An Extension of Economic Design of x-Bar Control Charts for Non Normally Distributed Data Under Weibull Shock Models

Three parameters—sample size, sampling intervals, and the control limits—must be determined when the x bar chart to monitor a manufacturing process. The constant sampling intervals were widely employed because of its administrative simplicity. However, the variable sampling interval (VSI) has recently been shown to give substantially faster detection of most process shifts than fixed-sampling-interval (FSI) for x-bar charts. In addition, these measurements in the subgroup are assumed to be normally distributed. That assumption may not be tenable. This investigation compares the economic design of x-bar control charts for non normal data under Weibull shock models with various sampling avenues.     

Economic design based on a multivariate Bayesian VSI chart with a dual control limit

In this paper, a multivariate Bayesian variable sampling interval (VSI) control chart for the economic design and optimization of statistical parameters is designed. Based on the VSI sampling strategy of a multivariate Bayesian control chart with dual control limits, the optimal expected cost function is constructed. The proposed model allows the determination of the scheme parameters that minimize the expected cost per time of the process. The effectiveness of the Bayesian VSI chart is estimated through economic comparisons with the Bayesian fixed sampling interval and the Hotelling's T² chart. This study is an in-depth study on a Bayesian multivariate control chart with variable parameter. Furthermore, it is shown that significant cost improvement may be realized through the new model.     

A control chart for a general Gaussian process

It has been repeatedly demonstrated that X-bar quality control charts perform poorly when the process subgroups being monitored are correlated. In this paper, we propose and investigate the performance of a control chart that accounts for subgroup correlations in a general Gaussian process. The time-series innovations algorithm is used to construct the desired chart from a set of one-step ahead predictions and prediction variances. The chart is applicable in both stationary and nonstationary settings. A simulation study shows that this ‘innovations’ chart performs as a traditional X-bar chart even when the correlation structure of the process must be estimated from a small number of subgroups. The innovations chart is then used to study a data set of motor shaft diameters which has correlated subgroups. The results here show that erroneous conclusions can be reached if subgroup correlations are ignored.     

Economic design of VSI X̄ control chart with correlated non-normal data under multiple assignable causes

It has been recently revealed that the Shewhart control charts with variable sampling interval (VSI) perform better than the traditional Shewhart chart with the fixed sampling interval in detecting shifts in the process. In most of these research works, the normality and independency of the process data or measurements are assumed and that the process is subjected to only one assignable cause. While, in practice, these assumptions usually do not hold, some recent studies are focused on working with only one or two of these violations. In this paper, the situation in which the process data are correlated and follow a non-normal distribution and that there is multiplicity of assignable causes in the process is considered. For this case, a cost model for the economic design of the VSI X? control chart is developed, where the Burr distribution is employed to represent the non-normal distribution of the process data. To obtain the optimal values of the design parameters, a genetic algorithm is employed in which the response surface methodology is applied. A numerical example is presented to show the applicability and effectiveness of the proposed methodology. Sensitivity analysis is also carried out to evaluate the effects of cost and input parameters on the performance of the chart.     

The economic design of multivariate binomial EWMA VSSI control charts

Since multi-attribute control charts have received little attention compared with multivariate variable control charts, this research is concerned with developing a new methodology to employ the multivariate exponentially weighted moving average (MEWMA) charts for m-attribute binomial processes; the attributes being the number of nonconforming items. Moreover, since the variable sample size and sampling interval (VSSI) MEWMA charts detect small process mean shifts faster than the traditional MEWMA, an economic design of the VSSI MEWMA chart is proposed to obtain the optimum design parameters of the chart. The sample size, the sampling interval, and the warning/action limit coefficients are obtained using a genetic algorithm such that the expected total cost per hour is minimized. At the end, a sensitivity analysis has been carried out to investigate the effects of the cost and the model parameters on the solution of the economic design of the VSSI MEWMA chart.     

Combined double sampling and variable sampling interval np chart

A combined double sampling and variable sampling interval (DSVSI) np chart is investigated in this study. The optimal design of the DSVSI np chart is based on minimizing the out-of-control average time to signal. From the numerical results, the DSVSI np chart performs reasonably well in comparison with the standard np chart, double sampling np chart, synthetic double sampling np chart, and other existing np type control charts for detecting increases in the process of fraction non conforming, based on the zero-state case. An example is provided to illustrate the application of the DSVSI np chart.     

Economic and economic statistical design of T 2 control chart with two adaptive sample sizes

The Hotelling's T ² control chart, a direct analogue of the univariate Shewhart X¯ chart, is perhaps the most commonly used tool in industry for simultaneous monitoring of several quality characteristics. Recent studies have shown that using variable sampling size (VSS) schemes results in charts with more statistical power when detecting small to moderate shifts in the process mean vector. In this paper, we build a cost model of a VSS T ² control chart for the economic and economic statistical design using the general model of Lorenzen and Vance [The economic design of control charts: A unified approach, Technometrics 28 (1986), pp. 3–11]. We optimize this model using a genetic algorithm approach. We also study the effects of the costs and operating parameters on the VSS T ² parameters, and show, through an example, the advantage of economic design over statistical design for VSS T ² charts, and measure the economic advantage of VSS sampling versus fixed sample size sampling.     

Double sampling |S| control chart with variable sample size and variable sampling interval

This study presents a control chart for monitoring shifts in the covariance matrix of a multivariate normally distributed process. This chart combines the double sampling, variable sample size and variable sampling interval features, and is called the DSVSSI |S| chart. A Markov chain approach is developed to design the DSVSSI |S| chart, by minimizing the average time to signal (ATS), for a specified shift size in the covariance matrix. The DSVSSI |S| chart has a better ATS performance compared to other existing charts. An example is given to illustrate the operation of the DSVSSI |S| chart.     

A single-featured EWMA-X control chart for detecting shifts in process mean and standard deviation

The combined EWMA-X chart is a commonly used tool for monitoring both large and small process shifts. However, this chart requires calculating and monitoring two statistics along with two sets of control limits. Thus, this study develops a single-featured EWMA-X (called SFEWMA-X) control chart which has the ability to simultaneously monitor both large and small process shifts using only one set of statistic and control limits. The proposed SFEWMA-X chart is further extended to monitoring the shifts in process standard deviation. A set of simulated data are used to demonstrate the proposed chart's superior performance in terms of average run length compared with that of the traditional charts. The experimental examples also show that the SFEWMA-X chart is neater and easier to visually interpret than the original EWMA-X chart.     

Adaptive Hotelling's T 2 Control Charts with Run Rules

This study investigates the statistical properties of the adaptive Hotelling's T ² charts with run rules in which the sample size and sampling interval are allowed to vary according on the current and past sampling points. The adaptive charts include variable sample size (VSS), variable sampling interval (VSI), and variable sample size and sampling interval (VSSI) charts. The adaptive Hotelling's T ² charts with run rules are compared with the fixed sampling rate Hotelling's T ² chart with run rules. The numerical results show that the VSS, VSI, and VSSI features improve the performance of the Hotelling's T ² chart with run rules.     

Hotelling’s T2 control chart with double warning lines

Recent studies have shown that the T ² control chart with variable sampling intervals (VSI) and/or variable sample sizes (VSS) detects process shifts faster than the traditional T ² chart. This article extends these studies for processes that are monitored with VSI and VSS using double warning lines (T ² —DWL). It is assumed that the length of time the process remains in control has exponential distribution. The properties of T ² —DWL chart are obtained using Markov chains. The results show that the T ² —DWL chart is quicker than VSI and/or VSS charts in detecting almost all shifts in the process mean.     

Design of adaptive s control charts

The Shewhart s chart has been widely used to monitor the standard deviation of a process. However, the main disadvantage of an s chart is its slowness to signal small increases in the variability. In this paper, ideas of adaptive control charts are extended to the Shewhart s chart for improving the efficiency in signalling increases in the standard deviation. A Markov chain model is applied to evaluate its performances and compares its performances with combined double sampling and variable sampling intervals s chart, variable parameters (VP) R chart, exponentially weighted moving average and Cusum charts. The statistical performances show that the VP s chart is more sensitive to increases in standard deviation.     

Robustness of the EWMA and the combined X¯–EWMA control charts with variable sampling intervals to non-normality

The exponentially weighted moving average (EWMA) control charts with variable sampling intervals (VSIs) have been shown to be substantially quicker than the fixed sampling intervals (FSI) EWMA control charts in detecting process mean shifts. The usual assumption for designing a control chart is that the data or measurements are normally distributed. However, this assumption may not be true for some processes. In the present paper, the performances of the EWMA and combined X¯–EWMA control charts with VSIs are evaluated under non-normality. It is shown that adding the VSI feature to the EWMA control charts results in very substantial decreases in the expected time to detect shifts in process mean under both normality and non-normality. However, the combined X¯–EWMA chart has its false alarm rate and its detection ability is affected if the process data are not normally distributed.     

Simultaneous confidence intervals on partial means of classes in the two-stage stratified sampling

Let’s consider a finite population of P units, each of them assumes a specific amount of the quantitative variable X. Moreover we assume that the range of values of X is subdivided into k classes and the sampling data come out from a two stage stratified sampling. The main purpose of the work is to determine the estimators, as well as their asymptotic distribution, of the partial means of classes, each of them is defined as a non linear function of the other parameters. Particularly, we are interested in determining the linear approximation estimators and, under convergence theorems, the asymptotic distribution. Afterwards we define the estimator of the vector of the partial means of classes and its asymptotic convergence to multivariate normal distribution is determined. These results are useful to develop simultaneous inferential procedures.     

A Phase I nonparametric Shewhart-type control chart based on the median

A nonparametric Shewhart-type control chart is proposed for monitoring the location of a continuous variable in a Phase I process control setting. The chart is based on the pooled median of the available Phase I samples and the charting statistics are the counts (number of observations) in each sample that are less than the pooled median. An exact expression for the false alarm probability (FAP) is given in terms of the multivariate hypergeometric distribution and this is used to provide tables for the control limits for a specified nominal FAP value (of 0.01, 0.05 and 0.10, respectively) and for some values of the sample size (n) and the number of Phase I samples (m). Some approximations are discussed in terms of the univariate hypergeometric and the normal distributions. A simulation study shows that the proposed chart performs as well as, and in some cases better than, an existing Shewhart-type chart based on the normal distribution. Numerical examples are given to demonstrate the implementation of the new chart.     

A modified economic-statistical design of the T2 control chart with variable sample sizes and control limits

Recent studies have shown that using variable sampling size and control limits (VSSC) schemes result in charts with more statistical power than variable sampling size (VSS) when detecting small to moderate shifts in the process mean vector. This paper presents an economic-statistical design (ESD) of the VSSC T² control chart using the general model of Lorenzen and Vance [22]. The genetic algorithm approach is then employed to search for the optimal values of the six test parameters of the chart. We then compare the expected cost per unit of time of the optimally designed VSSC chart with optimally designed VSS and FRS (fixed ratio sampling) T² charts as well as MEWMA charts.     

Bivariate control charts for testing x data patterns

A new control chart, called the θ chart, for monitoring the mean of a process with bivariate quality characteristics is proposed. It can identify a rotation, shift or alternation between the subgroups of the process mean. The conventional application of X² chart to identify a sudden shift of the process mean is also expanded to identify a change of the process mean or a change of the process dispersion. Furthermore, when used together, the θ and X² charts could provide further insight into the process.     

Optimal design of the variable sampling size and sampling interval variable dimension T2 control chart for monitoring the mean vector of a multivariate normal process

This paper develops the algorithm for the optimization designs of the adaptive T² Control Chart for Monitoring the Mean Vector of a Multivariate Normal Process. It includes the variable sample size, variable sampling interval and variable dimensional chart. The VDT² control chart performs well for moderate and large shifts in the mean vector. However, its performance for small shifts is poor. To improve the chart's performance in detecting such shifts, we propose the application of the variable sample size and sampling interval technique to the VDT² control chart, resulting in the VSSIVDT² control chart.     

Economic and economic–statistical designs of VSI Bayesian control chart using Monte Carlo method and ABC algorithm

The usual practice in using a Bayesian control chart to monitor a process is done by taking samples from the process with fixed sampling intervals. Recent studies on traditional control charts have shown that variable sampling interval (VSI) scheme compared to classical scheme (fixed ratio sampling, FRS) helps practitioners to detect process shifts more quickly. In this paper, the effectiveness of VSI scheme on performance of Bayesian control chart has been studied, based on economic (ED) and economic–statistical designs (ESD). Monte Carlo method and artificial bee colony algorithm have been utilized to obtain optimal design parameters of Bayesian control chart (sample size, sampling intervals, warning limit and control limit) since the statistic of this approach does not have any specified distribution. Finally, VSI Bayesian control chart has been compared to FRS Bayesian and VSI X-bar approaches based on ED and ESD, separately. According to the results, it has been found that the performance of VSI Bayesian scheme is better than FRS Bayesian and VSI X-bar approaches.     

The Variable Sampling Interval (VSI) in an Analysis of Taguchi's On-line Quality Monitoring Procedure for Variables

The procedure of on-line process control for variables proposed by Taguchi consists of inspecting the mth item (a single item) of every m items produced and deciding, at each inspection, whether the mean value is increased or not. If the value of the monitored statistic is outside of the control limits, one decides the process is out-of-control and the production is stopped for adjustment; otherwise, it continues. In this article, a variable sampling interval (with a longer L and a shorter m ≤ L) chart with two set of limits is used. These limits are the warning (±W) and the control (±C), where W ≤ C. The process is stopped for adjustment when an observation falls outside of the control limits or a sequence of h observations falls between the warning limits and the control limits. The longer sample interval is used after an adjustment or when an observation falls inside the warning limits; otherwise, the short sampling interval is used. The properties of an ergodic Markov chain are used to evaluate the time (in units) that the process remains in-control and out-of-control, with the aim of building an economic–statistical model. The parameters (the sampling intervals m and L, the control limits W and C and the length of run h) are optimized by minimizing the cost function with constraints on the average run lengths (ARLs) and the conformity fraction. The performance of the current proposal is more economical than the decision taken based on a sequence of length h = 1, L = m, and W = C, which is the model employed in earlier studies. A numerical example illustrates the proposed procedure.