• Journal of Korean Society for Quality Management
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• v.36 no.3
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• pp.34-44
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• 2008

This research proposes a design method based on the statistical characteristics of the Shewhart control chart incorporated with 2 of 2 and 2 of 3 runs rules respectively. A Markov chain approach is employed in order to calculate the in-control and out-of-control average run lengths(ARL). Two different control limit coefficients for the Shewhart scheme and the runs rule scheme are derived simultaneously to minimize the out-of-control average run length subject to the reasonable in-control average run length. Numerical examples show that the statistical performance of the hybrid control scheme are superior to that of the original Shewhart control chart.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.29 no.2
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• pp.97-103
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• 2006

Recently, the control chart is developed for monitoring processes with normal short production runs by the coefficient of variation(CV) characteristic for a normal distribution. This control chart does not work well in non-normal short production runs. And most of industrial processes are known to follow the non-normal distribution. Therefore, the control chart is required to be developed for monitoring the processes with non-normal short production runs by the CV characteristics for a non-normal distribution. In this paper, we suggest the control chart for monitoring the processes with a gamma short runs by the CV characteristics for a gamma distribution. This control chart is denoted by the gamma CV control chart. Futhermore evaluated the performance of the gamma CV control chart by average run length(ARL).

• Journal of Korean Institute of Industrial Engineers
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• v.34 no.3
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• pp.308-317
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• 2008

This research investigates economic characteristics of 2 of 2 runs rules under the Shewhart $\bar{X}$ control chart scheme. A Markov chain approach is employed in order to calculate the in-control average run length (ARL) and the average length of analysis cycle. States of the process are defined according to the process conditions at sampling time and transition probabilities are derived from the state definitions. A steady state cost function is constructed based on the Lorezen and Vance(1986) model. Numerical examples show that 2 of 2 runs rules are economically superior to the Shewhart $\bar{X}$ chart in many cases.

• Journal of the Korean Statistical Society
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• v.6 no.2
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• pp.97-102
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• 1977

For a sequence of Markov dependent Bernoulli trails, asymptotic normality of the number of success runs with length $r \geq 1$ is established, when the number of trials tends to infinity.

• Journal of Korean Society for Quality Management
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• v.40 no.2
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• pp.176-185
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• 2012

After a work of Derman and Ross(1997) that considered simple main runs rules and derived ARL (Average Run Length) using Markov chain modeling, $\bar{X}$ control chart based on diverse alternative main and supplementary runs rules that is the most popular control chart for monitoring the mean of a process are proposed. This paper reviews and discusses the-state-of-art researches for these runs rules and classifies according to several properties of runs rules. ARL derivation for a proposed runs rule is also illustrated.

• Journal of Korean Society for Quality Management
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• v.40 no.4
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• pp.467-480
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• 2012

Purpose: This paper introduces new 2-of-3 main and supplementary runs rules to increase the performance of the classical $\bar{X}$ control chart for detecting small process shifts. Methods: The proposed runs rules are compared with other competitive runs rules by numerical experiments. Nonlinear optimization problem to minimize the out-of-control ARL at a specified shift of process mean for determining action and warning limits at a time is formulated and a procedure to find two limits is illustrated with a numerical example. Results: The proposed 2-of-3 main and supplementary runs rules demonstrate an improved performance over other runs rules in detecting a sudden shift of process mean by simultaneous changes of mean and standard deviation. Conclusion: To increase the performance in the detection of small to moderate shifts, the proposed runs rules will be used with $\bar{X}$ control charts.

• Kyungpook Mathematical Journal
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• v.64 no.2
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• pp.287-301
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• 2024

From the mathematical and statistical point of view, a segment of a DNA strand can be viewed as a sequence of four-state (A, C, G, T) trials. Herein, we consider the distributions of runs and patterns related to the run lengths of multi-state sequences, especially for four states (A, B, C, D). Let X₁, X₂, . . . be a sequence of four state independent and identically distributed trials taking values in the set 𝒢 = {A, B, C, D}. In this study, we obtain exact formulas for the probability distribution function for the discrete distribution of runs of B's of order k. We obtain longest run statistics, shortest run statistics, and determine the distributions of waiting times and run lengths.

• Journal of the Korean Statistical Society
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• v.33 no.1
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• pp.35-58
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• 2004

For fixed positive integers and $\textit{k}\;(n\;{\geq}\;{\textit{k}}\;+\;2)$, the exact probability distributions of non-overlapping and overlapping patterns of two failures separated by (i) exactly $textsc{k}$ successes, (ii) at least $\textit{k}$ successes and (iii) at most $\textit{k}$ successes have been obtained for Bernoulli independent and Markov dependent trials by using combinatorial technique. The waiting time distributions for the first occurrence and the $r^{th}$ (r > 1) occurrence of the patterns have also been obtained.

• Journal of Korean Institute of Industrial Engineers
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• v.13 no.2
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• pp.97-103
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• 1987

Skip-Lot sampling plan is formulated in terms of renewal process. This approach facilitates studying the average outgoing quality in a short production run of length t, AOQ (t). By numerical studies it is found that the long-run average outgoing quality (AOQ) greatly overestimates AOQ (t) for short runs.

• The Korean Journal of Applied Statistics
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• v.27 no.1
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• pp.115-122
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• 2014

A test for randomness of the binary random sequence is proposed in this paper. The proposed test statistic is based on the mean length of runs distributed with truncated geometric distribution and asymptotically ${\chi}^2_2$-distributed when the size of the sequences is large. A small Monte Carlo simulation compared the size of the test with a significant level as well as evaluated the test power. We applied the proposed method to the sequence of yes or no numbers in Lotto 6/45 and concluded that the randomness of Lotto is retained.