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Process Capability Indices Dedicated to Bivariate Non Normal Distributions
Abstract
Purpose
Process capability indices (PCI) are frequently used in order to measure the performance of production processes. In their 2005 article, Castagliola and Castellanos proposed a new approach for the estimation of bivariate PCIs in the case of a bivariate normal distribution and a rectangular tolerance region. This paper proposes extending Castagliola and Garcia‐Castellanos's paper to the estimation of bivariate PCIs in the case of non‐normal bivariate distributions.
Design/methodology/approach
The proposed method is based on the use of Johnson's System of distributions/transformations in order to transform the bivariate non normal distribution into an approximate bivariate normal distribution. Numerical examples are presented and some criteria are given in order to choose the appropriate Johnson's distribution.
Research limitations/implications
The proposed method is only dedicated to the case of two quality characteristics and a rectangular tolerance region (the most common case).
Findings
The proposed method allows the evaluation of bivariate capability indices irrespective of the distribution of the data and thus allows obtaining more reliable estimates for these values.
Originality/value
The main originality of the method presented in this paper is its ability to compute bivariate capability indices when the distribution of the data is not a bivariate normal distribution, i.e. the general case.
... Thus MPCIs utilizes theoretical elliptical shape as the process region and compare so defined process area to the area of the specification area. When the assumption about the normal distribution can not be fulfilled, the data transformation is carried out to obtain normally distributed variables [1,3,2]. Many authors propose also a modification of the original specification limits. ...
... 1. the assumption, that all quality characteristics are normally distributed, 2. the limitation concerning a sample size, which should be large enough to guarantee the proper estimation of covariance matrix [19], 3. the complexity of methodology for assessing multivariate process capability indices [13]. ...
... 1. the probability of nonconforming items using distribution of the multivariate process, 2. the ratio of a tolerance region to a process region, 3. different approaches using loss functions, 4. geometric distance approach involving principle component analysis (PCA). ...
To evaluate the capability of manufacturing processes in satisfying the customer's needs, a variety of indices has been developed. Some of them are introduced by researchers to analyse the processes with multivariate quality characteristics. Most of the proposed in the literature multivariate capability indices are defined under assumption of normality distribution of the quality characteristics. Thus, the process region describing the variation of the data has an elliptical shape. In this paper, a multivariate process capability vector with three components is introduced, which allows to access the capability of a process with both normally and non-normally quality characteristics due to application of a pair of one-sided models as the process region shape. At the beginning, one-sided models are defined, next the proposed vector components are proposed and the methodology of their evaluation is presented. The methodology (which in fact could be also applied to both the correlated and non-correlated characteristics) is verified by applying simulation and real problems. The obtained results show that the proposed methodology performs satisfactorily in all considered cases. Copyright © 2014 John Wiley & Sons, Ltd.
... It is therefore important to develop control charts that are able to treat non-Gaussian data. In some particular cases, it is possible to transform non-Gaussian distribution in Gaussian distribution using Johnson tranformation system (see for example Castagliola and Garcia Castellanos [5]) before using a Hotelling T 2 rule. One another way is to use nonparametric control charts, for which no a priori information on the law of the observations is necessary. ...
... Remark 1 : The distribution of the data studied in this section, and more generally, the distributions often encountered in semiconductor manufacturing are multimodal. Thereby, methods such as Johnson transformation system, to transform non-Gaussian distributions in Gaussian distributions [5] are not effective. The Hotelling T 2 rule is therefore applied to the original data. ...
... As a consequence of systematic influences, the distribution of the process characteristics is non-normal. However, the scientific literature provides both univariate and multivariate process capability analyses based on non-normally distributed data (see e.g., Kotz and Johnson, 1993;Spedding and Rawlings, 1994;Somerville and Montgomery, 1996;Shore, 1998;Wang and Du, 2000;Castagliola and Castellanos, 2008). In any case, it is important to understand the reason for the non-normality of the process characteristics. ...
... Therefore, it is advisable to calculate a capability measure with both transformed and untransformed data. Second, it is possible to apply process capability indices dedicated to non-normally distributed process characteristics (see e.g., Wang and Du, 2000;Castagliola and Castellanos, 2008;Abbasi, 2009). ...
... It is therefore important to develop control charts that are able to treat non-Gaussian data. In some particular cases, it is possible to transform non-Gaussian distribution in Gaussian distribution using Johnson tranformation system (see for example Castagliola and Garcia Castellanos [5]) before using a Hotelling T 2 rule. One another way is to use nonparametric control charts, for which no a priori information on the law of the observations is necessary. ...
... Remark 1 : The distribution of the data studied in this section, and more generally, the distributions often encountered in semiconductor manufacturing are multimodal. Thereby, methods such as Johnson transformation system, to transform non-Gaussian distributions in Gaussian distributions [5] are not effective. The Hotelling T 2 rule is therefore applied to the original data. ...
In recent years, fault detection has become a crucial issue in semiconductor manufacturing. Indeed, it is necessary to constantly improve equipment productivity. Rapid detection of abnormal behavior is one of the primary objectives. Statistical methods such as control charts are the most widely used approaches for fault detection. Due to the number of variables and the possible correlations between them, these control charts need to be multivariate. Among them, the most popular is probably the Hotelling T<sup>2</sup> rule. However, this rule only makes sense when the variables are Gaussian, which is rarely true in practice. A possible alternative is to use nonparametric control charts, such as the k -nearest neighbor detection rule by He and Wang, in 2007, only constructed from the learning sample and without assumption on the variables distribution. This approach consists in evaluating the distance of an observation to its nearest neighbors in the learning sample constituted of observations under control. A fault is declared if this distance is too large. In this paper, a new adaptive Mahalanobis distance, which takes into account the local structure of dependence of the variables, is proposed. Simulation trials are performed to study the benefit of the new distance against the Euclidean distance. The method is applied on the photolithography step of the manufacture of an integrated circuit.
... • Grupo 2: estos índices se basan en la probabilidad de producto no conforme, como el índice propuesto por Wierda (1994), (Bothe, 1999), (Castagliola & Castellanos, 2008) y (de-Felipe & Benedito, 2017). ...
Los productos fabricados actualmente tienen varias características de calidad, todas tienen importancia para el cliente, y controlarlas y evaluarlas se ha convertido en una actividad de primer interés. La industria automotriz establece índices de capacidad para evaluar la capacidad de los procesos donde se tiene únicamente una variable de respuesta, y cuando estos son estables se recomienda utilizar el Cp y el Cpk como medibles de la capacidad del proceso de manufactura para manufacturar productos que cumplan con las especificaciones y sean catalogados como productos de calidad. Para evaluar la calidad completa de un producto, la cual depende de que se cumpla con varias características de calidad simultáneamente, existen propuestas en la literatura de cómo medir la capacidad de procesos multivariados; la mayoría de estas coinciden en que se debe establecer claramente una región de especificación que represente lo que establece el cliente y otra región que muestre la variación como medida de desempeño del proceso. En la definición de ambas regiones existe una gran controversia entre los autores, lo que lleva a presentar esta nueva propuesta, mediante la cual se definen de manera confiable estas dos regiones mencionadas, y al hacer una comparación entre ellas se pueden obtener los índices de capacidad multivariados Cpm y el Cpkm y el como una extensión de los índices univariados Cp y el Cpk. El documento incluye el análisis de datos de un proceso donde el producto, para ser de buena calidad, debe cumplir con dos características de calidad simultáneamente. Las mediciones obtenidas del proceso se pueden representar por una distribución normal multivariada, lo que permite medir la capacidad del proceso utilizando los índices propuestos y realizar una interpretación de estos en relación con el desempeño del proceso.
... In a similar study, Hosseinifard et al. (2009) examined the performance of the root transformation technique in comparison with the Box-Cox (1964) method and concluded that the root transformation technique is more efficient than the Box-Cox (1964) transformation in the procedure of computing the PCI. Castagliola and Castellanous (2008) used the Johnson transformation (1949) method for transforming the non-normally distributed quality characteristics to multivariate normally distributed data and then computed the multivariate process capability index for the transformed data. ...
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.
... Hosseinifard et al. [25] applied a similar procedure using other transformations such as Box–Cox [29]. The method presented by Castagliola and Castellanos [26] uses Johnson transformation; however, since this method has been only used for multivariate non-normal responses, its usage for transforming discrete data may be disputable. The reason is that Johnson transformation does not usually results in a desired transformation function with an acceptable P-value for discrete data. ...
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.
DOI: https://doi.org/10.26439/ing.ind2020.n038.4814
En el análisis de capacidad de procesos multivariados existen muchos índices que solo se aplican cuando los datos son normales y otros cuando los datos son no normales; lo mismo ocurre cuando las variables de calidad están correlacionadas y no correlacionadas. En este trabajo se propone un índice de capacidad multivariado CPME, desarrollado bajo el uso inicial de un índice univariado, según sea el caso normal o no normal, y cualquier correlación entre variables, para luego, a través de una función característica, extenderlo para el caso multivariado. Este índice puede ser aplicado para todos los casos anteriores. Como utilidad, presentamos ejemplos de aplicación de esta alternativa sobre un conjunto de datos reales y simulados, donde se encontró un amplio desempeño del índice propuesto frente a otros de capacidad similares.
En el análisis de capacidad multivariado existen muchos aspectos aún no resueltos en torno a algunos índices, como la no normalidad de los datos y si las variables de calidad están correlacionadas o no están correlacionadas. En este trabajo se pretendió proponer la elaboración de un índice de capacidad multivariado (ICPM) que funcione para ambos casos anteriores por medio de ejemplos de datos simulados. Cabe mencionar que se encontró un amplio desempeño del índice propuesto frente a otros índices de capacidad similares.
En el análisis de capacidad multivariado existen muchos aspectos aún no resueltos en torno a algunos índices, como la no normalidad de los datos y si las variables de calidad están correlacionadas o no están correlacionadas. En este trabajo se pretendió proponer la elaboración de un índice de capacidad multivariado (ICPM) que funcione para ambos casos anteriores por medio de ejemplos de datos simulados. Cabe mencionar que se encontró un amplio desempeño del índice propuesto frente a otros índices de capacidad similares.
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This paper offers a review of univariate and multivariate process capability indices (PCIs). PCIs are statistic indicators widely used in the industry to quantify the capability of production processes by relating the variability of the measures of the product characteristics with the admissible one. Univariate PCIs involve single-product characteristics while multivariate PCIs deal with the multivariate case. When analyzing the capability of processes, decision makers of the industry may choose one PCI among all the PCIs existing in the literature depending on different criteria. In this article, we describe, cluster, and discuss univariate and multivariate PCIs. To cluster the PCIs, we identify three classes of characteristics: in the first class, the characteristics related to the information of the process data input are included; the second class includes characteristics related to the approach used to calculate the PCIs; and in the third class, we find characteristics related to the information that the PCIs give. We discuss the strengths and weaknesses of each PCI using four criteria: calculation complexity, globality of the index, relation to proportion of nonconforming parts, and robustness of the index. Finally, we propose a framework that may help practitioners and decision makers of the industry to select PCIs.
There have been a number of process capability indices (PCIs) which are summary statistics to depict the process location and dispersion successfully proposed over the years for the purpose of assessing the capability of a process to meet certain specifications. Generally PCIs are used with symmetric tolerances when the target value (T) locates on the midpoint of the specification interval (m). Sometimes it is possible to meet the case target value does not located on mid-point (T ≠ m) which called asymmetric tolerances. To analyze the process which have asymmetric tolerances, PCIs with asymmetric tolerances have been developed and successfully applied. Process incapability index (C pp) is used both symmetric and asymmetric cases. In this paper, C pp with asymmetric tolerances (C" pp ) pp is analyzed by using fuzzy set theory to obtain a deep and flexible analysis. To produce fuzzy process incapability index with asymmetric tolerances (C̃" pp), fuzzy process mean μ̃ and fuzzy variance σ̃ 2 , which are obtained by using the fuzzy extension principle, are used together with fuzzy specification limits (SLs) and target value (T). The proposed index C̃" pp is applied in an automotive firm to analyze the capability of Volvo Marine motor's piston for diesel engine and the results prove that the fuzzy set theory brings an advantage which is including more information and flexibility to evaluate the process performance.
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.
Los productos manufacturados hoy en día poseen varias características de calidad que son importantes para el cliente y cuando no están correlacionadas, se usan varios índices de capacidad, tales como el Cp, Cpk y Cpm para evaluar la habilidad que tiene el proceso de producir productos de calidad. Para el caso contrario, se encontró en la revisión de literatura, índices de capacidad propuestos para medir la capacidad de un proceso cuando las características de calidad están correlacionadas. La mayoría de las propuestas coinciden en que debe definirse una región de especificación que represente lo que el cliente desea y otra región de variación del proceso que muestre el desempeño que tiene el proceso. Adicionalmente, como resultados de la revisión anterior, se observa que los autores difieren en la forma de definir ambas regiones, oportunidad que se aprovecha para presentar una nueva propuesta, mediante la cual se definen de una manera confiable ambas regiones y que al compararlas se obtienen un par de índices de capacidad multivariados CpM y CpkM, que son extensiones de los índices univariados Cp y Cpk. Por último, se utilizan los datos de un proceso, los cuales se pueden modelar por medio de una distribución normal bivariada para calcular los índices de capacidad propuestos en este trabajo, posteriormente se comparan con el valor de otros índices de capacidad similares, propuestos por otros autores, obteniéndose un desempeño por sobre otros índices de la literatura consultada.
The manufactured products have several quality characteristics that are important for the customer. When the quality characteristics are not correlated, using several indices, such as Cp, Cpk and Cpm to assess the ability of the process to produce quality products. For the opposite case was found in the literature review, proposed capability indices to measure the ability of a process when the quality characteristics are correlated. Most proposals agree that should define a specification region that represents what the customer wants, and another region of process variation that shows performance having the process. Additionally, as a result of the review, it was observed that different authors differ on how to define both regions, an opportunity that is used to present a new proposal, in which these regions are defined in a reliable manner and are compared, resulting in a pair of multivariate capability indices CpM and CpkM, which are extensions of univariate indexes Cp and Cpk. Finally, data from a process, which can be modeled by means of a bivariate normal distribution are used to calculate the capability indices proposed in this paper, then compared with the value of other similar capability indices, proposed by others authors give a performance over other indices of the literature.
Multivariate process capability analysis using the multivariate process capability vector. It allows to analyze a multivariate process with both normally and non-normally distributed and also with dependent and independent quality characteristics.
See: http://cran.r-project.org/package=mpcv
This paper concerns with the suppliers evaluation problem in a supply network. Companies are forced to re-evaluate the way they do business in today's highly competitive and global marketplace. Customers are no longer making buying decisions based solely on product quality and price, but are now also demanding high levels of service and flexibility. Therefore, a comprehensive approach for evaluating the suppliers is required. As service and flexibility are qualitative parameters, the aforementioned approach should be based on quality control disciplines. Here, we use control charts as a control tool for the suppliers' performance and Six Sigma criterions for more precise investigations. A 0/1 integer programming approach is applied to identify the optimal set of suppliers in the supply network. The efficiency of the proposed mechanism is presented using numerical illustrations.
Purpose
– The aim of this paper is to consider the effect of gauge measurement capability on the multivariate process capability index (MCp).
Design/methodology/approach
– With respect to measurement capability, the paper investigates the statistical properties of the estimated MCp and considers the effect of gauge measurement capability on the lower confidence bound, hypothesis testing, critical value and power of testing for MCp at the mentioned state.
Findings
– The results show that gauge measurement capabilities will notably change the results of estimating and testing the process capability index.
Originality/value
– The research would help quality experts to determine whether their processes meet the required capability, and to make more reliable decisions.
Process performance can be analyzed by using process capability indices (PCIs), which are summary statistics to depict the process’ location and dispersion successfully. Many PCIs have been proposed in the literature. Although they are very usable statistics to summarize process’ performance, they can give misleading results and can cause incorrect interpretation if the process parameters have a correlation. In this case, the new PCIs called robust PCIs (RPCIs) should be applied. In this paper RPCIs are obtained for a piston manufacturing company and the fuzzy set theory is incorporated to increase PCIs’ flexibility and sensitivity by defining specification limits and standard deviation as fuzzy numbers. Then fuzzy RPCIs are obtained to express the process performance more realistic for the piston manufacturing stage.
Within an industrial manufacturing environment, Process Capability Indices (PCIs) enable engineers to assess the process performance and ultimately improve the product quality. Despite the fact that most industrial products manufactured today possess multiple quality characteristics, the vast majority of the literature within this area primarily focuses on univariate measures to assess process capability. One particular univariate index, Cpm, is widely used to account for deviations between the location of the process mean and the target value of a process. While some researchers have sought to develop multivariate analogues of Cpm, modeling the loss in quality associated with multiple quality characteristics continues to remain a challenge. This paper proposes a multivariate PCI that more appropriately estimates quality loss, while offering greater flexibility in conforming to various industrial applications, and maintaining a more realistic approach to assessing process capability. Copyright © 2010 John Wiley & Sons, Ltd.
Process performance can be analyzed by using process capability indices (PCIs), which are summary statistics to depict the process location and dispersion successfully. Traditional PCIs are generally used for a process which has a symmetric tolerance when the target value (T) locates on the midpoint of the specification interval (m). When this is not the case (T≠m), there are serious disadvantages in the casual use and interpretation of traditional PCIs. To overcome these problems, PCIs with asymmetric tolerances have been developed and applied successfully. Although PCIs are very usable statistics, they have some limitations which prevent a deep and flexible analysis because of the crisp definitions for specification limits (SLs), mean, and variance. In this paper, the fuzzy set theory is used to add more information and flexibility to PCIs with asymmetric tolerances. For this aim, fuzzy process mean, μ˜ and fuzzy variance, σ˜2, which are obtained by using the fuzzy extension principle, are used together with fuzzy specification limits (SLs) and target value (T) to produce fuzzy PCIs with asymmetric tolerances. The fuzzy formulations of the indices C∼pk″,C∼pm∗,C∼pmk″, which are the most used PCIs with asymmetric tolerances, are developed. Then a real case application from an automotive company is given. The results show that fuzzy estimations of PCIs with asymmetric tolerances include more information and flexibility to evaluate the process performance when it is compared with the crisp case.
Process capability analysis often entails characterizing or assessing processes or products based on more than one engineering specification or quality characteristic. When these variables are related characteristics, the analysis should be based on a multivariate statistical technique. In this expository paper, three recently proposed multivariate methodologies for assessing capability are contrasted and compared. Through the use of several graphical and computational examples, the information summarized by these methodologies is illustrated and their usefulness is discussed.
Process capability indices have been widely used in the manufacturing industry, providing numerical measures on process precision, process accuracy, and process performance. Capability measures for processes with a single characteristic have been investigated extensively. However, capability measures for processes with multiple characteristics are comparatively neglected. In this paper, inspired by the approach and model of process capability index investigated by K. S. Chen et al. (2003) and A. B. Yeh et al. (1998), a note model of multivariate process capability index based on non-conformity is presented. As for this index, the data of each single characteristic don't require satisfying normal distribution, of which its computing is simple, and will not fell too theoretical. At last the application analysis is made.
The aim of this paper is to define two new capability indices BCP and BCPK dedicated to two quality characteristics, assuming a bivariate normal distribution and a rectangular tolerance region. These new capability indices are based on the computation of the theoretical proportion of non-conforming products over convex polygons. This computation is achieved by a new method of integration based on Green’s formula. The efficiency of the proposed capability indices is demonstrated by comparing our approach with others proposed previously, on simulated and real world industrial examples.
We provide a compact survey and brief interpretations and comments on some 170 publications on process capability indices which appeared in widely scattered sources during the years 1992 to 2000. An assessment of the most widely used process capability indices is also presented.
Process capability analysis often entails characterizing or assessing processes or products based on more than one engineering specification or quality characteristic. When these variables are related characteristics, the analysis should be based on a multivariate statistical technique. In this expository paper, three recently proposed multivariate methodologies for assessing capability are contrasted and compared. Through the use of several graphical and computational examples, the information summarized by these methodologies is illustrated and their usefulness is discussed.
We propose a multivariate process capability index (PCI) over a general tolerance zone which includes ellipsoidal and rectangular solid ones as special cases. Our multivariate PCI appears to be a natural generalization of the PCI C p for a univariate process to a multivariate process. Computing aspects of the proposed multivariate PCI are discussed in detail, especially for a bivariate normal process. It is noted that its distributional and inferential aspects are difficult to deal with. Resampling methods and a Monte Carlo procedure are suggested to overcome this difficulty. Some examples with a set of real data are presented to illustrate and examine the proposed multivariate PCI.
SUMMARY Measures of multivariate skewness and kurtosis are developed by extending certain studies on robustness of the t statistic. These measures are shown to possess desirable properties. The asymptotic distributions of the measures for samples
from a multivariate normal population are derived and a test of multivariate normality is proposed. The effect of nonnormality
on the size of the one-sample Hotelling's T2 test is studied empirically with the help of these measures, and it is found that Hotelling's T2 test is more sensitive to the measure of skewness than to the measure of kurtosis.
This paper presents simple criteria which can be used to select which of the three members of the Johnson System of distributions should be used for fitting a set of data. The paper also presents elementary formulas for estimating the parameters for each of the members of the family. Thus, many obstacles to the use of the Johnson System are resolved.
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.
[This abstract is based on the author's abstract.] A method is presented for evaluating the performance of a manufacturing process that produces parts described by two critical characteristics. Process performance is described most often either throug..
This article examines the performance of two families of non-Normal process capability indices, using a comprehensive simulation study with data generated from the Johnson family of distributions. This study shows that none of the three indices can be classified as good estimators when the process fallout or tail probabilities are used as the indicator for measuring product quality. As a result, the two families do not demonstrate a systematic pattern to deal with non-Normal process data. Clearly, the use of the sample range is not sufficiently precise to provide an appropriate denominator for these indices. More sophisticated estimators should be considered to improve the performance of these non-Normal capability indices.
Classical evaluation of process capability indices (PCI) supposes that the process is normally distributed with mean μ and standard deviation σ. If the process is not normal, then this evaluation may yield strange results. In order to evaluate non-normal process capability indices, we propose to use relations between process capability indices and the proportion of nonconforming items and to estimate this proportion using a fitting method based on Burr's distributions.
Many methods exist for measuring the capability of a single characteristic of a product (e.g., C p, Z min, or C pk). These metrics can also help estimate the percentage of parts that have this feature within its required tolerance. This knowledge is extremely useful for shop-floor personnel in determining what machines should run this part and for monitoring process improvements. However, a customer purchases an entire product which usually has many critical characteristics. All these features must be within their respective tolerances in order for the product to function correctly and, thus, satisfy the customer. A need exists to combine the usefulness of the capability indexes for individual characteristics with the goal of meeting a customer's expectation of receiving a quality product. This article presents a composite capability index for assessing the quality performance of an entire product. A product capability study first determines the probability that each characteristic of a product is within tolerance, then computes the combined probability that all are within tolerance. This combined result can be converted into a "Product" C pk index which measures total customer satisfaction, allows different products to be compared, and permits improvements in product capability to be tracked over time. The features may be characterized by either variable or attribute data and there is no limit to the number of characteristics that may be combined. The Product C pk concept can be extended to summarize the ability of an entire machining (or assembly) line to produce products meeting all quality requirements.
Process capability indices were originally defined under the assumption that process data can be adequately described by a normal distribution. Applied to non-normal processes, however, these indices can give misleading results. In this article, we outline the mechanics of using Johnson curves for modeling non-normal processes and for defining generalized capability indices. Compiling results from several sources, the article provides all necessary formulas for choosing a Johnson curve, estimating its parameters, performing probability calculations, plotting its density function, and defining generalized capability indexes. All calculations can easily be performed using statistical or spread-sheet software. Several examples are given to illustrate the calculations and versatility of Johnson curves.
In this paper a new process capability index (C
pq
) has been introduced, which is easy to compute and performs well when compared with its natural competitor (C
pm
).
Quality measures can be used to evaluate a process's performance. Analyzing related quality characteristics such as weight, width and height can be combined using multivariate statistical techniques. Recently, multivariate capability indices have been developed to assess the process capability of a product with multiple quality characteristics. This approach assumes multivariate normal distribution. However, obtaining these distributions can be a complicated task, making it difficult to derive the needed confidence intervals. Therefore, there is a need to develop one robust method to deal with the process performance on non-multivariate normal data. Principal component analysis (PCA) can transform the high-dimensional problems into lower dimensional problems and provide sufficient information. This method is particularly useful in analyzing large sets of correlated data. Also, the application of PCA does not require multivariate normal assumption. In this study, several capability indices are proposed to summarize the process performance using PCA. Also, the corresponding confidence intervals are derived. Real-world case studies will illustrate the value and power of this methodology.
Process capability calculations for non-normal distributions
- J Clements