Pro-rata Warranty Research Articles

Optimal preventive maintenance for repairable products with three types of failures sold under a renewable hybrid FRW/PRW policy

This paper proposes the optimal preventive maintenance policy for products under a warranty policy, which combines fully renewable free replacement warranty (FRW) and pro-rata warranty (PRW). Replacement age is divided into three separate periods: within FRW, during PRW and post-warranty. Moreover, it is assumed the product is subject to three types of failures arriving according to a non-homogeneous Poisson process. The first type can be fixed by minimal repair, and the third type is a catastrophic failure that results in product replacement. If the failure is of the second type and the product is at age t, it is either minimally repaired with probability p(t) or replaced by a new one with probability 1−p(t). The aim of this study is to develop the long-run expected cost rate for both warranted and non-warranted repairable products as well as to derive the optimal replacement time such that the incurred costs are minimized. The proposed approach provides valuable information for the product customers, managers, or engineers to make better decisions about the preventive replacement time of such products. The proposed model covers some preventive maintenance policies in the existing literature. Numerical computations are also performed to investigate the effect of the model’s parameters on the optimal solutions. Eventually, a real-life application is discussed to demonstrate the applicability of the proposed policy.

Non-parametric estimation of bivariate reliability from incomplete two-dimensional warranty data

A two-dimensional warranty coverage is available for those items for which age and usage jointly fall below the respective warranty limits, thus making the warranty field data incomplete. In this paper, the joint reliability function of age and usage from the observed incomplete two-dimensional warranty data has been estimated in terms of reliability function and density function of age and usage rate respectively by using non-parametric method under the assumption of independence between age and usage rate. Bivariate mean residual life for age and usage has also been determined using the bivariate reliability function. From the reliability engineering perspective, mean residual life provides valuable information for various decision making problems, such as optimizing burn-in test, extending warranty policy, and making maintenance decision. In this paper, the mean residual life has been predicted for formulating extended warranty policy. Pro-rata warranty policy has also been considered here using linear and non-linear pro-rata functions. The utilities of this research work have been demonstrated with the help of a real-life warranty data set.

Two-Dimensional Extended Warranty Strategy for an Economic Dependence Multi-Component Series System With Grid Search Algorithm

Extended warranty is an optional contract that often provides additional protection for users after the basic warranty expires. Considering the high warranty cost borne by manufacturers and the lack of system availability guarantee, the present study proposes a two-dimensional extended warranty approach for a multi-component series system considering economic dependence. Based on the imperfect preventive maintenance model and the combination of free warranty and pro-rata warranty policies, the manufacturer’s and user’s warranty cost model and availability model for a single component are established. Considering the economic dependence between multi-components, the preventive maintenance of each component is combined using the grouping maintenance approach. In the case study, we consider the reconnaissance equipment as the research object and use the grid search method to obtain the two-dimensional extended warranty scheme of reconnaissance equipment considering economic dependence. Comparative and sensitivity analyses verify that the approach can effectively reduce the extended warranty cost of the manufacturer and improve the availability of the multi-component system. Additionally, the approach can provide support for warranty cost calculation and system availability estimation under different extended warranty periods.

Determination of Bayesian optimal warranty length under Type-II unified hybrid censoring scheme

ABSTRACT We consider determination of optimal warranty length for the combined free replacement and pro-rata warranty (FRW-PRW) policy based on the Type-II unified hybrid censored data by Bayesian approach. A non-linear pro-rata rebate cost function is proposed based on which warranty cost is computed. It is assumed that the lifetime follows log normal distribution. The optimal warranty length is obtained by maximizing an expected utility function consisting of three cost functions such as economic benefit function, warranty cost function and dissatisfaction cost function. The expectation is taken with respect to the posterior predictive model for the time-to-failure data. It is observed that the non-linear pro-rata rebate cost function gives a larger warranty length with maximum profit as compared to linear pro-rata rebate cost function. A real-data set is analyzed in order to illustrate the proposed methodology of finding optimal warranty length.

Determination of the optimal warranty policy and period from the manufacturer’s/seller’s perspective

This study aims to determine the most appropriate warranty period and policy with the lowest warranty cost from the manufacturer/seller point of view. The warranty policy to be put forward and the warranty period to be offered under this policy should ensure customer satisfaction while at the same time causing a minimum cost to the manufacturer/seller. For this reason, the optimum warranty period within the scope of Free-Replacement Warranty (FRW) policy, Pro-Rata Warranty (PRW) policy, and Combined FRW/PRW policy which was the combination of those two warranty policies were tried to be determined. In the study which was done to determine the most appropriate warranty period, economic benefit and the benefit function obtained from warranty cost and dissatisfaction cost functions were used. While obtaining necessary formulas, exponential distribution, which was widely used in life tests as density function and gamma and inverse-gamma functions as prior distribution of the exponential distribution were used. To demonstrate the applicability and flexibility of the model, some typical sales models and numerical values for exponential lifetime distribution were used, and an algorithm was created to find the optimal solution. When all of the results were analyzed it was seen that Combined FRW/PRW policy was the most appropriate warranty cost as a result of testing the models obtained and it was also found that the warranty cost was almost 3% of the sale price. Within the frame of these obtained results, Combined FRW/PRW policy was suggested for the warranty service to be served with the product to the manufacturer/seller. If the warranty period number of revolutions were chosen almost the warranty cost will be in the minimum level and it will be almost 3% of the product sale price. The validity and accuracy of the proposed strategy was verified by a case study with a numerical sample providing a scientific basis.

A two-echelon supply chain model with price and warranty dependent demand and pro-rata warranty policy under cost sharing contract

In this article, a two-echelon supply chain model with a single-vendor a single-buyer is considered. The vendor's production process is imperfect and the market demand is assumed to be dependent on the buyer's selling price and warranty period. The vendor consents to return a definite portion of the buyer's purchase value, if any product is found defective within the length of warranty. The refund value or the warranty cost is considered as a function of the warranty period and the buyer's selling price of the item. This warranty cost is assumed to be fully borne by the vendor in the first model (Model I) while in the second model (Model II), it is assumed that the buyer agrees to bear a portion of the warranty cost. The proposed models are solved under decentralized scenario. We also derive and optimize the average total profit of the supply chain in order to obtain the optimal decisions of the centralized model. We consider a Stackelberg game between the vendor and the buyer in the decentralized scenario, where the vendor is assumed to be the leader and the buyer as the pursuer. Through numerical study, it is observed that, with respect to all the key decisions of the models, Model II provides better outcomes than Model I. Sensitivity analysis is also carried out to examine the impacts of changes of parameter-values on the optimum decisions.

Warranty Cost Models using Accelerated Life Tests on Dagum Distribution

Warranty one of the main factors which effect on the decision to purchase any product whether it is non repairable or repairable, therefore, in market the manufacturers can compete by using warranty service. The cost of warranty are wanted to predict which are mirrored on the price and profitability of products. To achieve this goal, the research is concerned with predicting the cost of the two common types of warranty models which are free rebate warranty and pro-rata rebate warranty when The lifetime of items is assumed to follow Dagum distribution. The constant stress partially accelerated life tests based on type II censoring is used. Maximum likelihood method is used to estimate the model parameters and acceleration factor of lifetime distribution from the test data. Confidence Interval for the model parameters are constructed using normal approximation and bootstrap method. Finally, Some numerical illustrations are provided.

Two-stage optimization model for renewing warranty considering warranty deadline

Renewing warranty can provide customers with better service, and thus help manufacturers to gain market opportunities. In engineering practice, the cost for replacement is usually higher than the cost for maintenance, hence manufacturers often face huge challenge to reduce the warranty service cost. With consideration of the warranty deadline, we propose a two-stage optimization model for renewing warranty. In the first stage, a renewing warranty with deadline (RWD) policy is implemented, where the deadline represents the cumulative uptime threshold. When the cumulative uptime exceeds the deadline, the product will be minimally repaired and kept to the residual warranty period. When RWD is expired, the replacement warranty with limited repairs (RWLR) policy is applied. Under the free replacement and pro-rata warranty policy, the corresponding two-stage cost optimization model is established from the manufacturer’s perspective, the aim is to minimize the cost rate and obtain the optimal warranty period. A numerical example is provided to illustrate the validity of the proposed model, and the sensitivity analysis is also carried out.

Hybrid NSGA-II for an imperfect production system considering product quality and returns under two warranty policies

This study presents a new bi-objective mathematical model for an imperfect production system (IPS) that accounts for product returns and quality levels under two different warranty policies. We considered uncertain customer demand as stochastic behavior and product price as a function of return compensation, product quality level, and warranty-period length. We simultaneously looked at the pro rata warranty and free replacement/repair warranty policies and assumed that customers can choose the desired policy. A return policy for an online distributor was also included and two objective functions were used to address the problem. The first objective function maximized the total expected revenue, and the second objective function maximized customer satisfaction. The non-dominated sorting genetic algorithm (NSGA-II) and two others, the basic bee metaheuristic and teaching-learning-based optimization algorithms, were used to generate the initial solution for use in the NSGA-II algorithm. The results from the hybrid algorithms revealed that the proposed method improves the performance of the NSGA-II algorithm. Finally, several specific sensitivity analyses were conducted to determine the effects of the problem parameters.

Post-warranty maintenance optimization for products with deterioration depending on aging and shock

ABSTRACTDue to the non-substitutability of the product, the monopoly manufacturer designs warranty to reduce maximally warranty cost. Although the monopoly manufacturer maintains responsibly product reliability in warranty, the consumer still confronts a problem about how to sustain product reliability after the designed warranty expires. Goal of this paper is, from the consumer’s perspective, to propose a reliability maintenance policy after expiry of the monopoly manufacturer’s warranty. First, a renewable repair-limit pro-rata replacement warranty (RRLPRRW) policy is considered from viewpoint of the monopoly manufacturer and used as a quality guarantee of the product with the deterioration affected by internal aging and external shock. Second, depreciation cost is earlier considered in the process of modeling life cycle cost, which differs from the traditional works on the consumer’s post-warranty reliability maintenance problem. Third, the consumer’s discounted cost rate model considering depreciation cost is derived to achieve optimal maintenance policy. Finally, a numerical example is provided to illustrate our approach. It was found that the proposed maintenance policy is more effective and the depreciation cost considered in life cycle cost has significant effect on extension of the post-warranty period.

A geometric process warranty model using a combination policy

ABSTRACTThis paper proposes a geometric process warranty model. Assume that a combination policy (W, T) is applied after selling a product, so that a free warranty is offered in [0, W), followed by a pro-rata warranty in [W, T). Assume further the successive operating times (repair times) of the product form a decreasing (increasing) geometric process. The average cost rate of the product to the manufacturer and a consumer can be derived respectively. For exponential distribution case, the explicit formulas of the average cost rate are obtained, and an finite algorithm for determination of an optimal combination policy is suggested.

Aggregate discounted warranty cost forecasting considering the failed-but-not-reported events

Warranty claims usually incur substantial costs to the manufacturers. In practice, it is not accurate or economical to reserve money for future claims based on the prediction of the overall (life cycle) warranty cost. As a more appropriate alternative, the aggregate warranty cost forecast technique can reduce the liquidity risks and improve the warranty service efficiency. In addition, when a product fails during the warranty period, a warranty claim will be counted only when the customer reports the failure to the manufacturer. This paper focuses on forecasting the discounted warranty cost, which depends on the product sales and failure processes, warranty terms, and customer behaviors, over arbitrary time interval. To characterize the failed-but-not-reported phenomenon, a flexible time-dependent function is proposed. We derive the mathematical formulations of related factors to discuss the modeling process of total discounted warranty cost over an arbitrary time interval (TDWCATI). The impacts of warranty length and customer reporting behavior are explored. The expectation and variance of the TDWCATI are obtained under the pro-rata warranty policy and the nonrenewable minimal-repair policy, which shows that the TDWCATI is useful in planning future warranty services and budgets over a specific time period.

One-Dimensional Renewable Warranty Management within Sustainable Supply Chain

Sensor embedded products utilize sensors implanted into products during their production process. Sensors are useful in predicting the best warranty policy and warranty period to offer a customer for remanufactured components and products. The conditions and remaining lives of components and products can be estimated prior to offering a warranty based on the data provided by the sensors. This helps reduce the number of claims during warranty periods, determines the right preventive maintenance (PM) policy, and eliminates unnecessary costs inflicted on the remanufacturer. The renewing, one-dimensional Free Replacement Warranty (FRW), Pro-Rata Warranty (PRW), and combination FRW/PRW policies’ costs for remanufactured products and components were evaluated with/without offering PM for different periods in this paper. To that end, the effect of offering renewable, one-dimensional, Free Replacement Warranty (FRW), or Pro-Rata Warranty (PRW), or combination FRW/PRW warranty policies for each disassembled component and sensor embedded remanufactured product was examined, and the impact of sensor embedded products on warranty costs was assessed. A case study and varying simulation scenarios is examined and presented to illustrate the model’s applicability.

Warranty Cost Modeling and Warranty Length Optimization Under Two Types of Failure and Combination Free Replacement and Pro-Rata Warranty

The product warranty has become an indispensable facet of business operations. Burn-in is effective at eliminating infant mortality and improving operational reliability levels for consumers. This paper considers the influence of different failure states and different phases of product reliability and warranty policies on warranty costs from predelivery inspection to the end of the warranty period. We then propose a comprehensive warranty cost model that considers burn-in, free replacement warranty and pro-rata warranty as three phases for repairable products presenting two types of failure (minimal and catastrophic failure) that involve minimal repair and replacement, respectively. Warranty costs are the result of a combination of the three phases, where two types of failure occur individually or simultaneously. Moreover, we developed a framework for the modeling process of warranty costs, and the effects of various parameters, such as the burn-in time, warranty period, and distribution function on warranty costs, were analyzed. Finally, a practical case was examined by using a warranty cost model, and through an after-sales service data analysis, we obtained the failure rate distribution and optimal warranty length by minimizing the average warranty cost, which can serve as a reference for manufacturers when developing warranty policies.

Developing a new PRW policy with limitation on each rectification cost

Warranty as a kind of service contract today plays a key role in business and legal transactions. Currently, a large number of products are being sold with warranty policies in the form of free replacement warranty, pro-rata warranty, and combined warranty policies. During PRW, when the product fails, the failed product is replaced with a new product with some cost to the consumer. Modeling failures and costs of PRW have complexities; because rectification cost of a product is in fact a random variable that interest rates and inflation will affect it. In this research we present a new pro rata warranty model. In this model the cost of individual claims to the manufacturer is limited to a predetermined fixed cost. Mathematical model for predicting expected costs to the manufacturer and buyer is developed at system level and analyzed by capturing the uncertainties of rectification costs over warranty period and considering the inflationary conditions. Finally, the model has been tested using numerical example.

A Bayesian Analysis in Determining Pro-rata Warranty Decision with Two-Dimensional Deterioration

A Bayesian Analysis in Determining Pro-rata Warranty Decision with Two-Dimensional Deterioration

Optimal pro-rata warranty decision with consideration of the marketing strategy under insufficient historical reliability data

Warranties play an important role in the market, not only safeguarding the rights and interests of consumers, but also promoting the sales and reputation of manufacturers. Due to their impacts on the market, manufacturers should take their warranty policies into consideration in developing an integrated marketing strategy which can promote a product by simultaneously considering pricing, production, and related post-sale services. Most of the studies regarding the issue of pro-rata warranties (PRW) only focus on cost analysis with the reliability estimation, and few consider how marketing strategies can be integrated in the warranty decision. Moreover, the manufacturer may not have sufficient historical data to estimate the deterioration of a newly developed product, and therefore, the results obtained from analytical models may not be reliable. A Bayesian analysis should be a reasonable approach in order to deal with such a problem for the situation of insufficient historical data because it can effectively assess the deterioration based on experts’ opinions and possibly few relevant data. Accordingly, this paper proposes a Bayesian decision-making model which considers the PRW policy with the pricing and the production strategy. Moreover, in order to deal with the complexity of computation of the proposed model, a heuristic algorithm is also provided, and finally, a practical application case is used to demonstrate its usefulness.

Warranty optimisation based on the prediction of costs to the manufacturer using neural network model and Monte Carlo simulation

Warranty is a powerful marketing tool, but it always involves additional costs to the manufacturer. In order to reduce these costs and make use of warranty's marketing potential, the manufacturer needs to master the techniques for warranty cost prediction according to the reliability characteristics of the product. In this paper a combination free replacement and pro rata warranty policy is analysed as warranty model for one type of light bulbs. Since operating conditions have a great impact on product reliability, they need to be considered in such analysis. A neural network model is used to predict light bulb reliability characteristics based on the data from the tests of light bulbs in various operating conditions. Compared with a linear regression model used in the literature for similar tasks, the neural network model proved to be a more accurate method for such prediction. Reliability parameters obtained in this way are later used in Monte Carlo simulation for the prediction of times to failure needed for warranty cost calculation. The results of the analysis make possible for the manufacturer to choose the optimal warranty policy based on expected product operating conditions. In such a way, the manufacturer can lower the costs and increase the profit.

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.

Modeling Total Expenditure on Warranty Claims

We approximate the distribution of total expenditure of a retail company over warranty claims incurred in a fixed period [0, T], say the following quarter. We consider two kinds of warranty policies, namely, the non-renewing free replacement warranty policy and the non-renewing pro-rata warranty policy. Our approximation holds under modest assumptions on the distribution of the sales process of the warranted item and the nature of arrivals of warranty claims. We propose a method of using historical data to statistically estimate the parameters of the approximate distribution. Our methodology is applied to the warranty claims data from a large car manufacturer for a single car model and model year.