Replacement Warranty Research Articles

Investigation of price, warranty length and production quantity on profit under free replacement warranty policy

Product pricing, production quantity and warranty length play an important role in maximizing the expected profit of any manufacturing Industry. The profit maximization models reported in the literature have considered only product price and warranty length as decision variables. It is found from the literature that the development of profit maximization models with the decision variables, namely price of the product, warranty length and also the quantity of production is scarcely reported in the literature. Hence, in this study, a profit maximization model for Free Replacement Warranty policy (FRW) under static sales market for a single period with price, warranty length and production quantity as decision variables has been proposed. A numerical illustration is provided to show the theoretical prediction and applicability of the model.

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.

Cost optimization model following extended renewing two-phase warranty

In this paper, we study an extended warranty model under which the customer is offered an additional warranty period after the original two-phase warranty expires. Under the original two-phase warranty, the warranty period is divided into two non-overlapping subintervals, one of which is for replacement warranty, and the other is for minimal repair warranty. If the system failure occurs during the original warranty period, the failed system is either replaced or minimally repaired by the manufacturer, and if the failure occurs during the extended warranty period, only the minimal repair is conducted. For the system failure during the replacement warranty period, the failed system is replaced by a new one, and the warranty term is renewed anew. Following the expiration of extended warranty, the customer is solely responsible for maintaining the system for a fixed length of time period and replaces the system at the end of such a maintenance period. During the maintenance period, only the minimal repair is given for each system failure. Such a maintenance model can be considered as a generalization of several existing maintenance models which can be obtained as special cases. The main purpose of this article is to determine, from the customer’s perspective, the optimal length of maintenance period after the extended warranty expires. As the criterion to determine the optimal replacement strategy, we adopt the expected cost rate per unit time during the life cycle of the system. Given the cost structures incurred during the life cycle of the system, we formulate the expected cost and the expected length of life cycle to obtain the expected cost rate. The uniqueness of optimal solution for the decision variable is verified when the life distribution of the system shows an increasing failure rate. Numerical examples are provided to illustrate the proposed optimal replacement strategy.

Optimal Overhaul-Replacement Policies for a Repairable Machine Sold with Warranty

This research deals with an overhaul-replacement policy for a repairable machine sold with a free replacement warranty (FRW), which will be used for a finite horizon T and evaluated at fixed interval s. At each evaluation point, the buyer considers three alternative decisions, i.e. (1) keep the machine, (2) overhaul it, or (3) replace it with a new identical one. If the machine fails during the warranty period, this is rectified at no cost to the buyer. Any failure occurring before and after the expiry of the warranty is restored by minimal repair. An overhaul-replacement policy was formulated using a dynamic programming approach. The results show that overhaul rejuvenation may extend the machine life cycle and delay the replacement decision. In contrast, the warranty stimulates early machine replacement and in so doing increases the replacement frequency for a certain range of replacement costs. This implies that in order to minimize the total ownership costs over T, the buyer needs to consider the minimal repair cost reduction due to the rejuvenation effect of an overhaul as well as the warranty benefit due to replacement. Numerical examples are presented for both illustrating the optimal policy and describing the behavior of the optimal solution.

A new replacement warranty policy indexed by the product’s correlated failure and usage time

We study a new warranty policy for non-repairable products which is indexed by two correlated random variables, age and usage and covers all failures in (0,t]. Two different warranty costs for the replacement of the failed product are considered, according to its usage being greater or less than a pre-specified level s>0. A bivariate probability distribution function is applied to incorporate the correlation effect of the two variables. Analytical expressions of the probability density function of the total warranty cost and its expected value, the probability distribution functions of the number of the failed products with usage greater or less than s and their corresponding expected values and costs are derived. Limit results are also obtained. The results obtained are useful measures in establishing the compensation policy and the evaluation of its performance under the proposed warranty. Illustrative numerical examples of the expected cost for Paulson, Pareto and Beta Stacy bivariate distributions are presented and discussed. In particular for Paulson’s bivariate probability distribution, closed form expressions for the expected costs are obtained.

A study on negative binomial inspection for imperfect production systems

Production systems continuously deteriorate with age and usage due to corrosion, fatigue, and cumulative wear in production processes, resulting in an increasing possibility of producing defective products. To prevent selling defective products, inspection is usually carried out to ensure that the performance of a sold product satisfies the customer requirements. Nevertheless, some defective products may still be sold in practice. In such a case, warranties are essential in marketing products and can improve the unfavorable image by applying higher product quality and better customer service. The purpose of this paper is to provide manufacturers with an effective inspection strategy in which the task of quality management is performed under the considerations of related costs for production, sampling, inventory, and warranty. A Weibull power law process is used to describe the imperfection of the production system, and a negative binomial sampling is adopted to learn the operational states of the production process. A free replacement warranty policy is assumed in this paper, and the reworking of defective products before shipment is also discussed. A numerical application is employed to demonstrate the usefulness of the proposed approach, and sensitivity analyses are performed to study the various effects of some influential factors.

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.

Pricing and inventory policies for Hi-tech products under replacement warranty

Companies, especially in the Hi-tech (high-technology) industry (such as computer, communication and consumer electronic products), often provide a replacement warranty period for purchased items. In reality, simultaneously determining the price and inventory decisions under warranty policy is an important issue. The objective of this paper is to develop a joint pricing and inventory model for Hi-tech products under replacement warranty policy. In the first model, we consider a Hi-tech product feature in which the selling price is declining in a trend. We determine the optimal inventory level for each period and retail price for the first period while maximising the total profit. In the second model, we further determine the optimal retail price and inventory level for each period in the dynamic demand market. This study develops solution approaches to solve the problems described above. Numerical analysis discusses the influence of system parameters on the company's decisions and behaviours. The results of this study could serve as a reference for business managers or administrators.

Optimal post-warranty maintenance policy with repair time threshold for minimal repair

In this paper, we consider a renewable minimal repair–replacement warranty policy and propose an optimal maintenance model after the warranty is expired. Such model adopts the repair time threshold during the warranty period and follows with a certain type of system maintenance policy during the post-warranty period. As for the criteria for optimality, we utilize the expected cost rate per unit time during the life cycle of the system, which has been frequently used in many existing maintenance models. Based on the cost structure defined for each failure of the system, we formulate the expected cost rate during the life cycle of the system, assuming that a renewable minimal repair–replacement warranty policy with the repair time threshold is provided to the user during the warranty period. Once the warranty is expired, the maintenance of the system is the user's sole responsibility. The life cycle of the system is defined on the perspective of the user and the expected cost rate per unit time is derived in this context. We obtain the optimal maintenance policy during the maintenance period following the expiration of the warranty period by minimizing such a cost rate. Numerical examples using actual failure data are presented to exemplify the applicability of the methodologies proposed in this paper.

A profit assessment model for equipment inspection and replacement under renewing free replacement warranty policy

Abstract For many types of equipment whose maintenance requires skilled labor or whose random failure may have serious consequences, such as aircraft engines, medical equipment and others, customers generally prefer to entrust maintenance activities to the manufacturer or any of its representatives. The latter must, in turn, provide the customer with a maintenance program that is economically viable for both parties. In this study, the customer and the manufacturer agree on what follows regarding a non-self-announcing failure equipment whose state is only known through inspection: during the warranty period, the manufacturer inspects the equipment according to a specified schedule which he establishes taking into account that the costs incurred for inspections and replacements are supported by him during the warranty period. After the expiration of the warranty, inspections and replacements are performed by the manufacturer and billed to the customer. Moreover, penalties related to inactivity periods between failures and their detection are always supported by the manufacturer. The warranty is applicable to any equipment replaced during the validity period of the service contract. This type of contract generates a profit for the manufacturer. In this paper, an analytical model taking into account the commitments of both parties has been developed. The model allows generating the instants (x1, x2,…, xn) at which the inspections must be performed and the corresponding expected profit for the manufacturer while considering the warranty period offered on the market for similar equipment. An algorithm has also been developed to generate the inspection instants given the costs structure and the lifetime probability distribution of the equipment. In a context where business models argue for a greater implication of suppliers towards their customers, the proposed decision model may be relevant and very useful.

Warranty servicing for discretely degrading items with non-zero repair time under renewing warranty

This paper considers a repair-replacement strategy for a special class of discretely degrading and repairable products under renewing free replacement warranty. Each product may experience N different working states with different exponential hazard functions, before the warranty contract expires. Once the item enters working state i (i=1,2,…,N), it can either fail or move to any of the subsequent working states with different probabilities, given that a transition has been made. In the former case, the best rectification action, replacement or minimal repair, regarding the failure status, i.e., the product degradation level at the failure time and the remaining warranty time, should be conducted to put the item into operation. It is assumed that replacement of the faulty item occurs instantly and the product warranty is renewed, but non-zero repair time has been included in the model. We derive the optimal replacement-repair policy to minimize the manufacturer’s expected warranty servicing cost per item sold. The Adomian decomposition method is used to find an analytic approximate solution for a special case with two working states. For N>2, a simulation based optimization method has been developed to analyze the expected warranty cost. Some numerical examples in each section are given to clearly demonstrate the application of this model.

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.

On optimal replacement-repair policy for multi-state deteriorating products under renewing free replacement warranty

In this paper a replacement-repair model is developed to study a renewing free replacement warranty (RFRW) for a class of multi-state deteriorating repairable products. After a replacement, the product warranty is renewed. Two parameters affect the manufacturer’s decision to minimally repair or replace a failed item: the product deterioration level at the time of failure and the remaining warranty time. We derive an optimal replacement-repair policy to minimise the expected warranty servicing cost per item sold. In the case of a product with two different working states, we are able to explicitly find the optimal values of the replacement-repair parameters using the Adomian decomposition method. In general, for N > 2 , it is impossible to find the optimal parameter values analytically. Hence, a computational procedure for finding these optimal values is proposed. For more clarification, numerical examples are also presented.

Hybrid minimal repair and age replacement policy for two-dimensional warranted products

In this paper, we investigate a hybrid minimal repair and age replacement policy for a repairable product sold with a two-dimensional non-renewing failure replacement warranty. We first model product failures using the one-dimensional approach and then use a more appropriate formulation to model the effect of age and usage to the product degradation. Under this policy, for a given usage rate y, the product is directly repaired minimally when it fails in (0, Sy] and replaced with the new one on the first failure in (Sy, ?y] or when its age reaches ?y, whichever occurs first. We obtain the global optimal solution of Sy, for a given ?y which minimises the expected cost per unit time to the buyer. We present numerical examples to illustrate the properties of the optimal solution.

An Optimal Age for Preventive Replacement Under Fully Renewing Combination Free Replacement and Pro Rata Warranty

In the present article, an age-replacement policy is considered for non repairable products under the fully renewing combination free replacement and pro-rata warranty (FRW/PRW) policy. For the fully renewing FRW/PRW policy, whenever a product fails within the warranty period, it is replaced by a new one and a new warranty issued. Meanwhile, if the failure occurs during the first time interval (i.e., the free-replacement warranty (FRW) interval), replacement is made at no cost to the buyer; however, if the failure occurs during the second time interval (i.e., the pro-rata warranty (PRW) interval), replacement is made at a pro-rata cost to the buyer. Pre-determining the age for preventive replacement to be in a different certain interval, cost models from the buyer/user perspective are developed, and the corresponding local optimal replacement ages are derived such that the long-run expected cost rate is minimized. Then, the global optimal replacement age (i.e., without any constraint) is determined. Structural properties of these optimal policies are obtained and presented. Special cases of the model are also discussed. Finally, numerical examples are given for illustration, which provides a comprehensive study about the behavior of the optimal solution.

Managing the Inventory of an Item with a Replacement Warranty

In this paper, we study a firm that faces demand from two sources: demand for new items and demand to replace failed items under warranty. We model this setting as a multiperiod single-product inventory problem where the demands for new items in different periods are independent and the demands for replacing failed items depend on the number and ages of the items under warranty. We consider backlogging and emergency supply cases, and study both discounted cost and average cost criteria. We prove the optimality of the w-dependent base-stock ordering policy where the base-stock level is a function of w, the vector representing the number of items at different ages currently under warranty. For the special case where the demand for new products is identically distributed, we prove the optimality of a stationary w-dependent base stock policy for the finite-horizon discounted and the infinite-horizon discounted and average cost cases. In our computational study, we find that an optimal w-dependent policy can lead to 69% average improvement in expected costs when compared to a policy that neglects demands from items under warranty.

Product warranty: modelling with 2D-renewal process

The study of product warranty is of interest to manufacturers for several reasons. Paid claims are a cost of doing business and a liability incurred by the manufacturer at the time of sale. For these reasons, estimating and forecasting warranty expenses is of interest. Also, warranty data provides information about the durability of products in the field and, therefore, is of interest to engineers. Here we emphasise two-dimensional warranties, like automotive warranties, that guarantee free repairs subject to both age and mileage limits. We aim to evaluate the average warranty expenses under a two-dimensional non-renewing free replacement warranty policy using real warranty data. We derive a numerical procedure for the evaluation of the two-dimensional renewal function and use it for the warranty cost analysis. In addition, we test whether the available data can be considered as coming from a two-dimensional renewal process.

On a conjecture of optimal repair-replacement strategies for warranted products

Under a free replacement warranty policy, the manufacturer has the option of rectifying a failure by either repairing the failed item or replacing it with a new one. In [N. Jack, F. Van der Duyn Schouten, Optimal repair-replacement strategies for a warranted product, Int. J. Prod. Econ. 67 (2000) 95–100] a conjecture is made on the optimality of an age-limit policy: there exists a control limit AL ( t ) , such that a minimal repair is performed on failure with t time units remaining in the warranty period if the age of the failing unit is less than or equal to AL ( t ) . In this paper, we prove that this conjecture is indeed correct. Moreover, we show that for a more general model where the repair cost is random, a generalized repair-cost-limit policy is optimal, and the age-limit policy becomes its degenerated form. Numerical examples are provided to illustrate the computational algorithm and to compare the performance of these policies.

Warranty cost analysis: quasi‐renewal inter‐repair times

PurposeTo provide a brief introduction to warranty analysis and a classification of general repairs. To introduce the notion of accelerated probability distribution and use it to model imperfect warranty repairs.Design/methodology/approachThe notion of accelerated probability distribution is discussed and its similarity with quasi‐renewal and geometric processes is observed. An approach to modeling imperfect warranty repairs based on the accelerated probability distributions is presented, and the corresponding expected warranty cost over the warranty period under non‐renewing free replacement warranty policy is evaluated.FindingsIt is observed that quasi‐renewal and the geometric processes are equivalent. Using data from an existing warranty database it is shown that the inter‐repair times form a quasi‐renewal process. The corresponding expected warranty cost over the warranty period under a non‐renewing free replacement warranty policy is evaluated.Research limitations/implicationsThis approach is applicable only if the cost of the warranty repair is an increasing function of the number of repairs.Practical implicationsProvides a useful approach to modeling inter‐repair times incorporating the idea of imperfect repairs in practice.Originality/valueProvides an approach to model imperfect warranty repairs and to evaluate the corresponding expected warranty cost.

Repair-replace strategies for two-dimensional warranty policies

For repairable items sold with free replacement warranty, the actions available to the manufacturer to rectify failures under warranty are to 1. (1) repair the failed item or 2. (2) replace it with a new one. A proper repair-replace strategy can reduce the expected cost of servicing the warranty. In this paper, we study repair-replace strategies for items sold with a two-dimensional free replacement warranty.