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.


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Contents

Chapter 1: Introduction.  An introduction to the process of optimization and an overview of the major topics covered in the course.  Last revision: December 2020 . Chapter 2: Introduction to Linear Programming.  The basic notions of linear programming and the simplex method. The simplex method is the easiest way to provide a beginner with a solid understanding of linear programming.  Last revision: December 2020. See animation  LP1 . A good  article on formulating LPs  by Gerry Brown and Rob Dell. Example questions with solutions . March 4, 2021. Chapter 3: Towards the Simplex Method for Efficient Solution of Linear Programs.  Cornerpoints and bases. Moving to improved solutions.  Last revision: December 2020 . See animation  LP2 . Chapter 4: The Mechanics of the Simplex Method.  The tableau representation as a way of illustrating the process of the simplex method. Special cases such as degeneracy and unboundedness.  Last revision: December 2020. See animation  LP3 . Example questions
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