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Transportation
Problems
Dr. Ron Tibben-Lembke
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Transportation Problems
Linear programming is good at solving
problems with zillions of options, and
finding the optimal solution. Could it work for transportation problems?
Costs are linear, and shipment quantities
are linear, so maybe so.
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Defining Variables
Define cij as the cost to ship one unit from
i to j.
Demand at location j is dj.
Supply at DC i is Si
Xij is the quantity shipped from DC i to
customer j.
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Formulation
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Transportation Method
You have 3 DCs, and need to deliver
product to 4 customers.
Find cheapest way to satisfy all demand
A 10
B 10
C 10
D 2
E 4
F 12
G 11
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Solving Transportation Problems
Trial and Error
Linear Programming
ooh, whats that?! Tell me more!
D E F G
A 10 9 8 7
B 10 11 4 5
C 8 7 4 8
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Setting up LP
Create a matrix of shipment costs (in grey inexample).
Create a matrix to hold the decision variables,shipment quantities (in yellow).
Sum amount sent to each destination.
Sum amount sent from each DC.
Enter demands and supplies at each location. Compute total cost of shipments (in blue).
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Using Solver
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If you dont check assume non-negative we get
the following results:
Solver doesnt converge to an optimal solution.
Why not?
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Inequalities
Use = for shipments to customers.
Do we really need to?
What do we do if supply is greater than
demand?
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Product Shortages
If total demand is greater than total supply,
what happens?
If demand in G is 15, we get this:
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Product Shortages
If demand at G is 15, there are no feasible
solutions, much less a best one.
We need to add a phantom source, Z, withhuge capacity. Think of it as a supplier
that ships empty boxes.
Now supply can satisfy total demand.
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Shortage Costs
What cost should we use for supplier Z?
It should be the last resort, so it should be higherthan any real costs.
The cost of a shipment from Z is really the costof shorting the customer.
If all customers are created equal, give them allthe same shortage cost.
If some are more important, give them highershortage costs, and well only short them as alast resort.
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Shortage Solution
Shortage is dealt with by shorting
customer A, and B.
Demand exceeds supply by 3 units. Ourfirst choice is to short A, because they are
the cheapest. We can only short them by
2, their total demand. Next, short B by 1 unit.
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