The multi-period pp-center problem with time-dependent travel times

This paper deals with an extension of the $p$-center problem, in which arc traversal times vary over time, and facilities are mobile units that can be relocated multiple times during the planning horizon. We investigate the relationship between this problem and its single-period counterpart. We also derive some properties and a special case. The insight gained with this analysis is then used to devise a two-stage heuristic. Computational results on instances based on the Paris (France) road graph indicate that the algorithm is capable of determining good-quality solutions in a reasonable execution time

Exact solution approaches for the discrete α\alpha-neighbor pp-center problem

The discrete $\alpha$-neighbor $p$-center problem (d-$\alpha$-$p$CP) is an emerging variant of the classical $p$-center problem which recently got attention in literature. In this problem, we are given a discrete set of points and we need to locate $p$ facilities on these points in such a way that the maximum distance between each point where no facility is located and its $\alpha$-closest facility is minimized. The only existing algorithms in literature for solving the d-$\alpha$-$p$CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d-$\alpha$-$p$CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures. We provide theoretical results on the strength of the formulations and convergence results for the lower bounds obtained after applying the lifting procedures or the variable fixing procedures in an iterative fashion. Based on our formulations and theoretical results, we develop branch-and-cut (B&C) algorithms, which are further enhanced with a starting heuristic and a primal heuristic. We evaluate the effectiveness of our B&C algorithms using instances from literature. Our algorithms are able to solve 116 out of 194 instances from literature to proven optimality, with a runtime of under a minute for most of them. By doing so, we also provide improved solution values for 116 instances