Inventory control of spares in a transport corporation — A case study

This paper reports on a study carried out in a transport organisation to find the optimal inventory of recoverable spares in a two-level inventory system consisting of a central depot and a set of 20 depots all over the territory of the organisation, with repair facilities. The problems of the present system are diagnosed and it is observed that the upper echelon's inability to meet the lower echelon's demand has resulted in an ever-increasing shortage. A mathematical model is developed to suit the system structure and a set of optimal inventory policies for recoverable spares at both the echelons is obtained for the case studied.     

Supply Performance in Multi‐Echelon Inventory Systems with Intermediate Product Demand: A Perspective on Allocation

In this article, we study the performance of multi‐echelon inventory systems with intermediate, external product demand in one or more upper echelons. This type of problem is of general interest in inventory theory and of particular importance in supply chain systems with both end‐product demand and spare parts (subassemblies) demand. The multi‐echelon inventory system considered here is a combination of assembly and serial stages with direct demand from more than one node. The aspect of multiple sources of demands leads to interesting inventory allocation problems. The demand and capacity at each node are considered stochastic in nature. A fixed supply and manufacturing lead time is used between the stages. We develop mathematical models for these multi‐echelon systems, which describe the inventory dynamics and allow simulation of the system. A simulation‐based inventory optimization approach is developed to search for the best base‐stock levels for these systems. The gradient estimation technique of perturbation analysis is used to derive sample‐path estimators. We consider four allocation schemes: lexicographic with priority to intermediate demand, lexiographic with priority to downstream demand, predetermined proportional allocation, and proportional allocation. Based on the numerical results we find that no single allocation policy is appropriate under all conditions. Depending on the combinations of variability and utilization we identify conditions under which use of certain allocation polices across the supply chain result in lower costs. Further, we determine how selection of an inappropriate allocation policy in the presence of scarce on‐hand inventory could result in downstream nodes facing acute shortages. Consequently we provide insight on why good allocation policies work well under differing sets of operating conditions.     

A hybrid inventory management system responding to regular demand and surge demand

This paper proposes a hybrid policy for a stochastic inventory system facing regular demand and surge demand. The combination of two different demand patterns can be observed in many areas, such as healthcare inventory and humanitarian supply chain management. The surge demand has a lower arrival rate but higher demand volume per arrival. The solution approach proposed in this paper incorporates the level crossing method and mixed integer programming technique to optimize the hybrid inventory policy with both regular orders and emergency orders. The level crossing method is applied to obtain the equilibrium distributions of inventory levels under a given policy. The model is further transformed into a mixed integer program to identify an optimal hybrid policy. A sensitivity analysis is conducted to investigate the impact of parameters on the optimal inventory policy and minimum cost. Numerical results clearly show the benefit of using the proposed hybrid inventory model. The model and solution approach could help healthcare providers or humanitarian logistics providers in managing their emergency supplies in responding to surge demands.     

Inventory Control with a Fixed Cost and a Piecewise Linear Convex Cost

This paper studies the optimal policy for a periodic‐review inventory system in which the production costs consist of a fixed cost and a piecewise linear convex variable cost. Such a cost function can arise from alternate sources of supply or from the use of overtime production. We fully characterize the structure of the optimal policy for the single‐period problem. For the multi‐period problem, the optimal policy can have disconnected production regions and complicated optimal produce‐up‐to levels, which implies that implementation of the optimal policy may not be practical. Fortunately, careful investigation shows that the optimal policy has some interesting properties. The structure of the optimal policy outlined by these properties leads to a practical and close‐to‐optimal heuristic policy. In an extensive numerical study, the average gap is only 0.02% and the worst gap is 1.37%.     

Dynamic Pricing and Inventory Management with Dual Suppliers of Different Lead Times and Disruption Risks

It is common for a firm to make use of multiple suppliers of different delivery lead times, reliabilities, and costs. In this study, we are concerned with the joint pricing and inventory control problem for such a firm that has a quick‐response supplier and a regular supplier that both suffer random disruptions, and faces price‐sensitive random demands. We aim at characterizing the optimal ordering and pricing policies in each period over a planning horizon, and analyzing the impacts of supply source diversification. We show that, when both suppliers are unreliable, the optimal inventory policy in each period is a reorder point policy and the optimal price is decreasing in the starting inventory level in that period. In addition, we show that having supply source diversification or higher supplier reliability increases the firm's optimal profit and lowers the optimal selling price. We also demonstrate that, with the selling price as a decision, a supplier may receive even more orders from the firm after an additional supplier is introduced. For the special case where the quick‐response supplier is perfectly reliable, we further show that the optimal inventory policy is of a base‐stock type and the optimal pricing policy is a list‐price policy with markdowns.     

Inventory Management for Customers with Alternative Lead Times

It is common for suppliers operating in batch‐production mode to deal with patient and impatient customers. This paper considers inventory models in which a supplier provides alternative lead times to its customers: a short or a long lead time. Orders from patient customers can be taken by the supplier and included in the next production cycle, while orders from impatient customers have to be satisfied from the on‐hand inventory. We denote the action to commit one unit of on‐hand inventory to patient or impatient customers as the inventory‐commitment decision, and the initial inventory stocking as the inventory‐replenishment decision. We first characterize the optimal inventory‐commitment policy as a threshold type, and then prove that the optimal inventory‐replenishment policy is a base‐stock type. Then, we extend our analysis to models to consider cases of a multi‐cycle setting, a supply‐capacity constraint, and the on‐line charged inventory‐holding cost. We also evaluate and compare the performances of the optimal inventory‐commitment policy and the inventory‐rationing policy. Finally, to further investigate the benefits and pitfalls of introducing an alternative lead‐time choice, we use the customer‐choice model to study the demand gains and losses, known as demand‐induction and demand‐cannibalization effects, respectively.     

THE VALUE OF SKU RATIONALIZATION IN PRACTICE (THE POOLING EFFECT UNDER SUBOPTIMAL INVENTORY POLICIES AND NONNORMAL DEMAND)

Several approaches to the widely recognized challenge of managing product variety rely on the pooling effect. Pooling can be accomplished through the reduction of the number of products or stock‐keeping units (SKUs), through postponement of differentiation, or in other ways. These approaches are well known and becoming widely applied in practice. However, theoretical analyses of the pooling effect assume an optimal inventory policy before pooling and after pooling, and, in most cases, that demand is normally distributed. In this article, we address the effect of nonoptimal inventory policies and the effect of nonnormally distributed demand on the value of pooling. First, we show that there is always a range of current inventory levels within which pooling is better and beyond which optimizing inventory policy is better. We also find that the value of pooling may be negative when the inventory policy in use is suboptimal. Second, we use extensive Monte Carlo simulation to examine the value of pooling for nonnormal demand distributions. We find that the value of pooling varies relatively little across the distributions we used, but that it varies considerably with the concentration of uncertainty. We also find that the ranges within which pooling is preferred over optimizing inventory policy generally are quite wide but vary considerably across distributions. Together, this indicates that the value of pooling under an optimal inventory policy is robust across distributions, but that its sensitivity to suboptimal policies is not. Third, we use a set of real (and highly erratic) demand data to analyze the benefits of pooling under optimal and suboptimal policies and nonnormal demand with a high number of SKUs. With our specific but highly nonnormal demand data, we find that pooling is beneficial and robust to suboptimal policies. Altogether, this study provides deeper theoretical, numerical, and empirical understanding of the value of pooling.     

带学习效应的双渠道供应链库存策略研究

针对具有学习行为的双渠道供应链问题,本文研究了两种分销渠道并存下的最优库存策略。有限计划期内,分销商通过传统销售和在线销售来满足下游顾客的需求。两种分销渠道下的销售单价为时变不减线性函数,当系统中各周期的生产订购固定成本以一定的概率具有学习效应行为时,分别建立了非变质产品生产存贮问题的混合整数约束优化模型以及易变质产品存贮问题的无约束混合整数优化模型,所建立模型的目标为极大化分销商总利润函数。对于这两类模型,通过分析其最优解的性质,利用将生产订购次数松弛为连续变量的技巧证明了最优解存在的唯一性。给出了最优策略的求解方法并比较了两类模型最优利润函数值的大小。最后通过数值算例对上述模型进行了验证,数值结果表明当供应链系统中存在学习效应行为时,该系统能够获得更多的利润。     

具有多元马氏需求的多产品多阶段库存优化模型

本文研究一类新的多产品库存控制策略,即具有多元马氏需求特征的多产品多阶段的订货点订货量(Q, R, SS)策略,该策略考虑市场需求在不同产品之间具有多元马氏转移特征,并考虑缺货因素设置安全库存。论文首先建立了多产品多阶段的多元马氏需求预测模型,并通过该模型确定了各种产品需求之间的关系。同时,在该模型的理论基础上,提出了多产品多阶段的总期望成本模型及其最优(Q, R, SS)策略,进而结合算例给出模型的最优策略的数值解。     

An inventory control policy for liquefied natural gas as a transportation fuel

In this paper, we study a novel stochastic inventory management problem that arises in storage and refueling facilities for Liquefied Natural Gas (LNG) as a transportation fuel. In this inventory problem, the physio-chemical properties of LNG play a key role in the design of inventory policies. These properties are: (1) LNG suffers from both quantity decay and quality deterioration and (2) the quality of on-hand LNG can be upgraded by mixing it with higher-quality LNG. Given that LNG quality can be upgraded, an inventory control policy for this problem needs to consider the removal of LNG as a decision variable. We model and solve the problem by means of a Markov Decision Process (MDP) and study the structural characteristics of the optimal policy. The insights obtained in the analysis of the optimal policy are translated into a simple, though effective, inventory control policy in which actions (i.e., replenishment and/or removal) are driven by both the quality and the quantity of the inventories. We assess the performance of our policy by means of a numerical study and show that it performs close to optimal in many numerical instances. The main conclusion of our study is that it is important to take quality into consideration when design inventory control policies for LNG, and that the most effective way to cope with quality issues in an LNG inventory system involves both the removal and the replenishment of inventories.     

Replenishment Policies for Multi‐Product Stochastic Inventory Systems with Correlated Demand and Joint‐Replenishment Costs

This study analyzes optimal replenishment policies that minimize expected discounted cost of multi‐product stochastic inventory systems. The distinguishing feature of the multi‐product inventory system that we analyze is the existence of correlated demand and joint‐replenishment costs across multiple products. Our objective is to understand the structure of the optimal policy and use this structure to construct a heuristic method that can solve problems set in real‐world sizes/dimensions. Using an MDP formulation we first compute the optimal policy. The optimal policy can only be computed for problems with a small number of product types due to the curse of dimensionality. Hence, using the insight gained from the optimal policy, we propose a class of policies that captures the impact of demand correlation on the structure of the optimal policy. We call this class (s, c, d, S)‐policies, and also develop an algorithm to compute good policies in this class, for large multi‐product problems. Finally using an exhaustive set of computational examples we show that policies in this class very closely approximate the optimal policy and can outperform policies analyzed in prior literature which assume independent demand. We have also included examples that illustrate performance under the average cost objective.     

面向批量订货和多需求类的生产与库存配给

针对一个面向两个需求类的生产企业,根据客户每次订货是否可分批交货,提出了当客户订货可分割和不可分割时供应商的最优生产和库存配给策略.分析表明,供应商的最优生产控制策略可用一个取决于系统状态的基准库存水平表示,最优的库存配给策略则用一个多层的取决于状态的配给水平向量表示.随后,该结论被推广至包含任意多个需求类的生产系统.数值分析验证了文中最优策略的有效性.     

VMI环境下库存竞争性产品的补货策略及最优货架空间的确定

本文在文献[1]的研究基础上,通过深入剖析FKO模型,运用动态规划方法,探讨了呈现特殊需求性质的库存竞争性产品在VMI环境下的多期动态补货策略,以及传统库存管理中不加考虑的库存空间优化问题。本文一方面是对FKO模型的补充与扩展,同时也证实了对于在特定的VMI补货环境下采用多期库存盘查模式的供应商来说,封顶式的补货策略是最优的,并且其关于最优补货水平的多期决策都是无远见的。因此收益分享合同下的单期最优库存水平可以作为设定最优货架展示空间的依据。     

Dynamic Pricing and Inventory Management with Regular and Expedited Supplies

We consider a periodic‐review inventory system with regular and expedited supply modes. The expedited supply is faster than the regular supply but incurs a higher cost. Demand for the product in each period is random and sensitive to its selling price. The firm determines its order quantity from each supply in each period as well as its selling price to maximize the expected total discounted profit over a finite or an infinite planning horizon. We show that, in each period if it is optimal to order from both supplies, the optimal inventory policy is determined by two state‐independent thresholds, one for each supply mode, and a list price is set for the product; if only the regular supply is used, the optimal policy is a state‐dependent base‐stock policy, that is, the optimal base‐stock level depends on the starting inventory level, and the optimal selling price is a markdown price that decreases with the starting inventory level. We further study the operational impact of such supply diversification and show that it increases the firm's expected profit, reduces the optimal safety‐stock levels, and lowers the optimal selling price. Thus that diversification is beneficial to both the firm and its customers. Building upon these results, we conduct a numerical study to assess and compare the respective benefit of dynamic pricing and supply diversification.     

A Bayesian Inventory Model Using Real‐Time Condition Monitoring Information

Lack of coordination between machinery fault diagnosis and inventory management for spare parts can lead to increased inventory costs and disruptions in production activity. We develop a framework for incorporating real‐time condition monitoring information into inventory decisions for spare parts. We consider a manufacturer who periodically replenishes inventory for a machine part that is subject to deterioration. The deterioration is captured via condition monitoring and modeled using a Wiener process. The resulting degradation model is used to derive the life distribution of a functioning part and to estimate the demand distribution for spare parts. This estimation is periodically updated, in a Bayesian manner, as additional information on part deterioration is obtained. We develop an inventory model that incorporates this updated demand distribution and demonstrate that a dynamic base‐stock policy, in which the optimal base‐stock level is a function of some subset of the observed condition monitoring information, is optimal. We propose a myopic critical fractile policy that captures the essence of the optimal policy, but is easier to compute. Computational experiments indicate that this heuristic performs quite well relative to the optimal policy. Adaptive inventory policies such as these can help manufacturers to increase machine availability and reduce inventory costs.     

冲击型负荷下的生产存储模型研究

建立了考虑冲击需求、变化生产率的最优生产存储库存策略. 其特点是:把一个生产周 期分为多个生产阶段,各个生产阶段时间可以不同;在每个生产阶段中,允许生产率小于需求 率,允许提前生产;而且,最大生产率要小于最大需求率. 研究发现,为满足冲击需求,提前存储 较多而启动机器数量较少,或提前存储较少而启动机器数量较多,不一定能够取得最佳效益, 要根据存储费用和启动费用共同确定. 在有限生产率的条件下,得出一种寻求最优生产- 库存 策略,为此类库存系统的管理决策提供了理论依据.     

Managing Retail Shelf and Backroom Inventories When Demand Depends on the Shelf‐Stock Level

Inventory displayed on the retail sales floor not only performs the classical supply function but also plays a role in affecting consumers’ buying behavior and hence the total demand. Empirical evidence from the retail industry shows that for some types of products, higher levels of on‐shelf inventory have a demand‐increasing effect (“billboard effect”) while for some other types of products, higher levels of on‐shelf inventory have a demand‐decreasing effect (“scarcity effect”). This suggests that retailers may use the amount of shelf stock on display as a tool to influence demand and operate a store backroom to hold the inventory of items not displayed on the shelves, introducing the need for efficient management of the backroom and on‐shelf inventories. The purpose of this study is to address such an issue by considering a periodic‐review inventory system in which demand in each period is stochastic and depends on the amount of inventory displayed on the shelf. We first analyze the problem in a finite‐horizon setting and show under a general demand model that the system inventory is optimally replenished by a base‐stock policy and the shelf stock is controlled by two critical points representing the target levels to raise up/drop down the on‐shelf inventory level. In the infinite‐horizon setting, we find that the optimal policies simplify to stationary base‐stock type policies. Under the billboard effect, we further show that the optimal policy is monotone in the system states. Numerical experiments illustrate the value of smart backroom management strategy and show that significant profit gains can be obtained by jointly managing the backroom and on‐shelf inventories.     

Lateral transshipment with partial request and random switching

A large body of inventory management research has been devoted to lateral transshipment. Most of the existent models assume that the unmet local demand will automatically request transshipment, and that the unmet local demand does not seek inventory at other locations within the same echelon. In contrast, we investigate a two-store retailer’s inventory replenishment and transshipment decisions when those two assumptions do not hold. Specifically, we use a fixed request rate to model partial demand for transshipment at the shortage store and a random switch rate to model the arrival of the unmet demand at the surplus store. We characterize the optimal transshipment and inventory replenishment policies. We find that it is not always in the best interest of the retailer to satisfy as much as possible the transshipment demand. In light of the switched demand flowing to the surplus store, the retailer may benefit from saving the leftover inventory at the surplus store for the switched demand. The optimal transshipment policy follows a double-threshold structure when the prospect of the switched demand is not large enough; and a transshipment quantity of zero becomes optimal otherwise. Through an extensive numerical analysis, we examine the impact of the request rate and the switch rate, together with other parameters. We also evaluate a few simple-to-use transshipment heuristics, including one that we devise based on the structure of the optimal transshipment policy. The consistent, near-optimal performance of the devised heuristic is a confirmation of the importance of our theoretical work on the optimal policy.     

基于库存分担策略的装配系统准时供货模型研究

本文考虑一个由多供应商和单制造商构成的装配系统。当市场的需求时间无法确定时,制造商通过对供应商设定合理的库存分担策略来降低自身成本,供应商则需要自行决定对制造商的补货时间并承担相应的库存持有成本和延迟惩罚成本。文章同时建立了供应商之间的纳什博弈模型和以制造商为主方的主从博弈模型,以找到供应商的最佳供货时间和制造商最优的库存承担时限。通过对比不同模式下供应链的整体绩效,找到实现供应链协调运作的必要条件,并通过数据分析进一步证明相关结论。     

基于系统柔性的MTS-MTO混合生产决策

研究生产商采用MTS、MTO混合作业的方式为不同客户提供产品和服务的策略。计划利用一组可灵活控制的动态设备处理那些不同需求的MTS和MTO生产业务,为此,我们开发了一个多服务台的排队模型,利用拟生灭过程和相位型分布得到了MTS、MTO排队系统平衡条件和稳态概率矩阵几何解。通过求解分块矩阵方程组,给出了系统队列长度、平均等待队长、顾客服务水平等绩效测度指标。建立了系统运作成本最优化的数学模型,采用搜索算法,确定了关键参数的边界值,找到了混合系统运作的最优策略。数值模拟和系统绩效比较分析结果显示：（1）动态切换策略能够更快速的帮助MTS恢复目标库存量,控制系统缺货风险,降低库存持有成本;（2）找到了满足顾客服务水平的最少的设备配置数量和库存成本最低的生产切换时间,且动态系统的平均队列长度低于静态系统;（3）混合运作策略减少了约2/3的静态系统平均队列长度,企业在队列长度减小的窗口期内可以接受更多订单和缩短MTO订单交货时间。