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What is the Optimization of Cost Problems?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
Optimization of Cost Problems is about finding the best possible way to do something so that the total cost is as low as possible. It means making smart choices to save money or resources while still achieving your goal.
Simple Example
Quick Example
Imagine you want to buy new school supplies but have a limited budget. You need to choose pens, notebooks, and a geometry box from different shops, each with different prices and combo offers. Optimizing cost here means finding the combination of items from various shops that gives you all you need for the lowest total price.
Worked Example
Step-by-Step
Problem: A small chai shop owner wants to buy tea powder and milk. Tea powder costs ₹150 per kg and milk costs ₹60 per litre. They need at least 5 kg of tea powder and 10 litres of milk. Their total budget is ₹1500. How much tea powder and milk can they buy to maximize their stock within budget?
Step 1: Understand the costs and limits. Tea powder (T) = ₹150/kg, Milk (M) = ₹60/litre. Minimum T = 5kg, Minimum M = 10L. Budget = ₹1500.
---Step 2: Calculate the cost of minimum required items. Cost for 5kg tea = 5 * ₹150 = ₹750. Cost for 10L milk = 10 * ₹60 = ₹600. Total minimum cost = ₹750 + ₹600 = ₹1350.
---Step 3: Calculate remaining budget. Remaining budget = ₹1500 - ₹1350 = ₹150.
---Step 4: Decide how to use the remaining budget. The owner can buy more tea powder or more milk with ₹150. Since tea powder is more expensive per unit, they can buy less of it. If they buy more milk, they can get ₹150 / ₹60 = 2.5 litres of milk. If they buy more tea powder, they can get ₹150 / ₹150 = 1 kg of tea powder.
---Step 5: To maximize stock, they should buy more of the item that gives more quantity for the remaining money. 2.5 litres of milk is more quantity than 1 kg of tea powder.
---Step 6: Final calculation. Total tea powder = 5 kg. Total milk = 10 L + 2.5 L = 12.5 L.
Answer: The owner can buy 5 kg of tea powder and 12.5 litres of milk to maximize their stock within the budget.
Why It Matters
Understanding cost optimization is super important in many fields. Engineers use it to design bridges or cars that are strong but not too expensive. Businesses use it to decide how much to produce and sell to make a profit. Even in space technology, ISRO scientists optimize fuel use to send rockets to Mars efficiently.
Common Mistakes
MISTAKE: Only looking at one cost factor (e.g., just the price per item) | CORRECTION: Always consider all relevant costs, like transport, discounts, and minimum purchase requirements, to get the true total cost.
MISTAKE: Ignoring constraints or limits (e.g., budget, minimum quantity needed) | CORRECTION: Always list all given constraints clearly and ensure your solution fits within them.
MISTAKE: Confusing cost minimization with quantity maximization | CORRECTION: Cost optimization is about achieving a goal (like getting enough items) for the *lowest cost*, not necessarily buying the most items if that increases the cost beyond the goal.
Practice Questions
Try It Yourself
QUESTION: A tiffin service needs 20 kg of rice. Shop A sells rice at ₹50/kg with a delivery charge of ₹100. Shop B sells rice at ₹55/kg with free delivery. Which shop should they choose to minimize cost? | ANSWER: Shop A. Cost from Shop A = (20 * 50) + 100 = 1000 + 100 = ₹1100. Cost from Shop B = (20 * 55) + 0 = ₹1100. Both are same, so either works, but if delivery was slightly less or rice slightly cheaper, A would be better.
QUESTION: A farmer wants to fence a rectangular field that needs to be 100 square meters in area. Fencing costs ₹20 per meter. What dimensions (length and width) should the field have to use the least amount of fencing? (Hint: A square uses the least perimeter for a given area) | ANSWER: Length = 10 meters, Width = 10 meters. Area = 10 * 10 = 100 sq m. Perimeter = 2 * (10 + 10) = 40 meters. Total cost = 40 * ₹20 = ₹800. Any other rectangle with area 100 (e.g., 20x5) will have a larger perimeter.
QUESTION: You are planning a school trip for 50 students. Bus rental costs ₹5000 per bus. Food costs ₹150 per student. If you need at least 2 buses and have a maximum budget of ₹15,000, how many buses can you rent and what is the maximum food budget per student you can afford? | ANSWER: You can rent 2 buses. Cost for 2 buses = 2 * ₹5000 = ₹10,000. Remaining budget for food = ₹15,000 - ₹10,000 = ₹5,000. Food budget per student = ₹5,000 / 50 students = ₹100 per student. So, 2 buses and ₹100 per student for food.
MCQ
Quick Quiz
What is the main goal of optimizing a cost problem?
To spend the most money possible
To find the cheapest way to achieve a goal
To ignore all expenses
To make the problem more complicated
The Correct Answer Is:
B
The core idea of cost optimization is to find the most economical or cheapest method to reach a specific objective or goal. Options A, C, and D contradict this fundamental principle.
Real World Connection
In the Real World
In India, companies like Flipkart and Amazon constantly optimize their delivery routes and warehouse locations to reduce shipping costs and deliver products faster. Swiggy and Zomato use complex algorithms to figure out which delivery partner should pick up which order to minimize fuel usage and delivery time, which ultimately saves them money and keeps food prices reasonable for us.
Key Vocabulary
Key Terms
OPTIMIZATION: Finding the best or most effective solution to a problem | COST: The amount of money or resources needed to achieve something | CONSTRAINT: A limit or restriction that must be followed | BUDGET: The total amount of money available | EFFICIENCY: Achieving maximum productivity with minimum wasted effort or expense
What's Next
What to Learn Next
Next, you can explore 'Linear Programming.' It's a powerful mathematical tool that helps solve more complex optimization problems with many variables and constraints, building directly on what you've learned here.
