Monte Carlo study of the sphere packing problem

We employ the Monte Carlo method to study a constrained optimization problem, that is packing spheres with unequal radii into a 3-D bounded region. Selection of the best fit solution is based on using the Boltzmann factor, e−ΔE/T to determine the transition probability, which allows us to search for the global optimal solution. We determined the least numbers of packed spheres that will occupy the largest volume. The optimal occupied volume found is around 44% of a bounded region volume, which is obtained within a relative short computing time. This suggests that our result could be able to give a good starting point for the radiosurgery treatment plan.