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Measuring process performance for multiple variables
Abstract
This article deals with a multivariate version of the Cpm, which is an extension of the work in the Quality Portfolio Chart article by Holmes. This multivariate measure, which we will refer to as the Crm, is comprised of two components - one of which me..
... 1 the indices which measure the ratio of a tolerance region to a process region, such as the research proposed by Wang et al. (2000), Shahriari et al. (1995), and Taam et al. (1993) 2 the indices based on proportion of conforming items, such as the research proposed by Polansky (2001), Pal (1999), and Chen (1994) 3 the indices based on principal components analysis (PCA), for instance the research proposed by Wang and Chen (1999) 4 the indices based on the extension of univariate PCI such as those introduced by Holmes and Mergen (1999) and Chen et al. (2003). ...
While most of the methods developed for computing process capability indices (PCI) concentrate on cases with normally or continuous non-normally distributed quality characteristics, computing this measure for processes with mixed distributed data has not been investigated so far. In this paper, a new method is proposed for computing (PCI) for mixed binary-normal quality characteristics. In the proposed method, first a mixed binary-normal distribution is considered to be fitted on the available data. Having estimated the unknown parameters of the fitted distribution using maximum likelihood estimation and genetic algorithm, the proportion of the conforming items of thecorresponding distribution is estimated by Monte Carlo simulation runs. Finally, the PCI is computed based on the relationship of PCI and proportion of conforming items. The performance of the proposed method is evaluated using simulation studies as well as a case study in a plastic injection moulding process.
... In the third category using principle component analysis both normal and non-normal cases have been studied for multivariate process capability indices; e.g., see Wang and Chen [20]. Some researchers introduced the fourth category based on the extension of univariate indices such as Chen et al. [21] and Holmes and Mergen [22]. Although several researches have been proposed for computing the capability indices for multivariate normal response processes, very few researches have discussed about computing PCI for multivariate non-normal response ones. ...
Most of the researches developed for single response and multi response optimization problems are based on the normality assumption of responses, while this assumption does not necessarily hold in real situations. In the real world processes, each product can contain correlated responses which follow different distributions. For instance, multivariate non-normal responses, multi-attribute responses or in some cases mixed continuous-discrete responses. In this paper a new approach is presented based on multivariate process capability index and NORTA inverse transformation for multi response optimization problem with mixed continuous-discrete responses. In the proposed approach, assuming distribution function of the responses is known in advance based on historical data; first we transform the multivariate mixed continuous-discrete responses using NORTA inverse transformation to obtain multivariate normal distributed responses. Then the multivariate process capability index is computed in each treatment. Finally, for determining the optimum treatment, the geometric mean value of multivariate Process Capability Index (PCI) is computed for each factor level and the most capable levels are selected as the optimum setting. The performance of the proposed method is verified through a real case study in a plastic molding process as well as simulation studies with numerical examples.
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.
Providing a single-valued assessment of the performance of a process is often one of the greatest challenges for a quality professional. Process Capability Indices (PCIs) precisely do this job. For processes having a single measurable quality characteristic, there is an ample number of PCIs, defined in literature. The situation worsens for multivariate processes, i.e., where there is more than one correlated quality characteristic. Since in most situations quality professionals face multiple quality characteristics to be controlled through a process, Multivariate Process Capability Indices (MPCIs) become the order of the day. However, there is no book which addresses and explains different MPCIs and their properties. The literature of Multivariate Process Capability Indices (MPCIs) is not well organized, in the sense that a thorough and systematic discussion on the various MPCIs is hardly available in the literature.
Handbook of Multivariate Process Capability Indices provides an extensive study of the MPCIs defined for various types of specification regions. This book is intended to help quality professionals to understand which MPCI should be used and in what situation. For researchers in this field, the book provides a thorough discussion about each of the MPCIs developed to date, along with their statistical and analytical properties. Also, real life examples are provided for almost all the MPCIs discussed in the book. This helps both the researchers and the quality professionals alike to have a better understanding of the MPCIs, which otherwise become difficult to understand, since there is more than one quality characteristic to be controlled at a time.
A multivariate exponentially weighted moving average (MEWMA) control chart is proposed for detecting process shifts during the phase II monitoring of simple linear profiles (SLPs) in the presence of within-profile autocorrelation. The proposed control chart is called MEWMA-SLP. Furthermore, two process capability indices are proposed for evaluating the capability of in-control SLP processes, and their utilization is demonstrated through examples. Intensive simulations reveal that the MEWMA-SLP chart is more sensitive than existing control charts in detecting profile shifts.
Most multivariate process capability indices proposed by other researchers provide quality measure in losses due to variability and being off-target. Some suggested loss function models that estimate losses due to rejection and quality costs but failed to consider the correlation between the multivariate quality characteristics. In this paper, we propose a novel new approach for estimating the total expected quality cost that provides quality cost measure not only in losses due to variability and being off-target but also losses due to rejection. The proposed model also estimates the quality costs for correlated quality characteristics with nominal specifications and provides the estimated costs due to individual characteristics; it helps to identify those process quality characteristics which have high variation when compared with their specification spread. The model will also improve the estimate of the rejection cost suggested by earlier researchers as it identifies the reject region as the region outside of the modified tolerance region. We present four different correlated bivariate normal process scenarios where the proposed method can be applicable and demonstrate that our approach provides a robust tool in estimating the risk of the product being rejected and performs as well as existing methods in estimating the total quality cost.
ultivariate process capability indices (MPCI), as an important means of statistical process control (SPC), can be used to ensure the high reliability of semiconductors manufacturing process. However, the reasonable sampling number is an important factor when considering MPCI values. As general, the large sample number requires much effort and time, or even cannot be achieved. In this paper, we evaluated the impact of different sample size on the calculations of multivariate process capability indices using simulation and analyses. After getting enough data and choosing disparate sample numbers, corresponding multivariate process capability indices can be obtained, which demonstrate the relationship between sampling numbers and calculation results. The conclusions have critical guiding significance for manufacturing semiconductors with high reliability requirement.
In some quality control applications, quality of a product or a process can be characterized by a profile defined as a functional relationship between a response variable and one or more explanatory variables. Many researchers have contributed to the development of linear and nonlinear profiles to monitor a process or product. However, less work has been devoted to the development of process capability indices in profile monitoring to evaluate process performance with respect to specification limits. This paper presents a process capability analysis when the quality characteristic of interest is represented by a linear profile. Simulation analyses along with a real case study in leather industry are used to evaluate the performance of the proposed method. Results indicate satisfactory performance.
Process Capability Indices, such as Cp and Cpk have been widely used as statistical tools to assess the manufacturing process performance. These indices provide numerical measures on process precision, process accuracy and process performance. Extensive researches have been done on the univariate process capability indices in the last two decades. However, process capability indices for processes with multiple quality characteristics have received little attention, comparatively. The multivariate process capability indices, which are used for evaluation of processes with correlated quality characteristics such as weight, height and width, could be investigated, in depth. In this research the concept of process capability and its relevant indices in univariate and multivariate cases are discussed. Based on the previous works, a new multivariate process capability vector (NMPCV) is introduced. This vector is based on a modification of the traditional multivariate process capability indices. The value and the power of this new index are evaluated using field and simulated data. The results of this research show that NMPCV is a better tool for judging the process accuracy, the process precision and the process performance. When the characteristics are highly correlated, the results are much better.
A new measure of the process capability (Cpm) is proposed that takes into account the proximity to the target value as well as the process variation when assessing process performance. The sampling distribution for an estimate of Cpm (Ĉpm) and some of its properties have been examined and an example of its application is included. The new index is easy to compute and, with the aid of the included tables, easy to analyze. Ĉpm has some more desirable statistical properties than Ĉp and Ĉpk,the estimates of the Cp and Cpk indices, respectivel
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.
In this paper we survey and critically comment on some basic concepts and notions relevant to the methodology dealing with process capability indices (PCI's). The emphasis is on distributional and inferential aspects. The intent is to contribute to the understanding and interpretation of these indices, if one is (unfortunately) required to use them. We regard them primarily as indicators of the expected proportion of nonconforming items, and not as having some relationship to arbitrary loss functions, as is currently the more fashionable viewpoint. New PCI's, related to the expected proportion of nonconforming items, are suggested.
In manufacturing industry, there is growing interest in quantitative measures of process variation under multivariate setting. This paper introduces a multivariate capability index and focuses its applications in geometric dimensioning and tolerancing. This index incorporates both the process variation and the process deviation from target. Two existing multivariate indices are compared with the proposed index.
The T2 control chart is commonly used for multivariate statistical process control. We propose a new method for computing the variance-covariance matrix. The proposed method eliminates some of the disadvantages observed in the application of the T2 cont..
The performance of a product often depends on several quality characteristics. These characteristics may have interactions. In answering the question “Is the process in control?”, multivariate statistical process control methods take these interactions into account. In this paper, we review several of these multivariate methods and point out where to fill up gaps in the theory. The review includes multivariate control charts, multivariate CUSUM charts, a multivariate MMA chart, and multivariate process capability indices. The most important open question from a practical point of view is how to detect the variables that caused an out-of-control signal. Theoretically, the statistical properties of the methods should be investigated more profoundly.
Decision Sciences in the College of Business at the Rochester Institute of Tech-nology, Rochester, NY. He holds a Ph.D. in administrative and engineering systems, an M.Sc. in industrial administra-tion, and a B.Sc. in management. He is a member of the American Society for Quality
- A Erhan Mergen
A. Erhan Mergen is a professor of Decision Sciences in the College of Business at the Rochester Institute of Tech-nology, Rochester, NY. He holds a Ph.D. in administrative and engineering systems, an M.Sc. in industrial administra-tion, and a B.Sc. in management. He is a member of the American Society for Quality. Downloaded by [Akdeniz Universitesi] at 11:34 23 December 2014