Process capability assessment for index C pk based on bayesian approach

Process capability indices have been proposed in the manufacturing industry to provide numerical measures on process reproduction capability, which are effective tools for quality assurance and guidance for process improvement. In process capability analysis, the usual practice for testing capability indices from sample data are based on traditional distribution frequency approach. Bayesian statistical techniques are an alternative to the frequency approach. Shiau, Chiang and Hung (1999) applied Bayesian method to index C_pm and the index C_pk but under the restriction that the process mean μ equals to the midpoint of the two specification limits, m. We note that this restriction is a rather impractical assumption for most factory applications, since in this case C_pk will reduce to C_p. In this paper, we consider testing the most popular capability index C_pk for general situation – no restriction on the process mean based on Bayesian approach. The results obtained are more general and practical for real applications. We derive the posterior probability, p, for which the process under investigation is capable and propose accordingly a Bayesian procedure for capability testing. To make this Bayesian procedure practical for in-plant applications, we tabulate the minimum values of Ĉ_pk for which the posterior probability p reaches desirable confidence levels with various pre-specified capability levels.     

Control Charts for One-sided Capability Indices

There are many industrial product characteristics are desired to be the bigger the best and the smaller the best. The two well-know processes capability indices C _pl and C _pu, which measure larger-the-better and smaller-the-better process capabilities. Obviously, the formulae for the two indices C _pl and C _pu are easy to understand and straightforward to apply. Thus, indices C _pl and C _pu have been utilized by a number of Japanese companies and the U.S. automotive industry by Ford Motor Company. Boyles (1991, Journal of Quality Technology. 23: 17–26) and Spring (1995, Total Quality Management 6(3): 427–438.) point out that as soon as and S control charts are in statistical control, the control charts of process capability indices can be used to monitor the quality of process. In the previous, we know that if the process is not in control, the process capability index control chart can be used to monitor the differences of process capability, and as soon as the process is in control the stable process capability can be identified. Therefore, process capability index control chart not only can be used to monitor the stability of process’s quality but also can be used to monitor the quality of process. Since Boyles (1991, Journal of Quality Technology 23: 17–26.) and Spiring (1995, Total Quality Management 6(1): 21–33.) had had research about control chart of the bilateral specification index C _pm., but there are many kinds of products, which meet unilateral quality specification. Therefore, we will construct the control chart of unilateral specification index C _pl and C _pu to monitor and evaluate the stability of process and process capability.     

Generalized confidence intervals for the process capability index C pm

Process capability indices have been proposed to the manufacturing industry for measuring process reproduction capability. The C _pm index takes into account the degree of process targeting (centering), which essentially measures process performance based on average process loss. To properly and accurately estimate the capability index, numerous conventional approaches have been proposed to obtain lower limits of the classical confidence intervals (CLCLs) for providing process capability information. In particular, lower confidence limits (LCLs) not only provide critical information regarding process performance but are used to determine if an improvement was made in reducing the nonconforming percent and the process expected loss. However, the conventional approach lacks for exact confidence intervals for C _pm involving unknown parameters which is a notable shortcoming. To remedy this, the method of generalized confidence intervals (GCIs) is proposed as an extension of classical confidence intervals (CCIs). For evaluating practical applications, two lower limits of generalized confidence intervals (GLCLs) for C _pm using generalized pivotal quantities (GPQs) are considered, (i) to assess the minimum performance of one manufacturing process/one supplier, and (ii) to assess the smallest performance of several manufacturing processes/several suppliers for equal as well as unequal process variances.     

Implementing lifetime performance index for the pareto lifetime businesses of the service industries

Process capability analysis is an effective means of measuring process performance and potential capability. In the service industries, process capability indices (PCIs) are utilized to assess whether business quality meets the required level. Hence, the performance index C _L is used as a means of measuring business performance, where L is the lower specification limit. In the technology of data transformation, this study constructs a uniformly minimum variance unbiased estimator (UMVUE) of C _L based on the right type II censored sample from the pareto distribution. The UMVUE of C _L is then utilized to develop a novel hypothesis testing procedure in the condition of known L. Finally, we give one practical example and the Monte Carlo simulation to assess the behavior of this test statistic for testing null hypothesis under given significance level. Moreover, the managers can then employ the new testing procedure to determine whether the business performance adheres to the required level.     

Yield measure for the process with multiple streams

Process capability indices, such as C _pk , have been widely used in the manufacturing industry to provide common quantitative measures for process performance. The index C _pk only provides an approximate rather than an exact measure of the process yield. To obtain an exact measure of the process yield, Boyles proposed a yield index S _pk . Capability measures for processes with single stream have been investigated extensively; however, multiple streams processes often occur in practice. Bothe presented a capability index for multiple streams process. In the present paper, a new index that is able to provide an exact measure of yield for a multiple streams process is developed. Three examples are given for illustration. From the results of the yield measure in the three examples, the conventional approach, using the arithmetic average of the estimated yield indices of all streams, will certainly over-estimate the process yield.     

Graphical Analysis of Capability of a Process Producing a Product Family

Statistical techniques are effective and powerful means of quantifying the variability of processes, analyzing this variability with reference to product requirements, and eliminating this variability in product manufacturing. Many process capability indices have been effectively and widely used to determine whether the quality of a process meets preset targets. However, conventional process capability indices cannot be applied to assess the entire process capability of a product family with nominal-the-best specifications. This work presents a novel process capability index (C_pp^T), which takes into account all family members. The index C_pp is a simple transformation from index C_pm, and C_pp^T provides additional, individual information concerning the accuracy and precision of a process. Vännman’s (δ, γ)-plot [Vännman and Deleryd, Quality and Reliability Engineering International 15(3): 213–217 (1999)] is revised to compare the process capabilities of family members under both 100% inspection and sampling plans. Examples are provided to demonstrate the method’s practical application.     

Measuring process capability index <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">pm</Emphasis></Subscript> with fuzzy data

The process capability index C _pm , which considers the process variance and departure of the process mean from the target value, is important in the manufacturing industry to measure process potential and performance. This paper extends its applications to calculate the process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } of fuzzy numbers. In this paper, the α-cuts of fuzzy observations are first derived based on various values of α. The membership function of fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } is then constructed based on the α-cuts of fuzzy observations. An example is presented to demonstrate how the fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} } is interpreted. When the quality characteristic cannot be precisely determined, the proposed method provides the most possible value and spread of fuzzy process capability index [(C)\tilde]_pm{\tilde {C}_{pm} }. With crisp data, the proposed method reduces to the classical method of process capability index C _pm .     

The communion bridge to Six Sigma and process capability indices

Six Sigma has already become an efficient improvement technique adopted by a great number of enterprises. Numbers of Sigma has become a tool of measuring process capability in some enterprises. But some of enterprises still use process capability indices (PCIs) to measure the process capability. So numbers of Sigma and PCIs both can be used to measure the process capability. The paper will research the relationship between PCIs and numbers of Sigma. In bilateral specifications, the paper will research the relationship between the PCIs which are C_p, C_pk, C_pm and C_pmk, S_pk and numbers of Sigma. In unilateral specifications, the paper will research the relationship between the PCIs which are C_pu and C_pl and numbers of Sigma. If supplier and buyer use different tools to measure the process capability, then the communion bridge to Six Sigma and PCIs can decrease the communicate noise.     

Distributional and Inferential Properties of the Estimated Precision Cp Based on Multiple Samples

Process precision index C_p has been widely used in the manufacturing industry to provide numerical measures on process potential. Pearn et al. (1998) considered an unbiased estimator of C_p for one single sample. They showed that the unbiased estimator is the UMVUE. They also proposed an efficient test for C_p based on one single sample, and showed that the test is the UMP test. In this paper, we consider an unbiased estimator of C_p for multiple samples. We show that the unbiased estimator is the UMVUE of C_p, which is asymptotically efficient. We also consider an efficient test for C_p, and show that the test is the UMPtest for multiple samples. The practitioners can use the proposed test on theirin-plant applications to obtain reliable decisions.     

Criteria of Determining the P/T Upper Limits of GR&;R in MSA

The MSA combines estimates of the variations of repeatability and reproducibility and is mainly analyzed by experimental design. The variations of personnel, measurement equipment, and the part itself can be analyzed via the data obtained to improve the capability of the measurement system. MSA of both QS9000 and ISO/TS16949 defines GR&R acceptable criteria. GR&R of the MSA in QS9000 is determined by the Precision-to-Tolerance (P/T) value, the percentage of the measurement system variations to the deviation during the manufacturing process or to the part tolerance. If the P/T value is less than 10%, the accuracy of the measurement system is acceptable. If the P/T value falls between 10 and 30%, acceptance of the accuracy of the measurement system is up to the company. When the P/T value is greater than 30%, precision of the measurement system will not be accepted. The aforementioned GR&R acceptance criteria were established by three major automobile companies of the US according to their past experiences. As the capability index C _pm reflects both process yield and process loss, we use C _pm to set a proper range of GR&R acceptable criteria. If the P/T value is not within the acceptable range, the measurement system is required for modification. If the P/T value is within the acceptable range, The process capability can be enhanced by improving the manufacturing process.     

Estimating and testing process accuracy with extension to asymmetric tolerances

Pearn et al. (Commun. Stat. Theory Methods, 27(4):985–1000, 1998) introduced the process accuracy index C _a to measure the degree of process centering, the ability to cluster around the center. In this paper, we derive an explicit form of the cumulative distribution function for the estimator [^(C)]_a{\hat{C}_a } with the case of symmetric tolerances. Subsequently, the distributional and inferential properties of the estimated process accuracy index C _a are provided. Calculations of the critical values, P-values, and lower confidence bounds are developed for testing process accuracy. Further, a generalization of C _a for the case with asymmetric tolerances is proposed to measure the process accuracy. Based on the results practitioners can easily perform the testing of the process accuracy, and make reliable decisions on whether actions should be taken to improve the process quality. An application is given to illustrate how we test the process accuracy using the actual data collected from the factory.     

The C”<Subscript>pk</Subscript> index for asymmetric tolerances: Implications and inference

The process capability index C_pk has been widely used in the manufacturing industry to provide numerical measures on process performance. Since C_pk 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 C_pk 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.     

Normal Approximation to the Distribution of the Estimated Yield Index Spk

Process yield is the most common criterion used in the manufacturing industry for measuring process performance. A measurement index, called S_pk, has been proposed to calculate the yield for normal processes. The measurement index _pk establishes the relationship between the manufacturing specifications and the actual process performance, which provides an exact measure on process yield. Unfortunately, the sampling distribution of the estimated _pk is mathematically intractable. Therefore, process performance testing cannot be performed. In this paper; we consider a normal approximation to the distribution of the estimated _pk, and investigate its accuracy computationally. We compare the critical values calculated from the approximate distribution with those obtained using the standard simulation technique, for various commonly used quality requirements. Extensive computational results are provided and analyzed. The investigation is useful to the practitioners for making decisions in testing process performance based on the yield.     

Process Capability Evaluation for the Process of Product Families

Process capability analysis is a highly effective means of assessing the process ability of manufacturing product that meets specifications. A larger process capability index (PCI) implies a higher process yield, and lower expected process loss. Many PCIs have been effectively and widely used to determine whether the quality of a process meets preset targets. However, those existing PCI cannot be applied to evaluate the process capability of a process producing a product family. This work presents a novel PCI (C _pm ^T ) which takes into account all family members and obtains the probability density function for the PCI for a product family. The relationship between the PCI and process yield is described. An example is provided to demonstrate the methodology for practical application.     

Estimating process capability index C′′pmk for asymmetric tolerances: Distributional properties

Pearn et al. (1999) considered a capability index C ^′′ _pmk, a new generalization of C _pmk, for processes with asymmetric tolerances. In this paper, we provide a comparison between C ^′′ _pmk and other existing generalizations of C _pmk on the accuracy of measuring process performance for processes with asymmetric tolerances. We show that the new generalization C ^′′ _pmk is superior to other existing generalizations of C _pmk. Under the assumption of normality, we derive explicit forms of the cumulative distribution function and the probability density function of the estimated index . We show that the cumulative distribution function and the probability density function of the estimated index can be expressed in terms of a mixture of the chi-square distribution and the normal distribution. The explicit forms of the cumulative distribution function and the probability density function considerably simplify the complexity for analyzing the statistical properties of the estimated index . Received April 2000     

Reliable inference for the Gini index

Although attention has been given to obtaining reliable standard errors for the plug-in estimator of the Gini index, all standard errors suggested until now are either complicated or quite unreliable. An approximation is derived for the estimator by which it is expressed as a sum of IID random variables. This approximation allows us to develop a reliable standard error that is simple to compute. A simple but effective bias correction is also derived. The quality of inference based on the approximation is checked in a number of simulation experiments, and is found to be very good unless the tail of the underlying distribution is heavy. Bootstrap methods are presented which alleviate this problem except in cases in which the variance is very large or fails to exist. Similar methods can be used to find reliable standard errors of other indices which are not simply linear functionals of the distribution function, such as Sen’s poverty index and its modification known as the Sen–Shorrocks–Thon index.     

Evaluating the Gauge Repeatability and Reproducibility for Different Industries

Measurement plays a significant role in Six sigma program. Usually, the gauge repeatability and reproducibility (GR&R) study needs to be conducted prior to the process capability analysis for verifying the accuracy of measuring equipments and helping organizations improve their product and service quality. Therefore, how to ensure the quality of measurement becomes an important task for quality practitioners. In performing the GR&R study, most industries are using the acceptance criteria of Precision to Tolerance(P/T) ratio as stipulated by QS9000. However, the adequacy of applying the same acceptance criteria to different manufacturing processes is very questionable. In this paper, a statistical method using the relationship between GR&R and process capability indices is proposed for evaluating the adequacy of the acceptance criteria of P/T ratio. Finally, a comparative analysis has also been performed for evaluating the accuracy of GR&R among three methods (ANOVA, Classical GR&R, and Long Form). Hopefully, the results of this research can provide a useful reference for quality practitioners in various industries.     

Unification of some multivariate process capability indices for asymmetric specification region

In manufacturing industries, it is often seen that the bilateral specification limits corresponding to a particular quality characteristic are not symmetric with respect to the stipulated target. A unified superstructure of univariate process capability indices was specially designed for processes with asymmetric specification limits. However, as in most of the practical situations a process consists of a number of inter‐related quality characteristics, subsequently, a multivariate analogue of , which is called C_M(u,v), was developed. In the present paper, we study some properties of C_M(u,v) like threshold value and compatibility with the asymmetry in loss function. We also discuss estimation procedures for plug‐in estimators of some of the member indices of C_M(u,v). Finally, the superstructure is applied to a numerical example to supplement the theory developed in this article.     

A new process capability index

In this paper a new process capability index (C _pq ) has been introduced, which is easy to compute and performs well when compared with its natural competitor (C _pm ). Work done while he was Lukacs Visiting Professor on leave from the University of Maryland.     

过程能力指数C_p、C_(pk)与过程性能指数P_p、P_(pk)

过程能力指数是一种广泛应用的产品质量评价工具。不同过程能力指数都有合理一面,但也都有不足之处,选用不当会产生误导。文章研究了过程能力指数C p、Cpk与过程性能指数Pp、Ppk的差别及原因,并分别给出其与不合格率的关系和相应的估计和置信区间。