Eugenio Kahn Epprecht

Shewhart control charts for dispersion adjusted for parameter estimation

ABSTRACT Several recent studies have shown that the number of Phase I samples required for a Phase II control chart with estimated parameters to perform properly may be prohibitively high. Looking for a more practical alternative, adjusting the control limits has been considered in the literature. We consider this problem for the classic Shewhart charts for process dispersion under normality and present an analytical method to determine the adjusted control limits. Furthermore, we examine the performance of the resulting chart at signaling increases in the process dispersion. The proposed adjustment ensures that a minimum in-control performance of the control chart is guaranteed with a specified probability. This performance is indicated in terms of the false alarm rate or, equivalently, the in-control average run length. We also discuss the tradeoff between the in-control and out-of-control performance. Since our adjustment is based on exact analytical derivations, the recently suggested bootstrap method is no longer necessary. A real-life example is provided in order to illustrate the proposed methodology.

Um método simples para o projeto ótimo de gráficos de X

A simple but efficient method has been developed for the design of -charts. The solutions provided are optimal in terms of the ratio between the number of items inspected per unit time and the detection speed (defined as the inverse of the average time to signal a shift in the mean). The method is superior to design procedures based exclusively on statistical properties of the chart (such as the ARL), and does not have the implementation drawbacks of economical design models. The results obtained differ from some results widely reported in the literature, which are shown to be sub-optimal as a consequence of their underlying model not considering some important aspects of the problem. An additional contribution of the method is the explicit treatment of optional constraints on the sample size and/or on the interval between samples. The ease of use of the method makes its implementation straightforward in any environment, and also makes it suitable for the classroom.

Statistical Process Control Based on Optimum Gages.

Classical statistical process control (SPC) by attributes is based on counts of nonconformities. However, process quality has greatly improved with respect to past decades, and the vast majority of samples taken from high-quality processes do not exhibit defective units. Therefore, control charts by variables are the standard monitoring scheme employed. However, it is still possible to design an effective SPC scheme by attributes for such processes if the sample units are classified into categories such as ‘large’, ‘normal’, or ‘small’ according to limits that are different from the specification limits. Units classified as ‘large’ or ‘small’ will most likely still be conforming (within the specifications), but such a classification allows monitoring the process with attributes charts. In the case of dimensional quality characteristics, gages can be built for this purpose, making inspection quick and easy and reducing the risk of errors. We propose such a control chart, optimize it, compare its performance with the traditional X¯ and S charts and with another chart in the literature that is also based in classifying observations of continuous variables through gaging, and present a brief sensitivity analysis of its performance. The new chart is shown to be competitive with the use of X¯–S charts, with the operational advantage of simpler, faster, and less costly inspection. Copyright

Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula

ABSTRACT Undergraduate and graduate students in a first-year probability (or a mathematical statistics) course learn the important concept of the moment of a random variable. The moments are related to various aspects of a probability distribution. In this context, the formula for the mean or the first moment of a nonnegative continuous random variable is often shown in terms of its c.d.f. (or the survival function). This has been called the alternative expectation formula. However, higher-order moments are also important, for example, to study the variance or the skewness of a distribution. In this note, we consider the rth moment of a nonnegative random variable and derive formulas in terms of the c.d.f. (or the survival function) paralleling the existing results for the first moment (the mean) using Fubinis theorem. Both nonnegative continuous and discrete integer-valued random variables are considered. These formulas may be advantageous, for example, when dealing with the moments of a transformed random variable, where it may be easier to derive its c.d.f. using the so-called c.d.f. method.

Phase II control charts for monitoring dispersion when parameters are estimated

ABSTRACT Shewhart control charts are among the most popular control charts used to monitor process dispersion. To base these control charts on the assumption of known in-control process parameters is often unrealistic. In practice, estimates are used to construct the control charts and this has substantial consequences for the in-control and out-of-control chart performance. The effects are especially severe when the number of Phase I subgroups used to estimate the unknown process dispersion is small. Typically, recommendations are to use around 30 subgroups of size 5 each.  We derive and tabulate new corrected charting constants that should be used to construct the estimated probability limits of the Phase II Shewhart dispersion (e.g., range and standard deviation) control charts for a given number of Phase I subgroups, subgroup size and nominal in-control average run-length (ICARL). These control limits account for the effects of parameter estimation. Two approaches are used to find the new charting constants, a numerical and an analytic approach, which give similar results. It is seen that the corrected probability limits based charts achieve the desired nominal ICARL performance, but the out-of-control average run-length performance deteriorate when both the size of the shift and the number of Phase I subgroups are small. This is the price one must pay while accounting for the effects of parameter estimation so that the in-control performance is as advertised. An illustration using real-life data is provided along with a summary and recommendations.

Optimum variable-dimension EWMA chart for multivariate statistical process control

ABSTRACT The variable-dimension T2 control chart (VDT2 chart) was recently proposed for monitoring the mean of multivariate processes in which some of the quality variables are easy and inexpensive to measure while other variables require substantially more effort or expense. The chart requires most of the times that only the inexpensive variables be sampled, switching to sampling all the variables only when a warning is triggered. It has good ARL performance compared with the standard T2 chart, while significantly reducing the sampling cost. However, like the T2 chart, it has limited sensitivity to small and moderate mean shifts. To detect such shifts faster, we developed an exponentially weighted moving average (EWMA) version of the VDT2 chart, along with Markov chain models for ARL calculation, and software (made available) for optimizing the chart design. The optimization software, which is based on genetic algorithms, runs in Windows© and has a friendly user interface. The performance analysis shows the great gain in performance achieved by the incorporation of the EWMA procedure.

In-control performance of the joint Phase II − S control charts when parameters are estimated

ABSTRACT The issue of the effects of parameter estimation on the in-control performance of control charts has motivated researchers for several decades. In this context, recently, acknowledging what has been called by some the practitioner-to-practitioner variability, a new perspective has been advocated, namely, the study of the conditional distribution of the in-control average run length (or the conditional false-alarm rate), which is more meaningful in practice. Adopting this new perspective, some authors have analyzed the conditional distribution of the false-alarm rate (or of the in-control average run length) of and of S charts separately. However, since the and S charts are not typically used separately but together or jointly in many applications, here we study the effects of parameter estimation on the performance of the two charts applied jointly (called the joint charts). For the joint charts, defining the joint false-alarm rate as the probability that at least one of the two charts ( and S) issues a false alarm, we obtain its conditional distribution, some quantiles of interest (upper prediction bounds for it) and the number of Phase I samples required to guarantee that with a high probability the conditional joint false alarm rate will not exceed a maximum tolerated value. We assume normality, consider Sp (the square root of the pooled variance) as the Phase I estimator of the process standard deviation, and consider two possibilities regarding the chart: (1) centered at and (2) centered at a specified target value. The results show (and we formally prove) that, whereas the required number of Phase I samples may be very large for the joint charts, interestingly, it lies between the corresponding numbers of samples required by the chart and by the S chart individually; so, considering the performance of the charts from the perspective of their joint use may slightly alleviate the required number of Phase I samples.

Otimização conjunta de gráficos de X-barra - S ou X-barra - R: um procedimento de fácil implementação

Resumo Este trabalho desenvolve um modelo para escolha otima dos parâmetros de operacao de graficos de e R (ou de e S) que minimiza a razao entre o custo de amostragem e a rapidez de deteccao de desvios na media ou aumentos na dispersao do processo. Admitem-se tres formas alternativas para o problema: minimizar os tempos medios de sinalizacao sob uma restricao ao custo de amostragem; minimizar esse custo sob uma restricao aos tempos de sinalizacao; e o problema multiobjetivo de minimizar o custo e os tempos de sinalizacao. Restricoes adicionais sao permitidas, para tratar de variantes do problema encontraveis na pratica. O procedimento evita a complexidade dos modelos de projeto economico usuais. Sao detalhados metodos para determinacao dos poucos parâmetros de especificacao e entrada exigidos pelo modelo. Um exemplo mostra que o procedimento e de facil aplicacao. Tudo isto aumenta sua aplicabilidade para um grande espectro de situacoes praticas tipicas. Palavras-chave Controle estatistico de processos, graficos de controle, projeto semi-economico, otimizacao, multiobjetivo

Statistical Control of Multiple-Stream Processes: A Literature Review

This paper presents a survey of the research on techniques for the statistical control of industrial multiple-stream processes—processes in which the same type of item is manufactured in several streams of output in parallel, or still continuous processes in which several measures are taken at a cross section of the product. The literature on this topic is scarce, with few advances since 1950, and experiencing a resurgence from the mid-1990s. Essential differences in the underlying models of works before and after 1995 are stressed, and issues for further research are pointed out.

Métodos de identificação de efeitos na dispersão em experimentos fatoriais não replicados

A identificacao de fatores que afetam a media e a dispersao de caracteristicas de qualidade e essencial para a otimizacao de produtos e processos produtivos. Para tal identificacao, costuma-se utilizar experimentos fatoriais. A analise de efeitos sobre a dispersao, contudo, usualmente demanda repeticoes do experimento sob as mesmas condicoes (replicacoes), que podem ser caras. Assim, varios metodos tem sido propostos na literatura para a identificacao de efeitos na dispersao a partir de experimentos fatoriais nao replicados. Neste artigo, analisamos alguns desses metodos, ilustrados com um exemplo de um processo produtivo, concluindo pela superioridade de um metodo iterativo baseado em modelos lineares generalizados. Finalmente, utilizando os modelos para a media e para a dispersao, fornecidos pelo metodo iterativo, procedemos a otimizacao do processo descrito no exemplo.