Lower confidence bounds for C PU and C PL based on multiple samples with application to production yield assurance

For stably normal processes with one-sided specification limits, capability indices C _PU and C _PL have been used to provide numerical measures on production yield assurance. Statistical properties of the estimators of C _PU and C _PL have been investigated extensively for cases with a single sample. It is shown that for multiple samples, the uniformly minimum-variance unbiased estimators of C _PU and C _PL are consistent and asymptotically efficient. Based on the uniformly minimum-variance unbiased estimators, an algorithm is developed with an efficient program using a direct search method to compute the lower confidence bounds for C _PU and C _PL. The lower confidence bounds convey critical information to the minimum capability of a process, providing a necessary yield assurance of production. The lower confidence bounds are tabulated for some commonly used capability requirement so that engineers/practitioners can use them for their in-plant applications. An example of a high-speed buffer amplifier is presented to illustrate the practicality of the approach to data collected from the factories for production yield assurance.     

An alternative approach to test process capability for unilateral specification with subsamples

Process capability indices are useful management tools, which provide common quantitative measures on manufacturing capability and production quality. The indices C_PU and C_PL are designed specifically for processes with one-sided manufacturing specifications. The majority of the results obtained so far related to the distributional properties of the estimated capability indices were derived based on the assumption of possessing a single sample. However, a common practice in process control is to estimate the process capability indices by using the past ‘in control’ data from several subsamples. In order to use previous in-control data from multiple subsamples to make correct decisions regarding process capability, the distribution of the estimated capability index based on multiple subsamples should be available. In this paper, we develop a capability testing procedure with one-sided specifications using a Bayesian approach based on subsamples collected over time from an in-control process. By applying the proposed testing procedure, the practitioners can make reliable decisions to determine whether their processes meet the pre-set capability requirement when a daily based or weekly based production control plan is implemented for monitoring process stability.     

Comparing Two Process Capability Indices under Balanced One‐Way Random Effect Model

Process capability indices such as C_p, C_pk, C_pmk and C_pm are widely used in manufacturing industries to provide a quantitative measurement of the performance of the products. In this article, we derived generalized confidence intervals for the difference between process capability indices for two processes under one‐way random effect model. Our study provides coverage probability close to the nominal value in almost all cases as shown via simulation. An example from industrial contexts is given to illustrate the results. Copyright © 2012 John Wiley & Sons, Ltd.     

Monitoring manufacturing quality for multiple Li-BPIC processes based on capability index C pmk

The multi-process performance analysis chart (MPPAC) based on process capability indices has been developed to analyse the manufacturing performance for multiple processes, which conveys critical information regarding the departure of the process mean from the target value, process variability, capability levels, which provides a guideline of directions for capability improvement. Existing MPPAC researches have plotted the sample estimates of the process indices on the chart. Conclusions were then made on whether processes meet the capability requirement and directions need to be taken for further quality improvement. Such an approach is highly unreliable since the sample point estimate is a random variable with no assessment of the sampling errors. Further, existing MPPAC researches only considered one single sample. Current quality control practice is to estimate process capability using multiple groups of control chart samples rather than one single sample. In this paper, we propose the C _pmk MPPAC combining the accuracy index C _a to access the performance of multiple manufacturing processes. Distributions of the estimated C _pmk and C _a are derived based on multiple control chart samples, and accurate lower confidence bounds are calculated. The lower confidence bounds of the estimated C _pmk and C _a are then employed to the MPPAC to provide reliable capability grouping for those multiple processes. A real-world example is presented to illustrate the applicability of the proposed MPPAC.     

Classical and Bayesian inference of Cpy for generalized Lindley distributed quality characteristic

The process capability index (PCI) is a quality control–related statistic mostly used in the manufacturing industry, which is used to assess the capability of some monitored process. It is of great significance to quality control engineers as it quantifies the relation between the actual performance of the process and the preset specifications of the product. Most of the traditional PCIs performed well when process follows the normal behaviour. However, using these traditional indices to evaluate a non‐normally distributed process often leads to inaccurate results. In this article, we consider a new PCI, C_py, suggested by Maiti et al, which can be used for normal as well as non‐normal random variables. This article addresses the different methods of estimation of the PCI C_py from both frequentist and Bayesian view points of generalized Lindley distribution suggested by Nadarajah et al. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, least square and weighted least square estimators, and maximum product of spacings estimators. Next, we consider Bayes estimation under squared error loss function using gamma priors for both shape and scale parameters for the considered model. We use Tierney and Kadane's method as well as Markov Chain Monte Carlo procedure to compute approximate Bayes estimates. Besides, two parametric bootstrap confidence intervals using frequentist approaches are provided to compare with highest posterior density credible intervals. Furthermore, Monte Carlo simulation study has been carried out to compare the performances of the classical and the Bayes estimates of C_py in terms of mean squared errors along with the average width and coverage probabilities. Finally, two real data sets have been analysed for illustrative purposes.     

Capability Testing Based on Cpm with Multiple Samples

Numerous process capability indices have been proposed in the manufacturing industry to provide unitless measures on process performance, which are effective tools for quality improvement and assurance. Most existing methods for capability testing are based on the distribution frequency approaches. Recently, Bayesian approaches have been proposed for testing capability indices C_p and C_pm but restricted to cases with one single sample. In this paper, we consider estimating and testing capability index C_pm based on multiple samples. We propose accordingly a Bayesian procedure for testing C_pm. Based on the Bayesian procedure, we develop a simple but practical procedure for practitioners to use in determining whether their manufacturing processes are capable of reproducing products satisfying the preset capability requirement. A process is capable if all the points in the credible interval are greater than the pre‐specified capability level. To make the proposed Bayesian approach practical for in‐plant applications, we tabulate the minimum values of for which the posterior probability p reaches various desirable confidence levels. Copyright © 2004 John Wiley & Sons, Ltd.     

A Bayesian procedure for process capability assessment

The usual practice of judging process capability by evaluating point estimates of some process capability indices has a flaw that there is no assessment on the error distributions of these estimates. However, the distributions of these estimates are usually so complicated that it is very difficult to obtain good interval estimates. In this paper we adopt a Bayesian approach to obtain an interval estimation, particularly for the index C_pm. The posterior probability p that the process under investigation is capable is derived; then the credible interval, a Bayesian analogue of the classical confidence interval, can be obtained. We claim that the process is capable if all the points in the credible interval are greater than the pre‐specified capability level ω, say 1.33. To make this Bayesian procedure very easy for practitioners to implement on manufacturing floors, we tabulate the minimum values of Ĉ_pm/ω, for which the posterior probability p reaches the desirable level, say 95%. For the special cases where the process mean equals the target value for C_pm and equals the midpoint of the two specification limits for C_pk, the procedure is even simpler; only chi‐square tables are needed. Copyright © 1999 John Wiley & Sons, Ltd.     

Measuring PPM Non‐conformities for Processes with Asymmetric Tolerances

Process yield has been the most basic and common criterion used in the manufacturing industry for evaluating process capability. The C_pk index has been used widely in the manufacturing industry. In this note, we considered a generalization of C_pk index which handles processes involving a target T with asymmetric tolerances. Particularly, we established a formula for measuring the PPM non‐conformities for given ratios of the two‐side tolerances. We proved the validity of the established formula and tabulated the upper bounds on PPM non‐conformities for various given C_pk index values and ratios of the two‐side tolerances. Copyright © 2012 John Wiley & Sons, Ltd.     

Developing New Multivariate Process Capability Indices for Autocorrelated Data

Traditionally, the process capability index is developed assuming that the process output data are independent and follow normal distribution. However, in most environmental cases, the process data have more than one quality characteristic and exhibit property of autocorrelation. We propose two novel multivariate process capability indices for autocorrelated data, NMAC_p and NMAC_pm, for the nominal‐the‐best case. For the smaller‐the‐better case, Γ(0) is used to modify the ND index and a new multivariate autocorrelated process capability index NMAC_pu is derived. Furthermore, a simulation study is conducted to compare the performance of the various multivariate autocorrelated indices. The simulation results show that the actual nonconforming rates can be correctly reflected by our proposed indices, which outperform the previous C_p^m, MC_p, MC_pm, NMC_p, NMC_pm, and ND indices under different time series models. Thus, our proposed capability indices can be used in evaluating the performance of multivariate autocorrelated processes. Finally, a realistic example in hydrological application further demonstrates the usefulness of our proposed capability indices. Copyright © 2013 John Wiley & Sons, Ltd.     

Re-evaluating the process capability indices for non-normal distributions

Classical process capability indices (PCIs) C _p , C _pu , C _pl and C _pk can indicate the potential process capability accurately when the quality characteristic of the product is normally distributed. When the process has a non-normal distribution, classical PCIs will be inappropriate and can misled the assessment of process capability. Zwick (1995), Schneider and Pruett (1995-96) and Tong and Chen (1998) proposed various PCIs for non-normal distributions. This paper compares the accuracy of these indices for several selected non-normal distributions based on the proportion of non-conformity of manufactured product. The results indicate that these PCIs lead to a larger number of errors in various combinations of shape parameters and specification limits. Therefore, this paper proposes three indices, S _pu , S _pl and S _pk , which can reflect accurately the proportion of nonconformity in either normal or non-normal distributions.     

Estimation of Cpm when Measurement Error is Present

In this paper we study the properties of the estimator of C_pm when the observations are affected by measurement errors. We compare the performances of the estimator in the error case with those of the estimator in the error‐free case. The results indicate that the presence of measurement errors in the data leads to different behavior of the estimator according to the entity of the error variability. We finally show how to use our results in practice. Copyright © 2005 John Wiley & Sons, Ltd.     

Computing process capability indices for non‐normal data: a review and comparative study

When the distribution of a process characteristic is non‐normal, C_p and C_pk calculated using conventional methods often lead to erroneous interpretation of the process's capability. Though various methods have been proposed for computing surrogate process capability indices (PCIs) under non‐normality, there is a lack of literature that covers a comprehensive evaluation and comparison of these methods. In particular, under mild and severe departures from normality, do these surrogate PCIs adequately capture process capability, and which is the best method(s) in reflecting the true capability under each of these circumstances? In this paper we review seven methods that are chosen for performance comparison in their ability to handle non‐normality in PCIs. For illustration purposes the comparison is done through simulating Weibull and lognormal data, and the results are presented using box plots. Simulation results show that the performance of a method is dependent on its capability to capture the tail behaviour of the underlying distributions. Finally we give a practitioner's guide that suggests applicable methods for each defined range of skewness and kurtosis under mild and severe departures from normality. Copyright © 1999 John Wiley & Sons, Ltd.     

Optimal process capability analysis for process design

Conventional process capability analysis is used to measure and control the quality level of a production process in real exercises for on-line quality management. There has been a deficiency in this type of management; namely, the defects which occur in the production process are only passively detected and modified afterwards. Additionally, conventional process capability expression has difficulty distinguishing between alternatives for process selection among possible candidates before process realisation. There is, therefore, considerable motivation for developing a process capability expression which can be used to evaluate alternatives at the beginning of the process design, i.e., off-line application. The conventional C_pm expression is built up by measuring mean deviation and process variances for on-line application. However, if C_pm is used for the process capability analysis for process design, an erroneous C_pm value is found and an inappropriate process design is ended. Thus, the proposed process capability expression revised from the conventional C_pm in consideration of the balance between tolerance cost and quality loss has been developed. This development is the main contribution of this research and, with this development, the appropriate mean and tolerance values can be determined simultaneously prior to the real production process so as to maximise the proposed process capability value. The production is then processed with the pre-determined mean and tolerance values in a real production process. The expectation after process realisation is that the produced responses will be the best of all the alternatives in terms of quality and cost, and that the process capability value obtained after the real production process will be close to the proposed process capability value maximised prior to the real production process.     

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

Process capability index C_pk is the most popular capability index widely used in the manufacturing industry. Existing research on the yield‐based measure index C_pk 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 C_pk, 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.     

Modified processes capability assessment with dynamic mean shift

Rapid advancements in manufacturing technology have seen electronic products becoming lighter and thinner. This has led to production yield requirements of sawing processes in the semiconductor industry becoming increasingly strict. For sawing processes requiring a very low fraction of defective, the process capability index C_pk is widely applied for production yield measurement under the assumption of process stability. However, in practice, even when the typical control charts are implemented, dynamic and unobserved natures of the actual process mean shift may occur. Since the trial control limits are established with reference to random samples, the associated performances are also with uncertainty because of sampling error. In this paper, we investigate the effects of parameter estimation error on the development of the phase I control chart and further compare three methods with various estimators for the process standard deviation. Then, we tabulate the modified adjustment magnitudes for the dynamic C_pk under various subgroup sizes and sample sizes. On the basis of the simulation results, the proposed method can avoid overestimating the true process capability and make more trustworthy decisions. Finally, for demonstration purposes, an application of process capability measures for a sawing process is presented.     

Process capability analysis chart with the application of Cpm

Process capability analysis (PCA) is a highly effective means of assessing the process ability of product that meets specifications. The process capability analysis chart (PCAC/C_pk ) evaluates the capabilities of multiprocess products together with nominal-the-best specifications, larger-the-better and smaller-the-better specifications. This study proposes process capability analysis chart (PCAC/C_pm ) to consider process yield and expected process loss. A new generated estimator for C_pm is proposed and the properties of statistical estimator and hypothesis test are discussed. A practical example was given for application.     

The New Nino Capability Index for Dynamic Process Capability Analysis

The process capability analysis is a crucial activity to evaluate if the process outcome meets the design specifications. Classically, such analysis is performed by verifying the in‐control condition of the process and evaluating suitable capability indices, by assuming the process in‐control steady‐state condition. However, the in‐control period of the process characterizes only a part of the system functioning cycle, the one with the lower defective rate. In particular, the system functioning cycle is also characterized by the out‐of‐control period, during which a greater defective rate is produced, and such increasing is not considered by the widely adopted capability indices. As consequence, the classical approaches to perform the process capability analysis involves an overestimation of the process capability level. For this reason, in order to overcome the previously described limitation, in the present paper it is proposed a new capability index based on the real defective rate of the process. Thus, such new index is able to estimate the real process capability level. Finally, in order to compare the new index to the conventional C_p capability index, a numerical comparison study related to a process capability analysis is carried out, and the related practical considerations are given. Copyright © 2013 John Wiley & Sons, Ltd.     

Non‐normal Capability Indices for the Weibull and Lognormal Distributions

Because the normal process capability indices (PCIs) C_p, C_pu, C_pl, and C_pk represent the times that the process standard deviation is within the specification limits; then, based on and by using the direct relations among the parameters of the Weibull, Gumbel (minimum extreme value type I) and lognormal distributions, the Weibull and lognormal PCIs are derived in this paper. On the other hand, because the proposed PCIs P_p, P_pu, P_pl, and P_pk were derived as a function of the mean and standard deviation of the analyzed process, they have the same practical meaning with those of the normal distribution. Results show that the proposed PCIs could be used as the standard C_p, C_pu, C_pl, and C_pk if a short‐term variance is analyzed. An application to a set of simulated data is presented. Copyright © 2015 John Wiley & Sons, Ltd.     

On the bootstrap confidence intervals of the process incapability index Cpp

The process incapability index C_pp is an indicator, introduced by Greenwich and Jahr-Schaffrath, for evaluating the capability of a process. When C_pp is applied to evaluate a process, estimating the confidence interval of C_pp 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 non-parametric, but computer intensive, estimation method. In the present paper we report the results of a simulation study on the behavior of four 95% bootstrap confidence intervals (i.e. standard bootstrap, percentile bootstrap, biased-corrected percentile bootstrap, and biased-corrected and accelerated bootstrap) for estimating C_pp when data are from a specific Burr distribution, which is used to represent various probability distributions. A detailed discussion of the simulation results is presented and some conclusions are provided.     

A Bayesian Approach to Capability Testing Based on Cpk with Multiple Samples

Process capability indices provide numerical measures to compare the output of a process to client's expectations. However, most of the existing researches have used traditional distribution frequency method by using a single sample due to assess process capability. An alternative to this approach is to use the Bayesian method. In this paper, we utilize a Bayesian approach based on subsamples to check process capability via capability index C_pk. As a new suggestion, we used the informative normal prior distribution and the characteristics of sufficient statistic of the parameter to drive the posterior distribution. The capability test is done, and the posterior probability p, for which the process under investigation is capable, is derived both based on the most popular index C_pk. Finally, a numerical example is given to clarify the method. Copyright © 2013 John Wiley & Sons, Ltd.