Optimal burn-in time under cumulative free replacement warranty

In this paper, optimal burn-in time to minimize the total mean cost, which is the sum of manufacturing cost with burn-in and cumulative warranty-related cost, is studied. When the products with cumulative warranty have high failure rate in the early period (infant mortality period), burn-in procedure is considered to eliminate the early product failures. After burn-in, the posterior product life distribution and the cumulative warranty-related cost are dependent on burn-in time; long burn-in period decreases the warranty-related cost, but it increases the manufacturing cost. The paper provides a methodology to obtain the total mean cost under burn-in and cumulative warranty. Properties of the optimal burn-in time are analyzed here. Numerical examples and sensitivity analysis are used to demonstrate the applicability of the methodology derived in the paper.     

Reliability modeling of surface controlled subsurface safety valves

The article discusses the consequences of choosing a Weibull life distribution instead of an exponential distribution. The discussions are based on a specific data set for surface controlled subsurface safety valves (SCSSVs) used in offshore oil and gas production wells. The estimates of the mean time to failure and the mean fractional dead-time based on the stochastic censored data set, are shown to be non-robust with respect to variations of the Weibull parameters. The availability of the SCSSV as a safety barrier against blowouts is discussed in relation to risk acceptance criteria and the principle of ‘as low as reasonably practicable’. Replacement policies to balance the risk and operational cost are proposed and discussed.     

Generalized renewal process for repairable systems based on finite Weibull mixture

Repairable systems can be brought to one of possible states following a repair. These states are: ‘as good as new’, ‘as bad as old’ and ‘better than old but worse than new’. The probabilistic models traditionally used to estimate the expected number of failures account for the first two states, but they do not properly apply to the last one, which is more realistic in practice. In this paper, a probabilistic model that is applicable to all of the three after-repair states, called generalized renewal process (GRP), is applied. Simplistically, GRP addresses the repair assumption by introducing the concept of virtual age into the stochastic point processes to enable them to represent the full spectrum of repair assumptions. The shape of measured or design life distributions of systems can vary considerably, and therefore frequently cannot be approximated by simple distribution functions. The scope of the paper is to prove that a finite Weibull mixture, with positive component weights only, can be used as underlying distribution of the time to first failure (TTFF) of the GRP model, on condition that the unknown parameters can be estimated. To support the main idea, three examples are presented. In order to estimate the unknown parameters of the GRP model with m-fold Weibull mixture, the EM algorithm is applied. The GRP model with m mixture components distributions is compared to the standard GRP model based on two-parameter Weibull distribution by calculating the expected number of failures. It can be concluded that the suggested GRP model with Weibull mixture with an arbitrary but finite number of components is suitable for predicting failures based on the past performance of the system.     

Optimal Burn-in Time to Minimize Cost for Products Sold Under Warranty

For products with an initially high failure rate, burn-in can be used to reduce the warranty cost. This paper examines the optimal burn-in time to achieve a trade-off between the reduction in the warranty cost and the increase in the manufacturing cost (as burn-in can be viewed as a part of the manufacturing process). Conditions for burn-in to be beneficial are derived for various warranty policies. A numerical example is presented.     

On preventive maintenance policy of a critical reliability level for system subject to degradation

Conventional preventive maintenance (PM) policies generally hold same time interval for PM actions and are often applied with known failure modes. The same time interval will give unavoidably decreasing reliabilities at the PM actions for degradation system with imperfect PM effect and the known failure modes may be inaccurate in practice. Therefore, field managers would prefer policy with an acceptable reliability level to keep system often at a good state.A PM policy with the critical reliability level is presented to address the preference of field managers. Through assuming that system after a PM action starts a new failure process, a parameter so-called degradation ratio is introduced to represent the imperfect effect. The policy holds a law that there is same number of failures in the time intervals of various PM cycles, and same degradation ratio for the system reliability or benefit parameters such as the optimal time intervals and the hazard rates between the neighboring PM cycles. This law is valid to any of the failure modes that could be appropriately referred as a ‘general isodegrading model’, and the degradation ratio as a ‘general isodegrading ratio’. In addition, life cycle availability and cost functions are derived for system with the policy. An analysis of the field data of a loading and unloading machine indicates that the reliability, availability and cost in life cycle might be well modeled by the present theory and approach.     

Optimal burn-in time and warranty length under fully renewing combination free replacement and pro-rata warranty

Warranty is an important factor of marketing products; however, warranty always involves additional costs to the seller and these costs usually depend on product reliability. Burn-in is considered as a part of the production process in which the manufactured products are operated under accelerated stresses for a short time period before their release and has been applied as a way for enhancing product reliability. In the present paper, a cost model is developed to determine the optimal burn-in time and warranty length for non-repairable products under the fully renewing combination free replacement and pro-rata warranty (FRW/PRW) policy. Denoting w as the warranty length of the product, under the FRW/PRW policy, the seller agrees to replace the product that fails prior to the time point w′, where w′<w, from the time of purchase with a new product at no cost to the buyer; meanwhile, any failure in the time interval from w′ to w results in a pro-rata replacement, i.e., any product is replaced with a new item at pro-rata cost to the buyer. Numerical examples are provided given that the failure time of the product follows either the mixed exponential distribution or the mixed Weibull distribution and four warranty policies are examined. Based on the results of analysis, it is noted that the fully renewing combination FRW/PRW is always better than the fully renewing policy in terms of cost.     

Optimal restarting distribution after repair for a Markov deteriorating system

We consider a repairable system such that different completeness degrees are possible for the repair (or corrective maintenance) that go from a ‘minimal’ up to a ‘complete’ repair. Our question is: to what extent must the system be repaired in case of failure for the long-run availability to be optimal? The system evolves in time according to a Markov process as long as it is running, whereas the duration of repairs follows general distributions. After repair, the system starts again in the up-state i with probability d(i). We observe from numerical examples that the optimal restarting distribution d^opt (such that the long-run availability is optimal) is generally random and does not correspond to a new start in a fixed up-state. Sufficient conditions under which the optimal restarting distribution is non-random are given. Also, the optimal restarting distribution is provided for two classical structures in reliability.     

Determination of the optimal burn-in time and cost using an environmental stress approach: a case study in switch mode rectifier

Stress burn-in is an effective burn-in means of screening out the infant mortality components of a system which are conducted under an extremely stressful environment. While investigating stresses, screening burn-in by thermal stress, voltage stress, or mechanical shock stress, most related studies failed to develop an effective method to determine the optimal burn-in time and burn-in cost for a practical operation. Therefore, this study presents an effective procedure that adopts robust design techniques and the accelerated stress test to determine the optimal burn-in time and burn-in cost. A case study of the production of switch mode rectifier demonstrates the proposed procedure's effectiveness. Moreover, the results show that the proposed procedure generalizes well, and can screen out the early failure from material and manufacturing process.     

Optimal structural design under stochastic uncertainty by stochastic linear programming methods

In the optimal plastic design of mechanical structures one has to minimize a certain cost function under the equilibrium equation, the yield condition and some additional simple constraints, like box constraints. A basic problem is that the model parameters and the external loads are random variables with a certain probability distribution. In order to get reliable/robust optimal designs with respect to random parameter variations, by using stochastic optimization methods, the original random structural optimization problem must be replaced by an appropriate deterministic substitute problem. Starting from the equilibrium equation and the yield condition, the problem can be described in the framework of stochastic (linear) programming problems with ‘complete fixed recourse’. The main properties of this class of substitute problems are discussed, especially the ‘dual decomposition’ data structure which enables the use of very efficient special purpose LP-solvers.     

Model uncertainty and model inaccuracy

The problem of model uncertainty versus model inaccuracy is examined in the light of the concept of the ‘probability of correctness of a model under a given context’ introduced by Apostolakis. To avoid possible difficulties linked with this concept, a distinction is introduced between ‘predictive’ models and ‘constitutive’ models, the former being generic in the sense that they can host the latter as submodels. A metric or distance between linear models as well as an objective of the model are introduced, from which we can give an operational definition of ‘model uncertainty’ (with respect to distribution of parameters of the associated constitutive models) and of ‘model accuracy’ with respect to a reference model. Finally the choice of a predictive model is linked to a loss function and a cost of using or defining a model.     

Optimisation of burn-in time considering the hidden loss of quality deviations in the manufacturing process

The burn-in test time is an important parameter of the complex batch processing machine scheduling problem. The omission of the loss of quality deviations in manufacturing generates a non-comprehensive and imperfect result in the optimisation of burn-in time, which hinders the identification of proactive and economical optimisation strategies to prevent infant failure in manufacturing. To solve this problem, this study visualises and quantifies for the first time the hidden loss caused by quality deviations in manufacturing and uses it as a newly added constraint to optimise the burn-in time. Firstly, a quality loss model composed of visible yield loss and warranty costs related to measurable but undetectable reliability vulnerabilities is defined. Secondly, the loss effects of growing defects are measured during the burn-in test, and the optimal burn-in time expressed by the proposed quality loss model is traded off between the additional burn-in cost and the decreased quality loss for an acceptable low infant failure rate. Finally, the effectiveness of the proposed optimisation approach is demonstrated using actual data from a control board with a high infant failure rate. Results show that the proposed method can systematically combine the fundamental loss of quality deviations in the optimisation of burn-in time, which supplements the commonly used optimality criteria, with the upstream loss of quality deviations in the form of manufacturing defects.     

Bayesian calculation of optimal burn‐in time using the joint criteria of cost and delivered reliability

Burn‐in is a quality control process used to minimize the warranty cost of a product by screening out defective products through prior operation for a period of time before sale. Two decision criteria used to determine the optimal burn‐in time are the maximization of the reliability of the delivered product and the minimization of the total cost, which are composed of the cost of burn‐in process and the cost of warranty claims. Because of uncertainty regarding the underlying lifetime distribution of the product, both the product reliability and the total cost are random variables. In this paper, the uncertainty in reliability and cost is quantified by use of Bayesian analysis. The joint distribution of reliability and cost is inferred from the uncertainty distribution of the parameters of the product lifetime distribution. To incorporate the uncertainty in reliability and cost as well as the tradeoff between them into the selection of optimal burn‐in time, the joint utility function of reliability and cost is constructed using the joint distribution of reliability and cost. The optimal burn‐in time is selected as the time that maximizes the joint utility function. Copyright © 2010 John Wiley & Sons, Ltd.     

Sequential degradation-based burn-in test with multiple periodic inspections

Burn-in has been proven effective in identifying and removing defective products before they are delivered to customers. Most existing burn-in models adopt a one-shot scheme, which may not be sufficient enough for identification. Borrowing the idea from sequential inspections for remaining useful life prediction and accelerated lifetime test, this study proposes a sequential degradation-based burn-in model with multiple periodic inspections. At each inspection epoch, the posterior probability that a product belongs to a normal one is updated with the inspected degradation level. Based on the degradation level and the updated posterior probability, a product can be disposed, put into field use, or kept in the test till the next inspection epoch. We cast the problem into a partially observed Markov decision process to minimize the expected total burn-in cost of a product, and derive some interesting structures of the optimal policy. Then, algorithms are provided to find the joint optimal inspection period and number of inspections in steps. A numerical study is also provided to illustrate the effectiveness of our proposed model.     

Selection of materials using compromise ranking and outranking methods

Selection of proper materials for different components is one of the most challenging tasks in the design and development of products for diverse engineering applications. Materials play a crucial and important role during the entire design and manufacturing process. Wrong selection of material often leads to huge cost involvement and ultimately drives towards premature component or product failure. So the designers need to identify and select proper materials with specific functionalities in order to obtain the desired output with minimum cost involvement and specific applicability. This paper attempts to solve the materials selection problem using two most potential multi-criteria decision-making (MCDM) approaches and compares their relative performance for a given material selection application. The first MCDM approach is ‘Vlse Kriterijumska Optimizacija Kompromisno Resenje’ (VIKOR), a compromise ranking method and the other one is ‘ELimination and Et Choice Translating REality’ (ELECTRE), an outranking method. These two methods are used to rank the alternative materials, for which several requirements are considered simultaneously. Two examples are cited in order to demonstrate and validate the effectiveness and flexibility of these two MCDM approaches. In each example, a list of all the possible choices from the best to the worst suitable materials is obtained taking into account different material selection criteria. The rankings of the selected materials almost corroborate with those as obtained by the past researchers.     

Optimal burn-in strategy for two-dimensional warranted products considering preventive maintenance

Burn-in and preventive maintenance (PM) are effective approaches to reduce the number of warranty claims and warranty cost during post-sale support. With harsher burn-in settings, early product defects can be removed, but at the same time product degradation is accelerated and more wear-out failures may be introduced. PM actions within warranty alleviate these negative effects. This paper proposes an optimal burn-in strategy for repairable products sold with a two-dimensional base warranty (BW) and an optional extended warranty (EW). Both performance-based and cost-based models incorporating PMs are developed to obtain optimal burn-in settings, including the burn-in duration and the burn-in usage rate, so as to minimise the expected number of warranty claims and total cost respectively. The impacts of different accelerated coefficients and PM degrees on the optimal burn-in strategy are analysed. In view of the performance and cost structures, we conduct numerical examples to illustrate the applicability of the proposed models. Practical implications from a sensitivity analysis for key parameters are also elaborated.     

Generalized renewal process for analysis of repairable systems with limited failure experience

Repairable systems can be brought to one of possible states following a repair. These states are: ‘as good as new’, ‘as bad as old’, ‘better than old but worse than new’, ‘better than new’, and ‘worse than old’. The probabilistic models traditionally used to estimate the expected number of failures account for the first two states, but they do not properly apply to the last three, which are more realistic in practice. In this paper, a robust solution to a probabilistic model that is applicable to all of the five after repair states, called generalized renewal process (GRP), is presented. This research demonstrates that the GRP offers a general approach to modeling repairable systems and discusses application of the classical maximum likelihood and Bayesian approaches to estimation of the GRP parameters. This paper also presents a review of the traditional approaches to the analysis of repairable systems as well as some applications of the GRP and shows that they are subsets of the GRP approach. It is shown that the proposed GRP solution accurately describes the failure data, even when a small amount of failure data is available.Recent emphasis in the use of performance-based analysis in operation and regulation of complex engineering systems (such as those in space and process industries) require use of sound models for predicting failures based on the past performance of the systems. The GRP solution in this paper is a promising and efficient approach for such performance-based applications.     

James-Stein estimators for failure rates and probabilities

A basic concept in this paper is that a group of components under study represents a sample from a common source (e.g. a factory) that has a failure rate distribution, with distribution parameters originally unknown. This concept is used to obtain generalized James-Stein estimators for individual component parameters. Such ‘shrinkage estimators’ are more accurate than conventional point estimates. They are developed here for component failure rate parameters based on Poisson data, and for failure probabilities based on binomial data. Optimal weights and shrinking factors are obtained as well as formulae for estimating the mean value and the variance of the source population, and the mean values and the variances of individual component estimates. Close connections to the empirical Bayesian estimation are shown     

A warranty forecasting model based on piecewise statistical distributions and stochastic simulation

This paper presents a warranty forecasting method based on stochastic simulation of expected product warranty returns. This methodology is presented in the context of a high-volume product industry and has a specific application to automotive electronics. The warranty prediction model is based on a piecewise application of Weibull and exponential distributions, having three parameters, which are the characteristic life and shape parameter of the Weibull distribution and the time coordinate of the junction point of the two distributions. This time coordinate is the point at which the reliability ‘bathtub’ curve exhibits a transition between early life and constant hazard rate behavior. The values of the parameters are obtained from the optimum fitting of data on past warranty claims for similar products. Based on the analysis of past warranty returns it is established that even though the warranty distribution parameters vary visibly between product lines they stay approximately consistent within the same product family, which increases the overall accuracy of the simulation-based warranty forecasting technique. The method is demonstrated using a case study of automotive electronics warranty returns. The approach developed and demonstrated in this paper represents a balance between correctly modeling the failure rate trend changes and practicality for use by reliability analysis organizations.     

Simulation modelling of repairable multi-component deteriorating systems for ‘on condition’ maintenance optimisation

The main purpose of this work is to model continuously monitored deteriorating systems by using Monte Carlo simulation and embedding the resulting model within an ‘on condition’ maintenance optimisation scheme that aims at minimising the expected total system cost over a given mission time. The simulation model is first introduced by considering a non-repairable single component subjected to stochastic degradation. The modelling is then generalised to multi-component repairable systems. To find the optimal degradation thresholds of maintenance intervention, the cost optimisation procedure employed is a simple search in the space of the maintenance thresholds. The sensitivity of the results to some of the driving cost parameters has also been examined.     

Improved quality of input data for maintenance optimization using expert judgment

Most maintenance optimization models need an estimate of the so-called ‘naked’ failure rate function as input. In practice it is very difficult to estimate the ‘naked’ failure rate, because overhauls and other preventive maintenance actions tend to ‘corrupt’ the recorded lifelengths. The purpose of this paper is to stress the importance of utilizing the knowledge of maintenance engineers, i.e., expert judgment, in addition to recorded equipment lifelengths, in order to get credible input data. We have shown that without utilizing expert judgment, the estimated mean time to failure may be strongly biased, often by a factor of 2–3, depending on the life distribution that is assumed. We recommend including a simple question about the mean remaining lifelength on the work-order forms. By this approach the knowledge of maintenance engineers may be incorporated in a simple and cost-effective way.