What is the Corner Point Method?

S6-SA1-0210

Grade Level:

Class 10

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

Definition

What is it?

The Corner Point Method is a technique used in Linear Programming to find the maximum or minimum value of an objective function. It involves evaluating the objective function at the 'corner points' or vertices of the feasible region, which is the area defined by the problem's constraints. The optimal solution (maximum or minimum) will always occur at one of these corner points.

Simple Example

Quick Example

Imagine you run a small chai stall and want to earn maximum profit. You have limits on how much milk and sugar you can buy (your constraints). The Corner Point Method helps you figure out the exact number of cups of 'kadak chai' and 'elaichi chai' you should sell to make the most money, by checking profit at different combinations of sales that fit your limits.

Worked Example

Step-by-Step

Let's say a factory makes two types of toys, A and B. Profit for toy A is Rs 7 and for toy B is Rs 5. They have two machines, M1 and M2. M1 can work for a maximum of 10 hours, and M2 for 12 hours. Toy A needs 1 hour on M1 and 2 hours on M2. Toy B needs 2 hours on M1 and 1 hour on M2. Find the number of toys of each type to maximize profit.

Let x be the number of toy A and y be the number of toy B.
Objective function (Profit): Z = 7x + 5y
Constraints:
M1: x + 2y <= 10
M2: 2x + y <= 12
x >= 0, y >= 0

1. Plot the inequalities and find the feasible region. The corner points are where the constraint lines intersect.
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2. Find the corner points:
a) (0,0) - Intersection of x=0 and y=0
b) (6,0) - Intersection of 2x + y = 12 and y=0 (If y=0, 2x=12, x=6)
c) (0,5) - Intersection of x + 2y = 10 and x=0 (If x=0, 2y=10, y=5)
d) Intersection of x + 2y = 10 and 2x + y = 12
Multiply first equation by 2: 2x + 4y = 20
Subtract second equation: (2x + 4y) - (2x + y) = 20 - 12 => 3y = 8 => y = 8/3
Substitute y = 8/3 into x + 2y = 10 => x + 2(8/3) = 10 => x + 16/3 = 10 => x = 10 - 16/3 = (30 - 16)/3 = 14/3
So, the point is (14/3, 8/3)
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3. Evaluate the objective function Z = 7x + 5y at each corner point:
a) At (0,0): Z = 7(0) + 5(0) = 0
b) At (6,0): Z = 7(6) + 5(0) = 42
c) At (0,5): Z = 7(0) + 5(5) = 25
d) At (14/3, 8/3): Z = 7(14/3) + 5(8/3) = 98/3 + 40/3 = 138/3 = 46
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4. Compare the Z values. The maximum value is 46.

Answer: The maximum profit is Rs 46, achieved by making 14/3 (approx 4.67) toys of type A and 8/3 (approx 2.67) toys of type B. (Note: In real-world problems, toy numbers would be integers, requiring integer programming, but for Corner Point Method, fractional values are accepted.)

Why It Matters

This method is crucial for making optimal decisions in various fields. Engineers use it to design efficient systems, while logistics companies use it to plan delivery routes for Zepto or Swiggy to save fuel and time. Even in medicine, it can help optimize resource allocation in hospitals, impacting careers in operations research, data science, and supply chain management.

Common Mistakes

MISTAKE: Not correctly identifying all corner points of the feasible region. | CORRECTION: Carefully plot all constraints and find the intersection points of the lines that form the boundaries of the feasible region. Don't forget (0,0) if it's part of the region.

MISTAKE: Evaluating the objective function at points that are not corner points or are outside the feasible region. | CORRECTION: Only evaluate the objective function at the vertices (corner points) of the feasible region. Any other point will not give the optimal solution.

MISTAKE: Making calculation errors when solving simultaneous equations to find intersection points. | CORRECTION: Double-check your algebra when solving for the coordinates of the intersection points. A small error here will lead to incorrect corner points and thus a wrong answer.

Practice Questions

Try It Yourself

QUESTION: For an objective function Z = 3x + 4y, find the maximum value if the corner points of the feasible region are (0,0), (5,0), (0,3), and (2,2). | ANSWER: Maximum Z = 14 (at point (2,2))

QUESTION: A baker makes two types of cookies, C1 and C2. Profit for C1 is Rs 10 and for C2 is Rs 8. Constraints are x + y <= 8 (total cookies) and 2x + y <= 10 (oven time). Find the maximum profit using corner points. | ANSWER: Maximum profit is Rs 70 (at point (2,6))

QUESTION: A farmer has 10 acres of land. He can grow wheat or rice. Wheat needs 1 day per acre and rice needs 2 days per acre. He has 16 days of labor available. Profit for wheat is Rs 5000 per acre and for rice is Rs 8000 per acre. What is the maximum profit? (Hint: Let x be acres of wheat, y be acres of rice.) | ANSWER: Maximum profit is Rs 64,000 (by growing 4 acres of wheat and 6 acres of rice)

MCQ

Quick Quiz

Which of the following is NOT a step in the Corner Point Method?

Graphing the feasible region

Finding the coordinates of the corner points

Evaluating the objective function at the center of the feasible region

Identifying the maximum or minimum value among the evaluated points

The Correct Answer Is:

The Corner Point Method specifically requires evaluating the objective function at the 'corner points' or vertices of the feasible region, not its center. The optimal solution is guaranteed to be at a vertex.

Real World Connection

In the Real World

Logistics companies like Delhivery or Ecom Express use principles similar to the Corner Point Method to optimize their delivery routes. They have constraints like vehicle capacity, delivery time windows, and fuel budget. By treating each delivery hub or customer location as a 'point' and using algorithms based on linear programming, they find the most efficient routes to minimize cost or maximize deliveries, ensuring your online orders reach you quickly.

Key Vocabulary

Key Terms

Objective Function: The function (like profit or cost) that you want to maximize or minimize. | Constraints: The limitations or restrictions (like available resources or time) expressed as inequalities. | Feasible Region: The area on a graph that satisfies all the given constraints. | Corner Points (Vertices): The points where the boundary lines of the feasible region intersect. | Linear Programming: A mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.

What's Next

What to Learn Next

Now that you understand the Corner Point Method, you're ready to explore more complex Linear Programming problems, including those with more variables. You can also learn about the Simplex Method, which is an advanced technique used for problems with many constraints and variables, building directly on the foundation you've built here!