Ming Ha Lee

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.

Optimal Statistical Design of a Multivariate EWMA Chart Based on ARL and MRL

Statistical design is applied to a multivariate exponentially weighted moving average (MEWMA) control chart. The chart parameters are control limit H and smoothing constant r. The choices of the parameters depend on the number of variables p and the size of the process mean shift δ. The MEWMA statistic is modeled as a Markov chain and the Markov chain approach is used to determine the properties of the chart. Although average run length has become a traditional measure of the performance of control schemes, some authors have suggested other measures, such as median and other percentiles of the run length distribution to explain run length properties of a control scheme. This will allow a thorough study of the performance of the control scheme. Consequently, conclusions based on these measures would provide a better and comprehensive understanding of a scheme. In this article, we present the performance of the MEWMA control chart as measured by the average run length and median run length. Graphs are given so that the chart parameters of an optimal MEWMA chart can be determined easily.

The design of the multivariate synthetic exponentially weighted moving average control chart

A multivariate synthetic exponentially weighted moving average (MSEWMA) control chart is presented in this study. The MSEWMA control chart consists of a multivariate exponentially weighted moving average (MEWMA) control chart and a conforming run length control chart. The average run length of the MSEWMA control chart is obtained using a Markov chain approach. From the numerical comparisons, it is shown that the MSEWMA control chart is more efficient than the multivariate synthetic T 2 control chart and the MEWMA control chart for detecting shifts in the process mean vector.

OPTIMAL STATISTICAL DESIGNS OF A MULTIVARIATE CUSUM CHART BASED ON ARL AND MRL

Optimal statistical designs of the multivariate CUSUM (MCUSUM) chart for multivariate individual observations based on ARL and MRL are proposed. Statistical design procedures refer to choices of the reference value, k and the control limit, H to ensure that the MCUSUM charts performance meets certain statistical criteria. The primary criterion is the average run length (ARL) which is the most commonly used measure of a control charts performance, while the median run length (MRL) which is the 50th percentage point of the run length distribution is suggested to be used as a potential alternative to the ARL or as a secondary criterion in the evaluation of the performance of the MCUSUM control chart. Although the MRL is used in the optimal design of the univariate EWMA and univariate CUSUM charts but the design of an optimal multivariate CUSUM chart based on MRL is not yet given in any literature. This paper also suggests a systematic approach of designing an optimal MCUSUM chart based on the average run length (ARL). The MRL profiles are considered as supplements to the ARL profiles for the control scheme. Examples of optimal designs of the MCUSUM chart based on both MRL and ARL are also presented. Tables are provided to determine the optimal chart parameters for the design of the chart based on both ARL and MRL.

Multivariate Synthetic |S| Control Chart with Variable Sampling Interval

A variable sampling interval (VSI) feature is introduced to the multivariate synthetic generalized sample variance |S| control chart. This multivariate synthetic control chart is a combination of the |S| sub-chart and the conforming run length sub-chart. The VSI feature enhances the performance of the multivariate synthetic control chart. The comparative results show that the VSI multivariate synthetic control chart performs better than other types of multivariate control charts for detecting shifts in the covariance matrix of a multivariate normally distributed process. An example is given to illustrate the operation of the VSI multivariate synthetic chart.

The Exact Run Length Distribution and Design of the Shewhart Chart with Estimated Parameters Based on Median Run Length

This paper investigates the Shewhart chart with estimated parameters. When parameters are unknown, we show that the average run length is a confusing performance measure and that the median run length, which is a better representation of the central tendency associated with different shapes of the run length distribution, is more intuitive. A new design model for the Shewhart chart with estimated parameters is developed. With this proposed design model, the Shewhart chart for both known and estimated parameters, even with a small number of Phase-I samples, will have an almost similar sensitivity for a specified shift.

Economically Optimal Design of a Multivariate Synthetic T 2 Chart

Economic and economic-statistical models are developed for the synthetic T 2 chart. The input parameters that result in larger cost and affect the optimal parameters are identified. The optimal parameters are quite robust toward changes in input parameters, except the number of variables and the Mahalanobis distance. Alternative choices of parameters, which result in minimal cost increase, can be chosen if it is infeasible to operate the chart optimally. The results are based on numerical examples and verified through simulation. The synthetic T 2 chart has better economic and economic-statistical performances than the Hotellings T 2 and MEWMA charts under most conditions.

Multivariate EWMA Control Chart with Adaptive Sample Sizes

Standard multivariate control charts usually employ fixed sample sizes at equal sampling intervals to monitor a process. In this study, a multivariate exponential weighted moving average (MEWMA) chart with adaptive sample sizes is investigated. Performance measure of the adaptive-sample-size MEWMA chart is obtained through a Markov chain approach. The performance of the adaptive-sample-size MEWMA chart is compared with the fixed-sample-size control chart in terms of steady-state average run length for different magnitude of shifts in the process mean. It is shown that the adaptive-sample-size chart is more efficient than the fixed-sample-size MEWMA control chart in detecting shifts in the process mean.

Synthetic Double Sampling X̄ Chart with Estimated Process Parameters

Abstract The synthetic double sampling (SDS) X̄ chart comprises the double sampling (DS) X̄ and conforming run length (CRL) sub-charts. The SDS X̄ chart has been studied under the assumption of known process parameters in the literature. Nevertheless, in practice, process parameters are usually unknown and are estimated from an in-control Phase I dataset. This paper investigates the performance of the SDS X̄ chart with estimated process parameters, in terms of the average run length (ARL), average number of observations to signal (ANOS) and standard deviation of the run length (SDRL). The performance of the SDS X̄ chart with estimated process parameters by minimizing the out-of-control ARL and the out-of-control ANOS is compared with the corresponding chart’s performance with known process parameters. In addition, the minimum number of Phase I samples required by the SDS X̄ chart with estimated process parameters so that it has approximately the same in-control ARL and ANOS performances as the chart with known process parameters is studied. The ARL, ANOS and SDRL properties of the SDS X̄ chart with estimated process parameters differ significantly from that of the chart with known process parameters. Therefore, suitable optimal charting parameters are introduced so that the SDS X̄ chart with estimated process parameters has an adequate performance as its known process parameters counterpart without having to use large number of Phase I samples and sample size.

A variable sampling interval synthetic Xbar chart for the process mean

The usual practice of using a control chart to monitor a process is to take samples from the process with fixed sampling interval (FSI). In this paper, a synthetic X¯ control chart with the variable sampling interval (VSI) feature is proposed for monitoring changes in the process mean. The VSI synthetic X¯ chart integrates the VSI X¯ chart and the VSI conforming run length (CRL) chart. The proposed VSI synthetic X¯ chart is evaluated using the average time to signal (ATS) criterion. The optimal charting parameters of the proposed chart are obtained by minimizing the out-of-control ATS for a desired shift. Comparisons between the VSI synthetic X¯ chart and the existing X¯, synthetic X¯, VSI X¯ and EWMA X¯ charts, in terms of ATS, are made. The ATS results show that the VSI synthetic X¯ chart outperforms the other X¯ type charts for detecting moderate and large shifts. An illustrative example is also presented to explain the application of the VSI synthetic X¯ chart.