The Simplex Method is a popular algorithm for solving Linear Programming (LP) problems, particularly those that involve maximizing or minimizing a linear objective function subject to a set of linear constraints. When dealing with a Standard Maximization Problem, the objective is to maximize a linear function subject to a set of linear inequalities (constraints) and non-negativity restrictions on the variables. Here are the typical steps involved in using the Simplex Method for such a problem:
1. Convert to Standard Form
First, ensure that the LP problem is in standard form:
- The objective function should be in maximization form.
- All constraints should be equalities (use slack variables to convert ≤ constraints into equalities, and surplus and artificial variables for ≥ constraints).
- All variables are non-negative.
2. Construct the Initial Simplex Tableau
Create the initial simplex tableau, which is a tabular representation of the objective function and constraints. The tableau includes the coefficients of the variables in the objective function and constraints, the right-hand side (RHS) values of the constraints, and the slack, surplus, and artificial variables introduced in step 1.
3. Identify the Entering Variable
Look at the bottom row (objective function row) excluding the RHS. The variable with the most negative coefficient in this row is selected as the entering variable (for maximization problems). This indicates which variable will enter the basis (set of variables used to express the solution).
4. Identify the Leaving Variable
For the column of the entering variable, calculate the ratio of the RHS to the coefficient of the entering variable in each constraint row (only consider positive coefficients). The variable corresponding to the smallest positive ratio is the leaving variable (it will leave the basis). This step ensures that all variables remain non-negative.
Perform row operations to make the entering variable have a coefficient of 1 in its row and 0 in all other rows. This process is called pivoting and involves using elementary row operations to modify the tableau.
6. Repeat Until Optimal
Repeat steps 3 to 5 until there are no more negative coefficients in the bottom row (objective function row), indicating that the current solution is