How can a factory use linear programming to choose quantities of two products to make?

To determine the optimal quantities of two products to manufacture, a factory can use linear programming to maximize profit while operating within resource constraints.12

The Mathematical Model

The process involves two primary components:

• Objective Function: This represents the goal, such as total profit. If a factory produces $x_1$ of the first product and $x_2$ of the second, and the profit on the second is twice that of the first, the objective function is $x_1+2x_2=k$, where $k$ is the total profit.12
• Constraints: These represent limitations such as raw materials, labor, or machine time. For example, if raw materials limit production to 8 units of $x_1$ and 5 units of $x_2$, and machine time limits the total batch to 10, the constraints are $x_1\le 8$, $x_2\le 5$, and $x_1+x_2\le 10$, alongside nonnegativity requirements ($x_1,x_2\ge 0$).12

Finding the Solution

For two variables, the factory can plot these constraints on a graph. The feasible solutions form a polygon. The optimal production quantity is found at a vertex (or "extreme point") of this region, where the objective function reaches its maximum value. For more complex problems, the simplex method is used to systematically test these vertices.123