Capability Index Cpk Research Articles

Revolutionary Hybrid RDS, Big Data and AI Quality Control: Cpk-Based Analysis

In this research, a combination of inspection and repeated deferred sampling is suggested utilizing the process capability index Cpk. This sampling approach can effectively evaluate the quality characteristics of normal distributions even when the mean and variance are unknown. Symmetrical and asymmetrical regions failing to meet optimal conditions for practical use are identified through specific tables Furthermore, the benefits of this proposed mixed sampling design are discussed. A comparison between the suggested policies and current practices is presented to illustrate the differences. A practical example is utilized to demonstrate the practicality of this hybrid sampling strategy. The optimal parameters can be determined using the two-point method of the operating characteristic curve.

Integrating capability index and generalized rule-switching mechanism for enhanced quick-switch sampling systems

Variables Quick-Switch Sampling Systems (VQSSs) that involve both normal and tightened inspections for lot dispositions have attracted significant attention and discussion. However, existing VQSS models exhibit limitations, particularly in their rule-switching mechanism. Practitioners may encounter situations where multiple rounds of tightened inspection are necessary to ensure submission quality when suspicions of process degradation arise. Current VQSS models do not effectively address this need. To overcome this limitation, we present VQSSG by integrating a VQSS with a generalized rule-switching mechanism and the widely used capability index Cpk. VQSSG offers practitioners the flexibility to specify the number of successive lot acceptances under tightened inspection before transitioning to normal inspection. Our research includes the derivation of operating characteristics and average sample number functions for VQSSG, based on the Cpk index. Through in-depth investigations, we construct nonlinear optimisation models to determine the optimal system design and identify distinct types and settings for VQSSG, highlighting its strengths and weaknesses. Additionally, we provide managerial insights and recommendations for the effective implementation and application of VQSSG. To validate the practicality of this system, an industrial case study is presented. This paper significantly enhances the efficiency and adaptability of sampling systems by introducing VQSSG. The findings and managerial suggestions contribute to a deeper understanding of potential real-world applications, thereby enhancing quality control and decision-making processes.

Robust confidence intervals for the process capability index Cpk with bootstrap improvement

The process capability index (PCI), C pk , one of the widely used tools for assessing the capability of a manufacturing process, expresses the deviation of the process mean from the midpoint of the specification limits. The C pk is known to perform well under the general assumption that the experimental data are normally distributed without contamination. Under this assumption, the sample mean and sample standard deviation are used for the estimation of the PCI. However, the sample mean and sample standard deviation are quite sensitive to data contamination and this will result in underperformance of C pk . Therefore, in this article, we propose alternatives to the conventional method by replacing the sample mean and sample standard deviation with robust location and scale estimators. We also propose a method for constructing a robust PCI C pk confidence interval which lends itself to robust statistical hypothesis testing. The robust hypothesis testing methods based on this confidence interval are shown to be quite efficient when the data are normally distributed yet also outperform the conventional method when data contamination exists.

Comparing the sequential and the repetitive Cpk sampling plans

AbstractThe index Cpk was created to quantify the ability of the process to produce products within the upper and lower specified limits (USL/LSL); but in the last two decades, the sample estimator of the capability index Cpk has also been used to decide whether a lot should be accepted or not; if the value of a sample of size n, randomly taken from the lot, is lower/larger than a threshold, then the lot is rejected/accepted. The sampling plan based on the value is known as the Cpk sampling plan. In recent publications, the Cpk sampling plan has been combined with the repetitive sampling scheme. In short, with the repetitive sampling scheme the Cpk sampling plan might be applied m times, consequently, the lot has m chances to be accepted; if the Cpk sampling plan is applied m times and the lot is not accepted, then the lot is finally rejected. The average number of inspected items per lot (ANI) the repetitive Cpk sampling plan (where m > 1) requires to decide whether the lot should be accepted or not is larger than the sample size n of the standard Cpk sampling plan (where m = 1), because of that the repetitive sampling strategy is not recommended. In this article, we propose the sequential Cpk sampling plan where the is compared with two thresholds (c1; c2) with c1 < c2; if the is lower (larger) than c1 (c2), then the lot is rejected (accepted); otherwise, the decision is left to the next sample. The ANI of the sequential Cpk sampling plan is a function of c1 and c2; depending on the choice of these two design parameters, the ANI of the sequential Cpk sampling plan is substantially lower than the sample size n of the standard Cpk sampling plan.

Improvement of production process variations of bolster spring of a train bogie manufacturing industry: a six-sigma approach

The need for improved productivity without sacrificing quality, which is in line the prime target of many manufacturing industries. The aim of this study is to investigate the causes of production variation: a case study of the rail manufacturing industry, South Africa. In this study, the six-sigma Define, Measure, Analyse, Improve and Control (DMAIC) phases were applied to enhance the process capability (long term) in the production of bolster compression springs in the main line of bogie secondary suspension system. In every phase of DMAIC method, a combination of both qualitative and quantitative techniques was utilized. First, process capability index Cpk of the current process was computed which was found less than 1. The results obtained indicated that the process capability index values were found to be 1 after the improvement phase. Hence, significant improvement was achieved in the area of reduction in process variation and product quality after taking corrective actions. From outcomes of the study, it can be concluded that process performance of a train manufacturing plant can be improved significantly by implementing six-sigma DMAIC methodology. The novelty of this study lies in the fact that the implementation of the six-sigma DMAIC phases to enhance the process capability (long term) and minimise variations in the production of bolster compression springs has not be sufficiently highlighted by the existing literature.

Determination of modified double sampling plan based on the process capability index

Sampling plans developed based on the process capability indices have great impact in assuring the quality of procured goods since such indices are widely applied in industries to monitor the process quality. In this manuscript, a modified double sampling plan is proposed for variables inspection based on the process capability index Cpk. The proposed sampling plan, an alternative to the existing double sampling plan, can be used with less complexity for the inspection of normally distributed quality characteristics when both mean and variance are unknown. To determine the optimal parameters of the proposed plan for symmetric and asymmetric cases of fraction nonconforming with double specification limits, we use the methodology of two points on the operating characteristic curve. It is shown that the proposed plan is more advantageous than other existing sampling plans under variables inspection. Finally, an economic aspect of design of the proposed sampling plan is also presented.

Process Capability Indices With Student Mixture Models Applied to Aircraft Engines GD&T

Abstract Usually, the process capability indices (PCIs) Cp and Cpk are used to quantify the production quality of Geometric Dimensioning & Tolerancing (GD&T). For the most popular PCIs, a strong assumption about normality is required. Many previous works provide information about misleading conclusions when we use usual PCIs without any verification of the dataset normality. In recent industrial developments, many processes increasingly complicate, and GD&T are mostly geometrical specifications. For that reason, most characteristics are not normally distributed. There are two issues when calculating PCIs with nonnormal distributions. First, we have to define PCIs that can deal with all distributions. Second, we need to determine the distribution of the dataset. To do so, we give a short review of the previous work about calculating PCIs with nonnormal distributions. Following that first part, we propose our method for nonnormal distributions. This method uses new PCIs based on the quantile of a standard normal distribution. Then, we consider an estimation method based on a Student mixture model. This allows handling nonnormal distributions, potentially with outliers, and mixture distributions, which are often treated as two different fields and are common in industrial datasets. Finally, we propose some use cases to benchmark our new PCIs and Student mixture model for aircraft engines GD&T applications.

Stage-independent multiple sampling plan by variables inspection for lot determination based on the process capability index Cpk

Multiple sampling plan (MSP) has been proved that the sample units required for inspection at each stage are usually smaller than the conventional single or double sampling. However, it is more complex to administer and difficult to derive the corresponding operating characteristic function since the judgment on the submitted lot under the MSP is not only dependent on the result of current sampling but also on previous sampling results. Thus, this paper attempts to provide a relaxed type of conventional MSP by assuming the sampling inspection at each stage is independent which is called variables stage-independent multiple sampling plan and integrated with the most widely-used process capability index Cpk . For the cost-efficient purpose, the plan parameters are solved under an optimisation model that minimises the average sample number by satisfying the required quality levels and tolerated risks. Finally, the applicability of the proposed plan is illustrated in a case study.

Verifying Valve Lash in Engine Assembly Using Tolerance Interval Method for Monitoring Process Performance

The process capability refers to the ability of a process to produce parts or provide services that are able to meet the given specification requirement. The process capability is commonly measured by process capability indices (PCIs) such as Cp and Cpk to evaluate process yield, process consistency, and statistical variation compared to the specification. Most of the component or hardware manufacturing industries require exhibiting a process capability index (Cpk) of 1.33 or above. This paper presents an alternate approach for monitoring a process to verify valve lash in engine assembly. Valve lash is the gap between the rocker arm and the valve spring in an engine and this gap must be tightly controlled due to the critical role valves play in engine performance. Achieving and maintaining process capability of valve lash of 1.33 is difficult and unrealistic due to the nature of the assembly process and multiple factors being involved. Using historical data collected from 1200 valves in 100 engines assembly, this paper illustrates the tolerance interval method for monitoring the process performance of valve lash in engine assembly is more practical and economical as compared to the conventional process capability.

Formability Prediction of Laser-Welded Stainless Steel AISI 304 and AISI 430

The effect of laser welding on the mechanical properties and the prediction of formability for austenitic stainless steel AISI 304 and ferritic steel AISI 430 when welded by a YLS-5000 fiber laser, were studied in the paper. The microstructure of the welded joint was analyzed using light microscopy. The mechanical properties were determined by static tensile testing. The forming limit diagrams were produced from notched samples at R5, R17, and R25 mm. The hardness values of the welded joint and the base material were determined using the Vickers method. Samples made of AISI 430 showed that the formability suffered due to laser welding. Longitudinal coarse ferrite grains were observed in the microstructure of the AISI 430 weld metal. The coarse-grained structure of the welded joint and the continuous interface along the centerline caused the failure of the AISI 430 laser-welded samples at significantly lower actual stress and strain values than were required to break the base material. No significant changes in the formability were observed in the AISI 304 samples after laser welding. The growth of dendrites was observed in the microstructure of the AISI 304 welded joint in a direction towards the centerline of the welded joint. A comparison of the experimentally determined FLD0 values and the values calculated from predictive equations showed that a better agreement was achieved for uniform elongation than for the strain hardening exponent. The manufacturability and economic efficiency of selected parts of an exhaust system by hydromechanical drawing were evaluated on the basis of the process capability index Cpk.

Performance assessment of surface-guided radiation therapy and patient setup in head-and-neck and breast cancer patients based on statistical process control.

To assess the effectiveness of SGRT in clinical applications through statistical process control (SPC). Taking the patients' positioning through optical surface imaging (OSI) as a process, the average level of process execution was defined as the process mean. Setup errors detected by cone-beam computed tomography (CBCT) and OSI were extracted for head-and-neck cancer (HNC) and breast cancer patients. These data were used to construct individual and exponentially weighted moving average (EWMA) control charts to analyze outlier fractions and small process shifts from the process mean. Using the control charts and process capability indices derived from this process, the patient positioning-related OSI performance and setup error were analyzed for each patient. Outlier fractions and small shifts from the process mean that are indicative of setup errors were found to be widely prevalent, with the outliers randomly distributed between fractions. A systematic error of up to 1.6 mm between the OSI and CBCT results was observed in all directions, indicating a significantly degraded OSI performance. Adjusting this systematic error for each patient using setup errors of the first five fractions could effectively mitigate these effects. Process capability analysis following adjustment for systematic error indicated that OSI performance was acceptable (process capability index Cpk = 1.0) for HNC patients but unacceptable (Cpk < 0.75) for breast cancer patients. SPC is a powerful tool for detecting the outlier fractions and process changes. Our application of SPC to patient-specific evaluations validated the suitability of OSI in clinical applications involving patient positioning.

Bayes estimator of process capability index Cpk with a specified prior mean

Process capability indices such as Cp , CpL , CpU , and Cpk have been used for evaluating the performance of a manufacturing process in statistical quality control and statistical process control fields. In this paper, a Bayes estimator of Cpk is constructed such that the prior mean is set to be equal to a specified value while the prior distribution is weakly informative. This estimator is compared with conventional Bayes and non Bayes estimators.

Study on Discrete Manufacturing Quality Control Technology Based on Big Data and Pattern Recognition

Aiming at the quality control problems in the discrete manufacturing process of large and superlarge equipment, which cannot meet the urgent needs of production, a quality control method based on big data and pattern recognition is proposed. A large amount of data is collected through the test equipment developed in the discrete manufacturing process; a database of typical working conditions and an information tracking system relying on the cloud platform were formed. The working conditions were divided by the principal component analysis (PCA) and improved K-means algorithm. The Markov prediction model predicts the working conditions, recognizes the pattern with typical working conditions, regulates the processing parameters, and achieves quality control. Taking the quality control of the hydraulic cylinder manufacturing process above 5 m as an example for experimental verification, the experiments indicated that working conditions can be automatically identified and classified through pattern recognition technology. The process capability index Cpk increased from 0.6 to 1, which proved the effectiveness of quality control and the improvement of processing capabilities.

Variables skip-lot sampling plans on the basis of process capability index for products with a low fraction of defectives

The skip-lot sampling plan (SkSP) is employed in supply chains to decrease the amount of inspection required for submitted lots when they have demonstrated a succession of lots with excellent quality. As only some fractions of lots are examined, the cost of inspection is reduced. With the current abundance of high-yield products, however, the majority of SkSP schemes have been utilized for attributes testing, which does not fully reveal the SkSP’s economic advantages. Thus, on the basis of the process capability index Cpk, the variables SkSP with single sampling as a reference plan (Cpk-SkSP-2) was developed. With management of the lot’s quality and tolerable risks agreeable to both the supplier and the buyer, the Cpk-SkSP-2 were incorporated with acceptance probabilities (rather than asymptotic approximations), which yielded the exact sampling distribution of the Cpk estimator at the specified quality standards. Furthermore, the equilibrium probability for the acceptance of Cpk-SkSP-2 was derived from a Markov chain technique. These treatments enable minimization of the average number of samples required to render more reliable and optimal plan parameters for the inspection of products with a low fraction of defectives. The results are compared with the variables Cpk-based single sampling plans. Finally, a graphical user interface was built on the basis of our proposed Cpk-SkSP-2 procedures and methodologies to facilitate data input, plan selection, criteria computation, and decision-making in practice.

Hypothesis testing of process capability index Cpk from the perspective of generalized fiducial inference

AbstractHypothesis testing of is an essential part for decision making on process capability. The difficulty to test the hypothesis of is the complexity of the natural estimator distribution even under the normal distribution. Thus, traditional methods of the hypothesis testing of based on the natural estimator only give the approximate test method under the normal distribution and seldom discuss the hypothesis testing under some commonly used but complex non‐normal distributions. The emerging generalized fiducial inference (GFI) in recent years is an effective method for statistical inference on complex statistics. Thus, we first propose a novel hypothesis testing method for based on generalized p‐value. For application, the mathematic expression of the proposed method for the commonly used normal distribution, Gamma distribution, Weibull distribution, and two‐parameter exponential distribution is derived in detail. Next, to study the performance of the proposed method in terms of the frequency property, the real probabilities of Type I error and Type II error are calculated through simulation. The calculated results show that the proposed method has satisfactory performance for different distributions. Finally, the implementation of the proposed method is illustrated by two real examples.

Combined attri-vari inspection policy for resubmitted lots based on the process capability index

In this paper, an attri-vari inspection policy for the resubmitted lots based on the process capability index Cpk is proposed. The proposed sampling plan comprises of both attribute and variables inspections for the resubmitted lots using single sampling plan. In the case of variables inspection, both symmetric and asymmetric fraction non conforming cases have been considered. Tables are developed and constructed for determining the optimal parameters of the proposed mixed plan based on two points on the operating characteristic curve approach by formulating the problem as a non linear programming. The industrial application of the proposed mixed sampling plan is explained with an illustrative real time example. Advantages of the proposed sampling plan are also discussed in terms of comparison with other existing sampling plans.

Optimization of Electro-Discharge Texturing Parameters for Steel Sheets' Finishing Rollers.

Exterior car-body parts are made of steel or aluminum sheets. Their formability and appearance after painting depends not only on the mechanical properties but also on their surface texture. The surface roughness characteristics, the roughness average Ra and the peak count Pc per centimeter depend on the texture of rolling mill’s finishing rollers, their wear and the degree of removal by the rolling mill. The research was carried out on heat-treated finishing rollers on the surface of which a controlled texture was created by changing the electro-discharge texturing (EDT) parameters. Parameters and the number of electro-discharge texturing experiments were optimized using full four-factor experiment techniques at the upper and lower levels of the parameters in the form of 24. The significance of the impact of individual EDT parameters and their interactions was identified based on the variance results. The ANOVA variance analysis results confirmed that the roughness Ra and the peak count Pc depend primarily on peak current (Ip), discharge peak voltage (Up), pulse on time (Pont) and pulse off time (Pofft). Optimization of the effect of the above parameters on the target roughness RaT,FR values and the peak count PcT,FR of finishing rollers was performed by the response surface methodology (RSM). Obtained regression models describe relationships between the input parameters of the electro-discharge texturing of finishing rollers and the output characteristics of the RaT,FR and the PcT,FR texture to a very high degree. The reliability of the electro-discharge texturing process of working rollers was assessed using the process capability index Cpk.

Modified processes capability assessment with dynamic mean shift

AbstractRapid advancements in manufacturing technology have seen electronic products becoming lighter and thinner. This has led to production yield requirements of sawing processes in the semiconductor industry becoming increasingly strict. For sawing processes requiring a very low fraction of defective, the process capability index Cpk is widely applied for production yield measurement under the assumption of process stability. However, in practice, even when the typical control charts are implemented, dynamic and unobserved natures of the actual process mean shift may occur. Since the trial control limits are established with reference to random samples, the associated performances are also with uncertainty because of sampling error. In this paper, we investigate the effects of parameter estimation error on the development of the phase I control chart and further compare three methods with various estimators for the process standard deviation. Then, we tabulate the modified adjustment magnitudes for the dynamic Cpk under various subgroup sizes and sample sizes. On the basis of the simulation results, the proposed method can avoid overestimating the true process capability and make more trustworthy decisions. Finally, for demonstration purposes, an application of process capability measures for a sawing process is presented.

Departmental action limits for TQA energy variations defined by means of statistical process control methods.

The purpose of this study is to define departmental action limits for energy percentage variation measured by means of step-wedge helical Tomotherapy quality assurance module. Individual charts using the Statistical Process Control techniques have been used to identify retrospectively out-of-control situations ascribable to documented actions performed on the Tomotherapy system. Using the in-control data of our analysis process capability indices (cp, cpk, cpm and cpmk) are calculated in order to document the real working condition of the Tomotherapy system. Our findings indicate use of an action limit of 1.0% for energy percentage variation difference between the measured and reference output is a good working condition of a Tomotherapy system. cp and cpk indices are suggested as good indices that correctly report the system capability. A method for calculating and reporting Tomotherapy action limits for the integrated self-checking TQA energy check was shown in this study. SPC technique has proven to be efficient in defining departmental action limits from retrospective data for TQA energy measurements, hence optimally enabling corrective improvements in the process of quality assurance.

Determination of multiple dependent state repetitive group sampling plan based on the process capability index

In this article, we propose a sampling plan called a multiple dependent state repetitive group sampling plan for variable inspection based on the process capability index Cpk. The proposed sampling plan is applicable for the inspection of normally distributed quality characteristics when both mean and variance are assumed to be unknown. This new plan under variable inspection will be very useful particularly in compliance testing. Tables are also constructed for the determination of optimal parameters for easy selection and implementation of the plan. The optimal parameters can be determined by using the approach of two points on the operating characteristic curve. Symmetric and asymmetric cases based on the fraction nonconforming by the lower and the upper specification limits are also considered. Advantages of the proposed plan are also discussed. It is also shown that the proposed plan outperforms other existing sampling plans under variable inspection.