A New Sum of Squares Exponentially Weighted Moving Average Control Chart Using Auxiliary Information

: The concept of control charts is based on mathematics and statistics to process forecast; which applications are widely used in industrial management. The sum of squares exponentially weighted moving average (SSEWMA) chart is a well-known tool for effectively monitoring both the increase and decrease in the process mean and/or variability. In this paper, we propose a novel SSEWMA chart using auxiliary information, called the AIB-SSEWMA chart, for jointly monitoring the process mean and/or variability. With our proposed chart, the attempt is to enhance the performance of the classical SSEWMA chart. Numerical simulation studies indicate that the AIB-SSEWMA chart has better detection ability than the existing SSEWMA and its competitive maximum EWMA based on auxiliary information (AIB-MaxEWMA) charts in view of average run lengths (ARLs). An illustrated example is used to demonstrate the efficiency of the proposed AIB-SSEWMA chart in detecting small process shifts.


Introduction
Statistical process control (SPC) is a statistical method application, which implies not only monitoring the status of a process, but also the ability to maintain process stabilization by distinguishing between common causes and assignable causes of variation in a process (Montgomery [1]). The control chart was first introduced by Walter A. Shewhart in the 1920s and widely used graphical tools in SPC, for monitoring production processes and improvement of process quality. For sophisticated production processes, the Shewhart control chart is insensitive in detecting small process shifts. To cope with this problem, Roberts [2] introduced the exponentially weighted moving average (EWMA) chart to improve the poor detection ability of the Shewhart control chart when small process shifts are of interest. Subsequently, various and extended EWMA charts are developed to deal shifts in process mean, process variability, and both. Such as: (I) Crowder [3], Ng and Case [4], Lucas and Saccucci [5], Steiner [6], and Sheu and Lin [7] developed extended EWMA charts to improve the performance of detecting small process mean shifts.
(III) Sweet [13] and Gan [14] developed two EWMA charts, one to detect mean shifts and the other to detect changes in variability. ( , ) ( 1) ; 1 where − Φ ⋅ 1 ( ) and V are both independent of the sample size n . Two EWMA statistics each for the mean and variance can be defined as: and , 0 1 , 1, 2 , Since t SE is nonnegative, the initial state of the SSEWMA chart requires only an upper control limit ( ) t UCL , which is given by: where ( ) and some action should be taken to identify and eliminate the assignable cause of the process. Xie [18] provides a simple and quick computation formula for t UCL , given as follows: The process is considered to be out of control when the statistic X t X t M N plots outside the circular control region centered at (0, 0) with a radius of t UCL . To avoid the problem of concentric circles, Xie [18] plots   , , X t X t M N on a circular control region centered at (0, 0) with a radius r of with its mean and variance given by X t X Var D n (9) We defined the following statistic for estimator 1 , X t D : follow a chi-squared distribution with − 1 n degrees of freedom when the process is in-control. Then, for an in-control process, is the distribution function of the standard normal distribution. Following Haq [34], the difference estimator of σ 2 X is defined by: where ρ * is the correlation coefficient between , X t V and , X t V . Hence, the mean and variance of 2 , X t D are: We defined the following statistic for estimator 2 , X t D : is also a standard normal random variable, that is, and The computation of * t UCL for the AIB-SSEWMA chart is similar to that of the SSEWMA chart by Xie [18]. It is noted that when correlation coefficient ρ = 0 , there does not exist a single correlated auxiliary variable Y , that is, the estimators of the mean and variance are only estimated by the study variable X . Then, the AIB-SSEWMA chart reduces to the SSEWMA chart, that is, they are identical when ρ = 0 .
The AIB-SSEWMA chart initiates an out-of-control signal when the statistic Each sample point   indicates that the change is likely caused by a shift in both, the process mean and variance.
The steps involved in constructing the AIB-SSEWMA chart for simultaneous monitoring of the process mean and variability are summarized as follows: (1) Choose a suitable λ (2) Use Equation (18) to compute the control limit, are computed by Equations (19) and (20), and plotted with coordinates   * * ( , ) A B . The circular control region is centered at (0, 0) and the radius is X t X t A B falls outside the circular control region given in Step (4). If an out-of-control point is detected, identify the source and direction of the shift corresponding to the position on the chart. Simple criteria and symbols are described in Table 1, and also depicted in Figure 1 for easy identification of the source and direction of an out-of-control signal. The symbols in Table 1 are defined as follows: " + m " and " − m " indicate an increase and a decrease in the process mean, respectively; " + v " and " − v " indicate an increase and a decrease in the process variance, respectively; " + + " indicates a simultaneous increase in the process mean and variance. Similar interpretations apply to the other three cases: " − + ", " − − ", and " + − ".
(6) Examine each of the out-of-control points.  Figure 1. Identification of the source and direction of an out of control signal for the auxiliary information-based (AIB)-sum of squares exponentially weighted moving average (SSEWMA) chart.

Evaluation and Performance Comparison
The performance of a control chart is generally measured in terms of its ARL and SDRL . An in-control ARL is expected to be sufficiently large to avoid frequent false alarms. However, when a present assignable cause displaces the process parameters, it results in an out-of-control process. Meanwhile, the out-of-control ARL and SDRL need to be sufficiently small to rapidly detect the process mean and/or variability shifts. A control chart with a smaller out-of-control ARL and SDRL for a given shift is generally indicated to have superior performance.
The ARL and SDRL values of the initial state SSEWMA and AIB-SSEWMA charts are computed through Monte Carlo simulations. In this study, the use of exact limits, instead of asymptotic limits, enables the initial state charts to quickly detect initial out-of-control signals effectively. An algorithm in R is developed to calculate ARL and SDRL values, which are an average of 50,000 run lengths. Without the loss of generality, the random samples ( , ) tj tj X Y , = 1, 2,..., j n , for , are drawn from a bivariate normal distribution, that is, where δ and τ are magnitudes of process mean and variability shifts, respectively.
, when δ = 0 and τ = 1 indicate that the underlying process is in-control. Subsequently the charting multiplier L and charting parameter λ are adjusted to achieve the desired in-control ARL . Table 2 Table 2 to achieve an in-control ARL of approximately 370. Note that ρ * is the correlation coefficient value between  Table 2, we observed that a small sample size n needs a large charting multiplier L to achieve the desired in-control ARL , which was obviously larger for ρ > 0.5 . For a fixed value of ρ , the values of L increased as the smoothing parameter λ increased. For a fixed value of λ , the values of L increased when 0.5 ρ > at 5 n = ; however, they decreased when 0.5 ρ > at = 10 n . Moreover, the difference between ' L s was large for a larger ρ value. For example, when = 5 n , the value of L at ρ = 0 for λ = 0.05 and 1.0 were 3.533 and 4.909, respectively; however, those at ρ = 0.95 are 3.544 and 5.503, respectively. Similar results were found regarding the large sample size = 10 n in Table 2. When δ ≠ 0 and/or τ = 1 , the underlying process containing assignable causes results in an out-of-control process. The initial state AIB-SSEWMA charts are maintained at the desired in-control ARL , and the out-of-control ARLs , and SDRLs were evaluated for specific process shifts. In this study, we considered the process mean shifts δ = respectively. In particular, the AIB-SSEWMA chart with ρ = 0 was identical to the SSEWMA chart proposed by Xie [18]. That is, the SSEWMA chart is a special case of the AIB-SSEWMA chart when ρ = 0 .  To ensure the accuracy of the simulation algorithm, the following example where ρ = 0 was used to compare our numerical results of the AIB-SSEWMA chart with those of the SSEWMA chart by Xie [18]. For λ = 0.05 , the control limit constants ( ) L K of both charts at in-control ARL of 370 were 3.533. This parameter combination gives the out-of-control ARL and SDRL of the initial state AIB-SSEWMA chart at δ = 0.25 and τ = 1.25 of 9.28 and 7.49, respectively (cf. Table 4).
Meanwhile, the out-of-control ARL and SDRL for the same parameter combination from Xie [18] were 9.28 and 7.49, respectively, which were similar to our simulated results. From Tables 3 and 4, it was observed that the performance of both the SSEWMA chart and the AIB-SSEWMA chart was better when detecting different shifts with smaller λ . Moreover, the AIB-SSEWMA chart performed uniformly better than the SSEWMA chart, which was obvious in detecting the small process mean and/or variability shifts. However, they performed almost equally in detecting large process shifts. As the value of ρ increased, the effectiveness of the The guidelines to count the number of out-of-control signals triggered by both charts have been reported by Haq [34]. According to Table 4, we considered the sample size = 5 n at in-control ≈ 370 ARL ; the parameter combinations of λ  Table 5 presents the number of out-of-control signals, which shows that the AIB-SSEWMA chart had better diagnostic abilities than that the SSEWMA chart. For example, in the case of an upward shift, there existed both process mean shift δ = 0.25 and variability shift τ = 1.25 . The SSEWMA and AIB-SSEWMA charts triggered 323 and 421 out-of-control " + + " signals out of 1000 signals, respectively.  00 0.25 0.50 1.00 1.50 2.00 3.00 0.00 0.25 0.50 1.00 1.50 2.00 3

Comparative Study
Haq [34] proposed the auxiliary information based (AIB) maximum EWMA chart, called the AIB-MaxEWMA chart. The effectiveness of the AIB-MaxEWMA chart in simultaneously detecting shifts in the process mean and variability encompasses the existing MaxEWMA chart. Haq [34] defines the plotting statistic of the AIB-MaxEWMA chart, say t H , based on * , X t A and * , X t B , and is given by: Since t H is a nonnegative quantity, the initial state of the AIB-MaxEWMA chart only requires an upper control limit ( ) t UCL , which is given by: where the value of control limit constant L helps in fixing the in-control ARL of the AIB-MaxEWMA chart to a desired level. Interested readers can refer to Haq [34] for further details. Below is a comprehensive comparison of the proposed AIB-SSEWMA and AIB-MaxEWMA charts for simultaneously detecting a small process mean and variability shifts. Haq [34] recommended using a smaller λ for effectively monitoring small process shifts. Accordingly, Table   6 lists the were extracted from Haq [34]. Comparing the AIB-MaxEWMA chart with the proposed AIB-SSEWMA chart, an interesting finding was that the AIB-MaxEWMA chart performed better than AIB-SSEWMA chart when detecting either only process mean shifts, or decreased in the process variance. However, the AIB-SSEWMA chart was more sensitive than the AIB-MaxEWMA chart in simultaneously detecting a small process mean and variability shifts.   To compare the performance of the sum of squares EWMA chart with and without the use of an auxiliary variable, the in-control ARL of the two charts were maintained as 370. According to  Table 7 and illustrated in Figure 2.  Table 7, respectively. Figure 2a,b plots the SSEWMA and AIB-SSEWMA charts, respectively, with the 30 samples from Table 7. The two charts show that the process remained in control for the first 10 samples. However, when an assignable cause produced small shifts in both the process mean and variability, the first out-of-control signal was detected with the SSEWMA chart on sample 24, while 20 samples were only required to enable the AIB-SSEWMA chart to signal the out-of-control. The remaining out-of-control samples of the AIB-SSEWMA chart in Figure 2b deviated far from the upper part of the * A  -axis and the right side of the * B  -axis, as they were shifted due to a joint increase in the process mean and variability. Similar results also occurred in the SSEWMA chart. Moreover, if the deviation of an out-of-control sample from the * A  or * B  axes was not obvious, Figure 1 provides a guide to identify the direction and source of a shift in the AIB-SSEWMA chart. For shifts in the process mean ( 0.25) δ = and variability ( 1.25) τ = , the illustration results indicate that the AIB-SSEWMA chart was more sensitive than the SSEWMA chart in detecting small process shifts.

Conclusions
As a single auxiliary quality characteristic (auxiliary variable) is available and accompanied with the quality characteristic (study variable), the performance of control charts is improved by the precision of process parameter estimators. In this article, we proposed a novel sum of squares exponentially weighted moving average chart, named as the AIB-SSEWMA chart, for effectively monitoring the small process mean and/or variability shifts. The study of the numerical simulations indicates that the AIB-SSEWMA chart performed uniformly better than the existing SSEWMA chart in detecting various kinds of process shifts. Additionally, the proposed chart was sensitive to small shifts in both the process mean and variability compared to its counterpart, the AIB-MaxEWMA chart. On the basis of the run length profiles, the study recommended that the AIB-SSEWMA chart with 0.05 λ = was more helpful in detecting the small process mean and variability shifts in an industrial production process. Future work was recommended to extend the AIB-SSEWMA chart by adding an adjustment parameter, a sum of squares generally weighted moving average (SSGWMA) chart using auxiliary information, namely the AIB-SSGWMA chart. Furthermore, the double technique could be adopted to develop the AIB-SSDGWMA chart to simultaneously monitor the process mean and variability shifts in a single chart.