1. Meaning of Linear Programming
Linear programming is a method used to make the best possible decision when there are limited resources. The idea is to choose values of certain variables so that a goal is achieved, like getting the maximum profit or the minimum cost.
The word linear means all equations or inequalities involved are straight-line relations. The word programming means planning or arranging something in the best way.
1.1. Why Linear Programming is Needed
In many real situations, resources such as time, money, materials, and labour are limited. Linear programming helps decide how to use these limited resources wisely.
- Choosing how many products to make when resources are limited.
- Deciding how to transport goods at the lowest cost.
- Allocating time among different tasks efficiently.
2. Key Elements of a Linear Programming Problem
Every linear programming problem has a few main parts. Understanding these makes it easier to form the mathematical model.
2.1. Decision Variables
These are the unknown values we want to determine. They usually represent quantities to be produced, used, or allocated.
For example, \( x \) and \( y \) may represent the number of two different products to make.
2.2. Objective Function
This is the expression we want to maximize or minimize. It could represent profit, cost, time, or any goal.
Example: maximize profit \( P = 5x + 7y \).
2.3. Constraints
Constraints are the conditions or limitations within which the decision must be made. They are usually inequalities like
\( 2x + y \le 10 \)
They express limits on resources such as materials, labour, or capacity.
2.4. Non-Negativity Conditions
Since we cannot produce negative quantities, we always include:
\( x \ge 0, \; y \ge 0 \)
3. Examples of Linear Programming Situations
Linear programming appears naturally in many simple day-to-day situations. A few examples make the idea very clear.
3.1. Resource Allocation Example
Suppose a workshop has limited hours for two different machines. The goal is to decide how many units of two products to manufacture so that the profit is maximum while staying within the machine time limits.
3.2. Cost Minimization Example
Choosing how to transport goods from warehouses to stores in such a way that the total transportation cost is as low as possible.
3.3. Diet or Mixture Example
Finding the combination of ingredients that meets nutrition requirements at the minimum cost.