Evaluating capability of a bivariate zero-inflated poisson process

A zero-inflated Poisson (ZIP) distribution is commonly used for modelling zero-inflated process data with single type of defect, and for developing appropriate tools for instituting statistical process control of manufacturing processes. However, in reality, such manufacturing scenarios are very common where more than one type of defect can occur. For example, occurrences of defects like solder short circuits (shorts) and absence of solder (skips) are very common on printed circuit boards. In literature, different forms of bivariate zero-inflated Poisson (BZIP) distributions are proposed, which can be used for modelling the manufacturing scenarios where two types of defects can occur. Control charts are designed for monitoring for such processes using BZIP models. Although evaluation of capability is an integral part of statistical process control of a manufacturing process, researchers have given very little effort on this aspect of zero-inflated processes. Only a few articles attempted to evaluate the capability of a univariate zero-inflated process and no work is reported on evbaluating capability of a bivariate zero-inflated process. In this paper, a methodology for measuring capability of a bivariate zero-inflated process is presented. The proposed methodology is illustrated using two case studies.&nbsp

A process capability index for zero-inflated processes

The proportion of zero defect (ZD) outputs is as an integral characteristic of a zero-inflated (ZI) process or high quality process. Different ZI processes can almost equally satisfy the same USL of number of defects but they can produce substantially different proportions of ZD products. The application of conventional method for process capability evaluation fails to discriminate these processes because in the conventional method, the process capability is evaluated taking into consideration the USL of number of defects only. In this paper, a new measure of process capability for ZI processes is proposed that can truly discriminate different ZI processes taking into account the USL of number of defects as well as the proportion of ZD units produced in these processes. In the proposed approach, at first a measure of process capability index (PCI) with respect to the USL is computed, and then the overall PCI is obtained by multiplying it with an appropriately defined multiplying factor. A real-life application is presented

Defects control charts for high-quality processes

The traditional C-chart by Shewhart has been widely applied for monitoring count data in industrial and nonindustrial processes. However, using C-chart always experiences an excessive amount of false alarms, since control limits of traditional C-chart are defined by impractical normal assumption. Specially, when we monitor two or more correlated characteristics of defects, C-chart becomes unsuitable. Thus, monitoring a process by traditional C-chart leads to the increase of unnecessary costs of inspection. There are many works that have attempted to improve C-charts. In this thesis, 11 selected improved versions of C-chart are presented. The performances of improved C-charts are evaluated in term of numerical results to demonstrate the sensitivity of the charts and costs of inspections. We also propose an optimal bivariate Poisson field chart to monitor two correlated characteristics of defects. Our chart is based on the optimization of bivariate Poisson confidence interval and projection of bivariate Poisson data in Poisson field. The detailed description of our proposed algorithm is presented by numerical data. The experimental results demonstrate improved performances regarding user-friendly visualization and false alarm rate Furthermore, we propose an optimal diagonal inflated bivariate Poisson field chart to monitor two over/under dispersed correlated count data. The detailed description of our chart will be presented. The experimental results demonstrate improved performances according to loss function and false alarm rate compared to other methods

On Shewhart Control Charts for Zero-Truncated Negative Binomial Distributions

The negative binomial distribution (NBD) is extensively used for thedescription of data too heterogeneous to be fitted by Poissondistribution. Observed samples, however may be truncated, in thesense that the number of individuals falling into zero class cannot bedetermined, or the observational apparatus becomes active when atleast one event occurs. Chakraborty and Kakoty (1987) andChakraborty and Singh (1990) have constructed CUSUM andShewhart charts for zero-truncated Poisson distribution respectively.Recently, Chakraborty and Khurshid (2011 a, b) have constructedCUSUM charts for zero-truncated binomial distribution and doublytruncated binomial distribution respectively. Apparently, very littlework has specifically addressed control charts for the NBD (see, forexample, Kaminsky et al., 1992; Ma and Zhang, 1995; Hoffman, 2003;Schwertman. 2005).The purpose of this paper is to construct Shewhart control chartsfor zero-truncated negative binomial distribution (ZTNBD). Formulaefor the Average run length (ARL) of the charts are derived and studiedfor different values of the parameters of the distribution. OC curvesare also drawn

Research on Scientific Derivation of Control Limits in Control Charts

Control Charts (CC) are the means to “manage the process behaviour” by analysing subsequent samples at regular intervals of time.: good decisions depend on Scientific analysis of data. Often, the data are considered Normally distributed; this is not completely right; data must be analysed according to their distribution: decisions are different with different distributions, because the Control Limits of the CC depend on the distribution. We compare our findings with Shewhart findings; later we extend the ideas to deal with “rare events”, with data not Normally distributed; we compare our results, found by RIT, for various cases in the literature: there is a big difference between the Shewhart CC and the Time Between Events CC; considering that, future decisions of Decision Makers will be both sounder and cheaper, when data are not normally distributed. ARL depends on the data distribution, not only on the “false alarm rate”. The novelty of the paper is due to the Scientific Way of Computing the Control Limits, both for the mean and for the variance

Control Chart for Correcting the ARIMA Time Series Model of GDP Growth Cases

The essential prerequisite for attending the G20 conference is a country's GDP because G20 members can significantly boost the economy and preserve the nation's financial stability. Time series data can be thought of as a country's Gross Domestic Product (GDP) at a particular point in time. In this research, the GDP numbers from five Southeast Asian nations that are attending the G20 fulfilling are used. The total was 47 observations made yearly, which extended from 1975 to 2001. A time series analysis was performed on the data gathered. The correctness of time series models is also evaluated using control charts based on this research. The control chart is constructed using the time series model's residuals as observations. After applying the IMR control chart for verification, the results revealed that the residuals, specifically the models for GDP in Malaysia, Singapore, and Thailand, are out of control. The white noise assumption is fulfilled by the time series model obtained for Brunei and Indonesia's GDP, but the residuals are out of control. Whether controlled residuals are used depends on the accuracy with which the time series model predicts the future. If the amount of residuals is under control, then the time series model produced is accurate and good enough for prediction. After using the IMR control chart to verify the residuals, the results indicate that the residuals, namely the models for GDP in Malaysia, Singapore, and Thailand, are not under control. The assumption of white noise is proved correct by the time series model obtained for the GDP of Brunei Darussalam and Indonesia. With that being said, the residuals are entirely out of control. The model must improve its ability to forecast various future periods. It is a consequence of the unmanageable residuals that the model contains. Even if the best available model has been obtained based on the criteria that have been defined, it is anticipated that the research findings will improve the theories that have previously been developed and raise knowledge regarding the usefulness of testing the time series model. In addition to all of that, it is intended that the research will produce a summary of cases of an increase in GDP from five Southeast Asian countries participating in the G20 conference.

Gráficos de control para procesos binomiales con exceso de ceros

Cuando se analizan datos de procesos en los que se observan muchas muestras conformes, es decir, muchas muestras sin defectos, la utilización del modelo ZIB (Zero Inflated binomial) puede ser una muy buena alternativa. Esto es especialmente cierto para el caso en que los datos muestran una frecuencia más elevada de ceros que la que se debería esperar si la muestra hubiera sido generada mediante una distribución binomial. Tradicionalmente estos procesos fueron monitoreados empleando la distribución binomial pero, bajo estas circunstancias, la distribución binomial tiende a subestimar la variabilidad del proceso. En esta situación, los gráficos de control tienen límites muy estrictos que determinan excesivas señales de falsa alarma, altos costos de inspección y frecuentes paradas del proceso. Cuando no se tiene en cuenta el exceso de ceros, se genera un modelo mal especificado y, en consecuencia, el gráfico de control resultante no cumple con la función para la cual ha sido construido En este trabajo se propone la utilización del modelo lineal generalizado para establecer la bondad de ajuste entre los datos observados y la distribución asumida para la construcción del gráfico. Se muestra además la aplicación del gráfico ZIB a un proceso con datos de una planta de autopartes, con análisis y discusión de los resultados.Fil: Joekes, Silvia. Universidad Nacional de Córdoba. Facultad de Ciencias Económicas. Instituto de Estadística y Demografía; Argentina.Fil: Smrekar, Marcelo. Universidad Nacional de Córdoba. Facultad de Ciencias Exactas, Físicas y Naturales. Laboratorio de Ingeniería y Mantenimiento Industrial; Argentina.Fil: Righetti, Andrea. Universidad Nacional de Córdoba. Facultad de Ciencias Económicas. Instituto de Estadística y Demografía; Argentina.Estadística y Probabilida