Chapter 11: Basic Linear Programming Concepts

Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a real-world problem can be represented accurately in the mathematical equations of a linear program, the method will find the best solution to the problem. Of course, few complex real-world problems can be expressed perfectly in terms of a set of linear functions. Nevertheless, linear programs can provide reasonably realistic representations of many real-world problems -- especially if a little creativity is applied in the mathematical formulation of the problem.

The subject of modeling was briefly discussed in the context of regulation. The regulation problems you learned to solve were very simple mathematical representations of reality. This chapter continues our trek down the modeling path. As we progress, the models will become more mathematical -- and more complex. The real world is always more complex than a model. Thus, as we try to represent the real world more accurately, our models will inevitably become more complex. You should remember the maxim discussed earlier that a model should only be as complex as is necessary in order to represent the real world problem reasonably well. Therefore, the added complexity introduced by using linear programming should be accompanied by some significant gains in our ability to represent the problem and, hence, in the quality of the solutions that can be obtained. You should ask yourself, as you learn more about linear programming, what the benefits of the technique are and whether they outweigh the additional costs.

The jury is still out on the question of the usefulness of linear programming in forest planning. Nevertheless, linear programming has been widely applied in forest management planning. Initial applications of the technique to forest management planning problems started in the mid 1960s. The sophistication of these analyses grew until, by the mid-1970s, the technique was being applied in real-world forest planning and not just in academic exercises. The passage of the Forest and Rangeland Renewable Resource Planning Act in 1974 created a huge demand for analytical forest planning methods, and linear programming was subsequently applied on almost every national forest in the country. The forest products industry has also adopted linear programming in their planning. Today, most large forest landowners use linear programming, or more advanced techniques similar to linear programming, in their forest management planning.

Linear programming (LP) is a relatively complex technique. The objective in this class is only to provide you with an introduction to LP and it's application in forest management planning. You should not expect to finish the course a linear programming expert. This is unnecessary, since few of you will ever need to formulate a forest planning LP in your careers. However, since so much forest planning today is based on LP techniques -- or, more generally, mathematical programming techniques -- it is very likely that you will need to understand at an intuitive level how mathematical programming is used in forest management planning. By the end of the course, you should have a basic understanding of how LP works; you should be able to formulate a small forest management planning problem as an LP; and you should be able to interpret the LP solution of a forest management planning problem. This background will help you understand modern forest planning better so you can be a better participant in planning processes and so you will feel more comfortable implementing forest plans that are based on mathematical programming techniques. Finally, you should understand the process of mathematical programming well enough to recognize some of the potential problems and pitfalls of applying these techniques.

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