In an earlier post, I showed how to set up a standard minimization problem from an application. In another post, I demonstrated how to find the corresponding dual maximization problem and to solve it with the Simplex Method. This algorithm is tedious and prone to arithmetic errors.
In the video below, the steps in the Simplex Methods are carried out in Google Sheets. The same steps can also be carried out in other spreadsheets such as Microsoft Excel.
Please note that there is a typo in the original matrix in the process. The cell in D2 should be a 1 instead of a zero to account for the slack variable in the first constraint.
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.
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?
As you may have noticed, some matrix operations are very tedious by hand. In particular, matrix multiplication and matrix inverses are much easier to do using a graphing calculator or WolframAlpha. In the Weekly Learning Plan, I put several handouts to help you use your calculator to do these problems.