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A Linear programming problem can be expressed in the following standard form: max z= c1x1+ Assumptions of Linear Programming 1 Examples of LP

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We now present examples of four general linear programming problems Under this assumption, the inequalities in constraints (2) and (3) must be satisfied

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EXAMPLE 1Giapetto's WoodcarvingIntroduction 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.

3.1What 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. SolutionIn developing the Giapetto model, we explore characteristics shared by all linear pro- gramming problems. Decision VariablesWe 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 x 1 number of soldiers produced each week x 2 number of trains produced each week Objective FunctionIn any linear programming problem, the decision maker wants to max- imize (usually revenue or proÞt) or minimize (usually costs) some function of the deci- sion variables. The function to be maximized or minimized is called the objective func- tion.For the Giapetto problem, we note that Þxed costs(such as rent and insurance) do not depend on the values of x 1 and x 2 . Thus, Giapetto can concentrate on maximizing (weekly revenues) (raw material purchase costs) (other variable costs). GiapettoÕs weekly revenues and costs can be expressed in terms of the decision vari- ables x 1 and x 2 . It would be foolish for Giapetto to manufacture more soldiers than can be sold, so we assume that all toys produced will be sold. Then

Weekly revenues weekly revenues from soldiers

weekly revenues from trains d so o l l d la ie rs r so w ld ee ie k rs d t o r l a l i a n rs t w ra e i e n k s 27x
1 21x
2 Also,

Weekly raw material costs 10x

1 9x 2

Other weekly variable costs 14x

1 10x 2

Then Giapetto wants to maximize

(27x 1 21x
2 ) (10x 1 9x 2 ) (14x 1 10x 2 ) 3x 1 2x 2 Another way to see that Giapetto wants to maximize 3x 1 2x 2 is to note that Weekly revenues weekly contribution to proÞt from soldiers weekly nonÞxed costs weekly contribution to proÞt from trains so w ld ee ie k rs t w ra e i e n k s Also,

27 10 14 3

21 9 10 2

Then, as before, we obtain

Weekly revenues weekly nonÞxed costs 3x

1 2x 2

Thus, GiapettoÕs objective is to choose x

1 and x 2 to maximize 3x 1 2x 2 . We use the vari- able zto denote the objective function value of any LP. GiapettoÕs objective function is

Maximize z3x

1 2x 2 (1) (In the future, we will abbreviate ÒmaximizeÓ by maxand ÒminimizeÓ by min.) The co- efÞcient of a variable in the objective function is called the objective function coefficient of the variable. For example, the objective function coefÞcient for x 1 is 3, and the objec- tive function coefÞcient for x 2 is 2. In this example (and in many other problems), the ob-

Contribution to proÞt

Contribution to proÞt

contribution to proÞt contribution to proÞt

50CHAPTER3Introduction to Linear Programming

jective function coefÞcient for each variable is simply the contribution of the variable to the companyÕs proÞt.

ConstraintsAs x

1 and x 2 increase, GiapettoÕs objective function grows larger. This means that if Giapetto were free to choose any values for x 1 and x 2 , the company could make an arbitrarily large proÞt by choosing x 1 and x 2 to be very large. Unfortunately, the values of x 1 and x 2 are limited by the following three restrictions (often called constraints): Constraint 1Each week, no more than 100 hours of Þnishing time may be used. Constraint 2Each week, no more than 80 hours of carpentry time may be used. Constraint 3Because of limited demand, at most 40 soldiers should be produced each week. The amount of raw material available is assumed to be unlimited, so no restrictions have been placed on this. The next step in formulating a mathematical model of the Giapetto problem is to ex- press Constraints 1Ð3 in terms of the decision variables x 1 and x 2 . To express Constraint

1 in terms of x

1 and x 2 , note that

Þni

s s o h l i d n i g er hrs. soldi w er e s ek made

Þnish

tr i a n i g n hrs. trai w ns ee m k ade 2(x 1 ) 1(x 2 ) 2x 1 x 2

Now Constraint 1 may be expressed by

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