Weibull Lifetime Distribution Research Articles

Optimal Designs of Multiple Step-Stress Accelerated Life Tests for One-Shot Devices With Weibull Lifetime Distributions

Optimal Designs of Multiple Step-Stress Accelerated Life Tests for One-Shot Devices With Weibull Lifetime Distributions

Inference of improved adaptive progressively censored competing risks data for Weibull lifetime models

Recently, an improved adaptive Type-II progressive censoring scheme is proposed to ensure that the experimental time will not pass a pre-fixed time and ends the test after recording a pre-fixed number of failures. This paper studies the inference of the competing risks model from Weibull distribution under the improved adaptive progressive Type-II censoring. For this goal, we used the latent failure time model with Weibull lifetime distributions with common shape parameters. The point and interval estimation problems of parameters, reliability and hazard rate functions using the maximum likelihood and Bayesian estimation methods are considered. Moreover, making use of the asymptotic normality of classical estimators and delta method, approximate intervals are constructed via the observed Fisher information matrix. Following the assumption of independent gamma priors, the Bayes estimates of the scale parameters have closed expressions, but when the common shape parameter is unknown, the Bayes estimates cannot be formed explicitly. To solve this difficulty, we recommend using Markov chain Monte Carlo routine to compute the Bayes estimates and to construct credible intervals. A comprehensive Monte Carlo simulation is conducted to judge the behavior of the offered methods. Ultimately, analysis of electrodes data from the life-test of high-stress voltage endurance is provided to illustrate all proposed inferential procedures.

Fatigue-life and stress distribution of a glass-ceramic under different loading conditions

This study aimed to evaluate the effect of different loading conditions on the mechanical behavior and stress distribution of a leucite-reinforced glass-ceramic. Plate-shaped ceramic specimens were obtained from leucite-reinforced glass-ceramic (1.5 × 8.4 × 8.3 mm) and adhesively cemented to a dentin analog substrate. Monotonic and cyclic contact fatigue tests were performed to simulate sphere-to-flat contact, using a 6 mm diameter spherical piston; and flat-to-flat contact, using a 3 mm diameter flat piston. For the monotonic test (n=20), a gradual compressive load (0.5 mm/min) was applied to the specimen using a universal testing machine. Failure load data were analyzed with Weibull statistics. The cyclic contact fatigue test was performed using protocols (load and a number of cycles) defined by the boundary technique (n=30). Fatigue data were analyzed using an inverse power law relationship and Weibull-lifetime distribution. The stress distribution was investigated using Finite Element Analysis (FEA). The monotonic and the fatigue Weibull modulus were similar among the two contact conditions. In fatigue, the slow crack growth exponent was greater for sphere-to-flat contact, which indicates that the load level had a greater effect on the specimen's probability of failure. In conclusion, FEA showed different stress distribution for the tested loading conditions. The stress distribution and probability of fatigue failure of specimens tested in sphere-to-flat contact showed greater dependency to load level.

Waste LEDs in China: Generation estimation and potential recycling benefits

With light emitting diode (LED) devices popularizing, waste LEDs are expected to increase rapidly. Nevertheless, recycling policies and chains lag, largely due to scattered knowledge of waste generation and recycling benefits. Herein, Holt's method is used to estimate the sales of LED devices in China from 2012-2025. Then, we employ the market supply A model with Weibull lifetime distribution embedded to quantify the waste generation and in-use stock of LEDs. On this basis, the potential economic and environmental benefits of recycling are explored. Results show that: (1) From 2012 to 2025, the in-use stock of LEDs will increase from 0.13 to 66.17 billion pieces, while waste generation will increase from 1.57 million to 4.79 billion pieces; (2) In 2025, the potential economic benefits of recycling will reach $11.99 billion, mainly from precious metals; (3) In 2025, the carbon emission reductions of recycling will reach 7.54 billion tons (CO2-eq.), derived from the substitution of recycling for other waste disposal options. To keep technologies, capacities, and collection systems relatively in sync with the growing waste LEDs, priority should be given to lifetime extension and then sustainable designs. Nowadays, incentives for lifetime extension and the utilization of recycling systems for other e-waste are beneficial. In the medium-term, designs of modularity, standardization, and generalization, as well as the integration of disassembly, pre-treatment, and resource extraction, would be the focus. In the long run, the authority and stakeholders should spend efforts on clarifying recycling responsibilities, developing tailed business models, and promoting cost-effective recycling technologies.

Economic Order Quantity (EOQ) for Perishable Goods with Weibull Distribution and Exponential Demand Rate Proportional to Price

Business organizations that deal with consumable and perishable items have consistently incurred enormous loss as a result of the nature of their goods. The losses have direct negative impact on revenues. Unplanned and lack of precise production prediction models are responsible for this. An appropriate prediction model, developed to guide production plan and processes will help manufacturers in deciding which product to make and in what quantity. In this study, the Economic Order Quantity (EOQ) for perishable goods with Weibull lifetime distribution and exponential demand rate proportional to price was developed for perishable goods. The differential equations governing the instantaneous state of inventory in the interval [0, t2] were obtained and solved for the equation of the quantity of inventory at time t. Using fixed parameters for the weibull and exponential distributions, simulation study was conducted on the derived EOQ model using R programming language. The simulation shows that the EOQ increases with increase in Weibull parameter. Real data on six loafs of bread obtained from Afe Babalola University bakery was used to illustrate how the model works. Result shows a good fit to the data and the average EOQ ranges from 60 to 400 loafs with ordering times of either 1 or two days interval. The pattern of EOQ varies between type of loafs of bread. The EOQ model developed is shown by this result to be appropriate for perishable goods with weibull lifetime distribution and exponential demand rate proportional to price.

English

Reliable data on e-waste generation is important for environmentally sound management systems. This study models e-waste generation from existing data on electrical electronics imports, consumption and e-waste generation from Nigerian households. Structured questionnaires were used to obtain information on Electrical Electronic Equipment (EEE) use, reuse, and disposal from households in Nigeria households. Data from placed on the market (POM) were obtained from United Nations University (UNU) for five EEE types (TV, DVD player, refrigerator, desktop and laptop) in Nigeria between 1995 and 2019 using the apparent consumption method. A forecast up to 2020 and backcasts to 1980 were made based on these data. The lifetime profile for these five EEEs was modeled using the Weibull distribution function characterized by a time-varying shape parameter and a scale parameter. The POM data from 1980 to 2020 and the lifetime of the selected EEE from households were analyzed and fit into the Weibull lifetime distribution functions. The differences between reuse and non-reuse options show that around 54 million units of DVD players; 106 million units of CRT TV; 22 million units of the refrigerator; 11 million units of laptops and 24 million units of desktop computers would have been delayed from transiting into e-waste stream between 1981 and 2020 through reuse options.   Key words: Electronic waste, electrical electronic equipment, lifetime distribution, re-use repair, recycling, Nigeria.

Reliability maximization of a load‐sharing system without redundant components

AbstractThis paper considers a problem of determining the optimal number of components for maximizing reliability of a load‐sharing series system in which no redundant components exist. Previously, this problem has been studied numerically, under the assumption that the load–life relationship is described by a power function, and the component follows an exponential or a Weibull lifetime distribution. In this paper, our objective is to obtain analytical results for the optimal number of components given that a general link function explains a load–life relationship and the component follows a general lifetime distribution. We show that the optimal number of components non‐decreases with the elasticity of the link function and the ratio of the component failures sensitive to the changing design load. We also show that the premature failures occurring early in the life of a component affect the optimal solution, depending on whether a link function is used for scaling the nominal failure rate independently of the mission time, or it is used for scaling the nominal life dependently with the mission time. We believe that the result is useful in selecting the number of the components to use in a system when studying design trade‐offs at an early design stage.

Statistical Inference of Jointly Type-II Lifetime Samples under Weibull Competing Risks Models

In this paper, we develop statistical inference of competing risks samples which are collected under a joint Type-II censoring scheme of products with Weibull lifetime distributions. These inferences are drawn from two independent fatal risks and come from two different lines of production with the same facility. The model parameters and the parameters of life (reliability and hazard rate functions) are estimated using maximum likelihood (ML), bootstrap and Bayes methods. Additionally, we constructed asymptotic ML confidence intervals, bootstrap confidence intervals and Bayes credible intervals. Furthermore, the theoretical results are assessed and compared through Monte Carlo simulations. Finally, one real data set is analyzed under the proposed model for illustrative purpose.

직관 퍼지 와이블 분포에서 알파 컷 셋을 이용한 시스템 신뢰성 분석

Purpose This study is to analyze system reliability using fuzzy Weibull lifetime distribution in which scale parameters are assumed to be Intuitionistic Fuzzy Set (IFS) with α-cut set due to uncertainties and inaccuracy of data. Methods The IFS is utilized to present the information of membership function and non-membership function with the three intervals that include belief, disbelief and unbelief functions. The paper discusses the intuitionistic fuzzy Weibull system reliability models to assess the membership function and non-membership function by considering the different types of component intuitionistic fuzzy scale parameter. Due to the existence of different types of uncertainties, the three types of component TIFS (Triangular IFS) and TrIFS (Trapezoidal IFS) with α-cut set were presented for computing intuitionistic fuzzy system reliability in both series structure and parallel structure. Results The validity of proposed model is demonstrated by analyzing intuitionistic fuzzy Weibull system reliabilities with α-cut set using different types of intuitionistic fuzzy scale parameter. A numerical example is indicated to discuss the usefulness and practicality of the proposed model. Conclusions When conducting a group decision making for evaluating the intuitionistic fuzzy Weibull system reliability, α-cut set models proposed in this paper can be used to reflect various experts’ judgments on agreement, disagreement and undecidedness when evaluating system reliability.

Experimental Design for the Lifetime Performance Index of Weibull Products Based on the Progressive Type I Interval Censored Sample

In this study, the experimental design is developed based on the testing procedure for the lifetime performance index of products following Weibull lifetime distribution under progressive type I interval censoring. This research topic is related to asymmetrical probability distributions and applications across disciplines. The asymptotic distribution of the maximum likelihood estimator of the lifetime performance index is utilized to develop the testing procedure. In order to reach the given power level, the minimum sample size is determined and tabulated. In order to minimize the total cost that occurred under progressive type I interval censoring, the sampling design is investigated to determine the minimum number of inspection intervals and equal interval lengths when the termination time of experiment is fixed or not fixed. For illustrative aims, one practical example is given for the implementation of our proposed sampling design to collect the progressive type I interval censored sample so that the users can use this sample to test if the lifetime performance index exceeds the desired target level.

Dynamic Warranty Reserve Planning Over a Finite Horizon

This article studies the warranty reserve planning problem for a firm that makes electronic products and maintains a reserve fund for warranty claims. For accounting purposes, the reserve fund is adjusted periodically. We model this problem in a multiperiod single-product setting.We propose a model that incorporates the randomness of product sales and failures to explicitly characterize the time-varying behavior of the warranty demand, which allows the firm to dynamically manage the reserve fund. Since the firm will eventually discontinue the product, we investigate a multiperiod dynamic reserve planning model to optimize warranty reserve levels over a finite horizon. We derive a closed-form expression of time-dependent warranty demand with general product sales and failure distributions. From the perspective of after-sales service, this result can be used to explain the dynamics involved in demand for warranty service. With the objective of minimizing the discounted total loss, we obtain the optimal reserve-setting policy and prove that the firm should set the reserve fund at either zero or a warranty-demand-dependent base stock level. To show the applicability and flexibility of our model, numerical experiments are conducted for some typical sales patterns and the Weibull lifetime distribution.

Joint Type-I Generalized Hybrid Censoring for Estimation Two Weibull Distributions

Products come from different lines with the same facility are tested under comparative life tests which known with the jointly censoring scheme. In this paper, two sets of products under the same facility have Weibull lifetime distributions are selected to test under Type- I generalized hybrid censoring scheme (GHCS). The observed censoring data are used to build the maximum likelihood (ML) estimators as well as approximate confidence intervals for the model parameters. Also, Bayes estimators with the help of MCMC methods are discussed. The analysis of simulated data set with Monte Carlo simulation study is used to illustrate and compare the theoretical results. Finally, a brief comment is summarized in concluding section.

Transmuted Topp-Leone Weibull Lifetime Distribution: Statistical Properties and Different Method of Estimation

In this work we focus on proposing a new lifetime Weibull type model called the transmuted Topp-Leone Weibull and studying its properties. We derive some new bivariate and multivariate transmuted Topp-Leone Weibull versions using “Farlie Gumbel Morgenstern (FGM) Copula”, “modified FGM Copula”, “Clayton Copula” and “Renyi's entropy Copula”. The estimation of its unknown parameters is carried out by considering different method of estimation. The statistical performances of all methods are studied by two real data sets and a numerical Monte Carlo simulation. The Cramer-Von Mises method is the best method for modeling the carbon fibers data. The maximum likelihood method is the best method for modeling the Leukemia data, however all other methods performed well.

RELIABILITY COMPARISON OF TWO UNIT REDUNDANCY SYSTEMS UNDER THE LOAD REQUIREMENT

AbstractThis paper compares the reliability functions of the cold standby, hot standby, and load-sharing redundancy configurations, each of which is composed of two identical components for meeting a given system requirement. Thus far, no research has been done into the conditions that make one configuration more reliable than another because their reliability functions have no closed forms even when the component follows a Weibull lifetime distribution. In this paper, two analytical results are obtained given that the reliability of each configuration is expressed in terms of the design and operational loads of the component. First, higher reliability can be achieved in a cold standby configuration than in a load-sharing configuration if the increase in the component reliability obtained from the reduction in the operational load is not significant. Second, a cold standby configuration exhibits better reliability and carries a higher load than a hot standby configuration if the design load can be increased with a less decrease in the component reliability.

Classical and Bayesian Inferences in Step-Stress Partially Accelerated Life Tests for Inverse Weibull Distribution Under Type-I Censoring

This paper deals with the classical and Bayesian estimations of step-stress partially accelerated life test model under type-I censoring for the inverse Weibull lifetime distribution. In classical estimation, the maximum likelihood estimates of the distribution parameters and the acceleration factor were obtained. In addition, approximate confidence intervals of the parameters were constructed based on the asymptotic distribution of the maximum likelihood estimators. Under Bayesian inference, besides the Lindley and Tierney–Kadane approximation posterior expectation methods, which yielded point estimates of the distribution parameters and the acceleration factors under square error loss function, we also applied the Gibbs sampling method, in order to construct credible intervals of these parameters together with their point estimates. Finally, Monte Carlo simulations were conducted to compare the performances of the above estimation methods.

Optimal progressive interval censoring plan under accelerated life test with limited budget

ABSTRACTIn this paper, we investigate some inference and design problems related to multiple constant-stress accelerated life test with progressive type-I interval censoring. A Weibull lifetime distribution at each stress-level combination is considered. The scale parameter of Weibull distribution is assumed to be a log-linear function of stresses. We obtain the estimates of the unknown parameters through the method of maximum likelihood, and also derive the Fisher's information matrix. The optimal number of test units, number of inspections, and length of the inspection interval are determined under D-optimality, T-optimality, and E-optimality criteria with cost constraint. An algorithm based on nonlinear mixed-integer programming is proposed to the optimal solution. The sensitivity of the optimal solution to changes in the values of the different parameters is studied.

Estimating Lifetimes and Stock Turnover Dynamics of Urban Residential Buildings in China

Building lifetime and stock turnover are both key determinants in modelling building energy and carbon. However in China, aside from anecdotal claims that urban residential buildings are generally short-lived, there are no recent official statistics, and empirical data are extremely limited. We present a system dynamics model where survival analysis is used to characterise the dynamic interplay between new construction, aging, and demolition of residential buildings in urban China. The uncertainties associated with building lifetime were represented using a Weibull distribution, whose shape and scale parameters were calibrated based on official statistics on floor area up to 2006. The calibrated Weibull lifetime distribution allowed us to estimate the dynamic stock turnover of Chinese urban residential buildings for 2007 to 2017. We find that the average lifetime of urban residential buildings was around 34 years, and the overall residential stock size reached 23.7 billion m2 in 2017. The resultant age-specific sub-stocks provide a baseline for the overall stock, which—along with the calibrated Weibull lifetime distribution—can be used in further modelling and for analysis of policies to reduce the whole-life embodied and operational energy and CO2 emissions in Chinese residential buildings.

Robustness of asymptotic accelerated life test plans to small‐sample settings

AbstractMuch of the statistical literature on optimal test planning for accelerated life testing utilize asymptotic methods to derive optimal test plans. While sufficient effort is made to assess the robustness of these test plans to the choice of design parameters and distribution assumptions, there is very little literature on the performance of asymptotic test plans relative to small samples (on the order of 10‐15 samples). An alternative concern is that the asymptotic test plans may not necessarily be the true “optimal” test plan for a given sample size. The purpose of this research is to present exact or “near‐exact” methods for developing test plans and compare the performance of these test plans with corresponding asymptotic test plans in small‐sample settings. The optimal location of design points and sample allocation is determined using each method for lognormal and Weibull lifetime distributions with both complete and Type 1 right‐censored data under two selected acceleration factor models. The investigations reveal that asymptotic test plans tend to corroborate quite well with exact test plans and thus are suitably robust to small‐sample settings in terms of optimal variance.

Optimum Periodic Component Reallocation and System Replacement Maintenance

In a repairable system with multiple functionally interchangeable components, the components may work in different environmental or operational stresses associated with their positions, and thus, experience different failure rates and nonhomogeneous degradation processes. For such a system, the system lifetime can be improved by reassigning the components among the positions in the system after a period of operation. In this sense, the component reallocation (CR) is an option of preventive maintenances. This paper studies a new maintenance policy, which performs periodic preventive system replacements and periodic preventive CRs between system replacements as well as minimal repairs for emergency failures. An optimization model is established to determine the time and assignment for CRs and the time for system replacements with the objective of minimizing expected annual system maintenance cost. The model also justifies if the CR is economically beneficial and deserved to be implemented. The proposed CR-based maintenance policy is further specified for exponential component lifetime distributions and Rayleigh lifetime distributions, and analytical results are derived. Finally, numerical examples for general Weibull lifetime distributions are presented to demonstrate the efficiency and insights of CRs in maintenance.

Mission Profile Based Reliability Evaluation of Capacitor Banks in Wind Power Converters

With the increasing penetration of wind power, reliable and cost-effective wind energy production is of more and more importance. The doubly-fed-induction-generator-based turbine system is widely used and dominates the wind market. In this paper, an analytical approach to assess reliability for power capacitors, both the dc-link capacitor bank and ac-side filter capacitor bank, is presented considering the annual mission profile. Based on the electrical behavior at various loading conditions, the lifecycle of the single power capacitor can be predicted through its electrothermal stresses. This percentile lifetime can be translated to the Weibull lifetime distribution of the power capacitor by considering the parameter uncertainties and tolerance variations. Thereafter, a reliability block diagram is used to bridge the reliability curves from the component-level of the individual capacitor to the system-level of the capacitor bank. A case study of a 2-MW wind turbine shows that the lifecycle is significantly reduced from the individual capacitor to the capacitor bank, where the dc-link capacitor bank dominates the lifetime consumption. Furthermore, the electrical stresses of the power capacitors are experimentally verified at a down-scaled 7.5 kW prototype.