Moving range EWMA control charts for monitoring the Weibull shape parameter

In this article, we propose an exponentially weighted moving average (EWMA) control chart for the shape parameter β of Weibull processes. The chart is based on a moving range when a single measurement is taken per sampling period. We consider both one-sided (lower-sided and upper-sided) and two-sided control charts. We perform simulations to estimate control limits that achieve a specified average run length (ARL) when the process is in control. The control limits we derive are ARL unbiased in that they result in ARL that is shorter than the stable-process ARL when β has shifted. We also perform simulations to determine Phase I sample size requirements if control limits are based on an estimate of β. We compare the ARL performance of the proposed chart to that of the moving range chart proposed in the literature.     

CUSUM Control Charts for Multivariate Poisson Distribution

A cumulative sum control chart for multivariate Poisson distribution (MP-CUSUM) is proposed. The MP-CUSUM chart is constructed based on log-likelihood ratios with in-control parameters, Θ₀, and shifts to be detected quickly, Θ₁. The average run length (ARL) values are obtained using a Markov Chain-based method. Numerical experiments show that the MP-CUSUM chart is effective in detecting parameter shifts in terms of ARL. The MP-CUSUM chart with smaller Θ₁ is more sensitive than that with greater Θ₁ to smaller shifts, but more insensitive to greater shifts. A comparison shows that the proposed MP-CUSUM chart outperforms an existing MP chart.     

CUSUM chart for detecting range shifts when monotonicity of likelihood ratio is invalid

It is often encountered in the literature that the log-likelihood ratios (LLR) of some distributions (e.g. the student t distribution) are not monotonic. Existing charts for monitoring such processes may suffer from the fact that the average run length (ARL) curve is a discontinuous function of control limit. It implies that some pre-specified in-control (IC) ARLs of these charts may not be reached. To guarantee the false alarm rate of a control chart lower than the nominal level, a larger IC ARL is usually suggested in the literature. However, the large IC ARL may weaken the performance of a control chart when the process is out-of-control (OC), compared with a just right IC ARL. To overcome it, we adjust the LLR to be a monotonic one in this paper. Based on it, a multiple CUSUM chart is developed to detect range shifts in IC distribution. Theoretical result in this paper ensures the continuity of its ARL curve. Numerical results show our proposed chart performs well under the range shifts, especially under the large shifts. In the end, a real data example is utilized to illustrate our proposed chart.     

A robust R control chart based on a two-step estimator of the process dispersion

ABSTRACT

Control charts are the frequently used tools for monitoring and controlling the processes. Classical control charts are sensitive to existing contaminated data which may be presented in the data collected from the processes. Thus, these charts are not able to control the processes precisely when the data are contaminated. Robust control charts are those which are less sensitive to contamination. Some robust control charts for monitoring the process variability were proposed in the past which are robust to some sorts of contamination. In this paper a new robust R control chart is proposed which is less sensitive to wide range of contaminations, i.e. general and local contaminations. Simulation studies are performed to compare the performance of the proposed control chart with some classical and robust control charts, using ARL and MSD as criteria for comparisons purposes. The simulation results show a very good performance of the proposed chart when both types of contaminations exist.     

Globally applicable control chart for online monitoring of stability of process mean

The shape features of run chart patterns of the most recent m observations arising from stable and unstable processes are different. Using this fact, a new monitoring statistic is defined whose value for given m depends on the pattern parameters but not on the process parameters. A control chart for this statistic for given m, therefore, will be globally applicable to normal processes. The simulation study reveals that the proposed statistic approximately follows normal distribution. The performances of the globally applicable control chart in terms of average run lengths (ARLs) are evaluated and compared with the X chart. Both in-control ARL and out-of-control ARLs with respect to different abnormal process conditions are found to be larger than the X chart. However, the proposed concept is promising because it can eliminate the burden of designing separate control charts for different quality characteristics or processes in a manufacturing set-up.     

Control chart for monitoring the Weibull shape parameter under two competing risks

ABSTRACT

In this paper, we propose a control chart to monitor the Weibull shape parameter where the observations are censored due to competing risks. We assume that the failure occurs due to two competing risks that are independent and follow Weibull distribution with different shape and scale parameters. The control charts are proposed to monitor one or both of the shape parameters of competing risk distributions and established based on the conditional expected values. The proposed control chart for both shape parameters is used in certain situations and allows to monitor both shape parameters in only one chart. The control limits depend on the sample size, number of failures due to each risk and the desired stable average run length (ARL). We also consider the estimation problem of the target parameters when the Phase I sample is incomplete. We assumed that some of the products that fail during the life testing have a cause of failure that is only known to belong to a certain subset of all possible failures. This case is known as masking. In the presence of masking, the expectation-maximization (EM) algorithm is proposed to estimate the parameters. For both cases, with and without masking, the behaviour of ARLs of charts is studied through the numerical methods. The influence of masking on the performance of proposed charts is also studied through a simulation study. An example illustrates the applicability of the proposed charts.     

A Multivariate Adaptive Exponentially Weighted Moving Average Control Chart

A multivariate extension of the adaptive exponentially weighted moving average (AEWMA) control chart is proposed. The new multivariate scheme can detect small and large shifts in the process mean vector effectively. The proposed scheme can be viewed as a smooth combination of a multivariate exponentially weighted moving average (MEWMA) chart and a Shewhart χ²-chart. The optimal design of the proposed chart is given according to a pre-specified in-control average run length and two shift sizes; a small and large shift each measured in terms of the non centrality parameter. The signal resistance of the newly proposed multivariate chart is also given. Comparisons among the new chart, the MEWMA chart, and the combined Shewhart-MEWMA (S-MEWMA) chart in terms of the standard and worst-case average run length profiles are presented. In addition, the three charts are compared with respect to their worst-case signal resistance values. The proposed chart gives somewhat better worst-case ARL and signal resistance values than the competing charts. It also gives better standard ARL performance especially for moderate and large shifts. The effectiveness of our proposed chart is illustrated through an example with simulated data set.     

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.     

A New Chart for Monitoring the Covariance Matrix of Bivariate Processes

In this article we consider a control chart based on the sample variances of two quality characteristics. The points plotted on the chart correspond to the maximum value of these two statistics. The main reason to consider the proposed chart instead of the generalized variance | S | chart is its better diagnostic feature, that is, with the new chart it is easier to relate an out-of-control signal to the variables whose parameters have moved away from their in-control values. We study the control chart efficiency considering different shifts in the covariance matrix. In this way, we obtain the average run length (ARL) that measures the effectiveness of a control chart in detecting process shifts. The proposed chart always detects process disturbances faster than the generalized variance | S | chart. The same is observed when the size of the samples is variable, except in a few cases in which the size of the samples switches between small size and very large size.     

Nonparametric control charts based on order statistics: Some advances

ABSTRACT

In this article, we introduce new nonparametric Shewhart-type control charts that take into account the location of two order statistics of the test sample as well as the number of observations in that sample that lie between the control limits. Exact formulae for the alarm rate, the run length distribution and the average run length (ARL) are all derived. A key advantage of the new charts is that, due to its nonparametric nature, the false alarm rate (FAR) and in-control run length distribution is the same for all continuous process distributions. Tables are provided for the implementation of the proposed charts for some typical FAR and ARL values. Furthermore, a numerical study carried out reveals that the new charts are quite flexible and efficient in detecting shifts to Lehmann-type out-of-control situations, while they seem preferable from a robustness point of view in comparison with the distribution-free control chart of Balakrishnan et al. (2009).     

Synthetic exponential control charts with unknown parameter

The existing synthetic exponential control charts are based on the assumption of known in-control parameter. However, the in-control parameter has to be estimated from a Phase I dataset. In this article, we use the exact probability distribution, especially the percentiles, mean, and standard deviation of the conditional average run length (ARL) to evaluate the effect of parameter estimation on the performance of the Phase II synthetic exponential charts. This approach accounts for the variability in the conditional ARL values of the synthetic chart obtained by different practitioners. Since parameter estimation results in more false alarms than expected, we develop an exact method to design the adjusted synthetic charts with desired conditional in-control performance. Results of known and unknown in-control parameter cases show that the control limit of the conforming run length sub-chart of the synthetic chart should be as small as possible.     

ARL Performance of the Tukey's Control Chart

A Tukey's control chart was designed to monitor single observation data. Its easy control-limits setting and simple statistical concept are the most important advantages. In this study, we setup the Tukey's control chart based on known probability distribution and construct its average run length calculator. ARL performance of the Tukey's control chart is evaluated under a normal distribution and non normal distributions, respectively. The comparison of ARL performance reveals that Tukey's control chart is not sensitive to shift detection when the process heavily violates the normal assumption. The number of observations needed for Tukey's control chart setup is less than Shewhart's control chart. Tukey's control chart is a better choice for the process mean monitoring.     

A Study on EWMA charts with runs rules—the Markov chain approach

ABSTRACT

Runs rules are usually used with Shewhart-type charts to enhance the charts' sensitivities toward small and moderate shifts. Abbas et al. in 2011 took it a step further by proposing two runs rules schemes, applied to the exponentially weighted moving average (EWMA) chart and evaluated their average run length (ARL) performances using simulation. They showed that the proposed schemes are superior to the classical EWMA chart and other schemes being investigated. Besides pointing out some erroneous ARL and standard deviation of the run length (SDRL) computations in Abbas et al., this paper presents a Markov chain approach for computing the ARL, percentiles of the run length (RL) distribution and SDRL, for the two runs rules schemes of Abbas et al. Using Markov chain, we also propose two combined runs rules EWMA schemes to quicken the two schemes of Abbas et al. in responding to large shifts. The runs rules (basic and combined rules) EWMA schemes will be compared with some existing control charting methods, where the former charts are shown to prevail.     

ARL-Design of S Charts with k-of-k Runs Rules

The standard S chart signals an out-of-control condition when one point exceeds a control limit. It can be augmented with runs rules to improve its performance in detecting assignable causes. A commonly used rule signals when k consecutive points exceed a control limit. This rule can be used alone or to supplement the standard chart. In this article we derive ARL expressions for charts with the k-of-k runs rule. We show how to design S charts with this runs rule, compare their ARL performance, and make a control chart recommendation when it is important to monitor for both increases and decreases in process dispersion.     

Optimal design of the adaptive exponentially weighted moving average control chart over a range of mean shifts

The adaptive exponentially weighted moving average (AEWMA) control chart is a smooth combination of the Shewhart and exponentially weighted moving average (EWMA) control charts. This chart was proposed by Cappizzi and Masarotto (2003) to achieve a reasonable performance for both small and large shifts. Cappizzi and Masarotto (2003) used a pair of shifts in designing their control chart. In this study, however, the process mean shift is considered as a random variable with a certain probability distribution and the AEWMA control chart is optimized for a wide range of mean shifts according to that probability distribution and not just for a pair of shifts. Using the Markov chain technique, the results show that the new optimization design can improve the performance of the AEWMA control chart from an overall point of view relative to the various designs presented by Cappizzi and Masarotto (2003). Optimal design parameters that achieve the desired in-control average run length (ARL) are computed in several cases and formulas used to find approximately their values are given. Using these formulas, the practitioner can compute the optimal design parameters corresponding to any desired in-control ARL without the need to apply the optimization procedure. The results obtained by these formulas are very promising and would particularly facilitate the design of the AEWMA control chart for any in-control ARL value.     

Designing a control chart of extended EWMA statistic based on multiple dependent state sampling

In this paper, we presented a memory type control chart (CC) based on multiple dependent state (MDS) sampling to pinpoint the slight variation in the process mean for the quality trait of normal distribution (ND). Two pairs of control limits denominated as internal and external control limits are derived using under control mean and variance. The essential steps are taken to get the value of average run length (ARL) for stable and disturb process. Various tables of ARLs are erected using different smoothing constants, shifts and MDS parameter. Comparisons are established to assess the effectiveness of initiated CC with the various existing CC in term of ARL. It has been ascertained that offered CC manifest the best performance in searching out the diminutive changes in the process mean. Two examples, one is based on simulation study and other is related to real-life data, have been discussed for its practical purpose.KEYWORDS: Multiple dependent state, normal distribution, smoothing constants, control chart     

Applying fast initial response features on GWMA control charts for monitoring autocorrelation data

ABSTRACT

A generally weighted moving average (GWMA) control chart with fast initial response (FIR) features is addressed to monitor an autoregressive process mean shift. Numerical simulations based on average run length (ARL) show that the GWMA control chart with additional FIR feature requires less time to detect small or moderate shifts than GWMA control chart at low level of autocorrelation; whereas these two control charts perform similarly at high level of autocorrelation. Regardless of any level of autocorrelation, GWMA control charts provided with additional FIR feature have a good performance in detecting large shifts during the initial stage.     

A weighted likelihood ratio test-based chart for monitoring process mean and variability

In this paper, a new single exponentially weighted moving average (EWMA) control chart based on the weighted likelihood ratio test, referred to as the WLRT chart, is proposed for the problem of monitoring the mean and variance of a normally distributed process variable. It is easy to design, fast to compute, and quite effective for diverse cases including the detection of the decrease in variability and individual observation case. The optimal parameters that can be used as a design aid in selecting specific parameter values based on the average run length (ARL) and the sample size are provided. The in-control (IC) and out-of-control (OC) performance properties of the new chart are compared with some other existing EWMA-type charts. Our simulation results show that the IC run length distribution of the proposed chart is similar to that of a geometric distribution, and it provides quite a robust and satisfactory overall performance for detecting a wide range of shifts in the process mean and/or variability.     

Design of an attribute np control chart for process monitoring based on repetitive group sampling under truncated life tests

In this paper, an attribute control chart under repetitive group sampling is designed for monitoring the production process where the lifetime of the product is considered as quality of the product. We assume that the lifetime follows the Pareto distribution of second kind with known shape parameter. The performance of the proposed chart is evaluated by average run length. The control limits coefficients as well as the repetitive group sampling parameter such as sample size are determined such that the in-control average run length is as close as to the specified average run length. Out-of-control average run length is also reported for different shift constants with corresponding optimal parameters. In addition, performance of proposed control chart is compared with the performance of existing chart. An economical designing of proposed control chart is also discussed.