Estimation of average backorders for an assemble‐to‐order system with random batch demands through extreme statistics

We investigate inventory management for a large‐scale multi‐product, multi‐component Assemble‐to‐Order system with general random batch demands. Results from extreme statistics theory are applied in developing approximation schemes for a widely used performance measure, customer backorders. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Signature based analysis of m‐Consecutive‐k‐out‐of‐n: F systems with exchangeable components

In this article, we study reliability properties of m‐consecutive‐k‐out‐of‐n: F systems with exchangeable components. We deduce exact formulae and recurrence relations for the signature of the system. Closed form expressions for the survival function and the lifetime distribution as a mixture of the distribution of order statistics are established as well. These representations facilitate the computation of several reliability characteristics of the system for a given exchangeable joint distribution or survival function. Finally, we provide signature‐based stochastic ordering results for the system's lifetime and investigate the IFR preservation property under the formulation of m‐consecutive‐k‐out‐of‐n: F systems. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Strategic capacity decision‐making in a stochastic manufacturing environment using real‐time approximate dynamic programming

In this study, we illustrate a real‐time approximate dynamic programming (RTADP) method for solving multistage capacity decision problems in a stochastic manufacturing environment, by using an exemplary three‐stage manufacturing system with recycle. The system is a moderate size queuing network, which experiences stochastic variations in demand and product yield. The dynamic capacity decision problem is formulated as a Markov decision process (MDP). The proposed RTADP method starts with a set of heuristics and learns a superior quality solution by interacting with the stochastic system via simulation. The curse‐of‐dimensionality associated with DP methods is alleviated by the adoption of several notions including “evolving set of relevant states,” for which the value function table is built and updated, “adaptive action set” for keeping track of attractive action candidates, and “nonparametric k nearest neighbor averager” for value function approximation. The performance of the learned solution is evaluated against (1) an “ideal” solution derived using a mixed integer programming (MIP) formulation, which assumes full knowledge of future realized values of the stochastic variables (2) a myopic heuristic solution, and (3) a sample path based rolling horizon MIP solution. The policy learned through the RTADP method turned out to be superior to polices of 2 and 3. © 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

Stein–Chen approximation and error bounds for order fill rates in assemble‐to‐order systems

Assemble‐to‐order (ATO) is an important operational strategy for manufacturing firms to achieve quick response to customer orders while keeping low finished good inventories. This strategy has been successfully used not only by manufacturers (e.g., Dell, IBM) but also by retailers (e.g., Amazon.com). The evaluation of order‐based performance is known to be an important but difficult task, and the existing literature has been mainly focused on stochastic comparison to obtain performance bounds. In this article, we develop an extremely simple Stein–Chen approximation as well as its error‐bound for order‐based fill rate for a multiproduct multicomponent ATO system with random leadtimes to replenish components. This approximation gives an expression for order‐based fill rate in terms of component‐based fill rates. The approximation has the property that the higher the component replenishment leadtime variability, the smaller the error bound. The result allows an operations manager to analyze the improvement in order‐based fill rates when the base‐stock level for any component changes. Numerical studies demonstrate that the approximation performs well, especially when the demand processes of different components are highly correlated; when the components have high base‐stock levels; or when the component replenishment leadtimes have high variability. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

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     

A heuristic for the two‐machine no‐wait openshop scheduling problem

In this paper the problem of minimizing makespan in a two‐machine openshop is examined. A heuristic algorithm is proposed, and its worst case performance ratio and complexity are analyzed. The average case performance is evaluated using an empirical study. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 129–145, 1999     

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 branch‐and‐price approach for the stochastic generalized assignment problem

In this article, we address a stochastic generalized assignment machine scheduling problem in which the processing times of jobs are assumed to be random variables. We develop a branch‐and‐price (B&P) approach for solving this problem wherein the pricing problem is separable with respect to each machine, and has the structure of a multidimensional knapsack problem. In addition, we explore two other extensions of this method—one that utilizes a dual‐stabilization technique and another that incorporates an advanced‐start procedure to obtain an initial feasible solution. We compare the performance of these methods with that of the branch‐and‐cut (B&C) method within CPLEX. Our results show that all B&P‐based approaches perform better than the B&C method, with the best performance obtained for the B&P procedure that includes both the extensions aforementioned. We also utilize a Monte Carlo method within the B&P scheme, which affords the use of a small subset of scenarios at a time to estimate the “true” optimal objective function value. Our experimental investigation reveals that this approach readily yields solutions lying within 5% of optimality, while providing more than a 10‐fold savings in CPU times in comparison with the best of the other proposed B&P procedures. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 131–143, 2014     

Fully sequential indifference‐zone selection procedures with variance‐dependent sampling

Fully sequential indifference‐zone selection procedures have been proposed in the simulation literature to select the system with the best mean performance from a group of simulated systems. However, the existing sequential indifference‐zone procedures allocate an equal number of samples to the two systems in comparison even if their variances are drastically different. In this paper we propose new fully sequential indifference‐zone procedures that allocate samples according to the variances. We show that the procedures work better than several existing sequential indifference‐zone procedures when variances of the systems are different. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Two‐machine flow shop no‐wait scheduling with a nonavailability interval

We study a two‐machine flow shop scheduling problem with no‐wait in process, in which one of the machines is not available during a specified time interval. We consider three scenarios of handing the operation affected by the nonavailability interval. Its processing may (i) start from scratch after the interval, or (ii) be resumed from the point of interruption, or (iii) be partially restarted after the interval. The objective is to minimize the makespan. We present an approximation algorithm that for all these scenarios delivers a worst‐case ratio of 3/2. For the second scenario, we offer a 4/3‐approximation algorithm. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Analysis of decentralized multi‐product pull systems with lost sales

We study a pull‐type, flexible, multi‐product, and multi‐stage production/inventory system with decentralized two‐card kanban control policies. Each stage involves a processor and two buffers with finite target levels. Production stages, arranged in series, can process several product types one at a time. Transportation of semi‐finished parts from one stage to another is performed in fixed lot sizes. The exact analysis is mathematically intractable even for smaller systems. We present a robust approximation algorithm to model two‐card kanban systems with batch transfers under arbitrary complexity. The algorithm uses phase‐type modeling to find effective processing times and busy period analysis to identify delays among product types in resource contention. Our algorithm reduces the effort required for estimating performance measures by a considerable margin and resolves the state–space explosion problem of analytical approaches. Using this analytical tool, we present new findings for a better understanding of some tactical and operational issues. We show that flow of material in small procurement sizes smoothes flow of information within the system, but also necessitates more frequent shipments between stages, raising the risk of late delivery. Balancing the risk of information delays vis‐à‐vis shipment delays is critical for the success of two‐card kanban systems. Although product variety causes time wasted in setup operations, it also facilitates relatively short production cycles enabling processors to switch from one product type to another more rapidly. The latter point is crucial especially in high‐demand environments. Increasing production line size prevents quick response to customer demand, but it may improve system performance if the vendor lead‐time is long or subject to high variation. Finally, variability in transportation and processing times causes the most damage if it arises at stages closer to the customer. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

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     

Steady‐state diffusion approximations for discrete‐time queue in hospital inpatient flow management

In this article, we analyze a discrete‐time queue that is motivated from studying hospital inpatient flow management, where the customer count process captures the midnight inpatient census. The stationary distribution of the customer count has no explicit form and is difficult to compute in certain parameter regimes. Using the Stein's method framework, we identify a continuous random variable to approximate the steady‐state customer count. The continuous random variable corresponds to the stationary distribution of a diffusion process with state‐dependent diffusion coefficients. We characterize the error bounds of this approximation under a variety of system load conditions—from lightly loaded to heavily loaded. We also identify the critical role that the service rate plays in the convergence rate of the error bounds. We perform extensive numerical experiments to support the theoretical findings and to demonstrate the approximation quality. In particular, we show that our approximation performs better than those based on constant diffusion coefficients when the number of servers is small, which is relevant to decision making in a single hospital ward.     

Multi‐item capacitated lot‐sizing problems with setup times and pricing decisions

We study a multi‐item capacitated lot‐sizing problem with setup times and pricing (CLSTP) over a finite and discrete planning horizon. In this class of problems, the demand for each independent item in each time period is affected by pricing decisions. The corresponding demands are then satisfied through production in a single capacitated facility or from inventory, and the goal is to set prices and determine a production plan that maximizes total profit. In contrast with many traditional lot‐sizing problems with fixed demands, we cannot, without loss of generality, restrict ourselves to instances without initial inventories, which greatly complicates the analysis of the CLSTP. We develop two alternative Dantzig–Wolfe decomposition formulations of the problem, and propose to solve their relaxations using column generation and the overall problem using branch‐and‐price. The associated pricing problem is studied under both dynamic and static pricing strategies. Through a computational study, we analyze both the efficacy of our algorithms and the benefits of allowing item prices to vary over time. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Approximation schemes for single‐machine scheduling with a fixed maintenance activity to minimize the total amount of late work

We consider the problem of scheduling n independent and simultaneously available jobs without preemption on a single machine, where the machine has a fixed maintenance activity. The objective is to find the optimal job sequence to minimize the total amount of late work, where the late work of a job is the amount of processing of the job that is performed after its due date. We first discuss the approximability of the problem. We then develop two pseudo‐polynomial dynamic programming algorithms and a fully polynomial‐time approximation scheme for the problem. Finally, we conduct extensive numerical studies to evaluate the performance of the proposed algorithms. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 172–183, 2016     

A computationally efficient approach for determining inventory levels in a capacitated multiechelon production‐distribution system

The system under study is a single item, two‐echelon production‐inventory system consisting of a capacitated production facility, a central warehouse, and M regional distribution centers that satisfy stochastic demand. Our objective is to determine a system base‐stock level which minimizes the long run average system cost per period. Central to the approach are (1) an inventory allocation model and associated convex cost function designed to allocate a given amount of system inventory across locations, and (2) a characterization of the amount of available system inventory using the inventory shortfall random variable. An exact model must consider the possibility that inventories may be imbalanced in a given period. By assuming inventory imbalances cannot occur, we develop an approximation model from which we obtain a lower bound on the per period expected cost. Through an extensive simulation study, we analyze the quality of our approximation, which on average performed within 0.50% of the lower bound. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 377–398, 2000     

Subspace dynamic‐simplex linear interpolation search for mixed‐integer black‐box optimization problems

Design and management of complex systems with both integer and continuous decision variables can be guided using mixed‐integer optimization models and analysis. We propose a new mixed‐integer black‐box optimization (MIBO) method, subspace dynamic‐simplex linear interpolation search (SD‐SLIS), for decision making problems in which system performance can only be evaluated with a computer black‐box model. Through a sequence of gradient‐type local searches in subspaces of solution space, SD‐SLIS is particularly efficient for such MIBO problems with scaling issues. We discuss the convergence conditions and properties of SD‐SLIS algorithms for a class of MIBO problems. Under mild conditions, SD‐SLIS is proved to converge to a stationary solution asymptotically. We apply SD‐SLIS to six example problems including two MIBO problems associated with petroleum field development projects. The algorithm performance of SD‐SLIS is compared with that of a state‐of‐the‐art direct‐search method, NOMAD, and that of a full space simplex interpolation search, Full‐SLIS. The numerical results suggest that SD‐SLIS solves the example problems efficiently and outperforms the compared methods for most of the example cases. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 305–322, 2017     

The stochastic joint replenishment problem: A new policy,analysis, and insights

In this study, we propose a new parsimonious policy for the stochastic joint replenishment problem in a single‐location, N‐item setting. The replenishment decisions are based on both group reorder point‐group order quantity and the time since the last decision epoch. We derive the expressions for the key operating characteristics of the inventory system for both unit and compound Poisson demands. In a comprehensive numerical study, we compare the performance of the proposed policy with that of existing ones over a standard test bed. Our numerical results indicate that the proposed policy dominates the existing ones in 100 of 139 instances with comparably significant savings for unit demands. With batch demands, the savings increase as the stochasticity of demand size gets larger. We also observe that it performs well in environments with low demand diversity across items. The inventory system herein also models a two‐echelon setting with a single item, multiple retailers, and cross docking at the upper echelon. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Some aspects in estimating warranty and post‐warranty repair demands

When customers buy a product, they are often eligible for free repairs for a certain warranty period. In this article, we study some important aspects, which are often overlooked in the literature but are of interest to the manufacturer, in estimating both warranty and post‐warranty repair demands. We consider that the installed base of the product (i.e., the number of units of the product actually in use) varies with time due to both new sales and units being taken out of service. When estimating warranty and post‐warranty repair demands, we explicitly address the fact that customers may not always request repairs for failed units. For the case where the product failure time distribution is exponential, we derive the closed‐form expressions for both types of repair demands of a single unit and of the time‐varying installed base. The insights into some risk‐related quantities are also presented. Furthermore, the proposed model is extended by considering delayed warranty claims that are frequently seen in practice. Numerical examples illustrate that understanding both types of repair demands and the related decision variables is important for managing the obligatory and profitable repair services. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 499–511, 2013     

On the first come–first served rule in multi‐echelon inventory control

A two‐echelon distribution inventory system with a central warehouse and a number of retailers is considered. The retailers face stochastic demand and replenish from the warehouse, which, in turn, replenishes from an outside supplier. The system is reviewed continuously and demands that cannot be met directly are backordered. Standard holding and backorder costs are considered. In the literature on multi‐echelon inventory control it is standard to assume that backorders at the warehouse are served according to a first come–first served policy (FCFS). This allocation rule simplifies the analysis but is normally not optimal. It is shown that the FCFS rule can, in the worst case, lead to an asymptotically unbounded relative cost increase as the number of retailers approaches infinity. We also provide a new heuristic that will always give a reduction of the expected costs. A numerical study indicates that the average cost reduction when using the heuristic is about two percent. The suggested heuristic is also compared with two existing heuristics. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007