Modeling lead-time demand for lumpy demand and variable lead time

Slow-moving items that occasionally exhibit large demand transactions are known as lumpy demand items. In modeling lumpy demand patterns, it is often assumed that the arrival of customer orders follows a Poisson process and that the order sizes are given by the geometric distribution. This gives rise to a stuttering Poisson (sP) model of lumpy demand. If lead times are constant, the result is a stuttering Poisson model of lead-time demand. Heretofore, authors such as Ward [18] and Mitchell, Rappold, and Faulkner [12] have assumed constant lead times and thus stopped at the sP model. We develop this model further by introducing the effect of lead-time variability. For illustration, we use the normal and the gamma distributions as characterizations of lead time. The resulting models of lead-time demand are referred to as the geometric Poisson normal (GPN) and the geometric Poisson gamma (GPG). For both these models, the article derives tractable expressions for calculating probabilities. Errors introduced by using the sP, constant lead-time model instead of the exact, variable lead-time model are also illustrated.     

The logarithmic poisson gamma distribution: A model for leadtime demand

This paper considers an inventory system in which demand occurrences arise according to a stationary Poisson process, demand sizes at each occurrence follow a logarithmic distribution, and leadtimes are random variables with the gamma distribution. Both the exact and approximate distribution for leadtime demand are derived and computations are performed which compare the approximation to the exact distribution. The results have application to both repairable and consumable item inventory systems.     

The (s,Q) inventory model with Erlang lead time and deterministic demand

We develop a simple, approximately optimal solution to a model with Erlang lead time and deterministic demand. The method is robust to misspecification of the lead time and has good accuracy. We compare our approximate solution to the optimal for the case where we have prior information on the lead‐time distribution, and another where we have no information, except for computer‐generated sample data. It turns out that our solution is as easy as the EOQ's, with an accuracy rate of 99.41% when prior information on the lead‐time distribution is available and 97.54–99.09% when only computer‐generated sample information is available. Apart from supplying the inventory practitioner with an easy heuristic, we gain insights into the efficacy of stochastic lead time models and how these could be used to find the cost and a near‐optimal policy for the general model, where both demand rate and lead time are stochastic. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Capacity expansion under a service‐level constraint for uncertain demand with lead times

For a service provider facing stochastic demand growth, expansion lead times and economies of scale complicate the expansion timing and sizing decisions. We formulate a model to minimize the infinite horizon expected discounted expansion cost under a service‐level constraint. The service level is defined as the proportion of demand over an expansion cycle that is satisfied by available capacity. For demand that follows a geometric Brownian motion process, we impose a stationary policy under which expansions are triggered by a fixed ratio of demand to the capacity position, i.e., the capacity that will be available when any current expansion project is completed, and each expansion increases capacity by the same proportion. The risk of capacity shortage during a cycle is estimated analytically using the value of an up‐and‐out partial barrier call option. A cutting plane procedure identifies the optimal values of the two expansion policy parameters simultaneously. Numerical instances illustrate that if demand grows slowly with low volatility and the expansion lead times are short, then it is optimal to delay the start of expansion beyond when demand exceeds the capacity position. Delays in initiating expansions are coupled with larger expansion sizes. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

The logistic–exponential survival distribution

For various parameter combinations, the logistic–exponential survival distribution belongs to four common classes of survival distributions: increasing failure rate, decreasing failure rate, bathtub‐shaped failure rate, and upside‐down bathtub‐shaped failure rate. Graphical comparison of this new distribution with other common survival distributions is seen in a plot of the skewness versus the coefficient of variation. The distribution can be used as a survival model or as a device to determine the distribution class from which a particular data set is drawn. As the three‐parameter version is less mathematically tractable, our major results concern the two‐parameter version. Boundaries for the maximum likelihood estimators of the parameters are derived in this article. Also, a fixed‐point method to find the maximum likelihood estimators for complete and censored data sets has been developed. The two‐parameter and the three‐parameter versions of the logistic–exponential distribution are applied to two real‐life data sets. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Location-allocation for distribution to a uniform demand with transshipments

This article analyzes the location-allocation problem for distribution from a single fixed origin via transshipment terminals to a continuous uniformly distributed demand. Distribution through terminals concentrates flows on the origin-to-terminal links and transportation economies of scale encourage the use of larger vehicles. Analytical expressions are derived for the optimal terminal locations, the optimal allocation of destinations to terminals, and the optimal transportation cost. Continuous analytic models assume either an allocation, by partitioning the service region into sectors, or terminal locations. This is unlikely to produce an optimal distribution system. The optimal cost is compared to the cost for suboptimal location-allocation combinations. Results indicate that the location decision is not too important if destinations are allocated optimally and that allocation to the nearest terminal may be poor, even with optimal locations. © 1992 John Wiley & Sons, Inc.     

Estimating the lead-time demand distribution for an autocorrelated demand by the pearson system and a normal approximation

The exact first four moments of lead-time demand L are derived for an AR(1) and a MA(1) demand structures where the arbitrary lead-time distribution is assumed to be independent of the demand structure. These moments then form a basis for the Pearson curve-fitting procedure for estimating the distribution of L. A normal approximation to L, a version of the central limit theorem, is obtained under some general conditions. Reorder points (ROPs) of an inventory system are then estimated based on the Pearson system and a normal approximation. Their performances are evaluated. Numerical investigation shows that the Pearson system performs extremely well. The normal approximation, however, is good only for some limited cases, and is sensitive to the choice of the lead-time distribution. A possible improvement is noted.     

Centralized inventory control in a two‐level distribution system with Poisson demand

This paper introduces a new replenishment policy for inventory control in a two‐level distribution system consisting of one central warehouse and an arbitrary number of nonidentical retailers. The new policy is designed to control the replenishment process at the central warehouse, using centralized information regarding the inventory positions and demand processes of all installations in the system. The retailers on the other hand are assumed to use continuous review (R, Q) policies. A technique for exact evaluation of the expected inventory holding and backorder costs for the system is presented. Numerical results indicate that there are cases when considerable savings can be made by using the new (α₀, Q₀) policy instead of a traditional echelon‐ or installation‐stock (R, Q) policy. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 798–822, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10040     

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     

Scheduling a production–distribution system to optimize the tradeoff between delivery tardiness and distribution cost

We consider a make‐to‐order production–distribution system with one supplier and one or more customers. A set of orders with due dates needs to be processed by the supplier and delivered to the customers upon completion. The supplier can process one order at a time without preemption. Each customer is at a distinct location and only orders from the same customer can be batched together for delivery. Each delivery shipment has a capacity limit and incurs a distribution cost. The problem is to find a joint schedule of order processing at the supplier and order delivery from the supplier to the customers that optimizes an objective function involving the maximum delivery tardiness and the total distribution cost. We first study the solvability of various cases of the problem by either providing an efficient algorithm or proving the intractability of the problem. We then develop a fast heuristic for the general problem. We show that the heuristic is asymptotically optimal as the number of orders goes to infinity. We also evaluate the performance of the heuristic computationally by using lower bounds obtained by a column generation approach. Our results indicate that the heuristic is capable of generating near optimal solutions quickly. Finally, we study the value of production–distribution integration by comparing our integrated approach with two sequential approaches where scheduling decisions for order processing are made first, followed by order delivery decisions, with no or only partial integration of the two decisions. We show that in many cases, the integrated approach performs significantly better than the sequential approaches. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Nonrobustness of the normal approximation of lead‐time demand in a (Q,R) system

For computing an optimal (Q, R) or kindred inventory policy, the current literature provides mixed signals on whether or when it is safe to approximate a nonnormal lead‐time‐demand (“LTD”) distribution by a normal distribution. The first part of this paper examines this literature critically to justify why the issue warrants further investigations, while the second part presents reliable evidence showing that the system‐cost penalty for using the normal approximation can be quite serious even when the LTD‐distribution's coefficient of variation is quite low—contrary to the prevalent view of the literature. We also identify situations that will most likely lead to large system‐cost penalty. Our results indicate that, given today's technology, it is worthwhile to estimate an LTD‐distribution's shape more accurately and to compute optimal inventory policies using statistical distributions that more accurately reflect the LTD‐distributions' actual shapes. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

Optimal pricing for a short life‐cycle product when customer price‐sensitivity varies over time

Technology products often experience a life‐cycle demand pattern that resembles a diffusion process, with weak demand in the beginning and the end of the life cycle and high demand intensity in between. The customer price‐sensitivity also changes over the life cycle of the product. We study the prespecified pricing decision for a product that exhibits such demand characteristics. In particular, we determine the optimal set of discrete prices and the times to switch from one price to another, when a limited number of price changes are allowed. Our study shows that the optimal prices and switching times show interesting patterns that depend on the product's demand pattern and the change in the customers' price sensitivity over the life cycle of the product. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

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     

Waiting time analysis for a queueing system with time‐limited service and exponential timer

We consider a single‐queue with exhaustive or gated time‐limited services and server vacations, in which the length of each service period at the queue is controlled by a timer, i.e., the server serves customers until the timer expires or the queue becomes empty, whichever occurs first, and then takes vacations. The customer whose service is interrupted due to the timer expiration may be attended according to nonpreemptive or preemptive service disciplines. For the M/G/1 exhaustive/gated time‐limited service queueing system with an exponential timer and four typical preemptive/nonpreemptive service disciplines, we derive the Laplace—Stieltjes transforms and the moment formulas for waiting times and sojourn times through a unified approach, and provide some new results for these time‐limited service disciplines. © John Wiley & Sons, Inc. Naval Research Logistics 48: 638–651, 2001.     

Polynomial‐time approximation scheme for concurrent open shop scheduling with a fixed number of machines to minimize the total weighted completion time

In this article, we consider the concurrent open shop scheduling problem to minimize the total weighted completion time. When the number of machines is arbitrary, the problem has been shown to be inapproximable within a factor of 4/3 ‐ ε for any ε > 0 if the unique games conjecture is true in the literature. We propose a polynomial time approximation scheme for the problem under the restriction that the number of machines is fixed. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

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     

Military Expenditure as a Proxy for State Power. The Case of France

This paper attempts to build a simple indicator of state power. Military expenditure is the paper’s point of departure, with the definitions given by NATO, SIPRI and others. This definition is discussed and a power version is build, using French budgetary data. Then a defence of the result against traditional or non-traditional critics is presented under an imperative of action. Finally, this concept is enlarged and a new concept of power expenditures is given, one more time using French budgetary data. The conclusion is that this large expenditure concept is an unbiased but imperfect indicator of the will to act, and has to be completed by GDP to indicate the capability to act in the long-run.     

The contexts of strategy as a guide for defense planning in the Philippines

Efforts toward developing an independent and credible Philippine defense policy were revived when Benigno Simeon C. Aquino III was elected President of the Philippines in 2010. President Aquino renewed the military modernization program in 2012 and emphasized the necessity of a “minimum credible defense posture.” Given the modernization efforts of the government, this article aims to contribute to the development of the concept of minimum credible defense posture by exploring how theory contributes to defense planning. It argues that the contexts of strategy are useful in planning for national defense because these contexts can provide a conceptual framework for defense planners. It proceeds in three parts: the opening section reviews the literature on defense planning; the second surveys the context of developing strategies as discussed in the General Theory of Strategy; and the last provides a definition of minimum credible defense and applies the context of developing strategies to the case of the Philippines.