Optimization problem, creating algorithm

Suppose that I have a big pipe and I want to put n small pipes in it, say 8 pipes 1 inch each. What is the smallest radius for the biggest pipe to contain all of the small ones? Constraint minimization, they must not be inside one another. How will I write the function, which will minimize this radius for the general case for a general n? I need to write an algorithm for n number of circles, when I change the n number the amount of the small circles will change and the radius of the biggest circle will change too.

Answer 1

[Walter Roberson]

If you could do that, then the values listed as "conjectured optimal" in the table at https://en.wikipedia.org/wiki/Circle_packing_in_a_circle would no longer be conjectures.

I would not even consider starting the coding without having read a few of the papers listed in the references of the above wikipedia link.

Unless, of course, you were just hoping for a "not bad" result that might not be optimal, such as if you were thinking of using PSO or GA.

[Gama Ali]

The article give different equation for different n numbers, what if I want 30 small circles in a bigger one? there is no formula mentioned for that in the article. http://www.hindawi.com/journals/aor/2009/150624/ This link is very close to what I am looking for, but I couldn't write an algorithm yet.

[Walter Roberson]

Did you not read the article?

"n fact, problems with circular objects are very hard optimization problems. They have a large number of variables and local minima. Thus, they require to be tackled with algorithms which mix local searches with heuristic procedures in order to widely explore the search space"

When you have a problem that difficult, there is no formula to solve it. Solving for n = 30 would probably take many months of computer time.