Optimal Warranty Period 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.

Empowering software reliability: Leveraging efficient fault detection and removal efficiency

Advancements in science and technology have led to the widespread use of computer systems in various applications, emphasizing the importance of software reliability. Software failures can have severe consequences, making thorough testing crucial. Software reliability growth models (SRGMs) play a significant role in enhancing reliability by predicting improvement over time. This article introduces a comprehensive approach to software reliability that incorporates a dynamic fault detection rate, along with fault removal efficiency. The fault detection rate measures the rate at which faults are identified during testing, reflecting the effectiveness of the testing process. By incorporating this dynamic component, the model provides a more accurate estimation of software reliability and enables adaptive testing strategies and resource allocation. Achieving a high fault detection rate is desirable, but organizations must consider the cost implications and strike a balance between reliability and time-to-market constraints. This article extends the analysis to calculate the optimal release time and optimal warranty period that minimize development costs, subject to the desired reliability. By considering these factors, development teams can make informed decisions regarding the timing of software release and the duration of the warranty period, optimizing both reliability and cost.

Optimal warranty period for the controllable deteriorating product under trade credit, inflation, and demand-dependent production rate

Optimal warranty period for the controllable deteriorating product under trade credit, inflation, and demand-dependent production rate

Optimization of Renewable Warranty by Considering System Reliability and Preventive Maintenance and Repairs

One of the effective ways to ensure the reliability of the product sold is to consider service and warranty contracts. Important decision variables in warranty policies include determining the optimal warranty period and the optimal number of maintenance activities. In this research, a model has been developed with the aim of achieving the lowest expected cost rate in the life of the device and the appropriate reliability by performing the optimal number of preventive maintenance measures. The warranty policy in this study is renewable in two dimensions, in which two dimensions of repair time and failure time are considered. Any damage leading to repair is done free of charge by the manufacturer and damage leading to replacement is done jointly between the manufacturer and the consumer by agreement. As a result, two algorithms of Genetic and Imperialist Competitive Algorithm have been developed to solve the model and have been compared with numerical example solving. Furthermore, the shelfs life of the system and the number of optimal repair and maintenance measures for maintaining reliability required by the buyer were obtained.

Optimal renewing free-maintenance warranty period and preventive replacement age for repairable products

This paper proposes a decision model for determining the optimal renewing free-maintenance warranty (RFMW) period and preventive replacement age of a general repairable product with a random workload. To model a maintenance action, if a failure occurs, either a corrective replacement or a minimal repair is performed, in case of catastrophic failure or minor failure, respectively. To avoid catastrophic failures occurring for a degraded product, a preventive replacement is predetermined for implementation at age T or at the completion of a working time, whichever comes last. During RFMW, any maintenance action (either repair or replacement) is free of charge to the customer. However, each catastrophic failure still results in a downtime cost to the customer, while a planned maintenance action increases the initial purchasing cost. The aim of this paper is to determine the optimal warranty period and preventive replacement age, from the collection viewpoint of the manufacturer as well as the customers, while minimising the corresponding expected cost rate functions. The effects of RFMW on the optimal preventive replacement model are further investigated by some comparative studies and demonstrated via a numerical example. It is shown that the proposed model provides a generalised application in maintenance theory and warranty strategy.

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.

Determination of Optimal Warranty Period with Preventive Maintenance Actions for Items from Heterogeneous Populations

In this study, we develop an optimal maintenance policy with replacement service and minimal repair service for items from heterogeneous populations and determine the optimal warranty length and repair time threshold. We consider the information-based repair-replacement policy model and develop the formula to evaluate the expected cost rate during the product life cycle. A general formulation is derived for the expected cost rate under a warranty policy for items of heterogeneous populations. When a replacement service and minimal repair service are provided for a failed item, then an item from a weak population has the property of an item from a weak population after service. Similarly, an item from a strong population has the property of an item from a strong population after service. We define the optimal maintenance strategies to minimize the expected cost rate with failure time and repair time for items with heterogeneous reliability characteristics. The effects of parameters of the intensity function for the failure times on the optimal length of the warranty period are studied numerically. Assuming that the product deteriorates, we illustrate the proposed approach using numerical applications and observe the impacts of relevant parameters on the optimal length of the warranty period.

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.

Optimizing a warranty–based sustainable product service system using game theory

In recent years, the concepts of sustainability and product service system have been closely associated with each other. In today’s competitive markets, due to Extended Producer Responsibility and customer environmental awareness, producers shift from ‘product-seller’ towards ‘product-service provider’ for using the environmental and economic advantages of coupling a product with services. This paper proposes a novel approach to determine the optimal warranty period and the out-of-warranty replacement period, from the point of view of the producer and the customer to minimise the total cost of usage and end of life of product. As regards the fact that adopted strategies by producers and customers sometimes are in conflict and it affects choosing the optimal product usage period, a game theory model was developed in this study. Finally, a case study with data from chain of local notebook service centres was applied to demonstrate some practical aspects of the developed model.

Optimal production run length and warranty period for an imperfect production system under selling price dependent on warranty period

This paper extends the extant literature to consider that the selling price of the product sold with warranty policy is a function of the length of the warranty period for an imperfect production system. A product under a free replacement/repair warranty policy is considered and the lifetime distribution of the product is assumed to be exponentially distributed. We seek to maximize the total profit per item through optimal determinations of the production run length and the warranty period. Based on the proposed mathematical model, we will derive some lemmas and the propositions to carry out an efficient solution procedure to determine the optimal warranty period and production run length. The numerical examples are given to illustrate the features of the proposed maximizing profit model and to show all of the possible cases of the solution procedure.

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.

Extended warranty policies with varying objectives

We consider a repairable system subject to two types of failures, minor and major requiring minimal repair and replacement, respectively, as maintenance actions. Extended warranty models for the system that includes a free replacement period and an extended warranty period are analysed. Consumers have several warranty options during the extended warranty period. Explicit expressions for the manufacturer’s long run average profit per unit time under these options are obtained. We enlarge the objective function by including the total sales volume. This is done by considering the demand as a displaced log linear function of the product price, fixed warranty period and the extended warranty period. Finally, we include the penalty cost of customer’s non-return into the objective function. Numerical illustrations are provided in each case to demonstrate optimal pricing and warranty period.

ECONOMIC PRODUCTION RUN LENGTH AND WARRANTY PERIOD FOR PRODUCT WITH WEIBULL LIFETIME

Traditional economic manufacturing quantity (EMQ) model addressed the perfect production for product. However, there possibly exists the defective product in the production process. In this paper, we present a modified EMQ model with shortage based on considering the product revenue and the inventory cost. The economic production run length, the maximum shortage level, and the optimal warranty period are simultaneously determined for obtaining the maximum expected total profit per year. The numerical example and the sensitivity analysis are further provided for illustration.

Optimum production run length and warranty period setting for the imperfect production system

Traditional economic manufacturing quantity (EMQ) model addressed that the perfect production for product. However, there possibly exists the defective product in the manufacturing process. Hence, it is necessary to consider the production process state in the modified EMQ model. Chen and Chung presented the quality selection problem to the imperfect production system for obtaining the optimum production run length and target level. Ladany and Shore proposed the problem of determining optimal warranty period of product in relation to the manufacturer’s lower specification limit. In this paper, we further integrate Chen and Chung’s and Ladany and Shore’s models in modified EMQ model for obtaining the optimum production run length and warranty period of product for the imperfect production system.

Optimal production, maintenance, and warranty strategies for item sold with rebate combination warranty

In this paper, a decision model is developed for an imperfect production system in which the demand for a product is assumed to be influenced by the warranty period offered to the customer. The production process may shift from an ‘in-control’ state to an ‘out-of-control’ state at any random time at which point non-conforming items are produced. To assess the state of the production process, periodic inspections are carried out during each production run. At the time of inspection, if the process is found to be in the ‘out-of-control’ state then restoration is done; otherwise, preventive maintenance is performed. The proposed model is formulated under a general process shift distribution and a rebate combination warranty. The expected cost per unit item is taken as the criterion for optimality. Some analytical results of the model are derived with restrictions. The optimal decisions (i.e. the optimal number of inspections, the optimal production time, and the optimal warranty period offered to the customer) are obtained numerically and the sensitivity of some model parameters is examined in a numerical example.

Profit Maximizing Warranty Period with Sales Expressed by a Demand Function

AbstractThe problem of determining the optimal warranty period, assumed to coincide with the manufacturer's lower specification limit for the lifetime of the product, is addressed. It is assumed that the quantity sold depends via a Cobb–Douglas‐type demand function on the sale price and on the warranty period, and that both the cost incurred for a non‐conforming item and the sale price increase with the warranty period. A general solution is derived using Response Modeling Methodology (RMM) and a new approximation for the standard normal cumulative distribution function. The general solution is compared with the exact optimal solutions derived under various distributional scenarios. Relative to the exact optimal solutions, RMM‐based solutions are accurate to at least the first three significant digits. Some exact results are derived for the uniform and the exponential distributions. Copyright © 2006 John Wiley & Sons, Ltd.

Determining the optimal software warranty period under various operational circumstances

Purpose – To determine the optimal software warranty period in continuous and discrete circumstances where the difference between the software testing environment and the operational environment can be characterised by an environment factor.Design/methodology/approach – Software reliability models based on continuous and discrete time non‐homogeneous Poisson processes are assumed to describe the failure occurrence phenomena under both environments. Based on the idea of accelerated life testing for hardware products, the operational profile of the software is modeled, and the total expected software cost incurred in both testing and operational phases is formulated.Findings – Under a milder condition, the optimal warranty period which minimizes the total software cost is derived analytically.Originality/value – This paper introduces the operational profile of software to model the difference between the testing environment and the operational environment.

OPTIMAL WARRANTY PERIOD AND OPTIMAL REPLACEMENT AGE OUT-OF-WARRANTY

This paper considers a general repairable product sold under a failure-free renewing warranty agreement. In the case of a general repairable model, there can be two types of failure: type Ⅰ failure (a minor failure), which can be rectified by minimal repairs; and type Ⅱ failure (a catastrophic failure), which can be removed only by replacement. After a minimal repair, the product is operational but the failure rate of the product remains unchanged. The aim of this paper is to determine the optimal warranty period and the optimal out-of-warranty replacement age, from the perspective of the seller (manufacturer) and the buyer (consumer), respectively, while minimizing the corresponding cost functions. And we can prove under some mild conditions, the optimal solution that minimizing the cost function is uniquely exists and finite. Finally, a numerical example is presented.

Determining optimal warranty periods from the seller's perspective and optimal out-of-warranty replacement age from the buyer's perspective

This paper considers a general repairable product sold under a failure-free renewing warranty agreement. In the case of a general repair model, there can be two types of failure: Type I failure (a minor failure), which can be rectified by minimal repairs; and type II failure (a catastrophic failure), which can be rectified only by replacement. After a minimal repair, the product is operational but the failure rate of the product remains unchanged. The aim of this paper is to determine the optimal warranty period and the optimal out-of-warranty replacement age, from the perspective of the seller (manufacturer) and the buyer (consumer), respectively, while minimizing the corresponding cost functions. We prove under mild conditions, that the optimal solution of minimizing the cost function exists and is finite. Further, a concise numerical example is demonstrated, and the sensitivity analysis of some of the parameters related to costs is carried out as well. Finally, some practical aspects of renewing the warranty policy for the practitioner and reader are addressed.

OPTIMAL WARRANTY PERIOD AND OPTIMAL REPLACEMENT AGE OUT-OF-WARRANTY

ABSTRACT This paper considers a general repairable product sold under a failure-free renewing warranty agreement. In the case of a general repairable model, there can be two types of failure: type I failure (a minor failure), which can be rectified by minimal repairs; and type II failure (a catastrophic failure), which can be removed only by replacement. After a minimal repair, the product is operational but the failure rate of the product remains unchanged. The aim of this paper is to determine the optimal warranty period and the optimal out-of-warranty replacement age, from the perspective of the seller (manufacturer) and the buyer (consumer), respectively, while minimizing the corresponding cost functions. And we can prove under some mild conditions, the optimal solution that minimizing the cost function is uniquely exists and finite. Finally, a numerical example is presented.