1. Summary of Simplex Procedure

1. Modify the constraints

 

 

Add a slack variable for each less-than-or-equal constraint. Add an artificial variable for each equality constraint. Add both an artificial and a surplus variable for each greater-than-or-equal constraint.

For each artificial variable, assign a very large cost (negative profit) in the objective function.

2. Identify the initial solution as composed of the slack and artificial variables.
3. Check for optimality

 

 

The current solution is optimal if: For maximisation, all coefficients in the, row are zero or negative. For minimisation, all coefficients in  row are zero or positive. If optimum has been reached, stop simplex procedure.

4. Entering variable

 

 

This is the variable associated with the largest positive (for maximisation) or largest negative (for minimisation) coefficient in the  row.

5. Leaving variable

 

 

For each row, calculate the ratio of the values in the "Solution Values" column divided by the coefficients in the entering variable column. Ignore any ratios that are negative. The leaving variable is the one associated with the row having the smallest ratio.

6. Re-solve the equation

1. Identify the pivot element as the coefficient in the entering variable column and leaving variable row.
2. Divide all the coefficients in the leaving variable row by the pivot element.
3. Modify the other rows, possibly including the objective function row, as:

1. Calculate the Z_j

 

 

For the j-th column: for each row, multiply the substitution coefficients by the c_j value for that row and sum. The total is Z_j. Repeat for all columns.

2. Calculate the 

9. Return to step 3