Search

Problem definition: We study profit allocation for a sourcing network, in which a buyer sources from a set of differentiated suppliers with limited capacity under uncertain demand for the final product. Whereas the buyer takes the lead in forming the sourcing network and designing the contract mechanism, due to their substantial bargaining power, the suppliers take the lead in determining the terms of the contract. Academic/practical relevance: We identify contracting mechanisms that will ensure the stability of the sourcing network in the long term, where a stable sourcing network requires an effective profit-allocation scheme that motivates all members to join and stay in the network. Methodology: We apply methods from game theory to model the network and analyze the Nash equilibrium of a noncooperative game under a proposed contracting mechanism. We then use a cooperative game model to study the stability of the resulting equilibrium. Results: We show that the optimal network profit, as a set function of the set of suppliers, is submodular, which allows us to demonstrate that the core of the cooperative game is not empty. We also establish a set of conditions that are equivalent to, but much simpler than, the original conditions for the core. We use these results to demonstrate that the proposed fixed-fee contracting mechanism can implement a stable network in the competitive setting by achieving a profit allocation that is in the core of the cooperative game. We also demonstrate that the grand coalition is stable in a farsighted sense under the Shapley value allocation. Managerial implications: Under the fixed-fee mechanism, the buyer’s decisions maximize the network profit, and each supplier earns a profit equal to its marginal contribution. When the aggregate capacity of the supplier network is high relative to demand, or demand is more likely to be small, the fixed-fee mechanism is likely to outperform the Shapley value allocation from the perspective of the buyer.

Problem definition: We study the optimal inventory ordering, expediting, and allocation decisions in a multiechelon supply chain over a finite horizon, in which customer orders are quoted with a fixed fulfillment time window, termed the service time target (STT). Academic/practical relevance: Service time target is commonly used as a marketing strategy to increase customer satisfaction and strengthen firms’ competitive edge. However, how to efficiently manage a multistage supply chain to meet the target has received relatively scant attention in the literature. Our study fills this gap. Methodology: We use dynamic programming to characterize the optimal policy. Results: We show that an echelon base stock policy and a rationing policy are optimal for inventory ordering and inventory allocation/expediting, respectively. We also develop a polynomial-time algorithm to compute the optimal policy. To derive these results, we uncover a new functional property named the decomposable of degree 2 property, which is a nontrivial generalization of the celebrated Clark-Scarf decomposition. This property further allows us to derive induced penalty and compensation to coordinate a decentralized serial system with STT and expediting. Managerial implications: Our result provides an efficient decision tool for managing centralized and decentralized serial supply chains with STT and expediting. Our model can be used to quantify the tradeoff between marketing and operational decisions, such as the impact of a marginal reduction in STT on system cost and expedition frequency, as explored in our numerical studies.

Problem definition: We consider an OEM supplying a single part to multiple buyers over an infinite horizon. We study how the OEM can digitize the spare parts supply chain by leveraging 3D printing via intellectual property (IP) licensing. Academic/practical relevance: Additive manufacturing, also known as 3D printing, has the potential to shift supply chains from global networks that rely on centralized production with traditional manufacturing technologies to mainly digital networks with distributed, local printing. Particularly well positioned to drive this transition are original equipment manufacturers (OEMs) who design and produce capital goods. Methodology: We first set up a benchmark model of the traditional physical supply chain with centralized production by the OEM. We then propose the OEM to act as an IP licensor by selling spare parts designs, rather than physical parts. With the license agreement, a buyer can print parts locally through a third-party printing service provider, enjoying a shorter lead time and lower setup cost. Given a license, each buyer chooses whether to switch to the IP licensing channel or stay in the traditional channel. The OEM selects the license terms to maximize the total profit across both channels. Results: We characterize the OEM’s optimal license and the resulting supply chain configuration. We show that 3D printing’s competency in price plays a dominant role in production decentral- ization. The conventional wisdom that 3D printing is suitable for slow moving parts holds only when it is competent along several dimensions. In a numerical experiment with realistic parameter settings, decentralized production occurs in a surprisingly large number of cases, and the proposed new business model can significantly increase the OEM’s profit. Managerial implications: Our results indicate that IP licensing by OEMs can become a major enabler in the transition to digital supply networks with distributed 3D printing, benefiting all parties involved.

We develop a game-theoretic model to explore why retail clusters are so popular in developing economies and when governments should facilitate the formation of retail clusters to improve social welfare. First, we find two determinants of retailer clusters: the valuation-cost ratio (consumers’ maximum valuation over retailers’ production cost) and retailer density (the number of retailers over unit transportation cost). The valuation-cost ratio and retailer density indicate retailers’ profit potential and competition intensity, respectively. Second, the equilibrium cluster size increases in the valuation-cost ratio. This finding explains the phenomenon that clusters are usually larger in developing economies (where numerous retailers sell unrecognized brands with low profit potential) than in developed economies. Third, when the retailer density of a product market exceeds a certain threshold, the market coverages of clusters overlap with each other (i.e., the overlapping case). Furthermore, when compared with the nonoverlapping case, the equilibrium cluster size in the overlapping case is larger for low-profit-potential products but smaller for high-profit-potential products. Together, valuation-cost ratio and retailer density define four types of clusters: overlapping massive clusters, nonoverlapping large clusters, nonoverlapping small clusters, and overlapping mini-clusters. Finally, the socially optimal cluster size is larger than the equilibrium cluster size, and the gap between these two cluster sizes decreases in the valuation-cost ratio. The online appendix is available at https://doi.org/10.1287/msom.2017.0663 . This paper has been accepted for the Manufacturing & Service Operations Management Special Issue on Value Chain Innovations in Developing Economies.

We study an inventory system with multiple supply sources and expediting options. The replenishment lead times from each supply source are stochastic, representing congestion and disruption. We construct a family of smart ordering and expediting policies that utilize real-time supply information. Such dynamic policies are generally difficult to evaluate, because the corresponding supply system is a tandem queue with state-dependent arrivals and routing, whose queue-length steady-state distribution is usually not in product form. Our main result is to identify two appealing special cases of the general policy, Policy-M and Policy-E, which possess simple product-form solutions and lead to closed-form performance measures. Policy-M retains full sourcing flexibility, but ignores upstream congestion in making expediting decisions. Policy-E only orders from the normal, farthest source, but makes expediting decisions based on both upstream and downstream information. A numerical study shows that the best Policy-M leads to a lower average cost than the best Policy-E in almost all cases. Also, implementing the best Policy-M parameters, the general policy only performs slightly better than Policy-M. These observations reveal the value of combining sourcing flexibility with some, but limited, dynamic expediting. Our findings are equally applicable to the equivalent tandem queue. They thus may aid dynamic routing and expediting decisions for online retailers and logistics providers, among others.

We consider a single-product, two-source inventory system with Poisson demand and backlogging. Inventory can be replenished through a normal supply source, which consists of a two-stage tandem queue with exponential production time at each stage. We can also place an emergency order by skipping the first stage, for a fee. There is no fixed order cost. There are linear order, holding, and back-order costs. Through a new approach, we obtain optimal ordering policies for the discounted or long-run average cost and also characterize near-optimal heuristic policies. The approach consists of four steps. The first step is to establish an equivalent system, in the sense that it has the same optimal policy as the original system. The second step is to construct a tandem queueing system, where costs are charged in accord with the equivalent system’s cost structure. The third step derives an optimal control of the service rate at each server so as to minimize the tandem queue’s system-wide cost. The fourth and final step is to translate the queue’s optimal policy to an optimal policy for the equivalent system and hence the original system.

Partial cross ownership (PCO) in a dyad supply chain refers to a situation where each firm holds a portion of its partner’s shares. We study this topic in push and pull supply chains, and prove that neither the supply chain’s nor any member’s profit changes with the percentage of the leader’s shares the follower holds. However, while the profits of the chain and the leader increase with the percentage of the follower’s shares held by the leader, the follower’s profit does not necessarily increase or decrease. As a result, both partners can always achieve a win–win by setting a proper price for transferring the follower’s shares to the leader. Moreover, the equilibrium wholesale price may be greater than the retail price in the push chain, but less than the marginal production cost in the pull chain, contradicting the usual results found in the literature. We also derive a necessary and sufficient condition on the structure of PCO allowing a pull chain to perform better than a push one. This extends Cachon (2004)’s result that a pull chain always performs better than a push one (without PCO). Finally, PCO can coordinate a chain if and only if each one holds half of the other’s shares.

Kuribo et al. (2019) studies a capacity provider and a group of independent buyers sharing a scarce but expensive-to-build capacity over a finite horizon. At the beginning of the horizon, the capac...

We consider single-item (r, q) and (s, T) inventory systems with integer-valued demand processes. While most of the inventory literature studies continuous approximations of these models and establishes joint convexity properties of the policy parameters in the continuous space, we show that these properties no longer hold in the discrete space, in the sense of linear interpolation extension and L♮-convexity. This nonconvexity can lead to failure of optimization techniques based on local optimality to obtain the optimal inventory policies. It can also make certain comparative properties established previously using continuous variables invalid. We revise these properties in the discrete space.

A supply stream is a continuous version of a supply chain. It is like a series inventory system, but stock can be held at any point along a continuum, not just at discrete stages. We assume stationary parameters and aim to minimize the long-run average total cost. We show that a stationary continuous-stage echelon base-stock policy is optimal. That is, at each geographic point along the supply stream, there is a target echelon inventory level, and the optimal policy at all times is to order and dispatch material so as to move the echelon inventory position as close as possible to this target. We establish this result by showing that the solutions to certain discrete-stage systems converge monotonically to a limit, as the distances between the stages become small, and this limit solves the continuous-stage system. With demand approximated by a Brownian motion, we show that, in the continuous-stage limit, the supply stream model is equivalent to one describing first-passage times. This linkage leads to some interesting and useful results. Specifically, we obtain a partial differential equation that characterizes the optimal cost function, and we find a closed-form expression for the optimal echelon base-stock levels in a certain special case, the first in the inventory literature. These expressions demonstrate that the well-known square-root law for safety stock does not apply in this context.

We present a general framework to study the design of spare parts logistics in the presence of three-dimensional (3-D) printing technology. We consider multiple parts facing stochastic demands and adopt procure/manufacture-to-stock versus print-on-demand to highlight the main difference of production modes featured in traditional manufacturing and 3-D printing. To minimize long-run average system cost, our model determines which parts to stock and which to print. We find that the optimal 3-D printer’s utilization increases as the additional unit cost of printing declines and the printing speed improves. The rate of increase, however, decays, demonstrating the well-known diminishing returns effect. We also find the optimal utilization to increase in part variety and decrease in part criticality, suggesting the value of 3-D technology in tolerating large part variety and the value of inventory for critical parts. By examining the percentage cost savings enabled by 3-D printing, we find that, although the reduction in printing cost continuously adds to the value of 3-D printing in a linear fashion, the impact of the improvement of printing speed exhibits S-shaped growth. We also derive various structural properties of the problem and devise an efficient algorithm to obtain near optimal solutions. Finally, our numerical study shows that the 3-D printer is, in general, lightly used under realistic parameter settings but results in significant cost savings, suggesting complementarity between stock and print in cost minimization. This paper was accepted by Victor Martínez-de-Albéniz, operations management.

This paper studies the optimization of the (S, T) inventory policy, where T is the replenishment interval and S is the order-up-to level. First, we demonstrate that the previously established joint convexity of the long-run average cost is false. Hence, the optimization is not straightforward. We then point out that the joint convexity concept depends on whether S and T are continuous or discrete variables, and in some situations it may not even be well defined. Nonetheless, we are able to identify several useful properties of the cost function, such as submodularity and coordinatewise convexity. Based on these properties, we develop efficient algorithms to compute the optimal policy for continuous and discrete demands.

We propose an approximation for the inverse first passage time problem. It is similar in spirit and method to the tangent approximation for the original first passage time problem. We provide evidence that the technique is quite accurate in many cases. We also identify some cases where the approximation performs poorly.

We consider a three-tier supply chain consisting of an original equipment manufacturer (OEM), a contract manufacturer (CM) and a supplier. We analyze and compare three outsourcing structures that are currently implemented by top-tier OEMs: (1) inhouse consignment, under which the OEM signs independent contracts with the CM and the supplier; (2) turnkey with integration, under which the OEM contracts with an alliance of the CM and the supplier; and (3) turnkey, under which the OEM contracts with the CM, and the CM then subcontracts with the supplier. The OEM is a Stackelberg leader who decides how much of the end product to produce. All parties use take-it-or-leave-it wholesale-price contracts. Both the CM and the supplier have private information about their own production costs. The OEM has prior information about these costs, but this prior information depends on the specific outsourcing structure. Each party's optimal decision is characterized. We then compare each party's profits across the three outsourcing structures and identify which benefits and when.

We study a single-item (r, q) inventory system, where r is the reorder point and q is the order quantity. The demand is a compound-Poisson process. We investigate the behavior of the optimal policy parameters and the long-run average cost of the system in response to stochastically shorter or less-variable lead times. We show that although some of the properties of the base-stock system can be extended to this more general model, some cannot. The same findings also apply when the comparison is conducted on the lead-time demand distributions.

Free riding in a multichannel supply chain occurs when one retail channel engages in the customer service activities necessary to sell a product, while another channel benefits from those activities by making the final sale. Although free riding is, in general, considered to have a negative impact on supply chain performance, certain recent industry practices suggest an opposite view: a manufacturer may purposely induce free riding by setting up a high-cost, customer service-oriented direct store to allow consumers to experience the product, anticipating their purchase at a retail store. This article examines how the free riding phenom- enon affects a manufacturer's supply chain structure decision when there are fixed plus incremental variable costs for operating the direct store. We consider factors such as the effort required to find and buy the product at a retail store after visiting the direct store, the existence of competing products in the market, and the extent of consumer need to obtain direct-store service. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 00: 000-000, 2009

This paper studies a periodic-review, serial supply chain in which materials are ordered and shipped according to (R,nQ) policies. Three information scenarios are considered, depending on the level of information available: echelon, local, and quasilocal. In the echelon scenario, each stage can access the inventory and cost information within its echelon (comprising the stage itself and all downstream stages); in the local scenario, each stage only accesses its own local information. Finally, in the quasilocal scenario, each stage knows its local information, plus the actual customer demands. We propose coordination schemes that regulate the stages to achieve the supply chain's optimal cost under each information setting. All these coordination schemes fit comfortably within an emerging practice called supply chain finance, which includes the organization and technology needed to implement them.

Consider an inventory system with multiple supply sources and Poisson demand. The replenishment lead times from each source are stochastic, representing congestion and disruption. We develop performance evaluation and optimization tools for a family of reasonable order policies. These policies take into account real-time supply information, which can be obtained through tracking technologies such as global positioning systems and radio frequency identification. Performance evaluation of such state-dependent policies is generally hard. The main thrust of this paper is to show that, under these policies, the supply system becomes a network of queues with a special routing mechanism called an overflow bypass. The solution has a simple product form. Thus, we obtain closed-form performance measures. These results reinterpret and extend the existing analysis of a system with two sources having deterministic lead times. We further extend the analysis to batch ordering policies, non-Poisson demand processes, and multiple demand classes.

Component commonality has been widely recognized as a key factor in achieving product variety at low cost. Yet the theory on the value of component commonality is rather limited in the inventory literature. The existing results were built primarily on single-period models or periodic-review models with zero lead times. In this paper, we consider a continuous-review system with positive lead times. We find that although component commonality is in general beneficial, its value depends strongly on component costs, lead times, and dynamic allocation rules. Under certain conditions, several previous findings based on static models do not hold. In particular, component commonality does not always generate inventory benefits under certain commonly used allocation rules. We provide insight on when component commonality generates inventory benefits and when it may not. We further establish some asymptotic properties that connect component lead times and costs to the impact of component commonality. Through numerical studies, we demonstrate the value of commonality and its sensitivity to various system parameters in between the asymptotic limits. In addition, we show how to evaluate the system under a new allocation rule, a modified version of the standard first-in-first-out rule.

We consider a multiproduct assemble-to-order (ATO) system, in which inventory is kept only at the component level and the finished products are assembled in response to customer demands. In addition to stochastic demand for finished products, the system experiences stochastic returns of subsets of components, which can then be used to satisfy subsequent demands. The system is managed over an infinite horizon using a component-level base-stock policy. We identify several ways in which returns complicate the behavior of the system, and we demonstrate how to handle these additional complexities when calculating or approximating key order-based performance metrics, including the immediate fill rate, the fill rate within a time window, and average backorders. We also present a method for computing a near-optimal base-stock policy. We use these results to address managerial questions on both operational and product-design levels. For example, we find that tracking product-based (as opposed to component-based) return information appears to provide much less value than tracking product-based demand information, and we explore the impact of the number of products, component lead times, and different patterns of component returns (joint versus independent returns, returns of common versus dedicated components) on the value of component commonality.

We consider a multiperiod inventory system of a perishable product with unobservable lost sales. Demand distribution parameters are unknown and are updated periodically using the Bayesian approach based on the censored historical sales data. We develop an explicit expression of the first-order condition for optimality that demonstrates the key trade-off of the problem. The result generalizes partial characterizations of this trade-off in the literature. It shows that the myopic solution is a lower bound on the optimal inventory level. It also enables us to quantify the expected marginal value of information.

The Internet has provided traditional retailers a new means with which to serve customers. Consequently, many “bricks-and-mortar” retailers have transformed to “clicks-and-mortar” by incorporating Internet sales. Examples of companies making such a transition include Best Buy, Wal-Mart, Barnes & Noble, etc. Despite the increasing prevalence of this practice, several fundamental questions remain: (1) Does it pay off to go online? (2) Which is the equilibrium industry structure? (3) What is the implication of this business model for consumers? We study these issues in an oligopoly setting and show that clicks-and-mortar arises as the equilibrium channel structure. However, we find that this equilibrium does not necessarily imply higher profits for the firms: in some cases, rather, it emerges as a strategic necessity. Consumers are generally better off with clicks-and-mortar retailers. If firms align with pure e-tailers to reach the online market, we show that a prisoner’s dilemma-type equilibrium may arise.

We consider two models of stochastic serial inventory systems with economies of scale for which the forms of optimal policies are known. In the first model, each stage has a fixed-order quantity, while in the second model, there is a fixed-order cost for external supplies. For each model, we show that the optimal policy parameters can be bounded and approximated by a series of independent, single-stage optimal policy parameters. We further construct closed-form bounds and approximations for the single-stage solutions and apply them to the serial systems. These results provide simple and effective solutions that will help to facilitate implementations in practice. They also allow us to see the connections between the serial and single-stage systems and sharpen our intuition on optimal policy parameters and system behavior.

We study channel coordination policies for products subject to midlife price declines during their short product life cycles. Using a two-period supply chain model consisting of one supplier and on...

We analyze a serial base-stock inventory model with Poisson demand and a fill-rate constraint. Our objective is to gain insights into the linkage between the stages to facilitate optimal system design and decentralized system control. To this end, we develop a closed-form approximation for the optimal base-stock levels. The development consists of two key steps: (1) convert the service-constrained model into a backorder cost model by imputing an appropriate backorder cost rate, and then adapt the single-stage approximation developed for the latter, and (2) use a logistic distribution to approximate the lead-time demand distribution in the single-stage approximation obtained in (1) to yield closed-form expressions. We then use the closed-form expressions to conduct sensitivity analyses and establish qualitative properties on system design issues, such as optimal total system stock, stock positioning, and internal fill rates. The closed-form approximation and most of the qualitative properties apply equally to the model with a backorder cost, although some differences do exist. Other results of this study include a bottom-up recursive procedure to evaluate any given echelon base-stock policy and lower bounds on the optimal echelon base-stock levels.

We consider a multiproduct assemble-to-order system. Components are built to stock with inventory controlled by base-stock rules, but the final products are assembled to order. Customer orders of each product follow a batch Poisson process. The leadtimes for replenishing component inventory are stochastic. We study the optimal allocation of a given budget among component inventories so as to minimize a weighted average of backorders over product types. We derive easy-to-compute bounds and approximations for the expected number of backorders and use them to formulate surrogate optimization problems. Efficient algorithms are developed to solve these problems, and numerical examples illustrate the effectiveness of the bounds and approximations. Contributed by the Supply Chains/Production-Inventory Systems Department

This paper analyzes a series inventory system with stationary costs and stochastic demand over an infinite horizon. A distinctive feature is that demand can be negative, representing returns from customers, as well as zero or positive. We observe that, as in a system with nonnegative demand, a stationary echelon base-stock policy is optimal here. However, the steady-state behavior of the system under such a policy is different from that in systems with nonnegative demands. We present an exact procedure and several approximations for computing the operating characteristics and system costs for any stationary echelon base-stock policy, and also describe an algorithm for computing a good policy. As an alternative to the echelon base-stock policy, we discuss a policy that uses only local information. Finally, we describe how to extend the analysis to the case where returns occur at multiple stages instead of just at the stage closest to demand, and the case where returns require a recovery lead time.

We study a multi-item stochastic inventory system in which customers may order different but possibly overlapping subsets of items, such as a multiproduct assemble-to-order system. The goal is to determine the right base-stock level for each item and to identify the key driving factors. We formulate a cost-minimization model with order-based backorder costs and compare it with the standard single-item, newsvendor-type model with item-based backorder cost. We show that the solution of the former can be bounded by that of the latter with appropriately imputed parameters. Starting with this upper bound, the optimal base-stock levels of the order-based problem can be obtained in a greedy fashion. We also show that the optimal base-stock levels increase in replenishment lead times but may increase or decrease in lead-time variability and demand correlation. Finally, we devise closed-form approximations of the optimal base-stock levels to see more clearly their dependence on the system parameters.

As widely accepted performance measures in supply chain management practice, frequency-based service levels such as fill rate and stockout rate are often considered in supply contracts under vendor-managed-inventory (VMI) programs. Using a decentralized two-party capacitated supply chain model consisting of one manufacturer and one supplier in a VMI environment, we demonstrate that supplier's service level is in general insufficient for the manufacturer to warrant the desired service level at the customer end. The method by which the supplier achieves her service level to the manufacturer also affects customer service level. By developing bounds on the customer service level, we show that the expected backorders at the supplier should also be taken into account. We suggest a supply contract that offers a menu of different combinations of supplier's service level and expected backorders according to a linear function. Under this contract, the manufacturer can control the end customer service regardless of how the supplier manages her inventory. The supplier has complete flexibility on which combination of the two quantities on the menu to choose according to her own cost functions. Because it does not require any detailed information on supplier's operational characteristics nor her costs, this kind of contract is expected to be easily implementable. In addition, we derive an estimate of the customer service level in terms of the new measures. Our findings have direct implications to supply chain metrics in general: The local service levels are insufficient measures to guarantee the system wide performance. Alternative local measures and/or coordination mechanisms should be employed to achieve desired system performance. Our analysis illustrates a possible way to explore such alternative measures.

This paper is concerned with the optimal location of a central warehouse, given a fixed number and the locations of the local warehouses. We investigate whether the solution determined by the traditional model that minimizes total transportation cost differs from the one determined by a model that also takes into account the inventory and service costs. We build simple models to address this question. Numerical results show that ignoring inventory costs in modeling location models may lead to inferior location solutions.

We consider an assemble-to-order (ATO) system: Components are made to stock by production facilities with finite capacities, and final products are assembled only in response to customers' orders. The key performance measures in this system, such as order fill rates, involve evaluation of multivariate probability distributions, which is computationally demanding if not intractable. The purpose of this paper is to develop computationally efficient performance estimates. We examine several ideas scattered in diverse literatures on approximations for multivariate probability distributions, and determine which approach is most effective in the ATO application. To do so, we first tailor different approximation ideas to the ATO setting to derive performance bounds, and then compare these bounds theoretically and numerically. The bounds also allow us to make connections between capacitated and uncapacitated ATO systems and gain various insights.

We study an assemble-to-order system with stochastic leadtimes for component replenishment. There are multiple product types, of which orders arrive at the system following batch Poisson processes. Base-stock policies are used to control component inventories. We analyze the system as a set of queues driven by a common, multiclass batch Poisson input, and derive the joint queue-length distribution. The result leads to simple, closed-form expressions of the first two moments, in particular the covariances, which capture the dependence structure of the system. Based on the joint distribution and the moments, we derive easy-to-compute approximations and bounds for the order fulfillment performance measures. We also examine the impact of demand and leadtime variability, and investigate the value of advance demand information.

We consider the classic N-stage serial supply systems with linear costs and stationary random demands. There are deterministic transportation leadtimes between stages, and unsatisfied demands are backlogged. The optimal inventory policy for this system is known to be an echelon base-stock policy, which can be computed through minimizing N nested convex functions recursively. To identify the key determinants of the optimal policy, we develop a simple and surprisingly good heuristic. This method minimizes 2Nseparate newsvendor-type cost functions, each of which uses the original problem data only. These functions are lower and upper bounds for the echelon cost functions; their minimizers form bounds for the optimal echelon base-stock levels. The heuristic is the simple average of the solution bounds. In extensive numerical experiments, the average relative error of the heuristic is 0.24%, with the maximum error less than 1.5%. The bounds and the heuristic, which can be easily obtained by simple spreadsheet calculations, enhance the accessibility and implementability of the multiechelon inventory theory. More importantly, the closed-form expressions provide an analytical tool for us to gain insights into issues such as system bottlenecks, effects of system parameters, and coordination mechanisms in decentralized systems.

We study a single-product assembly system in which the final product is assembled to order whereas the components (subassemblies) are built to stock. Customer demand follows a Poisson process, and replenishment lead times for each component are independent and identically distributed random variables. For any given base-stock policy, the exact performance analysis reduces to the evaluation of a set of M/G/∞ queues with a common arrival stream. We show that unlike the standard M/G/∞ queueing system, lead time (service time) variability degrades performance in this assembly system. We also show that it is desirable to keep higher base-stock levels for components with longer mean lead times (and lower unit costs). We derive easy-to-compute performance bounds and use them as surrogates for the performance measures in several optimization problems that seek the best trade-off between inventory and customer service. Greedy-type algorithms are developed to solve the surrogate problems. Numerical examples indicate that these algorithms provide efficient solutions and valuable insights to the optimal inventory/service trade-off in the original problems.

This paper considers a multistage serial inventory system with Markov-modulated demand. Random demand arises at Stage 1, Stage 1 orders from Stage 2, etc., and Stage N orders from an outside supplier with unlimited stock. The demand distribution in each period is determined by the current state of an exogenous Markov chain. Excess demand is backlogged. Linear holding costs are incurred at every stage, and linear backorder costs are incurred at Stage 1. The ordering costs are also linear. The objective is to minimize the long-run average costs in the system. The paper shows that the optimal policy is an echelon base-stock policy with state-dependent order-up-to levels. An efficient algorithm is also provided for determining the optimal base-stock levels. The results can be extended to serial systems in which there is a fixed ordering cost at stage N and to assembly systems with linear ordering costs.

We study an assemble-to-order inventory system. The stocks are held for components, with final products assembled only when customer orders are realized. Customer orders form a multivariate compound Poisson process, component replenishment leadtimes are constant, and demands in excess of inventory on hand are backlogged. The component inventories are controlled by (R, nQ) policies. We show that under certain general conditions the inventory position vector has a uniform equilibrium distribution. This result generalizes the corresponding single-item theory considerably. It allows us to express the key performance measures of the system, such as order fill rates and average order-based backorders, as the averages of their counterparts in the base-stock systems.

We study a multicomponent, multiproduct production and inventory system in which individual components are made to stock but final products are assembled to customer orders. Each component is produced by an independent production facility with finite capacity, and the component inventory is controlled by an independent base-stock policy. For any given base-stock policy, we derive the key performance measures, including the probability of fulfilling a customer order within any specified time window. Computational procedures and numerical examples are also presented. A similar approach applies to the generic multi-item make-to-stock inventory systems in which a typical customer order consists of a kit of items.

A customer order to a multi-item inventory system typically consists of several different items in different amounts. The probability of satisfying an arbitrary demand within a prespecified time window, termed the order fill rate, is an important measure of customer satisfaction in industry. This measure, however, has received little attention in the inventory literature, partly because its evaluation is considered a hard problem. In this paper, we study this performance measure for a base-stock system in which the demand process forms a multivariate compound Poisson process and the replenishment leadtimes are constant. We show that the order fill rate can be computed through a series of convolutions of one-dimensional compound Poisson distributions and the batch-size distributions. This procedure makes the exact calculation faster and much more tractable. We also develop simpler bounds to estimate the order fill rate. These bounds require only partial order-based information or merely the item-based information. Finally, we investigate the impact of the standard independent demand assumption when the demand is actually correlated across items.

This paper studies the impact of using delivery time guarantees as a competitive strategy in service industries where demands are sensitive to both price and delivery time. We assume that delivery reliability is crucial, and investment in capacity expansion is plausible in order to maintain a high probability of delivering the time guarantee. A mathematical framework is proposed to understand the interrelations among pricing, delivery time guarantee and capacity expansion decisions. Specifically, an optimization model is developed to determine the joint optimal selection of these three important decision variables, with an objective of maximizing the average net profit. We characterize the optimal decisions and study their qualitative behaviors as various parameters change. We further present a numerical example to illustrate how the results of our model can be used to provide useful managerial insights for selecting the best competing strategies for firms with different operating characteristics. Our model and results are also applicable to a make-to-order manufacturing environment.

We have read Ding et al. (2002), referred to hereafter as DPB, with great interest. DPB con ains two main results: Theorem A claims that the myopic inventory level is a lower bound on the optimal inventory level, and Theo rem B attempts to quantify the marginal expected value of information. However, there are mistakes in the proof of Theorem A. The result itself is correct; a proof is given in Lu et al. (2004). Furthermore, one of the erroneous expressions in DPB's proof leads to an error in the marginal expected value of information in Theorem B. We now provide more details. The equation numbers, unless specified otherwise, refer to the appendix of DPB, available at the Operations Research home page (http://or. pubs.informs.org). The main error in the proof of Theorem A is the follow ing. If a{x) and b{x) are two functions of the variable jc, then the derivative of a(x)b(x) with respect to x is d r / x, / m , / ,da{x) .db(x) ? [a(x)b(x)] = b(x) -?? + a(x) -^. DPB's mistake, in essence, is to omit the second term. More specifically, a key step in proving Theorem A is inequality (35)

This paper presents an inventory-control model which includes a Markovian model of the supply system. As that system evolves over time, so do the replenishment leadtimes. The optimal policy has the same structure as in standard models, but its parameters change dynamically to reflect current supply conditions. In this setting, contrary to conventional wisdom, a longer leadtime does not necessarily imply more inventory. The leadtime is important, but so is a concept we call order coverage.

How should inventory management respond when there is a possibility of imminent obsolescence (or, more generally, deteriorating demand)? We use an inventory-control model to address this question. The model incorporates a Markovian submodel to describe the uncertain events leading to obsolescence. These events and their uncertainties come in a variety of patterns. We devote considerable attention to specifying the submodel, and we compare a few alternatives numerically. Also, we compare optimal policies to simpler alternatives, and we investigate the response of the model to parameter changes. Generally, we find that obsolescence does (or should) have a substantial impact in the way inventories are managed. The nature of these effects, moreover, is fairly intricate. It appears that obsolescence cannot be captured in a simpler model through parameter adjustments. These conclusions presume that we cannot dispose of excess stock, either directly or through price promotions; we show also that the disposal option can make the problems of obsolescence more manageable.

We study a basic (r, q) system, in which the demand is a Poisson process and the leadtimes are independent, identically-distributed random variables. The key issue is the joint effect of the leadtime variance and the lot size q on performance. We know that, under a simple base-stock policy (with q = 1), the leadtime variance has no effect at all. We find here that, for larger q, the leadtime variance can have a significant adverse impact on performance. To explore this effect, we test two simple approximations. The simplest ignores the leadtime variance. The second approach is only a bit more complex; it captures the variance effect through a hybrid of two limiting approximations. Both methods provide useful information, but the second is more robust.

We noticed an error in the upper bound on the optimal system stock in Boyaci and Gallego (2001). We provide a procedure to compute the correct bound.

We study a basic continuous-time single-item inventory model where demands form a compound Poisson process and leadtimes are stochastic. The performance measure of interest is the long-run average cost. Order costs are linear, so a base-stock policy is optimal. We focus on the behavior of the optimal base-stock level in response to stochastically larger or more variable leadtimes. We also investigate the behavior of the corresponding long-run average costs. We show that a stochastically larger leadtime requires a higher optimal base-stock level. However, a stochastically larger leadtime may not necessarily result in a higher optimal average cost, because sometimes the variability effects may dominate. On the other hand, a more variable leadtime always leads to a higher optimal average cost. The effect of leadtime variability on optimal policies depends on the inventory cost structure: A more variable leadtime requires a higher optimal base-stock level if and only if the unit penalty (holding) cost rate is high (low). Similar results regarding the effect of leadtime demand are also discussed. In this latter case, the compound Poisson assumption is not necessary.

This paper explores qualitative effects of lead-time uncertainty in a basic continuous-time single-item inventory model with Poisson demand and stochastic lead times. The objective is to minimize the infinite-horizon expected total discounted cost. Order costs are linear, so a base-stock policy is optimal. Traditionally and intuitively, people believe that a (stochastically) longer lead time results in a larger lead-time demand and therefore that the system should have a higher optimal base-stock level. In fact, Song (1994) shows that this is true for models with the average cost criterion. Here, we show that this intuition is not always validated for models with the discounted cost criterion. We identify certain systems in which the lead times do have a monotonic impact on the optimal base-stock levels. Examples include systems with constant, geometric, gamma, or uniform lead-time distributions. In addition, we find that a shorter lead time does not necessarily result in a smaller optimal system cost.