Optimum step-stress accelerated degradation test for Wiener degradation process under constraints

To assess a product's reliability for subsequent managerial decisions such as designing an extended warranty policy and developing a maintenance schedule, Accelerated Degradation Test (ADT) has been used to obtain reliability information in a timely manner. In particular, Step-Stress ADT (SSADT) is one of the most commonly used stress loadings for shortening test duration and reducing the required sample size. Although it was demonstrated in many previous studies that the optimum SSADT plan is actually a simple SSADT plan using only two stress levels, most of these results were obtained numerically on a case-by-case basis. In this paper, we formally prove that, under the Wiener degradation model with a drift parameter being a linear function of the (transformed) stress level, a multi-level SSADT plan will degenerate to a simple SSADT plan under many commonly used optimization criteria and some practical constraints. We also show that, under our model assumptions, any SSADT plan with more than two distinct stress levels cannot be optimal. These results are useful for searching for an optimum SSADT plan, since one needs to focus only on simple SSADTs. A numerical example is presented to compare the efficiency of the proposed optimum simple SSADT plans and a SSADT plan proposed by a previous study. In addition, a simulation study is conducted for investigating the efficiency of the proposed SSADT plans when the sample size is small.     

Bayesian accelerated life test plans for series systems with Weibull component lifetimes

This article presents optimal Bayesian accelerated life test plans for series systems under Type-I censoring scheme. First, the component lifetimes are assumed to follow independent Weibull distributions. The scale parameters of Weibull lifetime distributions are related to the external stress variable through a general stress translation function. For a fixed number of design points, optimal Bayesian ALT plans are first obtained by solving constrained optimization problems under two different Bayesian design criteria. The global optimality of the resulting fixed-point optimal designs is then verified via the General Equivalence Theorem. This article also provides the optimized compromise ALT plans which are extremely useful in real-life applications. A detailed sensitivity analysis is then performed to find out the effect of various planning inputs on the resulting optimal Bayesian ALT plans. A simulation study is then conducted to visualize the resulting sampling variations from the optimal Bayesian ALT plans. Finally, this article considers a series system with dependent component lifetimes. Optimal ALT plans are obtained assuming a Gamma frailty model.     

Optimal Design of Accelerated Degradation Test based on Gamma Process Models

In this paper, optimal constant-stress accelerated degradation test plans are developed under the assumption that the degradation characteristic follows a Gamma processes. The test stress levels and the proportion of units allocated to each stress level are determined by D-criterion and V-criterion. The general equivalence theorem (GET) is used to verify that the optimized test plans are globally optimum. In addition, compromise test plans are also studied. Finally, an example is provided to illustrate the proposed method and a sensitivity analysis is conducted to investigate the robustness of optimal plans.     

Bayesian optimal life‐testing plan under the balanced two sample type‐II progressive censoring scheme

Joint progressive censoring schemes are quite useful to conduct comparative life‐testing experiment of different competing products. Recently, Mondal and Kundu (“A New Two Sample Type‐II Progressive Censoring Scheme,” Commun Stat‐Theory Methods; 2018) introduced a joint progressive censoring scheme on two samples known as the balanced joint progressive censoring (BJPC) scheme. Optimal planning of such progressive censoring scheme is an important issue to the experimenter. This article considers optimal life‐testing plan under the BJPC scheme using the Bayesian precision and D‐optimality criteria, assuming that the lifetimes follow Weibull distribution. In order to obtain the optimal BJPC life‐testing plans, one needs to carry out an exhaustive search within the set of all admissible plans under the BJPC scheme. However, for large sample size, determination of the optimal life‐testing plan is difficult by exhaustive search technique. A metaheuristic algorithm based on the variable neighborhood search method is employed for computation of the optimal life‐testing plan. Optimal plans are provided under different scenarios. The optimal plans depend upon the values of the hyperparameters of the prior distribution. The effect of different prior information on optimal scheme is studied.     

Planning accelerated life tests with random effects of test chambers

In accelerated life tests (ALTs), test units are often tested in multiple test chambers along with different stress conditions. The nonhomogeneity of test chambers precludes the complete randomized experiment and may affect the life‐stress relationship of the test product. The chamber‐to‐chamber variation should be taken into account for ALT planning so as to obtain more accurate test results. In this paper, planning ALTs under a nested experimental design structure with random test chamber effects is studied. First, by a 2‐phase approach, we illustrate to what extent different test chamber assignments to stress conditions may impact the estimation of unknown parameters. Then, D‐optimal test plans with 2 test chambers are considered. To construct the optimal design, we establish the generalized linear mixed model for failure‐time data and apply a quasi‐likelihood method, where test chamber assignments, as well as other decision variables that are required for planning ALTs, are simultaneously determined.     

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Step-stress accelerated degradation test (SSADT) is a useful tool for assessing the lifetime distribution of highly reliable products when the available test items are very few. In this paper, we discuss multiple-steps step-stress accelerated degradation models based on Wiener process, and we apply the objective Bayesian method for such analytically intractable models to obtain the noninformative priors (Jefferys prior and two Reference priors). Moreover, we show that their posterior distributions are proper, and we propose Gibbs sampling algorithms for the Bayesian inference based on the Jefferys prior and two Reference priors. Finally, we present some simulation studies to compare the objective Bayesian estimates with the other Bayesian estimate and the maximum likelihood estimates (MLEs). Simulation results demonstrate the superiority of objective Bayesian analysis method.     

Bayesian planning of optimal step-stress accelerated life test for log-location-scale distributions

This paper introduces some Bayesian optimal design methods for step-stress accelerated life test planning with one accelerating variable, when the acceleration model is linear in the accelerated variable or its function, based on censored data from a log-location-scale distributions. In order to find the optimal plan, we propose different Monte Carlo simulation algorithms for different Bayesian optimal criteria. We present an example using the lognormal life distribution with Type-I censoring to illustrate the different Bayesian methods and to examine the effects of the prior distribution and sample size. By comparing the different Bayesian methods we suggest that when the data have large(small) sample size B₁(τ) (B₂(τ)) method is adopted. Finally, the Bayesian optimal plans are compared with the plan obtained by maximum likelihood method.     

Using degradation models to assess pipeline life

Longitudinal inspections of thickness at particular locations along a pipeline provide useful information to assess the remaining life of the pipeline. In applications with different mechanisms of corrosion processes, we have observed various types of general degradation paths. We present two applications of fitting a degradation model to describe the corrosion initiation and growth behavior in a pipeline. We use a Bayesian approach for parameter estimation for the degradation model. The failure‐time and remaining lifetime distributions are derived from the degradation model, and we compute Bayesian estimates and credible intervals of the failure‐time and remaining lifetime distributions for both individual segments and for the entire pipeline circuit.     

Degradation modeling of 2 fatigue‐crack growth characteristics based on inverse Gaussian processes: A case study

Most modern products that are highly reliable are complex in their inner and outer structures. This situation indicates quality characterization by the interaction of multiple performance characteristics, which motivates the utilization of robust reliability models to obtain robust estimates. It is paramount to obtaining substantial information about a product's life cycle; therefore, when multiple performance characteristics are dependent, it is important to find models that address the joint distribution of performance degradation of such. In this paper, a reliability model for products with 2 fatigue‐crack growth characteristics related to 2 degradation processes is developed. The proposed model considers the dependence among degradation processes by using copula functions considering the marginal degradation processes as inverse Gaussian processes. The statistical inference is performed by using a Bayesian approach to estimate the parameters of the joint bivariate model. A time‐scale transformation is considered to assure monotone paths of the degradation trajectories. The comparison results of the reliability analysis, under both dependent and independent assumptions, are reported with the implementation of the proposed modeling in a case study, which consists of the crack propagation data of 2 terminals of an electronic device.     

Modeling of degradation data via wiener stochastic process based on acceleration factor constant principle

This paper proposes a systematic method of modeling accelerated degradation data based on the acceleration factor constant principle. Wiener stochastic process is considered because it is the most extensively used for degradation modeling. For the Wiener stochastic processes with three different time functions, the parameter relationships, which should be satisfied under any two different stress levels, are deduced according to the acceleration factor constant principle. The deduced parameter relationships indicate the stress-related parameters, which are applied to establish accurate accelerated degradation models. In addition, the deduced parameter relationships provide a guidance to test the consistency of the degradation mechanisms under different stress levels. A hypothesis method based on Analysis of Variance is adopted to identify the accelerated stress levels with different degradation mechanism. The degradation data under these stress levels should not be used to assess the product's reliability. The methods of validating accelerated degradation models and reliability assessments are also proposed. The simulation results prove the feasibility and effectiveness of the proposed methods. From the numerical example, it is concluded that the accelerated degradation model established based on the acceleration factor constant principle is more credible and accurate.     

A demand-driven method for scheduling optimal smooth production levels

This paper is concerned with demand-driven production scheduling in a commercial environment where smoothed production plans generation over a rolling horizon is desirable as new observations of demand are received through time. Demands are assumed to be normally distributed and dependent on the previous observed levels. The method of chance constraint of Charnes and Cooper is extended to multi-product production planning with variable workforce, back-ordered inventory, and nonstationary stochastic demand process. Bayesian procedures for revising the chance constraints and several variants of linear-programming-based production planning models are presented. In all cases the proposed methodology ensures that demands are satisfied, at a given level of confidence, while achieving smooth production.     

Optimal scaling of random-walk metropolis algorithms on general target distributions

One main limitation of the existing optimal scaling results for Metropolis–Hastings algorithms is that the assumptions on the target distribution are unrealistic. In this paper, we consider optimal scaling of random-walk Metropolis algorithms on general target distributions in high dimensions arising from practical MCMC models from Bayesian statistics. For optimal scaling by maximizing expected squared jumping distance (ESJD), we show the asymptotically optimal acceptance rate 0.234 can be obtained under general realistic sufficient conditions on the target distribution. The new sufficient conditions are easy to be verified and may hold for some general classes of MCMC models arising from Bayesian statistics applications, which substantially generalize the product i.i.d. condition required in most existing literature of optimal scaling. Furthermore, we show one-dimensional diffusion limits can be obtained under slightly stronger conditions, which still allow dependent coordinates of the target distribution. We also connect the new diffusion limit results to complexity bounds of Metropolis algorithms in high dimensions.     

Flexible capacity strategy with multiple market periods under demand uncertainty and investment constraint

We establish a flexible capacity strategy model with multiple market periods under demand uncertainty and investment constraints. In the model, a firm makes its capacity decision under a financial budget constraint at the beginning of the planning horizon which embraces n market periods. In each market period, the firm goes through three decision-making stages: the safety production stage, the additional production stage and the optimal sales stage. We formulate the problem and obtain the optimal capacity, the optimal safety production, the optimal additional production and the optimal sales of each market period under different situations. We find that there are two thresholds for the unit capacity cost. When the capacity cost is very low, the optimal capacity is determined by its financial budget; when the capacity cost is very high, the firm keeps its optimal capacity at its safety production level; and when the cost is in between of the two thresholds, the optimal capacity is determined by the capacity cost, the number of market periods and the unit cost of additional production. Further, we explore the endogenous safety production level. We verify the conditions under which the firm has different optimal safety production levels. Finally, we prove that the firm can benefit from the investment only when the designed planning horizon is longer than a threshold. Moreover, we also derive the formulae for the above three thresholds.     

The usefulness of Bayesian optimal designs for discrete choice experiments

Recently, the use of Bayesian optimal designs for discrete choice experiments, also called stated choice experiments or conjoint choice experiments, has gained much attention, stimulating the development of Bayesian choice design algorithms. Characteristic for the Bayesian design strategy is that it incorporates the available information about people's preferences for various product attributes in the choice design. This is in contrast with the linear design methodology, which is also used in discrete choice design and which depends for any claims of optimality on the unrealistic assumption that people have no preference for any of the attribute levels. Although linear design principles have often been used to construct discrete choice experiments, we show using an extensive case study that the resulting utility‐neutral optimal designs are not competitive with Bayesian optimal designs for estimation purposes. Copyright © 2011 John Wiley & Sons, Ltd.     

Estimation and Design of Sampling Plans for Monitoring Dependent Production Processes

We consider the problem of designing single and the double sampling plans for monitoring dependent production processes. Based on simulated samples from the process, Nelson proposed a new approach of estimating the characteristics of single sampling plans and, using these estimates, designing optimal plans. In this paper, we extend his approach to the design of optimal double sampling plans. We first propose a simple methodology for obtaining the unbiased estimators of various characteristics of single and double sampling plans. This is achieved by defining the various characteristics of sampling plans as explicit random variables. Some of the important properties of the double sampling plans are established. Using these results, an efficient algorithm is developed to obtain optimal double sampling plans. A comparison with a crude search shows that our algorithm leads to about 90% savings, on the average, in computational timings. The procedure is also explained through a suitable example for the ARMA(1,1) model. It is observed, for instance, that an optimal double sampling plan leads to about 23% reduction in average sample number, compared to an optimal single sampling plan. Tables for choosing the optimal plans for certain auto regressive moving average processes at some practically useful values of acceptable quality level and rejectable quality level are also presented.     

A mathematical model for identifying an optimal waste management policy under uncertainty

In the municipal solid waste (MSW) management system, there are many uncertainties associated with the coefficients and their impact factors. Uncertainties can be normally presented as both membership functions and probabilistic distributions. This study develops a scenario-based fuzzy-stochastic quadratic programming (SFQP) model for identifying an optimal MSW management policy and for allowing dual uncertainties presented as probability distributions and fuzzy sets being communicated into the optimization process. It can also reflect the dynamics of uncertainties and decision processes under a complete set of scenarios. The developed method is applied to a case study of long-term MSW management and planning. The results indicate that reasonable solutions have been generated. They are useful for identifying desired waste-flow-allocation plans and making compromises among system cost, satisfaction degree, and constraint-violation risk.     

EFFICIENCY IN GREAT BARRIER REEF WATER POLLUTION CONTROL: A CASE STUDY FOR THE DOUGLAS SHIRE

ABSTRACT. . Despite the lack of adequate economic evidence, water pollution from economic activities in Great Barrier Reef (GBR) catchments is often regarded excessive. In this paper marginal benefits from terrestrial (agricultural) water pollution and associated marginal marine costs from GBR degradation are estimated and used in an optimal control approach to determine optimal levels of water quality. Results, for a case study in the Wet Tropics in Australia, show that locally optimal levels of (fine sediment) water pollution are close to current levels, indicating that increased rates of (agricultural) water pollution lead to a decrease in local economic welfare. Globally optimal levels of water pollution are, however, below current levels, indicating that (inter‐) national compensation for beneficial spillovers from reduced GBR degradation can increase global economic welfare.     

A semiparametric Bayesian generalized linear mixed model for the reliability of Kevlar fibers

We analyze the reliability of NASA composite pressure vessels by using a new Bayesian semiparametric model. The data set consists of lifetimes of pressure vessels, wrapped with a Kevlar fiber, grouped by spool, subject to different stress levels; 10% of the data are right censored. The model that we consider is a regression on the log‐scale for the lifetimes, with fixed (stress) and random (spool) effects. The prior of the spool parameters is nonparametric, namely they are a sample from a normalized generalized gamma process, which encompasses the well‐known Dirichlet process. The nonparametric prior is assumed to robustify inferences to misspecification of the parametric prior. Here, this choice of likelihood and prior yields a new Bayesian model in reliability analysis. Via a Bayesian hierarchical approach, it is easy to analyze the reliability of the Kevlar fiber by predicting quantiles of the failure time when a new spool is selected at random from the population of spools. Moreover, for comparative purposes, we review the most interesting frequentist and Bayesian models analyzing this data set. Our credibility intervals of the quantiles of interest for a new random spool are narrower than those derived by previous Bayesian parametric literature, although the predictive goodness‐of‐fit performances are similar. Finally, as an original feature of our model, by means of the discreteness of the random‐effects distribution, we are able to cluster the spools into three different groups. Copyright © 2012 John Wiley & Sons, Ltd.     

STEM Teachers' Planned and Enacted Attempts at Implementing Engineering Design‐Based Instruction

This study investigates grades 5 and 6 science, technology, engineering, and mathematics (STEM) teachers' planned and actualized engineering design‐based instruction, the instruments used to characterize their efforts, and the implications this work has for teachers' implementations of an integrated approach to STEM education. Participants included 23 STEM teachers from six schools (three rural, two suburban, and one urban). Data were gathered via lesson implementation plans and classroom observations. Teachers demonstrated strength in planning for standards‐ and engineering design‐based lessons, incorporating engineering practices within their respective implementation plans, and aligning their plans with content and design process standards. Missing from their plans was attention to science concepts and their placement, use, and application within a design task. Classroom observations indicated that the teacher participants gave priority to “front loading,” the design process by concentrating more of their instructional time on problem identification and planning and less time on testing designs, communicating performance results, and redesigning. Measures utilized in this study provided insight into the content of teachers' planning and subsequent instruction and suggest potential for capturing content planning in the context of classrooms in which teachers are attempting to integrate novel curriculum, such as the new standards for engineering practices.