Chapter 12 · Linear Programming

Linear Programming

2 topics 66 practice questions

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Chapter Overview

Businesses, farms and kitchens constantly ask “how much of each should we make or buy to get the best result under our limits?” Linear programming answers such questions when the goal and the limits are linear. In this chapter you will learn the vocabulary, translate word problems into mathematical form, and solve two-variable problems by the graphical (corner-point) method.

Board focus: One 5-mark graphical LPP (formulate, draw, tabulate corner points, conclude) is expected every year, often with a case study or an MCQ on corner points.

Topics

1 Terminology and Formulation 24 questions
2 Graphical Method of Solution 42 questions

Key Concepts

An LPP optimises a linear objective function \(Z=px+qy\) subject to linear constraints and \(x,y\ge0\).
The set of points satisfying all constraints is the feasible region; its corners are the corner points.
Corner-point theorem: if an optimum exists, it occurs at a corner point of the feasible region.
A bounded region always has both a maximum and a minimum; for an unbounded region, test the open half-plane beyond the candidate value.

Formulas

Standard form of an LPP

Linear Programming · Terminology and Formulation

\[\text{Optimise } Z=px+qy\ \text{ subject to }\ a_ix+b_iy\ (\le,\ \ge)\ c_i,\quad x\ge0,\ y\ge0\]

Corner-point theorem

Linear Programming · Graphical Method of Solution

If an LPP has an optimal value, it is attained at a corner point of the feasible region. A bounded feasible region always has both a maximum and a minimum value of \(Z\).

Key Points

Testing a half-plane

Linear Programming · Graphical Method of Solution

To decide which side of \(ax+by=c\) to shade, substitute a point not on the line (usually the origin). If it satisfies the inequality, shade its side.

Multiple optimal solutions

Linear Programming · Graphical Method of Solution

If \(Z\) has the same optimal value at two adjacent corners, all points on the edge joining them are optimal.

Common Mistakes

Reversing inequalities

Linear Programming · Terminology and Formulation

“At least” means ≥ and “at most” means ≤. Reversing one inequality changes the whole feasible region.

Concluding too early on an unbounded region

Linear Programming · Graphical Method of Solution

On an unbounded region the largest value in the table need not be a maximum. Draw \(px+qy>M\) and check whether it meets the region.

Solved Examples

Potter’s problem

Linear Programming · Graphical Method of Solution

Maximise \(Z=60x+90y\) subject to \(x+2y\le16,\ x+y\le10,\ x,y\ge0\).

Corner points: \((0,0),(10,0),(4,6),(0,8)\). Values: 0, 600, 780, 720. The region is bounded, so \(Z_{\max}=₹780\) at \((4,6)\): make 4 mugs and 6 bowls.

Practice & Tests

Chapter Practice

Untimed mixed questions from all topics. Answers are revealed after you submit.

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Chapter Test

Timed test – about 15 questions in 30 minutes.

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Unit Test – Linear Programming

A 90-minute board-style test on the whole unit “Linear Programming”, drawn fresh from the question bank for every attempt.

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Linear Programming – Chapter Test

A board-style chapter test drawn fresh from the question bank each time: MCQs, assertion-reason, numericals, written answers you self-check against model solutions, and a case study.

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