Simplex Method Calculator

What is the Simplex Method?

The Simplex Method is a mathematical algorithm used to tackle linear programming problems. It’s a robust technique for optimising a linear objective function while adhering to a set of linear inequality or equality constraints. The method identifies the optimal solution by iterating through feasible solutions at the vertices of the feasible region until the best value for the objective function is found.

Linear programming problems often crop up in real-world situations like resource allocation, production scheduling, transportation, and finance. The Simplex Method offers a systematic way to solve these problems efficiently.

Features of the Simplex Method Calculator

• Allows users to input a linear objective function (e.g., 3x_1 + 4x_2).
• Supports inequality and equality constraints with options for ≤, =, and ≥.
• Enables users to select between maximisation and minimisation objectives.
• Offers two solution methods: Big M Method and Two-Phase Method.
• Displays step-by-step calculations, including intermediate tableaux and the final tableau.
• Visualises the feasible region and the optimal solution for 2D problems.

How to Use the Simplex Method Calculator

1. Enter the objective function in the provided field (e.g., 3x_1 + 4x_2).
2. Specify whether the problem is a maximisation or minimisation problem by checking or unchecking the "Maximise?" box.
3. Input constraints in the form of linear inequalities or equalities. For example:
   • 2x_1 + x_2 ≤ 100
   • x_1 + 2x_2 = 80
   Use the "+ Add Constraint" button to include additional constraints.
4. Choose the solution method (Big M Method or Two-Phase Method) from the dropdown menu.
5. Click "Calculate" to solve the problem. The results, including the optimal solution, final tableau, and visualisation, will be displayed.
6. If you want to reset the fields and start afresh, click the "Clear" button.

Example Usage

Objective: Maximise \(3x_1 + 4x_2\)

Constraints:

• \(2x_1 + x_2 ≤ 100\)
• \(x_1 + 2x_2 ≤ 80\)
• \(x_1, x_2 ≥ 0\)

Steps:

• Convert the inequalities into equalities by adding slack variables \(s_1\) and \(s_2\).
• Set up the initial simplex tableau with the coefficients of the variables and constraints.
• Iteratively solve the tableau by pivoting until the optimal solution is reached.
• The final solution is displayed along with the maximum value of the objective function.

Result: \(x_1 = 20\), \(x_2 = 30\), and the maximum value is \(180\).

FAQs

• What is linear programming?
  Linear programming is a mathematical method used to find the best possible outcome (like maximum profit or minimum cost) in a given mathematical model where the relationships are linear.
• What are the Big M Method and Two-Phase Method?
  The Big M Method adds artificial variables with large penalties (denoted as \(M\)) to ensure feasibility, while the Two-Phase Method solves the problem in two stages: first finding a feasible solution and then optimising the objective function.
• What does the "maximise" checkbox do?
  Checking this box solves the problem as a maximisation problem. If left unchecked, the calculator assumes a minimisation problem.
• Can the calculator handle non-linear problems?
  No, the calculator is specifically designed for linear programming problems where both the objective function and constraints are linear.
• What happens if the problem is unbounded?
  If the solution is unbounded, the calculator will show a message indicating that the problem does not have a finite optimal solution.

Benefits of Using the Simplex Method Calculator

• Saves time by automating tedious manual calculations.
• Provides a step-by-step breakdown, making it a valuable learning tool for students.
• Visualises feasible regions and solutions for better understanding.
• Handles complex problems efficiently with multiple constraints and variables.