J030 Sakshi Jadhav
J031 Naitik Jain
J038 Shanaya Karkhanis
J041 Ninad Khasale
Introduction:
Transportation problem (TP) is one of the predominant areas of Operations
Research and it has wide applications in inventory control, communication
networks, production planning, scheduling, personal allocation and so forth.
Transportation problems play an important role in logistics and supply chain
management for reducing costs and improving service.
Transportation problem is a special kind of Linear Programming Problem (LPP)
in which goods are transported from a set of sources to a set of destinations
subject to the supply and demand of the sources and destination respectively
such that the total cost of transportation is minimised. It is also sometimes
called the Hitchcock problem.
The solution procedure for the transportation problem consists of the following
phases:
• Phase 1: Mathematical formulation of the transportation problem.
• Phase 2: Finding an initial basic feasible solution.
• Phase 3: Optimize the initial basic feasible solution which is obtained in Phase
2.
Methodology: Allocation Table method
Algorithm for Allocation table method:
Step-1: Construct a Transportation Table (TT) from the given transportation
problem.
Step-2: Ensure whether the TP is balanced or not, if not, make it balanced.
Step-3: Select minimum odd cost (MOC) from all the cost cells of TT. If there
is no odd cost in the cost cells of the TT, keep on dividing all the cost cells by 2
(two) till obtaining at least an odd value in the cost cells.
Step-4: Form a new table which is to be known as allocation table (AT) by
keeping the MOC in the respective cost cell/cells as it was/were, and subtract
selected MOC only from each of the odd cost valued cells of the TT. Now all
the cell values are to be called as allocation cell value (ACV) in AT.
Step-5: First, start the allocation from the minimum of supply/demand. Allocate
this minimum of supply/ demand in the place of odd valued ACVs at first in the
AT formed in Step-4. If demand is satisfied, delete the column. If it is supplied,
delete the row.
Step-6: Now identify the minimum ACV and allocate minimum of
supply/demand at the place of selected ACV in the AT. In case of the same
ACVs, select the ACV where minimum allocation can be made. Again in case
of the same allocation in the ACVs, choose the minimum cost cell which is
corresponding to the cost cells of TT formed in Step-1 (i.e. this minimum cost
cell is to be found out from the TT which is constructed in Step-1). Again if the
cost cells and the allocations are equal, in such case choose the nearer cell to the
minimum of demand/supply which is to be allocated. Now if demand is satisfied
delete the column and if it is supply delete the row.
Step-7: Repeat Step-6 until the demand and supply are exhausted.
Step-8: Now transfer this allocation to the original TT.
Step-9: Finally calculate the total transportation cost of the TT. This calculation
is the sum of the product of cost and corresponding allocated value of the TT.
Example 1: Balanced problem
A mining company extracts gravel from three mines, L1, L2 and L3. The
weekly production of each mine is 75, 150 and 75 tons of gravel respectively.
The gravel has to be transported to five main consumers, K1, K2, K3, K4 and
K5 requiring for their needs 100, 60, 40, 75 and 25 tons of gravel per week
respectively. The problem that concerns the company's management is the
minimization of the required cost for the transportation of the product to the
consumers. For this purpose, a detailed cost analysis was carried out which gave
the results of the following table (the numbers denote the transportation cost in
$ per ton of gravel).
Transportation table as follows:
Finding the minimum odd cost, in this case it is 1. Subtracting it by all other odd
costs and hence creating an allocation table.
Allocating minimum possible units beginning with MOC, followed by
remaining minimum costs with minimum units.
Now transferring the same allocations on the original transportation table.
Calculating total cost:
3(25) +3(25) +1(25) +1(60) +2(15) +4(75) +1(75) =$640
Optimal solution using Excel:
Example 2: Unbalanced problem
Transportation Table
Balancing the table
Subtracting MOC (17) and creating an allocation table
Allocating units
Transferring the allocations on the original transportation table
Calculate total cost:
250(17) +250(26) +250(16) + 50(19) +200(0) = $15700
Optimal Solution using excel:
Example 3: Maximisation problem
Surya Roshni Ltd. has three factories - X, Y, and Z. It supplies goods to four
dealers spread all over the country. The production capacities of these factories
are 200, 500 and 300 per month respectively. Determine a suitable allocation to
maximize the total net return.
Initial table:
Convert to a minimisation problem table by subtracting each transportation cost
from maximum transportation cost.
Find the MOC (5) and make an allocation table. Start allocations:
Transfer the allocations onto the original maximisation table:
Calculate returns:
100(18) +100(25) +180(8) +220(7) +100(10) +300(20) = $11780
Optimal solution using excel:
Accuracy:
Method
ATM
VAM
LCM
NWCM
MODI (optimal
sol.)
Problem 1
$640
$690
$760
$765
$615
Problem 2
$15700
$15950
$16900
$16050
$15650
Problem 3
$11780
$15180
$14280
$13420
$15400
Conclusion:
In this paper we have compared Allocation table method with three other well
known methods to solve transportation problems namely- VAM, LCM, NWCM
with respect to three different type of problems. When it comes to balanced and
unbalanced problems the ATM method provides us with a solution closest to the
optimal solution. However, in the case of maximisation, it proves to be
significantly less optimal than the other methods. Seeing the simplicity of the
ATM algorithm it can be used to solve balanced and unbalanced problems with
ease. But the same is not recommended in case of maximisation problems.
References and Citations:
http://www.universalteacherpublications.com/univ/ebooks/or/Ch5/max.htm
http://www.universalteacherpublications.com/univ/ebooks/or/Ch5/unbal.htm
https://people.iee.ihu.gr/~vkostogl/en/Epixeirisiaki/Transportation%20problems
_en_29-5-2012.pdf
Research paper:
Ahmed, M. & Khan, Aminur & Uddin, Md & Ahmed, Faruque. (2016). A New
Approach to Solve Transportation Problems. Open Journal of Optimization. 05.
10.4236/ojop.2016.51003.
https://www.researchgate.net/publication/296335405_A_New_Approach_to_So
lve_Transportation_Problems