This article introduces POPMUSIC, a meta-heuristic that has been successfully applied to various combinatorial optimization problems. This meta-heuristic is especially useful for designing heuristic methods for large combinatorial problems that can be partially optimized. The basic idea is to optimize sub-parts of solutions until a local optimum is reached. Implementations of the technique to large centroid clustering and to the problem of balancing mechanical parts are shown to be very efficient.

We develop heuristics for a problem that models the static balancing of turbine fans: Load point masses at regularly spaced positions on the periphery of a circle so that the residual unbalance about the center --- which corresponds to the axis of rotation of the fan --- is as small as possible. We prove that our heuristics provide the same worst-case guarantee in terms of residual unbalance as does total enumeration. Furthermore, computational tests show that our heuristics are orders of magnitude faster and not far from optimum on average. The balancing of rotating elements in modern machinery is critical, and is done in some cases by sophisticated balancing machines. The presence of unbalance in a rotating machine results in vibrations and excess stresses on the bearings, both of which shorten its useful life. Static unbalance, the primary source of unbalance in narrow, disc-shaped rotors such as turbine fans, occurs when the center of gravity of the fan does not coincide wi...

Introduction Nonlinear Assignment Problems (NAP's) are NP-hard combinatorial optimization problems that are natural extensions of the classical Linear Assignment Problem (LAP). NAPs have numerous applications in different fields, ranging from the assignment of facilities to locations in location theory, to the tracking of elementary particles in high energy physics. We begin this section with the most basic of all assignment problems, the linear assignment problem. Consider that we are given n facilities and locations, and a matrix IR n\Thetan 3 C := (c ij ) where c ij represents the cost of placing facility i to location j. The objective is to find an optimal assignment of facilities to locations such that the total placement cost is minimized. Using the decision variables x ij 2 f0; 1