Linear Programming and the Simplex Method in .NET Integrated PDF417 in .NET Linear Programming and the Simplex Method

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10.8 Linear Programming and the Simplex Method using .net framework tocreate pdf417 on web,windows application Code 128 Code Set A Routine Implementing the Simplex Method The following routine i s based algorithmically on the implementation of Kuenzi, Tzschach, and Zehnder [4]. Aside from input values of M , N , m1 , m2 , m3 , the principal input to the routine is a two-dimensional array a containing the portion of the tableau (10.8.

18) that is contained between the double lines. This input occupies the M + 1 rows and N + 1 columns of a[1..


Note, however, that reference is made internally to row M + 2 of a (used for the auxiliary objective function, just as in 10.8.18).

Therefore the variable declared as float **a, must point to allocated memory allowing references in the subrange a[i][k], i = 1 . . .

m+2, k = 1 . . .

n+1 (10.8.20).

You will suffer endless VS .NET pdf417 2d barcode agonies if you fail to understand this simple point. Also do not neglect to order the rows of a in the same order as equations (10.

8.1), (10.8.

3), (10.8.4), and (10.

8.5), that is, objective function, -constraints, -constraints, =-constraints. On output, the tableau a is indexed by two returned arrays of integers.

iposv[j] contains, for j= 1 . . .

M , the number i whose original variable xi is now represented by row j+1 of a. These are thus the left-hand variables in the solution. (The rst row of a is of course the z-row.

) A value i > N indicates that the variable is a yi rather than an xi , xN+j yj . Likewise, izrov[j] contains, for j= 1 . .

. N , the number i whose original variable xi is now a right-hand variable, represented by column j+1 of a. These variables are all zero in the solution.

The meaning of i > N is the same as above, except that i > N + m1 + m2 denotes an arti cial or slack variable which was used only internally and should now be entirely ignored. The ag icase is set to zero if a nite solution is found, +1 if the objective function is unbounded, 1 if no solution satis es the given constraints. The routine treats the case of degenerate feasible vectors, so don t worry about them.

You may also wish to admire the fact that the routine does not require storage for the columns of the tableau (10.8.18) that are to the right of the double line; it keeps track of slack variables by more ef cient bookkeeping.

Please note that, as given, the routine is only semi-sophisticated in its tests for convergence. While the routine properly implements tests for inequality with zero as tests against some small parameter EPS, it does not adjust this parameter to re ect the scale of the input data. This is adequate for many problems, where the input data do not differ from unity by too many orders of magnitude.

If, however, you encounter endless cycling, then you should modify EPS in the routines simplx and simp2. Permuting your variables can also help. Finally, consult [5].

. #include "nrutil.h" #de fine EPS 1.0e-6 Here EPS is the absolute precision, which should be adjusted to the scale of your variables.

#define FREEALL free_ivector(l3,1,m);free_ivector(l1,1,n+1); void simplx(float **a, int m, int n, int m1, int m2, int m3, int *icase, int izrov[], int iposv[]) Simplex method for linear programming. Input parameters a, m, n, mp, np, m1, m2, and m3, and output parameters a, icase, izrov, and iposv are described above. { void simp1(float **a, int mm, int ll[], int nll, int iabf, int *kp,.

10. . Minimization or Maximization of Functions float *bmax); void simp .NET PDF-417 2d barcode 2(float **a, int n, int l2[], int nl2, int *ip, int kp, float *q1); void simp3(float **a, int i1, int k1, int ip, int kp); int i,ip,is,k,kh,kp,nl1; int *l1,*l3; float q1,bmax; if (m != (m1+m2+m3)) nrerror("Bad input constraint counts in simplx"); l1=ivector(1,n+1); l3=ivector(1,m); nl1=n; for (k=1;k<=n;k++) l1[k]=izrov[k]=k; Initialize index list of columns admissible for exchange, and make all variables initially right-hand. for (i=1;i<=m;i++) { if (a[i+1][1] < 0.

0) nrerror("Bad input tableau in simplx"); Constants bi must be nonnegative. iposv[i]=n+i; Initial left-hand variables. m1 type constraints are represented by having their slack variable initially left-hand, with no arti cial variable.

m2 type constraints have their slack variable initially left-hand, with a minus sign, and their arti cial variable handled implicitly during their rst exchange. m3 type constraints have their arti cial variable initially left-hand. } if (m2+m3) { Origin is not a feasible starting sofor (i=1;i<=m2;i++) l3[i]=1; lution: we must do phase one.

Initialize list of m2 constraints whose slack variables have never been exchanged out of the initial basis. for (k=1;k<=(n+1);k++) { Compute the auxiliary objective funcq1=0.0; tion.

for (i=m1+1;i<=m;i++) q1 += a[i+1][k]; a[m+2][k] = -q1; } for (;;) { simp1(a,m+1,l1,nl1,0,&kp,&bmax); Find max. coe . of auxiliary objecif (bmax <= EPS && a[m+2][1] < -EPS) { tive fn.

*icase = -1; Auxiliary objective function is still negative and can t be improved, hence no feasible solution exists. FREEALL return; } else if (bmax <= EPS && a[m+2][1] <= EPS) { Auxiliary objective function is zero and can t be improved; we have a feasible starting vector. Clean out the arti cial variables corresponding to any remaining equality constraints by goto one and then move on to phase two.

for (ip=m1+m2+1;ip<=m;ip++) { if (iposv[ip] == (ip+n)) { Found an arti cial variable for an simp1(a,ip,l1,nl1,1,&kp,&bmax); equality constraint. if (bmax > EPS) Exchange with column correspondgoto one; ing to maximum pivot element } in row. } for (i=m1+1;i<=m1+m2;i++) Change sign of row for any m2 conif (l3[i-m1] == 1) straints still present from the inifor (k=1;k<=n+1;k++) tial basis.

a[i+1][k] = -a[i+1][k]; break; Go to phase two. } simp2(a,m,n,&ip,kp); Locate a pivot element (phase one). if (ip == 0) { Maximum of auxiliary objective func*icase = -1; tion is unbounded, so no feasiFREEALL return; ble solution exists.

} one: simp3(a,m+1,n,ip,kp); Exchange a left- and a right-hand variable (phase one), then update lists..
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