José Jorge Muñoz, Manuel J. Campuzano, Verónica Deibe-Blanco
Abstract In this paper, a c-control chart using a triple sampling scheme (TS-c) is studied. The chart design, procedure, and a bi-objective optimization model are given to optimize the TS-c-chart. The Average Run Length for in-control and out-of-control ( ARL 1 mathrm{ARL}_{1} ), and Average Sample Number metrics are calculated. A Comparison among TS-c, Fixed parameters c (FP-c), VSS-c, EWMA-c, and Double Sampling c (DS-c) control charts are carried out in terms of ARL 1 mathrm{ARL}_{1} . The proposed TS-c-chart has lower ARL 1 mathrm{ARL}_{1} values for detecting small and moderate shifts in the mean number of non-conformities in control compared with FP-c, VSS-c, EWMA-c, and DS-c.
{"title":"Design and Optimization of c-Control Chart Using a Triple Sampling Scheme","authors":"José Jorge Muñoz, Manuel J. Campuzano, Verónica Deibe-Blanco","doi":"10.1515/eqc-2023-0012","DOIUrl":"https://doi.org/10.1515/eqc-2023-0012","url":null,"abstract":"Abstract In this paper, a c-control chart using a triple sampling scheme (TS-c) is studied. The chart design, procedure, and a bi-objective optimization model are given to optimize the TS-c-chart. The Average Run Length for in-control and out-of-control ( ARL 1 mathrm{ARL}_{1} ), and Average Sample Number metrics are calculated. A Comparison among TS-c, Fixed parameters c (FP-c), VSS-c, EWMA-c, and Double Sampling c (DS-c) control charts are carried out in terms of ARL 1 mathrm{ARL}_{1} . The proposed TS-c-chart has lower ARL 1 mathrm{ARL}_{1} values for detecting small and moderate shifts in the mean number of non-conformities in control compared with FP-c, VSS-c, EWMA-c, and DS-c.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"135 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76725330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract One of the most important issues in the operation of any one- or multi-variable control chart is to determine the design parameters. Because in practice, production processes are affected by several assignable causes, several papers have been published to determine the design parameters. In all the papers presented so far, it has been assumed that after the occurrence of one of the assignable causes until the issuance of the true alarm, no other assignable cause occurs. Contrary to popular opinion, this paper argues that the formulas presented under this assumption for the average cost and quality cycle time in previous papers are incorrect and shows how the formula can be corrected. Therefore, this paper theoretically and numerically examines the conditions of occurrence of this assumption and its relationship with the design parameters in the design of multivariate control charts. A new economic model for determining design parameters is also presented. Numerical results show that the old formulas have a significant under-estimation of the average cost per unit time of the quality cycle. Also, a numerical study for economic and economic-statistical design of T 2 {T^{2}} control chart is presented under the proposed model.
{"title":"An Extension of Yang and Rahim’s Model to Determine Design Parameters in Multivariate Control Charts Under Multiple Assignable Causes and Weibull Shock Model","authors":"Rahmat Shojaei, M. Bameni Moghadam","doi":"10.1515/eqc-2022-0053","DOIUrl":"https://doi.org/10.1515/eqc-2022-0053","url":null,"abstract":"Abstract One of the most important issues in the operation of any one- or multi-variable control chart is to determine the design parameters. Because in practice, production processes are affected by several assignable causes, several papers have been published to determine the design parameters. In all the papers presented so far, it has been assumed that after the occurrence of one of the assignable causes until the issuance of the true alarm, no other assignable cause occurs. Contrary to popular opinion, this paper argues that the formulas presented under this assumption for the average cost and quality cycle time in previous papers are incorrect and shows how the formula can be corrected. Therefore, this paper theoretically and numerically examines the conditions of occurrence of this assumption and its relationship with the design parameters in the design of multivariate control charts. A new economic model for determining design parameters is also presented. Numerical results show that the old formulas have a significant under-estimation of the average cost per unit time of the quality cycle. Also, a numerical study for economic and economic-statistical design of T 2 {T^{2}} control chart is presented under the proposed model.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"47 1","pages":"25 - 46"},"PeriodicalIF":0.0,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74257450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Reliability is a popular concept that has been used in the manufacturing industry. In this paper, we consider a parallel system containing n non-identical and independent components in which each component is repairable except when all components are failed. As a special case, estimating the reliability of the system with identical components is considered. In real life, the data obtained for repair rate and failure rate could be subject to uncertainty. Here, to address this situation, failure and repair rates are considered as fuzzy numbers to estimate the reliability of the system. Fuzzy system reliability is estimated using fuzzy failure and repair rates, which are obtained by using confidence intervals and point estimators of failure rate and repair rate.
{"title":"Reliability Estimation of Parallel Repairable System under Uncertainty in Lifetime Data","authors":"Sruthi K., Mahesh Kumar","doi":"10.1515/eqc-2022-0044","DOIUrl":"https://doi.org/10.1515/eqc-2022-0044","url":null,"abstract":"Abstract Reliability is a popular concept that has been used in the manufacturing industry. In this paper, we consider a parallel system containing n non-identical and independent components in which each component is repairable except when all components are failed. As a special case, estimating the reliability of the system with identical components is considered. In real life, the data obtained for repair rate and failure rate could be subject to uncertainty. Here, to address this situation, failure and repair rates are considered as fuzzy numbers to estimate the reliability of the system. Fuzzy system reliability is estimated using fuzzy failure and repair rates, which are obtained by using confidence intervals and point estimators of failure rate and repair rate.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"76 1","pages":"1 - 9"},"PeriodicalIF":0.0,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83842653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Independence between successive counts is not a sensible premise while dealing, for instance, with very high sampling rates. After assessing the impact of falsely assuming independent binomial counts in the performance of np-charts, such as the one with 3-σ control limits, we propose a modified np-chart for monitoring first-order autoregressive counts with binomial marginals. This simple chart has an in-control average run length (ARL) larger than any out-of-control ARL, i.e., it is ARL-unbiased. Moreover, the ARL-unbiased modified np-chart triggers a signal at sample t with probability one if the observed value of the control statistic is beyond the lower and upper control limits L and U. In addition to this, the chart emits a signal with probability γ L {gamma_{L}} (resp. γ U {gamma_{U}} ) if that observed value coincides with L (resp. U). This randomization allows us to set the control limits in such a way that the in-control ARL takes the desired value ARL 0 {operatorname{ARL}_{0}} , in contrast to traditional charts with discrete control statistics. Several illustrations of the ARL-unbiased modified np-chart are provided, using the statistical software R and resorting to real and simulated data.
例如,在处理非常高的采样率时,连续计数之间的独立性不是一个明智的前提。在评估了错误假设独立二项计数对np图性能的影响后,例如具有3 σ控制极限的np图,我们提出了一种改进的np图,用于监测具有二项边缘的一阶自回归计数。这个简单的图表有一个可控的平均运行长度(ARL)大于任何失控的ARL,也就是说,它是ARL无偏的。此外,如果控制统计量的观测值超出控制下限L和上限u,则arl无偏修正np图在样本t处触发一个概率为1的信号,此外,图表发出一个概率为γ L {gamma_{L}}的信号(resp。γ U {gamma_{U}}),如果观测值与L (resp。这种随机化允许我们以这样一种方式设置控制限制,即控制中的ARL取期望值ARL 0 {operatorname{ARL}_{0}},与具有离散控制统计的传统图表相反。利用统计软件R,结合真实数据和模拟数据,给出了arl无偏修正np图的几个例子。
{"title":"An ARL-Unbiased Modified np-Chart for Autoregressive Binomial Counts","authors":"M. Morais, P. Wittenberg, Camila Jeppesen Cruz","doi":"10.1515/eqc-2022-0052","DOIUrl":"https://doi.org/10.1515/eqc-2022-0052","url":null,"abstract":"Abstract Independence between successive counts is not a sensible premise while dealing, for instance, with very high sampling rates. After assessing the impact of falsely assuming independent binomial counts in the performance of np-charts, such as the one with 3-σ control limits, we propose a modified np-chart for monitoring first-order autoregressive counts with binomial marginals. This simple chart has an in-control average run length (ARL) larger than any out-of-control ARL, i.e., it is ARL-unbiased. Moreover, the ARL-unbiased modified np-chart triggers a signal at sample t with probability one if the observed value of the control statistic is beyond the lower and upper control limits L and U. In addition to this, the chart emits a signal with probability γ L {gamma_{L}} (resp. γ U {gamma_{U}} ) if that observed value coincides with L (resp. U). This randomization allows us to set the control limits in such a way that the in-control ARL takes the desired value ARL 0 {operatorname{ARL}_{0}} , in contrast to traditional charts with discrete control statistics. Several illustrations of the ARL-unbiased modified np-chart are provided, using the statistical software R and resorting to real and simulated data.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"52 1","pages":"11 - 24"},"PeriodicalIF":0.0,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78822353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Repairable systems are submitted to corrective maintenance and condition-based preventive maintenance actions. Condition-based preventive maintenance occurs at times which are determined according to the results of inspections and degradation or operation controls. The generalization of the models suggested makes it possible to integrate the dependence between corrective and preventive maintenances. In order to take into account this dependency and the possibility of imperfect maintenances, generalized competing risks models have been presented in Doyen and Gaudoin (2006). In this study, we revise the general case in which the potential times to next corrective and preventive maintenance are independent conditionally to the past of the maintenance process. We address the identifiability issue and we find a result similar to that of Zhou, Lu, Shi and Cheng (2018) for usual competing risks. We propose realistic models with exponential risks and derive their likelihood functions.
{"title":"General Independent Competing Risks for Maintenance Analysis","authors":"Makram Krit","doi":"10.1515/eqc-2022-0029","DOIUrl":"https://doi.org/10.1515/eqc-2022-0029","url":null,"abstract":"Abstract Repairable systems are submitted to corrective maintenance and condition-based preventive maintenance actions. Condition-based preventive maintenance occurs at times which are determined according to the results of inspections and degradation or operation controls. The generalization of the models suggested makes it possible to integrate the dependence between corrective and preventive maintenances. In order to take into account this dependency and the possibility of imperfect maintenances, generalized competing risks models have been presented in Doyen and Gaudoin (2006). In this study, we revise the general case in which the potential times to next corrective and preventive maintenance are independent conditionally to the past of the maintenance process. We address the identifiability issue and we find a result similar to that of Zhou, Lu, Shi and Cheng (2018) for usual competing risks. We propose realistic models with exponential risks and derive their likelihood functions.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"51 1","pages":"117 - 126"},"PeriodicalIF":0.0,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86564024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper analyses a stochastic volatility model generated by first order normal-Laplace autoregressive process. The model parameters are estimated by the generalized method of moments. A simulation experiment is carried out to check the performance of the estimates. Finally, a real data analysis is provided to illustrate the practical utility of the proposed model and show that it captures the stylized factors of the financial return series.
{"title":"On Normal-Laplace Stochastic Volatility Model","authors":"Shiji Kavungal, Rahul Thekkedath","doi":"10.1515/eqc-2022-0013","DOIUrl":"https://doi.org/10.1515/eqc-2022-0013","url":null,"abstract":"Abstract This paper analyses a stochastic volatility model generated by first order normal-Laplace autoregressive process. The model parameters are estimated by the generalized method of moments. A simulation experiment is carried out to check the performance of the estimates. Finally, a real data analysis is provided to illustrate the practical utility of the proposed model and show that it captures the stylized factors of the financial return series.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"65 1","pages":"127 - 136"},"PeriodicalIF":0.0,"publicationDate":"2022-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89613788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The process capability index (PCI), denoted by 𝐼, is a well-known characteristic in quality control analysis. Using Gini’s mean difference, we construct a new PCI, I G I_{G} say, assuming the two-parameter Weibull distribution (WD). In order to estimate the proposed I G I_{G} when the process follows the WD, we use five classical methods of estimation and compare the performance of the obtained estimators with respect to their mean squared errors (MSEs) through a simulation study. Confidence intervals for the proposed PCI are constructed based on five bootstrap confidence intervals (BCIs) methods. Monte Carlo simulation study has been carried out to compare the performance of these five BCIs in terms of average widths and coverage probabilities. Finally, three real data sets from electronic and food industries are employed for illustrating the effectiveness of the proposed study. All these data sets show that the width of bias-corrected accelerated bootstrap interval is minimum among all other considered BCIs.
过程能力指数(PCI)是质量控制分析中一个众所周知的特征,用𝐼表示。利用基尼均值差,假设双参数威布尔分布(WD),我们构造了一个新的PCI, I G I_{G}。为了在过程遵循WD时估计所提出的I G I_{G},我们使用了五种经典的估计方法,并通过仿真研究比较了所得到的估计器的均方误差(MSEs)的性能。基于五种自举置信区间(bci)方法构建了PCI的置信区间。通过蒙特卡罗模拟研究,比较了这五种bci在平均宽度和覆盖概率方面的性能。最后,采用来自电子和食品行业的三个真实数据集来说明所提出研究的有效性。所有这些数据集表明,在所有其他考虑的bci中,偏差校正加速自举间隔的宽度是最小的。
{"title":"Estimation and Confidence Intervals of Modified Process Capability Index Using Robust Measure of Variability","authors":"Mahendra Saha, S. Dey","doi":"10.1515/eqc-2022-0014","DOIUrl":"https://doi.org/10.1515/eqc-2022-0014","url":null,"abstract":"Abstract The process capability index (PCI), denoted by 𝐼, is a well-known characteristic in quality control analysis. Using Gini’s mean difference, we construct a new PCI, I G I_{G} say, assuming the two-parameter Weibull distribution (WD). In order to estimate the proposed I G I_{G} when the process follows the WD, we use five classical methods of estimation and compare the performance of the obtained estimators with respect to their mean squared errors (MSEs) through a simulation study. Confidence intervals for the proposed PCI are constructed based on five bootstrap confidence intervals (BCIs) methods. Monte Carlo simulation study has been carried out to compare the performance of these five BCIs in terms of average widths and coverage probabilities. Finally, three real data sets from electronic and food industries are employed for illustrating the effectiveness of the proposed study. All these data sets show that the width of bias-corrected accelerated bootstrap interval is minimum among all other considered BCIs.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"1 1","pages":"153 - 164"},"PeriodicalIF":0.0,"publicationDate":"2022-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79087138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Salmasnia, samrad Jafarian-Namin, Behnam Abdzadeh
Abstract Imperfect processes experience fault productions over time due to specific causes. Integrating the statistical process control, maintenance policy, and economic production quantity has led to more favorable results for the imperfect processes in literature. When monitoring a process, multiple assignable causes (ACs) may shift it to an out-of-control state. As indicated recently, if the interdependency of ACs is neglected, the total cost will be underestimated. Moreover, the mean and variance can simultaneously be affected by the occurrence of ACs. A non-central chi-square (NCS) chart was suggested for its decent performance against X-R chart in detecting the process disturbances and lowering quality loss cost. Besides, the increased occurrence rate of ACs over time leads to higher quality and maintenance costs. Employing a non-uniform sampling (NUS) scheme can significantly reduce costs. In the literature of modeling for imperfect processes under multiple ACs, all input parameters have always been fixed. The effectiveness of the models depends somewhat on the accurate estimates of these parameters. In reality, the estimation of parameters may be associated with uncertainty. For the first time, a robust design approach is proposed for designing NCS chart by considering the interval estimation of uncertain parameters. A particle swarm optimization (PSO) algorithm is used to present solutions. The proposed model is investigated through a real numerical example.
{"title":"Robust Optimization of an Imperfect Process when the Mean and Variance are Jointly Monitored under Dependent Multiple Assignable Causes","authors":"A. Salmasnia, samrad Jafarian-Namin, Behnam Abdzadeh","doi":"10.1515/eqc-2022-0018","DOIUrl":"https://doi.org/10.1515/eqc-2022-0018","url":null,"abstract":"Abstract Imperfect processes experience fault productions over time due to specific causes. Integrating the statistical process control, maintenance policy, and economic production quantity has led to more favorable results for the imperfect processes in literature. When monitoring a process, multiple assignable causes (ACs) may shift it to an out-of-control state. As indicated recently, if the interdependency of ACs is neglected, the total cost will be underestimated. Moreover, the mean and variance can simultaneously be affected by the occurrence of ACs. A non-central chi-square (NCS) chart was suggested for its decent performance against X-R chart in detecting the process disturbances and lowering quality loss cost. Besides, the increased occurrence rate of ACs over time leads to higher quality and maintenance costs. Employing a non-uniform sampling (NUS) scheme can significantly reduce costs. In the literature of modeling for imperfect processes under multiple ACs, all input parameters have always been fixed. The effectiveness of the models depends somewhat on the accurate estimates of these parameters. In reality, the estimation of parameters may be associated with uncertainty. For the first time, a robust design approach is proposed for designing NCS chart by considering the interval estimation of uncertain parameters. A particle swarm optimization (PSO) algorithm is used to present solutions. The proposed model is investigated through a real numerical example.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"38 1","pages":"137 - 151"},"PeriodicalIF":0.0,"publicationDate":"2022-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73617968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the statistical process control literature, counts of nonconforming items are frequently assumed to be independent and have a binomial distribution with parameters ( n , p ) (n,p) , where 𝑛 and 𝑝 represent the fixed sample size and the fraction nonconforming. In this paper, the traditional n p np -chart with 3-𝜎 control limits is reexamined. We show that, even if its lower control limit is positive and we are dealing with a small target value p 0 p_{0} of the fraction nonconforming ( p ) (p) , this chart average run length (ARL) function achieves a maximum to the left of p 0 p_{0} . Moreover, the in-control ARL of this popular chart is also shown to vary considerably with the fixed sample size 𝑛. We also look closely at the ARL function of the ARL-unbiased n p np -chart proposed by Morais [An ARL-unbiased n p np -chart, Econ. Qual. Control 31 (2016), 1, 11–21], which attains a pre-specified maximum value in the in-control situation. This chart triggers a signal at sample 𝑡 with probability one if the observed number of nonconforming items, x t x_{t} , is beyond the lower and upper control limits (𝐿 and 𝑈), probability γ L gamma_{L} (resp. γ U gamma_{U} ) if x t x_{t} coincides with 𝐿 (resp. 𝑈). A graphical display for the ARL-unbiased n p np -chart is proposed, taking advantage of the qcc package for the statistical software R. Furthermore, as far as we have investigated, its control limits can be obtained using three different search algorithms; their computation times are thoroughly compared.
在统计过程控制文献中,经常假定不合格品的数量是独立的,具有参数(n,p) (n,p)的二项分布,其中𝑛和𝑝代表固定的样本量和不合格品的比例。本文对传统的具有3- φ控制限的n ^ p np图进行了重新检验。我们表明,即使它的下控制极限是正的,并且我们处理的是不符合分数(p) (p) (p)的一个小目标值p0 p_{0},这个图表平均运行长度(ARL)函数在p0 p_{0}的左边达到最大值。此外,这个流行图表的控制ARL也显示出随固定样本量𝑛变化很大。我们还仔细研究了Morais提出的ARL-无偏n减去p - np -图的ARL函数[An ARL-无偏n减去p - np -图,经济学]。[q] . Control 31(2016), 1,11 - 21],在控制状态下达到预定最大值。如果观察到的不合格品数量x t x_{t}超出控制下限和上限(𝐿和𝑈),则该图表在样本𝑡触发一个信号,概率为1,概率为γ L gamma_{L} (resp. 1)。γ U gamma_{U}),如果x t x_{t}与𝐿(p。𝑈)。利用统计软件r的qcc包,提出了一种arl -无偏n≠p np -图的图形显示方法。此外,就我们所研究的,它的控制极限可以使用三种不同的搜索算法获得;它们的计算时间进行了彻底的比较。
{"title":"The 𝑛𝑝-Chart with 3-𝜎 Limits and the ARL-Unbiased 𝑛𝑝-Chart Revisited","authors":"M. Morais, P. Wittenberg, Camila Jeppesen Cruz","doi":"10.1515/eqc-2022-0032","DOIUrl":"https://doi.org/10.1515/eqc-2022-0032","url":null,"abstract":"Abstract In the statistical process control literature, counts of nonconforming items are frequently assumed to be independent and have a binomial distribution with parameters ( n , p ) (n,p) , where 𝑛 and 𝑝 represent the fixed sample size and the fraction nonconforming. In this paper, the traditional n p np -chart with 3-𝜎 control limits is reexamined. We show that, even if its lower control limit is positive and we are dealing with a small target value p 0 p_{0} of the fraction nonconforming ( p ) (p) , this chart average run length (ARL) function achieves a maximum to the left of p 0 p_{0} . Moreover, the in-control ARL of this popular chart is also shown to vary considerably with the fixed sample size 𝑛. We also look closely at the ARL function of the ARL-unbiased n p np -chart proposed by Morais [An ARL-unbiased n p np -chart, Econ. Qual. Control 31 (2016), 1, 11–21], which attains a pre-specified maximum value in the in-control situation. This chart triggers a signal at sample 𝑡 with probability one if the observed number of nonconforming items, x t x_{t} , is beyond the lower and upper control limits (𝐿 and 𝑈), probability γ L gamma_{L} (resp. γ U gamma_{U} ) if x t x_{t} coincides with 𝐿 (resp. 𝑈). A graphical display for the ARL-unbiased n p np -chart is proposed, taking advantage of the qcc package for the statistical software R. Furthermore, as far as we have investigated, its control limits can be obtained using three different search algorithms; their computation times are thoroughly compared.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"42 1","pages":"107 - 116"},"PeriodicalIF":0.0,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87634609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}