We begin by modeling this problem. Modeling a problem using linear programming involves writing it in the language of linear programming. There are rules about what you can and cannot do within linear programming. These rules are in place to make certain that the remaining steps of the process (solving and interpreting) can be successful.

Key to a linear program are the *decision variables*, *
objective*, and *constraints*.

**Decision Variables.** The decision variables represent (unknown)
decisions to be made. This is in contrast to *problem data*,
which are values that are either given or can be simply calculated
from what is given. For this problem, the decision variables are the
number of notebooks to produce and the number of desktops to produce.
We will represent these unknown values by and
respectively. To make the numbers more manageable, we will let
be the number of 1000 notebooks produced (so means a
decision to produce 5000 notebooks) and be the number of 1000
desktops. Note that a value like the quarterly profit is not (in this
model) a decision variable: it is an outcome of decisions and
.

**Objective.** Every linear program has an objective. This
objective is to be either minimized or maximized. This objective has
to be *linear* in the decision variables, which means it must be
the sum of constants times decision variables. is a
linear function. is not a linear function. In this case,
our objective is to maximize the function (what units
is this in?).

**Constraints.** Every linear program also has constraints limiting
feasible decisions. Here we have four types of constraints:
Processing Chips, Memory Sets, Assembly, and Nonnegativity.

In order to satisfy the limit on the number of chips available, it is
necessary that . If this were not the case (say
), the decisions would not be implementable (12,000 chips
would be required, though we only have 10,000). Linear programming
cannot handle arbitrary restrictions: once again, the restrictions
have to be *linear*. This means that a linear function of the
decision variables must be *related* to a constant, where *
related* can mean less than or equal to, greater than or equal to, or
equal to. So is a linear constraint, as is . is not a linear constraint, nor is . Our constraint for Processing Chips is a linear
constraint.

The constraint for memory chip sets is , a linear constraint.

Our constraint on assembly can be written , again a linear constraint.

Finally, we do not want to consider decisions like , where production is negative. We add the linear constraints , to enforce nonnegativity of production.

**Final Model.** This gives us the complete model of this problem:

*Formulating a problem as a linear program* means going through
the above process to clearly define the decision variables, objective,
and constraints.

Mon Aug 24 14:40:57 EDT 1998