Safety Stock Determination with Serially Correlated Demand in a Periodic-Review Inventory System

We consider a periodic-review inventory replenishment model with an order-up-to-R operating doctrine for the case of deterministic lead times and a covariance-stationary stochastic demand process. A method is derived for setting the inventory safety stock to achieve an exact desired stockout probability when the autocovariance function for Gaussian demand is known. Because the method does not require that parametric time-series models be fit to the data, it is easily implemented in practice. Moreover, the method is shown to be asymptotically valid when the autocovariance function of demand is estimated from historical data. The effects on the stockout rate of various levels of autocorrelated demand are demonstrated for situations in which autocorrelation in demand goes undetected or is ignored by the inventory manager. Similarly, the changes to the required level of safety stock are demonstrated for varying levels of autocorrelation.     

Customer differentiated end-of-life inventory problem

This paper deals with the service parts end-of-life inventory problem in a circumstance that demands for service parts are differentiated. Customer differentiation might be due to criticality of the demand or based on various service contracts. In both cases, we model the problem as a finite horizon stochastic dynamic program and characterize the structure of the optimal inventory policy. We show that when customers are differentiated based on the demand criticality then the optimal structure consists of time and state dependent threshold levels for inventory rationing. In case of differentiation based on service contracts, we show that in addition to rationing thresholds we also need contract extension thresholds by which the system decides whether to offer an extension to an expiring contract or not. By numerical experiments in both cases, we identify the value of incorporating such decisions in service parts end-of-life inventory management with customer differentiation. Moreover, we show that these decisions not only result in cost efficiency but also decrease the risk of part obsolescence drastically.     

在中心仓库有损失销售的二级分布库存管理模型的研究

通过对一个中心仓库和N个零售商的二级分布库存系统进行分析,采用基本(S-1,S)库存策略,综合运用了排队法和M ETR IC近似法,提出了一种在中心仓库有损失销售的二级库存管理模型,该模型描述在中心仓库缺货情况下,多数零售商不等待延期付货,而直接与供应商订货,导致中心仓库就会因损失销售而产生机会成本.该模型可达到二级分布库存系统的总成本最小.     

The M/M/1 queue with inventory, lost sale, and general lead times

We consider an M/M/1 queueing system with inventory under the $(r,Q)$ policy and with lost sales, in which demands occur according to a Poisson process and service times are exponentially distributed. All arriving customers during stockout are lost. We derive the stationary distributions of the joint queue length (number of customers in the system) and on-hand inventory when lead times are random variables and can take various distributions. The derived stationary distributions are used to formulate long-run average performance measures and cost functions in some numerical examples.     

服务水平约束下双需求类静态配给策略

考虑一个周期盘点、无限期、缺货回补、双需求类的库存系统,其中高优先级需求的目标服务水平较高。系统采用基准库存策略补充库存,依据静态配给策略分配库存,即优先满足高优先级需求,仅当持有库存水平不低于固定配给阈值时满足低优先级需求。优化目标是在服务水平约束下最小化期望库存持有量。为提升计算效率,引入“预留库存假设”,即允许通过提高低优先级需求缺货水平的方式补充库存,使得期末持有库存水平不低于本期高优先级需求缺货水平与固定配给阈值之和。基于预留库存假设,给出两类需求服务水平和期望库存持有量的解析表达式,证明上述绩效指标关于控制参数的单调性,刻画满足服务水平约束的控制参数可行域,得到原系统最优控制参数的近似求解算法。算例分析表明,基于预留库存假设的绩效衡量方法和参数求解算法准确性好且计算效率高。     

基于系统动力学的供应链库存策略仿真研究

针对一个动态、多级的供应链库存系统,应用系统动力学的方法,建立了供应链(s,S)库存策略下的物流成本模型,并通过动态仿真,分析了库存策略的变动对于供应链库存系统各级成员间库存供需的动态行为,提出了(s,S)策略下的供应链库存系统的有效管理方法.     

具有两类需求服务的随机库存系统控制策略研究

研究具有两类顾客排队需求服务的随机库存系统.系统采取(s,Q)补货策略且当库存水平下降到安全库存s时,到达的第二类顾客以概率P得到服务.首先,建立库存水平状态转移方程并通过递推算法求解获得库存水平稳态概率分布和系统稳态指标;接下来,构建库存成本函数;最后,采用数值试验的方法研究该库存系统的最优控制策略并考察系统参数的敏感性.     

Taming the bullwhip effect whilst watching customer service in a single supply chain echelon

We study a generalised order-up-to policy that has highly desirable properties in terms of order and inventory variance and customer service levels it generates. We quantify exactly the variance amplification in replenishment orders, i.e. the bullwhip effect, and the variance of inventory levels over time, for i.i.d. and the weakly stationary auto regressive (AR), moving average (MA) and auto regressive moving average (ARMA) demand processes. We demonstrate that high customer service as measured by fill-rate, and smooth replenishments need not increase inventory cost substantially. We observe that in some instances of the ARMA demand pattern this comes at the expense of a relatively small increase in safety stock, whilst in other instances inventory levels can actually be reduced.     

Managing inventory and service levels in a safety stock‐based inventory routing system with stochastic retailer demands

The inherent uncertainty in supply chain systems compels managers to be more perceptive to the stochastic nature of the systems' major parameters, such as suppliers' reliability, retailers' demands, and facility production capacities. To deal with the uncertainty inherent to the parameters of the stochastic supply chain optimization problems and to determine optimal or close to optimal policies, many approximate deterministic equivalent models are proposed. In this paper, we consider the stochastic periodic inventory routing problem modeled as chance‐constrained optimization problem. We then propose a safety stock‐based deterministic optimization model to determine near‐optimal solutions to this chance‐constrained optimization problem. We investigate the issue of adequately setting safety stocks at the supplier's warehouse and at the retailers so that the promised service levels to the retailers are guaranteed, while distribution costs as well as inventory throughout the system are optimized. The proposed deterministic models strive to optimize the safety stock levels in line with the planned service levels at the retailers. Different safety stock models are investigated and analyzed, and the results are illustrated on two comprehensively worked out cases. We conclude this analysis with some insights on how safety stocks are to be determined, allocated, and coordinated in stochastic periodic inventory routing problem. Copyright © 2017 John Wiley & Sons, Ltd.     

Multi-item inventory control with full truckloads: A comparison of aggregate and individual order triggering

In this paper, we consider the stochastic joint replenishment problem in an environment where transportation costs are dominant and full truckloads or full container loads are required. One replenishment policy, taking into account capacity restrictions of the total order volume, is the so-called QS policy, where replenishment orders are placed to raise the individual inventory positions of all items to their order-up-to levels, whenever the aggregate inventory position drops below the reorder level. We first provide a method to compute the policy parameters of a QS policy such that item target service levels can be met, under the assumption that demand can be modeled as a compound renewal process. The approximation formulas are based on renewal theory and are tested in a simulation study which reveals good performance. Second, we compare the QS policy with a simple allocation policy where replenishment orders are triggered by the individual inventory positions of the items. At the moment when an individual inventory position drops below its item reorder level, a replenishment order is triggered and the total vehicle capacity is allocated to all items such that the expected elapsed time before the next replenishment order is maximized. In an extensive simulation study it is illustrated that the QS policy outperforms this allocation policy since it results in lower inventory levels for the same service level. Although both policies lead to similar performance if items are identical, it can differ substantially if the item characteristics vary.     

An inventory sharing and allocation method for a multi-location service parts logistics network with time-based service levels

We consider a model to allocate stock levels at warehouses in a service parts logistics network. The network is a two-echelon distribution system with one central warehouse with infinite capacity and a number of local warehouses, each facing Poisson demands from geographically dispersed customers. Each local warehouse uses a potentially different base stock policy. The warehouses are collectively required to satisfy time-based service targets: Certain percentages of overall demand need to be satisfied from facilities within specified time windows. These service levels not only depend on the distance between customers and the warehouses, but also depend on the part availabilities at the warehouses. Moreover, the warehouses share their inventory as a way to increase achieved service levels, i.e., when a local warehouse is out of stock, demand is satisfied with an emergency shipment from another close-by warehouse. Observing that the problem of finding minimum-cost stock levels is an integer non-linear program, we develop an implicit enumeration-based method which adapts an existing inventory sharing model from the literature, prioritizes the warehouses for emergency shipments, and makes use of a lower bound. The results show that the proposed inventory sharing strategy results in considerable cost reduction when compared to the no-sharing case and the method is quite efficient for the considered test problems.     

Evaluating shortfall distributions in periodic inventory systems with stochastic endogenous demands and lead-times

This paper addresses a multi-period production/inventory problem with two suppliers, where demand sizes and supplier lead time are stochastic and correlated. A discrete time, single item inventory system is considered, where inventory levels are reviewed periodically and managed using a base-stock policy. At the end of each period, a replenishment order is placed, which enters a queue at the buffer stage and is consequently forwarded to the first available supplier. We present a mathematical model of this inventory system and determine optimal safety stock levels for it, in closed form, using matrix analytic techniques and the properties of phase type distributions. To account for the effect of order crossovers, which occur whenever replenishment orders do not arrive in the sequence in which they were placed, the inventory shortfall distribution is analyzed. Finally, a set of numerical experiments with a system with two suppliers is presented, where the proposed model is compared to other existing models.     

Optimal myopic policy for a stochastic inventory problem with fixed and proportional backorder costs

In this paper, we consider a single product, periodic review, stochastic demand inventory problem where backorders are allowed and penalized via fixed and proportional backorder costs simultaneously. Fixed backorder cost associates a one-shot penalty with stockout situations whereas proportional backorder cost corresponds to a penalty for each demanded but yet waiting to be satisfied item. We discuss the optimality of a myopic base-stock policy for the infinite horizon case. Critical number of the infinite horizon myopic policy, i.e., the base-stock level, is denoted by S. If the initial inventory is below S then the optimal policy is myopic in general, i.e., regardless of the values of model parameters and demand density. Otherwise, the sufficient condition for a myopic optimum requires some restrictions on demand density or parameter values. However, this sufficient condition is not very restrictive, in the sense that it holds immediately for Erlang demand density family. We also show that the value of S can be computed easily for the case of Erlang demand. This special case is important since most real-life demand densities with coefficient of variation not exceeding unity can well be represented by an Erlang density. Thus, the myopic policy may be considered as an approximate solution, if the exact policy is intractable. Finally, we comment on a generalization of this study for the case of phase-type demands, and identify some related research problems which utilize the results presented here.     

Inventory models with minimal service level constraints

This paper investigates inventory models in which the stockout cost is replaced by a minimal service level constraint (SLC) that requires a certain level of service to be met in every period. The minimal service level approach has the virtue of simplifying the computation of an optimal ordering policy, because the optimal reorder level is solely determined by the minimal SLC and demand distributions. It is found that above a certain “critical” service level, the optimal (s,S) policy “collapses” to a simple base-stock or order-up-to level policy, which is independent on the cost parameters. This shows the minimal SLC models to be qualitatively different from their shortage cost counterparts. We also demonstrate that the “imputed shortage cost” transforming a minimal SLC model to a shortage cost model does not generally exist. The minimal SLC approach is extended to models with negligible set-up costs. The optimality of myopic base-stock policies is established under mild conditions.     

(<Emphasis Type="Italic">Q,r</Emphasis>) Inventory Model with a Mixture of Lost Sales and Time-Weighted Backorders

This paper presents a stochastic inventory model for situations in which, during a stockout period, a fraction β of the demand is backordered and the remaining fraction 1 – β is lost. The model is suggested by the customers' different reactions to a stockout condition: during the stockout period, some patient customers wait until their demand is satisfied, while other impatient or urgent customers cannot wait and have to fill their demand from another source. The cost of a backorder is assumed to be proportional to the length of time for which the backorder exists, and a fixed penalty cost is incurred per unit of lost demand. Based on a heuristic treatment of a lot-size reorder-point policy, a mathematical model representing the average annual cost of operating the inventory system is developed. The optimal operating policy variables minimizing the average annual cost can be calculated iteratively. At the extremes β = 1 and β = 0, the model presented reduces to the usual backorders and lost sales case, respectively.     

Optimal inventory replenishment policy for a queueing system with finite waiting room capacity

We treat an inventory control problem in a facility that provides a single type of service for customers. Items used in service are supplied by an outside supplier. To incorporate lost sales due to service delay into the inventory control, we model a queueing system with finite waiting room and non-instantaneous replenishment process and examine the impact of finite buffer on replenishment policies. Employing a Markov decision process theory, we characterize the optimal replenishment policy as a monotonic threshold function of reorder point under the discounted cost criterion. We present a simple procedure that jointly finds optimal buffer size and order quantity.     

Age-based vs. stock level control policies for a perishable inventory system

In this study, we investigate the impact of modified lotsize-reorder control policy for perishables which bases replenishment decisions on both the inventory level and the remaining lifetimes of items in stock. We derive the expressions for the key operating characteristics of a lost sales perishable inventory model, operating under the proposed age-based policy, and examine the sensitivity of the optimal policy parameters with respect to various system parameters. We compare the performance of the suggested policy to that of the classical (Q,r) type policy through a numerical study over a wide range of system parameters. Our findings indicate that the age-based policy is superior to the stock level policy for slow moving perishable inventory systems with high service levels.     

Optimal models for a multi-stage supply chain system controlled by kanban under just-in-time philosophy

This research studies a multi-stage supply chain system that operates under a JIT (just-in-time) delivery policy. Kanbans play an important role in the information and material flows in a supply chain system. Thus, a kanban mechanism is employed to assist in linking different production processes in a supply chain system to implement the scope of JIT philosophy. For a multi-stage supply chain system, a mixed-integer nonlinear programming (MINLP) problem is formulated from the perspective of JIT delivery policy where a kanban may reflect to a transporter such as a truck or a fork-lifter. The number of kanbans, the batch size, the number of batches and the total quantity over one period are determined optimally. It is solved optimally by branch and bound method. A greedy heuristic to avoid the large computational time in branch-and-bound algorithm is developed for solving a large MINLP. Coupled with plant-wide efforts for cost control and management commitment, a logistic system for controlling the production as well as the supply chain is built, which results in minimizing the total cost of the supply chain system. The results show that the improvements in reduction of inventory, wasted labor and customer service in a supply chain are significantly accomplished through the kanban mechanism.     

An optimal replenishment policy for an EOQ model with partial backlogging

We study an inventory system where demand on the stockout period is partially backlogged. The backlogged demand ratio is a mixture of two exponential functions. The shortage cost has two significant costs: the unit backorder cost (which includes a fixed cost and a cost proportional to the length of time for which the backorder exists) and the cost of lost sales. A general procedure to determine the optimal policy and the minimum inventory cost for all the parameter values is developed. This model generalizes several inventory systems analyzed by different authors. Numerical examples are used to illustrate the theoretical results.     

Concepts for safety stock determination under stochastic demand and different types of random production yield

We consider a manufacturer’s stochastic production/inventory problem under periodic review and present methods for safety stock determination to cope with uncertainties that are caused by stochastic demand and different types of yield randomness. Following well-proven inventory control concepts for this problem type, we focus on a critical stock policy with a linear order release rule. A central parameter of this type of policy is given by the safety stock value. When non-zero manufacturing lead times are taken into account in the random yield context, it turns out that safety stocks have to be determined that vary from period to period. We present a simple approach for calculating these dynamic safety stocks for different yield models. Additionally, we suggest approaches for determining appropriate static safety stocks that are easier to apply in practice. In a simulation study we investigate the performance of the proposed safety stock variants.