Linear Programming Essay

2819 Words Aug 8th, 2011 12 Pages
Linear Programming (LP)

Linear programming, simply put, is the most widely used mathematical programming technique. It has a long history dating back to the 1930s. The Russian mathematical economist Leonid Kantorovich published an important article about linear programming in 1939. George Stigler published his famous diet problem in 1945 (“The Cost of Subsistence”). Of course, no one could actually solve these problems until George Dantzig developed the simplex method, which was published in 1951. Within a few years, a variety of American businesses recognized that they could save millions of dollars a year using linear programming models. And in the 1950s, that was a lot of money. In his book Methods of Mathematical Economics
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Now we have a linear objective function and all linear inequality/equality constraints, hence the method’s name, linear programming. The optimization problem can be written out in full as

[pic].

Observe that we have expressed all constraint inequalities in a single direction ([pic]). This is not necessary, but they make intuitive sense for a problem whose objective is to be maximized. In this case, the right-hand side values can be thought of as resources that are depleted in the course of maximizing the objective. In most well-posed problems, these resource constraints ensure that the maximum is bounded. For problems with two variables, we can even draw a picture of the feasible set (F) and come up with a solution.

A little visualization regarding the optimization (minimization or maximization) of a linear function over the set F should convince you that at least one optimal solution occurs at a vertex. A vertex is sometimes called a corner solution or an extreme point. This observation is the key to the simplex algorithm proposed by Dantzig in 1951 and subsequently coded into most commercial software packages such as Excel's Solver. In the picture above, you can solve for the intersection of the lines and determine the vertices. The vertices have coordinates of (30,10), (30,70), and (75,25). It is a simple matter to determine that (75,25) is the optimal solution. For most real world problems, you will need a computer. Fortunately,

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