Proportion Of Nonconforming Items Research Articles

Assessing the economic-statistical performance of an attribute SVSSI-np control chart based on genetic algorithms

The SVSSI scheme, featuring three variable sample sizes and two variable sampling intervals, considerably enhances the performance of control charts in detecting shifts. As a novel contribution, this study proposes an optimized design of the SVSSI-np control chart using genetic algorithms, illustrating its superior performance over other sampling schemes by considering economic and statistical perspectives in a unified model. The outcomes of implementing the SVSSI alongside other sampling schemes are assessed via a numerical example, demonstrating the influence of varying parameters such as process failure rate, shift value, and proportion of nonconforming items. Based on the obtained results, the SVSSI scheme outperforms other schemes by achieving over a 10% reduction in costs and over a 30% improvement in reducing time to signal and false alarm rates in most runs. The impact of inaccurately estimating the shift size is also investigated, revealing that incorrect estimates can lead to ineffective adjustments and improvements. Finally, optimal SVSSI designs are explored through comprehensive sensitivity analysis and robust optimization to address uncertainty, and the results of applying fractional factorial design are statistically analyzed. In summary, the SVSSI-np control chart demonstrates potential for enhanced cost reduction, process monitoring, and efficiency gains.

An index of capability for bivariate zero-inflated processes

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.

A closer look at the equivalence of Bernoulli and geometric CUSUM control charts

AbstractSome researchers have incorrectly concluded that the geometric CUSUM is superior to the Bernoulli CUSUM as a procedure for monitoring a repetitive process, even though the two procedures have been proved to be equivalent for detecting an upward shift in the proportion of nonconforming items. We use an exact Markov‐chain‐based methodology to re‐examine the relationship between geometric CUSUM and Bernoulli CUSUM control charts. Exact methods allow us to differentiate between similar but different‐valued quantities that have contributed to some misunderstandings in the literature. We show that for a random‐shift model, evaluations of steady‐state average number of inspected items until a signal (ANIs) are identical for both geometric CUSUMs and Bernoulli CUSUMs, provided the correct choices of return levels are made. We also show that a steady‐state geometric CUSUM based on a fixed‐shift model only uses the geometric CUSUM states, while a steady‐state geometric CUSUM based on a random‐shift model will reach the states of a Bernoulli CUSUM after a long series of zeros. We note that our conclusions are contrary to the published results of other researchers, and we examine these differences in detail.Layman's Abstract: Since, their introduction by Walter Shewhart in 1931, control charts such as the Shewhart p‐chart have had widespread application for monitoring the output quality from manufacturing processes, such as the proportion (p) in the stream of manufactured items that are nonconforming. But the Shewhart p‐chart is not very effective for monitoring processes when the proportion p is less than about 4 percent. Other charts, such as the geometric CUSUM (proposed in 1991) and the Bernoulli CUSUM (proposed in 1999) have been shown to be superior at identifying changes in p when p is small. Some researchers have relied on simulation‐based investigations to compare these two CUSUMs, and have incorrectly concluded that the geometric CUSUM is superior to the Bernoulli CUSUM, even though the two procedures have been proved to be equivalent for detecting an upward shift in the proportion p. Using exact, Markov‐chain‐based methodology, we re‐examine the relationship between the geometric CUSUM and the Bernoulli CUSUM control charts and demonstrate that these two charts are equivalent when evaluated correctly. Exact methods allow us to precisely compare these two monitoring schemes and correct some erroneous conclusions that have appeared in the literature.

Variable stage-independent double sampling plan with screening for acceptance quality loss limit inspection scheme

The quality loss suggested by Taguchi has been recognised as a new quality evaluation based on variable property instead of the quality evaluation based on traditional attribute property such as the proportion of nonconforming items. Some variable sampling inspection plans in order to assure the quality have been considered. As one of those sampling inspection plans, the single sampling plan with screening (SSPS) has been constructed for acceptance quality loss limit inspection scheme. Further, the repetitive group sampling plan with screening (RGSPS) has been developed for reducing the average total inspection (ATI). Although ATI by RGSPS has successfully been reduced in comparison with ATI by SSPS, RGSPS may sometimes increase the average sampling frequency (ASF) in return for reducing ATI. In this study, the stage-independent double sampling plan with screening (SIDSPS) based on the concept of the acceptance quality loss limit inspection scheme is proposed under the consideration of the trade-off between ATI and ASF. Through some numerical comparisons about ATI and ASF in SSPS, RGSPS, and SIDSPS, the usefulness of SIDSPS proposed in this study is confirmed.

Integrated production quality and condition-based maintenance optimisation for a stochastically deteriorating manufacturing system

This paper investigates the problem of optimally integrating production quality and condition-based maintenance in a stochastically deteriorating single- product, single-machine production system. Inspections are periodically performed on the system to assess its actual degradation status. The system is considered to be in ‘fail mode’ whenever its degradation level exceeds a predetermined threshold. The proportion of non-conforming items, those that are produced during the time interval where the degradation is beyond the specification threshold, are replaced either via overtime production or spot market purchases. To optimise preventive maintenance costs and at the same time reduce production of non-conforming items, the degradation of the system must be optimally monitored so that preventive maintenance is carried out at appropriate time intervals. In this paper, an integrated optimisation model is developed to determine the optimal inspection cycle and the degradation threshold level, beyond which preventive maintenance should be carried out, while minimising the sum of inspection and maintenance costs, in addition to the production of non-conforming items and inventory costs. An expression for the total expected cost rate over an infinite time horizon is developed and solution method for the resulting model is discussed. Numerical experiments are provided to illustrate the proposed approach.

Rectifying inspection for PAOSQLL scheme based on variable repetitive group sampling plan

The concept of the quality loss derived by Taguchi has been accepted as the evaluation measure of quality instead of the traditional attribute property such as the proportion of non-conforming items. Since the variable single sampling plan having the desired operating characteristics indexed by the quality loss was proposed in 1997, various kinds of sampling plans indexed by the quality loss have been considered. Among them, there are two kinds of rectifying variable single sampling (RVSS) plans indexed by the quality loss. In the RVSS plans, two inspection schemes called the acceptance quality loss limit (AQLL) scheme and the permissible average outgoing surplus quality loss limit (PAOSQLL) scheme have been formulated. Note that the concepts of the AQLL and PAOSQLL schemes in the RVSS plans indexed by the quality loss are equivalent to those of the lot tolerance percent defective (LTPD) and the average outgoing quality limit (AOQL) schemes in the traditional rectifying attribute single sampling plans, respectively. On the other hand, in the sampling plan having the desired operating characteristics, an attribute repetitive group sampling plan was proposed for the purpose of reducing the average sampling number in the inspection. Recently, by applying the repetitive group sampling to the rectifying variable sampling plan for the AQLL scheme, the rectifying variable repetitive group sampling (RVRGS) plan for the AQLL scheme has been considered for the purpose of reducing the average total inspection (ATI). However, the RVRGS plan for the PAOSQLL scheme has not been investigated yet. Accordingly, the RVRGS plan for the PAOSQLL scheme must be investigated to complete the RVRGS plans indexed by the quality loss. In this article, the RVRGS plan for the PAOSQLL scheme is addressed. Then, the design procedure in the RVRGS plan for the PAOSQLL scheme is proposed for the purpose of reducing ATI. Through some numerical investigations, the effectiveness to reduce ATI by the RVRGS plan for the PAOSQLL scheme is confirmed.

Comparison of Rigorous Design Procedure with Approximate Design Procedure for Variable Sampling Plans Indexed by Quality Loss

Traditional acceptance sampling plans have focused on the proportion of nonconforming items as an attribute criterion for quality. In today’s modern quality management under high quality production environments, the reduction of the deviation from a target value in a quality characteristic has become the most important purpose. In consequence, various inspection plans for the purpose of reducing the deviation from the target value in the quality characteristic have been investigated. In this case, a concept of the quality loss evaluated by the deviation from the target value has been accepted as the variable evaluation criterion of quality. Further, some quality measures based on the quality loss have been devised; e.g. the process loss and the process capability index. Then, as one of inspection plans based on the quality loss, the rigorous design procedure for the variable sampling plan having desired operating characteristics (VS-OC plan) indexed by the quality loss has been proposed by Yen and Chang in 2009. By the way, since the estimator of the quality loss obeys the non-central chi-square distribution, the rigorous design procedure for the VS-OC plan indexed by the quality loss is complicated. In particular, the rigorous design procedure for the VS-OC plan requires a large number of the repetitive and complicated numerical calculation about the non-central chi-square distribution. On the other hand, an approximate design procedure for the VS-OC plan has been proposed before the proposal of the above rigorous design procedure. The approximate design procedure for the VS-OC plan has been constructed by combining Patnaik approximation relating the non-central chi-square distribution to the central chi-square distribution and Wilson-Hilferty approximation relating the central chi-square distribution to the standard normal distribution. Then, the approximate design procedure has been devised as a convenient procedure without complicated and repetitive numerical calculations. In this study, through some comparisons between the rigorous and approximate design procedures, the applicability of the approximate design procedure has been confirmed.

Proposal of variable sequential sampling plan having desired operating characteristics indexed by quality loss

The proportion of non-conforming items has been traditionally utilised as an evaluation criterion for quality of items. However, the proportion of non-conforming items is not necessarily useful as a proper evaluation criterion for controlling high-quality manufacturing in recent years. Accordingly, in order to achieve further quality improvement and innovation, more careful quality evaluation has been required newly. Then, a concept of quality loss in the Taguchi methods has been devised as a severe criterion of quality evaluation. Hereby, a variable single sampling plan having desired operating characteristics (OCs) indexed by quality loss has been proposed in the area of statistical quality control. By the way, the most economical sampling inspection in the average sample number (ASN) is the sequential sampling plan based on the Wald’s sequential probability ratio test. Then, from the viewpoint of cost reduction, we discuss a variable sequential sampling plan having desired OC indexed by quality loss with the aim of expansion of the utility of variable sampling plan for quality loss. As the result, the design procedure of the sequential sampling plan for satisfying some required design conditions indexed by quality loss is provided. In addition, the effectiveness of the proposed sequential sampling plan is verified through some numerical examples.

Rectifying inspection for acceptable quality loss limit based on variable repetitive group sampling plan

It is usually assumed that a quality characteristic in an item obeys a normal distribution in the case that the quality of items is evaluated based on the variable property. Then, the concept of Taguchi’s quality loss has been accepted as the evaluation measure of quality instead of the traditional attribute property such as the proportion of nonconforming items. From this viewpoint, some variable sampling plans indexed by the quality loss have been investigated before now. As a study earliest among them, the variable single sampling plan based on operating characteristics (OC) indexed by the quality loss was considered. On the other hand, the attribute repetitive group sampling plan on OC was proposed for reducing the sampling number in the inspection. Recently, the variable repetitive group sampling (VRGS) plan on OC indexed by the quality loss has been considered. By the way, the rectifying inspection is known as one of the schemes of acceptance sampling inspection. Then, Dodge-Romig single sampling plans are known as the traditional rectifying inspection based on attribute sampling plans. Dodge-Romig rectifying attribute sampling plans provide the lot tolerance percent defective (LTPD) scheme on each lot and the average outgoing quality limit (AOQL) scheme for many lots. Furthermore, the rectifying variable single sampling (RVSS) plan indexed by the quality loss was investigated. In conformity with the traditional rectifying attribute sampling plans for the LTPD and AOQL schemes, the acceptance quality loss limit (AQLL) and specified permissible average outgoing surplus quality loss limit (PAOSQLL) schemes are respectively proposed in the RVSS plans indexed by the quality loss. In this article, we suppose that the quality characteristic in an item obeys a normal distribution. Under this condition, the rectifying variable repetitive group sampling (RVRGS) plan for AQLL is considered for the purpose of reducing the average total inspection (ATI). Specifically, the design procedure for finding out the required sample size and inspection criteria for satisfying the constraint of the quality assurance is derived. Lastly, it is shown that ATI of the RVRGS plan is reduced in comparison with that of the RVSS plan under the same condition.

Adjustment of Control Limits for Geometric Charts

The geometric chart has proven more effective than Shewhart p or np charts to monitor the proportion nonconforming in high-quality processes. Implementing a geometric chart commonly requires the assumption that the in-control proportion nonconforming is known or accurately estimated. However, accurate parameter estimation is very difficult and may require a larger sample size than that available in practice in high-quality process where the proportion of nonconforming items is very small. Thus, the error in the parameter estimation increases and may lead to deterioration in the performance of the control chart if a sample size is inadequate. We suggest adjusting the control limits in order to improve the performance when a sample size is insufficient to estimate the parameter. We propose a linear function for the adjustment constant, which is a function of the sample size, the number of nonconforming items in a sample, and the false alarm rate. We also compare the performance of the geometric charts without and with adjustment using the expected value of the average run length (ARL) and the standard deviation of the ARL (SDARL).

Variable sampling inspection plans with screening indexed by Taguchi’s quality loss for optimising average total inspection

There are some conventional attribute sampling inspection plans designed with respect to attribute property, such as the proportion of nonconforming items. Among them, the attribute sampling inspection plans with screening are defined under the lot tolerance percent defective (LTPD) inspection scheme and the average outgoing quality limit (AOQL) inspection scheme. On one hand, Taguchi has proposed the concept of quality loss as a measure to evaluate the quality of items based on the variable property instead of the traditional attribute quality evaluation. Hereby, in this article, the design procedures for variable sampling inspection plans with screening indexed by Taguchi’s quality loss in conformity with the conventional attribute sampling inspection plans with screening composed of the LTPD and AOQL inspection schemes are represented.

The Effect of Parameter Estimation on Upper‐sided Bernoulli Cumulative Sum Charts

The Bernoulli cumulative sum (CUSUM) chart has been shown to be effective for monitoring the rate of nonconforming items in high‐quality processes where the in‐control proportion of nonconforming items (p0) is low. The implementation of the Bernoulli CUSUM chart is often based on the assumption that the in‐control value p0 is known; therefore, when p0 is unknown, accurate estimation is necessary. We recommend using a Bayes estimator to estimate the value of p0 to incorporate practitioner knowledge and to avoid estimation issues when no nonconforming items are observed in phase I. We also investigate the effects of parameter estimation in phase I on the upper‐sided Bernoulli CUSUM chart by using the expected value of the average number of observations to signal (ANOS) and the standard deviation of the ANOS. It is found that the effects of parameter estimation on the Bernoulli CUSUM chart are more significant than those on the Shewhart‐type geometric chart. The low p0 values inherent to high‐quality processes imply that a very large, and often unrealistic, sample size may be needed to accurately estimate p0. A methodology to identify a continuous variable to monitor is highly recommended when the value of p0 is low and the required phase I sample size is impractically large. Copyright © 2012 John Wiley & Sons, Ltd.

Modelagem da fração de não-conformes em processos industriais

Em qualquer processo industrial, pode ser definido um conjunto de causas ou fatores que produzem determinado efeito sobre uma ou mais características de qualidade de um produto que pode ou não satisfazer às especificações do cliente, gerando a produção de produtos não-conformes. A modelagem da proporção ou fração de produtos não-conformes utilizando-se o modelo de regressão linear não é adequada por pelo menos duas razões: (i) pressupõe que as proporções seguem a distribuição normal, que não é correto; e (ii) possibilita a previsão de valores fora do intervalo [0,1]. Alternativas à modelagem da proporção de não-conformes são os Modelos Lineares Generalizados e os Modelos de Regressão Beta. O objetivo deste artigo é modelar a fração de não-conformes às especificações de uma indústria curtidora com enfoque nos Modelos de Regressão Beta e Modelo Linear Generalizado. Esses modelos podem ser estendidos a processos industriais que envolvam a produção de produtos não-conformes às especificações de manufatura.

Process Monitoring with Multivariate p-Control Chart

We assume that the operator is interested in monitoring a multinomial process. In this case the items are classified into (k+1) ordered distinct and mutually exclusive defect categories. The first category is used to classify the conforming defect-free items, while the remaining k categories are used to classify the nonconforming items in k defect grades, with increasing degrees of nonconformity. Usually the process is said to be capable if the overall proportion of nonconforming items is very small and remains low, or declines over time. Nevertheless, since we classify the nonconforming items into k distinct defect grades, the operator can also evaluate the overall level of defectiveness. This quality parameter depends on the k defect categories. Furthermore, we are interested in evaluating, over time, the proportion of nonconforming items in each category as well as the overall level of defectiveness. To achieve this goal, we propose (i) a normalized index that can be used to evaluate the capability of the process in terms of the overall level of defectiveness, and (ii) a two-sided Shewhart-type multivariate control chart to monitor the overall proportion of nonconforming items and the corresponding defectiveness level.

Capability Indices and the Proportion of Nonconforming Items

For nonnormal data, there exists no standard calculation for the process capability. Many different indices were developed and described in the literature, but none of them predict the proportion of nonconforming items well enough. This paper starts with a sharp upper bound for the proportion of nonconforming items if the distribution is in the class of unimodal distributions. This class contains the Student's t, the logistic, the gamma, the χ 2, the exponential, the log-normal, and the Weibull distributions. For a given C pk -value, the maximum proportions nonconforming for these parametric distributions are also determined and compared with the upper bound. These maximum proportions nonconforming are a first step to relate the process capability to the proportion of nonconforming items. Furthermore, two different capability indices for skewed data, developed by Munechika and Bai and Choi, are investigated on their ability to adapt to nonnormality. It turns out that for our selection of distributions, the Munechika capability index corrects the standard capability index C pk too much and the Bai and Choi capability does not compensate the C pk enough. Since this occurs for each of our skewed distributions, the true capability (based on the inverse normal distribution of the proportion nonconforming) lies in between these two capability indices. Therefore, we propose to calculate the maximum proportions nonconforming, the standard capability indices, and the indices of Munechika and Bai and Choi as a standard approach to determine the process capability. This approach maintains the simplicity of capability indices as a way to express a process in only a few numbers, and it indicates all aspects of the process capability. This method is illustrated in a real-life example.

Increasing the Sensitivity of Control Chart for Fraction Nonconforming

This paper discusses an easy approach for increasing the sensitivity of the p chart by incorporating runs rules. A method for calculating the exact control limits for the various standardized p chart schemes incorporating runs rules is shown. Simple mathematical calculations reveal that the performances of all p chart schemes incorporating runs rules are superior to that of the standard approach. These calculations enable practitioners to determine the average run length (ARL) value that corresponds to a certain level of fraction nonconforming, p 1, based on a standard or known predefined proportion of nonconforming items, p 0. Examples are given to illustrate the application of the procedures discussed.

A Nonparametric Multivariate Process Capability Index

For the multivariate manufacturing processes, tremendous difficulties are often encountered when one attempts to measure the process capability by directly extending the univariate approach, that is, comparing the specification limits (tolerance zones) with the actual process spread. In fact, the existing multivariate process capability indices developed along the same line are very complicated to apply even under the normality assumption. The authors of this paper propose a new multivariate process capability index that is directly related to the proportion of nonconforming items. Moreover, the new index is calculated in a nonparametric setting; hence, it does not rely on a particular distribution. The estimation methods of the new index are studied in detail. Simulations for the elliptical and rectangular tolerance zones under different bivariate distributions are carried out to illustrate the new approach. The applications of the new index to real-life examples are also presented.

Using dualnp-charts to detect changes

Attribute control charts are an important tool in detecting quality improvement or degradation. Ryan and Schwertman (J. Qual. Technol., 29, 86--98 (1997)) suggest that a carefully chosen sample size is needed to ensure that an attribute chart will have properties that are close to the properties assumed under normal theory. However, their recommendations require a considerable amount of flexibility in selecting the sample size. Unfortunately, such flexibility, particularly when there is a restriction on the maximum sample size, may not be possible. This is particularly true when the nominal proportion of non-conforming items is small. A dual control chart procedure is suggested, with one chart providing an early warning of quality deterioration and the other a cumulative chart having approximate normal theory properties in accordance with the suggested approach of Ryan and Schwertman. Copyright © 1999 John Wiley & Sons, Ltd.

Integrated production-inventory models with imperfect production processes and a limited capacity for raw materials

This paper deals with inventory models that unify the decisions for raw materials and the finished product for a single product manufacturing system. The product is manufactured in batches and raw materials are jointly replenished from outside suppliers. The system is assumed to deteriorate during the production process. As a result, some proportion of nonconforming items is produced. The objective is to minimize the total variable cost for the system. A solution procedure is developed to find a near optimal solution for the basic model. The analysis for the basic model is extended to cases where the proportion of defective items is not constant or the defective rate is a function of production setup cost.

Consequences of using the Cpk index to estimate the proportion of non-conforming items

The Cpk index is a common process performance standard which quantifies process location and variation. However, the index is also frequently used to estimate the proportion of non-conforming items without regard to the underlying restrictions or assumptions that apply. An example based on an actual industrial case study will serve to underscore the consequences of the indiscriminate use of the index for this purpose. Subsequently, additional experimentation and analyses will be conducted to explore the generality of the results obtained from the case study