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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.