*Example:
Product planning – LP model*
**1. Definition
of variables**

…
number of units of product A to be produced

… number
of units of product B to be produced

P … total profit

**2. Objective
function**
*Maximise: *

This equation states that the firm's total
profit is made up of the profit from product A (8 times the number sold)
plus the profit from product B (6 times the number sold).
**3. Constraints**

The first constraint relates to the availability
of time on the first machine. Each unit of product A uses three hours of
this machine, and each unit of product B uses two hours. Hence, the total
hours used is expressed by the left-hand side of the expression above.
This must be equal to or less than the total hours available on the first
machine - right-hand side = 24.
For the second machine, the constraint
is similar.

**4. Non-negativity**

*I*n addition, implicit in any linear
programming formulation are the constraints that restrict all variables
are non-negative. In terms of the problem, this means that the firm can
produce only zero or positive amounts.
Each LP model must have three parts 1-3
as above.

**5. The total
formulation**

Maximise:
subject to:

Precise definition of variables is necessary for
understanding and interpretation of the solution.

This example is simple, and one would not need linear
programming to solve it. However, problems involving dozens of products
and many different constraints cannot be solved intuitively, and linear
programming has proved valuable in these cases.