Design Optimization

Thanks to Frank Kampas for providing source code and animations for this challenge problem. Frank holds a Ph.D. in physics from Stanford University and can be reached at frank@physicistatlarge.com.

Attach:10_circle_packing_anim1.gif Δ

Download Sample MATLAB Code

MATLAB source code writes an APM optimization file and sends it to the APMonitor server for solution. After the script executes, a figure appears that shows the best and worst configurations.

• MATLAB Example Source Code

• Solve 4 Circle Packing Problem Online

• Solve 10 Circle Packing Problem Online

• Python Example Source Code for Multi-Start

Download Sample Python Code - Version 1

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Download Sample Python Code - Version 2

A second version solves multiple circle packing optimization problems with the same model using multi-dimensional arrays. Python source code re-writes the optimization file and sends it to the APM server for solution. After the script executes, a figure appears that shows the configuration that was produced.

• Python Example Source Code for 10 Circles

Attach:CirclePacking10.png Δ

Example Solution for 10 Circles

• Solve 10 Circle Packing Problem

(:html:) <iframe src="https://apmonitor.com/me575/uploads/Main/packcircles10.htm" width="500" height="1500" frameborder="1" marginheight="0" marginwidth="0">Loading...</iframe>

• 'containers' - in this case a circle that circumscribes all of the other circles

Python source code re-writes the optimization file and sends it to the APM server for solution. After the script executes, a figure appears that shows the configuration that was produced.

• Solve 4 Circle Packing Problem

<iframe src="https://apmonitor.com/me575/uploads/Main/packcircles.htm" width="500" height="600" frameborder="1" marginheight="0" marginwidth="0">Loading...</iframe>

<iframe src="https://apmonitor.com/me575/uploads/Main/packcircles.htm" width="450" height="600" frameborder="1" marginheight="0" marginwidth="0">Loading...</iframe>

<iframe src="https://docs.google.com/a/apmonitor.com/spreadsheet/embeddedform?formkey=dGp4bExwUU1wOVhOT3M1T2otY1g3LXc6MQ" width="550" height="900" frameborder="0" marginheight="0" marginwidth="0">Loading...</iframe>

<iframe src="https://docs.google.com/a/apmonitor.com/spreadsheet/embeddedform?formkey=dGp4bExwUU1wOVhOT3M1T2otY1g3LXc6MQ" width="400" height="600" frameborder="0" marginheight="0" marginwidth="0">Loading...</iframe>

<iframe src="https://docs.google.com/a/apmonitor.com/spreadsheet/embeddedform?formkey=dGp4bExwUU1wOVhOT3M1T2otY1g3LXc6MQ" width="500" height="800" frameborder="0" marginheight="0" marginwidth="0">Loading...</iframe>

(:html:) <iframe src="https://apmonitor.com/me575/uploads/Main/packcircles.htm" width="550" height="700" frameborder="1" marginheight="0" marginwidth="0">Loading...</iframe> (:htmlend:)

Background

Packing problems are a class of optimization problems in mathematics which involve attempting to pack objects together (often inside a container), as densely as possible. Many of these problems can be related to real life packaging, storage and transportation issues. In a packing problem, you are given:

• 'containers' - in this case a circle that circuscribes all of the other circles
• 'goods' (usually a single type of shape), some or all of which must be packed into this container

Usually the packing must be without overlaps between goods and other goods or the container walls. The aim is to find the configuration with the maximal density. In some variants the overlapping (of goods with each other and/or with the boundary of the container) is allowed but should be minimized.

Example Solution for 4 Circles

• Python Example Source Code

(:title Circle Packing Challenge Activity:) (:keywords project, nonlinear, optimization, engineering optimization, engineering design, interior point, active set, circle packing, packing optimization, python, matlab:) (:description A challenge problem to determine the minimum circumscribing radius for n circles with radius i^(-0.5) for i=1..n.:)

Challenge Problem in Global Optimization

Determine the minimum circumscribing radius for n circles with radius i^(-0.5) for i=1..n.

Download Sample Python Code

Record Your Best Answer

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