Progressive hedging-based metaheuristics for stochastic network design

We consider the stochastic variant of the fixed-charge capacitated multicommodity network design (CMND) problem in which demands are stochastic. We propose a two-stage stochastic programming formulation where design decisions make up the first stage, while a series of recourse decisions are made in the second stage to distribute the commodities according to observed demands. The overall objective is to optimize the cost of the first-stage design decisions plus the total expected distribution cost incurred in the second stage. To solve this formulation, we propose a metaheuristic framework inspired by the progressive hedging algorithm of Rockafellar and Wets. Following this strategy, scenario decomposition is used to separate the stochastic problem following the possible outcomes, or scenarios, of the random event. Each scenario subproblem then becomes a deterministic CMND problem to be solved. Our progressive hedging strategy takes advantage of specialized methods to solve effectively deterministic CMND problems. We also propose and compare different strategies aimed at penalizing non-consensus amongst scenario subproblems to approximate the global design. These strategies are then embedded into a parallel solution method, which is numerically qualified on a set of problem instances that also provide the means to examine the effects of demand correlations on the behavior of the proposed algorithm.