Michael B. C. Khoo

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.

Design of Runs Rules Schemes

Runs rules are often used to increase the sensitivity of a Shewhart control chart. In this work, plots of various runs rules schemes are given to simplify the determination of control limits based on a desired in-control average run length (ARL0).

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.

Optimization designs of the combined Shewhart-CUSUM control charts

This article presents the optimization design of the combined Shewhart chart and CUSUM chart (X@?&CUSUM chart in short) used in Statistical Process Control (SPC). While the optimization design effectively improves the overall performance of the X@?&CUSUM chart over the entire process shift range, it does not increase difficulties in understanding and implementing this combined chart. A new feature pertaining to an additional charting parameter w (the exponential of the sample mean shift) is also investigated, with the hope of further enhancing the detection effectiveness of the X@?&CUSUM chart. Moreover, this article provides the SPC practitioners with a design table to facilitate the designs of the X@?&CUSUM charts. From this design table, the users can directly find the optimal values of the charting parameters, according to the design specifications. The design table makes the design of an X@?&CUSUM chart as simple as the design of the simplest X@? chart. In general, this article will help to enhance the detection effectiveness of the X@?&CUSUM chart, and facilitate and promote its applications in SPC.

A combined synthetic&X chart for monitoring the process mean

The Shewhart X chart (or X chart) is widely used to monitor the mean of a quality characteristic x. This chart decides the process status based on the magnitude of the sample mean x and is effective for detecting large mean shifts. The synthetic chart is also a Shewhart type chart for monitoring the process mean, but it utilises the information about the time interval between two nonconforming samples. Here a sample is nonconforming if its x value falls beyond the predetermined warning limits. Unlike the X chart, the synthetic chart is more powerful to detect small shifts. The applications of the X and synthetic charts cover a wide variety of manufacturing processes and production lines, e.g., the monitoring of the mean values of the inside diameter of a piston-ring, the viscosity of aircraft paint, the resistivity of silicon wafers. This article proposes a combined scheme, the Syn-X chart, that comprises a synthetic chart and an X chart. The results of the performance studies show that the Syn-X chart always outperforms the individual X chart and synthetic chart under different conditions. It is more effective than the X chart and synthetic chart by 47% and 20%, respectively, over the wide range of mean shift values in different experiment runs.

Economic and economic statistical designs of the synthetic X¯ chart using loss functions

This paper proposes the economic and economic statistical designs of the synthetic X¯ chart. In the economic design, the optimal chart parameters that minimize the expected cost function are obtained, while in the economic statistical design, the optimal chart parameters are obtained by minimizing the expected cost function, subject to constraints on the in-control average run length (ARL0) and the out-of-control average run length (ARL1). A small increase in cost is incurred when the statistical constraints are added to the economic design, however, a significant improvement in statistical performance is attained. The sensitivity of the optimal cost and the chart parameters for different loss functions and input parameters is investigated. The effects of misspecification of the type of the loss function, and the Taguchi loss coefficient, as well as the risk aversion coefficient of the loss function, are also investigated. In addition, effects of the process capability index are studied. Based on numerical studies, comparisons are made between the synthetic X¯, Shewhart X¯, and EWMA charts.

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 chart and the conforming run length chart. It is known to outperform the Shewhart chart for all magnitudes of shifts and is also superior to the exponentially weighted moving average chart and the joint -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 sub-chart for skewed populations when process parameters are known and unknown. For symmetric populations, however, the limits of the new sub-chart are equivalent to that of the existing 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.

Two Improved Runs Rules for the Shewhart X¯ Control Chart

Runs rules are incorporated into the Shewhart X¯ control chart to increase its sensitivity in detecting small process shifts so that assignable causes can be detected more quickly. In this article, two improved runs rules are suggested. Average run length values for these new improved rules are computed and compared with that of the existing ones. The comparison shows that the new improved rules are superior in performance for large process average shifts, while maintaining the same sensitivity in the detection of small shifts.

The variable sampling interval X̄ chart with estimated parameters

The VSI chart has been investigated by many researchers under the assumption of known process parameters. However, in practice, these parameters are usually unknown and it is necessary to estimate them from the past data. In this paper, we evaluate and compare the performance of the VSI chart in terms of its average time to signal in the case where the process parameters are known and in the case where these parameters are estimated. We also provide new chart constants taking into account the number of phase I samples. Copyright

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.