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Introduction

How do farmers know how much of each crop to plant? How much of a product does a company know to make? Do they just guess and hope they get the most profit they can? No, it's all been carefully predetermined using a particular part of management science called "linear programming."

Linear programming (LP) takes all kinds of factors into consideration to determine the best combination of a purchasing or manufacturing process, to either maximize profit, minimize cost or some other goal. Just from that statement alone, you can tell that LP is a very important part of any business.

To help explain how this works, we are going to start with a graph of a simple linear programming problem:

Sample LP graph [1]

250px-Linear_programming_example_graph.png

To get an idea of the fundamental logic and processes of linear programming, it's best to see it graphically first. This problem appears to be a maximization problem of some sort; for this example, we'll say it's to maximize profit. The first part of solving a linear programming model is to set a goal, which we call the objective function. The next step is to specify the constraints of the problem, that is certain conditions that must be met or limited throughout the process of your business pertaining to the product you are gaining profit from, such as not exceeding a limit of a scarce resource, or having to meet a certain demand for a product.

On the graph above, let's assume the X axis represents one product, and the Y axis represents another and we want to produce the optimum number of those products that will give us the most profit. The other lines on the graph represent the constraints, and when they are all placed on the graph, a "feasible region" becomes apparent and contains all the values that satisfy the constraints. Now that we know which combinations of products are feasible, we want to find the best combination of those points that will give us the most profit possible from selling or making these products. We call this the "optimal solution." This point is found by analyzing the objective function to see where it fits best on the graph.

One thing to note from the example is that all equations, inequalities and functions must be linear, which means all values must have a power of 1 and they can only be added or subtracted, not multiplied or divided.

Now that you have an idea of the logic behind linear programming, browse the navigation to go more in-depth, learn how LP came about, and see some real-world cases of this method being used.


Who uses it?

There are many companies that actively use linear programming in their various decisions that they make. Many of them try to make very large decision models that can take many months to build, but once built they can save them a large amount of resources and money. With the market becoming more and more competitive, many companies are trying to sell or buy their linear programming models to other companies to stay ahead of the curve. [2]

One example of this is Refinery Consulting, a company that has a specialty in making linear programming models for manufacturing plants. They have seen that many businesses want more information about their process and operating conditions, so are hiring external consultants to help them in the decision process. They want a model that has elements from both yield predictions for their plant's and economical guidance for some of their business decisions. [2]

Companies that use these linear programming model's benefit from the information they give them. These models can vastly improve their planning predictions, as well as give them a better economical guidance in which resources they select for their different processes. In the case of Refinery Consulting, they have used their knowledge of these models which spans several decades to develop new ones in the ever changing world market. Most consultants that delve in this business also have access to their own software, which most of the time is built by the firm themselves. [2]

If you want to hire Refinery Consulting to help your manufacturing plant with their advanced linear programming consultants, click here, and then go to the "Help Us Help You" link in the top right corner!

Where to get the software?

There are many different types of software packages that you can use to solve linear programming problems, and make linear programming models. One of the simplest and most ready available one's is Solver. Solver is an add in tool that comes with Microsoft Excel, so if you have the latter, you will have Solver. It can solve most linear programming problems with ease, as long as it is not too complicated. For more information about Solver, visit the Overview section of this website, or go to Solver's home page.

If you prefer a more large scale linear programming package, then solver would not be the best choice. For more advanced problems with more variables and more constraints, there are a variety of different packages to choose from. One such example is the GNU Linear Programming Kit. This package is mainly intended for solving linear programming problems on a larger scale. Some of it's features include:[3]

* Revised simplex method
* Primal-dual interior point method
* Branch-and-bound method
* Translator for GNU MathProg
* Application program interface (API)
* Stand-alone LP/MIP solver[3]

This is a free software that anyone can download and use, and if you wish to download it, simply click here. More advanced linear programming suites are available out there, and are mostly used by the larger consulting firms. [3]


References

1. Patrick J. Cassidy, 1913. Answers.com 29 Apr. 2008. <http://www.answers.com/topic/inequality-1>
3. GNU Operating System. <http://www.gnu.org/prep/ftp.html>
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