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3 Introduction to Linear Programming

3 Introduction to Linear Programming - Introduction to...

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E X A M P L E 1 Giapetto’s Woodcarving Introduction to Linear Programming Linear programming (LP) is a tool for solving optimization problems. In 1947, George Dantzig de- veloped an efficient method, the simplex algorithm, for solving linear programming problems (also called LP). Since the development of the simplex algorithm, LP has been used to solve optimiza- tion problems in industries as diverse as banking, education, forestry, petroleum, and trucking. In a survey of Fortune 500 firms, 85% of the respondents said they had used linear programming. As a measure of the importance of linear programming in operations research, approximately 70% of this book will be devoted to linear programming and related optimization techniques. In Section 3.1, we begin our study of linear programming by describing the general char- acteristics shared by all linear programming problems. In Sections 3.2 and 3.3, we learn how to solve graphically those linear programming problems that involve only two variables. Solv- ing these simple LPs will give us useful insights for solving more complex LPs. The remainder of the chapter explains how to formulate linear programming models of real-life situations. 3.1 What Is a Linear Programming Problem? In this section, we introduce linear programming and define important terms that are used to describe linear programming problems. 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 manu- factured increases Giapetto’s variable labor and overhead costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each train built increases Giapetto’s variable la- bor and overhead costs by $10. The manufacture of wooden soldiers and trains requires two types of skilled labor: carpentry and finishing. A soldier requires 2 hours of finishing labor and 1 hour of carpentry labor. A train requires 1 hour of finishing and 1 hour of car- pentry labor. Each week, Giapetto can obtain all the needed raw material but only 100 fin- ishing hours and 80 carpentry hours. Demand for trains is unlimited, but at most 40 sol- diers are bought each week. Giapetto wants to maximize weekly profit (revenues costs). Formulate a mathematical model of Giapetto’s situation that can be used to maximize Gi- apetto’s weekly profit. Solution In developing the Giapetto model, we explore characteristics shared by all linear pro- gramming problems. Decision Variables We begin by defining the relevant decision variables. In any linear programming model, the decision variables should completely describe the decisions to be made (in this case, by Giapetto). Clearly, Giapetto must decide how many soldiers and trains should be manufactured each week. With this in mind, we define
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x 1 number of soldiers produced each week x 2 number of trains produced each week Objective Function In any linear programming problem, the decision maker wants to max- imize (usually revenue or profit) or minimize (usually costs) some function of the deci- sion variables. The function to be maximized or minimized is called the
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