Failure modeling and optimizing preventive maintenance strategy during two-dimensional extended warranty contracts

Offering extended warranty (EW) contracts for products such as automobiles is a good source of revenue for manufactures, insurers and third party companies. However, difficulties in the modeling of product’s failure process and assessing corrective and preventive maintenance actions’ effects on the reliability of product enforce the service providers to propose limited EW contracts with simple “minimal repair at failures” servicing strategy. In this paper for a product sold with a two-dimensional warranty, we model the failure process of product, the effect of imperfect preventive maintenance (PM) and corresponding servicing cost in terms of product’s age and usage. Then, we propose a mathematical optimization model to derive optimal number and degrees of preventive repairs to minimize the EW provider’s servicing cost. We also provide some guidelines to help the EW provider to design flexible EW contracts and determine their corresponding optimal maintenance strategies. To reproduce an illustrative numerical example, we use the failure history of a commercial vehicle produced in a plant in Iran. The provided results reveal that considering proper preventive maintenance strategy during the EW period may effectively reduces the cost of EW servicing.     

Designing and pricing of two-dimensional extended warranty contracts based on usage rate

For vehicles sold under a two-dimensional warranty, which consists of two-dimensional region of age and usage, automobile manufacturers or third-party after-sales departments offer free warranty service under the base warranty. Consumers require a longer and better developed warranty service because of the rapidly developing production technology and other reasons. In the topic of two-dimensional warranty, previous research has extended the coverage of the base warranty, which satisfies consumers’ requirements but places significant pressure on automobile manufacturers. This study proposes a designing and pricing model of two-dimensional extended warranty contracts. The mathematical model is divided into two categories: the purchase of two-dimensional extended warranty service at the point of sale and the expiration of the base warranty. Based on the maintenance strategy used during the extended warranty, we classify this study under the minimal repair and combined maintenance strategy models. With the use of the failure and usage rate distribution functions, the mathematical pricing models of each sub-case are presented. A numerical example of a vehicle made in China is given to illustrate the procedure of the models. The results of this study provide automobile manufacturers with guidelines on designing and pricing two-dimensional extended warranty contracts.     

Optimal burn-in strategy for two-dimensional warranted products considering preventive maintenance

Burn-in and preventive maintenance (PM) are effective approaches to reduce the number of warranty claims and warranty cost during post-sale support. With harsher burn-in settings, early product defects can be removed, but at the same time product degradation is accelerated and more wear-out failures may be introduced. PM actions within warranty alleviate these negative effects. This paper proposes an optimal burn-in strategy for repairable products sold with a two-dimensional base warranty (BW) and an optional extended warranty (EW). Both performance-based and cost-based models incorporating PMs are developed to obtain optimal burn-in settings, including the burn-in duration and the burn-in usage rate, so as to minimise the expected number of warranty claims and total cost respectively. The impacts of different accelerated coefficients and PM degrees on the optimal burn-in strategy are analysed. In view of the performance and cost structures, we conduct numerical examples to illustrate the applicability of the proposed models. Practical implications from a sensitivity analysis for key parameters are also elaborated.     

A two-dimensional preventive maintenance strategy for items sold with warranty

Conventional preventive maintenance (PM) strategies under two-dimensional (2D) warranties are usually age-based or usage-based, which means that the implementation of PM activities is based solely on item age or usage. In this paper, a new PM strategy, called 2D PM strategy, is proposed for items sold with a 2D warranty. Under this strategy, the item is preventively maintained every K units of age or L units of usage, whichever occurs first. The marginal approach is used to describe the effect of age and usage on item reliability by treating usage as a random function of age. Besides, the effect of PM is characterised by the reduction of virtual age. The objective of this study is to identify the optimal 2D PM strategy under fixed warranty terms so as to minimise the total expected warranty servicing cost from the manufacturer’s perspective. A numerical example is provided to demonstrate the effectiveness of the proposed PM strategy. It is shown that the 2D PM strategy contains the age-based and usage-based strategies as special cases, and outperforms them in terms of warranty servicing cost. Finally, how to implement the proposed PM strategy in practice is discussed with an illustrative case.     

Optimal preventive maintenance strategy for leased equipment under successive usage-based contracts

In the context of equipment leasing, maintenance service is usually bundled with the leased equipment and offered by the lessor as an integrated package under a lease contract. The lessor is then responsible to prescribe an effective maintenance policy to keep the equipment operational in an economical way. This paper investigates upgrade and preventive maintenance (PM) strategies for industrial equipment during successive usage-based lease contracts with consideration of a warranty period, from the lessor's perspective. The accelerated failure time model and age reduction model are adopted to capture the effect of usage rate and imperfect PM/upgrade on the equipment reliability, respectively. More importantly, since equipment usage rates may vary across different lease contracts, this study develops an age correspondence framework to characterise usage rate shifts between successive lease periods. The optimal upgrade degree and the optimal number and level of PM actions are progressively updated for each upcoming lease period to minimise the total expected lease servicing cost, by considering the usage rate and maintenance implementation history. Numerical studies show that under given cost structures, periodical PM activities within each lease period tends to outperform the pre-leasing upgrade actions, though both of them can reduce the lease servicing cost.     

基于寿命周期成本分析的产品延伸保修决策优化模型

随着用户对产品寿命周期成本的日益重视,延伸保修已成为生产商新的关注点。延伸保修对产品维修使用费用以及市场竞争力有很大影响,为得到产品延伸保修决策优化方案以降低寿命周期成本,在考虑预防性维修策略的基础上,分析了生产商延伸保修定价策略,构建出一种以用户在产品使用寿命期内单位时间总维修成本最小为目标的延伸保修决策优化模型,并提出了求解模型的网格搜索(grid search)算法。通过引入算例,借助Matlab仿真软件实现网格搜索算法,在保证生产商利益的基础上,有效的降低了产品单位时间总维修成本,延长了产品使用寿命周期长度,由此得到了用户最优延伸保修决策方案,验证了模型和算法的合理性和有效性。     

基于最小与缺陷维修的二维产品保证策略研究

通过对相关文献的研究,提出采用使用时间和使用程度二维变量把产品保证区域划分成三个子区域,对第二个子区域里发生的第一次故障采用缺陷维修策略,对在第一、第三个保证子区域里发生的故障和在第二个保证子区域内第一次故障后的所有故障都采用最小维修策略;通过建立该产品保证策略下产品保证成本模型并获得优化处理的结果,又与相关文献所提供数例处理结果比较后发现:当故障产品最小维修费用相对于更换维修费用较小时,本文提供的产品保证策略方案更优.     

Two-dimensional base warranty design based on a new demand function considering heterogeneous usage rate

Warranties are prevalent in the market. The manufacturer of products sold with warranties often faces a problem of balancing a trade-off between warranty cost and boosted demand by warranties when designing the warranties. For a two-dimensional base warranty with both age limit and usage limit, the heterogeneous usage rate of customers not only has an effect on warranty cost but also differentiates customers’ perceptions on warranty period. For example, the high-usage customers are more concerned about the usage limit of the warranty period, while the low-usage customers are more preferred to a longer age limit. This paper defines an attractiveness index based on the available warranty region to describe the extent of attractiveness of a two-dimensional warranty period to customers with heterogeneous usage rate. Moreover, a demand function based on the attractiveness index is proposed to describe the boosted demand of extending the warranty period. From the perspective of the manufacturer, the optimal two-dimensional base warranty period is designed by maximising the expected profit considering the trade-off. Both numerical examples and a case study are presented to illustrate the application of the proposed method.     

Optimal pricing and two-dimensional warranty policies for a new product

For commercial products, pricing and warranty are two crucial marketing strategies, which are used to promote the potential market share. The warranty policy adopted by most of the capital-intensive products, such as machines and automobiles, usually has two dimensions, i.e. warranty age and warranty usage. In this paper, we propose to investigate the profit-maximisation problem, in which the revenue and costs will be affected by the product price and the area of warranty region, for a new product sold under two-dimensional warranty. We assume that the product sales can be captured by a stochastic Bass model based on the nonhomogeneous Poisson process. The product follows a two-dimensional failure process and is covered by a non-renewable free minimal-repair warranty, with age and usage limits. We focus on three revenue/cost components, i.e. sales revenue, warranty cost and production cost, that will significantly affect a firm’s total profit. The profit is maximised by jointly optimising three decision variables, i.e. product price, warranty age limit and warranty usage limit. Numerical experiments are conducted to illustrate the effects of some key parameters, including product reliability, price elasticity, warranty elasticity and learning effect factor, on the optimal settings of the price and warranty region.     

二维质保管理研究综述与展望

除了低价格、高质量和卓越的性能外,完善的售后服务在刺激购买需求中起到了关键作用。在此背景下,越来越多的制造商开始提供质保服务。对于二维质保产品而言,产品的失效是使用时间和累积使用度共同作用的结果,例如汽车、重型机械等耐用消费品。本文通过对相关文献的整理,分别从产品现场可靠性建模、质保索赔预测和质保政策设计3个方面分别对有关二维质保管理的研究进行综述,并提出了二维质保管理领域进一步研究方向。     

A consumer incentive warranty policy and servicing strategy for products with uncertain quality

Owing to quality uncertainty in manufacturing, some of the items produced can be non-conforming. When items are sold with warranty, the warranty servicing cost for non-conforming items can be high. Life testing can be used to weed out such items and reduce the warranty cost. Such testing is expensive in terms of time and money. Releasing items with no testing can result in an increased number of early failures which in turn leads to consumer dissatisfaction and affects future profits, due to a loss of consumers. We propose a new policy which involves no testing and incorporates an incentive feature to reduce consumer dissatisfaction.     

A flexible two-dimensional basic warranty policy with two continuous warranty regions

Flexible warranty policies can satisfy customers by allowing them to select the warranty regions according to their usage rates. However, customers are usually unaware of their future usage rates when they purchase a product, which makes it difficult to make sound decisions. To address the problem, this paper proposes a flexible two-dimensional basic warranty policy with two continuous rectangular regions. Customers can gain experience of their usage rates through the first region, and thus make the second region more cost-effective. The total price of the two subregions is equal to the basic warranty price, which is prespecified by the manufacturer. The objective is to determine the optimal boundary between the two regions from the perspective of customers. First, the learning effect is used to depict the process of gaining experience from the first region. A warranty pricing model is then developed based on the polynomial failure intensity model. On that basis, the customer's expected repair cost over the warranty period is analyzed, and a numerical method is proposed for the optimization problem. A numerical example is presented to demonstrate the effectiveness of the proposed policy. The results show that customers can benefit more from the proposed policy than from the iso-cost warranty policy.     

考虑顾客不同使用强度的产品质保期和价格最优设计

站在制造商的角度进行一维质保期及产品价格策略的最优设计。考虑顾客不同使用强度对现场可靠性的影响,分别构建产品的失效率模型及最小维修策略时的质保成本模型。以期望利润最大化为目标,对最优的一维质保期和价格策略进行设计。最后,以某中央空调制造商作为算例,对中央空调的最优一维质保期和产品售价进行优化,并对主要参数进行敏感性分析。研究发现,特别是针对具有顾客不同使用强度的大型装置或系统,通过最优的质保期和价格策略设计可以有效地提升产品的利润期望值。     

基于服务价格和质量承诺的服务商竞争策略研究

现有研究仅关注单一服务商的质量承诺决策,而忽视多个服务商关于服务价格和质量承诺的竞争策略。为弥补这种研究不足,本文的研究并不仅停留在单一的服务承诺研究,而是扩展到服务价格层面进行综合研究,以此探讨服务商竞争策略。本文构建3种竞争模型,并运用最优化方法研究,得到结论如下。1) 在基于服务价格的竞争模型中,当质量承诺水平一定时,一旦有服务商涨价,其他服务商会制定相同的决策。然而,一旦有服务商提高质量承诺水平,其他服务商会制定降价决策予以应对。2) 在基于质量承诺水平的竞争模型中,当服务价格一定时,一旦有服务商提高质量承诺水平,其他服务商会适当地降低质量承诺水平来减少服务失败时的补偿。一旦有服务商从服务价格层面提升竞争力,那么,为应对这种竞争策略,其他服务商就会从质量承诺方面进行补偿。3) 本文基于服务竞争策略研究而建立了竞争模型,从模型研究中发现,各个服务商之间必然存在纳什均衡。本文对服务市场竞争环境下,关于服务价格和质量承诺的竞争策略的制定有一定的实践指导意义。     

Development of lifetime warranty policies and models for estimating costs

In today's fiercely competitive products market, product warranty has started playing an important role. The warranty period offered by the manufacturer/dealer has been progressively increasing since the beginning of the 20th Century. Currently, a large number of products are being sold with long-term warranty policies in the form of extended warranty, warranty for used products, service contracts and lifetime warranty policies. Lifetime warranties are relatively a new concept. The modelling of failures during the warranty period and the costs for such policies are complex since the lifespan in these policies are not defined well and it is often difficult to tell about life measures for the longer period of coverage due to usage pattern/maintenance activities undertaken and uncertainties of costs over the period. This paper focuses on defining lifetime, developing lifetime warranty policies and models for predicting failures and estimating costs for lifetime warranty policies.     

Investigating reliability improvement of second-hand production equipment considering warranty and preventive maintenance strategies

A cost model for optimal reliability improvement of warranted second-hand production equipment is developed. The second-hand production equipment of age x is subjected to an upgrade action of a certain level u before it is sold with a Free Repair Warranty. We look at determining the optimal upgrade level when not performing and when performing periodic preventive maintenance (PM) during the warranty period. Two different PM strategies are considered: (a) periodic PM actions having the same efficiency level; (b) periodic multi-phase PM actions with a maintenance efficiency level which varies according to the phase. The proposed model aims at helping the dealer to find the optimal upgrade level to perform before selling the second-hand equipment, and to assess whether performing PM actions during the warranty period, according to a specific maintenance strategy, is worthwhile in terms of cost reduction. Numerical experimentations considering each PM scenario are performed in order to investigate how each PM strategy impacts the improvement level to be performed and the associated total expected cost. The obtained results showed that the expected total cost incurred by the dealer is governed by a sensitive trade-off between the warranty servicing cost and the costs associated with the reliability improvement, and with the PM performed during the warranty period. It is also found that the proposed new periodic multi-phase PM policy with an increasing maintenance efficiency level yields lower upgrade levels, inducing lower costs for the dealer.     

Maintenance policies with two-dimensional warranty

A new maintenance policy which minimizes the total expected servicing cost for an item with two-dimensional warranty is proposed. An iterative procedure to estimate the item's failure rate function from historical observations and an optimization algorithm based on Monte Carlo simulation are applied to obtain the best maintenance policy. Numerical examples and sensitivity analysis are presented.     

Optimal reliability, warranty and price for new products

The success of a new product depends on both engineering decisions (product reliability) and marketing decisions (price, warranty). A higher reliability results in a higher manufacturing cost and higher sale price. Consumers are willing to pay a higher price only if they can be assured about product reliability. Product warranty is one such tool to signal reliability with a longer warranty period indicating better reliability. Better warranty terms result in increased sales and also higher expected warranty servicing costs. Warranty costs are reduced by improvements in product reliability. Learning effects result in the unit manufacturing cost decreasing with total sales volume and this in turn impacts on the sale price. As such, reliability, price and warranty decisions need to be considered jointly. The paper develops a model to determine the optimal product reliability, price and warranty strategy that achieve the biggest total integrated profit for a general repairable product sold under a free replacement-repair warranty strategy in a market and looks at two scenarios for the pricing and warranty of the product. The model assumes that the sale rate increases as the warranty period increases and decreases as the price increases. The maximum principle method is used to obtain optimal solutions for dynamic price and warranty situations. Finally, numerical examples are given to illustrate the proposed model.     

A repair-replace strategy based on usage rate for items sold with a two-dimensional warranty

When a repairable item fails under warranty, the manufacturer has the option of either repairing the failed item or replacing it with a new one. In this paper, a repair-replace strategy is discussed for the manufacturer of a product sold with a two-dimensional warranty. The strategy is based on a specified region of the warranty defined in terms of age and usage with the first failure in the region rectified by replacement and all other failures being minimally repaired. An accelerated failure time (AFT) model is used to allow for the effect of usage rate on item degradation.     

Optimal decision for warranty with consideration of marketing and production capacity

A successful marketing strategy should be managed to integrate the decisions, such as pricing, post-sale service and production in order to maximise profits but most manufacturers did not fully consider how to integrate them into an effective one in the past. Therefore, in this study, we proposed a mathematical model and a corresponding solution algorithm to assist the decision-makers systematically to decide the synthetic decision. The decision-makers can make warranty and marketing decision according to their production capacity. Besides, the impacts on the profit, the sale, the price and the warranty when the related costs were varying are also discussed in the study, so that the decision-maker can more easily handle the possible situations. The failure times of the product in our study were drawn from a non-homogeneous Poisson process with a specific intensity function (linear, exponential or power law). A preventive maintenance programme was also considered in the warranty policy. Finally, a numerical example was given to demonstrate the effectiveness of the proposed model.