The Recoverable Robust Two-Level Network Design Problem

Abstract

We consider a network design application which is modeled as the two level network design problem under uncertainty. In this problem, one of the two available technologies can be installed on each edge and all customers of the network need to be served by at least the lower level (secondary) technology. The decision maker is confronted with uncertainty regarding the set of primary customers, i.e., the set of nodes that need to be served by the higher level (primary) technology. A set of discrete scenarios associated to the possible realizations of primary customers is available. The network is built in two stages. In the first-stage the network topology must be determined. One may decide to install the primary technology on some of the edges in the first stage, or one can wait to see which scenario will be realized, in which case, edges with the installed secondary technology may be upgraded, if necessary to primary technology, but at higher recovery cost. The overall goal then is to build a spanning tree in the first stage that serves all customers by at least the lower level technology, and that minimizes the first stage installation cost plus the worst-case cost needed to upgrade the edges of the selected tree, so that the primary customers of each scenario can be served using the primary technology. Using the recently introduced concept of recoverable robustness, we address this problem of importance in the design of telecommunication and distribution networks, and provide mixed integer programming models and a branch-and-cut algorithm to solve it.

Combining two-stage stochastic programming and recoverable robustness to minimize the number of late jobs in the case of uncertain processing times

“…In the first stage, one can buy edges at a cheaper price, and when the sink gets revealed, one can buy the additionally needed edges at a higher price. This problem is closely related to the recoverable robust Two-level Network Design problem studied by Álvarez-Miranda et al (2015b). Given a graph with a root node r one must decide which edges to prepare, and whether this should be done by installing primary or secondary technology.…”

The recoverable robust facility location problem

“…Formulations and algorithms for different variants of the recoverable robust shortest path problem are given in [13]. Models, properties and exact algorithms for recoverable robust two-level network design problems are presented in [5]. A more general framework of the RRO is studied in [17] where multiple recovery stages are allowed.…”

“…Nonetheless, this additional robustness is obtained at the expenses of (i) an increase of the difficulty of the problem, since a larger search space must be considered, and (ii) an increase of the total solution cost, OP T RR , because more facilities have to be opened and more allocations have to be established in the first stage or because a new worst-case scenario induces a higher robust recovery cost (i.e., ω increases). The first of these effects has been coined as the Effort for Robustness in [5] called the Price of Robustness in [10].…”

Two-level rectilinear Steiner trees

“…To the best of our knowledge the two-level rectilinear Steiner tree problem has not been considered before despite its practical importance [14,15]. It is loosely related to the hierarchical network design problems [1,5,7] or multi-level facility location problems [3,4]. However, those problems are structurally different, typically considering problems in graphs, and do not apply to our case.…”