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About this program:   This program is for those who are familiar with the simplex method that uses negative slack variables when doing problems with mixed constraints or minimization.  You must enter the first tableau in matrix [A] with the proper slack variables and with the proper signs for the indicator row (objective function.)  The program then manipulates rows to give a first feasible solution and displays the solution in decimal form.  The solution may be displayed in fractional form, if appropriate, by pressing ENTER.  When you are finished with the answer, press ENTER because the program is STILL RUNNING in PAUSE Mode to permit scrolling the matrix.

Running the Program:
  Unlike several of my other programs, the student does not participate in this program other than to enter the matrix in position A and to formulate the first tableau with appropriate  slack variables.  I'll give you examples on this after the coding section. 

Memory used and entry time:  This program uses uses about 372 bits of memory.   I estimate that it will take about 15 minutes for an inexperienced programmer to enter this program by hand.

DISCLAIMER:  This program is free, and, therefore, I make no claims about it's efficacy, efficiency, or proper operation.  This is a new program as of 8/15/03.  If you find a problem with this program, or can suggest  improvements, please e-mail me at  .

Use of this Program:   You may use this program freely for your own personal use and for the use of other students, but use for publication or any means of profit in not allowed without my permission.






Program designation



Lbl 0


[B]T →[C]

Menu("SELECT", "STD MAX",1, "MIXED MAX", 2, "STD MIN",3)    

Lbl 2 :Lbl 3:Lbl 0


For (K, R, C-1)


Matr→List([B],K, L1)




If min(L1) = sum (L1)  and min(L1) = -1




min(L1)-->M: End

If M = -1    

For( N,1, R-1)


If Min=L1(N)


Then: N→I:End


[B]T →[C]


Matr→List([C],I, L1)






For (D,1, C-1)


If H=L1(D)

*row ((1/([B](I,J), I, J)→[B]    

For (E,1, R)


If E≠I and [B](E,J)≠0

Then: *Row+((-[B](E, J), [B], I,E)→[B]    





Lbl 1

Repeat P =0    
0→P: 0→K    
[B]T →[C]    
For (K,1,C-1)    
If [B](R,K)<0    
Then: 2→P:End    
If P=2    
Then: Matr→List([C], R,L1)    
0→L1 (C): min (L1)→M    
For (K,1,C-1)    
If M = L1(K)    
Then: K→J: End    
Matr→List([B], J,L1)    
For (K,1,R-1)    
If L1(K) > 0    
Then: L1(K)/[B](K,C)→L1(K): End    
If L1(K) < 0    
Then: 0→L1(K): End    
max( L1)→M    
If M>0    
Then: For (K,1, R-1)    
If M = L1(K)    
Then: K→I: End:End    
Else: 0→P: End: End    
If P≠0    
Then: *row(1/[B](I,J), [B], I)→[B]    
For (K,1,R)    
If  K≠I and [B](K,J)≠ 0    
Then: *row+(-[B](K,J), [B],I,K)→[B]    
Pause [B]    
Pause [B]►Frac    

Now for using it:  Let's do this a maximization problem:
Maximize P = 2x1 +4x2 +3x3
With these constraints:
x1 +3x2 + x3 6
2x1 +2x2 +x3 5
3x1 =x2 +4x3 7

Using slack variables, the first tableau will be this:
| 1   3     1  1  0  0  6|
|2    2     1  0  1  0  5 |
|3    1    4   0  0 1  7 |
|-2  -4  -3   0  0  0 0

Plug that into matrix A; go to your program and execute it.  When the Menu appears, enter 1.   After a few seconds your answer will be displayed.

Standard Minimization problem: (Be sure you know what standard minimization means.)
Minimize w= 3y1 +2y2
With these constraints:
y1 + 3y2 ≥6
2y1 + y2 ≥3

We can look at this problem like this:
Maximize:  z = -w= -3y1 - 2y2
Subject to:y1 + 3y2 ≥6
                   2y1 + y2 ≥3

So, our first tableau is this:
| 1   3    -1   0  |6
| 2   1     0  -1  |3
|3    2     0   0  |0

Enter that into matrix A; execute the program and enter 3 when the Menu appears. The display will be  this:
|  0    1  - 0.4   0.2    1.8 |
|  1   0    0.2  -0.6     0.6 |
|  0   0    0.2   1.4     -5.4|

Notice that -5.4 is the negative of the minimum, 5.4, and that y1 = 0.6 and y2 = 1.8.

Mixed ≤ and ≥ problem:
Maximize:    Z= 120x1 + 40x2 + 60x3
Subject To: 
x1 +x2 + x3 ≤ 100
x1 +x2 +x3 ≥60

Then our first tableau will be this:
| 1      1      1    1  0  0  |100
|400 160  280  0  1  0  |20000
|1        1    1     0  0 -1 | 60
|-120  -40  -60   0  0  0| 0

Enter that into matrix A; execute the program and enter 2 from the Menu.  The display will be  this:
| 0      0      0    1       0           1           40
| 1      0     0.5  0   4.1E-3    0.6666     43.3333
| 0      1     0.5  0    -4E-3     -1.666      16.666
| 0      0     20   0     0.333    13.333     5866.6


Last Revised: 8/23/03