Transportation Problems

Dr. Ron Lembke

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.

Transportation Problems

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

Defining Variables

shipment from source i to customer j

cost to ship one unit from i to j.

demand at destination j.

Supply at source i

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Formulation

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Demand mustbe satisfied

Shipments outcan‘t exceed supply

Transportation Costs

Need to know costs from each i, to each j

D E F G

A 10 9 8 7

B 10 11 4 5

C 8 7 4 8

ijc

Creating Transp LP in Excel

Demand and Supply Constraints

Overview – Setting up the LP

Create a matrix of shipment costs (in grey in example).

Create a matrix to hold the decision variables, shipment quantities (in yellow).

Compute total cost of shipments using SUMPRODUCT (in red).

Sum amount sent to each destination, enter demand at each destination

Sum amount sent from each DC, enter supplies available at each source

Adding Multiple Constraints at Once

LP Formulated

Solver Options

If you don’t check “assume non-negative” we get the following results:

Sending -15 units from A to D, Solverthinks we would have costs of 10*(-15).So we would earn $150 by sending -15units?

Bottom Line: Pretty much always click “Assume Non-negative.”

Optimal Solution

Inequality Notation

Use <= for shipments from Sources. Use >= for shipments to customers.

Do we really need to? What if supply is greater than demand?

No problem. Only send as much as you have to. Some units just don’t get sent.

Problem we just solved had this.

Product Shortages If total demand exceeds total supply? If demand in G is 15, we get this:

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, with huge capacity. Think of it as a supplier that ships empty boxes.

Now supply can satisfy total demand.

Shortage Costs

What cost should we use for supplier Z? It should be the last resort, so it should be higher

than any real costs. The cost of a shipment from Z is really the cost

of shorting the customer. If all customers are created equal, give them all

the same shortage cost. If some are more important, give them higher

shortage costs, and we’ll only short them as a last resort.

Shortage - Demand Exceeds Supply

To modify the LP formulation:-add row for Z costs (Row 6)-add row for Z shipments (Row 12)-change SumProduct (Total Costs) to include cost of shortages- In Solver dialog box, change variables- In Solver, change constraints to add Z

Shortage Solution

Shortage Solution

Shortage is dealt with by shorting customers D and E.

Demand exceeds supply by 3 units. Our first choice is to short D, because they are the cheapest. We can only short them by 2, their total demand.

Next, short E by 1 unit.

So What? Where do we use this? If we have finite supplies (always), and

estimates of demand (usually) and costs of each shipment (usually)

If you have the data, this finds the perfect answer

Former student working for 3PL had to figure out how to satisfy expected needs for this new software. How to get copies to distributors?

Summary of Transportation Problems Wrote LP to describe Transportation

Problem Formulated LP in Excel, solved with Solver Dealing with excess supply is no problem Dealing with shortage of supply, we have

to add imaginary source of “empty” boxes