Questions
1–2 questions in JEE Main/BITSAT
Difficulty
Easy
Importance
Low weightage but essential for easy scoring in boards and entrance exams.
Overview
Linear Programming Problem (LPP) is an optimization technique used to maximize or minimize a linear objective function subject to a set of linear constraints. In competitive exams, it focuses on identifying the feasible region and evaluating the optimal value at boundary vertices. Mastering this allows you to solve resource allocation and production planning problems efficiently.
Formulation of LPP
Formulation involves converting word problems into a mathematical model consisting of decision variables, an objective function, and structural constraints. The primary goal is to translate real-world limitations into linear inequalities while ensuring non-negativity restrictions are applied.
- Identify decision variables (x, y) clearly.
- Construct the objective function Z = ax + by.
- Formulate constraints as linear inequalities (≤, ≥, or =).
- Always state non-negativity constraints: x ≥ 0, y ≥ 0.
Graphical Solution Method
The graphical method is the standard approach for LPPs with two variables. The process involves plotting each constraint inequality as a line on a Cartesian plane and identifying the region that satisfies all constraints simultaneously.
- Draw equations by finding intercepts (x=0, y=0).
- Use the test point (0,0) to determine the side of the half-plane.
- The overlapping region of all inequalities is the Feasible Region.
- Constraints must be satisfied for all x, y in the feasible set.
Corner Point Theorem
The Corner Point Theorem is the backbone of LPP, stating that the optimal solution (maximum or minimum) of an LPP always occurs at a vertex of the feasible region. This transforms an infinite-search problem into a simple evaluation at a few critical points.
- Identify all vertices of the bounded feasible region.
- Calculate Z value at each corner point.
- Maximum value is the largest Z; minimum is the smallest Z.
- If the feasible region is unbounded, check if the optimal value exists.
Formula Sheet
Objective Function: Z = ax + by
Linear Inequality: ax + by ≤ c or ax + by ≥ c
Non-negativity constraint: x, y ≥ 0
Exam Tip
Always verify your intersection points by plugging them back into the boundary equations, as a single coordinate error will lead to an incorrect objective function value.
Common Mistakes
- Failing to shade the correct half-plane for inequalities, leading to an incorrect feasible region.
- Ignoring the non-negativity constraints (x≥0, y≥0) which limits the region to the first quadrant.
- Miscalculating the intersection points of boundary lines due to algebraic errors.
More Revision Notes
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