9781118041581

Linear Programs 709

saved. Third, constraints limit the possible values of the decision variables. For

example, limited labor supply may constrain the quantity of output. Similarly,

the federal agency cannot spend more than its available budget, and the con-

sumer-products firm must supply the quantities to its retail outlets subject to its

factories’ production capacities. Thus, in each case, the heart of the problem

is finding (calculating) values for the decision variables that best meet the given

objective while satisfying various constraints.

With respect to the first two features, decision variables and objectives, the

LP method resembles the optimization methods we have encountered already.

Like the pricing and output decisions of Chapters 2 and 3, LP decisions rely on

marginal analysis (of a special kind) for their solution. Unlike those decisions,

however, LP problems incorporate explicit resource constraints. The interplay

of these constraints creates new and interesting economic trade-offs.

In this chapter, we take a systematic approach to managerial decisions

involving economic constraints. First, we describe a number of constrained deci-

sion problems and show how they can be formulated and solved mathemati-

cally as linear programs. Next, we examine the important concept of shadow

prices for resources. Then we introduce more complex linear programming

decisions and illustrate the kinds of solutions furnished by computer programs.

LINEAR PROGRAMS

We can analyze a host of managerial decisions using linear programming. Here

is a representative example.

FINDING AN OPTIMAL COMPUTER MIX Consider a personal computer (PC)

manufacturer that produces two versions of its popular desktop computer. The

standard version has a high-capacity (80-gigabyte) hard disk, a conventional

disk drive, and a rewritable DVD drive. The economy version, which sells at a

lower price, has a 40-gigabyte hard drive and a conventional rewritable CD

drive. The prices, variable costs, and contributions of the models are shown in

the table.

Standard PC Economy PC

Price $1,600 $1,000

Variable cost 1,100 700

Contribution 500 300

The firm has ample components (such as monitors and keyboards) from

which to assemble PCs, but a limited capacity (given available factory space

c17LinearProgramming.qxd 9/26/11 11:05 AM Page 709