Approximation Algorithms for Stochastic Inventory Control Models (DRAFT)

We consider stochastic inventory control models in which the goal is to coordinate a sequence of orders of a single commodity, aiming to supply stochastic demands over a discrete, finite horizon with minimum expected overall ordering, holding and backlogging costs. In this paper, we address the longstanding problem of finding computationally efficient and provably good inventory control policies to these models in the presence of correlated and non-stationary (time-dependent) stochastic demands. This problem arises in many domains and has many practical applications in supply chain management. We consider two classical models, the periodic-review stochastic inventory control problem and the stochastic lot-sizing problem with correlated and non-stationary demands. Here the correlation is inter-temporal, i.e., what we observe in period s changes our forecast for the demand in future periods. We provide what we believe to be the first computationally efficient policies with constant worst-case performance guarantees; that is, there exists a constant C such that, for any instance of the problem, the expected cost of the policy is at most C times the expected cost of an optimal policy. The dominant paradigm in almost all of the existing literature has been to formulate these models using a dynamic programming framework. This approach has turned out to be very successful in characterizing the structure of the optimal policies, which follow simple forms of state-dependent base-stock policies and state-dependent (s, S) policies. However, in case the demands are non-stationary and correlated over time, computing these optimal policies is likely to be intractable. We present a new approach that leads to general approximation algorithms with constant performance guarantee for these classical models. Our approach is based on several novel ideas: we present a new (marginal) cost accounting for stochastic inventory models; we use cost-balancing techniques; and we consider non base-stock (order-up-to) policies that are extremely easy to implement on-line. Our results are valid for all of the currently known approaches in the literature to model correlation and nonstationarity of demands over time. More specifically, we provide a general 2-approximation algorithm for the periodic-review stochastic inventory control problem and a 3-approximation algorithm for the stochastic lot-sizing problem. That is, the constant guarantees are 2 and 3, respectively. For the former problem, we show that the classical myopic policy can be arbitrarily more expensive compared to the optimal policy. We also present an extended class of myopic policies that provides both upper and lower bounds on the optimal base-stock levels. ∗[email protected]. School of ORIE, Cornell University, Ithaca, NY 14853. Research supported partially by a grant from Motorola and NSF grants CCR-9912422& CCR-0430682. †[email protected] Dept. of Computer Science, Cornell University, Ithaca, NY 14853. ‡[email protected]. School of ORIE, Cornell University, Ithaca, NY 14853. Research supported partially by a grant from Motorola, NSF grant DMI-0075627, and the Querétaro Campus of the Instituto Tecnológico y de Estudios Superiores de Monterrey. §[email protected]. School of ORIE and Dept. of Computer Science, Cornell University, Ithaca, NY 14853. Research supported partially by NSF grants CCR-9912422&CCR-0430682.