What is Objective Function Maximisation?

S6-SA1-0560

Grade Level:

Class 10

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

Definition

What is it?

Objective Function Maximisation is about finding the best possible outcome or the highest value for a specific goal. We want to make something as big or as good as it can be, based on certain rules or limits. It's like trying to get the highest score in a game.

Simple Example

Quick Example

Imagine you want to score maximum marks in your Science exam. Your 'objective function' is your exam score. You maximise it by studying hard, understanding concepts, and practicing questions. The 'rules' are the syllabus and exam pattern.

Worked Example

Step-by-Step

Let's say a chaiwala wants to maximise his daily profit. He sells chai for ₹10 per cup. Each cup costs him ₹4 (milk, sugar, tea leaves). He can sell a maximum of 100 cups a day.

Step 1: Identify the objective. The objective is to maximise daily profit.
---Step 2: Define the profit per cup. Profit per cup = Selling Price - Cost Price = ₹10 - ₹4 = ₹6.
---Step 3: Define the objective function. If 'x' is the number of cups sold, then Daily Profit = ₹6 * x.
---Step 4: Identify the constraint. He can sell a maximum of 100 cups, so x <= 100.
---Step 5: To maximise profit, he needs to sell the maximum possible number of cups. So, x = 100.
---Step 6: Calculate the maximum profit. Maximum Profit = ₹6 * 100 = ₹600.

Answer: The chaiwala's maximum daily profit is ₹600.

Why It Matters

This concept helps engineers design faster rockets, doctors find the best medicine dosage, and businesses make more profit. It's crucial in AI to train intelligent systems and in space technology to plan efficient missions. Understanding this can open doors to exciting careers in science and technology.

Common Mistakes

MISTAKE: Confusing maximisation with minimisation (e.g., trying to get the lowest score instead of the highest). | CORRECTION: Always remember maximisation means aiming for the highest or best possible value.

MISTAKE: Ignoring the 'constraints' or limits given in a problem. | CORRECTION: Constraints are crucial. They define the boundaries within which you can maximise your objective. Always consider them.

MISTAKE: Not clearly defining what the 'objective function' actually is. | CORRECTION: Before solving, always write down exactly what you are trying to maximise (e.g., profit, speed, marks).

Practice Questions

Try It Yourself

QUESTION: A farmer wants to maximise the number of mangoes he harvests from his small orchard. If each tree gives 50 mangoes and he has 15 trees, what is the maximum number of mangoes he can harvest? | ANSWER: 750 mangoes (50 mangoes/tree * 15 trees)

QUESTION: A delivery driver earns ₹20 for each successful delivery. He can make a maximum of 30 deliveries in a day. What is the maximum amount of money he can earn in a day? | ANSWER: ₹600 (₹20/delivery * 30 deliveries)

QUESTION: A student is making a project for the science fair. She gets 10 points for creativity and 5 points for presentation. She wants to maximise her total score. If she can only work on either creativity or presentation for a total of 6 hours, and she spends 1 hour on creativity for every 2 hours on presentation, what is her maximum possible score? (Assume she gets full points for time spent). | ANSWER: This question is slightly tricky! If she spends 1 hour on creativity for every 2 hours on presentation, that means for every 3 hours she works, 1 is on creativity and 2 on presentation. In 6 hours, she would spend 2 hours on creativity (2*10 = 20 points) and 4 hours on presentation (4*5 = 20 points). Total score = 20 + 20 = 40 points. (If she spends all 6 hours on creativity she gets 6*10=60. If all 6 hours on presentation she gets 6*5=30. So she would maximise by spending all 6 hours on creativity, giving 60 points, if the 1:2 ratio is not a strict rule but just a typical pattern. If it's a strict rule, then 40 points.) Let's assume she can choose freely. So, 60 points.

MCQ

Quick Quiz

Which of the following best describes the goal of Objective Function Maximisation?

Finding the lowest possible value for a goal

Finding the average value of a goal

Finding the highest possible value for a goal, within given limits

Ignoring all limits to achieve any value

The Correct Answer Is:

Objective Function Maximisation specifically aims to find the greatest or highest value for a defined goal, while always considering any given constraints or limits. Options A and B are incorrect as they describe minimisation or averaging. Option D is incorrect because limits are crucial.

Real World Connection

In the Real World

When you use a navigation app like Google Maps or Ola, it calculates the 'fastest route' (maximising speed/minimising time) or 'shortest route' (minimising distance). Similarly, ISRO scientists use this concept to maximise fuel efficiency for rockets going to Mars, ensuring they get the best possible trajectory with available resources.

Key Vocabulary

Key Terms

OBJECTIVE FUNCTION: The goal or quantity you want to maximise or minimise | CONSTRAINT: A limit or rule that must be followed | OPTIMISATION: The process of finding the best solution (either maximum or minimum) | VARIABLE: A quantity that can change or take different values

What's Next

What to Learn Next

Now that you understand maximisation, explore 'What is Objective Function Minimisation?'. It's the opposite but equally important concept, where you try to achieve the smallest or lowest possible value for a goal. You'll see how both are part of a bigger topic called optimisation!