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Unformatted text preview: Introduction to Linear Programming Philip Marlowe Pizza Problem STOR 112 This document has been adapted from a technical note by Alan W. Neebe and David S. Rubin in 2002, and was updated to reflect the latest version of Microsoft Excel. Last updated August 2009. Contents 1 1 Formulating a Linear Program We will motivate our discussion of linear programming with the following hy- pothetical problem. The problem is large enough to be interesting, yet small enough so that most of the important linear programming concepts illustrated can be visualized geometrically. 1.1 Philip Marlowes Problem Philip Marlowe was considering the possibility of starting a gourmet pizza cater- ing service. While vacationing in Spiveys Corner, N.C., Marlowe learned the secret of making gourmet pizzas. He learned how to make a special pizza dough and discovered which select topping ingredients to use. The owner of a nearby convenience store agreed to distribute the pizzas. After sampling some of Mar- lowes wares, the owner quickly realized that he could easily sell all the pizzas which could be assembled in an afternoon. Marlowe figured that his rigorous schedule left him just enough time to devote Saturday afternoons to his enter- prise. He could mix up the dough, assemble the pizzas, deliver them to the store, and still have the evening left for other activities. Marlowe decided to start off distributing only two varieties of pizza: anchovy and cheese. He figured these two were guaranteed to sell, and he knew where he could obtain the best ingredients for these types. The main ingredients for his pizzas were pizza dough, anchovies, cheese, and tomato sauce. Pizza dough could be made up using locally available ingredients. He owned an old blender which was able to mix at most 40 pounds of dough in an afternoon. He arranged to have up to 6 pounds of fresh frozen Peruvian anchovies flown in weekly, while a cheese distributor in Wisconsin was willing to ship him up to 18 pounds of mozzarella weekly. Finally, the Red Brand Company agreed to provide up to 15 pounds of tomato sauce weekly. It might turn out to be advantageous to adjust these shipment schedules, but for the moment Marlowe decided to see what he could do with these amounts. All of Marlowes pizzas started off with one pound of dough, made into the traditional flat, circular shape. An anchovy pizza was topped with 4 ounces of anchovies, 4 ounces of cheese, and 8 ounces of tomato sauce, while a cheese pizza was topped with 8 ounces of cheese and 4 ounces of tomato sauce. Mar- lowe priced his pizzas so that, after accounting for all variable costs, his profit contribution was $1.50 per anchovy pizza and $1.00 per cheese pizza....
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- Spring '06