Question

Describe an application of linear programming you might find useful where you work, in your personal life or in a course you are taking. You will receive no credit for responses such as "There is no application where I work or in my personal life." In your post, demonstrate that you understand the goals of linear programming and how it can be applied in "the real world."

Answer 1

Linear programming is an optimization technique using a set of linear equations in order to maximize or minimize the objective function. Linear programming has been applied with great success in a variety of fields like military, agriculture, industry, transportation, health services, economics, even including social sciences and behavioural problems.

Let’s take a real life example.

Suppose I like making chocolates as a hobby and wish sell it. I make two types of chocolate. Let’s name it as A and B. To make each unit of chocolate, the following quantities are required:

Each unit of A requires 1 unit of Milk and 3 units of Cocoa. Each unit of B requires 1 unit of Milk and 2 units of Cocoa.

I have a total of 5 units of Milk and 12 units of Cocoa. By selling each unit of A and B, I want to make a profit of 3 $ and 5 $ respectively.

The first thing I’m going to do is represent the problem in a tabular form for better understanding.

	Milk	Cocoa	Profit per unit
A	1	3	3 $
B	1	2	5 $
Total units available	5	12

Let the total number of units produced by A be = X

Let the total number of units produced by B be = Y

Now, the total profit is represented by Z

The total profit I make is given by the total number of units of A and B produced multiplied by its per-unit profit of 3 $ and 5 $ respectively.

Profit: Max Z = 3X+5Y

Which means we have to maximize Z.

As per the above table, each unit of A and B requires 1 unit of Milk. The total amount of Milk available is 5 units. To represent this mathematically,

X+Y ≤ 5

Also, each unit of A and B requires 3 units & 2 units of Cocoa respectively. The total amount of Cocoa available is 12 units. To represent this mathematically,

3X+2Y ≤ 12

Also, the values for units of A and B can only be integers. So we have two more constraints,          

X ≥ 0 & Y ≥ 0

To make maximum profit, the above inequalities have to be satisfied. This is called formulating a real-world problem into a mathematical model.