Index Cpk Research Articles

UTILIZAÇÃO DO CONTROLE ESTATÍSTICO DE PROCESSOS (CEP) COMO INDICADOR DE VIOLAÇÃO EM PROGRAMAS DE PAGAMENTO DO LEITE PELA QUALIDADE

The aim of this study was to use the statistical process control (SPC) as a tool of recognition and rating for farms with larger probabilities of infraction of milk quality standards used by industry. Data of somatic cell count (SCC) and total bacterial count (TBC) from 452 farms were used. Proportion of infractions of quality standard was calculated considering limits of SCC = 400,000 cells/mL and TBC = 100,000 CFU/mL. Indices Cpk were calculated and farms were rated according to four categories of Cpk means and indices. The farms which presented larger means and standard deviation had larger frequencies of violation. Farms with means lower than limits proposed to SCC and TBC represented, respectively, 25.05 % and 97.78 % of farms. However, proportion of farms that provided milk consistently within the quality standards evaluated (Cpk ? 1) represented 4.65 % and 35.17 % of farms for SCC and TBC, respectively. Therefore, the Cpk index can be used by the industry associated with the current models to characterize milk quality from farms, since it identifies more consistent herds in producing quality-standard milk.KEYWORDS: bonus; Cpk index; somatic cell count; total bacterial count.

Variables sampling inspection scheme for resubmitted lots based on the process capability index Cpk

This paper attempts to develop a sampling inspection scheme by variables based on process performance index for product acceptance determination, which examines the situation where resampling is permitted on lots not accepted on original inspection. The equations for plan parameters, the required sample size and the corresponding critical value, are derived based on the exact sampling distribution rather than an approximation approach hence the decisions made are more accurate and reliable. Moreover, the efficiency of the proposed variables resubmitted sampling plan is evaluated and compared with the existing variables single sampling plan. For illustrative purpose, an example is presented to demonstrate the use of the derived results for making a decision on product acceptance determination.

Capability measures for Weibull processes with mean shift based on Erto's-Weibull control chart

Process capability indices (PCIs), which are effective tools for quality assurance and are guidance for process improvement, have been proposed in the manufacturing industry to provide numerical measures on process reproduction capability. PCIs are calculated under the assumption that the process is stable while the process mean and variation are not changeable. However, in practice, the process is dynamic. Under the Bothe’s adjustments, we showed the detection powers of the percentile-Weibull control chart, bootstrap-Weibull control chart, and the Bayes-Weibull control chart. It is realized that the Bothe’s adjustments are inadequate with data coming from Weibull processes. For this reason, the PCIs have to be adjusted. Bothe (2002) provided the adjustment method for normality processes. In this research, we consider Weibull processes, which cover a wide class of applications. We calculate the mean shift adjustments under various sample sizes n and Weibull parameter γ, with the power fixed to 0.5. Then, we implement the adjustments to accurately estimate capability index Cpk for Weibull processes with mean shift consideration. Finally, an application example is presented for illustration purpose. Key words: Dynamic Cpk, mean shift, process capability index, Weibull distribution,Weibull control chart.

Confidence Intervals for Process Capability Indices for the Unbalanced One‐Way Random Effect ANOVA Model

Confidence intervals for process capability index Cpk are developed for the unbalanced one‐way random effect model using Bissell's approximation method. The proposed limit is compared with the generalized lower confidence limit obtained using the generalized pivotal quantity method. To assess the accuracy of the method, a simulation study is presented. The results are illustrated with an industrial example. Copyright © 2011 John Wiley & Sons, Ltd.

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.

Capability assessment for processes with multiple characteristics: A generalization of the popular index Cpk

AbstractProcess capability index Cpk is the most popular capability index widely used in the manufacturing industry. Existing research on the yield‐based measure index Cpk to date is restricted to processes with single characteristics. However, many manufacturing processes are commonly described with multiple characteristics, for example, the gold bumping process in the TFT‐LCD (thin film transistor‐liquid crystal display) manufacturing industry. In the gold bumping process, gold bumps have multiple characteristics all having effects on the process yield. Obtaining accurate gold bumping manufacturing yield is very important for quality assurance and in providing guidance toward process improvement. To obtain accurate yield assessment for processes with multiple characteristics, we propose a new overall yield‐measure index C, which is a generalization of the index Cpk, and a natural estimator of C. For the purpose of making inferences on the process capability, we derive a quite accurate approximation of the distribution of since the distribution is analytically intractable. With this distribution, we tabulate the lower confidence bounds of the new index under various sample sizes for in‐plant applications. In addition, we construct a statistical test on the new yield‐measure index in order to examine whether the yield meets the customers' requirements. For illustration purpose, a real case in a gold bumping factory located in the Science‐based Industrial Park at Hsinchu, Taiwan is presented. Copyright © 2011 John Wiley & Sons, Ltd.

Developing a sampling plan based on Cpk—unknown variance

AbstractCommon acceptance sampling plans for variables do not take into account the process performance while determining the sample size needed. An attempt to overcome this gap was suggested by Negrin et al. (Quality Eng. 2009; 21: 306–318), where a sampling plan based on the Cpk index was developed. The plan is a multistage acceptance sampling plan based on Cpk for variables taken as a random sample from a lot of size N having (approximately) normal distribution with a known variance. In the current research, we relax the assumption of a known variance and develop a Cpk sampling plan based on unknown variance. We develop a sample size model and update the acceptance/rejection criteria of the previous sampling plan to the new, more realistic model with unknown variance. In addition we develop the operational characteristic curve (OCC). The sampling plan is compared with the commonly used plan MIL‐STD‐414 (Sampling Procedures and Tables for Inspection by Variables for Percent Defective. Department of Defense: Washington, DC, 1957) and it is found (via simulations) that the Cpk sampling plan has a smaller probability of accepting defective lots and the required sample size needed is found to be smaller for large lots. In addition, a comparison is made between the two OCCs and it is found that the OCC developed for the Cpk sampling plan shows a higher accuracy (in an order of magnitude) in classifying lots correctly. Copyright © 2010 John Wiley & Sons, Ltd.

Parameter Setting for Rod Push Production Process on Multiple Responses : A Case Study in Motorcycle Parts Factory

The aim of this research is to study appropriate setting of parameters in the production process ofRod Push which affect quality characteristics such as diameter and surface roughness. Cycle time is alsoconsidered. The problem was done at a study plant and it was found that the actual process capability index(Cpk) of two quality characteristics was less than the manufacturer’s standard of 1.33. Central composite designwas used to examine Rod Push produced by CNC with 5 controllable process factors including three factorsfrom rough-cutting process (spindle speed, feed rate, and depth of cut) and two factors from finish-cuttingprocess (spindle speed and feed rate). The nuisance factor of CNC was cutting tool wear. After collectingall data, the appropriate setting was investigated. The new results indicated that the new condition yieldedthe better process performance index and shorter cycle time. Keywords: DOE; Central Composite Design; Rod Push Process; CNC Turning Machine

High speed and high accuracy inspection of in-tray laser IC marking using line scan CCD with a new calibration model

This study used a high-speed high-resolution line scan CCD, and a Delta motor drive module to provide position feedback in a laser marking system. Based on a novel algorithm, image processing of the high-speed CCD scan was allowed the accurate determination of the laser marking location. This system was calibrated using a template, and a novel 1-D calibration model of the line scan CCD was developed. The relevant position of each IC in the tray was obtained based on the calibration algorithm. Gain and offset calibration, sub-pixel calculation, and normalized checks were performed in this automatic optical inspection system. The total processing time for laser correction marking, scanning and identification was about 2∼2.5 s, the positioning accuracy was×9 μm, and the industrial specification and process capability index Cpk reached over 1.2.

On the choice of a capability index with an application in carbon fibre manufacturing

Process capability indices Cp, Cpk, Cpm and Cpmk have been proposed to the manufacture industry to provide measures of process performance. The family C″p(u, v) depending on two parameters allows to generalise the previous indices for any position of the target in the tolerance interval. Grau (2010) studied the links between the value of the C″p(u, v) index and the maximum deviation of the process mean, as well as the maximum percentage of non-conforming items manufactured under the assumption of normality. From these results, we provide tables which allow the practitioner to choose a pair (u, v), so that the resulting index C″p(u, v) will meet his objectives best. We develop a decision making rule based on the natural estimator which can be used to test whether the process is capable or not. We then present a real study made on a carbon fibre manufacturing process to illustrate how this reasoning can be applied.

Robust design of railway vehicle suspension using a process capability index

Robust design for the primary suspension of a railway vehicle was performed according to the optimization of 10 dynamic responses representing driving safety and ride comfort, in which response surface models (RSMs) from the design of experiments (DOEs) were applied. To evaluate the probabilistic feasibility of robustness, an intensive computational process is mandatory. In the present study, the authors utilized the first-order Taylor series expansion to reduce the computational burden associated with a probabilistic feasibility evaluation, thus easily obtaining both an individual mean and variance of constraints. To overcome the difficulty of optimizing the mean and probabilistic variances for the 10 dynamic responses, a process capability index (Cpk) was introduced, which shows the mean value and scattering of the product quality to a certain extent and normalizes the objective functions irrespective of varying dimensions. Consequently, the robust design to optimize the 10 dynamic responses minimized the Cpk subjected to the constraint of Cpk ≥ 2, which satisfied 6σ. The proposed method improved the Cpk that violated the constraints obtained by the RSMs from DOEs and minimized the variance of the Cpk.

A Low Volume Production Process Capability Analysis

Statistical Process Control (SPC) and Process Capability Indices (PCI) are used extensively within manufacturing and service environments. This paper reports the results of a statistical quality improvement programme when applied to a low volume, high gauge frequency machining process. The resultant capability indices (Cp and Cpk) are calculated from acquired data both before and after process modification, and are discussed in relation to the specific process. Box Cox data transformations are utilised in order to centralise data and the applicability of the capability improvements are determined. Historically SPC and capability analysis are performed on high volume processes with data being gathered on a sample basis. This investigation applies the high volume theory to a low volume 100% data gathering process. The validity of capability analysis of this nature is discussed due to long cycle times and large gauge inspection frequency.

Developing a Sampling Plan Based on Cpk

ABSTRACT The common sampling plans for variables rely solely on process performance and do not take into account the engineering specifications while determining the sample size needed. When the process capability index Cpk is good, the limits of the distribution of the lot (the γ/2th percentile and the 1−γ/2th percentile) lie inside the engineering specifications. In this case the area of the tail of the normal distribution that lies outside the engineering specifications is very small, and hence the effect of the sampling error (which is the result of estimating the mean) on this area is negligible. However, when the limits of the distribution lie close to the engineering specifications (that is, when the Cpk is marginal), the area of the tail that lies outside the engineering specifications can be meaningful and therefore the sampling error may result in inaccuracies in estimating the number of defective items in the lot. The objective of this research is to develop a multistage sampling plan, based on Cpk and adjusted to acceptance sampling for variables, taken as a random sample from a lot of size N (given) having (approximately) a normal distribution with a known variance. The sampling plan was compared to MIL-STD-414 (1957) and it was found (via simulations) that the Cpk sampling plan has a smaller probability to reject good lots (0–0.03% vs. 0–0.27% for the MIL-STD-414) and a smaller probability to accept defective lots (0–0.01% vs. 5.2–8.37% for the MIL-STD-414). In addition, the required sample size is smaller for processes with high capability index.

Decision-making in testing process performance with fuzzy data

Over the years, numerous process capability indices (PCIs) have been proposed to the manufacturing industry to provide numerical measures of process performance. Most research efforts have focused on developing and investigating PCIs that assess process capability by precise measurements of output quality. However, real observations of continuous quantities are not precise numbers; in practice, they are more or less imprecise. Since observations of continuous random variables are imprecise the values of related test statistics become imprecise. Therefore, decision rules for statistical tests have to be adapted to this situation. This article presents a set of confidence intervals that produces triangular fuzzy numbers for the estimation of Cpk index using Buckley’s approach with some modification. Additionally, a three-decision testing rule and step-by-step procedure are developed to assess process performance based on fuzzy critical values and fuzzy p-values. This concept is also illustrated with an example for testing process performance.

Reducing defect generations in e-learning business: a Six Sigma approach

The process of learning, with significant technological development, has stepped forward towards e-learning not only for educational institutions but also for government and corporate businesses. The respective product development process of e-learning business often faces challenges in defect-free phases of the project life cycle across various types of projects. Higher the defect rate, higher is the reworking cost and lower is the productivity and customer dissatisfaction. This study attempts a disciplined, data-driven analytical Six-Sigma approach under DMAIC philosophy to reducing variation in one of such e-learning business processes, identifying controllable parameters for defect generation and taking necessary preventive actions. After taking actions on the critical activities, number of defects being generated and accordingly the amount of rework and efforts (man-months) consumed have been reduced satisfactorily. The statistical results show an improvement in the Pearson process capability index (Cpk) from 0.11 to 1.53.

New Process Capability Indices for One-Sided Tolerances

Capability indices are dimensionless quantities measuring the ability of a process to manufacture items whose characteristics must be within a specified tolerance range. In this case and for a normally distributed process, the indices Cp, Cpk, Cpm, and Cpmk are widely used. In this paper we study the case where a single tolerance is imposed because the shifts in the direction of this tolerance seem more serious than in the opposite direction. We propose a family of four indices having better properties than the existing indices. These new indices are created from the usual properties and interpretations of the indices Cp, Cpk, Cpm, and Cpmk, and can be used without difficulty in industry.

Generalized confidence intervals for assessing process capability of multiple production lines

AbstractProcess capability indices (PCIs) have become popular as unit‐less measures on whether a process is capable of reproducing items meeting the quality requirement. A reliable approach for testing process capability is to establish an interval estimate, for which we can assert that it contains the true PCI value with a reasonable degree of certainty. However, the construction of such an interval estimate is not trivial, since the distribution of the commonly used Cpk index involves unknown parameters. In this paper, we adopt the concept of generalized confidence intervals and generalized pivotal quantities to derive the generalized lower confidence bounds for providing critical information on process performance. Two practical applications in the area of process capability were considered, they include (i) assessing whether a process under investigation is capable and (ii) providing the lowest performance of the manufacturing processes from several production lines or several suppliers for quality assurance. The applicability of the derived results is also illustrated with examples. Copyright © 2008 John Wiley & Sons, Ltd.

Monitoring capability indices using run rules

PurposeThe purpose of this paper is propose a methodology for monitoring industrial processes that cannot be stabilized, but are nevertheless capable.Design/methodology/approachThe proposed procedure uses the CP(u,v) family of capability indices proposed by Vännman (including the indices CPK, CPM, CPMK) combined with one‐sided two‐out‐of‐three and three‐out‐of‐four run rules strategies.FindingsThis paper introduces a new strategy, where capability indices are monitored in place of the classical sample statistics like the mean, median, standard deviation or range.Practical implicationsWhen doing a capability analysis it is recommended to first check that the process is stable, e.g. by using control charts. However, there are occasions when a process cannot be stabilized, but is nevertheless capable. Then the classical control charts fail to efficiently monitor the process position and variability. The approach suggested in this paper overcomes this problem.Originality/valueThe experimental results presented in this paper demonstrate how the new proposed approach efficiently monitors capable processes by detecting decreases or increases of capability level.

A sensitivity study on the bootstrap confidence interval of the capability index Cpk

The capability index is a simple and quantitative way to describe process capability. The index Cpk is an indicator, involving specifications, process mean and standard deviation, for evaluating the capability of a process. When Cpk is applied to evaluate a process, estimating the confidence interval of Cpk is important for statistical inference on the process. Calculating the confidence interval for a process index usually needs the assumption about the underlying distribution. Bootstrapping is a nonparametric, but computer intensive, estimation method. Several types of bootstrap confidence intervals have been developed over the years. In the present paper we report the results of a simulation study on the behavior of the 95% biased-corrected and accelerated (BCa) percentile bootstrap confidence interval for estimating Cpk when the data are correlated and follow a Burr distribution, which is used to represent various probability distributions. Effects of sample size, process capability, non-normality and correlation on the BCa bootstrap interval of Cpk are simultaneously studied. A detailed discussion of the simulation results is presented and some conclusions are provided.

Process improvement: performance analysis of the setup time reduction-SMED in the automobile industry

Flexibility and responsiveness to customer demands are very important for success. Generally additional time is needed for setup caused by poor design of equipment. At this point the terms continuous process improvement and SMED (single minute exchange of dies) as an approach of lean manufacturing come into play. A lean manufacturing system is part of corporate culture, like tools and approaches. In this research work, the process capability analysis technique is implemented by using MINITAB14 software to investigate the relation between SMED methodology and equipment design. The index Cpk has been used in this application study to provide a quantitative measurement of the equipment design by applying the SMED methodology in automobile manufacturing. The results of this research study indicated that SMED in other words “quick changeover” is still a suitable method not only for manufacturing improvement but also for equipment/die design development.