In this article, we will explore linear programming through the lens of the Furniture Factory Problem. By formulating this problem as a linear programming model, we can maximize total revenue while respecting resource constraints. So let's dive into the details and learn how linear programming can be applied to solve real-world problems.
The Furniture Factory Problem
Imagine a furniture factory that produces chairs and tables. The goal is to develop a production plan that maximizes the total revenue while considering the available resources. The following information is provided:
Selling price of a chair: \$45
Selling price of a table: \$80
Available resources:
Mahogany: 400 units
Labor: 450 hours
The data scientist estimates the resource requirements for each product:
One chair requires 5 units of mahogany and 10 hours of labor.
One table requires 20 units of mahogany and 15 hours of labor.
Decision Variables
To create a production plan, we need to determine the number of chairs (x1) and the number of tables (x2) to produce. These decision variables represent the quantities we can control to maximize revenue. For this problem, x1 and x2 must be non-negative values (greater than or equal to 0).
Objective Function
The objective is to maximize the total revenue generated by the production of chairs and tables. The revenue from chairs can be calculated as 45x1, where 45 represents the selling price per chair. Similarly, the revenue from tables is 80x2, where 80 represents the selling price per table. Therefore, the objective function is:
Maximize: Revenue = 45x1 + 80x2
Resource Constraints
To ensure the production plan does not exceed the available resources, we impose constraints on mahogany and labor capacity.
Mahogany constraint
The total amount of mahogany consumed by chairs and tables combined should not exceed the available 400 units. Considering 5x1 as the mahogany consumption for chairs and 20x2 as the consumption for tables, the constraint can be expressed as:5x1 + 20x2 ≤ 400
Labor constraint
The total labor consumed by chairs and tables combined should not exceed the available 450 hours. With 10x1 representing the labor consumption for chairs and 15x2 for tables, the constraint can be expressed as:10x1 + 15x2 ≤ 450
Non-Negative Constraint
To maintain the feasibility of the production plan, the decision variables x1 and x2 must be non-negative:
x1, x2 ≥ 0
Final Linear Programming Formulation
Taking into account the objective function, resource constraints, and non-negative constraint, the Furniture Factory Problem can be formulated as a linear programming problem:
Maximize: Revenue = 45x1 + 80x2
Subject to:
5x1 + 20x2 ≤ 400 (Mahogany constraint)
10x1 + 15x2 ≤ 450 (Labor constraint)
x1, x2 ≥ 0 (Non-negative constraint)
Conclusion
Linear programming offers a systematic approach to solving optimization problems, such as the Furniture Factory Problem. By formulating the problem as a linear programming model, we can determine the optimal production plan that maximizes revenue while considering resource limitations. In this article, we introduced the concept of linear programming, explained the Furniture Factory Problem, and provided a step-by-step guide on formulating the problem mathematically. With this understanding, you can explore advanced techniques and algorithms to solve more complex real-world problems using linear programming.
Resources
Download the linear programming tutorial slides.
View the entire series:
Welcome: Linear Programming Tutorial
Chapter 1: Mathematical Programming
Chapter 2: Introduction to Linear Programming
Chapter 3: Mixed Integer Linear Programming Problems
Chapter 4: Furniture Factory Problem
Chapter 5: Simplex Method
Chapter 6: Modeling and Solving Linear Programming Problems
Chapter 7: Sensitivity Analysis of Linear Programming Problems
Chapter 8: Multiple Optimal Solutions
Chapter 9: Unbounded Linear Programming Problems
Chapter 10: Infeasible Linear Programming Problems
Chapter 11: Linear Programming Overview - Further Considerations
Chapter 12: Duality in Linear Programming
Chapter 13: Optimality Conditions
Chapter 14: Dual Simplex Method
