Economic design of an attribute control chart for monitoring mean life based on multiple deferred state sampling

In this paper, we design an attribute np control chart using multiple deferred state (MDS) sampling under Weibull distribution based on time truncated life test. This chart is constructed for monitoring the variation of mean life of the product in a manufacturing process. The optimal parameters of MDS sampling and the control limit coefficients are determined so that the in‐control average run length (ARL) is as close as to the target ARL. The optimal parameters of MDS sampling are sample size and number of successive subgroups required for declaring the current state of process. Out‐of‐control ARL is considered as a measure of the performance of proposed chart and reported with determined optimal parameters for various shift constants. The out‐of‐control ARL of the proposed chart obtained under various distributions is compared with each other. The performance of proposed control chart is compared with the performance of the existing control chart designed under single sampling. In addition, the economic design of proposed chart using variable sampling interval scheme is discussed, and sensitivity analysis on expected costs is also investigated.     

Adaptive sampling enhancement for Hotelling’s T charts

This paper makes a study of an adaptive sampling scheme useful to increase the power of the fixed sampling rate (FSR) T² control chart. In our study, the three parameters of T² control chart: the sample size, the sampling interval, and the upper percentage factor that is used for determining the action limit, vary between two values for a relaxed or tightened control, depending on the most recent T² value. The average time to signal (ATS) and adjusted average time to signal (AATS) a shift in the process mean vector for the new chart are derived and regarded as an objective function respectively to optimize its design parameters. With some minor changes, the new chart can be reduced to the variable sampling interval (VSI) T² chart, the sample size (VSS) T² chart, the variable sample size and sampling interval (VSSI) T² chart, or the FSR T² chart. Numerical comparisons among them are made and discussed. Furthermore, the effects of the initial sample number (use for estimating the in-control process parameters) upon the chart’s performance and adaptive design parameters are presented.     

Economic design of an adaptive <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> control chart

Varying three parameters of the Hotelling's T² control chart, namely, the sample size, the sampling interval length, and the action limit, results in a great improvement over the traditional T² control chart particularly for the small process shift. This paper presents the economic model for designing the variable parameters T² chart. In the economic design, a cost function is constructed, involving the cost of sampling and testing, the cost of false alarm, the cost to detect and remove the assignable cause, and the cost when the process is operating out-of-control. Also, a heuristic method of finding optimal values of adaptive design parameters for minimizing the cost function is presented. Variable and fixed parameters T² control charts are compared with respect to the expected loss per unit time. Furthermore, the effects of the initial sample number (used for estimating the parameters of F-distribution) upon the operating cost and adaptive design parameters are examined.     

Hotelling’s T charts with variable sample size and control limit

The ideas of variable sampling interval (VSI), variable sample size (VSS), variable sample size and sampling interval (VSSI), and variable parameters (VP) in the univariate case have been successfully applied to the multivariate case to improve the efficiency of Hotelling’s T² chart with fixed sampling rate (FSR) in detecting small process shifts. However, the main disadvantage in using most of these control schemes is an increasing in the complexity due to the adaptive changes in sampling intervals. In this paper, retaining the lengths of sampling intervals constant, a variable sample size and control limit (VSSC) T² chart is proposed and described. The statistical efficiency of the VSSC T² chart in terms of the average time to signal a shift in process mean vector is compared with that of the VP, VSSI, VSS, VSI, and FSR T² charts. From the results of comparison, it shows that the VSSC T² chart for a (very) small shift in the process mean vector gives a better performance than the VSSI, VSS, VSI, and FSR T² charts; meanwhile, it presents a similar performance to the VP T² chart. Furthermore, from the viewpoint of practicability, it is more convenient for administrating the control chart than the VSI, VSSI, and VP T² chart. Thus, it may provide a good option for quick response to small shifts in a multivariate process.     

Individuals control schemes for monitoring the mean and variance of processes subject to drifts

This paper investigates control chart schemes for detecting drifts in the process mean μ and/or process standard deviation σ when individual observations are sampled. Drifts may be due to causes such as gradual deterioration of equipment, catalyst aging, waste accumulation, or human causes, such as operator fatigue or close supervision. The standard Shewhart X chart and moving range (MR) chart are evaluated, as well as several types of exponentially weighted moving average (EWMA) charts and combinations of charts involving these EWMA charts. We show that the combinations of the EWMA charts detect slow-rate and moderate-rate drifts much faster than the combined X and MR charts. We also show that varying the sampling interval adaptively as a function of the process data results in notable reductions in the detection delay of drifts in μ and/or σ.     

Continuity Adjustment for Control Charts for Attributes

A unified approach is proposed for making a continuity adjustment on some control charts for attributes, e.g., np-chart and c-chart. through adding a uniform (0,1) random observation to the conventional sample statistic (e.g., npi and ci). The adjusted sample statistic then has a continuous distribution. Consequently, given any Type I risk a (the probability that the sample statistic is on or beyond the control limits), control charts achieving the exact value of a can be readily constructed. Guidelines are given for when to use the continuity adjustment control chart, the conventional Shewhart control chart (with ±3 standard deviations control limits), and the control chart based on the exact distribution of the sample statistic before adjustment.     

An extension of Banerjee and Rahim's model for economic design of moving average control chart for a continuous flow process

In this paper, we propose a model of a moving average control chart (MA control chart) with a Weibull failure mechanism from an economic viewpoint. When the process-failure mechanism follows a Weibull model or other models having increasing hazard rates, it is desirable to have the decreasing sampling interval with the age of the system. The MA control chart is used to monitor quality characteristics of raw material or products in a continuous process. A cost model utilizing a variable scheme instead of fixed sampling lengths in a continuous flow process is studied in this research. The variable sampling scheme is used to maintain a constant integrated hazard rate over each sampling interval. Optimal values for the design parameter, the moving subgroup size, the sampling interval, and the control limit coefficient are determined by minimizing the loss-cost model. The performance of the loss cost with various Weibull parameters is studied. A sensitivity analysis shows that the design parameters and loss cost depend on the model parameters and shift amounts.     

可变抽样区间和样本容量的非正态EWMA控制图经济设计

为了降低过程控制成本和提高监控效率,针对质量特性值不服从正态分布的情况,研究可变抽样区间和样本容量(VSSI)的指数加权移动平均(EWMA)控制图的经济设计问题。首先对监控非正态分布的EWMA控制图进行抽样区间和样本容量变化设计;其次建立VSSI非正态EWMA控制图的经济模型,通过使费用成本函数最小得到控制图的最优设计参数组合;然后给出工业中的一个例子,用遗传算法对经济模型搜寻最优解;接下来对经济模型进行灵敏度分析,得出控制图费用参数与设计参数之间的影响关系;最后通过最优性分析,得出所建立的VSSI非正态EWMA控制图的经济性优于VSI、VSS非正态EWMA控制图。     

Multivariate Bayesian process control for a finite production run

Most industrial products and processes are characterized by several, typically correlated measurable variables, which jointly describe the product or process quality. Various control charts such as Hotelling’s T², EWMA and CUSUM charts have been developed for multivariate quality control, where the values of the chart parameters, namely the sample size, sampling interval and the control limits are determined to satisfy given economic and/or statistical requirements. It is well known that this traditional non-Bayesian approach to a control chart design is not optimal, but very few results regarding the form of the optimal Bayesian control policy have appeared in the literature, all limited to a univariate chart design. In this paper, we consider a multivariate Bayesian process mean control problem for a finite production run under the assumption that the observations are values of independent, normally distributed vectors of random variables. The problem is formulated in the POMDP (partially observable Markov decision process) framework and the objective is to determine a control policy minimizing the total expected cost. It is proved that under standard operating and cost assumptions the control limit policy is optimal. Cost comparisons with the benchmark chi-squared chart and the MEWMA chart show that the Bayesian chart is highly cost effective, the savings are larger for smaller values of the critical Mahalanobis distance between the in-control and out-of-control process mean.     

Optimal average sample number of the SPRT chart for monitoring fraction nonconforming

The Sequential Probability Ratio Test (SPRT) control chart is a powerful tool for monitoring manufacturing processes. It is highly suitable for the applications where testing is destructive or very expensive, such as the automobile airbags test. This article studies the effect of the Average Sample Number (ASN) (i.e., the average sample size) on the chart’s performance. A design algorithm is proposed to develop the optimal SPRT chart for monitoring the fraction nonconforming p of Bernoulli processes. By optimizing the ASN and other charting parameters, the average detection speed of the SPRT chart is almost doubled. It is also found that the optimal SPRT chart significantly outperforms the optimal np and binomial CUSUM charts, in terms of Average Number of Defectives (AND), under different combinations of the design specifications. It is observed that the SPRT chart using a relatively smaller ASN and a shorter sampling interval (h) has a higher overall detection effectiveness.     

可变抽样区间图经济设计分析

传统控制图是静态的,其抽样区间为固定常数,往往不能及时发现生产过程异常。针对这种情况,本文在综合抽样成本、生产次品损失、误报警和漏报警损失、发现并纠正过程异常成本等基础上,提出可变抽样区间X图的经济设计方法。根据过程实际状态和由抽样结果采取的决策构建了一个二维时间离散的马尔可夫链,提出优化模型,并利用遗传算法寻找控制图参数的最优解。数值计算给出本文模型的具体求解过程。灵敏度分析研究了各参数对最优样本容量、控制限、警戒限、抽样区间及单位时间平均成本的影响。     

可变抽样区间几何EWMA控制图的经济设计

为了降低生产过程周期成本,本文对单位缺陷数服从几何分布时,可变抽样区间的指数加权移动平均(EWMA)控制图进行经济设计。首先建立可变抽样区间几何EWMA控制图的经济模型,使单位时间期望费用最小来确定参数的最优值;其次用遗传算法来寻找经济模型的最优解;最后对可变抽样区间几何EWMA控制图的经济模型进行灵敏度分析和最优性分析。研究结果表明单位时间期望费用分别随着异常原因发生的频率、过程失控时单位时间的质量费用、发现异常原因的时间期望值和纠正过程的时间期望值的增大而增大。     

Multivariate nonparametric control chart with variable sampling interval

This paper proposes an adaptive, multivariate, nonparametric, exponentially weighted moving average control chart with variable sampling interval. A number of studies have discussed multivariate nonparametric control charts. However, the proposed multivariate nonparametric control charts usually have strict requirements. In this paper, we construct a control chart for multivariate processes that is based on the Mahalanobis depth. Specifically, we use the concept of the Mahalanobis depth to reduce each multivariate measurement to a univariate index. It is worth mentioning that this approach is completely nonparametric. We also discuss the optimal strategy for the parameters. This chart is an adaptive chart and has a variable sampling interval. A simulation study demonstrates that the proposed chart is efficient in detecting various magnitudes of shifts. A gravel data and a wine quality detection example are given to introduce the proposed control chart.     

Optimal inspection policy for three-state systems monitored by control charts

This paper presents the formulations of the expected long-run cost per time unit for a system monitored by a static control chart and by an adaptive control chart respectively. The static chart has a fixed sampling interval and a fixed sample size. The adaptive chart has a fixed sample size but variable sampling intervals. The system is supposed to have three states, normal working state, failure delay time state, and failed state. Two levels of repair are used to maintain the system. A minor repair is used to restore the system if a detectable defect is confirmed by an inspection. A major repair will be performed if the system fails. The expected cost per time unit for maintaining such a system is obtained. The objective of such analysis is to find an optimal sampling policy for the inspection process. An artificially generated data example and a real data example are used to compare the expected cost per time unit for both the static and adaptive control charts.     

Cost-optimal multistage sampling plans in statistical quality control

Multistage Bayesian decision procedures in statistical quality control are known from attribute sampling. In this paper they are introduced in a more general framework occuring in lot-control by using the theory of Bayesian sequentially planned decision procedures. We show that under sufficiency and transitivity assumptions and monotonicity properties concerning the distribution and cost set-up these Bayes-procedures have (z, c ^–,c ⁺)-structure which, on one hand, generalizes results of K.-H. Waldmann and, on the other hand, reduces computational effort significantly. Finally, examples taken from attribute sampling and life testing for an outgoing lot are presented.The theory presented in this paper is taken from [11].     

Joint economic design of EWMA control charts for mean and variance

Control charts with exponentially weighted moving average (EWMA) statistics (mean and variance) are used to jointly monitor the mean and variance of a process. An EWMA cost minimization model is presented to design the joint control scheme based on pure economic or both economic and statistical performance criteria. The pure economic model is extended to the economic-statistical design by adding constraints associated with in-control and out-of-control average run lengths. The quality related production costs are calculated using Taguchi’s quadratic loss function. The optimal values of smoothing constants, sampling interval, sample size, and control chart limits are determined by using a numerical search method. The average run length of the control scheme is computed by using the Markov chain approach. Computational study indicates that optimal sample sizes decrease as the magnitudes of shifts in mean and/or variance increase, and higher values of quality loss coefficient lead to shorter sampling intervals. The sensitivity analysis results regarding the effects of various inputs on the chart parameters provide useful guidelines for designing an EWMA-based process control scheme when there exists an assignable cause generating concurrent changes in process mean and variance.     

在固定时间抽样的可变抽样区间的极值控制图

本文根据 Reynolds在固定时间抽样的可变抽样区间 (VSIFT)的 - x控制图 [1 ]的模型设计极值 (ζ)图。规定样本在相等间隔的固定时间点抽取 ,当过程有变化的迹象时 ,允许在两个固定时间之间抽取附加样本。若样本点超过控制限 ,则 VSIFT图同常规图一样发信号。本文计算了 VSIFTζ图的发信号前的平均时间 ,并同固定抽样区间 (FSI)的常规ζ图作比较。极值图不需计算 ,有关集中和分散的信息在一个图上给出 ,且可画上规格限 ,在实践中应用方便。本文设计的 VSIFTζ图能缩短过程失控时间从而减少不合格品数     

Maintenance models based on the <Emphasis Type="Italic">np</Emphasis> control charts with respect to the sampling interval

This paper develops models for the maintenance of a system based on np control charts with respect to the sampling interval. At any given time, the system is assumed to be in one of the three possible states; in-control, out-of-control and failure. If the control chart signals, suggesting the possibility of an out-of-control state, an investigation will be carried out. We assume that this investigation is perfect in that it reveals the true state of the system. If an assignable cause is confirmed by the investigation, a minor repair will be carried out to remove the cause. If the assignable cause is not attended to, it will gradually develop into a failure. When a failure occurs, the system cannot operate and a major repair is needed. We discuss three models depending on the assumptions related to the renewal mechanism, the occurrence of failures, and the time between minor repairs. The paper seeks to optimise the performance of such a system in terms of the sampling interval. Geometric processes are utilised for modelling the lifetimes between minor repairs if the minor repair cannot bring the system back to an as good as new condition. The expected cost per unit time for maintaining the systems with respect to the sampling interval of the control chart is obtained. Numerical examples are conducted to demonstrate the applicability of the methodology derived.     

可变抽样区间的非正态EWMA图的经济设计

本文主要讨论非正态总体下,可变抽样区间的EWMA图的经济设计问题。首先利用Burr分布近似各种非正态分布,建立可变抽样区间的非正态EWMA图的经济模型,使总期望费用最型小来确定参数的最优值;其次用遗传算法来寻找该经济模型的最优解,并给出工业中的一个例子;最后对可变抽样区间的非正态EWMA图的经济模型进行灵敏度分析。     

可变抽样区间的多变量自相关过程VAR控制图

基于批量-均值法的思想,向量自回归(VAR)控制图对多变量自相关过程的较小偏移可以进行有效控制。为了提高多变量自相关过程监控效率,本文研究可变抽样区间的VAR控制图。首先,对多变量自相关过程的VAR控制图进行可变抽样区间设计;然后,用蒙特卡洛模拟方法计算其平均报警时间;最后,以平均报警时间为评价准则,对所设计的可变抽样区间VAR控制图与固定抽样区间的VAR控制图进行比较研究。研究结果表明:所设计的可变抽样区间多变量自相关过程VAR控制图较固定抽样区间的多变量自相关过程VAR控制图能更好的监控过程的变化。