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.     

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.     

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.     

过程能力指数4个基本特性

根据过程能力指数Cp定义发现了过程能力指数的4个基本特性,即：过程能力指数的对半特性、可计量特性、零判据和基准。这4个特性为研究各种情况下过程能力指数公式提供了理论依据。     

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.     

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     

On a study of the exponential and Poisson characteristics of the Poisson process

Based on the exponential and Poisson characteristics of the Poisson process, in this work we present some characterizations of the Poisson process as a renewal process. More precisely, let γ_t be the residual life at time t of the renewal process A={A(t),t≥0 }, under suitable condition, we prove that if Var(γ_t)=E ² (γ_t),∀t≥0, then A is a Poisson process. Secondly, we show that if Var (A(t)) is proportional to E (A(t)), then A is a Poisson process also, and Var (A(t))=E (A(t)). Received: August 1999     

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.     

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.     

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。     

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.     

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 .     

Assessment of Interjudge Reliability in the Open-Ended Questions Coding Process

In the process of coding open-ended questions, the evaluation of interjudge reliability is a critical issue. In this paper, using real data, the behavior of three coefficients of reliability among coders, Cohen’s K, Krippendorff’s α and Perreault and Leigh’s I _r are patterned, in terms of the number of judges involved and the categories of answer defined. The outcome underlines the importance of both variables in the valuations of interjudge reliability, as well as the higher adequacy of Perreault and Leigh’s I _r and Krippendorff’s α for marketing and opinion research.     

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.     

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.     

A method to obtain new copulas from a given one

Given a strictly increasing continuous function φ from [0, 1] to [0, 1] and its pseudo-inverse φ^[−1], conditions that φ must satisfy for C_φ(x₁, . . . ,x_n)=φ^[−1](C(φ(x₁), . . . ,φ(x_n))) to be a copula for any copula C are studied. Some basic properties of the copulas obtained in this way are analyzed and several examples of generator functions φ that can be used to construct copulas C_φ are presented. In this manner, a method to obtain from a given copula C a variety of new copulas is provided. This method generalizes that used to construct Archimedean copulas in which the original copula C is the product copula, and it is related with mixtures     

Applying jump-diffusion processes to liquidate and convert venture capital

This study attempts to apply real options and expand the model designed by Lin and Huang [Lin, T.T., Huang, Y.T.: J. Technol. Manage. 8(3), 59–78 (2003)], which helps venture capital (VC) companies to optimize project exit decisions. The expected discounted factor and a jump-diffusion process combine to assess the value of a start-up company, and determine the threshold of the exit timing of liquidation or convertibility for establishing the optimal disinvestment evaluation model for VC companies. When the project value is below V_L^*V_L^\ast, the VC company carries out liquidation, but when the project value exceeds V_C^*V_C^\ast, the VC company performs convertibility. The project value is ranging between (V_L ^*,V_C^*)\left({V_L ^\ast,V_C^\ast}\right), and the best choice is holding the decision and waiting to carry out the rights of liquidation and convertibility next time. Besides, this work attempts to identify the expected discounted time in terms of the investment time for VC companies.     

An outlier test for linear processes — II. Large contamination

In Flak/Schmid (1993) an outlier test for linear processes was introduced. The test statistic bases on a comparison of each observation with a one-step predictor. It was assumed that an upper bound for the total number of outlierss _n is known, wheren denotes the sample size. The asymptotic distribution of the test statistic was derived under the assumption thats _n/n → 0 ands _n → ∞ asn → ∞. This note deals with the asymptotic behaviour of this quantity, ifs _n/n →p ₀ ∈ (0, 1).     

Nonparametric regression function estimation using interaction least squares splines and comlexity regularization

Let (X, Y) be a pair of random variables withsupp(X)⊆[0,1] ^l andEY ²<∞. Letm ^* be the best approximation of the regression function of (X, Y) by sums of functions of at mostd variables (1≤d≤l). Estimation ofm ^* from i.i.d. data is considered. For the estimation interaction least squares splines, which are defined as sums of polynomial tensor product splines of at mostd variables, are used. The knot sequences of the tensor product splines are chosen equidistant. Complexity regularization is used to choose the number of the knots and the degree of the splines automatically using only the given data. Without any additional condition on the distribution of (X, Y) the weak and strongL ₂-consistency of the estimate is shown. Furthermore, for everyp≥1 and every distribution of (X, Y) withsupp(X)⊆[0,1] ^l ,y bounded andm ^* p-smooth, the integrated squared error of the estimate achieves up to a logarithmic factor the (optimal) rate