Solving Linear Programming Using the Simplex Method

This article provides comprehensive details on implementing the simplex method for linear programming using MATLAB. We will demonstrate how to configure and execute the program, including key functions like setting up the initial tableau using linprog or custom matrix operations. The implementation covers the fundamental principles of the simplex algorithm, including pivot selection, basis management, and convergence checks through iterative calculations. We'll explore optimization techniques such as two-phase simplex method implementation and Bland's rule to prevent cycling, enhancing computational efficiency. The code structure handles constraint matrix formulation using sparse matrices for large-scale problems and incorporates sensitivity analysis through shadow price calculations. Practical examples will illustrate problem formulation, objective function coefficients input, and interpretation of output variables including optimal solutions and reduced costs. The MATLAB implementation utilizes vectorized operations for faster tableau updates and includes error handling for infeasible or unbounded cases.