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.
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:
The data scientist estimates the resource requirements for each product:
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).
The objective is to maximize the total revenue generated by the production of chairs and tables. The revenue from chairs can be calculated as 45×1, where 45 represents the selling price per chair. Similarly, the revenue from tables is 80×2, where 80 represents the selling price per table. Therefore, the objective function is:
Maximize: Revenue = 45×1 + 80×2
To ensure the production plan does not exceed the available resources, we impose constraints on mahogany and labor capacity.
To maintain the feasibility of the production plan, the decision variables x1 and x2 must be non-negative:
x1, x2 ≥ 0
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 = 45×1 + 80×2
Subject to:
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.
Download the linear programming tutorial slides.
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