A double exponentially weigiited moving average control procedure with variable sampling intervals

Shewhart, cumulative sum (CUSUM), and exponentially weighted moving average (EWMA) control procedures with variable sampling intervals (VSI) have been investigated in recent years for detecting shifts in the process mean. Such procedures have been shown to be more efficient when compared with the corresponding fixed sampling interval (FSI) charts with respect to the average time to signal (ATS) when the average run length (ARL) values of both types of procedures are held equal. Frequent switching between the different sampling intervals can be a complicating factor in the application of control charts with variable sampling intervals. In this article, we propose using a double exponentially weighted moving average control procedure with variable sampling intervals (VSI-DEWMA) for detecting shifts in the process mean. It is shown that the proposed VSI-DEWMA control procedure is more efficient when compared with the corresponding fixed sampling interval FSI-DEWMA chart with respect to the average time to signal (ATS) when the average run length (ARL) values of both types of procedures are held equal. It is also shown that the VSI-DEWMA procedure reduces the average number of switches between the sampling intervals and has similar ATS properties as compared to the VSI-EMTMA control procedure     

Improved switching rules in control procedures using variable sampling intervals

Some properties of control procedures with variable sampling intervals (VSI) have been investigated in recent years by Amin, Renolds et al, and others. Such procedures have been shown to be more efficient when compared to the corresponding fixed sampling interval (FSI) charts with respect to the Average Time to Signal (ATS) when the Average Run Length (ARL) values for both types of procedures are held equal. Frequent switching between the different sampling intervals can be a complicating factor in the application of control charts with variable sampling intervals (VSI). This problem is being addressed in this article, and improved switching rules are presented and evaluated for Shewhart, CUSUM, and EWMA control procedures. The proposed rules considerably reduce the average number of switches between the sampling intervals and also improve the ATS properties of the control procedures when compared to the conventional variable sampling interval procedures     

Two parameter cuscore qualty control procedures

This paper is concerned with the problem of simultaneously monitoring the process mean and process variability of continuous production processes using combined Shewhart-cumulative score (cuscore) quality control procedures developed by Ncube and Woodall (1984). Two methods of approach are developed and their properties are investigated. One method uses two separate Shewhart-cuscore control charts, one for determining shifts in the process mean and the other for detecting shifts in process variability. The other method uses a single combined statistic which is sensitive to shifts in both the mean and the variance. Each procedure is compared to the corresponding Shewhart schemes. It will be shown by average run length calculations that the proposed Shewhart- cuscore schemes are considerably more efficient than the comparative Shewhart procedures for certain shifts in the process mean and process variability for the case when the underlying process control variable is assumed to be normally distributed.     

Monitoring and diagnosing dependent process steps using VSI control charts

The paper proposes the variables sampling interval (VSI) scheme to monitor the means and the variances in two dependent process steps. The performance of the considered VSI control charts is measured by the adjusted average time to signal derived by a Markov chain approach. An example of the process control for the metallic film thickness of the computer connectors system shows the application and performance of the proposed VSI control charts in detecting shifts. Furthermore, the performance of the VSI control charts and the fixed sampling interval control charts are compared via the numerical analysis results. These demonstrate that the former is much faster in detecting shifts. Whenever quality engineers cannot specify the values of variable sampling intervals, the optimal VSI control charts are recommended. Furthermore, the impacts of misusing Shewhart charts to monitoring the process mean and variance in the second process step are also investigated.     

Optimal Variable Sample Size and Sampling Interval MSE Chart Based on Improved Square Root Transformation

In the field of statistical process control (SPC), control charts for attributes are widely used to detect the out-of-control condition by checking the number of nondefective units or nondefective in a sample. In this article, we use the average time to signal (ATS) and the average number of observations to signal (ANOS) to evaluate the performance of the optimal variable sample size and sampling interval (VSSI) improved square root transformation (ISRT) mean square error (MSE) (VSSI_ ISRT_ MSE) control chart for attribute data. In addition, this control chart will be used to monitor: (1) the difference between the process mean and the target value, and (2) the process variance shifts. We found that the optimal VSSI_ ISRT_ MSE chart performs better than the specific VSSI, the optimal variable sampling interval (VSI), and the fixed parameters (FP) ISRT_MSE charts. An example is given to illustrate this new proposed approach.     

Using a Single Average Loss Control Chart to Monitor Process Mean and Variability

A single chart, instead of X-bar and R charts or X-bar and S charts, to monitor simultaneously the process mean and the variability if found would cut down the time and effort. Some researches have been done in finding such charts. In reality, process target is more important than process mean. A much easier average loss chart is first proposed here to detect the increases in the difference of the process mean and the target and the variability simultaneously. An example of the customer complaint processing time of the customer service center of an IT company shows the application and the performance of the proposed average loss control chart. Furthermore, a more efficient optimal average loss chart with variable sampling intervals is proposed and performs better than the average loss chart with fixed sampling intervals and the Shewhart joint X-bar and S charts. Some numerical analyses demonstrated the findings.     

Weighted adaptive multivariate CUSUM charts with variable sampling intervals

The adaptive multivariate CUSUM (AMCUSUM) chart has received considerable attention because of its superior sensitivity against a range of mean shift sizes than that of the conventional non-adaptive multivariate CUSUM (MCUSUM) chart. Recently, weighted AMCUSUM (WAMCUSUM) charts with a fixed sampling interval (FSI) have been proposed, called the WAMCUSUM-FSI charts, which provide more sensitivity than the AMCUSUM-FSI charts. In this paper, we extend this work and propose WAMCUSUM charts with variable sampling interval (VSI), named the WAMCUSUM-VSI charts, for efficiently monitoring the mean of a multivariate normally distributed process. The Monte Carlo simulation method is used to compute the average time to signal (ATS) and the adjusted ATS (AATS) profiles of the existing and proposed charts. It is found that the WAMCUSUM-VSI charts perform substantially and nearly uniformly better than the WAMCUSUM-FSI charts in terms of the ATS and AATS performance criterion. An example is given to explain the implementation of the WAMCUSUM charts with fixed and VSIs.     

Shewhart control charts for the scale parameter of a Weibull control variable with fixed and variable sampling intervals

In this paper, we are concerned with pure statistical Shewhart control charts for the scale parameter of the three-parameter Weibull control variable, where, and are the location, the scale and the shape parameters, respectively, with fixed (FSI) and variable (VSI) sampling intervals. The parameters and are assumed to be known. We consider two-sided, and lower and upper one-sided Shewhart control charts and their FSI and VSI versions . They jointly control the mean and the variance of the Weibull control variable X. The pivotal statistic of those control charts is the maximum-likelihood estimator of for the Nth random sample XN=(X1N,X2N,...,XnN) of the Weibull control variable X. The design and performance of these control charts are studied. Two criteria, i.e. 'comparability criterion' (or 'matched criterion') under control and 'primordial criterion', are imposed on their design. The performance of these control charts is measured using the function average time to signal. For the VSI versions, the constant which defines the partition of the 'continuation region' is obtained through the 'comparability criterion' under control. The monotonic behaviour of the function average time to signal in terms of the parameters (magnitude of the shift suff ered by the target value 0), and is studied. We show that the function average time to signal of all the control charts studied in this paper does not depend on the value of the parameter or on 0, and, under control, does not depend on the parameter, when Delta (the probability of a false alarm) and n (sample size) are fixed. All control charts satisfy the 'primordial criterion' and, for fixed, on average, they all (except the two-sided VSI, for which we were not able to ascertain proof) are quicker in detecting the shift as increases. We conjecture - and we are not contradicted by the numerical example considered - that the same is true for the two-sided VSI control chart. We prove that, under the average time to signal criterion, the VSI versions are always preferable to their FSI versions. In the case of one-sided control charts, under the 'comparability criterion', the VSI version is always preferable to the FSI version, and this advantage increases with and the extent of the shift. Our one-sided control charts perform better and have more powerful statistical properties than does our two-sided control chart. The numerical example where n=5,0=1,=0.5, 1.0, 2.0, and Delta=1/370.4 is presented for the two-sided, and the lower and upper one-sided control charts. These numerical results are presented in tables and in figures. The joint influence of the parameters and in the function average time to signal is illustrated.     

Monitoring simple linear profiles using variable sample size schemes

ABSTRACT

Profile monitoring is one of the new research areas in statistical process control. Most of the control charts in this area are designed with fixed sampling rate which makes the control chart slow in detecting small to moderate shifts. In order to improve the performance of the conventional fixed control charts, adaptive features are proposed in which, one or more design parameters vary during the process. In this paper the variable sample size feature of EWMA₃ and MEWMA schemes are proposed for monitoring simple linear profiles. The EWMA₃ method is based on the combination of three exponentially weighted moving average (EWMA) charts for monitoring three parameters of a simple linear profile separately and the Multivariate EWMA (MEWMA) chart is based on the using a single chart to monitor the coefficients and variance of a general linear profile. Also a two-sided control chart is proposed for monitoring the standard deviation in the EWMA₃ method. The performance of the proposed charts is compared in terms of the average time to signal. Numerical examples show that using adaptive features increase the power of control charts in detecting the parameter shifts. Finally, the performance of the proposed variable sample size schemes is illustrated through a real case in the leather industry.     

Robustness of the EWMA and the combined X¯–EWMA control charts with variable sampling intervals to non-normality

The exponentially weighted moving average (EWMA) control charts with variable sampling intervals (VSIs) have been shown to be substantially quicker than the fixed sampling intervals (FSI) EWMA control charts in detecting process mean shifts. The usual assumption for designing a control chart is that the data or measurements are normally distributed. However, this assumption may not be true for some processes. In the present paper, the performances of the EWMA and combined X¯–EWMA control charts with VSIs are evaluated under non-normality. It is shown that adding the VSI feature to the EWMA control charts results in very substantial decreases in the expected time to detect shifts in process mean under both normality and non-normality. However, the combined X¯–EWMA chart has its false alarm rate and its detection ability is affected if the process data are not normally distributed.     

A modified economic-statistical design of the T2 control chart with variable sample sizes and control limits

Recent studies have shown that using variable sampling size and control limits (VSSC) schemes result in charts with more statistical power than variable sampling size (VSS) when detecting small to moderate shifts in the process mean vector. This paper presents an economic-statistical design (ESD) of the VSSC T² control chart using the general model of Lorenzen and Vance [22]. The genetic algorithm approach is then employed to search for the optimal values of the six test parameters of the chart. We then compare the expected cost per unit of time of the optimally designed VSSC chart with optimally designed VSS and FRS (fixed ratio sampling) T² charts as well as MEWMA charts.     

Estimation of the Change Point in Monitoring the Process Mean and Variance

Knowing the time of a process change could lead to quicker identification of the special cause and less process down time, as well as help to reduce the probability of incorrectly identifying the special cause. In this article, we propose the maximum likelihood estimator (MLE) for the process change point when a control chart with the fixed sampling rate (FSR) scheme or the variable sampling rate (VSR) scheme is used in monitoring a process to detect changes in the process mean and/or variance of a normal quality variable. We investigate the performance of this estimator when it is used in various types of control charts.     

The economic design of multivariate binomial EWMA VSSI control charts

Since multi-attribute control charts have received little attention compared with multivariate variable control charts, this research is concerned with developing a new methodology to employ the multivariate exponentially weighted moving average (MEWMA) charts for m-attribute binomial processes; the attributes being the number of nonconforming items. Moreover, since the variable sample size and sampling interval (VSSI) MEWMA charts detect small process mean shifts faster than the traditional MEWMA, an economic design of the VSSI MEWMA chart is proposed to obtain the optimum design parameters of the chart. The sample size, the sampling interval, and the warning/action limit coefficients are obtained using a genetic algorithm such that the expected total cost per hour is minimized. At the end, a sensitivity analysis has been carried out to investigate the effects of the cost and the model parameters on the solution of the economic design of the VSSI MEWMA chart.     

A new approach for monitoring process variance

ABSTRACT

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having non-normal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used in such circumstances. In this paper, we propose a new variance chart based on a simple statistic to monitor process variance shifts. We explore the sampling properties of the new monitoring statistic and calculate the average run lengths (ARLs) of the proposed variance chart. Furthermore, an arcsine transformed exponentially weighted moving average (EWMA) chart is proposed because the ARLs of this modified chart are more intuitive and reasonable than those of the variance chart. We compare the out-of-control variance detection performance of the proposed variance chart with that of the non-parametric Mood variance (NP-M) chart with runs rules, developed by Zombade and Ghute [Nonparametric control chart for variability using runs rules. Experiment. 2014;24(4):1683–1691], and the nonparametric likelihood ratio-based distribution-free exponential weighted moving average (NLE) chart and the combination of traditional exponential weighted moving average (EWMA) mean and EWMA variance (CEW) control chart proposed by Zou and Tsung [Likelihood ratio-based distribution-free EWMA control charts. J Qual Technol. 2010;42(2):174–196] by considering cases in which the critical quality characteristic has a normal, a double exponential or a uniform distribution. Comparison results showed that the proposed chart performs better than the NP-M with runs rules, and the NLE and CEW control charts. A numerical example of service times with a right-skewed distribution from a service system of a bank branch in Taiwan is used to illustrate the application of the proposed variance chart and of the arcsine transformed EWMA chart and to compare them with three existing variance (or standard deviation) charts. The proposed charts show better detection performance than those three existing variance charts in monitoring and detecting shifts in the process variance.     

Nonparametric quality control charts based on the sign statistic

Nonparametric control chart are presented for the problem of detecting changes in the process median (or mean), or changes in the process variability when samples are taken at regular time intervals. The proposed procedures are based on sign-test statistics computed for each sample, and are used in Shewhart and cumulative sum control charts. When the process is in control the run length distributions for the proposed nonparametric control charts do not depend on the distribution of the observations. An additional advantage of the non-parametric control charts is that the variance of the process does not need to be established in order to set up a control chart for the mean. Comparisons with the corresponding parametric control charts are presented. It is also shown that curtailed sampling plans can considerably reduce the expected number of observations used in the Shewhart control schemes based on the sign statistic.     

Economic design based on a multivariate Bayesian VSI chart with a dual control limit

In this paper, a multivariate Bayesian variable sampling interval (VSI) control chart for the economic design and optimization of statistical parameters is designed. Based on the VSI sampling strategy of a multivariate Bayesian control chart with dual control limits, the optimal expected cost function is constructed. The proposed model allows the determination of the scheme parameters that minimize the expected cost per time of the process. The effectiveness of the Bayesian VSI chart is estimated through economic comparisons with the Bayesian fixed sampling interval and the Hotelling's T² chart. This study is an in-depth study on a Bayesian multivariate control chart with variable parameter. Furthermore, it is shown that significant cost improvement may be realized through the new model.     

ON THE COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN REGRESSION AFTER VARIABLE SELECTION

This paper considers a linear regression model with regression parameter vector β. The parameter of interest is θ= a^Tβ where a is specified. When, as a first step, a data‐based variable selection (e.g. minimum Akaike information criterion) is used to select a model, it is common statistical practice to then carry out inference about θ, using the same data, based on the (false) assumption that the selected model had been provided a priori. The paper considers a confidence interval for θ with nominal coverage 1 ‐ α constructed on this (false) assumption, and calls this the naive 1 ‐ α confidence interval. The minimum coverage probability of this confidence interval can be calculated for simple variable selection procedures involving only a single variable. However, the kinds of variable selection procedures used in practice are typically much more complicated. For the real‐life data presented in this paper, there are 20 variables each of which is to be either included or not, leading to 2²⁰ different models. The coverage probability at any given value of the parameters provides an upper bound on the minimum coverage probability of the naive confidence interval. This paper derives a new Monte Carlo simulation estimator of the coverage probability, which uses conditioning for variance reduction. For these real‐life data, the gain in efficiency of this Monte Carlo simulation due to conditioning ranged from 2 to 6. The paper also presents a simple one‐dimensional search strategy for parameter values at which the coverage probability is relatively small. For these real‐life data, this search leads to parameter values for which the coverage probability of the naive 0.95 confidence interval is 0.79 for variable selection using the Akaike information criterion and 0.70 for variable selection using Bayes information criterion, showing that these confidence intervals are completely inadequate.     

A new double sampling control chart for monitoring process mean using auxiliary information

Statistical quality control charts have been widely accepted as a potentially powerful process monitoring tool because of their excellent speed in tracking shifts in the underlying process parameter(s). In recent studies, auxiliary-information-based (AIB) control charts have shown superior run length performances than those constructed without using it. In this paper, a new double sampling (DS) control chart is constructed whose plotting-statistics requires information on the study variable and on any correlated auxiliary variable for efficiently monitoring the process mean, namely AIB DS chart. The AIB DS chart also encompasses the classical DS chart. We discuss in detail the construction, optimal design, run length profiles, and the performance evaluations of the proposed chart. It turns out that the AIB DS chart performs uniformly better than the DS chart when detecting different kinds of shifts in the process mean. It is also more sensitive than the classical synthetic and AIB synthetic charts when detecting a particular shift in the process mean. Moreover, with some realistic beliefs, the proposed chart outperforms the exponentially weighted moving average chart. An illustrative example is also presented to explain the working and implementation of the proposed chart.