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Multivariate nonnormal process capability analysis
Abstract
There is a great deal of interest in the manufacturing industry for quantitative measures of process performance with multiple
quality characteristics. Unfortunately, multivariate process capability indices that are currently employed, except for a
handful of cases, depend intrinsically on the underlying data being normally distributed. In this paper, we propose a general
multivariate capability index based on the Mahanalobis distance, which is very easy to use. We also approximate the distribution
of these distances by the Burr XII distribution and then estimate its parameters using a simulated annealing search algorithm.
Finally, we give an example, based on real manufacturing process data, which demonstrates that the proportion of nonconformance
(PNC) using our proposed method is very close to the actual PNC value, which also justifies its adoption in this paper.
... Similar approaches of using distances and dimension reduction have been proposed in [14][15][16]. [14] recommended to combine correlated quality characteristics of any distribution using the generalised distance, GD variable, as follows: ...
... This section describes the development of the MTR and the TD methods based on the review done on past researches and its proposed models. The MTR and the TD methods presented in this study were developed by improving the models as presented in earlier studies [16], [21]. Both models were constructed and evaluated using the MATLAB R2014a software package. ...
... To reduce the dimension of the correlated quality characteristics, [16] proposed the computation of CD variable as in Eq. (6). The technique of dimension reduction has the advantage of reducing the dimensionality of the correlated multivariate data into a single variable however it requires data fitting and the definition of upper specification limit for the new variable. ...
In many cases, the quality of a manufactured product is determined by more than one characteristic and often, these quality characteristics are correlated. A number of methods for dealing with quality evaluation of multivariate processes have been proposed in the literature. However, some of these studies do not consider correlation among quality characteristics. In this paper, two new approaches for estimating the proportion of non-conformance for correlated multivariate quality characteristics with nominal specifications are proposed: (i) the modified tolerance region approach and (ii) the target distance approach. In the first approach, the p number of correlated variables are analysed based on the projected shadow of the p-dimensional hyper ellipsoid so that the ability to visualise the tolerance region and the process region for p > 2 is preserved. In the second approach, the correlated variables are combined and a new variable called the target distance is introduced. The proportion of non-conformance results estimated using both methods were used to compute the multi-variate capability index and the total expected quality cost. This study also suggest modification to the NMCp index as proposed in Pan and Lee (2010) such that the process capability for p > 2 can be measured correctly. The application of both approaches is demonstrated using two examples and it is shown that both methods i.e. the modified tolerance region and the target distance methods are capable of estimating the capability of multivariate processes.
... The capability index pc C [8] is used to summarize the performance for each geometric distance metric. Ahmad et al. [2] presented a capability analysis based on the statistical distance (SD) metric for multivariate normality data. However, the measurement data for a manufacturing product may include multivariate normality or multivariate non-normality, thus conventional-method application is limited. ...
... where GD MRD is the maximum radial distance from the target to the perimeter of the tolerance region, as defined by the Euclidean distance, and ( ) f g is the density function of the three-parameter Burr XII distribution. Using the one-to-one correspondence between the process-yield index and the process yield (Yield= 2 (3 ) 1 yield I ) where is the cumulative standard normal function, we obtain the process yield index, yield I (see Table 1 The SD metric for multivariate normality data proposed by Ahmad et al. [2] is the Mahalanobis distance, the distance of individual quality characteristics from their respective targets, scaled by their variance-covariance matrix. The SD metric is defined by ...
... where x is a vector ( 1 v ) of observations. 2 SD is equal to the maximum value of the squared weighted distance between the point X and its target T, scaled by its variance-covariance [6]. To justify the underlying distribution of this SD data, the normal distribution is selected to describe the SD data. ...
To evaluate the process yield for a manufactured product with multiple characteristics is an important task in the manufacturing industry. We present a general procedure based on the statistical distance metric and the geometric distance metric to evaluate the process yield for a manufactured product with multivariate normal/non-normal data. We select the normal distribution and the three-parameter Burr XII distribution to fit the statistical distance data and the geometric distance data, respectively. The process yield indices provide an exact measure of the overall process yield. This study uses three real examples to demonstrate the performance of the proposed approach. The results show that our procedure is an effective approach of evaluating the process yield for a manufactured product with multiple characteristics.
... However, such plug-in estimators are subject to sampling fluctuation and hence can not be considered as the substitute of the original PCIs unless their distributional and inferential properties are studied extensively. These properties of the PCIs in equation (1) have been studied extensively in literature (refer Kotz and Johnson [32], Pearn et al. [46] and the references there in). ...
... Finally, the capability assessments for multivariate processes with non-normal process distributions have been studied by Abbasi and Niaki [2], Ahmad et al. [1], Polansky [53] and so on. ...
In the context of statistical quality control, process capability index (PCI) is one of the widely accepted approaches for assessing the performance of a process with respect to the pre-assigned specification limits. The quality characteristic under consideration can have differnt types of specification limits like bilateral, unilateral, circular and so on. Use of single PCI for all the situations could be misleading. Hence appropriate PCIs need to be chosen based on the characteristics of the specification limits. Similar situations may arise for multivariate characteristics as well. In the present chapter, we have discussed about some of the PCIs for different specification limits including some PCIs for multivariate characteristics. A few numerical examples are given to suppliment our theoretical discussion.
... In handling the cases with multiple characteristics, multivariate methods for assessing process capability are proposed. These relevant multivariate capability indices can be found in [2]- [9]. Unfortunately, only multivariate capability indices under a mutually independent normal data can be used to obtain the process yield. ...
... Then, we propose the overall process yield index, TC pu;PC or TC pl;PC , for the multivariate normal data which is given by (9) where C pu;PC i or C pl;PC i represents the single measure of process yield index for the ith PC based on Kane's formula [11]. By analogy to (3), the overall process yield P PC can be obtained as ...
This paper proposes a procedure for estimating the process yield of multiple characteristics with one-sided specifications in a manufacturing process. The proposed process yield indices can be applied for a multivariate normality data or a multivariate non-normality data. These indices provide an exact measure of the overall process yield. Also, the approximate lower confidence bound for the true process yield is presented. Three examples are used to demonstrate the performance of the proposed approach. The results show that our procedure is an effective approach.
... They use the root transformation method to normalize nonnormal data and then use Monte Carlo simulation to estimate the PNC. Ahmad et al. [33] investigate multivariate non-normal process capability analysis based on the PNC. They reduce the dimension of the multivariate QCs using the covariance distance (CD). ...
Process capability indices (PCIs) have always been used to improve the quality of products and services. Traditional PCIs are based on the assumption that the data obtained from the quality characteristic (QC) under consideration are normally distributed. However, most data on manufacturing processes violate this assumption. Furthermore, the products and services of the manufacturing industry usually have more than one QC; these QCs are functionally correlated and, thus, should be evaluated together to evaluate the overall quality of a product. This study investigates and extends the existing multivariate non-normal PCIs. First, a multivariate non-normal PCI model from the literature is modeled and validated. An algorithm to generate non-normal multivariate data with the desired correlations is also modeled. Then, this model is extended using two different approaches that depend on the well-known Box–Cox and Johnson transformations. The skewness reduction is further improved by applying heuristics algorithms. These two approaches outperform the investigated model from the literature because they can provide more precise results regardless of the skewness type. The comparison is made based on the generated data and a case study from the literature.
... Com o uso de uma abordagem unificada para estudar o tamanho de percurso FMCI, propriedades para vários gráficos, o método pode ser aplicado para apresentar processos instáveis. Multivariate no normal process capability analysis ( AHMAD et al., 2009) Sistema de Medição de desempenho O trabalho utilizou a abordagem CD, dada com base no conceito de reduzir as dimensões multivariadas, transformando dados em variáveis correlacionadas por meio uma função de métrica. Esta abordagem produz uma melhoria significativa a mais de um método existente; usando o conjunto de dados escolhido, recomenda-se o método proposto para ser aplicado para estudos e comparações. ...
O estudo tem como objetivo identificar o que a literatura científica internacional aborda sobre o tema Avaliação de Desempenho na Gestão do Controle Estatístico de Processos (CEP), possibilitando, assim, a identificação de oportunidades de aperfeiçoamento. A pesquisa exploratória utilizou como instrumento de intervenção o ProKnow-C para a seleção do portfólio bibliográfico-PB e a análise das características deste fragmento da literatura. Os autores de destaque identificados foram Wen Lea Pearn, Shu-Ming Chung, com 5, 4, e 3 publicações respectivamente no PB. Foi possível também identificar os periódicos Quality and Reliability Engineering International e o Expert Systems With Applications como os dois com maior número de publicações. Já em relação às palavras-chaves presentes nos artigos do portfólio, as que se destacaram foram Process Capability Indices, Control Carts e Process Capability index. Os periódicos que apresentaram maior fator de impacto foram: European Jounal of Operational Research e Expert Systems with Applications. Constatou-se nos trabalhos a utilização de Sistemas de Mensuração de Desempenho nas atividades do controle estatístico de processos. Os resultados indicam ainda a utilização de indicadores, oriundos de modelos realistas de Avaliação de Desempenho, centrados na qualidade estatística sem ter em conta as necessidades, os valores e as preferências dos gestores dos contextos avaliados.
... Devido as especificações, sabe-se que as saídas dos processos, apresentam mais de uma característica de qualidade importante e que estas encontram-se correlacionadas. Portanto o meio industrial necessita de técnicas multivariadas, que abordem todas as especificações relevantes para serem analisadas, através dos índices de capabilidade de processos multivariados (AIAG, 2005;WANG, 2006;AHMAD et al., 2009). ...
... The vast majority of quality characteristics employed in industry are multivariate variables and they have been several multivariate PCIs introduced in the research literature. Except for the Geometric Distance and Covariance Distance PCIs introduced in Wang and Hubele (1999), Wang (2010) and Ahmad et al. (2009), respectively, most of these multivariate PCIs are designed in a similar way to the univariate PCIs by considering ratio of the volume of the tolerance region and that of the process region. They also assume that the variables are distributed under a multivariate normal distributions (cf. ...
Several measures of process yield, defined on univariate and multivariate normal process characteristics, have been introduced and studied by several authors. These measures supplement several well-known Process Capacity Indices (PCI) used widely in assessing the quality of products before being released into the marketplace. In this paper, we generalise these yield indices to the location-scale family of distributions which includes the normal distribution as one of its member. One of the key contributions of this paper is to demonstrate that under appropriate conditions, these indices converge in distribution to a normal distribution. Several numerical examples will be used to illustrate our procedures and show how they can be applied to perform statistical inferences on process capability.
... Some other industrial applications with individual observations can also be referenced in [9][10][11]. As a matter of fact, nonnormal processes commonly exist in industries [12][13][14][15][16][17][18][19][20][21][22]. Among those nonnormal applications, the gamma process is one of the most important cases in industry [12][13][14]16,[18][19][20]. ...
Statistical process control (SPC) charts are commonly used for detecting process disturbances. However, they do not provide enough information to identify the root causes of an out-of-control process. This difficulty can be overcome if we are able to promptly estimate the change point of a process, due to the fact that the change point usually reveals the most accurate information about root causes. As a consequence, this estimation becomes a very important research issue in SPC applications. Although recent studies have shown that the maximum likelihood estimation (MLE) estimator could be an effective estimate of the change point for a normal process, very little is known about the feasibility of using an MLE estimator for a gamma process with individual observations. In this study, our goal is to propose a fruitful approach to solving this problem. This study proposes the combination of MLE and the exponentially weighted moving average (EWMA) control charts to estimate the change point of a gamma process. We investigate various SPC modes and gamma process designs in this study, and the results show that an effective change point estimator could be achieved.
The evaluation of manufacturing processes aims to ensure that the processes meet the desired requirements. Therefore, process capability indexes are used to measure the capability of a process to meet customer requirements and/or engineering specifications. However, most of the manufacturing products have more than one quality characteristic (QC), in which case, the multivariate QCs should be evaluated together using a single capability index. The research in this article proposes a methodology for estimating the multivariate process capability index (PCI). First, the dimensions of the multivariate QCs are reduced into a new single variable using the proportion of the process specification region, by comparing each variable datapoint to its specification limits. Moreover, nonnormal data are transformed to normality using a root transformation algorithm. Then, a large data sample is generated using the parameters of the new variable. The generated data are compared to the specification limits to estimate the percent of nonconforming (PNC). Finally, the capability index of a given process datapoints is estimated using the PNC. Accordingly, managerial insights for the implementation of the proposed methodology in real industry are presented. The methodology was assessed by well-known multivariate samples from four different distributions, in which an algorithm was developed for generating these samples with their given correlations. The results show the effectiveness of the proposed methodology for estimating multivariate PCIs. Also, the results from this research outperform the previous published results in most cases.
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.
The main objective is to develop and test a novel SPC method to ensure the stability of final customer characteristics by controlling the upstream characteristics of manufacturing processes, considering their importance/contribution. The originality of the proposed method lies in the use of predictive models (multivariate regression), whose coefficients are used to weigh the contribution of each upstream characteristic to predict the final characteristics. In the context of continuous improvement environments, the application of different SPC approaches to manufacturing processes were compared. The results showed that the multivariate SPC method based on partial least squares regression was superior to traditional univariate and multivariate SPC methods in terms of the predictive precision to detect downstream faults in customer characteristics. However, the use of multiple linear regression may also be an option, since the identification of what upstream characteristic is causing the out-of-control signal is simpler than that of partial least squares regression, and predictive precision in the case of each of the two methods is comparable in practical terms.
There is a great challenge in carrying out multivariate process capability analysis and fault diagnostics on a high dimensional non-normal process, with multiple correlated quality characteristics, in a timely manner. This paper proposes a hybrid capable of performing process capability analysis and fault diagnostics on multivariate non-normal processes. The proposed hybrid first utilizes the Geometric Distance (GD) approach, to reduce dimensionality of the correlated data into fewer number of independent GD variables which can be assessed using univariate process capability indices (PCIs). This is followed by fitting Burr XII distribution to independent GD variables. The independent fitted distributions are used to estimate both yield and multivariate process capability in a time efficient way. Finally, machine learning approach, is deployed to carry out the task of fault diagnostic by identifying and ranking the correlated quality characteristics responsible for the poor performance of individual GD variables. The efficacy of the proposed hybrid is assessed through a real manufacturing example and four simulated scenarios. The results show that the proposed hybrid is robust in estimating both yield and multivariate process capability carrying out fault diagnostics beyond GD variables, and identifying the original characteristic responsible for poor performance.
Multivariate process capability analysis is needed to study the variation in manufacturing process relative to the designed tolerances when multiple quality characteristics are monitored simultaneously in practice. Due to the complicated joint probability density function between multiple quality characteristics, the dimensions reduction is used to solve the problems, such as strong correlation. In this paper principal component analysis method is used to reduce the dimensions of multiple quality characteristics at first. Based on the specification region, the specification center vector and target vector of principal components (PCs), three kinds of nonconforming proportion of multivariate process are proposed and defined, which are latent non-conforming, performance non-conforming and Taguchi non-conforming, respectively. Compared with the allowed process non-conforming by customers, these non-conforming calculations can be used to be the guideline on how to improve the manufacturing process for the engineers and operators. At last, a case study of a spindle motor process is presented.
This paper compares our proposed method of using the modified tolerance region (MTR) to the generalised distance (GD) approach for estimating the probability of non-conforming product. For correlated quality characteristics, the tolerance region needs to be redefined so that the model is more stringent in identifying potentially poor product quality, as perceived by customers. The paper shows that the MTR approach is more effective in estimating the proportion of non-conformance (PNC) for correlated quality characteristics as compared to the GD approach. We also compare the multivariate capability index computed using the PNC values obtained from both the GD and MTR methods. The estimation of PNC assists not only in the evaluation of process performance and capability but also in the quantification of cost of poor quality.
Purpose
– In the past few years, several capability indices have been developed for evaluating the performance of multivariate manufacturing processes under the normality assumption. However, this assumption may not be true in most practical situations. Thus, the purpose of this paper is to develop new capability indices for evaluating the performance of multivariate processes subject to non-normal distributions.
Design/methodology/approach
– In this paper, the authors propose three non-normal multivariate process capability indices (MPCIs) RNMCp, RNMCpm and RNMCpu by relieving the normality assumption. Using the two normal MPCIs proposed by Pan and Lee, a weighted standard deviation method (WSD) is used to modify the NMCp and NMCpm indices for the-nominal-the-best case. Then the WSD method is applied to modify the multivariate ND index established by Niverthi and Dey for the-smaller-the-better case.
Findings
– A simulation study compares the performance of the various multivariate indices. Simulation results show that the actual non-conforming rates can be correctly reflected by the proposed capability indices. The numerical example further demonstrates that the actual quality performance of a non-normal multivariate process can properly reflected by the proposed capability indices.
Practical implications
– Process capability index is an important SPC tool for measuring the process performance. If the non-normal process data are mistreated as a normal one, it will result in an improper decision and thereby lead to an unnecessary quality loss. The new indices can provide practicing managers and engineers with a better decision-making tool for correctly measuring the performance for any multivariate process or environmental system.
Originality/value
– Once the existing multivariate quality/environmental problems and their Key Performance Indicators are identified, one may apply the new capability indices to evaluate the performance of various multivariate processes subject to non-normal distributions.
Capability indices in both univariate and multivariate processes are extensively employed in quality control to assess the quality status of production batches before their release for operational use. It is traditionally a measure of the ratio of the allowable process spread and the actual spread. In this paper, we will adopt a bootstrap and sequential sampling procedures to determine the optimal sample size for estimating a multivariate capability index introduced by Pearns et. al. [12]. Bootstrap techniques have the distinct advantage of placing very minimum requirement on the distributions of the underlying quality characteristics, thereby rendering them more relevant under a wide variety of situations. Finally, we provide several numerical examples where the sequential sampling procedures are evaluated and compared.
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 performance indices generally rely on the assumption that the process follow normal distribution but in practice its non-normal with correlated characteristics patterns. This paper proposes two metaheuristic-based approaches to fit Burr distribution to such data; a single candidate model-based approach using a Simulated Annealing (SA) technique and a population based approach using a constraint-based Evolutionary Algorithm (EA). The fitted Burr distribution is then used to estimate the proportion of Non-conforming (PNC) which is then used to assess the efficacy of the proposed methods. The metaheuristic approaches are used to fit an appropriate Burr distribution to individual Geometric distance variables. Empirical performances of the proposed methods have been evaluated on real industrial data set using PNC criterion. Experimental results demonstrate that the new approach perform well than the existing.
Process capability indices can be considered as the effective and excellent means of measuring product quality and process performance. Now-a-day's it is impossible to insure the product quality without process capability analysis. This study measures the potentialities of a particular soap production process from a renowned soap manufacturer company namely Kohinoor Chemical Company Bangladesh Ltd (KCCBL). The sole concentrations were to measure the process indices namely process potential index, process performance index, process centering index etc. This paper work also shows the relationship between soap process parameters which help the manufacturers to find out the dependent process parameter when the impendent process parameters are known. The necessary preventions and recommendations for decreasing defects are shown in this study. This study considers thirty days for collecting data as its sample size.
The existing research of process capability indices of multiple quality characteristics mainly focuses on nonconforming of process output, the concept development of univariate process capability indices, quality loss function and various comprehensive evaluation methods. The multivariate complexity increases the computation difficulty of multivariate process capability indices(MPCI), which makes them hard to be used in practice. In this paper, a new PCA-based MPCI approach is proposed to assess the production capability of the processes that involve multiple product quality characteristics. This approach first transforms the original quality variables into standardized normal variables. MPCI measures are then provided based on the Taam index. Moreover, the statistical properties of these MPCIs, such as confidence intervals and lower confidence bound, are given to let the practitioners understand the capability indices as random variables instead of deterministic variables. A real manufacturing data set and a synthetic data set are used to demonstrate the effectiveness of the proposed method. An implementation procedure is also provided for quality engineers to apply our MPCI approach in their manufacturing processes. The case studies demonstrate the effectiveness and feasibility of this new kind of MPCI, which is easier to be used in production practice. The proposed research provides a novel approach of MPCI calculation.
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.
The popularity of the ear biometrics is due to its unique distinct structure and high user convenience. However, the presence of hair and other skin attributes makes the automatic detection of ear contour a real challenge for online applications. This paper presents an online biometric authentication using ear contours acquired from a robust peg free acquisition set up. The Gaussian classifiers are used to first segment the skin and non-skin areas in the ear images. Laplacian of Gaussian is then used to compute edges of the skin areas, which helped to get ear-ROI images. A localized region based active contour is finally located in the ear-ROI images. The ear-contours are then employed for the authentication using log Gabor and SIFT features. The experimental results carried out on 700 ear images confirm the utility of the proposed approach.
It is crucial than ever to measure manufacturing losses due to non-compliance of customer specifications. To assess these losses, industry is widely using proportion of non conformance PNC for performance evaluation of their manufacturing processes. Various methods have been proposed to estimate PNC for univariate quality characteristics, however estimating an accurate PNC for non-normal multivariate correlated quality characteristics is still a challenge for researchers. In this paper we review fitting Burr XII distribution to continuous positively skewed multivariate data using different search algorithm techniques. The proportion of nonconformance PNC for process measurements is then obtained by using only Burr XII distribution, rather than through the traditional practice of fitting different distributions to real data. We also employ artificial neural network based on Burr XII distribution to estimate PNC. The results based on the proposed methods are then compared with the exact proportion of nonconformance using real data from a manufacturing process.. Using the PNC criterion, the results show that the estimated PNC values obtained based on all three methods, simulated annealing, hybrid and artificial neural network are reasonably close to the actual PNC value. However, the estimated PNC based on the simulated annealing method is the closest to the actual PNC value.
This paper contains a bibliography of approximately 530 journal papers and books on process capability indices for the period 2000–2009. The related literature is classified into four major categories, namely, books, review/overview papers, theory- and method-related papers, and special applications. Theory- and method-related papers are further classified into univariate and multivariate cases, and special applications include acceptance sampling plans, supplier selection, and tolerance design and other optimizations. Copyright © 2010 John Wiley & Sons, Ltd.
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.
The concept of generalized order statistics was introduced by Kamps (1995) to unify several concepts that have been used in statistics such as order statistics, record values, and sequential order statistics. Estimation of the parameters of the Burr type XII distribution are obtained based on generalized order statistics. The maximum likelihood and Bayes methods of estimation are used for this purposes. The Bayes estimates are derived by using the approximation form of Lindley (1980). Estimation based on upper records from the Burr model is obtained and compared by using Monte Carlo simulation study. Our results are specialized to the results of AL-Hussaini and Jaheen (1992) which are based on ordinary order statistics.
There is a deep and useful connection between statistical mechanics (the behavior of systems with many degrees of freedom
in thermal equilibrium at a finite temperature) and multivariate or combinatorial optimization (finding the minimum of a given
function depending on many parameters). A detailed analogy with annealing in solids provides a framework for optimization
of the properties of very large and complex systems. This connection to statistical mechanics exposes new information and
provides an unfamiliar perspective on traditional optimization problems and methods.
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.
Pattern recognition or discrimination is about guessing or predicting the unknown nature of an observation, a discrete quantity such as black or white, one or zero, sick or healthy, real or fake. An observation is a collection of numerical measurements such as an image (which is a sequence of bits, one per pixel), a vector of weather data, an electrocardiogram, or a signature on a check suitably digitized. More formally, an observation is a d-dimensional vector x. The unknown nature of the observation is called a class. It is denoted by y and takes values in a finite set {1, 2,..., M}. In pattern recognition, one creates a function g(x): ℛ d → {1,..., M} which represents one’s guess of y given x. The mapping g is called a classifier. Our classifier errs on x if g(x) ≠ y.
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.
A new process capability index (Pel) is proposed. which takes into account possible asymmetry in the distribution of the measured process characteristic (X). The distribution of a natural estimator of this index is investigated.
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.
This paper describes mathematical and computational methodology for estimating the parameters of the Burr Type XII distribution by the method of maximum likelihood. Expressions for the asymptotic variances and covariances of the parameter estimates are given, and the modality of the log‐likelihood and conditional log‐likelihood functions is analyzed. As a result of this analysis for various a priori known and unknown parameter combinations, conditions are given which guarantee that the parameter estimates obtained will, indeed, be maximum likelihood estimates. An efficient numerical method for maximizing the conditional log‐likelihood function is described, and mathematical expressions are given for the various numerical approximations needed to evaluate the expressions given for the asymptotic variances and covariances of the parameter estimates. The methodology discussed is applied in a numerical example to life test data arising in a clinical setting.
Important features of multivariate process capability indices are comparability, interpretability and ease of implementation. When poor process capability is indicated by an index, the user should determine why the process is incapable (e.g. excessive variability or off-target process mean). One of the most used multivariate process capability indices is MCpm because it provides assessments of process precision and accuracy. In this work, we study and discuss a peculiarity of MCpm: processes that are equivalent in terms of precision, accuracy and MCpm index, after the occurrence of the same increase in the process variability, can have different values of the index. Because MCpm is often used for comparing processes, this behaviour may cause comparability difficulties. Therefore, we suggest how to take into account this specific behaviour for avoiding erroneous conclusions. Copyright © 2013 John Wiley & Sons, Ltd.
A set of families of distributions which might be useful for fitting data was described by Burr (1942). Special attention was focused on the family, Type XII, with generic distribution function $1 - (1 + x^c)^{-k} (x > 0)$ which yields a wide range of values of skewness, √ β1, and kurtosis, β2. The area in the (√β1, β2) plane corresponding to the Type XII distributions is derived and presented in two figures.
This paper considers the problem of minimizing a function F(x 1 , …, x n ) over a closed, bounded region S in n-dimensional space under the assumption that there exists a unique minimizing point (z 1 , …, z n )ϵS. In a previous paper I represented the coordinates of the minimizing point as the limit of a ratio of integrals. The same type of ratio appears, in a different context, in statistical mechanics where a Monte Carlo method has been developed, by Metropolis et al., for its numerical evaluation. The purpose of this paper is to point out the connection of Metropolis's method with the above type of minimisation problem. The idea of the method is to associate with the minimization problem a Markov chain whose sample averages converge with probability one to (approximately) the minimizing point (z 1 , …, z n ). The Markov chain should be easily realizable on a computer. An estimate of the error from sampling over a finite time period is given.
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.
Many widely-used statistical packages and recent authors addressing process capability apply Cpm in cases where tolerances are asymmetric. This can misrepresent process capability, andpoints to the need to apply appropriate capability measures in the asymmetric case. We compare andcontrast six relatively unfamiliar capability indices appropriate for asymmetric tolerances to givethem greater visibility and to help potential users choose among them. For some of these indicesmethods of estimation from process data have not been published; work on this is deferred to subsequent papers
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.
An impetus for the new revolution in quality technology has been Professor Genichi Taguchi's approach to quality engineering, best exemplified by his call for off-line quality control. However, much of the literature on this topic appears to be fragmented between engineering, statistics and quality control journals, each emphasizing a point of view that is pertinent to its readership. A consequence of the above is that there has been some difficulty in developing an appreciation for the totality of die approach, its key ingredients, and the several excellent contributions of many others in this important subject. In this paper, we attempt to help alleviate this difficulty by pointing out that an encompassing perspective on Taguchi's philosophy can be provided by statistical decision analysis. The subject deals with decision making in the face of partial or no information, and prescribes that an optimum decision is one that maximizes expected utility. The role of experimental design is to obtain partial information about the unknown quantities in an efficient manner. When viewed as such, much of what Taguchi prescribes, including his proposals for tolerance design, gets streamlined and integrated as a comprehensive package.
This paper is concerned with a general system of distributions , and the reciprocal transformation . Parameters c,k determine μ,σ and the standardized moments α3,α4. The purpose of this paper is to provide c,k values which cover a wide grid of α3,α4. values, for each c,k matching an α3,α4. point, μ and σ are also provided, The grid of α3,α4 values is sufficiently detailed that interpolation in moat cases appears unnecessary.
A majority of manufactured products have many quality characteristics that are important to the customer. To assess or evaluate the capability of a manufactured product with multiple characteristics, the quality characteristics need to be determined in advance and some characteristics can be correlated with each other. Also, the specification limits of the quality characteristics can be one-sided or two-sided. In order to deal with such multivariate data, there is a need to develop a new approach. In this study, the geometric distance variable is used to combine the correlated or uncorrelated quality characteristics. Then, the Luceño capability index is used to summarize the performance for each geometric distance variable. Finally, a composite capability index, MCpc, composed of several univariate capability indices, is proposed to analyze the capability of a manufactured product with multiple characteristics. In addition, the probability of the product non-conforming is also proposed. The application of the proposed methodology to a real-life case study is presented. Copyright © 2005 John Wiley & Sons, Ltd.
This paper proposes a novel modification of Clements’s method using the Burr XII distribution to improve the accuracy of estimates
of indices associated with one-sided specification limits for non-normal process data. This work proposes a novel Burr-based
method, and compares it with Clements’s method by simulation. Finally, an example application to semiconductor manufacturing
is presented.
In this paper, we propose a flow path to evaluate the process capability of an entire product composed of multiple process
characteristics. There are six steps in the flow path. Whether process data comply with a normal distribution or a non-normal
distribution, the flow path can be applied. Based on Cpu, Cpl, and Cpn, the research aims to develop a multi-process capability analysis chart (MPCAC) model to evaluate process capability in a
normal distribution. Similarly, the research aims to define non-normal multi-process capability analysis chart (NMPCAC) to
evaluate process capability in a non-normal distribution based on Npu, Npl, and Npn.
Weibull distribution plays an important role in failure distribution modeling in reliability studies. It is a hard work to estimate the parameters of Weibull distribution. This distribution has three parameters, but for simplicity, a parameter is omitted and as a result, the estimation of the others will be easily done. When the three-parameter distribution is of interest, the estimation procedure will be quite boring. Maximum likelihood estimation is a good method, which is usually used to elaborate on the parameter estimation. The likelihood function formed for the parameter estimation of a three-parameter Weibull distribution is very hard to maximize. Many researchers have studied this maximization problem. In this paper, we have briefly discussed this problem and proposed a new approach based on the simulated algorithm to solve that.
A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion.
A general method, suitable for fast computing machines, for investigating
such properties as equations of state for substances consisting of
interacting individual molecules is described. The method consists
of a modified Monte Carlo integration over configuration space. Results
for the two-dimensional rigid-sphere system have been obtained on
the Los Alamos MANIAC and are presented here. These results are compared
to the free volume equation of state and to a four-term virial coefficient
expansion. The Journal of Chemical Physics is copyrighted by The
American Institute of Physics.
Thesis (PH. D.) - University of Michigan, 1941. Reprinted from the Annals of mathematical statistics, vol. XIII, number 2, June, 1942. Bibliography: p. 232.
Summary This paper develops mathematical and computational methodology for fitting, by the method of maximum likelihood (ML), the
Burr Type XII distribution to multiply (or progressively) censored life test data. Mathematical expressions are given for
approximating the asymptotic variances and covariances of the ML estimates (MLEs) of the parameters of the Burr Type XII distribution.
A rigorous mathematical analysis is undertaken to investigate the existence and uniqueness of the MLEs for arbitrary sample
data. The methodology of this paper is applied to progressively censored sample data arising in a life test experiment.
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.
Cumulative frequency distribution
- Burr
A multivariate process capability index In: Trans American Society for Quality Control quality congress
- Wierda
A bivariate ca-pability vector Sta-tistics and design in process control: keeping pace with automated manufacturing
- N Hubele
- H Shahriari
- Cheng
Hubele N, Shahriari H, Cheng C (1991) A bivariate ca-pability vector. In: Keats JB, Montgomery DC (eds) Sta-tistics and design in process control: keeping pace with automated manufacturing. Marcel Dekker, New York, pp 299–310