Inventory Rationing in a Make‐to‐Stock System with Batch Production and Lost Sales

We address an inventory rationing problem in a lost sales make‐to‐stock (MTS) production system with batch ordering and multiple demand classes. Each production order contains a single batch of a fixed lot size and the processing time of each batch is random. Assuming that there is at most one order outstanding at any point in time, we first address the case with the general production time distribution. We show that the optimal order policy is characterized by a reorder point and the optimal rationing policy is characterized by time‐dependent rationing levels. We then approximate the production time distribution with a phase‐type distribution and show that the optimal policy can be characterized by a reorder point and state‐dependent rationing levels. Using the Erlang production time distribution, we generalize the model to a tandem MTS system in which there may be multiple outstanding orders. We introduce a state‐transformation approach to perform the structural analysis and show that both the reorder point and rationing levels are state dependent. We show the monotonicity of the optimal reorder point and rationing levels for the outstanding orders, and generate new theoretical and managerial insights from the research findings.     

Heuristics for Base‐Stock Levels in Multi‐Echelon Distribution Networks

We study inventory optimization for locally controlled, continuous‐review distribution systems with stochastic customer demands. Each node follows a base‐stock policy and a first‐come, first‐served allocation policy. We develop two heuristics, the recursive optimization (RO) heuristic and the decomposition‐aggregation (DA) heuristic, to approximate the optimal base‐stock levels of all the locations in the system. The RO heuristic applies a bottom‐up approach that sequentially solves single‐variable, convex problems for each location. The DA heuristic decomposes the distribution system into multiple serial systems, solves for the base‐stock levels of these systems using the newsvendor heuristic of Shang and Song (2003), and then aggregates the serial systems back into the distribution system using a procedure we call “backorder matching.” A key advantage of the DA heuristic is that it does not require any evaluation of the cost function (a computationally costly operation that requires numerical convolution). We show that, for both RO and DA, changing some of the parameters, such as leadtime, unit backordering cost, and demand rate, of a location has an impact only on its own local base‐stock level and its upstream locations’ local base‐stock levels. An extensive numerical study shows that both heuristics perform well, with the RO heuristic providing more accurate results and the DA heuristic consuming less computation time. We show that both RO and DA are asymptotically optimal along multiple dimensions for two‐echelon distribution systems. Finally, we show that, with minor changes, both RO and DA are applicable to the balanced allocation policy.     

Optimal Inventory Control with Retail Pre‐Packs

A pre‐pack is a collection of items used in retail distribution. By grouping multiple units of one or more stock keeping units (SKU), distribution and handling costs can be reduced; however, ordering flexibility at the retail outlet is limited. This paper studies an inventory system at a retail level where both pre‐packs and individual items (at additional handling cost) can be ordered. For a single‐SKU, single‐period problem, we show that the optimal policy is to order into a “band” with as few individual units as possible. For the multi‐period problem with modular demand, the band policy is still optimal, and the steady‐state distribution of the target inventory position possesses a semi‐uniform structure, which greatly facilitates the computation of optimal policies and approximations under general demand. For the multi‐SKU case, the optimal policy has a generalized band structure. Our numerical results show that pre‐pack use is beneficial when facing stable and complementary demands, and substantial handling savings at the distribution center. The cost premium of using simple policies, such as strict base‐stock and batch‐ordering (pre‐packs only), can be substantial for medium parameter ranges.     

Myopic Analysis for Multi‐Echelon Inventory Systems with Batch Ordering and Nonstationary/Time‐Correlated Demands

We provide an exact myopic analysis for an N‐stage serial inventory system with batch ordering, linear ordering costs, and nonstationary demands under a finite planning horizon. We characterize the optimality conditions of the myopic nested batching newsvendor (NBN) policy and the myopic independent batching newsvendor (IBN) policy, which is a single‐stage approximation. We show that echelon reorder levels under the NBN policy are upper bounds of the counterparts under both the optimal policy and the IBN policy. In particular, we find that the IBN policy has bounded deviations from the optimal policy. We further extend our results to systems with martingale model of forecast evolution (MMFE) and advance demand information. Moreover, we provide a recursive computing procedure and optimality conditions for both heuristics which dramatically reduces computational complexity. We also find that the NBN problem under the MMFE faced by one stage has one more dimension for the forecast demand than the one faced by its downstream stage and that the NBN policy is optimal for systems with advance demand information and stationary problem data. Numerical studies demonstrate that the IBN policy outperforms on average the NBN policy over all tested instances when their optimality conditions are violated.     

Dynamic Supply Risk Management with Signal‐Based Forecast,Multi‐Sourcing,and Discretionary Selling

We examine the critical role of advance supply signals—such as suppliers’ financial health and production viability—in dynamic supply risk management. The firm operates an inventory system with multiple demand classes and multiple suppliers. The sales are discretionary and the suppliers are susceptible to both systematic and operational risks. We develop a hierarchical Markov model that captures the essential features of advance supply signals, and integrate it with procurement and selling decisions. We characterize the optimal procurement and selling policy, and the strategic relationship between signal‐based forecast, multi‐sourcing, and discretionary selling. We show that higher demand heterogeneity may reduce the value of discretionary selling, and that the mean value‐based forecast may outperform the stationary distribution‐based forecast. This work advances our understanding on when and how to use advance supply signals in dynamic risk management. Future supply risk erodes profitability but enhances the marginal value of current inventory. A signal of future supply shortage raises both base stock and demand rationing levels, thereby boosting the current production and tightening the current sales. Signal‐based dynamic forecast effectively guides the firm's procurement and selling decisions. Its value critically depends on supply volatility and scarcity. Ignoring advance supply signals can result in misleading recommendations and severe losses. Signal‐based dynamic supply forecast should be used when: (a) supply uncertainty is substantial, (b) supply‐demand ratio is moderate, (c) forecast precision is high, and (d) supplier heterogeneity is high.     

Assemble‐to‐Order Inventory Management via Stochastic Programming: Chained BOMs and the M‐System

We study an inventory management mechanism that uses two stochastic programs (SPs), the customary one‐period assemble‐to‐order (ATO) model and its relaxation, to conceive control policies for dynamic ATO systems. We introduce a class of ATO systems, those that possess what we call a “chained BOM.” We prove that having a chained BOM is a sufficient condition for both SPs to be convex in the first‐stage decision variables. We show by examples the necessity of the condition. For ATO systems with a chained BOM, our result implies that the optimal integer solutions of the SPs can be found efficiently, and thus expedites the calculation of control parameters. The M system is a representative chained BOM system with two components and three products. We show that in this special case, the SPs can be solved as a one‐stage optimization problem. The allocation policy can also be reduced to simple, intuitive instructions, of which there are four distinct sets, one for each of four different parameter regions. We highlight the need for component reservation in one of these four regions. Our numerical studies demonstrate that achieving asymptotic optimality represents a significant advantage of the SP‐based approach over alternative approaches. Our numerical comparisons also show that outside of the asymptotic regime, the SP‐based approach has a commanding lead over the alternative policies. Our findings indicate that the SP‐based approach is a promising inventory management strategy that warrants further development for more general systems and practical implementations.     

Competing on Time: An Integrated Framework to Optimize Dynamic Time‐to‐Market and Production Decisions

This study develops a comprehensive framework to optimize new product introduction timing and subsequent production decisions faced by a component supplier. Prior to market entry, the supplier performs process design activities, which improve manufacturing yield and the chances of getting qualified for the customer's product. However, a long delay in market entry allows competitors to enter the market and pass the customer's qualification process before the supplier, reducing the supplier's share of the customer's business. After entering the market and if qualified, the supplier also needs to decide how much to produce for a finite planning horizon by considering several factors such as manufacturing yield and stochastic demand, both of which depend on the earlier time‐to‐market decision. To capture this dependency, we develop a sequential, nested, two‐stage decision framework to optimize the time‐to‐market and production decisions in relation to each other. We show that the supplier's optimal market entry and qualification timing decision need to be revised in real time based on the number of qualified competitors at the time of market‐entry decision. We establish the optimality of a threshold policy. Following this policy, at the beginning of each decision epoch, the supplier should optimally stop preparing for qualification and decide whether to enter the market if her order among qualified competitors exceeds a predetermined threshold. We also prove that the supplier's optimal production policy is a state‐dependent, base‐stock policy, which depends on the time‐to‐market and qualification decisions. The proposed framework also enables a firm to quantify how market conditions (such as price and competitor entry behavior) and operating conditions (such as the rate of learning and inventory/production‐related costs) affect time‐to‐market strategy and post‐entry production decisions.     

The Role of Contract Expirations in Service Parts Management

The majority of after‐sales service providers manage their service parts inventory by focusing on the availability of service parts. This approach, combined with automatic replenishment systems, leads to reactive inventory control policies where base stock levels are adjusted only after a service contract expires. Consequently, service providers often face excess stock of critical service parts that are difficult to dispose due to their specificity. In this study, we address this problem by developing inventory control policies taking into account contract expirations. Our key idea is to reduce the base stock level of the one‐for‐one policy before obsolescence (a full or partial drop in demand rate) occurs and let demand take away excess stock. We refer to this policy as the single‐adjustment policy. We benchmark the single‐adjustment policy with the multiple‐adjustment policy (allowing multiple base stock adjustments) formulated as a dynamic program and verify that for a wide range of instances the single‐adjustment policy is an effective heuristic for the multiple‐adjustment policy. We also compare the single‐adjustment policy with the world‐dependent base stock policy offered by Song and Zipkin (1993) and identify the parameter combinations where both policies yield similar costs. We consider two special cases of the single‐adjustment policy where the base stock level is kept fixed or the base stock adjustment is postponed to the contract expiration time. We find that the initial demand rate, contract expiration time, and size of the drop in demand rate are the three key parameters driving the choice between the single‐adjustment policy and its special cases.     

Multi‐Echelon Inventory Management under Short‐Term Take‐or‐Pay Contracts

We extend the Clark–Scarf serial multi‐echelon inventory model to include procuring production inputs under short‐term take‐or‐pay contracts at one or more stages. In each period, each such stage has the option to order/process at two different cost rates; the cheaper rate applies to units up to the contract quantity selected in the previous period. We prove that in each period and at each such stage, there are three base‐stock levels that characterize an optimal policy, two for the inventory policy and one for the contract quantity selection policy. The optimal cost function is additively separable in its state variables, leading to conquering the curse of dimensionality and the opportunity to manage the supply chain using independently acting managers. We develop conditions under which myopic policies are optimal and illustrate the results using numerical examples. We establish and use a generic one‐period result, which generalizes an important such result in the literature. Extensions to cover variants of take‐or‐pay contracts are included. Limitations are discussed.     

Optimal Ordering Policies for Inventory Problems With Dynamic Information Delays

Information delays exist when the most recent inventory information available to the Inventory Manager (IM) is dated. In other words, the IM observes only the inventory level that belongs to an earlier period. Such situations are not uncommon, and they arise when it takes a while to process the demand data and pass the results to the IM. We introduce dynamic information delays as a Markov process into the standard multiperiod stochastic inventory problem with backorders. We develop the concept of a reference inventory position. We show that this position along with the magnitude of the latest observed delay and the age of this observation are sufficient statistics for finding the optimal order quantities. Furthermore, we establish that the optimal ordering policy is of state‐dependent base‐stock type with respect to the reference inventory position (or state‐dependent (s, S) type if there is a fixed ordering cost). The optimal base stock and (s, S) levels depend on the magnitude of the latest observed delay and the age of this observation. Finally, we study the sensitivity of the optimal base stock and the optimal cost with respect to the sufficient statistics.     

Inventory Rationing for Multiple Class Demand under Continuous Review

We consider how a firm should ration inventory to multiple classes in a stochastic demand environment with partial, class‐dependent backlogging where the firm incurs a fixed setup cost when ordering from its supplier. We present an infinite‐horizon, average cost criterion Markov decision problem formulation for the case with zero lead times. We provide an algorithm that determines the optimal rationing policy, and show how to find the optimal base‐stock reorder policy. Numerical studies indicate that the optimal policy is similar to that given by the equivalent deterministic problem and relies on tracking both the current inventory and the rate that backorder costs are accumulating. Our study of the case of non‐zero lead time shows that a heuristic combining the optimal, zero lead time policy with an allocation policy based on a single‐period profit management problem is effective.     

Inventory Management for Customers with Alternative Lead Times

It is common for suppliers operating in batch‐production mode to deal with patient and impatient customers. This paper considers inventory models in which a supplier provides alternative lead times to its customers: a short or a long lead time. Orders from patient customers can be taken by the supplier and included in the next production cycle, while orders from impatient customers have to be satisfied from the on‐hand inventory. We denote the action to commit one unit of on‐hand inventory to patient or impatient customers as the inventory‐commitment decision, and the initial inventory stocking as the inventory‐replenishment decision. We first characterize the optimal inventory‐commitment policy as a threshold type, and then prove that the optimal inventory‐replenishment policy is a base‐stock type. Then, we extend our analysis to models to consider cases of a multi‐cycle setting, a supply‐capacity constraint, and the on‐line charged inventory‐holding cost. We also evaluate and compare the performances of the optimal inventory‐commitment policy and the inventory‐rationing policy. Finally, to further investigate the benefits and pitfalls of introducing an alternative lead‐time choice, we use the customer‐choice model to study the demand gains and losses, known as demand‐induction and demand‐cannibalization effects, respectively.     

Safety Stock,Excess Capacity or Diversification: Trade‐Offs under Supply and Demand Uncertainty

Firms mitigate uncertainty in demand and supply by carrying safety stock, planning for excess capacity and diversifying supply sources. In this study, we provide a framework to jointly optimize these three levers in a periodic review infinite horizon setting, and in particular we examine how one can reduce inventory and capacity investments through proper diversification strategies. Observing that a modified base‐stock inventory policy is optimal, we find that the capacity‐diversification problem is well behaved and characterize the optimal mix of safety stock, excess capacity and extra number of supply sources. We find that higher supply uncertainty results in higher safety stock, more excess capacity, and higher diversification. But safety stock and diversification are non‐monotonic in demand uncertainty. Our results can be extended to situations in which suppliers are heterogeneous, and can be used to develop effective heuristics.     

Cash‐Flow Based Dynamic Inventory Management

Small‐to‐medium‐sized enterprises (SMEs), including many startup firms, need to manage interrelated flows of cash and inventories of goods. In this study, we model a firm that can finance its inventory (ordered or manufactured) with loans in order to meet random demand which in general may not be time stationary. The firm earns interest on its cash on hand and pays interest on its debt. The objective is to maximize the expected value of the firm's capital at the end of a finite planning horizon. The firm's state at the beginning of each period is characterized by the inventory level and the capital level measured in units of the product, whose sum represents the “net worth” of the firm. Our study shows that the optimal ordering policy is characterized by a pair of threshold parameters as follows. (i) If the net worth is less than the lower threshold, then the firm employs a base stock order up to the lower threshold. (ii) If the net worth is between the two thresholds, then the firm orders exactly as many units as it can afford, without borrowing. (iii) If the net worth is above the upper threshold, then the firm employs a base stock order up to the upper threshold. Further, upper and lower bounds for the threshold values are developed using two simple‐to‐compute myopic ordering policies which yield lower bounds for the value function. We also derive an upper bound for the value function by considering a sell‐back policy. Subsequently, it is shown that policies of similar structure are optimal when the loan and deposit interest rates are piecewise linear functions, when there is a maximal loan limit and when unsatisfied demand is backordered. Finally, further managerial insights are provided with extensive numerical studies.     

MRP Planned Orders in a Multiple‐Supplier Environment with Differing Lead Times

This study examines a deterministic material requirements planning (MRP) problem where lead times at subsequent ordering moments differ. Adequate replenishment methods that can cope with lead time differences are lacking because of the order crossover phenomenon, that is, replenishment orders are not received in the sequence they are ordered. This study specifies how to handle order crossovers and recalculate planned order releases after an update of gross requirements. The optimal (s, S) policy is based on dynamic programing. The state space is kept to a minimum due to three fundamental insights. The performance of the optimal solution approach is compared with two heuristics based on relaxations and a benchmark approach in which order crossovers are ignored. A numerical analysis reveals that average cost savings up to 25% are possible if the optimal policy is used instead of the benchmark approach. The contribution of this study is threefold: (1) it generalizes theory on MRP ordering, allowing for lead time differences and order crossovers; (2) it develops new fundamental insights and an optimal solution procedure, leading to substantial cost saving; and (3) it provides good‐performing heuristics for a general and realistic replenishment problem that can replace the current replenishment methods within MRP.     

Managing Retail Shelf and Backroom Inventories When Demand Depends on the Shelf‐Stock Level

Inventory displayed on the retail sales floor not only performs the classical supply function but also plays a role in affecting consumers’ buying behavior and hence the total demand. Empirical evidence from the retail industry shows that for some types of products, higher levels of on‐shelf inventory have a demand‐increasing effect (“billboard effect”) while for some other types of products, higher levels of on‐shelf inventory have a demand‐decreasing effect (“scarcity effect”). This suggests that retailers may use the amount of shelf stock on display as a tool to influence demand and operate a store backroom to hold the inventory of items not displayed on the shelves, introducing the need for efficient management of the backroom and on‐shelf inventories. The purpose of this study is to address such an issue by considering a periodic‐review inventory system in which demand in each period is stochastic and depends on the amount of inventory displayed on the shelf. We first analyze the problem in a finite‐horizon setting and show under a general demand model that the system inventory is optimally replenished by a base‐stock policy and the shelf stock is controlled by two critical points representing the target levels to raise up/drop down the on‐shelf inventory level. In the infinite‐horizon setting, we find that the optimal policies simplify to stationary base‐stock type policies. Under the billboard effect, we further show that the optimal policy is monotone in the system states. Numerical experiments illustrate the value of smart backroom management strategy and show that significant profit gains can be obtained by jointly managing the backroom and on‐shelf inventories.     

Unsold Versus Unbought Commitment: Minimum Total Commitment Contracts with Nonzero Setup Costs

We study a minimum total commitment (MTC) contract embedded in a finite‐horizon periodic‐review inventory system. Under this contract, the buyer commits to purchase a minimum quantity of a single product from the supplier over the entire planning horizon. We consider nonstationary demand and per‐unit cost, discount factor, and nonzero setup cost. Because the formulations used in existing literature are unable to handle our setting, we develop a new formulation based on a state transformation technique using unsold commitment instead of unbought commitment as state variable. We first revisit the zero setup cost case and show that the optimal ordering policy is an unsold‐commitment‐dependent base‐stock policy. We also provide a simpler proof of the optimality of the dual base‐stock policy. We then study the nonzero setup cost case and prove a new result, that the optimal solution is an unsold‐commitment‐dependent (s, S) policy. We further propose two heuristic policies, which numerical tests show to perform very well. We also discuss two extensions to show the generality of our method's effectiveness. Finally, we use our results to examine the effect of different contract terms such as duration, lead time, and commitment on buyer's cost. We also compare total supply chain profits under periodic commitment, MTC, and no commitment.     

Lowest Cost May Not Lower Total Cost: Using “Spackling” to Smooth Mass‐Customized Production

Consider a manufacturer who mass customizes variants of a product in make‐to‐order fashion, and also produces standard variants as make‐to‐stock. A traditional manufacturing strategy would be to employ two separate manufacturing facilities: a flexible plant for mass‐customized items and an efficient plant for standard items. We contrast this traditional focus strategy with an alternative that better utilizes capacity by combining production of mass‐customized and standard items in one of two alternate spackling strategies: (1) a pure‐spackling strategy, where the manufacturer produces everything in a (single) flexible plant, first manufacturing custom products as demanded each period, and then filling in the production schedule with make‐to‐stock output of standard products; or (2) a layered‐spackling strategy, which uses an efficient plant to make a portion of its standard items and a separate flexible plant where it spackles. We identify the optimal production strategy considering the tradeoff between the cost premium for flexible (versus efficient) production capacity and the opportunity costs of idle capacity. Spackling amortizes fixed costs of capacity more effectively and thus can increase profits from mass customization vis‐à‐vis a focus strategy, even with higher cost production for the standard goods. We illustrate our framework with data from a messenger bag manufacturer.     

A Bayesian Inventory Model Using Real‐Time Condition Monitoring Information

Lack of coordination between machinery fault diagnosis and inventory management for spare parts can lead to increased inventory costs and disruptions in production activity. We develop a framework for incorporating real‐time condition monitoring information into inventory decisions for spare parts. We consider a manufacturer who periodically replenishes inventory for a machine part that is subject to deterioration. The deterioration is captured via condition monitoring and modeled using a Wiener process. The resulting degradation model is used to derive the life distribution of a functioning part and to estimate the demand distribution for spare parts. This estimation is periodically updated, in a Bayesian manner, as additional information on part deterioration is obtained. We develop an inventory model that incorporates this updated demand distribution and demonstrate that a dynamic base‐stock policy, in which the optimal base‐stock level is a function of some subset of the observed condition monitoring information, is optimal. We propose a myopic critical fractile policy that captures the essence of the optimal policy, but is easier to compute. Computational experiments indicate that this heuristic performs quite well relative to the optimal policy. Adaptive inventory policies such as these can help manufacturers to increase machine availability and reduce inventory costs.     

The Use of Switching Point and Protection Levels to Improve Revenue Performance in Order‐Driven Production Systems

In a multiproduct order‐driven production system, an organization has to decide how to selectively accept orders and allocate capacity to these orders so as to maximize total profit (TP). In this article, we incorporate the novel concept of switching point in developing three capacity‐allocation with switching point heuristics (CASP_a‐c). Our analysis indicates that all three CASP heuristics outperform the first‐come‐first‐served model and Barut and Sridharan's dynamic capacity‐allocation process (DCAP) model. The best model, CASP_b, has an 8% and 6% average TP improvement over DCAP using the split lot and whole lot policies, respectively. In addition, CASP_b performs particularly well under operating conditions of tight capacity and large price differences between product classes. The introduction of a switching point, which has not been found in previous capacity‐allocation heuristics, provides for a better balance between forward and backward allocation of available capacity and plays a significant role in improving TP.