A 2-Approximation Algorithm for Stochastic Inventory Control Models with Lost Sales

Abstract

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 dualbalancing 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 2 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 auto-regressive 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 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 as 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, Pál, Roundy and Shmoys [5]. 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 than 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 2. The analysis is based on several new ideas that we believe will contribute to future research in this domain.

Approximation Algorithms for Perishable Inventory Systems

“…The main idea of this approach is to associate the costs with each ordering decision instead of each period. However, the techniques developed for our worst-case performance analysis depart significantly from those in the previous studies (e.g., Levi et al 2007Levi et al , 2008a, which heavily rely on the existence of a one-to-one matching between the supply and demand units when the inventory units are consumed in a first-in-first-out manner. That is, when analyzing the performances of the approximation algorithms, all the previous studies "geometrically" match product units in a one-to-one manner for the systems operating under two different policies, and the costs for each pair of matched units can be readily compared.…”

Optimization via Simulation for Inventory Control Policies and Supplier Selection

“…At this point, balancing the two underlying opposing risks, the risk of underordering incurring stockout cost, and the risk of overordering incurring holding costs can be considered as one of the most important objectives. Also, Levi et al [10] presented that this objective works reasonably better than minimizing the sum of the two costs. However, total cost function per unit time which can be considered as a most common measure of effectiveness is generally used to determine the optimal (s, S) policies [11].…”

Selecting Ordering Policy and Items Classification Based on Canonical Correlation and Cluster Analysis

“…Huh et al (2009) compared the performance of base-stock policies in lost-sales and backorder models. Levi et al (2008) proposed a dual-balancing policy, in which the risk of ordering and holding are balanced. The authors proved that the expected total cost of this policy is at most twice the expected cost of the optimal policy.…”