The TI-83 calculator can be used to perform the row operations required in the Simplex algorithm. To do so, you must learn to use two of the functions provided in the matrix math menu:
Press 
and
and then go to
the Math menu.
This function multiplies all of the values in one row of a matrix by the same multiplier. It requires three arguments: the multiplier, the matrix, and the row number.
*row(multiplier, matrix, row)
For example, the function
*row(2, [A], 1)
multiplies each element in row 1 of matrix A by 2. If matrix A is given by
8 -1 0 7 3 -2 -3 5 1
then *row(2, [A], 1) would give this result:
16 -2 0 7 3 -2 -3 5 1
This function multiplies all of the values in one row of a matrix by the same multiplier, adds the resulting values to the elements in some other row, and stores the results in that other row. It requires four arguments: the multiplier, the matrix, and two row numbers. The values in the first row are multiplied by the multiplier and the products are added to the corresponding values in the second row.
*row+(multiplier, matrix, rowA, rowB)
For example, the function
*row+(8, [A], 2, 3)
multiplies the values in row 2 by 8 and adds the products to the corresponding values in row 3. If matrix A is given by
8 -1 0 7 3 -2 -3 5 1
then *row+(8, [A], 2, 3) would give this result:
8 -1 0 7 3 -2 53 29 -15
The first step is to enter the initial Simplex tableau as a matrix and store it as matrix A. In the illustrations, the column and row headings are added for clarity. They are not part of the matrix stored in the TI-83 calculator.
The pivot column is column 1 and the pivot row is row 1. We need to multiply the pivot row by the reciprocal of the pivot element (8, in this case) so that the pivot element becomes 1. In this case, the multiplier is 1/8 (the reciprocal of 8), the matrix is A, and the row is 1:
*row(1/8, [A], 1)yields
Now we need to perform row operations that will give us a value of 0 in the remaining rows of the pivot column. These operations are always of the same type:
Target Row = Target Row + Multiplier * Pivot Row
where the multiplier is the negative of the value at the intersection of the target row and the pivot column.
*row+(multiplier, matrix, pivot row, target row)
Let's begin with row 2. The value in row two, column 1 (the pivot column) is 2 so the multiplier is -2. Keeping in mind that the pivot row is row 1, the required row operation is given by:
Row 2 = Row 2 + (-2) * Row 1
This operation (and each operation that follows) is performed on the answer from
the previous matrix operation and not matrix A. Matrix A is just the initial
simplex tableau. As we apply the row operations in the Simplex algorithm, they
are always applied to the answer from the previous operation. To get the answer
from the previous operation use this keystroke combination:
One the TI-83 we enter the function arguments and tap the enter key button:
*row+(-2, Ans, 1, 2)
and get this result:
Now we need to perform the same type of operation on row three:
Row 3 = Row 3 + 90 * Row 1 *row+(90, Ans, 1, 3)
which yields this result:
At this point, we find the new pivot column and the new pivot row and repeat the process:
*row(2, Ans, 2)
Row 1 = Row 1 + (-0.25) * Row 2 *row+(-.25, Ans, 2, 1)
Row 3 = Row 3 + 2.5 * Row 2 *row+(2.5, Ans, 2, 3)
We are done because there are no negative values in the bottom row. The maximum profit is $4,600 when the company manufactures 40 tables and 40 chairs.