Lead Time Demand Distribution Research Articles

Inventory models with mixture of backorders involving reducible lead time and setup cost

Lead time and setup cost are controllable variables in our continuous review inventory model. In this study we assume that only the first and second moments, of the probability distribution of lead time demand, are known. Order quantity and backorder price discount are decision variables in our problem with mixture of backorders and lost sales. As a result of reducing lead-time we can have significant savings and it is made known through numerical examples.

Effective investment to reduce lost-sales rate in a periodic review inventory model

In 1973 Montgomery et al. proposed an (R, T) type of a periodic review inventory model in which the lost-sales rate caused by stockout is given. The purpose of this article is to investigate in this heuristic periodic review inventory model with partial lost-sales to effectively increase investment and to reduce the lost-sales rate. We discuss two models: complete (normal distribution) and partial (distribution free) information about the protection interval (i.e., review period plus lead time) demand distribution. For each model, two commonly used investment cost functional forms, logarithmic and power, are employed for lost-sales rate reduction. That is, for these two investment cost functions, we first assume that the protection interval demand follows a normal distribution, and then we assume that only first and second moments of the probability distribution of protected interval demand are known. Two algorithms are developed to find the optimal investment decision, and six numerical examples are given to illustrate the results.

Integrated vendor--buyer inventory system with sublot sampling inspection policy and controllable lead time

This article investigates the impact of inspection policy and lead time reduction on an integrated vendor--buyer inventory system. We assume that an arriving order contains some defective items. The buyer adopts a sublot sampled inspection policy to inspect selected items. The number of defective items in the sublot sampling is a random variable. The buyer's lead time is assumed reducible by adding crash cost. Two integrated inventory models with backorders and lost sales are derived. We first assume that the lead time demand follows a normal distribution, and then relax the assumption about the lead time demand distribution function and apply the minimax distribution-free procedure to solve the problem. Consequently, the order quantity, reorder point, lead time and the number of shipments per lot from the vendor to the buyer are decision variables. Iterative procedures are developed to obtain the optimal strategy.

A note on a (Q, r) model

Hadley and Whitin’s lot-size, reorder point model (Q, r) has been widely used in industry because the effects of uncertain demands during lead time is incorporated properly. However, there is no efficient solution procedure for computing the optimal policies. In the literature, an iterative solution procedure is proposed. To avoid iterative computations for an optimal policy, the (Q, r) model has been studied to derive an explicit formula for the optimal solution when lead time demand distribution possesses a certain property. Unfortunately, this result can not be applicable for the lead time demand distributions, Gamma or Normal, and they are sometimes considered the most appealing distributions for modeling the lead time demand. In this paper, we study the (Q, r) model assuming that the lead time demand is generally distributed. Some useful results are obtained to facilitate a robust and efficient solution procedure for the optimal policy. A numerical example is provided to illustrate the solution procedure for the optimal lot-size and reorder point policy.

A Closed-Form Approximation for Serial Inventory Systems and Its Application to System Design

We analyze a serial base-stock inventory model with Poisson demand and a fill-rate constraint. Our objective is to gain insights into the linkage between the stages to facilitate optimal system design and decentralized system control. To this end, we develop a closed-form approximation for the optimal base-stock levels. The development consists of two key steps: (1) convert the service-constrained model into a backorder cost model by imputing an appropriate backorder cost rate, and then adapt the single-stage approximation developed for the latter, and (2) use a logistic distribution to approximate the lead-time demand distribution in the single-stage approximation obtained in (1) to yield closed-form expressions. We then use the closed-form expressions to conduct sensitivity analyses and establish qualitative properties on system design issues, such as optimal total system stock, stock positioning, and internal fill rates. The closed-form approximation and most of the qualitative properties apply equally to the model with a backorder cost, although some differences do exist. Other results of this study include a bottom-up recursive procedure to evaluate any given echelon base-stock policy and lower bounds on the optimal echelon base-stock levels.

An analysis of the lead time demand distribution derivation in stochastic inventory systems

Among several approaches to derive the distribution of lead time demand, this paper concerns with a compound distribution approach. The mostly known analytic models using the compound distribution approach have been done by non-recognition of the compound nature of some components to reduce the analytic task. In order to provide some precautions to researchers and practitioners, this paper conducts a theoretical examination and clarifies the assumptions implicitly made by the analytic models. In addition, comparison experiments under various situations are also conducted.

An Excel based approach to optimise trade-offs between freight expediting and safety stock inventory costs

In this paper, we present an Excel based methodology for the computation of two optimal policy quantities, order size and reorder point of a (Q, r) inventory system under uncertainty when backorder costs are independent of the amount of the backorder and dependent on whether there is a shortage. It is shown that the total cost curve for the backorder model is a concave-convex function. Solving the problem using a spreadsheet solver would not necessarily yield the optimal solution. It is first necessary to identify the convex and concave domains prior to the application of Excel Solver to determine the optimal solution of the model. This approach is flexible and capable of handling different lead time demand distributions as well as the more realistic scenario when both the demand and lead times are randomly distributed. The methodology is illustrated with a simple problem.

Improved Base-Stock Approximations for Independent Stochastic Lead Times with Order Crossover

Order crossovers occur when replenishment orders arrive in a sequence that is different than the one in which they were placed. Order crossovers require that optimal reorder levels be set with regard to the inventory shortfall distribution rather than the lead-time demand distribution. Assuming periodic review and independent lead times, this paper suggests simple approximations of the shortfall distribution by showing that the variance of the number of orders outstanding is bounded above by the standard deviation of lead time divided by √3. Using this bound in a normal approximation improves significantly upon the common practice of basing policies on the lead-time demand distribution. A negative binomial approximation of the shortfall, based on its exact variance, offers even greater improvement, at the cost of some additional informational and computational requirements.

Inventory models with variable lead time and present value

The figures for inventory make up a huge proportion of a company's working capital. Because of this, we formulated the optimal replenishment policy considering the time value of money to represent opportunity cost. In this article, we provide a mixed inventory model, in which the distribution of lead time demand is normal, to consider the time value. First, the study tries to find the optimal reorder point and order quantity at all lengths of lead time with components crashed to their minimum duration. Secondly, we develop a method to insure the uniqueness of the reorder point to locate the optimal solution. Finally, some numerical examples are given to illustrate our findings.

Inventory control based on advanced probability theory, an application

Whenever stock is placed as a buffer between consumption and supply the decision when to replenish the stock is based on uncertain values of future demand and supply variables. Uncertainty exists about the replenishment lead time, about the number of demands and the quantities demanded during this period. We develop a new analytical expression for the reorder point, which is based on the desired service level and three distributions: the distribution of the quantity of single demands during lead time, the distribution of the lengths of time intervals between successive demands, and the distribution of the lead time itself. The distribution of lead time demand is derived from the distributions of individual demand quantities and not from the demand per period. It is not surprising that the resulting formulae for the mean and variance are different from those currently used. The theory developed is also applicable to periodic review systems. The system has been implemented at CERN and enables a significant enhancement of the service level, while reducing the average stock.

Proposta para a gestão de estoques de novos produtos: solução do modelo (Q,r) para a distribuição uniforme da demanda e do lead-time de suprimento

As premissas comumente consideradas nos modelos de estoques - aderência da demanda no lead-time de suprimento à Distribuição Normal, média e desvio-padrão conhecidos e lead-time de suprimento discreto - são freqüentemente irrealistas e podem implicar substanciais distorções na gestão dos estoques de novos produtos, sobretudo nos custos totais e nos indicadores de nível de serviço. Considerando que as empresas simultaneamente estocam os novos produtos e aprendem ao longo do tempo sobre as características da distribuição da demanda no lead-time de suprimento, é desenvolvida nesse artigo a solução do modelo (Q,r) - tamanho de lote e ponto de pedido - para o caso da demanda e do lead-time de suprimento com Distribuição Uniforme. Por ser definida por meio de dois parâmetros mais intuitivos que a média e o desvio-padrão - máximo e mínimo - e representar situações em que qualquer resultado tem igual probabilidade de ocorrência, a premissa da Distribuição Uniforme pode constituir uma primeira abordagem prática para a gestão de estoques de novos produtos.

Lot size effects on process lead time, lead time demand, and safety stock

A comprehensive process lot sizing/order point model is developed, in which queuing relationships are used to endogenously model the effects of lot size on the lead time demand distribution. The model is used to examine the effect of shortage penalty on the optimal lot size, when the model reflects job queuing dynamics arising from the lot size decision. A coordinated policy of safety stock adjustment and lead time reduction through lot size adjustment is shown be more efficient than responding to changes in the shortage penalty parameter solely through adjustment of safety stock.

The Effect of Lead Time Uncertainty on Safety Stocks

ABSTRACTThe pressure to reduce inventory investments in supply chains has increased as competition expands and product variety grows. Managers are looking for areas they can improve to reduce inventories without hurting the level of service provided. Two areas that managers focus on are the reduction of the replenishment lead time from suppliers and the variability of this lead time. The normal approximation of lead time demand distribution indicates that both actions reduce inventories for cycle service levels above 50%. The normal approximation also indicates that reducing lead time variability tends to have a greater impact than reducing lead times, especially when lead time variability is large. We build on the work of Eppen and Martin (1988) to show that the conclusions from the normal approximation are flawed, especially in the range of service levels where most companies operate. We show the existence of a service‐level threshold greater than 50% below which reorder points increase with a decrease in lead time variability. Thus, for a firm operating just below this threshold, reducing lead times decreases reorder points, whereas reducing lead time variability increases reorder points. For firms operating at these service levels, decreasing lead time is the right lever if they want to cut inventories, not reducing lead time variability.

A minimax distribution free procedure for mixed inventory model with backorder discounts and variable lead time

In this article, we investigate a continuous review inventory model with a mixture of backorders and lost sales in which the order quantity, the backorder discount and the lead time are considered as decision variables. It is assumed that the probability distribution of lead time demand is unknown but its first two moments are given, we apply the minimax distribution free procedure to solve this problem. An algorithm to find the optimal strategy is developed.

Managing waiting times of backordered demands in single‐stage (Q, r) inventory systems

AbstractWe present a service constrained (Q, r) model that minimizes expected holding and ordering costs subject to an upper bound on the expected waiting time of demands that are actually backordered. We show that, after optimizing over r, the average cost is quasiconvex in Q for logconcave continuous lead time demand distributions. For logconcave discrete lead time demand distributions we find a single‐pass efficient algorithm based on a novel search stopping criterion. The algorithm also allows for bounds on the variability of the service measure. A brief numerical study indicates how the bounds on service impact the optimal average cost and the optimal (Q, r) choice. The discrete case algorithm can be readily adapted to provide a single pass algorithm for the traditional model that bounds the expected waiting time of all demands (backordered or not). © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 557–573, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10028

Forecasting for inventory control with exponential smoothing

Exponential smoothing, often used in sales forecasting for inventory control, has always been rationalized in terms of statistical models that possess errors with constant variances. It is shown in this paper that exponential smoothing remains appropriate under more general conditions, where the variance is allowed to grow or contract with corresponding movements in the underlying level. The implications for estimation and prediction are explored. In particular, the problem of finding the predictive distribution of aggregate lead-time demand, for use in inventory control calculations, is considered using a bootstrap approach. A method for establishing order-up-to levels directly from the simulated predictive distribution is also explored.

A minimax distribution free procedure for mixed inventory models involving variable lead time with fuzzy lost sales

Recently, the continuous review inventory models with allowable shortages have been extended to include variable lead time, where a fixed fraction of the demand during the stockout period is backordered. However, in practice, the backorder (or lost sales) rate may change slightly due to some uncertainties. To incorporate this reality, this article attempts to apply the fuzzy set concepts to deal with the uncertain lost sales rate. For a situation where information about the lead time demand distribution is partial, we utilize the minimax distribution free procedure to find the optimal inventory strategy in the fuzzy sense. Two numerical examples are given to illustrate the results.

A minimax distribution free procedure for mixed inventory model involving variable lead time with fuzzy demand

In a recent paper, Ouyang and Wu applied the minimax decision approach to solve a continuous review mixed inventory model in which the lead time demand distribution information is unknown but the annual demand is fixed and given. However, in the practical situation, the annual demand probably incurs disturbance due to various uncertainties. In this article, we attempt to modify Ouyang and Wu's model by considering two fuzziness of annual demand (i.e., fuzzy number of annual demand and statistic-fuzzy number of annual demand) and to investigate a computing schema for the continuous review inventory model in the fuzzy sense. We give an algorithm procedure to obtain the optimal ordering strategy for each case. Scope and purpose In most of the early literature dealing with inventory problems, either using deterministic or probabilistic models, lead time is viewed as a prescribed constant or a stochastic variable. Recently, some researchers (e.g., Liao and Shyu, Ben-Daya and Raouf, and Ouyang and Wu) incorporated the crashing lead time idea to continuous review inventory models, in which the annual demand is given and fixed. However, in the real situation, the annual demand will probably have a little disturbance due to various uncertainties. The purpose of this article is to modify the Ouyang and Wu's model to accommodate this reality, specifically, we apply the fuzzy set concepts to deal with the uncertain annual demand. We first consider a case where the annual demand is treated as the triangular fuzzy number. Then, we employ the statistical method to construct a confidence interval for the annual demand, and through it to establish the corresponding fuzzy number (namely, the statistic-fuzzy number). For each fuzzy case, we investigate a computing schema for the new model and develop an algorithm to find the optimal ordering strategy.

Mixture inventory model involving variable lead time and controllable backorder rate

This paper allows the backorder rate as a control variable to widen applications of Ouyang et al.'s model [J. Oper. Res. Soc. 47 (1996) 829]. In this study, we assume that the backorder rate is dependent on the length of lead time through the amount of shortages. We discuss two models that are perfect and partial information about the lead time demand distribution, that is, we first assume that the lead time demand follows a normal distribution, and then remove this assumption by only assuming that the first and second moments of the probability distribution of lead time demand are known. For each case, we develop an algorithm to find the optimal ordering strategy. Three numerical examples are given to illustrate solution procedure.

Consequences of Order Crossover Under Order-Up-To Inventory Policies

Order crossover occurs whenever replenishment orders do not arrive in the sequence in which they were placed. This paper argues that order crossover is becoming more prevalent and analyzes the dangers of ignoring it. We present an exact iterative algorithm for computing the distribution of the number of orders outstanding, and formulae for the inventory shortfall distribution (the quantity of inventory in replenishment at the start of a period) and the more common lead-time demand distribution, which are different when order crossover is possible. The lead-time demand distribution can have much higher variability than the shortfall distribution. We show that basing inventory policies on the lead-time demand distribution—rather than the shortfall distribution—can lead to significantly higher inventory cost, even if the probability of order crossover is small. We give an alternative proof to that of Zalkind (1976), which shows that the variance of shortfall is less than the variance of the standard lead-time demand.