| --- |
| source: ncert |
| topic: algebra |
| class_level: class_12 |
| chapter: ch12 |
| difficulty: advanced |
| --- |
| |
| Class 12 | Ch12: Linear Programming |
| LINEAR PROGRAMMING PROBLEM (LPP): |
| Objective function: Z=ax+by (to maximise or minimise). |
| Constraints: inequalities (ax+by<=c, x>=0, y>=0 etc). |
| Feasible region: set of all points satisfying ALL constraints simultaneously. |
| Feasible solution: any point in feasible region. |
| Optimal solution: feasible solution that gives maximum/minimum value of Z. |
| CORNER POINT METHOD (THEOREM): |
| If optimal solution exists, it occurs at a CORNER POINT (vertex) of feasible region. |
| Steps: |
| 1. Graph all constraints. Find feasible region. |
| 2. Find all corner points (vertices) of feasible region. |
| 3. Evaluate Z at each corner point. |
| 4. Maximum/minimum value of Z is the optimal solution. |
| TYPES OF LPP: |
| Bounded feasible region: has maximum AND minimum. |
| Unbounded feasible region: may not have maximum (for maximise) or minimum (for minimise). |
| Infeasible: no feasible region (constraints contradictory). No solution. |
| FORMULATING LPP FROM WORD PROBLEMS: |
| Identify: variables (what to find), objective (what to optimise), constraints (conditions). |
| SOLVED EXAMPLES: |
| Example 1: Maximise Z=3x+4y subject to x+y<=450, 2x+y<=600, x>=0, y>=0. |
| Corner points: O(0,0), A(300,0), B(150,300), C(0,450). |
| Z at O=0. Z at A=900. Z at B=450+1200=1650. Z at C=1800. |
| Maximum Z=1800 at C(0,450). |
| Example 2: Minimise Z=5x+7y subject to 2x+y>=8, x+2y>=10, x>=0, y>=0. |
| Corner points: A(0,8), B(2,4), C(10,0). |
| Z at A=56. Z at B=10+28=38. Z at C=50. |
| Minimum Z=38 at B(2,4). |
| FINDING CORNER POINTS: solve pairs of boundary equations simultaneously. |
| Intersection of 2x+y=600 and x+y=450: subtract → x=150, y=300. |
| COMMON MISTAKES: |
| Evaluate Z at ALL corner points (not just one or two). |
| Feasible region must satisfy ALL constraints (including x>=0, y>=0). |
| For minimum in unbounded region: check if solution is truly minimum. |
