Explicit formulas for the order size and reorder point in certain inventory problems

This study concentrates on distributions of leadtime demand that permit explicit solution to the lot-size, reorder point model. The optimal order size for the general case is first expressed as a function of the economic order quantity and a quantity known as the “residual mean life” in reliability theory. The concept of “no aging” is then utilized to identify a broad class of distributions for which the optimal order size can be determined explicitly, independent of the reorder point.     

Degeneracy in inventory models

In order‐quantity reorder‐point formulations for inventory items where backordering is allowed, some of the more common ways to prevent excessive stockouts in an optimal solution are to impose either a cost per unit short, a cost per stockout occasion, or a target fill rate. We show that these popular formulations, both exact and approximate, can become “degenerate” even with quite plausible parameters. By degeneracy we mean any situation in which the formulation either cannot be solved, leads to nonsensical “optimal” solutions, or becomes equivalent to something substantially simpler. We explain the reasons for the degeneracies, yielding new insight into these models, and we provide practical advice for inventory managers. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 686–705, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10037     

Distribution free bounds for service constrained (Q,r) inventory systems

A classical and important problem in stochastic inventory theory is to determine the order quantity (Q) and the reorder level (r) to minimize inventory holding and backorder costs subject to a service constraint that the fill rate, i.e., the fraction of demand satisfied by inventory in stock, is at least equal to a desired value. This problem is often hard to solve because the fill rate constraint is not convex in (Q, r) unless additional assumptions are made about the distribution of demand during the lead‐time. As a consequence, there are no known algorithms, other than exhaustive search, that are available for solving this problem in its full generality. Our paper derives the first known bounds to the fill‐rate constrained (Q, r) inventory problem. We derive upper and lower bounds for the optimal values of the order quantity and the reorder level for this problem that are independent of the distribution of demand during the lead time and its variance. We show that the classical economic order quantity is a lower bound on the optimal ordering quantity. We present an efficient solution procedure that exploits these bounds and has a guaranteed bound on the error. When the Lagrangian of the fill rate constraint is convex or when the fill rate constraint does not exist, our bounds can be used to enhance the efficiency of existing algorithms. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 635–656, 2000     

On the use of a structural equation for determining inventory order quantities

This article considers a structural equation useful for characterizing the order quantity of several inventory models. A correct interpretation of this equation is provided and it is stressed that the equation should be used in conjunction with another equation for the reorder point. Failure to do so may give rise to improper interpretations and invalid conclusions. A specific case like this is cited for the sake of illustration.     

Error bounds for EOQ

In this article we explore the properties of the discounted total cost function for the economic order quantity. We show that it is convex. Furthermore, it is shown that the classical economic order quantity (based on Wilson's formula) is not less than the true optimum value based on discounting. Bounds for the discounted reorder interval (or order quantity) based on average cost analysis are also provided. Furthermore, we analytically show that larger the ratio of noncapital-related holding charges to the total holding charges, the more adverse is the effect on the accuracy of the average cost analysis.     

Use of dimensionless ratios to determine minimum-cost inventory quantities

The iteration usually necessary for simultaneous determination of minimum-cost order quantity and reorder point in (Q, r) inventory systems may be eliminated by a graphical technique employing dimensionless ratios. This technique is illustrated for three different types of stock-out penalty.     

More ado about economic order quantities (EOQ)

This paper is concerned with the determination of explicit expressions for economic order quantities and reorder levels, such that the cost of ordering and holding inventory is minimized for specific backorder constraints. Holding costs are applied either to inventory position or on-hand inventory, and the backorder constraint is considered in terms of the total number of backorders per year or the average number of backorders at any point in time. Through the substitution of a new probability density function in place of the normal p.d.f., explicit expressions are determined for the economic order quantities and the reorder points. The resulting economic order quantities are independent of all backorder constraints. It is also concluded that under certain conditions, the minimization of ordering costs and inventory holding costs (applied to inventory position), subject to a backorder constraint, is equivalent in terms of reorder levels to minimization of the safety level dollar investment subject to the same backorder constraint.     

Inventory competition with yield reliability improvement

This article studies the inventory competition under yield uncertainty. Two firms with random yield compete for substitutable demand: If one firm suffers a stockout, which can be caused by yield failure, its unsatisfied customers may switch to its competitor. We first study the case in which two competing firms decide order quantities based on the exogenous reliability levels. The results from the traditional inventory competition are generalized to the case with yield uncertainty and we find that quantity and reliability can be complementary instruments in the competition. Furthermore, we allow the firms to endogenously improve their yield reliability before competing in quantity. We show that the reliability game is submodular under some assumptions. The results indicate that the competition in quantity can discourage the reliability improvement. With an extensive numerical study, we also demonstrate the robustness of our analytical results in more general settings. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 107–126, 2015     

The relationship between decision variables and penalty cost parameter in (Q,R) inventory models

This paper is concerned with the optimum decision variables found using order quantity, reorder point (Q, R) inventory models. It examines whether the optimum variables (Q* and R*) are necessarily monotonic functions of the backorder cost parameter (or equivalently of the performance objective). For a general class of models it is proved that R* must increase as the performance objective is raised, and an inequality condition is derived which governs how Q* will change. Probability distributions of lead time demand are cited or found for which Q* increases, Q* decreases, and Q* is independent of increases in performance objectives or backorder cost parameter.     

A model for use in the rationing of inventory during lead time

While the traditional solution to the problem of meeting stochastically variable demands for inventory during procurement lead time is through the use of some level of safety stock, several authors have suggested that a decision be made to employ some form of rationing so as to protect certain classes of demands against stockout by restricting issues to other classes. Nahmias and Demmy [10] derived an approximate continuous review model of systems with two demand classes which would permit an inventory manager to calculate the expected fill rates per order cycle for high-priority, low-priority, and total system demands for a variety of parameters. The manager would then choose the rationing policy that most closely approximated his fill-rate objectives. This article describes a periodic review model that permits the manager to establish a discrete time rationing policy during lead time by prescribing a desired service level for high-priority demands. The reserve levels necessary to meet this level of service can then be calculated based upon the assumed probability distributions of high- and low-priority demands over lead time. The derived reserve levels vary with the amount of lead time remaining. Simulation tests of the model indicate they are more effective than the single reserve level policy studied by Nahmias and Demmy.     

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.     

Order-statistic calculation,costs, and service in an (s,Q) inventory system

A retailer or distributor of finished goods, or the manager of a spare-parts inventory system, must generally forecast the major portion of demand. A specific customer-service level p (fraction of replenishment intervals with no stockout) implies two challenges: achieve the service within a small interval plus or minus, and do so with a minimum-cost investment in inventory. The pth fractile of lead-time demand (LTD) is the reorder point (ROP) for this service measure, and is often approximated by that fractile of a normal distribution. With this procedure, it is easy to set safety stocks for an (s, Q) inventory system. However, Bookbinder and Lordahl [2] and others have identified cases where the normal approximation yields excessive costs and/or lower service than desired. This article employs an order-statistic approach. Using available LTD data, the ROP is simply estimated from one or two of the larger values in the sample. This approach is sufficiently automatic and intuitive for routine implementation in industry, yet is distribution free. The order-statistic method requires only a small amount of LTD data, and makes no assumptions on the form of the underlying LTD distribution, nor even its parameters μ and ρ. We compare the order-statistic approach and the normal approximation, first in terms of customer service and then using a model of expected annual cost. Based upon characteristics of the available LTD data, we suggest a procedure to aid a practitioner in choosine between the normal and order-statistic method. © 1994 John Wiley & Sons, Inc.     

A time dependent continuous review inventory model with compound poisson demand

We discuss a time dependent optimal ordering policy for a finite horizon inventory system for which the provision of service is essential and thus no stockout is allowed. It is assumed that the system can place an order at any point in time during the horizon when it cannot meet the customer's demand and that lead time is negligible. The demand is considered to be distributed as a compound Poisson process with known parameters and the functional equation approach of dynamic programming is used to formulate the objective function. An algorithm has been developed to obtain the solution for all the cases. In addition, analytical solutions of the basic equation under two limiting conditions are presented.     

(R.Q) inventory problem with unknown mean demand and learning

We address a single product, continuous review model with stationary Poisson demand. Such a model has been effectively studied when mean demand is known. However, we are concerned with managing new items for which only a Bayesian prior distribution on the mean is available. As demand occurs, the prior is updated and our control parameters are revised. These include the reorder point (R) and reorder quantity (Q). Deemer, taking a clue from some earlier RAND work, suggested using a model appropriate for known mean, but using a Compound Poisson distribution for demand rather than Poisson to reflect uncertainty about the mean. Brown and Rogers also used this approach but within a periodic review context. In this paper we show how to compute optimum reorder points for a special problem closely related to the problem of real interest. In terms of the real problem, subject to a qualification to be discussed, the reorder points found are upper bounds for the optimum. At the same time, the reorder points found can never exceed those found by the Compound Poisson (Deemer) approach. And they can be smaller than those found when there is no uncertainty about the mean. As a check, the Compound Poisson and proposed approach are compared by simulation.     

Inventory systems with imperfect demand information

An inventory system is described in which demand information may be incorrectly transmitted from the field to the stocking point. The stocking point employs a forwarding policy which attempts to send out to the field a quantity which, in general, is some function of the observed demand. The optimal ordering rules for the general n-period problem and the steady state case are derived. In addition orderings of the actual reorder points as functions of the errors are presented, as well as some useful economic interpretations and numerical illustrations.     

声自导雷平行齐射发现概率解析计算模型

为了准确快速地评估声自导鱼雷平行齐射的作战效能（发现概率）,从鱼雷、目标的相对运动入手,将目标位置及运动随机误差转移到鱼雷航向上;如此建立了计算声自导鱼雷平行齐射发现概率的一重积分解析模型（不考虑鱼雷航行性能误差）和四重积分解析计算模型（考虑鱼雷航行性能误差）。最后以统计模型为标准来验证两个解析模型的正确性。仿真结果显示,对于90%左右的考察态势,解析模型与统计模型的结果差的绝对值在1%之内,另外10%左右的考察态势,也在1%-3%之间。从而验证了所建解析模型的正确性和可行性。     

An inventory model of incentives for on-time delivery in just-in-time purchasing contracts

This article formulates an analytic model of just-in-time purchasing contracts and compares the minimum cost solution with the cost attainable through vertical integration. The models use standard inventory theory cost parameters and decision variables. The results quantify the increase in cost of buying an item rather than making it. Optimal incentives are characterized when JIT purchasing contracts are used. When JIT purchasing is implemented, buffer inventories are typically reduced. This inventory reduction makes on-time delivery critical to the buyer; yet timeliness is controlled by the supplier. As an incentive to provide on-time delivery, the buyer offers the supplier a bonus for on-time delivery. The supplier chooses a flow time allowance based upon the bonus offered. First- and second-order conditions are characterized in general, and examples are provided for exponentially and uniformly distributed flow times. The delivery timeliness obtainable in a vertically integrated firm is determined and compared with timeliness obtainable between separate firms. This comparison indicates that buyers who choose to purchase materials from a separate firm are more likely to experience late deliveries. The relationship between the value of the bonus and the proportion of on-time deliveries is also considered. The bonus required to achieve the same probability of on-time delivery as under vertical integration is also determined. © 1993 John Wiley & Sons, Inc.     

Models for multi-item continuous review inventory policies subject to constraints

Models are formulated for determining continuous review (Q, r) policies for a multiitem inventory subject to constraints. The objective function is the minimization of total time-weighted shortages. The constraints apply to inventory investment and reorder workload. The formulations are thus independent of the normal ordering, holding, and shortage costs. Two models are presented, each representing a convex programming problem. Lagrangian techniques are employed with the first, simplified model in which only the reorder points are optimized. In the second model both the reorder points and the reorder quantities are optimized utilizing penalty function methods. An example problem is solved for each model. The final section deals with the implementation of these models in very large inventory systems.     

Investment in the reduction of uncertainties in just-in-time purchasing systems

When the uncertainties of supply caused by lead-time variability, natural disasters, or other reasons are not considered as given, but as manageable, a company invests to reduce these uncertainties to save a company's operating cost. This article examines economic trade-offs of investments directed toward the reduction of uncertainties. Two models are presented. The first introduces the ability of a purchaser to reduce the variance of the lead time of its supplier. The second model adds the ability of the purchaser to not only influence the variance of its supplier's lead time, but to influence the proportion of defective units produced by the supplier. This article develops explicit solutions for jointly choosing optimal variances for both lead time and proportion of defective units within a reorder-point model. Two cases are considered; defective rate is independent of lead time, or defective rate depends on lead time. © 1994 John Wiley & Sons, Inc.     

Fuzzy-AHP法在潜射巡航导弹命中概率分析中的应用

为准确评价潜射巡航导弹的命中概率,对影响命中概率的因素进行了具体分析,建立了潜射巡航导弹命中概率综合评价的指标体系.应用模糊综合评判及层次分析法,给出了综合评价的数学模型.对算例的仿真计算,得出了较为可信的结果.结果表明应用此方法,可以更准确的对潜射巡航导弹的命中概率进行评估,为潜射巡航导弹的研发及使用保障提供决策参考.