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     

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     

A two product dynamic economic lot size production model with either-or production constraints

This paper considers the production of two products with known demands over a finite set of periods. The production and inventory carrying costs for each product are assumed to be concave. We seek the minimum cost production schedule meeting all demands, without backlogging, assuming that at most one of the two products can be produced in any period. The optimization problem is first stated as a nonlinear programming problem, which allows the proof of a result permitting the search for the optimal policy to be restricted to those which produce a product only when its inventory level is zero. A dynamic programming formulation is given and the model is then formulated as a shortest route problem in a specially constructed network.     

Optimal policy for a dynamic multi-echelon inventory model

A general multiperiod multi-echelon supply system consisting of n facilities each stocking a single product is studied. At the beginning of a period each facility may order stock from an exogenous source with no delivery lag and proportional ordering costs. During the period the (random) demands at the facilities are satisfied according to a given supply policy that determines to what extent stock may be redistributed from facilities with excess stock to those experiencing shortages. There are storage, shortage, and transportation costs. An ordering policy that minimizes expected costs is sought. If the initial stock is sufficiently small and certain other conditions are fulfilled, it is optimal to order up to a certain base stock level at each facility. The special supply policy in which each facility except facility 1 passes its shortages on to a given lower numbered facility called its direct supplier is examined in some detail. Bounds on the base stock levels are obtained. It is also shown that if the demand distribution at facility j is stochastically smaller (“spread” less) than that at another facility k having the same direct supplier and if certain other conditions are fulfilled, then the optimal base stock level (“virtual” stock out probability) at j is less than (greater than) or equal to that at facility k.     

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.     

Minimizing the false alarm rate in systems with transient abnormality

We consider a stochastic partially observable system that can switch between a normal state and a transient abnormal state before entering a persistent abnormal state. Only the persistent abnormal state requires alarms. The transient and persistent abnormal states may be similar in appearance, which can result in excess false alarms. We propose a partially observable Markov decision process model to minimize the false alarm rate, subject to a given upper bound on the expected alarm delay time. The cost parameter is treated as the Lagrange multiplier, which can be estimated from the bound of the alarm delay. We show that the optimal policy has a control‐limit structure on the probability of persistent abnormality, and derive closed‐form bounds for the control limit and present an algorithm to specify the Lagrange multiplier. We also study a specialized model where the transient and persistent abnormal states have the same observation distribution, in which case an intuitive “watchful‐waiting” policy is optimal. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 320–334, 2016     

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.     

Allocation policies and cost approximations for multilocation inventory systems

Consider a central depot that supplies several locations experiencing random demands. Periodically, the depot may place an order for exogenous supply. Orders arrive after a fixed leadtime, and are then allocated among the several locations. Each allocation reaches its destination after a further delay. We consider the special case where the penalty-cost/holding-cost ratio is constant over the locations. Several approaches are given to approximate the dynamic program describing the problem. Each approach provides both a near-optimal order policy and an approximation of the optimal cost of the original problem. In addition, simple but effective allocation policies are discussed.     

Optimal burn‐in procedure for periodically inspected systems

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we study burn‐in procedure for a system that is maintained under periodic inspection and perfect repair policy. Assuming that the underlying lifetime distribution of a system has an initially decreasing and/or eventually increasing failure rate function, we derive upper and lower bounds for the optimal burn‐in time, which maximizes the system availability. Furthermore, adopting an age replacement policy, we derive upper and lower bounds for the optimal age parameter of the replacement policy for each fixed burn‐in time and a uniform upper bound for the optimal burn‐in time given the age replacement policy. These results can be used to reduce the numerical work for determining both optimal burn‐in time and optimal replacement policy. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A single period model for a multiproduct perishable inventory system with economic substitution

This paper develops a single period model for a specific class of multiproduct perishable inventory systems where demands are interdependent. This class of inventory systems has the property that there is economic substitution between products. It is shown that the optimal policy has the economic substitution property, and that the rate of substitution is age dependent. The model serves as a generalization of a theorem discovered by Ignall and Veinott.     

Callable products with dependent demands

Capacity providers such as airlines often sell the same capacity to different market segments at different prices to improve their expected revenues. The absence of a secondary market, due to the nontransferability of airline tickets, gives rise to an opportunity for airlines to broker capacity between consumers with different willingness to pay. One way to broker capacity is by the introduction of callable products. The idea is similar to callable bonds where the issuer has the right, but not the obligation, to buy back the bonds at a certain price by a certain date. The idea of callable products was introduced before under the assumption that the fare-class demands are all independent. The independent assumption becomes untenable when there is significant demand recovery (respectively, demand cannibalization) when lower fares are closed (respectively, opened). In this case, consumer choice behavior should be modeled explicitly to make meaningful decisions. In this paper, we consider a general consumer choice model and develop the optimal strategy for callable products. Our numerical study illustrates how callable products are win-win-win, for the capacity provider and for both high and low fare consumers. Our studies also identify conditions for callable products to result in significant improvements in expected revenues.     

Stochastic regret minimization for revenue management problems with nonstationary demands

We study an admission control model in revenue management with nonstationary and correlated demands over a finite discrete time horizon. The arrival probabilities are updated by current available information, that is, past customer arrivals and some other exogenous information. We develop a regret‐based framework, which measures the difference in revenue between a clairvoyant optimal policy that has access to all realizations of randomness a priori and a given feasible policy which does not have access to this future information. This regret minimization framework better spells out the trade‐offs of each accept/reject decision. We proceed using the lens of approximation algorithms to devise a conceptually simple regret‐parity policy. We show the proposed policy achieves 2‐approximation of the optimal policy in terms of total regret for a two‐class problem, and then extend our results to a multiclass problem with a fairness constraint. Our goal in this article is to make progress toward understanding the marriage between stochastic regret minimization and approximation algorithms in the realm of revenue management and dynamic resource allocation. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 433–448, 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.     

Approximation algorithms for capacitated stochastic inventory systems with setup costs

We develop the first approximation algorithm with worst‐case performance guarantee for capacitated stochastic periodic‐review inventory systems with setup costs. The structure of the optimal control policy for such systems is extremely complicated, and indeed, only some partial characterization is available. Thus, finding provably near‐optimal control policies has been an open challenge. In this article, we construct computationally efficient approximate optimal policies for these systems whose demands can be nonstationary and/or correlated over time, and show that these policies have a worst‐case performance guarantee of 4. We demonstrate through extensive numerical studies that the policies empirically perform well, and they are significantly better than the theoretical worst‐case guarantees. We also extend the analyses and results to the case with batch ordering constraints, where the order size has to be an integer multiple of a base load. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 304–319, 2014     

Optimal nonmyopic gambling strategy for the generalized Kelly criterion

We consider the optimal wagers to be made by a gambler who starts with a given initial wealth. The gambler faces a sequence of two-outcome games, i.e., “win” vs. “lose,” and wishes to maximize the expected value of his terminal utility. It has been shown by Kelly, Bellman, and others that if the terminal utility is of the form log x, where x is the terminal wealth, then the optimal policy is myopic, i.e., the optimal wager is always to bet a constant fraction of the wealth provided that the probability of winning exceeds the probability of losing. In this paper we provide a critique of the simple logarithmic assumption for the utility of terminal wealth and solve the problem with a more general utility function. We show that in the general case, the optimal policy is not myopic, and we provide analytic expressions for optimal wager decisions in terms of the problem parameters. We also provide conditions under which the optimal policy reduces to the simple myopic case. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 639–654, 1997     

Dynamic inventory control with limited capital and short‐term financing

For most firms, especially the small‐ and medium‐sized ones, the operational decisions are affected by their internal capital and ability to obtain external capital. However, the majority of the literature on dynamic inventory control ignores the firm's financial status and financing issues. An important question that arises is: what are the optimal inventory and financing policies for firms with limited internal capital and limited access to external capital? In this article, we study a dynamic inventory control problem where a capital‐constrained firm periodically purchases a product from a supplier and sells it to a market with random demands. In each period, the firm can use its own capital and/or borrow a short‐term loan to purchase the product, with the interest rate being nondecreasing in the loan size. The objective is to maximize the firm's expected terminal wealth at the end of the planning horizon. We show that the optimal inventory policy in each period is an equity‐level‐dependent base‐stock policy, where the equity level is the sum of the firm's capital level and the value of its on‐hand inventory evaluated at the purchasing cost; and the structure of the optimal policy can be characterized by four intervals of the equity level. Our results shed light on the dynamic inventory control for firms with limited capital and short‐term financing capabilities.Copyright © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 184–201, 2014     

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     

Solving generalized transportation problems via pure transportation problems

This paper investigates certain issues of coefficient sensitivity in generalized network problems when such problems have small gains or losses. In these instances, it might be computationally advantageous to temporarily ignore these gains or losses and solve the resultant “pure” network problem. Subsequently, the optimal solution to the pure problem could be used to derive the optimal solution to the original generalized network problem. In this paper we focus on generalized transportation problems and consider the following question: Given an optimal solution to the pure transportation problem, under what conditions will the optimal solution to the original generalized transportation problem have the same basic variables? We study special cases of the generalized transportation problem in terms of convexity with respect to a basis. For the special case when all gains or losses are identical, we show that convexity holds. We use this result to determine conditions on the magnitude of the gains or losses such that the optimal solutions to both the generalized transportation problem and the associated pure transportation problem have the same basic variables. For more general cases, we establish sufficient conditions for convexity and feasibility. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 666–685, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10034     

Bounding the optimal burn‐in time for a system with two types of failure

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we consider the problem of determining bounds to the optimal burn‐in time and optimal replacement policy maximizing the steady state availability of a repairable system. It is assumed that two types of system failures may occur: One is Type I failure (minor failure), which can be removed by a minimal repair, and the other is Type II failure (catastrophic failure), which can be removed only by a complete repair. Assuming that the underlying lifetime distribution of the system has a bathtub‐shaped failure rate function, upper and lower bounds for the optimal burn‐in time are provided. Furthermore, some other applications of optimal burn‐in are also considered. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

基于年龄更换策略的威布尔分布备件需求计算方法

备件需求量预测在装备保障工作中发挥着重要作用。预防性维修可以消除装备潜在的故障或避免装备发生故障后所导致的严重后果,从而使装备始终处于期望状态。预防性维修策略下备件需求量建模能够以可预见的可靠性数据为基础进行,其关键问题是最佳更换间隔时间的确定。针对威布尔分布装备备件,采用年龄更换策略,确定最佳更换间隔时间,并在此基础上导出威布尔型备件需求计算模型,为备件需求量的预测提供了一种简单有效的方法。