How to Solve the Simplex Method Step by Step (With Examples)
Master the Simplex method for linear programming. Learn how to set up the initial tableau, choose pivot variables, and iterate towards the optimal solution.
Introduction to the Simplex Method
The Simplex method is a systematic procedure for solving linear programming problems. Developed by George Dantzig in 1947, it remains one of the most widely used algorithms for optimization.
Step 1: Standard Form
Before applying the Simplex method, you must convert the linear programming problem into standard form. This involves:
- Ensuring the objective is a maximization problem.
- Converting all inequalities into equalities by introducing slack variables (for ≤) or surplus variables (for ≥).
- Ensuring all variables are non-negative.
Step 2: The Initial Tableau
Once in standard form, construct the initial Simplex tableau. This matrix represents the system of equations. The bottom row typically represents the objective function (Z-row).
Step 3: Choosing the Pivot Element
Iteration is the core of the Simplex method:
- Entering Variable: Select the most negative value in the objective row. This identifies the pivot column.
- Leaving Variable: Perform the minimum ratio test (divide the RHS by the positive values in the pivot column). The smallest non-negative ratio determines the pivot row.
- The intersection is your pivot element.
Step 4: Row Operations
Use elementary row operations to make the pivot element 1 and all other elements in the pivot column 0. This gives you the next iteration's tableau.
Conclusion
Repeat steps 3 and 4 until there are no negative values in the objective row. At this point, you have reached the optimal solution!