Predisaster Preparation of Transportation Networks

We develop a new approach for a pre-disaster planning problem which consists in computing an optimal investment plan to strengthen a transportation network, given that a future disaster probabilistically destroys links in the network. We show how the problem can be formulated as a non-linear integer program and devise an AI algorithm to solve it. In particular, we introduce a new type of extreme resource constraint and develop a practically efficient propagation algorithm for it. Experiments show several orders of magnitude improvements over existing approaches, allowing us to close an existing real-world benchmark and to solve to optimality other, more challenging benchmarks. Earthquake Preparation We consider a real-world pre-disaster planning problem that was introduced in (Peeta et al. 2010). Given a transportation network, where links are subject to probabilistic failure, the task is to use a limited budget to decrease the failure probability on selected links such that the overall expected shortest-path distance between a number of origin/destination pairs (OD-pairs) is minimized. The original model assumes independence between the failures of links. Consequently, a priori an exact standard stochastic programming approach would need to consider an exponential number of scenarios. Therefore, the approach in (Peeta et al. 2010) solves the problem without guarantees on the quality of the solution. Exploiting value-interchangeability, a symmetry-notion from constraint programming, only very recently (Prestwich, Laumanns, and Kawas 2013; 2014) were able to reduce the number of scenarios massively to a point where the benchmark introduced in (Peeta et al. 2010) could be solved to optimality within a number of minutes. The idea of combining scenarios that all lead to the same value for the random variable considered (in our case: the sum of all expected shortest path lengths between all origin/destination pairs) is very interesting for scenario generation in general. We build on this idea and formulate the predisaster planning problem as an integer programming problem with a very simple constraint structure and a heavily ∗This research was supported through grant P22239-N13 by the Austrian Science Foundation FWF. non-linear objective. Due to the binary nature of investment decisions the problem is obviously not convex. Furthermore, we will show that it is hard to approximate and also neither subnor super-modular. We then introduce a new type of extreme resource constraint and develop a practically effective propagator which allows us to solve the existing benchmark instances in milliseconds, two to three orders of magnitude faster than existing methods. This then paves the way for tackling harder benchmark instances which consider several orders more non-interchangeable scenarios as well as correlated failure probabilities. Problem Definition and Complexity Definition 1. We are given an undirected graph G = (N,E) with a node set N = {1, . . . , n} and an edge set E = {{i, j} | i, j ∈ N, i < j}. Furthermore, we are given four functions. An a priori survival probability function s : E → [0, 1], a survival probability after investment function v : E → [0, 1], an edge length function l : E → IN, and a cost of investment function c : E → IN. Finally, we are given a budget limit C ∈ IN, as well as node pairs (o1, d1), . . . , (ok, dk) ∈ N and penalties Moh,dh ∈ IN for disconnecting the OD pair (oh, dh). Given a set I ⊆ E, we denote with pI : E → [0, 1], with pI(i, j) = v(i, j) if {i, j} ∈ I and pI(i, j) = s(i, j) otherwise, the survival probabilities after investing in links in I . We consider the probability space (Ω, PI) where Ω = {F | F ⊆ E} and PI is the unique probability measure with PI(F ) = ∏ {i,j}∈F pI(i, j) ∏ {i,j}/ ∈F (1− pI(i, j)). Finally, for all o, d ∈ N let us denote with SPLo,d : Ω → IN a random variable over (Ω, PI) such that SPLo,d(F ) returns the shortest-path length from o to d in the weighted graph (N,F, l), and Mo,d if no path from o to d exists in (N,F ) (we omit a formal definition of shortest paths due to limited