What is Objective Function Minimisation?

S6-SA1-0561

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

Objective Function Minimisation is a mathematical process where we try to find the smallest possible value of a specific function. This function, called the 'objective function,' represents something we want to make as low as possible, like cost, error, or time. Our goal is to find the right inputs that lead to this minimum output.

Simple Example

Quick Example

Imagine you are buying vegetables from the local mandi. You need 2 kg of tomatoes, 1 kg of potatoes, and 0.5 kg of onions. Each vendor has slightly different prices. Your objective is to spend the least amount of money. You would compare prices from different vendors and choose the one that offers the minimum total cost for all your vegetables.

Worked Example

Step-by-Step

Let's say a mobile data company wants to minimise the 'waiting time' for customers to load a webpage. They find that waiting time (W) depends on the number of servers (S) they use, and it can be roughly described by the formula: W = S^2 - 10S + 30. They want to find the number of servers (S) that gives the minimum waiting time.

Step 1: Understand the objective function: W = S^2 - 10S + 30. We want to find the value of S that makes W smallest.
---Step 2: We can test different values for S. Let's try S=1: W = (1)^2 - 10(1) + 30 = 1 - 10 + 30 = 21 minutes.
---Step 3: Try S=2: W = (2)^2 - 10(2) + 30 = 4 - 20 + 30 = 14 minutes.
---Step 4: Try S=3: W = (3)^2 - 10(3) + 30 = 9 - 30 + 30 = 9 minutes.
---Step 5: Try S=4: W = (4)^2 - 10(4) + 30 = 16 - 40 + 30 = 6 minutes.
---Step 6: Try S=5: W = (5)^2 - 10(5) + 30 = 25 - 50 + 30 = 5 minutes.
---Step 7: Try S=6: W = (6)^2 - 10(6) + 30 = 36 - 60 + 30 = 6 minutes.
---Step 8: We see that the waiting time starts increasing after S=5. So, the minimum waiting time is 5 minutes, achieved when S=5 servers are used.

Answer: The minimum waiting time is 5 minutes, achieved with 5 servers.

Why It Matters

This concept is super important in fields like AI/ML, where computers learn by minimising errors, or in engineering, to design products that use the least material. Doctors use it to find the minimum dose of medicine needed, and scientists use it to make chemical reactions as efficient as possible. It helps make things better, cheaper, and faster!

Common Mistakes

MISTAKE: Confusing minimisation with maximisation | CORRECTION: Minimisation aims for the smallest possible value, while maximisation aims for the largest. Always clearly identify what you are trying to minimise.

MISTAKE: Only checking a few random values and assuming the smallest found is the absolute minimum | CORRECTION: For simple functions, you might need to test values around the minimum point or use calculus (in higher grades) to find the exact minimum reliably.

MISTAKE: Not understanding what the 'objective function' represents | CORRECTION: Always clarify what the function's output (e.g., cost, error, time) means in the problem context before trying to minimise it.

Practice Questions

Try It Yourself

QUESTION: A small factory makes 'ladoos'. The cost (C) of making 'x' dozen ladoos is given by C = x^2 - 8x + 20. How many dozen ladoos should they make to minimise the cost? | ANSWER: 4 dozen ladoos (Cost = 4)

QUESTION: A student is trying to minimise the 'mistake points' (M) in a coding project, based on the number of lines of code (L). The formula is M = L^2 - 12L + 40. What is the minimum mistake points they can achieve? | ANSWER: 4 mistake points

QUESTION: An auto-rickshaw driver wants to minimise fuel consumption (F) on a 10 km trip. Fuel consumption depends on speed (S in km/h) as F = S^2 - 20S + 120. What speed should he drive at to minimise fuel consumption, and what is that minimum consumption? (Assume S can be any positive integer) | ANSWER: Speed = 10 km/h, Minimum Fuel Consumption = 20 units

MCQ

Quick Quiz

Which of the following is an example of objective function minimisation?

Finding the highest score in a video game

Calculating the average temperature of a city

Designing a bridge to use the least amount of steel

Measuring the total distance covered by a delivery person

The Correct Answer Is:

Option C aims to reduce or minimise the quantity of steel, which is a classic example of objective function minimisation (minimising material cost/usage). Options A, B, and D involve maximisation, averaging, or simple measurement, not minimisation.

Real World Connection

In the Real World

In India, companies like Swiggy or Zomato use objective function minimisation to find the shortest delivery routes for their riders, minimising fuel costs and delivery time. ISRO uses it to plan satellite orbits, ensuring minimum fuel is used to keep satellites in space. Even your local tailor might subconsciously minimise fabric waste when cutting clothes!

Key Vocabulary

Key Terms

OBJECTIVE FUNCTION: The specific mathematical formula or equation whose value we want to minimise or maximise | MINIMISATION: The process of finding the smallest possible value of a function | INPUTS: The values we change or choose to affect the output of the objective function | OPTIMAL SOLUTION: The specific input values that result in the minimum (or maximum) value of the objective function

What's Next

What to Learn Next

Great job understanding minimisation! Next, you can explore 'Objective Function Maximisation,' which is the opposite – finding the largest possible value. Then, you can learn about 'Optimisation Problems' which combine both to find the best possible solution in real-world scenarios.