Wai Chung Yeong

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.

Economically Optimum Design of a Synthetic X¯ Chart

This paper proposes an economic model for the synthetic chart. The synthetic chart is an integration of the chart and the CRL chart. A simplified algorithm to obtain the optimal parameters of the synthetic chart which minimizes the expected cost function is introduced. Numerical examples based on different values of input parameters are given, and sensitivity analyses of the parameters are performed. The input parameters which have a significant impact on the cost and choice of optimal parameters of the synthetic chart are identified. The effect of incorrect estimation of the input parameters is also investigated, and it is found that the optimal design parameters are quite robust to changes in the input parameters, except the shift parameter. It is also shown that if the chart cannot be operated at the economically optimal level, there is still a large choice of parameters to choose from which does not result in a large increase in cost. All the above analyses and results are based on numerical examples and verified through simulation over a wide range of parameter values. Comparisons are also performed among the synthetic Shewhart and EWMA charts. Based on the numerical examples and simulation over a wide range of parameter values, it is shown that the synthetic chart has better economic performance than the other two control charts under most conditions. Copyright

Economic‐Statistical Design of the Synthetic X¯ Chart with Estimated Process Parameters

In this paper, the effects of process parameter estimation on the cost of the synthetic chart are studied. We study the increase in cost when the optimal charts parameters corresponding to the known process parameters case are used to estimate the cost when the process parameters are actually just estimated. By studying the increase in cost, practitioners will be able to determine whether the optimal charts parameters, computed based on known process parameters, can still be used to reliably estimate the cost when the target values of the process mean and variance are estimated. We also look at the minimum number of preliminary subgroups needed for the estimation of process parameters so that the cost for the estimated process parameters case is almost the same as the cost for the known process parameters case. Furthermore, we also look at the cost savings, in the case of process parameter estimation when the optimal charts parameters are computed based on estimated process parameters, instead of adopting the charts parameters corresponding to known process parameters. This enables practitioners to determine the cost advantages of selecting the charts parameters that minimize the cost when process parameters are estimated, instead of adopting the optimal charting parameters corresponding to the case of known process parameters. Copyright

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.

Run-sum control charts for monitoring the coefficient of variation

The coefficient of variation (CV) is a unit-free and effective normalized measure of dispersion. Monitoring the CV is a crucial approach in Statistical Process Control when the quality characteristic has a distinct mean value and its variance is a function of the mean. This setting is common in many scientific areas, such as in the fields of engineering, medicine and various societal applications. Therefore, this paper develops a simple yet efficient procedure to monitor the CV using run-sum control charts. The run-length properties of the run-sum CV (RS-γ) charts are characterized by the Markov chain approach. This paper proposes two optimization algorithms for the RS-γ charts, i.e. by minimizing (i) the average run length (ARL) for a deterministic shift size and (ii) the expected ARL over a process shift domain. Performance comparisons under both the zero- and steady-state modes are made with the Shewhart-γ, Run-rules-γ and EWMA-γ charts. The results show that the proposed RS-γ charts outperform their existing counterparts for all or certain ranges of shifts in the CV. The application of the optimal RS-γ charts is illustrated with real data collected from a casting process.

Optimal Designs of the Variable Sample Size X¯Chart Based on Median Run Length and Expected Median Run Length

The variable sample size (VSS) X chart, devoted to the detection of moderate mean shifts, has been widely investigated under the context of the average run-length criterion. Because the shape of the run-length distribution alters with the magnitude of the mean shifts, the average run length is a confusing measure, and the use of percentiles of the run-length distribution is considered as more intuitive. This paper develops two optimal designs of the VSS X chart, by minimizing (i) the median run length and (ii) the expected median run length for both deterministic and unknown shift sizes, respectively. The 5th and 95th percentiles are also provided in order to measure the variation in the run-length distribution. Two VSS schemes are considered in this paper, that is, when the (i) small sample size (nS) or (ii) large sample size (nL) is predefined for the first subgroup (n1). The Markov chain approach is adopted to evaluate the performance of these two VSS schemes. The comparative study reveals that improvements in the detection speed are found for these two VSS schemes without increasing the in-control average sample size. For moderate to large mean shifts, the optimal VSS X chart with n1 =nL significantly outperforms the optimal EWMA X chart, while the former is comparable to the latter when n1 = nS. Copyright

A direct procedure for monitoring the coefficient of variation using a variable sample size scheme

ABSTRACT A variable sample size (VSS) scheme directly monitoring the coefficient of variation (CV), instead of monitoring the transformed statistics, is proposed. Optimal chart parameters are computed based on two criteria: (i) minimizing the out-of-control ARL (ARL1) and (ii) minimizing the out-of-control ASS (ASS1). Then the performances are compared between these two criteria. The advantages of the proposed chart over the VSS chart based on the transformed statistics in the existing literature are: the former (i) provides an easier alternative as no transformation is involved and (ii) requires less number of observations to detect a shift when ASS1 is minimized.

Economic and economic-statistical designs of the side sensitive group runs chart

A new optimization algorithm is proposed for the economic models of the SSGR chart.Optimal design parameters of the SSGR chart are computed based on ARL and EARL.Sensitivity analyses are conducted for various input parameters of the cost model.Effects of misspecification of the shift size on the performance of the SSGR chart are investigated.The performance of the SSGR chart is compared with that of the Shewhart X ? , synthetic, GR and EWMA charts. The idea of proposing the economic and economic-statistical designs of the side sensitive group runs (SSGR) chart is presented in this paper. In the economic design, a simplified algorithm is used to search for the optimal design parameters that minimize the expected hourly cost. Nevertheless, this design has a major weakness, where it overlooks the statistical performance of the control chart. Therefore, in order to improve the effectiveness of the control chart in detecting process shifts, the economic-statistical design takes into account the statistical properties while the cost is minimized by placing statistical constraints upon the cost model of the economic design. Besides formulating the economic and economic-statistical designs based on the average run length (ARL), the economic and economic-statistical designs of the SSGR chart are also formulated based on the expected average run length (EARL) since the process shift size is usually unknown in real situations. In this paper, the sensitivity analyses of the optimal cost and the optimal design parameters are implemented for various input parameters. The effects of misspecification of the shift size on the performance of the SSGR chart are also illustrated based on numerical examples for different input parameters. This paper will also look at whether the SSGR chart performs economically better than the Shewhart X ? , synthetic, group runs (GR) and EWMA charts in the economic-statistical design based on the EARL. From the results of comparison, it is shown that the economic performance of the SSGR chart is better than that of the other four control charts in most practical situations.

Monitoring the Coefficient of Variation Using a Variable Sampling Interval EWMA Chart

In recent years, the coefficient of variation (CV) chart is receiving increasing attention in quality control. A number of studies demonstrated that adaptive charts could detect process shifts faster than traditional charts. This paper proposes an EWMA chart with variable sampling interval (VSI) to monitor the CV. Formulas for computing the performance measures of the VSI EWMA-γ2 chart are derived using Markov chain, where γ2 denotes the CV squared. Comparative studies show that the VSI EWMA-γ2 chart significantly outperforms other competing charts. An example using real manufacturing data shows that the VSI EWMA-γ2 chart performs well in applications.

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

ABSTRACT This article proposes a CV chart by using the variable sample size and sampling interval (VSSI) feature to improve the performance of the basic CV chart, for detecting small and moderate shifts in the CV. The proposed VSSI CV chart is designed by allowing the sample size and the sampling interval to vary. The VSSI CV charts statistical performance is measured by using the average time to signal (ATS) and expected average time to signal (EATS) criteria and is compared with that of existing CV charts. The Markov chain approach is employed in the design of the chart.