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Unformatted text preview: Optimization Models Draft of August 26, 2005 I. Formulating an Optimization Model: An Introductory Example Robert Fourer Department of Industrial Engineering and Management Sciences Northwestern University Evanston, Illinois 602083119, U.S.A. (847) 4913151 [email protected] http://www.iems.northwestern.edu/ ˜ 4er/ Copyright c 1989–2005 Robert Fourer A–2 Optimization Models — § 1.0 Draft of August 26, 2005 A–3 1. A Simple Model To introduce the fundamentals of our subject, we begin with a simple exam ple, the “diet problem”: choosing from a menu of available foods to produce a diet that meets daily nutritional requirements. Since there are many different combinations of foods that meet the requirements, our goal will be to identify a combination that does the job at the lowest possible cost. This is the character istic goal of any optimization problem: to find a preferred arrangement among all that are acceptable. This chapter works through the basic steps of generating and solving a diet problem. In the process, we show how to formulate a mathematical “model” that describes diet problems in a general way, and how to submit a model and data to a software package that computes minimumcost diets and related in formation. Then in Chapter 2, your intuition for diets will enable you to see how our model must be refined to produce sensible results. You will also see how certain refinements can make the optimal diet harder to compute. These are characteristics that you will encounter repeatedly as you study different op timization problems and models for them. Diet problems are only one example of the general idea of a minimumcost input problem. Chapter 3 will take a tour through a variety of other applications of this idea, including blending, scheduling, and cutting. Then Chapter 4 will extend the idea even further to encompass output and inputoutput problems. 1.1 A small diet problem When approaching any unfamiliar kind of optimization problem, it’s best to start with a version that’s small and simple. Thus let’s begin by imagining that your diet is limited to a selection of items from a wellknown fast food restaurant. We’ll give each food a nickname to assist in referring to it: QP: Quarter Pounder FR: Fries, small MD: McLean Deluxe SM: Sausage McMuffin BM: Big Mac 1M: 1% Lowfat Milk FF: FiletOFish OJ: Orange Juice MC: McGrilled Chicken Suppose also that you are interested in providing your diet with appropriate amounts of seven “nutrients”: Prot: Protein Iron: Iron VitA: Vitamin A Cals: Calories VitC: Vitamin C Carb: Carbohydrates Calc: Calcium Your problem is to find the lowestcost combination of the foods that will pro vide a day’s requirements for the nutrients....
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 Optimization, diet problem

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