Solved Problems on Objective Function

Problem 1: A person wants some belts and wallets. He has total savings of Rs 6000 and wishes to spend all his savings on purchasing belts and wallets so that he can sell it later. The value of the wallet is Rs 20 and the value of the belt is Rs 10. He wants to store them in a cupboard and the maximum capacity of the cupboard is 50 units. He expects a profit of Rs 2 on the belt and Rs 3 on the wallet. Find the constraints and the resulting objective function.

Solution:

Let the x be the number of wallets to be purchased and y be the number of belts to be purchased. It is to be noted whenever maximum is mentioned in the problem we should use ‘⩽’ to find the constraints

The maximum investment is Rs 6000. The first constraint equation is

20x+10y⩽6000

The max storage capacity of the cupboard is 50

x+y⩽50

Here profit function is basically the objective function. Let this be denoted by P. Therefore the profit function is

P = 3x + 2y

Problem 2: Identify the constraint equations and the objective function from the given set

• 2x + 3y ⩾ 50
• x + y ⩽ 50
• 5x + 4y ⩽ 40
• Z = 7x + 8y

Where x and y are greater than 0.

Solution:

The constraints can be inequality or inequality format. But an objective function has always an equality symbol

Therefore the constraint equations are

2x + 3y ⩾ 50

x + y ⩽ 50

5x + 4y ⩽ 40

The objective equation is Z = 7x + 8y

Problem 3: A woman invests at most 7 hrs of time in making rotis and bread. She invests 2 hrs on rotis and 4 hr on bread. She targets to make at most 20 bread and rotis and wants to sell them and generate a profit of Rs 2 on roti and Rs 1 on bread. Find the objective function.

Solution:

Let x be the number of rotis and y be the number of bread.

A woman can invest a maximum of 7 hours by investing 2 hours on making a roti and 4 hour on making a bread. Therefore the first constraint equation is

2x + 4y ⩽ 7

The maximum number of bread and rotis she can make is 20

x + y ⩽ 20

Let the objective function be denoted by Z

Therefore Z = 2x + y.

Problem 4: The company wants to manufacture Product A and Product B. Product A requires 4 units of cocoa powder and 1 unit of milk powder Product B requires 3 units of cocoa powder and 2 units of milk powder. There are 87 units of cocoa powder available and 45 units of milk powder available. The profit to be earned on each product is $3 and $5 respectively. Find the objective function.

Solution:

Let x denote the number of Product A and y denote the number of items of type B.

The maximum quantity of cocoa powder is 87 units. So the first constraint equation is

4x + 3y ⩽ 87

The maximum amount of milk powder available is 45 units. So the second constraint equation is

x + 2y ⩽ 45

Here our aim is to maximize the profit. So our profit function is the Objective function. Let it be denoted by Z

Z = 3x + 5y

Problem 5: Two types of food packets A and B are to be generated which comprise of vitamins. There are at least 45 units of food packet A to be made available and the manufacture of both food packets should be at least 30. Generate the objective function to be generated where food packet A has 6 units of vitamins and food packet B has 8 units.

Solution:

Let x be the number of food packets A and y be the number of food packets B

At least 45 food packets are to be made available. Therefore the first constraint equation is

x ⩾ 45

The second constraint equation is

x + y ⩾ 30

The objective function is as follows:

Z = 6x + 8y

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”.