
S2
7
9
8
10
4
80
80
S3
9
3
7
60
10
6
10
70
S4
11
4
40
8
3
50
9
90
Demand
60
40
100
50
90
Table2.3
Total Transportation Cost:
(60 4) (0 4) (40 9) (80 4) (60 7)
(10 6) (40 4) (50 3) 1710
Optimal solution of Example2:
The optimal solution of Example1 determined by MODI is 1670.

Example3:
Consider a Mathematical Model of a transportation problem in given below:
Source
Destination
Supply
D1
D2
D3
D4
D5
S1
8
8
2
10
2
40
S2
11
4
10
9
4
70
S3
5
2
2
11
10
35
S4
10
6
6
5
2
90
S5
8
11
8
6
4
85
Demand
80
55
60
80
45
Table:3.1
Solution of Example3 Using Advanced Vogels Approximation Method (AVAM):
Source
Destination
Supply
D1
D2
D3
D4
D5
S1
8
8
2
40
10
2
40
S2
11
4
55
10
9
4
15
70
S3
5
15
2
2
20
11
10
35
S4
10
6
6
5
60
2
30
90
S5
8
65
11
8
6
20
4
85
Demand
80
55
60
80
45
Table:3.2 Total Transportation Cost:
(2 40) (4 55) (4 15) (5 15) (2 20)
(5 60) (2 30) (8 65) (6 20) 1475
Solution of Example3 Using Vogels Approximation Method (AVAM):
Source
Destination
Supply
D1
D2
D3
D4
D5
S1
8
8
2
40
10
2
40
S2
11
4
55
10
9
15
4
70
S3
5
15
2
2
20
11
10
35
S4
10
6
6
5
45
2
45
90
S5
8
65
11
8
6
20
4
85
Demand
80
55
60
80
45
Table:3.3 Total Transportation Cost:
(2 40) (4 55) (9 15) (515) (2 20)
(5 45) (2 45) (8 65) (6 20) 1505
Optimal solution of Example3:
The optimal solution of Example1 determined by MODI is 1475.


Result Analysis:
We observed in above examples that Advanced Vogels Approximation Method (AVAM) gives the lower feasible solution other than Vogels Approximation Method (VAM). Results using AVAM are very close to or equal to optimal solution. The comparison table of AVAM and VAM result are follows:
Methods
Transportation Problems
Optimal Solution
AVAM
VAM
Example1
1060
1085
1100
Example2
1670
1670
1710
Example3
1475
1475
1505
Table: 5

Conclusion
In this paper we fixed the computational error of Vogels Approximation Method (VAM) and proposed a new algorithm named Advanced Vogels Approximation Method (AVAM). From the above examples it is shown that AVAM gives the lower feasible solution than VAM and it is very close to optimal solution or equal to optimal solution.

References

Hamdy A. Taha. Operation Research: An Introduction,
Eighth Edition, ISBN13: 9780132555937

P. Rama Murthy. Operation Research, Second Edition,
ISBN (13): 9788122429442

TORA Optimizing System Software developed by Hamdy A Taha.

Hillier, F. S. and G. J. Lieberman. 1995. Introduction to Operations Research, 6th ed. New York: McGrawHill, Inc.

Md. Amirul Islam, Aminur Rahman Khan, M. Sharif Uddin and M. Abdul Malek Determination of Basic Feasible Solution of Transportation Problem: A New Aproach Jahangirnagar University Journal of Science, Vol. 35, No. 1, pp. 101 108, ISSN 10228594 (2012).

Aminur Rahman Khan A Resolution of the Transportation Problem: An Algorithmic Approach Jahangirnagar University Journal of Science, Vol. 34, No. 2, pp. 4962 ISSN 10228594 (2011).