Chao-Yu Chou

Economic-statistical design of X ¥ charts for non-normal data by considering quality loss

When the X ¥ control chart is used to monitor a process, three parameters should be determined: the sample size, the sampling interval between successive samples, and the control limits of the chart. Duncan presented a cost model to determine the three parameters for an X ¥ chart. Alexander et al. combined Duncans cost model with the Taguchi loss function to present a loss model for determining the three parameters. In this paper, the Burr distribution is employed to conduct the economic-statistical design of X ¥ charts for non-normal data. Alexanders loss model is used as the objective function, and the cumulative function of the Burr distribution is applied to derive the statistical constraints of the design. An example is presented to illustrate the solution procedure. From the results of the sensitivity analyses, we find that small values of the skewness coefficient have no significant effect on the optimal design; however, a larger value of skewness coefficient leads to a slightly larger sample size and sampling interval, as well as wider control limits. Meanwhile, an increase on the kurtosis coefficient results in an increase on the sample size and wider control limits.

Economic design of variable sampling intervals T2 control charts using genetic algorithms

Control charting is a graphical expression and operation of statistical hypothesis testing. In this paper, we develop the economic design of the variable sampling intervals (VSI) T^2 control chart to determine the values of the five test parameters of the chart (i.e. the sample size, the long sampling interval, the short sampling interval, the warning limit, and the control limit) such that the expected total cost, associated with the test procedure, is minimized. The genetic algorithm (GA) is employed to search for the optimal values of the five test parameters of the VSI T^2 chart, and an example is provided to illustrate the solution procedure. Sensitivity analysis is then carried out to investigate the effects of model parameters on the solution of the economic design.

Economic design of X charts for non-normally correlated data

When the X chart is applied to monitor a manufacturing process, three parameters should be determined: sample size, sampling interval between successive samples, and the control limits for the chart. In 1956, Duncan presented the first cost model to determine the three parameters for the X charts, which is called the economic design of X charts. This paper develops the economic design of X charts for non-normally correlated data. An example of juice production process is presented to illustrate the solution procedure. A sensitivity analysis is performed to show the effects of non-normality and correlation coefficient on the optimal design of the chart.

Economic design of autoregressive moving average control chart using genetic algorithms

When designing control charts, it is usually assumed that the observations from the process at different time points are independent. However, this assumption may not be true for some production processes, e.g., the continuous chemical processes. The presence of autocorrelation in the process data can result in significant effect on the statistical performance of control charts. Jiang, Tsui, and Woodall (2000) developed a control chart, called the autoregressive moving average (ARMA) control chart, which has been shown suitable for monitoring a series of autocorrelated data. In the present paper, we develop the economic design of ARMA control chart to determine the optimal values of the test and chart parameters of the chart such that the expected total cost per hour is minimized. An illustrative example is provided and the genetic algorithm is applied to obtain the optimal solution of the economic design. A sensitivity analysis shows that the expected total cost associated with the control chart operation is positively affected by the occurrence frequency of the assignable cause, the time required to discover the assignable cause or to correct the process, and the quality cost per hour while producing in control or out of control, and is negatively influenced by the shift magnitude in process mean.

Joint Economic Design of Variable Sampling Intervals (X) and R Charts Using Genetic Algorithms

Control charting is a graphical expression and operation of statistical hypothesis testing. In this article, we develop the joint economic design of the variable sampling intervals (VSI) and R charts to determine the values of the seven test parameters of the charts (i.e., the sample size, the long sampling interval, the short sampling interval, the warning limits of the chart, the control limits of the chart, the warning limit of the R chart, and the control limit of the R chart) such that the expected total cost associated with the test procedure is minimized. The genetic algorithm (GA) is employed to search for the optimal values of the seven test parameters of the VSI and R charts, and an example is provided to illustrate the solution procedure. A sensitivity analysis then is carried out to investigate the effects of cost and model parameters on the solution of the joint economic design.

Minimum-loss design of x-bar charts for correlated data

Abstract When the x-bar chart is used to monitor a production process, three parameters should be determined: the sample size, the sampling interval between successive samples, and the control limits for the chart. In 1956, Duncan presented a cost model to determine the three parameters, which is called the economic design of x-bar charts. In 1995, Alexander et al. combined Duncan’s cost model with the Taguchi’s quality loss function to present a loss model for determining the three parameters. When designing an x-bar chart, one usually assumes that the measurements within a sample are independent; however, this assumption may not be true for some processes. In this paper, we develop the minimum-loss design of x-bar charts for correlated measurements within a sample by incorporating the Taguchi’s quality loss function. An example of orange juice production process is presented to illustrate the solution procedure. From the results of sensitivity analysis, we find that as the measurements in the sample are positively correlated, highly correlated data result in a smaller sample size and a frequent sampling interval; however, as the measurements in the sample are negatively correlated, highly correlated data yield a smaller sample size and a narrower control limits.

On the bootstrap confidence intervals of the process incapability index CPP

The process incapability index Cpp is an indicator, introduced by Greenwich and Jahr-Schaffrath, for evaluating the capability of a process. When Cpp is applied to evaluate a process, estimating the confidence interval of Cpp is important for statistical inference on the process. Calculating the confidence interval for a process index usually needs the assumption about the underlying distribution. Bootstrapping is a non-parametric, but computer intensive, estimation method. In the present paper we report the results of a simulation study on the behavior of four 95% bootstrap confidence intervals (i.e. standard bootstrap, percentile bootstrap, biased-corrected percentile bootstrap, and biased-corrected and accelerated bootstrap) for estimating Cpp when data are from a specific Burr distribution, which is used to represent various probability distributions. A detailed discussion of the simulation results is presented and some conclusions are provided.

Robustness of the Variable Sample Size and Control Limit ¯X Chart to Non Normality

Abstract Recent studies demonstrated that the variable sample size (VSS) ¯X chart is quicker than the standard Shewhart (SS) ¯X chart in detecting small 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 tenable in some production processes. The Burr distribution has been used in the literature to represent various non normal distributions. In this article, the Burr distribution will be employed to evaluate the control charts for non normal populations. We first show the VSS and Shewhart ¯X charts are sensitive to non normality. Then we propose a method of varying the sample size and the control limits simultaneously. The variable sample size and control limit VSSCL ¯X chart is shown to be quicker than the VSS ¯X chart in detecting small and moderate shifts in the process. Most importantly, the risk of false alarm for the VSSCL ¯X chart can be substantially decreased. In addition, with proper selection of chart parameters, the VSSCL ¯X chart is more robust to non normality than the VSS and Shewhart ¯X charts.

Economic-statistical design of multivariate control charts for monitoring the mean vector and covariance matrix

Abstract When a control chart is used to monitor a process, three test parameters should be determined: the sample size, the sampling interval between successive samples, and the control limits or critical region of the chart. In this paper, we present the procedure to conduct the economic-statistical design of multivariate control charts for monitoring the process mean vector and covariance matrix simultaneously; i.e. to economically determine the optimum values of the three test parameters such that the statistical constraints (including the type I error probability and power) of the control chart can be satisfied. The test statistic −2l nL is applied to develop this procedure and the cost model is established based on the function given by Montgomery and Klatt. A numerical example is provided to illustrate the solution procedure of the design and then the effects of cost parameters on the optimal design are examined.

Application of computer simulation to the design of a traffic signal timer

In this article, we obtain a better timer design for a pretimed traffic signal of an intersection located in the urban area of Touliu, Taiwan, by using computer simulation. The objective of this study is to reduce the average waiting time per approaching vehicle during peak hours by redesigning the current signal timer. The validation procedure of a simulation model and the application of orthogonal arrays in experiments are briefly reviewed, and the data collection and output analysis procedure are presented. From the result of this study, a better signal timer design for this intersection is proposed. If the signal timer is appropriately set, the average waiting time per vehicle of this intersection during peak hours is expected to have a reduction of about 20.74%.