Increasing the Sensitivity of Control Chart for Fraction Nonconforming

This paper discusses an easy approach for increasing the sensitivity of the p chart by incorporating runs rules. A method for calculating the exact control limits for the various standardized p chart schemes incorporating runs rules is shown. Simple mathematical calculations reveal that the performances of all p chart schemes incorporating runs rules are superior to that of the standard approach. These calculations enable practitioners to determine the average run length (ARL) value that corresponds to a certain level of fraction nonconforming, p₁, based on a standard or known predefined proportion of nonconforming items, p₀. Examples are given to illustrate the application of the procedures discussed.     

The SPRT chart for monitoring a proportion

A control chart based on applying a sequential probability ratio test (SPRT) at each sampling point is considered for the problem of monitoring a process proportion p. This SPRT chart can be applied in situations in which items are inspected one by one, and the results of each inspection can be conveniently recorded before the next item is inspected. Some corrected diffusion theory approximations are given for the statistical properties of the SPRT and the SPRT chart. These approximations are very accurate and provide a simple method for designing an SPRT chart for practical applications. The sample size for the SPRT chart at a particular sampling time depends on the observations at that time, but the chart can be designed to have a specified average sampling rate when the process is in control. When there is a small shift in p, the average sampling rate per unit time will increase, but for a large shift in p the average sampling rate will decrease. For a given in-control average sampling rate and a given false alarm rate, the SPRT chart will detect changes in p much faster than the standard p-chart, which has traditionally been used for monitoring p. The SPRT chart will also detect changes in p much faster than the CUSUM chart for p. Thus, the SPRT chart can be used in place of traditional control charts to provide faster detection of changes in p or to reduce the sampling effort required to provide a given detection capability.     

CCC‐r charts' performance with estimated parameter for high‐quality process

CCC‐r charts are effective in detecting process shifts in the nonconforming rate especially for a high‐quality process. The implementation of the CCC‐r charts is usually under the assumption that the in‐control nonconforming rate is known. However, the nonconforming rate is never known, and accurate estimation is difficult. We investigate the effect of estimation error on the CCC‐r charts' performances through the expected value of the average number of observations to signal (EANOS) as well as the standard deviation of the average number of observations to signal (SDANOS). By comparing the in‐control performance of the CCC‐r charts, the CCC‐r chart with a larger value of r is more susceptible to the effects of parameter estimation. Meanwhile, the performance of the CCC‐r charts can converge when detecting upward shifts in p of out‐of‐control processes. We recommend the use of the CCC‐4 chart when considering its effectiveness in detecting shifts as well as its easier construction in practice. Furthermore, it is investigated that the CCC‐4 chart is less sensitive to parameter estimation while being more effective in detecting different process shifts when compared with Geometric CUSUM chart and synthetic chart.     

监控不合格品率的MEWMA-p控制图设计

传统Shewhart-p控制图只对单一属性的不合格品率进行监控,在过程发生偏移时有一定的滞后性。为提高不合格品率控制图的精度,提出一种多元指数加权移动平均不合格品率(multivariate exponentially weighted moving average p, MEWMA-p)控制图。该控制图将多个属性的不合格品率应用于多元指数加权移动平均控制图,可同时对多个属性进行监控,并且对于小范围的偏移更加敏感。对比分析同等偏移程度下指数加权移动平均不合格品率(exponentially weighted moving average p, EWMA-p)控制图与MEWMA-p控制图的平均运行长度(average run length,ARL)结果,并通过模拟仿真说明该方法的有效性。     

Properties of the Markov‐Dependent Attribute Control Chart with Estimated Parameters

The performance of attribute control charts that monitor Markov‐dependent data is usually evaluated under the assumption of known process parameters, that is, known values of a the probability an item is nonconforming given the previous item is conforming and b the probability an item is conforming given the previous item is nonconforming. In practice, these parameters are usually not known and are calculated from an in‐control Phase I‐data set. In this paper, a comparison of the in‐control ARL (average run length) properties of the attribute chart for Markov‐dependent data with known and estimated parameters is presented. The probability distribution of the estimators is developed and used to calculate the in‐control ARL and standard deviation of the run length of the chart with estimated parameters. For particular values of a and b, the in‐control ARL values of the charts with estimated parameters may be very different than those with known parameters. The size of the Phase‐I data set needed for charts with estimated parameters to exhibit the same in‐control ARL properties as those with known parameters may vary widely depending on the parameters of the process, but in general, large samples are needed to obtain accurate estimates. As the Phase‐I sample size increases, the in‐control ARL values of the charts with estimated parameters approach that of the known parameter case but not in a monotonic fashion as in the case of the X‐bar chart. Copyright © 2015 John Wiley & Sons, Ltd.     

Monitoring the process mean and variance using a weighted loss function CUSUM scheme with variable sampling intervals

Recent research has shown that adaptive control charts and the CUmulative SUM (CUSUM) schemes are quicker in detecting process shifts than traditional static Shewhart charts. This article proposes a weighted loss function CUSUM (WLC) scheme with Variable Sampling Intervals (VSI). It simultaneously monitors both mean shifts and an increasing variance shift by manipulating a single CUSUM chart. Most importantly, this VSI WLC scheme is much easier to operate and design than a VSI CCC scheme which comprises of three CUSUM charts (two of them monitoring the increasing and decreasing mean shifts and one monitoring the increasing variance shift). In terms of detection efficiency, the VSI WLC scheme is a much more powerful tool than the static X&S chart, the VSI X&S chart and the static WLC scheme. It is even more powerful than the VSI CCC scheme for many different combinations of mean and increasing variance shifts.     

Variable sampling interval exponentially weighted moving average median chart with estimated process parameters

The variable sampling interval exponentially weighted moving average median chart with estimated process parameters is proposed. The charting statistic, optimal design, performance evaluation, and implementation of the proposed chart are discussed. The average of the average time to signal (AATS) criterion is adopted to evaluate the performance of the proposed chart. The estimated process parameter‐based VSI EWMA median (VSI EWMA median‐e) chart is compared with the estimated process parameter‐based Shewhart median (SH median‐e), EWMA median (EWMA median‐e), and variable sampling interval run sum median (VSI RS median‐e) charts, in terms of the AATS criterion, where the VSI EWMA median‐e chart is shown to be superior. When process parameters are estimated, the standard deviation of the average time to signal (SDATS) criterion is used to evaluate the AATS performance of the VSI EWMA median‐e chart. Based on the SDATS criterion, the minimum number of phase‐I samples required by the VSI EWMA median‐e chart so that its performance is close to the known process parameters VSI EWMA median chart is recommended.     

Run sum X¯ control chart with estimated process parameters

The run sum $\overline{X}$ control chart is usually investigated under the assumption of known process parameters. In practice, process parameters are rarely known and they need to be estimated from an in‐control Phase I dataset. However, different practitioners use different numbers of Phase I samples to estimate the process parameters. As a result, the commonly used performance measure, ie, the average run length becomes a random variable. In this study, we present a run sum $\overline{X}$ control chart with estimated process parameters and use the standard deviation of the average run length to evaluate the average of the average run length performance of the run sum $\overline{X}$ chart when process parameters are estimated. Based on the standard deviation of the average run length criterion, the number of Phase I samples required by the estimated process parameter–based run sum $\overline{X}$ chart to have an average of the average run length performance close to that obtained under the assumption of known process parameters is recommended.     

An Efficient Nonparametric EWMA Wilcoxon Signed‐Rank Chart for Monitoring Location

The statistical performance of traditional control charts for monitoring the process shifts is doubtful if the underlying process will not follow a normal distribution. So, in this situation, the use of a nonparametric control charts is considered to be an efficient alternative. In this paper, a nonparametric exponentially weighted moving average (EWMA) control chart is developed based on Wilcoxon signed‐rank statistic using ranked set sampling. The average run length and some other associated characteristics were used as the performance evaluation of the proposed chart. A major advantage of the proposed nonparametric EWMA signed‐rank chart is the robustness of its in‐control run length distribution. Moreover, it has been observed that the proposed version of the EWMA signed‐rank chart using ranked set sampling shows better detection ability than some of the competing counterparts including EWMA sign chart, EWMA signed‐rank chart, and the usual EWMA control chart using simple random sampling scheme. An illustrative example is also provided for practical consideration. Copyright © 2016 John Wiley & Sons, Ltd.     

A New Exponentially Weighted Moving Average Control Chart for Monitoring Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely recognized as an advanced statistical process monitoring tool due to their excellent performance in detecting small to moderate shifts in process parameters. In this paper, we propose a new EWMA control chart for monitoring the process dispersion based on the best linear unbiased absolute estimator (BLUAE) obtained under paired ranked set sampling (PRSS) scheme, which we name EWMA‐PRSS chart. The performance of the EWMA‐PRSS chart is evaluated in terms of the average run length and standard deviation of run length, estimated using Monte Carlo simulations. These control charts are compared with their existing counterparts for detecting both increases and decreases in the process dispersion. It is observed that the proposed EWMA‐PRSS chart performs uniformly better than the EWMA dispersion charts based on simple random sampling and ranked set sampling (RSS) schemes. We also construct an EWMA chart based on imperfect PRSS (IPRSS) scheme, named EWMA‐IPRSS chart, for detecting overall changes in the process variability. It turns out that, with reasonable assumptions, the EWMA‐IPRSS chart outperforms the existing EWMA dispersion charts. A real data set is used to explain the construction and operation of the proposed EWMA‐PRSS chart. Copyright © 2014 John Wiley & Sons, Ltd.     

The Design of the ARL‐Unbiased S2 Chart When the In‐Control Variance Is Estimated

The S² chart has been known as a powerful tool to monitor the variability of the normal process. When the variance of the process is unknown, it needs to be estimated by Phase I samples. It is well known that there are serious effects of parameter estimation on the performance of the S² chart based on known parameter assumption. If the effects of parameter estimation are not considered, it can lead to an increase in the number of false alarms and a reduction in the ability of the chart to detect process changes except for very small shifts in the variance. Based on the criterion of average run length (ARL) unbiased, a S² control chart is developed when the in‐control variance is estimated. The performance of the proposed control chart is also evaluated in terms of the ARL and standard deviation of the run length. Finally, an example is used to illustrate the proposed control chart. Copyright © 2013 John Wiley & Sons, Ltd.     

Improved p charts to monitor process quality

It is a common practice to monitor the fraction p of non-conforming units to detect whether the quality of a process improves or deteriorates. Users commonly assume that the number of non-conforming units in a subgroup is approximately normal, since large subgroup sizes are considered. If p is small this approximation might fail even for large subgroup sizes. If in addition, both upper and lower limits are used, the performance of the chart in terms of fast detection may be poor. This means that the chart might not quickly detect the presence of special causes. In this paper the performance of several charts for monitoring increases and decreases in p is analyzed based on their Run Length (RL) distribution. It is shown that replacing the lower control limit by a simple runs rule can result in an increase in the overall chart performance. The concept of RL unbiased performance is introduced. It is found that many commonly used p charts and other charts proposed in the literature have RL biased performance. For this reason new control limits that yield an exact (or nearly) RL unbiased chart are proposed.     

Design of exponential control charts based on average time to signal using a sequential sampling scheme

Exponential charts based on time-between-events (TBE) data are widely investigated and applied in various fields. The average time to signal (ATS) is used instead of the average run length to evaluate the performance of TBE charts, since the ATS involves both the number and the time of samples inspected until a signal occurs. An ATS-unbiased exponential control chart is proposed when the in-control parameter is known. Considering the need in practice to start monitoring a production process as soon as possible, a sequential sampling scheme is adopted and the in-control parameter is estimated by an unbiased and consistent estimator. Some specific guidelines to stop updating control limits are obtained from the relationship between the phase I sample size and the actual false alarm rate. Finally, two real examples are given to illustrate the implementation and efficiency of the proposed method.     

The Negative Binomial Exponentially Weighted Moving Average Chart with Estimated Control Limits

The control chart based on the compound Poisson distribution (the negative binomial exponentially weighted moving average (EWMA) chart) has been shown to be more effective than the c‐chart to monitor the wafer nonconformities in semiconductor production process. The performance of the negative binomial EWMA chart is generally evaluated with the assumption that the process parameters are known. However, in many control chart applications, the process parameters are usually unknown and are required to be estimated. For an accurate parameter estimate, a very large sample size may be required, which is seldom available in the applications. This article investigates the effect of parameter estimation on the run length properties of the negative binomial EWMA charts. Using a Markov chain approach, we show that the performance of the negative binomial EWMA chart is affected when parameters are estimated compared with the known‐parameter case. We also provide recommendations regarding phase I sample sizes, smoothing constant and clustering parameter. The sample size must be quite large for the in‐control chart performance to be close to that for the known‐parameter case. Finally, a wafer process example has been used to highlight the practical implications of estimation error and to offer advice to practitioners when constructing/analysing a phase I sample. Copyright © 2013 John Wiley & Sons, Ltd.     

A control chart for monitoring the lognormal process variation using repetitive sampling

Control charts are developed to make the specific quality measures for a successful production process and follow normal distribution behaviors. But some real-life practices do not match such practices and exhibit some positively skewed behavior like lognormal distribution. The present study has considered this situation and proposed a monitoring control chart based on lognormal process variation using a repetitive sampling scheme. This concept proved better for detecting shifts as quickly as possible, and compared with the existing concept, results are elaborated through extensive tables. The average run lengths and standard deviations of the run lengths are being used as a performance evaluation measures and computed by using Monte Carlo simulations performed in R language. A real-life situation has been discussed in the example section to strengthen the proposed control chart concept in a real-life situation.     

A CCC‐r chart for high‐yield processes

The cumulative count of a conforming (CCC) chart is used to monitor high‐quality processes and is based on the number of items inspected until observing r non‐conforming ones. This charting technique is known as a CCC‐r chart. The function of the CCC‐r chart is the sensitive detection of an upward shift in the fraction defectives of the process, p. As r gets larger, the CCC‐r chart becomes more sensitive to small changes of upward shift in p. However, since many observations are required to obtain a plotting point on the chart, the cost is fairly high. For this trade‐off problem it is necessary to determine the optimal number of non‐conforming items observed before a point is plotted, the sampling (inspection) interval, and the lower control limit for the chart. In this paper a simplified optimal design method is proposed. For illustrative purposes, some numerical results for the optimal design parameter values are provided. The expected profits per cycle obtained using the proposed optimal design method are compared with those obtained using other misspecified parameter values. The effects of changing these parameters on the profit function are shown graphically. Copyright © 2001 John Wiley & Sons, Ltd.     

A Comparison of Inspector Performance Measures

Using experimental data from their signal-detection study, the authors examine traditional measures of inspector performance along with measures based on Bayes'rule. Results indicate that the inspector's decision process is affected by the a priori probability of nonconforming product in the inspection lot and that inspector decision error may result in a tighter OC curve for a sampling plan. The conclusions are that performance measures should be selected relative to performance evaluation objectives and that graphs describing acceptance sampling plans for ideal inspectors should be modified to reflect performance of real inspectors under actual conditions.     

On designing an efficient control chart to monitor fraction nonconforming

Process control measures are mostly applied in production and manufacturing industries. The most important tool used in these disciplines is control chart. In manufacturing and production processes, when the quality characteristic of interest cannot be directly measured, it becomes essential to apply attribute control charts. To monitor fraction nonconforming of the output, quality practitioners mostly prefer p-chart. In this article, a new progressive mean (PM) control chart is being proposed for monitoring drift in proportion of nonconforming products. The design evaluations of the proposed chart are made and compared through different properties of run length distribution, such as average run length (ARL), standard deviation of run length (SDRL), and some percentile points. The performance of the proposed chart is assessed under zero-state and steady-state scenarios. The proposed PM chart is compared with p-chart, moving average (MA) chart, optimal CUSUM chart, modified exponentially weighted moving average (EWMA) chart, and runs rules p-charts for monitoring fraction nonconforming. The proposed chart spots efficiently sustained disturbances in the process as compared with their existing counterparts. Two illustrative examples are also provided; one from real-life application of nonconforming bearing and seal assemblies data and the other from simulated data for the implementation of PM chart.     

Optimal design of one-sided exponential cumulative sum charts with known and estimated parameters based on the median run length

As a useful tool in statistical process control (SPC), the exponential control chart is more and more popular for monitoring high-quality processes. Considering both known and estimated parameter cases, the one-sided exponential cumulative sum (CUSUM) charts are studied in this paper through a Markov chain approach. Because the shape of the run length (RL ) distribution of the one-sided exponential CUSUM charts is skewed and it also changes with the mean shift size and the number of Phase I samples used to estimate the process parameter, the median run length (MRL ) is employed as a good alternative performance measure for the charts. The optimal design procedures based on MRL of the one-sided exponential CUSUM charts with known and estimated parameters are discussed. By comparing the MRL performance of the chart with known parameters with the one of the chart with estimated parameters, we investigate the effect of estimated process parameters on the properties of the chart. Finally, an application is illustrated to show the implementation of the chart.