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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. D2 A 10 E4 B 10 F 12 C 10 G 11 Find cheapest way to satisfy all demand Defining Variables x ij shipment from source i to customer j c ij cost to ship one unit from i to j. d j demand at destination j. S i Supply at source i Formulation M Min N c i 1 ij x ij j 1 N s .t . Shipments out can‘t exceed supply Demand must be satisfied x ij S i for i 1, , M j 1 M x ij d j for j 1, , N i 1 x ij 0 for all i , j Transportation Costs c ij 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 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, Solver thinks we would have costs of 10*(-15). So we would earn $150 by sending -15 units? 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