Twenty-Fifth Anniversary Multi-Echelon Inventory Conference Program Cornell University – The S. C. Johnson Graduate School of Management
June 16, 2002 7:30 AM – 8:30 AM: Continental Breakfast in the Dyson Atrium 8:30 AM – 10:15 AM – Service Parts Logistics and Networks in Room B09 Sage Hall Title Opening Remarks Optimal Stocking Policies for a Multi-Echelon Repairable Parts Logistics System A Multi-Echelon, Multi-Item Inventory Model for Service Parts Management with Generalized Service Level Constraints Optimal Inventory Placement in Networks with Commonality Author(s) Jack Muckstadt Ewing, Newman Caggiano, Jackson, Muckstadt, Rappold
Humair, Willems
10:15 AM – 10:30 PM: Break in the Dyson Atrium 10:30 AM – 12:00 Noon– Service Levels in Supply Chains in Room B09 Sage Hall Title Capacitated Serial Supply Systems: Approximations and Insights Fill Rates of Multi-Stage Supply Systems Optimal Policies for Single-Stage Inventory Systems with a Service Constraint Author(s) Gupta, Selvaraju Sobel Shang, Song
12:00 Noon – 1:00 PM: Lunch in the Dyson Atrium 1:00 PM – 2:30 PM –Multi-Echelon Inventory Policies in Room B09 Sage Hall Title Two-Echelon Inventory Control with Periodic Review, Lost Sales and Lead Times Optimal Control of Serial, Multi-Echelon Inventory/Production Systems with Periodic Batching Analysis of Ideal Assemble-To-Order Systems Author(s) Janakiraman and Muckstadt van Houtum, Scheller-Wolf, Yi de Kok
2:30 PM – 2:45 PM: Break in the Ramin Parlor 2:45 PM – 4:15 PM – Multi-Party Optimization in Multi-Echelon Settings in B09 Title A Real Beer Game: Make-To-Order Incentive Problem When to Commit in a Multi-Echelon Supply Chain with Partial Information Updating Inventory and Sourcing Strategies for Non-Commodities: The Role and the Extent of Use of Speculative Markets and Supply Contracts Author(s) Liu Fergusson, DeCroix, Zipkin Milner, Kouvelis
4:15 PM – 5:30 PM: Reception in the Dyson Atrium 5:30 PM – 7:00 PM: Banquet in the Dyson Atrium
Optimal Stocking Policies for a Multi-Echelon Repairable Parts Logistics System P. Lee Ewing Colorado School of Mines Alexandra Newman Colorado School of Mines We address an inventory problem faced by a major computer supplier that must maintain a competitive level of service on repair contracts for its parts. The company possesses a multi-echelon inventory system in which the lower-echelon warehouses provide customers directly with emergency repair parts while the upper-echelon warehouses repair parts returned from the lower-echelon and hold parts for replenishment of lower-echelon inventory. There are far fewer upper-echelon than lower-echelon warehouses. Inventory policies suggest where and how many critical repair parts the company should hold, and must balance the costs of holding inventory with customer service requirements. We expect new product introductions to be more reliable, resulting in reduced need for repair parts inventory. However, unit prices are expected to increase. Because repair parts can be extremely expensive, even minor reductions in repair parts inventory can save hundreds of millions of dollars over the next five years. We identify high cost, low demand parts that should be placed on a one-for-one, continuous review replacement policy, and develop the Analytical Stocking Model to determine inventory levels for these parts. Our model provides insights into relative quantities of parts that should be stored at both echelons, whether to store parts at some lower-echelon warehouses, and how to effectively use risk pooling. Specifically, we demonstrate that for the company we study, cost reductions of 60% are possible while maintaining the current level of service, or alternately, that system backorder reductions of over 97% are possible while keeping total inventory costs constant at their current level.
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A Multi-Echelon, Multi-Item Inventory Model for Service Parts Management with Generalized Service Level Constraints Kathryn E. Caggiano School of Business University of Wisconsin-Madison, Peter L. Jackson School of Operations Research and Industrial Engineering, Cornell University John A. Muckstadt School of Operations Research and Industrial Engineering Cornell University James A. Rappold School of Business University of Wisconsin-Madison In the realm of service parts management, customer relationships are often established through service agreements that extend over a period of months or years. These agreements typically apply to a product (or group of products) that the customer has purchased, and specify the type of service that will be provided, as well as the timing with which the service will take place. In the case of a customer that operates in multiple locations, service agreements may apply to several products across several locations. In this paper we describe a continuous review inventory model for a multi-item, multi-echelon distribution system for service parts in which service level constraints exist for general groups of items across multiple locations and distribution channels. In addition to instantaneous service level constraints, a special class of time-based service level constraints are also considered, in which the specified service times coincide with transport times from replenishment sites within the distribution network. We derive exact fill rate expressions for each item's distribution channel and describe a solution approach for determining optimal target inventory levels that meet all service level constraints at minimum investment.
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Optimal Inventory Placement in Networks with Commonality Salal Humair Optiant, Inc. Sean P. Willems Boston University
In this paper, we present an algorithm to optimize inventory placement in networks that have commonality. We define a cluster as a subgraph of k nodes that can have more than k-1 arcs connecting the nodes in the subgraph. A network is said to exhibit commonality if it contains one or more clusters. For networks where the clusters form a modified network that can be modeled as a spanning tree, we present an algorithm to optimally determine inventory levels in the network This work is an extension of the framework in Graves and Willems (2000) which presents an algorithm to optimize inventory placement in networks that can be modeled as spanning trees.
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Capacitated Serial Supply Systems: Approximations and Insights Diwakar Gupta Supply Chain and Operations Research Laboratory Department of Mechanical Engineering University of Minnesota N. Selvaraju Supply Chain and Operations Research Laboratory Department of Mechanical Engineering University of Minnesota Supply systems are often modeled as networks of queues and inventories. Each node in this network represents an activity that needs to be completed in order to process an order. Inventories are maintained in front of each processing stage; resulting in both semi-finished and finished goods inventories. These inventories are managed locally according to a base-stock control policy. It is difficult to obtain mathematically exact expressions for performance measures of such capacitated supply systems since the underlying queueing networks do not have a product-form structure except in some special cases. Approximations have been suggested in the literature to evaluate performance measures that are subsequently used to obtain optimal basestock levels and to gain insight into system behavior. In this work, we present a successive approximation scheme for serial supply systems that is more accurate than existing methods. We also provide a matrix-geometric computational procedure to determine nearly exact performance measures, which makes our comparisons with existing methods more accurate. The computational procedure is equivalent to assuming that each stage in the supply system operates according to a generalized kanban policy. We make use of the proposed approximation scheme to determine the optimal base-stock levels needed to minimize the total average cost of inventory and customer backorders. Through detailed numerical experiments, we report new insights into the system behavior. For example, contrary to an earlier conclusion that system cost is relatively insensitive to inventory positioning provided the overall stock level is about right, we observe this insensitivity only in the neighborhood of the optimal base-stock levels. In fact, in a two-stage system, the cost is sharply higher when stage-1 carries more than optimal amount of stock at the expense of smaller than optimal amount of stock at stage-2. Insights such as these help to sharpen some of the conclusions reported earlier in literature.
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Fill Rates of Multi-Stage Supply Systems Matthew J. Sobel Department of Operations Weatherhead School of Management Case Western Reserve University The fill rate, the average fraction of demand that is met from on-hand inventory, is an important measure of a supply system’s service quality. This talk presents formulas and bounds for the fill rate of periodic review supply systems having a series structure with buffer inventories between stages. Each period, quantities of items are released from the buffer inventories for further processing downstream. The system is capacitated if there are bounds on the quantities that can be processed during a period. Otherwise, it is uncapacitated. The released quantities are selected with a modified base-stock-level policy and end-item demands are nonnegative independent and identically distributed random variables (i.i.d.r.v.s). It is easy to calculate the fill rate and easier to calculate bounds on it when demand has an empirical discrete distribution. The research literature concerning fill rates concentrates on single stage supply systems, although many demand processes are met from inventories at the end of multi-stage systems. Several papers by Paul Glasserman and coauthors develop asymptotic bounds and approximations for the fill rate that are simple analytical expressions for capacitated multi-stage systems. In comparison, the fill rate formulas and bounds in this talk are exact but they are not simple analytical expressions. One set of byproducts of the analysis (for which there is no time in this talk) are useful fill rate results for single-stage models. These include fill rate formulas for general distributions of demand, gamma distributions of demand, and normal distributions of demand. In the normal case, a good approximation uses only the standard normal distribution function and an exact expression uses only the normal loss table. Another byproduct of the analysis is the strategically important conclusion that shorter supply chains have higher fill rates.
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Optimal Policies for Single-Stage Inventory Systems with a Service Constraint Kevin H. Shang Graduate School of Management University of California, Irvine Jing-Sheng Song Graduate School of Management University of California, Irvine It is well known that an (s, S) policy is optimal for periodic-review, single-stage inventory systems with a fixed plus linear order cost and linear holding and backorder costs. When there is no fixed order cost, the policy reduces to a base-stock policy with base-stock level S = s. The structure of the optimal policy is less known, however, if one replaces the backorder cost by a service level constraint. The purpose of the current study is to shed light on this issue. We consider a periodic-review, single-stage inventory system with random demands, a constant leadtime, a fixed plus linear order cost, and a linear holding cost. Instead of backorder costs, there is a fill-rate constraint in each period. All costs, fill-rate requirements, and demands may be non-stationary. Our objective is to minimize the expected total discounted cost for both finite and infinite horizons while satisfying the pre-specified fill-rate target. We show that the structure of the optimal policy is basically the same as in the backorder-cost model. When the order cost is linear in the quantity ordered, a base-stock policy is optimal. The optimal base-stock level is obtained by choosing the maximum value of two solutions: y, the minimum stocking level to satisfy the fill-rate requirement and S*, the stocking level that minimizes the expected total discounted cost without the fill-rate constraint of the current period. Similarly, when there is a fixed order cost, an (s, S) policy is optimal. Again, each policy parameter is obtained by taking the maximum value of two solutions: y and the corresponding parameter for the model ignoring the current period fill-rate constraint. Algorithms are also developed to compute the optimal policy.
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Two-Echelon Inventory Control with Periodic Review, Lost Sales and Lead Times Ganesh Janakiraman School of Operations Research and Industrial Engineering Cornell University John Muckstadt School of Operations Research and Industrial Engineering Cornell University We examine a two-echelon serial system, in which procurement decisions are made in each period of a finite planning horizon. There is a lead time of one period at both echelons. Excess demand is lost. Holding costs and lost-sales costs are charged at the end of every period. We are interested in analyzing the optimal inventory policy. Unfortunately, the optimal policy is quite complicated even for single location problems with lost-sales and lead times; some properties of the optimal policy and bounding functions are known. Consequently, it is clear that the structure of the optimal policy for the two-echelon problem would be complicated as well. In this paper, we formulate the problem as a stochastic dynamic program and prove basic properties of the optimal policy and the cost function. In addition, we compare the “lost sales problem” with the “backorder problem” and develop some insight and intuition to help understand why the former is difficult while the latter is structurally very simple.
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Optimal Control of Serial, Multi-Echelon Inventory/Production Systems with Periodic Batching Geert-Jan van Houtum Eindhoven University of Technology Alan Scheller-Wolf Carnegie Mellon University Yinxin Yi Carnegie Mellon University
Several results are known on optimal inventory control policies for single-item, multi-echelon inventory/production systems with linear inventory holding costs and either linear backorder costs or a service level constraint. For the serial multi-echelon system as studied by Clark and Scarf (1960), we know that: (i) basestock policies are optimal; (ii) the optimal basestock levels follow from Newsboy equations; (iii) algorithms are available for efficient exact computations. The same results are available for assembly systems. Distribution systems are more complicated and hence less is known for them. In real-life supply chains, batching may be needed for various reasons. In a supply chain, we distinguish transport nodes and production nodes. Transport nodes move materials from a stockpoint to a next downstream stockpoint by internal transport, trucks, ships or airplanes. At those nodes it may be important to have full pallets of products, full truckloads, and so on. Production nodes take materials from a stockpoint, apply certain production (or assembly) processes, and store the resulting materials in the next downstream stockpoint. Set-up or switching times often precede the real processing, and hence also in production nodes appropriate batching is required. Batching can be obtained in different ways. One extreme form is constituted by fixed batch sizes. This form has been investigated extensively in the literature on multi-echelon systems. Another extreme form is constituted by periodic batching, under which each stockpoint is allowed to place orders at equidistant time instants and the reorder interval may differ per stage. This form has hardly been studied in the literature, while it may be more appropriate for several real-life situations. In this talk, we consider the Clark-and-Scarf system with periodic batching, and we show that all results mentioned above can be generalized.
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Analysis of Ideal Assemble-To-Order Systems Prof. Dr. Ton G. de Kok Professor of Operations Planning and Control Director, European Supply Chain Forum Department of Technology Management Eindhoven In this presentation we consider two-stage assemble-to-order (ATO) systems, i.e. product demand is known before actual assembly of the product. Therefore only stocks of components are held. We assume that demand for individual products in consecutive time periods is i.i.d.. Demand for different products in the same period may be correlated. Lead times of components are assumed to be constant. The objective of the analysis is to derive closed-form expressions and accurate approximations for the service level, defined as the probability that demand for an end-product can be satisfied immediately, and average component physical stocks. By imposing a structural constraint on the ATO system we are able to characterize the service level as a general finite horizon ruin probability. The structural constraint states that long lead time components are more common then short lead time components. We call structures satisfying this constraint ideal. The reason to call these structures ideal is that if a component has a long lead time then from a cost perspective it is important that the lead time demand can be accurately forecast. This typically requires commonality, i.e. the forecast for the lead time demand is derived from aggregate product family forecasts, which are usually more accurate then forecast for single products. Various authors have shown that the service level can be calculated under the assumption of multivariate normal demand. However, fitting normal distributions to end-product demand requires low coefficients of variation of this demand, whereas ATO systems emerge when end-product demand is erratic. The ruin probability characterization mentioned above enables an approximate, yet accurate, analysis of ideal ATO systems under gamma-distributed demand. Based on this result we assess the impact of the multivariate normal assumption and we provide managerial insights into the relationship between product availability and component availability.
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A Real Beer Game: Make-To-Order Incentive Problem John J. Liu School of Business Administration University of Wisconsin-Milwaukee We study a JIT beer-brewery supply chain, where an authorized wholesaler submits periodic orders for bottled beer to a beer brewer who then produces to the order and delivers it within a fixed order cycle. Comparing to the pedagogical Beer Game as played in classrooms, the real “beer game” bears some differing features and issues: 1) There is a Stackelberg asymmetry in the order mechanism. For example, the “reward” for the wholesaler is induced via wholesaler’s order decisions, while the realization of the reward is up to how the brewer can deliver. Such reactive competition is referred as incentive problem in game theory. 2) Both parties can have access to the past data, based on which forecast is essential for both to make cycle-by-cycle order decisions. In this paper, we develop a single-product make-to-order incentive (MTOI) model, which incorporates both incentive control and state prediction. With incentive-controllability, we demonstrate and then argue that downstream information sharing can be unnecessary in incentive problems. The model is readily to be extended to multi-product MTOI problems. Keywords: Stochastic Incentive Game, feedback Stackelberg Solution, Kalman Filter
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When to Commit in a Multi-Echelon Supply Chain with Partial Information Updating Mark Ferguson Georgia Institute of Technology Greg DeCroix Duke University Paul Zipkin Duke University Negotiations between an end product manufacturer and a parts supplier often revolve around two main issues: the supplier's price and the length of time the manufacturer is contractually held to its order quantity, commonly termed the “commitment time frame”. Because actual demand is unknown, the specification of the commitment time frame determines how the demand risk is shared among the members of the supply chain. Casual observation indicates that most manufacturers prefer to delay commitments as long as possible while suppliers prefer early commitments. In this paper, we investigate whether these goals are always in the firm's best interest. In particular, we find that the manufacturer may sometimes be better off with a contract that requires an early commitment to its order quantity, before the supplier commits resources and the supplier may sometimes be better off with a delayed commitment. We also find that the preferred commitment time frame depends upon which member of the supply chain has the power to set their exchange price.
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Inventory and Sourcing Strategies for Non-Commodities: The Role and the Extent of Use of Speculative Markets and Supply Contracts Joseph M. Milner Olin School of Business Washington University Panos Kouvelis Olin School of Business Washington University The focus of our research is to better understand the role of speculators and their impact on suppliermanufacturer relationships. Speculators, using Business-to-Business exchanges (B2B) have attempted over the last several years to provide supply for non-commodity goods. These exchanges were intended to provide great benefits in supply chains by improving the matching of supply and demand. These efforts, to a great degree, have failed generally because many B2B exchanges ignored the increasing role that long term relationships play in supply chains. Through this research we hope to elucidate some aspects of how speculative markets for noncommodity goods might behave and the impact they will have on the use of long-term contracts for supply. We consider an N-firm model where supply is either purchased through a standing-order contract from a manufacturer or is purchased from a spot market where supply is held by speculators. Under the contract a fixed number of units are delivered to the firm each period. The price in the spot market is determined by considerations of competitive behavior among a set of speculators. We endogenize the spot market price by considering a market clearing price function determined by inventory cost parameters and expectations of future demand. The spot market price is then determined as the equilibrium price function under common demand beliefs and the future value of inventory. Using the model we investigate the degree to which firms should engage in long-term contracts vis-à-vis spot markets. We show that the standing order contract quantity should be chosen to equalize the marginal benefit of the contract over the spot market with the expected marginal cost of not utilizing the market because of excess inventory. As the number of firms participating in the market increases, we find that the speculators are able to exploit the benefits of inventory pooling and inventory imbalances. This leads to network benefits of decreasing prices and increasing reliance by firms on the spot. We also study how flexibility in supply contracts could be used to limit the extent of use of spot markets and increase the reliance on long-term contracts.
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