An optimal design of CUSUM control charts for binomial counts

The Shewhart p-chart or np-chart is commonly used for monitoring the counts of non-conforming items which are usually well modelled by a binomial distribution with parameters n and p where n is the number of items inspected each time and p is the process fraction of non-conforming items produced. It is well known that the Shewhart chart is not sensitive to small shifts in p. The cumulative sum (CUSUM) chart is a far more powerful charting procedure for detecting small shifts in p and only marginally less powerful in detecting large shifts in p. The choice of chart parameters of a Shewhart chart is well documented in the quality control literature. On the other hand, very little has been done for the more powerful CUSUM chart, possibly due to the fact that the run length distribution of a CUSUM chart is much harder to compute. An optimal design strategy is given here which allows the chart parameters of an optimal CUSUM chart to be determined easily. Optimal choice of n and the relationship between the CUSUM chart and the sequential probability ratio test are also investigated.     

The random intrinsic fast initial response of one-sided CUSUM charts

This article analyses the performance of a one-sided cumulative sum (CUSUM) chart that is initialized using a random starting point following the natural or intrinsic probability distribution of the CUSUM statistic. By definition, this probability distribution remains stable as the chart is used. The probability that the chart starts at zero according to this intrinsic distribution is always smaller than one, which confers on the chart a fast initial response feature. The article provides a fast and accurate algorithm to compute the in-control and out-of-control average run lengths and run-length probability distributions for one-sided CUSUM charts initialized using this random intrinsic fast initial response (RIFIR) scheme. The algorithm also computes the intrinsic distribution of the CUSUM statistic and random samples extracted from this distribution. Most importantly, no matter how the chart was initialized, if no level shifts and no alarms have occurred before time τ?>?0, the distribution of the run length remaining after τ is provided by this algorithm very accurately, provided that τ is not too small.     

The Continuous Run Sum Chart

The run sum chart is an effective two-sided chart that can be used to monitor for process changes. It is known that it is more sensible than the Shewhart chart with runs rules and its performance improves as the number of regions increases. However, as the number of regions increses the resulting chart has more parameters to be defined and its design becomes more involved. In this article, we introduce a one-parameter run sum chart. This chart accumulates scores equal to the subgroup means and signals when the cummulative sum exceeds a limit value. A fast initial response feature is proposed and its run length distribution function is found by a set of recursive relations. We compare this chart with other charts suggested in the literature and find it competitive with the CUSUM, the FIR CUSUM, and the combined Shewhart FIR CUSUM schemes.     

A Combination of CUSUM Charts for Monitoring a Zero-Inflated Poisson Process

The Zero-inflated Poisson distribution (ZIP) is used to model the defects in processes with a large number of zeros. We propose a control charting procedure using a combination of two cumulative sum (CUSUM) charts to detect increases in the parameters of ZIP process, one is a conforming run length (CRL) CUSUM chart and another is a zero truncated Poisson (ZTP) CUSUM chart. The control limits of the control charts are obtained using both Markov chain-based methods and simulations. Simulation experiments show that the proposed method outperforms an existing method. Finally, a real example is presented.     

Distribution-Free Exceedance CUSUM Control Charts for Location

Distribution-free (nonparametric) control charts can be useful to the quality practitioner when the underlying distribution is not known. A Phase II nonparametric cumulative sum (CUSUM) chart based on the exceedance statistics, called the exceedance CUSUM chart, is proposed here for detecting a shift in the unknown location parameter of a continuous distribution. The exceedance statistics can be more efficient than rank-based methods when the underlying distribution is heavy-tailed and/or right-skewed, which may be the case in some applications, particularly with certain lifetime data. Moreover, exceedance statistics can save testing time and resources as they can be applied as soon as a certain order statistic of the reference sample is available. Guidelines and recommendations are provided for the chart's design parameters along with an illustrative example. The in- and out-of-control performances of the chart are studied through extensive simulations on the basis of the average run-length (ARL), the standard deviation of run-length (SDRL), the median run-length (MDRL), and some percentiles of run-length. Further, a comparison with a number of existing control charts, including the parametric CUSUM chart and a recent nonparametric CUSUM chart based on the Wilcoxon rank-sum statistic, called the rank-sum CUSUM chart, is made. It is seen that the exceedance CUSUM chart performs well in many cases and thus can be a useful alternative chart in practice. A summary and some concluding remarks are given.     

A HEWMA-CUSUM control chart for the Weibull distribution

The Weibull distribution is one of the most popular distributions for lifetime modeling. However, there has not been much research on control charts for a Weibull distribution. Shewhart control is known to be inefficient to detect a small shift in the process, while exponentially weighted moving average (EWMA) and cumulative sum control chart (CUSUM) charts have the ability to detect small changes in the process. To enhance the performance of a control chart for a Weibull distribution, we introduce a new control chart based on hybrid EWMA and CUSUM statistic, called the HEWMA-CUSUM chart. The performance of the proposed chart is compared with the existing chart in terms of the average run length (ARL). The proposed chart is found to be more sensitive than the existing chart in ARL. A simulation study is provided for illustration purposes. A real data is also applied to the proposed chart for practical use.     

Conditional design of the CUSUM median chart for the process position when process parameters are unknown

In practice, different practitioners will use different Phase I samples to estimate the process parameters, which will lead to different Phase II control chart's performance. Researches refer to this variability as between-practitioners-variability of control charts. Since between-practitioners-variability is important in the design of the CUSUM median chart with estimated process parameters, the standard deviation of average run length (SDARL) will be used to study its properties. It is shown that the CUSUM median chart requires a larger amount of Phase I samples to sufficiently reduce the variation in the in-control ARL of the CUSUM median chart. Considering the limitation of the amount of the Phase I samples, a bootstrap approach is also used here to adjust the control limits of the CUSUM median chart. Comparisons are made for the CUSUM and Shewhart median charts with estimated parameters when using the adjusted- and unadjusted control limits and some conclusions are made.     

Comparison of different control charts for a Weibull process with type-I censoring

Some control charts have been proposed to monitor the mean of a Weibull process with type-I censoring. One type of control charts is to monitor changes in the scale parameter because it indicates changes in the mean. With this approach, we compare different control charts such as Shewhart-type and exponentially weighted moving average (EWMA) charts based on conditional expected value (CEV) and cumulative sum (CUSUM) chart based on likelihood-ratio. A simulation approach is employed to compute control limits and average run lengths. The results show that the CUSUM chart has the best performance. However, the EWMA-CEV chart is recommendable for practitioners with its competitive performance and ease of use advantage. An illustrative example is also provided.     

Economic Design of Cumulative Sum Control Charts for Monitoring a Process with Correlated Samples

The underlying assumption for the design of control charts is the measurements within a sample are independently distributed. However, there are many situations where the uncorrelation assumption may be unacceptable in practice. In this paper, the economic design of cumulative sum (CUSUM) control chart for correlated data within a sample is developed. The genetic algorithm is applied to find the optimal design parameters of the CUSUM control chart by minimizing the cost function. An illustrative example is given. A sensitivity analysis is then conducted to evaluate the effects of cost parameters, process parameters, and correlation coefficient on the economic design.     

A mixed GWMA–CUSUM control chart for monitoring the process mean

The memory-type control charts are widely used in the process and service industries for monitoring the production processes. The reason is their sensitivity to quickly react against the small process disturbances. Recently, a new cumulative sum (CUSUM) chart has been proposed that uses the exponentially weighted moving average (EWMA) statistic, called the EWMA–CUSUM chart. Similarly, in order to further enhance the sensitivity of the EWMA–CUSUM chart, we propose a new CUSUM chart using the generally weighted moving average (GWMA) statistic, called the GWMA–CUSUM chart, for efficiently monitoring the process mean. The GWMA–CUSUM chart encompasses the existing CUSUM and EWMA–CUSUM charts. Extensive Monte Carlo simulations are used to explore the run length profiles of the GWMA–CUSUM chart. Based on comprehensive run length comparisons, it turns out that the GWMA–CUSUM chart performs substantially better than the CUSUM, EWMA, GWMA, and EWMA–CUSUM charts when detecting small shifts in the process mean. An illustrative example is also presented to explain the implementation and working of the EWMA–CUSUM and GWMA–CUSUM charts.     

A CUSUM control chart for monitoring the variance when parameters are estimated

CUSUM control chart has been widely used for monitoring the process variance. It is usually used assuming that the nominal process variance is known. However, several researchers have shown that the ability of control charts to signal when a process is out of control is seriously affected unless process parameters are estimated from a large in-control Phase I data set. In this paper we derive the run length properties of a CUSUM chart for monitoring dispersion with estimated process variance and we evaluate the performance of this chart by comparing it with the same chart but with assumed known process parameters.     

On increasing the sensitivity of mixed EWMA–CUSUM control charts for location parameter

Control chart is an important statistical technique that is used to monitor the quality of a process. Shewhart control charts are used to detect larger disturbances in the process parameters, whereas cumulative sum (CUSUM) and exponential weighted moving average (EWMA) are meant for smaller and moderate changes. In this study, we enhanced mixed EWMA–CUSUM control charts with varying fast initial response (FIR) features and also with a runs rule of two out of three successive points that fall above the upper control limit. We investigate their run-length properties. The proposed control charting schemes are compared with the existing counterparts including classical CUSUM, classical EWMA, FIR CUSUM, FIR EWMA, mixed EWMA–CUSUM, 2/3 modified EWMA, and 2/3 CUSUM control charting schemes. A case study is presented for practical considerations using a real data set.     

CUSUM control schemes for Gaussian processes

A CUSUM control scheme for detecting a change point in a Gaussian process is derived. An upper and a lower bound for the distribution of the run length and for its moments is given. In a Monte Carlo study the average run length (ARL) of this chart is compared with the ARL of two other CUSUM procedures which are based on approximations to the sequential probability ratio, and, moreover, with EWMA schemes for autocorrelated data. Results on the optimal choice of the reference value are presented. Furthermore it is investigated how these charts behave if the model parameters are estimated.     

CUSUM Chart with Transformed Exponential Data

Time Between Events (TBE) charts were proposed to monitor the time between events occur based on exponential distribution, and have been shown to be more effective than monitoring the fraction non conforming directly. In this article, we consider monitoring the TBE data with CUSUM scheme by transformation. The idea behind it is to transform the TBE data to normal, and then apply the CUSUM scheme for the approximate normal data. Several simple transformation methods are examined. The calculation of Average Run Length (ARL) with Markov chain approach is described. Comparative studies on the ARL performance show that the transformed CUSUM is superior to the X-MR (Moving Range) chart with transformation, the Cumulative Quantity Control (CQC) chart, and have comparable performance with exponential CUSUM charts. The design procedures of optimal CUSUM chart are also presented. This study provides another possible alternative for monitoring TBE data with easy design procedures and relatively good performance.     

Cumulative sum schemes for surgical performance monitoring

Summary.  The standard cumulative sum (CUSUM), risk-adjusted CUSUM and Shiryayev–Roberts schemes for monitoring surgical performance are compared. We find that both CUSUM schemes are comparable in run length performance except when there is a high heterogeneity of surgical risks, in which case the risk-adjusted CUSUM scheme is more sensitive in detecting a shift in surgical performance. The Shiryayev–Roberts scheme is found to be less sensitive compared with the CUSUM schemes in detecting a deterioration in surgical performance. Using the Markov chain method, the exact average run length of a standard CUSUM scheme can be computed whereas the average run length of a risk-adjusted CUSUM scheme is approximated. For a risk-adjusted CUSUM scheme, the accuracy of the average run length depends on the fineness of the discretization of CUSUM values, which relies on the chart limit, shift to be detected optimally and in-control surgical risk distribution. A sensitivity analysis shows that the risk-adjusted CUSUM and Shiryayev–Roberts schemes still perform moderately well in detecting a deterioration and an improvement in surgical performances respectively even though there is a misspecification of the in-control surgical risk distribution. In general, the run length performance of the Shiryayev–Roberts scheme is comparatively less sensitive to a misspecification of the in-control surgical risk distribution.     

Measurement error effect on the CUSUM control chart

The performance of the cumulative sum (CUSUM) control chart for the mean when measurement error exists is investigated. It is shown that the CUSUM chart is greatly affected by the measurement error. A similar result holds for the case of the CUSUM chart for the mean with linearly increasing variance. In this paper, we consider multiple measurements to reduce the effect of measurement error on the charts performance. Finally, a comparison of the CUSUM and EWMA charts is presented and certain recommendations are given.     

Robust cusum: a robustness study for cusum quality control schemes

A standard CUSUM control scheme and four modified CUSUM control schemes are evaluated for robustness. The average run length (ARL) for each scheme is evaluated using a contaminated normal distribution, a distribution that has longer tails than the normal. A CUSUM control scheme that ignores the first suspected outlier, but gives an out-of-control signal for two successive outliers is found to perform well.     

Data-analytic aspects of the Shiryayev-Roberts control chart: surveillance of a non-homogeneous Poisson process

The Shiryayev-Roberts control chart has been proposed as a powerful competitor of the Shewhart control chart and the CUSUM procedure, on theoretical grounds. We demonstrate here the application of a Shiryayev-Roberts control chart to a non-homogeneous Poisson process. We show that, from a data-analytic point of view, the Shiryayev-Roberts surveillance scheme has several advantages over classical CUSUM charts. A case study of power failure times in a computer centre is used to illustrate our main points.     

Evaluation of CUSUM Charts for Finite-Horizon Processes

This article analyses and evaluates the properties of a CUSUM chart designed for monitoring the process mean in short production runs. Several statistical measures of performance that are appropriate when the process operates for a finite-time horizon are proposed. The methodology developed in this article can be used to evaluate the performance of the CUSUM scheme for any given set of chart parameters from both an economic and a statistical point of view, and thus, allows comparisons with various other charts.