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     

An optimal critical level policy for inventory systems with two demand classes

Traditional inventory systems treat all demands of a given item equally. This approach is optimal if the penalty costs of all customers are the same, but it is not optimal if the penalty costs are different for different customer classes. Then, demands of customers with high penalty costs must be filled before demands of customers with low penalty costs. A commonly used inventory policy for dealing with demands with different penalty costs is the critical level inventory policy. Under this policy demands with low penalty costs are filled as long as inventory is above a certain critical level. If the inventory reaches the critical level, only demands with high penalty costs are filled and demands with low penalty costs are backordered. In this article, we consider a critical level policy for a periodic review inventory system with two demand classes. Because traditional approaches cannot be used to find the optimal parameters of the policy, we use a multidimensional Markov chain to model the inventory system. We use a sample path approach to prove several properties of this inventory system. Although the cost function is not convex, we can build on these properties to develop an optimization approach that finds the optimal solution. We also present some numerical results. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

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.     

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     

Selling to strategic and loss‐averse consumers: Stocking,procurement, and product design policies

Motivated by the presence of loss‐averse decision making behavior in practice, this article considers a supply chain consisting of a firm and strategic consumers who possess an S‐shaped loss‐averse utility function. In the model, consumers decide the purchase timing and the firm chooses the inventory level. We find that the loss‐averse consumers' strategic purchasing behavior is determined by their perceived gain and loss from strategic purchase delay, and the given rationing risk. Thus, the firm that is cognizant of this property tailors its inventory stocking policy based on the consumers' loss‐averse behavior such as their perceived values of gain and loss, and their sensitivity to them. We also demonstrate that the firm's equilibrium inventory stocking policy reflects both the economic logic of the traditional newsvendor inventory model, and the loss‐averse behavior of consumers. The equilibrium order quantity is significantly different from those derived from models that assume that the consumers are risk neutral and homogeneous in their valuations. We show that the firm that ignores strategic consumer's loss‐aversion behavior tends to keep an unnecessarily high inventory level that leads to excessive leftovers. Our numerical experiments further reveal that in some extreme cases the firm that ignores strategic consumer's loss‐aversion behavior generates almost 92% more leftovers than the firm that possesses consumers’ loss‐aversion information and takes it into account when making managerial decisions. To mitigate the consumer's forward‐looking behavior, we propose the adoption of the practice of agile supply chain management, which possesses the following attributes: (i) procuring inventory after observing real‐time demand information, (ii) enhanced design (which maintains the current production mix but improves the product performance to a higher level), and (iii) customized design (which maintains the current performance level but increases the variety of the current production line to meet consumers’ specific demands). We show that such a practice can induce the consumer to make early purchases by increasing their rationing risk, increasing the product value, or diversifying the product line. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 435–453, 2015     

A partial characterization of the optimal ordering/rationing policy for a periodic review system with two demand classes and backordering

We consider a finite horizon periodic review, single product inventory system with a fixed setup cost and two stochastic demand classes that differ in their backordering costs. In each period, one must decide whether and how much to order, and how much demand of the lower class should be satisfied. We show that the optimal ordering policy can be characterized as a state dependent (s,S) policy, and the rationing structure is partially obtained based on the subconvexity of the cost function. We then propose a simple heuristic rationing policy, which is easy to implement and close to optimal for intensive numerical examples. We further study the case when the first demand class is deterministic and must be satisfied immediately. We show the optimality of the state dependent (s,S) ordering policy, and obtain additional rationing structural properties. Based on these properties, the optimal ordering and rationing policy for any state can be generated by finding the optimal policy of only a finite set of states, and for each state in this set, the optimal policy is obtained simply by choosing a policy from at most two alternatives. An efficient algorithm is then proposed. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

An application of servomechanisms to inventory

In recent years, some attention has been devoted to the application of techniques of control theory to inventory management. In particular, H. Vassian (1955) developed a model for a periodic review inventory system utilizing techniques of discrete variable servomechanisms to analyze the system in a cost-free structure. The resulting model is inherently deterministic, however, and emphasizes the control of inventory fluctuation about a safety level by selecting an appropriate order policy. Such an order policy is defined only up to an arbitrary method of forecasting customer demands. The present paper is a continuation of the model developed by Vassian in which exponential smoothing is used as a specific forecasting technique. Full recognition of the probabilistic nature of demand is taken into account and the requirement of minimizing expected inventory level is imposed. In addition, explicit formulas for the variance in inventory are derived as functions of the smoothing constant and the tradeoff between small variance and rapid system response is noted. Finally, in an attempt to remove the bias inherent in exponential smoothing, a modification of that technique is defined and discussed as an alternate forecasting method.     

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

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

Optimal inventory policies in contagious demand models

In the past, contagious distributions have been successfully applied in bacteriology, entomology, and accident statistics. This paper applies the notion of contagious distributions in the inventory control of new products and seasonal or style goods, which have an lying “true contagion” for their demands, namely, the influence of past demands on occurrence of demands. A contagious distribution is derived by assuming a nonstationary Poisson process where the demand rate at any instant depends on the past demands to that instant. Using this contagious distribution, an inventory model is developed seasonal goods and new product lines. Optimal order policies as a function of the initial level and the review period are derived.     

A class of inventory models with customer impatience

Classical inventory models generally assume either no backlogging of demands or unlimited backlogging. This paper treats the case wherein backlogged customers are willing to wait for a random period of time for service. A broad class of such models is discussed, with a more complete analysis performed on a simple subclass. Steady state equations are derived and solved assuming exponentially distributed interarrival times of customers, order delivery lead times, and customer patience.     

动态服务质量的多信道媒体接入控制传输机制

针对航空自组网在高负载下的服务质量及时延问题,提出一种动态服务质量的多信道媒体接入控制传输机制。以多信道检测统计为平台,结合优先级机制,通过在高负载网络中适当遏制低优先级业务,并且进行网络流量优化,保证高优先级业务的低时延发送;同时利用流量预测模型估计网络流量,通过粒子群优化算法进行优化,寻找合适的优先级门限值,确保高优先级业务接入率。通过计算机仿真可知,所设计的动态服务质量的多信道媒体接入控制传输机制,可在大负载网络中动态控制信道的接入,保持良好的网络吞吐量,其高优先级业务接入率达到99%以上,能有效解决航空数据链网络高业务量导致的服务质量及时延问题。     

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     

Optimal disposal policies for a single-item inventory system with returns

We consider a single-item inventory system in which the stock level can increase due to items being returned as well as decrease when demands occur. Returned items can be repaired and then used to satisfy future demand, or they can be disposed of. We identify those inventory levels where disposal is the best policy. It is shown that this problem is equivalent to a problem of controlling a single-server queue. When the return and demand processes are both Poisson, we find the optimal policy exactly. When the demand and return processes are more general, we use diffusion approximations to obtain an approximate model, which is then solved. The approximate model requires only mean and variance data. Besides the optimal policy, the output of the models includes such characteristics as the operating costs, the purchase rate for new items, the disposal rate for returned items and the average inventory level. Several numerical examples are given. An interesting by-product of our investigation is an approximation for the steady-state behavior of the bulk GI/G/1 queue with a queue limit.     

Value management of diagnostic equipment with cancelation,no‐show,and emergency patients

Magnetic resonance imaging and other multifunctional diagnostic facilities, which are considered as scarce resources of hospitals, typically provide services to patients with different medical needs. This article examines the admission policies during the appointment management of such facilities. We consider two categories of patients: regular patients who are scheduled in advance through an appointment system and emergency patients with randomly generated demands during the workday that must be served as soon as possible. According to the actual medical needs of patients, regular patients are segmented into multiple classes with different cancelation rates, no‐show probabilities, unit value contributions, and average service times. Management makes admission decisions on whether or not to accept a service request from a regular patient during the booking horizon to improve the overall value that could be generated during the workday. The decisions should be made by considering the cancelation and no‐show behavior of booked patients as well as the emergency patients that would have to be served because any overtime service would lead to higher costs. We studied the optimal admission decision using a continuous‐time discrete‐state dynamic programming model. Identifying an optimal policy for this discrete model is analytically intractable and numerically inefficient because the state is multidimensional and infinite. We propose to study a deterministic counterpart of the problem (i.e., the fluid control problem) and to develop a time‐based fluid policy that is shown to be asymptotically optimal for large‐scale problems. Furthermore, we propose to adopt a mixed fluid policy that is developed based on the information obtained from the fluid control problem. Numerical experiments demonstrate that this improved policy works effectively for small‐scale problems. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 287–304, 2016     

Integrating decisions with advance supply information in an assemble-to-order system

We study a periodic-review assemble-to-order (ATO) system with multiple components and multiple products, in which the inventory replenishment for each component follows an independent base-stock policy and stochastic product demands are satisfied according to a First-Come-First-Served rule. We assume that the replenishment for various component suffers from lead time uncertainty. However, the decision maker has the so-called advance supply information (ASI) associated with the lead times and thus can take advantage of the information for system optimization. We propose a multistage stochastic integer program that incorporates ASI to address the joint optimization of inventory replenishment and component allocation. The optimal base-stock policy for the inventory replenishment is determined using the sample average approximation algorithm. Also, we provide a modified order-based component allocation (MOBCA) heuristic for the component allocation. We additionally consider a special case of the variable lead times where the resulting two-stage stochastic programming model can be characterized as a single-scenario case of the proposed multistage model. We carry out extensive computational studies to quantify the benefits of integrating ASI into joint optimization and to explore the possibility of employing the two-stage model as a relatively efficient approximation scheme for the multistage model.     

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

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

Optimal policies for a multi-product inventory system with negotiable lead times

The objective of this paper is to determine the optimum inventory policy for a multi-product periodic review dynamic inventory system. At the beginning of each period two decisions are made for each product. How much to “normal order” with a lead time of λ_n periods and how much to “emergency order” with a lead time of λ_e periods, where λe = λ_n - 1. It is assumed that the emergency ordering costs are higher than the normal ordering costs. The demands for each product in successive periods are assumed to form a sequence of independent identically distributed random variables with known densities. Demands for individual products within a period are assumed to be non-negative, but they need not be independent. Whenever demand exceeds inventory their difference is backlogged rather than lost. The ordering decisions are based on certain costs and two revenue functions. Namely, the procurement costs which are assumed to be linear for both methods of ordering, convex holding and penalty costs, concave salvage gain functions, and linear credit functions. There is a restriction on the total amount that can be emergency ordered for all products. The optimal ordering policy is determined for the one and N-period models.     

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     

Service levels in distribution systems with random customer order size

An approximate method for measuring the service levels of the warehouse-retailer system operating under (s, S) policy is presented. All the retailers are identical and the demand process at each retailer follows a stationary stuttering Poisson process. This type of demand process allows customer orders to be for a random number of units, which gives rise to the undershoot quantity at both the warehouse and retailer levels. Exact analyses of the distribution of the undershoot quantity and the number of orders place by a retailer during the warehouse reordering lead time are derived. By using this distribution together with probability approximation and other heuristic approaches, we model the behavior of the warehouse level. Based on the results of the warehouse level and on an existing framework from previous work, the service level at the retailer level is estimated. Results of the approximate method are then compared with those of simulation. © 1995 John Wiley & Sons, Inc.     

An (s,S) inventory system with rest periods to the server

We study the (s,S) inventory system in which the server takes a rest when the level of the inventory is zero. The demands are assumed to occur for one unit at a time. The interoccurrence times between successive demands, the lead times, and the rest times are assumed to follow general distributions which are mutually independent. Using renewal and convolution techniques we obtain the state transition probabilities.