A survivable variant of the ring star problem

The Ring Star Problem consists in selecting a subset of nodes called hubs including the depot and linking them with a cycle, the remaining nodes being connected to exactly one hub, at minimum cost. We study a survivable variant of the Ring Star Problem where at most one node in a given subset of so-called uncertain nodes can fail if selected as a hub. We model this problem as an Integer Linear Program (ILP), that is also addressed with a Branch-and-Benders-cut decomposition. The Benders subproblem is turned into a linear program with the addition of new inequalities that are shown to be facet-defining, and several enhancements to both the ILP and Branch-and-Benders-cut algorithm are also presented. Both approaches are compared on the basis of extensive numerical experiments that bring the following conclusions. First, the survivable variant is shown to be much harder than the original Ring Star Problem, and the extra cost induced by survivability is significant. Second, the ILP formulation tends to produce tighter lower bounds but memory issues are frequent for large instances. Finally, the Branch-and-Benders-cut algorithm returns feasible solutions that are often of better quality than those produced by ILP, and is less frequently subjected to memory issues on the considered set of instances.