Linear Programming
Linear Programming (LP) ek tareeka hai jisme hum kisi quantity (jaise profit ya cost) ko kuch constraints (limits) ke under maximize ya minimize karte hain. Business, transport, diet planning — har jagah iska use hota hai.
1. Important Terms
- Objective function: jise optimize (max/min) karna hai, jaise \(Z = ax + by\).
- Constraints: linear inequalities jo variables par limit lagati hain.
- Decision variables: jinki value humein nikaalni hai (\(x, y\)).
- Non-negative restrictions: aam taur par \(x \ge 0,\ y \ge 0\).
- Feasible region: saari constraints ko satisfy karne wala common region.
- Feasible solution: feasible region ka koi bhi point.
2. Standard Form of an LPP
\[ \text{Maximize / Minimize } Z = ax + by \]
subject to constraints jaise \(a_1 x + b_1 y \le c_1\), \(\dots\), aur \(x, y \ge 0\).
3. Corner Point Method (Graphical)
Ek important fact: objective function \(Z\) ka maximum ya minimum hamesha feasible region ke kisi corner point (vertex) par hota hai.
Steps:
- Saari constraints ki lines banao aur feasible region shade karo.
- Feasible region ke corner points nikaalo.
- Har corner point par \(Z = ax + by\) ki value calculate karo.
- Jahan \(Z\) sabse zyada/kam ho, wahi optimal solution hai.
Example
Maximize \(Z = 3x + 2y\), subject to \(x + y \le 4\), \(x \ge 0\), \(y \ge 0\).
Feasible region ke corners: \((0,0), (4,0), (0,4)\). \(Z\) values:
\[ Z(0,0) = 0, \quad Z(4,0) = 12, \quad Z(0,4) = 8. \]
Maximum \(Z = 12\) point \((4, 0)\) par.
4. Bounded aur Unbounded Region
- Bounded: feasible region har taraf se band — max aur min dono milte hain.
- Unbounded: region khula hua — ho sakta hai max ya min exist na kare; extra check karna padta hai.
Key Takeaways
- LP = constraints ke under objective function \(Z = ax + by\) ko max/min karna.
- Feasible region = saari inequalities ka common region.
- Optimal value hamesha kisi corner point par milti hai → corner point method.
- Bounded region me max aur min dono; unbounded me extra check zaroori.