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.     

A CUSUM scheme with variable sample sizes for monitoring process shifts

The cumulative sum scheme (CUSUM) and the adaptive control chart are two approaches to improve chart performance in detecting process shifts. A weighted loss function CUSUM scheme (WLC) is able to monitor both the mean shift and the increasing variance shift by manipulating a single chart. This paper investigates the WLC scheme with a variable sample sizes (VSS) feature. A design procedure is firstly proposed for the VSS WLC scheme. Then, the performance of the chart is compared with that of four other competitive control charts. The results show that the VSS WLC scheme is more powerful than the other charts from an overall viewpoint. More importantly, the VSS WLC scheme is simpler to design and operate. A case study in the manufacturing industry is used to illustrate the chart application. The proposed VSS WLC scheme suits the scenario where the strategy of varying sample sizes is feasible and preferable to pursue a high capability of detecting process variations.     

A neural network-based control scheme for monitoring start-up processes and short runs

Traditional control charts are commonly used as a monitoring tool in long-run processes. However, such control charts, due to the need for phase I analysis, are not suitable for start-up processes or short runs. Q control charts have been developed to help monitor start-up processes and short runs. In this article, a back propagation network is proposed for detecting a mean shift in start-up processes and short runs. In-control run length distribution of the control scheme is estimated using simulation study to provide information about the possibility of a false alarm within a specified number of observations. Performance of the proposed control scheme is assessed using different performance measures. It is shown numerically that the proposed control scheme outperforms the CUSUM of Q charts in detecting small to moderate mean shifts.     

A special variable sample size and sampling interval Hotelling’s T 2 chart

Among the variable sampling interval (VSI), variable sample size (VSS), and variable sample size and sampling interval (VSSI) T ² charts that use two values of each adaptive parameter, a VSI T ² chart is the most efficient chart to detect large shifts in the process mean vector while a VSSI T ² chart is the most efficient one to detect small shifts. The statistical performance of the T ² chart proposed in this paper very closely matches that of such VSI and VSSI T ² charts for detecting the large and small shifts, respectively. The proposed chart uses two values of the sampling interval and three values of the sample size.     

Weighted-loss-function control charts

This article proposes a single weighted-loss-function chart (WL chart) for monitoring the process mean and variance simultaneously in statistical process control (SPC). The weights of the losses due to mean shift and variance shift in the loss function are balanced through a weighting factor λ, so that the WL chart is considerably more effective than the unadjusted loss function chart, the joint &S charts and many other charts as well. The improvement in detection effectiveness will directly lead to the reduction of a number of defects in the manufacturing process when out-of-control cases occur. Moreover, the WL chart provides a platform for the further development of the Cusum chart, runs rule, or adaptive chart for monitoring both process mean and variance.     

An EWMA t chart with variable sampling intervals for monitoring the process mean

This paper proposes a variable sampling interval (VSI) version of the fixed sampling interval (FSI) exponentially weighted moving average (EWMA) t chart developed by Zhang et al. for monitoring the changes in the process mean. An optimal design strategy based on the average time to signal (ATS) is presented. We determine the optimal parameters for the VSI EWMA t chart using a Markov chain approach so that the chart has the desired robustness property against errors in estimating the process standard deviation or changing standard deviation. Also, we explain how the various parameters of this VSI EWMA t chart can be computed and how the use of the VSI feature improves the statistical efficiency of the FSI EWMA t and FSI EWMA X-bar charts in terms of out-of-control ATS performances. Comparisons with the FSI EWMA t and FSI EWMA X-bar charts are performed.     

Statistical design of variable sample size and sampling interval $\bar{X}$\bar{X}control charts with run rules

In this paper, improved Shewhart control charts based on hybrid adaptive and run rule schemes are introduced to enhance the statistical performances of the traditional static scheme, designed with consideration given to the fixed values of sample size, the width of the control limits and the sampling frequency. The proposed hybrid adaptive schemes consider both variable sampling interval and variable sample size combined with run rules. The objective of this research is to develop a statistical comparison between adaptive schemes, charts with run rules and hybrid adaptive schemes with run rules to help decision-makers in the selection of the best performing chart for an expected value of shift in the mean of a controlled parameter. An extensive set of numerical results is presented to test the effectiveness of the proposed models in detecting small and moderate shifts in the process mean. The optimal statistical designs of the charts are obtained through a heuristic algorithm, properly modified to cope with the problem.     

Statistical design of variable sample size and sampling interval\bar Xcontrol charts with run rulescontrol charts with run rules

In this paper, improved Shewhart control charts based on hybrid adaptive and run rule schemes are introduced to enhance the statistical performances of the traditional static scheme, designed with consideration given to the fixed values of sample size, the width of the control limits and the sampling frequency. The proposed hybrid adaptive schemes consider both variable sampling interval and variable sample size combined with run rules. The objective of this research is to develop a statistical comparison between adaptive schemes, charts with run rules and hybrid adaptive schemes with run rules to help decision-makers in the selection of the best performing chart for an expected value of shift in the mean of a controlled parameter. An extensive set of numerical results is presented to test the effectiveness of the proposed models in detecting small and moderate shifts in the process mean. The optimal statistical designs of the charts are obtained through a heuristic algorithm, properly modified to cope with the problem.     

Parameterization of prismatic shapes and reconstruction of free-form shapes in reverse engineering

This article proposes an adaptive loss function (AL) control chart for statistical process control (SPC). This chart is able to monitor process shifts in the mean and variance simultaneously. It is appealing for its effectiveness in detecting process shifts and simplicity in operation under a computerized SPC environment. By using a fixed sample size and a fixed sampling interval, the AL chart can be operated as easily as the conventional ${\bar X}$ and S charts from the stand of the operators. Nevertheless, via the computer-aided adaption of some charting parameters, the AL chart is not only significantly more effective than the ${\bar X}$ and S charts, but it also outperforms the cumulative sum (CUSUM) charts for most of the combinations of mean and increasing variance shifts.     

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.     

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.     

Economic and economic-statistical designs of MEWMA control charts—a hybrid Taguchi loss, Markov chain, and genetic algorithm approach

Economic design of multivariate exponentially weighted moving average (MEWMA) control charts for monitoring the process mean vector involves determining economically the optimum values of the three control parameters: the sample size, the sampling interval between successive samples, and the control limits or the critical region of the chart. In the economic-statistical design, constraints (including the requirements of type I error probability and power) are added such that the statistical property of the chart is satisfied. In this paper, using the multivariate Taguchi loss approach, the Lorenzen–Vance (Technometrics 28:3-10, 1) cost function of implementing the control chart is extended to include intangible external costs along with the in-control average run length (ARL₀) and out-of-control average run length (ARL₁) as statistical constraints. A Markov chain model is then developed to estimate the ARLs and a genetic algorithm whose parameters are optimally obtained by design of experiments is used to solve the model and estimate the optimum values of the control chart parameters. A numerical example and a sensitivity analysis are provided to illustrate the solution procedure and to investigate the effects of cost parameters on the optimal designs. The results show that the proposed economic-statistical design of the chart has better statistical properties in comparison to the economic design while the difference between the costs is negligible.     

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.     

A new adaptive control chart for monitoring process mean and variability

Traditionally, an $\bar{X}$ chart is used to control the process mean, and an R chart is used to control the variance. However, these charts are not sensitive to the small shifts in the processes. The adaptive charts might be considered if the aim is to detect process changes quickly. In this paper, we propose a new adaptive single control chart which integrates the exponentially weighted moving average procedure with the generalized likelihood ratio test statistics for jointly monitoring both the process mean and variability. This new chart is effective in detecting the disturbances that shift the process mean, increase or decrease the process variance, or lead to a combination of both effects.     

一种变参数的控制图方法及其稳健分析

针对在实际应用中,现有的控制图仅仅对某个固定大小的偏移有效的问题,首先提出了一种变参数的控制图设计方法,并对这种控制图的平均运行长度的计算方法进行了探讨.然后讨论了变参数控制图的稳健设计方案,通过与常规控制图和EWMA图进行比较,说明提出的监控方法对过程中出现的不同程度的偏移都是有效的.     

An integrated optimization of cutting parameters and tool path generation in ultraprecision raster milling

This paper provides a new methodology for the integrated optimization of cutting parameters and tool path generation (TPG) based on the development of prediction models for surface roughness and machining time in ultraprecision raster milling (UPRM). The proposed methodology simultaneously optimizes the cutting feed rate, the path interval, and the entry distance in the feed direction to achieve the best surface quality in a given machining time. Cutting tests are designed to verify the integrated optimization methodology. The experimental results show that, in the fabrication of plane surface, the changing of entry distance improves surface finish about 40 nm (R _a ) and 200 nm (R _t ) in vertical cutting and decreases about 8 nm (R _a ) and 35 nm (R _t ) in horizontal cutting with less than 2 s spending extra machining time. The optimal shift ratio decreases surface roughness about 7 nm (R _a ) and 26 nm (R _t ) in the fabrication of cylinder surfaces, while the total machining time only increases 2.5 s. This infers that the integrated optimization methodology contributes to improve surface quality without decreasing the machining efficiency in ultraprecision milling process.     

The skipping strategy to reduce the effect of the autocorrelation on the <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> chart’s performance

In this article, we consider the T ² control chart for bivariate samples of size n with observations that are not only cross-correlated but also autocorrelated. The cross-covariance matrix of the sample mean vectors were derived with the assumption that the observations are described by a first-order vector autoregressive model—VAR (1). To counteract the undesired effect of autocorrelation, we build up the samples taking one item from the production line and skipping one, two, or more before selecting the next one. The skipping strategy always improves the chart’s performance, except when only one variable is affected by the assignable cause, and the observations of this variable are not autocorrelated. If only one item is skipped, the average run length (ARL) reduces in more than 30 %, on average. If two items are skipped, this number increases to 40 %.     

Double sampling \overline{X} control chart for a first-order autoregressive moving average process model

In this paper, we consider the double sampling (DS) $\overline{X} $ control chart for monitoring processes in which the observations can be represented as a first-order autoregressive moving average (ARMA(1, 1)) model. The properties of the DS $\overline{X} $ control chart with the sampling intervals driven by the rational subgroup concept are studied and compared with the Shewhart chart and the variable sample size (VSS) chart, both properly modified to account for the serial correlation. Numerical results show that the correlation within subgroups has a significant impact on the properties of the charts. For processes with low to moderate correlation levels, the DS $\overline{X} $ chart is substantially more efficient in detecting process mean shifts.     

Monitoring the coefficient of variation using a variable sample size control chart

This paper proposes an adaptive Shewhart control chart implementing a variable sample size strategy in order to monitor the coefficient of variation. The goals of this paper are as follows: (a) to propose an easy-to-use 3-parameter logarithmic transformation for the coefficient of variation in order to handle the variable sample size aspect; (b) to derive the formulas for computing the average run length, the standard deviation run length, and the average sample size and to evaluate the performance of the proposed chart based on these criteria; (c) to present ready-to-use tables with optimal chart parameters minimizing the out-of-control average run length as well as the out-of-control average sample size; and (d) to compare this chart with the fixed sampling rate, variable sampling interval, and synthetic control charts. An example illustrates the use of the variable sample size control chart on real data gathered from a casting process.     

The significance of load and rubbing time in the fourball lubricant test

Four-ball tests were run on compounded oils made up with the following additives: (a) di-t-octyl disulfide; (b) t-octyl chloride; (c) a dualcomponent disulfide-chloride mixture. Two test procedures were used for most of the work: 10 s runs with load as the variable experimental parameter and runs in which the load was held constant and the rubbing time was varied. Some runs were also made with a varied load for 2 s of running time. The wear scar dimensions were converted to wear volumes. None of the lubricants exhibited constant-rate load-governed wear behavior. The behavior of lubricants compounded with di-t-octyl disulfide or with t-octyl chloride was characterized by distinct load-governed transitions to increased wear; the dual-component additive did not show any sharp transitions but did depart from linear load-governed wear behavior. Transitions to increased wear at a load which was subtransitional for 10 s of rubbing could be induced by prolonging the time of rubbing. Suggestive parallelisms were observed in the effects of load and rubbing time as the experimental parameter. The role of load and rubbing time in the operation of the four-ball machine is examined and an approach to the rationale of evaluating lubricants by the four-ball test is suggested.