Optimal control of a resource-sharing multiprocessor with periodic maintenance

Shared resources and the processes that control them play a critical role in the functioning of concurrent systems. The article analyzes the production control of a workstation producing a number of products concurrently. The workstation is periodically stopped for maintenance. The objective of the production control is to minimize inventory and backlog costs over an infinite time horizon. Using the maximum principle and under the so-called agreeable cost structure, we derive the optimal production control. We prove that under this cost structure, the problem can be solved in polynomial time.     

Scheduling one-part-type serial manufacturing system under periodic demand: a solvable case

The paper studies one-part type, multiple-stage production system with periodic demands. A buffer of infinite capacity is placed after each machine. Inventory flow through buffers is controlled by machine production rates. The objective is to find a cyclic production rate, which minimizes all inventory-related expenses over an infinite planning horizon. With the aid of the maximum principle, optimal production policies are derived and the continuous-time scheduling problem is reduced to a discrete timing problem. As a result, a polynomial-time algorithm is suggested to calculate the optimal production rate. A numerical example is used to illustrate the algorithm.Scope and purposeNumerical and heuristic approaches have been suggested for production control of automated-serial-manufacturing systems. These approaches try to derive production control policies that would minimize overall costs related to inventory, backlog, and production. The quality of these approaches is often difficult to assess, and they can be time-consuming to implement. Therefore, increasing attention has been directed to optimal control policies of production systems that can be derived precisely and quickly. This paper addresses a special case of the production system manufacturing a single product type to meet a periodic demand. Given a certain assumption on cost relationship, we derive a fast and simple scheduling algorithm that calculates the optimal policy.     

Optimal production control in a discrete manufacturing system withunreliable machines and random demands

We consider a production control problem in a manufacturing system with a failure-prone machine and a stochastic demand. The objective is to minimize a discounted inventory holding and backlog cost over an infinite planning horizon. The optimal production control of continuous, stochastic manufacturing systems with a failure-prone machine and a constant demand has been considered in Akella and Kumar (1986). However, the problem of optimal production control for discrete stochastic manufacturing systems with uncertain demands remains open. In this paper, we investigate a case where the exogenous demand forms a homogeneous Poisson flow. Primarily, we show that the optimal production control for such a system is of the threshold control type. In addition, the explicit form of production control policy and the objective functions are provided. Numerical examples are included to demonstrate the results obtained in the paper and to compare with the one in Akella and Kumar     

Pricing and production lot-size/scheduling with finite capacity for a deteriorating item over a finite horizon

Although the lately evolved manufacturing technologies such as enterprise resource planning (ERP) provide a unified platform for managing and integrating core business processes within a firm, the decision-making between marketing and production planning still remains rather disjoint. It is due in large parts to the inherent weaknesses of ERP such as the fixed and static parameter settings and uncapacitated assumption. To rectify these drawbacks, we propose a decision model that solves optimally the production lot-size/scheduling problem taking into account the dynamic aspects of customer's demand as well as the restriction of finite capacity in a plant. More specifically, we consider a single product that is subject to continuous decay, faces a price-dependent and time-varying demand, and time-varying deteriorating rate, production rate, and variable production cost, with the objective of maximizing the profit stream over multi-period planning horizon. We propose both coordinated and decentralized decision-making policies that drive the solution of the multivariate maximization problem. Both policies are formulated as dynamic programming models and solved by numerical search techniques. In our numerical experiments, the solution procedure is demonstrated, comparative study is conducted, and sensitivity analysis is carried out with respect to major parameters. The numerical result shows that the solution generated by the coordinated policy outperforms that by the decentralized policy in maximizing net profit and many other quantifiable measures such as minimizing inventory investment and storage capacity.Scope and purposeWe consider a manufacturing firm who produces and sells a single product that is subjected to continuous decay over a lifetime, faces a price-dependent and time-varying demand function, shortages are allowed and a completely backlogged, and has the objective of determining price and production lot-size/scheduling so as to maximize the total profit stream over multi-period planning horizon. We develop a tactical-level decision model that solves the production scheduling problem taking into account the dynamic nature of customer's demand which is partially controllable through pricing schemes. As analogous to the sales and operations planning, the proposed scheme can be used as a coordination center of the APS system within a generic enterprise resource planning framework which integrates and coordinates distinct functions within a firm.This paper differs from the existing works in several ways. First, we propose a dynamic version of the joint pricing and lot-size/scheduling problem taking into account the capacitated constraint. Second, several key factors being considered in the model, such as the demand rate, deteriorating rate, production rate, and variable production cost are assumed time-varying that reflect the dynamic nature of the market and the learning effect of the production system. A third difference between the past research and ours is that the price can be adjusted upward or downward in our model, making the proposed pricing policy more responsive to the structural change in demand or supply.     

Grouping in decomposition method for multi-item capacitated lot-sizing problem with immediate lost sales and joint and item-dependent setup cost

This article examines a dynamic and discrete multi-item capacitated lot-sizing problem in a completely deterministic production or procurement environment with limited production/procurement capacity where lost sales (the loss of customer demand) are permitted. There is no inventory space capacity and the production activity incurs a fixed charge linear cost function. Similarly, the inventory holding cost and the cost of lost demand are both associated with a linear no-fixed charge function. For the sake of simplicity, a unit of each item is assumed to consume one unit of production/procurement capacity. We analyse a different version of setup costs incurred by a production or procurement activity in a given period of the planning horizon. In this version, called the joint and item-dependent setup cost, an additional item-dependent setup cost is incurred separately for each produced or ordered item on top of the joint setup cost.     

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.     

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.     

Joint optimisation of maintenance and production policies with subcontracting and product returns

This paper, an extension of our previous research, deals with the problem of jointly optimizing maintenance, production and inventory costs considering subcontracting and product returns. The manufacturing system, which fails randomly, has to satisfy a random product demand during a finite planning horizon under a required service level. The portion of products returned by the customers that are still in saleable condition are collected in the principle store from which customer demand is filled, while the portion that are non-conformal are collected in a second store and then remanufactured by a subcontractor. This study is validated by a real industrial case presented in this paper.     

Multiproduct production/inventory control under random demands

Studies the optimal production/inventory control policy for a single machine multiproduct production system. The machine produces to fill the end-product inventory stock and the demand is satisfied from the inventory when available; unsatisfied demand is backlogged until the product becomes available as the result of production. For each product, the demand follows a Poisson process and the unit processing time is known. When the machine switches production from one product to another, it incurs a set-up time and a set-up cost. The relevant costs include the set-up cost, a cost per unit time while the machine is running, and linear costs for inventory and backlogging. This problem is modeled as a semi-Markov decision process using the criterion of minimizing expected total cost with discounting over an infinite horizon. Procedures for computing near-optimal policies and their error bounds are developed. The error bound given by the authors' procedure is shown to be much tighter than the one given by the “norm-based” approach. Computational test results are presented to show the structure of the near-optimal policy and how its accuracy is affected by the system characteristics such as capacity utilization and set-up time     

Production control of a manufacturing system with multiple machinestates

The production control of a single-product manufacturing system with arbitrary number of machine states (failure modes) is discussed. The objective is to find a production policy that would meet the demand for the product with minimum average inventory or backlog cost. The optimal production policy has a special structure and is called a hedging-point policy. If the hedging points are known, the optimal production rate is readily specified. Assuming a set of tentative hedging points, the simple structure of the optimal policy is utilized to find the steady-state probability distribution of the surplus (inventory or backlog). Once this function is determined, the average surplus cost is easily calculated in terms of the values of the hedging points. The average cost is then minimized to find the optimum hedging points     

需求随机的变质性产品生产计划模型

研究了能力约束的有限计划展望期生产计划问题,各周期的需求随机,库存产品存在变质且变质率为常数。建立了问题的期望值模型,目标函数为极小化生产准备成本、生产成本、库存成本的期望值。提出了随机模拟、遗传算法和启发式算法相结合的求解算法。用数值实例对模型和算法进行了验证,优化结果表明模型和算法是有效的。     

A comparison of goal programming and simulation model results for a multiobjective multiperiod multiproduct manufacturing scheduling problem

Two modeling approaches, simulation modeling and goal programming, are applied to a multiobjective, multiperiod, multiproduct manufacturing scheduling problem. Results of the two monels are compared in terms of meeting four goals: 1) to operate within the limits of production capacity, 2) to meet the demand delivery schedule, 3) to minimize the total production and inventory costs, and 4) to minimize overtime hours. The comparison of the two models's estimates of production and inventory costs is carried out first as an overview of percent relative differences and then by inferential testing of the hypothesis that the two models cost estimates are not statististically different. Small sample t-testing and Wilcoxan rank sum tests results indicate that the simulation model and goal programming model production scheduling results are statistically different. Suggestions are presented to assist in a mangement decision as to which modeling approach leads to the least costly production schedule.     

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。     

Power-form demand and cost functions in inventory lot size models

This paper presents a continuous-time inventory model with known time-varying demand and carrying cost functions. Backlogs are prohibited, replenishments are assumed to be instantaneous and the planning horizon is finite. The problem is to find the optimal number and schedule of replenishments, i.e. the time intervals between consecutive orders, which minimizes total carrying and replenishment costs throughout the planning horizon.First, a necessary and sufficient condition on the optimal replenishment times is derived for general demand and carrying cost functions when the number of replenishments is fixed. Then, a complete solution (optimal number and schedule of replenishments) is given for the case of power-form demand and carrying cost functions. The asymptotic properties of the solution as the planning horizon tends to infinity are also investigated. The model lends itself to a tractable parametric analysis, and generalizes several special cases already known in the literature.     

Improving production planning by utilizing stochastic programming

The effects of fluctuating demand on production and inventory levels are important in manufacturing resource planning. Thus, the focus of this presentation is on aggregate production planning of manufacturing resources in order to satisfy stochastic demand for a family of products to minimize total costs that include production and inventory holding costs over a rolling horizon.

If it is assumed that, in a commercial setting, the demands are fixed, then the production plans generated by a mathematical programming procedure are not responsive to the actual fluctuations of stochastic demand in each time period.

The situation discussed here considers the case where demands are normally distributed with means and variances that are sequentially revised as new observations of demand are received over time. This assumption allows the probabilistic constraint to be converted to an equivalent linearly-constrained deterministic model. Extensions to the normality assumption are discussed. Also other ideas such as optimal control theory, learning and adaptive signal processing extensions are discussed as well.     

A tabu search procedure for coordinating production,inventory and distribution routing problems

This paper addresses the problem of optimally coordinating a production‐distribution system over a multi‐period finite horizon, where a facility production produces several items that are distributed to a set of customers by a fleet of homogeneous vehicles. The demand for each item at each customer is known over the horizon. The production planning determines how much to produce of each item in every period, while the distribution planning defines when customers should be visited, the amount of each item that should be delivered to customers and the vehicle routes. The objective is to minimize the sum of production and inventory costs at the facility, inventory costs at the customers and distribution costs. We also consider a related problem of inventory routing, where a supplier receives or produces known quantities of items in each period and has to solve the distribution problem. We propose a tabu search procedure for solving such problems, and this approach is compared with vendor managed policies proposed in the literature, in which the facility knows the inventory levels of the customers and determines the replenishment policies.     

Modelling the effect of demand variations on the performance of a production system

Modelling the effect of demand variations on a production system manufacturing multiple products is discussed. The various system costs involved in the production system, namely set-up cost and inventory cost incurred due to change in demands for the products with respect to products and planning periods are estimated. A statistical modelling is presented for determining the production capacity and inventory level requirement to satisfy the customer to a certain level decided by the management. Two important factors, (i) number of types of products and (ii) multiple planning horizons are considered to identify the costs as well as the production capacity and inventory level requirements. A statistical method, analysis of variance (ANOVA) is used to study the variations in the demands and costs involved. Finally, an example is presented to explain the application and the behaviour of the statistical model.     

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 heuristic solution procedure for the dynamic lot sizing problem with remanufacturing and product recovery

Here we discuss the lot sizing problem of product returns and remanufacturing. Let us consider a forecast of demands and product returns over a finite planning horizon — the problem is to determine an optimal production plan. This consists of either manufacturing new products or remanufacturing returned units, and in this way meets both demands at minimum costs. The costs of course are the fixed set-up expenses associated with manufacturing and/or remanufacturing lots and also the inventory holding costs of stocks kept on hand.In addition to showing that a general instance of this problem is NP-Hard, we develop an alternative mixed-integer model formulation for this problem and contrast it to the formulation commonly used in the literature. We show that when integrality constraints are relaxed, our formulation obtains better bounds. Our formulation incorporates the fact that every optimal solution can be decomposed into a series of well-structured blocks with distinct patterns in the way in which set-ups for manufacturing and remanufacturing occur. We then construct a dynamic programming based heuristic that exploits the block structure of the optimal solution. We also propose some improvement schemes as well. Finally, our numerical testing shows that the heuristic performs very well as intended.     

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.