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A novel approach to deriving the lower confidence limit of indices C pu , C pl , and C pk in assessing process capability
International Journal of Production Research
Abstract
Process capability indices (PCIs) are widely used as a measure of process potential and process performance. Unfortunately, the use of sample data to estimate PCIs means that any error in the sampling can introduce considerable uncertainty into the assessment of process capability. This necessitates the use of the lower confidence limit (LCL) in the estimation of minimum process capability. Furthermore, the complexity of sampling distributions of the PCIs greatly hinders interval estimation, such that only an approximate or asymptotic LCL can be achieved. This paper proposes a novel approach to deriving the
LCL of indices Cpu, Cpl and Cpk using Boole’s inequality and DeMorgan’s theorem. This approach is based on subsample data collected from a stable process. Hypothesis testing is also used to determine whether the process is capable of satisfying the quality requirements of customers. We calculated the critical values of the PCIs for various significance levels, capability requirements and sample sizes. Finally, we present analysis of two cases to demonstrate the applicability of the proposed approach.
... The authors of publications [58][59][60][61] pay special attention to the fact that process capability indices are an important tool in the area of quality management of manufacturing processes. The content of article [58] emphasizes that the use of process capability indices is characterized by a relatively high level of uncertainty, mainly due to the fact that any error in the sample can introduce significant uncertainty into the process capability assessment. ...
... The authors of publications [58][59][60][61] pay special attention to the fact that process capability indices are an important tool in the area of quality management of manufacturing processes. The content of article [58] emphasizes that the use of process capability indices is characterized by a relatively high level of uncertainty, mainly due to the fact that any error in the sample can introduce significant uncertainty into the process capability assessment. In publications [61,62], the authors emphasize that for the possible use of process capability indices for correct data Aleksy Kwilinski and Maciej Kardas Virtual Economics, Vol. 6, No. 4, 2023 analysis, the occurrence of a normal distribution in the collected values is necessary. ...
Process Capability Indices such as Process Capability Index (Cp) and Corrected Process Capability Index (Cpk), along with Process Performance Index (Pp) and Process Performance Corrected Index (Ppk), are most commonly used tools in quality management within manufacturing processes. They determine whether a process is capable of producing products within established tolerance limits, addressing both short-term and long-term variability. Despite widespread use, the analysis of Process Capability Indices often overlooks special variability within processes, which may lead to misleading interpretations of a process's capability, especially when the determination of tolerance limits is inadequately conducted. This article aims to verify the hypothesis that current analyses of Process Capability Indices fail to consider special variability, which could mislead the interpretation of a process's ability to meet its specifications. Statistical analyses were conducted using the Minitab software, based on dynamic viscosity measurements from the production process of solvent-based paint, to explore the implications of special variability on the interpretation of Process Capability Indices. The study revealed that while Process Capability Indices are useful for identifying quality management opportunities, their effectiveness is limited when special variability is present, often resulting in misinterpretations of a process's true capabilities. The findings highlight the need for methodologies that incorporate considerations of all forms of variability to ensure accurate process capability assessments. This gap in traditional analyses can affect the strategic decision-making in quality management, suggesting a critical area for further research and methodological development. The research confirms the need for a revised approach in analyzing Process Capability Indices, advocating for advanced methods that accurately reflect all forms of variability to improve quality management practices. Future research should focus on developing these methodologies to ensure more reliable and effective use of Process Capability Indices in quality management.
... PCIs are not only a tool for selecting outsourcers but also a convenient and effective communication tool for internal process engineers and quality control engineers, which can assist with handling various problems concerning process technology or quality arising in the production process of products [3][4][5][6][7][8]. Furthermore, the scheme of six-sigma quality improvement initiated by Motorola can assist corporations with their quality improvement as well as reduce the defect rate for their products [4,[9][10][11]. Therefore, the six-sigma quality improvement process contains significant implications for industry and is widely applied to manufacturing, aiming to enhance product quality levels as well as lower production defect rates [12][13][14][15]. ...
... In reality, process mean μ and process standard deviation σ are usually undetected. Therefore, they must be estimated from subsamples taken when the process is thought to be in control [11]. As noted by Montgomery [22], when ...
Whether it is important components of a machine tool itself or various important components processed by the machine tool, many vital quality characteristics mostly belong to the smaller-the-better type. When the process quality levels of these quality characteristics do not attain to the criteria, friction loss may increase during the machine operation, affecting not only the process precision and accuracy but also the lifetime of the product. Therefore, this study applied a smaller-the-better six-sigma quality index simultaneously demonstrating process quality level and process yield. Besides, in coping with statistical process control data, a one-tail confidence-interval-based fuzzy testing method was developed to evaluate process quality. Because this approach is built on the basis of confidence intervals, it can reduce the possibility of misjudgment resulting from sampling errors as well as integrate past experience to enhance the accuracy and precision of the assessment, and then it can grasp the timeliness of improvement.
... Six Sigma, developed in 1986 by Motorola, is widely applied to enhance product quality levels in the manufacturing industry [5,[14][15][16]. Several studies have investigated the correlations between PCIs and Six Sigma quality levels [9][10]17]. The wire-bonding process has two significant quality characteristics, both of which are the larger-the-better (LTB) type. ...
... Step 1: Calculate the mean of samples, the standard deviation of the samples, and index estimates of the 12 samples using Eqs. (16), (17), and (19). Step 3: Draw 12 axes at 30 degrees from each other, mark the fuzzy critical value ( F k ) on the axes of the radar chart, and connect the neighboring critical points to form a critical region in the form of a regular dodecagon, as shown below: ...
Taiwan is a world-leader in wafer foundry services and IC packaging and testing. Wire bonding is a crucial process in the overall IC-packaging industry chain. Thus, this paper proposes a process-quality evaluation model for wire bonding with multiple gold wires. We chose process quality indices as a tool of evaluation fully mirroring process yield and quality levels. These indices contain unknown parameters and thus require sample data to estimate. We first derived the uniformly minimum variance unbiased estimator of the indices and calculated the upper confidence limits of the indices based on DeMorgan’s theorem and Boole’s inequality. The upper confidence limits of the indices were then employed to create a confidence interval-based fuzzy membership function, in order to improve the accuracy of estimation as well as solve the problem of uncertainty of the measured data. Next, we obtained the fuzzy critical value and used index estimates and the fuzzy critical value to establish fuzzy test rules. Next, we marked the fuzzy critical value on the axes of a radar chart, which is a visualization evaluation tool, and connected neighboring critical points to create a critical region in the form of a regular polygon. The observed values of the indices were then marked on the axes to produce a visualized fuzzy radar evaluation chart. This fuzzy radar evaluation chart has a solid foundation in statistical inference, and evaluation rules were established using precise fuzzy test methods. Not only is this fuzzy radar evaluation chart easy to use, but it also reduces the chance of misinterpretations made by sampling errors, so that the accuracy of evaluation can be enhanced.
... Sci. 2019, 9,2623 2 of 11 quality reaches the customer's requirements [6][7][8][9][10][11][12][13][14][15]. A number of researchers have examined the relationship between PCIs and Six Sigma quality levels [5,[16][17][18]. ...
... Similar to [10], the observed value of the 100(1 − α)% upper confidence limit UQ PUh0 can be described as follows: ...
The electronics industry in Taiwan has achieved a complete information and communication technology chain with a firm position in the global electronics industry. The integrated-circuit (IC) packaging industry chain adopts a professional division of labor model, and each process (including wafer dicing, die bonding, wire bonding, molding, and other subsequent processes) must have enhanced process capabilities to ensure the quality of the final product. Increasing quality can also lower the chances of waste and rework, lengthen product lifespan, and reduce maintenance, which means fewer resources invested, less pollution and damage to the environment, and smaller social losses. This contributes to the creation of a green process. This paper developed a complete quality evaluation model for the IC packaging molding process from the perspective of a green economy. The Six Sigma quality index (SSQI), which can fully reflect process yield and quality levels, is selected as a primary evaluation tool in this study. Since this index contains unknown parameters, a confidence interval based fuzzy evaluation model is proposed to increase estimation accuracy and overcome the issue of uncertainties in measurement data. Finally, a numerical example is given to illustrate the applicability and effectiveness of the proposed method.
... Six Sigma quality levels are another quality evaluation method widely used in industry. Chen et al. (2017b) noted that many researchers have examined the application of PCIs in conjunction with Six Sigma to resolve practical engineering problems, including Chen et al. (2009), Huang et al. (2010), Chang et al. (2014), and Ouyang et al. (2014). Chen et al. (2017b) proposed a Six Sigma quality index that indicates the quality level and process yield directly. ...
... Chen et al. (2017b) noted that many researchers have examined the application of PCIs in conjunction with Six Sigma to resolve practical engineering problems, including Chen et al. (2009), Huang et al. (2010), Chang et al. (2014), and Ouyang et al. (2014). Chen et al. (2017b) proposed a Six Sigma quality index that indicates the quality level and process yield directly. That index provides values that are equivalent to quality levels and have a one-to-one relationship with process yield, as follows: ...
The operating performance index (OPI) was developed by Chen and Yang (J Comput Appl Math 343:737–747, 2018) from the Six Sigma process quality index. The fact that OPIs include unknown parameters means that they must be formulated using estimates based on sample data. Unfortunately, cost and effectiveness considerations in practice have led to sample size limitation and measurement uncertainty. In this study, we sought to enhance testing accuracy and overcome the uncertainties in measurement by applying confidence intervals of OPI to derive a fuzzy number and membership function for OPI. We developed a one-tailed fuzzy test method to determine whether performance reaches the required level. We also developed a two-tailed fuzzy testing method based on two OPIs to serve as a verification model for the effectiveness of improvement measures. Both fuzzy testing methods are proposed based on confidence intervals of the indices to reduce the risk of misjudgment caused by sampling errors and enhance testing accuracy.
... Chen et al. [7] claimed that several quality characteristics (QCs), composed of the smallerthe-better (STB), larger-the-better (LTB), and nominal-the-better (NTB) types, typically coexist in machine tool components. The tolerance for the NTB quality characteristic is given by T ± d, where T denotes the target value, d = USL − T = T − LSL, and USL and LSL represent the upper and lower specification limits, respectively. ...
The rapid progression of emerging technologies like the Internet of Things (IoT) and Big Data analytics for manufacturing has driven innovation across various industries worldwide. Production data are utilized to construct a model for quality evaluation and analysis applicable to components processed by machine tools, ensuring process quality for critical components and final product quality for the machine tools. Machine tool parts often encompass several quality characteristics concurrently, categorized into three types: smaller-the-better, larger-the-better, and nominal-the-better. In this paper, an evaluation index for the nominal-the-better quality characteristic was segmented into two single-sided Six Sigma quality indexes. Furthermore, the process quality of the entire component product was assessed by n single-sided Six Sigma quality indexes. According to numerous studies, machine tool manufacturers conventionally base their decisions on small sample sizes (n), considering timeliness and costs. However, this often leads to inconsistent evaluation results due to significant sampling errors. Therefore, this paper established fuzzy testing rules using the confidence intervals of the q single-sided Six Sigma quality indices, serving as the fuzzy quality evaluation model for components of machine tools.
... 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.
... From a statistical standpoint, point estimates mainly use single valueÎ ACC to estimate population parameter I ACC , whereas interval estimates use sample data to define a value range to infer unknown population parameter I ACC when the population is uncertain. There is a 100(1 − α)% probability that this range contains the actual value of the population parameter (Chen et al., 2017a(Chen et al., , 2017b. ...
Artificial intelligence (AI) assists in decision-making across various fields and industries. Diverse market needs have prompted the rapid evolution of AI learning algorithms. Deep learning networks (DLNs) process classification problems associated with perceptrons; this approach has become mainstream in the current AI era. To compare the classification and recognition performances of the designed DLN systems, most studies have applied confusion matrices as assessment tools and further computed accuracy, sensitivity, and specificity indices for judgment and analysis. However, the values of these indices change with the degree of learning achieved by the network system each time it is trained. Thus, using a single index value or mean value to determine recognition capabilities may lead to misjudgment. In view of this, we used accuracy to define a recognition performance index (RPI) IACC. Considering the unavoidable uncertainty in IACC, we further propose a triangular fuzzy number (TFN) for IACC. This is applied to develop a fuzzy test model for IACC to aid researchers in evaluating superiority among the designed DLN systems in terms of recognition capabilities. To demonstrate the applicability of the proposed approach, we implemented it on a LeNet-5 convolutional neural network system optimized using the Taguchi method for tomographic images of lung cancer provided by the 2015 International Society for Optics and Photonics (SPIE).
... Traditionally , , , and (Kane, 1986;Kotz and Johnson, 2002;Chen et al., 2017) most commonly known process capability indices for the univariate case. Traditionally, all these indices are developed assuming that the quality characteristic is a continuous variable. ...
Rapid technological advancement and implementation of automation and computerization in today's manufacturing set up resulted in many high quality processes, where defects are rarely observed. There are many high quality manufacturing processes where two or more types of defects may be generated from different types of equipment/process problems. The zeroinflated defects data containing two types of defects are commonly modeled by bivariate zero-inflated (BZI) Poisson distribution. Pal and Gauri (2022a) proposed a methodology for measuring capability of a BZI Poisson process. However, they ignored the count of zero defect (ZD) products produced in a BZI process. Because of that, Pal and Gauri (2022a) proposed approach fails to discriminate the BZI processes which produces different proportions of ZD units but having almost the same proportion of nonconforming items with respect to the USL of combined number of defects or USLs of individual defect types. In this paper, a new measure of process capability for BZI processes is proposed that can truly discriminate different BZI processes taking into account the USL of combined number of defects (or USLs of individual defect types) as well as the proportion of ZD units produced in these processes. The proposed methodology is illustrated using two case studies. The results of the case studies show that the proposed index well represents the true capability of BZI processes.
... Capabilities of processes are assessed in terms of different indices, e.g. , , , and (Kane, 1986;Kotz and Johnson, 2002;Chen et al., 2017;Polhemus, 2018). Historically, these indices are developed for a product characteristic that can be described as a continuous variable and follows normal distribution. ...
The proportion of zero defect (ZD) outputs is as an integral characteristic of a zero-inflated (ZI) process or high quality process. Different ZI processes can almost equally satisfy the same USL of number of defects but they can produce substantially different proportions of ZD products. The application of conventional method for process capability evaluation fails to discriminate these processes because in the conventional method, the process capability is evaluated taking into consideration the USL of number of defects only. In this paper, a new measure of process capability for ZI processes is proposed that can truly discriminate different ZI processes taking into account the USL of number of defects as well as the proportion of ZD units produced in these processes. In the proposed approach, at first a measure of process capability index (PCI) with respect to the USL is computed, and then the overall PCI is obtained by multiplying it with an appropriately defined multiplying factor. A real-life application is presented.
... Process Capability Indices (PCIs) are the most popular tool for process quality evaluation in the industry of machining and manufacturing [1][2][3][4]. They are a convenient device for evaluating and analyzing the process capabilities of products as well as a good tool bridging the gap between sales departments and customers [5][6][7][8][9][10][11][12][13]. Taiwan's output value and export volume of machine tools both are at the top of the list in the world, especially in the middle of Taiwan, a stronghold of machine tools and machining factories [14][15][16][17]. ...
Many studies have pointed out that the-smaller-the-better quality characteristics (QC) can be found in many important components of machine tools, such as roundness, verticality, and surface roughness of axes, bearings, and gears. This paper applied a process quality index that is capable of measuring the level of process quality. Meanwhile, a model of fuzzy quality evaluation was developed by the process quality index as having a one-to-one mathematical relationship with the process yield. In addition to assessing the level of process quality, the model can also be employed as a basis for determining whether to improve the process quality at the same time. This model can cope with the problem of small sample sizes arising from the need for enterprises’ quick response, which means that the accuracy of the evaluation can still be maintained in the case of small sample sizes. Moreover, this fuzzy quality evaluation model is built on the confidence interval, enabling a decline in the probability of misjudgment incurred by sampling errors.
... Numerous studies have pointed out that industries usually uses a statistical process control chart to monitor the quality of processes [26][27][28][29] as well as evaluate process capability when the process is stabilized [30][31][32][33][34]. Thus, this study employs control chart data to estimate the Taguchi cost loss index. ...
Process Capability Indices (PCIs) are not only a good communication tools between sales departments and customers but also convenient tools for internal engineers to evaluate and analyze process capabilities of products. Many statisticians and process engineers are dedicated to research on process capability indices, among which the Taguchi cost loss index can reflect both the process yield and process cost loss at the same time. Therefore, in this study the Taguchi cost loss index was used to propose a novel process quality evaluation model. After the process was stabilized, a process capability evaluation was carried out. This study used Boole’s inequality and DeMorgan’s theorem to derive the (1−α)×100% confidence region of (δ,γ2) based on control chart data. The study adopted the mathematical programming method to find the (1−α)×100% confidence interval of the Taguchi cost loss index then employed a (1−α)×100% confidence interval to perform statistical testing and to determine whether the process needed improvement.
... In addition, Wu et al. [4] suggested that when the value of this index is large enough, it can also reflect the process yield. Obviously, the Taguchi capability index reflects not only the process loss but also the process yield, which makes it a good indicator for evaluating the process capability [5][6][7][8]. It is expressed as follows: ...
The Taguchi capability index, which reflects the expected loss and the yield of a process, is a useful index for evaluating the quality of a process. Several scholars have proposed a process improvement capability index based on the expected value of the Taguchi loss function as well as the corresponding cost of process improvement. There have been a number of studies using the Taguchi capability index to develop suppliers’ process quality evaluation models, whereas models for evaluating suppliers’ process improvement potential have been relatively lacking. Thus, this study applies the process improvement capability index to develop an evaluation model of the supplier’s process improvement capability, which can be provided to the industry for application. Besides, owing to the current need to respond quickly, coupled with cost considerations and the limits of technical capabilities, the sample size for sampling testing is usually not large. Consequently, the evaluation model of the process improvement capability developed in this study adopts a fuzzy testing method based on the confidence interval. This method reduces the risk of misjudgment due to sampling errors and improves the testing accuracy because it can incorporate experts and their accumulated experiences.
... 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
... Furthermore, another widely used process control policy is based on PCR (Kane 1986;Chen, Wang, and Chang 2017), which is used extensively in manufacturing industries for process parameter and tolerance design (Wang, Mao, and Tu 2020). The classic PCR, C p , is defined as ...
Tolerances are critical to the product’s design, manufacturing, and quality. However, tolerances are often overlooked in a reverse engineering (RE) process for industrial applications, especially in legacy parts or spare parts remanufacturing. Ignoring tolerances could either unnecessarily call for high precision remanufacturing processes or make the reproduced parts unqualified. Additive manufacturing (AM) techniques are used in remanufacturing applications because of their ability to manufacturing intricated parts. Due to its layer-by-layer fabrication nature, the metrology for AM-created parts is drastically different when compared to traditional feature-generation processes. In this study, we first propose a novel way to classify manufacturing processes based on whether they directly identify or generate features, which could profoundly affect their metrology tools. Next, a systematic geometric inspection and tolerance estimation methodology for the RE system is proposed. A set of tools is developed to extract various geometric dimensional values from the point clouds based on their tolerancing types. Moreover, based on the domain knowledge in production process design and planning, methods are developed to estimate empirical tolerances from a small batch of legacy parts. Comparisons of empirical tolerances of real machined parts to their designed tolerances are presented to evaluate the performance of the proposed framework.
... 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). ...
... Mathematics 2020, 8,2129 2 of 17 and the standard deviation of the process is one-sixth of the tolerance, the quality level of the process is exactly 6 standard deviations, which means the Six Sigma quality index is exactly 6. ...
The quality characteristics with unilateral specifications include the smaller-the-better (STB) and larger-the-better (LTB) quality characteristics. Roundness, verticality, and concentricity are categorized into the STB quality characteristics, while the wire pull and the ball shear of gold wire bonding are categorized into the LTB quality characteristics. In terms of the tolerance, zero and infinity (∞) can be viewed as the target values in line with the STB and LTB quality characteristics, respectively. However, cost and timeliness considerations, or the restrictions of practical technical capabilities in the industry, mean that the process mean is generally far more than 1.5 standard deviations away from the target value. Researchers have accordingly proposed a process quality index conforming to the STB quality characteristics. In this study, we come up with a process quality index conforming to the LTB quality characteristics. We refer to these two types of indices as the unilateral specification process quality indices. These indices and the process yield have a one-to-one mathematical relationship. Besides, the process quality levels can be completely reflected as well. These indices possess unknown parameters. Therefore, sample data are required for calculation. Nevertheless, interval estimates can lower the misjudgment risk resulting from sampling errors more than point estimates can. In addition, considering cost and timeliness in the industry, samples are generally small, which lowers estimation accuracy. In an attempt to increase the accuracy of estimation as well as overcome the uncertainty of measured data, we first derive the confidence interval for unilateral specification process quality indices, and then propose a fuzzy membership function on the basis of the confidence interval to establish the two-tailed fuzzy testing rules for a single indicator. Lastly, we determine whether the process quality has improved.
... In the short term, the XS control chart can be employed to monitor whether the process is stable; after the process is stable, the chart data can be controlled to conduct a process capability evaluation. In the long term, the stable process capability can be evaluated by simple random sampling, in order to solve the problem of cost and timeliness [32][33][34][35]. Let n X X , , 1 be a random sample obtained from an in-control process with the normal distribution. ...
This study proposes the S
pa
-product capability analysis chart (S
pa
-PCAC), which can widely represent multiple process capabilities with asymmetric tolerances of Smaller-the-Better, Larger-the-Better, and Nominal-the-Best characteristics. Process capability index S
pa
is generated based on index S
pk
, which uses asymmetric tolerances to reasonably measure process capabilities. The interval estimates of the indices are derived to reliably assess process capabilities. The Six-Sigma-based quality-level and its connection with the process yield are introduced in the capability zone of S
pa
-PCAC to check if the process capabilities can meet the requirements. One example of an entire product is given for application.
... used C pm and change-of-variable technique to measure whether all the multi-quality characteristics of 95 unleaded gasoline satisfy specifications. Chen, Wang, and Chang (2017) applied the 100ð1 À αÞ% lower confidence limit (LCL) of C pu , C pl , and C pk to determine whether the multi-quality characteristics of ITO film are capable of satisfying the quality requirements of customers. ...
To reduce the pollution caused by the carbon emissions of automobiles and locomotive vehicles, many countries around the world have encouraged citizens to use bicycles for short-distance trips in recent years. Bicycles are comprised of many parts, and quick-release hubs are highly important for the fixtures on the front and rear wheel axles of a bicycle. The quick-release hub is a multi-quality characteristic product, including two larger-the-better quality characteristics (quick-release stroke and tensile strength) and three nominal-the-best quality characteristics (axis size, assembly distance, and the outer diameter of the spindle). To improve the quality of quick-release hubs, this study proposes a multi-quality characteristic analysis table (MQCAT) and a multi-quality characteristic analysis model (MQCAM). The proposed method can provide a valuable reference by which to guide efforts aimed at improvement for quick-release hub manufacturers. A quick-release hub manufacturer in central Taiwan is presented to illustrate the feasibility of the proposed method. In addition, a comparison with recent methods is provided to demonstrate the advantages of the proposed method. Finally, conclusions are made based on the research study findings.
... A process capability index (PCI) uses numerical quantification to examine the relationship between process performance and product specification. Many studies have used PCIs to judge whether a given process reaches the ability or quality demanded by customers [13][14][15][16][17][18][19][20]. At present, C pk is the most frequently used PCI for manufacturing industries [21]. ...
Supply chain management models integrate upstream and downstream organizations to enable rapid response to consumer needs. For the manufacturing industry, the process quality of suppliers is thus the foundation of sustainable growth for firms and an important indicator of whether a firm can effectively reduce waste and protect the environment. To this end, this paper proposes a model of supplier selection for manufacturers based on process quality assessment. First of all, Six Sigma quality index Qpk is adopted as the assessment tool to conveniently measure the quality level of process. Practical applications require estimates of Qpk from the data collected to analyze the process quality of each supplier. The fact that uncertainty is unavoidable in the collected data means that using the crisp estimate of Qpk can lead to misjudgment of the process quality. To enhance the reliability of evaluation and reduce the risk of misjudgment, the fuzzy number Q^˜pk is proposed to perform the fuzzy testing of two indices Qpk provided by suppliers with the intent of making reliable decisions on supplier selection.
... Process capability indices (PCIs) use numerical values to quantify and present the relationship between process performance and product specifications (Kane 1986;Chan et al. 1988;Pearn et al. 1992). PCIs have thus become an effective and convenient tool widely used in manufacturing to determine whether process performance meets quality requirements (Besseris 2019;Lepore et al. 2018;de-Felipe and Benedito 2017;Chen et al. 2002Chen et al. , 2017Lee et al. 2016;Gu et al. 2015;Chang et al. 2014). Most existing studies involving product quality assessment assumed that the quality characteristics of products follow a normal distribution; however, this assumption may not be applicable to every quality characteristic. ...
With intense competition in industry today, product quality has become a crucial factor influencing whether a firm can achieve sustainable operations and maintain competitiveness. Process capability indices are effective tools often used in manufacturing to determine whether products meets requirements, and most assume that the quality characteristics of products follow normal distributions. However, not all quality characteristics necessarily follow normal distributions; for example, product lifetime, a time-oriented quality characteristic, generally follows an exponential distribution or other associated non-normal distributions. The lifetime performance index was thus developed to gauge the lifetime performance of products, and most related studies use the precise values of time data to evaluate product lifetime. However, in practice, measurement errors may hinder the accuracy of the observed values of quality characteristics, and the time at which the lifetime of a product ends becomes imprecise, which may result in uncertainty in the evaluation method and lead to errors in judgment. For this reason, this study thus proposes a triangular shaped fuzzy number for to deal with imprecise data, and further develops a fuzzy testing model for lifetime performance index , to assist manufacturers in evaluating product lifetime performance more cautiously and precisely. Finally, we provide an illustration of how the proposed approach can be implemented through a numerical example.
... where I ∈ {U, L, K}, α = α/h, h = e + 2q, α is a given level of significance, e is the number of STB-or LTB-type unilateral quality characteristics, and q is the number of NTB-type bilateral quality characteristics for products comprising multiple quality characteristics of various type. According to Chou, Owen, and Borrego (1990) and Chen, Wang, and Chang (2017c), Equation (9) can be rewritten as follows: ...
Buyers are faced with selecting the optimal supplier, while suppliers are left to consider production costs. In this study, we developed a two-phase selection framework that allows buyers to evaluate the performance of suppliers while taking production costs into account for value maximisation. This scheme is a win-win solution capable of promoting long-term relationships between buyers and suppliers. Under the assumption of normality, the first phase involves constructing a new Six Sigma quality capability analysis chart (SSQCAC) which takes production costs into account. The objective is to evaluate all potential suppliers using the 100 × (1–α)% upper confidence limit (UCL) of an integrated Six Sigma quality index (SSQI) QPIh when dealing with products with smaller-the-better (STB), larger-the-better (LTB), or nominal-the-best (NTB) quality characteristics. According to interval estimation theory, this method can have a significant impact on the consumption of resources; i.e. the production costs of the supplier can be decreased by reducing the production quality to below that required by the buyer. The proposed method also filters out unsuitable suppliers in order to simplify the decision problem and reduce computational demands and operational risks/costs without compromising the quality of the final product. In the second phase, a detailed analysis is conducted using Euclidean distance measure to select the optimal supplier from among the remaining candidates. We conducted a real-world case study to evaluate the efficacy of the proposed method. We also conducted comparisons with existing methods to demonstrate the advantages of the proposed method and its managerial implications. Suggestions for future study are also provided.
Manufacturers aim to produce products that meet customer expectations. To this end, they must monitor and assess the machining quality of product to ensure high-quality reliable products. With the rise of consumer awareness, the products that customers can choose from are not only becoming more and more diverse, but also have multiple independent quality characteristics. Therefore, this study adopts the loss-based Six Sigma quality index [Formula: see text], which fully reflects quality level, loss, and yield for machining process of each characteristic l, to develop an assessment model for the machining quality of products with multiple independent quality characteristics of symmetric bilateral specifications. We also construct an analysis chart to simultaneously evaluate and determine the machining performance of all quality characteristics. In practice, [Formula: see text] must be estimated based on sample data expressed in precise values. However, uncertainty in sample data is natural and unavoidable in measurement, which increases the risk of misjudgment in assessing machining quality. In view of this, this study proposes the fuzzy statistical testing of [Formula: see text] based on the defined fuzzy estimation of [Formula: see text] to more accurately determine quality level of machining and reduce consumer risk for each characteristic. A real-world case involving five-way pipe product illustrates the applicability of the proposed technique.
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.
Multivariate auto‐correlated process control issues in industrial systems are a concern for statistical process monitoring (SPM). Traditional control charts produce large false alarms and/or miss timely detections of quality deterioration because they are unable to recognize the signals from multivariate auto‐correlated response variables. To track multivariate auto‐correlated processes, this paper presents a new residual‐based mixed multivariate control chart using cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) approaches. Using in‐control data, the multi‐output least square support vector regression model's optimal hyper‐parameters are determined, and a bootstrap method is used to estimate the upper control limit of the proposed control chart. The suggested control chart has strong detection performance for a small magnitude mean vector shift based on the average run length (ARL) performance for a particular range of shifts. Experimental result elaborates that the proposed control chart is more sensitive to detecting the mean vector shift compared with the existing commonly used models, such as multivariate CUSUM and multivariate EWMA control charts. The proposed control chart model and corresponding computational algorithm are successfully applied to SPM in an electronic conductor production line with multivariate auto‐correlated attributes.
Amidst fierce competition and diversifying customer needs in today’s markets, product development and design have shifted toward a more user-oriented model. This makes it common for products to have multiple quality characteristics. However, simply meeting customer needs is no longer sufficient; quality is also a key factor of a consumer’s willingness to buy a product. To this end, process capability indices have been used to measure the relationship between manufacturing specifications and processing performance, as well as serve as a bridge of communication between manufacturers and clients. In view of this, the loss-based capability index [Formula: see text] that fully reflects process loss and yield is employed in this study to analyze the process performance of each quality characteristic. However, [Formula: see text] must be estimated based on collected samples, in which the measured values of all sample data are expressed with precise values. Uncertainty and imprecision in collected data increase the risk of misjudgment. To reduce this risk, the [Formula: see text] confidence interval of [Formula: see text] is first derived to define fuzzy estimations of both the critical value and index, and develop a fuzzy process capability analysis model for a machined product with multiple quality characteristics of symmetric tolerance. Finally, an industrial example involving a five-way pipe product is presented to illustrate the applicability of the proposed approach. The results show that the proposed fuzzy analysis model makes determination of the process capability of each quality characteristic more reliable and rigorous to ensure that manufactured products meet requirements.
Aiming to reduce the social-economic loss caused by deficient products, the processing target value setting requirements must be achieved through multiple manufacturing processes. Meanwhile, the processing target value setting for each step must properly match its own manufacturing process capability so as to lower the ratios of process scrap and rework. Consequently, the quality of the final product can be ensured to reduce social-economic loss, such as carbon emissions caused by faulty maintenance after sales. By means of the production data analysis as well as quality engineering, this study discovered the best conditions for multiple manufacturing processes and then adopted the genetic neural networks method to estimate the optimal stable-process standard deviation for each processing step. Next, according to the aforementioned optimal process standard deviation, the reasonable target value of each processing step was determined by the processing target value and the process capability set in the final product. Finally, this study provided the multiple-process capability analysis chart to evaluate the process capability after experiment. The processing of the green electric vehicle motor was taken as an example to explain the application of the reasonable processing target value setting model mentioned in this study.
Maintaining high levels of process quality is crucial to the competitiveness of manufacturing firms in today's increasingly global marketplace. To ensure the quality of manufactured products meets customer needs, process capability indices (PCIs) are widely used to analyze the process performance of various processing characteristics. Products characterise by processing characteristics of both unilateral and bilateral specifications are common in the current sales market. Manufacturing firms must often adopt multiple PCIs to analyze the process performance of a single product, which is inefficient in practical applications and management. Yield-based index (Formula presented.) is not subject to this limitation. For this reason, we employed (Formula presented.) to evaluate process performance and the effectiveness of improvement measures. In practice, (Formula presented.) is estimated from samples, which means that misjudgment may occur in the assessment of process performance and improvement effectiveness due to sampling errors. We therefore derived the (Formula presented.) confidence interval of (Formula presented.) and, based on the producer's perspective, used the upper confidence limit to evaluate improvement effectiveness. To lower the risk of misjudgment and increase the reliability of improvement effectiveness in the case of data uncertainty, this paper further proposes fuzzy estimation using the right-sided confidence interval of (Formula presented.) and develops the fuzzy judgement model.
The high-quality processes usually have more count of zeros than are expected under chance variation of its underlying Poisson or other count distribution. Therefore, these processes are usually referred to as zero-inflated processes. The zeroinflated processes are commonly modelled by zero-inflated Poisson (ZIP) or zero-inflated negative binomial (ZINB) distribution. In a manufacturing set up, the evaluation of process capability index of a zero-inflated process can be useful in many ways, e.g. i) predicting how well the process will hold the specifications, ii) selecting between competing vendors, and iii) assisting product developers/designers in modifying the process, etc. However, researchers have given very little attentions on this aspect of zero-inflated processes. Only one such attempt is reported in literature. But, it does not always represent the true capabilities of zero-inflated processes, and sometimes it may give very misleading impression about the capability of the concerned process. In this article, the concept of Borges and Ho (2001) is applied to zero-inflated processes and a new approach for computation of process capability index of zero-inflated processes is developed. The proposed method reveals the true capabilities of zero-inflated processes consistently. Application of the proposed approach and its effectiveness are illustrated using two datasets published by past researchers.
The quick-switch sampling (QSS) systems based on process capability index (PCI) have been proved to be useful for inspecting processes streaming at a low level of defects because of their small sizes required and dynamic responsiveness to problems. However, most of the recent studies have aimed at deriving the mathematical model and comparing performances for the acceptance criteria-type QSS system, and only a few of them have an interest in the development of the required sample-size-type QSS system; their studies have limited simultaneous discussions on both systems’ advantages and disadvantages, not to mention the lack of investigations on the impact of the joint application of the two QSS system types. In this paper, an integrated QSS (IQSS) system based on the most popular PCI was proposed; the system can accommodate two existing types of PCI-based QSS systems and further boost the sampling performance and discriminatory power. Moreover, by operating suitable types of PCI-based QSS systems in different stages of the supplier-buyer partnership, a progressive lot-disposition strategy was introduced to construct a solid supplier-buyer relationship. We also developed a web-based tool to accurately and efficiently execute all types’ PCI-based QSS systems. Finally, the industrial applicability was demonstrated in a case study.
Process quality and yield are always crucial determinants of a firm's competitiveness; therefore, monitoring and assessment for process quality are vital aspects of their sustainable development. The Six Sigma quality control is based on defect rate and nonconforming units to eliminate factors that cause process defects. Thus, it has become a powerful instrument for manufacturers to achieve the quality goals of only 3.4 defects per million opportunities. However, the originated Six Sigma program is based on the 1.5σ shift in the process mean to recognize the tendency of processes to shift over the long term and achieve this goal. With the production environment of intelligent manufacturing today, manufacturers keep up to date with production performance and then monitor the distance between the process mean shift and the target value at all times. This makes it possible to adopt a more rigorous standard by decreasing in allowable shift to define the quality level in Six Sigma. For this reason, this paper proposes the redefined Six Sigma quality indexes as an assessment tool and has developed a modified approach for Six Sigma quality assessment of products with multiple characteristics. The proposed method uses more rigorous standards to analyze process quality, thereby allowing manufacturers to provide industry partners or consumers with products of higher quality, meet their needs, and even enhance their own industry competitiveness to continue progressing toward sustainability. We present a real-world example to demonstrate the practical applicability of the proposed approach. © 2020 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959
Product lifetime is one of the most important issues associated with circular and sharing economies. In the face of global warming, increasing product quality and product lifetime can enhance product value and industry competitiveness. Many electronic products use designs with spare electronic components to enhance their reliability. To prevent the loss of generality, we assumed that an electronic product has m electronic components, that is, one primary electronic component and m-1 spare electronic components. When the primary electronic component fails, then the product will immediately switch to a spare. Under the assumption that the warranty period is L units of time, we proposed a lifetime performance index for electronic products. With an identical mean time between failure (MTBF) for each electronic component, we investigated the relationships between the number of electronic components and the lifetime performance index, failure rate, and reliability of the product. We next developed uniformly minimum variance unbiased (UMVU) estimator for the lifetime performance index of electronic products and its probability density function. Then we established the uniformly most powerful (UMP) test and the power of the test for the lifetime performance index for electronic products. Finally, we presented a numerical example to demonstrate the applicability of the proposed method and model.
In the face of increased competition and globalization, supply chain management has become a crucial aspect of securing competitive advantage. Within this, the quality offered by suppliers is vital. Thus, in this paper, we develop a model to select the best supplier based on process quality from the perspective of the buyer. We first employ the process capability index Spk, which can provide an exact measure of process yield, to evaluate the process quality of suppliers. However, Spk must be estimated from preliminary samples or obtained subsamples. This increases uncertainty and can lead to miscalculation. To ensure the reliability of assessment, we derive the lower confidence limit of Spk to serve as a standard of process quality assessment and construct a process capability analysis chart to select the best supplier. The chart also provides recommendations for process improvement to serve as a reference for suppliers with poor performance. Finally, we present two real-world case studies to demonstrate the practical applicability of the proposed method.
Literatürde ilk önerilmiş olan süreç yetenek indeksleri, süreç verilerinin normal dağılması, kalite karakteristiklerine ait toleransların simetrik olması ve sürecin kontrol altında olması varsayımları altında çalışmaktadır. İlerleyen çalışmalarda, süreç verilerinin normal dağıldığı ve toleransların asimetrik olduğu, süreç verilerinin asimetrik bir dağılıma uyduğu ve toleransların simetrik olduğu durumlar için bazı yetenek indeksleri önerildiği görülmektedir. Bu çalışmanın amacı, literatürdeki diğer çalışmalardan farklı olarak, toleransların asimetrik ve süreç verilerinin dağılımının normal olmadığı durumlar için yeni bir süreç yetenek indeksi önermektir. Asimetrik toleranslı ve asimetrik dağılımlı süreçlerin tanımlanması için Pearson dağılım ailesi ile çalışılmıştır. Önerilen indekste baz dağılım ile süreç dağılımının dört temel bileşeni karşılaştırılmaktadır. Dağılımın hedef değerden uzaklığı, basıklık ve çarpıklık değerlerindeki değişimi ile alt ve üst spesifikasyon limitlerine olan yakınlığı dikkate alınmıştır. Önerilen indeksin çeşitli durumlarda gösterdiği performans örneklerle incelenmiştir. Sonuç olarak indeks değeri, süreç dağılımının basıklık değeri arttıkça ve süreç dağılım verileri hedef değere yaklaştıkça artmakta, çarpıklık değerindeki artışlarda azalmaktadır.
Supplier selection is a practical problem in supply chain management and quality is the most important criterion in supplier selection. In this study, we developed a supplier selection model based on process quality, in which the Six Sigma quality index [Formula: see text] is used as a tool to assess the process quality provided by suppliers. Note that index estimation based on sample data is prone to uncertainty in the assessment of process quality. Therefore, we derived the confidence interval of [Formula: see text] via mathematical programming to reduce the likelihood of assessment miscalculations, and then used this interval to perform a pairwise comparison of suppliers. Our goal was to identify criteria that can be used to select the optimal suppliers for long-term collaborations and sustainable partnerships. A case study is also presented to demonstrate the practical implementation of the proposed method.
Machine tools are fundamental equipment in industrial production, and their processing quality exerts a direct impact on the quality of the component product that they process. Thus, machine tool manufacturers develop various machine tools depending on market needs and processing functions, and the processed component products generally possess multiple smaller-the-better, larger-the-better, and nominal-the-best quality characteristics at the same time. For this reason, this study employed the widely used process capability indices, [Formula: see text], [Formula: see text], and [Formula: see text] to develop a model that can evaluate the process quality of component products and analyze the processing quality of various machine tools. We first converted the process capability indices into functions of the accuracy and precision indices and constructed a multi-characteristic quality analysis chart that can identify the reason for poor process quality in a quality characteristic. Furthermore, considering the fact that the process capability indices can only be estimated, which may lead to misjudgment in the evaluation of process quality, we derived the [Formula: see text] upper confidence limits of indices and the coordinates formed by the corresponding accuracy and precision indices. Manufacturers can then evaluate the process quality levels of the quality characteristics based on where the coordinates falls in the multi-characteristic quality analysis chart. This can more reliably assist manufacturers in monitoring the processing quality of their machine tools and providing feedback to the machine tool manufacturers for machine improvement.
Outsourcing has become a competitive business model for the global manufacturing market, enabling firms to increase their overall production efficiency and enhance their own competitiveness in manufacturing. In view of this, we took into account the two important factors of process quality and manufacturing time to develop an optimal contractor selection model. To effectively assess process quality, we employed the Six Sigma quality index (SSQI) as our assessment tool. Based on the concept that manufacturing time that is too short may affect manufacturing quality and manufacturing time that is too long may affect delivery, we proposed an index to evaluate the performance of manufacturing time for contractors. Index values require estimation of sample data prior to assessment. Thus, sampling errors can result in uncertainty and misjudgment in performance evaluation. To overcome this limitation, we calculated the 1001−α% upper confidence limits of the SSQI and the manufacturing time index as standards for the evaluation of contractor process performance. Finally, we developed a contractor selection matrix with manufacturing time index θw as the horizontal axis and SSQI as the vertical axis. We applied the proposed approach to a case involving grinding of a gear product to demonstrate its efficacy and applicability.
Performance evaluation matrix (PEM) has captured many scholars’ attention and has been widely used to a variety of industry fields to evaluate their efficiency performance. The data mainly come from the collection of questionnaires in which customers’ varied opinions are enquired; however, Likert scale will be modified to fuzzy linguistic scale due to some uncertainty aspect in the linguistic data. In contrast, the major flaw of some researches was that they failed to present a verification approach with statistical inference to confirm the effectiveness of the improvement. Regarding to the related issues, Chen et al. (2018a) use discrimination index in reliance section to establish fuzzy membership function; subsequently, they propose fuzzy hypothesis testing to locate critical-to-quality (CTQ) so as to make further improvement. Following the same logic, this paper uses the identical process to develop the fuzzy verification method of the improvement efficiency for CTQ. Finally, this paper puts forward a numerical example to demonstrate application of the proposed verification model. The result means that there is sufficient evidence showing that performance improvement is poor and non-effective. Thus, manager should adopt otherwise measures to improve. Obviously, the method proposed particularly highlights the reduction of sampling error, shies away from the complexity inherent in fuzzy semantic collection, overcomes the uncertainty in the collected data, and increases the reliability of evaluation method.
As the Internet-of-Things matures, technologies for the measurement and collection of production data are also improving. What is needed next is an effective information analysis model to aid in the timely adjustment of manufacturing parameters to optimize production. Production data analysis models must be continuously and properly utilized to monitor and maintain process quality. Improving product quality helps to lengthen service life, decrease scrap and rework, and reduce the social losses caused by malfunctions and maintenance. The Taguchi capability index C pm can fully reflect the losses and yield of processes and is a convenient and effective tool to evaluate and analyze process data in the industry. As it contains unknown parameters, we derived the upper confidence limit (UCL) of C pm based on collected production data. Due to the fuzzy uncertainties that are common in measurement data, we then used the UCL of the index to construct a fuzzy membership function and propose a fuzzy testing decision-making model to determine whether processes are in need of improvement. Before the proposed fuzzy test methods became full-fledged, we used the concept of sample size and the rules of statistical testing to explain the motivation underlying those methods. In fact, the sample size influences the risk of misjudgment in UCL, and in practice, sample sizes are rarely large due to cost and time considerations, thereby they produce larger UCL with a corresponding decrease in accuracy and increase in risk of misjudgment. The fuzzy test methods proposed in this study are based on statistical inference, and judgement is aided by expertise, thus are capable of solving the problems associated with larger UCL. Therefore, the theoretical foundation of this fuzzy testing decision-making model is the UCL, it can lower the chance of misjudgment caused by sampling errors and increase evaluation accuracy.
Quality data fraud not only destroys the trust between suppliers and customers but also misleads the decision-making when choosing suppliers. Thus, it is preferred to use the quality data measured by customers to evaluate the manufacturing process capability indexes (PCIs). In practice, the suppliers always conduct a preliminary internal inspection to eliminate the nonconforming items before selling products, and quality data measured by the customers are truncated by the specification limits, which makes it difficult to measure the PCIs. This paper proposes a novel method to estimate the PCIs based on the truncated data. First, we propose a new data filling method called the QA-EM by integrating the EM and quantile-filling algorithms. Consequently, the truncated data can be converted into pseudo-complete data. A comparison study with other methods is further carried out to demonstrate the superiority of our proposed method. Then, various interval methods for estimating PCIs are applied to calculate the lower confidence limits of Cpk based on the pseudo-complete data. We investigate the performance of different methods in terms of coverage rate. The results indicate that the generalised confidence interval method performs better than the competitors. Finally, an industrial example is presented to illustrate the application of our method.
Purpose
The purpose of this paper is to establish mechanisms for process improvement so that production efficiency and product quality can be expected, and create a sustainable development in terms of circular economy.
Design/methodology/approach
The authors obtain a critical value from statistical hypothesis testing, and thereby construct a process capability indices chart, which both lowers the chance of quality level misjudgment caused by sampling error and provides reference for the processes improvement in poor quality levels. The authors used the bottom bracket of bicycles as an example to demonstrate the model and methods proposed in this study.
Findings
This approach enables us to plot multiple quality characteristics, despite varying attributes and specifications, onto the same process capability analysis chart. And it therefore increases accuracy and precision to reduce rework and scrap rates (reduce), increase product availability, reduce maintenance frequency and increase reuse (reuse), increase the recycle rates of components (recycle) and lengthen service life, which will delay recovery time (recovery).
Originality/value
Parts manufacturers in the industry chain can upload their production data to the cloud platform. The quality control center of the bicycle manufacturer can utilized the production data analysis model to identify critical-to-quality characteristics. The platform also offers reference for improvement and adds the improvement achievements and experience to its knowledge management to provide the entire industry chain. Feedback is also given to the R&D department of the bicycle manufacturer as reference for more robust product designs, more reasonable tolerance designs, and selection criteria for better parts suppliers, thereby forming an intelligent manufacturing loop system.
In this papers, we address a double samplimg Cpm control chart. It is well known that a double sampling scheme can reduce average sampling number in comparison with a single sampling scheme in sampling inspection. However, it is complicated and difficult to design the double sampling Cpm control chart because a judgment rule of the 2nd sampling stage depends on a record of 1st sampling stage. Therefore, this paper proposes a double sampling Cpm control chart incorporating the feature that a judgment rule of 2nd sampling stage is independent of a record of 1st sampling. The design algorithm for the proposed double sampling Cpm control chart is constructed by taking the economical operation of this control chart into the consideration. That is, the economic design of the double sampling Cpm control chart is addressed. Through some numerical comparison, it has been confirmed that the proposed double sampling Cpm control chart has an advantage in the expected total operating cost over the traditional single sampling Cpm control chart.
By taking process targeting as well as process variability into consideration, the Taguchi capability index Cpm gives a reasonable indication of process loss. This makes it an ideal tool for practical applications that depend on the evaluation of process quality. The ability of Six Sigma quality management to reduce process defect rates has led a number of researchers to investigate the relationship between Cpm and Six Sigma quality levels. Unfortunately, previous efforts indicate the quality level using only a range, rather than a specific value. Our objective in this study was to develop a Taguchi Six Sigma quality index Qpm that retains the advantages of Cpm in the assessment of process performance, while providing a specific value for the quality level associated with the process in question. To ensure rigorous quality assessments, we employed the upper confidence limit of Qpm in the design of a testing model for use by manufacturers. Fuzziness and stochastic uncertainty are unavoidable aspects of data collection. We, therefore, adopted a right half triangular-shaped fuzzy number for Qˆpm to deal with imprecise data. We also developed a method of the fuzzy hypothesis testing for Qpm to make reliable decisions for process quality assessment.
This paper proposes an operating performance index (OPI) based directly on the Six Sigma process quality index. We considered the following case: the number of customers arriving at a store follows a Poisson process of rate λ and the sequence of inter-arrival times is an exponential random variable with mean τ, where τ = 1/λ. In operating performance assessments, using the statistical testing to evaluate the OPI can lower the risk of misjudgment caused by sampling error. A smaller sample size entails a larger sampling error, while a larger sample size entails a smaller sampling error. Thus, using critical value for statistical testing can result in inconsistencies for different sample size. We therefore referred to the one-tailed fuzzy method proposed by Buckley (2005) for the parameters of population distributions and developed a two-tailed fuzzy test for the OPI to assess operating performance. The advantage of this approach is that conventional measurement methods can be used during data collection, and then the confidence interval of the OPI can be used to construct a fuzzy membership function for fuzzy testing. This not only lowers the risk of misjudgment caused by sampling error, but also enhances testing accuracy.
Under the impact of global warming, enterprises must grow, take care of the protection of the green environment and reduce the impact on society. Therefore, 4R (Reduce, Reuse, Recycle, Recovery) is an important green concept. For sustainable production, improving product quality is one of the most crucial strategies, because enhancing product quality can reduce process scrap and the rate of Reduce, increase product availability, prolong maintenance intervals, and increase the rate of Reuse for maintenance as well as the rate of Recycle for the reproduction of all components. Meanwhile, owing to the increase of the service life, the time of Recovery will be postponed. Taiwan's electronics industry has developed from semiconductor technology to peripheral electronic components with a complete industrial ecosystem and a highly specialized division of labor. In the entire specialized and cooperative industrial chain, suppliers must be carefully selected to ensure the quality of each component and the final product quality. To sum up, this paper can fully reflect the six sigma indicators of the process yields and quality levels to construct a complete green supplier fuzzy selection model. The fuzzy evaluation model developed in this paper analyzes whether the method of collecting the required data is consistent with the traditional method of collecting data, which can increase the practical convenience. Besides, regardless of the sample sizes, more accurate judgments can be made to select suppliers, assisting the businesses and all suppliers to improve process quality (technical upgrades) and performance and thereby increasing 4R benefits as well as moving towards the goal of industry sustainability.
Gears are among the most crucial components in the transmission systems of machine tools. Gear manufacturing includes a number of processing procedures. The grinding process is an important procedure involving high precision and fairly small grinding surfaces. For this reason, this study aimed at developing a quality assessment model for the internal cylindrical grinding process of gears. The Six Sigma quality indices (SSQIs) were used to directly assess the quality of the internal cylindrical grinding process due to their ability to directly reflect quality level and process yield. Since the process may include nominal-the-best (NTB), larger-the-better (LTB) and smaller-the-better (STB) quality characteristics, so we used the variable transformation method to normalise the specifications of each quality characteristic for the convenient and effective management and analysis of process performance for multiple quality characteristics. We then constructed a multi-characteristic process quality analysis chart (MPQAC) to simultaneously assess the quality levels of various quality characteristics. Furthermore, the MPQAC can provide references for process improvement. This ensures the quality of internal cylindrical grinding and enhances the quality of gear and machine tool products. Finally, a real-world application and numerical experiments demonstrate the effectiveness and practical applicability of the proposed method. © 2019
Depending on the quality characteristic, a process capability index (PCI) can be used for one-sided specifications or for bilateral specifications. A number of researchers have investigated the statistical properties of one-sided specification indices and proposed methods for applications. The later introduction of the Six Sigma approach also assisted many firms in effectively enhancing their production capacities, reducing waste, and increasing effectiveness. Chen et al. (2017a) modified the PCI for one-sided specifications and proposed the Six Sigma Quality Index (SSQI), which coincidently equals the quality level and has a one-to-one relationship with yield. However, uncertainty in quality characteristic measurements is common in practice, which can lead to judgment errors in conventional process capability assessment methods. This study therefore developed an SSQI for one-sided specifications based on the fuzzy testing method created by Buckley (2005) and developed a Six Sigma fuzzy evaluation index and testing model. In addition to having a simpler calculation procedure, the model takes the process capability and Six Sigma quality level into consideration and can process the uncertainties in the data to make it more convenient for the industry to solve engineering issues. Finally, we presented a practical example to demonstrate the applications. The model proposed in this study can provide the industry with a practical approach to assess process quality in a fuzzy environment.
In the face of fierce global competition, firms are outsourcing important but nonessential tasks to external professional companies. Corporations are also turning from competitive business models to cooperative strategic partnerships in hopes of swiftly responding to consumer needs and enhancing overall efficiency and industry competitiveness. This research developed an outsourcing partner selection model in hopes of helping firms select better outsourcing partners for long-term collaborations. Process quality and manufacturing time are vital when evaluating outsourcing partner. We therefore used process capability index Cpm and manufacturing time performance index Ih in the proposed model. Sample data from random samples are needed to calculate the point estimates of indices, however, it is impossible to obtain a sample with a structure completely identical to that of the population, which means that sampling generates unavoidable sampling errors. The reliability of point estimates are also uncertain, which inevitably leads to misjudgment in some cases. Thus, to reduce estimate errors and increase assessment reliability, we calculated the 100(1 - α)% confidence intervals of the indices Cpm and Ih, then constructed the joint confidence region of Cpm and Ih to develop an outsourcing partner selection model that will help firms select better outsourcing partners for long-term collaborations. We also provide a case as an illustration of how the proposed selection model is implemented.
Process capability analysis are widely used in industry to achieve and maintain a high-quality level of manufactured items. Various indices have been proposed, but the most widely used are Cp, Cpk, Cpm and Cpmk. This paper gives the calculation method of the process capability index, discusses the process capability analysis and evaluation method, and analyzes in detail the process countermeasures of the process capability index is too large or too small. Through examples, the normality test method and controlled state test method for sample data are given. That is, the Q–Q probability plot and the Anderson–Daling test method are used to determine whether the sample sequence is normal distribution. The Xbar-R control chart is used to determine whether the sample sequence is in a process-controlled state. Using Minitab software, control charts and process capability diagrams were plotted, and process capability indices Cp and Cpk are calculated and analyzed. Finally, according to the quality data of customer feedback, the process capability level and state of the processing process are determined, so that the process capability can meet the customer’s requirements.
Process capability analysis is a very well-known and widely accepted method of assessing the ability of a process to produce items within pre-assigned specification limits. Most of the process capability indices (PCI) available in literature are formulated in terms of the parameters of the concerned quality characteristics. However, since the actual values of these parameters are often unknown, their estimated values are used to evaluate the estimated capability of a process. One such estimation procedure may be to use the estimates of these parameters obtained from the corresponding control charts used to check the stability of the said process. In this article, we used this approach to redefine plug-in (natural) estimators of the two most famous PCI’s for unilateral specification limits viz., CPU and CPL. We formulated the corresponding unbiased estimators and uniformly minimum variance unbiased estimators (UMVUE), wherever possible, and their distributions as well. We also designed the process capability control charts of CPU and CPL based on these UMVUEs. For constructing these control charts, we used the estimators of the parameters of the quality characteristics as obtained from the corresponding and charts. These charts can be used to check the consistency of capability of a process and also to keep a constant vigil on the process. Two numerical examples have been discussed and it has been observed that our proposed process capability control charts are more efficient to detect changes in process capability than those already available in literature.
Process capability index C pk has been the most popular one used in the manufacturing industry dealing with problems of measuring reproduction capability of processes to enhance product development with very low fraction of defectives. In the manufacturing industry, lower confidence bound (LCB) estimates the minimum process capability providing pivotal information for quality engineers to monitoring the process and assessing process performance for quality assurance. The main objective of this paper is to compare and contrast the LCBs on C pk using two approaches, Classical method and Bayesian method.
A key issue faced within the manufacturing industry is determining how to measure quality characteristics and prioritise improvements to be made to all substandard quality characteristics of a product with respect to resource requirements and performance improvement potential. This study proposes a QCAC–Entropy–TOPSIS approach in order to address this issue. It combines the Quality Characteristic Analysis Chart (QCAC), entropy method and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). The proposed method is not only helpful to measure and determine whether the quality characteristics meet 6σ, 5σ, 4σ or 3σ but also to rank improvements in all substandard quality characteristics of a product in light of resource requirements and potential for performance improvements simultaneously, making it suitable to all manufacturing industries. Moreover, it also can be a powerful tool for analysing the problems that lead to substandard quality characteristics due to poor accuracy and/or precision. Firstly, using the QCAC, the substandard quality characteristics of the product can be determined and the corresponding values of the Discrimination Distance (DD) can then be computed. Subsequently, all the substandard quality characteristics can be regarded as alternatives when conducting entropy and TOPSIS analyses. Secondly, the weights of the evaluation criteria can be calculated by using the entropy method. Lastly, the weights of the evaluation criteria and the values of DD can be substituted into the TOPSIS method. The manufacturer can then categorically prioritise improvement options for all substandard quality characteristics with respect to resource requirements, and consider potential for performance improvements simultaneously. An example is provided for a bicycle quick release manufacturer to illustrate in detail the calculation process of the developed approach. Finally, the advantages of the proposed method are also given through comparisons with Process Capability Analysis Chart and TOPSIS methods.
Six Sigma has already become an efficient improvement technique adopted by a great number of enterprises. Numbers of Sigma
has become a tool of measuring process capability in some enterprises. But some of enterprises still use process capability
indices (PCIs) to measure the process capability. So numbers of Sigma and PCIs both can be used to measure the process capability.
The paper will research the relationship between PCIs and numbers of Sigma. In bilateral specifications, the paper will research
the relationship between the PCIs which are Cp, Cpk, Cpm and Cpmk, Spk and numbers of Sigma. In unilateral specifications, the paper will research the relationship between the PCIs which are Cpu and Cpl and numbers of Sigma. If supplier and buyer use different tools to measure the process capability, then the communion bridge
to Six Sigma and PCIs can decrease the communicate noise.
KeywordsSix Sigma-Process capability indices-Bilateral specifications-Unilateral specifications-Communion
Welded bond quality is a very important working aspect for the manufacturing and construction industries. A product is affected by quality of welding. A set of evaluation methods for welding operators is proposed in this paper, based on a theoretical analysis, to assure the quality of product. A new index using a categorical quantity expression to evaluate the welding performance is proposed. The mathematical relationship of welding performance, number of bad welds and the maximum tolerable number of bad welds are derived. Then, the best estimators and the deviations of this index are statistically inferred. Finally, a procedure and criteria are proposed to evaluate the performance of a welding operator. This evaluation method can be used in decision-making for the selection of a welding operator. Besides, it can also be used as an analytical tool to evaluate and improve the welding ability of a welding technician and the welding performance of a welding operator.
The liquid crystal displays (LCDs) possess the most mature technology and best consuming competitiveness in the flat panel display (FPD) industries. Of all the LCDs, thin film transistor-liquid crystal display (TFT-LCD) keeps some real advantages. It is thinner, smaller and lighter than other displays and has low power consumption, low-radiation, high-contrast and high-dpi at the same time. Hence, the TFT-LCD panel is widely applied in daily electronic products, and the demand for the TFT-LCD panel increases. The product life cycle of the TFT-LCD panel is gradually getting into the mature stage, and meanwhile, Taiwans firms face fierce competition from South Korea and Japan. Thus, at present, continual cost reduction is an all-out pursuing topic for Taiwans panel manufacturers. If the quality and yield of the TFT-LCD panel can be effectively enhanced, the non-conforming rate and the cost of the TFT-LCD panel will be reduced. In this study, TFT-LCD panel manufacturing process is discussed, and then five critical-to-quality (CTQ) characteristics (or sub-processes) are identified and summarized. At the early stages, the assessing model of process quality index (Cpm) and the MAIC (i.e., measure-analyze-improve-control) approach of Six Sigma are used to measure and analyze the CTQ characteristics that make TFT-LCD panel unqualified and incapable. During the later stages, Hartleys homogeneity test and joint confidence intervals were conducted to determine the optimal parameter setting of the critical factor in TFT-LCD panel manufacturing process. By using these optimal settings, Six Sigma quality level can be achieved.
The capability indices Cp, CPU, CPL, k and Cpk are presented and related to process parameters. These indices are shown to form a complementary system of measures of process performance, and can be used with bilateral and unilateral tolerances, with or without target values. A number of Japanese industries currently use the five indices and the U.S. automotive industry has started using these measures in a number of areas. Various applications of the indices are discussed along with statistical sampling considerations.
Process capability analysis is a vital part of an overall quality improvement programme. Numerous techniques and tools have been proposed for process capability analysis. Among these, indices and charts of process capability are simple and effective tools and widely used in the manufacturing industry. Many scholars have revealed numerous valuable aspects of previously developed tools and methods. Due to the rising demands of product quality, the current tools and methods are insufficient for enabling managers to make informed decisions. To address this gap, this study proposes a hypothesis testing procedure which determines whether the process capabilities satisfy the target level. Furthermore, this study proposes an integrated quality test chart (IQTC), which can display the process potential and performance for an entire product with smaller-the-better, larger-the-better and nominal-the-best specifications. The proposed procedure and IQTC incorporate the quality-level concept of the Six Sigma model and can be used to quantitate the relationships among the quality level, capability indices and process yield. They can be applied to assist managers in measuring, monitoring, analysing and improving process performance in a timely manner which will help ensure that the quality levels of their products meet customer demands. Finally, an example is provided to illustrate how to use the proposed procedure and IQTC.
The proportion of conformance of a process, i.e. the probability of producing within the specification area, is related to the majority of the most commonly used process capability indices (PCIs). In this article, the relationship of the four most widely used indices, i.e. Cp, Cpk, Cpm and Cpmk to the proportion of conformance is examined in the case of normal processes. Results known in the literature are presented along with several new ones. Various plots, revealing interesting aspects of these relationships are also provided.
Process capability indices (PCIs) are highly effective means for assessing the process potential and performance in manufacturing industry. However, the approaches developed for process capability indices (PCIs) can not analyse multi-process products. Some PCIs cause either the problems of sampling error of point estimates or full inspection. This paper aims to develop an advanced multi-process performance analysis chart with asymmetric tolerances (AMPPAC-AT) to assess the process performance of multi-process products. This approach takes into account interval estimates of the PCIs to overcome the shortcomings of full inspection or sampling error of point estimates. It can precisely measure an entire product composed of symmetric tolerances, asymmetric tolerances, larger-the-better, and smaller-the better characteristics. For application study, T-bar ceiling suspension system is handled to demonstrate the implementation process of AMPPAC-AT approach. We believe that our approach helps continuous improvement on overall assembly quality with asymmetric tolerances.
Lower confidence limits are derived for the common measures of process capability, usually indicated by Cp, CPU, CPL, and Cpk. The measures are estimated based on a random sample of observations from the process when the process is assumed to be normally distributed and has reached a state of statistical control.
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.
The indices Cp, Cpk, and Cpm are widely used to estimate whether a process is capable. Recently, techniques and tables were developed to construct lower 95% confidence limits for each index. These techniques, however, assume the underlying process is normally distributed, and processes that are modestly nonnormal do occur and can be hard to detect. Therefore, three nonparametric bootstrap lower confidence limits are proposed for each of these indices. A simulation using three distributions (one normal and two nonnormal) was conducted, and a comparison was made of the performances of the bootstrap and the parametric estimates. The simulation demonstrated that in the normal process environment the bootstrap confidence limits perform comparably to confidence limits based on normality and in nonnormal process environments the bootstrap estimates perform significantly better.
Capability indices compare the actual performance of a manufacturing process to the desired performance. In practice these indices are estimated using sample data, often with quite small sample sizes. Thus, it is of interest to obtain confidence limits (in particular, lower bounds) for the actual capability index given a sample estimate. Several methods for computing lower confidence bounds for these indices are described. A geometric view of the problem is presented and used as a basis for evaluating the performance of the various methods.
We provide a compact survey and brief interpretations and comments on some 170 publications on process capability indices which appeared in widely scattered sources during the years 1992 to 2000. An assessment of the most widely used process capability indices is also presented.
Rapid developments in advanced manufacturing techniques and information technology have resulted in significant increases in the functionality, precision, and complexity of products. Most products have multiple quality characteristics, which may include multiple unilateral and bilateral specifications. The process capability for any quality characteristic must be satisfied if customers are to accept a product. The production process of complex products generally consists of several stages and processes. The qualities of these processes can be independent or interdependent. This paper develops an integrated assessment model for the manufacture of multi-process, multi-characteristic products based on the process capability index and process yield rate. The proposed method assess six-sigma improvement using the 'define, measure, analyse, design, verify' model. This study investigates the manufacturing processes of 12.7 mm ordinary-grade machine gun cartridges in order to increase their shooting precision. The proposed assessment model identifies the critical-to-quality processes that affect the precision, and then applies redesign and experimental design to obtain the optimal combination of process parameters. Finally, a mathematical programming method verifies the results, confirming an improvement in shooting precision. The resulting cartridges meet the customer demand for match quality.
Effective management of process quality is crucial for attracting customers and maintaining their loyalty, and the quantification of quality for processes with multiple characteristics has recently been receiving considerable attention. This paper proposes a comprehensive process quality index to provide numerical measures for the precision, accuracy, and performance of processes in the manufacturing of golf clubshafts, using both smaller-the-better type and symmetric nominal-the-best type characteristics. The point estimates of these indices were replaced with joint confidence blocks in order to overcome a lack of reliability. Based on this index, we developed a control chart with process capability zones describing joint confidence blocks of all characteristics related to golf club-shafts within a single chart. Finally, we present an assessment procedure and illustrated examples with which to evaluate the process quality for the manufacture of golf club-shaft. The results reveal that the straightness and torsion of the club-shaft are "out of control" and unqualified, whereas the length, tip outer circumference, and butt outer circumference of the club-shaft are categorized as "excessive." The application of this chart enables engineers involved in the manufacturing of golf equipment to simultaneously monitor and control the quality characteristics of golf club-shafts.
A multiple stream process consists of several identical process streams. We present the process yield index for multiple stream processes with individual observations. An approximate distribution of the estimator of is derived. A simulation study is conducted to evaluate the performance of the confidence interval using the proposed method and the existing method. The simulation results show that the proposed method outperforms the existing method regarding interval length. We extend this process yield index for the case of subsamples. An approximate distribution for the estimator of is also derived. Two real examples are used to demonstrate the performance of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.
Process capability indices have been widely used in the manufacturing industry to provide numerical measures for process potential and process performance. However, from the perspective of improving process quality, there is a significant amount of process information that cannot be conveyed with a single index. Therefore, a single index does not have the ability to represent distinct problems in process performance or to provide the production department with sufficient information to make improvements. Thus, we applied an accuracy index δ and a precision index γ capable of reflecting the degree of deviation from target values and the degree of variance and incorporated the quality-level concept of the six-sigma model to develop a process quality-level analysis chart capable of analyzing the process capabilities of multiple quality characteristics. In addition to being able to directly identify the quality levels for various quality characteristics, the process quality-level analysis chart also provides recommendations for improvement of all quality-level regions to serve as a reference for production departments. Mathematical programming was used to develop a statistical hypothesis testing model to assist production departments in confirming the effectiveness of implemented improvements. This was achieved through derivation of a joint confidence interval for δ and γ from Boole's inequality. From this, we obtained the upper and lower limits of the (1 - α) × 100% for δ and γ. Finally, we also applied the proposed approach to a case study of a five-way pipe process.
The evolution of technology has increased the demand for convenient and user-friendly human-machine interfaces. This demand has driven the rapid development of the touch panel industry. Functioning as a conductive medium, indium tin oxide (ITO) film is a key factor in the quality of touch screens or panels. Because the sputtering process of ITO film includes both STB- and NTB-type quality characteristics, the analysis methodology proposed in the past cannot represent distinct influences or provide the production department with sufficient reference for improvements. Therefore, the purpose of this paper is to apply the bilateral index Cpm, which fully reflects process loss and yield, to develop an effective assessment model for the sputtering process with one-sided and two-sided specifications. Traditionally, Cpm can only be used to measure the capability of processes with two-sided specifications. We used the variable transformation method to develop the multi-process capability analysis chart. This chart is capable of measuring process precision and accuracy in relation to both types of quality characteristics. When process capability falls short of required standards, manufacturers can investigate the reasons according to the position of each quality characteristic on the chart. These data provide an important point of reference for proposing improvements to the sputtering process to enhance the quality of ITO film products and increase the competitive advantage and productivity of manufacturers.
Quality engineers use capability (or precision) indices to indicate the extent to which a process or machine can satisfy a specification. Like all other parameter estimates based on sampling, such indices are subject to uncertainty. The paper presents simple but effective approximations for standard errors and confidence intervals for some widely used indices. The presentation covers both American and Britisch/European terminology.
The capability indices are widely used by quality professionals as an estimate of process capability. Many process indices have been proposed and developed with Cp, Cpk and Cpm among the most widely used. More recently, techniques have been developed to construct lower 95 percent confidence limits for each index. These techniques are based on the assumption that the underlying process is normally distributed. The non-parametric but computer intensive method called Bootstrap is utilized and the Bootstrap confidence limits are calculated for these indices. A simulation using three distributions (normal, log-normal and chi-squared) was conducted and a comparison was made of the performances of the Bootstrap and the parametric estimates.
The common measures of process capability, usually indicated by Cp, CPU, CPL, and Cpk are considered. Tables of lower confidence limits are given on these measures where the sample mean, and the sample range (properly adjusted by d2) are substituted for the population mean and population standard deviation in the definition formulas.
The interpretation of Cpk:, a common measure of process capability and confidence limits for it, is based on the assumption that the process is normally distributed. The non-parametric but computer intensive method called Bootstrap is introduced and three Bootstrap confidence interval estimates for C^ are defined. An initial simulation of two processes (one normal and the other highly skewed) is presented and discussed
In today's world there is increasing emphasis on personal style and individuality, and hence a preference for semi-finished, customizable products over finished products. Consequently, sales of do-it-yourself products are increasing, including the high-quality equipment necessary to complete semi-finished goods. A representative example is the sewing machine; sewing machines must have high process yield to keep after-sales service costs low. In the current paper, a key component of sewing machines, the upper axle bearing, is used to present an implementation of Six Sigma via the measure-analyse-improve-control (MAIC) approach. Four bearing characteristics are identified as being critical to quality and are used to develop process capability indices for evaluating bearing quality. Then, a multi-characteristic product capability analysis chart (MPCAC) is used to identify and analyse the factors affecting bearing quality. Finally, the results of experiments and statistical tests using control charts to identify the optimum process levels are presented. The results show that the MAIC approach will help manufacturers and suppliers of sewing machines achieve Six Sigma quality.
The process capability indices Cp,Cp∗, CPL, CPU, CPL∗, CPU∗CpkCpk∗Cpm and C∗pm are presented and related to process parameters. These indices form a complementary system of measures of performance and are used by a number of U.S. and Japanese industries. Some properties of an estimator of Cpm,Ĉpm are given, and the operating characteristic curve approach is used to analyze Ĉp,Ĉpm and Ĉpm∗ where Ĉp and Ĉpm∗ are estimators of Ĉp and Cpm∗ respectively. Two tables for the analysis and use of Ĉp and Ĉpm are included, and an example of application of Ĉp,Ĉpm, and Ĉpk is provided, where Ĉpk is an estimator of Ĉpk. Finally, criticisms and benefits of the use of process capability indices are presented.
Practitioners of industrial statistics are generally familiar with the common C
p
and C
pk
process capability indices. However, many additional indices have been proposed, and knowledge of these is less widespread.
More importantly, information regarding the indices' comparative behavior is lacking. This paper compares the behavior of
various indices under shifting process conditions. Both useful and misleading characteristics of the indices are identified.
We begin with a short history of process capability measures. Several process capability indices are reviewed. Application
areas for capability indices are also summarized. The indices are grouped according to the loss functions which are used in
their interpretation. Characteristics of the various indices are discussed. Finally, recommendations are made for selection
of indices at differing levels of process performance.
The process capability indices C(p) and C(pk) are widely used to provide
unitless measures of process potential and performance. These indices do
not adequately address the issue of process centering. An alternative
definition of C(p) advocated by Taguchi (1985, 1986) addresses this
issue directly. Later authors introduced the name C(pm) for the Taguchi
index and examined statistical properties of an inefficient estimator
under the assumption that the process mean coincides with the target
value. The present paper presents statistical procedures based on the
original Taguchi estimator which require no assumptions on the process
mean. C(pm) and C(pk) are compared and contrasted to dispel the notion
that C(pk) measures process centering.
Process capability indices are useful management tools, which provide common quantitative measures on manufacturing capability and production quality. The indices CPU and CPL are designed specifically for processes with one-sided manufacturing specifications. The majority of the results obtained so far related to the distributional properties of the estimated capability indices were derived based on the assumption of possessing a single sample. However, a common practice in process control is to estimate the process capability indices by using the past ‘in control’ data from several subsamples. In order to use previous in-control data from multiple subsamples to make correct decisions regarding process capability, the distribution of the estimated capability index based on multiple subsamples should be available. In this paper, we develop a capability testing procedure with one-sided specifications using a Bayesian approach based on subsamples collected over time from an in-control process. By applying the proposed testing procedure, the practitioners can make reliable decisions to determine whether their processes meet the pre-set capability requirement when a daily based or weekly based production control plan is implemented for monitoring process stability.
Chen and Pearn (2001) proposed a new generalization of process capability indices (PCIs) for processes with asymmetric tolerances. (u, v), is superior to the original index Cp (u, v) and other existing generalizations by being closely related to actual the process yield, more sensitive to the process centering for given values of μ and σ, and the on-target process characteristic with the maximal value. In this article, Cp″ (u, v) is presented as the function of the accuracy index δ″ and the precision index γ″. We investigate the relationships of δ″ and γ″ with the process yield. We obtain the exact cumulative distribution functions and explicit forms of probability density functions of the natural estimators of δ″ and Cp″ (u, v) based on small subsamples data collecting from past “in-control” and S control charts. In addition, we derive the rth moments of and (u, v) and the expected values and the variances for , , and (u, v). We also analyze the statistical properties of the estimated indices , , and (u, v) assuming the process is normally distributed.
Estimators for σ, I/σ and the process capability index based on m preliminary samples, each of size n, are derived and their properties are discussed. A test of hypothesis and a confidence bound for CP are developed.
Under the assumption of normality, the distribution of estimators of a class of capability indices, containing the indices , , and , is derived when the process parameters are estimated from subsamples. The process mean is estimated using the grand average and the process variance is estimated using the pooled variance from subsamples collected over time for an in-control process. The derived theory is then applied to study the use of hypothesis testing to assess process capability. Numerical investigations are made to explore the effect of the size and number of subsamples on the efficiency of the hypothesis test for some indices in the studied class. The results for and indicate that, even when the total number of sampled observations remains constant, the power of the test decreases as the subsample size decreases. It is shown how the power of the test is dependent not only on the subsample size and the number of subsamples, but also on the relative location of the process mean from the target value. As part of this investigation, a simple form of the cumulative distribution function for the non-central -distribution is also provided. Copyright © 2003 John Wiley & Sons, Ltd.
A range of process capability indices is widely used to measure process performance. The simplicity of the formulae for these capability indices is both a strength and a weakness. The underlying assumptions behind capability indices are frequently overlooked. Capability studies usually result in single point estimates which may result in misleading assessments of process performance. Point estimates ignore sampling error, and safer estimates can be obtained by constructing confidence intervals. The construction of confidence intervals is considered in some detail. Testing or measurement variability give rise to additional uncertainty in process capability assessments. Inaccurate assessments of process performance can result if the basic assumptions and sources of uncertainty are overlooked.
Process precision index Cp has been widely used in the manufacturing industry for measuring process potential and precision. Estimating and testing process precision based on one single sample have been investigated extensively. In this paper, we consider the problem of estimating and testing process precision based on multiple samples taken from (
[`(x)]\bar{x}
,R)or (
[`(x)]\bar{x}
,S)control chart. We first investigate the statistical properties of the natural estimator of Cp and implement the hypothesis testing procedure. We then develop efficient MAPLE programs to calculate the lower confidence bounds, critical values, and p-values based on m samples of size n. Based on the test, we develop a step-by-step procedure for practitioners to use in determining whether their manufacturing processes are capable of reproducing products satisfying the preset precision requirement.
Process capability indices (PCIs) have been widely used to measure the actual process information with respect to the manufacturing specifications, and become the common language for process quality between the customer and the supplier. Most of existing research works for capability testing are based on the traditional frequentist point of view and statistical properties of the estimated PCIs are derived based on the assumption of one single sample. In this paper, we consider the problem of estimating and testing process capability using Bayesian approach based on subsamples collected over time from an in-control process. The posterior probability and the credible interval for the most popular index Cpk under a non-informative prior are derived. The manufacturers can use the presented approach to perform capability testing and determine whether their processes are capable of reproducing product items satisfying customers’ stringent quality requirements when a daily-based or weekly-based production control plan is implemented for monitoring process stability.
Process capability indices (PCIs), Cp, Ca, Cpk, Cpm, and Cpmk have been developed in certain manufacturing industry as capability measures based on various criteria, including process consistency, process departure from a target, process yield, and process loss. It is noted in certain recent quality assurance and capability analysis works that the three indices, Cpk, Cpm, and Cpmk provide the same lower bounds on the process yield. In this paper, we investigate the behavior of the actual process yield, in terms of the number of non-conformities (in ppm), for processes with fixed index values of Cpk=Cpm=Cpmk, possessing different degrees of process centering. We also extend Johnson's [1992. The relationship of Cpm to squared error loss. Journal of Quality Technology 24, 211–215] result formulating the relationship between the expected relative squared loss and PCIs. Also a comparison analysis among PCIs is carried out based on various criteria. The result illustrates some advantages of using the index Cpmk over the indices Cpk and Cpm in measuring process capability (yield and loss), since Cpmk always provides a better protection for the customers. Additionally, several extensions and applications to real world problem are also discussed. The paper contains some material presented in the Kotz and Johnson [2002. Process capability indices—a review, 1992–2000. Journal of Quality Technology 34(1), 1–19] survey but from a different perspective. It also discusses the more recent developments during the years 2002–2006.
In the trend toward the development of electronic products that are compact and lightweight, as portable consumer electronic products, such as the cell phone, Bluetooth, GPS, W-LAN, digital camera, wireless phone, and notebook computer, increase in demand, the frequency control components needed for communications related industries receive increased attention. The crystal oscillator is widely used as a frequency selective passive component in communications related industries because of excellent characteristics, such as temperature stability and a low loss. A crystal oscillator consists mainly of a quartz crystal and an IC that controls the oscillation circuits, and is applied to high precision communications products, requiring high frequency accuracy. A crystal oscillator with an output frequency that deviates or is unstable will seriously degrade the quality and functionality of an expensive communications product.This present research investigates the crystal oscillator manufacturing processes, developing risk priority number analysis specifically for critical-to-quality processes and identifying the optimum priority for improvements in the process quality. Using Taguchi experimental design techniques the optimal parameter design is determined for quality characteristics and a mathematic programming method establishes an objective mode for monitoring quality. Lastly, the present research uses a real case to verify the modes proposed in this project, to enhance customer satisfaction, and produce crystal oscillators with a competitive advantage.
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.
A review of the four basic process capability indices has been made. The interrelationship among these indices has been highlighted. Attention has been drawn to their drawbacks. The relation of these indices to the proportion nonconforming has been dwelt upon and the requirement of the adequate sample size has been emphasized. Cautionary remarks on the use of these indices in the case of nonnormal distributions, skewed distributions, and autocorrelated data are also presented. The effect of measurement error on process capability indices has been dealt with in great detail. Copyright (c) 2008 The Author. Journal compilation (c) 2008 International Statistical Institute.
Statistical Properties of Capability Indices
- W D Heavlin
Heavlin, W. D. 1988. Statistical Properties of Capability Indices. Technical Report 320. Sunnyvale, CA: Technical Library, Advanced
Micro Devices.