Process performance evaluation using evolutionary algorithm

Nowadays every business is using different quantitative measures and techniques to assess performance of their products / services. It is well known that different manufacturing processes very often manufacture products with quality characteristics that do not follow normal distribution. In such cases, fitting a known non-normal distribution to these quality characteristics would lead to erroneous results. Furthermore, there is always more than one characteristic Critical to Quality (CTQ) in the process outcomes and very often these quality characteristics are correlated with each other. In this paper, we assess performance of such a bivariate process data which is non-normal as well as correlated. We will use the geometric distance approach to reduce the dimension of the correlated non-normal bivariate data and then fit Burr distribution to the geometric distance variable. The optimal parameters of the fitted Burr distribution are estimated using Evolutionary Algorithm (EA). The results are compared with those using Simulated Annealing (SA) algorithm. The proportion of nonconformance (PNC) for process measurements is then obtained by using the fitted Burr distributions based on the two methods. The results based on both search algorithms are then compared with the exact proportion of nonconformance of the data. Finally, a case study using real data is presented

Aplicação do controle estatístico de processo em uma indústria automobilística / Application of statistical process control in an automobile industry

O cenário competitivo exige melhoria contínua dos processos. Evidentemente isto implica a redução de variabilidade. Com isso, cada vez mais se observa a necessidade de melhorar continuamente os processos. Tendo como objetivo a competitividade e a melhoria de seus processos produtivos as empresas, vem em crescente evolução utilizando-se de ferramentas estatísticas, tanto para o desenvolvimento quanto para o monitoramento de seus processos produtivos. Neste contexto encontra-se a empresa foco deste estudo, no qual o objetivo é utilizar ferramentas estatísticas para explorar os índices de capacidade de um processo de corte em uma indústria automobilística para análise da capacidade do processo, já que um processo é capaz se seus índices de capacidade estiverem dentro das especificações da empresa e do cliente. A utilização da metodologia de Controle Estatístico de Processo (CEP) será usada para quantificar quão bem o processo atende aos requisitos do cliente. O estudo assume as formas de pesquisa bibliográficas, estudo de caso e levantamento de coleta de dados, portanto classifica-se como exploratória e descritiva quanto aos seus objetivos. Em suma, o propósito deste artigo é mostrar de forma sucinta a capacidade do processo e por fim apresentar que a mesma poderia aproveitar-se da oportunidade de reduzir perdas em sua produção através do uso do controle estatístico da qualidade. 

Controle estatístico do processo de torneamento duro na presença de variável de ruído

The hard turning process has characteristics that favor its choice in relation to traditional turning. It presents better results of surface integrity, short process time, possibility of working without the use of lubricating fluid. Surface integrity results are the focus of studies of machining processes in general. The surface roughness is observed in relation to several controllable input parameters. Among the inputs can also be identified noise variables. These variables are uncontroled, but they affect process responses. In addition, there is more than one roughness response. Multivariate studies allow the simultaneous analysis of correlated responses. Thus, this research aims to perform univariate and multivariate analysis by principal components of the hard turning process. This analysis seeks to observe the effect of the noise variable on the responses and verify process stability and capability. For such analysis a hybrid method of experiment design and statistical process control was used. The AISI 52100 hardened steel turning was selected for the study application. Process scenarios were developed with a factorial arrangement. For this scenarios, regression models, control charts, process performance, sigma level and parts per million estimation were calculated. The performance (Ppk) results for the multivariate scenario were between 0.18 to 1.11. The best scenario presented had a cutting speed of 170m/min and a flow of lubricating fluid of 3 l/min.Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPqO processo de torneamento duro do aço ABNT 52100 tem características que favorecem sua seleção por apresentar resultados satisfatórios de integridade da superfície, menores tempo de processo, possibilidade de usinagem sem uso de fluido refrigerante. Pesquisas indicam que resultados de integridade da superfície são foco de estudos de processos por torneamento em geral. As rugosidades superficiais são observadas em relação a diversos parâmetros controláveis de entrada. Dentre as entradas pode se identificar também variáveis de ruído. Essas variáveis não estão sob controle, mas afetam as respostas do processo. Além disso, podem existir mais de uma resposta de rugosidade. Para isso estudos multivariados permitem a análise simultânea das respostas correlacionadas. Desse modo, essa pesquisa objetiva realizar análises univariada e multivariada por meio de componentes principais do processo de torneamento duro. Essa análise busca observar o efeito da variável de ruído nas respostas e verificar a estabilidade e desempenho do processo. Para tal análise um método híbrido de projeto de experimentos e controle estatístico do processo o torneamento do aço ABNT 52100 foi selecionado para aplicação do estudo. Com um arranjo fatorial foram desenvolvidos e comparados cenários do processo. Para esses cenários foram calculados modelos de regressão, cartas controle, performance do processo, estimativa de nível sigma e defeitos por milhão. Os resultados de performance (Ppk) para o cenário multivariado foram de 0.18 a 1.11. O melhor cenário apresentado teve velocidade de corte 170m/min e vazão de fluido refrigerante de 3l/min

Development and application of process capability indices

In order to measure the performance of manufacturing processes, several process capability indices have been proposed. A process capability index (PCI) is a unitless number used to measure the ability of a process to continuously produce products that meet customer specifications. These indices have since helped practitioners understand and improve their production systems, but no single index can fully measure the performance of any observed process. Each index has its own drawbacks which can be complemented by using others. Advantages of commonly used indices in assessing different aspects of process performance have been highlighted. Quality cost is also a function of shift in mean, shift in variance and shift in yield. A hybrid is developed that complements the strengths of these individual indices and provides the set containing the smallest number of indices that gives the practitioner detailed information on the shift in mean or variance, the location of mean, yield and potential capability. It is validated that while no single index can fully assess and measure the performance of a univariate normal process, the optimal set of indices selected by the proposed hybrid can simultaneously provide precise information on the shift in mean or variance, the location of mean, yield and potential capability. A simulation study increased the process variability by 100% and then reduced by 50%. The optimal set managed to pick such a shift. The asymmetric ratio was able to detect both the 10% decrease and 20% increase in µ but did not alter significantly with a 50% decrease or a 100% increase in σ, which meant it was not sensitive to any shift in σ. The implementation of the hybrid provides the quality practitioner, or computer-aided manufacturing system, with a guideline on prioritised tasks needed to improve the process capability and reduce the cost of poor quality. The author extended the proposed hybrids to fully measure the performance of a process with multiple quality characteristics, which follow normal distribution and are correlated. Furthermore, for multivariate normal processes with correlated quality characteristics, process capability analysis is not complete without fault diagnostics. Fault diagnostics is the identification and ranking of quality characteristics responsible for multivariate process poor performance. Quality practitioners desire to identify and rank quality characteristics, responsible for poor performance, in order to prioritise resources for process quality improvement tasks thereby speeding up the process and minimising quality costs. To date, none of the existing commonly used source identification approaches can classify whether the process behaviour is caused by the shift in mean or change in variance. The author has proposed a source identification algorithm based on mean and variance impact factors to address this shortcoming. Furthermore, the author developed a novel fault diagnostic hybrid based on the proposed optimal set selection algorithm, principal component analysis, machine learning, and the proposed impact-factor. The novelty of this hybrid is that it can carry out a full multivariate process capability analysis and provides a robust tool to precisely identify and rank quality characteristics responsible for the shifts in mean, variance and yield. The fault diagnostic hybrid can guide the practitioners to identify and prioritise quality characteristics responsible for the poor process performance, thereby reducing the quality cost by effectively speeding up the multivariate process improvement tasks. Simulated scenarios have been generated to increase/decrease some components of the mean vector (µ2/µ4) and in increase/reduce the variability of some components (σ1 reduced to close to zero/σ6 multiplied by 100%). The hybrid ranked X2 and X6 as the most contributing variables to the process poor performance and X1 and X4 as the major contributors to process yield. There is a great challenge in carrying out process capability analysis and fault diagnostics on a high dimensional multivariate non-normal process, with multiple correlated quality characteristics, in a timely manner. The author has developed a multivariate non-normal fault diagnostic hybrid capable of assessing performance and perform 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. 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 the least performing GD variable. 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. The novelty of the proposed non-normal fault diagnostic hybrid is that it considers quality characteristics related to the least performing GD variable, instead of investigating all the quality characteristics of the multivariate non-normal process. The efficacy of the proposed hybrid is assessed through a real manufacturing examples and simulated scenarios. Variables X1,, X2 and X3 shifted away from the target by 25%, 15% and 35%, respectively, and the hybrid was able to select variables X3 to be contributing the most to the corresponding geometric distance variable's poor performance

Process capability assessment for univariate and multivariate non-normal correlated quality characteristics

In today's competitive business and industrial environment, it is becoming more crucial than ever to assess precisely process losses due to non-compliance to customer specifications. To assess these losses, industry is extensively using Process Capability Indices for performance evaluation of their processes. Determination of the performance capability of a stable process using the standard process capability indices such as and requires that the underlying quality characteristics data follow a normal distribution. However it is an undisputed fact that real processes very often produce non-normal quality characteristics data and also these quality characteristics are very often correlated with each other. For such non-normal and correlated multivariate quality characteristics, application of standard capability measures using conventional methods can lead to erroneous results. The research undertaken in this PhD thesis presents several capability assessment methods to estimate more precisely and accurately process performances based on univariate as well as multivariate quality characteristics. The proposed capability assessment methods also take into account the correlation, variance and covariance as well as non-normality issues of the quality characteristics data. A comprehensive review of the existing univariate and multivariate PCI estimations have been provided. We have proposed fitting Burr XII distributions to continuous positively skewed data. The proportion of nonconformance (PNC) for process measurements is then obtained by using Burr XII distribution, rather than through the traditional practice of fitting different distributions to real data. Maximum likelihood method is deployed to improve the accuracy of PCI based on Burr XII distribution. Different numerical methods such as Evolutionary and Simulated Annealing algorithms are deployed to estimate parameters of the fitted Burr XII distribution. We have also introduced new transformation method called Best Root Transformation approach to transform non-normal data to normal data and then apply the traditional PCI method to estimate the proportion of non-conforming data. Another approach which has been introduced in this thesis is to deploy Burr XII cumulative density function for PCI estimation using Cumulative Density Function technique. The proposed approach is in contrast to the approach adopted in the research literature i.e. use of best-fitting density function from known distributions to non-normal data for PCI estimation. The proposed CDF technique has also been extended to estimate process capability for bivariate non-normal quality characteristics data. A new multivariate capability index based on the Generalized Covariance Distance (GCD) is proposed. This novel approach reduces the dimension of multivariate data by transforming correlated variables into univariate ones through a metric function. This approach evaluates process capability for correlated non-normal multivariate quality characteristics. Unlike the Geometric Distance approach, GCD approach takes into account the scaling effect of the variance-covariance matrix and produces a Covariance Distance variable that is based on the Mahanalobis distance. Another novelty introduced in this research is to approximate the distribution of these distances by a Burr XII distribution and then estimate its parameters using numerical search algorithm. It is demonstrates that the proportion of nonconformance (PNC) using proposed method is very close to the actual PNC value

Multivariate process variability monitoring for high dimensional data

In today’s competitive market, the quality of a product or service is no longer measured by a single variable but by a number of variables that define the quality of the final product or service. It is known that these quality variables of products or services are correlated with each other, and it is therefore important to monitor these correlated quality characteristics simultaneously. Multivariate quality control charts are capable of such monitoring. Multivariate monitoring of industrial or clinical procedures often involves more than three correlated quality characteristics, and the status of the process is judged using a sample of one size. The majority of existing control charts for monitoring multivariate process variability for individual observations are capable of monitoring up to three quality characteristics. One of the hurdles in designing optimal variability control charts for large dimension data is the enormous computing resources and time that is required by the simulation algorithm to estimate the charts parameters. In this research, a novel algorithm based on the parallelised Monte Carlo simulation has been developed to improve the ability of the Multivariate Exponentially Weighted Mean Squared Deviation (MEWMS) and Multivariate Exponentially Weighted Moving Variance (MEWMV) charts to monitor multivariate process variability with a greater number of quality characteristics. Different techniques have been deployed to reduce computing space and the time complexity taken by the algorithm. The novelty of this algorithm is its ability to estimate the optimal control limit L (optimal L) for any given number of correlated quality characteristics, size of the shifts to be detected based on the smoothing constant, and the given in-control average run length in a computationally efficient way. The optimal L for the MEWMS and MEWMV charts to detect small, medium and large shifts in the covariance matrix of up to fifteen correlated quality characteristics has been provided. Furthermore, utilising the large number of optimal L values generated by the algorithm has enabled us to develop two mathematical functions that are capable of predicting L values for MEWMS and MEWMV charts. This would eliminate the need for further execution of the parallelised Monte Carlo simulation for high dimension data. One of the main challenges in deploying multivariate control charts is to identify which characteristics are responsible for the out-of-control signal detected by the charts, and what is the extent of their contribution to the signal. In this research, a smart diagnostic technique has been developed by using a hybrid of the wrapper filter approach to effectively identify the variables that are responsible for the process faults and to classify the percentage of their contribution to the faults. The robustness of the proposed techniques has been demonstrated through their application to a range of clinical and industrial multivariate processes where the percentage of correct classifications is presented for different scenarios. The majority of the existing multivariate control charts have been developed to monitor processes that follow multivariate normal distribution. In this thesis, the author has proposed a control chart for a non-normal high dimensional multivariate process based on the percentile point of Burr XII distribution. Geometric distance variables are fitted to the subset of correlated quality characteristics to reduce the dimension of the data, which is then followed by fitting the Burr XII distribution to each geometric distance variable. Since individual distance variables are independent, each can be monitored by individual control charts based on the percentile points of the fitted Burr XII distributions. A simulated annealing approach is used to estimate parameters of the Burr XII distribution. The proposed hybrid is utilised to identify and rank the variables responsible for the out-of-control signals of geometric distance variables

Análise de Capabilidade de Processos Multivariados usando o Método dos Componentes Principais Ponderados.

A análise de capabilidade de processos tem sido usada para quantificar quão bem o processo atende aos requisitos dos clientes e para identificar e reduzir a variabilidade. No entanto, existe uma lacuna para se quantificar quando o processo apresenta características de qualidade correlacionadas, situação comum nos processos de fabricação. Considerando que os estudos desenvolvidos para dita análise em relação às métricas e propriedades estatísticas são em sua maioria para processos univariados, e que os métodos multivariados existentes apresentam restrições que fazem que sua aplicação seja limitada. Portanto, a principal contribuição deste estudo é a proposta de um método para análise multivariada de capabilidade, denominado Componentes Principais Ponderados (WPC). Este utiliza como resposta do modelo os escores dos componentes principais, ponderados por seus autovalores ou pela porcentagem de explicação de cada componente. O método proposto por Liao (2005) tem sido usado na otimização de processos com múltiplas respostas e foi aplicado por Peruchi et al. (2013) na determinação de índices de avaliação de sistemas de medição. No que diz respeito a esta nova abordagem, o WPC não foi simplesmente aplicado para se determinar os índices multivariados dos estimadores clássicos de capabilidade Cp, Cpk, Cpm e Cpmk; mas também para se estimar os índices de desempenho Pp, Ppk, Ppm e Ppmk, os intervalos de confiança das estimativas, a proporção de não conformes em PPM e o nível sigma. A eficiência do método se demonstrou usando dados da literatura, experimentais e simulados; neste último caso foi testado em diferentes níveis de desempenho de processo e graus de correlação. Em todos esses casos as estimativas do WPC foram comparadas com as de outros três métodos baseados em Análise de Componentes Principais (PCAM, PCAX e PCAW), e julgadas com intervalos de confiança univariados para se determinar se eram adequadas. O WPC se mostrou mais robusto do que os métodos PCAM e PCAW; além disso, exibiu um decrescimento das estimativas de desempenho com o aumento da correlação, tendência que já tem sido comprovada pelas pesquisas de Tano e Vännman (2013) e Guevara e Vargas (2007), e que é oposta aos métodos apresentados baseados em PCA. Isto faz presumir que o método proposto representa melhor o efeito de correlação entre as características, embora os métodos PCAX e PCAM tenham um bom comportamento perante dos intervalos de confiança univariados

Process capability estimation for non-normal quality charactersitics using Clement, Burr and Box--Cox methods

In today's competitive business environment, it is becoming more crucial than ever to assess precisely process losses due to non-compliance to customer specifications. To assess these losses, industry is widely using process capability indices for performance evaluation of their processes. Determination of the performance capability of a stable process using the standard process capability indices requires that the underlying process data should follow a normal distribution. However, if the data is non-normal, measuring process capability using conventional methods can lead to erroneous results. Different process capability indices such as Clements percentile method and data transformation method have been proposed to deal with the non-normal situation. Although these methods are practiced in industry, there is insufficient literature to assess the accuracy of these methods under mild and severe departures from normality. This article reviews the performances of the Clements non--normal percentile method, the Burr based percentile method and Box--Cox method for non-normal cases. A simulation study using Weibull, Gamma and lognormal distributions is conducted. Burr's method calculates process capability indices for each set of simulated data. These results are then compared with the capability indices obtained using Clements and Box--Cox methods. Finally, a case study based on real world data is presented. References I. W. Burr (1942). Cumulative frequency distribution. Ann Math Stat 13: 215--232. http://www.ams.org/mathscinet/pdf/6644.pdf I. W. Burr (1973). Parameters for a general system of distributions to match a grid of $\alpha _3$ and $\alpha _4$. Commun Stat 2:1--21. http://www.zentralblatt-math.org/zmath/search/?q=an:03413665&type=pdf&format=complete G. E. P. Box and D. R. Cox (1964). An analysis of transformation. J Roy Stat Soc B 26:211--252. http://www.jstor.org/view/00359246/di993152/99p02493/0 J. A. Clements (1989). Process capability calculations for non-normal distributions. Quality Progress 22:95--100. http://www.asq.org/qic/display-item/index.html?item=14059 N. L. Johnson (1949). System of frequency curves generated by methods of translation. Biometrika 36:149--176. http://www.jstor.org/view/00063444 /di 992300/99 p 0220l/0 S. Kotz and C. R. Lovelace (1998). Process capability indices in theory and practice. Arnold, London. http://www.amazon.com/Process-Capability-Indices-Theory-Practice/dp/0340691778 S. Kotz and N. L. Johnson (1993). Process capability indices. New York: Chapman and Hall. http://www.amazon.com/Process-Capability-Indices-Samuel-Kotz/dp/041254380X L. A. R. Rivera, N. F. Hubele and F. D. Lawrence (1995). $C_{pk}$ index estimation using data transformation. Comput Ind Engng 29: 55-58. http://www.ingentaconnect.com /content/els/03608352/1995/00000029/00000001/art00045 L. C. Tang, S. E. Than (1999). Computing process capability indices for non-normal data : a review and comparative study. Qual Reliab Engng Int 15: 339--353. doi:CCC 0748-8017/99/050339 D. Montgomery (1996). Introduction to Statistical Quality Control. 5th edition. Wiley, New York. http://bcs.wiley.com/he-bcs/Books?action=index&bcsId=2077&itemId=0471656313 J. O'Connell and Q. Shao (2004). Further investigation on a new approach in analyzing extreme events. CSIRO Mathematical and Information Sciences Report No. 04/41. http://www.cmis.csiro.au/techreports/docs/x0000ihw.pdf Pei--Hsi Liu and Fei--Long Chen (2006). Process capability analysis of non-normal process data using the Burr XII distribution. Int J Adv Manuf Technol 27: 975--984. doi:10.1007/s 00170-004-2263-8 S. Somerville and D. Montgomery (1996). Process capability indices and non-normal distributions. Quality Engineering 19(2):305--316. doi:10.1080/08982119608919047 F. K. Wang (2006). Quality evaluation of a manufactured product with multiple characteristics. Qual Relib Engng Int 22: 225--236. http://doi.wiley.com/10.1002/qre.712 H. H. Wu, J. S. Wang and T. L. Liu (1998). Discussions of the Clements-based process capability indices. In: Proceedings of the 1998 CIIE National Conference pp.561--566. doi:10.1007/s 00170-004-2263-