1. QUANTITATIVE ANALYSIS FOR MANAGERS TRANSPORTATION MODEL
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2. Learning Objectives When you complete this chapter, you should be able
to identify or define:
Transportation modelling
to explain or use:
Northwest-corner method
Least Cost method
Vogel’s Approximation method
Stepping-stone method
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3. Outline Transportation Modelling Developing an Initial Solution
The Northwest-Corner Method
The Least-Cost Method
The Vogel’s Approximation Method
The Stepping-Stone Method
Special issues in modelling
- Demand Not Equal to Supply
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4. Transportation Problem
How much should be shipped from several sources to several destinations
Sources: Factories, warehouses, etc.
Destinations: Warehouses, stores, etc.
Transportation models
Find lowest cost shipping arrangement
Used primarily for existing distribution systems
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5. A Transportation Model Requires
The origin points, and the capacity or supply per period at each
The destination points and the demand per period at each
The cost of shipping one unit from each origin to each destination
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6. Transportation Problem Solutions steps
Define problem
Set up transportation table (matrix)
Summarizes all data
Keeps track of computations
Develop initial solution
Northwest corner Method
Vogel’s Approximation Method
Find optimal solution
Stepping stone method
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7. Methods for finding Initial Solution
1. North-West Corner Method (NWCM)
Begin in the upper-left-hand corner of the transportation table for a shipment and allocate as many units as possible equal to minimum between available capacity and requirement; i.e. min (a1, b1).
Allocate the maximum that is possible, min (100, 90) = 90. Now move horizontally to the second column in the first row.
Repeat the above steps

8. Example 1:
A company has three factories F1, F2 and F3 with production capacity 100, 250 and 150 units per week respectively. These units are to be shipped to four warehouses W1, W2, W3 and W4 with requirement of 90, 160, 200 and 50 units per week respectively. The transportation costs (in Rs) per unit between factories and warehouses are given as follows:
Warehouse W1 W2 W3 W4
Capacity F1 30 25 40 20
100 F2 29 26 35
250 F3 31 33 37
150
Requirement 90
160
200 50
Factory

9. The total cost of transportation is obtained as follows:
Route Units per unit
From To shipped X cost (Rs) = Total Cost
F1 W1
F1 W2
F2 W2
F2 W3
F3 W3
F3 F4
Total =

10. 2. The Least Cost Method
Identify the cell with the lowest cost. Arbitrarily break any ties for the lowest cost.
Allocate as many units as possible to that cell without exceeding the supply or demand. Then cross out that row or column (or both) that is exhausted by this assignment.
Find the cell with the lowest cost from the remaining cells.
Repeat steps 2 & 3 until all units have been allocated.

11. Total Transportation cost = Rs 15, 020 Condition:
LCM
Total Transportation cost = Rs 15, 020
Condition:
Occupied shipping routes = (no. of rows +
no. of columns) – 1
= –1
= 6
This cost is less than the cost determined by
NWCM. Therefore, this method is preferred over
the NWCM.

12. 3. Vogel’s Approximation Method (VAM)
This method is preferred over the other two methods
because the initial basic feasible solution obtained is
either optimal or very close to the optimal solution.
For each row and column, find the difference between the two lowest unit shipping costs.
Identify the row or column with the greatest opportunity cost or difference.
Assign as many units as possible to the lowest-cost square in the row or in the column selected.

13. W1 W2 W3 W4
Capacity F1 30 25 40 20
100 F2 29 26 35
250 F3 31 33 37
150
Requirement 90
160
200 50
500

14. Stepping-Stone Method
Optimality Test
Stepping-Stone Method
This method starts with an evaluation of
each of the unoccupied cells to decide
whether it would be economical to introduce
any of these cells into the current solution.

15. The Stepping Stone Method
Apply any of the three methods to obtain the initial basic feasible solution
Select any unused cell to be evaluated
Begin at this cell. Trace a closed path back to the original cell via cells that are currently being used (only horizontal or vertical moves allowed)
Place + in unused cell; alternate - and + on each corner cell of the closed path
Calculate improvement index: add together the unit cost figures found in each cell containing a +; subtract the unit cost figure in each cell containing a -.
Repeat steps 1-4 for each unused square
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16. Stepping Stone Method contd.
Check the sign of each of the net change in the unit transportation costs. If all net changes are plus (+) or zero, then an optimal solution has been achieved, otherwise go to next step
Select the unoccupied cell with most negative net change among all unoccupied cells. If two minus values are equal, select that one which will result in moving as many units as possible into the selected unoccupied cell with the minimum cost
Assign the maximum unit that can be shipped on the new route. This is done by looking at the closed paths (-) sign and we select the smallest number found in the cells with (-) signs and make the transfer

17. Example 2:
A company is spending Rs1000 on transportation of its units from three plants to four distribution centres. The availabilities and requirements of units with units cost of transportations are given as:
Distribution Centres D1 D2 D3 D4
Availabilities P1 19 30 50 12 7 P2 70 40 60 10 P3 20 18
Requirements 5 8 15
What can be the maximum saving for the company by optimum distribution?

18. Special Issues in the Transportation Model
Demand not equal to supply
Called ‘unbalanced’ problem
Add dummy source if demand > supply
Add dummy destination if supply > demand
Degeneracy in Stepping Stone Method
Too few shipping routes (cells) used
Number of occupied cells should be: m + n - 1
Create artificially occupied cell (0 value)
Represents fake shipment
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