Optimal FCFS allocation rules for periodic‐review assemble‐to‐order systems

In Assemble‐To‐Order (ATO) systems, situations may arise in which customer demand must be backlogged due to a shortage of some components, leaving available stock of other components unused. Such unused component stock is called remnant stock. Remnant stock is a consequence of both component ordering decisions and decisions regarding allocation of components to end‐product demand. In this article, we examine periodic‐review ATO systems under linear holding and backlogging costs with a component installation stock policy and a First‐Come‐First‐Served (FCFS) allocation policy. We show that the FCFS allocation policy decouples the problem of optimal component allocation over time into deterministic period‐by‐period optimal component allocation problems. We denote the optimal allocation of components to end‐product demand as multimatching. We solve the multi‐matching problem by an iterative algorithm. In addition, an approximation scheme for the joint replenishment and allocation optimization problem with both upper and lower bounds is proposed. Numerical experiments for base‐stock component replenishment policies show that under optimal base‐stock policies and optimal allocation, remnant stock holding costs must be taken into account. Finally, joint optimization incorporating optimal FCFS component allocation is valuable because it provides a benchmark against which heuristic methods can be compared. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 158–169, 2015     

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

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

Approximation methods for the analysis of a multicomponent,multiproduct assemble‐to‐order system

In this article, we consider the performance evaluation of a multicomponent, multiproduct assemble‐to‐order (ATO) system. Each component is managed independently using a base‐stock policy at a supply facility with limited production capacity and an infinite buffer. The arrivals of demands follow a multivariate Poisson process and unfilled demands are backlogged. Because exact analysis of the proposed system is not feasible, we propose two approximation methods which provide upper and lower bounds for various performance measures such as fill rate, average waiting time, and average number of backorders of the proposed system. Our computational experiments demonstrate the effectiveness of the two approximation methods under various system settings. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

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     

Irreducibility of joint inventory positions in an assemble‐to‐order system under (r,nQ) policies

In a typical assemble‐to‐order system, a customer order may request multiple items, and the order may not be filled if any of the requested items are out of stock. A key customer service measure when unfilled orders are backordered is the order‐based backorder level. To evaluate this crucial performance measure, a fundamental question is whether the stationary joint inventory positions follow an independent and uniform distribution. In this context, this is equivalent to the irreducibility of the Markov chain formed by the joint inventory positions. This article presents a necessary and sufficient condition for the irreducibility of such a Markov chain through a set of simultaneous Diophantine equations. This result also leads to sufficient conditions that are more general than those in the published reports. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Rolling‐horizon replenishment: Policies and performance analysis

We consider a rolling‐horizon (RH) replenishment modeling framework under which a buyer can update demand information and inventory status, modify order quantities committed previously, place an advanced order for a new period at the end of the RH, and move along in time seamlessly. We show that the optimal order policy for the two‐period RH problem is a dual‐threshold type for updating period(s) plus a base‐stock type for the advanced order. We provide analytical formulas and algorithms to compute the optimal thresholds and the optimal base‐stock level exactly. With our analytical results and numerical procedures, we demonstrate the significant value of RH replenishment in matching supplies to demands more closely. We also show that with RH updating (flexibility), the value of additional demand information beyond the RH diminishes quickly. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Dynamic capacity reservation and due date quoting in a make‐to‐order system

We consider a make‐to‐order manufacturer facing random demand from two classes of customers. We develop an integrated model for reserving capacity in anticipation of future order arrivals from high priority customers and setting due dates for incoming orders. Our research exhibits two distinct features: (1) we explicitly model the manufacturer's uncertainty about the customers' due date preferences for future orders; and (2) we utilize a service level measure for reserving capacity rather than estimating short and long term implications of due date quoting with a penalty cost function. We identify an interesting effect (“t‐pooling”) that arises when the (partial) knowledge of customer due date preferences is utilized in making capacity reservation and order allocation decisions. We characterize the relationship between the customer due date preferences and the required reservation quantities and show that not considering the t‐pooling effect (as done in traditional capacity and inventory rationing literature) leads to excessive capacity reservations. Numerical analyses are conducted to investigate the behavior and performance of our capacity reservation and due date quoting approach in a dynamic setting with multiple planning horizons and roll‐overs. One interesting and seemingly counterintuitive finding of our analyses is that under certain conditions reserving capacity for high priority customers not only improves high priority fulfillment, but also increases the overall system fill rate. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Fixed‐interval joint‐replenishment policies for distribution systems with multiple retailers and stochastic demand

We consider a distribution system consisting of a central warehouse and a group of retailers facing independent stochastic demand. The retailers replenish from the warehouse, and the warehouse from an outside supplier with ample supply. Time is continuous. Most previous studies on inventory control policies for this system have considered stock‐based batch‐ordering policies. We develop a time‐based joint‐replenishment policy in this study. Let the warehouse set up a basic replenishment interval. The retailers are replenished through the warehouse in intervals that are integer multiples of the basic replenishment interval. No inventory is carried at the warehouse. We provide an exact evaluation of the long‐term average system costs under the assumption that stock can be balanced among the retailers. The structural properties of the inventory system are characterized. We show that, although it is well known that stock‐based inventory control policies dominate time‐based inventory control policies at a single facility, this dominance does not hold for distribution systems with multiple retailers and stochastic demand. This is because the latter can provide a more efficient mechanism to streamline inventory flow and pool retailer demand, even though the former may be able to use more updated stock information to optimize system performance. The findings of the study provide insights about the key factors that drive the performance of a multiechelon inventory control system. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 637–651, 2013     

A produce‐to‐stock system with advance demand information and secondary customers

We consider a manufacturer (i.e., a capacitated supplier) that produces to stock and has two classes of customers. The primary customer places orders at regular intervals of time for a random quantity, while the secondary customers request a single item at random times. At a predetermined time the manufacturer receives advance demand information regarding the order size of the primary customer. If the manufacturer is not able to fill the primary customer's demand, there is a penalty. On the other hand, serving the secondary customers results in additional profit; however, the manufacturer can refuse to serve the secondary customers in order to reserve inventory for the primary customer. We characterize the manufacturer's optimal production and stock reservation policies that maximize the manufacturer's discounted profit and the average profit per unit time. We show that these policies are threshold‐type policies, and these thresholds are monotone with respect to the primary customer's order size. Using a numerical study we provide insights into how the value of information is affected by the relative demand size of the primary and secondary customers. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

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

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

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     

Managing components for assemble‐to‐order products with lead‐time‐dependent pricing: The full‐shipment model

We study a component inventory planning problem in an assemble‐to‐order environment faced by many contract manufacturers in which both quick delivery and efficient management of component inventory are crucial for the manufacturers to achieve profitability in a highly competitive market. Extending a recent study in a similar problem setting by the same authors, we analyze an optimization model for determining the optimal component stocking decision for a contract manufacturer facing an uncertain future demand, where product price depends on the delivery times. In contrast to our earlier work, this paper considers the situation where the contract manufacturer needs to deliver the full order quantity in one single shipment. This delivery requirement is appropriate for many industries, such as the garment and toy industries, where the economies of scale in transportation is essential. We develop efficient solution procedures for solving this optimization problem. We use our model results to illustrate how the different model parameters affect the optimal solution. We also compare the results under this full‐shipment model with those from our earlier work that allows for multiple partial shipments. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

The one‐warehouse multiretailer problem with an order‐up‐to level inventory policy

We consider a two‐level system in which a warehouse manages the inventories of multiple retailers. Each retailer employs an order‐up‐to level inventory policy over T periods and faces an external demand which is dynamic and known. A retailer's inventory should be raised to its maximum limit when replenished. The problem is to jointly decide on replenishment times and quantities of warehouse and retailers so as to minimize the total costs in the system. Unlike the case in the single level lot‐sizing problem, we cannot assume that the initial inventory will be zero without loss of generality. We propose a strong mixed integer program formulation for the problem with zero and nonzero initial inventories at the warehouse. The strong formulation for the zero initial inventory case has only T binary variables and represents the convex hull of the feasible region of the problem when there is only one retailer. Computational results with a state‐of‐the art solver reveal that our formulations are very effective in solving large‐size instances to optimality. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

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.     

Coherent systems based on sequential order statistics

The sequential order statistics (SOS) are a good way to model the lifetimes of the components in a system when the failure of a component at time t affects the performance of the working components at this age t. In this article, we study properties of the lifetimes of the coherent systems obtained using SOS. Specifically, we obtain a mixture representation based on the signature of the system. This representation is used to obtain stochastic comparisons. To get these comparisons, we obtain some ordering properties for the SOS, which in this context represent the lifetimes of k‐out‐of‐n systems. In particular, we show that they are not necessarily hazard rate ordered. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Production‐inventory systems with imperfect advance demand information and updating

We consider a supplier with finite production capacity and stochastic production times. Customers provide advance demand information (ADI) to the supplier by announcing orders ahead of their due dates. However, this information is not perfect, and customers may request an order be fulfilled prior to or later than the expected due date. Customers update the status of their orders, but the time between consecutive updates is random. We formulate the production‐control problem as a continuous‐time Markov decision process and prove there is an optimal state‐dependent base‐stock policy, where the base‐stock levels depend upon the numbers of orders at various stages of update. In addition, we derive results on the sensitivity of the state‐dependent base‐stock levels to the number of orders in each stage of update. In a numerical study, we examine the benefit of ADI, and find that it is most valuable to the supplier when the time between updates is moderate. We also consider the impact of holding and backorder costs, numbers of updates, and the fraction of customers that provide ADI. In addition, we find that while ADI is always beneficial to the supplier, this may not be the case for the customers who provide the ADI. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

A branch and bound algorithm for computing optimal replacement policies in consecutive k‐out‐of‐n‐systems

This paper presents a branch and bound algorithm for computing optimal replacement policies in a discrete‐time, infinite‐horizon, dynamic programming model of a binary coherent system with n statistically independent components, and then specializes the algorithm to consecutive k‐out‐of‐n systems. The objective is to minimize the long‐run expected average undiscounted cost per period. (Costs arise when the system fails and when failed components are replaced.) An earlier paper established the optimality of following a critical component policy (CCP), i.e., a policy specified by a critical component set and the rule: Replace a component if and only if it is failed and in the critical component set. Computing an optimal CCP is a optimization problem with n binary variables and a nonlinear objective function. Our branch and bound algorithm for solving this problem has memory storage requirement O(n) for consecutive k‐out‐of‐n systems. Extensive computational experiments on such systems involving over 350,000 test problems with n ranging from 10 to 150 find this algorithm to be effective when n ≤ 40 or k is near n. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 288–302, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10017     

Technical note: Worst‐case performance of power‐of‐two policies for serial inventory systems with incremental quantity discounts

This study presents power‐of‐two policies for a serial inventory system with constant demand rate and incremental quantity discounts at the most upstream stage. It is shown that an optimal solution is nested and follows a zero‐inventory ordering policy. To prove the effectiveness of power‐of‐two policies, a lower bound on the optimal cost is obtained. A policy that has a cost within 6% of the lower bound is developed for a fixed base planning period. For a variable base planning period, a 98% effective policy is provided. An extension is included for a system with price dependent holding costs. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Higher‐order Markov chain models for categorical data sequences*

In this paper we study higher‐order Markov chain models for analyzing categorical data sequences. We propose an efficient estimation method for the model parameters. Data sequences such as DNA and sales demand are used to illustrate the predicting power of our proposed models. In particular, we apply the developed higher‐order Markov chain model to the server logs data. The objective here is to model the users' behavior in accessing information and to predict their behavior in the future. Our tests are based on a realistic web log and our model shows an improvement in prediction. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

On the economic application of airlift to product distribution and its impact on inventory levels

In many resupply situations, the decisionmaker has the option of “purchasing” faster replenishment leadtimes. For example, a premium may be paid for delivery by parcel post rather than slower but less expensive delivery by railway express. It may be economically advantageous to pay shipment premiums for faster leadtimes when considering the possible cost reductions in pipeline (on-order) inventory and safety stock levels. This paper develops a decision rule which, for any given item, will indicate whether it is economically advantageous to purchase a faster leadtime. The general methodology is then applied to a peacetime military resupply operation involving several million items, each requiring a decision as to whether the item should be shipped by air or sea.