The Corner Point Method

Name: The Corner Point Method
Author: StemPy Lab

A profit that grows along a straight line must be largest at the edge of what is allowed. Use that idea to find the maximum or minimum of an objective function at the corner points. See why unbounded regions need an extra test, what happens when there is no feasible solution or many, and check each answer with scipy.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
The objective function and the corner point method
The quantity we want as large or as small as possible is the objective function, such as the profit $Z = 2x + 3y$ . The lines $Z = k$ for different $k$ are parallel. As $k$ grows, the line slides across the feasible region, and the last point it touches is a corner. So on a bounded feasible region the maximum and the minimum of $Z$ are found by working out $Z$ at every corner point and comparing the values.
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 2about 25 min
Unbounded regions and when an optimum exists
When the feasible region has no outer edge, the corner values alone do not prove an optimum. Take the best corner value M and draw the open half-plane that would beat it. If this half-plane has no point in common with the region, M is the optimum. If it has a common point, the optimum does not exist. For a cost on a region with $\ge$ rules, the minimum usually exists and the maximum does not.
1 simulation1 practice question1 Python program4 problems
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Lesson 3about 25 min
No solution, many solutions and a check with linprog
Three special cases can happen. If the rules clash, the feasible region is empty, so there is no feasible solution. If the lines $Z = k$ are parallel to an edge, the optimum is reached at both ends of that edge and at every point between them, so there are many optimal solutions. And on an unbounded region an optimum may not exist. The function linprog in scipy solves a problem and reports which case it found, so it checks a hand solution.
1 simulation1 practice question1 Python program3 problems1 Python lab1 question set1 note
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The Corner Point Method: Timed Test

Board exam format15 questions · 20 min

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The Corner Point Method

Course outline (3 lessons)

The objective function and the corner point method

Unbounded regions and when an optimum exists

No solution, many solutions and a check with linprog

Test Your Learning

Your teacher

Discussion (0)