Linear Optimization

Linear programming (LP; also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming.

To get started with LP, I recommend the following reading:

The examples on this page will show you how to find a graphical solution to a simple LP problem and how to derive an optimum solution using Excel.

Examples:

thinning example (with videos)
bird habitat and timber production

Click on the links to get to the related content directly.


Linear Optimization Example 01 – Thinning from above, thinning from below.

In this example you will learn how to

  • find a graphical solution for a linear optimization problem
  • use the Excel Solver Add-in to find an optimum solution for a simplified forest management problem

Problem Description

A forest owner, who owns a 10 ha forest would like to maximize economic output. The stand is a 40 year old spruce stand, so this year (t=0) she would like to carried out a thinning. She has a budget of 11,500 CAD available for her thinning activities. The options she has regarding thinning are to thin the stand from below (thinning type 1) or to thin the stand from above (thinning type 2). Both activities are related with specific risks, returns and costs (see below). In 20 years (t=20) she will carry out the final harvest. Depending on her activities today, the forest will develop until year 20.

The Question:
On how many hectares should she carry out thinning from below, and on how many hectares should she carry out thinning from above to maximize the return from her forest?

We can use linear optimization to find the optimum solution for this decision problem. To fully understand what a linear optimization is, it is best to start with a manual approach.
So, get pen and paper ready. Download the WorkSheet. Watch the tutorial and work along with it.

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Linear Optimization
Finding a Graphical Solution for our thinning problem

After learning how to  find a graphical solution for our allocation problem let’s have a look at how to use the Excel Solver Add-In to find a solution.

Picture11) Download the EXCEL file for this tutorial here: Example_THINNING_with Solution

2) Watch the video that walks you through the solution using the Excel solver

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This example is based on an example described in the book
Knoke, T.;  Griess, V.C. et al. (2012): Forstbetriebsplanung. Ulmer. 405p. (in GERMAN)


Linear Optimization Example 02 – Bird Habitat & Timber Production

In this example you will learn how to

  • find a graphical solution for a linear optimization problem
  • use the Excel Solver Add-in to find an optimum solution for a simplified forest management proble

Download the LP_example_BirdHabitat PDF file. It contains a detailed description and walks you through the steps of finding a solution for a typcal allocation problem using Excel.
You can use the LP_example_BirdHabitat_SolverWorksheet to try and find your own solution using the Solver.