Rebate Warranty Research Articles

Quality assurance using accelerated life testing via rebate warranty

Quality assurance using accelerated life testing via rebate warranty

Acceptance Sampling Scheme with Warranty for Life Testing Using Bayesian Inference Methods

Abstract We propose an empirical Bayesian (EB) procedure with a rebate warranty policy to search for optimal sampling plans for three-parameter Burr-type XII distribution (3pBXIID) under truncated life testing. The proposed method requires fewer assumptions than its competitors to search for optimal sampling plans when the lifetimes of products follow a 3pBXIID. The Newton–Raphson method using the quasi-Newton (QN) algorithm, the evolution algorithm method using the particle swarm optimization (PSO) algorithm and genetic algorithm (GA) are used in the proposed EB procedure to search for reliable estimates of the parameters in the Bayesian model, and the obtained estimates are denoted as QN-EB, PSO-EB, and GA-EB, respectively. The performance of the QN-EB, PSO-EB, and GA-EB estimates were evaluated by performing Monte Carlo simulations. The simulation results show that the PSO-EB method outperforms the GA-EB and QN-EB methods in obtaining reliable estimates of the parameters in terms of bias and mean squared error. A grid-searching algorithm with the plug-in parameters of PSO-EB estimates to search for the optimal EB sampling plans is presented. Two examples pertaining to the lifetime of an oil-well pump in a sucker-rod oil pumping system and the first failure time of a small electric cart are used to illustrate the application of the proposed EB method.

Optimal maintenance strategies for warranty products with limited repair time and limited repair number

Nowadays, to protect the interests of customers, many warranty policies take into account the limit of repair time and the limit of repair number. Under a rebate warranty policy, the seller has to give a refund to the customer according to the rebate warranty contract if the number of product failures or a single repair time for failure exceeds some certain limits. Minimal repair is conducted when the product fails, and the repair cost is charged to seller during the warranty period. After warranty expires, the product continues operating for a period of time before being replaced. The repair cost during the post-warranty period is undertaken by the customer. From the customer’s perspective, the expected life cycle cost rate of the warranty product is derived. By minimizing the expected cost rate, the existence and uniqueness of the optimal maintenance strategy for the warranty product are proven theoretically. Finally, as an example, the warranty car under the protection of “lemon laws” is studied to illustrate the proposed rebate warranty model.

Optimal replacement strategies for warranty products with multiple failure modes after warranty expiry

The optimal replacement problem for a warranty product subject to (M + 1) types of mutually exclusive failure modes, including M repairable failure modes and a catastrophic failure mode, is investigated in this paper. When the product fails with a repairable failure mode, a minimal repair will be conducted to remove the failure. However, if a catastrophic failure mode occurs, the failure product can only be replaced by a new one. Within the warranty period, the seller needs to complete each failure repair within a repair time limit. Once a failure cannot be repaired within the stipulated time, under the rebate warranty, the customer is refunded a proportion of the sale price of the product based on its working time, and the warranty terminates simultaneously; or under the free replacement warranty, the seller is obligated to provide a new product at no charge to the customer. Under both the warranty policies, the repair cost is free of charge to the customer within the warranty time, while the customer is responsible for all the cost for repairs within the post-warranty time. From the customer’s point of view, the expected life cycle cost rate of the product under the rebate and free replacement warranty are respectively derived. The existence and uniqueness of the optimal replacement strategies after warranty expiring are also proved theoretically by minimizing the cost rate. Finally, we give numerical examples to illustrate the two warranty models and show the effectiveness of proposed strategies.

Optimum reliability acceptance sampling plan using Type-I generalized hybrid censoring scheme for products under warranty

PurposeThe purpose of this paper is to design an optimal reliability acceptance sampling plan (RASP) using the Type-I generalized hybrid censoring scheme (GHCS) for non-repairable products sold under the general rebate warranty. A cost function approach is proposed for products having Weibull distributed lifetimes incorporating relevant costs.Design/methodology/approachFor Weibull distributed product lifetimes, acceptance criterion introduced by Lieberman and Resnikoff (1955) is derived for Type-I GHCS. A cost function is formulated using expected warranty cost and other relevant cost components incorporating the acceptance criterion. The cost function is optimized following a constrained optimization approach to arrive at the optimum RASP. The constraint ensures that the producer's and the consumer's risks are maintained at agreed-upon levels.FindingsOptimal solution using the above approach is obtained for Type-I GHCS. As a special case of Type-I GHCS, the proposed approach is also used to arrive at the optimal design for Type-I hybrid censoring scheme as shown in Chakrabarty et al. (2019). Observations regarding the change in optimal design and computational times between the two censoring schemes are noted. An extensive simulation study is performed to validate the model for finite sample sizes and the results obtained are found to be in strong agreement. In order to analyze the sensitivity of the optimal solution due to misspecification of parameter values and cost components, a well-designed sensitivity analysis is carried out using a real-life failure data set from Lawless (2003). Interesting observations are made regarding the change in optimal cost due to change in parameter values, the impact of warranty cost in optimal design and change in optimal design due to change in lot sizes.Originality/valueThe research presents an approach for designing optimal RASPs using Type-I generalized hybrid censoring. The study formulates optimum life test sampling plans by minimizing the average aggregate costs involved, which makes it valuable in dealing with real-life problems pertaining to product quality management.

Design and Analysis of Accelerated Life Testing and its Application Under Rebate Warranty

The study deals with the advancement of accelerated life testing in the field of product warranty. The expected total cost and expected cost rate for age replacement is estimated under warranty policy using accelerated life testing (ALT) plans. Under constant stress, the lifetimes of the units are assumed to follow generalised exponential distribution. The estimation process is carried through maximum likelihood estimation method. Also, the Age Replacement Policy under Pro-rate Rebate Warranty is discussed. Finally, an application example is presented to illustrate the theoretical results.

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.

Optimum life test plan for Type-I hybrid censored Weibull distributed products sold under general rebate warranty

The selection of an appropriate life test plan is extremely important for any product as it not just improves the quality of the product but also reduces testing costs. In this approach, however, the choice of suitable costs play an important role. In this paper, a decision model is developed to determine the optimal life testing plan (LTP) by minimising the relevant costs involved for non-repairable products sold under the general rebate warranty. LTP are developed in the presence of Type-I hybrid censoring for products having Weibull distributed lifetimes. A constrained optimisation approach is followed considering both producer's and consumer's risk and suitable analysis techniques are employed in obtaining the optimal solution. Monte Carlo simulation is conducted for numerical illustration. In order to study the sensitivity of the optimal solution due to mis-specification of parameter values and cost components, a well-designed sensitivity analysis is incorporated using a real-life Type-I hybrid censored data set.

Empirical Bayesian Strategy for Sampling Plans with Warranty Under Truncated Censoring

To reach an optimal acceptance sampling decision for products, whose lifetimes are Burr type XII distribution, sampling plans are developed with a rebate warranty policy based on truncated censored data. The smallest sample size and acceptance number are determined to minimize the expected total cost, which consists of the test cost, experimental time cost, the cost of lot acceptance or rejection, and the warranty cost. A new method, which combines a simple empirical Bayesian method and the genetic algorithm (GA) method, named the EB-GA method, is proposed to estimate the unknown distribution parameter and hyper-parameters. The parameters of the GA are determined through using an optimal Taguchi design procedure to reduce the subjectivity of parameter determination. An algorithm is presented to implement the EB-GA method. The application of the proposed method is illustrated by an example. Monte Carlo simulation results show that the EB-GA method works well for parameter estimation in terms of small bias and mean square error.

Analysis of a hybrid warranty policy for discrete-time operating products

This article considers a warranty policy consisting of a renewable free-replacement period and a rebate period for products operating in discrete time. Under such a Hybrid Warranty Policy (HWP), if a product fails during the first N periods (from period 1 to period N), it is replaced with either a new or a repaired unit for free (or at the manufacturer's expense) and the policy is renewed; if it fails in the next W − N periods (from period N + 1 to period W), then the manufacturer refunds a pre-specified proportion of the sales price to the buyer. A pure rebate warranty policy and a pure renewable replacement policy can be considered as special cases of the HWP. With the HWP, customer service and warranty cost can be traded off. The conditions under which the HWP is more cost-effective are derived from the perspective of the manufacturer or seller. Some structural properties of the HWP are examined. Furthermore, how to choose between using new and repaired products to replace failed products is discussed. A numerical example is presented to demonstrate the computation of the optimal N*, the decision variable of the HWP, and a sensitivity analysis is performed to gain some managerial insights for practitioners.

An application of Prekopa's theorem on the logconcavity of warranty reserves in reliability engineering

The log-concavity of the expected refund rate for an item with a nonrenewing pro-rata rebate warranty policy, and of the expected total reserve to service the warranty over the item life cycle under the same policy are proved. Upper bounds of warranty reserves are derived. The results are extended to other measures for spares and repairs under other warranty policies. Applications are given to ranking items' prices and production and inventory control.

Accelerated life testing and age-replacement policy under warranty on exponentiated Pareto distribution

In This Paper, we focus on estimating the expected total cost and expected cost rate for age replacement under warranty policy. We describe how to use design of Accelerated life testing (ALT) plans by assuming constant stress and life times are independently distributed as Exponentiated Pareto Distribution (EP) for predicting the cost of age replacement under warranty policy. This problem is studied when maximum likelihood estimators (MLE) are obtained. Also, the Age Replacement Policy under Pro- rate Rebate Warranty is displayed. Finally, numerical examples are carried out to illustrate theoretical results.

Risk-embedded Bayesian acceptance sampling plans via conditional value-at-risk with Type II censoring

An acceptance sampling plan is usually determined by minimizing the expectation of the sum of the relevant costs involved. This expected cost minimization approach, however, could result in a great cost at a probability that is unacceptable to a decision maker. This study therefore develops a risk-embedded model via conditional value-at-risk that allows a decision maker to choose an acceptance sampling plan with minimal expected excess cost in accordance with his or her attitude towards risk. We focus in this study on Bayesian acceptance sampling under Type II censoring for Weibull distributed product lifetime with known shape parameter and products being sold with a general rebate warranty. We explore, through numerical analysis, the individual effects and cross-effects of a decision maker’s attitude towards risk and various unit costs on the Bayesian acceptance sampling plans, aiming to gain insights into the role of a decision maker’s risk aversion in the determination of Bayesian acceptance sampling plans.

Customized warranty offering for configurable products

Although configurable products have been common in the market, buyers often receive standard warranty offerings rather than customized ones from manufacturers. In this study, a reliability based method is proposed to specify warranties for certain product configuration schemes. A continuous time Markov model is adopted to measure the vulnerabilities of components involved in configuration. Then the warranty claim probability of a product with multiple components is calculated, and the warranty costs in varied warranty periods are estimated. Under a minimal repair warranty in which only failed components are replaced free of charge, warranty plans can be customized for different configuration schemes on the basis of warranty costs. Next, an example of laptop computer is introduced to illustrate the procedure and effectiveness of the proposed method. Impacts of operating frequency and operating time of products on warranty costs are also measured. In addition, some recommendations are provided in order to deal with issues of replacement policy, pro-rata rebate warranty (PRW) and compatibility between components.

The Effect of a Pro-Rata Rebate Warranty on the Age Replacement Policy With Salvage Value Consideration

This paper presents the effects of salvage value on the optimal age-replacement policy for non-repairable products sold with a pro rata rebate warranty (PRRW). Cost models from the customer's perspective are developed for both warranted, and non-warranted products. The corresponding optimal replacement ages are derived such that the long-run expected cost rate is minimized. Under the increasing failure rate assumption, the existence and uniqueness of the optimal age for preventive replacement are shown, and the impacts of a PRRW on the optimal age-replacement policy are investigated analytically. Finally, a numerical example is provided for the optimal policy illustration and verification.

Modeling and analysis of effective ways for improving the reliability of second-hand products sold with warranty

Often, customers are uncertain about the performance and durability of the used/second-hand products. The warranties play an important role in reassuring the buyer. Offering the warranty implies that the dealer incurs additional costs to service any claims made by the customers. Reducing warranty costs is an issue of great interest to dealers. One way of improving the reliability and reducing the warranty servicing cost for second-hand items is through actions such as overhaul and upgrade which are carried out by the dealer or a third party. Improving actions allow the dealer to offer better warranty terms and to sell the item at a higher price. This paper deals with two effective approaches (virtual age approach and screening test approach) to decide on the reliability improvement strategies for second-hand products sold under various warranty policies (failure-free, rebate warranty, and a combination of free replacement and lump sum). A numerical example illustrates that from a dealer’s point of view, it is beneficial to carry out an improvement action only if the reduction in the warranty servicing cost is greater than the extra cost incurred due to this improvement action.

BAYESIAN SAMPLING PLANS WITH PROGRESSIVE CENSORING AND WARRANTY POLICY

The paper investigates the design of Bayesian sampling plan for the exponential lifetime model under progressive type-II censoring, in which items are manufactured in batches and sold to consumers with a general rebate warranty policy. Assume that the mean lifetime of items is random and varies from lot to lot. A cost model consists of the cost per item on test, the cost per item of test time and the costs of rejecting and accepting an item is established, and an algorithm is provided to determine the optimal Bayesian sampling plan which minimizes the expected average cost per lot. The use of the proposed method is illustrated by numerical results and an example. A sensitivity study is conducted to evaluate the influences of the removal scheme and using incorrect estimates for the hyper-parameters on the proposed sampling plans. The proposed method can be extended to the Weibull lifetime model as its shape parameter is known.

Optimal spare ordering policy under a rebate warranty

This paper develops, from the customer’s perspective, the optimal spare ordering policy for a non-repairable product with a limited-duration lifetime and under a rebate warranty. The spare unit for replacement is available only by order and the lead time for delivery follows a specified probability distribution. Through evaluation of gains due to the rebate and the costs due to ordering, shortage, and holding, we derive the expected cost per unit time and cost effectiveness in the long run and examine the optimal ordering time by minimizing or maximizing these cost expressions. We show that there exists a unique optimum solution under mild assumptions. We provide a numerical example and illustrate sensitivity analysis.

Reliability sampling plans for Weibull distribution with limited capacity of test facility

This paper establishes reliability sampling plans for the Weibull lifetime distribution based on type II censored data with limited capacity of test facility. The products are sold under a general rebate warranty policy. It is also assumed that the shape parameter of the lifetime distribution is known, and the scale parameter is a random variable varying from lot to lot. A cost model is established which contains the cost per unit on test, the cost per unit time for life test, and the costs of rejecting and accepting a unit. An algorithm for determining the optimal reliability sampling plans which minimize the expected average cost per lot is provided. Some numerical results to illustrate the use of the proposed method are studied.

ECONOMIC DESIGNS OF LIFE TEST SAMPLING PLANS FOR REPAIRABLE PRODUCTS

Economic designs of life test sampling plans are considered for repairable products which are sold under a general rebate warranty policy. It is assumed that the products in each lot have the same failure intensity function, but the parameters of the function vary from lot to lot according to a known prior distribution. Cost models are constructed which involve three cost components; test cost, accept cost, and reject cost, Methods of finding optimal sampling plans which minimize the expected average cost per lot are presented. Numerical examples are given and sensitivity analyses for parameters of the prior distribution are performed