Optimal control of a production‐inventory system with both backorders and lost sales

We consider the optimal control of a production inventory‐system with a single product and two customer classes where items are produced one unit at a time. Upon arrival, customer orders can be fulfilled from existing inventory, if there is any, backordered, or rejected. The two classes are differentiated by their backorder and lost sales costs. At each decision epoch, we must determine whether or not to produce an item and if so, whether to use this item to increase inventory or to reduce backlog. At each decision epoch, we must also determine whether or not to satisfy demand from a particular class (should one arise), backorder it, or reject it. In doing so, we must balance inventory holding costs against the costs of backordering and lost sales. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. We show that the optimal policy can be described by three state‐dependent thresholds: a production base‐stock level and two order‐admission levels, one for each class. The production base‐stock level determines when production takes place and how to allocate items that are produced. This base‐stock level also determines when orders from the class with the lower shortage costs (Class 2) are backordered and not fulfilled from inventory. The order‐admission levels determine when orders should be rejected. We show that the threshold levels are monotonic (either nonincreasing or nondecreasing) in the backorder level of Class 2. We also characterize analytically the sensitivity of these thresholds to the various cost parameters. Using numerical results, we compare the performance of the optimal policy against several heuristics and show that those that do not allow for the possibility of both backordering and rejecting orders can perform poorly.© 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

Minimizing the false alarm rate in systems with transient abnormality

We consider a stochastic partially observable system that can switch between a normal state and a transient abnormal state before entering a persistent abnormal state. Only the persistent abnormal state requires alarms. The transient and persistent abnormal states may be similar in appearance, which can result in excess false alarms. We propose a partially observable Markov decision process model to minimize the false alarm rate, subject to a given upper bound on the expected alarm delay time. The cost parameter is treated as the Lagrange multiplier, which can be estimated from the bound of the alarm delay. We show that the optimal policy has a control‐limit structure on the probability of persistent abnormality, and derive closed‐form bounds for the control limit and present an algorithm to specify the Lagrange multiplier. We also study a specialized model where the transient and persistent abnormal states have the same observation distribution, in which case an intuitive “watchful‐waiting” policy is optimal. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 320–334, 2016     

A novel gradient approach for optimal design and sensitivity analysis of EWMA control charts

Conventional control charts are often designed to optimize out‐of‐control average run length (ARL), while constraining in‐control ARL to a desired value. The widely employed grid search approach in statistical process control (SPC) is time‐consuming with unsatisfactory accuracy. Although the simulation‐based ARL gradient estimators proposed by Fu and Hu [Manag Sci 45 (1999), 395–413] can alleviate this issue, it still requires a large number of simulation runs to significantly reduce the variance of gradient estimators. This article proposes a novel ARL gradient estimation approach based on integral equation for efficient analysis and design of control charts. Although this article compares with the results of Fu and Hu [Manag Sci 45 (1999), 395–413] based on the exponentially weighted moving average (EWMA) control chart, the proposed approach has wide applicability as it can generally fit into any control chart with Markovian property under any distributions. It is shown that the proposed method is able to provide a fast, accurate, and easy‐to‐implement algorithm for the design and analysis of EWMA charts, as compared to the simulation‐based gradient estimation method. Moreover, the proposed gradient estimation method facilitates the computation of high‐order derivatives that are valuable in sensitivity analysis. The code is written in Matlab, which is available on request. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 223–237, 2014     

基于马尔科夫决策的目标选择策略

目标选择是军事计划的关键要素之一。基于马尔科夫决策方法,解决具有复杂目标间关联的多阶段目标选择问题。使用与或树描述目标体系各层状态间的影响关联,并以目标体系整体失效为求解目的,建立了基于离散时间MDP的多阶段打击目标选择模型。在LRTDP算法基础上提出一种启发式方法,通过判断从当前目标体系状态到达体系失效状态的演化过程中的可能资源消耗和失败概率,来提供对当前状态的评估值,该方法能有效排除问题搜索空间中不能到达体系失效目的的中间状态,压缩了由于目标间复杂关联而增长的巨大状态空间。用实验验证了该方法有效性,实验结果表明,该方法直观实用,对目标间具有复杂关联关系的目标打击决策有一定参考价值。     

Limiting behavior of the stochastic sequential assignment problem

The stochastic sequential assignment problem (SSAP) considers how to allocate available distinct workers to sequentially arriving tasks with stochastic parameters such that the expected total reward obtained from the sequential assignments is maximized. Implementing the optimal assignment policy for the SSAP involves calculating a new set of breakpoints upon the arrival of each task (i.e., for every time period), which is impractical for large‐scale problems. This article studies two problems that are concerned with obtaining stationary policies, which achieve the optimal expected reward per task as the number of tasks approaches infinity. The first problem considers independent and identically distributed (IID) tasks with a known distribution function, whereas in the second problem tasks are derived from r different unobservable distributions governed by an ergodic Markov chain. The convergence rate of the expected reward per task to the optimal value is also obtained for both problems. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

基于BER和CTMC的航天器发射组织过程可靠度模型

在航天器发射工程中,存在组织过程可靠性评估难以量化的问题。基于航天器发射组织过程的时间特性呈现多个子过程并发执行且子过程具有Markov性和齐次性的特点,使用连续时间Markov链(CTMC)建立多吸收态的组织过程可靠度模型,利用互模拟等价关系(BER)简化组织过程的状态空间,并进一步给出各种状态转移率和组织过程可靠度计算方法。最后,通过数据分析说明模型在航天器发射工程进度计划评估中的作用和意义。     

运动模糊图像的运动模糊方向鉴别

曝光瞬间造成图像模糊的运动通常作为直线运动近似处理,若能找出模糊图像的运动模糊方向,并将之旋转到水平轴,则二维问题可简化为一维来处理,大大简化由模糊图像估计出运动模糊点扩散函数以及图像恢复的过程,并为图像恢复的并行计算创造有利条件。由于运动模糊降低了运动方向上图像的高频成分,沿着运动方向实施高通滤波(方向微分),可保证微分图像灰度值(绝对值)之和最小。基于此,本文利用双线性插值的方法,固定并适当选取方向微分的微元大小,构造出3×3方向微分乘子,得到了高效高精度的自动鉴别运动模糊方向的新方法,并通过数值实验进行了验证。     

多层弹道导弹防御体系射击策略研究

主要利用离散马尔可夫链(DTMC)对多层弹道导弹防御系统反导作战射击策略问题进行了研究.定义了多层弹道导弹防御体系防御层识别矩阵、拦截矩阵、综合防御能力矩阵等描述多层弹道导弹防御体系防御层作战能力的相关概念,建立了多层弹道导弹防御体系的DTMC模型.在此基础上,应用多层弹道导弹防御体系的DTMC模型对多层弹道导弹防御体系射击策略对作战效能的影响进行了预测分析,最后应用实例验证了模型的有效性及实用性.研究结果可为制定多层弹道导弹防御体系射击策略问题提供决策依据.     

基于马尔可夫预测的低消耗弹药库存仿真建模分析

应用马尔可夫方法的无后效性原理,建立了低消耗弹药库存的预测模型,为该类弹药的年度供应提供依据。在模型建立的基础上,对其仿真算法进行了描述,并应用计算机进行了仿真实现。     

On parallel machine replacement problems with general replacement cost functions and stochastic deterioration

The parallel machine replacement problem consists of finding a minimum cost replacement policy for a finite population of economically interdependent machines. In this paper, we formulate a stochastic version of the problem and analyze the structure of optimal policies under general classes of replacement cost functions. We prove that for problems with arbitrary cost functions, there can be optimal policies where a machine is replaced only if all machines in worse states are replaced (Worse Cluster Replacement Rule). We then show that, for problems with replacement cost functions exhibiting nonincreasing marginal costs, there are optimal policies such that, in any stage, machines in the same state are either all kept or all replaced (No‐Splitting Rule). We also present an example that shows that economies of scale in replacement costs do not guarantee optimal policies that satisfy the No‐Splitting Rule. These results lead to the fundamental insight that replacement decisions are driven by marginal costs, and not by economies of scale as suggested in the literature. Finally, we describe how the optimal policy structure, i.e., the No‐Splitting and Worse Cluster Replacement Rules, can be used to reduce the computational effort required to obtain optimal replacement policies. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005