Effects of imprecise measurement on the two dependent processes control for the autocorrelated observations

The observations from the process output are always assumed to be independent when using a control chart to monitor a process. However, for many processes the observations are autocorrelated and including the measurement error due to the measurement instrument. This autocorrelation and measurement error can have a significant effect on the performance of the process control. This paper considers the problem of monitoring the mean of a quality characteristic X on the first process, and the mean of a quality characteristic Y on the second process, in which the observations X can be modeled as an ARMA model and observations Y can be modeled as an transfer function of X since the state of the second process is dependent on the state of the first process. To distinguish and maintain the state of the two dependent processes with measurement errors, the Shewhart control chart of residuals and the cause-selecting control chart, based on residuals, are proposed. The performance of the proposed control charts is evaluated by the rate of true or false alarms. By numerical analysis, it shows that the performance of the proposed control charts is significantly influenced by the variation of measurement errors. The application of the proposed control charts is illustrated by a numerical example .     

An approach to controlling two dependent process steps with autocorrelated observations

The observations from the process output are always assumed independent when using a control chart to monitor a process. However, for many processes the process observations are autocorrelated. This autocorrelation can have a significant effect on the performance of the control chart. This paper considers the problem of monitoring the mean of a quality characteristic X on the first process step and the mean of a quality characteristic Y on the second process step, in which the observations X can be modeled as an AR(1) model and observations Y can be modeled as a transfer function of X since the state of the second process step is dependent on the state of the first process step. To effectively distinguish and maintain the state of the two dependent process steps, the Shewhart control chart of residual and the cause-selecting control chart are proposed. The proposed control charts’ performance is measured by the rate of alarm on the proposed charts. From numerical analysis, it shows that the performance of the proposed control charts is much better than the misused Hotelling T² control chart and the individual Shewhart X and Y control charts.     

用残差CUSUM控制图检测自相关过程中的故障

由于累积和控制图(cumulative sum,简称CUSUM)算法应用的基本假设是从过程得到的观测值彼此独立,而工业过程通常是自相关过程,数据自相关性会影响CUSUM控制图的性能.针对这一问题,利用数学推理的方法分析了数据自相关性对CUSUM控制图性能的不利影响,正自相关会使CUSUM控制图的平均运行长度变短,可能引起假警报;负自相关会使CUSUM控制图的平均运行长度变长,可能造成故障漏报.在此基础上,提出用残差CUSUM控制图来检测自相关过程中的故障,运用时间序列模型的残差检测自相关过程中的故障,消除数据自相关性对控制图性能的不利影响.结果表明,该方法具有较好的鲁棒性和较高的准确率.     

Adaptive sampling interval cause-selecting control charts

This article considers the statistical adaptive process control for two dependent process steps. We construct an adaptive sampling interval Z _X control chart to monitor the quality variable produced by the first process step, and use the adaptive sampling interval Z _e control chart to monitor the specific quality variable produced by the second process step. By using the proposed adaptive sampling interval control charts, we can quickly detect and distinguish which process step is out of control. The performance of the proposed adaptive sampling interval control charts is measured by the adjusted average time to signal (AATS), which was derived by a Markov chain approach, for an out-of-control process. An empirical automobile braking system example shows the application and the performance of the proposed adaptive sampling control charts in detecting shifts in process means. Some numerical results obtained demonstrated that the performance of the proposed adaptive sampling cause-selecting control charts outperforms the fixed sampling interval cause-selecting control charts.     

Design and application of exponential chart for monitoring time-between-events data under random process shift

Exponential charts based on time-between-events (TBE) data were developed for monitoring high-yield process like the process which has achieved six-sigma quality level and has recently shown to be very useful in manufacturing systems, in reliability and maintenance monitoring, and also in service-related applications in general. This article develops an economic model of the exponential chart (known as TBE_random chart) for monitoring time-between-events data; the design algorithm considers the random characteristic of the process shifts and therefore better reflects the real process conditions. The probability distribution of the random process shift is modeled by a Rayleigh distribution based on the sample data acquired during the operation of the control chart. The design of the proposed control chart scheme is demonstrated, and the properties are compared with that of other exponential charts. The results of the numerical studies show that the consideration of the random process shift in designing an exponential chart provides an excellent in-control stability of the charting scheme, which in turn helps in saving time and cost for searching the false alarms. The proposed control chart is easy to understand and operate, and thus the floor operators can utilize and understand it as easily as with a traditional charting scheme.     

Control charts for individual observations of a bivariate Poisson process

In this paper, three single-control charts are proposed to monitor individual observations of a bivariate Poisson process. The specified false-alarm risk, their control limits, and ARLs were determined to compare their performances for different types and sizes of shifts. In most of the cases, the single charts presented better performance rather than two separate control charts (one for each quality characteristic). A numerical example illustrates the proposed control charts.     

Using genetic algorithm and response surface methodology for statistically constrained optimization of VSI X-bar control charts under multiple assignable causes and non-normality

Variable sampling interval (VSI) control charts have been introduced with the aim of improving performance of traditional control charts. Usually, in the economic–statistical design of the VSI $ \overline{X} $ control charts, it is assumed that observations are normally distributed and process is subjected to only one assignable cause. However, in practice these assumptions could easily fail to hold, and results no longer could be realistic. This paper considers non-normal observations for the case of multiple assignable causes to develop a cost model for the economic design of VSI $ \overline{X} $ control chart. Being more applicable for all types of distributions, Burr distribution is employed for representing the distribution of non-normal process data. Since the proposed design consists of a complex nonlinear cost function that cannot be solved using a classical optimization method, genetic algorithm (GA) searching method as an efficient famous metaheuristic is employed to find the optimal values for the design parameters. Moreover, to improve the performances, response surface methodology is employed to calibrate GA parameters. The effectiveness of the proposed scheme is evaluated through a numerical example. Sensitivity analysis is also carried out to show the effects of cost and process parameters on the outputs of the model. Results show that in all cases, presented VSI model has better economical and statistical performances than its corresponding fixed sampling interval scheme.     

On-machine wire electrical discharge dressing (WEDD) of metal-bonded grinding wheels

Traditional multivariate control charts such as Hotelling’s χ ² and T ² control charts are designed to monitor vectors of variable quality characteristics. However, in certain situations, data are expressed in linguistic terms and, under these circumstances, variable or attribute multivariate control charts are not suitable choices for monitoring purposes. Fuzzy multivariate control charts such as fuzzy Hotelling’s T ² could be considered as efficient tools to overcome the problems of linguistic observations. The purpose of this paper is to develop a fuzzy multivariate exponentially weighted moving average (F-MEWMA) control chart. In this paper, multivariate statistical quality control and fuzzy set theory are combined to develop the proposed method. Fuzzy sets and fuzzy logic are powerful mathematical tools for modeling uncertain systems in industry, nature, and humanity. Through a numerical example, the performance of the proposed control chart was compared to the fuzzy Hotelling’s T ² control chart. Results indicate uniformly superior performance of the F-MEWMA control chart over Hotelling’s T ² control chart.     

Estimation in X-bar control charts: effects and corrections

This paper presents an approach for improving the control limits of $ \overline{X} $ control charts when the parameters of the process are estimated and the control chart is in operation. In these conditions, the observed average run length (ARL) may be very different from the planned ARL since the parameter estimates may have a larger error. To minimize this problem, the data collected in effective control (phase 2) will be used to re-estimate the parameters with a precision greater than that obtained in phase 1. Thus, we defined a minimum sample size of observations of phase 2, which is constituted of a mixture of two normal distributions that should be used to re-estimate the process parameters. The proposal is illustrated with numerical example.     

Performance evaluation of control chart for multiple assignable causes using genetic algorithm

With a view to monitoring and controlling manufacturing processes in industries, control charts are widely used and needed to be designed economically to achieve minimum quality costs. Many authors have studied the economic design of the $ \overline{X} $ control chart after Duncan (J Am Stat Assoc 51(274):228–242, 1956) first proposed the economic model of the $ \overline{X} $ control chart for a single assignable cause. But, in practice, multiple assignable causes are more logical and realistic. Moreover, the economic design does not consider statistical properties like bound on type I and type II error, and average time to signal (ATS). This paper focuses on evaluating the performance of genetic algorithm (GA) in pure economic and economic statistical design of the $ \overline{X} $ control chart for multiple assignable causes. The performances of GA are demonstrated by comparing its result with the previously proposed grid search technique for a numerical example. The Duncan model of multiple assignable causes is adopted to formulate objective function, and the computation is achieved by approximation through a numerical method named Simpson's 1/3 rule. Comparison distinctly shows the superiority of GA over grid search results for economic statistical design.     

Optimization design of a CUSUM control chart based on taguchi’s loss function

The CUSUM charts have been widely used in statistical process control (SPC) across industries for monitoring process shifts and supporting online measurement and distributed computing. This paper proposes an algorithm for the optimimal design of a CUSUM control chart detecting process shifts in the mean value. The algorithm optimizes the sample size, sampling interval, control limit and reference parameter of the CUSUM chart through minimizing the overall mean value (ML) of a Taguchi’s loss function over the probability distribution of the random process mean shift. A new feature related to the exponential of the sample mean shift is elaborated. Comparative studies reveal that the proposed ML-CUSUM chart is considerably superior to the Shewhart ML- $\overline{X} $ chart and the conventional CUSUM chart in terms of the overall loss of ML.     

数控机床旋转轴误差的快速测量与辨识新方法

根据齐次变换理论推导出旋转轴基本几何误差辨识模型,在此基础上,提出了一种基于球杆仪的旋转轴基本几何误差快速测量和辨识新方法,将球杆仪一端的中心座分别安装在旋转工作台的3个不同位置,通过联动控制球杆仪另一端球心按圆形轨迹运动,分别测量旋转轴圆周每个离散位置点在X、Y、Z方向上的偏差,并根据所建立的辨识模型,辨识出旋转轴的6项基本几何误差。同时,提出了基于系数矩阵灵敏度分析的方法,用于指导测量点的合理分布,减少测量误差的影响,从而提高误差辨识精度。     

The variable sample size t control chart for monitoring short production runs

Starting the online monitoring of a quality characteristic by means of a control chart at the beginning of a short production run is often a challenging issue for quality practitioners: in fact, the frequent absence of preliminary information prevents from getting a precise estimate of the characteristic mean and standard deviation. Furthermore, for short runs having a finite rolling horizon, the number of inspections scheduled within the run can be too small to get sufficient samples allowing the phase I implementation of the chart to be completed. Recently, t control charts have been proposed as efficient means to overcome this problem because they do not need any phase I tentative control limits definition or preliminary process knowledge. In this paper, a variable sample size (VSS) version of the t chart is proposed. Adaptive control charts have been implemented with success in long runs: here, the performance of the variable sample size strategy is investigated for a chart used in a short run. The statistical performance of the VSS t chart is compared with the one of the fixed-parameter (FP) t chart for both scenarios of fixed and unknown shift size, with the latter situation being frequent in short-run manufacturing environments. An extensive numerical investigation reveals the potential benefits of the proposed chart. When the statistical design is optimized with respect to a fixed value of the shift size δ, the VSS t chart has a better statistical performance than the FP t chart for moderate to large values of δ. Conversely, for the unknown shift size condition, the VSS t chart always outperforms the FP t chart for in-control average sample sizes ASS₀?>?7. An illustrative example shows the implementation of the VSS during the production of a finite lot of mechanical parts.     

Identifying the mean step change of UBM chart

The traditional control charts are developed based on the assumption that the successive observations are independent and identically distributed. In some processes, the independence assumption is violated when there is autocorrelation between observations. To solve this problem, two methods, classified as model-based and model-free, could be applied. When a control chart alarms an assignable cause, it is essential to detect the process change point in order to remove the root cause. In the presence of autocorrelated data, different methods for change-point identification have been applied only for model-based methods. Hence, this is considered as the research gap and an attempt is made to fill this gap by applying maximum likelihood function in unweighted batch mean control chart, one of the most applied model-free methods. In this article, an estimator is presented to determine the change point for the first-order autoregressive process, AR(1). When a real change occurs, the performance of proposed estimator is evaluated through simulation.     

The performance of the Shewhart sign control chart for finite horizon processes

In many manufacturing environments, the production horizon of the same part code between two consecutive set-ups should be limited to a few hours or shifts. When 100 % sampling is not possible, on-line quality control on a quality characteristic should be immediately started by means of a control chart. In this paper, we investigate the statistical performance of a nonparametric (distribution-free) Shewhart Sign (SN) control chart for monitoring the location of a quality characteristic in a production process with a finite horizon and a small number of scheduled inspections. The observations taken from the process are assumed to be continuous random variables. By implementing a SN control chart, any model assumption about the distribution of observations is needless to guarantee a nominal in-control (IC) performance: after each process set-up, this overcomes the important problem of lack of information about the distribution of the observations collected for the quality characteristic to be monitored. An extensive simulation study is conducted to compare the statistical performance of the distribution-free Shewhart SN control chart to the normal theory-based Shewhart Student’s t control chart: several types of distributions of observations and different numbers of scheduled inspections are considered to show the advantages related to the implementation of the Shewhart SN control chart. An illustrative example presents the implementation of the Shewhart SN control chart on a real data set collected in a beverage company.     

Optimal control policy for dependent process steps with over-adjusted means and variances

Over-adjustment to processes may result in shifts in process mean and variance, ultimately affecting the quality of products. An economic adjustment model is developed for the joint design of X̄-S² control charts and ē-S_e² cause-selecting control charts to control both means and variances of two dependent process steps using the Markov chain approach. The objective is to determine the optimal control policy of the proposed control charts, which effectively detect and distinguish the shifts of means and variances on the dependent process steps and minimize the total quality control cost. Application of the proposed control charts is illustrated through a numerical example.     

The run sum t control chart for monitoring process mean changes in manufacturing

The $ \overline{X} $ type charts are not robust against estimation errors or changes in process standard deviation. Thus, the t type charts, like the t and exponentially weighted moving average (EWMA) t charts, are introduced to overcome this weakness. In this paper, a run sum t chart is proposed, and its optimal scores and parameters are determined. The Markov chain method is used to characterize the run length distribution of the run sum t chart. The statistical design for minimizing the out-of-control average run length (ARL₁) and the economic statistical design for minimizing the cost function are studied. Numerical results show that the t type charts are more robust than the $ \overline{X} $ type charts for small shifts, in terms of ARL and cost criteria, with respect to changes in the standard deviation. Among the t type charts, the run sum t chart outperforms the EWMA t chart for medium to large shifts by having smaller ARL₁ and lower minimum cost. The run sum t chart surpasses the $ \overline{X} $ type charts by having lower ARL₁ when the charts are optimally designed for large shifts but the run sum $ \overline{X} $ and EWMA $ \overline{X} $ prevail for small shifts. In terms of minimum cost, the $ \overline{X} $ type charts are superior to the t type charts. As occurrence of estimation errors is unpredictable in real process monitoring situations, the run sum t chart is an important and useful tool for practitioners to handle such situations.     

Bootstrap method approach in designing multi-attribute control charts

In a production process, when the quality of a product depends on more than one correlated characteristic, multivariate quality control techniques are used. Although multivariate statistical process control is receiving increased attention in the literature, little work has been done to deal with multi-attribute processes. In monitoring the quality of a product or process in multi-attribute environments in which the attributes are correlated, several issues arise. For example, a high number of false alarms (type I error) occur and the probability of not detecting defects (type II error) increases when the process is monitored by a set of independent uni-attribute control charts. In this paper, to overcome these problems, first we develop a new methodology to derive control limits on the attributes based on the bootstrap method in which we build simultaneous confidence intervals on the attributes. Then, based upon the in-control and out-of-control average run length criteria we investigate the performance of the proposed method and compare it with the ones from the Bonferroni and Sidak’s procedure using simulation. The results of the simulation study show that the proposed method performs better than the other two methods. At the end, we compare the bootstrap method with the T ² control chart for attributes.     

A new link function in GLM-based control charts to improve monitoring of two-stage processes with Poisson response

In this paper, a new procedure is developed to monitor a two-stage process with a second stage Poisson quality characteristic. In the proposed method, log and square root link functions are first combined to introduce a new link function that establishes a relationship between the Poisson variable of the second stage and the quality characteristic of the first stage. Then, the standardized residual statistic, which is independent of the quality characteristic in the previous stage and follows approximately standardized normal distribution, is computed based on the proposed link function. Then, Shewhart and exponentially weighted moving average (EWMA) cause-selecting charts are utilized to monitor standardized residuals. Finally, two examples and a case study with a Poisson response variable are investigated, and the performance of the charts is evaluated by using average run length (ARL) criterion in comparison with the best literature method.     

Integrated control chart system for time-between-events monitoring in a multistage manufacturing system

This article develops a control chart system consisting of several individual time-between-events (TBE) charts, each of which is used to monitor the time between successive events at different process stages in the manufacturing of a product in a multistage manufacturing system. The design algorithm considers all the TBE charts within a system in an integrative and optimal manner. Numerical studies show that the proposed design algorithm improves the performance characteristics of the chart system significantly and thus the product quality is further guaranteed. The proposed control chart system is easy to understand and operate; thus, the floor operators can utilize and understand it as easily as for the traditional system.