Stochastic optimization in supply chain networks: averaging robust solutions

Abstract

We propose a novel robust optimization approach to analyze and optimize the expected performance of supply chain networks. We model uncertainty in the demand at the sink nodes via polyhedral sets which are inspired from the limit laws of probability. We characterize the uncertainty sets by variability parameters which control the degree of conservatism of the model, and thus the level of probabilistic protection. At each level, and following the steps of the traditional robust optimization approach, we obtain worst case values which directly depend on the values of the variability parameters. We go beyond the traditional robust approach and treat the variability parameters as random variables. This allows us to devise a methodology to approximate and optimize the expected behavior via averaging the worst case values over the possible realizations of the variability parameters. Unlike stochastic analysis and optimization, our approach replaces the high-dimensional problem of evaluating expectations with a low-dimensional approximation that is inspired by probabilistic limit laws. We illustrate our approach by finding optimal base-stock and affine policies for fairly complex supply chain networks. Our computations suggest that our methodology (a) generates optimal base-stock levels that match the optimal solutions obtained via stochastic optimization within no more than 4 iterations, (b) yields optimal affine policies which often times exhibit better results compared to optimal base-stock policies, and (c) provides optimal policies that consistently outperform the solutions obtained via the traditional robust optimization approach.