Approximation Algorithms for Capacitated Stochastic Inventory Control Models

We consider two classical stochastic inventory control models, the periodic-review stochastic inventory control problem and the stochastic lot-sizing problem. 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 important problem of finding computationally efficient and provably good inventory control policies for 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.Our approach is based on a new marginal cost accounting scheme for stochastic inventory control models combined with novel cost-balancing techniques. Specifically, in each period, we balance the expected cost of over ordering (i.e, costs incurred by excess inventory) against the expected cost of under ordering (i.e., costs incurred by not satisfying demand on time). This leads to what we believe to be the first computationally efficient policies with constant worst-case performance guarantees for a general class of important stochastic inventory models. 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. In particular, we provide worst-case guarantee of 2 for the periodic-review stochastic inventory control problem and a worst-case guarantee of 3 for the stochastic lot-sizing problem. Our results are valid for all of the currently known approaches in the literature to model correlation and non-stationarity of demands over time.

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“…In a subsequent papers [16,15], we consider two generalizations of the model considered in this paper:…”

Section: Discussionmentioning

confidence: 99%

“…As we shall show, the simple idea of balancing is powerful and leads to policies that have constant expected worst-case performance guarantees. We again believe that the balancing idea will have more applications in constructing and analyzing algorithms for other stochastic inventory control models (see [16] and [15] for follow-up work).…”

Section: Introductionmentioning

confidence: 99%

Approximation Algorithms for Stochastic Inventory Control Models

et al. 2007

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“…These ideas have been extended to capacitated models [7] and multi-echelon models [6], again with backlogged demands. These dual-balancing policies are computationally efficient and have a worstcase performance guarantee of 2 for the respective models under general assumptions on the demand structure and the cost parameters.…”

Section: Mathematics Of Operations Research Xx(x) Pp Xxx-xxx C 200mentioning

confidence: 99%

“…Next we discuss the case where the demands are integer-valued random variables and the order quantity in each period is restricted to be an integer. We briefly describe a randomized dual-balancing policy using ideas identical to ones used in [5,7].…”

Section: Dual-balancing Policymentioning

confidence: 99%

“…We first describe a marginal cost accounting scheme for lost-sales models with capacity constraints on the size of the order in each period. This scheme follows closely the ideas introduced by Levi, Roundy, Shmoys and Truong in the context of capacitated models with backlogged demand [7]. The idea is to associate with the decision of how many units to order in each period s, all the future lost-sales costs that are forced by this decision.…”

Section: Capacitated Modelmentioning

confidence: 99%

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

Levi¹,

Janakiraman

Nagarajan

2008

Mathematics of OR

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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.

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Approximation Algorithms for Stochastic Optimization Problems in Operations Management

Shi

2011

Wiley Encyclopedia of Operations Research and Management Science

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This article provides an introduction to approximation algorithms in stochastic optimization models arising in various application domains, including central areas of operations management, such as scheduling, facility location, vehicle routing problems, inventory and supply chain management, and revenue management. Unfortunately, these models are very hard to solve to optimality in both theory and practice. We will survey recent development on approximation algorithms for these stochastic optimization models and their performance analysis techniques with worst‐case performance guarantees.

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Approximation Algorithms for Capacitated Stochastic Inventory Control Models

Cited by 75 publications

References 23 publications

Approximation Algorithms for Stochastic Inventory Control Models

Approximation Algorithms for Stochastic Inventory Control Models

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

Approximation Algorithms for Stochastic Optimization Problems in Operations Management

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