📌 1. Introduction

Linear Programming helps us find the best possible solution under given conditions.

👉 Example:

• Maximizing profit
• Minimizing cost
• Efficient use of resources

---

📖 2. Basic Terminology

---

🔹 1. Decision Variables

The variables whose values we need to determine.

Example:$$x, y$$x,y

---

🔹 2. Objective Function

A linear function that needs to be maximized or minimized.

---

$Z = ax + by$Z=ax+by

---

🔹 3. Constraints

Restrictions on variables, expressed as linear equations or inequalities.

Example:$$ax + by \leq c$$ax+by≤c

---

🔹 4. Non-Negativity Constraints

$$x \geq 0, \quad y \geq 0$$x≥0,y≥0

---

📊 3. Mathematical Formulation of LPP

General form:

Maximize/Minimize:$$Z = ax + by$$Z=ax+by

Subject to:$$a_1x + b_1y \leq c_1$$a1​x+b1​y≤c1​ $$a_2x + b_2y \leq c_2$$a2​x+b2​y≤c2​ $$x, y \geq 0$$x,y≥0

---

📐 4. Graphical Method

The graphical method is used to solve LPP involving two variables.

---

📌 Steps:

1. Convert inequalities into equations

2. Plot lines on graph

3. Identify feasible region

4. Find corner points

5. Evaluate objective function

---

📊 5. Feasible Region

The region satisfying all constraints is called the feasible region.

---

🔹 Types:

• Bounded region
• Unbounded region

---

📌 6. Optimal Solution

The best value (maximum or minimum) of objective function.

👉 Occurs at corner points of feasible region.

---

📐 7. Corner Point Method

Steps:

1. Find vertices of feasible region
2. Evaluate objective function at each vertex
3. Choose maximum/minimum value

---

📊 8. Special Cases

---

🔹 1. Unique Solution

Only one optimal solution

---

🔹 2. Infinite Solutions

More than one optimal solution

---

🔹 3. Unbounded Solution

No maximum or minimum

---

🔹 4. Infeasible Solution

No feasible region

---

📌 9. Important Concepts

🔹 Convex Set

A set where line joining any two points lies completely inside the set.

---

🔹 Corner Point Theorem

Optimal solution occurs at extreme points.

---

📊 10. Applications of LPP

• Production planning
• Transportation
• Diet problems
• Resource allocation
• Business optimization

---

❗ Common Mistakes

• Wrong graph plotting
• Incorrect feasible region
• Ignoring non-negativity
• Calculation errors

---

🧠 Exam Tips

• Draw graph carefully
• Mark feasible region clearly
• Check all corner points
• Practice NCERT problems

---

📚 Practice Questions

1. Solve LPP graphically
2. Find feasible region
3. Determine optimal solution
4. Identify special cases

---

🎯 Conclusion

Linear Programming is a practical and scoring chapter. It helps in solving optimization problems efficiently. Mastering graphical methods and understanding feasible regions will help you perform well in exams.