Linear Programming (LP) is a powerful tool in Operations Research (OR) used to solve optimization problems. It helps in finding the best possible outcome (like maximizing profit or minimizing cost) under given constraints. This guide breaks down the formulation of LP problems into simple steps with relatable examples and clear illustrations.
What is Linear Programming?
Linear Programming involves:
- Objective Function: The goal you want to achieve (maximize or minimize).
- Decision Variables: The quantities you control to achieve the objective.
- Constraints: The restrictions or limitations on the decision variables.
- Non-Negativity Restrictions: The decision variables cannot be negative (in most real-world scenarios).
Steps to Formulate an LP Problem
Here’s a step-by-step approach to formulating LP problems:
Step 1: Understand the Problem
Identify the key components:
- What is the goal (maximize or minimize)?
- What decisions need to be made?
- What are the restrictions?
Step 2: Define the Decision Variables
Choose variables that represent the quantities you want to determine. Use symbols like for clarity.
Step 3: Write the Objective Function
Express the goal as a mathematical equation using the decision variables.
Step 4: Formulate the Constraints
Translate each restriction into a mathematical inequality or equation.
Step 5: Add Non-Negativity Restrictions
State that the decision variables must be ≥ 0.
Example: A Candy Factory Problem
Scenario
A candy factory produces two types of candies: Type A and Type B. Each type requires sugar and chocolate in varying amounts. The factory’s goal is to maximize profit while staying within the available resources.
Data
- Profit per candy:
- Type A: $5
- Type B: $3
- Resource requirements per candy:
- Sugar (kg): Type A: 2, Type B: 1
- Chocolate (kg): Type A: 3, Type B: 2
- Available resources:
- Sugar: 100 kg
- Chocolate: 120 kg
Formulation
- Decision Variables:
- Let = number of Type A candies produced.
- Let = number of Type B candies produced.
- Objective Function: Maximize profit:
- Constraints:
- Sugar constraint:
- Chocolate constraint:
- Non-Negativity Restrictions:
Illustration: Graphical Representation
- Plot the Constraints: Draw the lines for each constraint on a graph where the x-axis represents (Type A) and the y-axis represents (Type B).
- For :
- If , .
- If , .
- For :
- If , .
- If , .
- For :
- Shade the Feasible Region: The feasible region is where all constraints overlap, and it represents all possible solutions.
- Evaluate the Objective Function: Identify the corner points of the feasible region and calculate for each. The maximum gives the optimal solution.
Tips for Formulating LP Problems
- Clearly define your decision variables; they should represent what you control.
- Write the objective function and constraints step by step to avoid confusion.
- Ensure all units are consistent (e.g., kilograms, hours, dollars).
- Use a graphical approach for two-variable problems to visualize the solution.
- For larger problems, consider using software like Excel Solver, MATLAB, or Python’s PuLP library.
Common Applications of LP
- Manufacturing: Optimizing production schedules, resource allocation.
- Transportation: Minimizing shipping costs.
- Diet Planning: Designing low-cost, nutritious meal plans.
- Finance: Portfolio optimization.
By breaking the problem into manageable steps and using simple examples, anyone can learn to formulate LP problems. Start with small scenarios, practice regularly, and soon you’ll be an LP pro!
Photo by Christina Morillo: https://www.pexels.com/photo/two-women-looking-at-the-code-at-laptop-1181263/