An integrated replenishment and production control policy under inventory inaccuracy and time-delay

Inventory inaccuracy has a significant negative impact on the performance of raw materials replenishment and production control. It usually leads to high inventory holding cost or large backlog penalty. To hedge against it, we investigate a replenishment and production control problems for a multiple machines and multiple product-types production/inventory system with inventory inaccuracy. The objective is to minimize the average production cost, including the inventory holding cost and the backlog penalty. In addition to inventory inaccuracy, the lead-time of raw materials replenishment and the unreliability of machines are also taken into consideration. In light of the principle of dynamic programming, a simplified optimal replenishment and production control policy is constructed based on the assumption that the records of inventories are completely accurate. To overcome the shortcomings of the simplified policy in hedging against inventory inaccuracy, a conditional expectation-based replenishment and production control policy is developed based on the fundamental structure of the simplified optimal replenishment and production control policy, and the conditional probability of the physical inventory level. Numerical experiments are conducted to examine the performance of the proposed policy for hedging against inventory inaccuracy, and sensitivity analysis is carried out to study how the parameters of the production/inventory system and the proposed policy affect the average production cost.     

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     

A simulation optimization based control policy for failure prone one-machine,two-product manufacturing systems

This paper presents the optimal flow control for a one-machine, two-product manufacturing system subject to random failures and repairs. The machine capacity process is assumed to be a finite state Markov chain. The problem is to choose the production rates so as to minimize the expected discounted cost of inventory/backlog over an infinite horizon. We first show that for constant demand rates and exponential failure and repair time distributions of the machine, the hedging point policy is optimal. Next, the hedging point policy is extended to non-exponential failure and repair time distributions models. The structure of the hedging point policy is parameterized by two factors representing the thresholds of involved products. With such a policy, simulation experiments are coupled with experimental design and response surface methodology to estimate the optimal control policy. Our results reveal that the hedging point policy is also applicable to a wide variety of complex problems (i.e. non-exponential failure and repair time distributions) where analytical solutions may not be easily obtained.     

Production rate control for failure-prone production systems withno backlog permitted

In this paper, we consider systems in which backlog is not allowed. We show that the hedging point policy is still optimal. For systems with backlog, it is usually quite straightforward to show that their optimal cost-to-go functions are convex-a key property that is needed for the hedging point policy to be optimal. With no backlog permitted, it becomes much more difficult to establish the convexity property, and the explicit formulas for the optimal hedging point and the optimal cost-to-go functions have to be obtained, based on which the convexity property can then be verified. The method we use in this paper to derive these explicit formulas is mainly based on an interesting relationship between the inventory process of the system under the hedging point policy and some stochastic processes which are well studied in queueing theory     

A simulation optimization approach in production planning of failure prone manufacturing systems

In this paper, the implementation of a new method to control the production rate of manufacturing systems, based on the combination of stochastic optimal control theory, discrete event simulation, experimental design and response surface methodology is outlined. The system under study consists of several parallel machines, multiple-product manufacturing system. Machines are subject to failures and repairs and their capacity process is assumed to be a finite state Markov chain throughout the analytical control model. The problem is to choose the production rates so as to minimize the expected discounted cost of inventory/backlog over an infinite horizon. We first show that, for constant demand rates and exponential failure and repair times distributions of the machines, the hedging point policy is optimal. The structure of the hedging point policy is then parameterized by factors representing the thresholds of involved products. With such a policy, simulation experiments are combined to experimental design and response surface methodology to estimate the optimal control policy. We obtain that the hedging point policy is also applicable to a wide variety of complex problems including non-exponential failure and repair times distributions and random demand rates. Analytical solutions may not be easily obtained for such complex situations.     

Optimal cycle production of a manufacturing system subject to deterioration

A single part type, single machine, fluid model manufacturing system is considered. The machine is subject to a deterministic deterioration process, which depends on the operation rate. The objective is to minimize a long term average cost index which penalizes inventory surplus and backlog. The optimal policy determined presents some similarities with the policy conjectured optimal in the past for a similar Markovian formulation. The considered problem can be applied to several real contexts, and not just from the manufacturing domain, as briefly discussed in the paper.     

Stohastic optimal production control problem with corrective maintenance

We consider a production control problem in a manufacturing system with failure-prone machines and a constant demand rate. The objective is to minimise a discounted inventory holding and backlog cost over an infinite planning horizon. The availability of the machines is improved through a corrective maintenance strategy. The decision variables are the production and the machine repair rates, which influence the inventory levels and the system capacity, respectively. It is shown that, for constant demand rates and exponential failure and repair times distributions of the machines, the hedging point policy is optimal. Such a policy is modified herein and parameterised by factors representing the thresholds of involved products and switching inventory levels for corrective maintenance. With the obtained policy, simulation experiments are combined to experimental design and response surface methodology to estimate the optimal production and corrective maintenance policies, respectively. The usefulness of the proposed approach is illustrated through a numerical example.     

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。     

On the ordering of optimal hedging points in a class ofmanufacturing flow control models

The optimal flow control policy of a single-product unreliable manufacturing system that must meet a constant demand rate is known to be a threshold type policy: safety production surplus levels called hedging points (thresholds) are associated with each discrete stochastic capacity state of the system and serve to protect the production process from uncertainty in future capacity availability. This correspondence extends and generalizes previous results on the ordering of optimal hedging points. The authors' method is based on examining special properties of the Bellman optimality conditions of the underlying stochastic control problem     

Optimal control structure of an unreliable manufacturing systemwith random demands

This paper considers an unreliable manufacturing system with random processing times and random product demands. The system can produce many part types and its total capacity is constrained by a fixed constant. The objective is to find an optimal service-rate allocation policy between different part types by minimizing the expected discounted inventory and backlog cost. Structural properties of the optimal control policy are investigated. It is shown that the optimal policy is of a switching structure. For producing a one part type case, the optimal control is a threshold policy. For producing a two part types case, the optimal control can be described by three monotone switching curves and its asymptote properties are derived. Numeric examples are given to illustrate the results     

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.     

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     

Hedging policies for failure-prone manufacturing systems:optimality of JIT and bounds on buffer levels

The authors consider multiple part-type single-machine manufacturing systems subject to failures. They provide necessary and sufficient conditions for the optimality of just-in-time (JIT) manufacturing, and further show that the necessary and sufficient conditions coincide when the backlog costs for all the parts are equal. In general, finding optimal hedging policies for multiple part-type problems is extremely difficult. They provide bounds on the optimal hedging points and also prove that, under an optimal hedging policy, the buffer levels will be upper bounded     

制造系统的递阶滚动优化调度模型

研究的对象是只有一台不可靠（failure-prone)机器的非完全柔性制造系统，该系统能生产多种产品，但在同一时刻只能生产一种产品，并且当机器由生产一种产品向生产另一种产品切换时，需要考虑setup时间及其成本，待决策变量是setup序列及产品生产率，本文基于非完全柔性制造系统的特点，引入递阶层控的思想，采用新的递阶结构框架和阈值控制策略，对问题进行分解，建立了考虑setup时间及成本的递阶流率控制最优化调度模型，并给出了递阶的滚动优化算法，仿真结果表明，这种调度策略更易于工程实现。     

Hedging-point production control with multiple failure modes

We consider the control of a production facility subject to multiple failure modes. Motivated by a work of Akella-Kumar (1986) and Bielecki-Kumar (1988) on single-failure-mode models, we study hedging-point policies, in which production is controlled to its maximum rate whenever inventory is below a critical level and set to zero whenever inventory is above that level. The maximum production rate varies with the state of the machine. Assuming that the machine state is governed by a semi-Markov process, we evaluate average and discounted inventory costs for any hedging point, thus providing a simple mechanism for identifying optimal hedging points. Our most explicit results require that intervals in which demand exceeds production are exponentially distributed. We drop the exponential assumption at the expense of obtaining asymptotics rather than exact results     

Maintenance scheduling and production control of multiple-machine manufacturing systems

This paper deals with the production and preventive maintenance control problem for a multiple-machine manufacturing system. The objective of such a problem is to find the production and preventive maintenance rates for the machines so as to minimize the total cost of inventory/backlog, repair and preventive maintenance. A two-level hierarchical control model is presented, and the structure of the control policy for both identical and non-identical manufacturing systems is described using parameters, referred to here as input factors. By combining analytical formalism with simulation-based statistical tools such as experimental design and response surface methodology, an approximation of the optimal control policies and values of input factors are determined. The results obtained extend those available in existing literature to cover non-identical machine manufacturing systems. A numerical example and a sensitivity analysis are presented in order to illustrate the robustness of the proposed approach. The extension of the proposed production and preventive maintenance policies to cover large systems (multiple machines, multiple products) is discussed.     

Pade approximants for the transient optimization of hedging controlpolicies in manufacturing

Part production is considered over a finite horizon in a single-part multiple-failure mode manufacturing system. When the rate of demand for parts is constant, for Markovian machine-mode dynamics and for convex running cost functions associated with part inventories or backlogs, it is known that optimal part-production policies are of the so-called hedging type. For the infinite-horizon case, such policies are characterized by a set of constant critical machine-mode dependent inventory levels that must be aimed at and maintained whenever possible. For the finite-horizon (transient) case, the critical levels still exist, but they are now time-varying and in general very difficult to characterize. Thus, in an attempt to render the problem tractable, transient production optimization is sought within the (suboptimal) class of time-invariant hedging control policies, a renewal equation is developed for the cost functional over finite horizon under an arbitrary time-invariant hedging control policy     

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.     

基于马尔科夫决策过程的ATO系统独立组件与产品双需求最优决策研究

研究多维组件, 单一产品的双需求型面向订单装配(Assemble-to-order, ATO)系统. 产品需求为延期交货型, 当其不被满足时将产生缺货等待成本; 而独立组件需求为销售损失型, 其不被满足时将产生缺货损失成本. 该问题可以抽象成一个动态马尔科夫决策过程(Markov decision process, MDP), 通过对双需求模型求解得到状态依赖型最优策略, 即任一组件的最优生产--库存策略由系统内其他组件的库存水平决定. 研究解决了多需求复杂ATO系统的生产和库存优化控制问题. 提出在一定条件下, 组件的基础库存值可以等价于最终产品需求的库存配给值. 组件的基础库存值与库存配给值随系统内其他组件库存的增加而增加, 而产品需求的库存配给值随系统组件库存和产品缺货量的增加而减少. 最后通过数值实验分析缺货量及组件库存对最优策略结构的影响, 并得到了相应的企业生产实践的管理启示.     

Dynamic setup scheduling and flow control in manufacturing systems

We propose a method for flow control of parts in a manufacturing system with machines that require setups. The setup scheduling problem is investigated in the context of a multilevel hierarchy of discrete events with distinct frequencies. The higher level of the hierarchy calculates a target trajectory in the surplus/backlog space of the part types which must be tracked at the level of setups. We consider a feedback setup scheduling policy which usescorridors in the surplus/backlog space of the part types to determine the timing of the set-up changes in order to guide the trajectory in the desired direction. An interesting case in which the trajectory leads to a target point (e.g., a hedging point) is investigated in detail. It is shown that in this case the surplus/backlog trajectory at the setup level can lead to a limit cycle. Conditions for linear corridors which result in a stable limit cycle are determined.