Linear Programming — Part 2: The corner-point method: visual lesson, CBSE Class 12 Mathematics

Lesson

Linear Programming, Part 2: The corner-point method

The best plan always sits at a corner — if a best plan exists.

In this lesson you will

State the Corner Point Theorem for bounded regions
Solve two-variable linear programming problems by the corner-point method
Use the half-plane test on unbounded regions
Recognise multiple optimal solutions and problems with no solution

$x + y = 8$
$x + 2y = 10$
$Z = 15$
$Z = 28$

Parallel lines $3x + 5y = k$: the last region point touched is a corner.

Concept

What you will be able to do

State the Corner Point Theorem
Solve LPPs with the corner-point method
Apply the half-plane test on unbounded regions
Recognise multiple optimal solutions
Recognise problems with no solution

Concept

Corner Point Theorem

If the feasible region is bounded (and non-empty), $Z = ax + by$ has both a maximum and a minimum, and each occurs at a corner point. If the region is unbounded, an optimum, if it exists, also occurs at a corner.

$x + y = 8$
$x + 2y = 10$
$Z = 15$
$Z = 28$

Parallel lines $3x + 5y = k$: the last region point touched is a corner.

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