Determining the optimal run time for EPQ model with scrap,rework, and stochastic breakdowns

This paper is concerned with determination of optimal run time for an economic production quantity (EPQ) model with scrap, rework, and stochastic machine breakdowns. In real life manufacturing systems, generation of defective items and random breakdown of production equipment are inevitable. In this study, a portion of the defective items is considered to be scrap, while the other is assumed to be repairable. Total production-inventory cost functions are derived respectively for both EPQ models with breakdown (no-resumption policy is adopted) and without breakdown taking place. These cost functions are integrated and the renewal reward theorem is used to cope with the variable cycle length. Theorems on conditional convexity of the integrated overall costs and bounds of the production run time are proposed and proved. We conclude that the optimal run time falls within the range of bounds and it can be pinpointed by the use of the bisection method based on the intermediate value theorem. Numerical example is provided to demonstrate its practical usages.     

Determining the optimal production policy for a system with imperfect production and shortage

The classic EPQ model assumes that items are produced of perfect quality and no shortage is permitted. In the real world situation, however, due to process deterioration or other factors, the occurrence of imperfect quality items is inevitable. This paper develops an extended economic production quantity (EPQ) model with imperfect production, shortage, and imperfect rework. We assume that the quality scan is conducted during the production. The scanned imperfect items are classified as the repairable and scrap. We consider that not all of the repairable items can be restored to meet the specified quality standard. Only some portion of defective items can be restored as normal items, the other results in defective, due to repair failure, can be sold at a discounted price to a secondary market. The renewal reward theorem is utilized to deal with the variable cycle length. The production quantity and the shortage level are determined in an optimal manner so as to minimize the average system cost. A numerical example is used to demonstrate its practical usage. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal maintenance policy in a multistate deteriorating standby system

An optimal maintenance policy for a multistate deteriorating standby system is proposed in this study. Traditionally, a system could only presume two operational states: success or failure, and the maintenance policy is to determine the optimal number of standby components, subject to factors such as maintenance capability, cost of the standby items, etc., so as to minimize the operational cost. This study considers a more general production system in which progressive deterioration is incurred during the operating time, hence resulting in degrading performance. By modeling the system as a multistate deteriorating system, an optimal maintenance policy is obtained by determining the optimal number of standby components required in the system and the optimal state in which the replacement of deteriorating components shall be made.     

An extended replacement policy for a deteriorating system with multi-failure modes

In this paper, the maintenance problem for a deteriorating system with k + 1 failure modes, including an unrepairable failure (catastrophic failure) mode and k repairable failure (non-catastrophic failure) modes, is studied. Assume that the system after repair is not “as good as new” and its deterioration is stochastic. Under these assumptions, an extended replacement policy N is considered: the system will be replaced whenever the number of repairable failures reaches N or the unrepairable failure occurs, whichever occurs first. Our purpose is to determine an optimal extended policy N^∗ such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal extended policy N^∗ can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results of the repair model proposed in this paper.     

有两种故障状态的两部件串联系统的预防维修策略

本文研究的是由两个部件串联组成且有两种故障状态的系统的预防维修策略, 当系统的工作时间达到T时进行预防维修, 预防维修使部件恢复到上一次故障维修后的状态。每个部件发生故障都有两种状态, 可维修和不可维修。当部件的故障为可维修故障时, 修理工对其进行故障维修, 且每次故障维修后的工作时间形成随机递减的几何过程, 每次故障后的维修时间形成随机递增的几何过程。当部件发生N次可维修故障或一次不可维修故障时进行更换。以部件进行预防维修的间隔和更换前的可维修故障次数N组成的二维策略(T, N) 为策略, 利用更新过程和几何过程理论求出了系统经长期运行单位时间内期望费用的表达式, 并给出了具体例子和数值分析。     

Optimal production run time for a deteriorating production system under an extended inspection policy

A deteriorating production system is subjected to random deterioration from an in-control state to an out-of-control state with a general shift distribution. In order to reduce the defective items, part inspection policy, under which production inspections are performed only at the end of the production run, and full inspection policy are both considered in the literature. Moreover, the former dominates the latter. Since the product produced towards the end of a production cycle are more likely to be defective, it can further economize the inspection costs that they are directly reworked without inspection. In this paper, we propose an extended product inspection policy for a deteriorating production system. Product inspections are performed in the middle of a production cycle, and after the inspection, all products produced until the end of the production run are fully reworked. Based on the model, we show that there exists a production run time and a corresponding unique inspection policy such that the expected total cost per item per cycle is minimized. Finally, numerical examples are provided to illustrate our extended inspection policy, and indicate that such product inspection model will reduce the quality-related cost than part inspection does.     

An optimization problem of manufacturing systems with stochastic machine breakdown and rework process

This paper is concerned with optimization of production run time that takes stochastic breakdown and the reworking of defective items into consideration. In a real‐life manufacturing process, production of imperfect quality items as well as random breakdowns of production equipment is inevitable. All defective items produced are assumed to be repairable through a rework process right after the regular production stops in each cycle. This research starts with derivations of the cost functions for production systems with breakdown (no‐resumption policy is considered) and without breakdown taking place, respectively. Then cost functions of both cases are integrated. Theorems on conditional convexity of the overall cost function and bounds for optimal production run time are proposed and proved. This study concludes that although the optimal run time cannot be expressed in a closed form, it falls within the range of bounds. Hence, it can be pinpointed by the use of the bisection method based on the intermediate value theorem. A numerical example is provided to demonstrate its practical usages. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal replenishment policy for imperfect quality EMQ model with rework and backlogging

This paper derives the optimal replenishment policy for imperfect quality economic manufacturing quantity (EMQ) model with rework and backlogging. The classic EMQ model assumes that all items produced are of perfect quality. However, in real‐life manufacturing settings, generation of imperfect quality items is almost inevitable. In this study, a random defective rate is assumed. All items produced are inspected and the defective items are classified as scrap and repairable. A rework process is involved in each production run when regular manufacturing process ends, and a rate of failure in repair is also assumed. Unit disposal cost and unit repairing and holding costs are included in our mathematical modelling and analysis. The renewal reward theorem is employed in this study to cope with the variable cycle length. The optimal replenishment policy in terms of lot‐size and backlogging level that minimizes expected overall costs for the proposed imperfect quality EMQ model is derived. Special cases of the model are identified and discussed. Numerical example is provided to demonstrate its practical usage. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimal production run length for products sold with warranty

This article studies the optimal production run length for a deteriorating production system in which the products are sold with free minimal repair warranty. The deterioration process of the system is characterized by a two-state continuous-time Markov chain. For products sold with free minimal repair warranty, we show that there exists a unique optimal production run length such that the expected total cost per item is minimized. Since there is no closed form expression for the optimal production run length, an approximate solution is derived. In addition, three special cases which provide bounds for searching the optimal production run length are investigated and some sensitivity analysis is carried out to study the effects of the model parameters on the optimal production run length. Finally, a numerical example is given to evaluate the performance of the optimal production run length.     

Optimal economic production quantity policy for imperfect process with imperfect repair and maintenance

This study integrates maintenance and production programs with the economic production quantity (EPQ) model for an imperfect process involving a deteriorating production system with increasing hazard rate: imperfect repair and rework upon failure (out of control state). The imperfect repair performs some restorations and restores the system to an operating state (in-control state), but leaves its failure until perfect preventive maintenance (PM) is performed. There are two types of PM, namely imperfect PM and perfect PM. The probability that perfect PM is performed depends on the number of imperfect maintenance operations performed since the last renewal cycle. Mathematical formulas are obtained for deriving the expected total cost. For the EPQ model, the optimum run time, which minimizes the total cost, is discussed. Various special cases are considered, including the maintenance learning effect. Finally, a numerical example is presented to illustrate the effects of PM, setup, breakdown and holding costs.     

Optimal production time and number of maintenance actions for an imperfect production system under equal-interval maintenance policy

This paper deals with the optimal production/maintenance (PM) policy for a deteriorating production system which may shift from the in-control state to the out-of-control state while producing items. The process is assumed to have a general shift distribution. Under the commonly used maintenance policy, equal-interval maintenance, the joint optimizations of the PM policy are derived such that the expected total cost per unit time is minimized. Different conditions for optimality, lower and upper bounds and uniqueness properties on the optimal PM policy are provided. The implications of another commonly used policy, to perform a maintenance action only at the end of the production run, are also discussed. Structural properties for the optimal policy are established so that an efficient solution procedure is obtained. In the exponential case, some extensions of the results obtained previously in the literature are presented. A numerical example is provided to illustrate the solution procedure for the optimal production and maintenance policy.     

生产线随机崩坍情形下的变质品EPQ策略

变质品生产过程,可能率先出现"次品"的不稳定生产情形,随后机器崩坍;生产状态稳定性迁移时机、机器崩坍时间、维修时间皆乃随机变量;同时,企业无法观测当期需求,只能根据前期需求而随机地安排启动生产时刻.理论模型及数值算例皆表明,此种情况下,企业可以非等周期生产,存在组织生产次数(N)与生产率(P)的优解.敏感度分析看出,当需求拖后率增加、变质率+次品率降低时,企业成本显著降低,但首期生产启动时刻、生产率几乎没有变化.     

Optimal production run length for deteriorating production system with a two-state continuous-time Markovian processes under allowable shortages

The paper studies optimal production run length for a deteriorating production system in which the shortages are allowed and the deterioration processes are characterized by a two-state continuous-time Markov chain. We show that there exists a unique optimal production run length to minimize the expected total relevant cost. In addition, bounds for the optimal production run length are provided to develop the solution procedure. Finally, a numerical example is given to illustrate the results and sensitivity analysis is also performed     

Economic production quantity model with repair failure and limited capacity

We develop an economic production quantity (EPQ) model with random defective items and failure in repair. The existence of only one machine results with limited production capacity and shortages. The aim of this research is to derive the optimal cycle length, the optimal production quantity and the optimal back ordered quantity for each product so as to minimize the total expected cost (holding, shortage, production, setup, defective items and repair costs). The convexity of the model is derived and the objective function is proved convex. Two numerical examples illustrate the practical usage of the proposed method.     

Performance modeling of state dependent system with mixed standbys and two modes of failure

The present investigation deals with a multicomponent repairable system with state dependent rates. For smooth functioning of the system, mixed standbys (warm and cold) are provided so that the failed units are immediately replaced by standbys if available. To prevent congestion in the system due to failure of units, permanent along with additional repairmen are provided to restore the failed units. It is assumed that the units may fail in two modes. The units have exponential life time and repair time distributions. The failed unit may balk in case of heavy load of failed units. The failed units may also wait in the queue and renege on finding the repairmen busy according to a pre-specified rule. The Chapman–Kolmogorov equations, governing the model in the form of matrix are constructed using transition flow rates of different states. The steady state solution of queue size distribution is derived using product formula. A cost function is suggested to determine the optimal number of warm and cold standbys units required for the desired level of quality of service. The numerical illustrations are carried out to explore the effect of different parameters on performance measures.     

Interrupt and Opportunistic Replacement Strategies for Systems of Deteriorating Components

In some factories production epochs occur that allow system components to be replaced at reduced cost. Over a long production run the unit cost of replacing these stochastically deteriorating components can be controlled by decisions which govern when production is to be interrupted for component replacement and when components are to be replaced at the reduced cost replacement opportunities. This paper develops and analyses models for optimizing "interrupt and opportunistic" replacement strategies in simple systems. Numerical results are given that illustrate the advantages of combining interrupt replacement with opportunistic replacement.     

Optimal repair–replacement policies for a system with two types of failures

In this paper, the optimal replacement problem is investigated for a system with two types of failures. One type of failure is repairable, which is conducted by a repairman when it occurs, and the other is unrepairable, which leads to a replacement of the system at once. The repair of the system is not “as good as new”. The consecutive operating times of the system after repair form a decreasing geometric process, while the repair times after failure are assumed to be independent and identically distributed. Replacement policy N is adopted, where N is the number of repairable failures. The system will be replaced at the Nth repairable failure or at the unrepairable failure, whichever occurs first. Two replacement models are considered, one is based on the limiting availability and the other based on the long-run average cost rate of the system. We give the explicit expressions for the limiting availability and the long-run average cost rate of the system under policy N, respectively. By maximizing the limiting availability A(N) and minimizing the long-run average cost rate C(N), we theoretically obtain the optimal replacement policies N^∗ in both cases. Finally, some numerical simulations are presented to verify the theoretical results.     

泊松冲击下单部件混合贮备系统可靠性分析

研究的是泊松冲击下由三同型部件组成的单部件混合贮备系统可靠性.在系统三部件分别为工作部件、温贮备部件和冷贮备部件假定下,且假定三部件在工作过程中都是接着工作、修理、冷贮备、温贮备、工作依次进行状态转移,同时假设部件寿命和维修时间都服从指数分布,利用基本概率理论和马尔可夫过程理论分别对不可修和可修两类系统进行讨论得到相关可靠性指标.     

Joint Determination of Production Cycle and Inspection Intervals in a Deteriorating Production System

This paper considers production-maintenance policy for the deteriorating production system which can go ‘out of control’ while producing items. Once out of control, the production process produces some proportion of defective items. The defective items are reworked at some cost before being shipped, or, if passed to the customer, incur much larger warranty cost. Thus, to operate this system economically, periodic inspection and restoration of the process are needed. A mathematical model representing the expected annual cost is developed to determine the production cycle and process inspection intervals jointly. A case of equally spaced inspection intervals is solved by using an approximation to the cost function.     

Mathematical modelling for determining economic batch size and optimal number of deliveries for EPQ model with quality assurance

The classic economic production quantity (EPQ) model assumes a continuous inventory-issuing policy for satisfying product demand and a perfect production for all items produced. However, in a real-life vendor–buyer integrated system, a multi-delivery policy is often used in lieu of continuous issuing policy and it is inevitable to generate defective items during a production run. This study addresses these issues by incorporating multiple deliveries of the finished batch, customer's inventory-holding cost and manufacturer's quality assurance cost into an EPQ model with the imperfect reworking of random defective items. Mathematical modelling and analyses are employed. Convexity of the long-run expected cost function is proved by the use of Hessian matrix equations, and the closed-form solutions in terms of the optimal lot size and optimal number of deliveries are obtained. The research results are demonstrated with a numerical example with a discussion on its practical usage.