x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 34 | 30 | 37 | 1 | 0 | 0 | 0 | 0 | 888 |
s2 | 11 | 2 | -15 | 0 | 1 | 0 | 0 | 0 | 33 |
s3 | -16 | 9 | -2 | 0 | 0 | 1 | 0 | 0 | -48 |
s4 | -14 | -74 | 31 | 0 | 0 | 0 | 1 | 0 | -304 |
8 | -4 | -14 | 0 | 0 | 0 | 0 | 1 | 0 |
We can do this with the TI-83 calculators and the SM3 program.
Although the program will allow us to enter a base matrix to represent the
constraints, it is easier to construct a temporary matrix first and then
give SM3 the name of the matrix when it is needed.
For this example, we will create the basic matrix in matrix [A] using the
matrix editor. The basic matrix has a row for every constraint and a column for every
variable along with one for the constants in the inequalities.
However, the SM3 program assumes that all inequalities have been converted to
≤ type inequalities. Thus, we need to rewrite the inequalities as
11 x1 + 2 x2 + -15 x3 ≤ 33 -16 x1 + 9 x2 + -2 x3 ≤ -48 -14 x1 + -74 x2 + 31 x3 ≤ -304 ![]() ![]() ![]() The SM3 program now needs the coeeficients of the variables for the objective function. In this case we need to enter the coeeficients, in the right order, as part of a list, as shown in: ![]() The next question from the SM3 program relates to the choice between finding the maximum or minimum. The prompt and the user response are on the following screen: ![]() ![]() |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 34 | 30 | 37 | 1 | 0 | 0 | 0 | 0 | 888 |
s2 | 11 | 2 | -15 | 0 | 1 | 0 | 0 | 0 | 33 |
s3 | -16 | 9 | -2 | 0 | 0 | 1 | 0 | 0 | -48 |
s4 | -14 | -74 | 31 | 0 | 0 | 0 | 1 | 0 | -304 |
8 | -4 | -14 | 0 | 0 | 0 | 0 | 1 | 0 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 34 | 30 | 37 | 1 | 0 | 0 | 0 | 0 | 888 |
s2 | 11 | 2 | -15 | 0 | 1 | 0 | 0 | 0 | 33 |
s3 | -16 | 9 | -2 | 0 | 0 | 1 | 0 | 0 | -48 |
s4 | -14 | -74 | 31 | 0 | 0 | 0 | 1 | 0 | -304 |
8 | -4 | -14 | 0 | 0 | 0 | 0 | 1 | 0 |
The SM3 program arrives at this point of the process and displays:
![]() Once we press the ENTER key on the calculator, the SM3 program will first use the *row operation to make the pivot hold the value 1 and then use the *row+ operation to make the 0 value in the rest of that column. This web page shows the entire process below; we will resume tracing the SM3 program after a few steps. |
Use the elementary row operation *row to change the pivot item to be 1. That is, multiply row 4 by -1 / 74.
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 34 | 30 | 37 | 1 | 0 | 0 | 0 | 0 | 888 |
s2 | 11 | 2 | -15 | 0 | 1 | 0 | 0 | 0 | 33 |
s3 | -16 | 9 | -2 | 0 | 0 | 1 | 0 | 0 | -48 |
s4 | 7/37 | 1 | -31/74 | 0 | 0 | 0 | -1/74 | 0 | 152/37 |
8 | -4 | -14 | 0 | 0 | 0 | 0 | 1 | 0 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 1048/37 | 0 | 1834/37 | 1 | 0 | 0 | 15/37 | 0 | 28296/37 |
s2 | 393/37 | 0 | -524/37 | 0 | 1 | 0 | 1/37 | 0 | 917/37 |
s3 | -655/37 | 0 | 131/74 | 0 | 0 | 1 | 9/74 | 0 | -3144/37 |
s4 | 7/37 | 1 | -31/74 | 0 | 0 | 0 | -1/74 | 0 | 152/37 |
324/37 | 0 | -580/37 | 0 | 0 | 0 | -2/37 | 1 | 608/37 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 1048/37 | 0 | 1834/37 | 1 | 0 | 0 | 15/37 | 0 | 28296/37 |
s2 | 393/37 | 0 | -524/37 | 0 | 1 | 0 | 1/37 | 0 | 917/37 |
s3 | -655/37 | 0 | 131/74 | 0 | 0 | 1 | 9/74 | 0 | -3144/37 |
x2 | 7/37 | 1 | -31/74 | 0 | 0 | 0 | -1/74 | 0 | 152/37 |
324/37 | 0 | -580/37 | 0 | 0 | 0 | -2/37 | 1 | 608/37 |
The SM3 program jumps from telling us the location of the pivot point
to showing us the results of using that pivort point:
![]() |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 1048/37 | 0 | 1834/37 | 1 | 0 | 0 | 15/37 | 0 | 28296/37 |
s2 | 393/37 | 0 | -524/37 | 0 | 1 | 0 | 1/37 | 0 | 917/37 |
s3 | -655/37 | 0 | 131/74 | 0 | 0 | 1 | 9/74 | 0 | -3144/37 |
x2 | 7/37 | 1 | -31/74 | 0 | 0 | 0 | -1/74 | 0 | 152/37 |
324/37 | 0 | -580/37 | 0 | 0 | 0 | -2/37 | 1 | 608/37 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 1048/37 | 0 | 1834/37 | 1 | 0 | 0 | 15/37 | 0 | 28296/37 |
s2 | 393/37 | 0 | -524/37 | 0 | 1 | 0 | 1/37 | 0 | 917/37 |
s3 | -655/37 | 0 | 131/74 | 0 | 0 | 1 | 9/74 | 0 | -3144/37 |
x2 | 7/37 | 1 | -31/74 | 0 | 0 | 0 | -1/74 | 0 | 152/37 |
324/37 | 0 | -580/37 | 0 | 0 | 0 | -2/37 | 1 | 608/37 |
Here is the SM3 presentation of the same information:
![]() |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 1048/37 | 0 | 1834/37 | 1 | 0 | 0 | 15/37 | 0 | 28296/37 |
s2 | 393/37 | 0 | -524/37 | 0 | 1 | 0 | 1/37 | 0 | 917/37 |
s3 | 1 | 0 | -1/10 | 0 | 0 | -37/655 | -9/1310 | 0 | 24/5 |
x2 | 7/37 | 1 | -31/74 | 0 | 0 | 0 | -1/74 | 0 | 152/37 |
324/37 | 0 | -580/37 | 0 | 0 | 0 | -2/37 | 1 | 608/37 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 0 | 0 | 262/5 | 1 | 0 | 8/5 | 3/5 | 0 | 3144/5 |
s2 | 0 | 0 | -131/10 | 0 | 1 | 3/5 | 1/10 | 0 | -131/5 |
s3 | 1 | 0 | -1/10 | 0 | 0 | -37/655 | -9/1310 | 0 | 24/5 |
x2 | 0 | 1 | -2/5 | 0 | 0 | 7/655 | -8/655 | 0 | 16/5 |
0 | 0 | -74/5 | 0 | 0 | 324/655 | 4/655 | 1 | -128/5 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 0 | 0 | 262/5 | 1 | 0 | 8/5 | 3/5 | 0 | 3144/5 |
s2 | 0 | 0 | -131/10 | 0 | 1 | 3/5 | 1/10 | 0 | -131/5 |
x1 | 1 | 0 | -1/10 | 0 | 0 | -37/655 | -9/1310 | 0 | 24/5 |
x2 | 0 | 1 | -2/5 | 0 | 0 | 7/655 | -8/655 | 0 | 16/5 |
0 | 0 | -74/5 | 0 | 0 | 324/655 | 4/655 | 1 | -128/5 |
The calculator display at this point is
![]() |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 0 | 0 | 262/5 | 1 | 0 | 8/5 | 3/5 | 0 | 3144/5 |
s2 | 0 | 0 | -131/10 | 0 | 1 | 3/5 | 1/10 | 0 | -131/5 |
x1 | 1 | 0 | -1/10 | 0 | 0 | -37/655 | -9/1310 | 0 | 24/5 |
x2 | 0 | 1 | -2/5 | 0 | 0 | 7/655 | -8/655 | 0 | 16/5 |
0 | 0 | -74/5 | 0 | 0 | 324/655 | 4/655 | 1 | -128/5 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 0 | 0 | 262/5 | 1 | 0 | 8/5 | 3/5 | 0 | 3144/5 |
s2 | 0 | 0 | -131/10 | 0 | 1 | 3/5 | 1/10 | 0 | -131/5 |
x1 | 1 | 0 | -1/10 | 0 | 0 | -37/655 | -9/1310 | 0 | 24/5 |
x2 | 0 | 1 | -2/5 | 0 | 0 | 7/655 | -8/655 | 0 | 16/5 |
0 | 0 | -74/5 | 0 | 0 | 324/655 | 4/655 | 1 | -128/5 |
Here is the program display of the same information:
![]() |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 0 | 0 | 262/5 | 1 | 0 | 8/5 | 3/5 | 0 | 3144/5 |
s2 | 0 | 0 | 1 | 0 | -10/131 | -6/131 | -1/131 | 0 | 2 |
x1 | 1 | 0 | -1/10 | 0 | 0 | -37/655 | -9/1310 | 0 | 24/5 |
x2 | 0 | 1 | -2/5 | 0 | 0 | 7/655 | -8/655 | 0 | 16/5 |
0 | 0 | -74/5 | 0 | 0 | 324/655 | 4/655 | 1 | -128/5 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 0 | 0 | 0 | 1 | 4 | 4 | 1 | 0 | 524 |
s2 | 0 | 0 | 1 | 0 | -10/131 | -6/131 | -1/131 | 0 | 2 |
x1 | 1 | 0 | 0 | 0 | -1/131 | -8/131 | -1/131 | 0 | 5 |
x2 | 0 | 1 | 0 | 0 | -4/131 | -1/131 | -2/131 | 0 | 4 |
0 | 0 | 0 | 0 | -148/131 | -24/131 | -14/131 | 1 | 4 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 0 | 0 | 0 | 1 | 4 | 4 | 1 | 0 | 524 |
x3 | 0 | 0 | 1 | 0 | -10/131 | -6/131 | -1/131 | 0 | 2 |
x1 | 1 | 0 | 0 | 0 | -1/131 | -8/131 | -1/131 | 0 | 5 |
x2 | 0 | 1 | 0 | 0 | -4/131 | -1/131 | -2/131 | 0 | 4 |
0 | 0 | 0 | 0 | -148/131 | -24/131 | -14/131 | 1 | 4 |
Here is the SM3 version of the tableau.
![]() |
The SM3 program announces the same thing as:
![]() ![]() |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 0 | 0 | 0 | 1 | 4 | 4 | 1 | 0 | 524 |
x3 | 0 | 0 | 1 | 0 | -10/131 | -6/131 | -1/131 | 0 | 2 |
x1 | 1 | 0 | 0 | 0 | -1/131 | -8/131 | -1/131 | 0 | 5 |
x2 | 0 | 1 | 0 | 0 | -4/131 | -1/131 | -2/131 | 0 | 4 |
0 | 0 | 0 | 0 | -148/131 | -24/131 | -14/131 | 1 | 4 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 0 | 0 | 0 | 1 | 4 | 4 | 1 | 0 | 524 |
x3 | 0 | 0 | 1 | 0 | -10/131 | -6/131 | -1/131 | 0 | 2 |
x1 | 1 | 0 | 0 | 0 | -1/131 | -8/131 | -1/131 | 0 | 5 |
x2 | 0 | 1 | 0 | 0 | -4/131 | -1/131 | -2/131 | 0 | 4 |
0 | 0 | 0 | 0 | -148/131 | -24/131 | -14/131 | 1 | 4 |
The program displays the same information as
![]() |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 0 | 0 | 0 | 1/4 | 1 | 1 | 1/4 | 0 | 131 |
x3 | 0 | 0 | 1 | 0 | -10/131 | -6/131 | -1/131 | 0 | 2 |
x1 | 1 | 0 | 0 | 0 | -1/131 | -8/131 | -1/131 | 0 | 5 |
x2 | 0 | 1 | 0 | 0 | -4/131 | -1/131 | -2/131 | 0 | 4 |
0 | 0 | 0 | 0 | -148/131 | -24/131 | -14/131 | 1 | 4 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s1 | 0 | 0 | 0 | 1/4 | 1 | 1 | 1/4 | 0 | 131 |
x3 | 0 | 0 | 1 | 5/262 | 0 | 4/131 | 3/262 | 0 | 12 |
x1 | 1 | 0 | 0 | 1/524 | 0 | -7/131 | -3/524 | 0 | 6 |
x2 | 0 | 1 | 0 | 1/131 | 0 | 3/131 | -1/131 | 0 | 8 |
0 | 0 | 0 | 37/131 | 0 | 124/131 | 23/131 | 1 | 152 |
x1 | x2 | x3 | s1 | s2 | s3 | s4 | z | ||
s2 | 0 | 0 | 0 | 1/4 | 1 | 1 | 1/4 | 0 | 131 |
x3 | 0 | 0 | 1 | 5/262 | 0 | 4/131 | 3/262 | 0 | 12 |
x1 | 1 | 0 | 0 | 1/524 | 0 | -7/131 | -3/524 | 0 | 6 |
x2 | 0 | 1 | 0 | 1/131 | 0 | 3/131 | -1/131 | 0 | 8 |
0 | 0 | 0 | 37/131 | 0 | 124/131 | 23/131 | 1 | 152 |
The program displays the tableau at this point as
![]() |
The program announces the end of the process:
![]() |
The SM3 presents the solution as
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
©Roger M. Palay Saline, MI 48176 October, 2010