Common Types of Objective Functions
There are two types of objective functions.
- Maximization Objective Function
- Minimization Objective Function
Let’s discuss these two types in detail as follows:
Maximization Objective Function
In this type, we usually aim to maximize the objective function. The vertices that are found after graphing the constraints have a tendency to generate the maximum value of the objective function. Let us illustrate with the help of an example
Example: A man invests at most 8 hrs of time in making wallets and school bags. He invests 2 hrs in making wallets and 4 hr in school bags. He targets to make at most 5 wallets and school bags and wants to sell them and generate a profit of Rs 20 on a wallet and Rs 100 on a school bag. Find the objective function.
Solution:
Let x be the number of rotis and y be the number of bread.
A man can invest a maximum of 8 hours by investing 2 hours on making a wallet and 4 hour on making a school bag. Therefore the first constraint equation is
2x + 4y ⩽ 8
⇒ x + 2y ⩽ 4
The maximum number he can make is 5
x+y ⩽ 5
Let the objective function be denoted by Z
Therefore Z = 20x + 100y
Minimization Objective Function
In this type, we usually aim to minimize the objective function. The vertices that are found after graphing the constraints have a tendency to generate the minimum value of the objective function. Let us illustrate with the help of an example
Example: Given the sum of the two variables is at least 20. It is given one variable is greater than equal to 9. Derive the objective function if the cost of one variable is 2 units and the cost of another variable is 9 units.
Solution:
Let x and y be the two variables. It is given sum of the two variables should be at least 20.
x+y ⩾ 20
and x ⩾ 9
Above two inequalities are constraints for the following objective function.
Let the objective function be denoted by Z. Therefore Z is
Z = 2x + 9y
Objective Function
Objective Function is the objective of the Linear Programming Problem as the name suggests. In linear programming or linear optimization, we use various techniques and methods to find the optimal solution to the linear problem with some constraints. The technique can also include inequality constraints as well. The objective function in Linear Programming is to optimize to find the optimum solution for a given problem.
In this article, we will learn all about the Objective Function including its definition, types, how to formulate an objective function for any given problem, etc. We will also learn various representations of Objective Functions such as Linear Objective Functions or Non-linear objective functions. So, let’s start learning about this fundamental concept in Linear Programming i.e., “Objective Function”.
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