Eugenio K. Epprecht

Economic design of a Vp chart

Abstract We develop an economic model for X control charts having all design parameters varying in an adaptive way, that is, in real time considering current sample information. In the proposed model, each of the design parameters can assume two values as a function of the most recent process information. The cost function is derived and it provides a device for optimal selection of the design parameters. Through a numerical example one can foresee the savings that the developed model possibly provides.

Adaptive Control Charts for Attributes

We develop a general model for adaptive c , np , u and p control charts in which one, two or three design parameters (sample size, sampling interval and control limit width) switch between two values, according to the most recent process information. For a given in-control average sampling rate and a given false alarm rate, the adaptive chart detects changes in the process much faster than a chart with fixed parameters. Moreover, this study also offers general guidance on how to choose an effective design.

Constrained optimization model for the design of an adaptive X chart

An economic-statistical model is developed for variable parameters (VP) X charts in which all design parameters vary adaptively, that is, each of the design parameters (sample size, sampling interval and control-limit width) vary as a function of the most recent process information. The cost function due to controlling the process quality through a VP X chart is derived. During the optimization of the cost function, constraints are imposed on the expected times to signal when the process is in and out of control. In this way, required statistical properties can be assured. Through a numerical example, the proposed economicstatistical design approach for VP X charts is compared to the economic design for VP X charts and to the economic-statistical and economic designs for fixed parameters (FP) X charts in terms of the operating cost and the expected times to signal. From this example, it is possible to assess the benefits provided by the proposed model. Varying some input parameters, their effect on the optimal cost and on the optimal values of the design parameters was analysed.

ADAPTIVE SAMPLE SIZE CONTROL CHARTS FOR ATTRIBUTES

Theoretically, the X chart with variable sample sizes (VSS) is faster than the traditional X chart for detecting moderate shifts in the process. The idea is to vary sample sizes according to what is observed from the process. If the current X value ..

Monitoring the process mean and variance using a synthetic control chart with two-stage testing

In this paper, we propose a synthetic control chart with two-stage testing (SyTS chart) to control the process mean and variance. As in the case of Shewhart control charts, samples are taken from the process at regular time intervals; however, testing is performed in two stages. During the first stage, one item of the sample is inspected; if its value is close to the target value of the process mean, then this terminates testing. Otherwise, the testing goes on to the second stage, where the remaining items are inspected and a non-central chi-square statistic is computed taking into account all items of the sample. When this statistic is larger than a specified value, the sample is classified as nonconforming. According to the synthetic procedure, the signal is based on the conforming run length (CRL). A comparative study shows that the SyTS chart and the joint and S charts with double sampling are very similar in performance. However, from a practical viewpoint, it is more convenient to monitor the process by looking at only one chart rather than looking at two charts separately. Comparisons with the joint and S charts and with several CUSUM schemes show that the SyTS chart has better overall performance.

Double-sampling control charts for attributes

In this article, we propose a double-sampling (DS) np control chart. We assume that the time interval between samples is fixed. The choice of the design parameters of the proposed chart and also comparisons between charts are based on statistical properties, such as the average number of samples until a signal. The optimal design parameters of the proposed control chart are obtained. During the optimization procedure, constraints are imposed on the in-control average sample size and on the in-control average run length. In this way, required statistical properties can be assured. Varying some input parameters, the proposed DS np chart is compared with the single-sampling np chart, variable sample size np chart, CUSUM np and EWMA np charts. The comparisons are carried out considering the optimal design for each chart. For the ranges of parameters considered, the DS scheme is the fastest one for the detection of increases of 100% or more in the fraction non-conforming and, moreover, the DS np chart is easy to operate.

Effect of the Amount of Phase I Data on the Phase II Performance of S2 and S Control Charts

Studies on the effect of parameter estimation on control-chart performance have mostly focused on the marginal (unconditional) run length (RL) distribution and some associated characteristics. However, once process parameters are estimated from an in-control (IC) reference sample, the RL follows its conditional distribution given the parameter estimates. With this in mind, our focus is the conditional RL distribution of the S and S2 charts. First, we concentrate on the IC conditional RL distribution, which is geometric with parameter (probability of success) αTRUE, the unknown attained false-alarm probability, given the estimate of the process standard deviation. We obtain and examine the distribution of αTRUE as a function of the number and the size of the reference samples. We consider a one-sided prediction interval for αTRUE and, for several sample sizes, obtain the minimum number of reference sample observations that guarantees, with a specified probability, that αTRUE will not exceed the nominal value of the false-alarm probability, αNOM, by more than a prespecified percentage. Next, we argue that (and demonstrate why), in the particular case of S and S2 charts, there is no practical need of a similar analysis of the effects of parameter estimation on their out-of-control performance, and draw some conclusions in terms of guidance for the user.

Performance Comparisons of EWMA Control Chart Schemes

Abstract Combining an EWMA chart with a Shewhart chart is traditionally recommended as a means of providing good protection against both small and large shifts in the process mean. Capizzi and Masarotto have proposed an EWMA chart with a variable smoothing constant (AEWMA) for the same purpose. In this paper, we optimize the designs of the AEWMA and of the combined EWMA-Shewhart schemes with regard to pairs of shifts in the process mean and compare their performances. When the schemes are optimized for the same pair of shifts, their ARL profiles practically coincide. The choice between the AEWMA and the EWMA-Shewhart scheme then becomes a matter of personal preference. We also explore using an AEWMA together with a Shewhart chart, but we find no performance improvement. An additional contribution of this paper is the tables of optimal designs of each scheme for several pairs of shifts.

Optimum Multiple and Multivariate Poisson Statistical Control Charts

This paper deals with the simultaneous statistical process control of several Poisson variables. The practitioner of this type of monitoring may employ a multiple scheme, i.e. one chart for controlling each variable, or may use a multivariate scheme, based on monitoring all the variables with a single control chart. If the user employs the multivariate schemes, he or she can choose from, for example, three options: (i) a control chart based on the sum of the different Poisson variables; (ii) a control chart on the maximum value of the different Poisson variables; and (iii) in the case of only two variables, a chart that monitors the difference between them. In this paper, the previous control charts are studied when applied to the control of p = 2, 3 and 4 variables. In addition, the optimization of a set of univariate Poisson control charts (multiple scheme) is studied. The main purpose of this paper is to help the practitioner to select the most adequate scheme for her/his production process. Towards this goal, a friendly Windows© computer program has been developed. The program returns the best control limits for each control chart and makes a complete comparison of performance among all the previous schemes. Copyright

T2 Control Charts With Variable Dimension

This paper proposes a new multivariate T2 control chart, termed the variable dimension T2 (VDT2) chart, in which the number of monitored quality characteristics is variable. When there are p related variables to be monitored, the approach is useful if there is a subset of p1 variables that are easy and/or inexpensive to measure, while the remaining variables are difficult and/or expensive to measure but provide additional useful information on whether the process mean has shifted. The VDT2 chart adaptively determines the number of variables to be monitored (either p1 or p) depending on the last value of the charted statistic. We provide Windows®-based software to optimize the parameters of the control chart using a Markov chain model to compute the performance measures and a genetic algorithm to conduct the optimization. The optimized VDT2 chart is quite powerful, and we demonstrate the somewhat surprising result that it detects mean shifts faster than a standard T2 chart for which all p variables are always measured, while also reducing sampling costs.