SIMPLEX S,M1,M2,M3,L
solves a linear programming problem presented by the matrix file S
with M1+M2+M3 constraints and lists the results from line L
(optional) onwards.
The ordinary simplex algorithm is used.
The solution vector and the values of the M1+M2 slack variables
will be given as results. These vectors are also saved in matrix files
SIMPLEX.M and SLACK.M, respectively.
The Simplex Output Table is saved as matrix file TSIMPLEX.M .
The algorithm has been taken from
Numerical Recipes by Press, Flannery, Teukolsky and Vetterling.
The structure of the problem is given on the next page:
The problem to be solved is:
Maximize
Z=A(0,1)*X1+A(0,2)*X2+...+A(0,N)*XN
subject to the primary constraints
X1>=0, X2>=0, ... , XN>=0
and simultaneously subject to M=M1+M2+M3 additional constraints,
M1 of them of the form
A(I,1)*X1+A(I,2)*X2+...+A(I,N)*XN <= B(I), B(I)>=0, I=1,...,M1
M2 of them of the form
A(I,1)*X1+A(I,2)*X2+...+A(I,N)*XN >= B(I) >= 0, I=M1+1,...,M1+M2
and M3 of them of the form
A(I,1)*X1+A(I,2)*X2+...+A(I,N)*XN = B(I) >= 0, I=M1+M2+1,...,M
The matrix of coefficients S with M+1 rows and N+1 columns has the form
0 A(0,1) A(0,2) ... A(0,N)
B(1) -A(1,1) -A(1,2) ... -A(1,N)
B(2) -A(2,1) -A(2,2) ... -A(2,N)
... ... ... ... ...
B(M) -A(M,1) -A(M,2) ... -A(M,N)
and it must be saved in a MAT file before activating SIMPLEX.
Example 1:
Maximize Z=X1+X2+3*X3-0.5*X4 with all the X's non-negative and
also with X1+2*X3 <= 740
2*X2-7*X4 <= 0
X2-X3+2*X4 >= 0.5
X1+X2+X3+X4 = 9 We have M1=2, M2=1, M3=1
This problem is described and solved by:
1 *
2 *MATRIX S
3 */// C X1 X2 X3 X4
4 *Z 0 1 1 3 -0.5
5 *Z1 740 -1 0 -2 0
6 *Z2 0 0 -2 0 7
7 *Z3 0.5 0 -1 1 -2
8 *Z4 9 -1 -1 -1 -1
9 *
10 *MAT SAVE S / Saving the matrix
11 *SIMPLEX S,2,1,1,12
12 *
Example 2: B1 B2 B3 B4
Solving a two-person zero-sum game: A1 3 6 1 4
A2 5 2 4 2
A3 1 4 3 5
2 *MATRIX A
3 */// C A1 A2 A3 V
4 *C 0 0 0 0 1
5 *B1 0 3 5 1 -1
6 *B2 0 6 2 4 -1
7 *B3 0 1 4 3 -1
8 *B4 0 4 2 5 -1
9 *V 1 -1 -1 -1 0
10 *
11 *MAT SAVE A
12 *SIMPLEX A,4,0,1,END+2 / gives the optimal mixed strategy for player A
13 *MAT B=A' / *B~A' 5*6
14 *MAT DIM B /* rowB=5 colB=6
15 *MAT B(1,colB)=-1
16 *SIMPLEX B,0,3,1,END+2 / gives the optimal mixed strategy for player B
M = More information on mathematical operations
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