Optimal Policies for a Multi-Echelon Inventory Problem

doi:10.1287/MNSC.1040.0265

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Optimal Policies for a Multi-Echelon Inventory Problem

Journal Article•DOI•

Optimal Policies for a Multi-Echelon Inventory Problem

Andrew J. Clark, Herbert E. Scarf¹•Institutions (1)

Stanford University¹

01 Jul 1960-Management Science (INFORMS)-Vol. 50, Iss: 12, pp 1782-1790

TL;DR: The problem of determining optimal purchasing quantities in a multi-installation model of this type, which arises when there are several installations, is considered.

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Abstract: In the last several years there have been a number of papers discussing optimal policies for the inventory problem. Almost without exception these papers are devoted to the determination of optimal purchasing quantities at a single installation faced with some pattern of demand. It has been customary to make the assumption that when the installation in question requests a shipment of stock, this shipment will be delivered in a fixed or perhaps random length of time, but at any rate with a time lag which is independent of the size of the order placed. There are, however, a number of situations met in practice in which this assumption is not a tenable one. An important example arises when there are several installations, say 1, 2,..., N, with installation 1 receiving stock from 2, with 2 receiving stock from 3, etc. In this example, if an order is placed by installation 1 for stock from installation 2, the length of time for delivery of this stock is determined not only by the natural lead time between these two sites, but also by the availability of stock at the second installation. In this paper we shall consider the problem of determining optimal purchasing quantities in a multi-installation model of this type.

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Journal Article•DOI•

Information distortion in a supply chain: the bullwhip effect

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Hau L. Lee¹, Vineet Padmanabhan¹, Seungjin Whang¹•Institutions (1)

Stanford University¹

01 Apr 1997-Management Science

TL;DR: The authors analyzes four sources of the bullwhip effect: demand signal processing, rationing game, order batching, and price variations, and shows that the distortion tends to increase as one moves upstream.

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Abstract: (This article originally appeared in Management Science, April 1997, Volume 43, Number 4, pp. 546-558, published by The Institute of Management Sciences.) Consider a series of companies in a supply chain, each of whom orders from its immediate upstream member. In this setting, inbound orders from a downstream member serve as a valuable informational input to upstream production and inventory decisions. This paper claims that the information transferred in the form of "orders" tends to be distorted and can misguide upstream members in their inventory and production decisions. In particular, the variance of orders may be larger than that of sales, and distortion tends to increase as one moves upstream-a phenomenon termed "bullwhip effect." This paper analyzes four sources of the bullwhip effect: demand signal processing, rationing game, order batching, and price variations. Actions that can be taken to mitigate the detrimental impact of this distortion are also discussed.

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4,124 citations

Book Chapter•DOI•

Supply Chain Coordination with Contracts

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Gérard P. Cachon¹•Institutions (1)

University of Pennsylvania¹

01 Jan 2003-Supply Chain Management

TL;DR: This chapter extends the newsvendor model by allowing the retailer to choose the retail price in addition to the stocking quantity, and discusses an infinite horizon stochastic demand model in which the retailer receives replenishments from a supplier after a constant lead time.

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Abstract: Publisher Summary This chapter reviews the supply chain coordination with contracts. Numerous supply chain models are discussed. In each model, the supply chain optimal actions are identified. The chapter extends the newsvendor model by allowing the retailer to choose the retail price in addition to the stocking quantity. Coordination is more complex in this setting because the incentives provided to align one action might cause distortions with the other action. The newsvendor model is also extended by allowing the retailer to exert costly effort to increase demand. Coordination is challenging because the retailer's effort is noncontractible—that is, the firms cannot write contracts based on the effort chosen. The chapter also discusses an infinite horizon stochastic demand model in which the retailer receives replenishments from a supplier after a constant lead time. Coordination requires that the retailer chooses a large basestock level.

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2,626 citations

Cites background from "Optimal Policies for a Multi-Echelo..."

...As in Ha (1996), the contract designer has a prior belief regarding the other ...rm’s cost and that ...rm knows his cost precisely. They propose a quantity discount schedule, which is like a contract menu: there is a unique per unit price for each quantity the buyer may choose. As in Ha (1996) supply chain coordination is not achieved if there is asymmetric information....
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...…incurs no cost associated with retail backorders.65 Lee and Whang (1999) base their coordination scheme on the work by Clark and Scarf (1960).66 Clark and Scarf (1960), which focuses only on system wide performance, demonstrates that base stock policies are optimal and they can be evaluated…...
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...…of the decentralized game, it is useful to evaluate the optimal base stock policy without that traditional cost reallocation.63 Given ¦(sr; ss) is continuous, any optimal policy with ss > 0 must set the following two 62 Clark and Scarf (1960) pioneered that approach for the …nite horizon problem....
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...…that ¯s = 0; i.e., the supplier incurs no cost associated with retail backorders.65 Lee and Whang (1999) base their coordination scheme on the work by Clark and Scarf (1960).66 Clark and Scarf (1960), which focuses only on system wide performance, demonstrates that base stock policies are optimal…...
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...Representative papers in this area include Lal and Staelin (1984), Monohan (1984), Lee and Rosenblatt (1986), Dada and Srikanth (1987) and Weng (1995). These papers are not included primarily because there already exist several excellent reviews: see Dolan and Frey (1987) and Boyaci and Gallego (1997). In addition, the quantity discount literature primarily focuses on models with deterministic demand, whereas the recent work in supply chain contracting primarily focuses on models with stochastic demand (Bernstein and Federgruen 1999, and Chen, Federgruen, Zheng 2001, are exceptions)....
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Book•

Lectures on Stochastic Programming: Modeling and Theory

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Alexander Shapiro, Darinka Dentcheva, Andrzej Ruszczyński

24 Sep 2009

TL;DR: The authors dedicate this book to Julia, Benjamin, Daniel, Natan and Yael; to Tsonka, Konstatin and Marek; and to the Memory of Feliks, Maria, and Dentcho.

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Abstract: List of notations Preface to the second edition Preface to the first edition 1. Stochastic programming models 2. Two-stage problems 3. Multistage problems 4. Optimization models with probabilistic constraints 5. Statistical inference 6. Risk averse optimization 7. Background material 8. Bibliographical remarks Bibliography Index.

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2,443 citations

Cites methods from "Optimal Policies for a Multi-Echelo..."

...The optimality of the basestock policy for the multi-stage inventory model was first proved in Clark and Scarf [44]....
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Journal Article•DOI•

Supply Chain Inventory Management and the Value of Shared Information

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Gérard P. Cachon¹, Marshall L. Fisher¹•Institutions (1)

University of Pennsylvania¹

01 Aug 2000-Management Science

TL;DR: In traditional supply chain inventory management, orders are the only information firms exchange, but information technology now allows firms to share demand and inventory data quickly and inexpensively, and it is concluded that implementing information technology to accelerate and smooth the physical flow of goods through a supply chain is significantly more valuable.

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Abstract: In traditional supply chain inventory management, orders are the only information firms exchange, but information technology now allows firms to share demand and inventory data quickly and inexpensively. We study the value of sharing these data in a model with one supplier, N identical retailers, and stationary stochastic consumer demand. There are inventory holding costs and back-order penalty costs. We compare a traditional information policy that does not use shared information with a full information policy that does exploit shared information. In a numerical study we find that supply chain costs are 2.2% lower on average with the full information policy than with the traditional information policy, and the maximum difference is 12.1%. We also develop a simulation-based lower bound over all feasible policies. The cost difference between the traditional information policy and the lower bound is an upper bound on the value of information sharing: In the same study, that difference is 3.4% on average, and no more than 13.8%. We contrast the value of information sharing with two other benefits of information technology, faster and cheaper order processing, which lead to shorter lead times and smaller batch sizes, respectively. In our sample, cutting lead times nearly in half reduces costs by 21% on average, and cutting batches in half reduces costs by 22% on average. For the settings we study, we conclude that implementing information technology to accelerate and smooth the physical flow of goods through a supply chain is significantly more valuable than using information technology to expand the flow of information.

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1,790 citations

Journal Article•DOI•

Supply chain design and analysis:: Models and methods

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Benita M. Beamon¹•Institutions (1)

University of Cincinnati¹

15 Aug 1998-International Journal of Production Economics

TL;DR: In this article, the authors provide a focused review of literature in multi-stage supply chain modeling and define a research agenda for future research in this area, which is largely a result of the rising costs of manufacturing, the shrinking resources of manufacturing bases, shortened product life cycles, the leveling of the playing field within manufacturing, and the globalization of market economies.

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1,717 citations

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References

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Open Access

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The optimality of (s,s) policies in the dynamic inventory problem

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Herbert E. Scarf

10 Apr 1959

837 citations

"Optimal Policies for a Multi-Echelo..." refers background or methods in this paper

...) Then there exists a sequence of critical numbers Sn sn so that in period n it is optimal to order only if x1+· · ·+ w −1 < sn and if we do order, we order an amount Sn− x1+· · ·+w −1 (Scarf 1960)....
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...An extensive discussion of this model is given in Karlin and Scarf (1958), and we shall summarize here that material which will be of use to us....
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...Then there exists a sequence of critical numbers Sn sn so that in period n it is optimal to order only if x1+· · ·+ w −1 sn and if we do order, we order an amount Sn− x1+· · ·+w −1 (Scarf 1960)....
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...Using the type of reasoning employed in Karlin and Scarf (1958), this sequence of functions may be shown to satisfy the following functional equation:...
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Book•

Studies in the Mathematical Theory of Inventory and Production

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J. Johnston, Kenneth J. Arrow, Samuel Karlin, Herbert E. Scarf

01 Dec 1958

TL;DR: The Assessment Division of The Applied Physics Laboratory has undertaken new responsibilities and is expanding its Senior Analytical Staff as discussed by the authors, which will provide guide lines for the hardware research of future years, including analyses of tactical situations, the employment of future weapon systems and the application of the most recent advances in science and technology.

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Abstract: The Assessment Division of The Applied Physics Laboratory has undertaken new responsibilities and is expanding its Senior Analytical Staff. Senior Scientists in such fields as Mathematics, Physics and Physical Chemistry have in the past proven very effective in solving the types of problems involved which include analyses of tactical situations, the employment of future weapon systems and the application of the most recent advances in science and technology. Performance of the work requires close association with scientists of other laboratories, operations research personnel of all branches of the Armed Services, and with senior military and civilian personnel. Studies undertaken by this group will provide guide lines for the hardware research of future years. Staff members are expected to initiate ideas in support of a broad program of National Defense needs and carry them through appropriate analyses with assurance that sound results will be given consideration by the responsible agencies. The Laboratory's locale, equidistant between Baltimore and Washington, D. C , allows staff members to select urban, suburban or rural living and either of these two outstanding centers of culture as a focal point for fine living. These appointments offer exceptional opportunities. For information and arrangements for interview, write in confidence to:

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568 citations

Journal Article•DOI•

Studies in the Mathematical Theory of Inventory and Production

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Kenneth J. Arrow, Samuel Karlin, Herbert E. Scarf

01 Mar 1959-A Quarterly Journal of Operations Research

68 citations

"Optimal Policies for a Multi-Echelo..." refers background in this paper

...In the last several years there have been a number of papers (Arrow et al. 1958) discussing optimal policies for the inventory problem....
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...Introduction In the last several years there have been a number of papers (Arrow et al. 1958) discussing optimal policies for the inventory problem....
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