Mixed-integer linear programming approaches for nested $p$-center problems with absolute and relative regret objectives

TL;DR

This paper introduces the nested p-center problem, a multi-period extension with nesting constraints, and develops mixed-integer programming formulations and algorithms to optimize absolute and relative regret objectives over time.

Contribution

It presents new formulations and solution algorithms for the nested p-center problem with two regret-based objectives, incorporating nesting properties and valid inequalities.

Findings

Nesting reduces solution costs and facility openings.

Formulations effectively solve instances from literature.

Nesting ensures solution consistency over time.

Abstract

We introduce the nested $p$-center problem, which is a multi-period variant of the well-known $p$-center problem. The use of the nesting concept allows to obtain solutions, which are consistent over the considered time horizon, i.e., facilities which are opened in a given time period stay open for subsequent time periods. This is important in real-life applications, as closing (and potential later re-opening) of facilities between time periods can be undesirable. We consider two different 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 absolute regret objective and two formulations for the version with 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 also begins heuristics and primal heuristics. We conducted a computational study on instances from the literature for the $p$-center problem, which we adapted to our problems. We also analyse the effect of nesting on the solution cost and the number of open facilities.