Samarthya

What is Linear Programming?

Linear programming is a mathematical technique used to find the best possible outcome (maximum profit or minimum cost) in a situation where the relationships between the factors involved are linear. It’s like finding the sweet spot where you get the most out of what you have, given certain limitations.

Example: A Furniture Company

Let’s say a furniture company makes chairs and tables. Each chair takes 2 hours to assemble and 1 hour to finish. Each table takes 3 hours to assemble and 2 hours to finish. The company has a total of 40 hours available for assembly and 30 hours for finishing each week. If each chair sells for $50 and each table sells for $75, how many chairs and tables should the company make each week to maximize its revenue?

Components of Linear Programming

1. Decision Variables: These are the things you’re trying to decide. In our example:
   • x = number of chairs to make
   • y = number of tables to make
2. Objective Function: This is what you’re trying to maximize or minimize. Here, it’s the revenue:
   • Revenue = 50x + 75y (We want to make this as big as possible)
3. Constraints: These are the limitations you have. In our case:
   • Assembly time: 2x + 3y ≤ 40 (Can’t use more than 40 hours of assembly time)
   • Finishing time: 1x + 2y ≤ 30 (Can’t use more than 30 hours of finishing time)
   • Non-negativity: x ≥ 0, y ≥ 0 (Can’t make a negative number of chairs or tables)