Two-dimensional Warranty Research Articles

Optimal preventive maintenance strategy for repairable items under two-dimensional warranty

For repairable items under the two-dimensional warranty, the manufacturer expenses warranty costs for rectifying the failures occurring over the warranty coverage. Preventive maintenance actions reduce the occurrence intensity of item failures, and hence the warranty cost resulted from corrective maintenance, but bring additional preventive maintenance cost. In this paper, we investigate a periodic and imperfect preventive maintenance strategy for an item covered by a fixed and combined base warranty and extended warranty region from the manufacturer׳s perspective. After the base warranty expires, the buyer purchases an extended warranty service contract from the same manufacturer for extra protection. Over the whole warranty coverage, the item failures are minimally repaired by the manufacturer at no cost to the buyer. The effect of preventive maintenance action is characterized by reducing the item׳s failure intensity proportionally to the maintenance degree at each preventive maintenance action. Then a mathematical model is proposed to derive the optimal preventive maintenance strategy so as to minimize the total expected warranty servicing cost to the manufacturer. A numerical example is presented to illustrate the application of the proposed model and evaluate the impact of different model parameters on the optimal solutions and the corresponding warranty servicing costs.

Cost analysis of two-dimensional warranty for products with periodic preventive maintenance

This study uses a bivariate approach, which simultaneously considers the time and usage of a repairable product, and takes into account periodic preventive maintenance to develop a two-dimensional warranty policy for the repairable product. The proposed model is based on the bivariate Weibull process to analyze the breakdown process of a repairable product simultaneously in terms of time and usage. A repairable product may be differently utilized by various customers. We consider two types of customers: (1) customers whose product warranty is terminated because the warranty time limit has reached first; (2) customers whose product warranty is terminated because the warranty usage limit has reached first, to perform cost analyses for determining an appropriate warranty policy. However, since it is often the case that the repairable product may be equally likely purchased by the two types of customers who are equivalently important to the product manufacturer; the optimal warranty policy would thus be the equilibrium solution of the cost analyses which pay the same attention to both types of customers in order to maximize the total profit of the manufacturer.

Optimal warranty policies considering repair service and replacement service under the manufacturer’s perspective

In this paper, under the manufacturer’s point of view, a warranty cost model is proposed in consideration of both repair service and replacement service simultaneously upon the system failure to find the optimized warranty period in terms of an expected cost rate during the warranty cycle. If a failed system can’t be repaired by the threshold warranty servicing time in the warranty service center, the replacement service is provided for the failed system instead of continuing to provide the repair service. Under such circumstance, new warranty cost models are developed and the renewable warranty policy is investigated for free and pro-rata two-dimensional warranty dependent on both failure time and warranty servicing time. From the manufacturer’s point of view, the warranty period is newly defined starting from the system’s purchasing time and ending at the expiration time of the warranty. This paper determines the optimal warranty period by applying Nelder–Mead downhill simplex method to minimize the expected cost rate during the warranty period. The real application is discussed adopting the proposed approach based on the actual field data and numerical examples are presented to exemplify the applicability of the methodologies derived in this paper.

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.

Approximation of the Bivariate Renewal Function

Recently, Kambo and his co-researchers (2012) proposed a method of approximation for evaluating the one-dimensional renewal function based on the first three moments. Their method is simple and elegant, which gives exact values for well-known distributions. In this article, we propose an analogous method for the evaluation of bivariate renewal function based on the first two moments of the variables and their joint moment. The proposed method yields exact results for certain widely used bivariate distributions like bivariate exponential distribution, bivariate Weibull distributions, and bivariate Pareto distributions. An illustrative example in the form of a two-dimensional warranty problem is considered and comparisons of our method are made with the results of other models.

Warranty forecasting from incomplete two-dimensional warranty data

Warranty modelling with incomplete data is a major issue in reliability analysis. The incomplete failure region characterized by warranty field data may be classified into several domains representing failures from manufacturing/assembly defects, usage or fatigue. In the present paper a data driven approach has been suggested to demark the regions optimally through estimation of the change point in a hazard function. In the perspective of bivariate warranty analysis, as relevant in automobiles, we have assumed the lifetime distribution to be a mixture of distributions corresponding to the burn-in period and the useful life period. The proportions of observations in different regions demarketed by the warranty policy in the bivariate plane have been estimated by considering mileage along with age. The estimation scheme has been verified and validated through extensive simulation studies. The utilities of the results have been demonstrated by addressing several issues through a real life synthetic warranty data set from a large automobile company.

Two-Dimensional Warranty With Reliability-Based Preventive Maintenance

In dealing with post-sale warranties, considering only operating time may not be sufficient because products often deteriorate through usage and over time, which both influence product reliability. A two-dimensional warranty is thus more reasonable in practice for both manufacturers and customers. Periodic preventive maintenance within the warranty term seems more prevalent in practice due to its convenience and operability for manufacturers and consumers. However, increases in product deterioration may cause breakdowns to occur more often, even after periodic preventive maintenance has just been performed. In this study, a two-dimensional warranty policy is examined with consideration of non-periodic preventive maintenance to determine the optimal two-dimensional warranty term with a constraint that the product reliability should be above a certain threshold. A numerical application is provided to demonstrate the effectiveness of the proposed approach, with sensitivity analyses being conducted to investigate the robustness of the derived optimal warranty policy.

Optimal burn-in for repairable products sold with a two-dimensional warranty

Warranty data analyses reveal that products sold with two-dimensional warranties may have significant infant mortalities. To deal with this problem, this article proposes and studies a new burn-in modeling approach for repairable products sold with a two-dimensional warranty. More specifically, two types of failures are characterized—i.e., normal and defect failures—and performance and cost-based burn-in models are developed under the non-renewing free-repair warranty policy. The proposed models subsume the special cases of a one-dimensional warranty, allow different failure modes to have distinct accelerated relationships, and take consumer usage heterogeneity into consideration. Under mild assumptions, it is established that the optimal burn-in usage rate should be as high as possible, provided that no extraneous failure modes are introduced. Furthermore, It is shown that the optimal burn-in duration determined from the performance-based model is not shorter than that from the cost-based model. Numerical examples are used to demonstrate the benefits of burn-in. In addition, some practical implications from a sensitivity analysis are elaborated.

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.

A New Warranty Policy With Failure Times and Warranty Servicing Times

We develop a new warranty policy with respect to the failure time and warranty servicing time, where those two variables are statistically correlated in bivariate distributions. Based on the developed approaches, we investigate the properties of the bivariate function, and obtain the number of warranty services in a warranty period using the field data. The warranty service includes the repair service, and replacement service. An approach of analyzing the warranty cost is proposed for a product in which the failure time and warranty servicing time are used simultaneously to determine the eligibility of a warranty claim. In the vast literature of the warranty modeling, the two-dimensional warranty is commonly consisting of the usage, and the age. In this study, we focus on the failure time and warranty servicing time as two factors under the warranty policy. Using the field data, the parameters are calculated, and the warranty model is investigated using a bivariate exponential distribution (BED) which was the best fit distribution based on the field data. Numerical examples are discussed to demonstrate the applicability of the methodology derived in the paper. The results of our research in this paper indicate that more sophisticated bivariate distributions other than the BED is worth the effort, as a future research problem, or using other data sets in a general industrial application setting.

Warranty Cost Analysis for $k\hbox{-}out\hbox{-}of\hbox{-}n$ Systems With 2-D Warranty

This paper presents new warranty cost models subject to two types of warranty periods (warranty and postwarranty periods), minimal repairs, and various types of warranty policies including free repair/replacement warranty and prorata warranty for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -out-of- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> systems. Two-dimensional warranty is applied to investigate the warranty cost using repair times and failure times. We obtain the long-run expected cost functions per unit time with respect to both the warranty and postwarranty periods by considering the aspects of both the manufacturer and the customer. This paper considers a periodic preventive maintenance (PM) policy with both corrective maintenance and PM and also determines three decision variables including warranty period, repair time limit, and periodical maintenance cycles, which minimize the long-run expected cost. Numerical examples are discussed to demonstrate the applicability of the methodologies, and results including the three optimum decision-variable solution and their corresponding long-run expected cost per unit are derived.

An imperfect repair strategy for two-dimensional warranty

The objective of this study is to develop a warranty servicing strategy, based on imperfect and minimal repairs, that minimises the total warranty servicing cost. For pre-specified degrees of repair, the expected warranty servicing cost is minimised by optimally partitioning a two-dimensional rectangular warranty region into n disjoint subregions. We numerically illustrate the imperfect repair strategy and provide a comparison with previously studied warranty repair strategies.

WARRANTY SERVICING WITH A BROWN–PROSCHAN REPAIR OPTION

Reducing warranty servicing costs are of great interest to product manufacturers or sellers who are contractually bound to provide post-sales support, up to a specified warranty period, usually in the form of some remedial action that restores a failed item to a functioning condition. Here, in the spirit of Jack et al. (2009) strategy based on partitioning the effective warranty period into three intervals, we consider and analyze the cost of a new two-dimensional warranty servicing strategy, that probabilistically exercises a choice between a replacement and a minimal repair to rectify the first failure if any, in the middle interval. The behavior of the expected cost is numerical illustration with Weibull failure model.

A decision model for adopting an extended two-dimensional warranty considering different maintenance policies with variable customer's behavior

In this paper we study the opportunity provided by adopting a two-dimensional extended warranty for the buyer as well as for the consumer of randomly failing products. The buyer has to decide to pay or not for the extended warranty when purchasing the product. The total expected cost incurred by each side during the product's life cycle is expressed in order to determine the maximum extra cost the consumer should pay and the minimum price at which the manufacturer should sell the extended warranty. This is done under different options in terms of maintenance strategies adopted during the product's lifecycle.

Warranty Costs Assessment for Automotive Applications

The article deals with current problems of warranty costs in automotive applications. The two-dimensional warranty widely used in automotive industry is presented. The relationship between warranty costs and product reliability is also presented. The article analyses problems with estimation of a number of failure occurrence over a warranty period and estimation of costs related to repair failures. The mathematical model enables to predict warranty costs for two-dimensional warranty is presented. The article also deals with problems of estimation of an instant of warranty period expiration and demonstrates its estimation on an example of a vehicle.

Analysis of a Two-dimensional Warranty Servicing Strategy with an Imperfect Repair Option

Items sold under warranty often require post sale support in terms of repair or replacement. Providence of such support costs the warrantor considerably and affects his profit margin. Thus reducing warranty servicing costs is an issue of great interest in warranty studies. In this paper, we look at two variations of a servicing strategy in two-dimensional warranty regimes involving minimal and imperfect repairs. Our work demonstrates the modeling and analysis of costs under these servicing strategies and compares their performance to other strategies present in the literature.

A two-dimensional warranty servicing strategy based on reduction in product failure intensity

The cost of servicing a warranty depends, amongst other factors, on the type of repair performed under warranty. Although “all minimal repair” and “all replacement” policies are easy to implement and analyze, they are not always feasible and/or practical. Having a combination of different types of repair often leads to lower warranty servicing costs. In this article, to reduce the warranty servicing cost, we study a servicing strategy that involves performing imperfect repairs in place of some of the minimal repairs of an “all minimal repair” strategy; the effect of an imperfect repair is characterized by a drop in the conditional intensity function of the failure process. We consider both fixed and random degrees of repair. For a given type of product, we partition the warranty region so that the expected total warranty servicing cost is minimized. We provide a numerical illustration and a comparison with previously-studied repair–replacement strategies.

Warranty Analysis Based on Different Lengths of Warranty Periods

Global companies can sell their products with dierent warranty periods based on location and times. Customers can select the length of warranty on their own if they pay an additional fee. In this paper, we consider the warranty period and the repair time limit as random variables. A two-dimensional warranty policy is considered with repair times and failure times. The repair times are considered within the repair time limit and the failure times are considered within the warranty period. Under the non-renewable warranty policy, we obtain the expected number of warranty services and their variances in the censored area by warranty period and repair time limit to conduct a warranty cost analysis. Numerical examples are discussed to demonstrate the applicability of the methodologies and results using field data based on the proposed approach in the paper.

Two-Dimensional Warranty Cost Analysis for Second-Hand Products

In spite of the recent steady increase of the volume of the second-hand markets, often customers remain in doubt regarding the quality and durability of the second-hand products. Aiming to reduce and share this uncertainty, dealers offer warranty on their products. Offering warranty for second-hand products is a relatively new marketing strategy employed by dealers of used electronic equipment, furniture, automobiles, etc. Usually, for used products, the dealer's expected warranty cost is a function of product reliability, past age and usage, servicing strategy and conditions and terms of the warranty policy/contract. Sometimes the offered policy is limited by two parameters, typically the product age and usage after the sale. This type of policies is referred to as two-dimensional warranty policies. In this article, we develop statistical models for estimating the dealer's expected warranty cost for second-hand products sold with two-dimensional free repair/replacement warranty.

Warranty cost analysis for second-hand products using bivariate approach

This research aims to develop two stochastic models, based on the characterisation of the system failure and to compute the expected warranty cost for a second-hand product, when the products are sold with free replacement and/or pro rata warranties. A two-dimensional warranty, which focuses on both age and usage of the system, has been considered for the above warranty models. Further, a numerical example is provided to demonstrate the methodology of the developed model, using bivariate Weibull and exponential distributions. Also, the sensitivity analysis performed exhibits the effect of the parameters on the cost functions.