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Performance evaluation of a bivariate normal process
Abstract
[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 section demonstrates the application of both the MTR and TD methods using data acquired from the literature. Subsection 4.1 presents the P NC results for bivariate and trivariate cases using the data available in [22][23], respectively. Using the estimated P NC values, we calculated the capability indices MC pk in Eq. (2) and these indices were compared to the values obtained using NMC p and NMC pm , respectively. ...
... In this study, the calculations and analysis for both the MTR and TD approaches is achieved using the combination of the statistical package Minitab 16 and the computing software MATLAB R2014a. [22] presented an example of a bivariate manufacturing process where the quality of bobbin is characterised by the its height (BH) and weight (BW). A sample of size is provided and the data is assumed to follow a bivariate normal distribution. ...
... The specification limits of the process are given as ,( ) -. The non-conformance percentage and capability index as calculated in Pal (1999) is 0.00106 and 0.976, respectively. ...
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.
... Taam et al. [4] and later Pal [5] proposed the index C pb = S R /A P , where S R represents the surface defined by the rectangle of the upper and lower tolerances and A P is the process surface including 99.73% of the process data. Wang and Chen [6] proposed multivariate equivalents for C p , C pk , C pm , and C pmk based on principal component analysis decomposition. ...
Process capability is determined by comparing the actual performance of a process with required specifications. Several indices have been proposed to report the capability of a process in the univariate case. When performance is tracked in several dimensions, an extension of these indices is required. We present several multivariate capability indices, and extend the idea of multivariate tolerance regions for assessing the capability of a process.
Keywords:
process capability;
specification limits;
multivariate statistical process control;
Mahalanobis T2 charts;
multivariate tolerance regions
... Taam et al. [4] and later Pal [5] proposed the index C pb = S R /A P , where S R represents the surface defined by the rectangle of the upper and lower tolerances and A P is the process surface including 99.73% of the process data. Wang and Chen [6] proposed multivariate equivalents for C p , C pk , C pm , and C pmk based on principal component analysis decomposition. ...
Process capability is determined by comparing the actual performance of a process with required specifications. Several indices have been proposed to report the capability of a process in the univariate case. When performance is tracked in several dimensions, an extension of these indices is required. We present several multivariate capability indices, and extend the idea of multivariate tolerance regions for assessing the capability of a process.
... In this way, Taam, Subbaiah and Liddy [13] generated the first multivariate capability index for the bivariate case. Pal [14] proposed the index. Bothe [15] proposed a method in order to compute the multivariate C pk index. ...
In many industrial instances product quality depends on a multitude of dependent characteristics and as a
consequence, attention on capability indices shifts from univariate domain to multivariate domain. In this
research fuzzy inference system is used to determine the process capability index. Fuzzy sets can represent
imprecise quantities as well as linguistic terms. Fuzzy inference system (FIS) is a method, based on the
fuzzy theory, which maps the input values to the output values. The mapping mechanism is based on some
set of rules, a list of if-then statements. In this research Mamdani fuzzy inference system is used to derive
the overall output process capability when subjected to six crisp input and one output. This paper deals with
a novel approach to evaluating process capability based on readily available information using fuzzy
inference system.
... In this way, Taam, Subbaiah and Liddy [13] generated the first multivariate capability index for the bivariate case. Pal [14] proposed the index. Bothe [15] proposed a method in order to compute the multivariate C pk index. ...
In many industrial instances product quality depends on a multitude of dependent characteristics and as a
consequence, attention on capability indices shifts from univariate domain to multivariate domain. In this
research fuzzy inference system is used to determine the process capability index. Fuzzy sets can represent
imprecise quantities as well as linguistic terms. Fuzzy inference system (FIS) is a method, based on the
fuzzy theory, which maps the input values to the output values. The mapping mechanism is based on some
set of rules, a list of if-then statements. In this research Mamdani fuzzy inference system is used to derive
the overall output process capability when subjected to six crisp input and one output. This paper deals with
a novel approach to evaluating process capability based on readily available information using fuzzy
inference system
... 1 the indices which measure the ratio of a tolerance region to a process region, such as the research proposed by Wang et al. (2000), Shahriari et al. (1995), and Taam et al. (1993) 2 the indices based on proportion of conforming items, such as the research proposed by Polansky (2001), Pal (1999), and Chen (1994) 3 the indices based on principal components analysis (PCA), for instance the research proposed by Wang and Chen (1999) 4 the indices based on the extension of univariate PCI such as those introduced by Holmes and Mergen (1999) and Chen et al. (2003). ...
While most of the methods developed for computing process capability indices (PCI) concentrate on cases with normally or continuous non-normally distributed quality characteristics, computing this measure for processes with mixed distributed data has not been investigated so far. In this paper, a new method is proposed for computing (PCI) for mixed binary-normal quality characteristics. In the proposed method, first a mixed binary-normal distribution is considered to be fitted on the available data. Having estimated the unknown parameters of the fitted distribution using maximum likelihood estimation and genetic algorithm, the proportion of the conforming items of thecorresponding distribution is estimated by Monte Carlo simulation runs. Finally, the PCI is computed based on the relationship of PCI and proportion of conforming items. The performance of the proposed method is evaluated using simulation studies as well as a case study in a plastic injection moulding process.
... Chen 12 proposed a multivariate PCI using the concept of a tolerance zone, which allows flexible specifications and no assumptions on the process distribution. Pal 13 proposed a bivariate PCI as the ratio of the area of the specification rectangle and the 99.73% area of the process region, similar to the index proposed by Taam et al. 10 Bothe 14 proposed a multivariate C pk index defined as Z P /3, where P is the conforming proportion and Z P is the Pth quantile of the standard normal distribution. Wang et al. 15 compared three multiple PCIs proposed by Taam et al., 10 Chen, 12 and Shahriari et al., 9 respectively, via graphical and computational examples. ...
Process capability indices (PCIs) have been widely used in industries for assessing the capability of manufacturing processes. Castagliola and Castellanos (Quality Technology and Quantitative Management 2005, 2(2):201–220), viewing that there were no clear links between the definition of the existing multivariate PCIs and theoretical proportion of nonconforming product items, defined a bivariate Cpk and Cp (denoted by BCpk and BCp, respectively) based on the proportions of nonconforming product items over four convex polygons for bivariate normal processes with a rectangular specification region. In this paper, we extend their definitions to MCpk and MCp for multivariate normal processes with flexible specification regions. To link the index to the yield, we establish a ‘reachable’ lower bound for the process yield as a function of MCpk. An algorithm suitable for such processes is developed to compute the natural estimate of MCpk from process data. Furthermore, we construct via the bootstrap approach the lower confidence bound of MCpk, a measure often used by producers for quality assurance to consumers. As for BCp, we first modify the original definition with a simple preprocessing step to make BCp scale-invariant. A very efficient algorithm is developed for computing a natural estimator of BCp. This new approach of BCp can be easily extended to MCp for multivariate processes. For BCp, we further derive an approximate normal distribution for , which enables us to construct procedures for making statistical inferences about process capability based on data, including the hypothesis testing, confidence interval, and lower confidence bound. Finally, the proposed procedures are demonstrated with three real data sets. Copyright © 2012 John Wiley & Sons, Ltd.
... Pal [20] proposed the index C pb D S R =A P , where S R represents the surface defined by the rectangle of the upper and lower tolerances and A P is the process surface including 99.73% of the process data. Wang and Chen [21] proposed multivariate equivalents for C p , C pk , C pl , and C pu that are based on a principal component analysis decomposition. ...
The semiconductor industry ranges from the design and production of semiconductors on silicon wafers to automatic placement robots that insert semiconductor devices on hybrid microcircuits.Wafers consist of electronic circuits or chips that are characterized by electrical and mechanical characteristics. Process modeling and simulations provide predictions of geometries and material properties of semiconductor devices and wafer structures and help design and improve manufacturing processes such as photolithography, etching, deposition, and ion implantation. In this paper, we focus on three application areas of industrial statistics to the semiconductor industry. These are: (1) process capability indices, (2) process monitoring, and (3) multivariate statistical process control. We refer to two case studies that set a context and provide examples to the presented techniques. Copyright © 2012 John Wiley & Sons, Ltd.
... See, for instance, the indices introduced by Wang et al. [14], Shahriari et al. [15] and Taam et al. [16]. The second category computes the indices based on the probability of nonconforming items such as studies by Pal [17], Chen [18] and Polansky [19] . In the third category using principle component analysis both normal and non-normal cases have been studied for multivariate process capability indices; e.g., see Wang and Chen [20]. ...
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.
... Philippe Castagliola and Jose-Victor Castellanos 5 defined two new capability indices BC p and BC pk dedicated to two quality characteristics, based on the computation of the theoretical Nomenclature: det(.) determinant F (v, n-v) Snedecor's F distribution with v and n-v degrees freedom LPL i lower process limit (modified) for variable i, i = 1, 2 LSL i lower specification limit for variable i, i = 1, 2 MPCIs Multivariate process capability indices PCIs Process capability indices R rotation matrix S sample covariance matrix S -1 inverse of sample covariance matrix sgn signum function T 2 Hotelling's T-square statistic UPL i upper process limit ( 6 proposed an index. Bothe 7 proposed a method in order to compute the multivariate C pk index. ...
Process capability indices are intended to provide a single-number assessment of the consistency of a manufacturing process relative to the engineering specification limits on quality characteristics. In many industrial instances product quality depends on a multitude of dependent characteristics and as a consequence, attention on capability indices shifts from univariate domain to multivariate domain. In this paper five different multivariate methodologies are used for measuring and comparing capability of a ceramic table-ware manufacturing process. Based on their multivariate process capability index values and expected rejection rate, the result shows that Castagliola's index is the best and followed by Chen, Taam, Shahriari and Braun respectively in this case.
En las compañías manufactureras, es indispensable conocer la capacidad que tienen los procesos de cumplir especificaciones o metas relacionadas con la eficiencia operativa, ya sea al planear las condiciones de calidad en manufactura o al momento de evaluar la gestión a través de los sistemas de gestión integrados. En las décadas recientes, ha surgido el concepto de capacidad del proceso o desempeño del proceso, que proporciona una estimación cuantitativa de qué tan conforme es un proceso. Este trabajo ilustra una metodología para calcular un indicador de capacidad de proceso multivariado validado en una compañía productora de bebidas gaseosas, el cual resume el comportamiento del sistema de gestión integrado y orienta a los administradores de procesos a tomar decisiones estratégicas sobre el control y la mejora de los procesos con base en la identificación de variables claves de procesos pertenecientes a los diferentes sistemas de gestión, basados históricamente con valores variables analizadas de manera univariada, recurriendo a análisis densos y sin percepción de las correlaciones posibles entre los diferentes factores de los sistemas integrados de gestión de calidad. La metodología está basada en el análisis de la base de datos correspondiente a los resultados de los indicadores de gestión de los diferentes sistemas de calidad, obtenidos históricamente y almacenados en los sistemas de información de la compañía. Estos datos se trataron como variables aleatorias distribuidas normalmente y agrupadas matemáticamente como variables con comportamientos distribuidos con chi – cuadrado, estableciendo metas o valores nominales de resultados de los sistemas de calidad. De estos cálculos resultaron valores apropiados a un desarrollo estable del sistema de calidad, logrando disminuir la dispersión a través del cálculo del indicador de capacidad y reflejando la maduración del sistema integral de gestión.
We present a two-phase methodology based on the concept of depth to measure the capability of processes characterized by the functional relationship of multivariate nonlinear profile data, treated as multivariate functional observations. In the first phase, the modified tolerance region is estimated using a historical data set, while in the second, a current process is assessed using the proposed three-component vector, where the first component measures the volume ratio between the current process region and the modified tolerance region; the second measures the probability that the median of the current process is within the modified tolerance region, and the third measures the probability that the current process region is inside the modified tolerance region. To facilitate interpretation, a single index is derived from this capability vector. A simulation study is carried out to assess the performance of the proposed method. An real example illustrates the applicability of this approach.
Providing a single-valued assessment of the performance of a process is often one of the greatest challenges for a quality professional. Process Capability Indices (PCIs) precisely do this job. For processes having a single measurable quality characteristic, there is an ample number of PCIs, defined in literature. The situation worsens for multivariate processes, i.e., where there is more than one correlated quality characteristic. Since in most situations quality professionals face multiple quality characteristics to be controlled through a process, Multivariate Process Capability Indices (MPCIs) become the order of the day. However, there is no book which addresses and explains different MPCIs and their properties. The literature of Multivariate Process Capability Indices (MPCIs) is not well organized, in the sense that a thorough and systematic discussion on the various MPCIs is hardly available in the literature.
Handbook of Multivariate Process Capability Indices provides an extensive study of the MPCIs defined for various types of specification regions. This book is intended to help quality professionals to understand which MPCI should be used and in what situation. For researchers in this field, the book provides a thorough discussion about each of the MPCIs developed to date, along with their statistical and analytical properties. Also, real life examples are provided for almost all the MPCIs discussed in the book. This helps both the researchers and the quality professionals alike to have a better understanding of the MPCIs, which otherwise become difficult to understand, since there is more than one quality characteristic to be controlled at a time.
A multivariate exponentially weighted moving average (MEWMA) control chart is proposed for detecting process shifts during the phase II monitoring of simple linear profiles (SLPs) in the presence of within-profile autocorrelation. The proposed control chart is called MEWMA-SLP. Furthermore, two process capability indices are proposed for evaluating the capability of in-control SLP processes, and their utilization is demonstrated through examples. Intensive simulations reveal that the MEWMA-SLP chart is more sensitive than existing control charts in detecting profile shifts.
In this paper, a process capability index for two correlated quality characteristics jointly following bivariate exponential distribution has been proposed. The expectation and sampling variance of the estimated index have been derived. Choice of the natural process interval corresponding to a specified coverage probability has been discussed.
Multivariate capability analysis has been the focus of study in recent years, during which many authors have proposed different multivariate capability indices. In the operative context, capability indices are used as measures of the ability of the process to operate according to specifications. Because the numerical value of the index is used to conclude about the capability of the process, it is essential to bear in mind that almost always that value is obtained from a sample of process units. Therefore, it is really necessary to know the properties that the indices have when they are calculated on sampling information, in order to assess the goodness of the inferences made from them. In this work, we conduct a simulation study to investigate distributional properties of two existing indices: NMCpm index based on ratio of volumes and Mp2 index based on principal component analysis. We analyze the relative bias and the mean square error of the estimators of the indices, and we also obtain their empirical distributions that are used to estimate the probability that the indices classify correctly a process as capable or as incapable. The results allow us to recommend the use of one of these indices, as it has shown better properties. Copyright
With respect to the problem that traditional Wiener process can't characterize the difference between the population and the individual, this paper proposes a bootstrap method to solve the problem. First of all, we use normal distribution to describe the drift parameter and volatility parameter of Wiener process. Then, we generate the bootstrap sample through bootstrap estimation of the parameter. Finally, based on the bootstrap sample, the population degradation parameters are obtained and the lifetime of product is predicted. A long life product in spaceflight is presented as an example to validate the method.
In the context of process capability analysis, the results of most processes are dominated by two or even more quality characteristics, so that the assessment of process capability requires that all of them are considered simultaneously. In recent years, many researchers have developed different alternatives of multivariate capability indices using different approaches of construction.
In this paper, four of them are compared through the study of their ability to correctly distinguish capable processes from incapable processes under a diversity of simulated scenarios, defining suitable minimum desirable values that allow to decide whether the process meets or does not meet specifications. In this sense, properties analyzed can be seen as sensitivity and specificity, assuming that a measure is sensitive if it can detect the lack of capability when it actually exists and specific if it correctly identifies capable processes. Two indices based on ratios of regions and two based on the principal component analysis have been selected for the study. The scenarios take into account several joint distributions for the quality variables, normal and non‐normal, several numbers of variables, and different levels of correlation between them, covering a wide range of possible situations.
The results showed that one of the indices has better properties across most scenarios, leading to right conclusions about the state of capability of processes and making it a recommendable option for its use in real‐world practice. Copyright © 2015 John Wiley & Sons, Ltd.
Process capability indices measure the ability of a process to provide products that meet certain specifications. Few references deal with the capability of a process characterized by a functional relationship between a response variable and one or more explanatory variables, which is called profile. Specifically, there is not any reference analysing the capability of processes characterized by multivariate nonlinear profiles. In this paper, we propose a method to measure the capability of these processes, based on principal components for multivariate functional data and the concept of functional depth. A simulation study is conducted to assess the performance of the proposed method. An example from the sugar production illustrates the applicability of this approach.
Most multivariate process capability indices proposed by other researchers provide quality measure in losses due to variability and being off-target. Some suggested loss function models that estimate losses due to rejection and quality costs but failed to consider the correlation between the multivariate quality characteristics. In this paper, we propose a novel new approach for estimating the total expected quality cost that provides quality cost measure not only in losses due to variability and being off-target but also losses due to rejection. The proposed model also estimates the quality costs for correlated quality characteristics with nominal specifications and provides the estimated costs due to individual characteristics; it helps to identify those process quality characteristics which have high variation when compared with their specification spread. The model will also improve the estimate of the rejection cost suggested by earlier researchers as it identifies the reject region as the region outside of the modified tolerance region. We present four different correlated bivariate normal process scenarios where the proposed method can be applicable and demonstrate that our approach provides a robust tool in estimating the risk of the product being rejected and performs as well as existing methods in estimating the total quality cost.
ultivariate process capability indices (MPCI), as an important means of statistical process control (SPC), can be used to ensure the high reliability of semiconductors manufacturing process. However, the reasonable sampling number is an important factor when considering MPCI values. As general, the large sample number requires much effort and time, or even cannot be achieved. In this paper, we evaluated the impact of different sample size on the calculations of multivariate process capability indices using simulation and analyses. After getting enough data and choosing disparate sample numbers, corresponding multivariate process capability indices can be obtained, which demonstrate the relationship between sampling numbers and calculation results. The conclusions have critical guiding significance for manufacturing semiconductors with high reliability requirement.
In some quality control applications, quality of a product or a process can be characterized by a profile defined as a functional relationship between a response variable and one or more explanatory variables. Many researchers have contributed to the development of linear and nonlinear profiles to monitor a process or product. However, less work has been devoted to the development of process capability indices in profile monitoring to evaluate process performance with respect to specification limits. This paper presents a process capability analysis when the quality characteristic of interest is represented by a linear profile. Simulation analyses along with a real case study in leather industry are used to evaluate the performance of the proposed method. Results indicate satisfactory performance.
Process capability indices (PCIs) are used in statistical process control to evaluate the capability of the processes in satisfying the customer's needs. In the past two decades varieties of PCI are introduced by researchers to analyze the process capability with univariate or multivariate quality characteristics. To the best of our knowledge, most famous multivariate capability indices are proposed when the quality characteristics have both upper and lower specification limits. These indices are incapable to assess the multivariate processes capability with unilateral specification. In this article, we propose a new multivariate PCI to analyze the processes with one or more unilateral specification limits. This new index also accounts for all problems in the best PCIs of the literature. The performance of the proposed index is evaluated by real cases under different situations. The results show that the proposed index performs satisfactorily in all cases considered. Copyright © 2012 John Wiley & Sons, Ltd.
In the chapter, the term “manufacturing process” refers to the set of relevant engineering processes together with all the other factors—such as raw material and manufacturing environment—that enter the process of translating product design into the finished product. The interaction between the product design and the manufacturing process cannot be ignored. These issues highlight the challenge of quantifying the capability of industrial processes. This chapter provides an overview of the progress that has been made so far in meeting this challenge. As is frequently the case in scientific investigations, the first attempts at quantifying the capability of industrial processes were made in the context of a highly simplified model—the model of statistical control in one dimension with the univariate normal distribution lurking in the background. A large part of the process capability literature is concerned with this model and/or its variations and extensions, and this literature has found useful applications in industry. The chapter provides an overview of the literature from the point of view of applications and/or theoretical developments. It includes estimation and testing for the various process control indices. The chapter considers the Bayesian approach to quantifying process capability. The remarkable progress in Markov chain Monte Carlo techniques has opened new vistas for the application of the Bayesian paradigm.
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.
Process Capability Indices, such as Cp and Cpk have been widely used as statistical tools to assess the manufacturing process performance. These indices provide numerical measures on process precision, process accuracy and process performance. Extensive researches have been done on the univariate process capability indices in the last two decades. However, process capability indices for processes with multiple quality characteristics have received little attention, comparatively. The multivariate process capability indices, which are used for evaluation of processes with correlated quality characteristics such as weight, height and width, could be investigated, in depth. In this research the concept of process capability and its relevant indices in univariate and multivariate cases are discussed. Based on the previous works, a new multivariate process capability vector (NMPCV) is introduced. This vector is based on a modification of the traditional multivariate process capability indices. The value and the power of this new index are evaluated using field and simulated data. The results of this research show that NMPCV is a better tool for judging the process accuracy, the process precision and the process performance. When the characteristics are highly correlated, the results are much better.
A new measure of the process capability (Cpm) is proposed that takes into account the proximity to the target value as well as the process variation when assessing process performance. The sampling distribution for an estimate of Cpm (Ĉpm) and some of its properties have been examined and an example of its application is included. The new index is easy to compute and, with the aid of the included tables, easy to analyze. Ĉpm has some more desirable statistical properties than Ĉp and Ĉpk,the estimates of the Cp and Cpk indices, respectivel
Preface. Part A. Set Theory. 1. Basic Concepts of Set Theory. 2. Relations and Functions. 3. Properties of Relations. 4. Infinities. Appendix A1. Part B. Logic and Formal Systems. 5. Basic Concepts of Logic. 6.Statement Logic. 7. Predicate Logic. 8. Formal Systems, Axiomatization, and Model Theory. Appendix B1. Appendix BII. Part C. Algebra. 9. Basic Concepts of Algebra. 10. Operational Structures. 11. Lattices. 12. Boolean and Heyting Algebras. Part D. English as a Formal Language. 13. Basic Concepts of Formal Languages. 14. Generalized Quantifiers. 15. Intensionality. Part E. Languages, Grammars, and Automata. 16. Basic Concepts of Languages, Grammars, and Automata. 17. Finite Automata, Regular Languages and Type 3 Grammars. 18. Pushdown Automata, Context-Free Grammars and Languages. 19. Turing Machines, Recursively Enumberable Languages, and Type 0 Grammars. 20. Linear Bounded Automata, Context-Sensitive Languages and Type 1 Grammars. 21. Languages Between Context-Free and Context-Sensitive. 22. Transformational Grammars. Appendix EI. Appendix EII. Review Problems. Index.
The capability indices Cp, CPU, CPL, k and Cpk are presented and related to process parameters. These indices are shown to form a complementary system of measures of process performance, and can be used with bilateral and unilateral tolerances, with or without target values. A number of Japanese industries currently use the five indices and the U.S. automotive industry has started using these measures in a number of areas. Various applications of the indices are discussed along with statistical sampling considerations.
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 bivariate normal distribution function may be expressed as the product of a marginal normal distribution times a conditional distribution. By approximating this conditional distribution, we obtain a simple method for approximating bivariate normal probabilities. When the correlation falls in the interval (-5, .5), the maximum absolute error in our approximation is always less than .0008. The conditional distribution that we approximate is referred to as a 'normal conditioned on a truncated normal' distribution and is related to screening and selection problems.
In this paper we examine the behaviour of bivariate generalizations of the process capability indices Cp, CPU and CPL. It is shown that when the characteristics of interest have a bivariate normal distribution the distribution of is related to the bivariate χ distribution and that of is related to the bivariate noncentral t distribution. Since the process depends on two variables, it seems reasonable to require that BOTH variables conform with the definition of capability; that is, the process will be considered not capable if either of the indices is below a prescribed value. Tests of hypotheses concerning this requirement on the indices, CP1 and CP2 , are developed. Critical points, based on the bivariate χ distribution, for testing these hypotheses are tabulated.
Distributions in Statistics National Bureau of Standards. Tables of the Bivariate Nor-mal Distribution Function and Related Functions
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Probability Functions
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