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

Abstract

We study solution approaches for the design of reliably connected networks. Specifically, given a network with arcs that may fail at random, the goal is to select a minimum cost subset of arcs such the probability that a connectivity requirement is satisfied is at least 1− , where is a risk tolerance. We consider two types of connectivity requirements. We first study the problem of requiring an s-t path to exist with high probability in a directed graph. Then we consider undirected graphs, where we require the graph to be fully connected with high probability. We model each problem as a stochastic integer program with a joint chance constraint, and present two formulations that can be solved by a branch-and-cut algorithm. The first formulation uses binary variables to represent whether or not the connectivity requirement is satisfied in each scenario of arc failures and is based on inequalities derived from graph cuts in individual scenarios. We derive additional valid inequalities for this formulation and study their facetinducing properties. The second formulation is based on probabilistic graph cuts, an extension of graph cuts to graphs with random arc failures. Inequalities corresponding to probabilistic graph cuts are sufficient to define the set of feasible solutions and can be separated efficiently at integer solutions, allowing this formulation to be solved by a branch-and-cut algorithm. Computational results demonstrate that the approaches can effectively solve instances on large graphs with many failure scenarios. In addition, we demonstrate that, by varying the risk tolerance, our model yields a rich set of solutions on the efficient frontier of cost and reliability.

A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support

“…The details are beyond the scope of this paper, but we conjecture that this algorithm could be shown to converge finitely provided that a constraint qualification holds at every nonlinear programming problem solved in the algorithm, an assumption that is standard for convergence of the LP/NLP algorithm. Many special cases of (1) are known to be N P-hard [20,12,50]. Therefore, we cannot expect a polynomial-time algorithm for (1), which is why we propose a branch-and-cut algorithm.…”

“…In this context, the costs are incurred when resources are acquired, but the allocation of resources to demands incurs no additional cost. A small list of example applications of CCMPs that fit the structure of (1) includes multicommodity flow [7], optimal vaccination planning [8], air quality management [9], aquifer remediation [10], and reliable network design [11,12].…”

Topological optimization of reliable networks under dependent failures

“…These methods give approximate solutions without any guarantee of convergence to optimality. A recent approach, proposed in [22], employs a sample of failure event scenarios to determine the optimal topology. This technique, called the sample average approximation (SAA), converges to an optimal solution when the number of samples is sufficiently large.…”

“…This model is related to the ideas of [22], which is based on a sample average approximation (SAA) of the probability that the network is…”

“…Variables u s e represent whether or not link e is operative under scenario s. To do so, constraints (4) and (5) [22]. However, we adhere to the model including explicitly the u s e variables because we will discuss some extensions in which these variables are required in the model.…”