Conforming Run Length Research Articles

Optimal design of the synthetic chart for the process mean based on median run length

Control charts are usually designed using average run length as the criterion to be optimized. The shape of the run length distribution changes according to the shift in the mean, from highly skewed for an in-control process to approximately symmetric when the mean shift is large. Since the shape of the run length distribution changes with the mean shift, the Median Run Length (MRL) provides a more meaningful interpretation for in-control and out-of-control performances of the charts, and it is readily understood by practitioners. This article presents an optimal design procedure for a synthetic chart able to monitor the mean based on the MRL under the zero- and steady-state modes. The synthetic chart integrates and conforming run length charts. Pseudocodes and Mathematica programs are presented for the computation of the optimal parameters of the synthetic chart based on a desired in-control MRL (MRL(0)), a given sample size, and a specified mean shift for which a quick detection is needed. [Supplementary materials are available for this article. Go to the publisher's online edition of IIE Transactions for additional example, additional performance study, proof, tables, and figures.]

Using Runs Rules to Monitor an Attribute Chart for a Markov Process

For some repetitive production processes, the quality measure taken on the output is an attribute variable. An attribute variable classifies each output item into one of a countable set of categories. One of the simplest and most commonly used attribute variables is the one that classifies an item as either “conforming” or “nonconforming.” A tool used with a considerable amount of success in industry for monitoring the quality of a production process is the quality control chart. In this paper, a sequence of random variables, Xi, i = 1, 2, … , is used to classify an item as conforming or nonconforming under a stationary Markov chain model and under 100% sequential sampling. The sequence of random variables to be plotted on the control chart is Y, i = 1, 2, … , where Y1 counts the number of conforming items before the first nonconforming item and Yi, i = 2, 3,. counts the number of conforming items between the (i-l)th and the ih nonconforming items. In the literature, Yi is called the CRL (Conforming Run Length). The contribution of this paper to the literature is to present runs rules for an attribute control chart of this type. The efficiency of these charts is evaluated using the average run length (ARL) of the charts. The supplemental runs rules that are presented are two out of three values of Yi, i = 1,2, … falling below determined lower limits, four out of five values of Yi, i = 1,2, … falling below determined lower limits, and eight out of eight values of Yi, i = 1,2,... falling below determined lower limits.

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.

Nonparametric Synthetic Control Charts for Process Variation

In this article, we propose nonparametric synthetic and side‐sensitive synthetic control charts for controlling fraction nonconforming due to increase in the process variation. Synthetic control chart is a combination of sign and conforming run length control charts. We compare performance of the proposed control charts with the Shewhart sign and S2 charts. Our performance study shows that the proposed control charts have a higher power of detecting out‐of‐control signal. We also study the steady‐state behavior of a nonparametric synthetic control chart. We present a Markov chain model to evaluate the steady‐state average run length of the synthetic and side‐sensitive synthetic control charts. Copyright © 2011 John Wiley & Sons, Ltd.

The synthetic [Xbar] chart with estimated parameters

A synthetic [Xbar] chart consists of an integration of a Shewhart [Xbar] chart and a conforming run length chart. This type of chart has been extensively used to detect a process mean shift under the assumption of known process parameters. However, in practice, the process parameters are rarely known and are usually estimated from an in-control Phase I data set. The goals of this article are to (i) evaluate (using a Markov chain model) the performances of the synthetic [Xbar] chart when the process parameters are estimated; (ii) compare it with the case where the process parameters are assumed known to demonstrate that these performances are quite different when the number of samples used during Phase I is small; and (iii) suggest guidelines concerning the choice of the number of Phase I samples and to provide new optimal constants, especially dedicated to the number of samples used in practice.

A synthetic double sampling control chart for the process mean

This article proposes a synthetic double sampling chart that integrates the Double Sampling (DS) chart and the conforming run length chart. The proposed procedure offers performance improvements in terms of the zero-state Average Run Length (ARL) and Average Number of Observations to Sample (ANOS). When the size of a mean shift δ (given in terms of the number of standard deviation units) is small (i.e., between 0.4 and 0.6) and the mean sample size n= 5, the proposed procedure reduces the out-of-control ARL and ANOS values by nearly half, compared with both the synthetic and DS charts. In terms of detection ability versus the Exponentially Weighted Moving Average (EWMA) chart, the synthetic DS chart is superior to the synthetic or even the DS chart, as the former outperforms the EWMA chart for a larger range of δ values compared to the latter. The proposed procedure generally outperforms the EWMA chart in the detection of a mean shift when δ is larger than 0.5 and n= 5 or 10. Although the proposed procedure is less sensitive than the EWMA chart when δ is smaller than 0.5, this may not be a setback as it is usually not desirable, from a practical viewpoint, to signal very small shifts in the process to avoid too frequent process interruptions. Instead, under such circumstances, it is better to leave the process undisturbed.

A Nonparametric Shewhart-Type Synthetic Control Chart

In this article, we provide a nonparametric Shewhart-type synthetic control chart based on the signed-rank statistic to monitor shifts in the known in-control process median. The synthetic control chart is a combination of a signed-rank chart due to Bakir (2004) and a conforming run length chart due to Bourke (1991). The operation and design of the chart are discussed and the performance of the chart has been studied. The chart has an attractive average run length behavior as compared to the parametric control chart for a class of symmetric continuous process distributions. The proposed chart performs better than the nonparametric signed-rank chart given by Bakir (2004) and Chakraborti and Eryilmaz (2007).

A generalized Conforming Run Length control chart for monitoring the mean of a variable

The attribute Conforming Run Length ( CRL) control chart has attracted increasing research interests in Statistical Process Control (SPC). It decides the process status based on the interval or distance between two nonconforming units. This article proposes a Generalized CRL chart (namely GCRL chart) for monitoring the mean of a measurable quality characteristic x under 100% inspection. To run a GCRL chart, each unit will be classified as a passing or nonpassing unit depending on whether the sample value of x falls within or beyond a pair of lower and upper inspection limits LIL and UIL. When a nonpassing unit is detected, the GCRL chart checks the distance between the current and last nonpassing units in order to determine the process status (in control or out of control). The inspection limits LIL and UIL are determined by an optimization design. The GCRL chart not only solves a dead-corner problem suffered by the conventional CRL chart, but also considerably outperforms the latter for detecting mean shifts. The most interesting finding is that the attribute GCRL chart excels the variable X chart to a significant degree in SPC for variables. It suggests that the simple attribute chart may replace the variable chart in some SPC applications. The design of the GCRL chart has to be carried out by a computer program, but the design can be completed almost in no time in a personal computer.

Monitoring the process mean and variance using a synthetic control chart with two-stage testing

In this paper, we propose a synthetic control chart with two-stage testing (SyTS chart) to control the process mean and variance. As in the case of Shewhart control charts, samples are taken from the process at regular time intervals; however, testing is performed in two stages. During the first stage, one item of the sample is inspected; if its value is close to the target value of the process mean, then this terminates testing. Otherwise, the testing goes on to the second stage, where the remaining items are inspected and a non-central chi-square statistic is computed taking into account all items of the sample. When this statistic is larger than a specified value, the sample is classified as nonconforming. According to the synthetic procedure, the signal is based on the conforming run length (CRL). A comparative study shows that the SyTS chart and the joint and S charts with double sampling are very similar in performance. However, from a practical viewpoint, it is more convenient to monitor the process by looking at only one chart rather than looking at two charts separately. Comparisons with the joint and S charts and with several CUSUM schemes show that the SyTS chart has better overall performance.

A Multivariate Synthetic Control Chart for Monitoring the Process Mean Vector of Skewed Populations Using Weighted Standard Deviations

This article proposes a multivariate synthetic control chart for skewed populations based on the weighted standard deviation method. The proposed chart incorporates the weighted standard deviation method into the standard multivariate synthetic control chart. The standard multivariate synthetic chart consists of the Hotelling's T 2 chart and the conforming run length chart. The weighted standard deviation method adjusts the variance–covariance matrix of the quality characteristics and approximates the probability density function using several multivariate normal distributions. The proposed chart reduces to the standard multivariate synthetic chart when the underlying distribution is symmetric. In general, the simulation results show that the proposed chart performs better than the existing multivariate charts for skewed populations and the standard T 2 chart, in terms of false alarm rates as well as moderate and large mean shift detection rates based on the various degrees of skewnesses.

A SYNTHETIC CONTROL CHART FOR MONITORING THE PROCESS MEAN OF SKEWED POPULATIONS BASED ON THE WEIGHTED VARIANCE METHOD

A synthetic control chart for detecting shifts in the process mean integrates the Shewhart [Formula: see text] chart and the conforming run length chart. It is known to outperform the Shewhart [Formula: see text] chart for all magnitudes of shifts and is also superior to the exponentially weighted moving average chart and the joint [Formula: see text]-exponentially weighted moving average charts for shifts of greater than 0.8σ in the mean. A synthetic chart for the mean assumes that the underlying process follows a normal distribution. In many real situations, the normality assumption may not hold. This paper proposes a synthetic control chart to monitor the process mean of skewed populations. The proposed synthetic chart uses a method based on a weighted variance approach of setting up the control limits of the [Formula: see text] sub-chart for skewed populations when process parameters are known and unknown. For symmetric populations, however, the limits of the new [Formula: see text] sub-chart are equivalent to that of the existing [Formula: see text] sub-chart which assumes a normal underlying distribution. The proposed synthetic chart based on the weighted variance method is compared by Monte Carlo simulation with many existing control charts for skewed populations when the underlying populations are Weibull, lognormal, gamma and normal and it is generally shown to give the most favourable results in terms of false alarm and mean shift detection rates.

A Multivariate Synthetic Control Chart for Monitoring Process Mean Vector

In this article, a multivariate synthetic control chart is developed for monitoring the mean vector of a normally distributed process. The proposed chart is a combination of the Hotelling's T 2 chart and Conforming Run Length chart. The operation, design, and performance of the chart are described. Average run length comparisons between some other existing control charts and the synthetic T 2 chart are presented. They indicate that the synthetic T 2 chart outperforms Hotelling's T 2 chart and T 2 chart with supplementary runs rules.

A Multivariate Synthetic Control Chart for Process Dispersion

A multivariate synthetic control chart for monitoring the covariance matrix of a normally distributed process is proposed in this paper. The proposed chart is a combination of the traditional generalized sample variance |S| chart and the conforming run length chart. The operation, design and performance of this chart are described. Average run length comparisons between some other existing control charts and the multivariate synthetic control chart are presented. It is observed that the multivariate synthetic control chart outperforms the traditional |S| chart for monitoring the covariance matrix of a normally distributed process.

A synthetic control chart for monitoring process dispersion with sample range

A synthetic control chart for monitoring the changes in the standard deviation of a normally distributed process is proposed in this paper. The synthetic chart consists of the sample range (R) chart and the conforming run-length (CRL) chart. The R chart can be viewed as a special case of the synthetic chart. The operation, design and performance of this chart are described. Average run- length comparisons between other procedures and the synthetic chart are presented. It indicates that the synthetic chart is a good alternative for monitoring process dispersion. The variable sampling interval (VSI) schemes, as an enhancement to the synthetic chart, are discussed to further improve the chart performance. An example is presented to illustrate the application of synthetic chart and its VSI scheme.

A synthetic control chart for monitoring process dispersion with sample standard deviation

Some factors in manufacturing such as faulty raw material, unskilled/careless operators, and loosening of machine settings may lead to a change in process dispersion without necessarily influencing the level of the process mean. This paper proposes a synthetic control chart for monitoring the changes in the standard deviation of a normally distributed process. The synthetic chart is a combination of the sample standard deviation ( S ) chart and the conforming run length (CRL) chart. The S chart can be regarded as a special case of the synthetic chart. The operation, design, and performance of this chart are described. Average run length comparisons between other procedures and the synthetic chart are presented. It indicates that the synthetic chart is a good alternative for monitoring process dispersion. The variable sampling interval (VSI) schemes as an enhancement to the synthetic chart are discussed to further improve the chart performance. An example is presented to illustrate the application of synthetic chart and its VSI scheme.

A Group Runs Control Chart for Detecting Shifts in the Process Mean

Wu and Spedding [Journal of Quality Technology 32: 32-38, 2000] have proposed the synthetic control chart to detect small shifts in process mean. In this paper, for the same problem, we propose a control chart called ’Group Runs’ (GR) control chart, which is a combination of Shewhart's chart and an extended version of conforming run length chart. It is numerically verified that, the GR chart gives a significant reduction in out of control ATS as compared to those of the synthetic control chart and chart when in-control ATS is not smaller than a certain number. Steady state performance of the GR chart is also better. Hence, GR chart turns out to be economical to detect small shifts in the process mean.

Combined X-bar and CRL Charts for the Gamma Process

This paper (i) discusses the combined $\bar x$ and conforming run length (CRL) charts under the assumption that the quality characteristic under study follows a Gamma(?, ?, β) distribution with known parameters ?, ? and β, (ii) examines the performance of the combined $\bar x$ and CRL charts for the Exp(?, β) process, when its parameters ? and β are unknown and must be estimated from preliminary samples taken during the trial period, and (iii) obtains the adjusted control limits of the combined $\bar x$ and CRL charts that have a specified in-control average run length value for samples taken from the Exp(?, β) process with unknown parameters. Studies of the average run lengths reveal that the combined $\bar x$ and CRL charts outperform the Shewhart $\bar x$ chart with either asymmetric probability limits or 3-sigma control limits.

THE ROBUSTNESS OF THE SYNTHETIC CONTROL CHART TO NON-NORMALITY

The Synthetic control chart has recently been introduced as an improvement over the standard Shewhart control chart for detecting changes in the mean of a normally distributed process. The salient features of the Shewhart chart and the Conforming Run Length chart are integrated to produce the Synthetic control chart. In many practical instances, the Synthetic chart is superior to the Shewhart chart in terms of quicker detection of out-of-control status when the process data are normally distributed. The robustness of the Synthetic chart to violations of the normality assumption is the central theme of this study. We find that in-control average run lengths for the Synthetic chart are reasonably close to the normal theory values when there is moderate nonnormality or when the sample size n is large. Additionally, out-of-control average run lengths are comparable to the corresponding normal theory values for a variety of non-normal distributions.

A Synthetic Control Chart for Detecting Fraction Nonconforming Increases

In this paper we propose a new control chart, the synthetic control chart, for the detection of increases in the fraction nonconforming (p). The operation, design, and performance of this control chart are described. The synthetic chart is a combination of the np chart and the conforming run length chart. It provides the user with some freedom in adjusting the control chart parameters so that the out-of-control average time to signal can be minimized. Numerical tests have indicated that the synthetic chart has a higher power for detecting process shifts in the fraction nonconforming than both the np chart and the conforming run length chart. The improvement is significant for all cases, resulting in a 50% or greater reduction of the out-of-control average time to signal. The application of the synthetic chart may therefore substantially enhance the effectiveness of statistical process control for attributes.

A comparative study of the CRL-type control charts

The CRL (Conforming Run Length) type control charts have attracted increasing interest recently for attribute Statistical Process Control (SPC). The two most promising charts of this type are identified as the CRL-CUSUM chart and the SCRL (Sum of CRLs) chart. This article compares the operating characteristics of these two charts in a comprehensive manner. The general findings reveal that the CRL-CUSUM chart excels the SCRL chart in detecting downward (decreasing) fraction nonconforming (p) shifts and large-scale upward (increasing) p shifts. However, the SCRL chart is superior to the CRL-CUSUM chart in detecting the small and moderate scale upward p shifts, especially when the normal p value is small. The information acquired in this study will provide Quality Assurance (QA) engineers with useful guidance for selecting and applying the CRL-type control charts. Copyright © 2000 John Wiley & Sons, Ltd.