Treffer: Mixed‐Integer Linear Programming Approaches for Nested p‐Center Problems With Absolute and Relative Regret Objectives.

We introduce the nested p$$ p $$‐center problem, a multi‐period variant of the well‐known p$$ p $$‐center problem. Using the nesting concept allows us to obtain solutions consistent over the considered time horizon, that is, facilities that are opened in a given time period stay open for subsequent periods. This consistency is important in real‐life applications, as closing (and potentially later re‐opening) of facilities between time periods can be undesirable. We consider two versions of our problem, with the difference being the objective function. The first version considers the sum of the absolute regrets (of nesting) over all time periods, and the second version considers minimizing the maximum relative regret over the time periods. We present three mixed‐integer programming formulations for the version with an absolute regret objective and two formulations for the version with a relative regret objective. For all the formulations, we present valid inequalities. Based on the formulations and the valid inequalities, we develop branch‐and‐bound/branch‐and‐cut solution algorithms. These algorithms include a preprocessing procedure that exploits the nesting property and starting heuristics and primal heuristics. We conducted a computational study on instances from the literature for the p$$ p $$‐center problem, which we adapted to our problems. We also analyze the effect of nesting on the solution cost and the number of open facilities. [ABSTRACT FROM AUTHOR]

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