Repairable Products Research Articles

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.

Warranty Cost Analyses Using Quasi-Renewal Processes for Multicomponent Systems

In this paper, warranty cost models are presented based on the quasi-renewal processes and exponential distribution. Cost analyses are conducted for various systems under the basic assumption that a repair service is imperfect. We develop warranty cost models, reliability, and other measures for several systems, including multicomponent systems. This paper focuses on warranty cost analysis, including repairable products with a given warranty period considering conditional probabilities and renewal theory. The exponential distribution is used to analyze and obtain the warranty cost. Numerical examples are discussed to demonstrate the applicability of the methodology derived in this paper.

A mixed inspection policy for CSP-2 and precise inspection under inspection errors and return cost

This article deals with a mixed policy between precise inspection and continuous sampling plan-2 (CSP-2) with inspection errors (Types I and II) and return cost. The optimal mixed inspection policy is determined by using the criterion of maximizing the unit net profit. Specifically, with respect to non-repairable and repairable products, the following decision variables are determined such that the unit net profit is maximal: (1) the optimal sampling plan for CSP-2 and (2) the optimal proportions for which precise inspection should be performed for the non-inspected items in the procedure of CSP-2, the non-defective items, and the defective items identified by CSP-2. Overall, the analytical results indicate that, depending on the nine parameters (the process defective fraction, Type I error, Type II error, the selling price of an item, the unit repair cost, the unit return cost, the unit opportunity cost, the unit inspection cost, and the unit cost of precise inspection), both non-repairable and repairable products have two main optimal inspection policies for CSP-2: “do not inspect” and “do 100% inspection”; also, for the non-inspected items, the non-defective items, and the defective items identified in CSP-2, there are two main proportions on which precise inspection should be performed: 100% and 0%. A numerical example is provided to illustrate the proposed method. Besides, a comparison with CSP-2 is also made to evaluate the effectiveness of the proposed model.

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.

The Optimal Maintenance Period for Repairable Product after Warranty Expiration under Renewing Warranty

A model of the optimal maintenance period for repairable product after warranty expiration under renewing warranty is proposed. In the process to fulfilling warranty, the failures of product within warranty period happen randomly, the replacements of products successively failed with warranty period form a renewing process. Given failure rate of product, the period of fulfilling warranty is function of warranty period. User purchasing a product is considered an investment; the period of fulfilling warranty and the maintenance period after warranty make up the life cycle of product investment. Taking various cost factors into account, making model of the cost rate in life cycle of product investment under renewing warranty, the cost rate is function of maintenance period. Give the failure rate of product is an increasing function, it is derived that there is unique optimal maintenance period minimizing the cost rate. Finally, numerical example is given for illustration.

Optimal preventive maintenance warranty policy for repairable products with periodically increasing failure rate

In this paper, a preventive maintenance warranty (PMW) policy for repairable products is investigated. The objective is to determine the optimal number of preventive maintenance (PM) actions, the degree of maintenance and the length of each PM interval; while minimising the total expected warranty cost. The failure rate of repairable products is assumed to be periodically increasing over time. A unique relationship between successive PM intervals is determined. For a special case of failure rate and minimal repair cost, a relationship among the PM intervals is derived. A unique property is inferred from this special case, which relates the maximum failure level in the adjoining PM intervals. The derived relations are illustrated by a numerical example. Yeh and Lo presented a mathematical model for the optimal determination of a PMW policy under similar periodic failure rates. The present study is an extension of their work and shows that unequal PM intervals provide an economic benefit.

On prorated servicing costs for two-dimensional warranties with combined repair-replacement strategies

We study a non-renewing two-dimensional warranty policy for repairable products with pro-rated cost for rectifying failures within a warranty coverage period defined by a rectangular region. Product failures in different warranty regions have different cost implications based on repair-replacement strategy employed by the manufacturer and the time and usage at the event of product failure. We obtain expressions for expected cost to the manufacturer based on two possible repair-replacement strategies which can be employed towards servicing the product in the event of failures and present the respective costs associated with them.

Optimal warranty policies for systems with imperfect repair

We investigate a system whose basic warranty coverage is minimal repair up to a specified warranty length. An additional service is offered whereby first failure is restored up to the consumers’ chosen level of repair. The problem is studied under two system replacement strategies: periodic maintenance before and after warranty. It turns out that our model generalizes the model of Rinsaka and Sandoh [K. Rinsaka, H. Sandoh, A stochastic model with an additional warranty contract, Computers and Mathematics with Applications 51 (2006) 179–188] and the model of Yeh et al. [R.H. Yeh, M.Y. Chen, C.Y. Lin, Optimal periodic replacement policy for repairable products under free-repair warranty, European Journal of Operational Research 176 (2007) 1678–1686]. We derive the optimal maintenance period and optimal level of repair based on the structures of the cost function and failure rate function. We show that under certain assumptions, the optimal repair level for additional service is an increasing function of the replacement time. We provide numerical studies to verify some of our results.

Dynamic pricing and warranty policies for products with fixed lifetime

Abstract We consider a repairable product with known market entry and departure times. A warranty policy is offered with product purchase, under which a customer can have a failed item repaired free of charge in the warranty period. It is assumed that customers are heterogeneous in their risk attitudes toward uncertain repair costs incurred after the warranty expires. The objective is to determine a joint dynamic pricing and warranty policy for the lifetime of the product, which maximizes the manufacturer’s expected profit. In the first part of the analysis, we consider a linearly decreasing price function and a constant warranty length. We first study customers’ purchase patterns under several different pricing strategies by the manufacturer and then discuss the optimal pricing and warranty strategy. In the second part, we assume that the warranty length can be altered once during the product lifetime in developing a joint pricing and warranty policy. Numerical studies show that a dynamic warranty policy can significantly outperform a fixed-length warranty policy.

A mixed inspection policy for CSP-1 and precise inspection under inspection errors and return cost

This paper is to deal with a mixed policy between precise inspection and CSP-1 with inspection errors (Types I and II) and return cost. First, the concept of a renewal reward process is employed to obtain the long-term net profit of the mixed inspection policy. Then, with respect to non-repairable and repairable products, we determine the following four decision variables such that the unit net profit is maximal: (1) the optimal clearance number, (2) the optimal sampling frequency, and (3) the proportions which should be taken precise inspection for the non-inspected items in the procedure of CSP-1 and the non-defectives identified by CSP-1. Overall, the analytical results indicate that depending on seven parameters (Type I error, Type II error, the selling price of an item, the unit repair cost, the unit return cost, the unit cost of precise inspection, and the process defective fraction), there are three possible optimal inspection policies for CSP-1: “Do not inspect”, “Do 100% inspection”, or any setting for ( i, s); for both of the non-defectives identified in CSP-1 and the non-inspected items, there are three possible proportions which should be performed a precise inspection: all, none, or any proportion.

Development of a new reliability prediction model for predicting the reliability of repairable products

An accurate product reliability prediction model can not only predict and track the product's reliability and operational performance, but also offer useful information for managers to take follow-up actions to improve the product quality and cost. The time series decomposition approach is widely used for prediction, but is not used for reliability prediction. This study proposes a new method for predicting the reliability of repairable products. The novel method constructs a predictive model by employing the time series decomposition approach. Finally, one real case study is presented to illustrate the feasibility of the proposed method.

Optimal price and pro rata decisions for combined warranty policies with different repair options

A repairable product under a non-renewing combined warranty policy that is subject to a displaced log-linear demand function of the product's price and pro rata period length is considered. Expressions for the manufacturer's long-run average profit per unit time under replacement, minimal and general repair options are obtained. In addition, expressions for the stationary points and second-order conditions of the profit function are presented. Numerical illustrations that demonstrate optimal product pricing, pro rata length determination, and repair option selection to maximize the manufacturers, profit are given.

A new warranty strategy: Combining a renewing free‐replacement warranty with a rebate policy

AbstractIn this paper, a new warranty policy is proposed and analyzed for repairable products. This policy combines a renewing free‐replacement warranty with a rebate policy. Under this combined policy, the warranty period is partitioned into two intervals. If a product fails during the first interval, the seller replaces the product with a new or repaired product, and the warranty begins anew for the replacement. If a product fails in the second interval, the seller refunds a pre‐specified proportion of the sales price. We derive conditions under which the combined policy is cost‐effective from the perspective of the seller. A numerical example is presented and a sensitivity analysis is performed. Our results imply that reduced demand due to higher prices can be offset with a longer warranty period. Finally, the future extension as well as the practical application of this warranty policy is discussed through an informal survey. Copyright © 2008 John Wiley & Sons, Ltd.

Optimal two-dimensional warranty repair strategy

For repairable products, the warrantor has options in choosing the type of repair performed to an item that fails within the warranty period. The focus is on a particular warranty repair strategy, related to the degree of the warranty repair, under a non-renewing two-dimensional warranty policy that is free of charge to the consumer. A rectangular warranty region, as in the automotive industry, is considered and partitioned into disjoint subregions. Each of these subregions has a preassigned degree of repair for a faulty item. First, for a partition of size n, an expression is derived for the associated expected warranty servicing cost per item sold. Second, using an example, for a given discretization of the warranty period, the way in which the number of subregions and their shape can be determined, so that the expected warranty servicing cost per item sold is minimum, is demonstrated.

Optimal age-replacement policy for renewing warranted products

Preventive replacement is a well-known topic in the literature of operational research and reliability. The effects of a renewing free-replacement warranty (RFRW) on the age replacement policy for a repairable product with a general failure model are discussed. There are two types of failure when the product fails. One is type I failure, which can be removed by a minimal repair, and the other is type II failure, which can be removed only by a replacement. After a minimal repair, the product is operational but the failure rate of the product remains unchanged. For both warranted and non-warranted products, cost models are developed and the corresponding optimal replacement ages are derived based on the minimized long run expected cost rate. The impacts of the RFRW on the optimal replacement age are investigated analytically. Finally, numerical examples are given for the purposes of illustration.

Warranty cost analysis for products with a dormant state

Most building services products are installed while a building is constructed, but they are not operated until the building is commissioned. The warranty of the products may cover the time starting from their installation to the end of the warranty period. Prior to the commissioning of the building, the products are at a dormant mode (i.e., not operated) but protected by the warranty. For such products, both the usage intensity and the failure patterns are different from those with continuous usage intensity and failure patterns. This paper develops warranty cost models for repairable products with a dormant mode from both the manufacturer’s and buyer’s perspectives. Relationships between the failure patterns at the dormant mode and at the operational mode are also discussed. Numerical examples and sensitivity analysis are used to demonstrate the applicability of the methodology derived in the paper.

A Bayesian decision analysis in determining the optimal policy for pricing, production, and warranty of repairable products

A successful industry strategy should be managed to integrate the decisions, such as pricing, production, and customer services, in order to maximize profits. In fact, some research has been carried out to cope with the multiple considerations for the case in which sufficient historical data are available. However, if sufficient historical data cannot be gathered to confidently estimate the deterioration of a new product, then the solution may not be assertively reliable. In dealing with such a problem for the situation of scarce historical data, a Bayesian analysis should be suitable because it can effectively assess the deterioration based on experts' opinions and possibly few relevant data. In this paper, we employed a mathematical programming approach along with a Bayesian updating process to tackle such a complex decision problem, and the optimal prior and posterior decisions of pricing scheme, production plan, and warranty policy can thus be determined simultaneously. In addition, we provided a computerized architecture to help decision makers in implementing the proposed approach. Finally, a practical application case was used to demonstrate the usefulness of the proposed model.

Optimal age-replacement policy for repairable products under renewing free-replacement warranty

This paper investigates the effects of a renewing free-replacement warranty (RFRW) on the age replacement policy for a repairable product with a general failure model. In the general model, there are two types of failure when the product fails. One is type I failure (minor failure), which can be removed by a minimal repair; and the other is type II failure (catastrophic failure), which can be removed only by a replacement. After a minimal repair, the product is operational but the failure rate of the product remains unchanged. For both warranted and nonwarranted products, cost models are developed, and the corresponding optimal replacement ages are derived such that the long-run expected cost rate is minimized. The impacts of the RFRW on the optimal replacement age are investigated analytically. Finally, numerical examples are given for illustration.

Optimal reliability, warranty and price for new products

The success of a new product depends on both engineering decisions (product reliability) and marketing decisions (price, warranty). A higher reliability results in a higher manufacturing cost and higher sale price. Consumers are willing to pay a higher price only if they can be assured about product reliability. Product warranty is one such tool to signal reliability with a longer warranty period indicating better reliability. Better warranty terms result in increased sales and also higher expected warranty servicing costs. Warranty costs are reduced by improvements in product reliability. Learning effects result in the unit manufacturing cost decreasing with total sales volume and this in turn impacts on the sale price. As such, reliability, price and warranty decisions need to be considered jointly. The paper develops a model to determine the optimal product reliability, price and warranty strategy that achieve the biggest total integrated profit for a general repairable product sold under a free replacement-repair warranty strategy in a market and looks at two scenarios for the pricing and warranty of the product. The model assumes that the sale rate increases as the warranty period increases and decreases as the price increases. The maximum principle method is used to obtain optimal solutions for dynamic price and warranty situations. Finally, numerical examples are given to illustrate the proposed model.