What is Feasible Region? | Simple Explanation for Class 10

S3-SA5-0144

What is the Feasible Region in Optimisation?

Grade Level:

Class 10

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

Definition

What is it?

The Feasible Region in Optimisation is the set of all possible solutions that satisfy all the given conditions (constraints) of a problem. Think of it as the 'safe zone' where all your rules are followed. Any point within this region is a valid solution, even if it's not the best one.

Simple Example

Quick Example

Imagine you want to buy samosas and jalebis. You have Rs. 100. Each samosa costs Rs. 10 and each jalebi costs Rs. 20. Also, you want to buy at least 2 samosas. The 'feasible region' would be all the combinations of samosas and jalebis you can buy without spending more than Rs. 100 and buying at least 2 samosas.

Worked Example

Step-by-Step

Let's say you're planning a small party and need to buy chips and cold drinks.

Conditions:
1. You have a budget of Rs. 60.
2. A packet of chips costs Rs. 10.
3. A bottle of cold drink costs Rs. 15.
4. You must buy at least 1 packet of chips.
5. You must buy at least 1 bottle of cold drink.

Let 'x' be the number of chips packets and 'y' be the number of cold drink bottles.

---Step 1: Write down the constraints as inequalities.
Budget: 10x + 15y <= 60
Chips minimum: x >= 1
Cold drink minimum: y >= 1
Also, since you can't buy negative items: x >= 0, y >= 0 (though x>=1 and y>=1 already cover this).

---Step 2: List possible combinations (x, y) that satisfy x >= 1 and y >= 1 and are whole numbers (you can't buy half a chip packet).
Possible (x, y) combinations: (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1), (4,2), (5,1)

---Step 3: Check each combination against the budget constraint (10x + 15y <= 60).
(1,1): 10(1) + 15(1) = 25 <= 60 (YES)
(1,2): 10(1) + 15(2) = 40 <= 60 (YES)
(1,3): 10(1) + 15(3) = 55 <= 60 (YES)
(2,1): 10(2) + 15(1) = 35 <= 60 (YES)
(2,2): 10(2) + 15(2) = 50 <= 60 (YES)
(2,3): 10(2) + 15(3) = 65 (NO - too expensive)
(3,1): 10(3) + 15(1) = 45 <= 60 (YES)
(3,2): 10(3) + 15(2) = 60 <= 60 (YES)
(4,1): 10(4) + 15(1) = 55 <= 60 (YES)
(4,2): 10(4) + 15(2) = 70 (NO - too expensive)
(5,1): 10(5) + 15(1) = 65 (NO - too expensive)

---Step 4: Identify the feasible region.
The feasible region consists of all the (x, y) points that satisfy all conditions. In this case, the integer points are: (1,1), (1,2), (1,3), (2,1), (2,2), (3,1), (3,2), (4,1).

Why It Matters

Understanding the feasible region helps scientists and engineers find the best way to do things, like making the fastest computer chip or designing the most fuel-efficient rocket. It's crucial in fields like AI/ML to train models efficiently and in economics to make smart business decisions, helping people in careers from data scientists to logistics managers.

Common Mistakes

MISTAKE: Including points that violate one or more constraints in the feasible region. | CORRECTION: Every single constraint must be satisfied for a point to be in the feasible region. Check all conditions carefully for each potential solution.

MISTAKE: Confusing the feasible region with the optimal solution. | CORRECTION: The feasible region is the *set of all possible solutions*. The *optimal solution* is just one specific point within that region that gives the best (maximum or minimum) value for your goal.

MISTAKE: Forgetting that variables like 'number of items' must be non-negative (x >= 0, y >= 0). | CORRECTION: Always remember to include non-negativity constraints for real-world problems unless specified otherwise, as you cannot have negative quantities.

Practice Questions

Try It Yourself

QUESTION: You are making a rangoli design. You have red powder (R) and blue powder (B). You can use at most 10 spoons of powder in total. You must use at least 2 spoons of red powder. Which of these combinations is NOT in the feasible region: (3R, 5B), (1R, 8B), (7R, 2B)? | ANSWER: (1R, 8B) is not in the feasible region because it violates the condition 'at least 2 spoons of red powder'.

QUESTION: A mobile game gives you coins for completing tasks. Task A gives 5 coins, Task B gives 10 coins. You need to complete at least 2 Task A and at least 1 Task B. You can complete a maximum of 6 tasks in total. List all feasible combinations of (Task A, Task B) completions. | ANSWER: Feasible combinations: (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (4,1), (4,2), (5,1)

QUESTION: A small factory produces two types of toys, cars (C) and dolls (D). Each car takes 2 hours to make, and each doll takes 3 hours. The factory has 12 hours available per day. They must produce at least 1 car and at least 1 doll. Also, the number of dolls produced cannot be more than twice the number of cars. Which of these points are feasible: (2C, 3D), (3C, 2D), (4C, 1D)? | ANSWER: (2C, 3D) is feasible (2*2 + 3*3 = 13 > 12, so no, this is not feasible). (3C, 2D) is feasible (2*3 + 3*2 = 12 <= 12; 2 <= 2*3, so yes). (4C, 1D) is feasible (2*4 + 3*1 = 11 <= 12; 1 <= 2*4, so yes). So, (3C, 2D) and (4C, 1D) are feasible.

MCQ

Quick Quiz

Which statement best describes the Feasible Region?

It is the single best solution to an optimisation problem.

It is the area where all conditions of a problem are met.

It is the area where no conditions are met.

It is always a straight line on a graph.

The Correct Answer Is:

The feasible region includes all points that satisfy every constraint or condition of the problem. Option A describes the optimal solution, not the feasible region. Options C and D are incorrect.

Real World Connection

In the Real World

Think about how delivery apps like Zomato or Swiggy decide which delivery partner gets which order. They use optimisation! The 'feasible region' for a delivery partner would be all the orders they can pick up and deliver on time, given their current location, traffic, and vehicle type. The app then finds the 'optimal' (best) order for them to maximise efficiency.

Key Vocabulary

Key Terms

OPTIMISATION: The process of finding the best possible solution to a problem given a set of constraints. | CONSTRAINTS: The conditions or limitations that must be satisfied in a problem. | SOLUTION: A set of values for the variables that satisfy all the constraints. | INEQUALITY: A mathematical statement showing that two expressions are not equal (e.g., x < 5 or y >= 2).

What's Next

What to Learn Next

Now that you understand the feasible region, the next step is to learn about finding the 'Optimal Solution'. This involves using the feasible region to find the single best answer that either maximises profit or minimises cost, which is super useful!