Capacity expansion with lead times and autocorrelated random demand

The combination of uncertain demand and lead times for installing capacity creates the risk of shortage during the lead time, which may have serious consequences for a service provider. This paper analyzes a model of capacity expansion with autocorrelated random demand and a fixed lead time for adding capacity. To provide a specified level of service, a discrete time expansion timing policy uses a forecast error‐adjusted minimum threshold level of excess capacity position to trigger an expansion. Under this timing policy, the expansion cost can be minimized by solving a deterministic dynamic program. We study the effects of demand characteristics and the lead time length on the capacity threshold. Autocorrelation acts similarly to randomness in hastening expansions but has a smaller impact, especially when lead times are short. However, the failure either to recognize autocorrelation or to accurately estimate its extent can cause substantial policy errors. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

Optimal machine capacity expansions with nested limitations under stochastic demand

This paper studies capacity expansions for a production facility that faces uncertain customer demand for a single product family. The capacity of the facility is modeled in three tiers, as follows. The first tier consists of a set of upper bounds on production that correspond to different resource types (e.g., machine types, categories of manpower, etc.). These upper bounds are augmented in increments of fixed size (e.g., by purchasing machines of standard types). There is a second‐tier resource that constrains the first‐tier bounds (e.g., clean room floor space). The third‐tier resource bounds the availability of the second‐tier resource (e.g., the total floor space enclosed by the building, land, etc.). The second and third‐tier resources are expanded at various times in various amounts. The cost of capacity expansion at each tier has both fixed and proportional elements. The lost sales cost is used as a measure for the level of customer service. The paper presents a polynomial time algorithm (FIFEX) to minimize the total cost by computing optimal expansion times and amounts for all three types of capacity jointly. It accommodates positive lead times for each type. Demand is assumed to be nondecreasing in a “weak” sense. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

On the (S − 1, S) lost sales inventory model with priority demand classes

In this paper an inventory model with several demand classes, prioritised according to importance, is analysed. We consider a lot‐for‐lot or (S ? 1, S) inventory model with lost sales. For each demand class there is a critical stock level at and below which demand from that class is not satisfied from stock on hand. In this way stock is retained to meet demand from higher priority demand classes. A set of such critical levels determines the stocking policy. For Poisson demand and a generally distributed lead time, we derive expressions for the service levels for each demand class and the average total cost per unit time. Efficient solution methods for obtaining optimal policies, with and without service level constraints, are presented. Numerical experiments in which the solution methods are tested demonstrate that significant cost reductions can be achieved by distinguishing between demand classes. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 593–610, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10032     

An adaptive forecasting algorithm and inventory policy for products with short life cycles

Products with short life cycles are becoming increasingly common in many industries, such as the personal computer (PC) and mobile phone industries. Traditional forecasting methods and inventory policies can be inappropriate for forecasting demand and managing inventory for a product with a short life cycle because they usually do not take into account the characteristics of the product life cycle. This can result in inaccurate forecasts, high inventory cost, and low service levels. Besides, many forecasting methods require a significant demand history, which is available only after the product has been sold for some time. In this paper, we present an adaptive forecasting algorithm with two characteristics. First, it uses structural knowledge on the product life cycle to model the demand. Second, it combines knowledge on the demand that is available prior to the launch of the product with actual demand data that become available after the introduction of the product to generate and update demand forecasts. Based on the forecasting algorithm, we develop an optimal inventory policy. Since the optimal inventory policy is computationally expensive, we propose three heuristics and show in a numerical study that one of the heuristics generates near‐optimal solutions. The evaluation of our approach is based on demand data from a leading PC manufacturer in the United States, where the forecasting algorithm has been implemented. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

To wait or not to wait: Optimal ordering under lead time uncertainty and forecast updating

There has been a dramatic increase over the past decade in the number of firms that source finished product from overseas. Although this has reduced procurement costs, it has increased supply risk; procurement lead times are longer and are often unreliable. In deciding when and how much to order, firms must consider the lead time risk and the demand risk, i.e., the accuracy of their demand forecast. To improve the accuracy of its demand forecast, a firm may update its forecast as the selling season approaches. In this article we consider both forecast updating and lead time uncertainty. We characterize the firm's optimal procurement policy, and we prove that, with multiplicative forecast revisions, the firm's optimal procurement time is independent of the demand forecast evolution but that the optimal procurement quantity is not. This leads to a number of important managerial insights into the firm's planning process. We show that the firm becomes less sensitive to lead time variability as the forecast updating process becomes more efficient. Interestingly, a forecast‐updating firm might procure earlier than a firm with no forecast updating. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Revenue sharing contracts in a supply chain with uncontractible actions

We consider a supplier–customer relationship where the customer faces a typical Newsvendor problem of determining perishable capacity to meet uncertain demand. The customer outsources a critical, demand‐enhancing service to an outside supplier, who receives a fixed share of the revenue from the customer. Given such a linear sharing contract, the customer chooses capacity and the service supplier chooses service effort level before demand is realized. We consider the two cases when these decisions are made simultaneously (simultaneous game) or sequentially (sequential game). For each game, we analyze how the equilibrium solutions vary with the parameters of the problem. We show that in the equilibrium, it is possible that either the customer's capacity increases or the service supplier's effort level decreases when the supplier receives a larger share of the revenue. We also show that given the same sharing contract, the sequential game always induces a higher capacity and more effort. For the case of additive effort effect and uniform demand distribution, we consider the customer's problem of designing the optimal contract with or without a fixed payment in the contract, and obtain sensitivity results on how the optimal contract depends on the problem parameters. For the case of fixed payment, it is optimal to allocate more revenue to the supplier to induce more service effort when the profit margin is higher, the cost of effort is lower, effort is more effective in stimulating demand, the variability of demand is smaller or the supplier makes the first move in the sequential game. For the case of no fixed payment, however, it is optimal to allocate more revenue to the supplier when the variability of demand is larger or its mean is smaller. Numerical examples are analyzed to validate the sensitivity results for the case of normal demand distribution and to provide more managerial insights. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Performance evaluation of a production/inventory system with periodic review and endogenous lead times

This paper considers a discrete time, single item production/inventory system with random period demands. Inventory levels are reviewed periodically and managed using a base‐stock policy. Replenishment orders are placed with the production system which is capacitated in the sense that there is a single server that sequentially processes the items one at a time with stochastic unit processing times. In this setting the variability in demand determines the arrival pattern of production orders at the queue, influencing supply lead times. In addition, the inventory behavior is impacted by the correlation between demand and lead times: a large demand size corresponds to a long lead time, depleting the inventory longer. The contribution of this paper is threefold. First, we present an exact procedure based on matrix‐analytic techniques for computing the replenishment lead time distribution given an arbitrary discrete demand distribution. Second, we numerically characterize the distribution of inventory levels, and various other performance measures such as fill rate, base‐stock levels and optimal safety stocks, taking the correlation between demand and lead times into account. Third, we develop an algorithm to fit the first two moments of the demand and service time distribution to a discrete phase‐type distribution with a minimal number of phases. This provides a practical tool to analyze the effect of demand variability, as measured by its coefficient of variation, on system performance. We also show that our model is more appropriate than some existing models of capacitated systems in discrete time. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

An exact analysis of a joint production‐inventory problem in two‐echelon inventory systems

We consider a two‐echelon inventory system with a manufacturer operating from a warehouse supplying multiple distribution centers (DCs) that satisfy the demand originating from multiple sources. The manufacturer has a finite production capacity and production times are stochastic. Demand from each source follows an independent Poisson process. We assume that the transportation times between the warehouse and DCs may be positive which may require keeping inventory at both the warehouse and DCs. Inventory in both echelons is managed using the base‐stock policy. Each demand source can procure the product from one or more DCs, each incurring a different fulfilment cost. The objective is to determine the optimal base‐stock levels at the warehouse and DCs as well as the assignment of the demand sources to the DCs so that the sum of inventory holding, backlog, and transportation costs is minimized. We obtain a simple equation for finding the optimal base‐stock level at each DC and an upper bound for the optimal base‐stock level at the warehouse. We demonstrate several managerial insights including that the demand from each source is optimally fulfilled entirely from a single distribution center, and as the system's utilization approaches 1, the optimal base‐stock level increases in the transportation time at a rate equal to the demand rate arriving at the DC. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Service design at diagnostic service centers

We study a service design problem in diagnostic service centers, call centers that provide medical advice to patients over the phone about what the appropriate course of action is, based on the caller's symptoms. Due to the tension between increased diagnostic accuracy and the increase in waiting times more in‐depth service requires, managers face a difficult decision in determining the optimal service depth to guide the diagnostic process. The specific problem we consider models the situation when the capacity (staffing level) at the center is fixed, and when the callers have both congestion‐ and noncongestion‐related costs relating to their call. We develop a queueing model incorporating these features and find that the optimal service depth can take one of two different structures, depending on factors such as the nurses' skill level and the maximum potential demand. Sensitivity analyses of the two optimal structures show that they are quite different. In some situations, it may (or may not) be optimal for the manager to try to expand the demand at the center, and increasing skill level may (or may not) increase congestion. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

The (s,Q) inventory model with Erlang lead time and deterministic demand

We develop a simple, approximately optimal solution to a model with Erlang lead time and deterministic demand. The method is robust to misspecification of the lead time and has good accuracy. We compare our approximate solution to the optimal for the case where we have prior information on the lead‐time distribution, and another where we have no information, except for computer‐generated sample data. It turns out that our solution is as easy as the EOQ's, with an accuracy rate of 99.41% when prior information on the lead‐time distribution is available and 97.54–99.09% when only computer‐generated sample information is available. Apart from supplying the inventory practitioner with an easy heuristic, we gain insights into the efficacy of stochastic lead time models and how these could be used to find the cost and a near‐optimal policy for the general model, where both demand rate and lead time are stochastic. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Inventory management with advance capacity information

An important aspect of supply chain management is dealing with demand and supply uncertainty. The uncertainty of future supply can be reduced if a company is able to obtain advance capacity information (ACI) about future supply/production capacity availability from its supplier. We address a periodic‐review inventory system under stochastic demand and stochastic limited supply, for which ACI is available. We show that the optimal ordering policy is a state‐dependent base‐stock policy characterized by a base‐stock level that is a function of ACI. We establish a link with inventory models that use advance demand information (ADI) by developing a capacitated inventory system with ADI, and we show that equivalence can only be set under a very specific and restrictive assumption, implying that ADI insights will not necessarily hold in the ACI environment. Our numerical results reveal several managerial insights. In particular, we show that ACI is most beneficial when there is sufficient flexibility to react to anticipated demand and supply capacity mismatches. Further, most of the benefits can be achieved with only limited future visibility. We also show that the system parameters affecting the value of ACI interact in a complex way and therefore need to be considered in an integrated manner. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Coordination of pricing and inventory control across products

We consider the joint pricing and inventory‐control problem for a retailer who orders, stocks, and sells two products. Cross‐price effects exist between the two products, which means that the demand of each product depends on the prices of both products. We derive the optimal pricing and inventory‐control policy and show that this policy differs from the base‐stock list‐price policy, which is optimal for the one‐product problem. We find that the retailer can significantly improve profits by managing the two products jointly as opposed to independently, especially when the cross‐price demand elasticity is high. We also find that the retailer can considerably improve profits by using dynamic pricing as opposed to static pricing, especially when the demand is nonstationary. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Optimal and asymptotically optimal policies for assemble‐to‐order n‐ and W‐systems

We consider two specially structured assemble‐to‐order (ATO) systems—the N‐ and W‐systems—under continuous review, stochastic demand, and nonidentical component replenishment leadtimes. Using a hybrid approach that combines sample‐path analysis, linear programming, and the tower property of conditional expectation, we characterize the optimal component replenishment policy and common‐component allocation rule, present comparative statics of the optimal policy parameters, and show that some commonly used heuristic policies can lead to significant optimality loss. The optimality results require certain symmetry in the cost parameters. In the absence of this symmetry, we show that, for systems with high demand volume, the asymptotically optimal policy has essentially the same structure; otherwise, the optimal policies have no clear structure. For these latter systems, we develop heuristic policies and show their effectiveness. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 62: 617–645, 2015     

Requirements planning for modular products

The focus of this paper is on determining the requirements of different component options of a modular end‐product in an uncertain environment. We explicitly model two distinct sources of uncertainty: stochastic end‐product demand and unknown market proportions for the different product options available. Our cost minimizing model focuses on determining the optimal requirements policies for component options that meet a pre‐set service level. We show that simple common‐sense requirements policies are not generally optimal; there is a non‐linear connection between service level and component requirements that is hard to characterize without a detailed analysis. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Exact analysis of (R,s, S) inventory control systems with lost sales and zero lead time

We study an (R, s, S) inventory control policy with stochastic demand, lost sales, zero lead‐time and a target service level to be satisfied. The system is modeled as a discrete time Markov chain for which we present a novel approach to derive exact closed‐form solutions for the limiting distribution of the on‐hand inventory level at the end of a review period, given the reorder level (s) and order‐up‐to level (S). We then establish a relationship between the limiting distributions for adjacent values of the reorder point that is used in an efficient recursive algorithm to determine the optimal parameter values of the (R, s, S) replenishment policy. The algorithm is easy to implement and entails less effort than solving the steady‐state equations for the corresponding Markov model. Point‐of‐use hospital inventory systems share the essential characteristics of the inventory system we model, and a case study using real data from such a system shows that with our approach, optimal policies with significant savings in inventory management effort are easily obtained for a large family of items.     

Optimal inventory control policy for periodic‐review inventory systems with inventory‐level‐dependent demand

We consider a setting in which inventory plays both promotional and service roles; that is, higher inventories not only improve service levels but also stimulate demand by serving as a promotional tool (e.g., as the result of advertising effect by the enhanced product visibility). Specifically, we study the periodic‐review inventory systems in which the demand in each period is uncertain but increases with the inventory level. We investigate the multiperiod model with normal and expediting orders in each period, that is, any shortage will be met through emergency replenishment. Such a model takes the lost sales model as a special case. For the cases without and with fixed order costs, the optimal inventory replenishment policy is shown to be of the base‐stock type and of the (s,S) type, respectively. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Stochastic production capacity: A bane or a boon for quick response supply chains?

The quick response (QR) system that can cope with demand volatility by shortening lead time has been well studied in the literature. Much of the existing literature assumes implicitly or explicitly that the manufacturers under QR can always meet the demand because the production capacity is always sufficient. However, when the order comes with a short lead time under QR, availability of the manufacturer's production capacity is not guaranteed. This motivates us to explore QR in supply chains with stochastic production capacity. Specifically, we study QR in a two-echelon supply chain with Bayesian demand information updating. We consider the situation where the manufacturer's production capacity under QR is uncertain. We first explore how stochastic production capacity affects supply chain decisions and QR implementation. We then incorporate the manufacturer's ability to expand capacity into the model. We explore how the manufacturer determines the optimal capacity expansion decision, and the value of such an ability to the supply chain and its agents. Finally, we extend the model to the two-stage two-ordering case and derive the optimal ordering policy by dynamic programming. We compare the single-ordering and two-ordering cases to generate additional managerial insights about how ordering flexibility affects QR when production capacity is stochastic. We also explore the transparent supply chain and find that our main results still hold.     

Base‐stock policies in capacitated assembly systems: Convexity properties

We study an assembly system with a single finished product managed using an echelon base‐stock or order‐up‐to policy. Some or all operations have capacity constraints. Excess demand is either backordered in every period or lost in every period. We show that the shortage penalty cost over any horizon is jointly convex with respect to the base‐stock levels and capacity levels. When the holding costs are also included in the objective function, we show that the cost function can be written as a sum of a convex function and a concave function. Throughout the article, we discuss algorithmic implications of our results for making optimal inventory and capacity decisions in such systems.© 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Optimal control of a capacitated inventory system with multiple demand classes

This article studies the optimal control of a periodic‐review make‐to‐stock system with limited production capacity and multiple demand classes. In this system, a single product is produced to fulfill several classes of demands. The manager has to make the production and inventory allocation decisions. His objective is to minimize the expected total discounted cost. The production decision is made at the beginning of each period and determines the amount of products to be produced. The inventory allocation decision is made after receiving the random demands and determines the amount of demands to be satisfied. A modified base stock policy is shown to be optimal for production, and a multi‐level rationing policy is shown to be optimal for inventory allocation. Then a heuristic algorithm is proposed to approximate the optimal policy. The numerical studies show that the heuristic algorithm is very effective. © 2011 Wiley Periodicals, Inc. Naval Research Logistics 58: 43–58, 2011     

Risk‐sensitive sizing of responsive facilities

We develop a risk‐sensitive strategic facility sizing model that makes use of readily obtainable data and addresses both capacity and responsiveness considerations. We focus on facilities whose original size cannot be adjusted over time and limits the total production equipment they can hold, which is added sequentially during a finite planning horizon. The model is parsimonious by design for compatibility with the nature of available data during early planning stages. We model demand via a univariate random variable with arbitrary forecast profiles for equipment expansion, and assume the supporting equipment additions are continuous and decided ex‐post. Under constant absolute risk aversion, operating profits are the closed‐form solution to a nontrivial linear program, thus characterizing the sizing decision via a single first‐order condition. This solution has several desired features, including the optimal facility size being eventually decreasing in forecast uncertainty and decreasing in risk aversion, as well as being generally robust to demand forecast uncertainty and cost errors. We provide structural results and show that ignoring risk considerations can lead to poor facility sizing decisions that deteriorate with increased forecast uncertainty. Existing models ignore risk considerations and assume the facility size can be adjusted over time, effectively shortening the planning horizon. Our main contribution is in addressing the problem that arises when that assumption is relaxed and, as a result, risk sensitivity and the challenges introduced by longer planning horizons and higher uncertainty must be considered. Finally, we derive accurate spreadsheet‐implementable approximations to the optimal solution, which make this model a practical capacity planning tool.© 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008