Abstract
The objective of this paper is to quantify the effect of autocorrelation coefficients, shift magnitude, types of control charts, types of controllers, and types of monitored signals on a control system. Statistical process control (SPC) and automatic process control (APC) were studied under non-stationary stochastic disturbances characterized by the integrated moving average model, ARIMA . A process model was simulated to achieve two responses, mean squared error (MSE) and average run length (ARL). A factorial design experiment was conducted to analyze the simulated results. The results revealed that not only shift magnitude and the level of autocorrelation coefficients, but also the interaction between these two factors, affected the integrated system performance. It was also found that the most appropriate combination of SPC and APC is the utilization of the minimum mean squared error (MMSE) controller with the Shewhart moving range (MR) chart, while monitoring the control signal (X) from the controller. Therefore, integrating SPC and APC can improve process manufacturing, but the performance of the integrated system is significantly affected by process autocorrelation. Therefore, if the performance of the integrated system under non-stationary disturbances is correctly characterized, practitioners will have guidelines for achieving the highest possible performance potential when integrating SPC and APC.
1. Introduction
Better quality leads to cost reduction. The two major techniques used to monitor and reduce the variation in manufacturing processes are statistical process control (SPC) and automatic process control (APC). The SPC method separates assignable causes from common causes. Shewhart control charts are traditional SPC tools based on the assumption that each observation is uncorrelated. However, the independence assumption is violated in many scenarios, especially in continuous process industries where advanced measurement technologies and shortened sampling intervals are used. Under normal, uncorrelated conditions, the process model has a fixed mean (), and the fluctuation around the mean is the result of white noise (). A process model of SPC can be expressed as follows:
However, when observations are correlated, the correlation structure and drift in the mean are characterized by disturbances. If process observations vary around a fixed mean and have a constant variance, this type of variability is called the stationary behaviour. Otherwise, the behaviour is non-stationary. MacGregor [1] indicated that there are two types of disturbances, deterministic and stochastic disturbances. Stochastic disturbances are random and might be stationary or non-stationary, while deterministic disturbances are a step shift or ramp in the process mean. Box and Jenkins [2] introduced a stochastic difference equation that can model stochastic disturbances. This equation has been used to forecast one-step ahead disturbances, according to the data characteristics of stationary or non-stationary. The stochastic difference equation is expressed in the form of an autoregressive integrated moving average model, ARIMA (p, d, q), as follows: The ARIMA (p, d, q) model indicates as the order of the autoregressive part, d as the amount of difference, and as the order of the moving average part. Mostly, non-stationary disturbances are modelled using the integrated moving average equations, ARIMA (0, 1, 1) or IMA (1, 1), as recommended by Montgomery et al. [3] and Box and Luceno [4]. IMA (1, 1) is a special form of the ARIMA (1, 1, 1) model, with . The ARIMA model is considered a powerful method to solve the correlation problem, especially when it is applied to improve the capability of control charts to monitor the autocorrelated processes.
Application of an ARIMA model to SPC was reported by Lu and Reynolds [5]. Autocorrelated observations were characterized by an ARIMA model and an exponentially weighted moving average (EWMA) chart was utilized to monitor the residuals, based on the ARIMA model forecast values. Similarly, English et al. [6] compared the performance of the and the EWMA chart under an autocorrelated process. Moreover, whether the SPC control chart should be used to monitor the error signal () or the control signal () is debatable, as highlighted by Montgomery [7].
An alternative to SPC, automatic process control (APC), was developed to monitor and control processes irrespective of the pattern of observed data. For APC, frequent process adjustments keep the output on the desired target. Taguchi et al. [8] suggested that the uniform quality of products is achieved by making adjustments according to process information. One example is the error of a watch, which is a function of time. Since both good and poor quality watches can be calibrated as needed, there is no difference between the quality of these two watches. Therefore, quality is not only a function of design, but also a function of the control system. According to Box and Luceno [4], two types of controllers, minimum mean squared error (MMSE) and proportional integral (PI), have been widely used in manufacturing processes to reduce variation due to autocorrelated disturbances. The control action from these controllers is known to whiten disturbances and only leaves random residuals. A PI controller is more robust than an MMSE controller, since it can compensate for unexpected mean shifts. However, Jiang and Tsui [9] claimed that the robustness of a PI controller makes it difficult to detect a mean shift in an autocorrelated process. For this reason, the selection of an appropriate controller is still an open issue for discussion.
While APC effectively reduces predictable quality variations, SPC monitors the process to detect unexpected causes of variation. APC responds to regulate the output response when there is a deviation from the target. Therefore, integrating SPC and APC could lead to a substantial improvement in process quality, since they complement each other. Numerous efforts have been attempted to assess the performance of an integrated system. Montgomery et al. [3] used a Shewhart chart, with EWMA and cumulative sum (CUSUM) charts to monitor the output deviation from the target when there are different sizes of process mean shifts. The MMSE controller was used to compensate for disturbances. The performance measurement was the average run length (ARL) and the average squared deviation from the target measured performance. Nembhard [10] utilized state-space equations to model dynamic processes while disturbances were characterized by ARIMA model. The integrated system was the combination of PI controller with Shewhart and EWMA charts. For a case study, Capilla et al. [11] deployed an MMSE controller with CUSUM and EWMA charts to monitor polymer viscosity in a reactor in which AR and ARMA (1, 1) models were utilized to represent disturbances. The results showed that the integrated system can effectively minimize the variability of viscosity, while process shift can be detected rapidly.
According to Gultekin et al. [12], the combination use of PI controller and control charts (Shewhart and CUSUM) was initiated to reduce the output variation of continuously stirred tank reactors caused by deterministic disturbances and random input disturbances. By using a simulation, the integration was proved that it can reduce the mean squared error by 81% compared to the utilization of PI controller alone. Runger et al. [13] compared the performance of MMSE and PI controller when EWMA chart was utilized. The mathematical models of each control policy were derived and the comparison was done analytically. The disturbances were represented by IMA (1, 1) model and different shift magnitudes. Kandananond [14] performed the economic analysis to determine the optimal combination of controller and SPC chart which minimized the mean squared error of the output and average run length. For different models of disturbances, Kandananond [15] conducted the experimental analysis to assess the performance of the integrated system when disturbances followed ARMA (1, 1) model. For related publications, see Jiang and Tsui [9], MacGregor [16], Harris and Ross [17], Janakiram and Keats [18], Duffuaa et al. [19], Bisgaard and Kulachi [20], and Triantafyllopoulos et al. [21].
According to the above discussion, a limited amount of research has been conducted to characterize the statistical performance of integrated systems. The purpose of this paper is to study the effects of assignable causes (shift), levels of disturbance coefficients, types of controllers, types of control charts, and types of monitored signals on autocorrelated processes by using factorial experimental design. With this research, practitioners will have guidelines to achieve the highest performance of integrating SPC and APC.
2. Process Background
The basis of the analysis in this paper is a mathematical model used to study process autocorrelation effects on the integrated process control system performance. The autocorrelation level of process outputs was controlled by adjusting the IMA coefficient. The autocorrelation behaviour is known to significantly downgrade the performance of control charts since the control limits of control charts are narrower than expected and might signal false alarms more frequently. Moreover, the situation could be more complicated when a shift occurs in the autocorrelated process. Two types of responses, mean squared error (MSE) and average run length (ARL), are obtained in order to assess the sensitivity and adjust the ability of the integrated system when there is a shift in the autocorrelated process.
2.1. Process Description
The autocorrelated process used was a continuous process with only one quality characteristic, represented by . Adjustments were made automatically by two types of controllers, minimum mean squared error (MMSE) and proportional integral (PI), in order to keep the process mean as close as possible to the target (). As a shift occurred in the process, the moving range (MR) chart and exponentially weighted moving average (EWMA) were utilized to monitor the individual measurement of process mean to detect a shift.
2.2. Process Model
The observation of a process is considered from period 1 to 100 (t = 1, 2, 3, …, 100) and the process description for observation t+1 () equals The source of autocorrelation is process disturbances, characterized by the integrated moving average model, ARIMA (0, 1, 1), as follows: where are disturbances at time and respectively, are random errors at time and t, respectively, and θ is the moving average (MA) parameter which ranged from 1 to 1.
After an assignable cause occurs in the process, a shift of size in the form of a step function is injected into a process as follows: where is the magnitude of a shift, is the time that a shift occurs.
When the process mean is off target, each type of controller is utilized to compensate for disturbances and shifts. The derivation of the MMSE controller proceeds as follows.
Rearrange (3) as follows: The deviation from the target () should be minimized, so the possible value of follows (7) as follows: However, the future value of δ(t) and the value of disturbance in the next time frame are unknown, so these values are set to their minimum MSE forecasts. As a result, the value of should be chosen such that where is the MMSE forecast of the disturbance. According to Box and Jenkins [2], the predicted disturbance equals Substituting the disturbance , Since the value of (t) is unknown, it is set to zero. For the MMSE controller, the adjustment signal () for disturbances following the ARIMA (0, 1, 1) model is expressed as follows: For PI controller, the optimal control signal is Since the target (T) in this study is equal to zero, equation (12) is simplified to The values of G and P are numerically calculated to minimize the output response and control signal variances by following this constraint where is the variance of output error, is the process output.
After the adjustment has been performed, the error signal () is given by Both control signal () and error signal () are monitored by a Shewhart moving range (MR) chart and an exponentially weighted moving average (EWMA) chart. The control limits for a moving range chart are where is the process mean and equals , , is the average of moving average, .
For an EWMA chart, the control limits are expressed as follows: where is the average of preliminary data, is the width of control limits, and is the weight assigned to the observation. The values of and used were recommended by Lucas and Saccucci [22].
The mean squared error (MSE) is a method for evaluating a controller and is expressed as follows: at the end of observation period (). On the other hand, the performance of a control chart is measured by the average run length (ARL), which is the expected number of samples taken before a shift is detected. Figure 1 shows the schematic presentation of the process model.
3. Simulation Modelling and Experiment
A factor screening experiment was designed using a statistical package, Design Expert Version 7.1, to analyze the effect of the autocorrelation and other factors on the responses. The selected design was a 25 full factorial design with a total of 32 runs. The selected factors were the MA parameters (θ), shift sizes, types of controllers, types of control charts, and types of signals. The two responses were mean squared error (MSE) and average run length (ARL). Each factor was set to high and low levels as shown in Table 1. θ was set to 0.9 and 0.9 for low and high values, respectively, since θ in the ARIMA (0, 1, 1) model is restricted to the range 1 < θ < 1. Additionally, shift sizes of 0.5 σa and 3.5 σa were utilized to represent small and large shifts. Two types of control charts, a Shewhart moving range (MR) chart and an exponentially weighted moving average (EWMA) chart, were deployed in order to monitor control signal () and error signal () after the adjustment by the MMSE and PI controllers.
Regarding the simulation, each run was composed of 10,000 iterations which have been accomplished by using Palisade’s @Risk Version 5.0. The random errors () from each period were simulated by following normal distribution with zero mean and a constant variance as follows: The simulation results and the analysis of each response are shown in the following section.
4. Performance Analysis
The performance analysis was conducted to examine the effect of input factors on the responses. The statistical design is 25 factorial and the design matrix for all factors and the corresponding responses are shown in Table 2. For each response, the analysis of variance (ANOVA) approach was utilized to reveal the significant factors and their interactions. Model adequacy checking was performed to ensure that ANOVA assumptions were not violated and that there were no residual outliers.
4.1. Analysis of Mean Squared Errors (MSE)
According to the half normal plot in Figure 2, the types of controllers (C) contribute the highest effect on the average MSE, followed by shift sizes (B), θ (A) and types of signals (E) in that order. Moreover, on the basis of the analysis of variance (ANOVA) in Table 3, the interaction effects exist and are based mostly on the above factors, with the highest-order term being BCDE. As a result, types of charts (D) and the interactions BD, BE, CD, CE, BCD, BCE, BDE, and CDE are included in the model even though they do not have small p-values. The hierarchical principle indicates that if there is a high-order term in the model, it will contain all the lower-order terms which compose it.
As shown in Figure 3, at the high level shift size (3.5), the average MSE is maximized when θ is highly negative () while the minimum average MSE is achieved at . However, there is no significant difference in the average MSE at the low level shift size (0.5). The contour plot in Figure 4 also points out that there is a strong interaction between shift sizes and the level of autocorrelation (θ). The average MSE is maximized only when shift size is large (3.5) and θ is low (). At the high θlevel (0.9), it is interesting to note that the average MSE is minimized even when the shift size is large (3.5).
Due to the cube plot in Figure 5, the average MSE when the MMSE controller is utilized is significantly lower than the ones when the PI controller is a part of the integrated system at both the low and high shift levels (0.5 and 3.5). Therefore, the MMSE controller should be the most appropriate controller to keep the process mean on the target. This result has been confirmed by the interaction plot between types of controllers and types of charts in Figure 6. The best result (minimum average MSE) is achieved when MMSE controller is selected.
According to the cube plots (Figure 7), the interaction between shift magnitude (B), controllers (C), and control charts (D) is shown at different levels of θ, when the effect from types of signals (E) is averaged. At the low level of θ (), the minimum average MSE (1.02902) is obtained at the low level of shift magnitude when MR chart is integrated with MMSE controller. At the high level shift size (3.5), MR chart with MMSE controller is still an appropriate choice to minimize average MSE, since the average MSE (2.47177) is lower than those occurring when a PI controller or an EWMA chart is utilized (2.48747, 2.50857, and 2.93032, resp.). At the high θ level (0.9), when an integrated system is composed of an MMSE controller and MR chart, the average MSE is still minimized and disregards shift magnitude.
4.2. Analysis of Average Run Length (ARL)
The half-normal plot (Figure 8) indicates that θ (A), shift sizes (B), types of signals (E) contribute the significant main effects, and there are interactions involving some of these factors with the highest order term being BCE. Results from the analysis of variance (ANOVA) in Table 4 also confirm that these three factors and the interaction BCE are statistically significant, since their -values are small. Similar to the analysis of MSE, the main effect from controller () and interactions BC and CE are included in the model because of the hierarchical principle.
As shown in Figure 9, the cube plot shows the ARLs at different levels of shift sizes (B), types of controllers, (C) and types of signals (E).
According to the plot, the ARLs (14.1441, 2.1723, 2.81313, 2.82013) when control signal (X) is monitored are shown to be significantly lower than those (36.8581, 23.9772, 8.62977, 45.383) when the error signal (e) is observed. These results signify that it is correct to assign the control signal to be monitored by the integrated system.
Another point of interest is that ARL is shown to be sensitive to types of controllers. At the low level shift (0.5), when MMSE controller is utilized and control signal is monitored, the ARL is only 2.81313. However, the effect tends to increase to 14.1441 when all situations are the same, but the PI controller is integrated with SPC. However, there is no significant difference at the high level of shift (2.82013 and 2.1723). This indicates that the MMSE controller should be selected as a part of the integrated system because it can keep ARL at a low level.
Another factor which supports the utilization of MMSE controller is robustness. As shown in Figure 10, when the PI controller is utilized, the ARL is sensitive to changes in θ. As a result, the ARL significantly increases as the value of θ changes from 0.9 to 0.9. Moreover, at the low θlevel (0.9), the ARL is considerably lower than the one at the high θ level (0.9). However, there is no significant difference in the value of ARL when MMSE controller is integrated with the SPC system since the MMSE controller is not just only robust to the autocorrelation change but also outperforms the PI controller in term of ARL minimization. In addition, both controllers cause no difference in the ARL when θ is low. However, when θ is high, the MMSE controller causes a lower ARL than the one when the PI controller is utilized.
5. Summary and Conclusions
This paper focuses on the in-depth analysis of an integrated control system in order to quantify the effects of the selected factors on autocorrelated process. According to the analysis, the disturbance model, ARIMA (0, 1, 1), effects on the responses have been quantified. In addition, appropriate types of controllers, types of control charts, and types of monitored signals have been determined. In summary, the above analysis is concluded as follows.(1)For ARL response, the MMSE controller should be utilized since it is robust to the change in θ, that is, the ARLs are not significantly different at the low and high level θ. Moreover, at the different levels of shift sizes and θ, the MMSE controller has an equivalent or better potential to minimize ARL than the PI controller.(2)The performance of the integrated system to minimize ARL will be significantly improved if an SPC chart is utilized to monitor control signal rather than error signal. (3)The types of control charts, MR or EWMA, utilized in the integrated system have no significant effect on ARL.(4) The minimization of MSE is not only affected by the shift magnitude but also by the level of correlation (θ).(5)The most appropriate combination of the integrated system to minimize MSE is the utilization of Shewhart MR chart with an MMSE controller.
The analysis performed in this research has suggested essential guidelines to implement the integrated SPC effectively. An extension of the research might include the utilization of different process models, for example, ARIMA (1, 0, 0), which represent stationary disturbances.
6. Discussion
Because of the limitation of SPC, APC methodology, which focuses on the adjustment of the process with the frequency that ensures the lowest deviation from the target, is integrated with SPC to solve the correlation and assignable cause problems. According to APC, the correlation embedded in the observations will be predicted by fitting the appropriate forecasting model to the correlated data. The integrated moving average (IMA) is a class of forecasting models for monitoring correlated observations and it is proved to best represent process disturbances because of its flexibility. In this study, a simulation model was developed to represent system performance in terms of the mean squared error (MSE) of the resulting output and the average run length (ARL) of the SPC chart utilized. Simulated results were analyzed to identify influential factors likely to affect the system performance. In practical, if the process considered was correctly characterized by the integrated moving average (IMA) model. The integrated SPC and APC is a powerful technique to maintain the process mean on the target. However, the continuous improvement of the model is required in order to ensure the accuracy of the IMA model, since the process model has no specific pattern over periods of time.