What is the Objective Function? | Simple Explanation for Class 10

S3-SA5-0145

What is the Objective Function in Optimisation?

Grade Level:

Class 10

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Objective Function in optimisation is the main mathematical equation that you want to either maximise (make as big as possible) or minimise (make as small as possible). It represents the goal of your problem, like maximising profit or minimising cost. The value of this function tells you how good your chosen solution is.

Simple Example

Quick Example

Imagine you are selling samosas and want to earn the most profit. Your profit depends on how many samosas you sell and how much each costs to make. The equation that calculates your total profit (e.g., Profit = (Selling Price - Cost Price) * Number of Samosas) is your Objective Function, because you want to make this profit as high as possible.

Worked Example

Step-by-Step

Let's say a mobile data company wants to minimise the cost of providing 100 GB of data. They have two options for data packs: Pack A costs Rs. 5 per GB and Pack B costs Rs. 4 per GB, but Pack B can only provide a maximum of 60 GB. Let 'x' be the GB from Pack A and 'y' be the GB from Pack B.

Step 1: Identify the goal. The goal is to minimise the total cost.
---Step 2: Define the variables. x = GB from Pack A, y = GB from Pack B.
---Step 3: Write the expression for total cost. Cost = (Cost per GB of A * x) + (Cost per GB of B * y). So, Cost = 5x + 4y.
---Step 4: This expression, 5x + 4y, is our Objective Function because we want to minimise its value.
---Step 5: Identify constraints (rules). We need 100 GB total: x + y = 100. Pack B has a limit: y <= 60. Also, x >= 0 and y >= 0.
---Step 6: To minimise the cost, we should use as much of the cheaper Pack B as possible, up to its limit. So, let y = 60.
---Step 7: Calculate x using the total GB constraint: x + 60 = 100, so x = 40.
---Step 8: Substitute x=40 and y=60 into the Objective Function: Cost = 5(40) + 4(60) = 200 + 240 = 440.

Answer: The minimum cost is Rs. 440, and the Objective Function used to calculate this was Cost = 5x + 4y.

Why It Matters

Understanding the Objective Function is crucial in fields like AI/ML, where it helps train models to make accurate predictions or decisions by minimising errors. Engineers use it to design efficient structures or systems, while economists use it to model optimal resource allocation. It's the core of solving real-world problems to achieve the best possible outcome.

Common Mistakes

MISTAKE: Confusing the Objective Function with constraints (rules). | CORRECTION: The Objective Function is what you want to maximise/minimise. Constraints are the limits or conditions that must be met.

MISTAKE: Not clearly defining whether the goal is to maximise or minimise the function. | CORRECTION: Always state whether you are trying to find the highest (maximise) or lowest (minimise) possible value of the Objective Function.

MISTAKE: Using variables that don't directly impact the goal. | CORRECTION: Ensure all variables in your Objective Function are relevant to the quantity you are trying to optimise (e.g., profit, cost, distance).

Practice Questions

Try It Yourself

QUESTION: A farmer wants to maximise the number of mangoes (M) harvested. The number of mangoes depends on the number of trees (T) and the amount of fertiliser (F). Which of these is most likely the Objective Function? A) T + F = 100, B) M = 5T + 2F, C) T <= 50 | ANSWER: B) M = 5T + 2F

QUESTION: You are building a fence around a rectangular garden and want to minimise the total length of the fence. If the length is 'L' and width is 'W', what would be your Objective Function? | ANSWER: Objective Function: Length of fence = 2L + 2W (You want to minimise this value).

QUESTION: A tiffin service makes two types of meals: Deluxe (D) and Standard (S). Deluxe meals give Rs. 30 profit each, and Standard meals give Rs. 20 profit each. The service wants to maximise total profit. Write the Objective Function for total profit. | ANSWER: Objective Function: Total Profit = 30D + 20S

MCQ

Quick Quiz

Which of the following best describes the role of an Objective Function in optimisation?

It lists all the possible solutions to a problem.

It is the mathematical expression we aim to maximise or minimise.

It sets the limits or conditions for the variables.

It is always a straight line equation.

The Correct Answer Is:

The Objective Function is the core equation that represents the goal you want to achieve, either making it as large (maximise) or as small (minimise) as possible. Options A, C, and D describe other aspects or are incorrect.

Real World Connection

In the Real World

When you use a navigation app like Google Maps or Ola/Uber, it finds the 'best' route for you. The app's underlying optimisation algorithm uses an Objective Function to minimise travel time or distance, considering current traffic and road conditions. This helps delivery services like Swiggy or Zomato plan the fastest routes for their riders.

Key Vocabulary

Key Terms

OPTIMISATION: The process of finding the best possible solution to a problem, either by maximising or minimising something. | MAXIMISE: To make something as large as possible. | MINIMISE: To make something as small as possible. | VARIABLES: Quantities that can change or vary in a mathematical problem. | CONSTRAINTS: The rules or limits that must be followed when solving an optimisation problem.

What's Next

What to Learn Next

Now that you understand the Objective Function, the next step is to learn about 'Constraints in Optimisation'. Constraints are the rules or limits that restrict the values your variables can take, and they work hand-in-hand with the Objective Function to find the best possible solution.