Note: Dual sourcing with nonidentical suppliers

We analyze a dual-sourcing inventory model with exponential lead times and constant unit demand in which the order quantity is split in some proportion between two sources of supply. Unlike earlier studies, we do not require that the two sources be identical in terms of the lead-time parameters or the supply prices. We compare the expected total annual costs for the two-source and the traditional single-source models over a wide range of parameter values. We confirm the findings of earlier studies that, under stochastic lead times, dual sourcing yields savings in holding and shortage costs that could outweigh the incremental ordering costs. With this more general model, we demonstrate that savings from dual sourcing are possible even where the mean or the variability of the second source is higher. © 1993 John Wiley & Sons, Inc.     

Dual sourcing: Responsive hedging against correlated supply and demand uncertainty

This article analyzes dual sourcing decisions under stochastically dependent supply and demand uncertainty. A manufacturer faces the trade‐off between investing in unreliable but high‐margin offshore supply and in reliable but low‐margin local supply, where the latter allows for production that is responsively contingent on the actual demand and offshore supply conditions. Cost thresholds for both types of supply determine the optimal resource allocation: single offshore sourcing, single responsive sourcing, or dual sourcing. Relying on the concept of concordance orders, we study the effects of correlation between supply and demand uncertainty. Adding offshore supply to the sourcing portfolio becomes more favorable under positive correlation, since offshore supply is likely to satisfy demand when needed. Selecting responsive capacity under correlated supply and demand uncertainty is not as straightforward, yet we establish the managerially relevant conditions under which responsive capacity either gains or loses in importance. Our key results are extended to the broad class of endogenous supply uncertainty developed by Dada et al. [Manufact Serv Operat Mange 9 (2007), 9–32].© 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Optimal material control in an assembly system with component commonality

Allocation of scarce common components to finished product orders is central to the performance of assembly systems. Analysis of these systems is complex, however, when the product master schedule is subject to uncertainty. In this paper, we analyze the cost—service performance of a component inventory system with correlated finished product demands, where component allocation is based on a fair shares method. Such issuing policies are used commonly in practice. We quantify the impact of component stocking policies on finished product delays due to component shortages and on product order completion rates. These results are used to determine optimal base stock levels for components, subject to constraints on finished product service (order completion rates). Our methodology can help managers of assembly systems to (1) understand the impact of their inventory management decisions on customer service, (2) achieve cost reductions by optimizing their inventory investments, and (3) evaluate supplier performance and negotiate contracts by quantifying the effect of delivery lead times on costs and customer service. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48:409–429, 2001     

Supplier certification and quality investment in supply chains

Global sourcing has made quality management a more challenging task, and supplier certification has emerged as a solution to overcome suppliers' informational advantage about their product quality. This article analyzes the impact of certification standards on the supplier's investment in quality, when a buyer outsources the production process. Based on our results, deterministic certification may lead to under‐investment in quality improvement technology for efficient suppliers, thereby leading to potential supply chain inefficiency. The introduction of noisy certification may alleviate this under‐investment problem, when the cost of information asymmetry is high. While allowing noisy certification always empowers the buyer to offer a menu to screen among heterogeneous suppliers, the buyer may optimally choose only a limited number of certification standards. Our analysis provides a clear‐cut prediction of the types of certifiers the buyer should use for heterogeneous suppliers, and we identify the conditions under which the supplier benefits from noisy certification. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Oligopolistic contracting: Channel coordination under competition

We study contracts between a single retailer and multiple suppliers of two substitutable products, where suppliers have fixed capacities and present the retailer cost contracts for their supplies. After observing the contracts, the retailer decides how much capacity to purchase from each supplier, to maximize profits from the purchased capacity from the suppliers plus his possessed inventory (endowment). This is modeled as a noncooperative, nonzero‐sum game, where suppliers, or principals, move simultaneously as leaders and the retailer, the common agent, is the sole follower. We are interested in the form of the contracts in equilibrium, their effect on the total supply chain profit, and how the profit is split between the suppliers and the retailer. Under mild assumptions, we characterize the set of all equilibrium contracts and discuss all‐unit and marginal‐unit quantity discounts as special cases. We also show that the supply chain is coordinated in equilibrium with a unique profit split between the retailer and the suppliers. Each supplier's profit is equal to the marginal contribution of her capacity to supply chain profits in equilibrium. The retailer's profit is equal to the total revenue collected from the market minus the payments to the suppliers and the associated sales costs.     

Inventory models of future supply uncertainty with single and multiple suppliers

We consider order-quantity/reorder-point inventory models where the availability of supply is subject to random fluctuations. We use concepts from renewal reward processes to develop average cost objective function models for single, two, and multiple suppliers. Identifying the regenerative cycle for each problem aids the development of the cost function. In the case of two suppliers, spectral theory is used to derive explicit expressions for the transient probabilities of a four-state continuous-time Markov chain representing the status of the system. These probabilities are used to compute the exact form of the average cost expression. For the multiple-supplier problem, assuming that all the suppliers have similar availability characteristics, we develop a simple model and show that as the number of suppliers becomes large, the model reduces to the classical EOQ model. © 1996 John Wiley & Sons, Inc.     

A continuous-review inventory system with lost sales and variable lead time

Using a system-point (SP) method of level crossings, we derive the stationary distribution of the inventory level (stock on hand) in a continuous-review inventory system with compound Poisson demand, Erlang as well as hyperexponentially distributed lead times, and lost sales. This distribution is then used to formulate long-run average cost functions with/without a service level constraint. Some numerical results are also presented, and compared with the Hadley and Whitin heuristic. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 259–278, 1998     

Inventory control and periodic price discounting campaigns

This paper develops an inventory model that determines replenishment strategies for buyers facing situations in which sellers offer price‐discounting campaigns at random times as a way to drive sales or clear excess inventory. Specifically, the model deals with the inventory of a single item that is maintained to meet a constant demand over time. The item can be purchased at two different prices denoted high and low. We assume that the low price goes into effect at random points in time following an exponential distribution and lasts for a random length of time following another exponential distribution. We highlight a replenishment strategy that will lead to the lowest inventory holding and ordering costs possible. This strategy is to replenish inventory only when current levels are below a certain threshold when the low price is offered and the replenishment is to a higher order‐up‐to level than the one currently in use when inventory depletes to zero and the price is high. Our analysis provides new insight into the behavior of the optimal replenishment strategy in response to changes in the ratio of purchase prices together with changes in the ratio of the duration of a low‐price period to that of a high‐price period. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007.     

Transshipment through crossdocks with inventory and time windows

A major challenge in making supply meet demand is to coordinate transshipments across the supply chain to reduce costs and increase service levels in the face of demand fluctuations, short lead times, warehouse limitations, and transportation and inventory costs. In particular, transshipment through crossdocks, where just‐in‐time objectives prevail, requires precise scheduling between suppliers, crossdocks, and customers. In this work, we study the transshipment problem with supplier and customer time windows where flow is constrained by transportation schedules and warehouse capacities. Transportation is provided by fixed or flexible schedules and lot‐sizing is dealt with through multiple shipments. We develop polynomial‐time algorithms or, otherwise, provide the complexity of the problems studied. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

When is a base stock policy optimal in recovering disrupted cyclic schedules?

The extended economic lot scheduling problem (EELSP) is concerned with scheduling the production of a set of items in a single facility to minimize the long-run average holding, backlogging, and setup costs. Given an efficient cyclic production schedule for the EELSP, called the target schedule, we consider the problem of how to schedule production after a single schedule disruption. We propose a base stock policy, characterized by a base stock vector, that prescribes producing an item until its inventory level reaches the peak inventory of the target schedule corresponding to the item's position in the production sequence. We show that the base stock policy is always successful in recovering the target schedule. Moreover, the base stock policy recovers the target schedule at minimal excess over average cost whenever the backorder costs are proportional to the processing times. This condition holds, for example, when the value of the items is proportional to their processing times, and a common inventory carrying cost and a common service level is used for all the items. Alternatively, the proportionality condition holds if the inventory manager is willing to select the service levels from a certain set that is large enough to guarantee any minimal level of service, and then uses the imputed values for the backorder costs. When the proportionality condition holds we provide a closed-form expression for the total relevant excess over average cost of recovering the target schedule. We assess the performance of the base stock policy when the proportionality condition does not hold through a numerical study, and suggest some heuristic uses of the base stock policy. © 1994 John Wiley & Sons, Inc.     

Safety stock reduction by order splitting

In this article we consider an item for which a continuous review, reorder point, order quantity inventory control system is used. The amount of safety stock required depends upon, among other factors, the average value and variability of the length of the replenishment lead time. One way to reduce these quantities is to split orders among two or more vendors. In this article the random lead times are assumed to have Weibull distributions. This permits the development of analytic expressions for the reduction in the expected value and variability of total demand until the critical first (earliest) delivery received from a vendor. An expression is also obtained for the reorder point that provides a given probability of no stockout prior to the first delivery. Lower bounds are given on the order quantity so as to ensure that the probability of a stockout before any one of the later (second, third, etc.) deliveries is sufficiently small to be considered negligible. The analytic and tabular results can be used to estimate the benefits (reduced carrying costs and/or increased service level) of order splitting.     

Strategic dynamic sourcing from competing suppliers with transferable capacity investment

We study the supplier relationship choice for a buyer that invests in transferable capacity operated by a supplier. With a long‐term relationship, the buyer commits to source from a supplier over a long period of time. With a short‐term relationship, the buyer leaves open the option of switching to a new supplier in the future. The buyer has incomplete information about a supplies efficiency, and thus uses auctions to select suppliers and determine the contracts. In addition, the buyer faces uncertain demand for the product. A long‐term relationship may be beneficial for the buyer because it motivates more aggressive bidding at the beginning, resulting a lower initial price. A short‐term relationship may be advantageous because it allows switching, with capacity transfer at some cost, to a more efficient supplier in the future. We find that there exists a critical level of the switching cost above which a long‐term relationship is better for the buyer than a short‐term relationship. In addition, this critical switching cost decreases with demand uncertainty, implying a long‐term relationship is more favorable for a buyer facing volatile demand. Finally, we find that in a long‐term relationship, capacity can be either higher or lower than in a short‐term relationship. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Capacity expansion under a service‐level constraint for uncertain demand with lead times

For a service provider facing stochastic demand growth, expansion lead times and economies of scale complicate the expansion timing and sizing decisions. We formulate a model to minimize the infinite horizon expected discounted expansion cost under a service‐level constraint. The service level is defined as the proportion of demand over an expansion cycle that is satisfied by available capacity. For demand that follows a geometric Brownian motion process, we impose a stationary policy under which expansions are triggered by a fixed ratio of demand to the capacity position, i.e., the capacity that will be available when any current expansion project is completed, and each expansion increases capacity by the same proportion. The risk of capacity shortage during a cycle is estimated analytically using the value of an up‐and‐out partial barrier call option. A cutting plane procedure identifies the optimal values of the two expansion policy parameters simultaneously. Numerical instances illustrate that if demand grows slowly with low volatility and the expansion lead times are short, then it is optimal to delay the start of expansion beyond when demand exceeds the capacity position. Delays in initiating expansions are coupled with larger expansion sizes. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Production/distribution system design with inventory considerations

This study addresses the design of a three‐stage production/distribution system where the first stage includes the set of established retailers and the second and third stages include the sets of potential distribution centers (DCs) and potential capacitated suppliers, respectively. In this problem, in addition to the fixed location/operating costs associated with locating DCs and suppliers, we consider the coordinated inventory replenishment decisions at the located DCs and retailers along with the appropriate inventory costs explicitly. In particular, we account for the replenishment and holding costs at the retailers and selected DCs, and the fixed plus distance‐based transportation costs between the selected plants and their assigned DCs, and between the selected DCs and their respective retailers, explicitly. The resulting formulation is a challenging mixed‐integer nonlinear programming model for which we propose efficient heuristic solution approaches. Our computational results demonstrate the performance of the heuristic approaches as well as the value of integrated decision‐making by verifying that significant cost savings are realizable when the inventory decisions and costs are incorporated in the production distribution system design. © 2012 Wiley Periodicals, Inc. Naval Research Logistics 59: 172–195, 2012     

A multi-item EOQ model with inventory cycle balancing

This article concerns a multi-item, infinite-horizon, lot-sizing problem, where the objective is to minimize a total cost function made up of reordering cost, holding cost, and a cost determined by peak inventory levels. By spreading inventory replenishments over the reordering cycle, the peak inventory level can be reduced. The model permits the derivation of simultaneously optimal solutions for the length of the cycle and the individual item replenishment times within the cycle. An alternative formulation, in which total storage capacity is modeled as a constraint, is also solved.     

Optimal control of a production‐inventory system with both backorders and lost sales

We consider the optimal control of a production inventory‐system with a single product and two customer classes where items are produced one unit at a time. Upon arrival, customer orders can be fulfilled from existing inventory, if there is any, backordered, or rejected. The two classes are differentiated by their backorder and lost sales costs. At each decision epoch, we must determine whether or not to produce an item and if so, whether to use this item to increase inventory or to reduce backlog. At each decision epoch, we must also determine whether or not to satisfy demand from a particular class (should one arise), backorder it, or reject it. In doing so, we must balance inventory holding costs against the costs of backordering and lost sales. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. We show that the optimal policy can be described by three state‐dependent thresholds: a production base‐stock level and two order‐admission levels, one for each class. The production base‐stock level determines when production takes place and how to allocate items that are produced. This base‐stock level also determines when orders from the class with the lower shortage costs (Class 2) are backordered and not fulfilled from inventory. The order‐admission levels determine when orders should be rejected. We show that the threshold levels are monotonic (either nonincreasing or nondecreasing) in the backorder level of Class 2. We also characterize analytically the sensitivity of these thresholds to the various cost parameters. Using numerical results, we compare the performance of the optimal policy against several heuristics and show that those that do not allow for the possibility of both backordering and rejecting orders can perform poorly.© 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

Optimal control policies for a periodic review inventory system with emergency orders

We describe a periodic review inventory system where emergency orders, which have a shorter supply lead time but are subject to higher ordering cost compared to regular orders, can be placed on a continuous basis. We consider the periodic review system in which the order cycles are relatively long so that they are possibly larger than the supply lead times. Study of such systems is important since they are often found in practice. We assume that the difference between the regular and emergency supply lead times is less than the order-cycle length. We develop a dynamic programming model and derive a stopping rule to end the computation and obtain optimal operation parameters. Computational results are included that support the contention that easily implemented policies can be computed with reasonable effort. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 187–204, 1998     

Supply management in assembly systems

We consider the case when n components are needed to assemble a given product. Components are provided by suppliers, and the period between the order time and the time a component is available (i.e., the lead time) is a random variable with a known distribution. The due date for the assembled product is also known. The costs to be taken into account are the inventory costs of the components and the backlogging cost of the assembled product. We propose an iterative algorithm which leads to the optimal order instants of the components. © 1993 John Wiley & Sons, Inc.     

A model for use in the rationing of inventory during lead time

While the traditional solution to the problem of meeting stochastically variable demands for inventory during procurement lead time is through the use of some level of safety stock, several authors have suggested that a decision be made to employ some form of rationing so as to protect certain classes of demands against stockout by restricting issues to other classes. Nahmias and Demmy [10] derived an approximate continuous review model of systems with two demand classes which would permit an inventory manager to calculate the expected fill rates per order cycle for high-priority, low-priority, and total system demands for a variety of parameters. The manager would then choose the rationing policy that most closely approximated his fill-rate objectives. This article describes a periodic review model that permits the manager to establish a discrete time rationing policy during lead time by prescribing a desired service level for high-priority demands. The reserve levels necessary to meet this level of service can then be calculated based upon the assumed probability distributions of high- and low-priority demands over lead time. The derived reserve levels vary with the amount of lead time remaining. Simulation tests of the model indicate they are more effective than the single reserve level policy studied by Nahmias and Demmy.     

A (Q,R) inventory model with lost sales and erlang-distributed lead times

The exact expression is derived for the average stationary cost of a (Q,R) inventory system with lost sales, unit Poisson demands, Erlang-distributed lead times, fixed order cost, fixed cost per unit lost sale, linear holding cost per unit time, and a maximum of one order outstanding. Explicit expressions for the state probabilities and a fast method of calculating them are obtained for the case of Q greater than R. Exponential lead times are analyzed as a special case. A simple cyclic coordinate search procedure is used to locate the minimum cost policy. Examples of the effect of lead time variability on costs are given.