Optimal impulse control on an unbounded domain with nonlinear cost functions

In this paper we consider the optimal impulse control of a system which evolves randomly in accordance with a homogeneous diffusion process in ℜ¹. Whenever the system is controlled a cost is incurred which has a fixed component and a component which increases with the magnitude of the control applied. In addition to these controlling costs there are holding or carrying costs which are a positive function of the state of the system. Our objective is to minimize the expected discounted value of all costs over an infinite planning horizon. Under general assumptions on the cost functions we show that the value function is a weak solution of a quasi-variational inequality and we deduce from this solution the existence of an optimal impulse policy. The computation of the value function is performed by means of the Finite Element Method on suitable truncated domains, whose convergence is discussed. Mathematics Subject Classification: 49J40, 60G40, 65N30     

Optimal policy for brownian inventory models with general convex inventory cost

We study an inventory system in which products are ordered from outside to meet demands, and the cumulative demand is governed by a Brownian motion. Excessive demand is backlogged. We suppose that the shortage and holding costs associated with the inventory are given by a general convex function. The product ordering from outside incurs a linear ordering cost and a setup fee. There is a constant leadtime when placing an order. The optimal policy is established so as to minimize the discounted cost including the inventory cost and ordering cost.     

求解硬时间窗约束下随机需求库存-路径问题的优化算法

随机需求库存-路径问题(Stochastic Demand Inventory Routing Problem, SDIRP)即考虑随机需求环境下供应链中库存与配送的协调优化问题,是实施供应商管理库存策略过程中的关键所在,也是典型的NP难题之一。文章以具有硬时间窗约束的随机需求库存-路径问题(Stochastic Demand Inventory Routing Problem with Hard Time Windows, SDIRPHTW)为研究对象,将SDIRPHTW分解为直接配送的随机库存-路径问题和具有硬时间窗约束的路径优化问题两个子问题,并以最小化系统运行成本和用车数量为目标,设计了一个基于(s,S)库存策略和修正C-W节约法的启发式算法。最后,通过相应的数值算例验证了算法的有效性。     

Portfolio optimization with uncertain exit time in infinite-time horizon

In this paper, we study infinite-period mean-variance formulations for portfolio selections with an uncertain exit time. We employ the convergence control method together with the dynamic programming algorithm to derive analytical expressions for the optimal portfolio policy and the mean-variance efficient frontier under certain conditions. We illustrate these results by an numerical example.     

快速补货模式下的动态定价与库存联合策略

在供应有限的情况下,研究常规补货和快速补货下商品动态定价问题.首先,建立了动态规划模型,理论证明了最优库存策略是基于(s,S)策略下改进的基本库存策略.其次,提出了一种启发式策略求复杂系统的最优策略,启发式算法能够求出最优价格和最优库存水平.最后,数值算例研究表明,库存管理中采用快速补货提高了零售商的利润;初始库存水平越高零售商的利润越高.     

How useful is commonality? Inventory and production decisions to maximize survival probability in start-ups

This paper deals with component commonality in start-up manufacturingfirms. We present a two-product Markov decision model that examinesthe implications of the inventory and production strategiesfor the survival probability of the firm. The advantage of usingcomponent commonality is studied for varying costs, demand correlationsand order replenishment lead times. Optimal policies are derived,and minimum stock levels for survival are obtained. Moreover,we state the conditions under which simplified production decisionscan be made. It is shown that commonality is not only usefulas a way of dealing with demand uncertainty, but that its increaseduse is preferred for strongly substitutable products, and shorterreplenishment lead times.     

复合Poisson 模型带比例与固定交易费用的最优分红与注资

研究了复合Poisson 模型带比例与固定费用的最优分红与注资问题. 每次分红与注资时, 存在比例及固定的交易费用. 通过控制分红与注资的时刻以及分红及注资量,实现破产前分红减注资的折现期望的最大化. 由于存在固定交易费用, 问题为一个脉冲控制问题. 根据问题的参数不同, 问题的解可分为两大类. 一类解为只进行最优分红不需要注资, 而另一类情况需要注资. 需要注资时, 最优注资策略由最优注资上界以及最优注资下界描述. 当赤字小于最优注资下界的绝对值时, 进行注资. 最后, 在理赔为指数分布时明确地给出了两类共七种最优策略以及值函数的形式. 从而彻底地解决了该问题.     

一类跳扩散需求存贮系统(s,S)库存控制策略研究

考虑的是连续检查库存,需求为一个常时间函数和-个复合Poison跳扩散随机过程的和的存贮系统最优库存控制问题.基于期望折扣成本最小建立了无穷时间区间具有固定订购成本的最优库存模型,确定可采用(s,S)策略进行库存控制,给出了最优(s,S)策略的充要条件--HJB方程Ⅰ、Ⅱ.我们采用猜测的方法确定了最优(s,S)策略对应的值函数形式,建立了确定库存参数的最优化模型.     

An Inventory System with Postponed Demands

Abstract

In this article we consider a continuous review perishable inventory system in which the demands arrive according to a Markovian arrival process (MAP). The items in the inventory have shelf life times that are assumed to follow an exponential distribution. The inventory is replenished according to an (s, S) policy and the replenishing times are assumed to follow a phase type distribution. The demands that occur during stock out periods either enter a pool which has capacity N (<∞) or leave the system. Any demand that arrives when the pool is full and the inventory level is zero, is also assumed to be lost. The demands in the pool are selected one by one, if the replenished stock is above s, with interval time between any two successive selections is distributed as exponential with parameter depending on the number of customers in the pool. The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady state case. The measures of system performance in the steady state are derived and the total expected cost rate is also calculated. The results are illustrated numerically.     

Optimal control of a facility with periodic interrupted demand

The system investigated consists of a stochastic periodic stream of raw material, a continuous processing operation with controllable deterministic service rates, and a storage facility. The arrival stream is periodically interrupted and divided into alternating on-off intervals of fixed length. The processing facility is allowed to operate during the off-interval. Superimposed on this system is a cost structure composed of processing and holding costs. Such operations may be found in manufacturing as well as service systems (for example, dry cleaners, machine shops, repair and maintenance shops, printers, information processing centers, etc). A service rate control rule that minimizes the infinite-horizon discounted expected total cost is found. Existence and uniqueness of long-term optimal cost and policy functions is shown. Since the optimal policy cannot be expressed explicitly, an approximate solution was obtained. An error bound on the optimal cost associated with this solution is exhibited. The approximate solution is characterized by a service rate control rule that is a linear function of the level of inventory at the start of each on-interval and a piecewise linear function of inventory at the start of each off-interval. The optimal discounted expected total cost is quadratic in the inventory level at the start of each interval. Computational results indicate relative cost errors in the order of 2–3 percent.This research was performed at the Sanitary Engineering Research Laboratory and Operations Research Center of the University of California, Berkeley. It was made possible by US Public Health Research Grant UI-00547 from the Environmental Control Administration-Bureau of Solid Waste Management and by National Science Foundation Grant GK-1684.The author thanks Professor C. R. Glassey for not only suggesting this research, but for his constant encouragement and suggestions throughout its duration. He also thanks Professors W. S. Jewell and P. H. McGauhey whose comments on the draft were very helpful.     

Optimal control of diffusion processes with reflection

In this paper, we consider the problem of optimally controlling a diffusion process on a closed bounded region ofR ⁿ with reflection at the boundary. Employing methods similar to Fleming (Ref. 1), we present a constructive proof that there exists an optimal Markov control that is measurable or lower semicontinuous. We prove further that the expected cost function corresponding to the optimal control is the unique solution of the quasilinear parabolic differential equation of dynamic programming with Neumann boundary conditions and that there exists a diffusion process (in the sense of Stroock and Varadhan) corresponding to the optimal control.This work was partially supported by the National Science Foundation, Grant No. GK-18339, by the Office of Naval Research, Grant No. NR-042-264, and by the National Research Council of Canada, Grant No. A3609.The author would like to thank S. R. Pliska, J. Pisa, and N. Trudinger for helpful suggestions. He is especially grateful to Professor A. F. Veinott, Jr., for help and advice in the preparation of the doctoral dissertation, on which part of this paper is based. Finally, he wishes to thank one of the referees for the careful reading and constructive comments on an earlier version of this paper.     

固定时间域可变采样间隔的统计控制图

本文从提高统计控制图对过程波动检测能力和方便管理的角度出发 ,对可变采样间隔(VSI)控制图进行改进 ,提出了针对连续过程质量控制应用需要的固定时间域可变采样间隔(VSIFT)控制图。文章详细介绍了VSIFT均值极差控制图、VSIFTEWMA控制图的设计 ,并分别评价了它们对过程异常状态的检测能力     

Impulsive control in management: Prospects and applications

An outline of impulsive control and its applications in management is reviewed. Conditions for optimal impulsive controls when the dynamic process is given by a stochastic differential equation with Wiener and jump processes are given. Applications to inventory control, capacity expansion, vehicle dispatching, maintenance-replacement-inspection, and pricing problems are formulated, and optimality conditions found. Although there are many other applications that can be outlined, these provide some motivation for further study in this emerging field.This work was supported by the European Institute for Advanced Studies in Management, Brussels, Belgium.     

Index Heuristics for Multiclass M/G/1 Systems with Nonpreemptive Service and Convex Holding Costs

We consider the optimal service control of a multiclass M/G/1 queueing system in which customers are served nonpreemptively and the system cost rate is additive across classes and increasing convex in the numbers present in each class. Following Whittle's approach to a class of restless bandit problems, we develop a Langrangian relaxation of the service control problem which serves to motivate the development of a class of index heuristics. The index for a particular customer class is characterised as a fair charge for service of that class. The paper develops these indices and reports an extensive numerical investigation which exhibits strong performance of the index heuristics for both discounted and average costs.     

Average Cost Optimality in Inventory Models with Markovian Demands

This paper is concerned with long-run average cost minimization of a stochastic inventory problem with Markovian demand, fixed ordering cost, and convex surplus cost. The states of the Markov chain represent different possible states of the environment. Using a vanishing discount approach, a dynamic programming equation and the corresponding verification theorem are established. Finally, the existence of an optimal state-dependent (s, S) policy is proved.     

Convex duality for finite-fuel problems in singular stochastic control

Upon introducing a finite-fuel constraint in a stochastic control system, the convex duality formulation can be set up to represent the original singular control problem as a minimization problem over the space of vector measures at each level of available fuel. This minimization problem is imbedded tightly into a related weak problem, which is actually a mathematical programming problem over a convex,w*-compact space of vector-valued Radon measures. Then, through the Fenchel duality principle, the dual for the finite-fuel control problems is to seek the maximum of smooth subsolutions to a dynamic programming variational inequality. The approach is basically in the spirit of Fleming and Vermes, and the results of this paper extend those of Vinter and Lewis in deterministic control problems to the finite-fuel problems in singular stochastic control. Meanwhile, we also obtain the characterization of the value function as a solution to the dynamic programming variational inequality in the sense of the Schwartz distribution.The author is much indebted to Professor Wendell H. Fleming for his constant support and many helpful discussions during the preparation of this paper.     

On the computation of the optimal cost function for discrete time Markov models with partial observations

We consider several applications of two state, finite action, infinite horizon, discrete-time Markov decision processes with partial observations, for two special cases of observation quality, and show that in each of these cases the optimal cost function is piecewise linear. This in turn allows us to obtain either explicit formulas or simplified algorithms to compute the optimal cost function and the associated optimal control policy. Several examples are presented.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-86-0029, in part by the National Science Foundation under Grant ECS-8617860, in part by the Advanced Technology Program of the State of Texas, and in part by the DoD Joint Services Electronics Program through the Air Force Office of Scientific Research (AFSC) Contract F49620-86-C-0045.     

Stochastic impulse control with regime switching for the optimal dividend policy when there are business cycles,taxes and fixed costs

We consider the problem of finding the optimal dividend policy for a company whose cash reserve follows a Brownian motion with drift and volatility modulated by an observable finite-state continuous-time Markov chain. The Markov chain represents the regime of the economy. We allow fixed costs and taxes associated with the dividend payments. This optimization problem generates a stochastic impulse control problem with regime switching. We solve this problem and obtain the first analytical solutions for the optimal dividend policy when there are simultaneously fixed costs, taxes and business cycles. Our results show that the optimal dividend policy depends strongly on the regime of the economy, on fixed costs and on taxes.     

Control Policy for a Manufacturing System with Random Yield and Rework

We develop a production policy that controls work-in-process (WIP) levels and satisfies demand in a multistage manufacturing system with significant uncertainty in yield, rework, and demand. The problem addressed in this paper is more general than those in the literature in three aspects: (i) multiple products are processed at multiple workstations, and the capacity of each workstation is limited and shared by multiple operations; (ii) the behavior of a production policy is investigated over an infinite-time horizon, and thus the system stability can be evaluated; (iii) the representation of yield and rework uncertainty is generalized. Generalizing both the system structure and the nature of uncertainty requires a new mathematical development in the theory of infinite-horizon stochastic dynamic programming. The theoretical contributions of this paper are the existence proofs of the optimal stationary control for a stochastic dynamic programming problem and the finite covariances of WIP and production levels under the general expression of uncertainty. We develop a simple and explicit sufficient condition that guarantees the existence of both the optimal stationary control and the system stability. We describe how a production policy can be constructed for the manufacturing system based on the propositions derived.