Product's Lifetime Distribution Research Articles

Degradation analysis on trend gamma process

AbstractManufactures always face a big challenge to obtain the sufficient reliability information for high‐quality products when only a relatively short time period is available for an internal life testing. Fortunately, if QCs exists, whose degradation over time can be related to reliability, an approach is suggested to collect sufficient degradation data for estimating the product's lifetime distribution more accurately. In numerical applications, the gamma process (GP) is commonly used when the degradation path is strictly increasing. However, in some circumstances, the GP is not able to successfully capture the degradation path. One resolution to tackle this problem is to consider the random effect into the GP model. In this paper, we propose a trend gamma process (TGP), which integrated the merits of trend function into a GP model, as an alternative approach to overcome this obstacle. The proposed TGP model is a generalized formulation of GP model, and it is attempted to transform a monotonic stochastic process into a GP. The results based on simulation studies and real data examples illustrate that our proposed TGP model is suitable for widely use when the degradation path is monotonically increasing.

Lifetime performance evaluation model based on quick response thinking

In practice, lifetime performance index CL has been a method commonly applied to the evaluation of quality performance. L is the upper or lower limit of the specification. The product lifetime distribution is mostly abnormal distribution. This study explored that the lifetime of commodities comes from exponential distribution. Complete data collection is the primary goal of analysis. However, the censoring type is one of the most commonly used methods due to considerations of manpower and material cost or the timeliness of product launch. This study adopted Type-II right censoring to find out the uniformly minimum variance unbiased (UMVU) estimator of the lifetime performance index CL and its probability density function. Afterward this study obtained the 100×(1-α)% confidence interval of the lifetime performance index CL as well as created the uniformly most powerful (UMP) test and the power of the test for the product lifetime performance index. Last, this study came up with a numerical example to demonstrate the suggested method as well as the application of the model.

Reliability analysis of wind turbine based on degradation threshold

The time-dependent reliability model of the structure was established by using the Gamma process and combined with the stress-strength interference model to describe the product strength degradation process. The characteristics of the failure threshold based on strength degradation was analyzed, and the life distribution of the product can be calculated under the condition of given degradation threshold was pointed out. Taking a vertical axis wind turbine as an example, ANSYS finite element analysis software was used to calculate the load at the stress concentration of the shaft end of the wind turbine. According to the distribution of load and based on the threshold of product degradation, the product lifetime distribution subject to Gamma degradation was analyzed, the variation law of reliability at the shaft end of the wind turbine with time was studied, and the probability distribution of lifetime under a given threshold of degradation was solved. By using this method, the product reliability subject to Gamma degradation and the lifetime distribution under a given threshold could be obtained, which provides a method for product reliability evaluation with difficulty in obtaining reliability data.

Profile optimum planning for degradation analysis

AbstractDegradation tests are widely used to assess the lifetime information of highly reliable products. Before conducting a degradation test, the fundamental issues in regard to decision variables are to determine how many sample sizes are needed, how long samples need to be tested, and how many measurements are taken, particularly in early experiments with limited budgets. By minimizing the approximate variance of the estimated qth quantile of the product's lifetime distribution with a cost constraint, profile optimum planning (POP) is proposed to provide a systematic solution to these decision variables under mild conditions. Based on the derived theoretical results, the use of flow charts efficiently simplifies complex optimization problems in practical applications. In addition, sensitivity analysis is studied to elucidate the effect of how the uncertainty in POP can be divided and allocated to the experimental costs and model parameters. Technical derivations, flow charts, and analysis results are available online as Supporting Information.

Step-stress accelerated degradation test planning based on Wiener process with correlation

ABSTRACTTo assess the lifetime distribution of highly reliable or expensive product, one of the most commonly used strategies is to construct step-stress accelerated degradation test (SSADT) which can curtail the test duration and reduce the test cost. In reality, it is not unusual for a unit with a higher degradation rate which exhibits a more volatile degradation path. Recently, Ye, Chen, and Shen [(2015). A new class of Wiener process models for degradation analysis. Reliability Engineering and System Safety, 139, 58–67] proposed a Wiener process to capture the positive correlation between the drift rate and the volatility. In this paper, an optimal SSADT plan is developed under the assumption that the underlying degradation path follows the Wiener process with correlation. Firstly, the stochastic diffusion process is introduced to model a typical SSADT problem. Then the design variables, including the sample size, the measurement frequency and the numbers of measurements under each stress level, are optimised by minimising the asymptotic variance of the estimated p-percentile of the product's lifetime distribution subject to the total experimental cost not exceeding a pre-specified budget. Finally, a numerical example is presented to illustrate the proposed method.

Estimating Fatigue Life of Structural Components from Accelerated Data via a Birnbaum-saunders Model with Shape and Scale Stress Dependent Parameters

The Birnbaum-Saunders model is widely applied to model fatigue failures caused by cyclic stresses both in the case of standard and accelerated life tests. This latter kind of tests are adopted when the product of interest is very reliable, in order to obtain failure data in a reasonably short amount of time. Estimates of the product's reliability or the long-term performances at normal use condition are then obtained, from accelerated failure data, adopting functional relationships accounting for the effect of the accelerating variables on the product's lifetime distribution. Customarily these models are formulated assuming that the accelerating variables affect the values of the lifetime distribution parameters, and not its form. In particular, in literature the Birnbaum-Saunders distribution is usually applied to accelerated data under the hypothesis that the scale parameter only depends on the stress conditions, while the shape parameter doesn’t depend on them. In this paper, an applicative example is presented in which this standard model does not work satisfactorily. In fact, it is shown as, in the case of the considered real set of accelerated fatigue failure data, the Birnbaum-Saunders distribution, in which both scale and shape parameters depend on the stress conditions, fits the data significantly better than the abovementioned standard option. Difference among lifetime distribution estimates provided by the two different considered models are highlighted and discussed.

Lifetime Inference for Highly Reliable Products Based on Skew-Normal Accelerated Destructive Degradation Test Model

The accelerated destructive degradation test (ADDT) method provides an effective way to assess the reliability information of highly reliable products whose quality characteristics degrade over time, and can be taken only once on each tested unit during the measurement process. Conventionally, engineers assume that the measurement error follows the normal distribution. However, degradation models based on this normality assumption often do not apply in practical applications. To relax the normality assumption, the skew-normal distribution is adopted in this study because it preserves the advantages of the normal distribution with the additional benefit of flexibility with regard to skewness and kurtosis. Here, motivated by polymer data, we propose a skew-normal nonlinear ADDT model, and derive the analytical expressions for the product's lifetime distribution along with its corresponding $100p$ th percentile. Then, the polymer data are used to illustrate the advantages gained by the proposed model. Finally, we addressed analytically the effects of model mis-specification when the skewness of measurement error are mistakenly treated, and the obtained results reveal that the impact from the skewness parameter on the accuracy and precision of the prediction of the lifetimes of products is quite significant.

On Studying Partially Accelerated Life Tests Under Progressive Stress

Abstract For highly reliable products, a progressive stress accelerated life test has been proposed to obtain timely information of the product's lifetime distribution. This article considers a progressive stress partially accelerated life test model when the lifetime of a product under use condition follows Weibull distribution. It is assumed that the progressive stress is directly proportional to time. The statistical properties of the maximum likelihood (ML) estimators of the model parameters such as existence, uniqueness, and invariance are studied. The biases and mean square errors of the maximum likelihood estimators are computed to assess their performances in the presence of the stress method developed in this article through a Monte Carlo simulation study.

Bootstrap and Maximum Entropy Based Small-sample Product Lifetime Probability Distribution

How to build the small-sample data distribution is a major issue to be addressed for analyzing large power equipment lifetime. To find a better product lifetime probability distribution function, a new method based on bootstrap and maximum entropy is proposed to study on the problem of small-sample product life probability distribution. The result shows that this new method is more accurate and feasible compared with bootstrap and maximum entropy respectively. Finally, an actual example of small-sample product lifetime distribution shows that this new approach has better guiding significance for practice.

Equivalent step-stress accelerated life tests with log-location-scale lifetime distributions under Type-I censoring

Accelerated Life Testing (ALT) is used to provide timely estimates of a product's lifetime distribution. Step-Stress ALT (SSALT) is one of the most widely adopted stress loadings and the optimum design of a SSALT plan has been extensively studied. However, few research efforts have been devoted to establishing the theoretical rationale for using SSALT in lieu of other types of stress loadings. This article proves the existence of statistically equivalent SSALT plans that can provide equally precise estimates to those derived from any continuous stress loading for the log-location-scale lifetime distributions with Type-I censoring. That is, for any optimization criterion based on the Fisher information matrix, SSALT is identical in comparison to other continuous stress loadings. The Weibull and lognormal distributions are introduced as special cases. For these two distributions, the relationship among statistical equivalencies is investigated and it is shown that two equivalent ALT plans must be equivalent in terms of the strongest version of equivalency for many objective functions. A numerical example for a ramp-stress ALT, using data from an existing study on miniature lamps, is used to illustrate equivalent SSALT plans. Results show that SSALT is not only equivalent to the existing ramp-stress test plans but also more cost-effective in terms of the total test cost.

Forecasting Replacement Demand of Durable Goods and the Induced Secondary Material Flows: A Case Study of Automobiles

SummaryThe aim of this article is to propose a method for forecasting future secondary material flows by combining a product lifetime distribution analysis with a waste input‐output analysis and present a simple case study of automobiles. The case study demonstrates that the proposed method enables us to estimate replacement demand of new vehicles, number of end‐of‐life (EOL) vehicles arising from the aging of vehicles, volume of shredder scraps recovered from EOL vehicles, and volume of shredder scraps required to meet final consumption in the future.

Statistical Lifetime Inference With Skew-Wiener Linear Degradation Models

Degradation models are widely used to assess the lifetime information of highly reliable products possessing quality characteristics that both degrade over time and can be related to reliability. The performance of a degradation model largely depends on an appropriate model description of the product's degradation path. Conventionally, the random or mixed-effect model is one of the most well-known approaches presented in the literature in which the normal distribution is commonly adopted to represent unit-to-unit variability in the degradation model. However, this assumption may not appropriately signify accurate projections for practical applications. This paper is motivated by laser data wherein the normal distribution is relaxed with a skew-normal distribution that consequently provides greater flexibility as it can capture a broad range, non-normal, asymmetric behavior in unit-to-unit variability. Based on the proposed degradation model, we first derive analytical expressions for a product's lifetime distribution along with its corresponding mean-time-to-failure ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MTTF</i> ). We then utilize the laser data to illustrate advantages gained by the proposed model. Finally, we address effects from the skewness parameter with regard to the accuracy of both a product's <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MTTF</i> and its <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> th failure quantile; especially when the underlying skew-normal distribution is mis-specified as a normal distribution. The result demonstrates that effects from the skewness parameter on the tail probabilities of a product's lifetime distribution are not negligible when the random effect of the true degradation model follows a skew-normal distribution.

A new approach to assess product lifetime performance for small data sets

Because of cost and time limit factors, the number of samples is usually small in the early stages of manufacturing systems, and the scarcity of actual data will cause problems in decision-making. In order to solve this problem, this paper constructs a counter-intuitive hypothesis testing method by choosing the maximal p-value based on a two-parameter Weibull distribution to enhance the estimate of a nonlinear and asymmetrical shape of product lifetime distribution. Further, we systematically generate virtual data to extend the small data set to improve learning robustness of product lifetime performance. This study provides simulated data sets and two practical examples to demonstrate that the proposed method is a more appropriate technique to increase estimation accuracy of product lifetime for normal or non-normal data with small sample sizes.

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

AbstractBurn‐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 &amp; Sons, Ltd.

Progressive-Stress Accelerated Degradation Test for Highly-Reliable Products

For highly reliable products with very few test units, a progressive-stress accelerated degradation test (PSADT) has been proposed in the literature to obtain timely information of the product's lifetime distribution. The results, however, are restricted to the case where the product's degradation path follows a Wiener process (Brownian motion) with a linear drift rate. But in practical applications, the product's mean degradation path may be non-linear. Hence, how to address the lifetime distribution in this situation is a worthy topic for reliability analysts. In this paper, a PSADT with a non-linear degradation path is constructed using the cumulative exposure model. Then the product's lifetime distribution can be analytically obtained by the first passage time of its degradation path. Furthermore, we derive an exact relationship between the lifetime distributions of the PSADT, and the conventional constant-stress degradation test (CSDT), which allows us to extrapolate the product's lifetime distribution under typical stress. Finally, the usage of the proposed model, and the efficiency of PSADT to reduce the product's life testing time are demonstrated in the example.

Mis-Specification Analysis of Linear Degradation Models

Degradation models are widely used to assess the lifetime information of highly reliable products if there exists quality characteristics whose degradation over time can be related to reliability. The performance of a degradation model depends strongly on the appropriateness of the model describing a product's degradation path. In this paper, motivated by laser data, we propose a general linear degradation path in which the unit-to-unit variation of all test units can be considered simultaneously with the time-dependent structure in degradation paths. Based on the proposed degradation model, we first derive an implicit expression of a product's lifetime distribution, and its corresponding mean-time-to-failure (MTTF). By using the profile likelihood approach, maximum likelihood estimation of parameters, a product's MTTF, and their confidence intervals can be obtained easily. In addition, laser degradation data are used to illustrate the proposed procedure. Furthermore, we also address the effects of model mis-specification on the prediction of the product's MTTF. It shows that the effect of the model mis-specification on the predictions of a product's MTTF is not critical under the case of large samples. However, when the sample size and the termination time are not large enough, a simulation study shows that these effects are not negligible.

Optimal Design for Accelerated Degradation Tests with Several Stresses based on Wiener Process

Abstract Design for accelerated degradation tests (ADTs) in most literature concentrates on the case of just one stress, but the performance of many highly reliable products are affected by several stresses simultaneously, so it is necessary to study the optimal models for the ADTs with several stresses. This article first introduces a basic modeling framework of degradation path and then proposes an optimal model for ADTs with several stresses based on Wiener process. Under the constraint of total experimental cost, sample size, measurement times, and measurement frequency are obtained by minimizing the asymptotic variance of the estimated 100pth percentile of the product's lifetime distribution. Finally, a numerical example is presented to illustrate the proposed model.

Simulations of Thermomechanical Stresses and Optical Misalignment in 1550-nm Transmitter Optoelectronic Modules Using FEM and Process Dispersions

Expertise of packaging for optoelectronic components requires the solution of optical, mechanical, and electrical problems in the same way. The purpose of this paper is to present three-dimensional simulations using finite-element method (FEM) of thermomechanical stresses and strains in 1550-nm laser modules induced by Nd:YAG crystal laser welds and thermal cycles on main subassembly laser submount. Nonlinear FEM computations, taking into account of experimental sigma(epsiv) measured curves, show that the laser welding process can induce high level of strains in columns of the laser platform, bearing the laser diode, responsible of an optical axis shift and a gradual drop of the optical power in relation with relaxation of accumulated stresses in the subassembly (W. M. Sherry et al., ldquoHigh performance optoelectronic packaging for 2.5 and 10 Gb/s laser modules,rdquo in Proc. Electron. Compon. Technol. Conf., 1996, pp. 620-627). Typical stresses are close to 160 MPa with drift about 5 MPa with the dispersion of energy level of the laser Nd: YAG beam. The introduction of both material and process dispersion in order to evaluate their impact on product lifetime distribution has been taking into account. In the case of thermal cycles, stresses can occur on elements sensitive to coefficient of thermal expansion mismatches such as solder joints between the laser platform and thermoelectric cooler and as fiber glued into the pigtail leading to crack propagation with sudden drop of optical power. A previous paper demonstrated that laser submount is the most sensitive part of optical system (Deshayes, et al., ldquoThree-dimensional FEM simulations of thermal mechanical stresses in 1.55 mum laser modules,rdquo Microelectron. Rel., vol. 43, no. 7, pp. 1125 -1136, Jul. 2003). Experimental analyses were also conducted to correlate simulation results and monitor the output optical power of laser modules after 500 thermal cycles ( -40degC/ + 85degC VRT).

Forecasting quantities of disused household CRT appliances – A regional case study approach and its application to Baden-Württemberg

Due to special requirements regarding logistics and recycling, disused cathode ray tube (CRT) appliances are handled in some countries as a separate waste fraction. This article presents a forecast of future household waste CRT quantities based on the past and present equipment of households with television sets and computer monitors. Additional aspects taken into consideration are the product life time distribution and the ongoing change in display technology. Although CRT technology is fading out, the findings of this forecast show that quantities of waste CRT appliances will not decrease before 2012 in Baden-Württemberg, Germany. The results of this regional case study are not quantitatively transferable without further analysis. The method provided allows analysts to consider how the time shift between production and discard could impact recycling options, and the method could be valuable for future similar analyses elsewhere.

Optimal Design for Step-Stress Accelerated Degradation Tests

Today, many products are designed to function for a long period of time before they fail. For such highly-reliable products, collecting accelerated degradation test (ADT) data can provide useful reliability information. However, it usually requires a moderate sample size to implement an ADT. Hence, ADT is not applicable for assessing the lifetime distribution of a newly developed or very expensive product which only has a few available test units on hand. Recently, a step-stress ADT (SSADT) has been suggested in the literature to overcome the above difficulty. However, in designing an efficient SSADT experiment, the issue about how to choose the optimal settings of variables was not discussed, such as sample size, measurement frequency, and termination time. In this study, we first use a stochastic diffusion process to model a typical SSADT problem. Next, under the constraint that the total experimental cost does not exceed a predetermined budget, the optimal settings of these variables are obtained by minimizing the asymptotic variance of the estimated 100p/sup th/ percentile of the product's lifetime distribution. Finally, an example is used to illustrate the proposed method.