A fixed interval ordering policy for joint optimization of age replacement and spare part provisioning

A new policy is presented for the joint optimization of age replacement and spare provisioning. The policy, referred to as a fixed interval ordering policy, is formulated by combining an age replacement policy with a periodic review ( t₀,q) type inventory policy, where t0 is the order interval and q is the order quantity. It is generally applicable to any operating system with either a single item or a number of identical items. A SLAM based simulation model has been developed to determine the optimal values of the decision variables by minimizing the total cost of replacement and inventory. The behaviour of the policy has been studied for a number of case problems specifically constructed by five-factor second-order rotatory design and the effects of different cost elements and item failure characterisics have been highlighted. The performance of the proposed policy has also been compared with that of the stocking policy which incorporates a continuous review ( s, S) type of inventory policy, where s is the stock reorder level and S is the maximum stock level. Simulation results clearly indicate that the optimal fixed interval ordering policy is less expensive than the optimal stocking policy when the system consists of a large number of operating units.     

Bullwhip reduction for ARMA demand: The proportional order-up-to policy versus the full-state-feedback policy

A ‘proportional’ order-up-to policy reacting to ARMA demand is analyzed using stochastic optimal control theory. This policy is compared with a full-state-feedback order-up-to policy. Necessary conditions for an optimum of a weighted sum of the inventory and the ordering variances for both policies are formulated. Based on this a relatively simple expression for the ‘full-state’ policy is derived. The comparison between the two policies demonstrates that the ‘intuitively’ designed proportional policy does not fulfill the objective of controlling both the inventory and ordering variance for all parameter values of the demand model as well as the full-state-feedback policy. The full-state-feedback policy outperforms the proportional policy in several aspects.     

需求随机依赖库存环境下的订货仿真优化模型

需求的随机性和依赖库存性是库存问题的特点之一,在需求以泊松分布的形式随机依赖库存的条件下讨论了（Q,T）型库存控制问题。为了评估库存控制策略的平均盈利水平,建立了该库存问题的离散事件系统仿真模型,设计了一种基于仿真的种群重叠、遗传操作非重叠的进化算法,用以优化库存控制策略,类似设计了基于仿真的模拟退火和粒子群优化算法进行比较。通过实例分析了不同参数的变化对模型最优解的影响,灵敏度分析表明需求依赖库存效应越明显时,利润水平越高,最优订货策略越倾向于高库存、短周期和现货销售。仿真实例说明了基于仿真的优化算法的可行性、有效性。     

按单装配系统中组件生产和库存分配控制策略研究

针对由两种组件、三类顾客需求组成的按单装配系统, 本文研究了其中的组件生产控制与库存分配问题. 在各类顾客需 求是泊松到达过程, 各种组件加工时间服从指数分布的假设下, 我们运用马尔科夫决策理论建立了无限期折扣总成本模型, 根据Lippman转换得到了相应归一化后的离散最优方程, 在此基础之上分析了生产和库存分配联合最优控制策略的结构性质. 本文证明了最优策略是依赖于系统状态的动态策略. 组件的最优生产策略是动态基库存策略, 其中基库存水平是关于系统中其他组件库存水平的非减函数. 而最优的分配策略是动态的阈值策略, 对于只需一种组件构成的顾客需求, 组件的分配阈值是系统中另一组件库存水平的增函数; 而对于同时需要两种组件组成的顾客需求, 其各组件的分配阈值是另一组件库存水平的减函数. 最后通过数值试验给出了各个参数对联合最优控制策略的影响, 并得到了相应的管理启示.     

A simple and robust batch-ordering inventory policy under incomplete demand knowledge

Generally, the derivation of an inventory policy requires the knowledge of the underlying demand distribution. Unfortunately, in many settings demand is not completely observable in a direct way or inventory records may be inaccurate. A variety of factors, including the potential inaccuracy of inventory records, motivate managers to seek replenishment policies where the inventory is reviewed periodically and a fixed quantity Q is ordered once the inventory level is found to be under a certain point r. To apply such a policy, however, firms must derive the values r and Q without a clear understanding of the demand distribution. We develop estimators of the first two moments of the (periodic) demand by means of renewal theoretical concepts and a regression-based method, and use these estimators in conjunction with the Power Approximation (PA) method of Ehrhardt and Mosier (1984) to obtain an (r, Q) replenishment policy. The proposed methodology is robust and easy to code into a spreadsheet application. A series of numerical studies are carried out to evaluate the accuracy and precision of the estimators, and to investigate the impact of the estimation on the optimality of the inventory policies. Our experiments indicate that the proposed (r, Q) policy is very close, with regard to the expected total cost per period, to the (s, S) policy obtained via the PA method when the demand process is fully observable and inventory records are accurate.     

Issuing for perishable inventory management with a minimum inventory volume constraint

We study a perishable inventory system that requires to maintain a minimum inventory volume at all times, where the minimum amount in the constraint is significant with respect to regular market demand and the traditional Economic Manufacturing Quantity (EMQ) models do not suffice. The problem is motivated by applications in homeland security, which are related to the management of pharmaceuticals in the Strategic National Stockpile (SNS) for emergency preparedness. We use a modified EMQ model to represent the system and consider the issuing policy given a fixed ordering quantity, as well as the joint ordering and issuing policy problem. We aim to maximize the profit of the system under a linearly-decreasing price structure assumption. We first present the optimal issuing policy with a given ordering quantity for this modified EMQ model. Then we demonstrate that maximizing the revenue of the ordering policy with FIFO (first-in, first-out) and LIFO (last-in, first-out) issuing policies can be formulated as a non-convex non-smooth unconstrained optimization problem. The properties of both problems (e.g. optimizing issuing policy with a fixed ordering quantity and optimizing ordering policy with a fixed issuing policy) is analyzed and an efficient exact algorithm is presented to solve the joint ordering and issuing problem. We show that the LIFO issuing policy is optimal with a linearly-decreasing price structure that deteriorates with the age of the item.     

需求和回收品均随机的制造-再制造系统生产控制

需求及回收品数量和时间的不确定性, 导致制造-再制造系统的库存管理非常困难. 为控制库存尽可能位于某一合理区间内, 在假设库存水平变化由无负跳跃Lévy 过程描述条件下, 利用更新过程和鞅理论, 构建了系统期望折扣总费用模型, 并采用交叉熵法确定最优的生产速率和调整阈值. 最后, 通过仿真实验分析了回收品、需求和系统参数对最优控制策略和期望折扣费用的影响.     

Joint management of finished goods inventory and demand process for a make-to-stock product: a computational approach

In this paper, we consider the joint management of finished goods inventory and demand for a product in a make-to-stock production system. The production process is random with controllable mean rate, and the demand process is stochastic with changeable mean rate dependent on the sale price being high or low. The management issue is how to dynamically adjust the production rate and the sale price to maximize the long run total discounted profit. We show that: 1) the optimal management of the finished goods inventory follows a base stock policy: when the inventory is above certain base stock level, the production is halted; otherwise the maximum production rate is deployed to raise the inventory to the base stock level; and 2) the optimal management of the demand process follows a price switch threshold policy: when the inventory is above the threshold, the low sale price is chosen to sell the product; and below it the high price is chosen to reduce the demand. We provide an algorithm to compute the base stock level and price switch threshold. Extension to multiple price choices is given with proofs highlighted.     

A belief-rule-based inventory control method under nonstationary and uncertain demand

This paper is devoted to investigating inventory control problems under nonstationary and uncertain demand. A belief-rule-based inventory control (BRB-IC) method is developed, which can be applied in situations where demand and demand-forecast-error (DFE) do not follow certain stochastic distribution and forecasting demand is given in single-point or interval styles. The method can assist decision-making through a belief-rule structure that can be constructed, initialized and adjusted using both manager’s knowledge and operational data. An extended optimal base stock (EOBS) policy is proved for initializing the belief-rule-base (BRB), and a BRB-IC inference approach with interval inputs is proposed. A numerical example and a case study are examined to demonstrate potential applications of the BRB-IC method. These studies show that the belief-rule-based expert system is flexible and valid for inventory control. The case study also shows that the BRB-IC method can compensate DFE by training BRB using historical demand data for generating reliable ordering policy.     

Supplier selection and procurement decisions with uncertain demand,fixed selection costs and quantity discounts

In this paper, we study the supplier selection and procurement decision problem with uncertain demand, quantity discounts and fixed selection costs. In addition, a holding cost is incurred for the excess inventory if the buyer orders more than the realized demand and the shortage must be satisfied by an emergent purchase at a higher price otherwise. The objective is to select the suppliers and to allocate the ordering quantity among them to minimize the total cost (including selecting, procurement, holding and shortage costs, etc.). The problem is modeled as a Mixed Integer Programming (MIP) and is shown to be NP-hard. Some properties of the optimal policy are provided and an optimal algorithm is proposed based on the generalized Bender's decomposition. Numerical experiments are conducted to show the efficiency of the algorithm and to obtain some managerial insights.     

A Bayesian approach to a dynamic inventory model under an unknown demand distribution

In this paper, the Bayesian approach to demand estimation is outlined for the cases of stationary as well as non-stationary demand. The optimal policy is derived for an inventory model that allows stock disposal, and is shown to be the solution of a dynamic programming backward recursion. Then, a method is given to search for the optimal order level around the myopic order level. Finally, a numerical study is performed to make a profit comparison between the Bayesian and non-Bayesian approaches, when the demand follows a stationary lognormal distribution. A profit comparison is also made between the stationary and non-stationary Bayesian approaches to observe whether the Bayesian approach incorporates non-stationarity in the demand. And, it is observed whether stock disposal reduces the losses due to ignoring non-stationarity in the demand.Scope and purposeIn the context of inventory models, one of the crucial factors to determine an optimal inventory policy, is the accurate forecasting or estimation of the demand for items in the inventory. The assumption of a constant demand is seriously questioned in recent times, since in reality the demand is generally uncertain and may even vary with time. For instance, the demand for new products, spare parts, or style goods, is likely to fluctuate widely, the average demand is quite likely to be low, and may exhibit a trend. In such situations, the Bayesian approach is a very useful tool for demand estimation, which is applicable even when past observations are scarce. In this paper, we use this approach to estimate the demand for an item, and obtain the expressions for finding the optimal inventory policies. We give a simpler method to find the optimal inventory policy, since the procedure to obtain the optimal inventory policy in the Bayesian framework, is quite tedious especially for long planning horizons, and in cases where the future demand becomes unpredictable. To widen the application of the method, we have given a general procedure which is not restricted to any particular probability distribution for the demand. We compare the Bayesian approach with the corresponding non-Bayesian approach, in terms of the optimum expected profits, when the demand follows a lognormal distribution. We also investigate how well the Bayesian approach incorporates non-stationarity in the demand.     

Combining an alternative multi-delivery policy into economic production lot size problem with partial rework

This paper combines an alternative multi-delivery policy into an imperfect economic production quantity (EPQ) model with partial rework, with the purpose of reducing supplier’s stock holding cost. We extend the problem examined by Chiu et al. [Chiu, Y.-S. P, Chiu, S. W., Li, C.-Y., & Ting, C.-K. (2009). Incorporating multi-delivery policy and quality assurance into economic production lot size problem. Journal of Scientific and Industrial Research, 68(6), 505-512] by considering an n + 1 delivery policy in lieu of n multi-delivery plan for the specific EPQ model with partial rework. Under such a policy, an initial delivery of perfect (finished) items is distributed to customer for satisfying product demand during manufacturer’s regular production time and rework time. At the end of rework, fixed quantity n installments of the finished products are delivered to customer at a fixed interval of time. As a result, a closed-form optimal replenishment batch size solution to the problem is obtained. A numerical example with analysis and comparison is provided to show practical usage of the proposed model and demonstrate its significant savings in stock holding cost.     

基于混合需求的供应链多级库存协同订货模型

针对供应链多级库存系统存在混合需求的情况,建立了基于混合需求的多级库存协同订货模型。该模型假设在供应链中只有最下游节点面临的需求是独立需求,而其他上游节点面临的需求都是与之相关的相关需求。由于相关需求是一种块状需求,其库存成本构成与独立需求明显不同。因此,通过对多级库存系统的库存成本构成进行重新分析,分别给出了需求确定时不允许缺货和允许缺货的协同订货模型。另外,还通过对安全库存的分析给出了需求不确定时的协同订货模型。最后,给出了模型求解的遗传算法,并进行了实例仿真分析,展示了这种协同订货模型在混合需求的供应链中的实用性。     

An optimal inventory solution for Weibull demand distributions

The problem considered here is an infinite stage inventory decision process in which the penalty cost for failing to have in stock sufficient items to satisfy demand at the next stage includes a fixed administrative cost (red tape) and a cost proportional to the deficiency in stock. It is shown here that when the demand follows a Weibull distribution, optimal constant stock level solution exists. A simple computer program is given to calculate this optimal solution.     

An EOQ model of homogeneous products while demand is salesmen’s initiatives and stock sensitive

The author develops an inventory model to determine the retailer’s optimal order quantity for homogeneous products. It is assumed that the amount of display space is limited and the demand of the products is dependent on the display stock level, where a huge stock of one product has a negative effect on the other product. Also, the replenishment rate depends on the level of stock of the items. The objective of the model is to maximize the profit function, considering the effect of inflation and time value of money, by Pontryagin’s Maximal Principles. The stability analysis of the concerned dynamical system has been carried out analytically as well as numerically.     

On the performance of the base-stock inventory system under a compound Erlang demand distribution

In this paper, we propose a new method for determining the optimal base-stock level in a single echelon inventory system where the demand is a compound Erlang process and the lead-time is constant. The demand inter-arrival follows an Erlang distribution and the demand size follows a Gamma distribution. The stock is controlled according to a continuous review base-stock policy where unfilled demands are backordered. The optimal base-stock level is derived based on a minimization of the total expected inventory cost. A numerical investigation is conducted to analyze the performance of the inventory system with respect to the different system parameters and also to show the outperformance of the approach that is based on the compound Erlang demand assumption as compared to the classical Newsboy approach. This work allows insights to be gained on stock control related issues for both slow and fast moving stock keeping units.     

Production/inventory systems with a stochastic production rate under a continuous review policy

A single production facility is dedicated to producing one product with completed units going directly into inventory. The unit production time is a random variable. The demand for the product is given by a Poisson process and is supplied directly from inventory when available, or is backordered until it is produced by the production facility. Relevant costs are a linear inventory holding cost, a linear backorder cost, and a fixed setup cost for initiating a production run. The objective is to find a control policy that minimizes the expected cost per time unit.The problem may be modeled as an M/G/1 queueing system, for which the optimal decision policy is a two-critical-number policy. Cost expressions are derived as functions of the policy parameters, and based on convexity properties of these cost expressions, an efficient search procedure is proposed for finding the optimal policy. Computational test results demonstrating the efficiency of the search procedure and the behavior of the optimal policy are presented.     

缺货不补的随机存储系统的建模与仿真

基于(s,S)策略,讨论了缺货不通过到达的订货进行补充的随机存储系统.假定需求量、需求发生的间隔、以及提前时间都为随机变量,建立了系统的数学模型.给出库存总费用和缺货率的计算公式,将二者加权平均得到策略的综合系数.设计了系统仿真运行框图,在各种费用参数给定后,利用计算机仿真和综合系数来选定最优存储策略.     

A perishable inventory system with modified (S−1,S) policy and arbitrary processing times

In this article we analyze a lost sales (S−1,S) perishable system, under Poisson demands and exponential lifetimes, in which the reorders are placed at every demand epoch so as to take the inventory position back to its maximum level S. The items are replenished one at a time and the resupply time has arbitrary distribution. The various operating characteristics are obtained using Markov renewal techniques. A matrix recursive scheme is developed to determine the stationary distribution of the underlying Markov chain. The efficiency of this procedure in the determination of optimal S that minimizes the long run expected cost rate is discussed. Sensitivity analysis of various system parameters is also carried out.Scope and purposeThe analysis of perishable inventory systems, which has potential applications in various sectors of industry, is far more difficult than their infinite lifetime counterparts. In the study of perishable systems under traditional (S−1,S) policy, reorders for items are placed at demand as well as failure epochs. However, it will be more meaningful not to place orders at failure epochs, which is also practical and economical since it avoids continuous monitoring of items. This paper deals with a continuous review perishable system under a more realistic base stock policy, with variable ordering quantity and arbitrary unit resupply times. The operating characteristics of this complex model are derived using the techniques of semi-regenerative processes. A matrix recursive scheme developed to determine the stationary distribution has enabled us to code an efficient numerical program for cost optimization.     

Determination Of Outdate And Shortage Quantities In The Inventory Problem With Fixed Lifetime

The fixed lifetime perishable inventory system is examined and an estimator for the probability that an item will be sold in a period is obtained. The estimator for the probability that an item will be sold is given by \vartheta / S where \vartheta is the average demand per period and S is the inventory level at the start of a period. The probability is used to derive outdate and shortage quantities. Furthermore, shortage-outdate operating curve is obtained. The probability of running out of stock is obtained from the demand distribution. The operating characteristics obtained in this paper are very important because, for practical problems, available mathematically optimal solutions to the fixed lifetime perishable inventory problem cannot be realized due to their computational complexity. Guidelines for managing a real fixed lifetime perishable inventory system are given. A computer program has also been developed to compute the operating characteristics.