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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