Linear Programming — Lesson
1) Hook — A Fun Real-Life Example
Imagine you are the manager of a small sweets shop in Kolkata. You make two popular sweets: Rasgulla and Sandesh. Each sweet requires a certain amount of milk and sugar, and you want to maximize your profit given limited resources. How many Rasgullas and Sandesh should you make daily to earn the highest profit without running out of ingredients? This is a classic problem solved by Linear Programming.
2) Core Concepts
Linear Programming (LP) is a mathematical method to find the best outcome (maximum or minimum) of a linear objective function, subject to a set of linear inequalities called constraints.
Key components of an LP problem:
- Decision Variables: Unknowns to determine (e.g., number of Rasgullas and Sandesh).
- Objective Function: Linear function to maximize or minimize (e.g., profit).
- Constraints: Linear inequalities representing resource limits (e.g., milk, sugar availability).
- Feasible Region: Set of all points satisfying constraints, usually shown on a graph.
Example:
| Sweet | Milk (litres) | Sugar (kg) | Profit (₹ per unit) |
|---|---|---|---|
| Rasgulla (x) | 0.5 | 0.2 | ₹10 |
| Sandesh (y) | 0.4 | 0.3 | ₹12 |
Available resources daily: 20 litres of milk and 12 kg of sugar.
Formulating the problem:
- Decision variables: x = number of Rasgullas, y = number of Sandesh
- Objective function (profit): Maximize Z = 10x + 12y
- Constraints:
- Milk: 0.5x + 0.4y ≤ 20
- Sugar: 0.2x + 0.3y ≤ 12
- Non-negativity: x ≥ 0, y ≥ 0
Graphical method: Plot constraints on xy-plane, find feasible region, and evaluate Z at corner points.
3) Key Formulas / Rules
Standard Form of Linear Programming Problem:
Maximize or Minimize:
Z = c₁x₁ + c₂x₂ + ... + cₙxₙ
Subject to constraints:
a₁₁x₁ + a₁₂x₂ + ... + a₁ₙxₙ ≤ b₁
a₂₁x₁ + a₂₂x₂ + ... + a₂ₙxₙ ≤ b₂
...
x₁, x₂, ..., xₙ ≥ 0
Steps to Solve Graphically (2 variables):
- Plot each inequality as an equation on the xy-plane.
- Identify the feasible region satisfying all constraints.
- Find corner points (vertices) of the feasible region.
- Calculate the objective function value at each vertex.
- The maximum or minimum value of Z occurs at one of these vertices.
4) Did You Know?
The Linear Programming method was first developed during World War II to optimize resource allocation for military logistics. Today, it helps Indian farmers decide the best crop mix to maximize profits given limited land and water, or companies like Tata Steel optimize production costs!
5) Exam Tips
- Always write down decision variables clearly. Define what x and y represent.
- Convert constraints into inequalities carefully. Check signs (≤, ≥) and constants.
- Draw neat and accurate graphs. Label axes and shade feasible region properly.
- Calculate corner points precisely. Solve simultaneous equations where constraints intersect.
- Check non-negativity constraints. Negative values for x or y are not acceptable.
- Practice previous years’ questions: ICSE often asks to formulate LP problems, graph constraints, find feasible region, and compute maximum/minimum values.
Previous Year Question Pattern Example:
- Formulate the LP problem from a word problem. (5 marks)
- Graph the constraints and shade the feasible region. (5 marks)
- Find the maximum/minimum value of the objective function. (5 marks)
Linear Programming — Mcq
Linear Programming — Mnemonic
Mnemonic 1: "LP PLAN" for steps of Linear Programming 📋✏️
- L – List variables (निर्धारित करो वेरिएबल्स)
- P – Put constraints (सीमाएँ लगाओ)
- P – Plot feasible region (क्षेत्र बनाओ)
- L – Locate corner points (कोना बिंदु खोजो)
- A – Apply objective function (लक्ष्य सूत्र लगाओ)
- N – Note maximum/minimum (अधिकतम/न्यूनतम निकालो)
“LP PLAN से LP आसान!” 😄
Mnemonic 2: Funny Hindi rhyme for constraints and objective 🎯
"सीमाएँ हों tight,
लक्ष्य हो bright,
Linear Programming से,
मिले solution right!"
😊
Mnemonic 3: Acronym "F.O.C.C" for key terms in Linear Programming 🧠
- F – Function (Objective Function)
- O – Optimise (Maximize or Minimize)
- C – Constraints (सीमाएँ)
- C – Corner points (कोना बिंदु)
“F.O.C.C करो, solution पक्के करो!” 😎
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