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A Multivariate Measure of Process Capability
Abstract
A measure of process capability for the multivariate normal case is proposed. This measure takes into account both proximity to the target and the variation observed in the process. The result is analogous to the univariate measure of process capability referred to as Cpm. Some statistical properties associated with the measure are examined. Multivariate specification limits and their creation are also discussed.
... Adicionalmente, esta nueva metodología permitirá su aplicación cuando las condiciones de las variables sean independientes, el histórico de datos por indicador supere el número total de variables y no exista homogeneidad en las muestras; lo cual la diferencia de las metodologías descritas por otros autores como Chan et al. (1991) donde se basa en una comparación de esperanzas de distribuciones, Taam et al. (1993) quienes proponen un índice construido mediante la comparación de los volúmenes de las regiones de tolerancia natural del proceso y de especificación, y Wang y Du (2000) que propone llevar a cabo el análisis de capacidad multivariado sobre una transformación de los datos originales, haciendo uso de la técnica multivariada de componentes principales. ...
En las compañías manufactureras, es indispensable conocer la capacidad que tienen los procesos de cumplir especificaciones o metas relacionadas con la eficiencia operativa, ya sea al planear las condiciones de calidad en manufactura o al momento de evaluar la gestión a través de los sistemas de gestión integrados. En las décadas recientes, ha surgido el concepto de capacidad del proceso o desempeño del proceso, que proporciona una estimación cuantitativa de qué tan conforme es un proceso. Este trabajo ilustra una metodología para calcular un indicador de capacidad de proceso multivariado validado en una compañía productora de bebidas gaseosas, el cual resume el comportamiento del sistema de gestión integrado y orienta a los administradores de procesos a tomar decisiones estratégicas sobre el control y la mejora de los procesos con base en la identificación de variables claves de procesos pertenecientes a los diferentes sistemas de gestión, basados históricamente con valores variables analizadas de manera univariada, recurriendo a análisis densos y sin percepción de las correlaciones posibles entre los diferentes factores de los sistemas integrados de gestión de calidad. La metodología está basada en el análisis de la base de datos correspondiente a los resultados de los indicadores de gestión de los diferentes sistemas de calidad, obtenidos históricamente y almacenados en los sistemas de información de la compañía. Estos datos se trataron como variables aleatorias distribuidas normalmente y agrupadas matemáticamente como variables con comportamientos distribuidos con chi – cuadrado, estableciendo metas o valores nominales de resultados de los sistemas de calidad. De estos cálculos resultaron valores apropiados a un desarrollo estable del sistema de calidad, logrando disminuir la dispersión a través del cálculo del indicador de capacidad y reflejando la maduración del sistema integral de gestión.
... The PCIs are extensively utilized to gauge the capability of the production process within the specified tolerance pre-set by the product designers or customers when the production process is in a perfect state of statistical control. Among the plethora of the suggested PCIs, the most frequently used PCIs in the literature are C p , C pk , C pm , C * pmk , C pmk and C p (u, v) [see, Juran (1974); Kane (1986); Hsiang and Taguchi (1985); Chan et al. (1988); Pearn et al. (1992), Vannman (1995)]. These indices, whose computation is meaningful only for normally distributed processes described by one characteristic. ...
Process capability index (PCI) is used to quantify the relation between the actual performance of the process and the pre-set specifications of the product. In this article, we utilize bootstrap re-sampling simulation method to construct bootstrap confidence intervals, namely, standard bootstrap (SB) and percentile bootstrap (PB) of the difference between two generalized process capability indices (δ = C py1 − C py2) to select the better of two processes or manufacturer's/supplier's through simulation when the underlying distribution is a normal distribution. The distribution of δ, the difference between two processes, or manufacturer's/supplier's capability indices, cannot be inferred statistically. Thus, we use the bootstrap re-sampling simulation technique to construct bootstrap confidence intervals of δ. The maximum likelihood method of estimation is used to estimate the parameters of the model. The proposed two bootstrap confidence intervals can be effectively employed to determine which one between the two processes or manufacturer's/supplier's has a better process capability. Monte Carlo simulations are performed to compare the performances of the proposed bootstrap confidence intervals for δ in terms of their estimated average widths and corresponding coverage probabilities. Simulation results showed that the average widths of the PB confidence interval perform better than their counterparts. Finally, two real data sets are presented to illustrate the bootstrap confidence intervals of the difference between two process capability indices.
This book is about modern industrial statistics and it applications using R, MINITAB and JMP. It is an expanded second edition of a book entitled Modern Industrial Statistics: Design and Control of Quality and Reliability, Wadsworth Duxbury Publishing, 1998, Spanish edition: Estadistica Industrial Moderna: Diseño y Control de Calidad y la Confiabilidad, Thomson International, 2000. Abbreviated edition: Modern Statistics: A Computer-based Approach, Thomson Learning, 2001. Chinese edition: China Statistics Press, 2003 and Softcover edition, Brooks-Cole, 2004.
We present a two-phase methodology based on the concept of depth to measure the capability of processes characterized by the functional relationship of multivariate nonlinear profile data, treated as multivariate functional observations. In the first phase, the modified tolerance region is estimated using a historical data set, while in the second, a current process is assessed using the proposed three-component vector, where the first component measures the volume ratio between the current process region and the modified tolerance region; the second measures the probability that the median of the current process is within the modified tolerance region, and the third measures the probability that the current process region is inside the modified tolerance region. To facilitate interpretation, a single index is derived from this capability vector. A simulation study is carried out to assess the performance of the proposed method. An real example illustrates the applicability of this approach.
In this article we have proposed multivariate cumulative sum control chart based on bivariate ranked set schemes for quick identification of small variation in the process mean vector. Also, we have offered multivariate measure of process capability based on bivariate ranked set schemes for testing the customer requirements. In the construction of control chart, we have designed plotting statistic, and derived control limit. Regarding the multivariate measure of process capability, we have defined an estimator and then computed the critical values for inference purposes. In order to compare the performance of existing and proposed control charts, we have obtained various performance measures. Results reveal that performance of proposed control chart based on bivariate ranked set schemes depends on the choice of the factors such as sampling scheme, sample size, magnitude of association between concomitant variable and study variables, and size of the shift. Furthermore, comparative analysis shows that the performance of the proposed control chart based on bivariate ranked set schemes outperforms the existing methods. Finally, real life example is included in which we have applied proposed and existing control charts for monitoring calcium–magnesium and residual sodium contents in irrigation water. In addition, the implementation of the proposed multivariate measure of process capability ensures the level of calcium–magnesium and residual sodium contents in irrigation water to satisfy the requirements of customers or engineering tolerance.
Process capability indices like Cp, Cpk, Cpm and Cpmk are commonly used in industry to assess the competence of a process to conform the control limits on quality parameters of process study. As product designs are complicated to meet international standards and consumers’ requirements are changeable rapidly, quality characteristics must be analyzed simultaneously to improve product’s quality and also to consider the aspect of multicollinearity among different quality characteristics. The objective of this paper is to measure univariate and multivariate capability indices to assess performance of multistage processes of paper manufacturing at Paswara Papers Ltd., Meerut. The paper manufacturing has various stages, namely pulping, screening, cleaning, deinking, refining, pressing, steam drying, bleaching, steam drying and size press. Different capability indices are measured for various processes of paper manufacturing. The results reveal that multivariate process capability indices, introduced by Shahriari and Lawrence (fourth industrial engineering research conference, pp 304–309, 1995, [6]) [Cpm = 0.6197], gives higher capability as compared to Taam and Liddy (J Appl Stat 20:12, 1993, [5]) [MCpm = 0.1408], and Pan and Lee (Qual Reliab Eng Int 26:3–15, 2010, [7]) [NMCpm = 0.0130].
The capability indices Cp, CPU, CPL, k and Cpk are presented and related to process parameters. These indices are shown to form a complementary system of measures of process performance, and can be used with bilateral and unilateral tolerances, with or without target values. A number of Japanese industries currently use the five indices and the U.S. automotive industry has started using these measures in a number of areas. Various applications of the indices are discussed along with statistical sampling considerations.
A recent awareness of the need to compete more effectively with Eastern industry has led Western manufacturing companies to take more notice of the potential benefits of quality tools, such as statistical process control. This paper outlines the use of traditional SPC techniques, i.e. control chart design and usage, the importance of process capability as an integral part of SPC, and how these techniques can lead to misinterpretation when applied to modern-day manufacturing practices. Small-batch manufacturing and computer-integrated manufacturing often call for 100 per cent inspection, which introduces the possibility of significant correlation in serial data. Also discussed are process variability and the importance of being able to distinguish between within-sample and between-sample variation.
Over the years, Japanese managers, engineers, and workers have been very successful in reducing manufacturing costs by adopting more enlightened quality thinking and by applying more technical quality methods. Since 1980, US automotive companies and their suppliers have made dramatic improvements in quality. Much of this improvement has been due to more intensive application of traditional quality methods. Although statistical methods are uniform throughout the world, they are applied very differently in the East and West. Of foremost importance is the new definition of 'manufacturing' quality as product uniformity around the target rather than conformance to specification limits. This difference in thinking results in totally different operational and engineering approaches to quality and productivity. This article explores the distinctions in terms of process variations, capability index, relationships between cost and quality, and related subjects.
The multivariate acceptance sampling problem is discussed and a solution for a particular sub-class of the general problem developed. This is where the related quality variables have a multivariate normal distribution with known covariance matrix, and where the specification region may be set in the form of a general ellipsoid. Single sampling acceptance procedures by variables for fraction defective are developed. Methods are given for determining the sample size n and acceptance number c so as to achieve desired operating characteristics.