Stochastic analysis via robust optimization

Abstract

To evaluate the performance and optimize systems under uncertainty, two main avenues have been suggested in the literature: stochastic analysis and optimization describing the uncertainty probabilistically and robust optimization describing the uncertainty deterministically. Instead, we propose a novel paradigm which leverages the conclusions of probability theory and the tractability of the robust optimization approach to approximate and optimize the expected behavior in a given system. Our framework models the uncertainty via polyhedral sets inspired by the limit laws of probability. We characterize the uncertainty sets by variability parameters that we treat as random variables. We then devise a methodology to approximate and optimize the average performance of the system via a robust optimization formulation. Our framework (a) avoids the challenges of fitting probability distributions to the uncertain variables, (b) eliminates the need to generate scenarios to describe the states of randomness, and (c) demonstrates the use of robust optimization to evaluate and optimize expected performance. We illustrate the applicability of our methodology to analyze the performance of queueing networks and optimize the inventory policy for supply chain networks. In Part I, we study the case of a single queue. We develop a robust theory to study multi-server queues with possibly heavy-tailed primitives. Our methodology (a) provides approximations that match the diffusion approximations for light-tailed queues in heavy traffic, and (b) extends the framework to analyze the transient behavior of heavy-tailed queues. In Part II, we study the case of a network of queues. Our methodology provides accurate approximations of (a) the expected steady-state behavior in generalized queueing networks, and (b) the expected transient behavior in feedforward queueing networks. Our approach achieves significant computational tractability and provides accurate approximations relative to simulated values. In Part III, we study the case of a supply chain network. Our methodology (a) obtains optimal base-stock levels that match the optimal solutions obtained via stochastic optimization, (b) yields optimal affine policies which oftentimes 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.