Dissertations / Theses on the topic 'XBART'

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO
Os gráficos de controle estão entre as ferramentas indispensáveis para monitorar o desempenho de um processo em várias indústrias. Quando estimativas de parâmetros são necessárias para projetar esses gráficos, seu desempenho é afetado devido aos erros de estimação. Para resolver esse problema, no passado, pesquisadores avaliavam o desempenho desses métodos em termos do valor esperado do número médio de amostras até um alarme falso condicionado às estimativas dos parâmetros (denotado por 0). No entanto, esta solução não considera a grande variabilidade do 0 entre usuários. Então, recentemente, surgiu a ideia de medir o desempenho dos gráficos de controle usando a probabilidade de o 0 ser maior do que um valor especificado – que deve estar próximo do desejado nominal. Isso é chamado de Exceedance Probability Criterion (EPC). Para aplicar o EPC, a função de distribuição acumulada (c.d.f.) do 0 é necessária. No entanto, para um dos gráficos de controle mais utilizados, o gráfico Xbarra, também conhecido como gráfico x (sob a suposição de distribuição normal), a expressão matemática da c.d.f. não está disponível na literatura. Como contribuição nesse sentido, o presente trabalho apresenta a derivação exata da expressão matemática da c.d.f. do 0 para três possíveis casos de estimação de parâmetros: (1) quando a média e o desvio-padrão são desconhecidos, (2) quando apenas a média é desconhecida e (3) quando apenas o desvio-padrão é desconhecido. Assim, foi possível calcular o número mínimo de amostras iniciais, m, que garantem um desempenho desejada do gráfico em termos de EPC. Esses resultados mostram que m pode assumir valores consideravelmente grandes (como, por exemplo, 3.000 amostras). Como solução, duas novas equações são derivadas aqui para ajustar os limites de controle garantindo assim um desempenho desejado para qualquer valor de m. A vantagem dessas equações é que uma delas fornece resultados exatos enquanto a outra dispensa avançados softwares de computador para os cálculos. Um estudo adicional sobre o impacto desses ajustes no desempenho fora de controle (OOC) fornece tabelas que ajudam na decisão do melhor tradeoff entre quantidade adequada de dados e desempenhos IC e OOC preferenciais do gráfico. Recomendações práticas para uso desses resultados são aqui também fornecidas.
Control charts are among the indispensable tools for monitoring process performance in various industries. When parameter estimation is needed to design these charts, their performance is affected due to parameter estimation errors. To overcome this problem, in the past, researchers have evaluated the performance of control charts and designed them in terms of the expectation of the realized in-control (IC) average run length (0). But, as pointed recently, this solution does not account for what is known as the practitioner-to-practitioner variability (i.e., the variability of 0). So, a recent idea emerged where control chart performance is measured by the probability of the 0 being greater than a specified value - which must be close to the nominal desired one. This is called the Exceedance Probability Criterion (EPC). To apply the EPC, the cumulative distribution function (c.d.f.) of the 0 is required. However, for the most well-known control chart, named the two-sided Shewhart Xbar (or simply X) Chart (under normality assumption), the mathematical c.d.f. expression of the 0 is not available in the literature. As a contribution in this respect, the present work presents the derivation of the exact c.d.f. expression of the 0 for three cases of parameters estimation: (1) when both the process mean and standard deviation are unknown, (2) when only the mean is unknown and (3) when only the standard deviation is unknown. Using these key results, it was possible to calculate the exact minimum number of initial (Phase I) samples (m) that guarantees a desired in-control performance in terms of the EPC. These results show that m can be prohibitively large (such as 3.000 samples). As a solution to this problem, two new equations are derived here to adjust the control limits to guarantee a desired in-control performance in terms of the EPC for any given value of m. The advantage of these equations (compared to the existing adjustments methods) is that one provides exact results and the other one does not require too many computational resources to perform the calculations. A further study about the impact of these adjustments on the out-of-control (OOC) performance provides useful tables to decide the appropriate amount of data and the adjustments that corresponds to a user preferred tradeoff between the IC and OOC performances of the chart. Practical recommendations for using these findings are also provided in this research work.

abstract: Bayesian Additive Regression Trees (BART) is a non-parametric Bayesian model that often outperforms other popular predictive models in terms of out-of-sample error. This thesis studies a modified version of BART called Accelerated Bayesian Additive Regression Trees (XBART). The study consists of simulation and real data experiments comparing XBART to other leading algorithms, including BART. The results show that XBART maintains BART’s predictive power while reducing its computation time. The thesis also describes the development of a Python package implementing XBART.
Dissertation/Thesis
Masters Thesis Statistics 2019

碩士
輔仁大學
數學系
83
Economic models for the design of control charts based on Duncan's approach have been studied in the recent past.However, the economic design of control charts has not been developed in a systematic manner so far.Consequently, various assumptions and approaches have been made. The most researchers consider the independence of occurrence for the multiple assignable cau- ses,but to confine in a short time interval the probablity is zero for the common occurrence of the assignable causes. The structure of this study is : (1) To flex the confinement of the independence assumption, we allow the occurence for the multip- le assignable causes in the short time interval. (2) To apply the $\bar{X}$ - control charts to the generalized process model . (3) To obtain the derived cost function. It is believed that the expected cycle time and expected cycle cost are more easily obtained by the proposed Markov chain method than by extending the Duncan's approach and others approaches. The design method can be applied to mul - tiple process variables and to a variety of control charts.

碩士
逢甲大學
工業工程與系統管理學研究所
96
Numerous methods, including Cumulative Sum(CuSum) , Exponentially Weighted Moving Average(EWMA), variable sample size, Variable Sampling Interval(VSI) , variable Control limit coefficient, have been proposed to improve the capability of monitoring process in Shewhart control chart. In practice, these methods aren’t simple, resulting in mistakes and loading. The capability of the control chart monitoring process is represented in two ARL (average run length). Essentially, the ARL is the average number of points that must be plotted before a point indicated an out-of control condition. When the process is in-control, the ARL of the control chart is the larger the better. Adversely, the process is out-of-control, the ARL of the control chart is the smaller the better. The ARL is affected by control chart parameters: sample size, sampling interval, control limit coefficient, out-of-control probability, the process shift amount and the rules of judgment. In this study, the central composite design (CCD) was used to allocate factor-level and data was obtained by computer simulation for the aim that the optimal parameters of Shewhart Xbar-R control chart were obtained.

碩士
國立中山大學
企業管理研究所
82
.Xbar 管制圖利用基本的統計理念,以簡單的圖表及相關的判斷準則來顯 示品質狀況,是製程管制的主要工具之一。以往皆以人工繪製及判讀 .Xbar管制圖,耗時且易誤判,例如;因為趨勢與連串異常模式不易被發現。 將專家系統融入.Xbar管制圖,解決上述問題的可行方案。本研究以半導體 複晶蝕刻製程為對像,和廠商製程管制工程師、製程工程師及現場操作人 員共同合作,藉由瞭解製造程序、廠商使用的管制圖異常模式判讀與製程 異常分析步驟,及藉SmartQ--通用型管制圖專家系統建構工具(Shell),分 析三個月的製程歷史資料,歸納出五個常見異常模式。然後與合作廠商利 用魚骨圖、柏拉圖等品質改善之技巧,找出五個異常模式的相對應成因與 改善方法。再以此五個異常規則修改SmartQ之智庫,建構了第一個複晶蝕 刻.Xbar管制圖專家系統(First-Q)。最後藉由軟體測試方法--隨機測試, 瞭解First-Q專家系統的能力。發現它的可靠度達99%並且在降低異常製程 的發生上具有相當的效用。 .Xbar control chart uses fundamental statistics concepts, simple charting and correlative judgement rules to express states of product and service quality. Traditionally,.Xbar control chart are graphed and interpreted by human experts. It is time consuming and prone to be erroneous. Abnormal process models,e.g.,trends and runs,can not be eaily detected. Expert systems technique is one of the most effective methods to solve these problems. The underlying study is poly etching process in semiconductor industry. While knowledge for control chart interpretation was extracted from statistical process control textbook and related literatures,diagnostic rules were obtained by working with process control engineers,process engineers and shop floor operators. First,three-month historical process data are analyzed and five out-of-control patterns of .Xbar control chart are repetitive occurred. Then,cause and effect diagram, Pareto diagram are used to look for preventive method and the causes of abnormal process models. Finally,First-Q the first control chart expert system in poly etching process,are constructed by implementing five more rules for abnormal process diagnosis into Smart-Q,which is a general purpose quality control expert system shell. By utilizing a software reliability technique,random method and new data from processes, the reliability of First-Q expert system is 99%. So,First-Q is a powerful in correcting abnormal process and can be used to assist shop floor operators.

碩士
國立高雄大學
統計學研究所
97
This thesis proposes a skewness and kurtosis correction (SKC) method to set up Xbar and R control charts for process monitoring. The developed SKC control limits take account of the skewness and kurtosis of process distribution, and are simply adjustments of the conventional Shewhart control charts. Type I risks of the proposed control charts are compared with those of some existing control charts when the underlying distribution is Weibull, lognormal, and Burr. Simulation results show that if the underlying distribution is asymmetric and leptokuric, then our SKC method offers considerable improvement over the existing control charts when it is desirable for Type I risks to close to 0.27% .

碩士
中原大學
工業工程研究所
94
This thesis is to explore the control chart design for application to the corre-lated data. We assume that the quality characteristic X follows an ARTA(p) process, which is a covariance stationary process with an arbitrary marginal distribution. A considerable number of previous works have been dealing with the control chart de-sign for correlated normality data, little has been endeavored to the effect of the cor-related non-normality data. In this thesis, we assume that each item is inspected and its quality characteristic is measured at the moment it is produced. Hence, we consider only two parameters: sample size m and control-limit factor k for the design chart. We develop an op-timization model for the control chart design with correlated non-normality data. The rationale for the model is to look for the optimal design parameters so as to maximize the out-of-control performance with specific mean shift, under a fixed performance of in-control process. We propose a procedure of simulation, incorporated with retro-spective approximation, to find the optimal design of control chart. In addition, we also examine the effects of correlated non-normality data on the design parameters. From the results of simulation experiments, our findings reveal that the control chart design requires larger sample size as the data get higher positively correlated. For stronger negative correlation, the sample size does not change monotonically with the magnitude of correlation. Additionally, a larger sample is required for non-normal marginal distributions, such as exponential, lognormal, and t distributions, than for the normal marginal distribution with same lag-p autocorrelations. (Our empirical results only show p=1.) In addition, we compare the chart to the ARMAST chart in different corre-lated data. The data processes we selected are AR(1) process and ARTA(1) with U(0,1) marginal distribution. The results indicate that the application of chart is more robust than ARMAST chart with non-normality marginal distribution.

碩士
中原大學
工業工程研究所
94
We propose variance reduction methods to reduce the variance for estimating the average run length in Shewart control chart. When the data { } are not independently, analytical computation of ARL may be difficult. In this case, the simulation approach is an obvious choice, but the variance of ARL is large. In the Shewhart chart, the in-control run length N for independent data is a geometric random variable with mean and variance , where is the probability that an point falls outside the control limits when the process is in control. Since is usually small, the resulting ARL (= ) is large and the standard deviation of the run length is almost as big as the ARL. This proposal considers variance reduction techniques to improve the simulation efficiency in estimating E(N) when analytical computation of E(N) is not possible. For our simulated experiments, we consider the correlated data process such as ARTA process to reduce the variance in estimating ARL. ARTA process is a correlated stationary process that takes a base AR (Autoregressive) process with a desired marginal distribution and can be applied widely (Cario and Nelson 1998). General variance reduction techniques (VRTs) include common random numbers, antithetic variates, control variates, important sampling, etc. (Law and Kelton 2000). Three cases are considered: (i) the data are independent, (ii) the ARTA lag p=1 and the sample size n=1, and (iii) . For case (i), we propose an importance sampling VRT. For case (ii), we propose the Markov’s approximation as a numerical approximation. We also propose the control variate VRTs for case (ii) if the simulation approach is used. For case (iii) we use the AR(1) ARL estimator as the control variate for variance reduction. According to our experiment results, we find that: (1) the effect of variance reduction in symmetric distribution is better than in non- symmetric distribution when the data process is independent and using the random-hazard VRT to correlation data. (2) The effect of variance reduction in low correlation is better than in high correlation using the random-hazard VRT to correlation data. (3) The effect of variance reduction is decreasing when the sample size is increasing using control-variates VRT to correlation data.

碩士
中原大學
工業工程研究所
95
This work compares the Xbar and R charts performance for the symmetric and asymmetric limits. The Shewhart X and R control charts are control schemes used commonly in statistical process control. A conventional way of setting the control limits is to choose a set of symmetric limits, e.g., the 3-sigma control limits. Despite literature of constructing asymmetric control limits for skewed distributions exists, none has compared these two kinds of limits on an equal basis. In addition, this thesis proposes a new method for constructing the asymmetric control limits for quality measurements are autocorrelated. We compare the out-of-control ARL (average run length) for symmetric and asymmetric limits while keeping their in-control ARL values the same. In our experiments, we use three and five testing examples for the X and R charts, respectively. Three testing examples are used for X control chart. The first two examples assume that the quality measurements are independent with gamma, and Johnson unbounded distributions, respectively. The thrid assumes that the quality measurements are autocorrelated with a exponential marginal distribution. Five testing examples are used for R control chart. The first three examples assume that the quality measurements are independent with uniform, exponential, and Johnson unbounded distributions, respectively. The forth and fifth assumes that the quality measurements are autocorrelated with a exponential and uniform marginal distribution, respectively. Our empirical study that the X chart performance for the symmetric and asymmetric control limits depends on the shift direction and the skewness of the X distribution (as well as the population skewness). When the quality characteristic has a rightly skewed distribution, the symmetric control limits perform better than the asymmetric control limits when the process mean shifts to the right and worse when the process mean shifts to the left. Analogously, when the skewness is negative, the asymmetric control limits perform better than the symmetric limits when the process mean shifts to the right and worse when the process mean shifts to the left. For the R chart, our empirical study shows similar results as for the X chart. If the range has a rightly skewed distribution, the symmetric limits are more powerful in detecting an increase in the process standard deviation. Analogously, if the range has a left skewed distribution, the asymmetric limits perform better than the symmetric.

碩士
國立屏東科技大學
工業管理系所
100
In recent years, internet services become popular and application styles grow diversity. Different types of examples exist including Yahoo, Google; Facebook; Booking Services, Library Electronic Databases; and Amazon’s S3 Storage Cloud, providing Internet users with access to many facilities and business opportunities. In order to provide the above-mentioned types of network services, service companies must set up network servers for client connection. However, hackers take advantage of some loopholes in server's operating system to attack service host or design attack programs for other users. In order to prevent hacker attacks and trade secrets, many network security companies and system development institutions develop various types of network traffic and fire detection system. However, when network traffic is particularly heavy, the detecting system will become the networking bottleneck and degrade the transmission performance. It also increases the workload for mangers to analyze the log. In this study, we use the concept of statistical process control combined with network flow data characteristics to propose Xbar and EWMA control charts for monitoring network traffic and detecting abnormal network behavior. Moreover, we use the network simulator NS2 to simulate normal and abnormal traffic data to evaluate the charts’ performance according to three criteria: F measure, minimizing false negative rate under acceptable false positive rate and average first detected run length. It is reasonable to suggest that L=9.5~10 for Xbar control chart and (λ, L)=(0.9, 2) for EWMA control chart, respectively. Finally, network managers can use the control charts to assist monitoring network traffic.

碩士
中原大學
工業工程研究所
92
This thesis considers the economic-statistical design of ‾ X and R control chart assuming that the quality characteristic measurement (i.e. observations) are nonnormal and the in control time is Weibull. When designing a control chart, four parameters—the sample size n, successive sampling time h, control limit factor k1 for ‾ X chart, and control limit factor k2 for R chart—must be determined. In economicstatistical design, the four parameters are chosen so that the expected cost per hour is minimized under constraints on Type I and Type II error probabilities. The expected cost function is computed by the modfy Costa cost model. In reality manufacture process, there is not only one cause to produce the fail products. Hence, if we only consider the shift in mean and use ‾ X chart to detect, it is not reasonable in reality manufacture process. Because both the process mean and variance may change simultaneoisly during a production cycle, therefore both ‾ X and R charts are employed to monitor the variations in the process parameters simultaneously. In this study, we propose a cost model which can be used in nonnormal data and Weibull in-control time. We modify the Costa fully economic design model to conform the general case. We choose the economic-statistical design because it controls the probability of having Type I and II errors probabilities and the expected cost increases marginally. We assume that the quality characteristic measurement are sampled independently from a Johnson distribution. The Johnson distribution is general in that it can be modeled to fit all possible values of the skewness andkurtosis. Type I and Type II error probabilities are difficult to compute and need to be estimated via simulation experiment. We perform the sensitivity analysis to study the effects of nonnormality and the amount of shift in mean and variance. The optimal value of {n, h, k1, k2} are computed by the grid search method. The results show that: When the kurtosis increases, the sample size n, control limit k1 for ‾ X chart, and the expected cost decrease and sampling interval h increaes. When the amount shift of mean or variance increases, the sample size n and sampling interval h decrease but control limit k1 for ‾ X chart and control limit k2 for R chart increases for Johnson unbounded distribution. When variance shift occurs frequently, the control limit coefficient for ‾ X chart, k1, tends to values that render the ‾ X chart unnecessary. When mean shift occurs frequently, the control limit coe?cient for R chart, k2, tends to unnecessary for Johnson unbounded distribution but still remains for Johnson bounded distribution. When the variance shifts more frequently than the mean, the expected cost increases.

博士
中原大學
工業與系統工程研究所
98
This thesis considers the design problem of the Xbar chart with symmetric control limits for autocorrelated data processes. The Xbar chart, proposed by W.A. Shewhart in 1931, is a useful tool to detect a shift in the process mean. The design problem of the Xbar chart is to determine the sample size n of Xbar and the control-limit factor k (number of standard deviations away from the center line). A conventional assumption of the Xbar chart is that the sample means are independent and normally distributed. Nevertheless, in practice the quality characteristic measurements may be autocorrelated and nonnormally distributed. Furthermore, when the standard deviation of Xbar is unknown but in-control observations are given, the standard deviation of Xbar needs to be estimated and it is random. The autocorrelation, nonnormality, and parameter estimation affect the values of the design parameters n and k for meeting specified control chart performances, e.g., average run length. This thesis consists of four subproblems. Subproblem 1 considers the effects of the nonnormality, autocorrelation, and parameter estimation on the Xbar chart's performance. Subproblems 2 to 4 consider the design problem of the Xbar chart assuming that the data process is known (denoted as model-based design), has known autocovariance structure but unknown marginal distribution (denoted as distribution-free design), and is unknown (denoted as model-free design), respectively. For Subproblem 1, we show that (1) The nonnormality has nonmonotonic effects on the mean and standard deviation of the run length. (2) When the number m of in-control Xbar observations increases, the mean and standard deviation of run length decrease and converge to the corresponding values of m=infinity. (3) As the lag-1 autocorrelation phi of sample means increases, P{N_0=2} (where N_0 is the in-control run length) decreases and the tail probabilities increase. Therefore, the ARL (Average Run Length) and standard deviation of run length increase. (4) For the simultaneous effect of autocorrelation and estimation, the distribution of the run length has a heavier tail as |phi| increases, m decreases, or |phi| increases and m decreases simultaneously. We also study the asymptotic result of Var(S_{Xbar}^2) to recommend the m. (5) For the simultaneous effect of nonnormality and estimation, for small shift magnitude or shift magnitude = 0., the difference of the ARL between the nonnormal and estimation case with the normal and known variance case increases as m increases. For a large shift magnitude (>=2), the difference of the ARL between the nonnormal and estimation case with the normal and known variance case decreases as m increases. (6) For the simultaneous effect of autocorrelation, nonnormality, and estimation: we conclude that the three-dimensional effect is nonmonotic on the mean and standard deviation of run length. For Subproblem 2, we propose an algorithm to compute values of the sample size n and control-limit factor k that minimize the out-of-control ARL while keeping the in-control ARL at a specified value. Simulation experiments are run to compare the proposed Xbar chart design with the EWMAST (Exponentially Weighted Moving Average STationary), SCC (Special Cause Control), and ARMAST (Auto-Regressive Moving-Average STationary) charts. When the data process is ARMA(1, 1) (AutoRegressive of order 1 and Moving Average of order 1) process, the ARMAST chart outperforms the proposed Xbar chart, EWMA, and SCC because the ARMAST chart is designed based on the ARMA(1, 1) process. When the data process is autocorrelated with nonnormal distribution, the proposed Xbar chart often performs better than the ARMAST chart. For Subproblem 3, we propose two distribution-free methods (denoted as Methods 1 and 2) to compute the sample size n and control-limit factor k that minimize the out-of-control ARL while keeping the in-control ARL at a specified value. Methods 1 and 2 are based on the assumptions that sample means are independently and normally distributed and that sample means follow an AR(1) (AutoRegressive of order 1) process, respectively. We further modify Methods 1 and 2 by setting the sample size to be at least 30 to meet the central limit theorem. The modified Method 2 outperforms the modified Method 1. We compare the modified Method 2 with R&W (Runger and Willemain) and DFTC (Distribution-Free Tabular CuSum) charts. The R&W and DFTC charts perform better when the autocorrelation is small to moderate and the shift is small or large. Our modified Method 2 performs better when the autocorrelation is high and the shift is moderate to large. For Subproblem 4, we propose a model-free Xbar chart design for unknown autocorrelated data processes, when M observations of in-control quality characteristic measurements are given. The standard deviation of Xbar is estimated by the NBM (nonoverlapping batch means) method. The Xbar chart design parameters n and k and the NBM batch size omega are determined to minimize the out-of-control ARL while keeping the in-control ARL at a specified value. During the search of n, k, and omega, the ARL values are computed assuming that the sample means and batch means are independently and normally distributed. The computed optimal values of n and omega are further modified to be at least 30 for meeting the central limit theorem. Simulation experiments are run to evaluate the performance of the proposed model-free Xbar chart using AR(1) processes and ARTA(1) (AutoRegressive To Anything of order 1) processes with t and exponential marginal distributions as testing examples. The experimental results show that our model-free method performs well in general; it performs better when the autocorrelation is small or when the autocorrelation is large but the shift is small or moderate. The proposed model-free Xbar chart performs worse when the data process is ARTA(1) with an exponential marginal distribution.

碩士
中原大學
工業工程研究所
95
In this thesis, we consider the joint economic–statistical design ‾X and R control charts which is based on the framework of the Costa and Chen et al. According to the cost model we assume that the quality characteristic are autocorrelated and the in–control time follows the Weibull distribution. There are four design parameters : the sample size n, sampling interval h and the control limit factors k1 and k2 are determined to design control charts ‾X and R, respectively. At present, most literatures research either the economic–statistical design in which the data is satisfying independent assumption or the control chart in which the data is satisfying autocorrelated. But there are a few literatures in researching of the combination of these two concepts. Consequently, we investigate these combined themes in this thesis. In this thesis, we use the grid search method by Monte Cario simulation to find the optimal design parameters to minimize the expected cost per hour. We also perform the sensitivity analysis to study the effects of autocorrelation, the amount of shift in mean and variance and the Weibull scale parameters. As the result, when the autocorrelation increases positively and then we would find that the sample size n and the sampling interval h increase, but control limit factors k1 and k2 decrease. The expected hourly cost increases by the autocorrelation either positively and nagetively increase. When the amount shift of mean increases, n, h and k2 decrease and k1 and the expected hourly cost increase. When the amount shift of variance increases, n, h and k1 decrease and k2 and the expected hourly cost increase. When 1 increase, n, h and k1 decrease and k2 and the expected hourly cost increase, but in the shift of the combination ( , )=(1,2), k1 decrease and n, h ,k2 and the expected hourly cost increase.

碩士
中原大學
工業工程研究所
91
This thesis considers the economical-statistical design of Xbar-control chart assuming that the quality characteristic measurement (i.e. observations) are nonnormal and the in-control time is Weibull. When designing a control chart, three parameters-the sample size n, time h between successive samples, and control-limit factor k-must be determined. In economic-statistical design, the three parameters are chosen so that the expected cost per hour is minimized under constraints on Type I and Type II error probabilities. The cost function is computed by the McWilliams cost model. Although some nonnormal literature exists, the assumption is made on the distribution of sample average, which depends on the unknown sample size. We assume that the quality characteristic measurement are sampled independently from a Johnson distribution. The Johnson distribution is general in that it can be modeled to fit all possible values of the skewness and kurtosis. This is a stochastic optimization problem because Type I and Type II error probabilities is difficult to compute and need to be estimated via simulation experiment. Hence, a stochastic optimization algorithm is needed. We propose an algorithm of stochastic optimization, revised retrospective optimization flexible tolerance method, to solve our optimization problem including one discrete and two continuous decision variable vector X ={n, h, k}. Empirical results show that the solution of Revised RO-FT is very close to the true optimum in our testing problems. Finally, we perform the sensitivity analysis of the nonnormality and Weibull effect on the optimal values of {n, h, k}. Conclusion of sensitivity analysis are as follows. When skewness is constant and equal to zero, an increase on kurtosis leads to increases on sample size n, decreases on time h between successive samples, as well as a wider control-limit factor k. When skewness is constant and not only greater than but also close to zero (as well as far away zero), an increase on kurtosis leads to increases on n, decreases on h, as well as a wider k. When kurtosis is constant and close to normal (as well as a little far away normal), an increase on skewness leads to decreases on n, increases on h, as well as a narrower k. When kurtosis is constant and far away normal, an increase on skewness leads to no signidicant effect on n, increases on h, as well as a narrower k. When an increase both on skewness and kurtosis leads to increases on n and h, as well as a wider k. Weibull shape parameter heta and scale parameter lambda are sensitive when shape parameter heta<=1 on the decision variable time h between successive samples. When an increase on shape parameter heta leads to increases on h. Keywords: Economic-Statistical Design; Expected Cost Per Hour; Revised Retrospective Optimization Flexible Tolerance Method; Nonnormality; Xbar Control Chart.