Mathematical Programming Algorithms for Two-Path Routing Problems with Reliability Considerations

Abstract

Most traditional routing problems assume perfect operability of all arcs and nodes. However, when independent arc failure probabilities exist, a secondary objective must be present to retain some measure of expected functionality, introducing nonlinear, nonconvex constraints. We examine the Robust Two-Path Problem, which seeks to establish two paths between a source and destination node wherein at least one path must remain fully operable with some threshold probability. We consider the case where both paths must be arc-disjoint and the case where arcs can be shared between the paths. We begin by proving the NP-hardness of these problems, and then examine various strategies for solving the resulting nonlinear integer program, including pruning, coefficient tightening, lifting, and branch-and-bound partitioning schemes. We discuss the advantages and disadvantages of these methods, and conclude with computational results.

Branch-and-cut approaches for chance-constrained formulations of reliable network design problems

“…To emphasize that these are cuts derived from the deterministic graphs associated with individual scenarios, we call the s-t cuts from scenario graphs G k , k ∈ N , scenario-based s-t cuts. Applying the max-flow min-cut theorem to the extended formulation defined by (3) leads to the following integer programming (IP) formulation for (2):…”

“…In contrast, we directly require that the graph be connected with high probability, and do not assume that arc failures are independent. Andreas and Smith [2] study an extension of the two arc-disjoint paths model to a probabilistic setting. They also assume that arcs fail independently, and the solution approach depends on this assumption.…”

“…If z k = 0, the right-hand side of (3a) becomes 0, so it is feasible to assign y k a = 0 for all a ∈ A k , and hence the constraints corresponding to scenario k do not constrain the choice of arcs. Thus, replacing (2c) with (3) in (2) yields a valid MIP formulation of (2). We refer to this MIP formulation as the extended formulation of (2).…”

“…Thus, replacing (2c) with (3) in (2) yields a valid MIP formulation of (2). We refer to this MIP formulation as the extended formulation of (2).…”

“…First observe that (1) is equivalent to formulation (2): for every Q feasible to (1), ∃ (x, z) feasible to (2) with the same objective value, and conversely, for every (x, z) feasible to (2) there is a Q feasible to (1) with the same objective value. Therefore, it is sufficient to prove the problem formulated in (2) is N P-hard. By letting q = |N | , the decision version of (2) can be stated as follows.…”