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.     

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.     

Combining process capability indices from a sequence of independent samples

Process capability indices such as C_p, C_pk and C_pm have been widely used in manufacturing for process assessment and for evaluation of purchasing decisions. Quality engineers and process operators are often asked to summarize the quality of a process by combining a sequence of C_pk or C_pm. This paper compares three different methods for combining the estimates of the process capability indices C_pk, C_pm and C_pmk from a sequence of independent samples. The criterion of minimum mean-squared-error (MSE) is applied. We find that, in general, methods for the combination of sample process capability indices based on the overall sample mean and pooled sample variance give smaller MSE than those based on weighted averages of the estimates of process capability indices.     

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.     

Thermal diffusivity measurement by the transient hot-wire technique: A reappraisal

The theory of the transient hot-wire technique for thermal conductivity measurements is reassessed in the special context of thermal diffusivity measurements. A careful examination of the working equation and an error analysis are employed to identify the principal sources of error. Notwithstanding earlier claims to the contrary, the best precision that can be attained in thermal diffusivity measurements is of the order of ±3%, while the accuracy is inevitably poorer. Experimental evidence is adduced from two different instruments that supports the analysis given here. Although the technique cannot yield values of the thermal diffusivity, k, as accurate as can be achieved by the use of the best possible individual values of ,, and C _p in the relation k=/C _p, the simplicity of the technique makes it attractive for many purposes. It is even possible to derive values of the isobaric heat capacity C _p for many fluids not available from other methods.     

On the essential work of fracture of linear low-density-polyethylene. I. Precision of the testing method

The precision (i.e. the repeatability) of the essential work of fracture (EWF) method in determining the fracture parameters of a highly extendible linear low-density-polyethylene film is investigated. In order to minimize any interference from external variables, a random data collection procedure is adopted to extract, from a large data set, various EWF samples with sizes ranging from 11 to 150 data points. Two different notching procedures have been considered, involving different tools (scalpel or razor blade) and cutting methodologies.The notching procedure has only a marginal influence in terms of the correlation coefficient of the linear regression and standard error on the specific essential work of fracture (w_e). However, the mean of w_e values is markedly affected by the notching procedure, being its influence on the specific non-essential work of fracture (βw_p) parameter relatively lower. The dispersion of the w_e and βw_p data around their mean values decreases as the sample size increases, with a trend clearly affected by the notching procedure.     

A sample-saving method for heat capacity measurements on powders using relaxation calorimetry

An experimental method is described for determining the low-temperature heat capacity (C_p) of mg-sized powder samples using the Quantum Design “Physical Properties Measurement System” (PPMS). The powder is contained in an Al pan as an ∼1 mm thick compressed layer. The sample is not mixed with Apiezon N grease, as compared to other methods. Thus, it is not contaminated and can be used for further study. This is necessary for samples that are only available in tiny amounts. To demonstrate the method various samples, all insulating in nature, were studied including benzoic acid, sapphire and different silicate minerals. The measurements show that the method has an accuracy in C_p to better than 1% at T above 30–50 K and ±3–5% up to ±10% below. The experimental procedure is based on three independent PPMS and three independent differential scanning calorimetry (DSC) measurements. The DSC C_p data are used to slightly adjust the PPMS C_p data by a factor . This is done because heat capacities measured with a DSC device are more accurate around ambient T (?0.6%) than PPMS values and is possible because the deviation of PPMS heat capacities from reference values is nearly constant between about 50 K and 300 K. The resulting standard entropies agree with published reference values within 0.21% for the silicates, by 0.34% for corundum, and by 0.9% for powdered benzoic acid. The method thus allows entropy determinations on powders with an accuracy of better than 1%. The advantage of our method compared to other experimental techniques is that the sample powder is not contaminated with grease and that heat capacity values show less scatter at high temperatures.     

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.     

A graphical approach to obtaining confidence limits of Cpk

The process capability index C_pk has been widely used as a process performance measure. In practice this index is estimated using sample data. Hence it is of great interest to obtain confidence limits for the actual index given a sample estimate. In this paper we depict graphically the relationship between process potential index (C_p), process shift index (k) and percentage non-conforming (p). Based on the monotone properties of the relationship, we derive two-sided confidence limits for k and C_pk under two different scenarios. These two limits are combined using the Bonferroni inequality to generate a third type of confidence limit. The performance of these limits of C_pk in terms of their coverage probability and average width is evaluated by simulation. The most suitable type of confidence limit for each specific range of k is then determined. The usage of these confidence limits is illustrated via examples. Finally a performance comparison is done between the proposed confidence limits and three non-parametric bootstrap confidence limits. The results show that the proposed method consistently gives the smallest width and yet provides the intended coverage probability. © 1997 John Wiley & Sons, Ltd.     

Lower confidence bounds with sample size information for Cpm applied to production yield assurance

The process capability index C _pm , sometimes called the Taguchi index, has been proposed to the manufacturing industry as providing numerical measures on process performance. A lower confidence bound estimates the minimum process capability, conveying critical information regarding product quality, which is essential to quality assurance. The sample size determination is directly related to the cost of the data collection plan. The purpose of this paper is to provide explicit formulas with efficient algorithms to obtain the lower confidence bounds and sample sizes required for specified precision of the estimation on C _pm using the maximum likelihood estimator (MLE) of C _pm . We also provide tables for the engineers/practitioners to use for their in-plant applications. A real-world example taken from a microelectronics manufacturing process is investigated to illustrate the applicability of the proposed approach. The implementation of existing statistical theory for capability assessment bridges the gap between the theoretical development and factory applications.     

Sequential Optimum Procedures for Unbiased Estimation of a Binomial Parameter

Let x ₁, x ₂, … x _n, … be a sequence of independent random variables with a common density function P(x = 1) = p, P(x = 0) = 1 – p, 0 < p < 1. This paper considers the non-randomized sequential procedures δ's for estimating p and the following three problems on choice of δ. (i) Choose δ to minimize E _v (Z _δ – P)² subject to E _p N _δ ≤ m where m ≥ 1 and Z _δ is an unbiased estimate of p; (ii) Choose δ to minimize E_p N _δ subject to E _p (Z _δ – p)² ≤ α where α is a real positive number; (iii) choose δ to minimize C E_p N _δ + E _p (Z _δ – p)² where C is the cost of an observation. In each case the minimization is to be done uniformly in p if possible; otherwise the supremum over p of the risk in question is to be minimized. A procedure is constructed for problem (i) when m is not an integer. A fixed sample size procedure is shown to be admissible and minimax for problem (ii). A procedure is constructed which is asymptotically uniformly better than the fixed sample size for problem (ii). Furthermore, for problem (iii) some optimum procedures are constructed.     

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.     

Confidence intervals and sample size determination for Cpm

The capability index, C_pm, sometimes called the Taguchi index, has the desirable characteristic of being sensitive to both dispersion and deviation of the process average from the engineering target. As a result, the proposed estimators of C_pm have a sampling distribution that is dependent on the non‐central chi‐square distribution. Hence, constructing confidence intervals, performing hypothesis testing or estimating sample size requirements necessitates manipulation of a rather complex functional expression, typically beyond the capabilities of practitioners who need readily available tools. Here, a simple graphical procedure is proposed and illustrated for obtaining exact confidence intervals for C_pm. The graphical procedure allows the user to simply enter the graph with an estimate of the index and a value of the non‐centrality parameter for a given sample size to arrive at end‐points of 90%, 95% or 99% one‐sided or two‐sided confidence intervals. Detailed tables are also provided to assist the user for a wider range of sample values and sample sizes. In addition, a procedure is also presented for determining the minimum sample size required for attaining a pre‐specified level of accuracy of the C_pm. Extensive tables are provided for the user with a simple example illustrating the facility of the technique. Copyright © 2001 John Wiley & Sons, Ltd.     

Mechanical and photoelastic properties of oriented poly-4-methyl-1-pentene investigated by Brillouin spectroscopy

The elastic constants C ₁₁, C ₁₃, C ₃₃ and C ₄₄ of oriented poly-4-methyl-1-pentane (P4M1P) films were measured with Brillouin scattering. The photoelastic constants p ₁₁, p ₁₃, p ₃₁ and p ₃₃ were measured by evaluating the integral intensities of the phonon lines. The correlation of the photoelastic constants p ₁₃ and p ₃₁ with the degree of stretching was determined by evaluating the relative integral intensities of the Brillouin lines with the phonon propagation vector into and perpendicular to the stress direction. The other photoelastic constants were measured by comparing the angle-dependent relative integral intensities of the transverse and longitudinal Brillouin lines. The elastic constants, as well as birefringence, were surprisingly found to be unaffacted by further stretching of the sample above the necking region. The density was constant for all degrees of stretching. The photoelastic constants p ₁₁ and p ₃₃ showed significant variation during deformation. The results have been compared with former measurements of polypropylene (PP) by Cavanaugh and Wang. Because the elastic constants are unchanged during deformation, the modulation of the intensities is due to the variation of the photoelastic constants for this material.     

The Effect of Parameter Estimation on Upper‐sided Bernoulli Cumulative Sum Charts

The Bernoulli cumulative sum (CUSUM) chart has been shown to be effective for monitoring the rate of nonconforming items in high‐quality processes where the in‐control proportion of nonconforming items (p₀) is low. The implementation of the Bernoulli CUSUM chart is often based on the assumption that the in‐control value p₀ is known; therefore, when p₀ is unknown, accurate estimation is necessary. We recommend using a Bayes estimator to estimate the value of p₀ to incorporate practitioner knowledge and to avoid estimation issues when no nonconforming items are observed in phase I. We also investigate the effects of parameter estimation in phase I on the upper‐sided Bernoulli CUSUM chart by using the expected value of the average number of observations to signal (ANOS) and the standard deviation of the ANOS. It is found that the effects of parameter estimation on the Bernoulli CUSUM chart are more significant than those on the Shewhart‐type geometric chart. The low p₀ values inherent to high‐quality processes imply that a very large, and often unrealistic, sample size may be needed to accurately estimate p₀. A methodology to identify a continuous variable to monitor is highly recommended when the value of p₀ is low and the required phase I sample size is impractically large. Copyright © 2012 John Wiley & Sons, Ltd.     

A Double Sampling Scheme for Estimating from Misclassified Multinomial Data with Applications to Sampling Inspection

In some situations, it is desired to estimate multinomial proportions from data which have been misclassified. One such area is the sampling inspection area of quality control. In this paper, it is assumed that two measuring devices are available to classify units into one of r mutually exclusive categories. The first device is an expensive procedure which classifies units correctly; the second device is a cheaper procedure which tends to misclassify units. In order to estimate the proportions p_i (i = 1,2, …, r) a double sampling scheme is presented. At the first stage, a sample of N units is taken and the fallible classifications are obtained; at the second stage a subsample of n units is drawn from the main sample and the true classifications are obtained. The maximum likelihood estimates of the p_i are derived along with their asymptotic variances. Optimum values of n and N which minimize the measurement costs for a fixed precision of estimation and which minimize the precision for fixed cost are derived. The procedures of estimation and allocation of sample size are illustrated by an example in quality control.     

A new apparatus for the measurement of the isobaric heat capacity of liquid refrigerants

A new automated adiabatic flow calorimeter was developed which enables one to measure the isobaric heat capacity, C _p, of pure fluids and their mixtures in the liquid phase. The calorimeter has been carefully designed to keep the heat loss from the sample fluid as small as possible being regarded as negligible. The experimental apparatus constitutes a closed circuit of the sample circulation using a combination of two mounted metallic bellows and a metering pump. The present apparatus is designed to measure C _p at temperatures to 500 K and pressures to 15 MPa and is also applicable to measurements in the critical region as well as the region near the saturated liquid state because of its excellent mass flow rate control stability and the high adiabatic efficiency of the calorimeter. The C _p of liquid refrigerant 114 (R114) has been measured at temperatures from 275 to 415 K and pressures up to 3.2 MPa including the critical region with experimental uncertainty of less than ±0.4%. The heat capacity of saturated liquid R114 has also been derived from the data measured in the single phase.Paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.     

Schnellbewertung von Dispersionsgefüge und mechanischen Eigenschaften karbidbildender Stähle

Fast Evaluation of dispersed Microstructure and mechanical Properties of Carbide forming Steels A possibility of a fast evaluation concerning the dispersed microstructure and the properties of C-, Cr- and Mn-alloy steels is presented. The dispersed microstructure was adjusted by precipitation and coarsening from the martensitic initial condition. The quick recording coercive field strength H_C and microstructural parameters obtained from scanning electronmicroscopic investigations were used in the relationship H_C-particle diameter d_p relative to a microstructure fast evaluation. Due to the repeated nucleation the M₃C carbides grow and coarse parallel and in particle generations. Thus always a particle size spectrum is obtained. A fast evaluation of properties is realized by the connection H_C – yield strength R_p0.2 / tensile strength R_m / hardness HV. Since H_C reacts on the size of precipitated particles, and the course H_C – time t / d_p / particle volume (d_p/2)³ encloses two maxima, which both separate the precipitation (A) and coarsening stage (V), at any H_C-value one of the attaching possibilities (A or V) must be excluded. This can be reached by considering the obtained temperature and time limiting values. The magnetic measuring method with the structure-sensitive quantity H_C yields to an assertion relative to the microstructual condition within one minute. The error amounts to ≤0.5%. The application of the H_C-measurement is time- and cost-economic. A special sample preparing is not necessary. The condition of the surface does not enter into the measuring result. The yielded connections H_C – R_p0.2 / R_m / HV can be used like a standard referred to the tested grades of steel.     

A Note on the Cp Statistic

C. Mallows proposed a statistic, C_p , for variable selection in multiple regression. Gorman and Toman (1966) published the statistic, its derivation, and several examples of its use. The purpose of this note is to show that there is a one-to-one correspondence between C_p and the older, well established criterion known as adjusted R ², e.g. Ezekiel (1930).