### LP1

# 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:

#### Variables:

- 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

#### Constraints:

- 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.