• The Korean Journal of Applied Statistics
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• v.33 no.2
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• pp.161-170
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• 2020

The combined ${\bar{X}}-S^2$ chart is a traditional control chart for simultaneously detecting mean and variance. Control limits for the combined ${\bar{X}}-S^2$ chart are determined so that each chart has the same individual false alarm rate while maintaining the required false alarm rate for the combined chart. In this paper, we provide flexibility to allow the two charts to have different individual false alarm rates as well as evaluate the effect of flexibility. The individual false alarm rate of the ${\bar{X}}$ chart is taken to be γ times the individual false alarm rate of the S² chart. To evaluate the effect of selecting the value of γ, we use the out-of-control average run length and relative mean index as the performance measure for the combined ${\bar{X}}-S^2$ chart.

• Journal of Korean Society for Quality Management
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• v.32 no.4
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• pp.156-168
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• 2004

This research investigates the statistical efficiency of variable sampling size & sampling interval(VSSI) $\bar{X}$ charts under two assignable causes. Algorithms for calculating the average run length(ARL) and average time to signal(ATS) of the VSSI $\bar{X}$ chart are proposed by employing Markov chain method. States of the process are defined according to the process characteristics after the occurrence of an assignable cause. Transition probabilities are carefully derived from the state definition. Statistical properties of the proposed chart are also investigated. A simple procedure for designing the proposed chart is presented based on the properties. Extensive sensitivity analyses show that the VSSI $\bar{X}$ chart is superior to the VSS or VSI $\bar{X}$ chart as well as to the Shewhart $\bar{X}$ chart in statistical sense, even tinder two assignable causes.

• Journal of Korean Society for Quality Management
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• v.28 no.3
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• pp.18-30
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• 2000

Recent studies have shown that the $\bar{X}$ chart with variable sampling intervals(VSI) and the $\bar{X}$ chart with variable sample size(VSS) are much quicker than Shewhart $\bar{X}$ chart in detecting shiks in the process. Shewhart $\bar{X}$ chart has been beneficial to detect large shifts but it is hard to apply Shewhart $\bar{X}$ chart in detecting moderate shifts in the process mean. In this article the $\bar{X}$ chart using variable sample size(VSS) and variable sampling Intervals(VSI) has been proposed to supplement the weak point mentioned above. So the purpose of this paper is to consider finding the design parameters which minimize expected loss costs for unit process time and measure the performance of VSSI(variable sample size and sampling interval) $\bar{X}$ chart. It is important that assignable causes be detected to maintain the process controlled. This paper has been studied under the assumption that one cycle is from starting of the process to eliminating the assignable causes in the process. The other purpose of this article is to represent the expected loss costs in one cycle with three process parameters(sample size, sampling interval and control limits) function and find the three parameters.

• Journal of Korean Society for Quality Management
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• v.44 no.4
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• pp.751-770
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• 2016

Purpose: Two-stage ${\bar{X}}$ chart is a useful tool for process control when a surrogate variable may be used together with a performance variable. This paper extends the two-stage ${\bar{X}}$ chart to a three stage version by decomposing the first stage into the preliminary stage and the main stage. Methods: The expected cost function is derived using Markov-chain approach. The optimal designs are found for numerical examples using a genetic algorithm combined with a pattern search algorithm and compared to those of the two-stage ${\bar{X}}$ chart. Sensitivity analysis is performed to see the parameter effects. Results: The proposed design outperforms the optimal design of the two-stage ${\bar{X}}$ chart in terms of the expected cost per unit time unless the correlation between the performance and surrogate variables is modest and the shift in process mean is smallish. Conclusion: Three-stage ${\bar{X}}$ chart may be a useful alternative to the two-stage ${\bar{X}}$ chart especially when the correlation between the performance and surrogate variables is relatively high and the shift in process mean is on the small side.

• Journal of Korean Society for Quality Management
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• v.45 no.4
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• pp.943-956
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• 2017

Purpose: This paper proposes a VSS(Variable Sample Size) ${\bar{X}}$ control chart using surrogate variable and shows its effectiveness compared with FSS(Fixed Sample Size) ${\bar{X}}$ control chart using either performance variable or surrogate variable. Methods: The expected cost function of VSS ${\bar{X}}$ control chart is derived. The optimal designs are then found for numerical examples using a GA(genetic algorithm) and compared to those of the FSS ${\bar{X}}$ control charts. Results: Computational results show that VSS ${\bar{X}}$ control chart using surrogate variables is superior to FSS ${\bar{X}}$ control chart using either performance variable or surrogate variable from the economic view points. Conclusion: The proposed VSS ${\bar{X}}$ control chart will be useful in industry fields where a performance variable is not avaliable or too costly.

• Journal of the Korean Data and Information Science Society
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• v.27 no.6
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• pp.1487-1498
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• 2016

In monitoring a process, in-control process parameters must be estimated from the Phase I data. When we design the control chart based on the estimated process parameters, the control limits are usually chosen to satisfy a specific in-control average run length (ARL). However, as the run length distribution is skewed when the process is either in-control or out-of-control, the median run length (MRL) can be used as alternative measure instead of the ARL. In this paper, we evaluate the performance of Shewhart $\bar{X}$ chart with estimated parameters in terms of the average of median run length (AMRL) and the standard deviation of MRL (SDMRL) metrics. In simualtion study, the grand sample mean is used as a process mean estimator, and several competing process standard deviation estimators are used to evaluate the in-control performance for various amounts of Phase I data.

• Journal of Korean Society for Quality Management
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• v.31 no.2
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• pp.117-130
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• 2003

This paper investigates the economic design of synthetic control charts. The synthetic control chart has been proven to be statistically superior to the $\bar{X}$-control chart, but its economic characteristics have not been known. We develop an economic model of the synthetic control chart, based on Duncan's model. The synthetic chart has one more decision variable, the lower control limit for the conforming run length. In addition to this, the significance level and the power of the synthetic chart are more complicated than those of the $\bar{X}$-chart. These features make the optimization problem more difficult. We propose an optimization algorithm by adapting the congruent gradient algorithm. We compare the optimal cost of the synthetic chart with that of (equation omitted)-control chart, under the same input parameter set of Duncan’s. For all cases investigated, the synthetic chart shows superior to the $\bar{X}$-chart. The synthetic control chart is easy to implement, and it has better characteristics than the $\bar{X}$-chart in economical sense as well as in statistical sense, so it will be a good alternative to the traditional control charts.

• Journal of Korean Society for Quality Management
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• v.28 no.3
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• pp.44-52
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• 2000

In this paper we apply VSS X chart to the case when two assignable causes effect the process mean shift. We compare the case when two compound assignable causes effect the process mean with another one which effects individually. For the practical use of VSS $\bar{X}$ chart we propose the sample size which minimizes the average number of samples until the signal is given in out-of-control state.

• Journal of the Korean Operations Research and Management Science Society
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• v.39 no.1
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• pp.1-12
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• 2014

This article proposes a variable sampling interval (VSI) $\bar{X}$ control chart using weighted standard deviation (WSD) method for skewed populations. The WSD method decomposes the standard deviation of a quality characteristic into upper and lower deviations and adjusts control limits and warning limits of a control chart in accordance with the direction and degree of skewness. A control chart constant is derived for estimating the standard deviation of skewed distributions with the mean of sample standard deviations. The proposed chart is compared with the conventional VSI $\bar{X}$ control chart under some skewed distributions. Simulation study shows that the proposed WSD VSI chart can control the in-control average time to signal (ATS) as an adequate level better than the conventional VSI chart, and the proposed chart can detect a decrease in the process mean of a quality characteristic following a positively skewed distribution more quickly than the standard VSI chart.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.37 no.1
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• pp.151-156
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• 2014

Control charts are generally used for process control, but the role of traditional control charts have been limited in case of a contaminated process. Traditional $\bar{x}$ control charts have not been activated well for such a problem because of trying to control processes as center line and control limits changed by the contaminated value. This paper is to propose robust $\bar{x}$ control charts which is considering a location parameter in order to respond to contaminated process. In this paper, we consider $\bar{x}_{\alpha}$, that is trimmed rate; typically ten percent rate is used. By comparing with p, ARL value, the responding results are decided. The comparison resultant results of proposed two control charts are shown and are well contrasted.