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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The model sheet and LP reports for the banking example will be constructed in class.

Sensitivity Analysis

Sensitivity analysis involves a variety of “what if” scenarios. For example, (1) Suppose you change the loan requirement from 30 to 40 million. How does this affect your bottom line? (2) Suppose you change the total amount available for investment from 100 million to 110 million. How does this affect your bottom line? (3) Suppose you change the return on securities from 5% to 7.5%. How does this affect your investment solution? We will answer these questions in class using Excel’s sensitivity report.

Blending Problems

One of the most common uses of linear programming is to determine an optimal blend. Although the earliest applications dealt with blending fuels, people quickly realized that everything from sausages to investments could be blended with an appropriately specified LP model. The idea behind all of these problems is to mix multiple products so that the blend meets minimum requirements on desirable dimensions but does not exceed maximum levels on undesirable dimensions. For example, in constructing an optimal diet from a menu of possible sources, one wants to meet minimum requirements on things like calcium and vitamin C without exceeding maximum levels on things like sodium and cholesterol. Since there are typically an infinite number of blends that satisfy the requirements, one selects the blend that achieves some additional objective (e.g.,