A combined synthetic and np scheme for detecting increases in fraction nonconforming

The applications of attribute control charts cover a wide variety of manufacturing processes in which quality characteristics cannot be measured on a continuous numerical scale or even a quantitative scale. The np control chart is an attribute chart used to monitor the fraction nonconforming p of a process. This chart is effective for detecting large process shifts in p. The attribute synthetic chart is also proposed to detect p shifts. It utilizes the information about the time interval or the Conforming Run Length (CRL) between two nonconforming samples. During the implementation of a synthetic chart, a sample is classified as nonconforming if the number d of nonconforming units falls beyond a warning limit. Unlike the np chart, the synthetic chart is more powerful to detect small and moderate p shifts. This article proposes a new scheme, the Syn-np chart, which comprises a synthetic chart and an np chart. Since the Syn-np chart has both the strength of the synthetic chart for quickly detecting small p shifts and the advantage of the np chart of being sensitive to large p shifts, it has a better and more uniform overall performance. Specifically, it is more effective than the np chart and synthetic chart by 73% and 31%, respectively, in terms of Weighted Average of Average Time to Signal (WAATS) over a wide range of p shifts under different conditions.     

Extended maximum generally weighted moving average control chart for monitoring process mean and variability

In this paper, we propose an extended control chart, called the maximum generally weighted moving average (MaxGWMA) control chart, to simultaneously detect both increases and decreases in the mean and/or variability of a process. Simulations are performed to evaluate the average run length, standard deviation of the run length, and diagnostic abilities of the MaxGWMA and maximum exponentially weighted moving average (MaxEWMA) charts. An extensive comparison reveals that the MaxGWMA control chart is more sensitive than the MaxEWMA control chart.     

A sum of squares double exponentially weighted moving average chart

The sum of squares double exponentially weighted moving average (SS-DEWMA) chart is proposed to improve the performance of the single SS-EWMA chart, in the detection of initial out-of-control signals. The SS-DEWMA chart uses the sum of squares statistic and it simultaneously monitors the process mean and variance in a single chart. A simulation study is conducted to show that the optimal SS-DEWMA chart provides better zero state average run length (ARL) and standard deviation of the run length (SDRL) performances than the optimal SS-EWMA chart. In addition, as suggested by one of the reviewers, the cyclical steady state ARLs and SDRLs of the SS-DEWMA and SS-EWMA charts are compared, where it is found that the former did not perform as well as the latter. Note that to the best of the authors’ knowledge, a study on DEWMA type charts’ steady state ARL and SDRL performances has yet to be made in the literature. A situation in which the SS-DEWMA chart could be more useful than the SS-EWMA chart is explained in Sections 4 and 6.     

A study of process monitoring based on inverse Gaussian distribution

The inverse Gaussian distribution has considerable applications in describing product life, employee service times, and so on. In this paper, the average run length (ARL) unbiased control charts, which monitor the shape and location parameters of the inverse Gaussian distribution respectively, are proposed when the in-control parameters are known. The effects of parameter estimation on the performance of the proposed control charts are also studied. An ARL-unbiased control chart for the shape parameter with the desired ${\mathit{ARL}}_0$, which takes the variability of the parameter estimate into account, is further developed. The performance of the proposed control charts is investigated in terms of the ARL and standard deviation of the run length. Finally, an example is used to illustrate the proposed control charts.     

Optimal design of the double sampling X¯ chart with estimated parameters based on median run length

The double sampling (DS) $\overline{X}$ chart when the process parameters are unknown and have to be estimated from a reference Phase-I dataset is studied. An expression for the run length distribution of the DS $\overline{X}$ chart is derived, by conditioning and taking parameter estimation into account. Since the shape and the skewness of the run length distribution change with the magnitude of the mean shift, the number of Phase-I samples and sample sizes, it is shown that the traditional chart’s performance measure, i.e. the average run length, is confusing and not a good representation of a typical chart’s performance. To this end, because the run length distribution is highly right-skewed, especially when the shift is small, it is argued that the median run length (MRL) provides a more intuitive and credible interpretation. From this point of view, a new optimal design procedure for the DS $\overline{X}$ chart with known and estimated parameters is developed to compute the chart’s optimal parameters for minimizing the out-of-control MRL, given that the values of the in-control MRL and average sample size are fixed. The optimal chart which provides the quickest out-of-control detection speed for a specified shift of interest is designed according to the number of Phase-I samples commonly used in practice. Tables are provided for the optimal chart parameters along with some empirical guidelines for practitioners to construct the optimal DS $\overline{X}$ charts with estimated parameters. The optimal charts with estimated parameters are illustrated with a real application from a manufacturing company.     

Enhancing the performance of CUSUM scale chart

Control charts act as the most important statistical process monitoring tool, widely used for the purpose of identifying unusual variations in process parameters. Researchers have implemented different rules to increase the sensitivity of Shewhart, CUSUM and EWMA control charts for the detection of small shifts in process location. However, for the monitoring of process scale, the use of such rules has been limited to Shewhart charts. This study proposes the implementation of sensitizing rules in CUSUM scale charts to enhance their ability to detect smaller changes in process variability. The performance of the proposed schemes is evaluated and compared with the simple scale CUSUM scheme, the EWMS chart, the M-EWMS chart and the COMB chart, in terms of run length characteristics such as average run length (ARL) and standard deviation of the run length distribution (SDRL). Control chart coefficients to set the ARL at the desired level are also provided. Two numerical examples are given to illustrate the application of the proposed schemes on practical data sets.     

A new monitoring design for uni-variate statistical quality control charts

In this research, an iterative approach is employed to analyze and classify the states of uni-variate quality control systems. To do this, a measure (called the belief that process is in-control) is first defined and then an equation is developed to update the belief recursively by taking new observations on the quality characteristic under consideration. Finally, the upper and the lower control limits on the belief are derived such that when the updated belief falls outside the control limits an out-of-control alarm is received. In order to understand the proposed methodology and to evaluate its performance, some numerical examples are provided by means of simulation. In these examples, the in and out-of-control average run lengths (ARL) of the proposed method are compared to the corresponding ARL’s of the optimal EWMA, Shewhart EWMA, GEWMA, GLR, and CUSUM[11] methods within different scenarios of the process mean shifts. The simulation results show that the proposed methodology performs better than other charts for all of the examined shift scenarios. In addition, for an autocorrelated AR(1) process, the performance of the proposed control chart compared to the other existing residual-based control charts turns out to be promising.     

Trend Autoregressive Model Exact Run Length Evaluation on a Two-Sided Extended EWMA Chart

The Extended Exponentially Weighted Moving Average (extended EWMA) control chart is one of the control charts and can be used to quickly detect a small shift. The performance of control charts can be evaluated with the average run length (ARL). Due to the deriving explicit formulas for the ARL on a two-sided extended EWMA control chart for trend autoregressive or trend AR(p) model has not been reported previously. The aim of this study is to derive the explicit formulas for the ARL on a two-sided extended EWMA control chart for the trend AR(p) model as well as the trend AR(1) and trend AR(2) models with exponential white noise. The analytical solution accuracy was obtained with the extended EWMA control chart and was compared to the numerical integral equation (NIE) method. The results show that the ARL obtained by the explicit formula and the NIE method is hardly different, but the explicit formula can help decrease the computational (CPU) time. Furthermore, this is also expanded to comparative performance with the Exponentially Weighted Moving Average (EWMA) control chart. The performance of the extended EWMA control chart is better than the EWMA control chart for all situations, both the trend AR(1) and trend AR(2) models. Finally, the analytical solution of ARL is applied to real-world data in the health field, such as COVID-19 data in the United Kingdom and Sweden, to demonstrate the efficacy of the proposed method.     

Measuring the performance improvement of a double generally weighted moving average control chart

This study extends the sum of squares generally weighted moving average (SS-GWMA) control chart by using the double generally weighted moving average (DGWMA) technique. The proposed expanded chart is called the sum of squares double generally weighted moving average (SS-DGWMA) control chart. Simulations are performed to evaluate the average run length (ARL) and standard deviation of run length (SDRL) of the SS-DGWMA, SS-DEWMA, and SS-GWMA charts. An extensive comparison shows that the optimal SS-DGWMA chart is superior to the optimal SS-GWMA and SS-DEWMA charts in all studied scenarios. The SS-DGWMA chart is also easy to implement and to interpret the abnormal signals.     

A multivariate synthetic double sampling T2 control chart

In this article, we propose a multivariate synthetic double sampling T² chart to monitor the mean vector of a multivariate process. The proposed chart combines the double sampling (DS) T² chart and the conforming run length (CRL) chart. On the whole, the proposed chart performs better than its standard counterparts, namely, the Hotelling’s T², DS T², and synthetic T² charts, in terms of the average run length (ARL) and average number of observations to sample (ANOS). The proposed chart also outperforms the multivariate exponentially weighted moving average (MEWMA) chart for moderate and large shifts but the latter is more sensitive than the former towards small shifts. For a variable sample size chart, like the synthetic DS T² chart, ANOS is a more meaningful performance measure than ARL. ANOS relates to the actual number of observations sampled but ARL merely deals with the number of sampling stages taken. Interpretation based on ARL is more complicated as either n₁ or n₁ + n₂ observations are taken in each sampling stage.     

基于协变量信息的自适应MEWMA控制图

在统计过程控制中,质量变量通常会受到许多协变量因素的影响,充分利用协变量的有用信息可以进一步提高控制图的灵敏度,因此提出一种新的自适应多元EWMA控制图,并计算ARL进行比较。在MEWMA控制图的基础上引入一个权重函数,根据协变量的有用信息自适应的调节统计量的加权参数：当收集到的协变量信息发生偏移时,选择较大的加权参数,更关注当前和附近时间点观测值的偏移程度;反之则选择较小的参数。大量数值模拟分析表明,充分利用协变量中的有用信息之后,监控效果明显优于MEWMA、MCUSUM控制图。     

Comparative study of the performance of the CuSum and EWMA control charts

This work presents a comparative study of the performance of the cumulative sum (CuSum), as well as the exponentially weighted moving average (EWMA) control charts. The objective of this research is to verify when CuSum and EWMA control charts do the best control region, in order to detect small changes in the process average. Starting from the data of a productive process, several series were simulated. CuSum and EWMA control charts were used to determine the average run length (ARL) to detect a condition out of control. ARL found by each chart which was then, compared. It was observed that the CuSum control chart practically did not sign points out of control for the levels of variation between ±1.0 standard deviation. For these variation levels the EWMA control chart was more efficient than CuSum. Among the parameters EWMA control chart the ones with constant λ=0.10 and 0.05, with the respective control limits L=2.814 and 2.625, were the ones that detected larger number of altered positions.     

Cumulative quantity control chart for the mixture of inverse Rayleigh process

The engineering processes are made up of a number of the phenomenons working together that may lead to defects with multiple causes. In order to model such types of multiple cause defect systems we may not rely on simple probability models and hence, the need arises for mixture models. The commonly used control charts are based on simple models with the assumption that the process is working under the single cause defect system. This study proposes a control chart for the two component mixture of inverse Rayleigh distribution. The proposed chart namely IRMQC chart is based on mixture cumulative quantity using the quantity of product inspected until specified numbers of defects are observed. The single cause chart is also discussed as a special case of the proposed mixture cumulative quantity chart. The control structure of the proposed chart is designed, and its performance is evaluated in terms of some useful measures, including average run length (ARL), expected quality loss (EQL) and relative ARL (RARL). An illustrative example along a case study, is also given to highlight the practical aspects of the proposal.     

On efficient use of auxiliary information for control charting in SPC

In Statistical Process Control (SPC), monitoring of the process dispersion has a major impact on the performance of processes like manufacturing, management and services. Control charts act as the most important SPC tool, used to differentiate between common and special cause variations in the process. The use of auxiliary information can enhance the detection ability of control charts and hence an efficient monitoring of process parameter(s) can be done. This study deals with the Shewhart type variability control charts based on auxiliary characteristics for the non-cascading processes, assuming stability of auxiliary parameters. The control chart structures of these variability charts are provided and their performance evaluations are carried out in terms of average run length (ARL), relative average run length (RARL) and extra quadratic loss (EQL) under the normal and t distributed process environments. The comparisons have been made among different variability charts and superiorities are established based on their detection abilities for different amounts of shifts in process dispersion. An illustrative example is also provided in support of the theory, and finally the study ends with concluding remarks and suggestions for future research.     

Optimal linear combination of Poisson variables for multivariate statistical process control

In this paper we analyze the monitoring of p Poisson quality characteristics simultaneously, developing a new multivariate control chart based on the linear combination of the Poisson variables, the LCP control chart. The optimization of the coefficients of this linear combination (and control limit) for minimizing the out-of-control ARL is constrained by the desired in-control ARL. In order to facilitate the use of this new control chart the optimization is carried out employing user-friendly Windows© software, which also makes a comparison of performance between this chart and other schemes based on monitoring a set of Poisson variables; namely a control chart on the sum of the variables (MP chart), a control chart on their maximum (MX chart) and an optimized set of univariate Poisson charts (Multiple scheme). The LCP control chart shows very good performance. First, the desired in-control ARL (ARL₀) is perfectly matched because the linear combination of Poisson variables is not constrained to integer values, which is an advantage over the rest of charts, which cannot in general match the required ARL₀ value. Second, in the vast majority of cases this scheme signals process shifts faster than the rest of the charts.     

A control chart based on likelihood ratio test for detecting patterned mean and variance shifts

Control charts based on generalized likelihood ratio test (GLRT) are attractive from both theoretical and practical points of view. Most of the existing works in the literature focusing on the detection of the process mean and variance are almost based on the assumption that the shifts remain constant over time. The case of the patterned mean and variance changes may not be well discussed. In this research, we propose a new control chart which integrates the exponentially weighted moving average (EWMA) procedure with the GLRT statistics to monitor the process with patterned mean and variance shifts. The attractive advantage of our control chart is its reference-free property. Due to the good properties of GLRT and EWMA procedures, our simulation results show that the proposed chart provides quite effective and robust detecting ability for various types of shifts. The implementation of our proposed control chart is illustrated by a real data example from chemical process control.     

New EWMA control charts for monitoring process dispersion

There exist two EWMA-type dispersion charts for monitoring dispersion increases in the literature. One resets the EWMA statistic to zero whenever it is below zero. The other one truncates negative normalized observations to zero in the EWMA statistic. This paper proposes two one-sided EWMA charts for detecting dispersion increases and decreases, respectively, and one two-sided EWMA chart for monitoring dispersion increases or decreases simultaneously. Simulation studies show that the proposed upper-sided EWMA chart performs better than the two existing counterparts for detecting increases in dispersion, and that the proposed lower-sided EWMA chart significantly outperforms the two lower-sided EWMA charts developed similar to their two existing upper-sided EWMA charts for detecting decreases in dispersion. Moreover, the proposed two-sided EWMA chart provides much better sensitivity than the two two-sided EWMA charts generalized from the two existing upper-sided EWMA charts for detecting overall changes in dispersion.     

On the performance of the conditional decision procedure in geometric charts

To improve the performance of control charts the conditional decision procedure (CDP) incorporates a number of previous observations into the chart’s decision rule. It is expected that charts with this runs rule are more sensitive to shifts in the process parameter. To signal an out-of-control condition more quickly some charts use a headstart feature. They are referred as charts with fast initial response (FIR). The CDP chart can also be used with FIR. In this article we analyze and compare the performance of geometric CDP charts with and with no FIR. To do it we model the CDP charts with a Markov chain and find closed-form ARL expressions. We find the conditional decision procedure useful when the fraction p of nonconforming units deteriorates. However the CDP chart is not very effective for signaling decreases in p.