Process Capability Indices Based on the Highest Density Interval

For process capability indices (PCIs) of non‐normal processes, the natural tolerance is defined as the difference between the 99.865 percentile and the 0.135 percentile of the process characteristic. However, some regions with relatively low probability density may still be included in this natural tolerance, while some regions with relatively high probability density may be excluded for asymmetric distributions. To take into account the asymmetry of process distributions and the asymmetry of tolerances from the viewpoint of probability density, the highest density interval is utilized to define the natural tolerance, and a family of new PCIs based on the highest density interval is proposed to ensure that regions with high probability density are included in the natural tolerance. Some properties of the proposed PCIs and two algorithms to compute the highest density interval are given. A real example is given to show the application of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

双侧公差限产品的合格概率计算方法

为了定量分析产品的合格性,在正态分布假设前提下,考虑参数测量结果测量不确定度的影响,建立了双侧公差限产品的合格概率计算模型,为批量产品和单件产品合格概率的定量计算提供了理论依据,同时给出了测量分析计算实例。结果表明:产品合格概率的定量计算结果,对于分析产品质量问题成因和提高产品质量具有重要的指导意义,给出的方法对单侧公差限设计要求的产品合格概率的计算同样具有参考价值。     

基于阈值超越概率的结构耐久性与损伤容限综合分析方法

飞机结构耐久性分析和损伤容限分析都是以裂纹扩展分析为基础的,该文根据二者之间的内在联系,将耐久性分析与损伤容限分析进行综合考虑。该方法以损伤容限设计手册中规定的非概率初始裂纹尺寸作为阈值裂纹尺寸,根据裂纹尺寸超过阈值的概率以及耐久性分析结果确定概率损伤容限分析时的初始裂纹尺寸分布。裂纹尺寸超过阈值的概率可以通过检测数据给出,评估结果较为可靠。在概率损伤容限分析时,为了避免多重积分的计算,给出了结构安全余量的显式表达式以及一次二阶矩方法的计算公式,提高了可靠度计算效率。该文结合工程实例并与现有方法进行了对比分析,结果表明该方法是合理的、有效的。     

A new system formulation for the tolerance analysis of overconstrained mechanisms

The goal of tolerance analysis is to verify whether design tolerances enable a mechanism to be functional. The current method consists in computing a probability of failure using Monte Carlo simulation combined with an optimization scheme called at each iteration. This time consuming technique is not appropriate for complex overconstrained systems. This paper proposes a transformation of the current tolerance analysis problem formulation into a parallel system probability assessment problem using the Lagrange dual form of the optimization problem. The number of events being very large, a preliminary selective search algorithm is used to identify the most contributing events to the probability of failure value. The First Order Reliability Method (FORM) for systems is eventually applied to compute the probability of failure at low cost. The proposed method is tested on an overconstrained mechanism modeled in three dimensions. Results are consistent with those obtained with the Monte Carlo simulation and the computing time is significantly reduced.     

A novel statistical tolerance analysis method for assembled parts

Tolerance analysis of an assembly is an important issue for mechanical design. Among various tolerance analysis methods, statistical analysis is the most commonly employed method. However, the conventional statistical tolerance method is often based on the normal distribution. It fails to predict the resultant tolerance of an assembly of parts with non-normal distributions. In this paper, a novel method based on statistical moments is proposed. Tolerance distributions of parts are first transferred into statistical moments that are then used for computing tolerance stack-up. The computed moments, particularly the variance, the skewness and the kurtosis, are then mapped back to probability distributions in order to calculate the resultant tolerance of the assembly. The proposed method can be used to analyse the resultant tolerance specification for non-normal distributions with different skewness and kurtosis. Simulated results showed that tail coefficients of different distributions with the same kurtosis are close to each other for normalised probabilities between ?3 and 3. That is, the tail coefficients of a statistical distribution can be predicted by the coefficients of skewness and kurtosis. Two examples are illustrated in the paper to demonstrate the proposed method. The predicted resultant tolerances of the two examples are only 0.5% and 1.5% differences compared with that by the Monte Carlo simulation for 1,000,000 samples. The proposed method is much faster in computation with higher accuracy than conventional statistical tolerance methods. The merit of the proposed method is that the computation is fast and comparatively accurate for both symmetrical and unsymmetrical distributions, particularly when the required probability is between ±2σ and ±3σ.     

Optimal Design of Fibre-reinforced Laminated Plates Accounting for Manufacturing Uncertainty

A procedure to design symmetrically laminated plates under buckling loads for minimum weight with manufacturing uncertainty (tolerance) in the ply angle and plate thickness, which are the design variables, is described. A minimum buckling load capacity is the design constraint implemented. It is assumed that the probability of any tolerance value occurring within the tolerance band, compared with any other, is equal, and thus the approach is a worst case scenario approach. The effects of bending–twisting coupling are neglected in implementing the procedure, and the Downhill Simplex method is used as the search technique, but the methodology is flexible and allows any appropriate problem formulation and search algorithm to be substituted. Two different tolerance scenarios are used for the purposes of illustrating the methodology, and plates with varying aspect ratios and loading ratios are optimally designed and compared. The results demonstrate the importance of carrying out design optimisation of composite structures with the effects of manufacturing tolerances included.     

Tolerance‐based reliability and optimization design of switched‐mode power supply

Due to its high efficiency and low power consumption, switched‐mode power supply (SMPS) represents the development trend of the stabilized voltage power supply. However, tolerance has become one of the key factors in the design of SMPS because of the process fluctuation of electronic components, unstable input parameters of the circuit system, influence of working conditions and environment, and the effect of aging. In order to improve the reliability of SMPS and reduce the manufacturing cost, this paper proposes a reliability analysis and optimization design method based on the tolerance and sensitivity analysis. Finally, this method is applied to the tolerance design for the positive switching power supply of the SMPS circuits, and the optimal tolerance design scheme is obtained. Furthermore, the reliability and probability density curves are evaluated.     

CMM在产品检验中的不确定度分析与合格性判定

采用测量系统分析方法,基于量值特性指标对三坐标测量机在产品检验中的不确定度进行评定。以测量不确定度作为产品检验的质量指标,结合产品设计精度,建立产品合格判定的误判率与产品设计公差、产品检验结果估计值和不确定度的数学模型。通过三坐标测量机对零件进行产品检验实验,对测量系统的量值特性给出合理评价,在产品检验不确定度评定基础上对产品的合格性做出判定,并计算可能存在的误判率,以验证理论方法的可行性与可操作性。     

Development and evaluation of an uncertainty importance measure in fault tree analysis

In a fault tree analysis, an uncertainty importance measure is used to identify those basic events that significantly contribute to the uncertainty of the top-event probability. This paper defines an uncertainty importance measure of a basic event or of a group of basic events, and develops a two-stage procedure for experimentally evaluating the measure under the assumption that the probability of each basic event follows a lognormal distribution. The proposed method utilizes the Taguchi tolerance design technique with modifications. Then, the so-called contribution ratios which evaluate the main and/or interaction effects of the uncertainties of log-transformed basic-event probabilities on the uncertainty of the log-transformed top-event probability are calculated. The contribution ratios are used to estimate the defined uncertainty importance measure of a basic event or of a group of basic events. The proposed method consists of two stages for computational efficiency. In the first stage, the basic events with negligible effects on the uncertainty of the log-transformed top-event probability are screened out, and more detailed analyses are conducted in the second stage with a substantially smaller number of basic events. In addition, this paper presents an analysis method to quantify the percentage reduction in the uncertainty of the log-transformed top-event probability when the uncertainty of each basic-event probability is reduced.     

On the Determination of a Safe Life for Distributions Classified by Failure Rate

If the distribution of life length is a member of a specified subset of distributions which have increasing failure rates, we find a method of determining functions of the sample data which can be used to provide lower tolerance bounds of given confidence or a service life with specified assurance of no failures among a given number of items to be produced.

The method of finding such functions of the observations is shown to be optimal in a sense and the calculation of a lower bound for the probability of no failure in the future production is carried out when such functions are used. The confidence in the tolerance bound and the assurance of no failure in a production lot of specified size are compared when using bounds obtained from these functions.     

Discussion: Fuzzy Thinking

This article investigates computation of pointwise and simultaneous tolerance limits under the logistic regression model for binary data. The data consist of n binary responses, where the probability of a positive response depends on covariates via the logistic regression function. Upper tolerance limits are constructed for the number of positive responses in m future trials for fixed as well as varying levels of the covariates. The former provides pointwise upper tolerance limits, and the latter provides simultaneous upper tolerance limits. The upper tolerance limits are obtained from upper confidence limits for the probability of a positive response, modeled using the logistic function. To compute pointwise upper confidence limits for the logistic function, likelihood-based asymptotic methods, small sample asymptotics, as well as bootstrap methods are investigated and numerically compared. To compute simultaneous upper tolerance limits, a bootstrap approach is investigated. The problems have been motivated by an application of interest to the U.S. Army, dealing with the testing of ballistic armor plates for protecting soldiers from projectiles and shrapnel, where the success probability depends on covariates such as the projectile velocity, size of the armor plate, etc. Such an application is used to illustrate the tolerance interval computations in the article. We provide the R codes used for the calculations presented in the examples in the article as supplementary material, available online.     

Tolerance design for the fit between bore and shaft for precision assemblies with significant error-scaling problems

An inspection procedure may be necessary when a process does not have the capability to produce parts that meet specifications. After inspection, the accepted parts then have a probability density function which is truncated. In a previous paper, the authors presented a method for the computation of tolerances for the fit between a bore and a shaft, based on the construction of a probabilistic model of the fit, using truncated probability density functions for the bore and shaft. In this paper, a method of design is presented for the computation of tolerances for the bore and shaft that are necessary to satisfy the tolerance of the fit and at the same time minimize the probability of rejection for the processes that produce the bore and shaft. Application is made to the situation where the variations in the dimensions of the parts may be large compared with their nominal dimension or the fit requirement of the assembly. Such a situation will be denoted as the error-scaling problem.     

Multivariate tolerance design using quality loss

The determination of tolerance allocations among design parameters is an integral phase of product/process design. Such allocations are often necessary to achieve desired levels of product performance. We extend our prior research on tolerance allocation by developing both parametric and nonparametric methods for a multivariate set of performance measures that are functions of a common set of design parameters. The parametric method is novel and assumes full information about the probability distribution of design parameter processes. The proposed nonparametric method assumes that only partial information is available and significantly extends prior research by considering a more contemporary and realistic model for manufacturer costs. For both methods we derive economically based models that represent the costs, both internal (supplier) and external (manufacturer), of tolerance allocation under several different process scenarios. These scenarios are based on the manner of disposition of nonconforming product. For the parametric methods we derive tolerance allocation solutions that jointly minimize expected total cost of the supplier and manufacturer. For the nonparametric methods we derive solutions for tolerance allocation that jointly minimizes the maximum expected total cost. An example in the fabrication of a rubber tread compound is used to: (i) demonstrate the implementation of our proposed methodologies for tolerance allocation; (ii) illustrate and compare the nonparametric and parametric methods; land iii) assess the sensitivity of optimal tolerance allocations to changes in process model types, cost coefficient estimates, and manner of disposition of nonconforming product.     

Nonlinear logistic regression mixture experiment modeling for binary data using dimensionally reduced components

This article motivates, presents, and illustrates an approach using nonlinear logistic regression (NLR) for modeling binary response data from a mixture experiment when the components can be partitioned into groups used to form dimensionally reduced components (DRCs). A DRC is formed from a linear combination of the components in a group having similar roles and/or effects of the same sign, where the linear combinations over all groups are normalized so that the DRC proportions sum to one. Linear combinations of a particular form provide for quantifying the effects of the remaining components in a group relative to a chosen component. This reason, plus dimensional reduction, are the primary motivations for the proposed DRC mixture experiment modeling approach. NLR is required because models expressed in terms of the DRCs are nonlinear in the parameters that specify the linear combinations. A method for obtaining nonparametric tolerance limits on the probability of a “success” for the binary response variable using a bootstrap approach is also presented. Finally, the article shows how DRCs provide for visualizing data and modeling results that otherwise would be impossible. A real database on the presence or absence of nepheline crystals in simulated nuclear waste glass is used to illustrate the NLR modeling and nonparametric tolerance limit approaches. The methodology is general and can be applied to other applications.     

The optimal schedule to produce a given number of acceptable parts with a specified confidence level

This paper deals with the problem of finding the optimal schedule for producing, with a probability α, a finite number H, of parts which have a diameter within specified tolerance limits. It is assumed that the diameter is a normally distributed variable that exhibits a linear trend in the process mean. The solution involves determining the optimal run size(s), as well as the specific number of runs of each size, required to produce at least H parts, with probability α, at minimum cost. A solution algorithm is provided and computational experience reported.     

Optimal stopping in a cumulative damage model

Summary We consider the problem of optimal stopping in a cumulative damage model in which a prescribed level may be surpassed only with small probability, but should be approached as precise as possible. Questions of this kind occur, e.g., in the determination of tolerance levels of medical treatments and in coping with metal fatigue. A control limit policy is proved to be optimal.     

面向质量目标的Q控制图设计

针对大批量生产开始阶段的过程监控,提出了一种基于预定质量目标的Q控制图监控方法.其基本思路是利用面向质量目标的统计公差技术与Q控制图相结合应用,以实现大批量过程开始阶段均值和方差未知时面向质量目标的过程监控.基于质量目标建立统计公差(CP*,k*),并将该统计公差转化为基于给定置信概率的对CP和k的估计值的判定条件.通过案例分析,面向质量目标的Q控制图能够在过程保持受控状态的前提下以一定置信概率保证质量目标.     

A Bayesian method for robust tolerance control and parameter design

This paper proposes a Bayesian method to set tolerance or specification limits on one or more responses and obtain optimal values for a set of controllable factors. The existence of such controllable factors (or parameters) that can be manipulated by the process engineer and that affect the responses is assumed. The dependence between the controllable factors and the responses is assumed to be captured by a regression model fit from experimental data, where the data are assumed to be available. The proposed method finds the optimal setting of the control factors (parameter design) and the corresponding specification limits for the responses (tolerance control) in order to achieve a desired posterior probability of conformance of the responses to their specifications. Contrary to standard approaches in this area, the proposed Bayesian approach uses the complete posterior predictive distribution of the responses, thus the tolerances and settings obtained consider implicitly both the mean and variance of the responses and the uncertainty in the regression model parameters.     

Tolerance chart optimization

A tolerance chart is a graphical representation of a process plan and a manual procedure for controlling tolerance stackup when the machining of a component involves interdependent tolerance chains. This heuristic, experience-based method of allocating tolerances to individual cuts of a process plan can be embodied in a computer-based module. This paper introduces a graph theoretic representation for the tolerance chart. A special path tracing algorithm is used to identify tolerance chains from this graph. Optimal tolerance allocation among individual cuts is achieved using a linear goal programming model instead of existing heuristic methods. A more comprehensive mixed integer programming model is developed to incorporate linear tolerance cost functions and alternative process selection.     

钢框架结构基于变形可靠度的全概率抗震设计

我国抗震设计规范的抗震设计方法并不是真正意义上的概率极限状态设计,结构可靠度的应用也没有体现出结构的体系可靠度设计水平,因此该文提出了基于变形可靠度验算的二阶段抗震设计方法.该方法采用结构可靠度的数值模拟方法,通过验算小震作用下结构构件承载能力极限状态下的抗震可靠度,验算结构小震作用下正常使用极限状态下和大震作用下侧向...