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What is a Feasible Region in Linear Programming?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
A Feasible Region in Linear Programming is like a special area on a graph where all the possible solutions to a problem can be found. It's the region that satisfies all the given conditions (called constraints) at the same time. Any point inside this region is a 'feasible' or possible solution.
Simple Example
Quick Example
Imagine you have to buy pens and pencils for school, but you can spend only Rs 50. Also, you need at least 3 pens and at least 2 pencils. The 'feasible region' would be all the combinations of pens and pencils you can buy that fit all these rules (staying within Rs 50, buying at least 3 pens, and at least 2 pencils).
Worked Example
Step-by-Step
Let's say a bakery makes two types of biscuits: Butter Biscuits (B) and Chocolate Biscuits (C).
Condition 1: They can make a maximum of 100 biscuits in total (B + C <= 100).
Condition 2: They must make at least 20 Butter Biscuits (B >= 20).
Condition 3: They must make at least 30 Chocolate Biscuits (C >= 30).
Step 1: Draw a graph with Butter Biscuits (B) on the x-axis and Chocolate Biscuits (C) on the y-axis.
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Step 2: Draw the line B + C = 100. Since B + C <= 100, the feasible region will be below or to the left of this line.
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Step 3: Draw the line B = 20. Since B >= 20, the feasible region will be to the right of this line.
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Step 4: Draw the line C = 30. Since C >= 30, the feasible region will be above this line.
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Step 5: The area on the graph where all three shaded regions overlap is the Feasible Region. It's usually a polygon (like a triangle or a square with slanted sides).
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Answer: The Feasible Region is the area enclosed by the lines B=20, C=30, and B+C=100, where B and C are non-negative.
Why It Matters
Understanding feasible regions helps people make the best decisions when resources are limited, like a factory deciding how many products to make to maximize profit. This is crucial in fields like AI, Data Science, and Economics, helping engineers and business analysts solve complex problems and optimize outcomes.
Common Mistakes
MISTAKE: Not considering all the given conditions (constraints) when drawing the region. | CORRECTION: Always check that every single condition is represented on your graph and that the shaded region satisfies ALL of them.
MISTAKE: Shading the wrong side of a line (e.g., shading above for '<=' or below for '>='). | CORRECTION: Pick a test point (like (0,0) if it's not on the line) and substitute its coordinates into the inequality. If it satisfies the inequality, shade the side containing the test point; otherwise, shade the opposite side.
MISTAKE: Forgetting that variables like 'number of items' cannot be negative. | CORRECTION: Always remember that in most real-world problems, quantities cannot be negative, so the feasible region is usually restricted to the first quadrant (where x >= 0 and y >= 0).
Practice Questions
Try It Yourself
QUESTION: A farmer has 10 acres of land. He wants to plant wheat (W) and rice (R). He must plant at least 2 acres of wheat and at least 3 acres of rice. Write down the inequalities that define the feasible region for planting. | ANSWER: W + R <= 10, W >= 2, R >= 3
QUESTION: If a shopkeeper sells 'x' mobile covers and 'y' screen protectors. He must sell at least 5 mobile covers (x >= 5) and at least 10 screen protectors (y >= 10). The total items sold cannot exceed 30 (x + y <= 30). Is the point (6 mobile covers, 12 screen protectors) within the feasible region? | ANSWER: Yes. (6 >= 5 is true, 12 >= 10 is true, and 6 + 12 = 18 <= 30 is true).
QUESTION: Draw the feasible region for the following conditions: x >= 1, y >= 2, x + y <= 6. What kind of shape is the feasible region? | ANSWER: The feasible region is a triangle with vertices at (1,2), (1,5), and (4,2).
MCQ
Quick Quiz
Which of the following describes a Feasible Region?
The area where no solutions are possible.
The area that satisfies only one condition.
The area that satisfies all given conditions simultaneously.
The area where the problem has no answer.
The Correct Answer Is:
C
A feasible region is defined as the set of all points that satisfy every single constraint (condition) of a linear programming problem at the same time. Options A, B, and D are incorrect because they describe areas that either don't satisfy all conditions or have no solutions.
Real World Connection
In the Real World
Delivery companies like Zepto or Dunzo use feasible regions to plan the most efficient routes for their riders, considering factors like delivery time, fuel cost, and rider availability. This helps them deliver items quickly and save money. Even in ISRO, scientists use similar concepts to optimize rocket fuel usage and trajectory.
Key Vocabulary
Key Terms
LINEAR PROGRAMMING: A method to find the best outcome (like maximum profit or minimum cost) in a mathematical model | CONSTRAINT: A condition or restriction that must be satisfied by the variables in a problem | OPTIMIZATION: The process of finding the best possible solution to a problem | INEQUALITY: A mathematical statement showing that two expressions are not equal (e.g., x < 5 or y >= 10) | VERTEX: A corner point of the feasible region, where two or more boundary lines intersect.
What's Next
What to Learn Next
Now that you understand feasible regions, your next step is to learn about the 'Objective Function' and 'Optimal Solution'. You'll see how to use the feasible region to find the very best solution for a problem, like finding the maximum profit or minimum cost!
