How Do You Solve a Standard Minimization Problem?

In an earlier post, we wrote out a standard minimization problem from an application. Now we’ll solve that problem by finding the dual standard maximization problem and applying the Simplex Method.

This process is covered in Section 4.4 of the textbook. For complete details on the process, consult this section and the many examples contained in this section. I will assume that you have looked over this section and are familiar with carrying out row operations on a matrix.

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How Do You Write Out a Standard Minimization Problem?

Let’s take a look at a problem that requires a bit of ingenuity to put into standard minimization form.

Problem – Garton’s Seeds has a seed mixture containing three types of seeds: bluegrass, rye, and Bermuda. The cost per pound of the three seeds are 16 cents, 14 cents and 12 cents. Bluegrass seed must be at least 25% of the each batch. The amount of Bermuda must be no more than 2/3 the amount of rye in each batch. To fill current orders, Garton’s must make at least 6000 lbs of the mixture. How many pounds of each seed should be in the batch so that the cost of the batch is minimized?

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Using the Simplex Method to Solve A Standard Maximization Problem

The basic algorithm for solving a standard maximization problem is covered in Section 4.3. This process, called the Simplex Method, uses matrices and row operations to gauge whether an objective function is maximized at corner points.

In the example below, I write out a standard maximization problem from an application and then solve it with the Simplex Method.

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Relative vs Absolute Extrema

Although a relative extrema may seem to be very similar to an absolute extrema, they are actually quite different. The term “relative” means compared to numbers nearby…so a relative extrema is either a bump or a dip on the function.

The term “absolute” means the most extreme on the entire function. An absolute extrema is the very highest or lowest point on the function. This may occur at a bump or a dip. They may also occur at the ends of the function if it is defined on a closed interval.

The MathFAQ below illustrates how to find these points on a function.

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