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What is the Objective Function Maximisation?
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 value for a specific goal. Imagine you have a target, like getting the highest marks in an exam; this concept helps you figure out the best way to achieve that target.
Simple Example
Quick Example
Think about a chai shop owner. Their objective is to maximise their daily profit. They need to decide how many cups of masala chai and how many cups of ginger chai to sell to make the most money, considering the cost of ingredients and how much they can sell.
Worked Example
Step-by-Step
Let's say a student wants to maximise their total marks in two subjects: Math and Science. They know studying Math for 1 hour gives them 10 marks, and studying Science for 1 hour gives them 8 marks. They have a total of 5 hours to study.
---Step 1: Define the objective function. Let M be hours for Math, S be hours for Science. Total Marks (Objective) = 10*M + 8*S.
---Step 2: Define the constraints. Total study time is 5 hours, so M + S <= 5. Also, M >= 0 and S >= 0 (cannot study negative hours).
---Step 3: Consider different combinations of study hours within the 5-hour limit.
---Step 4: If M=5, S=0: Total Marks = 10*5 + 8*0 = 50.
---Step 5: If M=0, S=5: Total Marks = 10*0 + 8*5 = 40.
---Step 6: If M=3, S=2: Total Marks = 10*3 + 8*2 = 30 + 16 = 46.
---Step 7: If M=4, S=1: Total Marks = 10*4 + 8*1 = 40 + 8 = 48.
---Step 8: Comparing the total marks, studying Math for 5 hours and Science for 0 hours gives the highest marks.
Answer: The maximum total marks are 50, achieved by studying Math for 5 hours.
Why It Matters
This concept is super important in fields like AI/ML, where algorithms learn to make the best decisions, and in engineering, to design the most efficient structures. Doctors use it to find the best treatment plans, and ISRO scientists use it to plan rocket trajectories for maximum fuel efficiency.
Common Mistakes
MISTAKE: Only looking at one possible solution and assuming it's the best. | CORRECTION: Always explore several valid options or combinations to ensure you find the absolute highest value.
MISTAKE: Forgetting about the 'constraints' or limits given in the problem. | CORRECTION: Always list down all conditions and limits (like total time, budget, available resources) before trying to find the maximum.
MISTAKE: Confusing maximisation with minimisation. | CORRECTION: Remember, maximisation means aiming for the highest possible value (like maximum profit), while minimisation means aiming for the lowest (like minimum cost).
Practice Questions
Try It Yourself
QUESTION: A baker makes two types of cookies: Chocolate Chip (gives 5 rupees profit) and Butter (gives 3 rupees profit). They can make a maximum of 10 cookies in total. How many of each should they make to maximise profit? | ANSWER: 10 Chocolate Chip cookies, 0 Butter cookies (Profit = 50 rupees)
QUESTION: A student has 6 hours to prepare for a debate and a presentation. Each hour of debate prep gives 7 points, and each hour of presentation prep gives 5 points. To maximise total points, how should they divide their time? | ANSWER: 6 hours for debate prep, 0 hours for presentation prep (Total points = 42)
QUESTION: A farmer wants to plant tomatoes and potatoes. Tomatoes give 1200 rupees profit per acre, and potatoes give 900 rupees profit per acre. They have 5 acres of land. Due to water availability, they can plant a maximum of 3 acres of tomatoes and a maximum of 4 acres of potatoes. How many acres of each should they plant to maximise profit? | ANSWER: 3 acres of tomatoes, 2 acres of potatoes (Profit = 1200*3 + 900*2 = 3600 + 1800 = 5400 rupees)
MCQ
Quick Quiz
Which of the following is an example of an objective function that needs to be maximised?
Minimising the cost of a product
Finding the shortest route to a destination
Achieving the highest possible score in a game
Reducing the amount of waste produced
The Correct Answer Is:
C
Maximisation means aiming for the highest possible value. 'Achieving the highest possible score in a game' directly reflects this goal. The other options are examples of minimisation.
Real World Connection
In the Real World
In India, companies like Swiggy or Zomato use objective function maximisation to figure out the best delivery routes for their riders, aiming to deliver food as fast as possible to maximise customer satisfaction. Even the Indian Premier League (IPL) teams use similar ideas to pick players for their squad to maximise their chances of winning.
Key Vocabulary
Key Terms
OBJECTIVE FUNCTION: The mathematical expression of the goal you want to maximise or minimise. | CONSTRAINTS: The limits or restrictions within which you must operate. | OPTIMISATION: The process of finding the best possible solution (either maximum or minimum). | VARIABLE: A quantity that can change or vary in a problem.
What's Next
What to Learn Next
Next, you can explore 'Objective Function Minimisation'. It's the flip side of this concept, where instead of finding the highest value, you aim for the lowest, like minimising costs or time. Understanding both will give you a complete picture of optimisation!
