In-plant Applications Research Articles

Production Yield for Multiple Line Processes: Product Acceptance Determination

Abstract Product inspection and quality testing are very important, particularly for product sentencing in product acceptance determination. Because the production yield requirements for most high-tech processes are very stringent, processes with a very low fraction of defectives are common. In such cases, process capability indices have been widely applied to provide numerical evaluations in process capability, which are effective tools for quality assurance. In product acceptance determination, methods of evaluating yield for processes with a single manufacturing line have been investigated extensively, but processes with multiple lines have been comparatively neglected. However, multiple manufacturing lines are popular in many factory applications because of high demand. Consequently, in this paper, we consider the product acceptance sampling plan based on CMpk to deal with product acceptance determination for processes with multiple lines. The required sample sizes and the critical acceptance values for various levels of producers’ and customers’ risks with various values of acceptable quality level (AQL) and lot tolerance percent defective (LTPD) are tabulated. In addition, a Monte Carlo simulation is conducted and the coverage rates of producers’ and customers’ risks for various values of AQL and LTPD are provided. The product acceptance determination based on CMpu for processes with one-sided specification limits and multiple lines are also discussed. The computational results are useful to the practitioners for their in-plant applications.

A Bayesian approach for group supplier selections based on the popular process-capability-index Cpk

Supplier selection is an important part of supply chain management. In the initial stage of production setting, the decision-maker usually faces the problem of selecting the “best” one(s) from available manufacturing material suppliers. For this purpose, process capability indices (PCIs) are commonly used in the literature to rank the suppliers under selection; and among these PCIs, Cpk could be the most popular one. This problem of selecting the best supplier(s) has received considerable attention in the literature but mainly from the frequentist point of view. In this paper, we tackle the so-called group supplier selection problem via the Bayesian approach, namely selecting a group of suppliers that would include the supplier of the largest Cpk value with a high level of confidence in a Bayesian sense. Based on the observed data and available prior information, we develop a practical procedure for the group supplier selection, which is useful for practitioners operating in-plant applications.

Analytic Solution to Product Acceptance Determination for Gold Bumping Process With Multiple Manufacturing Lines

In recent years, gold bumping process has been applied extensively for the package technology of liquid crystal display driver integrated circuit, which is an essential component in portable devices. Because the increasing requirement of high-definition display devices, the gold bumping process has become more difficult and it is requested to be of high quality with very low fraction of defectives. Unfortunately, conventional methods for product acceptance determination no longer work because any sample of reasonable size probably contains no defective gold bump product items. In addition, in the globally competitive manufacturing environment, gold bumping processes involving multiple manufacturing lines are quite common in the Science-Based Industrial Park in Hsinchu, Taiwan, because of economic scale considerations. In this paper, we provide analytical solutions to gold bump product acceptance determination, which provide both manufacturers and customers to reserve their own rights by compromising on a rule for gold bumping process with multiple manufacturing lines. For the convenience of inplant applications, we tabulate the number of required inspection units, the critical acceptance values for various manufacturer's risks and consumer's risks, and various number of manufacturing lines. For illustration purpose, a real application in a gold bumping factory, which is located in the Science-Based Industrial Park in Hsinchu, Taiwan, is included.

Process Capability Evaluation for Square Bumps with Mean Shift

Abstract Process capability indices have been widely used in manufacturing industries to provide numerical measures of process potential and performance. In in-plant applications, some inevitable process mean shift may be undetected when the statistical process control charts are applied, particularly, in the bumping process for square bumps. Bothe in 2002 provided a statistical reason for considering such a shift in the process mean to accommodate the undetected mean shifts for normally distributed processes when evaluating process capability. In this paper, we consider a bumping process for square bumps in which the data can be formulated as a non-central chi-square distribution, a class of non-normal distributions. To evaluate the popular yield-based process capability Cpk more accurately, we accommodate the magnitudes of undetected mean shifts using the modified capability evaluation formula and tabulate the detection powers under various subgroup sizes and non-central chi-square parameters. Based on the modified process capability evaluation, we can provide a more reliable process capability evaluation of capability index Cpk for square bumps and make more correct decisions. For illustration purpose, a real application in a bumping factory which is located on the Science-based Industrial Park in Hsinchu, Taiwan, is presented.

Statistical inference on the multiple-stream yield index

Process capability indices have been used in manufacturing industries to provide a quantitative measure of process performance. In this paper, we proposed a multiple-stream yield index , and developed a procedure to provide practitioners an efficient tool for interval estimation and hypothesis testing on the yield index . Numerical examples are presented to illustrate how we may apply the procedure to actual data collected in the factory. The procedure is easy to use, and should be expected for in-plant applications.

Assessing process capability based on the lower confidence bound of Cpk for asymmetric tolerances

Abstract It has been proved that process capability indices provide very efficient measures of the capability of processes from many different perspectives. At the present time, the C pk index is used more than any other index for measuring process capability. However, most existing research works for capability testing have focused on processes with symmetric tolerances, but not for asymmetric tolerances. 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, which provides the sample sizes necessary to achieve a desired lower confidence bound, is directly related to the cost of the data collection plan. This paper provides explicit formulas with efficient algorithms to obtain the lower confidence bounds and sample sizes required for specified precision of the estimation on C pk for processes with asymmetric tolerances. A Matlab computer program using a binary search method is developed. For the practitioners to use in their in-plant applications, we tabulate lower confidence bounds for some commonly used capability requirement and the sampling accuracy of C pk for sample sizes determination. A realistic example of forging process is presented to illustrate the applicability of the proposed method.

A Hypothesis Testing Procedure on Assessing Process Performance for Asymmetric Tolerances

The capability index Cpm takes into account the proximity of the process mean to the target and as well as the process variation. However, Cpm index has been shown to be inappropriate for measuring process performance with asymmetric tolerances. Several modifications of Cpm have been proposed to handle asymmetric tolerances, and has been shown to be superior to other existing generalizations. In this article, we implement the theory of hypothesis testing to develop a decision making procedure for assessing process performance based on index with critical values. The behavior of the critical values versus the unknown distribution parameter is also investigated to obtain reliable critical values for testing. To make the proposed procedure practical for in-plant applications, tables of critical values for various parameters are provided and an application example is demonstrated to evaluate the process performance for illustration.

Testing process precision for truncated normal distributions

Process precision index C p has been widely used in the manufacturing industry to provide numerical measures of process precision, which essentially reflects product quality consistency. Precision measures using C p for normal processes, contaminated normal processes, have been investigated extensively, but are neglected for truncated normal processes. Truncated normal processes are common in the manufacturing industry, particularly, for factories equipped with automatic machines handling fully inspections, and scrap/rework products falling outside the specification limits. If the processes follow the normal distribution, then the inspected products (processes) must follow the truncated normal distribution. In this note, we consider the precision measure for truncated normal processes. We investigate the analytically intractable sampling distribution of the estimated C p, and obtain a rather accurate approximation. Using the results, we develop a practical testing procedure for practitioners to use in their in-plant applications.

Assessing process accuracy - an inferential method about the process accuracy index

Process accuracy index Ca has been developed in the manufacturing industry to provide numerical measures on evaluating process centering. Distributional and inferential properties of the estimated accuracy index were investigated in the previous work. In this paper, a reliable hypotheses testing procedure on process centering for in-plant applications is constructed. Useful critical values to ensure the process reaching a certain desirable level of the time are tabulated. A practical example is also presented to illustrate how the proposed reliable procedure may be applied.

Production quality and yield assurance for processes with multiple independent characteristics

Process capability indices have been widely used in the manufacturing industry providing numerical measures on process potential and process performance. Capability measure for processes with single characteristic has been investigated extensively, but is comparatively neglected for processes with multiple characteristics. In real applications, a process often has multiple characteristics with each having different specifications. Singhal [Singhal, S.C., 1990. A new chart for analyzing multiprocess performance. Quality Engineering 2 (4), 397–390] proposed a multi-process performance analysis chart (MPPAC) for analyzing the performance of multi-process product. Using the same technique, several MPPACs have been developed for monitoring processes with multiple independent characteristics. Unfortunately, those MPPACs ignore sampling errors, and consequently the resulting capability measures and groupings are unreliable. In this paper, we propose a reliable approach to convert the estimated index values to the lower confidence bounds, then plot the corresponding lower confidence bounds on the MPPAC. The lower confidence bound not only gives us a clue minimum actual performance which is tightly related to the fraction of non-conforming units, but is also useful in making decisions for capability testing. A case study of a dual-fiber tip process is presented to demonstrate how the proposed approach can be applied to in-plant applications.

Bayesian approach for measuring EEPROM process capability based on the one-sided indices CPU and CPL

The purpose of process capability analysis is to provide numerical measures on whether a process is capable of reproducing items meeting the manufacturing specifications. Capability analyses have received consider- able recent research attention and increased usage in process assessments and purchasing decisions. Most ex- isting research works on capability analysis focus on estimating and testing process capability based on the tra- ditional distribution frequency approach. In this paper, we propose a Bayesian approach based on the indices CPU and CPL to measure EEPROM process capability, in which the specifications are one-sided rather than two-sided. We obtain the credible intervals of CPU and CPL and develop a Bayesian procedure for capability testing. The posterior probability p, for which the process under investigation is capable, is derived. The credible interval is a Bayesian analog of the classical lower confidence interval. A process satisfies the manufacturing capability requirements ifall the points in the credible interval are greater than the pre- specified capability level w. To make this Bayesian pro- cedure practical for in-plant applications, a real example of an EEPROM manufacturing process is investigated, demonstrating how the Bayesian procedure can be applied to actual data collected in the factories.

A decision-making procedure on process centering: a lower bound of the estimated accuracy index

Process accuracy index Ca has been proposed in the manufacturing industry to provide numerical measures on assessing process performance with respect to the accuracy. Investigations on process centering (the ability to cluster around the specification midpoint) in existing engineering statistics and quality assurance literature reveal conservatively rare. In this paper, a natural estimator of the accuracy index is considered under the normality assumption. The exact probability density function and the r-th moment raised from a non-central chi-squared distribution of the estimated index are derived. A decision-making procedure on process centering for in-plant applications is constructed. Values in lower bound of the estimated index required to ensure the process reaching a desirable accuracy level of the time are also tabulated. A practical example is presented to illustrate how the proposed reliable procedure may be applied.

Testing manufacturing performance based on capability index Cpm

The loss-based process capability index Cpm, sometimes called the Taguchi index, has been proposed to the manufacturing industry to measure process performance. In this paper, we propose a method to assess the performance of a normally distributed process. We implement the theory of testing hypothesis using the natural estimator of Cpm, and provide an efficient program to calculate the p-values. We also provide tables of the critical values for some commonly used capability requirements. Based on the test, we develop a simple step-by-step procedure for in-plant applications.

A review and extensions of sample size determination for estimating process precision and loss with a designated accuracy ratio

Numerous process capability indices, including Cp, Cpk, Cpm, and Cpmk, have been proposed in the manufacturing industry providing numerical measures of process capability based on various criteria. The index Cp provides a measure of process precision, which reflects the consistency of product quality. The index Cpm, also called the Taguchi index, essentially measures process loss. Lower confidence bounds estimate the minimum process capability conveying critical information regarding product quality, which is essential to quality assurance. Existing research works have focused on constructing lower confidence bounds, but investigation on the sample sizes required for a specified accuracy ratio of the estimation has been comparatively neglected. The sample size determination is important, as it is directly related to the cost of the data collection plan. In this paper, we review some existing formulas for lower confidence bounds that can be used to determine the sample size required for a given estimating accuracy ratio. We then present a different approach, with efficient MATLAB programs, to obtain the sample sizes using the UMVUE (uniformly minimum variance unbiased estimator) of Cp, and the MLE (maximum likelihood estimator) of Cpm. We also provide tables of the sample size information for the engineers/practitioners to use for their in-plant applications.

A random interval estimation of the estimated process accuracy index

Process accuracy index Ca has been proposed in the manufacturing industry to provide a numerical measure to assess process performance with respect to accuracy. Information regarding process centering (the ability to cluster around the specification mid-point) in existing engineering statistics and quality assurance literature is relatively rare. In this paper, a natural estimator of the accuracy index is considered under the normality assumption. The exact probability density function and the rth moment raised from a non-central chi-squared distribution of the estimated process accuracy index are derived. A decision-making procedure based on a random interval estimation of the considered index is also constructed. A practical example is demonstrated to illustrate how the proposed reliable procedure may be applied for in-plant applications.

Process capability assessment for index Cpkbased 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.

Evaluating process centering with large samples – an approximate lower bound on the accuracy index

Process accuracy index Ca has been proposed in the manufacturing industry to provide numerical measures on process centering (the ability to cluster around the specification mid-point). Investigations on process accuracy in existing engineering statistics and quality assurance literature are all focused on small samples with a normality assumption. In this paper, a natural estimator of process accuracy index is considered. The limiting distribution and related large sample properties of the considered estimator are studied under general populations having fourth central moment exists. A decision-making procedure based on an approximate 100(1-α)% lower bound of the process accuracy index is also constructed. A practical example is demonstrated to illustrate how the proposed procedure may be applied for in-plant applications to judge whether the process runs under the desirable accuracy requirement.

Sample size determination for estimating process precision and process loss based on multiple samples with designated accuracy ratio

Numerous process capability indices including Cp, Cpk, Cpm, and Cpmk have been proposed in the manufacturing industry providing numerical measures on process capability based on various criteria. The index Cp provides measures on process precision, which reflect the consistency of product quality. The index Cpm, also called the Taguchi index, essentially measures process loss. Lower confidence bounds estimate the minimum process capability conveying critical information regarding product quality, which is essential to quality assurance. Existing research works have focused on constructing lower confidence bounds, but investigation on the sample sizes required for specified accuracy ratio of the estimation has been comparatively neglected. The sample size determination is important directly related to the cost of data collection plan. Furthermore, existing researches considered solely one single sample. Current quality control practice, however, is to estimate process capability using multiple groups of control chart samples rather than one singl sample. In this paper, based on multiple samples collected from past in-control and S control chart data, the sampling distributions of the estimated Cp and Cpm are derived with efficient Matlab programs that can be used to determine the sample size required for given estimating accuracy ratio. We also provide tables of the sample size information for the engineers/practitioners to use for their in-plant applications.

Testing process performance based on capability index Cpk with critical values

Process capability index Cpk has been widely used in the manufacturing industry as a process performance measure. In this paper, we investigate the natural estimator of the index Cpk, and show that under the assumption of normality its distribution can be expressed as a mixture of the chi-square and the normal distributions. We also implement the theory of hypothesis testing using the natural estimator of Cpk, and provide efficient Maple programs to calculate the p-values as well as the critical values for various values of @a-risk, capability requirements, and sample sizes. The behavior of the p-values and critical values as functions of the distribution parameters are investigated to obtain tight critical values for reliable testing. Based on the test, we develop a simple and practical procedure for in-plant applications. The practitioners can use the proposed procedure to determine whether their process meets the preset capability requirement, and make reliable decisions.

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.