Synthetic Chart Research Articles

Some theoretical comments regarding the run-length properties of the synthetic and runs-rules monitoring schemes – part 2: steady-state

In this paper, the long term (also known as the steady-state mode) run-length theoretical properties of the four different types of synthetic and runs-rules monitoring schemes, that were empirically analysed in part 1 of this research work. That is, using the Markov chain imbedding technique (thoroughly discussed in Part 1 of this work), the closed-form expressions of the steady-state initial probabilities and average run-length (ARL) vectors are derived; so that the corresponding steady-state ARL and overall performance expressions, of each of the four different types of the synthetic and runs-rules monitoring schemes, may be formulated. Since there is very little literature on steady-state analysis of the synthetic and runs-rules charts, the closed-form expressions derived here will ease the understanding and implementation of the different types synthetic and runs-rules schemes in practice and in further academic research.

Some theoretical comments regarding the run-length properties of the synthetic and runs-rules monitoring schemes – part 1: zero-state

In this paper, we discuss the short-term (also known as zero-state mode) run-length theoretical properties of the four different types of synthetic and runs-rules monitoring schemes that were empirically analyzed in another paper. That is, we provide and point out how each corresponding type of the 2-of-(H + 1) runs-rules and synthetic charts’ transition probabilities matrices (TPMs) differ from each other in zero-state, for any positive integer H. Next, using these general TPMs and the standard Markov chain formulae, we derive the general form of the average run-length (ARL) vectors and the corresponding zero-state ARL expressions for any shift value for each of the four different types of the synthetic and runs-rules monitoring schemes. Finally, we provide expressions to calculate the overall run-length performance for each of the schemes. While there is lots of literature available on empirical analysis of zero-state synthetic and runs-rules charts, there is very little on the corresponding theoretical analysis. We believe this paper will, in some part, fill this gap and encourage more research in this area.

Combined double sampling and variable sampling interval np chart

ABSTRACTA combined double sampling and variable sampling interval (DSVSI) np chart is investigated in this study. The optimal design of the DSVSI np chart is based on minimizing the out-of-control average time to signal. From the numerical results, the DSVSI np chart performs reasonably well in comparison with the standard np chart, double sampling np chart, synthetic double sampling np chart, and other existing np type control charts for detecting increases in the process of fraction non conforming, based on the zero-state case. An example is provided to illustrate the application of the DSVSI np chart.

Synthetic exponential control charts with unknown parameter

ABSTRACTThe existing synthetic exponential control charts are based on the assumption of known in-control parameter. However, the in-control parameter has to be estimated from a Phase I dataset. In this article, we use the exact probability distribution, especially the percentiles, mean, and standard deviation of the conditional average run length (ARL) to evaluate the effect of parameter estimation on the performance of the Phase II synthetic exponential charts. This approach accounts for the variability in the conditional ARL values of the synthetic chart obtained by different practitioners. Since parameter estimation results in more false alarms than expected, we develop an exact method to design the adjusted synthetic charts with desired conditional in-control performance. Results of known and unknown in-control parameter cases show that the control limit of the conforming run length sub-chart of the synthetic chart should be as small as possible.

Phase II synthetic exponential charts and effect of parameter estimation

Exponential charts for monitoring time-between-events data are widely studied recently as it has shown a wide range in different applications. To improve the sensitivity of the simple exponential chart, this paper develops a synthcontrol chart to monitor the two-sided shifts in the exponential mean. This synthetic chart consists of an exponential sub-chart and a conforming run length (CRL) sub-chart and it is designed from two new perspectives. Previous synthetic charts are constructed based on the optimization program for a specified shift size, which proves to be difficult to predict in practice. In this paper, the synthetic chart is constructed based on a new design method which takes the mean (ARL) and the standard deviation (SDRL) of the run length into consideration. The other improvement is that the exponential sub-chart is designed to ensure that the CRL sub-chart only needs a lower control limit and that the proposed synthetic chart is ARL-unbiased. The effect of parameter estimation on the proposed synthetic chart is also studied in this paper. Performance comparisons show that our proposed chart has a better performance in general.

Distribution-free synthetic and runs-rules control charts combined with a Mann-Whitney chart

Control chart is one of the most important tools used in statistical process control and monitoring (SPCM) to detect changes in quality characteristics. This paper investigates the performance of the improved modified distribution-free synthetic and runs-rules charts combined with a Mann-Whitney (MW) chart, in terms of the average run length (ARL) and median run length (MRL) through intensive simulation. It is observed that the newly proposed improved modified synthetic and runs-rules MW charts present very attractive run-length properties and outperform the competing charts under non-normal underlying distributions. Numerical examples are given as illustration of the design and implementation of the proposed charts.

A modified side-sensitive synthetic chart to monitor the process mean

In this paper, we propose a modified side-sensitive (MSS) synthetic chart which signals only if all the consecutive plotting statistics that lead to an out-of-control event fall on one side of the centre line, unlike the non-side-sensitive, standard and revised side-sensitive synthetic charts that also signal even when some of the plotting statistics fall on opposite sides of the centre line. Moreover, we use the Markov chain imbedding technique to study and compare the zero-state and steady-state average run-length (ARL), extra quadratic loss, average ratio of the ARLs and performance comparison index of the proposed MSS chart with other Shewhart-type synthetic and runs-rules charts. The synthetic chart with this MSS feature has a better overall zero-state and steady-state performance than the existing synthetic charts and hence makes it a strong contender in many applications where existing synthetic charts are currently used.

Economic-statistical design of synthetic max chart

This paper studies the economic-statistical design of a synthetic max chart. The synthetic max chart consists of a combination of a max sub-chart and a conforming run length sub-chart. The design of the chart is developed by considering the minimization of the cost function, subject to constraints on the average run length. A sensitivity analysis is performed to investigate the effects of the cost and process parameters on the solution of the economic-statistical design. Furthermore, the economic-statistical designs of the synthetic max chart and the max chart are compared in this study. The results show that the synthetic max chart outperforms the max chart from an economic-statistical viewpoint. Although the economic-statistical design of the synthetic max chart has the disadvantage of a slight increase in cost over the economic design, the economic-statistical design gives better in-control statistical performance, where the desirable statistical properties are justified. This study is also extended to the multivariate case.

Control charts for the coefficient of variation

The coefficient of variation (CV) has received an extensive attention in quality control. The existing control charts for monitoring the two-sided changes in the CV do not consider the average run length (ARL) unbiased criterion. In practice, the practitioners prefer the ARL-unbiased control charts. This paper proposes an ARL-unbiased Shewhart-type control chart to monitor the two-sided shifts in the CV. Then an ARL-unbiased synthetic chart in which the conforming run length sub-chart only needs a lower control limit is proposed to improve the performance of the ARL-unbiased Shewhart-type chart. The comparison results show that the proposed synthetic chart outperforms the proposed ARL-unbiased control chart in terms of the ARL. The effects of the parameter estimation on the proposed charts are also studied. Finally, a real example is used to illustrate the proposed control charts.

Short note explaining the error in ‘Machado M.A.G. and Costa A.F.B. (2014) “A side-sensitive synthetic chart combined with an X chart.” International Journal of Production Research 52 (11): 3404–3416’

National Research Foundation [Thuthuka Program: TTK14061168807; grant number 941]; National Research Foundation and Department of Science and Technology [Innovation Doctoral Scholarship SFH14081591713].

Synthetic charts to control bivariate processes with autocorrelated data

In this study, we propose the use of simultaneous X‾ charts to control bivariate processes with autocorrelated data. The first set of X‾ charts is side-sensitive with regard to the same variable (SV X‾ charts) and the second one is side-sensitive with regard to both variables (BV X‾ charts). The Markov chain approach was used to obtain the steady-state properties of the X‾ charts. In comparison with the standard synthetic T2 chart, the SV and the BV charts signal faster in a wide variety of disturbances, except when the variables are high correlated. The BV charts are simpler and signal faster than the SV charts.

The Case Against the Use of Synthetic Control Charts

The synthetic chart principle proposed by Wu and Spedding (2000) initiated a stream of publications in the control charting literature. Originally, it was claimed that the new chart has superior average run length (ARL) properties. Davis and Woodall (2002) indicated that the synthetic chart is nothing else than a particular runs-rule chart. Moreover, they criticized the design of the performance evaluation and advocated use of the steady-state ARL. The latter measure was used then, e.g., in Wu et al. (2010). In most of the papers on synthetic charts that actually used the steady-state framework, it was not rigorously described. See Khoo et al. (2011) as an exception, where it was revealed that the cyclical steady-state design was considered. The aim of this paper is to carefully analyze the steady-state (cyclical and the more popular conditional) for the synthetic chart, the original “2 of L + 1” (L < 1) runs-rule chart, and competing EWMA charts with two types of control limits. It turns out that the EWMA chart has a uniformly (over a large range of potential shifts) better steady-state ARL performance than the synthetic chart. Furthermore, the synthetic control chart exhibits the poorest performance among all considered competitors. Thus, we advise not applying synthetic control charts.

A new synthetic control chart for monitoring process mean using auxiliary information

ABSTRACTQuality control charts have been widely recognized as a potentially powerful statistical process monitoring tool in statistical process control because of their superior ability in detecting shifts in the process parameters. Recently, auxiliary-information-based control charts have been proposed and shown to have excellent speed in detecting process shifts than those based without it. In this paper, we design a new synthetic control chart that is based on a statistic that utilizes information from both the study and auxiliary variables. The proposed synthetic chart encompasses the classical synthetic chart. The construction, optimal design, run length profiles, and the performance evaluation of the new chart are discussed in detail. It turns out that the proposed synthetic chart performs uniformly better than the classical synthetic chart when detecting different kinds of shifts in the process mean under both zero-state and steady-state run length performances. Moreover, with reasonable assumptions, the proposed chart also surpasses the exponentially weighted moving average control chart. An application with a simulated data set is also presented to explain the implementation of the proposed control chart.

Standard Deviation of the Run Length (SDRL) and Average Run Length (ARL) Performances of EWMA and Synthetic Charts

Statistical process control (SPC) techniques are applied to monitor a process. The control chart is a valuable tool in SPC. the shewhart control chart was the first chart proposed in the literature of SPC and it is still used in process monitoring in today's manufacturing and service industries. both the exponentially weighted moving average (EWMA) and synthetic charts outperform the shewhart control chart for detecting shifts in the process mean. Since both the EWMA and synthetic charts provide better process mean shifts detection performance, a study on which chart to use in process monitoring under different situations is the aim of this work. this paper compares the average run length (ARL) and standard deviation of the run length (SDRL) profiles of the EWMA and synthetic charts. comparisons are made based on the normality assumption. The mathematica programs are used to compute the ARLs and SDRLs of the EWMA and synthetic charts. the ARL results indicate that the EWMA chart is superior to the synthetic chart for detecting small mean shifts, but the latter prevails for detecting moderate and large shifts. however, in terms of the SDRL, the EWMA chart surpasses the synthetic chart for small and moderate shifts. Index Terms—Average run length (ARL), exponentially weighted moving average (EWMA) chart, synthetic chart, standard deviation of the run length (SDRL).

Economic design of the upper-sided synthetic chart with measurement errors

Considering a quadratic Taguchi loss function for poor quality products, as well as measurement errors in the process characteristic, this paper proposes an economic model for the upper-sided synthetic chart allowing multiple measurements for each sample item. The steady-state ARL of this chart is computed using a Markov chain method and the optimal number of measurements are obtained through the minimisation of the total cost model. For the proposed economic and economic statistical model, numerical examples are presented to evaluate the effect of measurement errors on the upper-sided synthetic chart. Finally, the sensitivity analyses of the optimal chart parameters and the cost of the upper-sided synthetic chart are also investigated using different input parameter values. All the above results are verified through a wide range of the input parameter values. Two illustrative examples are also provided.

A side-sensitive synthetic chart combined with a VSS [formula omitted] chart

In independent investigations, the variable sample size (VSS) scheme and the side-sensitive synthetic (SS) rule proved to reduce the delay with which the X‾ chart signals. Based on these findings we investigate the X‾ chart’s performance with the joint use of the VSS scheme and the side-sensitive synthetic rule. The SSVSS X‾ chart outperforms the pure VSS X‾ chart, especially when the risk of false alarms and the rate of inspected items per sampling cannot be high. In these cases, the VSS X‾ chart detects moderate mean shifts (around one standard deviation) in half of the time thanks to the side-sensitive synthetic rule.

Optimal design of a synthetic chart for monitoring process dispersion with unknown in-control variance

For a stable manufacturing process, quality problems are often caused by changes in process dispersion. Although there have been plenty of research on the monitoring of process dispersion, the existing studies of synthetic charts for monitoring process dispersion only focus on the upward shift monitoring. However, the decrease shift monitoring is also necessary and important. In this paper, a synthetic S2 chart is proposed to simultaneously monitor both upward and downward shifts and it consists of a Shewhart-type two-sided S2 sub-chart and a conforming run length sub-chart. In the known in-control variance case, the conforming run length sub-chart only needs a lower control limit and the proposed synthetic S2 chart is shown to be average run length (ARL) unbiased. The effect of parameter estimation on the proposed synthetic S2 chart is also investigated as it is an important issue especially in the real manufacturing processes. Considering that the in-control variance is usually unknown and needs to be estimated by Phase I samples in practice, a new synthetic S2 chart in which conforming run length sub-chart also only needs a lower control limit is developed when the in-control variance is estimated. Furthermore, optimal designs for both known and unknown parameter cases are studied. The advantage of the proposed chart in performance is shown in the results of the comparison with the ARL-unbiased S2 chart. Also, an example illustrates the construction and application procedure of this proposed chart.

Modified Synthetic Control Chart for One-Step Markov-Dependent Processes

Quality measurement by the attributes method often does not result in independent random variables for the count of non conformities or non conforming units. A model based on Markovian dependence can be considered as a practical option in such situations. For a Markovian production process, Adnaik et al. (in press) have proposed the “Single Attribute Control Charts” (SACCs) using two criteria. The first one is by controlling the error probabilities being (α*, β*) and the second, by using the ATS criterion. In this article, we propose a “Modified Synthetic Single Attribute Control Chart” for one-step Markovian Dependent process using ATS criterion (MSyn-SACC-MD).Zero state as well as the steady-state performance of the chart is studied. It is numerically illustrated that the proposed chart is more efficient than SACC-MD-A as proposed by Adnaik et al. (in press).

New Synthetic Control Charts for Monitoring Process Mean and Process Dispersion

A statistical quality control chart is widely recognized as a potentially powerful tool that is frequently used in many manufacturing and service industries to monitor the quality of the product or manufacturing processes. In this paper, we propose new synthetic control charts for monitoring the process mean and the process dispersion. The proposed synthetic charts are based on ranked set sampling (RSS), median RSS (MRSS), and ordered RSS (ORSS) schemes, named synthetic‐RSS, synthetic‐MRSS, and synthetic‐ORSS charts, respectively. Average run lengths are used to evaluate the performances of the control charts. It is found that the synthetic‐RSS and synthetic‐MRSS mean charts perform uniformly better than the Shewhart mean chart based on simple random sampling (Shewhart‐SRS), synthetic‐SRS, double sampling‐SRS, Shewhart‐RSS, and Shewhart‐MRSS mean charts. The proposed synthetic charts generally outperform the exponentially weighted moving average (EWMA) chart based on SRS in the detection of large mean shifts. We also compare the performance of the synthetic‐ORSS dispersion chart with the existing powerful dispersion charts. It turns out that the synthetic‐ORSS chart also performs uniformly better than the Shewhart‐R, Shewhart‐S, synthetic‐R, synthetic‐S, synthetic‐D, cumulative sum (CUSUM) ln S2, CUSUM‐R, CUSUM‐S, EWMA‐ln S2, and change point CUSUM charts for detecting increases in the process dispersion. A similar trend is observed when the proposed synthetic charts are constructed under imperfect RSS schemes. Illustrative examples are used to demonstrate the implementation of the proposed synthetic charts. Copyright © 2014 John Wiley &amp; Sons, Ltd.

The Effect of Measurement Errors on the Synthetic Chart

Measurement errors often exist in quality control applications. In this paper, the performance of the synthetic chart is investigated when measurement errors exist using a linearly covariate error model. It is shown that the performance of the synthetic chart is significantly affected in the presence of measurement errors. The effect of taking multiple measurements for each item in a subgroup on the performance of synthetic chart is also investigated in this paper. An example is provided in order to illustrate the application of the synthetic chart with measurement errors. Copyright © 2014 John Wiley &amp; Sons, Ltd.