Note on “replenishment run time problem with machine breakdown and failure in rework”

A recursive search algorithm along with mathematical analysis was used in a recent paper (Chiu, Y.-S.P., Chen, K.-K., Ting, C.-K., 2012. Replenishment run time problem with machine breakdown and failure in rework. Expert Systems with Applications, 39, 1291–1297) to determine the optimal operating policy for a specific replenishment run time problem. In its optimization procedure a conditional convexity theorem on the cost function was adopted. This paper presents a direct proof of convexity of the function to enhance quality of their solution process and eliminate unnecessary computational efforts in verifying the conditional convexity.     

带返工的定点装配车间生产计划与调度集成优化

针对定点装配车间不合格品返工导致生产计划及调度不可行的情况,提出一种基于返工延后处理的定点装配车间生产计划与调度集成优化方法.首先制定返工延后处理的粗生产计划;下达计划生产后,将每周期产生的不合格品放入缓冲区,在下周期初时重新调整生产计划并求解新计划下的最优调度,判断其是否满足装配班组负载率要求,不断交替迭代生产计划与调度直至达到计划与调度的均衡和优化.最后通过算例验证了所提出方法的可行性和有效性.     

Optimization of the finite production rate model with scrap,rework and stochastic machine breakdown

This study employs mathematical modeling along with a recursive searching algorithm to determine the optimal run time for an imperfect finite production rate model with scrap, rework, and stochastic machine breakdown. In real-life manufacturing systems, generation of defective items and machine breakdown are inevitable. The objective of this paper is to address these issues and to be able to derive the optimal production run time. It is assumed that the proposed manufacturing system produces defective items randomly, a portion of them is considered to be scrap, and the other portion can be repaired through rework. Further, the proposed system is subject to random breakdown and when it occurs, the abort/resume (AR) policy is adopted. Under such an inventory control policy, the production of the interrupted lot will be resumed immediately when machine is fixed and restored. Mathematical modeling along with a recursive searching algorithm is used for deriving the replenishment policy for such a realistic production system.     

The correct solution procedure on the replenishment run time problem with machine breakdown and failure in rework for supply chain management

Recently, Lin and Chiu [Lin, H. D., & Chiu, Y. S. P. (2012). Note on “Replenishment run time problem with machine breakdown and failure in rework. Expert Systems with Applications, <http://dx.doi.or10.1016/j.eswa.2012.05.001g/>] show that the long-run average production-inventory costs per unit time is a convex function. The main purpose not only reveals that their cost function is not a convex function but also presents the correct solution procedure to locate the optimal solution for Lin and Chiu (2012).     

An economic production quantity model with random defective rate,rework process and backorders for a single stage production system

This paper revisits the economic production quantity (EPQ) model with rework process at a single-stage manufacturing system with planned backorders. It is well known that any imperfect production system of real life has random defective rates. In this direction, this paper extends an inventory model to allow random defective rates. Basically, three different inventory models are developed for three different distribution density functions such as uniform, triangular, and beta. The analytical derivation provides closed-form solution for each inventory model. We have made comparison tables of optimal results among the distribution functions. Some numerical examples and sensitivity analysis are given to illustrate the inventory models.     

An economic production quantity model for deteriorating items with preventive maintenance policy and random machine breakdown

In recent years, many researches on economic production quantity (EPQ) models with machine breakdown and preventive maintenance have been developed, but few of them have developed integrated models for deteriorating items. In this study, we develop EPQ models for deteriorating items with preventive maintenance, random machine breakdown and immediate corrective action. Corrective and preventive maintenance times are assumed to be stochastic and the unfulfilled demands are lost sales. Two EPQ models of uniform distribution and exponential distribution of corrective and maintenance times are developed. An example and sensitivity analysis is given to illustrate the models. For the exponential distribution model, it is shown that the corrective time parameter is one of the most sensitive parameters to the optimal total cost.     

Maximising profits for an EPQ model with unreliable machine and rework of random defective items

This article deals with an economic production quantity (EPQ) model in an imperfect production system. The production system may undergo in ‘out-of-control’ state from ‘in-control’ state, after a certain time that follows a probability density function. The density function varies with reliability of the machinery system that may be controlled by new technologies, investing more costs. The defective items produced in ‘out-of-control’ state are reworked at a cost just after the regular production time. Occurrence of the ‘out-of-control’ state during or after regular production-run time is analysed and also graphically illustrated separately. Finally, an expected profit function regarding the inventory cost, unit production cost and selling price is maximised analytically. Sensitivity analysis of the model with respect to key parameters of the system is carried out. Two numerical examples are considered to test the model and one of them is illustrated graphically.     

Robust scheduling optimization for flexible manufacturing systems with replenishment under uncertain machine failure disruptions

This paper studies the scheduling problem for the flexible manufacturing systems (FMSs) under uncertain machine failure disruptions, where machine allocations and job schedules need to be determined to achieve a set of production due-date requirements as well as possible. A robust scheduling optimization model is proposed based on the concept of threshold scenario, bounded by which the due-dates are guaranteed to be achieved. It is shown that the associated stochastic scheduling problem can be equivalently solved by computing the solution of a mixed-integer linear program (MILP). Computational results show that our proposed model performs well in achieving the planned due-dates under uncertainty when compared to various standard approaches. The practical applicability of our approach is verified using a real stamping industry application, in which the stamping parts are various types of voice coil motor yokes used in commercial hard disk drive actuators. Apart from FMSs, the proposed approach can also be applied to various other industries including project scheduling, airline scheduling, transportation scheduling.     

炼油企业生产计划和能量系统集成优化研究

炼油企业通常由生产过程和公用工程两个主要部分组成。将生产计划和公用工程集成优化,不仅获得真正的全局最优解,而且克服了传统生产计划人为制定生产过程与公用工程之间中间物流（各压力等级的蒸汽,瓦斯等）价格的缺陷,避免了蒸汽减温减压和放空,以及炼厂气放空的经济损失和环境破坏。首先采用IDEF0方法建立了生产计划与公用工程的功能模型,描述了两者之间的相互关系。接着建立了生产计划与公用工程集成的混合整数线性规划模型。模型描述了装置能耗不仅与加工量相关,而且与装置的生产方案相关,更真实地反映了生产过程与能量系统之间的定量关系。模型还在全厂范围内进行各压力等级蒸汽和燃料的平衡。最后,将建立的全厂集成优化模型应用于中国北方的某大型炼油厂,验证了模型的有效性。     

Robust rendezvous for multi-robot system with random node failures: an optimization approach

In this paper, we consider the problem of designing distributed control algorithms to solve the rendezvous problem for multi-robot systems with limited sensing, for situations in which random nodes may fail during execution. We first formulate a distributed solution based upon averaging algorithms that have been reported in the consensus literature. In this case, at each stage of execution a one-step sequential optimal control (i.e., näive greedy algorithm) is used. We propose a distributed stochastic optimal control algorithm that minimizes a mean–variance cost function for each stage, given that the probability distribution for possible node failures is known a priori, as well as a minimax version of the problem when the prior probability distribution is not known. We demonstrate via extensive numerical simulations that our proposed algorithm provides statistically better rendezvous task performance than contemporary algorithms in cases for which failures occur.     

Optimal maintenance policy for equipment subject to random deterioration and random failure A modern control theory approach

The stochastic version of Pontryagin's maximum principle is applied to determine an optimal maintenance policy of equipment subject to random deterioration. The deterioration of the equipment with age is modelled as a random process. Next the model is generalized to include random catastrophic failure of the equipment. The optimal maintenance policy is derived for two special probability distributions of time to failure of the equipment, namely, exponential and Weibull distributions Both the salvage value and deterioration rate of the equipment are treated as state variables and the maintenance as a control variable. The result is illustrated by an example.     

A multi objective optimization approach for flexible job shop scheduling problem under random machine breakdown by evolutionary algorithms

This paper addresses the stable scheduling of multi-objective problem in flexible job shop scheduling with random machine breakdown. Recently, numerous studies are conducted about robust scheduling; however, implementing a scheme which prevents a tremendous change between scheduling and after machine breakdown (preschedule and realized schedule, respectively) can be critical for utilizing available resources. The stability of the schedule can be detected by a slight deviation of start and completion time of each job between preschedule and realized schedule under the uncertain conditions. In this paper, two evolutionary algorithms, NSGA-II and NRGA, are applied to combine the improvement of makespan and stability simultaneously. A simulation approach is used to evaluate the state and condition of the machine breakdowns. After the introduction of the evaluation criteria, the proposed algorithms are tested on a variety of benchmark problems. Finally, through performing statistical tests, the algorithm with higher performance in each criterion is identified.     

The machine breakdown paradox: How random shifts in the production rate may increase company profits

The causes and effects of machine breakdowns have frequently been investigated in the past. One popular stream of research studies technical errors in production and analyzes their impact on the inventory policy of the company. In this paper, we show that random shifts in the production rate of a machine, which may occur, for example, due to technical defects, may lead to a reduction in total cost and therewith to an increase in profit. This obvious paradox may lead to situations where it is economically rational for the company to sustain a technically inefficient situation, or even to take measures to intentionally induce a shift in the production rate, for example by damaging the machine on purpose. In this paper, we illustrate this paradox by referring to an existing inventory model, and trace it back to common assumptions made in the literature.     

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.     

Robust production and maintenance planning in stochasticmanufacturing systems

In this paper the authors consider a preventive maintenance and production model of a flexible manufacturing system with machines that are subject to breakdown and repair. The preventive maintenance can be used to reduce the machine failure rates and improve the productivity of the system. The control variables are the rate of maintenance and the rate of production; the objective is to choose a control process that optimizes a robust cost of inventory/shortage, production, and maintenance. The value function is shown to be locally Lipschitz and to satisfy a Hamilton-Jacobi-Isaacs equation. A sufficient condition for optimal control is obtained. Finally, an algorithm is given for solving the optimal control problem numerically     

Robust optimization of constrained mechanical system with joint clearance and random parameters using multi-objective particle swarm optimization

Joint clearance and uncertainty are inevitable in mechanical systems due to design tolerance, abrasion, manufacture error, assembly error and imperfections. In this study, kinematic analysis and robust optimization of constrained mechanical systems with joint clearance and random parameters were performed. Joint clearance was modeled by Lankarani-Nikravesh contact force model, and probability space was applied for characterizing uncertain parameters. A kinematic analysis method based on Baumgarte approach and confidence region method was presented to predict kinematic error of the mechanical system. Slider-crank mechanism, an illustrative example was presented to show the influence of clearance and uncertainty on the kinematic accuracy. Then, a novel multi-objective robust optimization methodology was presented for kinematic accuracy robust optimization design of the constrained mechanical system. In this approach, a multi-objective robust optimization model derived from 95% confidence region is constructed to reduce the effects of clearance and parameter uncertainty on 95% confidence region of kinematic error. The robust optimization model is a double-loop process. A multi-objective robust optimization strategy, combing Kriging surrogate model, multi-objective particle swarm optimization, confidence region and Monte Carlo methods, was proposed to search the design variables for minimizing the optimization objectives derived from confidence region while balancing computational accuracy and efficiency of the optimization process. The optimal results of the slider-crank mechanism demonstrated the validity and feasibility of the proposed robust optimization method.     

Multi-objective optimization by genetic algorithm of structural systems subject to random vibrations

This paper deals with a multi-objective optimization criterion for linear viscous-elastic device utilised for decreasing vibrations induced in mechanical and structural systems by random loads. The proposed criterion for the optimum design is the minimization of a vector objective function. The multi-objective optimization is carried out by means of a stochastic approach. The design variables are the device frequency and the damping ratio. As cases of study, two different problems are analysed: the base isolation of a rigid mass and the tuned mass damper positioned on a multi degree of freedom structural system subject to a base acceleration. The non-dominated sorting genetic algorithm in its second version (NSGA-II) is adopted to obtain the Pareto sets and the corresponding optima for different characterizations of the system and input.     

Optimizing a production system with rework and equal sized batch shipments

In several cases producing new or recovering defective products takes place on a common facility, with these activities are carried out in lots. Consequently, there is a necessity to coordinate the production and rework activities with respect to the timing of operations and also with regard to appropriate lot sizes for both processes while completely satisfying a given demand. Thereby, a decision has to be made on how to assign units completed at one stage to partial lots—called batches—for shipment to the next operation. In this paper we present a lot size model which addresses all of these aspects. Based on total relevant costs per unit time, an optimization method is developed to determine the economic production and rework quantity as well as the corresponding batch sizes for both activities. The algorithm is illustrated by a numerical example followed by a sensitivity analysis of the models behavior under different problem parameters.     

A random search heuristic for a multi-objective production planning

Real world production planning is involved in optimizing different objectives while considering a spectrum of parameters, decision variables, and constraints of the corresponding cases. This comes from the fact that production managers desire to utilize from an ideal production plan by considering a number of objectives over a set of technological constraints. This paper presents a new multi-objective production planning model which is proved to be NP-Complete. So a random search heuristic is proposed to explore the feasible solution space with the hope of finding the best solution in a reasonable time while extracting a set of Pareto-optimal solutions. Then each Pareto-optimal solution is considered as an alternative production plan in the hand of production manager. Both the modeling and the solution processes are carried out for a real world problem and the results are reported briefly. Also, performance of the proposed problem-specific heuristic is verified by comparing it with a multi-objective genetic algorithm on a set randomly generated test data.     

A volume flexible economic production lot-sizing problem with imperfect quality and random machine failure in fuzzy-stochastic environment

An “economic production lot size” (EPLS) model for an item with imperfect quality is developed by considering random machine failure. Breakdown of the manufacturing machines is taken into account by considering its failure rate to be random (continuous). The production rate is treated as a decision variable. It is assumed that some defective units are produced during the production process. Machine breakdown resulting in idle time of the respective machine which leads to additional cost for loss of manpower is taken into account. It is assumed that the production of the imperfect quality units is a random variable and all these units are treated as scrap items that are completely wasted. The models have been formulated as profit maximization problems in stochastic and fuzzy-stochastic environments by considering some inventory parameters as imprecise in nature. In a fuzzy-stochastic environment, using interval arithmetic technique, the interval objective function has been transformed into an equivalent deterministic multi-objective problem. Finally, multi-objective problem is solved by Global Criteria Method (GCM). Stochastic and fuzzy-stochastic problems and their significant features are illustrated by numerical examples. Using the result of the stochastic model, sensitivity of the nearer optimal solution due to changes of some key parameters are analysed.