Warranty Servicing Research Articles

Two-dimensional warranty policy design with the consideration of protection term

In practice, car products are sold with 2-dimensional (2D) warranty policies, on which existing literature usually assumes that the products are repaired upon failures throughout the warranty period. However, this is not always the case in real life; for example, car manufacturers may need to replace defective cars during an early period of warranty, when the total time on repairing the failures reaches a specified repair time constraint or the number of repair attempts reaches a specified repair number constraint. Motivated by this, this paper divides a warranty period into two stages: the initial protection term and the subsequent repair stage. During the initial protection term, the manufacturer needs to provide a replacement service when a specified replacement condition is satisfied; while during the subsequent repair stage, the manufacturer should provide repair services upon product failures. Taking the warranty servicing cost into account, a profit model is built. The expected total profit of the product is derived, and the optimal warranty period and product price are designed to maximize the manufacturer’s profit. Finally, numerical experiments on car examples are provided to illustrate the considered warranty policy with protection term.

Joint optimization of inspection and condition-based maintenance for a deteriorating product under extended warranty

Condition-based maintenance (CBM) has been widely applied in engineering practice, but it has been rarely integrated into extended warranty policies. When the inspection cost is high, it becomes important to schedule the inspection plan when making CBM decisions. This paper jointly optimizes inspection and CBM for a deteriorating product under extended warranty. The deterioration of the product is characterized as a non-stationary gamma process. At each inspection epoch, the decision maker needs to determine whether to repair the product and when to perform the next inspection. The objective is to simultaneously optimize the inspection policy and the maintenance policy that minimize the warranty servicing cost. The joint optimization problem is formulated by stochastic dynamic programming. The optimal policy is proven to be of a control-limit form. A backward induction algorithm integrating the obtained structural properties is proposed to find the optimization results. A numerical example is conducted to illustrate the effectiveness of the proposed approach. A graphical representation of the obtained policy is provided to support practical maintenance implementation. The comparison with a popular CBM policy confirms the superiority of the proposed policy.

Designing a two-dimensional maintenance policy for products sold under a two-dimensional warranty by considering both warranty servicing cost and lost opportunity

ABSTRACT Based on the manufacturer-consumer context, a two-dimensional preventive maintenance policy is developed for sold products under a two-dimensional warranty by pursuing two objectives: (1) minimization of the imposed warranty servicing cost on the manufacturer and (2) minimization of lost opportunity cost resulting from the product downtime. In addition, in order to adapt the model to real applications, we depart from the literature by taking into consideration the continuous nature of the usage rate variable. To illustrate the advantages of the model, three types of couple-based comparative models are made. Type I consists of two models with a one-dimensional preventive maintenance policy, while type II includes a model with no maintenance activity. Finally, type III involves two single-objective models. The results obtained from the comparison of the suggested model with the type I and II models confirm the significant effect of employing a two-dimensional preventive maintenance policy in improving both manufacturers’ and customers’ satisfaction. In addition, comparisons of the suggested bi-objective model with three types of models indicate the importance of taking into account both the warranty servicing cost and loss opportunity.

The bivariate exponentiated additive Weibull distribution and its multivariate extension with applications

Recently, Abd EL-Baset and Ghazal (2020) introduced the exponentiated additive Weibull distribution, which is very useful in modeling the reliability analysis of a product, such as reliability of decision-making and cost–benefit analysis. The products inherently decay with usage and age, resulting in more extreme failures. Fixing a failed product may take longer warranty servicing time, indicating that failure and warranty servicing times correlate to bivariate distributions. This paper introduces a bivariate exponentiated additive Weibull distribution such that the marginals have exponentiated additive Weibull distributions. Statistical properties, parameter estimation using the maximum likelihood, and the bootstrap confidence interval are presented. We also generalize the results for the multivariate case. A simulation study has been executed to evaluate the performance of the parameter estimation via mean square error and absolute bias. Finally, we use three datasets failure and warranty servicing times, American football league, and the air quality to compare the proposed bivariate model to other models. The results revealed that the proposed bivariate model delivers a better fit than other models.

Warranty service contracts design for deteriorating products with maintenance duration commitments

With the increasing diversification of customers’ demand and purchasing behaviors, more and more manufacturers have focused their attention on the warranty service contracts design. The maintenance duration of the sold product, which plays an important role in the normal production and operation process of the user, is frequently taken into consideration in warranty contracts. In this study, we design different warranty contracts with various combinations of maintenance duration and availability requirements. The manufacturer commits to compensate for each overdue repair or failing to satisfy the availability target. The customers’ choice behavior is described by the multinomial logit (MNL) model, and customers often form their own minimum acceptable levels (also referred to as reference points) of maintenance duration and availability when making purchasing decisions, which have an impact on the contract choice. The expected warranty servicing profit is maximized to determine the optimal price, maintenance duration and availability. Finally, the proposed warranty contracts are demonstrated by numerical examples. We find that the maintenance duration affects not only the warranty cost but also the customer choice, which further affects the optimal contract pricing and profits.

Design of a performance-based warranty policy with replacement–repair strategy and cumulative cost threshold

Performance-based warranty policies have become increasingly popular because they can not only help customers hedge against uncertainty about product performance but also help manufacturers build credibility and drive sales. However, adopting a performance-based warranty policy imposes a greater financial burden on the manufacturer because performance guarantee level is typically higher than the physical failure threshold. Finding a balance between the increased willingness to purchase and the associated cost is a major challenge for manufacturers. In this study, we focus on restorable products and propose a performance-based warranty policy that includes a replacement–repair strategy and cumulative cost limit. The warranty terminates either when the product reaches an age limit or when the cumulative maintenance cost exceeds a predetermined limit. We have divided the warranty period into two compensation regions (i.e, replacement and repair region) to increase customer satisfaction and reduce the risk of breakdown. We also consider both positive and negative effects of imperfect repairs on performance deterioration. We compute the warranty servicing cost based on a product performance degradation model. The optimal pricing and warranty decisions can then be obtained by maximizing the overall profit. Moreover, we conduct comprehensive simulation studies to demonstrate the effectiveness of the proposed policy. The results prove that our proposed policy is win-win for both manufacturers and customers. Managerial suggestions are also given to provide guidance for implementing the proposed performance-based warranty policy.

Optimal condition-based warranty policy for multi-state products with three guarantees service

ABSTRACT Within a certain period of time after products are sold, many manufacturers offer free repairing, replacing and refunding for the defective products, referred to as three guarantees service. Under the assumption that the working states of the product can be partitioned into three non-overlapping sets and its degradation can be modeled by homogeneous Markov chain, a three-stage and conditioned-based warranty model is proposed. Under the model, the whole warranty period is divided into a compulsory replacement period, a condition-based maintenance period, and a minimal repair period. Over the condition-based maintenance period, depending on the state set that the working state before the failure belongs to, failed products are rectified minimally, imperfectly, replaced by an identical one and the warranty terms are renewed, respectively. Markov process and renewal process theories are used to analyze the failure process and the warranty servicing cost over the whole warranty period. Based on double Riemann sum, a recursive algorithm for finding the optimal partition method of working states and the warranty period is proposed. A numerical example is given to illustrate the validity of the warranty strategy.

Optimization of Preventive Maintenance and Replacement Strategies for Nonrenewing Two-Dimensional Warranty Products Experiencing Degradation and External Shocks

“Vehicle damage insurance” in China, “Protection Plus Mobile Elite” launched by Samsung, “iPhone Apple Care +” offered by Apple, and so on are special warranty services aiming at external shocks. These warranty terms, named accident insurance, exist widely in the marketplace. Considering the effect of external shocks on degradation, a two-dimensional preventive maintenance and replacement strategy for products, sold with a nonrenewing and two-dimensional rectangle warranty service, is proposed in this paper. Under this strategy, preventive maintenance actions are scheduled based on units of age or usage, which occurs first. There is a reduction in the intensity function after a preventive maintenance action. Each shock before the Nth shock causes a failure of the product or an increase in product failure rate. The product is replaced by a new one on the Nth shock if it survives the N − 1 shocks. From the view of manufacturers, the mean warranty servicing cost over the whole warranty region is obtained by using the renewal theory. Based on direct numerical Riemann–Stieltjes integration, an approximation algorithm of the cost is also given. The mean cost is minimized by the optimization of preventive maintenance interval and the number of shocks N. A numerical example is given to illustrate the feasibility of the proposed strategy. The effects of maintenance cost, the arrival rate of the shocks, and other model parameters on the optimal strategy are also investigated numerically.

Comparison of three preventive maintenance warranty policies for products deteriorating with age and a time-varying covariate

The development of condition monitoring technology initiates a new tendency towards incorporating condition-based maintenance (CBM) into warranty policies to reduce the manufacturer's expense over the warranty period. However, CBM is not always more cost-effective than time-based maintenance (TBM), or vice versa. This paper investigates three preventive maintenance (PM) policies for a product sold with a free repair warranty and examines under which situation each policy is optimal. The product deteriorates with both age and a time-varying covariate and its failure rate is described by Cox's proportional hazards model. The covariate can only be revealed by condition monitoring. In policy 1, no inspection is performed and PM actions are carried out periodically. Continuous inspection is performed in policy 2 where a PM action is recommended once continuing operation is not cost-effective. Policy 3 applies periodic inspection to select appropriate maintenance action at each inspection epoch. The warranty servicing costs are analyzed in the Markov decision process framework based on a discretization method. The policy structures of the CBM policies are proved to be of control-limit forms. A practical numerical example is provided to illustrate the proposed approach. The results can help the manufacturer design effective warranty policies.

Optimisation of two-dimensional warranty region under preventive maintenance over product lifetime by considering both manufacture and consumer's point of views

An appropriate warranty period plays an essential role in consumer satisfaction and manufacturer costs. Manufacturer and consumer are two critical parts of any transaction; therefore, inattention to each of them causes dissatisfaction and financial loss. Therefore, this research proposes a model to determine an optimal two-dimensional warranty period to minimise both manufacturer's warranty servicing cost and consumer's post-warranty cost. Furthermore, to reduce the frequency of product failures, a preventive maintenance program is implemented during the warranty and post-warranty periods, respectively, by the manufacturer and the consumer. Finally, two comparisons are made to show the model effectiveness. The first one shows the superiority of the joint minimisation of the manufacturer's warranty cost and the consumer's post-warranty cost over the single minimisation. The second one shows that a proper preventive maintenance program can effectively reduce the costs of both the manufacturer and the consumer.

A performance-based warranty for products subject to competing hard and soft failures

This article studies a performance-based warranty for products subject to competing hard and soft failures. The two failure modes are competing in the sense that either one, on a “whichever-comes-first” basis, can cause the product to fail. A performance-based warranty not only covers the repair or replacement of any defect, but also guarantees the minimum performance level throughout the warranty period. In this article, we propose three compensation policies—that is, free replacement, penalty, and full refund, when a product’s performance fails to meet the guaranteed level. The expected warranty servicing costs for the three policies are derived, based on the competing risks concept. A warranty design problem is further formulated to simultaneously determine the optimal product price, warranty length, and performance guarantee level so as to maximize the manufacturer’s total profit. Numerical studies are conducted to demonstrate and compare the three performance-based compensation policies. It is shown that the full refund policy always leads to the lowest total profit, whereas neither of the other two policies can dominate each other in all scenarios. In particular, the free replacement policy results in a higher total profit than the penalty policy when the replacement cost is low, the penalty cost coefficient is high, and/or the product reliability is high.

Multi-phase reliability growth test planning for repairable products sold with a two-dimensional warranty

For products sold with a two-dimensional warranty policy, the warranty servicing cost can be reduced through reliability growth during development. This paper investigates a multi-phase reliability growth test program for repairable products with independent competing failure modes. Considering a test-find-test scheme, an accelerated reliability growth model is developed, allowing different failure modes having distinct accelerated relationships. Taking usage heterogeneity and mode-specific failure learning effect into account, the reliability growth is achieved through periodic fixes on a phase-by-phase basis. From the manufacturer’s perspective, the main objective is to achieve the optimal trade-off between the warranty cost and the reliability growth test cost by determining the optimal test program that minimizes the expected total cost per product sold, ensuring that the pre-specified reliability growth requirement is met. Assuming the tri-Weibull product failure distribution, we illustrate the proposed optimization model numerically and study the effect of relevant parameters on the optimal reliability growth test program. The results show that the proposed approach yields significant cost reduction and reliability improvement for the examples studied in this paper, especially when the manufacturer possesses high failure learning ability, and the product has expensive repair cost per a warranty failure and extensive warranty coverage.

Bi-objective optimization of non-periodic preventive maintenance strategy by considering time value of money

Recently, design of preventive maintenance (PM) policies during the warranty period has attracted the attention of researchers. The methods mainly design warranty servicing strategies in a way that reduce the cost imposed on the manufacturer without considering the impact of customer dissatisfaction. While dissatisfaction with a product is an important issue which may result in the loss of potential buyers and switching existing buyers to competitors. Therefore, this study develops a bi-objective model which simultaneously minimizes the manufacturer and the buyer cost under a Non-homogeneous Poisson Process framework. Also, a non-periodic preventive maintenance strategy is presented in which PM actions are performed at discrete time instants in a way that the expected number of failures remains a constant value over all PM intervals. Furthermore, it is a known fact that the value of money reduces over time due to different reasons and has a significant impact on long-term contracts. Since PMs and repairs are conducted at different times, the time value of money is considered to estimate the cost more accurately. A comparative study is conducted to support this claim that the presented non-periodic reliability-based PM policy has a better performance in comparison with a periodic PM policy.

Warranty cost analysis with an alternating geometric process

In this study, we model the warranty claims process and evaluate the warranty servicing costs under non-renewing, renewing and restricted renewing free repair warranties. We assume that the repair time for rectifying the claims is non-zero and the repair cost is a function of the length of the repair time. To accommodate the ageing of the product and repair equipment, we use a decreasing geometric process to model the consecutive operational times and an increasing geometric process to model the consecutive repair times. We identify and study the alternating geometric process, which is an alternating process with cycles consisting of the item’s operational time followed by the corresponding repair time. We derive new results for the alternating geometric process in a finite horizon and use them to evaluate the warranty costs over the warranty period and over the life cycle of the product under a non-renewing free repair warranty, a renewing free repair warranty and a restricted renewing free repair warranty. Properties of the model are demonstrated using a simulation study and by fitting the models to real data from an automotive manufacturer.

On optimal upgrade strategy for second-hand multi-component systems sold with warranty

Reliability improvement strategies such as upgrade, reconditioning and remanufacturing have been extensively adopted by dealers of second-hand systems to improve the system reliability and reduce the warranty servicing cost. However, most existing studies on this topic do not consider the multi-component structures of complex second-hand systems, and either treat them as black-box systems by ignoring their internal structures or simply deal with individual components. In this paper, a new upgrade model is developed for complex second-hand systems sold with a non-renewing free repair/replacement warranty, by explicitly considering their multi-component configurations. Two types of components, i.e. repairable and non-repairable components, are taken into account. During the upgrade process, non-repairable components can be upgraded only by replacement (if necessary), while repairable ones may be imperfectly upgraded with various degrees. The main objective of the dealer is to determine which components to upgrade and the corresponding upgrade degrees, to minimise the total expected servicing cost. In view of the problem structure, a marginal analysis based algorithm is presented. It is shown that the proposed upgrade strategy contains the ‘no upgrade’ strategy and the ‘component-level perfect upgrade/replacement’ strategy as special cases, and outperforms them. Finally, several extensions of the proposed upgrade model are discussed.

Warranty cost analysis: Increasing warranty repair times

In this study, we model the warranty servicing costs under nonrenewing and renewing free repair warranties. We assume nonzero increasing repair times with the warranty cost depending on the length of the repair time. An increasing geometric process is used to model the consecutive repair times. We introduce the generalised alternating renewal process, which is an alternating process with cycles consisting of an item's operational time followed by the corresponding repair time. We derive analytical results for a generalised alternating renewal process with a finite time horizon and use them to evaluate the warranty costs over the warranty period and over the life cycle of the product under the nonrenewing free repair warranty and renewing free repair warranty. Properties of the model are demonstrated through a simulation study and through the application to warranty claims data from an automotive manufacturer.

An availability-based warranty policy considering preventive maintenance and learning effects

By considering learning effects during the process of maintenance, this article investigates an availability-based warranty policy from the manufacturer’s perspective. Imperfect preventive maintenance activities are implemented to reduce the number of equipment failures so as to meet the product’s availability requirement and minimize warranty servicing cost during the warranty period. Hybrid hazard rate model is used to describe imperfect preventive maintenance actions; maintenance plan and warranty servicing cost are optimized by considering learning effects. Preventive maintenance schedule within a warranty period is optimized cycle by cycle, and the warranty cost rate is minimized under the requirement of availability within each cycle. Numerical example of a wind turbine gearbox is provided to verify the effectiveness of the proposed availability-based warranty policy. Sensitivity analysis shows that by considering learning effects on maintenance, the gearbox can maintain a higher availability level with a lower warranty cost than the case without consideration of learning effects.

A mean-variance optimisation approach to collectively pricing warranty policies

Warranty policy can influence the profit and cost of a product. In practice, a manufacturer commonly produces more than one product, or a portfolio of products, and provides warranty servicing for them. Many authors have attempted to optimise warranty policy to maximise the profit or minimise the cost of each individual product. Warranty claims of the products produced by the same manufacturer, however, may be due to common causes, since the products may be designed by the same engineer team or using the same type of components. This implies that the numbers of warranty claims of different products may be related, and optimisation of warranty policies for each individual product may therefore cause biased decisions. To overcome this disadvantage, this paper aims to collectively optimise a manufacturer's total profit for a portfolio of different products by using a mean-variance optimisation approach. A tool from the probability theory, copulas, is used to depict the dependence among the warranty claims of different products. Numerical examples are provided to illustrate the application of the proposed methods.

Game theoretic modelling of service agent warranty fraud

When a warranty provider outsources warranty servicing to an external service agent this agent may act in a fraudulent manner. In this paper we consider a special case of service agent fraud—with the service agent overbilling the warranty provider for some of the warranty claims. A detailed inspection of a claim may be made to identify whether or not the service agent has committed fraud, but this inspection involves an additional cost to the warranty provider. This cost may be recovered by imposing a penalty on the service agent whenever a fraud is committed and it is detected. This penalty is specified in the maintenance service contract. A game theoretic approach is used to find the optimal overbilling strategy for the service agent and the optimal inspection strategy for the warranty provider. The optimal solution is the mixed strategy Nash equilibrium of a static game between the two parties.

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.