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.

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.

Formulation

Transportation Method You have 3 DCs, and need to deliver product to 4 customers. Find cheapest way to satisfy all demand D 2 A 10 E 4 B 10 F 12 C 10 G 11

Solving Transportation Problems Trial and Error Linear Programming – ooh, what’s that?! Tell me more! D E F G A 10 9 8 7 B 11 4 5 C

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

Using Solver

If you don’t check “assume non-negative” we get the following results: Solver doesn’t converge to an optimal solution. Why not?

Inequalities Use <= for shipments from DCs. Use >= for shipments to customers. Do we really need to? What do we do if supply is greater than demand?

Product Shortages If total demand is greater than total supply, what happens? 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 Solution Shortage is dealt with by shorting customer A, and B. Demand exceeds supply by 3 units. Our first 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.