Comparison of standard and individual limits Phase I Shewhart $\overline{X}$, $R$, and $S$ charts

Phase I control charts are used as aids in bringing a process into and defining the meaning of a process being in a state of statistical in‐control. This is done by looking at the data in retrospect to answer the question ‘were these data taken from an in‐control process?’ Typically the data is collected as independent random samples taken periodically from the output of the process. Commonly it is recommended that each sample be compared via a statistic(s) to a control limit(s) that is (are) function(s) of all the data. We refer to these charts as ‘standard limits’ charts. An alternative method is to compare each sample with all or some subcollection of the remaining samples. These charts are referred to as ‘individual’ limits charts. The individual limits charts appear to be attractive in detecting a sample from an out‐of‐control process when the other samples were taken when the process was in‐control. We demonstrate using a simple probability analysis that a standard limits Phase I Shewhart chart performs better than the individual limits Phase I Shewhart chart. Furthermore, it is shown that the individual limits Phase I Shewhart charts can be designed to be equivalent. This also holds for the individual and standard limits Phase I Shewhart charts. Copyright © 2003 John Wiley & Sons, Ltd.     

A sensitivity analysis of an integrated model for joint determination of economic design of $\overline{x}$‐control charts,economic production quantity and production run length for a deteriorating production system

This study presents an integrated model for the joint economic design of ‐control charts and maintenance schedules and, simultaneously, determines the economic production quantity and production run length for a deteriorating production system. The operating state of the production process is classified as either in control or out of control. In the latter state, the process produces some defective items. An ‐control chart is used to monitor the process mean. Both uniform and non‐uniform inspection schemes are adopted. Inspection and maintenance are performed simultaneously. Replacement cost is assumed to be very high. The process failure mechanism is assumed to follow a general probability distribution with an increasing failure rate. The concept of a truncated production cycle is introduced. The production cycle begins when a new component is installed and ends with a repair after the detection of a failure or after a specified number of inspection intervals, , whichever occurs first. The effects of preventive maintenance on quality control are discussed. Numerical examples are provided to evaluate the performance of the model. Sensitivity analyses are conducted to study the effects of various model parameters. Copyright © 2002 John Wiley & Sons, Ltd.     

The reliability analysis of balking and reneging in a repairable system with warm standbys

We study the reliability analysis of a repairable system with operating units, warm standby units and repairmen in which the balking and reneging of units are considered. It is assumed that failed units balk (refuse to join) with a constant probability ( ) and renege (leave the queue after joining) according to a negative exponential distribution with parameter . The failure times of the operating unit and of a standby unit are assumed to be exponentially distributed with parameters λ and α, respectively. The repair time distribution is also exponential. The repairable system is out of , in which . The reliability characteristics, such as the system reliability and the mean time to system failure (MTTF), in a repairable system are derived. Several cases are analysed graphically to study the effect of various parameters on the reliability and MTTF of the system. Copyright © 2002 John Wiley & Sons, Ltd.     

On the distribution of the estimated process yield index Spk

This paper considers an asymptotic distribution for an estimate of the process yield index proposed by Boyles (1994). The asymptotic distribution of is useful in statistical inferences for . An illustrative example is given for hypothesis testing and for interval estimation on the yield index . Copyright © 2002 John Wiley & Sons, Ltd.     

An Improved R (Range) Control Chart for Monitoring the Process Variance

An R chart is often used to monitor shifts in the process variability. However, the range, , statistics from a sampling distribution are highly skewed. Hence, the classical R chart based on the control limits will not give an in‐control average run length of approximately 370, or equivalently a type I error, . In this paper, an approach is shown to obtain the control limits of an improved R chart based on a desired type I error from the density function of the R_i statistics. Copyright © 2004 John Wiley & Sons, Ltd.     

On the reliability of the estimated incapability index

Greenwich and Jahr‐Schaffrath (1995) introduced the process incapability index , which provides an uncontaminated separation between information concerning the process precision and process accuracy . In this paper, we consider the three indices, and investigate the statistical properties of their natural estimators. For the three indices, we obtain their UMVUEs and MLEs, and compare the reliability of the two estimators based on the relative mean square errors. In addition, we construct 90%, 95%, and 99% upper confidence limits, and the maximum values of for which the process is capable 90%, 95%, and 99% of the time. The results obtained in this paper are useful to the practitioners in choosing good estimators and making reliable decisions on judging process capability. Copyright © 2001 John Wiley & Sons, Ltd.     

A Bayesian Approach to Obtain a Lower Bound for the Cpm Capability Index

The Taguchi capability index C_pm, which incorporates the departure of the process mean from the target value, has been proposed to the manufacturing industry for measuring manufacturing capability. A Bayesian procedure has been considered for testing process performance assuming , which was generalized without assuming . Statistical properties of the natural estimator of the index C_pm for normal processes have been investigated extensively. However, the investigation was restricted to processes with symmetric tolerances. Recently, a generalized C_pm, referred to as , was proposed to cover processes with asymmetric tolerances. Under the normality assumption, the statistical properties of the estimated including the exact sampling distribution, the rth moment, expected value, variance, and the mean‐squared error were obtained. In this paper, we use a Bayesian approach to obtain the interval estimation for the generalized Taguchi capability index . Consequently, the manufacturing capability testing can be performed for quality assurance. Copyright © 2005 John Wiley & Sons, Ltd.     

A fraction defective based capability index

Two problems greatly affect the use of capability indices such as , and : the lack of affinity with the process fraction defective π and the difficulty of dealing with the sampling distributions of their natural estimators. In this paper, a capability index which is in one‐to‐one correspondence with π is introduced and simple inferential procedures under a Bayesian perspective are developed to facilitate its use in industrial applications. Copyright © 2001 John Wiley & Sons, Ltd.     

Estimated incapability index: reliability and decision making with sample information

The process incapability index , which provides an uncontaminated separation between information concerning the process precision and process accuracy, has been proposed to measure process performance for industry applications. In this paper, we investigate the reliability of the natural estimator computationally, based on the ‐level confidence relative error for various sample sizes. We also develop a decision‐making procedure for judging if the process satisfies the preset quality requirement. The investigation is useful to the practitioners in determining the sample sizes required in their applications for the decisions reliable to the desired level. Copyright © 2002 John Wiley & Sons, Ltd.     

Scalable Chi‐Square Distance versus Conventional Statistical Distance for Process Monitoring with Uncorrelated Data Variables

Multivariate statistical process control charts are often used for process monitoring to detect out‐of‐control anomalies. However, multivariate control charts based on conventional statistical distance measures, such as the one used in the Hotelling's control chart, cannot scale up to large amounts of complex process data, e.g. data with a large number of variables and a high rate of data sampling. In our previous work we developed a multivariate statistical process monitoring procedure based on a more scalable chi‐square distance measure and tested this procedure for detecting out‐of control anomalies—intrusions—in a computer process using computer audit data. The testing results demonstrated the comparable performance of the scalable chi‐square procedure to that of Hotelling's control chart. To establish the chi‐square procedure as a generic, viable multivariate statistical processing monitoring procedure, we conduct a series of further studies to understand the detection power and limitations of the chi‐square procedure for processes with various kinds of data and various types of out‐of‐control anomalies in addition to the scalability and demonstrated performance of the chi‐square procedure for computer intrusion detection. This paper reports on one of these studies that investigates the effectiveness of the scalable chi‐square procedure in detecting out‐of‐control anomalies in processes with uncorrelated data variables, each of which has a normal probability distribution. The results of this study indicate that the chi‐square procedure is at least as effective as Hotelling's control chart for monitoring processes with uncorrelated data variables. Copyright © 2003 John Wiley & Sons, Ltd.     

A Comparative Study of Joint Monitoring Schemes for APC Processes

To monitor processes under automatic process control (APC), several joint monitoring schemes have recently been proposed, such as the U statistics, and , that originate from the uniformly most powerful test. It is known that the chart is sensitive to large shift detection whereas the chart is sensitive to small shift detection. To take advantage of both charts, this paper examines a combined use of the two U charts. A comparative study indicates that the combined procedure improves over a single U chart and performs better than other joint monitoring schemes, including Hotelling's chart for APC processes. Copyright © 2006 John Wiley & Sons, Ltd.     

An adaptive chart for monitoring the process mean and variance

Traditionally, an chart is used to control the process mean and an R chart is used to control the process variance. However, these charts are not sensitive to small changes in the process parameters. The adaptive and R charts might be considered if the aim is to detect small disturbances. Due to the statistical character of the joint and R charts with fixed or adaptive parameters, they are not reliable in identifying the nature of the disturbance, whether it is one that shifts the process mean, increases the process variance, or leads to a combination of both effects. In practice, the speed with which the control charts detect process changes may be more important than their ability in identifying the nature of the change. Under these circumstances, it seems to be advantageous to consider a single chart, based on only one statistic, to simultaneously monitor the process mean and variance. In this paper, we propose the adaptive non‐central chi‐square statistic chart. This new chart is more effective than the adaptive and R charts in detecting disturbances that shift the process mean, increase the process variance, or lead to a combination of both effects. Copyright © 2006 John Wiley & Sons, Ltd.     

A CUSUM scheme with variable sample sizes and sampling intervals for monitoring the process mean and variance

The adaptive control feature and CUSUM chart are two monitoring schemes that are much more effective than the traditional static Shewhart chart in detecting process shifts in mean and variance. However, the designs and analyses of the adaptive CUSUM chart are mathematically intractable and the operation is very laborious. This article proposes a VSSI WLC scheme, which is a weighted‐loss‐function‐based CUSUM (WLC) scheme using variable sample sizes and sampling intervals (VSSI). This scheme detects the two‐sided mean shift and increasing standard deviation shift based on a single statistic WL (the weighted loss function). Most importantly, the VSSI WLC scheme is much easier to operate and design than a VSSI CCC scheme which comprises three individual CUSUM charts (two of them monitoring the increasing and decreasing mean shifts and one monitoring the increasing variance shift). Overall, the VSSI WLC scheme is much more effective than the static &S charts (by 72.36%), the VSSI &S charts (by 30.97%) and the static WLC scheme (by 50.94%) for detection. It is even more effective than the complicated VSSI CCC scheme for most cases. Copyright © 2006 John Wiley & Sons, Ltd.     

Distributional properties of estimated capability indices based on subsamples

Under the assumption of normality, the distribution of estimators of a class of capability indices, containing the indices , , and , is derived when the process parameters are estimated from subsamples. The process mean is estimated using the grand average and the process variance is estimated using the pooled variance from subsamples collected over time for an in‐control process. The derived theory is then applied to study the use of hypothesis testing to assess process capability. Numerical investigations are made to explore the effect of the size and number of subsamples on the efficiency of the hypothesis test for some indices in the studied class. The results for and indicate that, even when the total number of sampled observations remains constant, the power of the test decreases as the subsample size decreases. It is shown how the power of the test is dependent not only on the subsample size and the number of subsamples, but also on the relative location of the process mean from the target value. As part of this investigation, a simple form of the cumulative distribution function for the non‐central ‐distribution is also provided. Copyright © 2003 John Wiley & Sons, Ltd.     

Capability measures for processes with multiple characteristics

Process capability indices, such as , , and , have been widely used in the manufacturing industry providing numerical measures on process precision, process accuracy, and process performance. Capability measures for processes with a single characteristic have been investigated extensively. However, capability measures for processes with multiple characteristics are comparatively neglected. In this paper, we consider a generalization of the yield index proposed by Boyles, for processes with multiple characteristics. We establish a relationship between the generalization and the process yield. We also develop a control chart based on the proposed generalization, which displays all the characteristic measures in one single chart. Using the chart, the engineers can effectively monitor and control the performance of all process characteristics simultaneously. Copyright © 2003 John Wiley & Sons, Ltd.     

Attribute Charts for Monitoring a Dependent 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 which classifies an item as either ‘conforming’ or ‘non‐conforming’. A tool used with a considerable amount of success in industry for monitoring the quality of a production process is the quality control chart. Generally a control charting procedure uses a sequence, of the quality measures to make a decision about the quality of the process. How this sequence is used to make a decision defines the control chart. In order to design a control chart one must consider how the underlying sequence, is modeled. The sequence is often modeled as a sequence of independent and identically distributed random variables. For many industrial processes, this model is appropriate, but in others it may not be. In this paper, a sequence of random variables, is used to classify an item as conforming or non‐conforming under a stationary Markov chain model and under 100% sequential sampling. Two different control charting schemes are investigated. Both schemes plot a sequence of measures on the control chart, that count the number of conforming items before a non‐conforming item. The first scheme signals as out‐of‐control if a value of falls below a certain lower limit. The second scheme signals as out‐of‐control if two out of two values of fall below a certain lower limit. The efficiency of both of the control charts is evaluated by the average run length (ARL) of the chart and the power of the chart to detect a shift in the process. The two out of two scheme is shown to have high power and a large ARL given certain parameter values of the process. An example of the two out of two scheme is provided for the interested reader. Copyright © 2006 John Wiley & Sons, Ltd.     

Robustness properties of multivariate EWMA control charts

In this paper, the robustness of the multivariate exponentially weighted moving average (MEWMA) control chart to non‐normal data is examined. Two non‐normal distributions of interest are the multivariate distribution and the multivariate gamma distribution. Recommendations for constructing MEWMA control charts when the normality assumption may be violated are provided. Copyright © 2002 John Wiley & Sons, Ltd.     

Statistical analysis of process capability indices with measurement errors

Process capability indices (PCIs) have been widely used in manufacturing industries to provide a quantitative measure of process potential and performance. While some efforts have been dedicated in the literature to the statistical properties of PCIs estimators, scarce attention has been given to the evaluation of these properties when sample data are affected by measurement errors. In this work we deal with the problem of measurement‐error effects on the performance of PCIs. The analysis is illustrated with reference to and , i.e. the two most common measures suggested to evaluate process capability. Copyright © 2002 John Wiley & Sons, Ltd.     

Some notes on the construction and evaluation of supersaturated designs

Supersaturated designs offer a potentially useful way to investigate many factors with few experiments. A super‐saturated design evaluates factors with experiments, where . In some supersaturated designs the number of factors being investigated may exceed the number of experiments by a factor of three or more. The problem of developing a supersaturated design is described in computational complexity terms and methods of construction are discussed. Using standard combinatorial formulation may permit researchers to use additional methods to develop additional designs. Designs are generated with a modified method and compared with published designs using existing criteria and a new criterion. Copyright © 2002 John Wiley & Sons, Ltd.     

Control charts for positively‐skewed populations with weighted standard deviations

This paper proposes a heuristic method of constructing , cumulative sum and exponentially weighted moving average control charts for skewed populations with weighted standard deviations obtained by decomposing the standard deviation into upper and lower deviations adjusted in accordance with the direction and degree of skewness. These control charts, however, reduce to standard control charts when the underlying distribution is symmetric. Simple formulae are derived to estimate unknown process parameters from means and ranges of subgroups. The false alarm rates of these control charts are compared with those of existing control charts when the underlying distribution is Weibull, gamma, or lognormal. Simulation results show that considerable improvement over the existing methods can be achieved when the underlying distribution is skewed. Copyright © 2001 John Wiley & Sons, Ltd.