What is an Optimisation Problem Graphically?

S3-SA5-0320

Grade Level:

Class 9

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

Definition

What is it?

An optimisation problem, when solved graphically, means finding the best possible solution (like maximum profit or minimum cost) for a situation by looking at its graph. We draw lines and regions based on given conditions, and the 'best' point is usually at a corner of the valid region.

Simple Example

Quick Example

Imagine you run a small chai stall and want to make the most profit. You can sell two types of chai: Masala Chai and Ginger Chai. You have limited milk and sugar. Graphically, you'd draw lines representing your milk and sugar limits. The area where both conditions are met is your 'feasible region'. You then find the point in this region that gives you the highest profit.

Worked Example

Step-by-Step

Let's say a bakery makes two types of biscuits: Butter Biscuits (x) and Chocolate Chip Biscuits (y). They have 100g of butter and 120g of flour. Each Butter Biscuit needs 10g butter and 10g flour. Each Chocolate Chip Biscuit needs 5g butter and 15g flour. They want to maximise the total number of biscuits.

Step 1: Write down the inequalities. Butter: 10x + 5y <= 100. Flour: 10x + 15y <= 120. Also, x >= 0 and y >= 0 (cannot make negative biscuits).
---Step 2: Convert inequalities to equations for plotting. 10x + 5y = 100 (or 2x + y = 20) and 10x + 15y = 120 (or 2x + 3y = 24).
---Step 3: Find points for each line. For 2x + y = 20: If x=0, y=20 (0,20). If y=0, x=10 (10,0). For 2x + 3y = 24: If x=0, y=8 (0,8). If y=0, x=12 (12,0).
---Step 4: Plot these lines on a graph. Shade the region below both lines and in the first quadrant (where x and y are positive). This is the feasible region.
---Step 5: Identify the corner points of this feasible region. These are (0,0), (10,0), (0,8), and the intersection of 2x + y = 20 and 2x + 3y = 24.
---Step 6: Solve the system of equations to find the intersection point: Subtract (2x + y = 20) from (2x + 3y = 24) to get 2y = 4, so y = 2. Substitute y=2 into 2x + y = 20, so 2x + 2 = 20, 2x = 18, x = 9. The intersection point is (9,2).
---Step 7: The objective is to maximise total biscuits, which is P = x + y. Calculate P at each corner point: P(0,0) = 0, P(10,0) = 10, P(0,8) = 8, P(9,2) = 9 + 2 = 11.
---Step 8: The maximum value is 11 at (9,2).
Answer: The bakery should make 9 Butter Biscuits and 2 Chocolate Chip Biscuits to maximise total biscuits to 11.

Why It Matters

Understanding optimisation helps engineers design more efficient machines, data scientists build better prediction models, and economists allocate resources wisely. It's used in logistics to find the shortest delivery routes for companies like Flipkart, and in manufacturing to reduce waste and save money.

Common Mistakes

MISTAKE: Not shading the correct feasible region based on the inequalities (e.g., shading above a 'less than or equal to' line) | CORRECTION: Always test a point (like (0,0) if it's not on the line) in each inequality to see which side of the line satisfies the condition.

MISTAKE: Forgetting to include x >= 0 and y >= 0 as constraints, especially in real-world problems where quantities cannot be negative | CORRECTION: Always assume non-negativity for physical quantities unless stated otherwise, which limits the solution to the first quadrant.

MISTAKE: Incorrectly finding the coordinates of the corner points, especially the intersection points of lines | CORRECTION: Double-check your algebraic calculations when solving simultaneous equations for intersection points. Plotting accurately helps verify.

Practice Questions

Try It Yourself

QUESTION: A small factory produces two items, A and B. It takes 1 hour to make item A and 2 hours to make item B. The factory has a total of 10 working hours per day. Write an inequality representing this constraint. | ANSWER: x + 2y <= 10 (where x is item A and y is item B)

QUESTION: If the profit for item A is Rs 50 and for item B is Rs 70, write the objective function to maximise profit. | ANSWER: P = 50x + 70y

QUESTION: A farmer wants to plant corn (x) and wheat (y). He has 10 acres of land. Planting corn costs Rs 1000 per acre and wheat costs Rs 800 per acre. He has a budget of Rs 9000. Write the two inequalities for land and budget. | ANSWER: Land: x + y <= 10. Budget: 1000x + 800y <= 9000 (or 10x + 8y <= 90)

MCQ

Quick Quiz

Which of the following is typically NOT a corner point of the feasible region in a graphical optimisation problem?

The origin (0,0)

An intersection of two constraint lines

A point far outside the shaded feasible region

An intersection of a constraint line with an axis

The Correct Answer Is:

The feasible region is the area where all conditions are met. Corner points are always on the boundary of this region, including the origin, axis intersections, and intersections of constraint lines. A point far outside the shaded region would not satisfy all conditions.

Real World Connection

In the Real World

Logistics companies like Delhivery or Ecom Express use graphical optimisation (or more complex versions of it) to plan delivery routes for their drivers. They need to deliver many parcels to different addresses with limited fuel and time. They use algorithms that graphically find the shortest or most efficient path, ensuring your online orders from Amazon or Myntra reach you on time while saving the company money.

Key Vocabulary

Key Terms

OPTIMISATION: Finding the best possible solution (maximum or minimum) | FEASIBLE REGION: The area on a graph where all given conditions are met | OBJECTIVE FUNCTION: The equation (e.g., for profit or cost) that you want to maximise or minimise | CONSTRAINTS: The limitations or conditions (like limited resources) expressed as inequalities

What's Next

What to Learn Next

Now that you understand optimisation graphically, you can explore the Simplex Method, which is an algebraic way to solve more complex optimisation problems with many variables. It builds on the same core ideas of finding the best solution under constraints.