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notes_1 - Notes On Optimization Theory & Analysis (ESE...

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Notes On Optimization Theory & Analysis (ESE 304) Michael A. Carchidi September 9, 2010 Chapter 1 - An Introduction to Model Building The following notes are based on the text entitled: Operations Research by Wayne L. Winston (4th edition), and these can be viewed at https://courseweb.library.upenn.edu/ after you log in using your PennKey user name and Password. 1 .1AnIntroduct iontoMode l ing (OR) is simply a scienti f c approach to decision making that seeks to best design and operate a system, usually under conditions requiring the allocation of scarce resources. A system is an organization of interdependent components that work together to accomplish some goal. A good part of operations research is taking the verbal conditions of a problem and constructing a mathematical model consistent with these verbal conditions. We illustrate this with several examples throughout this chapter, while introducing some terminology in-between.
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Example : (Giapetto’s Woodcarving) Giapetto’s Woodcarving Inc., manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier that is manufactured increases Giapetto’s variable labor and overhead costs by $14 . A train sells for $21 and uses $9 worth of raw materials. Each train that is manufactured increases Giapetto’s variable labor and overhead costs by $10 . The manufacture of wooden soldiers and trains requires two types of labor: f nishing and carpentry. A soldier requires 2 hours of f nishing labor and 1 hour of carpentry labor. A train requires 1 hour of f nishing labor and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw materials but only 100 f nishing hours and 80 carpentry hours. Demand for trains is unlimited, but at most 40 soldiers are bought each week. Giapetto wants to maximize weekly pro f ts (revenues minus costs). Formulate a mathematical model of Giapetto’s situation that can be used to maximize Giapetto’s weekly pro f t. We may summarize much of this information in the following table. Soldiers Trains Selling Price $27 $21 Cost of Raw Materials $10 $9 Variable Labor and Overhead Cost $14 $10 Hours of Finishing Labor 2 1 Hours of Carpentry Labor 1 1 In developing the Giapetto model, we explore characteristics shared by many of these types of problems. Decision Variables : Clearly, Giapetto must decide how many soldiers and how many trains should be manufactured each week, and the decision vari- ables must represent these decisions. Consequently, we have two decision variables x 1 = the number of soldiers produced each week, and x 2 = thenumbero ftra insproducedeachweek . Objective Function : The decision maker wants to maximize (usually pro f t) or minimize (usually cost) some function of the decision variables. This 2
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function that is to be maximized (or minimized) is called the objective function. In the Giapetto problem, we note that f xed costs (such as rent and insurance) do not depend on the decision variables since these costs will be present for any values of the decision variables. Therefore Giapetto can
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notes_1 - Notes On Optimization Theory & Analysis (ESE...

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