Linear Programming: Simplex Method Solver

📌 Project Overview

This project features a Python-based implementation of the Simplex Algorithm, a popular method for solving linear programming problems (LPP). The solver is designed to find the optimal solution (maximization) for objective functions subject to a set of linear inequality constraints.

🧮 Mathematical Approach

The solver processes problems in their standard form:

• Objective Function: Maximize $Z = c_1x_1 + c_2x_2 + ... + c_nx_n$
• Constraints: $Ax \le b$, where $x \ge 0$

The algorithm utilizes:

1. Slack Variables: To convert inequalities into equalities.
2. Initial Tableau: Constructing the starting matrix for the iterative process.
3. Pivoting: Systematically selecting entering and leaving variables based on the optimality condition and the ratio test.
4. Optimality Check: Iterating until no further improvements to the objective function can be made.

🚀 Key Features

• Step-by-Step Execution: The Jupyter Notebook format allows for a clear, cell-by-cell walkthrough of the optimization process.
• Tableau Visualization: Displays the transformation of the simplex tableau at each iteration.
• Optimization Logic: Handles the identification of pivot rows and columns to find the basic feasible solution.
• Python-Powered: Utilizes numpy for efficient matrix operations and linear algebra.

📂 File Structure

• Simplex.ipynb – The primary Jupyter Notebook containing the algorithm implementation, logic, and example problems.

🛠 Setup and Usage

1. Requirements: Ensure you have Python and Jupyter installed (via Anaconda or pip).
2. Dependencies:

   pip install numpy
   pip install matplotlib