Non-stationary inventory position processes

This paper investigates single-product non-stationary inventory problems associated with non-stationary demand processes, backorders allowed, stochastically independent inter-review-period demands, and review periods not necessarily of equal length. Sufficient conditions are obtained for the convergence in distribution of inventory positions in 〈R, r, T〉 inventory systems; in particular, the uniform long-run distribution is found robust in 〈nQ, r, T〉 systems for a large class of demand processes. It is also shown that in 〈R, r, T〉 systems the subsequences of cyclic non-stationary inventory position process associated with cyclic behavior of demand patterns converge in distribution.     

Supply chain inventory optimization with two customer classes in discrete time

In this paper we consider a single-item inventory system where two demand classes with different service requirements are satisfied from a common inventory. A critical level, reorder point, order quantity or (s, q, k) policy is in use. The time axis is divided into discrete time units, which is a common characteristic of many real-life supply-chain processes. The inventory process within the lead time of a replenishment order is modelled as a sequence of (1) an ordinary renewal process and (2) two alternating renewal processes. Approximations are developed for the demand class-specific fill rates and the probability distribution of the waiting time of low priority customer orders. This waiting time distribution is used for the inventory allocation in a two-stage supply chain.     

A partial replenishment model for an inventory with constant demand

An inventory with constant demand is considered. The inventory is checked according to a Poisson process and replenished either fully or partially when the stock is below a threshold. We obtained the stationary distribution of the level of the inventory. After assigning several costs to the inventory, we also derived the long-run average cost per unit time. A numerical example is studied to find the optimal values of the checking rate and threshold, which minimize the long-run average cost.     

变换核估计在未知需求分布函数条件下多阶段库存控制策略中的应用研究

传统的多阶段库存控制主要致力于库存持有以及过多库存的经济性研究.随机库存模型经常假定需求分布已知,这样可以产生容易解决的方案.但随着销售信息的不断更新,需求分布函数的参数常常未知.这样传统的多阶段库存模型很难产生最优的库存控制策略.当前文献对未知需求分布函数条件下的多阶段库存管理问题研究得不多,当需求分布函数随时间变化,是个多阶段随机规划问题,通常情况难以直接进行求解.针对一般非平稳需求,还缺少有效的库存管理方法.本文致力于变换核估计和优化理论相结合的方法研究未知需求分布函数条件下多阶段库存控制策略,提供一条多阶段库存控制的新思路.可以很好地确定各阶段的最优订货点、最高库存、最低库存等来达到整个系统的最优,从而节省更多的成本,达到营运资本的永久性减少、更高的销售量和客户满意度,从而增加企业的竞争力.     

Stability analysis and optimization of an inventory system with bounded orders

This paper addresses the control of a one-item inventory system subject to random order lead time and random demand. The key parameter of the control policy is the objective inventory. In each period, the order to be placed brings the inventory position as close as possible to the objective inventory. The order of each period is kept between a lower bound and an upper bound. We show that the distribution of the inventory level converges to its stationary distribution provided that the lower bound is smaller than the average demand, the upper bound is greater than the average demand and some regularity conditions hold. The average inventory cost is shown to be a convex function of the objective inventory level. A simulation-based approach is proposed for the determination of the optimal objective inventory. A method of bisection with derivative is then used to determine the optimal objective inventory. The derivatives needed in various iterations of this method are estimated using a single sample path with respect to a given objective inventory. Numerical results are provided.     

Inventory Problems with Partially Observed Demands and Lost Sales

This paper considers the case of partially observed demand in the context of a multi-period inventory problem with lost sales. Demand in a period is observed if it is less than the inventory level in that period and the leftover inventory is carried over to the next period. Otherwise, only the event that it is larger than or equal to the inventory level is observed. These observations are used to update the demand distributions over time. The state of the resulting dynamic program consists of the current inventory level and the current demand distribution, which is infinite dimensional. The state evolution equation for the demand distribution becomes linear with the use of unnormalized probabilities. We study two demand cases. First, the demands evolve according to a Markov chain. Second, the demand distribution has an unknown parameter which is updated in the Bayesian manner. In both cases, we prove the existence of an optimal feedback ordering policy. Permanent address of J. Adolfo Minjárez-Sosa: Departamento de Matemáticas, Universidad de Sonora, Hermosillo, Sonora, México. This project was partially supported by NSF Grant 0509278, ARPATP Grant 009741-0019-2006, and CONACYT (Mexico) Grant 46633-F.     

Base stock policy with retrial demands

In this work, we consider a continuous review base stock policy inventory system with retrial demands. The maximum storage capacity is S. It is assumed that primary demand is of unit size and primary demand time points form a Poisson process. A one-to-one ordering policy is adopted. According to this policy, orders are placed for one unit, as and when the inventory level drops due to a demand. We assume that the demands occur during the stock-out periods enter into the orbit of infinite size. The lead time is assumed to be exponential. The joint probability distribution of the inventory level and the number of demands in the orbit are obtained in the steady state case. Various system performance measures in the steady state are derived. The results are illustrated with suitable numerical examples.     

Inventory System Subject to Disaster with General Interarrival Times

Abstract

We discuss a single commodity continuous review (s, S) inventory system in which commodities get damaged due to external disaster. Shortages are not permitted and lead time is assumed to be zero. The interarrival times of demands constitute a family of i.i.d. random variables with a common arbitrary distribution. The quantity demanded at a demand epoch is arbitrarily distributed which depends only on the time elapsed since the last demand epoch. Transient and steady state probabilities of the inventory levels are derived by identifying suitable semi-regenerative process. In the case when the demand is for unit item and the disaster affects only an exhibiting item, the steady state probability distribution is obtained as uniform. An optimization problem is discussed and numerical examples are provided.     

联合订货下基于ø-散度的多产品库存鲁棒优化模型

针对多产品联合库存决策问题,在市场需求不确定条件下,建立了考虑联合订货成本的多产品库存鲁棒优化模型。针对不确定市场需求,采用一系列未知概率的离散情景进行描述,给出了基于最小最大准则的鲁棒对应模型,并证明了(s,S)库存策略的最优性。进一步,在仅知多产品市场需求历史数据基础上,采用基于ø-散度的数据驱动方法构建了满足一定置信度要求的关于未知需求概率分布的不确定集。在此基础上,为获得(s,S)库存策略的相关参数,运用拉格朗日对偶方法将所建模型等价转化为易于求解的数学规划问题。最后,通过数值计算分析了Kullback-Leibler散度和Cressie-Read散度以及不同的置信水平下的多产品库存绩效,并将其与真实分布下应用鲁棒库存策略得到的库存绩效进行对比。结果表明,需求分布信息的缺失虽然会导致一定的库存绩效损失,但损失值很小,表明基于文中方法得到的库存策略能够有效抑制需求不确定性扰动,具有良好的鲁棒性。     

基于MCMC的随机库存系统中提前期需求分布的计算

在需求和提前期均是随机的库存系统中,提前期需求的分布是由提前期分布与需求分布复合而成的,这个复合分布的计算通常是困难的。本文采用基于Gibbs抽样的马尔科夫链蒙特卡洛(MCMC,Markov chain Monte Carlo)方法,抽取条件分布样本作为提前期需求分布的样本,通过样本来计算提前期需求分布密度、服务水平和损失函数。这种方法避免了直接求解复杂积分计算上的困难,也克服了近似分布拟合偏差过大的问题,有效地解决了随机需求与随机提前期的复杂库存系统中提前期需求确定问题。理论与数值分析结果表明:与现有方法相比较,基于MCMC的方法具有计算简便、拟合精度高、通用性好等特点。     

On inventory problems with arbitrary cost pattern,demand pattern and demand distribution

A detailed analysis of inventory models without setup costs, arbitrary demand distribution and arbitrary demand and cost pattern is given. First it is shown that the corresponding one-period model without ordering costs may be reduced to another simpler one with appropriately modified demand distribution. Several representations are given for the modified demand distribution. As one of the main results this reduction turns out to be robust in most cases. In a final chapter the results are applied to the determination of an optimal policy for a class of N-period inventory models with convex holding-and shortage costs and without setup costs.     

考虑疫情风险与双重时效性的生鲜品供应中断库存控制策略研究

为解决重大疫情引发的供应中断导致生鲜品库存水平振荡,在市场需求随机变化的情形下,建立一个由供应商、配送中心和零售商组成的三级生鲜供应链库存系统,考虑由疫情引起的三种风险情景,引入系统动力学模型对零售商动态库存系统运营进行仿真分析。研究发现：受疫情风险传导系数和变质率两个序参量影响,供应链库存呈现振荡趋势;通过确定不同供应中断时长下的疫情风险情景提出优化保鲜投入策略、安全库存策略、以及共享库存联合提前转运策略有效降低零售端库存水平振荡并使其呈现渐稳趋势,实现产品在交付过程中的双重时效性,达到供需匹配,缓解疫情风险带来的影响,为相关零售企业提供决策支持。     

Bundle pricing of inventories with stochastic demand

We consider a retailer selling a fixed inventory of two perishable products over a finite horizon. Assuming Poisson arrivals and a bivariate reservation price distribution, we determine the optimal product and bundle prices that maximize the expected revenue. Our results indicate that the performances of mixed bundling, pure bundling and unbundled sales strategies heavily depend on the parameters of the demand process and the initial inventory levels. Bundling appears to be most effective with negatively correlated reservation prices and high starting inventory levels. When the starting inventory levels are equal and in excess of average demand, most of the benefits of bundling can be achieved through pure bundling. However, the mixed bundling strategy dominates the other two when the starting inventory levels are not equal. We also observe that an incorrect modeling of the reservation prices may lead to significant losses. The model is extended to allow for price changes during the selling horizon. It is shown that offering price bundles mid-season may be more effective than changing individual product prices.     

Analysis of a two-sided production policy with inventory-level-dependent production rates

In this paper we analyse a stochastic production/inventory problem with compound Poisson demand and state (i.e. inventory level) dependent production rates. Customers arrive according to a Poisson process where the amount demanded by each customer is assumed to have a general distribution. When the inventory W(t) falls below a critical level m, production is started at a rate of r[W(t)], i.e. production rate dynamically changes as a function of the inventory level. Production continues until a level M (œ w m) is reached. Excess demand is assumed to be lost. We identify a dam content process X that is a dual for the inventory level W and develop the stationary distribution for the X process. To achieve this we use tools from renewal and level crossing theories. The two-sided (m, M) policy is optimized using the expected cost obtained from the stationary density of W and a conditional (on w) expected cost function for this process. For a special case, we obtain explicit results for all the relevant expressions. Numerical examples are provided for several test problems. © 1996 John Wiley & Sons, Ltd.     

Estimating the demand pattern for C category items

This paper presents a methodology for estimating the demand pattern for the slowest-moving C category inventory items. The methodology uses an aggregation-by-items scheme and a forecasting procedure based on conditional demand analysis whereby aggregate demand is assumed to be an arbitrarily mixed, heterogeneous Poisson distribution. Practical aspects of demand heterogeneity, parameter estimation and model implementation are illustrated using a case study in retail inventory planning and control.     

Decision making on stock levels in cases of uncertain demand rate

Inventory control is a typical problem of decision making. In this paper a periodic replenishment of stock, the spare parts being of one kind, is discussed for some cases when the demand rate is uncertain. The first decision, before all others in the sequence, is done by assuming an a priori distribution of demand rate. In time, as the demand process goes on, corrections of parameters of the a priori distribution are made according to the accumulated knowledge about past demand. This Bayesian approach to decision making based on learning about the uncertain demand rate is known for the case when the demand rate is unknown but constant. It is shown that this same approach can be used in some cases when the demand rate is unknown and not constant. Results are given and used for inventory control.     

On Approximating Lead Time Demand Distributions Using the Generalised λ-type Distribution

In this paper the use of the generalised λ-type distribution (GLD) is proposed for the analysis of standard inventory problems. Using this distribution to approximate the lead time demand distribution we analyse the generalised newsboy problem and a (Q, r) policy. The standard inventory measures like optimal order size, reorder level, average demand lost, etc. are obtained under the GLD and are compared with those given by Shore's approximation and also under exact distributional assumptions. Through a numerical study the various inventory measures are compared using the GLD and Shore's approximation with the exact distributions. The comparison reveals that the GLD approximation is better suited than Shore's approximation to model the lead time demand.     

Optimal production and rationing policies of a make-to-stock production system with batch demand and backordering

In this paper, we consider the stock rationing problem of a single-item make-to-stock production/inventory system with multiple demand classes. Demand arrives as a Poisson process with a randomly distributed batch size. It is assumed that the batch demand can be partially satisfied. The facility can produce a batch up to a certain capacity at the same time. Production time follows an exponential distribution. We show that the optimal policy is characterized by multiple rationing levels.     

Estimating the lead-time demand distribution when the daily demand is non-normal and autocorrelated Hon-Shiang LAU

This paper develops convenient formulas for estimating the probability distribution of lead time demand when the inventory item's daily demand follows a probability distribution of any arbitrary shape, and when simple univariate models can be found for the autocorrelated daily demand series. Numerical examples are presented to illustrate the use of these formulas. The numerical results also indicate that significant error may be incurred when inventory decisions are made without proper consideration of the autocorrelations or the arbitrary distribution shapes of daily demands.     

VMI条件下具有复合二项随机需求的销售商库存策略研究

考虑一个典型的单一产品的二级供应链系统:单供应商对单销售商,假定系统中销售商的需求分布为复合二项分布,未满足的需求机会损失;补货间隔时间为一随机变量.本文采用概率方法对销售商的需求分布、期望缺货、期望库存周期及库存的稳定性分布进行研究的基础上,构建了使单位时间内销售商的期望库存成本费用最小的库存模型,由此模型便可确定VMI模式下供应商对销售商的库存补货参数s和S,并且给出了在补货响应时间为泊松分布的情况下模型的求解算法,还给出了及时补货响应情况下的5个算例.为补货策略的实施提供了一种简单易于控制的思路和方法.