1. Transportation Problem

2. Concepts

3. Concepts
What is the Transportation Problem? The transportation problem is a special type of linear programming problem where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations.

4. Transportation Example

5. Transportation Example
We have 3 factories and 3 warehouses Decision: How much to ship from each origin to each destination? Objective: Minimize transportation cost.
Transportation Cost per Unit
Factory
Bhiwandi Warehouse
(300 units)
Palghar Warehouse
(200 units)
Uran Warehouse
Dadar (100 units)
Rs. 5
Rs. 4
Rs. 3
Thane (300 units)
Rs. 8
Vasai (300 units)
Rs. 9
Rs. 7

6. Warehouses (Destination)
Factories (Origin)
Transportation Example
100 units - Dadar
Bhiwandi – 300 units
300 units - Thane
Palghar – 200 units
Possible Routes
300 units - Vasai
Uran – 200 units

7. Transportation Cost per Unit
Transportation Example
Decision Variables:
Xij = number of units transported from factory “i” to warehouse “j”
Objective Function:
Min 5Xdb + 4Xdp + 3Xdu
+ 8Xtb + 4Xtp + 3Xtu
+ 9Xvb + 7Xvp + 5Xvu
Transportation Cost per Unit
Factory
Bhiwandi Warehouse
(300 units)
Palghar Warehouse
(200 units)
Uran Warehouse
Dadar
(100 units)
Rs. 5
Rs. 4
Rs. 3
Thane
Rs. 8
Vasai (300 units)
Rs. 9
Rs. 7
Constraints:
(supply nodes)
- (Xdb + Xdp + Xdu) = - 100
- (Xtb + Xtp + Xtu) = - 300
- (Xvb + Xvp + Xvu) = - 300
Constraints:
(demand nodes)
Xdb + Xtb + Xvb = 300
Xdp + Xtp + Xvp = 200
Xdu + Xtu + Xvu = 200

8. The Transportation Method

9. The Transportation Method
Broadly the method involves 3 steps:
Obtaining Initial Basic Feasible Solution
North-West Corner Rule
Least Cost Method
Vogel’s Approximation Method
Testing the Optimality
Stepping Stone Method
Modified Distribution Method (MODI)
Improving the Solution

10. Unbalanced Problems
If (Total Supply) > (Total Demand), then for each supply node: (outflow) < (supply)
Excess supply is assumed to go to the inventory and costs nothing for shipping. Dummy destination column is added, assuming requirement equal to amount of excess supply and zero transportation cost
If (Total Supply) < (Total Demand), then for each demand node: (inflow) < (demand)
Dummy origin row is added, assuming availability equal to demand-supply gap and zero transportation cost

11. Prohibited Routes Examples:
In some situations some specific routes may not be available resulting in prohibited routes.
Examples:
Unfavorable weather conditions
Strike on a particular route
To cope with prohibited routes, the cost of using that route is taken extremely large (which is written as “M”)

12. Transportation Method The Initial Basic Feasible Solution

13. Steps for Initial Basic Feasible Solution:
North West Corner Rule
Steps for Initial Basic Feasible Solution:
Start with the North West Corner of the transportation table and choose the cell in first row & first column.
Compare the supply available at source and demand at destination, corresponding to the chosen cell and allocate lower of the two values.
Calculate the remaining quantity at the respective destination and source and strike out the fully allocated row or column
Continue in zigzag manner until the last source and last destination are covered, so that the south-east corner is reached.

14. Numerical

15. North West Corner Rule Type
Limitations of NorthWest Corner Rule
Although this method is relatively simple, it is not efficient in terms of cost minimizing. Because it takes into account only the available supply and demand requirements in making assignments and takes no account of the transportation cost involved.

16. Vogel’s Approximation Method
Steps for Initial Basic Feasible Solution:
RIM – RDBMS Interface Module RIM conditions - refers to all supply meeting all demand requirement for a particular transportation problem

17. Numerical

18. Vogel’s Approximation Method
Unbalanced Type

19. Penalty Type

20. Case of Degeneracy
Degeneracy: exists in a transportation problem when the number of filled cells is less than the number of rows (origins ‘m’) plus the number of columns (destinations ‘n’) minus one
(m + n - 1).
Degeneracy can occur at 2 stages:
During the initial feasible solution
During the optimality test
Degeneracy requires some adjustment in the matrix to evaluate the solution achieved. The form of this adjustment involves inserting some value in an empty cell so a closed path can be developed to evaluate other empty cells. This value may be thought of as an infinitely small amount, having no direct bearing on the cost of the solution.

21. Modified Distribution Method (MODI) - for testing optimality

22. The Transportation Method