Standard Deviation Of Run Length Research Articles

Joint Monitoring of Process Location and Dispersion Based on CUSUM Procedure and Generalized Likelihood Ratio in the Presence of Measurement Errors

In this paper, we propose a new methodology based on the combination of cumulative sum procedure and generalized likelihood ratio statistic for joint monitoring of the process location and dispersion. Then, we explore the effect of measurement errors on detecting ability of the proposed control chart when (i) the variance of measurement error is constant (ii) the variance of measurement error increases linearly as the level of the process mean increases. We also utilize multiple measurements on each sample point in order to decrease the adverse effects of measurement errors on the performance of the proposed control charts. Two numerical examples based on simulation studies are given to evaluate the ability of the proposed methods in terms of average run length, median run length, standard deviation of run length, and the first and third quantile points of the run length distribution (Q1 and Q3). Finally, a real life example is given to illustrate the application of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.

High quality process monitoring using a class of inter-arrival time distributions of the renewal process

For high-quality processes where the defect rate is very low, e.g., parts per million (ppm), time-between-events (TBE) control charts have several advantages over the ordinary control charts. Most existing TBE control charts are based on the homogeneous Poisson process assumption, so that the distribution of TBE can only be exponential. However, the exponential distribution is not suitable in many applications, especially when the failure rate is not constant. In this article, we introduce a new TBE control chart, based on the renewal process, where the distribution of the TBE belongs to a parametric class of absolutely continuous distributions, which includes some well-known and commonly used lifetime distributions, i.e., exponential, Rayleigh, Weibull, Burr type XII, Pareto and Gompertz. The control structure of the proposed chart is derived analytically in general and numerical examples are presented for the Weibull distribution, due to its relevance in reliability. The performance of the proposed control charts is evaluated in terms of some standard useful measures, including average run length (ARL), the standard deviation of run length, the coefficient of variation of run length, expected quality loss (EQL) and relative ARL (RARL). The effect of parameter estimation, using both maximum likelihood and Bayesian methods, is also discussed. This study also presents an illustrative example and four case studies to highlight the practical aspects of the new TBE chart.

On designing a new Tukey-EWMA control chart for process monitoring

Exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts are used to detect smaller shifts in process parameters. The usual EWMA and CUSUM charts depend on the normality assumption for a better detection ability. This study proposes an efficient EWMA control chart based on the spirit of Tukey control chart, especially designed for skewed distributions. The performance of the proposed and the competing charts is measured using different length properties such as average run length (ARL), standard deviation of run length (SDRL), and median run length (MDRL). We have observed that the proposed chart is quite efficient at detecting process shifts of smaller magnitude, especially for skewed distributions. For practical considerations, the proposed chart is implemented at aerospace manufacturing data on industrial production index.

Monitoring the coefficient of variation using a variable sample size control chart

This paper proposes an adaptive Shewhart control chart implementing a variable sample size strategy in order to monitor the coefficient of variation. The goals of this paper are as follows: (a) to propose an easy-to-use 3-parameter logarithmic transformation for the coefficient of variation in order to handle the variable sample size aspect; (b) to derive the formulas for computing the average run length, the standard deviation run length, and the average sample size and to evaluate the performance of the proposed chart based on these criteria; (c) to present ready-to-use tables with optimal chart parameters minimizing the out-of-control average run length as well as the out-of-control average sample size; and (d) to compare this chart with the fixed sampling rate, variable sampling interval, and synthetic control charts. An example illustrates the use of the variable sample size control chart on real data gathered from a casting process.

Enhanced cumulative sum charts for monitoring process dispersion.

The cumulative sum (CUSUM) control chart is widely used in industry for the detection of small and moderate shifts in process location and dispersion. For efficient monitoring of process variability, we present several CUSUM control charts for monitoring changes in standard deviation of a normal process. The newly developed control charts based on well-structured sampling techniques - extreme ranked set sampling, extreme double ranked set sampling and double extreme ranked set sampling, have significantly enhanced CUSUM chart ability to detect a wide range of shifts in process variability. The relative performances of the proposed CUSUM scale charts are evaluated in terms of the average run length (ARL) and standard deviation of run length, for point shift in variability. Moreover, for overall performance, we implore the use of the average ratio ARL and average extra quadratic loss. A comparison of the proposed CUSUM control charts with the classical CUSUM R chart, the classical CUSUM S chart, the fast initial response (FIR) CUSUM R chart, the FIR CUSUM S chart, the ranked set sampling (RSS) based CUSUM R chart and the RSS based CUSUM S chart, among others, are presented. An illustrative example using real dataset is given to demonstrate the practicability of the application of the proposed schemes.

Fraction nonconforming control charts with m-of-m runs rules

In this article, we proposed m-of-m control chart for fraction nonconforming to increase sensitivity of the standard p-chart. The m-of-m control chart for fraction nonconforming is described, and the 3-of-3 control chart is discussed in detail as an illustration of the m-of-m control chart. The performance of proposed control charts is measured in terms of average run length, standard deviation of run length and quartiles under zero-state and steady-state modes. The Markov chain approach is used to compute average run length, standard deviation of run length and quartiles. Comparison study revealed that the performance proposed m-of-m control chart with m = 2, 3 is significantly better than the standard p-chart. Average run length values of the proposed control charts are at least 15 % less than that of the standard p-chart values for small to moderate shifts. Standard deviation of run length values of the proposed control charts is also at least 18 % less than the standard p-chart. An example is given to illustrate an application of procedures discussed.

Run Length Distribution of CUSUM Control Schemes for Negative Binominal Processes

The CUSUM control scheme is an important tool in detecting the quality of goods or services. It is used in many industries for rapid detection of many industrial processes. The average run length (ARL) and distribution of run length and standard deviation of run length (SDRL) are the essential characteristics in constructing a control chart and are also important indexes for comparing performance with other control charts. Until now, no theoretic derivation for the run length and its standard deviation underlying negative binominal distribution for the CUSUM scheme have been developed. In this paper, an analytic method for steady state average run length as well as SDRL and two-sided ARL and SDRL with different control statistics is developed for negative binomial CUSUM (NB CUSUM). To decrease computer memory requirements and computing time, a sparse matrix operation method is also proposed to obtain the run length when conducting the Markov chain method. This analytic method makes the computations of run length for one-sided and two-sided NB CUSUM schemes easier. A real numerical example also shows the analytic results are consistent with the simulation results.

A Sum of Squares Generally Weighted Moving Average Control Chart

In this article, we propose a new control chart called the sum of squares generally weighted moving average (SS-GWMA) control chart to simultaneously detect both the increase and decrease in the mean and/or variability. This new scheme is compared with the sum of squares exponentially weighted moving average (SS-EWMA) control chart. A simulation study is conducted to show that SS-GWMA control charts outperform SS-EWMA charts, in terms of the average run length (ARL), standard deviation of run length (SDRL), and diagnostic abilities. The design of SS-GWMA control charts is also discussed.

A sensitive non-parametric EWMA control chart

In practice, we may not always have normally distributed quality characteristics of interest. This leads to the need for non-parametric techniques which are not dependent on the assumptions about the parent distribution. This study develops a non-parametric exponentially weighted moving average (EWMA) chart (namely the NPSSEWMA chart) for an improved monitoring of process location. The proposal is based on the use of sign statistics on a moving pattern in an EWMA setup. The design structure of the proposed chart is developed and its performance is evaluated in terms of different properties including average run length (ARL), standard deviation run length (SDRL), percentiles, relative ARL (RARL), extra quadratic loss (EQL), and performance comparison index (PCI). The proposal is compared with recently developed non-parametric counterparts namely NPSEWMA, NPASEWMA, and NPSCUSUM charts. It is observed that the design structure of the proposed NPSSEWMA chart outshines the existing counterparts. An application example is also included in the study for practical demonstration.

Combined Application of Shewhart and Cumulative Sum R Chart for Monitoring Process Dispersion

This study analyzes the performance of combined applications of the Shewhart and cumulative sum (CUSUM) range R chart and proposes modifications based on well-structured sampling techniques, the extreme variations of ranked set sampling, for efficient monitoring of changes in the process dispersion. In this combined scheme, the Shewhart feature enables quick detection of large shifts from the target standard deviation while the CUSUM feature takes care of small to moderate shifts from the target value. We evaluate the numerical performance of the proposed scheme in terms of the average run length, standard deviation of run length, the average ratio average run length, and average extra quadratic loss. The results show that the combined scheme can detect changes in the process that were small or large enough to escape detection by the lone Shewhart R chart or CUSUM R chart, respectively. We present a comparison of the proposed schemes with several dispersion charts for monitoring changes in process variability. The practical application of the proposed scheme is demonstrated using real industrial data.

The Steady-State Performance of Cumulative Count of a Conforming Control Chart Based On Runs Rules

Cumulative count of conforming control chart is usually used to monitor fraction nonconforming in high-yield processes. In this article, we propose m-of-m control chart based on cumulative count of conforming units for high-yield processes. The steady-state properties of the m-of-m control chart are investigated. We compare performance of the m-of-m control chart with control chart based on cumulative count of conforming units. We present Markov chain model of the m-of-m control chart to evaluate average run length, standard deviation of run length and quartiles.

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.

Performance of Tukey's and Individual/Moving Range Control Charts

This paper compares two control charts: Tukey (TCC) and individual/moving range (XmR) control charts. Both are designed to examine single observation per time period, but little is known about which one is more efficient and under what conditions. We simulated data from different distributions and examined the performance of the two control charts on these data. Performance was assessed using the of average run length, extra quadratic loss, median run length, standard deviation run length, performance comparison index, and relative average run length. Overall, TCC was more efficient than XmR, when observations had binomial, Rayleigh, logistic, lognormal, Maxwell, normal, Poisson, Weibull (with α = 10, β = 1), and Student's t (30 and 10 degrees of freedom) distributions. XmR was more efficient when observations had Student's t (with 4 degrees of freedom) and gamma (with α = 4, β = 1) distributions. These results suggest that improvement teams could reach faster conclusions if they use TCC in most common situations. Copyright © 2014 John Wiley & Sons, Ltd.

New Exponentially Weighted Moving Average Control Charts for Monitoring Process Mean and Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely accepted because of their excellent performance in detecting small to moderate shifts in the process parameters. In this paper, we propose new EWMA control charts for monitoring the process mean and the process dispersion. These EWMA control charts are based on the best linear unbiased estimators obtained under ordered double ranked set sampling (ODRSS) and ordered imperfect double ranked set sampling (OIDRSS) schemes, named EWMA-ODRSS and EWMA-OIDRSS charts, respectively. We use Monte Carlo simulations to estimate the average run length, median run length, and standard deviation of run length of the proposed EWMA charts. We compare the performances of the proposed EWMA charts with the existing EWMA charts when detecting shifts in the process mean and in the process variability. It turns out that the EWMA-ODRSS mean chart performs uniformly better than the classical EWMA, fast initial response-based EWMA, Shewhart-EWMA, and hybrid EWMA mean charts. The EWMA-ODRSS mean chart also outperforms the Shewhart-EWMA mean charts based on ranked set sampling (RSS) and median RSS schemes and the EWMA mean chart based on ordered RSS scheme. Moreover, the graphical comparisons of the EWMA dispersion charts reveal that the proposed EWMA-ODRSS and EWMA-OIDRSS charts are more sensitive than their counterparts. We also provide illuminating examples to illustrate the implementation of the proposed EWMA mean and dispersion charts. Copyright © 2014 John Wiley & Sons, Ltd.

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.

An Improved Maximum Exponentially Weighted Moving Average Control Chart for Monitoring Process Mean and Variability

Maximum exponentially weighted moving average (MaxEWMA) control charts have gained considerable attention for detecting changes in both process mean and process variability. In this paper, we propose an improved MaxEWMA control charts based on ordered ranked set sampling (ORSS) and ordered imperfect ranked set sampling (OIRSS) schemes for simultaneous detection of both increases and decreases in the process mean and/or variability, named MaxEWMA‐ORSS and MaxEWMA‐OIRSS control charts. These MaxEWMA control charts are based on the best linear unbiased estimators of location and scale parameters obtained under ORSS and OIRSS methods. Extensive Monte Carlo simulations have been used to estimate the average run length and standard deviation of run length of the proposed MaxEWMA control charts. These control charts are compared with their counterparts based on simple random sampling (SRS), that is, MaxEWMA‐SRS and MaxGWMA‐SRS control charts. The proposed MaxEWMA‐ORSS and MaxEWMA‐OIRSS control charts are able to perform better than the MaxEWMA‐SRS and MaxGWMA‐SRS control charts for detecting shifts in the process mean and dispersion. An application to real data is provided to illustrate the implementation of the proposed MaxEWMA control charts. Copyright © 2013 John Wiley & Sons, Ltd.

Improved Exponentially Weighted Moving Average Control Charts for Monitoring Process Mean and Dispersion

Exponentially weighted moving average (EWMA) control charts are mostly used to monitor the manufacturing processes. In this paper, we propose some improved EWMA control charts for detecting the random shifts in the process mean and process dispersion. These EWMA control charts are based on the best linear unbiased estimators obtained under ordered ranked set sampling (ORSS) and ordered imperfect ranked set sampling (OIRSS), named EWMA‐ORSS and EWMA‐OIRSS charts, respectively. Monte Carlo simulations are used to estimate the average run length, median run length and standard deviation of run length of the proposed EWMA control charts. It is observed that the EWMA‐ORSS mean control chart is able to detect the random shifts in the process mean substantially quicker than the Shewhart‐cumulative sum and the Shewhart‐EWMA control charts based on the RSS scheme. Both EWMA‐ORSS and EWMA‐OIRSS location charts perform better than the classical EWMA, hybrid EWMA, Shewhart‐EWMA and fast initial response‐EWMA charts. The EWMA‐ORSS dispersion control chart performs better than the simple random sampling based CS‐EWMA and other EWMA control charts in efficient detection of the random shifts that occur in the process variability. An application to real data is also given to explain the implementation of the proposed EWMA control charts. Copyright © 2013 John Wiley & Sons, Ltd.

New Exponentially Weighted Moving Average Control Charts for Monitoring Process Dispersion

Exponentially weighted moving average (EWMA) control charts have received considerable attention for detecting small changes in the process mean or the process variability. Several EWMA control charts are constructed using logarithmic and normalizing transformations on unbiased sample variance for monitoring changes in the process dispersion. In this paper, we propose new EWMA control charts for monitoring process dispersion based on the best linear unbiased absolute estimators obtained under simple random sampling (SRS) and ranked set sampling (RSS) schemes, named EWMA‐SRS and EWMA‐RSS control charts. The performance of the proposed EWMA control charts is evaluated in terms of the average run length and standard deviation of run length, estimated by using Monte Carlo simulations. The proposed EWMA control charts are then compared with their existing counterparts for detecting increases and decreases in the process dispersion. It turns out that the EWMA‐RSS control chart performs uniformly better than its analogues for detecting overall changes in process dispersion. Moreover, the EWMA‐SRS chart significantly outperforms the existing EWMA charts for detecting increases in process variability. A real data set is also used to explain the construction and operations of the proposed EWMA control charts. Copyright © 2013 John Wiley & Sons, Ltd.

A New Cumulative Sum Quality Control Scheme for Monitoring the Process Mean

A control chart is a powerful statistical process monitoring tool that is frequently used in many industrial and service organizations to monitor in‐control and out‐of‐control performances of the manufacturing processes. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts have been recognized as potentially powerful tool in quality and management control. These control charts are sensitive to both small and moderate changes in the process. In this paper, we propose a new CUSUM (NCUSUM) quality control scheme for efficiently monitoring the process mean. It is shown that the classical CUSUM control chart is a special case of the proposed controlling scheme. The NCUSUM control chart is compared with some of the recently proposed control charts by using characteristics of the distribution of run length, i.e. average run length, median run length and standard deviation of run length. It is worth mentioning that the NCUSUM control chart detects the random shifts in the process mean substantially quicker than the classical CUSUM, fast initial response‐based CUSUM, adaptive CUSUM with EWMA‐based shift, adaptive EWMA and Shewhart–CUSUM control charts. An illustrative example is given to exemplify the implementation of the proposed quality control scheme. Copyright © 2013 John Wiley & Sons, Ltd.

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.