A comparison of adaptive and stationary repair-cost limit criteria for a producer's warranty

A producer sells a product together with a warranty valid fora specified duration.Whenever the product fails during the warrantyperiod, the producer is responsible for making the product functional,either by replacing or repairing the equipment,according towhether the anticipated repair cost exceeds or does not exceeda specified repair-cost limit. It is assumed that repairs areminimal repairs, and that the distribution of repair costs isknown, as is the time-to-failure distribution of the product.In this paper, we compare two models in discrete time. The firstone involves an adaptive repair-cost limit which is set dynamicallyaccording to the age of the product, and the length of warrantyremaining. The second model assumes a constant or stationaryrepair-cost limit throughout the warranty period. Empiricalresults are summarized.     

Optimal age-replacement time with minimal repair based on cumulative repair-cost limit for a system subject to shocks

An operating system is subject to random shocks that arrive according to a non-homogeneous Poisson process and cause the system failed. System failures experience to be divided into two categories: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. An age-replacement model is studied by considering both a cumulative repair-cost limit and a system’s entire repair-cost history. Under such a policy, the system is replaced at age T, or at the k-th type-I failure at which the accumulated repair cost exceeds the pre-determined limit, or at any type-II failure, whichever occurs first. The object of this article is to study analytically the minimum-cost replacement policy for showing its existence, uniqueness, and the structural properties. The proposed model provides a general framework for analyzing the maintenance policies, and presents several numerical examples for illustration purposes.     

A graphical method to repair-cost limit replacement policies with imperfect repair

In this paper, we consider a different type of repair-cost limit replacement problem with imperfect repair from earlier models. We focus on the problem of determining the timing to stop repairing a unit after it fails, and propose a nonparametric method to estimate the optimal repair-cost limit which minimizes the total expected cost per unit time in the steady-state, by applying the total time on test (TTT) concept. Through a numerical example, the optimal policy is calculated from the repair-cost data directly, and the benefit of the proposed method is shown.     

Optimal replacement model with age-dependent failure type based on a cumulative repair-cost limit policy

This paper presents a replacement model with age-dependent failure type based on a cumulative repair-cost limit policy, whose concept uses the information of all repair costs to decide whether the system is repaired or replaced. As failures occur, the system experiences one of the two types of failures: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. A critical type-I failure means a minor failure at which the accumulated repair cost exceeds the pre-determined limit for the first time. The system is replaced at the nth type-I failure, or at a critical type-I failure, or at first type-II failure, whichever occurs first. The optimal number of minimal repairs before replacement which minimizes the mean cost rate is derived and studied in terms of its existence and uniqueness. Several classical models in maintenance literature are special cases of our model.     

Probability distributions of cost,revenue and profit over a warranty cycle

A warranty cycle starts with the sale of a new item and terminates when the warranty servicing obligation of the manufacturer created by this sale ends. Cost and profit measures over a warranty cycle are of interest with regards to accounting, pricing, product quality, warranty policy design, and other issues. For most predictive and decision making purposes, these variables cannot be described adequately by their means alone. In this paper, we obtain the probability distributions of the manufacturer's rebate, cost, revenue and profit during a cycle, under a (combination) freereplacement/pro-rata warranty policy, with the incorporation of the customer repurchase behavior under warranty. We first derive the joint distribution of the numbers of free and pro-rata replacements during a cycle. We then construct and compute the distributions of interest by conditioning on the numbers of such replacements. In so doing, we take into account the warranty policy, in terms of the lengths of the free and pro-rata periods, the cost and rebate profiles, customer repurchase probabilities during a pro-rata period, and the failure time distribution of the product. We provide several numerical examples, using data that are typical of battery warranties.     

Maintenance optimization for second‐hand products following periodic imperfect preventive maintenance warranty period

The maintenance policy for a product's life cycle differs for second‐hand and new products. Although several maintenance policies for second‐hand products exist in the literature, they are rarely investigated with reference to periodic inspection and preventive maintenance action during the warranty period. In this research, we study an optimal post‐warranty maintenance policy for a second‐hand product, which was purchased at age x with a fixed‐length warranty period. During the warranty period, the product is periodically inspected and maintained preventively at a prorated cost borne by the user, while any product failure is only minimally repaired by the dealer. After the warranty expires, the product is self‐maintained by the user for a fixed‐length maintenance period and the costs incurred during this time are fully borne by the user. At the end of the maintenance period, the product is replaced with a product of the user's choice. This study is focused on the determination of an optimal length for the maintenance period after the warranty expiration. As a criterion for the optimality, we adopt the long‐run mean cost during the second‐hand product's life cycle from the user's perspective. Finally, our results are analyzed numerically for sensitive analysis of several relevant factors, assuming that the failure distribution follows a Weibull distribution.     

Diagnostics in Birnbaum–Saunders accelerated life models with an application to fatigue data

In industrial statistics, there is great interest in predicting with precision lifetimes of specimens that operate under stress. For example, a bad estimation of the lower percentiles of a life distribution can produce significant monetary losses to organizations due to an excessive amount of warranty claims. The Birnbaum–Saunders distribution is useful for modeling lifetime data. This is because such a distribution allows us to relate the total time until the failure occurs to some type of cumulative damage produced by stress. In this paper, we propose a methodology for detecting influence of atypical data in accelerated life models on the basis of the Birnbaum–Saunders distribution. The methodology developed in this study should be considered in the design of structures and in the prediction of warranty claims. We conclude this work with an application of the proposed methodology on the basis of real fatigue life data, which illustrates its importance in a warranty claim problem. Copyright © 2012 John Wiley & Sons, Ltd.     

按比例保修策略下多保修期差别定价研究

保修策略是联系厂商和顾客的一种合同责任,它往往用于促进产品的销售。产品保修期长短与其定价有着紧密的联系。本文在考虑一个厂商采用按比例保修策略及设定多个保修期的基础上,以产品的保修期和价格为决策变量,以厂商收益最大化为目标,研究了产品故障次数成指数分布下最优保修期和价格的组合策略,并通过算例比较了差别定价和单一定价下厂商的收益。最后,通过灵敏度分析,研究了消费者保修期偏移弹性对厂商收益所产生的影响。     

Warranty cost analysis for products with a dormant state

Most building services products are installed while a building is constructed, but they are not operated until the building is commissioned. The warranty of the products may cover the time starting from their installation to the end of the warranty period. Prior to the commissioning of the building, the products are at a dormant mode (i.e., not operated) but protected by the warranty. For such products, both the usage intensity and the failure patterns are different from those with continuous usage intensity and failure patterns. This paper develops warranty cost models for repairable products with a dormant mode from both the manufacturer’s and buyer’s perspectives. Relationships between the failure patterns at the dormant mode and at the operational mode are also discussed. Numerical examples and sensitivity analysis are used to demonstrate the applicability of the methodology derived in the paper.     

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本文考虑了一个其产品保修期内免费小修的退化 生产系统的定期检修策略. 系统的退化过程包括三个状态: 可控制状态, 不可控制状态, 故障状态. 过程呆在可控制状态和不可控制状态的时间假设都服从指数分布. 生产系统在固定的时刻t或发生故障时进行检修, 两者以先发生为准. 本文讨论了使单位产品每周期期望成本最小的最优定期检修时间本文考虑了一个其产品保修期内免费小修的退化生产系统的定期检修策略.系统的退化过程包括三个状态:可控制状态,不可控制状态,故障状态.过程呆在可控制状态和不可控制状态的时间假设都服从指数分布.生产系统在固定的时刻t﹡或发生故障时进行检修,两者以先发生为准.本文讨论了使单位产品每周期期望成本最小的最优定期检修时间t﹡,三种特殊情况显示了最优值t的性质.此外,灵敏性分析和数字实例说明了模型中的参数对最优定期检修策略的影响.     

Evaluation of the warranty cost of a product with type III stochastic dependence between components

This paper analyses respectively the expected warranty costs from the perspectives of the manufacturer and the consumer. For a two-component series system with stochastic dependence between components, both the non-renewing free replacement policy and the renewing replacement policy are examined. It is assumed that whenever component 1 fails, a random damage to component 2 is occurred while a component 2 failure causes the system failure. Component 2 fails when its total accumulative damage exceeds a pre-determined level L. By considering the consumer’s behavior and the product service time, the warranty costs allocations between the manufacturer and the consumer are presented. Numerical examples are given to demonstrate the applicability of the methodology. It is proved that, independent of the type of the warranty policy, the failure interaction between components impacts the manufacturer profits and the consumer costs. The initial warranty length has also an impact on the product quality preferences to both the consumer and the manufacturer.     

Optimal warranty policies for systems with imperfect repair

We investigate a system whose basic warranty coverage is minimal repair up to a specified warranty length. An additional service is offered whereby first failure is restored up to the consumers’ chosen level of repair. The problem is studied under two system replacement strategies: periodic maintenance before and after warranty. It turns out that our model generalizes the model of Rinsaka and Sandoh [K. Rinsaka, H. Sandoh, A stochastic model with an additional warranty contract, Computers and Mathematics with Applications 51 (2006) 179–188] and the model of Yeh et al. [R.H. Yeh, M.Y. Chen, C.Y. Lin, Optimal periodic replacement policy for repairable products under free-repair warranty, European Journal of Operational Research 176 (2007) 1678–1686]. We derive the optimal maintenance period and optimal level of repair based on the structures of the cost function and failure rate function. We show that under certain assumptions, the optimal repair level for additional service is an increasing function of the replacement time. We provide numerical studies to verify some of our results.     

Optimal pricing,EOL (end of life) warranty,and spare parts manufacturing strategy amid product transition

We study firm’s strategy to determine its product price and warranty period, and plan the spare parts manufacturing so as to maximize its profit and at the same time to fulfill its commitment to providing the customer with the key part continuously over the relevant decision time horizon, i.e., the product’s life cycle plus its EOL service (warranty) period. To examine the research question, we develop and solve a two-stage optimal control theory model. From the numerical analysis, we infer as follows. It is not always true that the longer the EOL warranty period, the better for the company’s profitability, implying there exists an optimal EOL warranty period that balances all the relevant forces like market demand and cost structures. The relationship between optimal EOL warranty period and failure rate (defect rate) is concave: when the defect rate is moderate, the company has to increase its EOL warranty period as the defect rate increases so as to compensate for the deteriorating quality; but, when the defect rate is beyond a threshold level, the company needs to curtail its EOL warranty commitment as the defect rate increases in order to avoid excessive cost to service the failed parts. By depicting key dynamics in this managerial problem, this paper sheds light on how to make decision for optimal pricing and warranty when the product life cycle is finite and the company is obliged to provide after-sales services to customers for an extended period of time after the current product is no longer produced.     

基于风险调整控制图的产品保证索赔监控和预警研究

为了解决传统索赔监控与预警模型存在的未考虑索赔率大小对预警能力的影响、模型对保证数据精度要求高、难以处理索赔率较低的失效模式等问题,基于非参数估计和泊松分布对单位时间内产品保证索赔记录进行了拟合,根据各监控阶段的可用信息量大小构造了分阶段风险调整控制图对保证索赔进行监控和预警,并采用了在指定监控时间内对所有销售产品进行统一监控的监控方式。模型的有效性利用实际算例从误报警率、报警能力和响应速度三个指标进行了分析。     

Optimal preventive-maintenance warranty policy for repairable products

This paper investigates preventive-maintenance warranty (PMW) policies for repairable products. When the length of a warranty period is pre-specified, the optimal number of preventive maintenance actions, corresponding maintenance degrees, and the maintenance schedule are jointly determined. We show that there exists a unique optimal policy such that the expected total warranty cost is minimized. Under some reasonable conditions, structural properties of the optimal policy are obtained. Using these properties, an algorithm is provided to efficiently search for the optimal policy. In the case where the failure density is Weibull, a closed form expression of the optimal policy is derived. Finally, the impact of providing preventive maintenance within a warranty period is illustrated through a numerical example.     

Lot sizing and testing for items with uncertain quality

We consider the problem where items are produced in lots and sold with warranty. Due to manufacturing variability, some items do not conform to the design specifications and their performance is inferior (for example, have higher failure rate). The warranty servicing cost for these is much higher than for those which conform. Two approaches have been advocated for reducing the warranty cost per item released and in both it is achieved at the expense of increased manufacturing cost. The first involves life testing to weed out nonconforming items and the second involves strategies to reduce nonconforming items being produced. In this paper, the authors develop a model which combines both approaches and quality control decisions are made optimally to minimize the total (manufacturing and warranty) cost. It extends the earlier models of the authors which deals with quality decisions based solely on either the first or the second approach.     

关于汽车追尾冲击模型的维修策略

讨论了关于汽车追尾的冲击模型的可修系统．在系统不能修复如新的条件下,假定汽车运行时间构成随机递减的几何过程,逐次追尾后的维修时间构成随机递增的几何过程．分别考虑汽车按比例保修和免费保修条件下,以汽车追尾次数N为策略,以车主在汽车长期运行单位时间内的期望费用为目标函数,导出目标函数的解析表达式P1（N）与P2（N）．最后,通过实例分析,求出最优策略N＊,使得车主在汽车长期运行单位时间内的期望费用最小．     

Optimal maintenance strategy under renewable warranty with repair time threshold

In this paper, we consider a periodic preventive maintenance model, from the manufacturer's perspective, which can be implemented to reduce the maintenance cost of a repairable product during a given warranty period. The product is assumed to deteriorate with age and the warranty policy we adopt in this paper takes into account the two factors of failure time and repair time of the product when the product failure occurs. Under the proposed two-factor warranty, a repair time threshold is pre-determined and if the repair takes more time than that of the threshold, the failed product is replaced with a renewed warranty policy. Otherwise, the product is only minimally repaired to return to the operating state. During such a renewable warranty period, preventive maintenance is conducted to reduce the rate of degradation periodically while the product is in operation. By assuming certain cost structures, we formulate the expected warranty cost during the warranty period from the manufacturer's perspective when a periodic preventive maintenance strategy is adapted. Although more frequent preventive maintenance increases the warranty cost, the chance of product failures would be reduced. The main aim of this paper is to accomplish the optimal trade-off between the warranty cost and the preventive maintenance period by determining the optimal preventive maintenance period that minimizes the total expected warranty cost during the warranty period. Assuming the power law process for the product failures, we illustrate our proposed maintenance model numerically and study the impact of relevant parameters on the optimal preventive maintenance policy.     

Modeling and analysis of a warranty policy using new and reconditioned parts

Remanufacturing processes such as refurbishing and reconditioning can extend the life of a product returned from the field. This provides financial opportunities and allows manufacturers to engage in sustainable practices. However, the inability to access a sufficient quantity of reconditioned components from end‐of‐life products can force the concurrent utilization of new components. This paper deals with the determination of an optimal warranty policy where a mixture of new and reconditioned components are used to carry out replacements upon failure for products under warranty. A mathematical optimization model is developed to maximize the manufacturer's expected total profit based on four decision variables: the warranty length, the sale price, the age of reconditioned components, and the proportion of reconditioned components to be used. A numerical procedure is used to compute the optimal solution. Numerical results are provided and discussed to demonstrate the validity and the added value of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Cost analysis on renewable full-service warranties for multi-component systems

This paper presents a novel warranty policy named ‘full-service warranty’ (FSW) for repairable multi-component systems under which the failed component(s) or subsystem(s) will be replaced, in addition, a (perfect) maintenance action will be performed to reduce the chance of future system failure, both free of charge to consumers. Such a policy is desirable for both consumers and manufacturers since consumers receive better warranty service compared to the traditional free repair policy, at the same time, manufacturers may enjoy increase in sale as well as cost-saving due to improved product reliability by the maintenance action. Under the renewable FSW policy, from manufacturers’ point of view, cost models for complex systems with series, parallel, series–parallel (s–p) and parallel–series (p–s) structure are developed. Exact expressions for the first and second centered moments of warranty cost per product sold are obtained. Sensitivity analysis is performed based on a numerical example.