9781118041581

Linear Programs 711

For instance, producing 100 of each model per week would require a total of

(80)(100) (40)(100) 12,000 gigabytes of capacity, which is safely within

the 20,000 gigabytes available. Finally, consider the labor constraint. Here, the

firm uses a total of 5S 5E hours of labor. The total amount of labor cannot

exceed 2,000.

The complete mathematical description of the problem consists of the

objective function (to be maximized), the three resource constraints, and two

nonnegativity constraints, S 0 and E 0. These last two constraints simply

reflect the impossibility of producing negative quantities. Although obvious

(even trivial) to the decision maker, we must include them to get the right

answer. (Computer programs don’t have the same intuition as managers.)

We now have a fully formed linear program. What makes it linear? All of the

expressions, for both the objective function and the constraints, are linear. In a

linear expression, we can multiply the variable by a constant, we can add or sub-

tract these multiplied variables, and we can add a constant to the expression. But

this is all; we cannot multiply two variables together, we cannot raise variables to

powers, we cannot take square roots of variables, or anything else. Roughly speak-

ing, the linearity assumption requires that the key quantities in the actual man-

agerial problem—revenues, costs, and profits—vary proportionally with changes

in the firm’s decision variables. For instance, if the firm can sell its product at

fixed prices, its revenue is proportional to output, thus satisfying linearity.

However, if the firm faces a downward-sloping demand curve (the usual cir-

cumstance in Chapters 2 and 3), revenue is a nonlinear function of output; that

is, the revenue graph is curved. The linear programming method cannot handle

this case. (Instead, we can use a related method, nonlinear programming.)

Graphing the LP Problem

We can solve small-scale linear programs, like the PC example, using graphical

methods. This approach provides the numerical solution and offers insight

into the factors that determine the optimal decision. The method consists of

the following steps:

1. Construct a graph, placing a decision variable on each axis.


2. Graph each constraint as though it were binding, that is, as if it held
   with strict equality.


3. Find the feasible region, the area of the graph that simultaneously
   satisfies all constraints.


4. Superimpose contours of the objective function on the feasible region
   to determine the optimal corner of the region.


5. Solve the appropriate equations of the LP problem to determine the
   optimal values of the decision variables at the corner solution.

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