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In this paper, we consider the stock rationing problem of a single-item make-to-stock production/inventory system with multiple demand classes. Demand arrives as a Poisson process with a randomly distributed batch size. It is assumed that the batch demand can be partially satisfied. The facility can produce a batch up to a certain capacity at the same time. Production time follows an exponential distribution. We show that the optimal policy is characterized by multiple rationing levels.

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... Most of studies assumed that the demand arrival pattern followed a Poisson process and the production time followed either exponential or Erlang distribution. Ha [14] modeled an [21] Melchiors et al. [19] Deshpande et al. [6] Moon and Kang [20] Melchiors [19] Dekker et al. [5] Arslan et al. [1] Ha [14,15] Ha [16] de Véricourt et al. [2,3] Gayon et al. [12] Xu et al. [27] Evans [8] Kaplan [17] Sobel and Zhang [23] Frank et al. [11] Veinott [26] Topkis [25] Ding et al. [7] This Study M/M/1 make-to-stock queue with multiple demand classes and lost sales. He proved that the optimal production policy was a base stock policy and the optimal allocation policy was a rationing policy. ...

... Later, Gayon et al. [12] further extended the study to the case of an M/E k /1, maketo-stock queue with multiple demand classes and backorders. Xu et al. [27] extended Ha's model [15] in another direction. In their case, they assumed demands arrived with a randomly distributed batch size and could be partially satisfied. ...

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

... de Vericourt et al. (2002) consider an extension of Ha (1997b) with multiple-demand class. Huang and Iravani (2008) and Xu et al. (2010) further extend the de Vericourt et al. (2002) model to more general settings, such as multiple demand classes and batch arrival. Benjaafar et al. (2010) study a production/inventory system with a single product and two customer classes where both backorders and lost sales are permitted. ...

... Following a similar analysis as in Xu et al. (2010) and Çil et al. (2009), we can show that T ia ...

The rationing literature has been so far oblivious to the fact that sellable items may not only be finished products, but also intermediate products to manufacture subsequent sellable products. We address this gap by considering an inventory rationing problem in a two-product tandem make-to-stock production/inventory system. Bulk demand arrives to a partial-batch production system with exponentially-distributed production time for each batch. The management has to decide whether to run or stop production and whether the various classes of demand for both products—intermediate product (IP) and finished product or end product (EP)—have to be satisfied from available inventory or not—in which case demand is lost—in order to maximize the firm's expected profit. We present the corresponding dynamic programming formulation and characterize the optimal policy. The resulting policy depends on dynamic switching curves, which define a) thresholds to continue or discontinue production, and b) thresholds to satisfy or turn down incoming orders. We identify the key value drivers and compare various heuristic policies through extensive numerical analyses.

... It is well-established in the supply chain literature that the demand process can exhibit high variability and unpredictability (Germain et al. 2008), time-dependency (Neale and Willems 2009), inter-arrival auto-correlation (Lee and Chew 2005), and/or batch demand (Lian and Liu 2001;Lu 2008;Xu et al. 2010;Xu et al. 2011). Accounting for any of these characteristics on their own would complicate the analysis of the inventory system. ...

Modeling the behavior of customer demand is a key challenge in inventory control, where an accurate characterization of the demand process often involves accounting for a wide range of statistical descriptors. This motivates the use of Markovian processes, due to their proven versatility in matching key components of point processes, to capture the behavior of customer demand. Accordingly, this work presents computational frameworks for continuous inventory models with a batch Markovian demand. A Markovian formulation of the system state-space is presented along with computational approaches to obtain key inventory performance measures. Compact matrix representations are considered for the steady-state solution of the system performance measures. The transient and non-stationary behavior of the inventory system is calculated by numerically integrating the corresponding set of Kolmogorov forward equations. A byproduct of this work is explicitly expressing the solution of the moments of the batch Markovian counting process by a compact matrix exponential equation. Numerical examples illustrate the computational efficiency of the mathematical frameworks when evaluating and comparing the performance of different re-ordering policies.

... Such demand is called a single demand. However, the batch demands are very common in supply chain management, examples can be found in [8][9][10]. ...

This paper studies an M/M/1 queueing-inventory system with batch demands. Customers arrive in the system according to a compound Poisson process, where the size of the batch demands for each arrival is a random variable that follows a geometric distribution. The inventory is replenished according to the standard (s,S) policy. The replenishment time follows an exponential distribution. Two models are considered. In the first model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer takes away all the items in the inventory, and a part of the customer’s batch demands is lost. In the second model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer leaves without taking any item from the inventory, and all of the customer’s batch demands are lost. For these two models, the authors derive the stationary conditions of the system. Then, the authors derive the stationary distributions of the product-form of the joint queue length and the on-hand inventory process. Besides this, the authors obtain some important performance measures and the average cost functions by using these stationary distributions. The results are illustrated by numerical examples.

... Fadiloglu and Bulut (2010) propose a dynamic rationing policy for the continuous-review inventory systems based on the status of outstanding orders. Moreover, both Huang and Iravani (2008) and Xu et al. (2010) consider stock rationing problems with batch demand while Pang et al. (2014) study inventory rationing in the make-to-stock systems with batch ordering and lost sales. ...

This paper studies the joint stock and production capacity rationing polices for a make-to-stock system with two partially substitutable products manufactured at two factories, respectively. Customers are classified into three types with each type further segmented into multiple classes. Type I and Type II customers are selective and accept their favorite products only. Type III accepts either of the two products and hence its demand is regarded as the flexible demand. The management has to decide whether to run or stop production and whether the various classes of demand for each type of customers have to be satisfied from available inventory or not — in which case demand is lost — in order to minimize the firm’s inventory and production costs. We formulate the problem as a make-to-stock queue system and characterize the optimal joint stock and production rationing policies as monotone switching curves. Numerical examples are given to illustrate the optimal policies and demonstrate the value of flexible demand. Moreover, we develop a simple heuristic policy and compare the costs associated with the optimal and heuristic policies for different parameters.

... This is not the only available option. As it can be seen from the literature on inventory management, possible alternatives would consider back-order costs depending on backorder duration or depending on the stock-out frequency (Xu et al., 2010;Daqin Wang et al., 2013a;Deshpande et al., 2003). Although the analytics becomes more complex, MLQ could also be devised for the second case. ...

Supply chain effectiveness and costs are affected by the demand variability, especially in the upstream echelons. The propagation of demand variability moving upstream in the supply chain has been widely studied in the literature and it is known as the Bullwhip Effect phenomenon. In this paper, the bullwhip effect in a European automotive spare parts chain is identified, with the aim of shedding some light on how demand variability propagates in different groups of products. We considered more than 30,000 products, characterised by different technical characteristics, demand classes and planning parameters.The results showed that the considered supply chain is affected by the bullwhip effect. Additional analyses suggested that the bullwhip effect is larger for fast moving products rather than for slow movers. Hence, dealers tends to decouple supply and demand and, when they are given incentives to forward-buy, they may prefer to forward-buy fast moving items. Moreover, as the dealers tend to exploit the promotional benefits by forward-buying in the final period of promotions, frequent switches from promotional to non promotional periods tend to increase the propagation of the demand variability.

... This is not the only available option. As it can be seen from the literature on inventory management, possible alternatives would consider back-order costs depending on backorder duration or depending on the stock-out frequency (Xu et al., 2010;Daqin Wang et al., 2013a;Deshpande et al., 2003). Although the analytics becomes more complex, MLQ could also be devised for the second case. ...

In spare part distribution, contracts characterized by different unit prices, lead times, minimum order quantities are usually offered to different types of customers. In this context, one of the main problems is the decision on the allocation of the on–hand inventory to the different customers in order to minimize the total back–order cost. In this paper, we propose two allocation policies that use a dynamic critical level to reserve on-hand inventory for high priority customers. The first policy sets the critical level according to a target service level for high priority demand, while the other sets the critical level according to the back-order costs for the different classes of customers. The effectiveness of the two policies with respect to no rationing or stock piling polices is shown using the data of a European spare part company. Moreover, the relationship between the two policies is evaluated from a managerial perspective.

... This is not the only available option. As it can be seen from the literature on inventory management, possible alternatives would consider back-order costs depending on backorder duration or depending on the stock-out frequency (Xu et al., 2010;Daqin Wang et al., 2013a;Deshpande et al., 2003). Although the analytics becomes more complex, MLQ could also be devised for the second case. ...

The increase of demand variability along the supply chain has a huge impact in companies, where demand fluctuations make the forecasting and planning processes harder, especially in the upstream levels. We present an empirical analysis on the Bullwhip Effect (BE) identification on more than 30,000 SKUs of a three-echelon European automotive spare parts supply chain. With our analyses, we aim at shedding some light on the incentives that lead to the smoothing/amplification of demand variability.
We analyse the bullwhip effect on two different aggregation levels. First, we evaluate the global BE along the supply chain, calculated on the total demand series. We aim at identifying the intensity of the BE in each level and at investigating if the empirical context reflects the literature results concerning a monotonic demand variability amplification through the echelons. Secondly, we increase the level of detail and we study the BE on the single SKUs. The objective is to measure how the intensity of the BE varies across products. Also we want to investigate what product characteristics tend to make the bullwhip effect stronger.
Preliminary analyses confirm the presence of bullwhip when considering the total demand series. Demand variability amplifies through each echelon, especially moving from dealers to local warehouses. This first outcome provides us with insights on the incentives structure of the company. In fact, dealers are independent entities. The strong increase of demand variability from independent nodes to the distribution company is probably due to the incentive structures given to dealers. In particular, demand at the top level is found to be more than twice variable than the final sell-out. Further analyses will be devoted to understanding the final effect of such a great demand variability on the suppliers’ behaviour. Also, the comparison of the BE distribution among different demand classes calculated on single SKUs gives us a very interesting perspective. The BE is higher for fast moving products than for slow movers. The combination of the two results suggests that dealers tend to decouple demand and supply for fast movers. Thus, when they are given an incentive to forward buy, they prefer to forward buy fast moving items. Further research will be devoted to prove the validity of our hypothesis on dealers’ behaviour and investigate the contribution of other product characteristics on the BE.

We analyze a manufacturing system that produces a single type of product to serve multiple classes of customers with seasonal demands. The manufacturer has the flexibility to vary production rates to adapt to seasonal demands. We model the seasonal demand using a Markov modulated Poisson process, and analyze the underlying Markov decision problem to derive optimal production and inventory control policies for the manufacturer. We show that the optimal policy is characterized by a season dependent inventory threshold vector for adjusting production rates and rationing inventory. Further, we impose service level constraints according to the customer classes in each season, and analyze the impact of the service level constraints on the optimal policy and cost under different seasonal demand conditions.

When many weights are stochastic, the flow allocation of paths is a key stage of the transportation network's efficiency. Currently, a variety of methods (or heuristics) have been proposed to solve this complex optimization problem, with good results in some cases just with limitations in the special fields. On this basis, we develop an algorithm for model multiobjective that combines ideas under stochastic weight. The method performs well even when the order of magnitude and/or the range of the parameters were unknown. It refines iteratively a sequence of parameter distributions through preference combined with partial exampling from a historical prior defined over the support of all previous iterations. Using the simulated and real experimental data, we exemplify our method with multi-objective improved models--estimated the weight efficiently even in the absence of the weight.

In a profit-oriented organisation with limited resources, the effective allocation of limited capacities to meet demands at the right time to maximise profits is called yield management or revenue management. For a make-to-order manufacturer, products are made only after a received order is confirmed by the customer. The decision to accept or reject the order inquiry is called capacity rationing. This paper investigates three applications extended from the dynamic stochastic capacity rationing (DSCR) procedure proposed previously. The first application extension considers the problem environment, where products are classified into a discrete number of profit classes. Numerical results show that the total profit obtained by the discrete-class DSCR is significantly higher compared with other previous approaches. In the second extension, a boundary profit is calculated and used as a reference value for a price quotation/negotiation in response to a customer order inquiry, which includes a requested quantity of products and delivery date. The third extension allows partial order fulfilment, which is more realistic to several industrial environments. Numerical experiments are also conducted to validate the proposed application extensions of DSCR.

The lack of homogeneity in the product (LHP) is defined as the lack of uniformity required by the customer in the products. The LHP appears in companies where the final products obtained are not homogeneous, leading to the existence of different references (subtypes) of the same product. This lack of homogeneity is a problem when the client needs to be served through homogeneous units of a product and commit orders are based on planned quantities, whose final homogeneity characteristics are unknown at the time of acquiring the customer commitments. The frequent discrepancies caused by the LHP between planned homogeneous amounts and those actually obtained and available, can prevent the delivery of committed orders. To solve this problem, we propose a mathematical programming model for the reallocation of inventory in Make to Stock (MTS) ceramic tile companies characterized by the LHP that combines multiple objectives. The proposed mathematical model has been validated by its application to a real case of a ceramic company. The analysis of the obtained results indicates significant improvements in the number of orders completed on time and in sales revenue achieved.
La Falta de Homogeneidad en el Producto (FHP), se define como la carencia de la homogeneidad requerida por el cliente en los productos. La FHP aparece en empresas en las que los productos finales obtenidos no son homogéneos, dando lugar a la existencia de diferentes referencias (subtipos) de un mismo producto final. Esta falta de homogeneidad supone un problema cuando el cliente requiere ser servido a través de unidades homogéneas de un mismo producto y sus pedidos se comprometen en base cantidades planificadas, cuyas características de homogeneidad finales se desconocen en el momento de adquirir los compromisos con el cliente. Las constantes discrepancias provocadas por la FHP entre las cantidades planificadas y las realmente obtenidas y disponibles, pueden impedir servir pedidos comprometidos previamente. Para resolver este problema, se propone un modelo de programación matemática que permite reasignar el inventario en empresas caracterizadas por la FHP que fabrican contra almacén (Make to Stock: MTS) que combina varios objetivos. El modelo matemático propuesto se ha validado mediante su aplicación a un caso real de una empresa cerámica. El análisis de los resultados indica la obtención de mejoras considerables en la cantidad de pedidos completados a tiempo y en los ingresos por ventas.

We study the 'Creation of Pooling in Inventory and Queueing Models'. This research consists of the study of sharing a scarce resource (such as inventory, server capacity, or production capacity) between multiple customer classes. This is called pooling, where the goal is to achieve cost or waiting time reductions. For the inventory and queueing models studied, both theoretical, scientific insights are generated, as well as strategies which are applicable in practice.
This monograph consists of two parts: pooling and polling . In the first part, pooling is applied to multi-location inventory models. It is studied how cost reduction can be achieved by the use of stock transfers between local warehouses, so-called lateral transshipments. In this way, stock is pooled between the warehouses. The setting is motivated by a spare parts inventory network, where critical components of technically advanced machines are kept on stock, to reduce down time durations. We create insights into the question when lateral transshipments lead to cost reductions, by studying several models.
Firstly, a system with two stock points is studied, for which we completely characterize the structure of the optimal policy, using dynamic programming. For this, we formulate the model as a Markov decision process. We also derived conditions under which simple, easy to implement, policies are always optimal, such as a hold back policy and a complete pooling policy. Furthermore, we identified the parameter settings under which cost savings can be achieved. Secondly, we characterize the optimal policy structure for a multi-location model where only one stock point issues lateral transshipments, a so-called quick response warehouse. Thirdly, we apply the insights generated to the general multi-location model with lateral transshipments. We propose the use of a hold back policy, and construct a new approximation algorithm for deriving the performance characteristics. It is based on the use of interrupted Poisson processes. The algorithm is shown to be very accurate, and can be used for the optimization of the hold back levels, the parameters of this class of policies. Also, we study related inventory models, where a single stock point servers multiple customers classes.
Furthermore, in the first part, the pooling of server capacity is studied. For a two queue model where the head-of-line processor sharing discipline is applied, we derive the optimal control policy for dividing the servers attention, as well as for accepting customers. Also, a server farm with an infinite number of servers is studied, where servers can be turned off after a service completion in order to save costs. We characterize the optimal policy for this model.
In the second part of the thesis, polling models are studied, which are queueing systems where multiple queues are served by a single server. An application is the production of multiple types of products on a single machine. In this way, the production capacity is pooled between the product types. For the classical polling model, we derive a closed-form approximation for the mean waiting time at each of the queues. The approximation is based on the interpolation of light and heavy traffic results. Also, we study a system with so-called smart customers, where the arrival rate at a queue depends on the position of the server. Finally, we invent two new service disciplines (the gated/exhaustive and the k-gated discipline) for polling models, designed to yield 'fairness and efficiency' in the mean waiting times. That is, they result in almost equal mean waiting times at each of the queues, without increasing the weighted sum of the mean waiting times too much.

This paper addresses a rationing problem in a two-level, vertically integrated distribution system composed of one manufacturer and several retail points. The motivating case, developed in the vending machine sector and modeled as a newsvendor-like problem, is representative of many real settings where short-term changes in demand can be substantial while capacity modification is not a viable option. The paper provides an analytical discussion of the problem from two different standpoints: a pure profit-maximization perspective and a minimum service-level perspective, both subject to a product availability constraint that affects the service level the company can provide, and the related expected profit. By analyzing the Lagrangean formulation of the problem, we devise efficient computational procedures based on dichotomy search to find the optimal allotments to retailers, maximizing the expected profit and ensuring a minimum service level. Then, we extend the analysis to the evaluation of the highest service level that can be provided, under a product availability constraint. We identify conditions such that the proposed search procedures succeed in finding the optimal solutions, as well as bounds for the search domains. The proposed approach is legitimated under several demand distribution functions subject to a few commonly adopted restrictions that encompass many of the usually adopted continuous distributions. Finally, the paper presents a three-step decision-making framework using the proposed procedures, summarizing the decision paths the manufacturer might follow in order to optimize the allocation decision.

This paper examines dynamic selling (DS) problems under demand uncertainties. Quality-graded products with fully downward substitutable demands are considered. Downward demand substitution indicates that demands for lower quality grade products can be fulfilled by either designated or higher quality grade products. In this dynamic selling problem, decision makers need to choose an optimal selling policy in each decision epoch. The objective is to identify an optimal policy for the dynamic selling of quality-graded inventory.DS problems are formulated as a discrete-time Markov decision process (MDP) model. In the MDP model, demand type and inventory levels are state variables. The objective is to maximize expected profits. In such a multi-dimensional dynamic decision problem, computational complexity is a chief concern. This study proves the structure of optimal policies that significantly reduce computational complexity. Performance of optimal dynamic selling policies is evaluated in detailed numerical studies.

This paper considers inventory systems which maintain stocks to meet various demand classes with different priorities. We use the concept of a support level control policy. That is rationing is accomplished by maintaining a support level, say K, such that when on hand stock reaches K, all low priority demands are backordered. We develop four analytical and simulation models to improve the existing models. Firstly, multiple support levels are used instead of using a single support level. Secondly, a simulation model with a more realistic assumption on the demand process has been provided. Thirdly, a single period deterministic cost minimisation model has been developed analytically. Finally, we address a continuous review (Q, r) model with a compound Poisson process.

We consider a capacitated supply system that produces a single item that is demanded by several classes of customers. Each customer class may have a different backorder cost, so stock allocation arises as a key decision problem. We model the supply system as a multi customer make-to-stock queue. Using dynamic programming, we show that the optimal allocation policy has a simple and intuitive structure. In addition, we present an efficient algorithm to compute the parameters of this optimal allocation policy. Finally, for a typical supply chain design problem, we illustrate that ignoring the stock allocation dimension---a frequently encountered simplifying assumption---can lead to incorrect managerial decisions.

This paper considers an inventory system which maintains stock to meet both high and low priority demands. This model is suggested by the operation of a spare parts pool in a military depot: high priority demands are those which might result in the grounding of an aircraft, for example, while low priority demands are those which arise from the routine restocking of base level inventories. We analyze the following type of control policy: there is a support level, say K > 0, such that when the level of on hand stock reaches K, all low priority demands are backordered while high priority demands continue to be filled. Both continuous review and periodic review systems are considered. The objective of the analysis is to develop methods for comparing fill rates when there is rationing and when there is no rationing for specified values of the reorder point, order quantity and support level.

We consider the optimal production and inventory control of an assemble-to-order system with m components, one end-product, and n customer classes. A control policy specifies when to produce each component and, whenever an order is placed, whether or not to satisfy it from on-hand inventory. We formulate the problem as a Markov decision process and characterize the structure of an optimal policy. We show that a base-stock production policy is optimal, but the base-stock level for each component is dynamic and depends on the inventory level of all other components (more specifically, it is nondecreasing). We show that the optimal inventory allocation for each component is a rationing policy with different rationing levels for different demand classes. The rationing levels for each component are dynamic and also nondecreasing in the inventory level of all other components. We compare the performance of the optimal policy to heuristic policies, including the commonly used base-stock policy with fixed base-stock levels, and find them to perform surprisingly well, especially for systems with lost sales.

Motivated by a study of the logistics systems used to manage consumable service parts for the U.S. military, we consider a static threshold-based rationing policy that is useful when pooling inventory across two demand classes characterized by different arrival rates and shortage (stockout and delay) costs. The scheme operates as a (Q, r) policy with the following feature. Demands from both classes are filled on a first-come- first-serve basis as long as on-hand inventory lies above a threshold level K.O nce on-hand inventory falls below this level, low priority (i.e., low shortage cost) demand is backordered while high priority demand continues to be filled. We analyze this sta- tic policy first under the assumption that backorders are filled according to a special threshold clearing mechanism. Structural results for the key performance measures are established to enable an efficient solution algorithm for computing stock control and rationing parameters (i.e., Q, r, and K). Numerical results confirm that the solution under this special threshold clearing mechanism closely approximates that of the pri- ority clearing policy. We next highlight conditions where our policy offers significant savings over traditional 'round-up' and 'separate stock' policies encountered in the mil- itary and elsewhere. Finally, we develop a lower bound on the cost of the optimal rationing policy. Numerical results show that the performance gap between our static threshold policy and the optimal policy is small in environments typical of the military and high technology industries.

We consider a periodic review inventory system with two priority demand classes, one deterministic and the other stochastic. The deterministic demand must be met immediately in each period. However, the units of stochastic demand that are not satisfied during the period when demand occurs are treated as lost sales. At each decision epoch, one has to decide not only whether an order should be placed and how much to order, but also how much demand to fill from the stochastic source. The firm has the option to ration inventory to the stochastic source (i.e., not satisfy all customer demand even though there is inventory in the system).We first characterize the structure of the optimal policy. We show that, in general, the optimal order quantity and rationing policy are state dependent and do not have a simple structure. We then propose a simple policy, called ( s, k, S) policy, wheres andS (ordering policy) determine when and how much to order, whilek (rationing policy) specifies how much of the stochastic demand to satisfy. We report the results of a numerical study, which shows that this simple policy works extremely well and is very easy to compute.

This paper examines the impact of customer order sizes on a make-to-stock system with multiple demand classes. We first characterize the manufacturer's optimal production and rationing policies when the demand is nonunitary and lost if unsatisfied. We also investigate the optimal policies of a backorder system with two demand classes and fixed order sizes. Through a numerical study, we show the effects of batch orders on the manufacturer's inventory cost as well as on the benefit of optimal stock rationing. It is shown that batch ordering may reduce the manufacturer's overall cost if carefully introduced in a first-come-first-served (FCFS) system. With the same effective demand rates, the customers' order sizes also have a strong impact on the benefit of optimal stock rationing. Subject classifications: batch ordering; make-to-stock; stock rationing. Area of review: Manufacturing, Service, and Supply Chain Operations. History: Received October 2004; revisions received July 2006, June 2007; accepted July 2007.

In this chapter we discuss inventory systems where several demand classes may be distinguished. In particular, we focus on single-location inventory systems and we analyse the use of a so-called critical level policy. With this policy some inventory is reserved for high-priority demand. A number of practical examples where several demand classes naturally arise are presented, and the implications and modelling of the critical level policy in distribution systems are discussed. Finally, an overview of the literature on inventory systems with several demand classes is given. Keywords. Inventory, demand classes, rationing, critical level 1 Introduction Inventory systems often face customer demand for many different products. The demand characteristics may vary from product to product and therefore an inventory manager will generally apply a customised policy for every product. However, in most cases, all customer demand for a single product is handled in a uniform way. Although t...

A multiperiod single product nonstationary inventory problem is studied in which the system is reviewed at the beginning of each of a sequence of periods of equal length. The model has the following features. There are several classes of demand for the product in each period. The demands in different periods are independent but not necessarily identically distributed. The cost structure is nonstationary with the ordering cost being proportional to the amount ordered. Conditions are given that ensure that the base stock ordering policy is optimal and that the base stock levels in each period are easy to calculate. The results are based on the work given in the literature and on properties of stochastically ordered distributions that are developed in the paper. The case of a linear holding cost and a linear shortage cost is studied in detail.

This paper considers the stock rationing problem of a single-item, make-to-stock production system with several demand classes and lost sales. When demand is Poisson and processing time has an Erlang distribution, we show that a single-state variable called work storage level can be employed to completely capture the information regarding inventory level and the status of current production. The optimal rationing policy can be characterized by a sequence of monotone critical work storage levels. For each demand class, there exists a work storage level at or below which it is optimal to start rejecting the demand of this class in anticipation of future arrival of higher-priority demands. The optimal production policy can also be characterized by a critical work storage level. Our numerical examples indicate that a critical stock level policy, which ignores information on the status of current production, performs very well.

The author considers the problem of controlling M/M/c queueing systems. By providing a new definition of the time of transition, the standard set of decision epochs is enlarged and a preferred version of the n-period problem in which the times between transitions are exponential random variables with constant parameter is obtained. Using this new device, it is possible to utilize the inductive approach in a manner characteristic of inventory theory.

This article considers the problem of production control and stock rationing in a make-to-stock production system with two priority customer classes and backordering. The problem is formulated as a queueing control model. With Poisson arrivals and exponential production times, we show that the optimal production control and stock-rationing policies can be characterized by a single, monotone switching curve. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 457–472, 1997

This paper considers the stock rationing problem of a single-item, make-to-stock production system with several demand classes and lost sales. For the case of Poisson demands and exponential production times, we show that the optimal policy can be characterized by a sequence of monotone stock rationing levels. For each demand class, there exists a stock rationing level at or below which it is optimal to start rejecting the demand of this class in anticipation of future arrival of higher priority demands. A simple queueing model is analyzed to compute the operating cost of a rationing policy. In a numerical study, we compare the optimal rationing policy with a first-come first-served policy to investigate the benefit of stock rationing under different operating conditions of the system.

We consider the problems associated with an inventory system in which demands for stock are of n classes of varying importance. When demand from a given class arrives one must decide whether to satisfy it or to not satisfy it and conserve stock for possible use later to satisfy demand from a more important class. Conditions are given under which the optimal rationing policy between successive procurements of new stock is determined by a set of critical rationing levels such that at a given time one satisfies demand of a given class only if no demand of a more important class remains unsatisfied and as long as the stock level does not fall below the critical rationing level for that class at that time. Conditions are also given under which the optimal procurement policy at a given time is determined by a single critical level in the usual manner. Further conditions are given which assure that the optimal rationing and procurement policies may be determined myopically.

A single product system with linear costs is considered when customers are of two different types. Penalty costs for lost sales differ for the two types of customers and; therefore, optimal system control requires different shortage probabilities for the two classes. In one case, high penalty customers independently arrive while at the other extreme it is assumed that there is only one such customer. Single critical number policies are optimal in the simple situations. If the priority type customers become active and register displeasure by changing their demand pattern when their demands are not satisfied, simple convexity or Polya frequency function arguments are not sufficient to guarantee that the optimal policy remains one of the single critical number variety.

An increasing number of manufacturers have started to pursue a strategy that promotes inventory sharing among the dealers in their distribution network. In this paper we analyze a decentralized dealer network in which each independent dealer is given the flexibility to share his inventory. We model inventory sharing as a multiple demand classes problem in which each dealer faces his own customer demand with high priority, and inventory-sharing requests from other dealers with low priority. Assuming that each dealer uses a base-stock and threshold-rationing policy for his inventory-stocking and inventory-sharing decisions, we explicitly model the interactions between the dealers through inventory sharing and obtain a closed-form cost function for each dealer based on the steady-state distribution of the inventory levels at the two dealers. We then provide a detailed supermodularity analysis of the inventory-sharing and inventory-rationing game in which each dealer has a two-dimensional strategy set (stocking level and rationing level). We show that the full-sharing game (in which dealers precommit to sharing all of their on-hand inventory) and the fixed-sharing-level game (in which dealers precommit to sharing a portion of their on-hand inventory) are supermodular, and thus a pure-strategy Nash equilibrium is guaranteed to exist. For the rationing game (in which dealers precommit to their stocking levels), we show that there exists a dominant strategy equilibrium on the dealers' sharing (rationing) levels. Finally, a comprehensive computational study is conducted to highlight the impact of the manufacturer's incentives, subsidies, and/or transshipment fees on the dealers' sharing behavior.

When a customer requests a discount fare, the airline must decide whether to sell the seat at the requested discount or to hold the seat in hope that a customer will arrive later who will pay more. We model this situation for a single-leg flight with multiple fare classes and customers who arrive according to a semi-Markov process (possibly nonhomogeneous). These customers can request multiple seats (batch requests) and can be overbooked. Under certain conditions, we show that the value function decreases as departure approaches. If each customer only requests a single seat or if the requests can be partially satisfied, then we show that there are optimal booking curves which decrease as departure approaches. We also provide counterexamples to show that this structural property of the optimal policy need not hold for more general arrival processes if the requests can be for more than one seat and must be accepted or rejected as a whole.

We consider a single-product inventory system that serves multiple demand classes, which differ in their shortage costs or service level requirements. We assume a critical-level control policy, and show the equivalence between this inventory system and a serial inventory system. Based on this equivalence, we develop a model for cost evaluation and optimization, under the assumptions of Poisson demand, deterministic replenishment lead-time, and a continuous-review (Q, R) policy with rationing. We propose a computationally-efficient heuristic and develop a bound on its performance. We provide a numerical experiment to show the effectiveness of the heuristic and the value from a rationing policy. Finally, we describe how to extend the model to permit service times, and to embed within a multi-echelon setting.

Out of tune: IPod shortage rocks apple

- Wingfield

N. Wingfield, Out of tune: IPod shortage rocks apple, The Wall Street Journal
16 (December) (2004).

A single-product inventory model for multiple demand classes

- Arslan