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A Bibliography of the Literature on Process Capability Indices: 2000-2009
Abstract
This paper contains a bibliography of approximately 530 journal papers and books on process capability indices for the period 2000–2009. The related literature is classified into four major categories, namely, books, review/overview papers, theory- and method-related papers, and special applications. Theory- and method-related papers are further classified into univariate and multivariate cases, and special applications include acceptance sampling plans, supplier selection, and tolerance design and other optimizations. Copyright © 2010 John Wiley & Sons, Ltd.
... Process capability analysis focuses on evaluating the ability of process that makes products or services to meet given specifications. Such an ability is usually measured by process capability indices [22][23]. Certain scholars suggested that process capability can be estimated when PCI is introduced into the parameter and tolerance design stages [24][25][26]. ...
... In recent decades, the PCI was an important approach used in on-line quality management. Various PCIs have been proposed from the viewpoints of product specification and quality loss [22][23]. Juran [1] first introduced the concept of capability ratio and proposed the index p C to compare the evaluation of process output with the tolerance range of design. ...
The process capability index (PCI) is widely used in on-line quality control stage for measuring and controlling the quality level of a production process. The calculation of PCI requires a large number of samples, but in the off-line quality control stage, a certain production process in off-line quality control stage only has a few individual observations. From the perspective of quality loss and tolerance cost, this paper proposes a parameter and tolerance economic design approach for multivariate quality characteristics based on the modified PCI with individual observations. The response surface models of mean and variance are constructed using individual observations, and exponential models are fitted according to the tolerance cost data of design variables. A modified PCI is proposed with consideration of three types of quality characteristics. The optimal design variables and tolerances are obtained by a comprehensive optimization model that is constructed based on the proposed PCI. An example of an isobutylene-isoprene rubber (IIR) inner tube is used to: (i) demonstrate the implementation of our proposed approach, (ii) improve the PCI value and reflect the sensitivity of the deviation between process mean and specification, and (iii) reduce the risk of increasing cost of quality caused by replicated experimental design and some other unknown reasons.
... Yum and Kim [76] compiled a list of approximately 530 journal articles and books on process capability assessment published between 2000 and 2009. The development between 2002 and 2006 was described in their paper by Wu et al. [77]. ...
This study aims to highlight the importance of a systematic approach to process capability assessment and the importance of following a sequence of steps. Statistical process control provides several different ways of assessing process capability. This study evaluates the process capability of crown cap manufacturing through capability indices. In addition to calculating the indices, the evaluation involves extensive data analysis. Before calculating the capability indices, the assumptions for their correct selection and use were also verified. Several statistical tests were used to verify each assumption. The research value of the study lies in pointing out that not all tests led to the same conclusions. It highlights the importance of selecting the appropriate test type for the evaluated process quality characteristics.
... (Wu et al., 2009) discussed the developments on PCIs between the years 2002 and 2006. (Yum and Kim, 2011) provided a bibliography of the literature on PCIs for the period 2000-2009. A comprehensive study is also performed by (de-Felipe and Benedito, 2017) for univariate and multivariate PCIs. ...
When a process is statistically under control, one may be interested in assessing the process performance based on the specification limits provided by the customer. This evaluation is referred to as process capability analysis. Manufacturing operations are often involved with multistage processes, in which the output of a stage is the input of its subsequent stage. This property is known as the cascade property. Existing methods in capability analysis studies are not applicable when a process or product is represented by profiles. This study presents a method to conduct process capability analysis in a multistage process when quality of a product or process is characterized by a simple linear profile. The performance of the proposed method for a two-stage process is evaluated by numerical simulation using an example from the literature. The results indicate that the proposed method eliminates the effect of the cascade property for different shift sizes and autocorrelations.
... Dey et al. (2023) introduced a novel process capability index denoted as ′ pmc , which relies on an asymmetric loss function (linear exponential) and a tolerance cost function. For a more comprehensive bibliography of the literature on process capability indices, readers are referred to Yum and Kim (2011) and Yum (2023). ...
We consider the process capability index C pmc when a tolerance cost function is introduced. It is well known that C pmc performs well under the general assumption that the data is not contaminated. Under this assumption, the standard sample mean and sample variance are used to estimate C pmc. However, it is also well known that this estimate is extremely sensitive to data contamination since the sample mean and sample variance have a zero breakdown point. This in turn makes any constructed confidence interval (CI) also very sensitive to data contamination. In this paper, we develop robust estimators of the process capability index along with robust CIs. We compare the performance of the proposed estimators of C pmc using the notion of statistical power and receiver operating characteristic curves. Finally, we investigate the use of bootstrapping approaches for improving power of associated hypothesis tests. The results clearly indicate that, when data contamination exists, the methods together with bootstrapping outperform the conventional method.
... This assessment can be conducted by performing a capability analysis [29]. The fields of application are vast, as described in three reviews presenting the references dealing with capability analysis from 1992 to 2000 (170 references) [30], from 2000 to 2009 (530 references) [31] and from 2010 to 2021 (1080 references) [32]. The increasing number of references shows the importance of the subject of capability indices. ...
The performance assessment of additive manufacturing (AM) printers is still a challenge since no dedicated standard exists. This paper proposes a systematic method for evaluating the dimensional and geometrical performance of such machines using the concept of machine performance. The method was applied to an Ultimaker 2+ printer producing parts with polylactic acid (PLA). The X and Y axes of the printer were the most performant and led to narrower potential and real tolerance intervals than the Z axis. The proposed systematic framework can be used to assess the performance of any material extrusion printer and its achievable tolerance intervals.
... The classical and Bayes estimates of PCIs are obtained by [12,13] for generalized exponential and normal distributions, respectively. For an expository review, the reader may follow the following articles of bibliography of the literature on PCIs, viz., [25], [38], [48] and [1]. ...
A meaningful subject of discourse in manufacturing industries is the assessment of the lifetime performance index. In manufacturing industries, the lifetime performance index is used to measure the performance of the product. A generalized lifetime performance index (GLPI) is defined by taking into consideration the median of the process measurement when the lifetime of products follow a parametric distribution may serve better the need of quality engineers and scientists in industry. The present study constructs various point estimators of the GLPI based on progressive type II right censored data for the Lindley distributed lifetime in both classical and Bayesian setup. We perform Monte Carlo simulations to compare the performances of the maximum likelihood and Bayes estimates with a gamma prior of C Y (L) under progressive type-II right censoring scheme. Finally, the validity of the model is adjudged through analysis of a data set.
... To assess a process's capability, industrialists use process capability analysis, which involves using PCIs to provide a numerical representation of the process's capability within specified specification limits (Kotz & Johnson, 2002;Yum & Kim, 2011). Numerous studies have been conducted on PCIs, including an example such as quality characteristic X following a normal distribution with lower and upper specification limits (i.e., LSL and USL). ...
Acceptance sampling plans are a commonly used statistical technique in quality control and assurance applications, enabling producers and consumers to make informed decisions regarding the acceptance or rejection of product lots. In contrast to traditional lot-by-lot sampling plans, skip-lot sampling plans only inspect a fraction of submitted lots and have been shown to be an efficient sampling strategy, particularly when product lots from suppliers exhibit consistent stability and high yield. This study proposes a skip-lot sampling plan by variables inspection using an advanced process capability index that combines the benefits of yield-based and loss-based indices. A mathematical model is developed to determine the plan’s parameters, with the goal of minimizing the average sample number (ASN) required for inspection while accounting for quality and risk constraints specified by both business partners. The proposed plan’s operating characteristics curve and ASN curve are thoroughly examined and compared to traditional plans. Additionally, a case study is presented to demonstrate the practical application of the proposed plan. This study also highlights the benefits of the proposed plan and emphasizes the importance of optimizing the sampling strategy for lot sentencing to improve quality control and reduce inspection costs.
... A literature review of PCIs is given by Kotz and Johnson (2002), while Anis (2008) provides an expository review of PCI. Spiring et al. (2003) and Yum and Kim (2011) provides some helpful bibliography. Ali and Riaz (2014) studied generalized PCI from Bayesian point of view under a simple and mixture of generalized lifetime models. ...
... Therefore, numerous ASPs by variables inspection have been developed for customized circumstances in the past decades to assist examiners in conducting suitable and reliable judgments on submissions (Lam et al., 2006;Santos-Fernández et al. 0.2014;Balamurali & Usha, 2015;Yen et al., 2015;Usha & Balanurali, 2018;Ramyamol et al. 2019). Besides, process capability indices (PCIs) are popular-used tools in industries for measuring the performance or process yield, which is a function of process information and customized specifications (Pearn & Kotz, 2006;Wu et al., 2009;Yum et al. 2011). Accordingly, PCI-based ASPs have been well-studied for various business environments, such as in Aslam et al. (2013a), Lee et al. (2016), Arizono et al. (2020), Marziyeh et al. (2021), Wang and Wu (2021), Liu et al., (2022), Wu et al. (2023), and Wang et al. (2022). ...
As the process yield has significantly raised because of the advanced development of manufacturing technology today, engineers would logically attempt to inspect fewer sample items for the quality evaluation of processes or products. Therefore, in this paper, an efficient sampling inspection method based on the process yield index Spk is developed for lot sentencing, wherein the inspection is performed only on a fractional submitted lot rather than examining every following submission. Both the average sample number (ASN) and operating characteristic (OC) functions of the proposed method are derived on the basis of the Markov chain technique. Further, an optimization model that minimizes the ASN and constrains two OC functions restricted to given quality requirements and tolerable risks is constructed. Performance comparisons in terms of economy and discriminatory power are analyzed by contrasting ASN and OC curves with existing Spk-based methods under the same quality conditions to emphasize the superiority of the proposed method. For easy implementation, we prove the applicability of the proposed method by demonstrating a case study taken from an integrated circuit packaging company.
... Computation of all these indices requires assumption that the quality characteristic is a continuous variable and follows normal distribution. The details about these indices are available in Kane (1986), Kotz and Johnson (2002), Yum and Kim (2011), Chen et al. (2017) and Yum (2023). The generalization of these indices for continuous but non-normal variables is suggested by Clements (1989), Chen (2000), Kovarik andSarga (2014), andSafder et al. (2019). ...
For assessing capability of a normal process with upper specification limit (USL) conventionally Cpu index is estimated to facilitate better decision making in product and process management. But, in practice, many quality characteristics having USL only, e.g. count data, proportion defective etc. are discrete and follow Poisson or binomial distributions. Some unconventional indices (e.g. Cu , Cfu ¸ Cpcu and Cpyu) are proposed in literature for assessing capability of Poisson or binomial processes. Due to legacy of usages of Cpu index and its interpretations, a user of an unconventional index often tends to interpret its values with reference to the values of Cpu for the bad, good or highly capable normal processes, and get a false impression about the capability of the concerned Poisson or binomial process. In this paper, the key features of those unconventional indices are highlighted and then some numerical analysis is carried out for assessing the interpretation issues associated with these unconventional indices. The results of these analyses reveal that although there is no interpretation issue for the unconventional index Cu , there are serious interpretation issues with all other unconventional indices. The mathematical relationships of estimates of other unconventional indices with the estimate of Cu index are established. It is recommended to convert the estimates of other unconventional indices into estimated Cu value using those relationships before any decision making. Otherwise, users of the other unconventional indices may inadvertently be led to erroneous decision making.
... In this context, given that there are eight estimators to estimate the same parameter, it is important that these estimators are accurate, efficient and consistent, which are desirable properties of estimators. Among the identified works, in addition to others listed by Yum & Kim (2011), no articles were identified that analyzed and compared the consistency of process capability indices with normal distribution. ...
Capability analysis seeks to estimate the probability that a process will produce compliant products. The capability indices are dimensionless parameters that measure how well the process can meet specifications. In the literature, eight capability indices are listed, among others, considering a stable process under statistical control and based on the normal probability distribution,
defined by: Cp, Pp, Cpk, Ppk, Cpm, Ppm, Cpmk, and Ppmk. Basically, the index formulas differ in the calculations of the variability within and total, and of the shifts of the mean in relation to the nominal value and the nearest specification limit. The objective of this article was to compare these capacity indexes, and for that, it was chosen the most consistent estimator, that is, the one that improved the
accuracy and efficiency as the number of observations increased. Thus, a simulation of 30,000 values of a normal random variable with a mean equal to zero and a standard deviation equal to one was performed. This made it possible to sample this process 1,000 times using 5, 10, 15, 20, 25, and 30 rational subgroups with individual observations or sample elements. Subsequently, 20 mean
shifts were provoked, with values ranging from 0.1 to 2 and varying by 0.1 unit. According to the results, it was concluded that the indexes Cpk and Ppk were the most consistent in presenting higher accuracy and efficiency for at least 15 rational subgroups or sample elements, regardless of the magnitude of the mean displacement in relation to the nominal value.
Keywords: Capability index; Estimator; Quality control
... For more details, see the references. [5][6][7][8][9][10][11][12][13] More recent studies are considered by the references. [14][15][16][17] To model PCIs with uncertainty, Yalçin and Kaya 14 incorporated the fuzzy set theory. ...
We consider the process capability index, a widely-used quality-related statistic used to assess the quality of products and performance of monitored processes in various industries. It is widely known that the conventional process capability indices perform well when the quality process being monitored has a normal distribution. Unfortunately, using the indices to evaluate a non-normally distributed process often leads to inaccurate results. In this article, we consider a new process capability index, , that can be used in both normal and non-normal scenarios. The objective of this article is threefold: (i) We provide a corrected form of the confidence interval for. (ii) We compare the performance of three non-parametric bootstrap confidence intervals for. Specifically, the standard bootstrap, percentile bootstrap, and bias-corrected percentile bootstrap. Under various distributional assumptions such as the normal, chi-square, Student , Laplace, and two-parameter exponential distributions , the estimated coverage probabilities and average width of the confidence intervals and bootstrap confidence intervals for are compared. (iii) The power of the respective bootstrap approaches is evaluated by using the equivalence relation between confidence interval construction and two-sided hypothesis testing. We also provide the receiver operating characteristic curves to evaluate their performance. Finally, as an illustrative example, an actual data set related to groove dimensions (in inches) measured from a manufacturing process of ignition keys is re-analyzed to illustrate the utility of the proposed methods.
... One can find extensive literature dealing with PCIs. For example, Kotz and Johnson (1993), Tang and Than (1999), Spiring et al. (2003), Mathew et al. (2007) and Yum and Kim (2011) for a bibliography of papers up to the year 2010. For brevity of space, the reader is referred to Anis (2008) for a comprehensive review of the four basic indices, namely C p , C pk , C pm and C pmk . ...
The process capability index (PCI) has been one of the most useful indicators for evaluating the capability of a manufacturing process. Since PCI is based on sample observations, it is essentially an estimated value. Hence, it is natural to think of a confidence interval (CI) of the PCI. In this paper, bootstrap confidence intervals and highest posterior density (HPD) credible intervals of non-normal PCIs, C N pmk , CNpm, C N pk and CNp are studied through simulation when the underlying distribution is two parameter logistic-exponential (LE). First, maximum likelihood method is used to estimate the model parameters and then three boot-strap confidence intervals (BCIs) are considered for obtaining CIs of non-normal PCIs, C N pmk , CNpm, C N pk and CNp. Next, Bayesian estimation is studied with respect to symmetric (squared error) loss function using gamma priors for the model parameters. In order to assess the performance of BCIs and HPD credible intervals of C N pmk , CNpm, C N pk and CNp with respect to average width, coverage probabilities and relative coverage, Monte Carlo simulations are conducted. Finally, a real data set, related to weight of the rubber edge of the speaker driver has been analyzed for illustrative purpose.
... The idea of 101-2 quality loss was used in the capability index; it takes into account production item variation in relation to the target value and manufacturing requirements. More details for the readers who are interested could be found in Yum and Kim (2011) Wu et al., (2009) and Kotz and Johnson (2002). Therefore, to validate the flexibility of the SkSP-2 plan and to address the question of process centering, this article incorporate the SkSP-2, as variable-type, with the recognized Taguchi capability index ( ) using single sampling plan as reference plan, to meet the essential quality characteristic of products with bilateral specification limits, where the suggested sampling scheme are symbolised by -SkSP-2. ...
... Other discussions focusing on the generalized PCIs could be applied to both normal and non-normal cases, see Erfanian and Gildeh, 7 Lovelace and Swain, 17 Yum and Kim, 18 Pina-Monarrez et al., 19 Shi et al., 20 Saha et al., 21 and many others. In addition, there were also some scholars who discussed the robust estimation of PCIs, Abu-Shawiesh et al. 22 compared the performances of modified confidence intervals based on robust scale estimators with classical confidence interval for the process capability index when the process has a non-normal distribution. ...
Since sampling variation would lead to the inaccurate assessment of process capability indices (PCIs), the interval estimation of PCIs has received considerable attention recently. The coverage probabilities (CPs) of the widely used bootstrap confidence intervals (BCIs) of PCIs are not sufficiently close to their nominal confidence level. Moreover, the bootstrap method is time‐consuming. This paper develops a procedure for constructing generalized confidence intervals (GCIs) of two widely used percentile‐based PCIs for the Birnbaum–Saunders (BS) distribution. A simulation study is conducted and the results indicate that the proposed GCI outperforms its bootstrap counterparts in terms of the CPs, the average widths (AWs) of the confidence intervals, and the variability of the interval widths. Finally, two real examples are used to illustrate the implementation of the proposed procedure.
... Several PCIs have been recommended in literature to provide quantitative measures on process capability and performance. To date, several review studies have been published on the PCIs, see, for example, Kotz et al. (2002); Spiring et al. (2003); Wu, Pearn, and Kotz (2009) ;Yum and Kim (2011). The readers may also refer to a review paper by De-Felipe and Benedito (2017) on univariate and multivariate PCIs. ...
Process capability index Cpm, sometimes called Taguchi index, is a widely used measure to assess the ability of a process to cluster around the target. In calculating C_pm, it is assumed that the process data are independent, however in many processes this assumption is violated. Because the exact distribution of estimator Cpm of Cpm
for autocorrelated data is unknown, in this paper we derive explicit expressions for the moments of ~ Cpm and discuss its statistical properties in terms of these moments. The study shows that the estimator ~ Cpm is, in general, positively biased. Further, the exact
values of mean and standard deviations of ~ Cpm are compared with the corresponding approximated values provided by earlier studies and it is found that the earlier studies overemphasize the e�ect of autocorrelation on ~ Cpm, especially for smaller sample sizes, used for calculating ~ Cpm.
... The most widely used other indices are C pk (Kane, 1986), C pm (Hsing and Taguchi, 1985;Chen et al., 2008) and C pmk (Choi and Owen, 1990;Pearn et al., 2005). More detailed information on these indices are available in Kotz and Johnson (1993), English and Taylor (1993), Kotz and Lovelace (1998), Kotz and Johnson (2002), Wu et al. (2009), Yum and Kim (2011), Chen et al. (2017), Polhemus (2017) and De-Felipe and Benedito (2017). Historically, all these indices are developed for a product characteristic that can be described as a continuous variable and follows normal distribution. ...
Many product characteristics are qualitative in nature, e.g. colour, brightness, surface finish etc. The manufacturing process of such products is usually described in terms of fraction nonconforming or conforming which is assumed to follow binomial distribution. Measuring capability of a binomial process implies assessing to what extent the fraction nonconforming or conforming in the continuous stream of lots conform to the specification limits. The Cp or Cpl of a binomial process can be estimated using several approaches. However, these approaches generally give widely varying assessment about the capability of a given binomial process. Consequently, a user of the index may inadvertently be led to erroneous decision making based on an inaccurate estimate of the index. In this paper, a procedure is proposed for assessing accuracies of estimates of Cpu or Cpl obtained by different methods. Subsequently, the best method for evaluating capability of a binomial process is identified based on analysis of multiple case studies, and also the methods giving inaccurate estimates are highlighted. Keywords: Process capability index, binomial process, fraction nonconforming, nonconforming lot (NL), predicted NL%, prediction error
... All these indices are developed for a product characteristic that can be described as a continuous variable and follows normal distribution. The details about these indices are available in Kane (1986), Kotz and Johnson (1993), English and Taylor (1993), Kotz and Johnson (2002), Vannman (2006), Chen et al. (2008), Wu et al. (2009), Yum and Kim (2011), Grau (2012) and Chen et al. (2017). The generalization of these indices for continuous non-normal variables are suggested by Clements (1989), Pearn and Kotz (1994), Pearn and Chen (1995), Shore (1998), Chen (2000), Goswami and Dutta (2013), Wang et al. (2016), Shi et al, (2016), Polhemus (2018) and Chen et al. (2019). ...
... Excellent reviews on them are given by Kotz and Johnson (2002). In addition, Spring et al. (2003) and Yum and Kim (2010) provide an extensive bibliography on process capability indices. ...
... To analyze quality of the product by using manufacturing technology, the process capability analysis can be measured through various stages of the manufacturing process including process, product design, manufacturing, and manufacturing planning; for more details, see Statisti and Tehnike. 1 A number of process capability indices have been proposed to determine whether the manufactured product is capable or not and also product meets the specifications given by the company or satisfies the customer. More literature review on process capability indices can be found in Kotz and Johnson, 2 Spiring et al., 3 Sappakitkamjorn and Niwitpong, 4 Yum and Kim, 5 and Piña-Monarrez et al. 6 Zhang 7 rightly pointed out that the two most frequently used indices are among the available numerous process capability indices. ...
The Gamma distribution has been applied in research in several areas of knowledge, due to its good flexibility and adaptability nature. Process capacity indices like are widely used when the measurements related to the data follow a normal distribution. This article aims to estimate the index for non-normal data using Gamma distribution. We discuss maximum likelihood estimation and a Bayesian analysis through the Gamma distribution using an objective prior, known as matching prior that can return Bayesian estimates with good properties for the. A comparative study is made between classical and Bayesian estimation. The proposed Bayesian approach is considered with the Markov Chain Monte Carlo method to generate samples of the posterior distribution. A simulation study is carried out to verify whether the posterior distribution presents good results when compared with the classical approach in terms of the mean relative errors and the mean square errors, which are the two commonly used metrics to evaluate the parameter estimators. Based on the real data set, Bayesian estimates and credibility intervals for unknown parameters and the prior distribution are achieved to verify if the process is under control.
... Besides, mathematical programming is also engaged to obtain the importance experts' weight, that is why mathematical programming can make a valid output that is optimum in decision-making problems. 37,38 The organization of the current paper is structured as follows. In Section 2, a new framework is provided to show how subjective safety and reliability analysis can put forth the valid results on the basis of multiobjective functions. ...
In the current study, we aim to solve a group decision‐making problem on the basis of expert judgment with consideration of both global and local ignorance in system safety assessment. When data are acquired from various sources, especially subjective ones, the estimation of the data's precise value is challenging. In such a case, the basic belief assignment (bba) function is utilized to cope with the decision‐making procedure on the basis of Dempster‐Shafer theory (DST). DST, as an effective tool for manipulating inaccurate values, stills is not adequate to obtain the optimum values under the consideration of global and local subjective ignorance. To address the aforementioned subjects in this study, an innovative heuristic approach using a multiobjective programming model is introduced to derive the optimal values (bba) in the system safety assessment. An illustrative example is provided in detail to show the applicability and feasibility of the introduced heuristic approach.
... de-Felipe and Benedito (2017) reviewed the literature on univariate and multivariate process capability indices and identified three clusters with which univariate and multivariate process capability indices can be categorized. Yum and Kim (2011) provided a bibliographical study on process capability studies. Wu et al. (2009) reviewed the literature pertaining to the theoretical backgrounds and the applications of process capability indices with a focus on the area of quality assurance. ...
Statistical Process Control (SPC) methods can significantly increase organizational efficiency if appropriately used. The primary goal of process capability studies is to obtain critical information about processes to render them even more effective. This paper proposes a comprehensive framework for proper implementation of SPC studies, including design of the sampling procedure and intervals as well as process capability indices. Some of the most essential process capability indices in the literature were reviewed to develop a methodology to utilize process capability indices within the SPC framework. The current study presents an efficiency-oriented criterion designed for measuring SPC implementation productivity. The framework is applied to the windshield installation process of an Iranian automobile assembly line. The process was sampled in various sessions. Results verify that the implemented SPC framework could successfully improve the process and that the proposed framework could significantly address bottleneck in the process and enhance the quality of the process from the level of satisfactory to excellent according to reference values of process capability indices. Managerial insights are also drawn from results.
... Non-normally distributed processes are not uncommon in practice. Combining this fact with the misleading results of applying basic PCIs to non-normal processes while treating them as normal distributions forced academicians and practitioners to investigate the characteristics of process capability indices with non-normal data (Kotz and Johnson, 2002, Spiring, et al., 2003and Yum and Kim, 2010. There are two approaches adopted in estimating PCI for non-normal process situation include: ...
This study compares the performances of Gini Mean, Clements and Box-Cox transformation methods for estimating process capability Indices when the distribution of the process data is (skewed) non-normal. The use of process performance index (PPI) is implored for process capability analysis (PCA) using Weibull distribution. Data was simulated using R software with a decision interval (target point) of 1.0 and 1.5. Performance assessment was carried out using Boxplots, descriptive statistics and the root mean square deviation. It is observed that Gini mean difference based process capability indices performs best in estimating the process capability indices closest to a set target for varying distribution parameters at different sample sizes, followed by Clements and lastly, the Box-Cox transformation method.
... [23][24][25] In order to formulate the aggregation procedure about the opinions obtained by exports, a mathematical programming can make valid outputs, which are feasible and optimum. 26,27 In this accordance, the main aim of this study is to introduce a heuristic approach based on a mathematical programming as well as an extension to fuzzy set theory. The introduced method has enough capability to eliminate the shortages of available multi-objective decision-making methods and does not contain any of abovementioned disadvantages. ...
Probabilistic risk assessment techniques are the important tools which can considerably improve the safety performance of the studied system and reduce the risk to an acceptable level. Typically, decision‐making process is an important part of risk assessment methods that accordingly bring the ambiguity inside. Decision makers as experts commonly express their subjective opinions about the occurrence of the root events in order to obtain the probability of the undesired event. Subsequently, the critical root events are identified, and possible intervention is performed to reduce the probability of the critical events. However, the serious point is the viability of the obtained probabilities and priority ranking of the critical events. In this study, a heuristic optimization model of linear mathematical programming using triangular intuitionistic fuzzy number (TrIFN) is proposed to obtain the feasible, optimum, and reliable results compared with available methods. The Spearman correlation is performed to examine the reliability and behavior of the proposed model. In order to show the effectiveness of the proposed approach, it is applied on a real case study. The application of the model confirms its robustness to prioritize critical root events over the conventional one.
... The process capability index (PCI) is a quantitative technique widely used in quality control and assurance for assessing process performance and its ability to meet manufacturing tolerances. Numerous types of PCIs have been developed for different industries [15][16][17][18]. The two indices below belong to the first generation of PCIs [19]: ...
Acceptance sampling is a practical quality tool for lot sentencing and is used primarily for incoming or outgoing inspections. The repetitive group sampling plan (RGSP) has been developed and shown to be superior to the traditional single sampling plan. However, RGSP’s lot disposition does not consider the valuable information from preceding lots, which may reduce its discriminatory power and sampling efficiency. Therefore, we propose a modified RGSP for variables inspection using the loss-based capability index. It uses not only the current sample information but also the results of preceding lots to sentence the current lot. The proposed plan’s operating characteristic (OC) function is derived using the estimated index’s exact sampling distribution. An optimization model is formulated to solve the plan parameters by minimizing the average sample number (ASN) required for inspection. The proposed plan’s advantages over the traditional sampling plans in terms of the OC curve and the ASN are addressed. Moreover, plan parameters are tabulated under various conditions and an application example is presented to illustrate its applicability.
... Extensive literature is available on PCI. A detailed review paper by [6] and a bibliography of process capability papers by [7] and Yum and Kim [8] are excellent sources. For further details on PCI see [9]- [23]. ...
This study purposed a process capability index based control chart under the new extended form of multiple dependent state sampling (MDS) named as generalized multiple dependent state sampling (GMDS). The scheme is based on inner and outer control limits and utilizes the previous state of the samples. The performance comparisons of the proposed chart with the existing charts are made by using out-of-control ARL. The simulation study showed the superiority of the proposed chart over the existing PCI based control charts under Shewhart and MDS schemes. An empirical illustration is also given to demonstrate the application of the proposed chart.
Purpose
The paper aims to identify the effect of ignorance of correlatedness among process observations and to implement new sampling schemes; skip and mixed sampling, in order to reduce the effect of autocorrelation on process capability index (PCI) C p m .
Design/methodology/approach
Autocorrelated observations are generated using autoregressive process of order two (AR (2)) using Monte Carlo simulations. The PCI is computed based on these observations assuming the independence. The skip and mixed sampling schemes are then used to form sub-groups among correlated observations. The PCI obtained using sub-groups from skip and mixed sampling schemes are assessed using sample mean and sample standard deviation.
Findings
The paper provides empirical insights into how the effect of autocorrelation decreases in the estimated value of PCI C p m . The use of new sampling schemes, skip and mixed sampling, reduces the effect of autocorrelation on estimates of PCI C p m .
Originality/value
This paper fulfills an identified need to study how to reduce the effect of autocorrelation on PCI C p m .
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 paper, 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.
This is the author's second bibliography on process capability indices (PCIs) and contains approximately 1080 journal papers and books for the period 2010–2021. The related literature is classified into six major categories, namely, books, review/overview papers, theory‐ and method‐related papers, special applications, software packages, and papers omitted in the author's previous bibliography. Theory‐ and method‐related papers are further classified into univariate, multivariate, and functional PCI‐related papers. Special applications include acceptance sampling, control charts, supplier selection, and tolerance design and other optimizations. The present bibliography consists of two parts. Part I contains books, review/overview papers, and univariate PCI‐related papers, while Part II includes multivariate and functional PCI‐related papers, special applications, software packages, and papers omitted in the author's previous bibliography.
This is the second part of the bibliography on process capability indices (PCIs) for the period 2010–2021, and includes multivariate and functional PCI‐related papers, special applications, software packages, and papers omitted in the author's previous bibliography.
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.
In this paper, we shall discuss some statistical properties of the estimator of [Formula: see text] when sample observations are autocorrelated and affected by measurement errors. The presence of autocorrelation in production units is very common in many industries like chemical, food processing, pharmaceutical, paper, and mineral. At the same time some amount of measurement error is invariably present in the sample observations due to inaccurate measurement process. In this paper, we discuss the case of a first-order stationary autoregressive process where measurement error follows a Gaussian distribution. The comparison of the statistical properties of the estimator in this case with the error-free case is the subject matter of this paper.
Statistical quality control is the most useful tool to monitor the process, whether it is the control or not under certain conditions, quality control technique for monitoring the production process through balanced data. In some cases, there are some uncertainties, and unbalanced data are presented. In these circumstances, obtaining the control limit is ineffective for the control charts. This paper deals with obtaining a fuzzy X¯−S chart by using trapezoidal fuzzy number (TpFN). The unbalanced TpFN information for each sample is converted and deals with the fuzzy X¯−S chart. The proposed technique is considered how to deal with the unbalanced control limits. Instead, we conducted a fuzzy process capability performance to measure the process performance. Lastly, we illustrate with an example to show the efficiency of the proposed fuzzy chart for uncertain information.
After a product has undergone a manufacturing process, it usually has several important quality characteristics. When the process quality of all quality characteristics meets the requirements of the quality level, the process quality of the product can be guaranteed to satisfy customers’ needs. A large number of studies have pointed out that good process quality can raise product yield and product value; at the same time, it can reduce the ratio of rework and scrap, achieve the effect of energy saving and waste reduction, and contribute to the sustainable operation of enterprises as well the environment. Since the six sigma method combines the statistical analysis method of manufacturing cost and production data, it is a useful tool for process improvement and process quality enhancement. Therefore, this paper adopted the six sigma-define, measure, analyze, improve and control (DMAIC) improvement process to lift the manufacturing process quality of multi-characteristic products. Besides, the Taguchi process capability index is one of the commonly used tools for quality assessment in the industry. Not only can it reflect the process loss, but it also can ensure the process yield when the index value is large enough. Consequently, this paper discussed the relationship between the Taguchi process capability index and the six sigma quality level. Meanwhile, the entire six sigma DMAIC improvement process was built on the basis of the process capability index and developed by the method of statistical quality control. Hence, the proposed method is very convenient for process engineers to apply, as well as is helpful for enterprises to move toward the goal of smart manufacturing and sustainability.
Process yield has been a standard metric for measuring the capability and performance of manufacturing processes. Process capability index Spk, a concise unit-less gauge with yield-sensitive functionality, communicates succinctly the genuine process yield for normally distributed processes. However, in frequentist statistics, the exact sampling distribution of Spk’s natural estimator is intractable. Various frequentist approaches have attempted to address its wide-scale accuracy in statistical inference. Among them, the approach of generalized confidence interval (GCI) has been demonstrated superiority. In this paper, we incorporate Markov chain Monte Carlo (MCMC) algorithms to introduce a Bayesian-type approach. Extensive simulations in comparison of accuracy and precision performances between the Spk’s frequentist and Bayesian inferences are conducted. Concerning coverage rates and average interval widths of the inferential criteria, Spk’s Bayesian MCMC credibility intervals perform better than frequentist GCIs in most cases, particularly, the cases with only a few samples of information acquired from the manufacturing process.
Acceptance sampling, the most widely used technique in practice, provides rules for making decisions on product acceptance with specified quality standards. A quick-switching sampling (QSS) system, which involves a Normal sampling plan and a Tightened sampling plan with switching rules between them, has recently been discussed. This paper proposes a variables QSS system with parameters ( nN, nT, k) based on a third-generation capability index, which takes process yield and quality loss into consideration. A nonlinear optimization model is formulated to determine the designed plan parameters by minimizing average sample number and subject to the quality and risk levels specified by both the producer and the consumer. The comparison study between the designed sampling system and conventional sampling plan is also conducted in terms of common metrics including operating characteristic curve, average sample number, and average run length. Finally, a practical example is presented along with a step-by-step description of the procedure.
Purpose
The purpose of the paper is the construction of confidence intervals for the ratio of the values of process capability index C pm for two processes. These confidence intervals can be used for comparing the capability of any pair of competitive processes.
Design/methodology/approach
Two methods for constructing confidence intervals for the ratio of the values of process capability index C pm for two processes are proposed. The suggested techniques are based on a two-step approximation of the doubly non-central F distribution. Their performance is tested via simulation.
Findings
The performance of the suggested techniques seems to be rather satisfactory even for small samples, as illustrated through the use of simulated data.
Practical implications
The practical implication of the suggested techniques is that they can be implemented in real-world applications, since they can be used for comparing the capability of any pair of competitive processes.
Originality/value
The paper presents two new methods for constructing confidence intervals for the ratio of the values of process capability index C pm for two processes.
This paper covers a hitherto unresolved problem, the search for a practical method for the Cpk^ control chart. The simplicity of the proposed method facilitates its immediate application. This control chart method reduces the complexity of the current paradigm for capability and quality control studies, while increasing sensitivity to out-of-control signals and improving the early detection aspect, as the ARL results show. The concept of distance implicit in Cpk definition is used to derive a new method based on chi-square distribution. The goodness of fit of the derived expression, which approximates the distribution of the Cpk^ statistic of the proposed method to the reference distribution of the simulated values and to the exact probability density function, was confirmed graphically, through the comparison of the histograms and the probability density functions, and analytically, using the F-test.
In today’s manufacturing environment, the rate of defective products has been continuously decreasing; thus, variables sampling plans with process capability indices (PCIs) have been recommended to gather more information about a manufacturing process and reduce required sample sizes for inspection. In particular, skip-lot sampling plan (SkSP) is suitable for a series of lots having stable and excellent product quality. Moreover, the concept of repetitive group sampling (RGS), which can allow the use of less samples to maintain desired protection to producers and consumers, is especially appropriate where inspection or testing is costly or destructive. This study, by incorporating the advantages of PCIs, SkSP, and RGS, constructs a variables SkSP with RGS as the reference plan (called SkSP-RGS) based on one-sided PCIs for products with a unilateral specification limit. The proposed plan reduces the sample size while achieving a similar discriminatory power, compared with a conventional variables single sampling plan (SSP), a RGS plan (RGSP), and a SkSP of type 2 (SkSP-2). Tables of plan parameters are provided for frequently applied quality and risk requirements so that practitioners can easily apply the proposed plan.
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.
Statistical process control is an effective quality control technique to monitor a production process with balanced data under certain conditions. However, there are some situations where dealing with uncertainty and unbalanced data is considered. In such situations, the traditional statistical control charts are not effective to obtain control limits. The aim of this paper is fourfold. First of all, the collected unbalanced data are converted to triangular fuzzy numbers for each sample. Second, this paper develops a fuzzy X¯-S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bar{X} - S $$\end{document} control chart while dealing with unbalanced fuzzy data. Third, a proposed approach is presented on how to deal with unbalanced fuzzy data for calculations of control limits. Besides, fuzzy process capability analyses are conducted to measure process performance. Finally, an illustrative example is conducted to show the effectiveness of the proposed fuzzy X¯-S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bar{X} - S $$\end{document} control chart with unbalanced data for uncertainty.
Process capability indices are widely used to check quality standards both at the production level and for business activity. They consider the location and the deviation from specification limits and targets. The literature contains many contributions on this topic both in the univariate and the multivariate context. Motivated by a real semiconductor case study, we discuss the role of rational subgroups and the challenge they present in the computation of capability indices, especially when data refer to lots of products. In addition, our context involves a mix of problems: unilateral specification limit, nonsymmetric distribution of the data, evidence of data from a mixture of distributions, and the need to filter one component of the mixture. After solving the previous issues and because of the peculiar characteristics of semiconductor processes based on the so called “wafers,” we contribute to the literature a proposal on how to compute capability indices in the case of heteroscedastic spatial processes. With a generalized additive model, we show that it is possible to estimate a capability surface that allows the identification of regions expected to not be fully compliant with the desired quality standards.
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.
Capability analysis allows evaluating the conformity of the production to the project specifications in industrial processes. Different indices can be used to assess the process capability, among them the Cpm (or Taguchi) index. In this work we propose the estimation of Cpm for normally distributed processes using ranked set sampling (RSS) and two extensions: pair ranked set sampling (PRSS), as an economical alternative; and double ranked set sampling (DRSS), as a more efficient (and expensive) strategy. Also, three different Cpm estimators were considered. Their performances regarding bias, mean squared error, and relative efficiency were evaluated through Monte Carlo simulation. The results indicated that: (i) There was a substantial variation in performances for different Cpm estimators, particularly for small samples; (ii) RSS based estimators outperformed their simple random sampling counterparts; (iii) DRSS estimator presented the lowest mean square error; and (iv) PRSS estimator showed competitive performance to its counterparts in different scenarios.
The process capability analysis is a crucial activity to evaluate if the process outcome meets the design specifications. Classically, such analysis is performed by verifying the in-control condition of the process and evaluating suitable capability indices, by assuming the process in-control steady-state condition. However, the in-control period of the process characterizes only a part of the system functioning cycle, the one with the lower defective rate. In particular, the system functioning cycle is also characterized by the out-of-control period, during which a greater defective rate is produced, and such increasing is not considered by the widely adopted capability indices. As consequence, the classical approaches to perform the process capability analysis involves an overestimation of the process capability level. For this reason, in order to overcome the previously described limitation, in the present paper it is proposed a new capability index based on the real defective rate of the process. Thus, such new index is able to estimate the real process capability level. Finally, in order to compare the new index to the conventional Cp capability index, a numerical comparison study related to a process capability analysis is carried out, and the related practical considerations are given.
Evaluating the capability of a manufacturing process is an important initial step in any quality improvement program. Methodologies have been reported in literature for evaluating capability of processes with quantitative characteristics of different natures. However, quantifying capability of a process with qualitative responses remains a difficult task. In this paper, a novel method is presented for measuring capability of a process with ordinal responses involving three ordered categories. The proposed method makes use of the process area of proportions and specification area of proportions defined in a two-dimensional plane. Extensive simulation studies reveal that in general, the defined process area does not always follow a normal distribution. However, by using Johnson transformation, it can be transformed to normal distribution and thus capability of a process with ordinal responses can be evaluated easily. Application of the proposed method to a real life problem is presented.
In this manuscript, we propose a sampling plan called multiple dependent state repetitive group sampling plan for variables 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 variables 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 non-conforming 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 compared to other existing sampling plans under variables inspection.
Simultaneous Perturbation Stochastic Approximation (SPSA) algorithms are alternative methods for optimizing systems where the relationship between the dependent variables and independent variables of a process is unknown. The objective of this research is to determine the optimum succession measure of SPSA that maximizes the Process Capability Index (PCI) through second order regression models by means of experimental simulation. The results show that three out of the ten combinations of the succession measures evaluated in SPSA yield optimum values that maximize the PCI according to the Six Sigma Methodology (DMAIC—Define, Measure, Analyze, Improve, and Control), this because the values have behaviors classified as world-class, this is, processes that generate less than 3.4 defects per million opportunities, which improves customer satisfaction and reduces cycle time and defects.
Cost is a major driver in many, if not all, of the decisions that a firm must make. Thus, in determining a level of quality improvement (as measured by process capability), cost is a major consideration—especially for those firms that produce limited amounts of a particular product, have a product that is late in the product life cycle, or have budgetary constraints. This article addresses the question of how much should be spent on process improvement with respect to improvement in process capability. Using the Taguchi loss function, a cost/savings model is developed as a decision-making tool for firms with product life-cycle and budgetary constraints.
Process capability analysis often entails characterizing or assessing processes or products based on more than one engineering specification or quality characteristic. When these variables are related characteristics, the analysis should be based on a multivariate statistical technique. In this expository paper, three recently proposed multivariate methodologies for assessing capability are contrasted and compared. Through the use of several graphical and computational examples, the information summarized by these methodologies is illustrated and their usefulness is discussed.
Most products have multiple characteristics, and customers accept products whenever all process capabilities of each characteristic satisfy preset specifications. Obviously, univariate process capability indices cannot meet the requirements stated as above. And Chen et al. (2001) used point estimation to evaluate process capability, even though the method agrees with 100% tests. If the check is sampling, the process capability must consider sampling error. So the research constructs a measuring model of process capability to consider sampling error to evaluate process capability for a multi-process produce based on Cpk which was proposed by Kane (1986).
Research efforts in the area of process capability have largely been devoted to finding a better process capability index (PCI) and to a lesser extent on the stochastic behavior of the estimated PCIs [1], [2]. Much of this development has gone unused for many reasons including a) a plethora of indices, b) interpretation, c) software support, d) standards and e) dissemination. The addition of the indices appears to have had little impact on the practitioners. C p and C pk (including C pl and C pu) [3] continue to be the most heavily used indices with C pm [4] and C pmk [5] occurring occasionally. The addition of stochastic assessments for estimated PCIs is a positive development, however statistical developments have frequently lacked background knowledge and implementation ease, hindering use by practitioners. A case study from the printing industry will be used to draw attention to areas impacting the practical use of PCIs. Concepts including a) establishing effective tolerance limits, b) the ongoing assessment and interpretation of PCIs and c) ongoing improvement will be presented. We will use the case to a) illustrate a strategy followed by practitioners using PCIs as a quality management tool, b) to draw attention to gaps that exist in the practical use of PCIs, c) to illustrate how some of these difficulties were overcome and d) to highlight research areas in the practical use of PCIs. A variety of quality tools including flowcharts/process documentation, control charts, process capability indices and experimental design are illustrated throughout the manuscript. Data used in the analyses has been included where permitted. Although drawn from the printing industry, the tools used, lessons learned and generic achievements are applicable to the wider area of product design and manufacturing, particularly where customers have unique requirements for a common product.
We discuss the pitfalls in using the well known process capability indices like C p , C pk , etc., with angular data. A few new process capability indices suitable for use with angular data have been proposed and their utility is illustrated using a real life example.
The indices Cp and Cpk are extensively used to assess process capability. However, they only take into account the process mean and standard deviation, but not the proximity of the process mean to the target value, T, of the process characteristic. Cpm does take into account the proximity of the process mean to the target value. We propose a method for selecting or judging the better of two suppliers or processes based on a confidence interval for the ratio Cpm1/Cpm2. Four methods of obtaining approximate confidence intervals are presented and compared, one based on the statistical theory given in Boyles (1991) and three based on the bootstrap, (referred to as SB (standard bootstrap), PB (percentile bootstrap), and BCPB (biased-corrected percentile bootstrap)). The performance was compared using simulation, which showed that, in two independent and normal process environments, Boyles's (1991) confidence interval and the SB confidence interval are more reliable than the PB and BCPB methods. A sample size of greater than 50 is recommended for selecting the most capable of two suppliers or processes.
A process is usually defined to be capable if the process capability index exceeds a stated threshold value, e.g., Cpm > 4/3. This inequality can be expressed graphically as a region in the plane defined by the process parameters (μ,σ). In the obtained plot special regions can be plotted to test for process capability. These regions are similar to confidence regions for (μ,σ). This idea of using regions in process capability plots to assess the capability is developed further for the capability index Cpm. A new circular region is constructed that can be used, in a simple graphical way, to draw conclusions about the capability of the process at a given significance level. Using circular regions several characteristics with different specification limits and different sample sizes can be monitored in the same plot. Under the assumption of normality the suggested method is investigated with respect to power as well as compared to other existing graphical methods for drawing inference about process capability.
Engineers are challenged with determining whether a process is capable of producing a product that meet tolerances as specified in the product's design. Process capability studies provide the practitioner two measurements to determine if the process can produce to specifications. This paper uses two case studies to examine the appropriate use of the process capability ratio (Cp) and the process capability index (Cpk). The first case determined if a flexographic printing press was capable of accurately holding consecutive repeats that conformed to specifications. Although successfully achieving good Cp values with a specification range of +/- 1/8 of an inch, the Cp values for +/- 1/16 of an inch indicated that the press was not capable of meeting specifications. The second case determines assignable cause for burn-in failure for two prototype industrial grade compressors. Although the Cp indicated that the process was capable, Cpk's for the target settings were not being calculated and nominal means were consistently being set larger for shafts and smaller for bores than specified.
Process capability indices have been widely used in the manufacturing industry, providing numerical measures on process precision, process accuracy, and process performance. Capability measures for processes with a single characteristic have been investigated extensively. However, capability measures for processes with multiple characteristics are comparatively neglected. In this paper, inspired by the approach and model of process capability index investigated by K. S. Chen et al. (2003) and A. B. Yeh et al. (1998), a note model of multivariate process capability index based on non-conformity is presented. As for this index, the data of each single characteristic don't require satisfying normal distribution, of which its computing is simple, and will not fell too theoretical. At last the application analysis is made.
Complete characterization of a complex part or process requires many capability indexes. As such, one can reduce the number of capability indexes of a complex part to one. This is an effective way to measure producibility and single out the part's worst feature.
Process incapability index Cpp has been introduced to the manufacturing industry to measure process performance. But all existing methods for testing Cpp require further estimation of the distribution parameters when calculating the p-values and critical values causing additional sampling errors, which is unreliable. In this paper, an efficient SAS computer program is provided to calculate the p-values. Extensive calculations were performed to examine the behavior of the p-values and the critical values c 0 against the distribution parameter. Useful critical values for commonly used capability requirements are also tabulated. A simple but practical step-by-step hypothesis-testing procedure is developed for in-plant applications.
The vast majority of research on capability indices has assumed that the data consists of one large, representative sample. In practice, and in much of the quality control literature, process data are collected over time in subsamples representing rational subgroups. In this paper we examine the statistical behavior of two C pm estimators based on this more realistic data structure. The estimators correspond to pooled and un-pooled variance estimators. The theoretical findings are applied to hypothesis testing and power calculations. The power functions of the tests based on the two estimators are used to determine the minimum number of subsamples needed to meet a threshold requirement that power exceeds 0.80. Extensive tables of the recommended number of subsamples are provided with comments on their usage.
Huang et al. (2002) proposed a multi-process capability analysis chart (MPCAC) based on process capability indices Spk, Cpl, and Cpu to evaluate the integrated process capability for the entire product with the nominal-the-best, the larger-the-better, and the smaller-the-better characteristics. The MPCAC conveys critical information of multiple characteristics of the entire product regarding the process accuracy, the process precision, and the process capability zones from one single chart, which is an effective tool for evaluating product quality. However, from a practical perspective, the MPCAC chart did not consider sampling errors hence the capability information provided from this chart is often unreliable and misleading. In this paper, a manufacturing application involving multiple characteristics of low dropout voltage regulator (LDOVR) is investigated. We consider the sampling error by obtaining the lower confidence bounds (LCBs) of Spk, Cpl, and Cpu. In order to exactly measure the degree of process centering, the LCB of the accuracy index Ca is also considered in this MPCAC. The LCB presents the minimum true capability of the process, which is essential to product capability assurance. We employ the LCBs to MPCAC to provide more reasonably reliable capability assurance for the product with multiple characteristics. Significance: The proposed MPCAC is useful for manufacturing capability assurance of the product with multiple characteristics. The MPCAC prioritizes the order of the processes for further capability improvement effort should focus on, either to move the process mean or reduce the process variation. The developed lower confidence bounds (LCBs) can be used to construct accurate MPCAC providing information regarding the true capabilities, and the corresponding process yields. The MPCAC incorporating with the LCB is applied to the low dropout voltage regulator (LDVOR) for monitoring/controlling product quality.
The statistical comparison of competing manufacturing processes is an important aspect of statistical quality control that aids quality managers In the choice of potential suppliers of products, manufacturing methodologies, or proposed adjustments to the manufacturing process. The selection criteria, in the absence of considerations such as cost, is based on a quality metric, such as the capability of each manufacturing process. Because quality metrics are estimated based on sample process data, the inherent variability in the estimates must be accounted for when selecting the best manufacturing process. In this paper, we consider solutions to this problem based on permutation testing methodology. In the case of two processes, the methodology is based on i simple permutation test of the null hypothesis that the two processes have equal capability. In the case of more than two processes, multiple-comparison techniques are used in conjunction with the proposed permutation tests. The advantage of using the permutation methods is that the significance levels of the permutation tests are exact regardless of the distribution of the process data. The methodology is demonstrated using several examples, and the potential performance of the methods are investigated empirically.
One element of tolerance analysis is the formula used to relate product tolerances to component tolerances. This paper discusses deficiencies with traditional tolerancing, outlines a simple procedure for converting process capability information into an improved tolerancing formula tailored to a specific class of products, and describes how this analysis can contribute to substantive improvements in profits by improving tolerancing of future products and helping to identify improvement opportunities in production of current products.
A program written in ANSI C that computes a kernel based estimate of the proportion nonconforming is presented. The bandwidth for the kernel estimate is computed using several multistage plug-in methods that have been shown to perform well in practice. These bandwidths, along with estimates of the process fallout rate, are provided by the program.
The factors that govern the tool wear vary with the time rather being constant. In this paper, a model for the residual tool wear process is developed using Glejser test for determining the variations in heteroskedasticity of the process. On the basis of this test, a control chart of the tool wear is developed and process capability indices are calculated. The tool quality life is determined once the minimum process capability indices are known. A method is also developed to determine the tool quality life of automatic machine tools by integrating tool cutting life and knowing the desired quality life.
We suggest that one should focus on the process parameters (y, a) instead of the capability index alone by using process capability plots, both when defining what is meant by a capable process and when deciding whether a process can be considered capable or not. First we introduce the so called the (S, y)-plot, to define a capability region. The (S, y)-plot is based on a capability index but focuses on the parameters (µ, a). We have earlier defined a class of capability indices, containing the indices C
13
, C
pk
, C
pm
, and C
pm
k. By varying the parameters of the class various indices as well as various capability regions with different properties are obtained.Under the assumption of normality we suggest two process capability plots to be used when interpreting the results of the estimated indices in the above mentioned class for deciding whether a process can be considered capable or not. One of these plots is the so called (b“, ÿ)-plot, which is based on the estimated index and is a generalization of the (S, y)-plot. The other is the so called confidence rectangle plot. This plot is based on a confidence region for (S, y), which is plotted in the (S,y)-plot. These plots are compared and discussed from different aspects. An advantage with using a process capability plot, compared to using the capability index alone, is that we will instantly get visual information, simultaneously about the location and spread of the studied characteristics, as well as information about the capability of the process. When the process is non-capable, the plots are helpful when trying to understand if it is the variability, the deviation from target, or both that need to be reduced to improve the capability. In this way the proposed graphical methods give a clear direction for quality improvement.
Most commonly seen process capability indices (PCIs) such as C p, Cpk and Cpm are appropriate only for symmetric specification limits and should not be used for the process with asymmetric tolerances. For asymmetric tolerances, it seems that Cpmk has been the best alternative heretofore. However, Cpmk may not be applicable to the search for the prior candidate to improve. In addition, when the specification limits can be changed, we may not compare the process capability properly between before and after the change point with Cpmk. In this paper a new measure is proposed for asymmetric specification limits, which modifies Cpmk with the numerator min[USL-E(X),E(X)-LSL] replaced by E[min(USL-X,X-LSL)]. The new process capability index, C pio, adds up the contribution of individual observations to the manufacturing process. The properties of C pio are investigated and its estimator, Ĉ pio, is provided. It is proved that √n( Ĉ pio-Cpio) has an asymptotic normal distribution under the assumption that the fourth moment is finite. Finally two examples of its application are included.
For the multivariate manufacturing processes, tremendous difficulties are often encountered when one attempts to measure the process capability by directly extending the univariate approach, that is, comparing the specification limits (tolerance zones) with the actual process spread. In fact, the existing multivariate process capability indices developed along the same line are very complicated to apply even under the normality assumption. The authors of this paper propose a new multivariate process capability index that is directly related to the proportion of nonconforming items. Moreover, the new index is calculated in a nonparametric setting; hence, it does not rely on a particular distribution. The estimation methods of the new index are studied in detail. Simulations for the elliptical and rectangular tolerance zones under different bivariate distributions are carried out to illustrate the new approach. The applications of the new index to real-life examples are also presented.
In today's competitive business environment, it is becoming more crucial than ever to assess precisely process losses due to non-compliance to customer specifications. To assess these losses, industry is widely using process capability indices for performance evaluation of their processes. Determination of the performance capability of a stable process using the standard process capability indices requires that the underlying process data should follow a normal distribution. However , if the data is non-normal, measuring process capability using conventional methods can lead to erroneous results. Different process capability indices such as Clements percentile method and data transformation method have been proposed to deal with the non-normal
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 C(p), C(pk), C(pm), and C(pmk) 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 C(p), C(pk), C(pm), and C(pmk), and can be used without difficulty in industry.
Chen et al. [5] proposed a process capability analysis chart (PCAC) based on process capability indices Cpk, Cpl, and Cpu to evaluate the integrated process capability for the entire product with the nominal-the-best, the larger-the-better, and the smaller-the better characteristics. The PCAC conveys critical information of multiple characteristics of the entire product regarding accuracy, precision, and the process capability zones from one single chart, which is an effective tool for evaluating product quality/reliability. However, from a practical perspective, the PCAC chart is never considered sampling errors hence the capability information provided from this chart is often unreliable and misleading. In this paper, a manufacturing application involving multiple characteristics of precision voltage reference (PVR) is investigated. We consider the sample error by obtaining the lower confidence bounds (LCBs) of Cpk, Ca, Cpl, and Cpu. The lower confidence bound presents the minimum true capability information of the process, which is essential to product reliability assurance. We employ the lower confidence bounds to PCAC to provide more reasonably reliable capability information for the product with multiple characteristics.
Process capability indices (PCIs) for processes with symmetric tolerances have received substantial research attention. But, PCIs for processes with asymmetric tolerances have been comparatively neglected. Recently, Boyles (1994) reviewed the existing PCI literature and proposed several new indices to handle processes with asymmetric tolerances. In this paper we analyze PCIs based on various process characteristics, then introduce a new class of capability indices to handle processes with asymmetric tolerances. The proposed new indices are compared with existing PCIs in terms of process yield, process centering, and process characteristic related to loss functions. The results indicate that the new indices are superior to the existing capability indices, and provide greater accuracy in current applications using PCIs to measure process potential and performance.
In this research, we consider the maximization of process capability as the criterion in product/process design that is used for selecting preferred design factor levels and propose several approaches for single and multiple response performance measure designs. All of these approaches assume that the relationship between a process performance measure and a set of design factors is represented via an estimate of a response surface function. In particular, we develop; (i) criteria for selecting an optimal design, which we call MCpk and MCpm; (h) mathematical programming formulations for maximizing MCpk and MCpm, including formulations for maximizing the desirability index (Harrington, 1965) and for maximizing the standardized performance criteria (Barton and Tsui, 1991) as special cases of the formulation for maximizing MCpk, (iii) formulations for considering cost when maximizing MCpk and MCpm, (iv) a means for assessing propagation of error; (v) a robust design method for assessing design factor effects on residual variance; (vi) a means for assessing the optimality of a proposed solution: and (vii) an original application in the screening of printed circuit board panels.
Chen (1998) proposed an adjusted incapability index Cnpp to handle processes with both symmetric and asymmetric tolerances. The incapability index Cnpp provides an uncontaminated separation between information concerning the process accuracy and the process precision. In order to test whether a normally distributed process is capable or not using the theory of testing statistical hypothesis, we first derive explicit form for the probability density function of the natural estimator of the index Cnpp. Then, we develop a decision-making rule based on the estimated index Ĉnpp.
In this paper, we study the properties of the sequential fixed-width estimation of the quality index Cpm where 2d is the width of the fixed-width interval. We study the properties of the fixed-width stopping rule and the sequential estimator of the quality index Cpm. We show that this fixed-width rule terminates finitely when d > 0. Also, under some reasonable conditions, we show that this fixed-width sequential estimation is asymptotically efficient and asymptotically consistent as d tends to 0.
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.
Theoretical and simulation results are derived for the posterior distribution of the process capability index C pk under two different but related prior distributions. For the conventional prior, the exact posterior moments of the index are calculated. Using these moments, Pearson curve and Cornish-Fisher approximations to the posterior distribution are obtained for a real problem. Gibbs sampling is used in the case of the reference (probability matching) prior to obtain the unconditional posterior distribution of C pk . Finally, it is shown that the probability matching prior for the distribution function (a function of the index) is the same as the probability matching prior for the index itself.
Several process capability indices have been proposed to provide a unitless measure to detect whether a process is capable of producing items satisfying the quality requirement preset by the product designer or consumer. In this paper, we develop a procedure of entire product capability indices for measuring a product capability with many quality characteristics. Also, we provide a simple and convenient minimum value of the estimator of PCI, for which process is capable at least 100(1-α)% of the time, and justifying whether the process's potentiality and performance reaching consumer's expected specification.
Service quality is important when selecting a banking service. Many factors affecting service quality are not measurable, and it is difficult to consistently evaluate front-line personal operation performance in the banking system. Consequently, most of the banks use Methods Time Measurement (MTM-C) to improve banking operation quality and reduce customer service time.
From the perspective of consumers, standardizing operation time is an important criterion when selecting a banking service, and the specification of banking service quality is in accordance with the process capability index Cpu applied in the manufacturing industry. The Cpu index is a powerful and effective tool for evaluating product quality or process performance. Hence, this paper attempts to measure banking operation quality via the process capability index Cpu, by setting standard operation times as an upper specification limits for each front-line operation item, and to provide useful information for the operations performance management in the banking system
A process capability index is a function of some parameters of some involved distribution and it specifies a quality characteristic. The process capability indices C p , C pk and C pm have been used in manufacturing industries to provide a quantitative measurement of the potential performance of the manufacturing. These indices are easy to apply. Most studies on process capability indices are focusing on point estimates, which may result in misleading assessments of process performance. In this paper, we consider C p , C pk and C pm , and develop a simple and convenient procedure for decision making in assessing process capability.
In the banking service system, the uncertainty of customer demands, the service encounters, and customer queries are the major causes that affect and interfere the stability of tellers’ performance. The causes are classified as “noise” to enlarge the variability of tellers’ performance and these are the sources why the service time in the system for customers is fluctuant. One of the ways to enhance the service quality of the banking system is customer satisfaction; and the customer satisfaction can be achieved by way of both shortening the expected service time in the system and realizing the operation efficiency and stability of the service system. Good personnel, advanced hardware and effective evaluation tools are to sustain the stable and efficient service of the banking system. These special quality attributes are similar to the concept of the signal-to-noise ratio. Based on the ratio, we propose a service index to evaluate the operational effectiveness of the tellers and the entire banking system. Procedures are also provided. Through simple calculation of the average and standard deviation of the service time, and referring to the tables, the operational effectiveness of the tellers and the entire banking system are evaluated. The subjective and efficient methodology that we propose can be easily applied to the daily operation management of the banking system.
Process Capability Indices (PCIs) have proliferated in both use and variety during the last decade. Many statisticians and quality control engineers studied the indices of processes so that the precision of assessing the quality and efficiency of a process can be enhanced. However, these studies depend heavily on the assumption of normal variability. Process data do not always follow a normal distribution. A one-sided specification limit is an immediate clue that the data might be non-normal. If the underlying distributions are non-normal, then the capability calculations are highly unreliable since the conventional estimator S 2 of σ 2 is sensitive to departures from normality, and estimators of those indices are calculated using S 2 . Therefore, those basic indices are inappropriate for processes with non-normal distributions. Thus we propose an index C Npl to evaluate the case, where the underlying distributions may not be normal, provide comparisons between C pl and C Npl indices and present a case study to illustrate how the index C Npl may be applied to actual data collected from factories.
Process Capability Analysis (PCA) is a powerful tool to assess the process capability for products that meet certain specifications. K. S. Chen et al. [Int. J. Prod. Res. 39, 4077-4087 (2001)] modify S. C. Singhal’s [Qual. Eng. 4, 75-81 (1991)] method to propose a Process Capability Analysis Chart (PCAC) to evaluate process capability for an entire product composed of smaller-the-better specifications, larger-the-better specifications and nominal-the-best specifications. However, C pk cannot reflect the specific process yield. In this paper, the index C ps is selected to replace C pk , reflecting an existing one-to-one mathematical relationship with the process yield. The integrated process capability index for the entire product is proposed. An example is given to illustrate the application of the modified PCAC.
The rapidity and quality of official documents process in a government unit is one of the most important characteristics relating to its quality and efficiency of administrative management and performance. We always use the speed and quality of document-process to assess a government unit if it is capable and efficient. In this paper, the document-process means the whole procedure: receiving, dealing, transmitting, and filing a document. We propose an efficiency index to evaluate the efficiency assessment of the document-process. We study the characteristics of the proposed efficiency index and derive the uniformly minimum variance unbiased estimator (UMVUE) of the efficiency index. We show that the UMVUE of the efficiency index is distributed as a non-central t-distribution random variable multiplied by a constant when sampling from the normal distribution. In addition, based on the theory of statistical testing hypothesis we develop a step-by-step procedure using the UMVUE of the efficiency index for practitioners to use in judging whether a document-process meets the efficiency requirement.
Process capability index Cpk is important in manufacturing industry. This paper extends its applications to calculate the process capability index [Ctilde]pk of fuzzy numbers. Unlike previous researches, the α -cuts of fuzzy observations are first derived based on various values of α . The membership function of fuzzy process capability index [Ctilde]pk is then constructed based on the α -cuts of fuzzy observations. Two examples are used to illustrate how to interpret the fuzzy process capability index [Ctilde]pk . When the quality characteristic can not be precisely determined, the proposed method not only provides the most possible value and spread of fuzzy process capability index [Ctilde]pk , but also can be easily applied to the fuzzy number with different types of membership functions. With crisp data, the proposed method reduces to the classical method of process capability index Cpk .
From Taylor’s first order estimate and from expectancy and variance adjustments we obtain confidence intervals for the family of capability indices introduced by K. Vännmen [Comm. Stat., Theory Methods 26, No. 1, 159–180(1997; Zbl 0900.62552); ibid., No. 8, 2049–2072 (1997; Zbl 0955.62648)]. The evaluation of confidence interval accuracy is achieved from a simulation study in which the covering percentage and the confidence interval length for the asymptotic, bootstrap percentile and bootstrap-t methods are analysed.
The two well-known unilateral specification process capability indices CPL and CPU are popularly used for measuring the process capability of larger-the-better type or smaller-the-better type products in the manufacturing industry. The practitioners can use them to determine whether their processes meet the capability requirement and select the better supplier. Under normal assumption, Chou (1994) developed an effective procedure using estimators of CPL and CPU for practitioners to determine whether two processes have equal capability or not. His works will be extended to the case of k processes. We take the tests of testing the equality of coefficients of variation from k populations and a new proposed test, the Modified-Bartlett's test, to determine whether k processes have equal capability or not. By Monte Carlo experiments, we have demonstrated that Modified-Bartlett's test has appropriate empirical size and has higher power than others. We also present a testing procedure that is convenient for practitioners to select the better supplier step by step.
Process capability indices have proliferated in both use and variety during the last decade. They can provide the manufacturers with a means to monitor the quality levels of the procedures in process. However, most of the studies associated with analyzing the quality and efficiency of a process are limited to discussing one single quality characteristic of a process capability. In general, quality characteristics can be categorized into three types, the nominal-the-best, the smaller-the-better, and the larger-the-better types. Typical quality characteristics of the larger-the-better type include the hardness, the tensile and the compressive strengths. In manufacturing practices, a process usually contains several models, which result in different specifications. However, it seems that there is still no academic report published for evaluating the process capability of a product family of the larger-the-better type with several models. The objective of this paper is thus aimed at proposing a process capability index for evaluating a product family of the larger-the-better type. The relationship between the index and the yield as well as some statistical features of the index will also be discussed in this paper. Finally, a set of simple procedures for evaluation purpose will also be provided to the industry for reference.
The process capability index C pm is an effective tool to evaluate process capability since the index can reflect a centering process capability adequately. A neuro-fuzzy inference method is proposed in this study to evaluate the process capability based on an estimated process capability index of C pn calculated from sampled data. This method combines the advantages of fuzzy systems and neuro-networks such that a grade instead of sharp evaluation result can be obtained. Inference is based on a fuzzy statement associated with a statistical testing hypothesis formed with α-risk and p-values that is related to the estimated process capability index and a product designer’s concerns. Furthermore, a neuro-network is used to refine the membership function and to generate a Sugeno if-then rule in a systematic way. Results of the inference are described as a score value of the process capability index and are used to represent the grade of the process capability. An illustrated example of ball-point pens demonstrates that the presented method is effective for evaluation of centering process capabilities.
For stably normal processes with one-sided specification limits, the capability indices C PU and C PL have been used to provide numerical measures for product quality assurance from manufacturing perspectives. Statistical properties of the estimators of C PU and C PL have been investigated extensively for cases with one single sample. We consider testing product quality assurance for cases of several groups of samples with unequal sizes. We obtain uniformly minimum variance unbiased estimators (UMVUEs) of C PU and C PL , and develop a powerful test for that purpose. We also implement Fortran programs to compute the p-values, critical values, for testing product quality assurance. A practical procedure using the UMVUEs is provided to assist the practitioners judging whether their processes are capable of reproducing reliable products. An example of voltage limiting amplifiers (VLA) is presented to illustrate the practicality of our approach to actual data collected from real-world applications.
Process capability indices (PCIs) can be viewed as effective and excellent means of measuring product quality and performance. In recent years, most studies are limited to test the process capability of one single quality characteristic, but they are not sufficient for testing the process capability of an entire product with multi-characteristics. In this paper, we apply the well-known incex C pm to propose an integrated product capability index which considers numerical quality characteristics with bilateral specifications. We further show a theorem, which helps us to develop a testing procedure of the integrated product capability index forjudging whether the process capabilities of total quality characteristics meet the customer’s demands. In addition, the relationship between the integrated product capability index and the yield of the entire product are introduced.
We consider a generalization of the process capability index C pk proposed by W. L. Pearn and K. S. Chen [J. Appl. Stat. 25, No. 6, 801–810 (1998; Zbl 0945.62127)] for asymmetric tolerances. We investigate the distribution of the new estimator, and derive its limiting distribution under general conditions for which P{μ≥T}=p is given.