Graphical Solution of a 2-Variable Linear Program

See Chapter 2: Introduction to Linear Programming.

The Acme Bicycle Company Example (see Chapter 1) is summarized below:


x₁: mountain bikes to produce (bikes/day)

x₂: racers to produce  (bikes/day)

Variable non-negativity:

x₁ ≥ 0,  x₂ ≥ 0

Objective Function:

Maximize daily profit ($/day):  Max Z =  15 x₁ + 10 x₂


Mountain bike production limit (bikes/day): x₁ ≤ 2

Racer production limit (bikes/day): x₂ ≤ 3

Metal finishing machine production limit (bikes/day): x₁ + x₂ ≤ 4

Note that units are specified for everything: variables, the objective function, and the constraints. This is important.

This problem has only two variables so it can be sketched on a plane. The animation below graphs the problem in order to find the point that maximizes the daily profit rate, i.e. the optimum point.