Capability Index Cpk Research Articles

Capability measures for Weibull processes with mean shift based on Erto's-Weibull control chart

Process capability indices (PCIs), which are effective tools for quality assurance and are guidance for process improvement, have been proposed in the manufacturing industry to provide numerical measures on process reproduction capability. PCIs are calculated under the assumption that the process is stable while the process mean and variation are not changeable. However, in practice, the process is dynamic. Under the Bothe’s adjustments, we showed the detection powers of the percentile-Weibull control chart, bootstrap-Weibull control chart, and the Bayes-Weibull control chart. It is realized that the Bothe’s adjustments are inadequate with data coming from Weibull processes. For this reason, the PCIs have to be adjusted. Bothe (2002) provided the adjustment method for normality processes. In this research, we consider Weibull processes, which cover a wide class of applications. We calculate the mean shift adjustments under various sample sizes n and Weibull parameter γ, with the power fixed to 0.5. Then, we implement the adjustments to accurately estimate capability index Cpk for Weibull processes with mean shift consideration. Finally, an application example is presented for illustration purpose. Key words: Dynamic Cpk, mean shift, process capability index, Weibull distribution,Weibull control chart.

Confidence Intervals for Process Capability Indices for the Unbalanced One‐Way Random Effect ANOVA Model

Confidence intervals for process capability index Cpk are developed for the unbalanced one‐way random effect model using Bissell's approximation method. The proposed limit is compared with the generalized lower confidence limit obtained using the generalized pivotal quantity method. To assess the accuracy of the method, a simulation study is presented. The results are illustrated with an industrial example. Copyright © 2011 John Wiley & Sons, Ltd.

Process Capability Evaluation for Square Bumps with Mean Shift

Abstract Process capability indices have been widely used in manufacturing industries to provide numerical measures of process potential and performance. In in-plant applications, some inevitable process mean shift may be undetected when the statistical process control charts are applied, particularly, in the bumping process for square bumps. Bothe in 2002 provided a statistical reason for considering such a shift in the process mean to accommodate the undetected mean shifts for normally distributed processes when evaluating process capability. In this paper, we consider a bumping process for square bumps in which the data can be formulated as a non-central chi-square distribution, a class of non-normal distributions. To evaluate the popular yield-based process capability Cpk more accurately, we accommodate the magnitudes of undetected mean shifts using the modified capability evaluation formula and tabulate the detection powers under various subgroup sizes and non-central chi-square parameters. Based on the modified process capability evaluation, we can provide a more reliable process capability evaluation of capability index Cpk for square bumps and make more correct decisions. For illustration purpose, a real application in a bumping factory which is located on the Science-based Industrial Park in Hsinchu, Taiwan, is presented.

Capability assessment for processes with multiple characteristics: A generalization of the popular index Cpk

AbstractProcess capability index Cpk is the most popular capability index widely used in the manufacturing industry. Existing research on the yield‐based measure index Cpk to date is restricted to processes with single characteristics. However, many manufacturing processes are commonly described with multiple characteristics, for example, the gold bumping process in the TFT‐LCD (thin film transistor‐liquid crystal display) manufacturing industry. In the gold bumping process, gold bumps have multiple characteristics all having effects on the process yield. Obtaining accurate gold bumping manufacturing yield is very important for quality assurance and in providing guidance toward process improvement. To obtain accurate yield assessment for processes with multiple characteristics, we propose a new overall yield‐measure index C, which is a generalization of the index Cpk, and a natural estimator of C. For the purpose of making inferences on the process capability, we derive a quite accurate approximation of the distribution of since the distribution is analytically intractable. With this distribution, we tabulate the lower confidence bounds of the new index under various sample sizes for in‐plant applications. In addition, we construct a statistical test on the new yield‐measure index in order to examine whether the yield meets the customers' requirements. For illustration purpose, a real case in a gold bumping factory located in the Science‐based Industrial Park at Hsinchu, Taiwan is presented. Copyright © 2011 John Wiley & Sons, Ltd.

Parameter Setting for Rod Push Production Process on Multiple Responses : A Case Study in Motorcycle Parts Factory

The aim of this research is to study appropriate setting of parameters in the production process ofRod Push which affect quality characteristics such as diameter and surface roughness. Cycle time is alsoconsidered. The problem was done at a study plant and it was found that the actual process capability index(Cpk) of two quality characteristics was less than the manufacturer’s standard of 1.33. Central composite designwas used to examine Rod Push produced by CNC with 5 controllable process factors including three factorsfrom rough-cutting process (spindle speed, feed rate, and depth of cut) and two factors from finish-cuttingprocess (spindle speed and feed rate). The nuisance factor of CNC was cutting tool wear. After collectingall data, the appropriate setting was investigated. The new results indicated that the new condition yieldedthe better process performance index and shorter cycle time. Keywords: DOE; Central Composite Design; Rod Push Process; CNC Turning Machine

High speed and high accuracy inspection of in-tray laser IC marking using line scan CCD with a new calibration model

This study used a high-speed high-resolution line scan CCD, and a Delta motor drive module to provide position feedback in a laser marking system. Based on a novel algorithm, image processing of the high-speed CCD scan was allowed the accurate determination of the laser marking location. This system was calibrated using a template, and a novel 1-D calibration model of the line scan CCD was developed. The relevant position of each IC in the tray was obtained based on the calibration algorithm. Gain and offset calibration, sub-pixel calculation, and normalized checks were performed in this automatic optical inspection system. The total processing time for laser correction marking, scanning and identification was about 2∼2.5 s, the positioning accuracy was×9 μm, and the industrial specification and process capability index Cpk reached over 1.2.

On the choice of a capability index with an application in carbon fibre manufacturing

Process capability indices Cp, Cpk, Cpm and Cpmk have been proposed to the manufacture industry to provide measures of process performance. The family C″p(u, v) depending on two parameters allows to generalise the previous indices for any position of the target in the tolerance interval. Grau (2010) studied the links between the value of the C″p(u, v) index and the maximum deviation of the process mean, as well as the maximum percentage of non-conforming items manufactured under the assumption of normality. From these results, we provide tables which allow the practitioner to choose a pair (u, v), so that the resulting index C″p(u, v) will meet his objectives best. We develop a decision making rule based on the natural estimator which can be used to test whether the process is capable or not. We then present a real study made on a carbon fibre manufacturing process to illustrate how this reasoning can be applied.

Robust design of railway vehicle suspension using a process capability index

Robust design for the primary suspension of a railway vehicle was performed according to the optimization of 10 dynamic responses representing driving safety and ride comfort, in which response surface models (RSMs) from the design of experiments (DOEs) were applied. To evaluate the probabilistic feasibility of robustness, an intensive computational process is mandatory. In the present study, the authors utilized the first-order Taylor series expansion to reduce the computational burden associated with a probabilistic feasibility evaluation, thus easily obtaining both an individual mean and variance of constraints. To overcome the difficulty of optimizing the mean and probabilistic variances for the 10 dynamic responses, a process capability index (Cpk) was introduced, which shows the mean value and scattering of the product quality to a certain extent and normalizes the objective functions irrespective of varying dimensions. Consequently, the robust design to optimize the 10 dynamic responses minimized the Cpk subjected to the constraint of Cpk ≥ 2, which satisfied 6σ. The proposed method improved the Cpk that violated the constraints obtained by the RSMs from DOEs and minimized the variance of the Cpk.

A Low Volume Production Process Capability Analysis

Statistical Process Control (SPC) and Process Capability Indices (PCI) are used extensively within manufacturing and service environments. This paper reports the results of a statistical quality improvement programme when applied to a low volume, high gauge frequency machining process. The resultant capability indices (Cp and Cpk) are calculated from acquired data both before and after process modification, and are discussed in relation to the specific process. Box Cox data transformations are utilised in order to centralise data and the applicability of the capability improvements are determined. Historically SPC and capability analysis are performed on high volume processes with data being gathered on a sample basis. This investigation applies the high volume theory to a low volume 100% data gathering process. The validity of capability analysis of this nature is discussed due to long cycle times and large gauge inspection frequency.

Developing a Sampling Plan Based on Cpk

ABSTRACT The common sampling plans for variables rely solely on process performance and do not take into account the engineering specifications while determining the sample size needed. When the process capability index Cpk is good, the limits of the distribution of the lot (the γ/2th percentile and the 1−γ/2th percentile) lie inside the engineering specifications. In this case the area of the tail of the normal distribution that lies outside the engineering specifications is very small, and hence the effect of the sampling error (which is the result of estimating the mean) on this area is negligible. However, when the limits of the distribution lie close to the engineering specifications (that is, when the Cpk is marginal), the area of the tail that lies outside the engineering specifications can be meaningful and therefore the sampling error may result in inaccuracies in estimating the number of defective items in the lot. The objective of this research is to develop a multistage sampling plan, based on Cpk and adjusted to acceptance sampling for variables, taken as a random sample from a lot of size N (given) having (approximately) a normal distribution with a known variance. The sampling plan was compared to MIL-STD-414 (1957) and it was found (via simulations) that the Cpk sampling plan has a smaller probability to reject good lots (0–0.03% vs. 0–0.27% for the MIL-STD-414) and a smaller probability to accept defective lots (0–0.01% vs. 5.2–8.37% for the MIL-STD-414). In addition, the required sample size is smaller for processes with high capability index.

Reducing defect generations in e-learning business: a Six Sigma approach

The process of learning, with significant technological development, has stepped forward towards e-learning not only for educational institutions but also for government and corporate businesses. The respective product development process of e-learning business often faces challenges in defect-free phases of the project life cycle across various types of projects. Higher the defect rate, higher is the reworking cost and lower is the productivity and customer dissatisfaction. This study attempts a disciplined, data-driven analytical Six-Sigma approach under DMAIC philosophy to reducing variation in one of such e-learning business processes, identifying controllable parameters for defect generation and taking necessary preventive actions. After taking actions on the critical activities, number of defects being generated and accordingly the amount of rework and efforts (man-months) consumed have been reduced satisfactorily. The statistical results show an improvement in the Pearson process capability index (Cpk) from 0.11 to 1.53.

A sensitivity study on the bootstrap confidence interval of the capability index Cpk

The capability index is a simple and quantitative way to describe process capability. The index Cpk is an indicator, involving specifications, process mean and standard deviation, for evaluating the capability of a process. When Cpk is applied to evaluate a process, estimating the confidence interval of Cpk is important for statistical inference on the process. Calculating the confidence interval for a process index usually needs the assumption about the underlying distribution. Bootstrapping is a nonparametric, but computer intensive, estimation method. Several types of bootstrap confidence intervals have been developed over the years. In the present paper we report the results of a simulation study on the behavior of the 95% biased-corrected and accelerated (BCa) percentile bootstrap confidence interval for estimating Cpk when the data are correlated and follow a Burr distribution, which is used to represent various probability distributions. Effects of sample size, process capability, non-normality and correlation on the BCa bootstrap interval of Cpk are simultaneously studied. A detailed discussion of the simulation results is presented and some conclusions are provided.

On the bootstrap confidence intervals of the capability index Cpk for multiple process streams

PurposeThe present paper aims to present the results of a simulation study on the behavior of the four 95 percent bootstrap confidence intervals for estimating Cpk when collected data are from a multiple streams process.Design/methodology/approachA computer simulation study is developed to present the behavior of four 95 percent bootstrap confidence intervals, i.e. standard bootstrap (SB), percentile bootstrap (PB), biased‐corrected percentile bootstrap (BCPB), and biased‐corrected and accelerated (BCa) bootstrap for estimating the capability index Cpk of a multiple streams process. An analysis of variance using two factorial and three‐stage nested designs is applied for experimental planning and data analysis.FindingsFor multiple process streams, the relationship between the true value of Cpk and the required sample size for effective experiment is presented. Based on the simulation study, the two‐stream process always gives a higher coverage percentage of bootstrap confidence interval than the four‐stream process. Meanwhile, BCPB and BCa intervals lead to better coverage percentage than SB and PB intervals.Practical implicationsSince a large number of process streams decreases the coverage percentage of the bootstrap confidence interval, it may be inappropriate to use the bootstrap method for constructing the confidence interval of a process capability index as the number of process streams is large.Originality/valueThe present paper is the first work to explore the behavior of bootstrap confidence intervals for estimating the capability index Cpk of a multiple streams process. It is concluded that the number of process streams definitively affects the performance of bootstrap methods.

Measuring process capability index Cpk with fuzzy data

Process capability index Cpk is important in manufacturing industry. This paper extends its applications to calculate the process capability index [Ctilde]pk of fuzzy numbers. Unlike previous researches, the α -cuts of fuzzy observations are first derived based on various values of α . The membership function of fuzzy process capability index [Ctilde]pk is then constructed based on the α -cuts of fuzzy observations. Two examples are used to illustrate how to interpret the fuzzy process capability index [Ctilde]pk . When the quality characteristic can not be precisely determined, the proposed method not only provides the most possible value and spread of fuzzy process capability index [Ctilde]pk , but also can be easily applied to the fuzzy number with different types of membership functions. With crisp data, the proposed method reduces to the classical method of process capability index Cpk .

Capability Index for Relational Functions: Inferential Procedures

ABSTRACT Capability indices such as Cp and Cpk are used to evaluate the ability of a process to produce according to specified requirements. When a variable of interest Y is not an observable one, but it is a function of a set of independent input variables, inferential procedures cannot be carried out. In this article, methods to obtain approximated confidence intervals for the variance () of the output Y are used to make inference on capability index Cp and Cpk of the variable of interest Y. Numerical examples illustrate the proposed methods, and simulation studies were conducted to evaluate the coverage proportions of the proposed methods.

Testing process performance based on capability index Cpk with critical values

Process capability index Cpk has been widely used in the manufacturing industry as a process performance measure. In this paper, we investigate the natural estimator of the index Cpk, and show that under the assumption of normality its distribution can be expressed as a mixture of the chi-square and the normal distributions. We also implement the theory of hypothesis testing using the natural estimator of Cpk, and provide efficient Maple programs to calculate the p-values as well as the critical values for various values of @a-risk, capability requirements, and sample sizes. The behavior of the p-values and critical values as functions of the distribution parameters are investigated to obtain tight critical values for reliable testing. Based on the test, we develop a simple and practical procedure for in-plant applications. The practitioners can use the proposed procedure to determine whether their process meets the preset capability requirement, and make reliable decisions.

The C?pk index for asymmetric tolerances: Implications and inference

The process capability index Cpk has been widely used in the manufacturing industry to provide numerical measures on process performance. Since Cpk is a yield-based index which is independent of the target T, it fails to account for process centering with symmetric tolerances, and presents an even greater problem with asymmetric tolerances. Pearn and Chen (1998) considered a new generalization C”pk which was shown to be superior to other existing generalizations of Cpk for processes with asymmetric tolerances. In this paper, we investigate the relation between the fraction nonconforming and the value of C”pk. Furthermore, we derive explicit forms of the cumulative distribution function and the probability density function for the natural estimator ”pk, under the assumption of normality. We also develop a decision making rule based on the natural estimator ”pk, which can be used to test whether the process is capable or not.

Using normal approximation for calculating the p-value in assessing process capability index Cpk

When a process is in a perfect state of statistical control, the process capability indices are widely used to measure the capability of the process to manufacture items within the required tolerance. Most evaluations on process capability indices focus on point estimates, which may result in unreliable assessments of process performance. The index Cpk has been used in various manufacturing industries to provide a quantitative measurement of the performance of the manufacturing. In this paper, based on the theory of statistical hypothesis testing, we propose a procedure on assessing Cpk index for practitioners to use in determining whether a given process as capable. Applying Hamaker’s approximation, the testing procedure can be adequately derived from a normal distribution approximation, which avoids the more complicated non-central t-distribution. The provided approach is easy to use and the decision making is more reliable.

Manufacturing capability information for multiple product quality characteristics: a case study to precision voltage reference

Chen et al. [5] proposed a process capability analysis chart (PCAC) based on process capability indices Cpk, Cpl, and Cpu to evaluate the integrated process capability for the entire product with the nominal-the-best, the larger-the-better, and the smaller-the better characteristics. The PCAC conveys critical information of multiple characteristics of the entire product regarding accuracy, precision, and the process capability zones from one single chart, which is an effective tool for evaluating product quality/reliability. However, from a practical perspective, the PCAC chart is never considered sampling errors hence the capability information provided from this chart is often unreliable and misleading. In this paper, a manufacturing application involving multiple characteristics of precision voltage reference (PVR) is investigated. We consider the sample error by obtaining the lower confidence bounds (LCBs) of Cpk, Ca, Cpl, and Cpu. The lower confidence bound presents the minimum true capability information of the process, which is essential to product reliability assurance. We employ the lower confidence bounds to PCAC to provide more reasonably reliable capability information for the product with multiple characteristics.

Uniformly Most Powerful Test for Process Capability Index Cpk

Process capability indices are important measures in statistical science and index Cpk is the most commonly used index in managerial decisions. In statistical inferences for Cpk, testing problems are relatively less explored than point and interval estimating problems. Tests for Cpk are often based on confidence intervals obtained from central limit theorems. Unfortunately, most tests are lack of theoretical justifications. In this article, we first propose a test for Cpk that is asymptotically equivalent to the likelihood ratio test. The above results then lead to an asymptotically UMP test for Cpk. In practice, UMP test provides a wise way to reduce the risk of making wrong decisions and save resources.