MATLAB Implementation of the Simplex Algorithm

The simplex algorithm is a widely-used efficient method for solving linear programming problems. Implementing the simplex algorithm in MATLAB enables efficient solutions to optimization problems, particularly those involving linear constraints and objective functions.

The fundamental approach of the simplex algorithm involves iteratively moving along the vertices of the feasible solution polyhedron to gradually approach the optimal solution. Core algorithmic steps include: initialization, selecting entering variables, selecting leaving variables, and updating the basis. In MATLAB, we can leverage matrix operations to simplify these computational steps, making the algorithm more efficient.

First, we need to initialize the problem parameters, including the objective function coefficient matrix, constraint coefficient matrix, and right-hand side constraint values. We then determine whether to apply the two-phase method or Big-M method by checking if the initial basis is feasible, which addresses artificial variable issues.

During each iteration, the algorithm selects the non-basic variable that yields the greatest improvement in the objective function to enter the basis. The leaving variable is determined using the minimum ratio test to ensure the new solution remains feasible. This process continues until no further improvement in the objective function is possible, indicating an optimal solution has been reached.

MATLAB's implementation advantage lies in its powerful matrix computation capabilities, enabling efficient handling of large-scale linear programming problems. Additionally, MATLAB provides built-in functions like `linprog` that can further simplify the linear programming solution process.

In summary, implementing the simplex algorithm in MATLAB not only facilitates rapid solutions to linear programming problems but also provides a clear computational framework for learning optimization algorithms.