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and necessary equipment) for producing DVD drives and hard-disk drives. The

firm’s maximum weekly outputs are 200 DVD drives and 20,000 gigabytes of

hard-drive capacity. The firm can split its hard-drive capacity in any way between

the two models. For instance, it could devote all of its hard-drive capacity to

250 standard models or, alternatively, to 500 economy models. Or it could pro-

duce other combinations—for instance, 200 standard models and 100 econ-

omy models.

In addition, the firm assembles computers using a 50-person labor force

that supplies 2,000 hours of labor per week. The two models require roughly

equal assembly time—an average of 5 labor-hours each. How many computers

of each type should the firm produce to maximize its profit? Answering this

question requires two steps: (1) formulating the firm’s decision as a linear pro-

gram; that is, a set of mathematical equations that precisely describe the firm’s

available options; and (2) solving these mathematical equations.

The formulation stage begins with the identification of the relevant deci-

sion variables. The firm must determine two key quantities: the number of stan-

dard models (S) and the number of economy models (E). The firm seeks to

maximize the total contribution () it obtains from these products. We can

express this contribution algebraically as

[OF]

The goal to be maximized—in this instance, total contribution—is called the

objective function (OF).

Next we identify the production constraints. The company cannot produce

an unlimited number of computers. It faces three principal constraints. First,

the firm can produce only 200 DVD drives a week. This means that, at most, it

can produce 200 standard models. Also, it can produce a maximum of 20,000

gigabytes of hard drives. Finally, the firm has only 2,000 hours of labor to devote

to production of PCs. The algebraic representations of these constraints are

[D]

[H]

[L]

As the labels in brackets indicate, the inequalities correspond to the DVD drive,

hard-disk drive, and labor constraints, respectively. The right-hand side of each

inequality lists the total capacity (or supply) of the particular input. The left-hand

side shows the total amount used of each resource if the firm produces quanti-

ties S and E of the models. For instance, producing S standard models requires

S number of DVD drives—one drive per machine. Thus, according to the first

constraint, DVD capacity limits the weekly output of standard models to 200.

Next consider the hard-disk constraint. Production of the models in the

quantities S and E together requires 80S 40E gigabytes of hard-disk capacity.

5S5E2,000.

80S40E 20,000

S 200

500S300E.

710 Chapter 17 Linear Programming

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