#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <string.h>
#include <assert.h>
#define M 20
#define N 20
static const double epsilon = 1.0e-8;
int equal(double a, double b) { return fabs(a-b) < epsilon; }
typedef struct {
int m, n; // m=rows, n=columns, mat[m x n]
double mat[M][N];
} Tableau;
void nl(int k){ int j; for(j=0;j<k;j++) putchar('-'); putchar('\n'); }
void print_tableau(Tableau *tab, const char* mes) {
static int counter=0;
int i, j;
printf("\n%d. Tableau %s:\n", ++counter, mes);
nl(70);
printf("%-6s%5s", "col:", "b[i]");
for(j=1;j<tab->n; j++) { printf(" x%d,", j); } printf("\n");
for(i=0;i<tab->m; i++) {
if (i==0) printf("max:"); else
printf("b%d: ", i);
for(j=0;j<tab->n; j++) {
if (equal((int)tab->mat[i][j], tab->mat[i][j]))
printf(" %6d", (int)tab->mat[i][j]);
else
printf(" %6.2lf", tab->mat[i][j]);
}
printf("\n");
}
nl(70);
}
/* Example input file for read_tableau:
4 5
0 -0.5 -3 -1 -4
40 1 1 1 1
10 -2 -1 1 1
10 0 1 0 -1
*/
void read_tableau(Tableau *tab, const char * filename) {
int err, i, j;
FILE * fp;
fp = fopen(filename, "r" );
if( !fp ) {
printf("Cannot read %s\n", filename); exit(1);
}
memset(tab, 0, sizeof(*tab));
err = fscanf(fp, "%d %d", &tab->m, &tab->n);
if (err == 0 || err == EOF) {
printf("Cannot read m or n\n"); exit(1);
}
for(i=0;i<tab->m; i++) {
for(j=0;j<tab->n; j++) {
err = fscanf(fp, "%lf", &tab->mat[i][j]);
if (err == 0 || err == EOF) {
printf("Cannot read A[%d][%d]\n", i, j); exit(1);
}
}
}
printf("Read tableau [%d rows x %d columns] from file '%s'.\n",
tab->m, tab->n, filename);
fclose(fp);
}
void pivot_on(Tableau *tab, int row, int col) {
int i, j;
double pivot;
pivot = tab->mat[row][col];
assert(pivot>0);
for(j=0;j<tab->n;j++)
tab->mat[row][j] /= pivot;
assert( equal(tab->mat[row][col], 1. ));
for(i=0; i<tab->m; i++) { // foreach remaining row i do
double multiplier = tab->mat[i][col];
if(i==row) continue;
for(j=0; j<tab->n; j++) { // r[i] = r[i] - z * r[row];
tab->mat[i][j] -= multiplier * tab->mat[row][j];
}
}
}
// Find pivot_col = most negative column in mat[0][1..n]
int find_pivot_column(Tableau *tab) {
int j, pivot_col = 1;
double lowest = tab->mat[0][pivot_col];
for(j=1; j<tab->n; j++) {
if (tab->mat[0][j] < lowest) {
lowest = tab->mat[0][j];
pivot_col = j;
}
}
printf("Most negative column in row[0] is col %d = %g.\n", pivot_col, lowest);
if( lowest >= 0 ) {
return -1; // All positive columns in row[0], this is optimal.
}
return pivot_col;
}
// Find the pivot_row, with smallest positive ratio = col[0] / col[pivot]
int find_pivot_row(Tableau *tab, int pivot_col) {
int i, pivot_row = 0;
double min_ratio = -1;
printf("Ratios A[row_i,0]/A[row_i,%d] = [",pivot_col);
for(i=1;i<tab->m;i++){
double ratio = tab->mat[i][0] / tab->mat[i][pivot_col];
printf("%3.2lf, ", ratio);
if ( (ratio > 0 && ratio < min_ratio ) || min_ratio < 0 ) {
min_ratio = ratio;
pivot_row = i;
}
}
printf("].\n");
if (min_ratio == -1)
return -1; // Unbounded.
printf("Found pivot A[%d,%d], min positive ratio=%g in row=%d.\n",
pivot_row, pivot_col, min_ratio, pivot_row);
return pivot_row;
}
void add_slack_variables(Tableau *tab) {
int i, j;
for(i=1; i<tab->m; i++) {
for(j=1; j<tab->m; j++)
tab->mat[i][j + tab->n -1] = (i==j);
}
tab->n += tab->m -1;
}
void check_b_positive(Tableau *tab) {
int i;
for(i=1; i<tab->m; i++)
assert(tab->mat[i][0] >= 0);
}
// Given a column of identity matrix, find the row containing 1.
// return -1, if the column as not from an identity matrix.
int find_basis_variable(Tableau *tab, int col) {
int i, xi=-1;
for(i=1; i < tab->m; i++) {
if (equal( tab->mat[i][col],1) ) {
if (xi == -1)
xi=i; // found first '1', save this row number.
else
return -1; // found second '1', not an identity matrix.
} else if (!equal( tab->mat[i][col],0) ) {
return -1; // not an identity matrix column.
}
}
return xi;
}
void print_optimal_vector(Tableau *tab, char *message) {
int j, xi;
printf("%s at ", message);
for(j=1;j<tab->n;j++) { // for each column.
xi = find_basis_variable(tab, j);
if (xi != -1)
printf("x%d=%3.2lf, ", j, tab->mat[xi][0] );
else
printf("x%d=0, ", j);
}
printf("\n");
}
void simplex(Tableau *tab) {
int loop=0;
add_slack_variables(tab);
check_b_positive(tab);
print_tableau(tab,"Padded with slack variables");
while( ++loop ) {
int pivot_col, pivot_row;
pivot_col = find_pivot_column(tab);
if( pivot_col < 0 ) {
printf("Found optimal value=A[0,0]=%3.2lf (no negatives in row 0).\n",
tab->mat[0][0]);
print_optimal_vector(tab, "Optimal vector");
break;
}
printf("Entering variable x%d to be made basic, so pivot_col=%d.\n",
pivot_col, pivot_col);
pivot_row = find_pivot_row(tab, pivot_col);
if (pivot_row < 0) {
printf("unbounded (no pivot_row).\n");
break;
}
printf("Leaving variable x%d, so pivot_row=%d\n", pivot_row, pivot_row);
pivot_on(tab, pivot_row, pivot_col);
print_tableau(tab,"After pivoting");
print_optimal_vector(tab, "Basic feasible solution");
if(loop > 20) {
printf("Too many iterations > %d.\n", loop);
break;
}
}
}
Tableau tab = { 4, 5, { // Size of tableau [4 rows x 5 columns ]
{ 0.0 , -0.5 , -3.0 ,-1.0 , -4.0, }, // Max: z = 0.5*x + 3*y + z + 4*w,
{ 40.0 , 1.0 , 1.0 , 1.0 , 1.0, }, // x + y + z + w <= 40 .. b1
{ 10.0 , -2.0 , -1.0 , 1.0 , 1.0, }, // -2x - y + z + w <= 10 .. b2
{ 10.0 , 0.0 , 1.0 , 0.0 , -1.0, }, // y - w <= 10 .. b3
}
};
int main(int argc, char *argv[]){
if (argc > 1) { // usage: cmd datafile
read_tableau(&tab, argv[1]);
}
print_tableau(&tab,"Initial");
simplex(&tab);
return 0;
}
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <string.h>
#include <assert.h>

#define M 20
#define N 20

static const double epsilon   = 1.0e-8;
int equal(double a, double b) { return fabs(a-b) < epsilon; }

typedef struct {
  int m, n; // m=rows, n=columns, mat[m x n]
  double mat[M][N];
} Tableau;

void nl(int k){ int j; for(j=0;j<k;j++) putchar('-'); putchar('\n'); }

void print_tableau(Tableau *tab, const char* mes) {
  static int counter=0;
  int i, j;
  printf("\n%d. Tableau %s:\n", ++counter, mes);
  nl(70);

  printf("%-6s%5s", "col:", "b[i]");
  for(j=1;j<tab->n; j++) { printf("    x%d,", j); } printf("\n");

  for(i=0;i<tab->m; i++) {
    if (i==0) printf("max:"); else
    printf("b%d: ", i);
    for(j=0;j<tab->n; j++) {
      if (equal((int)tab->mat[i][j], tab->mat[i][j]))
        printf(" %6d", (int)tab->mat[i][j]);
      else
        printf(" %6.2lf", tab->mat[i][j]);
      }
    printf("\n");
  }
  nl(70);
}

/* Example input file for read_tableau:
     4 5
      0   -0.5  -3 -1  -4 
     40    1     1  1   1 
     10   -2    -1  1   1 
     10    0     1  0  -1  
*/
void read_tableau(Tableau *tab, const char * filename) {
  int err, i, j;
  FILE * fp;

  fp  = fopen(filename, "r" );
  if( !fp ) {
    printf("Cannot read %s\n", filename); exit(1);
  }
  memset(tab, 0, sizeof(*tab));
  err = fscanf(fp, "%d %d", &tab->m, &tab->n);
  if (err == 0 || err == EOF) {
    printf("Cannot read m or n\n"); exit(1);
  }
  for(i=0;i<tab->m; i++) {
    for(j=0;j<tab->n; j++) {
      err = fscanf(fp, "%lf", &tab->mat[i][j]);
      if (err == 0 || err == EOF) {
        printf("Cannot read A[%d][%d]\n", i, j); exit(1);
      }
    }
  }
  printf("Read tableau [%d rows x %d columns] from file '%s'.\n",
    tab->m, tab->n, filename);
  fclose(fp);
}

void pivot_on(Tableau *tab, int row, int col) {
  int i, j;
  double pivot;

  pivot = tab->mat[row][col];
  assert(pivot>0);
  for(j=0;j<tab->n;j++)
    tab->mat[row][j] /= pivot;
  assert( equal(tab->mat[row][col], 1. ));

  for(i=0; i<tab->m; i++) { // foreach remaining row i do
    double multiplier = tab->mat[i][col];
    if(i==row) continue;
    for(j=0; j<tab->n; j++) { // r[i] = r[i] - z * r[row];
      tab->mat[i][j] -= multiplier * tab->mat[row][j];
    }
  }
}

// Find pivot_col = most negative column in mat[0][1..n]
int find_pivot_column(Tableau *tab) {
  int j, pivot_col = 1;
  double lowest = tab->mat[0][pivot_col];
  for(j=1; j<tab->n; j++) {
    if (tab->mat[0][j] < lowest) {
      lowest = tab->mat[0][j];
      pivot_col = j;
    }
  }
  printf("Most negative column in row[0] is col %d = %g.\n", pivot_col, lowest);
  if( lowest >= 0 ) {
    return -1; // All positive columns in row[0], this is optimal.
  }
  return pivot_col;
}

// Find the pivot_row, with smallest positive ratio = col[0] / col[pivot]
int find_pivot_row(Tableau *tab, int pivot_col) {
  int i, pivot_row = 0;
  double min_ratio = -1;
  printf("Ratios A[row_i,0]/A[row_i,%d] = [",pivot_col);
  for(i=1;i<tab->m;i++){
    double ratio = tab->mat[i][0] / tab->mat[i][pivot_col];
    printf("%3.2lf, ", ratio);
    if ( (ratio > 0  && ratio < min_ratio ) || min_ratio < 0 ) {
      min_ratio = ratio;
      pivot_row = i;
    }
  }
  printf("].\n");
  if (min_ratio == -1)
    return -1; // Unbounded.
  printf("Found pivot A[%d,%d], min positive ratio=%g in row=%d.\n",
      pivot_row, pivot_col, min_ratio, pivot_row);
  return pivot_row;
}

void add_slack_variables(Tableau *tab) {
  int i, j;
  for(i=1; i<tab->m; i++) {
    for(j=1; j<tab->m; j++)
      tab->mat[i][j + tab->n -1] = (i==j);
  }
  tab->n += tab->m -1;
}

void check_b_positive(Tableau *tab) {
  int i;
  for(i=1; i<tab->m; i++)
    assert(tab->mat[i][0] >= 0);
}

// Given a column of identity matrix, find the row containing 1.
// return -1, if the column as not from an identity matrix.
int find_basis_variable(Tableau *tab, int col) {
  int i, xi=-1;
  for(i=1; i < tab->m; i++) {
    if (equal( tab->mat[i][col],1) ) {
      if (xi == -1)
        xi=i;   // found first '1', save this row number.
      else
        return -1; // found second '1', not an identity matrix.

    } else if (!equal( tab->mat[i][col],0) ) {
      return -1; // not an identity matrix column.
    }
  }
  return xi;
}

void print_optimal_vector(Tableau *tab, char *message) {
  int j, xi;
  printf("%s at ", message);
  for(j=1;j<tab->n;j++) { // for each column.
    xi = find_basis_variable(tab, j);
    if (xi != -1)
      printf("x%d=%3.2lf, ", j, tab->mat[xi][0] );
    else
      printf("x%d=0, ", j);
  }
  printf("\n");
}

void simplex(Tableau *tab) {
  int loop=0;
  add_slack_variables(tab);
  check_b_positive(tab);
  print_tableau(tab,"Padded with slack variables");
  while( ++loop ) {
    int pivot_col, pivot_row;

    pivot_col = find_pivot_column(tab);
    if( pivot_col < 0 ) {
      printf("Found optimal value=A[0,0]=%3.2lf (no negatives in row 0).\n",
        tab->mat[0][0]);
      print_optimal_vector(tab, "Optimal vector");
      break;
    }
    printf("Entering variable x%d to be made basic, so pivot_col=%d.\n",
      pivot_col, pivot_col);

    pivot_row = find_pivot_row(tab, pivot_col);
    if (pivot_row < 0) {
      printf("unbounded (no pivot_row).\n");
      break;
    }
    printf("Leaving variable x%d, so pivot_row=%d\n", pivot_row, pivot_row);

    pivot_on(tab, pivot_row, pivot_col);
    print_tableau(tab,"After pivoting");
    print_optimal_vector(tab, "Basic feasible solution");

    if(loop > 20) {
      printf("Too many iterations > %d.\n", loop);
      break;
    }
  }
}

Tableau tab  = { 4, 5, {                     // Size of tableau [4 rows x 5 columns ]
    {  0.0 , -0.5 , -3.0 ,-1.0 , -4.0,   },  // Max: z = 0.5*x + 3*y + z + 4*w,
    { 40.0 ,  1.0 ,  1.0 , 1.0 ,  1.0,   },  //    x + y + z + w <= 40 .. b1
    { 10.0 , -2.0 , -1.0 , 1.0 ,  1.0,   },  //  -2x - y + z + w <= 10 .. b2
    { 10.0 ,  0.0 ,  1.0 , 0.0 , -1.0,   },  //        y     - w <= 10 .. b3
  }
};

int main(int argc, char *argv[]){
  if (argc > 1) { // usage: cmd datafile
    read_tableau(&tab, argv[1]);
  }
  print_tableau(&tab,"Initial");
  simplex(&tab);
  return 0;
}
1. Tableau Initial:
----------------------------------------------------------------------
col: b[i] x1, x2, x3, x4,
max: 0 -0.50 -3 -1 -4
b1: 40 1 1 1 1
b2: 10 -2 -1 1 1
b3: 10 0 1 0 -1
----------------------------------------------------------------------
2. Tableau Padded with slack variables:
----------------------------------------------------------------------
col: b[i] x1, x2, x3, x4, x5, x6, x7,
max: 0 -0.50 -3 -1 -4 0 0 0
b1: 40 1 1 1 1 1 0 0
b2: 10 -2 -1 1 1 0 1 0
b3: 10 0 1 0 -1 0 0 1
----------------------------------------------------------------------
Most negative column in row[0] is col 4 = -4.
Entering variable x4 to be made basic, so pivot_col=4.
Ratios A[row_i,0]/A[row_i,4] = [40.00, 10.00, -10.00, ].
Found pivot A[2,4], min positive ratio=10 in row=2.
Leaving variable x2, so pivot_row=2
3. Tableau After pivoting:
----------------------------------------------------------------------
col: b[i] x1, x2, x3, x4, x5, x6, x7,
max: 40 -8.50 -7 3 0 0 4 0
b1: 30 3 2 0 0 1 -1 0
b2: 10 -2 -1 1 1 0 1 0
b3: 20 -2 0 1 0 0 1 1
----------------------------------------------------------------------
Basic feasible solution at x1=0, x2=0, x3=0, x4=10.00, x5=30.00, x6=0, x7=20.00,
Most negative column in row[0] is col 1 = -8.5.
Entering variable x1 to be made basic, so pivot_col=1.
Ratios A[row_i,0]/A[row_i,1] = [10.00, -5.00, -10.00, ].
Found pivot A[1,1], min positive ratio=10 in row=1.
Leaving variable x1, so pivot_row=1
4. Tableau After pivoting:
----------------------------------------------------------------------
col: b[i] x1, x2, x3, x4, x5, x6, x7,
max: 125 0 -1.33 3 0 2.83 1.17 0
b1: 10 1 0.67 0 0 0.33 -0.33 0
b2: 30 0 0.33 1 1 0.67 0.33 0
b3: 40 0 1.33 1 0 0.67 0.33 1
----------------------------------------------------------------------
Basic feasible solution at x1=10.00, x2=0, x3=0, x4=30.00, x5=0, x6=0, x7=40.00,
Most negative column in row[0] is col 2 = -1.33333.
Entering variable x2 to be made basic, so pivot_col=2.
Ratios A[row_i,0]/A[row_i,2] = [15.00, 90.00, 30.00, ].
Found pivot A[1,2], min positive ratio=15 in row=1.
Leaving variable x1, so pivot_row=1
5. Tableau After pivoting:
----------------------------------------------------------------------
col: b[i] x1, x2, x3, x4, x5, x6, x7,
max: 145 2 0 3 0 3.50 0.50 0
b1: 15 1.50 1 0 0 0.50 -0.50 0
b2: 25 -0.50 0 1 1 0.50 0.50 0
b3: 20 -2 0 1 0 0 1 1
----------------------------------------------------------------------
Basic feasible solution at x1=0, x2=15.00, x3=0, x4=25.00, x5=0, x6=0, x7=20.00,
Most negative column in row[0] is col 2 = 0.
Found optimal value=A[0,0]=145.00 (no negatives in row 0).
Optimal vector at x1=0, x2=15.00, x3=0, x4=25.00, x5=0, x6=0, x7=20.00,