![IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. IV (Jan. - Feb. 2017), PP 40-43
www.iosrjournals.org
DOI: 10.9790/5728-1301044043 www.iosrjournals.org 40 | Page
A New Method for Solving Transportation Problems
Considering Average Penalty
S.M. Abul Kalam Azad1
, Md. Bellel Hossain2
1
Department of Mathematics, Rajshahi University of Engineering and Technology, Bangladesh
2
Department of Mathematics, Rajshahi University of Engineering and Technology, Bangladesh
Abstract: Vogel’s Approximation Method (VAM) is one of the conventional methods that gives better Initial
Basic Feasible Solution (IBFS) of a Transportation Problem (TP). This method considers the row penalty and
column penalty of a Transportation Table (TT) which are the differences between the lowest and next lowest
cost of each row and each column of the TT respectively. In a little bit different way, the current method
consider the Average Row Penalty (ARP) and Average Column Penalty (ACP) which are the averages of the
differences of cell values of each row and each column respectively from the lowest cell value of the
corresponding row and column of the TT. Allocations of costs are started in the cell along the row or column
which has the highest ARP or ACP. These cells are called basic cells. The details of the developed algorithm
with some numerical illustrations are discussed in this article to show that it gives better solution than VAM and
some other familiar methods in some cases.
Keywords: VAM, IBFS, TP, TT, ARP, ACP
I. Introduction
The optimal cost is desirable in the movement of raw materials and goods from the sources to
destinations. Mathematical model known as transportation problem tries to provide optimal costs in
transportation system. Some well known and long use algorithms to solve transportation problems are Vogel’s
Approximation Method (VAM), North West Corner (NWC) method, and Matrix Minima method. VAM and
matrix minima method always provide IBFS of a transportation problem. Afterwards many researchers provide
many methods and algorithms to solve transportation problems. Some of the methods and algorithms that the
current research has gone through are: ‘Modified Vogel’s Approximation Method for Unbalance Transportation
Problem’ [1] by N. Balakrishnan; Serder Korukoglu and Serkan Balli’s ‘An Improved Vogel’s Approximation
Method (IVAM) for the Transportation Problem’ [2]; Harvey H. Shore’s ‘The Transportation Problem and the
Vogel’s Approximation Method’ [3]; ‘A modification of Vogel’s Approximation Method through the use of
Heuristics’ [4] by D.G. Shimshak, J.A. Kaslik and T.D. Barelay; A. R. Khan’s ‘A Re-solution of the
Transportation Problem: An Algorithmic Approach’ [5]; ‘A new approach for finding an Optimal Solution for
Trasportation Problems’ by V.J. Sudhakar, N. Arunnsankar, and T. Karpagam [6]. Kasana and Kumar [7]
bring in extreme difference method calculating the penalty by taking the differences of the highest cost and
lowest cost in each row and each column. The above mentioned algorithms are beneficial to find the IBFS to
solve transportation problems. Besides, the current research also presents a useful algorithm which gives a
better IBFS in this topic.
II. Algorithm
Step 1 Subtract the smallest entry from each of the elements of every row of the TT and place them on the
right-top of corresponding elements.
Step 2 Apply the same operation on each of the columns and place them on the right-bottom of the
corresponding elements.
Step 3 Place the Average Row Penalty (ARP) and the Average Column Penalty (ACP) just after and below
the supply and demand amount respectively within first brackets, which are the averages of the
right-top elements of each row and the right-bottom elements of each column respectively of the TT.
Step 4 Identify the highest element among the ARPs and ACPs, if there are two or more highest elements;
choose the highest element along which the smallest cost element is present. If there are two or
more smallest elements, choose any one of them arbitrarily.
Step 5 Allocate ijx = min ( ji ba , ) on the left top of the smallest entry in the ),( ji th of the TT.
Step 6 If ji ba , leave the ith row and readjust jb as ijj abb /
.
If ji ba , leave the jth column and readjust ia as jii baa /
.](https://image.slidesharecdn.com/h1301044043-170708044954/85/A-New-Method-for-Solving-Transportation-Problems-Considering-Average-Penalty-1-320.jpg)
![A New Method for Solving Transportation Problems Considering Average Penalty
DOI: 10.9790/5728-1301044043 www.iosrjournals.org 43 | Page
Solution:
The row differences and column differences are:
Factories
Showrooms
Supply
D1 D2 D3
S1
1
04 0
03 2
15 9
S 2
2
26 1
25
0
04 8
S 3
1
48 3
710 0
37 10
Demand 7 12 8 27
Table: 3.2
The allocations with the help of ARP and ACP are:
Factories
Showrooms Supply
D1 D2 D3 ARP
S1 4 9
3 5 9 (1) - -
S2 6 3
5 5
4 8 (1) (1) (1)
S3
7
8 10 3
7 10 (1.3) (1.3) (0.5)
Demand 7 12 8 27
ACP
(2) (3) (1.3)
(1) (2.5) (1.5)
(1) - (1.5)
Table: 3.3
The transportation cost is
n
j
ijij
m
i
xcz
11
3778543593 z
= 139 units.
IV. Comparison of Results
The developed algorithm in the current work gives optimal or near optimal solution. However, a comparison of
the current work with the three existing conventional methods is presented in case of the three above
illustrations.
Methods Solutions
Illustration – 1 Illustration – 2 Illustration – 3
Current Method 248 68 139
North-West Corner Method 320 93 150
Matrix Minima Method 248 79 145
VAM 248 68 150
Optimal Solution 240 68 139
Table: 4
V. Conclusion
The current method considers all the opportunity costs or penalty in a transportation table by taking
averages of the penalties. On the other hand, some other methods take some of the penalties only (ie. the lowest
and the next lowest, the highest and the lowest etc.). The outcomes of the present algorithm are optimal or near
optimal solutions while several examples were tested.
References
[1] N. Balakrishnan, ‘Modified Vogel’s Approximation Method for Unbalance Transportation Problem,’ Applied Mathematics
Letters 3(2), 9,11,1990.
[2] Serdar Korukoglu and Serkan Balli, ‘An Improved Vogel’s Approximation Method for the Transportation Problem’, Association
for Scientific Research, Mathematical and Computational Application Vol.16 No.2, 370-381, 2011.
[3] H.H. Shore, ‘The Transportation Problem and the Vogel’s Approximation Method’, Decision Science 1(3-4), 441-457, 1970.
[4] D.G. Shimshak, J.A. Kaslik and T.D. Barelay, ‘A modification of Vogel’s Approximation Method through the use of Heuristics’,
Infor 19,259-263, 1981.
[5] Aminur Rahman Khan, ‘A Re-solution of the Transportation Problem: An Algorithmic Approach’ Jahangirnagar University
Journal of Science, Vol. 34, No. 2, 49-62, 2011.
[6] V.J. Sudhakar, N. Arunnsankar, T. Karpagam, ‘A new approach for find an Optimal Solution for Trasportation Problems’,
European Journal of Scientific Research 68 254-257, 2012.
[7] H.S. Kasana and K.D. Kumar, ‘Introductory Operation Research: Theory and Applications’, Springer PP, 509-511, 2004.](https://image.slidesharecdn.com/h1301044043-170708044954/85/A-New-Method-for-Solving-Transportation-Problems-Considering-Average-Penalty-4-320.jpg)
A New Method for Solving Transportation Problems Considering Average Penalty
- 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. IV (Jan. - Feb. 2017), PP 40-43 www.iosrjournals.org DOI: 10.9790/5728-1301044043 www.iosrjournals.org 40 | Page A New Method for Solving Transportation Problems Considering Average Penalty S.M. Abul Kalam Azad1 , Md. Bellel Hossain2 1 Department of Mathematics, Rajshahi University of Engineering and Technology, Bangladesh 2 Department of Mathematics, Rajshahi University of Engineering and Technology, Bangladesh Abstract: Vogel’s Approximation Method (VAM) is one of the conventional methods that gives better Initial Basic Feasible Solution (IBFS) of a Transportation Problem (TP). This method considers the row penalty and column penalty of a Transportation Table (TT) which are the differences between the lowest and next lowest cost of each row and each column of the TT respectively. In a little bit different way, the current method consider the Average Row Penalty (ARP) and Average Column Penalty (ACP) which are the averages of the differences of cell values of each row and each column respectively from the lowest cell value of the corresponding row and column of the TT. Allocations of costs are started in the cell along the row or column which has the highest ARP or ACP. These cells are called basic cells. The details of the developed algorithm with some numerical illustrations are discussed in this article to show that it gives better solution than VAM and some other familiar methods in some cases. Keywords: VAM, IBFS, TP, TT, ARP, ACP I. Introduction The optimal cost is desirable in the movement of raw materials and goods from the sources to destinations. Mathematical model known as transportation problem tries to provide optimal costs in transportation system. Some well known and long use algorithms to solve transportation problems are Vogel’s Approximation Method (VAM), North West Corner (NWC) method, and Matrix Minima method. VAM and matrix minima method always provide IBFS of a transportation problem. Afterwards many researchers provide many methods and algorithms to solve transportation problems. Some of the methods and algorithms that the current research has gone through are: ‘Modified Vogel’s Approximation Method for Unbalance Transportation Problem’ [1] by N. Balakrishnan; Serder Korukoglu and Serkan Balli’s ‘An Improved Vogel’s Approximation Method (IVAM) for the Transportation Problem’ [2]; Harvey H. Shore’s ‘The Transportation Problem and the Vogel’s Approximation Method’ [3]; ‘A modification of Vogel’s Approximation Method through the use of Heuristics’ [4] by D.G. Shimshak, J.A. Kaslik and T.D. Barelay; A. R. Khan’s ‘A Re-solution of the Transportation Problem: An Algorithmic Approach’ [5]; ‘A new approach for finding an Optimal Solution for Trasportation Problems’ by V.J. Sudhakar, N. Arunnsankar, and T. Karpagam [6]. Kasana and Kumar [7] bring in extreme difference method calculating the penalty by taking the differences of the highest cost and lowest cost in each row and each column. The above mentioned algorithms are beneficial to find the IBFS to solve transportation problems. Besides, the current research also presents a useful algorithm which gives a better IBFS in this topic. II. Algorithm Step 1 Subtract the smallest entry from each of the elements of every row of the TT and place them on the right-top of corresponding elements. Step 2 Apply the same operation on each of the columns and place them on the right-bottom of the corresponding elements. Step 3 Place the Average Row Penalty (ARP) and the Average Column Penalty (ACP) just after and below the supply and demand amount respectively within first brackets, which are the averages of the right-top elements of each row and the right-bottom elements of each column respectively of the TT. Step 4 Identify the highest element among the ARPs and ACPs, if there are two or more highest elements; choose the highest element along which the smallest cost element is present. If there are two or more smallest elements, choose any one of them arbitrarily. Step 5 Allocate ijx = min ( ji ba , ) on the left top of the smallest entry in the ),( ji th of the TT. Step 6 If ji ba , leave the ith row and readjust jb as ijj abb / . If ji ba , leave the jth column and readjust ia as jii baa / .
- 2. A New Method for Solving Transportation Problems Considering Average Penalty DOI: 10.9790/5728-1301044043 www.iosrjournals.org 41 | Page If ji ba , leave either ith row or j-th column but not both. Step 7 Repeat Steps 1 to 6 until the rim requirement satisfied. Step 8 Calculate m i n j ijij xcz 1 1 , z being the minimum transportation cost and ijc are the cost elements of the TT. III. Numerical Illustrations Illustration 01 The per unit transportation cost (in thousand dollar) and the supply and demand (in number) of motor bikes of different factories and showrooms are given in the following transportation table. Factories Showrooms Supply ( ia ) D1 D2 D3 D4 W1 9 8 5 7 12 W2 4 6 8 7 14 W3 5 8 9 5 16 Demand ( jb ) 8 18 13 3 42 Table: 1.1 We want to solve the transportation problem by the current algorithm. Solution The row differences and column differences are: Factories Showrooms Supply D1 D2 D3 D4 W1 4 59 3 28 0 05 2 27 12 W2 0 04 2 06 4 38 3 27 14 W3 0 15 3 28 4 49 0 05 16 Demand 8 18 13 3 42 Table: 1.2 The allocations with the help of ARP and ACP are: Factories Showrooms Supply D1 D2 D3 D4 ARP W1 9 8 12 5 7 12 (2.2) - - - W2 8 4 6 6 8 7 14 (2.2) (2.2) (1) (1) W3 5 12 8 1 9 3 5 16 (1.7) (1.7) (2.3) (0.5) Demand 8 18 13 3 42 ACP (2) (1.3) (2.3) (1.3) (0.5) (1) (0.5) (1) - (1) (0.5) (1) - (1) (0.5) - Table: 1.3 The transportation cost is n j ijij m i xcz 11 35191286684125 z = 248 $ Illustration 02 A company manufactures Toy Robots for children and it has three factories S1, S2 and S3 whose weekly production capacities are 3, 7 and 5 hundred pieces respectively. The company supplies Toy Robots to its four showrooms located at D1, D2, D3 and D4 whose weekly demands are 4, 3, 4 and 4 hundred pieces respectively. The transportation costs per hundred pieces of Toy Robots are given below in the Transportation Table:
- 3. A New Method for Solving Transportation Problems Considering Average Penalty DOI: 10.9790/5728-1301044043 www.iosrjournals.org 42 | Page Factories Showrooms Supply iaD1 D2 D3 D4 S1 2 2 2 1 3 S2 10 8 5 4 7 S3 7 6 6 8 5 Demand jb 4 3 4 4 15 Table: 2.1 We want to solve the transportation problem by the current algorithm. Solution: The row differences and column differences are: Factories Showrooms Supply ia D1 D2 D3 D4 S1 1 02 1 02 1 02 0 01 3 S2 6 810 4 68 1 35 0 34 7 S3 1 57 0 46 0 46 2 78 5 Demand jb 4 3 4 4 15 Table: 2.2 The allocations with the help of ARP and ACP are: Factories Showrooms Supply iaD1 D2 D3 D4 ARP S1 3 2 2 2 1 3 (0.7) - - - S2 10 8 3 5 4 4 7 (2.7) (2.7) (2.6) - S3 1 7 3 6 1 6 8 5 (0.7) (0.7) (0.3) (0.3) Demand jb 4 3 4 4 15 ACP (4.3) (3.3) (2.3) (3.3) (1.5) (1) (0.5) (2) (1.5) (1) (0.5) - - - - - Table: 2.3 The transportation cost is n j ijij m i xcz 11 16361737443532 z = 68 units. Illustration 03 A company manufactures toilet tissues and it has three factories S1, S2 and S3 whose weekly production capacities are 9, 8 and 10 thousand pieces of toilet tissues respectively. The company supplies tissues to its three showrooms located at D1, D2 and D3 whose weekly demands are 7, 12 and 8 thousand pieces respectively. The transportation costs per thousand pieces are given in the next Transportation Table: Factories Showrooms Supply iaD1 D2 D3 S1 4 3 5 9 S 2 6 5 4 8 S 3 8 10 7 10 Demand jb 7 12 8 27 Table: 3.1
- 4. A New Method for Solving Transportation Problems Considering Average Penalty DOI: 10.9790/5728-1301044043 www.iosrjournals.org 43 | Page Solution: The row differences and column differences are: Factories Showrooms Supply D1 D2 D3 S1 1 04 0 03 2 15 9 S 2 2 26 1 25 0 04 8 S 3 1 48 3 710 0 37 10 Demand 7 12 8 27 Table: 3.2 The allocations with the help of ARP and ACP are: Factories Showrooms Supply D1 D2 D3 ARP S1 4 9 3 5 9 (1) - - S2 6 3 5 5 4 8 (1) (1) (1) S3 7 8 10 3 7 10 (1.3) (1.3) (0.5) Demand 7 12 8 27 ACP (2) (3) (1.3) (1) (2.5) (1.5) (1) - (1.5) Table: 3.3 The transportation cost is n j ijij m i xcz 11 3778543593 z = 139 units. IV. Comparison of Results The developed algorithm in the current work gives optimal or near optimal solution. However, a comparison of the current work with the three existing conventional methods is presented in case of the three above illustrations. Methods Solutions Illustration – 1 Illustration – 2 Illustration – 3 Current Method 248 68 139 North-West Corner Method 320 93 150 Matrix Minima Method 248 79 145 VAM 248 68 150 Optimal Solution 240 68 139 Table: 4 V. Conclusion The current method considers all the opportunity costs or penalty in a transportation table by taking averages of the penalties. On the other hand, some other methods take some of the penalties only (ie. the lowest and the next lowest, the highest and the lowest etc.). The outcomes of the present algorithm are optimal or near optimal solutions while several examples were tested. References [1] N. Balakrishnan, ‘Modified Vogel’s Approximation Method for Unbalance Transportation Problem,’ Applied Mathematics Letters 3(2), 9,11,1990. [2] Serdar Korukoglu and Serkan Balli, ‘An Improved Vogel’s Approximation Method for the Transportation Problem’, Association for Scientific Research, Mathematical and Computational Application Vol.16 No.2, 370-381, 2011. [3] H.H. Shore, ‘The Transportation Problem and the Vogel’s Approximation Method’, Decision Science 1(3-4), 441-457, 1970. [4] D.G. Shimshak, J.A. Kaslik and T.D. Barelay, ‘A modification of Vogel’s Approximation Method through the use of Heuristics’, Infor 19,259-263, 1981. [5] Aminur Rahman Khan, ‘A Re-solution of the Transportation Problem: An Algorithmic Approach’ Jahangirnagar University Journal of Science, Vol. 34, No. 2, 49-62, 2011. [6] V.J. Sudhakar, N. Arunnsankar, T. Karpagam, ‘A new approach for find an Optimal Solution for Trasportation Problems’, European Journal of Scientific Research 68 254-257, 2012. [7] H.S. Kasana and K.D. Kumar, ‘Introductory Operation Research: Theory and Applications’, Springer PP, 509-511, 2004.