Linear Programming
Board: CBSE | Class: Class 12
Comprehensive study notes for Linear Programming by Ajay Yadav.
Key Concepts
Linear Programming Problem
Optimize (maximize/minimize) a linear objective function Z = ax+by subject to linear constraints. Feasible region: set of points satisfying all constraints.
Graphical Method
(1) Plot constraints as equations. (2) Identify feasible region (shaded). (3) Find corner points. (4) Evaluate Z at each corner. (5) Optimal = max/min among corners.
Types of Solutions
Bounded feasible region: optimal exists at corner. Unbounded: max/min may not exist. Infeasible: no point satisfies all constraints. Multiple: Z same at multiple corners (entire edge).
Corner Point Method
If feasible region is bounded, the optimal solution of LPP occurs at a corner point. If unbounded and Z has a max/min, it occurs at a corner.
Important Formulas
| Objective | Z = ax + by (maximize/minimize) |
| Constraints | Linear inequalities in x,y |
| Optimal | At corner points of feasible region |
Solved Examples
Example 1: Maximize Z=3x+2y subject to x+y≤4, x≥0, y≥0.
Solution: Corners: (0,0)→0, (4,0)→12, (0,4)→8. Max=12 at (4,0).
Example 2: Minimize Z=2x+3y subject to x+y≥6, x+2y≥8, x,y≥0.
Solution: Corners: (6,0)→12, (0,6)→18, (4,2)→14. Min=12 at (6,0).
Practice Questions
- Max Z=5x+3y subject to x+y≤10, 2x+y≤16, x,y≥0.
- Min Z=x+2y subject to 2x+y≥8, x+3y≥9, x,y≥0.
- A manufacturer has constraints. Formulate LPP and solve.
- Solve graphically: Max Z=4x+2y, x+2y≤6, 3x+y≤8, x,y≥0.
- When does LPP have no feasible solution? Explain.
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