随机需求不可靠制造系统的最优服务率分配策略

本文讨论了一类随机需求不可靠制造系统的最优服务率控制问题。所研究的系统能同时生产多类产品，但生产能力受常数限制。目标是通过最小化库顾和欠缺的期望折扣费用，寻找最优服务率分配策略，本文证明了最优策略具有开关结构，并针对生产单类和两类产品的系统详细研究了最优控制策略的结构性质，最后以数值例子验证了本文的结论。     

Optimal hedging point control for a failure-prone manufacturing system

This paper considers a manufacturing system with multiple operational modes producing one part type. The part processing time at each operational mode is exponentially distributed and its rate is controllable. The demand arrival is random and described by a Poisson process. By minimizing an infinite-horizon discounted expected cost function, the optimal service rate control is derived. It is proved that the optimal policy is of a hedging point structure by examining the properties of the optimal cost function such as convexity, monotonicity and asymptotic behaviours. The hedging points at different operational modes can be ordered according to their production capacities. The relationships of the hedging points with some system parameters are presented. These structural properties of the optimal control policy are helpful in finding simple and realistic suboptimal policies for practical manufacturing systems. A numerical example is given to demonstrate our results.     

不完全柔性制造系统的最优控制

考虑具有随机需求的不完全柔性制造系统的最优控制，系统在各种产品间的切换时间是不可忽略的。运用马尔可夫最优决策过程归纳方法，导出机器服务率的最优控制策略。通过分析最优值函数的性质，证明最优策略具有简单的阈值结构，从而可得到次优生产策略--阈值控制策略。     

Optimal manufacturing flow controllers: zero-inventory policies andcontrol switching sets

A continuous-flow control model of a single workstation with multiple failure modes and part types is considered. Although the general model is intractable, properties of the optimal control policy are identified that can be used to help formulate heuristic policies. Control switching sets are described and shown to have a threshold form. For a single machine with two part-types, conditions are found under which no inventory is held, analogous to the single-part-type result of Bielecki and Kumar (1988)     

Switching curve policy for manufacturing systems with no backlogpermitted

For failure-prone manufacturing systems with deterministic machine up time, the switching curve policy, which has a special property and can be characterized by a single parameter, was shown to be optimal. In previous work, the problem of optimal production rate control was studied under the assumption that backlog is allowed. In this paper, the authors consider manufacturing systems in which backlog is not permitted. The authors show that the switching curve policy is still optimal in this case. They also consider the system model in which one machine has three states: down, idle, and operational. This model can reduce maintenance times of the machine, and then reduce its maintenance costs     

Optimal control of a queueing system with two interacting servicestations and three classes of impatient tasks

The problem of task selection and service priority is studied for a queueing network with two interacting service stations and three classes of impatient tasks. By using stochastic dynamic programming, a functional equation for the optimal, state-dependent priority assignment policy is derived. Properties of the optimal cost-to-go functions and the optimal policy are established through inductive proofs. It is shown that the optimal policy is governed by two switching surfaces in the three-dimensional state space (one dimension for each task class). For the infinite-time-horizon case, the optimal policy is shown to be stationary. In this case, the optimal cost-to-go function and switching surfaces are obtained numerically by using the overrelaxed Gauss-Seidel method. Sensitivities of the optimal policy with respect to key system parameters are also investigated     

Optimal and suboptimal control policies for a solar collector system

An optimal control problem to maximize the net energy gathered by a flat-plate solar collector system by controlling the collector fluid flow rate is investigated. The problem is formulated in terms of a distributed parameter system and solved using the method of characteristics. It is shown that if the pump of the collector loop is such that its pumping power is greater than a linear function of the fluid velocity, then the optimal control policy is one in which the fluid flow is instantly switched between zero and maximum rates. Necessary conditions that determine the optimal switching times are derived. Because the resultant switching function of the optimal policy is shown to be decomposable into two parts, one that depends on the state of the system and another that requires a priori knowledge of the solar intensity over the entire period of operation, a suboptimal control policy that can be implemented by an on/off feedback controller with hysteresis is proposed. When this suboptimal policy is compared with the optimal policy, it is shown that on a clear day with sufficient solar insolation to dictate a two-switch optimal policy, the two policies are identical. Under other weather conditions, the feedback suboptimal controller will keep the pump off for a slightly shorter period of time than the time dictated by the optimal control.     

Optimal Integrated Ordering and Production Policy in a Supply Chain With Stochastic Lead-Time, Processing-Time, and Demand

This paper seeks the optimal integrated ordering and production control in a supply chain by minimizing the expected sum of material and product holding costs and demand backordering costs subject to finite capacitated warehouses. With the assumptions of exponential replenishment lead-times, exponential processing-times, Poisson demand arrivals, and at most one outstanding order with its size changeable at any time, it is shown that the optimal integrated policy can be characterized by two monotonic switching curves. The optimal ordering decision follows a set of order-up-to-point policies, while the optimal production decision follows a set of base-stock policies. Based on the monotonic and asymptotic properties of the switching curves, a simple linear switching threshold policy is proposed, which performs extremely well in the experiments. The key assumptions are then relaxed and numerical examples illustrate that the main structural properties of the optimal policy are preserved.     

Optimal average cost manufacturing flow controllers: convexity anddifferentiability

The authors consider the control of a production facility consisting of a single workstation with multiple failure modes and part types using a continuous flow control model. Technical issues concerning the convexity and differentiability of the differential cost function are investigated. It is proven that under an optimal control policy the differential cost is C¹ on attractive control switching boundaries     

线性混杂系统优化控制的Monte Carlo统计预测方法

针对含扩散项的线性混杂切换系统优化控制问题, 为降低优化求解的计算复杂性, 提出了Monte Carlo统计预测方法. 首先通过数值求解技术把连续时间优化控制问题转化为离散时间的Markov决策过程问题; 然后在若干有限状态子空间内, 利用反射边界技术来求解相应子空间的最优控制策略; 最后根据最优控制策略的结构特性, 采用统计预测方法来预测出整个状态空间的最优控制策略. 该方法能有效降低求解涉及大状态空间及多维变量的线性混杂切换系统优化控制的计算复杂性, 文末的仿真结果验证了方法的有效性.     

Adaptive predictive control policy for nonlinear stochastic systems

The problem of optimal adaptive predictive control for nonlinear stochastic systems is considered. A classification of various suboptimal approaches is given. A new approximation for the probability measure for the extended state of the system is suggested to derive a new suboptimal control law. It is assumed for this approximation that the system operates in closed-loop feedback mode for one part of the extended state vector and in open feedback loop mode for the other part of this vector. The certainty equivalence (CE) assumption is used only for the first part of the extended state vector. An analytical comparison for the suggested control policy shows its superiority in control quality compared with that of the open-loop optimal control policy in the case of an exactly observed first part of the extended state vector. The upper bound of the performance index is determined for this case. The suggested control policy has a simple form for linear systems with unknown parameters. A simulated example is used to demonstrate the potential of the suggested method and its superiority over the usually applied CE policy     

Production control of a pull system with production and demanduncertainty

We consider a continuous material-flow manufacturing system with an unreliable production system and a variable demand source which switches randomly between zero and a maximum level. The failure and repair times of the production system and the switching times of the demand source are assumed to be exponentially distributed random variables. The optimal production flow control policy that minimizes the expected average inventory carrying and backlog costs is characterized as a double-hedging policy. The optimal hedging levels are determined analytically by minimizing the closed-form expression of the cost function. We investigate two approximate single hedging policies. It is empirically shown that an approximate policy that uses a single hedging level which is the sum of a production uncertainty term and a demand uncertainty term gives accurate results for the expected average cost     

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.     

Minimax control of switching systems under sampling

We consider a general class of systems subject to two types of uncertainty: A continuous deterministic uncertainty that affects the system dynamics, and a discrete stochastic uncertainty that leads to jumps in the system structure at random times, with the latter described by a continuous-time finite state Markov chain. When only sampled values of the system state is available to the controller, along with perfect measurements on the state of the Markov chain, we obtain a characterization of minimax controllers, which involves the solutions of two finite sets of coupled PDEs, and a finite dimensional compensator. For the linear-quadratic case, a complete characterization is given in terms of coupled generalized Riccati equations, which also provides the solution to a particular H^∞ optimal control problem with randomly switching system structure and sampled state measurements.     

Optimal Control of Switching Times in Switched Stochastic Systems

In this paper, we solve an optimal control problem for switched stochastic systems using calculus of variations, where the objective is to minimize a cost functional defined on the state and the switching times are the sole control variables. In particular, we focus on the problem in which a pre‐specified sequence of active subsystems is given. For one switching time case, the derivative of the cost functional with respect to the switching time is derived, which has an especially simple form and can be directly used in gradient descent algorithms to locate the optimal switching instant. Then, we propose an approach to deal with the problem with multi‐switching times case. Finally, two numerical examples are given, highlighting the viability and advantages of the proposed methodology.     

Sufficient optimality conditions for controlling switched systems

The optimal control problem for a switched system whose state vector includes both continuous and discrete components is considered. The continuous part of the system is governed by differential equations, while its discrete part, which models the operation of a switching device with memory, is governed by recurrence inclusions. The discrete part switches the operation modes of the continuous part of the system, and is itself affected by the continuous part. The switching times and their number are not specified in advance. They are found as a result of optimizing a performance index. This problem is a generalization of a classical optimal control problem for continuous–discrete systems. It is shown that the cost function is constructed from certain auxiliary functions—the so-called conditional cost functions. Equations for the conditional cost functions are derived and sufficient optimality conditions are proved. The application of these conditions is demonstrated by the example of designing a time optimal switched system that is similar to A.A. Fel’dbaum’s classical example of the time optimal continuous system.     

Optimal control for systems with deterministic production cycles

Considers a failure prone production system with deterministic production cycles. The objective is to find the optimal production rate to minimize long-run average cost. The authors first show that the optimal control policy belongs to a type of switching curve policy which has a special structural property and can be characterized by a single parameter. The authors then establish a relationship between the surplus process of the system under the optimal control policy and the workload process of a D/G/1 queue, based on which the existing results in queueing theory which can be applied to obtain the steady-state probability distribution of the surplus process under the optimal control policy     

Use of optimal control theory for computing optimal injection policies for enhanced oil recovery

The theory of optimal control of distributed parameter systems has been applied to the problem of determining an optimum injection policy for a chemical flooding enhanced oil recovery process. The optimization criterion is to maximize the difference between gross revenue and cost of chemicals injected. Necessary conditions for optimality are obtained through application of the calculus of variations and the distributed weak minimum principle. A gradient method is used for the computation of optimal injection policies. The performance of the algorithm was examined for the surfactant injection in a one-dimensional flooding problem. Two types of interfacial tension behaviour were considered: a type ‘A’ system in which the interfacial tension is a monotically decreasing function with solute concentration and a type ‘B’ system in which a minimum in the interfacial tension occurs at a nominal surfactant concentration. Different initial values were used in order to establish the convergence of the computational algorithm. For a type ‘A’ system, the shape of the optimum injection strategy was not unique; however, there is a unique optimum for the amount of surfactant needed. For a type ‘B’ system, the shape of the optimum injection was unique as well as the amount injected. The results of this work show that given the properties of an oil reservoir and the properties of a surfactant solution, an optimum injection policy which minimizes a specific economic objective functional can be obtained using distributed parameter control theory.     

Joint admission,production sequencing,and production rate control for a two-class make-to-order manufacturing system

Today's manufacturing industry faces a number of challenges related to the rapid delivery of products with a high degree of variety. Striking a balance between the effectiveness in capacity utilization and the rapidness in order-fulfillment is a substantial challenge for manufacturing companies. This work aims to provide a theoretical basis from which to address this practical question. To this end, we address the problem of coordinating the admission, production sequencing, and production rate controls for a two-class make-to-order manufacturing system. Formulating the problem as a Markov decision process model, we identify the structural properties of optimal control policies under both discounted and average profit criteria. We show that the cμ rule is optimal for production sequencing and the optimal admission and production rate control policies can be characterized by the state-dependent threshold levels, provided that the production times are not associated with customer class. We also show that the optimal production rates are monotone in the system state, as in the case of a single class queueing system, and that the lower priority class can be preferred to the higher priority class in order admission under a certain condition on the system parameters. Our numerical study demonstrates that a considerable economic benefit can be achieved if the production rate is dynamically controlled between the minimum and maximum rates rather than fixed to the mean rate of these values. Finally, we present a heuristic policy that is described by linear switching functions for the control of order admission and a selection rule for the control of production rate. We compare the performance of our heuristic to the optimal policy using a numerical experiment, revealing that the heuristic provides near optimal solutions to test example problems and is robust to the system parameters.