Class 12 Maths Linear Programming Notes

12.1 Introduction

Linear Programming (L.P.) is a mathematical technique to optimize a linear objective function subject to linear constraints.

• Widely used in business, economics, engineering, and operations research
• Helps in maximizing profit or minimizing cost
• Solutions are generally obtained using graphical methods or simplex method

---

12.2 Linear Programming Problem and Its Mathematical Formulation

A Linear Programming Problem (LPP) consists of:

1. Objective Function – The function to maximize or minimize:

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

• Example: Maximize profit or minimize cost

2. Constraints – Linear inequalities that restrict the solution:

$$\begin{cases} x + y \le 10 \\ 2x + y \le 15 \\ x \ge 0, y \ge 0 \end{cases}$$⎩⎨⎧​x+y≤102x+y≤15x≥0,y≥0​

3. Non-Negativity Conditions – Variables must be greater than or equal to zero:

$$x \ge 0, \, y \ge 0$$x≥0,y≥0

Steps to Formulate LPP

1. Define variables clearly
2. Formulate objective function in terms of variables
3. Write constraints as linear inequalities
4. Include non-negativity conditions

---

🔹 Key Notes

• LPP is widely applicable in production planning, transportation, resource allocation, and diet problems
• The solution lies at the vertices (corner points) of the feasible region
• Graphical method is used for two-variable LPPs
• For higher dimensions, simplex method is applied

---

✅ Key Points to Remember

• LPP involves objective function + constraints + non-negativity conditions
• Feasible solutions satisfy all constraints simultaneously
• Optimization occurs at corner points of feasible region
• Linear programming is a powerful tool in management and economics