In this paper, we describe the first computationally efficient policies for stochastic inventory models with lost sales and replenishment lead times that admit worst-case performance guarantees. In particular, we introduce dual-balancing policies for lost-sales models that are conceptually similar to dual-balancing policies recently introduced for a broad class of inventory models in which demand is backlogged rather than lost. That is, in each period, we balance two opposing costs: the expected marginal holding costs against the expected marginal lost-sales cost. Specifically, we show that the dual-balancing policies for the lost-sales models provide a worst-case performance guarantee of two under relatively general demand structures. In particular, the guarantee holds for independent (not necessarily identically distributed) demands and for models with correlated demands such as the AR 1 model and the multiplicative autoregressive demand model. The policies and the worst-case guarantee extend to models with capacity constraints on the size of the order and stochastic lead times. Our analysis has several novel elements beyond the balancing ideas for backorder models. Key words: inventory; approximation; dual balancing; algorithms; lost sales MSC2000 subject classification: Primary: 90B05; secondary: 68W25 OR/MS subject classification: Primary: inventory/production, approximation/heuristics; secondary: production/scheduling, approximation/heuristics History: Received November 2, 2005; revised November 1, 2006 and March 11, 2007. 1. Introduction. In this paper, we address one of the fundamental problems in stochastic inventory theory, the single-item, single-location, periodic-review, stochastic inventory control problem with lost sales, which we refer to as the lost-sales problem. This problem has challenged researchers and practitioners for over five decades because very little is known about the structure of the optimal policy, and there are no known provably good heuristics, even for the simplest settings. We build on ideas first proposed by Levi et al. [6]. They proposed what are called dual-balancing policies for a class of inventory models where unsatisfied demand is backlogged rather than lost. These policies have worst-case performance guarantees-that is, for each instance of the problem, the expected cost of the policy is guaranteed to be at most C times the optimal expected cost (for some constant C). In this paper, we discuss the implementation and the worst-case analysis of dual-balancing policies applied to inventory models with lost sales. These models have mathematical characteristics that are very different from the models in which excess demand is backlogged and thus require a fundamentally different and novel worst-case analysis. In particular, we shall describe the first computationally efficient policies for inventory models with lost sales that have a worst-case performance guarantee of two. The analysis is based on several new ideas that we believe will contribute to future research in this domain. Stochastic inventory theory provides streamlined models with the following common setting. The goal is to coordinate a sequence of orders over a planning horizon of finitely many discrete periods, aiming to supply a sequence of random demands over the planning horizon with minimum expected cost. The cost consists of a per-unit ordering cost for ordering supply units at the beginning of each period (with or without capacity constraints), a per-unit holding cost for carrying excess inventory from one period to the next, and a per-unit penalty cost for not satisfying demand on time. The dynamics of these models is as follows. At the beginning of each period, before the demand in this period is observed, a nonnegative procurement order is placed with an outside supplier incurring a cost proportional to the number of units ordered. This order will arrive and become available after a lead time of several periods. The demand in that period is then observed and is satisfied to the maximum extent possible from the current inventory on hand. At the end of the period, two possible costs are incurred: Excess units of inventory incur a proportional holding cost and unsatisfied units of demand incur a proportional penalty cost. The goal is to find an ordering policy that minimizes the overall expected costs over the entire horizon.
Levi et al. REF study production and lost sales over multiple periods focusing on the performance of a simple heuristic policy that balances the inventory holding cost and lost-sales penalty in each period, with the aim of establishing its robustness relative to the optimal policy in a very general context.
A 2-Approximation Algorithm for Stochastic Inventory Control Models with Lost Sales
Math. Oper. Res.
Math. Oper. Res.
Mathematics Computer Science