You'll find the "best" answer only through a bruteforce search of all solutions. This is not an "AI" problem, though you can use some AI techniques to figure it out. At the end of the day, if you don't have too many people/places and need the optimal solution, I'd go with brute force. Using random methods won't help you arrive at a solution any more efficiently. Linear programming lends it self to these types of problems, but you may not want to have to learn it, and it will still need brute force to find the best soln :)...
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