Maysa S. De Magalhães

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.

An adaptive chart for monitoring the process mean and variance

Traditionally, an chart is used to control the process mean and an R chart is used to control the process variance. However, these charts are not sensitive to small changes in the process parameters. The adaptive and R charts might be considered if the aim is to detect small disturbances. Due to the statistical character of the joint and R charts with fixed or adaptive parameters, they are not reliable in identifying the nature of the disturbance, whether it is one that shifts the process mean, increases the process variance, or leads to a combination of both effects. In practice, the speed with which the control charts detect process changes may be more important than their ability in identifying the nature of the change. Under these circumstances, it seems to be advantageous to consider a single chart, based on only one statistic, to simultaneously monitor the process mean and variance. In this paper, we propose the adaptive non-central chi-square statistic chart. This new chart is more effective than the adaptive and R charts in detecting disturbances that shift the process mean, increase the process variance, or lead to a combination of both effects. Copyright

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.

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.

Joint economic model for totally adaptive and R charts

Recent studies have shown that the amount of cost savings produced by a Vp X chart designed economically is higher than the fixed parameter X chart. This paper extends these studies for processes that are monitored jointly by X and R charts having all design parameters varying adaptively. We develop a joint expected cost model for a process whose mean is controlled by a Vp X chart and whose variance is controlled by a Vp R chart. The cost function due to controlling the process quality through these Vp charts is derived. This model provides a tool for optimal selection of the design parameters, in this way the minimum cost can be obtained. The possible savings provided by the developed model can be assessed.

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.

Gráfico de controle de VMAX para o monitoramento da matriz de covariâncias

Neste artigo e proposto, para o monitoramento de processos normais bivariados, um grafico de controle baseado nas variâncias amostrais de duas caracteristicas de qualidade. Os pontos plotados no grafico correspondem ao valor da maior variância amostral. O grafico proposto, denominado grafico de VMAX, tem um desempenho superior ao do grafico da variância amostral generalizada |S| e, alem disso, tem uma melhor capacidade de diagnostico, ou seja, com ele e mais facil identificar a variavel que teve sua variabilidade alterada pela ocorrencia da causa especial. Quando a amostragem dupla esta em uso o grafico proposto tambem tem um desempenho superior ao do grafico de |S|, exceto em alguns casos em que o tamanho da segunda amostra e muito grande.

Economic-statistical design of variable parameters non-central chi-square control chart

Production processes are monitored by control charts since their inception by Shewhart (1924). This surveillance is useful in improving the production process due to increased stabilization of the process, and consequently standardization of the output. Control charts keep track of a few key quality characteristics of the outcome of the production process. This is done by means of univariate or multivariate charts. Small improvements in control chart methodology can have significant economic impact in the production process. In this investigation, we propose the monitoring of a single variable by means of a variable parameter non-central chi-square control chart. The design of the chart is accomplished by means of optimizing a cost function. We use here a simulated annealing optimization tool, due to the difficulty of classical gradient based optimization techniques to handle the optimization of the cost function. The results show some of the drawbacks of using this model.