We consider two fundamental stochastic optimization problems that arise in the context of supply-chain models, the single-period newsvendor problem and its multiperiod extension with independent demands. These problems are among the most well-studied stochastic optimization problems in the Operations Research literature. Most commonly, these problems are studied from the perspective that the input probability distributions are given in terms of specific probability distribution functions that are computationally tractable; under this assumption, both problems can be solved efficiently. Unfortunately, this information is unlikely to be available in practice, and hence we make the more realistic assumption that the probability distribution is given by a "black box" from which independent samples can be drawn. We give the first fully polynomial randomized approximation schemes for these two problems in this sampling-based model.Our work provides new insights into the power of two of the most often-used approaches to solving stochastic optimization problems, the sample average approximation (SAA) and stochastic dynamic programming. For the newsvendor problem, we show that by taking a polynomial number of samples and then solving the newsvendor problem with respect to the resulting approximation to the true distribution, we obtain provably near-optimal solution. This significantly extends the class of problems for which the SAA is known to yield a scheme. Finally, we show how to adapt the framework of stochastic dynamic programming to yield an approximation scheme for the multiperiod newsvendor problem with independent demands. We believe that this is an interesting first step towards the goal of providing a mechanism for deriving efficient approximate stochastic dynamic programming methods for a wide range of multistage stochastic optimization problems.

We consider several classical models in deterministic inventory theory: the single-item lot-sizing problem, the joint replenishment problem, and the multi-stage assembly problem. These inventory models have been studied extensively, and play a fundamental role in broader planning issues, such as the management of supply chains. We shall give a novel primal-dual framework for designing algorithms for these models that significantly improve known results in several ways: the performance guarantees for the quality of the solutions improve on or match previously known results; the performance guarantees hold under much more general assumptions about the structure of the costs, and the algorithms and their analysis are significantly simpler than previous known results. Finally, our primal-dual framework departs from the structure of previously studied primal-dual approximation algorithms in significant ways, and we believe that our approach may find application in other settings.We provide 2-approximation algorithms for the joint replenishment problem and for the assembly problem, and solve the single-item lot-sizing problem to optimality. The results for the joint replenishment and the lot-sizing problems also hold for their generalizations with back orders allowed. As a byproduct of our work, we prove known and new upper bounds on the integrality gap of the LP relaxations for these problems.