How does an airline allocate aircraft and crew across hundreds of airports to maximize multi-million-dollar profits while strictly satisfying thousands of pilot rest hours and fuel capacity limits? Linear Programming Problems (LPP) optimize real-world operations.
Why This Chapter Matters
In Class 12 Mathematics, "Linear Programming" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.
Before You Begin (Prerequisites)
Linear inequalities from Class 11.
Plotting straight lines in 2D.
Convex polygonal regions.
What You Will Learn (Core Objectives)
Formulate mathematical Linear Programming Problems: Objective Function ($Z = ax + by$), Decision Variables, and Constraints.
Identify Feasible Region, Infeasible Region, Bounded vs Unbounded regions.
Apply the Corner Point Theorem to find optimal solutions (Maximum or Minimum).
Solve manufacturing, diet, and transportation optimization problems graphically.
Evaluate unbounded feasible regions using half-plane tests.
Chapter Roadmap & Progression
11. LPP Formulation & Corner Point T...
Complete Concept Guide (100% Curriculum Coverage)
1. LPP Formulation & Corner Point Theorem
Objective Function: $Z = ax + by$ to be maximized or minimized.
Constraints: Linear inequalities expressing resource limits ($x \ge 0, y \ge 0$ non-negative constraints).
Feasible Region: Common intersection region determined by all constraints simultaneously.
Corner Point Theorem: The optimal value (maximum or minimum) of $Z$ occurs strictly at one of the Corner Points (Vertices) of the feasible region!
Linear Programming - Key Conceptual & Analytical Model
Chapter Summary & 10 Key Takeaways
Takeaway 1
Objective Function: Mathematical function $Z = ax + by$ to be optimized.
Takeaway 2
Decision Variables: Non-negative unknowns $x, y \ge 0$ representing production quantities.
Takeaway 3
Feasible Polygon: Convex region satisfying all simultaneous linear constraints.
Takeaway 4
Corner Point Theorem: Extreme values occur strictly at polygon corner vertices.
Takeaway 5
Iso-Profit Line: Linear contour of constant objective value.
Check Your Understanding (Diagnostic Practice Questions)
Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.
1
Solve graphically: Maximize $Z = 4x + y$ subject to constraints: $x + y \le 50, 3x + y \le 90, x \ge 0, y \ge 0$.
Reveal Answer & Explanation
Answer: Corner points of the feasible region are: $O(0, 0)$, $A(30, 0)$, $B(20, 30)$ (intersection of $x+y=50$ and $3x+y=90$), and $C(0, 50)$. Values of $Z$: $Z(O) = 0$, $Z(A) = 4(30) + 0 = 120$, $Z(B) = 4(20) + 30 = 110$, $Z(C) = 4(0) + 50 = 50$. Maximum value of $Z$ is 120 at corner point $(30, 0)$. Max Z = 120 at (30, 0).
2
What is a 'Feasible Region' in a Linear Programming Problem?
Reveal Answer & Explanation
Answer: The common region determined by the intersection of all the linear constraints, including non-negative constraints ($x \ge 0, y \ge 0$), where every point satisfies all problem requirements. Common region satisfying all constraints.
3
State the Corner Point Theorem for linear programming.
Reveal Answer & Explanation
Answer: Let $R$ be the feasible region (convex polygon) for an LPP and let $Z = ax + by$ be the objective function. When $R$ is bounded, $Z$ has both a maximum and minimum value on $R$ and each of these occurs at an extreme point (corner point/vertex) of $R$. Optimal values occur at polygon vertices.
4
Minimize $Z = 3x + 5y$ such that $x + 3y \ge 3, x + y \ge 2, x \ge 0, y \ge 0$.
Reveal Answer & Explanation
Answer: Corner points of feasible region: $A(0, 2), B(1.5, 0.5)$ (intersection), and $C(3, 0)$. Values of $Z$: $Z(A) = 3(0) + 5(2) = 10$, $Z(B) = 3(1.5) + 5(0.5) = 4.5 + 2.5 = 7$, $Z(C) = 3(3) + 5(0) = 9$. Minimum value is 7 at $(1.5, 0.5)$. Min Z = 7 at (1.5, 0.5).
5
Can an objective function have multiple optimal solutions in an LPP? Explain.
Reveal Answer & Explanation
Answer: Yes. If two corner points produce the same optimal maximum (or minimum) value of $Z$, then every point on the line segment connecting those two corner points also provides the same optimal value (infinitely many optimal solutions). Yes, all points on boundary segment between twin optimal corners.
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