1. Transportation problems Operational Research Level 4
Prepared by T.M.J.A.Cooray
Department of Mathematics
MA 4020-Transportation problems

2. MA 4020-Transportation problems
Introduction
Transportation problem is a special kind of LP problem in which goods are transported from a set of sources to a set of destinations subject to the supply and demand of the source and the destination respectively, such that the total cost of transportation is minimized.
MA 4020-Transportation problems

3. MA 4020-Transportation problems
Examples:
Sources
factories,
finished goods warehouses ,
raw materials ware houses,
suppliers etc.
Destinations
Markets
Finished goods ware house
raw materials ware houses,
factories,
MA 4020-Transportation problems

4. MA 4020-Transportation problems
A schematic representation of a transportation problem is shown below D1 a1 a2 ai am S1 b1 b2 bj bn D2 Si Dj Dn Sm
MA 4020-Transportation problems

5. MA 4020-Transportation problems
m- number of sources
n- number of destinations
ai- supply at source I
bj – demand at destination j
cij – cost of transportation per unit from source i to destination j
Xij – number of units to be transported from the source i to destination j
MA 4020-Transportation problems

6. MA 4020-Transportation problems
Destination j
j n
Supply a1 a2 ai am 1 2 i m
c11
c12
c1j
c1n
ci1
ci2
cij
cin
cm1
cm2
cmn S O U R C E i
MA 4020-Transportation problems
Demand b1 b bj bn

7. Transportation problem: represented as a LP model
MA 4020-Transportation problems

8. MA 4020-Transportation problems
The ideal situation is shown below.,with equalities instead of inequalities. There are “mn” unknown variables and m+n-1 independent equations.
MA 4020-Transportation problems

9. MA 4020-Transportation problems
When solving the transportation problem ,the number of possible routes should be  m+n-1.
If it is <m+n-1, it is called a degenerate solution.
In such a case evaluation of the solution will not be possible.
In order to evaluate the cells /routes (using the u-v method or the stepping stone method ) we need to imagine/introduce some used cells/routes carrying / transporting a very small quantity, say . That cell should be selected at the correct place.
MA 4020-Transportation problems

10. MA 4020-Transportation problems
Example: Consider a transportation problem involving 3 sources and 3 destinations.
Source 1 2 3
Demand
Destination
Supply
200
300
500
1000 20 10 15 12 9 25 30 18
400
MA 4020-Transportation problems

11. Types of transportation problems
Balanced transportation problems
Unbalanced transportation problems
Include a dummy source or a dummy destination having a supply “d” or demand “d” to convert it to a balanced transportation problem.
Where d=
MA 4020-Transportation problems

12. MA 4020-Transportation problems
Example
Plant 1 2 3 4 5
Demand 10 15 9 25 30 14 7 20 13 - 8 W
A 1 R
E 2 H
O 3 U
S 4 E
Supply
MA 4020-Transportation problems

13. Solution of transportation problems
Two phases:
First phase:
Find an initial feasible solution
2nd phase:
Check for optimality and improve the solution
MA 4020-Transportation problems

14. Find an initial feasible solution
North west corner method
Least cost method
Vogel’s approximation method
MA 4020-Transportation problems

15. Checking for optimality
U-V method
Stepping-Stone method
MA 4020-Transportation problems

16. Example-( having a degenerate solution) Introduce  to for phase 2..
Destinations 1 2 3
Supply 25 5 6 15 4 20 7 10 30
Sources S1 S2 S3 S4
Demand
MA 4020-Transportation problems

17. MA 4020-Transportation problems
Transshipment models.
In transportation problems ,shipments are sent directly from a particular source to a particular destination to minimize the total cost of shipments.
It is sometimes economical if the shipment passes through some transient nodes in between the sources and destinations.
In transshipment models it is possible for a shipment to pass through one or more intermediate nodes before it reaches its destination.
MA 4020-Transportation problems

18. MA 4020-Transportation problems
Transshipment problem with sources and destinations acting as transient nodes
Number of starting nodes as well as the number of ending nodes is the sum of number of sources and the number of destinations of the original problem.
Let B=
be the buffer stock and it is added to all the starting nodes and all the ending nodes.
MA 4020-Transportation problems

19. MA 4020-Transportation problems
… …
a1+B
aj+B
am+B B S1 B
b1+B
bn+B S1 Sj Sj Sm Sm D1 D1 Dn Dn
MA 4020-Transportation problems

20. MA 4020-Transportation problems
Destinations D1,D2,….Dn are included as additional starting nodes mainly to act as transient nodes.they don’t have any original supply and the supply of these nodes should be at least B.
The sources S1,S2,….Sm are included as additional ending nodes mainly to act as transient nodes.these nodes are not having any original demand.But each of these transient nodes is assigned with B units as the demand value.
MA 4020-Transportation problems

21. MA 4020-Transportation problems
We need to know the transshipment cost between the sources ,between the destinations and between sources and destinations .
MA 4020-Transportation problems

22. MA 4020-Transportation problems
Example
Supplies at the sources are 100,200,150 and 350 and Demand at the destinations are 350 and 450 respectively. S1 S2 S3 S4 D1 D2 4 20 5 25 12 10 6 15 8 45 7 30 18 60 23
MA 4020-Transportation problems

23. MA 4020-Transportation problemsD1 D2 4 20 5 25 12
=900 10 6
=1000 15 8 45 7
=950 30
=1150 18 60
800 23
=1250
Same algorithms can be used to solve this transshipment
problem.
MA 4020-Transportation problems

24. MA 4020-Transportation problems
Transportation problem with some transient nodes between sources and destination.
Consider the case where the shipping items are first sent to intermediate finished goods ware houses from the supply points/factories and then to the destinations.
To solve these problems the capacity at each transient node is made equal to B.
Where B =
MA 4020-Transportation problems

25. MA 4020-Transportation problems
Example
Multi plant organization has 3 plants and three market places.
The goods from the plants are sent to market places through two intermediate finished goods warehouses.
Cost of transportation per unit between plants and warehouses and warehouses to market places and also supply values of plants and demand values of the markets are shown in the table.
MA 4020-Transportation problems

26. MA 4020-Transportation problemsW1 W2
SUPPLY P1  15 30
200 P2 28 10
300 P3
400 40 20 25 35
DEMAND
100
900
900
900
900
900
Solution of the problem is same as Ordinary transportation Problems.
MA 4020-Transportation problems

27. MA 4020-Transportation problems