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.     

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.     

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.     

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.     

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 .     

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.     

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.     

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     

Estimating and Testing Process Precision with Presence of Gauge Measurement Errors

Process capability indices have been widely used in the manufacturing industry. Those capability indices, quantifying process potential and performance, are important for any successful quality improvement activities and quality program implementation. Because of the simplicity and easy of understanding, the precision index C_p has gained its popularity for measuring process consistency. However, the quality of data on the process characteristics relies very much on the gauge measurement. Conclusions about capability of the process just only based on the single numerical value of the index are not reliable. In this paper, we not only conduct the performance of the index C_p with gauge measurement errors, but also present adjusted confidence interval bounds and critical values for capability testing purpose of C_p with unavoidable measurement errors. Our research would help practitioners to determine whether the factory processes meet the capability requirement, and make more reliable decisions.     

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.     

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.     

On the properties of the hotelling��s T2 control chart with variable sampling intervals

When T ² control chart is used to monitor a process, it is usually assumed that the samples of size n ₀ is taken at constant intervals t ₀ . In this paper, we investigate the T ² control chart for monitoring the process mean vector when the sampling intervals are variable. Recent studies have shown that the variable sampling interval (VSI) scheme helps practitioners detect process shifts more quickly than the classical scheme Fixed Ratio Sampling (FRS). In this paper, it is assumed that the length of time the process remains in control is exponentially distributed.     

Monitoring process mean level using auxiliary information

In this study, a Shewhart‐type control chart is proposed for the improved monitoring of process mean level (targeting both moderate and large shifts which is the major concern of Shewhart‐type control charts) of a quality characteristic of interest Y. The proposed control chart, namely the M_r chart, is based on the regression estimator of mean using a single auxiliary variable X. Assuming bivariate normality of (Y, X), the design structure of M_r chart is developed for phase I quality control. The comparison of the proposed chart is made with some existing control charts used for the same purpose. Using power curves as a performance measure, better performance of the proposedM_r chart is observed for detecting the shifts in mean level of the characteristic of interest.     

Response of double-auction markets to instantaneous Selling�CBuying signals with stochastic Bid�CAsk spread

Statistical properties of order-driven double-auction markets with Bid–Ask spread are investigated through the dynamical quantities such as response function. We first attempt to utilize the so-called Madhavan–Richardson–Roomans model (MRR for short) to simulate the stochastic process of the price-change in empirical data sets (say, EUR/JPY or USD/JPY exchange rates) in which the Bid–Ask spread fluctuates in time. We find that the MRR theory apparently fails to simulate so much as the qualitative behaviour (‘non-monotonic’ behaviour) of the response function R(l) (l denotes the difference of times at which the response function is evaluated) calculated from the data. Especially, we confirm that the stochastic nature of the Bid–Ask spread causes apparent deviations from a linear relationship between the R(l) and the auto-correlation function C(l), namely, R(l) μ -C(l){R(l) \propto -C(l)}. To make the microscopic model of double-auction markets having stochastic Bid–Ask spread, we use the minority game with a finite market history length and find numerically that appropriate extension of the game shows quite similar behaviour of the response function to the empirical evidence. We also reveal that the minority game modeling with the adaptive (‘annealed’) look-up table reproduces the non-linear relationship R(l) μ -f(C(l)){R(l) \propto -f(C(l))} (f(x) stands for a non-linear function leading to ‘λ-shapes’) more effectively than the fixed (‘quenched’) look-up table does.     

Hotelling’s <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> control chart with two adaptive sample sizes

Some quality control schemes have been developed when several related quality characteristics are to be monitored. The familiar multivariate process monitoring and control procedure is the Hotelling’s T ² control chart for monitoring the mean vector of the process. It is a direct analog of the univariate shewhart [`(x)]{\bar{x}} chart. As in the case of univariate, the ARL improvements are very important particularly for small process shifts. In this paper, we study the T ² control chart with two-state adaptive sample size, when the shift in the process mean does not occur at the beginning but at some random time in the future. Further, the occurrence time of the shift is assumed to be exponentially distributed random variable.     

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.     

On the monitoring of multi-attributes high-quality production processes

Over the last decade, there have been an increasing interest in the techniques of process monitoring of high-quality processes. Based upon the cumulative counts of conforming (CCC) items, Geometric distribution is particularly useful in these cases. Nonetheless, in some processes the number of one or more types of defects on a nonconforming observation is also of great importance and must be monitored simultaneously. However, there usually exist some correlations between these two measures, which obligate the use of multi-attribute process monitoring. In the literature, by assuming independence between the two measures and for the cases in which there is only one type of defect in nonconforming items, the generalized Poisson distribution is proposed to model such a problem and the simultaneous use of two separate control charts (CCC & C chats) is recommended. In this paper, we propose a new methodology to monitor multi-attribute high-quality processes in which not only there exist more than one type of defects on the observed nonconforming item but also there is a dependence structure between the two measures. To do this, first we transform multi-attribute data in a way that their marginal probability distributions have almost zero skewnesses. Then, we estimate the transformed mean vector and covariance matrix and apply the well-known χ² control chart. In order to illustrate the proposed method and evaluate its performance, we use two numerical examples by simulation and compare the results. The results of the simulation studies are encouraging.     

A Bayesian-like estimator of the process capability index Cpmk

Pearn et al. (1992) proposed the capability index C_pmk, and investigated the statistical properties of its natural estimator for stable normal processes with constant mean μ. Chen and Hsu (1995) showed that under general conditions the asymptotic distribution of is normal if μ≠m, and is a linear combination of the normal and the folded-normal distributions if μ=m, where m is the mid-point between the upper and the lower specification limits. In this paper, we consider a new estimator for stable processes under a different (more realistic) condition on process mean, namely, P (μ≥m)=p, 0≤p≤1. We obtain the exact distribution, the expected value, and the variance of under normality assumption. We show that for P (μ≥m)=0, or 1, the new estimator is the MLE of C_pmk, which is asymptotically efficient. In addition, we show that under general conditions is consistent and is asymptotically unbiased. We also show that the asymptotic distribution of is a mixture of two normal distributions. RID="*" ID="*"  The research was partially supported by National Science Council of the Republic of China (NSC-89-2213-E-346-003).     

On the advantages of economically designed variable sample sizes and sampling intervals T2 control chart: double warning lines scheme

Faraz and Parsian (Stat Pap 47:569–593, 2006) have shown that the double warning lines (DWL) scheme detects process shifts more quickly than the other variable ratio sampling schemes such as variable sample sizes (VSS), variable sampling intervals (VSI) and variable sample sizes and sampling intervals (VSSVSI). In this paper, the DWL T² control chart for monitoring the process mean vector is economically designed. The cost model proposed by Costa and Rahim (J Appl Stat 28:875–885, 2001) is used here and is minimized through a genetic algorithm (GA) approach. Then the effects of the model parameters on the chart parameters and resulting operating loss is studied and finally a comparison between all possible variable ratio sampling (VRS) schemes are made to choose the best option economically.