This project is a Python-based tool designed to solve linear programming problems using two approaches:
- Simplex Method: A numerical algorithm to find the optimal solution to linear programming problems.
- Graphical Method: A visual representation of the problem, highlighting the feasible region, constraints, and optimal solution.
- Dual Calculation:
- Solves problems using the Simplex method for exact numerical results.
- Provides graphical representation to visualize the feasible region and optimal solution.
- Dynamic Visualization:
- Highlights the feasible region clearly.
- Displays constraints with clear labels.
- Marks the optimal solution explicitly with annotations.
- Plots the optimal objective function line dynamically.
- Comparison:
- Outputs results from both the Simplex and Graphical methods for verification and understanding.
The constraints represent the problem's limitations and are defined as inequalities or equalities involving variables (x_1, x_2, \ldots). Each constraint specifies:
- Coefficients: The weights of the variables in the equation.
- Operator: The type of relationship ((\leq, \geq, =)).
- Value: The constant on the right-hand side of the equation.
Example:
(6x_1 + 4x_2 \leq 24)
The objective function is the formula to be optimized (maximized or minimized), typically represented as:
[ z = c_1x_1 + c_2x_2 + \ldots ]
- Coefficients: Represent the weights or contributions of each variable to the function.
- Objective: The target to either maximize or minimize.
Example:
Maximize (z = 5x_1 + 4x_2)
The graphical method includes:
- Constraints:
- Plotted as lines with distinct colors for easy differentiation.
- Labeled directly on the graph for clarity.
- Feasible Region:
- Highlighted with a transparent fill to clearly show the valid solution space.
- Optimal Solution:
- Marked with a bold red dot.
- Annotated with the coordinates and objective value.
- Objective Function:
- Displayed as a dashed line passing through the optimal solution.
- Annotated with its value.
The Simplex method outputs:
- Optimal Solution: Values of the variables ((x_1, x_2, \ldots)) that optimize the objective function.
- Objective Value: The optimized value of (z).
The graphical method displays:
- A clear visualization of constraints.
- The feasible region and its boundaries.
- The optimal solution and the corresponding objective function.
This tool is ideal for:
- Teaching and learning linear programming concepts.
- Visualizing and solving small-scale optimization problems.
- Comparing numerical and graphical approaches for understanding and verification.
This project is licensed under the MIT License. Use it freely for educational and professional purposes.
Special thanks to the creators and maintainers of matplotlib and numpy for providing the tools used to develop this solver.