Linear Programming, Lecture 1. Introduction, simple models, graphic solution

Introduction

This tutorial provides a comprehensive introduction to linear programming, focusing on simple models and graphic solutions. Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model. It is widely used in fields such as economics, business, engineering, and military applications. By following this guide, you will gain an understanding of the basics of linear programming and how to visualize solutions graphically.

Step 1: Understanding Linear Programming Concepts

• Definition: Linear programming is a technique for optimizing a linear objective function, subject to linear equality and inequality constraints.
• Key Components:
  • Objective Function: This is the function you want to maximize or minimize. For example, maximizing profit or minimizing cost.
  • Constraints: These are the restrictions or limitations on the variables. They can be equations or inequalities.
  • Decision Variables: These are the variables that will be adjusted to achieve the best outcome.

Practical Tip

Ensure that the objective function and constraints are linear. Non-linear relationships do not fit into the linear programming framework.

Step 2: Formulating a Linear Programming Problem

1. Identify the Objective: Clearly define what you want to maximize or minimize.
2. Determine the Decision Variables: Choose the variables that will influence your objective.
3. Set Up Constraints: Establish the limits on your decision variables based on real-world restrictions.
   • Example: If you are managing a budget, your constraints may include total spending limits.

Common Pitfall

Avoid vague definitions of your objective or constraints. Be specific to ensure clarity in your model.

Step 3: Graphical Representation of Solutions

• Coordinate System: Use a Cartesian coordinate system to plot the constraints.
• Plotting Constraints:
  • Convert each constraint into an equation.
  • Find the intercepts and plot them on the graph.
• Shading Feasible Region: Identify the area where all constraints overlap. This region represents all possible solutions.

Practical Tip

Use different colors to shade the feasible region clearly, making it easier to visualize and analyze.

Step 4: Identifying the Optimal Solution

• Corner Point Method: The optimal solution for a linear programming problem lies at one of the vertices (corner points) of the feasible region.
• Evaluating Corner Points: Calculate the value of the objective function at each corner point.
• Select Optimal Value: The corner point that produces the highest (or lowest) value for the objective function is your solution.

Real-World Application

This method is applicable when planning resource allocation in businesses, such as maximizing production output or minimizing costs.

Conclusion

In this tutorial, we've covered the basic principles of linear programming, including formulating problems, graphical representation, and identifying optimal solutions. To deepen your understanding, consider practicing with real-world scenarios and more complex models. You can explore software tools and applications that specialize in linear programming for advanced analysis and solutions.