Optimal shift coloring of pharmacies

The pharmacy service requires that some pharmacies are always available and shifts have to be organized: a shift corresponds to a subset of pharmacies that must be open 24 hours a day on a particular week. Under the requirement that each pharmacy belongs to exactly one shift and the assumption that users minimize the distance to the closest open pharmacy during each shift, we want to determine a partition of the pharmacies into a given number of shifts, such that the total distance covered by users is minimized. It may be also required that shift cardinalities are balanced. We discuss different versions and the related computational complexity, showing that the problem is NP-Hard in general. A set packing formulation is presented and solved by branch-and-price, together with a fast solution technique based on a tabu search. They have been applied to real and random instances showing that (i) the set packing formulation is very tight and often exhibits no integrality gap; (ii) the branch-and-price solves problems of practical relevance to optimality in a reasonable amount of time (order of minutes); (iii) the tabu search finds optimal or near-optimal solutions in order of seconds.