Linear Programming — Lesson
1) Hook — A Fun Real-Life Example
Imagine you run a small sweet shop in Chennai. You make two popular sweets: ladoos and jalebis. Each ladoo requires 2 units of sugar and 3 units of flour, while each jalebi requires 4 units of sugar and 2 units of flour. You have only 100 units of sugar and 90 units of flour available daily. You earn ₹10 per ladoo and ₹15 per jalebi. How many ladoos and jalebis should you make to maximize your profit?
This problem is a classic example of Linear Programming, where we find the best outcome (maximum profit) under given constraints (available sugar and flour).
2) Core Concepts — What is Linear Programming?
Linear Programming (LP) is a method to achieve the best outcome (maximum or minimum) in a mathematical model whose requirements are represented by linear relationships.
- Variables: Quantities to determine (e.g., number of ladoos = x, number of jalebis = y)
- Objective Function: The function to maximize or minimize (e.g., Profit = 10x + 15y)
- Constraints: Linear inequalities representing limits (e.g., sugar and flour availability)
- Feasible Region: The set of all possible solutions satisfying constraints
- Optimal Solution: The point in the feasible region giving the best value of the objective function
Step-by-step approach:
| Step | Description |
|---|---|
| 1 | Define variables (x, y) |
| 2 | Write constraints as inequalities |
| 3 | Formulate objective function |
| 4 | Graph constraints and find feasible region |
| 5 | Find corner points of feasible region |
| 6 | Evaluate objective function at corner points to find optimum |
Example: Sweet Shop Problem
Variables: Let x = number of ladoos, y = number of jalebis.
Constraints:
- Sugar: 2x + 4y ≤ 100
- Flour: 3x + 2y ≤ 90
- Non-negativity: x ≥ 0, y ≥ 0
Objective function: Maximize profit, Z = 10x + 15y.
Graphical representation:
| Constraint | Intercepts | Inequality |
|---|---|---|
| 2x + 4y ≤ 100 | x=50 (y=0), y=25 (x=0) | Below or on the line |
| 3x + 2y ≤ 90 | x=30 (y=0), y=45 (x=0) | Below or on the line |
| x, y ≥ 0 | First quadrant only | |
Corner points of feasible region:
- (0, 0)
- (0, 25)
- (18, 18) — intersection of 2x + 4y = 100 and 3x + 2y = 90
- (30, 0)
Evaluate Z = 10x + 15y at each point:
| Point (x, y) | Z = 10x + 15y |
|---|---|
| (0, 0) | 0 |
| (0, 25) | 375 |
| (18, 18) | 10×18 + 15×18 = 180 + 270 = 450 |
| (30, 0) | 300 |
Maximum profit ₹450 occurs at (18, 18). So, make 18 ladoos and 18 jalebis.
3) Key Formulas / Rules
Linear Programming Problem (LPP) consists of:
- Objective function: Z = ax + by (to maximize or minimize)
- Constraints: Linear inequalities such as px + qy ≤ r
- Non-negativity constraints: x ≥ 0, y ≥ 0
- Feasible region: Intersection of all constraints on graph
- Optimal solution: Found at corner points of feasible region
4) Did You Know?
Linear Programming was first developed during World War II to optimize resource allocation for military logistics. Today, it is widely used in industries across India — from agriculture planning in Punjab to manufacturing in Tamil Nadu — to maximize profits and reduce costs efficiently!
5) Exam Tips — Score High in Board Exams
- Always define variables clearly. Write what x and y represent before starting.
- Check inequality signs carefully. Mistakes in ≤ or ≥ change the feasible region drastically.
- Plot constraints accurately. Use intercepts to draw straight lines on graph paper.
- Shade the feasible region properly. It must satisfy all constraints simultaneously.
- Calculate corner points precisely. Solve equations of intersecting lines carefully.
- Evaluate objective function at all corner points. Don’t miss any point.
- Remember non-negativity constraints (x, y ≥ 0). Only first quadrant is valid.
- Common mistake: Forgetting to include non-negativity constraints or misinterpreting inequalities.
- Board pattern: Usually, 6-8 marks question with stepwise solution — variables, constraints, objective, graph, corner points, and conclusion.
Linear Programming — Mcq
Linear Programming — Mnemonic
Mnemonic 1: "LP PLAN" for steps of Linear Programming 📋✏️
- L – List variables (like x and y)
- P – Put constraints (inequalities)
- P – Plot the graph (draw lines)
- L – Locate feasible region (where all constraints meet)
- A – Analyse corner points (vertices)
- N – Name optimum (max or min value)
Remember: "LP PLAN" = Your step-by-step plan to solve Linear Programming problems! 😊
Mnemonic 2: Hindi rhyme for constraints and objective function 🎯
“अंक लिखो, शर्तें जोड़ो, रेखा खींचो, क्षेत्र खोजो, कोना पकड़ो, मान निकालो।”
Translation: Write variables, add constraints, draw lines, find region, pick corners, calculate value.
This simple Hindi rhyme helps you remember all steps in a fun way! 🎉
Mnemonic 3: Funny acronym "COPS" for key Linear Programming terms 🚓
- C – Constraints (शर्तें)
- O – Objective function (लक्ष्य कार्य)
- P – Plot graph (रेखा खींचना)
- S – Solution region (हल क्षेत्र)
Think of "COPS" as the police who catch the best solution in Linear Programming! 🚨😄
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