Max-Half-Mchart: A Simultaneous Control Chart Using a Half-Normal Distribution for Subgroup Observations

A Simultaneous control chart is a well-known tool for monitoring the process mean and process variability with a single chart. In recent decades, many researchers have been interested in developing simultaneous control charts. The Shewhart chart is the most common and simple simultaneous control chart. The Multivariate Maximum control chart (Max-Mchart) is a type Shewhart chart that simultaneously monitors the process of multivariate data. This paper proposes a new transformation using a half-normal distribution to improve the Max-chart performance for subgroup observations. The new proposed chart is called Max-Half-Mchart. The Average Run Length (ARL) results show that the proposed Max-Half-Mchart outperforms the Max-Mchart. Additionally, in real data scenarios, the proposed Max-Half-Mchart is consistent with the statistic in the Hotelling T2 chart and the Generalized Variance (GV) chart.


I. INTRODUCTION
A simultaneous chart is a single chart that simultaneously monitors the process mean and process variability. Many researchers have proposed simultaneous control charts. For univariate cases, the simultaneous control chart was first introduced by White and Schroeder [1] using the boxplot method. Since then, many researchers have developed univariate simultaneous control charts. Chen and Cheng [2] developed the Maximum Chart (Max-Chart) that combined statistics X and statistics S. A study conducted by Chao and Cheng [3] developed the semicircle control chart using a half-circle graph. The simultaneous Exponentially Weighted Moving Average (EWMA), the so-called omnibus EWMA, was introduced by Domangue and Patch [4]. Zhang et al. [5] developed the simultaneous univariate EWMA using the Generalized Likelihood Ratio (GLR). Chen et al. [6] developed a simultaneous EWMA chart using X and statistics S. Chen and Thaga [7] developed the Maximum Cumulative Sum (Max-CUSUM) control chart for autocorrelated data. Thaga [8] created the Sum of Square Cumulative Sum (SS-CUSUM) chart based on the sum of squares of the maximum standard CUSUM statistics.
The associate editor coordinating the review of this manuscript and approving it for publication was Yiqi Liu.
Researchers are extending the work on univariate cases. A multivariate simultaneous control chart was also developed. Spiring and Cheng [9] developed the Alternate Variable control chart, which is presented in a boxplot form. Chen et al. [10] developed the Maximum Multivariate EWMA (Max-MEWMA) to handle cases with changing sample sizes. The Maximum Multivariate CUSUM (Max-MCUSUM) was introduced by Cheng and Thaga [11] using a standardized mean vector and the covariance matrix for the statistics. The Max-MCUSUM for autocorrelated data was developed by Khusna et al. [12]. The bivariate Max-Chart was developed by combining the Hotelling T 2 and Generalized Variance (GV) statistics [13]. Then, Thaga and Gabaitiri [14] extended The Bivariate Max-Chart [13] to the Maximum Multivariate chart (Max-Mchart) combined the Hotelling T 2 and GV statistics using the normal standard distribution. Sabahno et al. [15] expanded Max-Mchart using the gamma distribution in GV for monitoring process variability. Max-Mchart for individual observation was developed by Kruba et al. [16] using normal standard and half-normal distributions, called the Max-Half-Mchart.
From among the studies mentioned above, Max-Mchart is well known. However, in some cases, Max-Mchart [14] produces an incorrect result. Furthermore, the Max-Mchart [14] effective only when detecting large shifts. To overcome this issue and improve the Max-Mchart's performance during the monitoring process, this paper proposes an improvement on Max-Mchart [14] using a half-normal distribution in terms of transformation. The Max-Half-Mchart for subgroup observations proposed here is the extension of Kruba et al. [16], which focused on individual observations. This paper also proposes different calculations of the Upper Control Limit (UCL), since the statistics of the proposed Max-Half-Mchart have an unknown specific distribution.

II. NEW SIMULTANEOUS CONTROL CHART
Let X ij denote multivariate observations, with i representing the subgroup order and j representing the observation order in each group. In this research, Max-Mchart carries out the following transformations [14]: And x and S i can be computed by the following equation: The µ and are defined as follows: (.) denotes the normal standard distribution function, H (.) denotes the Chi-square distribution function, n is the number of subgroups, m is the order in a subgroup, and p is the number of quality characteristics. The Max-Mchart is defined as: In some cases, monitoring the mean process and variability process using conventional Max-Mchart as equation (7) produces incorrect results. This fact is empirically proven by applying the conventional Max-Mchart to Woven Poly Propylene (WPP) production i.e., plastic bag production [17]. Figure 1 presents Hotelling's T 2 statistic, and Figure 2 depicts the GV statistic. Figure 3 shows that, in the conventional Max-Mchart the 6 th , 16 th , 20 th , and 46 th samples are out of control, even though they are in-control in terms of the mean    Table 1.  As reported in Table 1, the 6 th , 16 th , 20 th , and 46 th samples are detected as out-of-control because the cumulative distribution function of the Chi-square is too small (close to zero). Therefore, when its value is transformed to the quantile of the standard normal distribution, it produces a large negative value. To refine this inaccurate conclusion obtained by Max-Mchart, this study adopts a quantile approach similar to Max-Half-Mchart for individual observations [16]. Kruba et al. [16] proposed a half-normal distribution approach since it has a positive domain that is the same as the cumulative distribution function of the Chi-square distribution. This new Max-Mchart is the so-called Max-Half-Mchart as it is developed with transformation using a half-normal distribution. The Half normal distribution is used because it only has the positive domain that will lead to the correct result even when the cumulative distribution function of the Chisquare is too small. The statistics can be calculated using the following equation: The difference between equation (7) and equation (8) is the absolute notation. Equation (8) has no absolute sign because the half-normal only has the positive domain.
. . , n (9) and  (9), statistic V H i by using equation (10) and C H i using equation (8) for each subgroup observation. 2. Determine the upper control limit (UCL) using Algorithm 2, which is presented in the next section. 3. If C H i > UCL then the process shift occurs according to the following details: this indicates a shift in the process means.  (9) and equation (10). Then, calculate the UCL for control limit of the proposed Max-Half-Mchart. For the UCL will be discussed in thext section using Algorithm 2. After determining the UCL, the statistic C H i will be compared with the UCL with criteria as point 3 at the Algoritm 1 above.

III. BOOTSTRAP CONTROL LIMIT
The control limit of the conventional Max-Mchart and proposed Max-Half-Mchart use only UCL (Upper Control Limit) since the statistics of both control charts are greater (C i > 0 and C H i > 0). The UCL of Max-Mchart can be calculated with an analytical formula as in Thaga and Gabaitiri [14]. However, the proposed Max-Half-Mchart has an unknown specific distribution, and the bootstrap control limit is therefore used. In this paper, the results of Max-Mchart and Max-Half-Mchart are compared. To provide an equivalent comparison for both charts, the bootstrap approach is used to calculate the UCL for the conventional Max-Mchart and the proposed Max-Half-Mchart.
In this paper, the control limit for the proposed chart is free of distribution. The bootstrap approach defines the control limit. Many types of research have been conducted using the bootstrap approach to determine the control limit. Park [18] calculated the control limit for the median control chart using the bootstrap approach. Khusna et al. [19] proposed the bootstrap for Max-MCUSUM for autocorrelated data. Ahsan et al. [20] also built a control limit in the T 2 control chart for detection instructions. The Bootstrap control limit for conventional Max-Mchart was also developed by Kruba et al. [21] when monitoring fertilizer. The control limit for the proposed control chart in this paper is calculated using Algorithm 2.

Algorithm 2 Bootstrap Control Limit
1. Set α, µ and . 2. For iteration l = 1 : 1000 do the following • Generate data following a multivariate normal distribution N p ( µ, ) • Calculate statistics using equation (9) for the process mean, equation (10) for the process variability, and statistics of the proposed Max-Half-Mchart using equation (8). Then, for the conventional Max-Mchart, calculation statistic use equation (1) for the process mean, equation (2) for the process variability, and equation (7) for the statistics of the proposed chart • Resampling bootstrap the statistic equation (8) and equation (7)  The UCL can be obtained using Algoritmt 2. First, set the α µ and . In this paper, the author set α = 0.00275 and the µ and the as in-control phase (Phase 1). Second, for 1000 iteration generate 5000 data following a multivariate normal distribution N p ( µ, ). Then, calculate statistic of the proposed Max-Half-Mchart as explain in Algorithm 1. For, the statistics of conventional Max-Mchart (C i ) can be calculated as equation (7). To obatain equation (7), the statistic of M i for the process mean in equation (1) and V i for the process variability in equation (2) must be calculated first since the equation (7) consist of the equation (1) and equation (2). Then, resampling bootsrap the statistics of the proposed Max-Half-Mchart (C H i ) and the conventional Max-Mchart (C i ) as many as 1000 times to obatain 1000 statistics of both charts. From those statistic, calculate the precentile from 1000 samples using C H (100(1−α)),l and C (100(1−α)),l . The last, the control limit can be calculated as UCL h = for the conventional Max-Mchart.

IV. PERFORMANCE COMPARISON OF MAX-MCHART AND MAX-HALF-MCHART FOR SUBGROUP OBSERVATIONS
ARL is the average number of observations that are first found to be out-of-control. The ARL serves to measure the effectiveness of a control chart when detecting changes in a process. In this paper, a simulation is carried out with 5000 samples that follow N p µ g = 0, g = I . Then, 5000 samples are assigned into 1000 groups, each of which contains five samples. The ARL1 is obtained by shifting the mean vector a + µ g and shifting the covariance matrix b g , where the index g indicates good conditions (in-control). Thus, ARL0 is the value of ARL when a = 0 and b = 1. The performance of the proposed Max-Half-Mchart is compared to that of the conventional Max-Mchart using the ARL value. The calculation of the control limits for both charts uses the bootstrap approach with α = 0.00275. The ARL calculation results are explained for the following scenario. The scenario is a combination of correlation ρ = 0.3 with two quality characteristics of p = 4 and p = 5. The ARL can be calculated using algorithm 3.

Algorithm 3 Average Run Length
1. Set the UCL as calculated by Algorithm 2. 2. Set µ and . 3. Generate data following a multivariate normal distribution N p ( µ, ). 4. Calculate the statistics using equation (9) for the process mean, equation (10) for the process variability, and equation (8) Table 2 presents the ARL of the proposed Max-Half-Mchart with p = 4 and ρ = 0.3. When the mean vector shifts from a = 0 to a = [0.5, 0.5, 0.5, 0.5] and the covariance matrix is an in-control condition, the ARL reduces from 368 to 142. However, the ARL also decreases from 368 to 24. If the covariance matrix shifts from b = 1 to b = 1.5, the mean vector remains in control. Table 2 explains that the proposed Max-Half-Mchart is faster when detecting the covariance matrix. Additionally, the proposed Max-Half-Mchart also demonstrates better performance when monitoring the mean vector than the covariance matrix because the ARL of the shift in the mean vector drops faster and becomes close to 1. Table 2 Table 3. When the mean vector shifts from a = 0 to a = [4.5, 4.5, 4.5, 4.5, 4.5] and the covariance matrix is an in-control condition; the ARL reduces from 374 to 1. The ARL also decreases from 374 to 1. If the covariance matrix shifts from b = 1 to b = 1.5, the mean vector remains in control. Table 2        that both control charts exhibit better performance when the number of quality characteristics becomes higher (p = 4 top = 5) during detection of shifts in the mean vector.  Table 3. For each data set, 175 random samples are generated that follow N 5 µ g + a, b g , where the values of a and b determined by the scenario described in Table 4. µ g is the mean vector that is calculated from the in-control phase (Phase I). The g is the covariance matrix that is also calculated from the in-control phase. The 175 random samples are divided into 35 subgroups, where each subgroup contains five observations. The first half of the data remains in control (Phase I). The second half of the data (Phase II) represents the shift in the mean vector, shift in the covariance matrix, or shift in both the mean vector and the covariance matrix.
Plots of the data set used in each simulation are shown in Figure 6. If the statistics of the data simulation are shifted in the vector mean, then they are marked by M ++ . If they have shifted in the covariance matrix, then they are marked with V ++ . However, if they have shifted in both the vector mean and covariance matrix, then they are marked by B ++ . The upper control limit (UCL) is obtained using Algorithm 2. All data sets are generated in two sections except Data set 1, for which they are generated using the in-control vector mean ( a = [0, 0, 0, 0, 0]) and in-control covariance (b = 1) for both sections. Therefore, in Figure 7(a), Figure 8(a), Figure 9(a) and Figure 10(a) no out-of-control signal is detected. Figure 7 shows the simulation of the Hotelling T 2 control chart. The Hotelling T 2 control chart is a chart that is only monitoring the shift in the process mean. Figure 7 The out-of-control signals for both charts are detected from the 16 th sample since the statistic of the Hotelling T 2 (T i ) exceeds the UCL. The out-of-control signals detected in Figure 7(b) and Figure 7(d) are caused only by the change in the mean vector. Consequently, in Figure 7(c) no out-ofcontrol signal has been detected since the Data set 3 generates only shifts in covariance matrix by setting b = 2.7. Figure 8 illustrates the simulation of the GV control chart. In Figure 8(b), no out-of-control has been detected since the GV control chart only monitoring the shift in the covariance matrix. Figure 8(c) and Figure 8(d) are the monitoring results for Data set 3 and Data set 4, respectively. The second half of Data set 3 is generated with the shift in the covariance matrix (b = 2.7) and the mean vector is in-control. The second half of Data set 4 is generated by setting a = [1.7, 1.7, 1.7, 1.7, 1.7] and b = 2.7. The outof-control signals for both charts are detected from the 16 th sample since the statistic of the GV control chart exceeds the UCL. As shown in Figure 8(c) and Figure 8(d), the out-ofcontrol signals detected are caused only by the change in the covariance matrix. Figure 9 provides the simulation of the proposed Max-Half-Mchart using the half-normal distribution. Figure 9(b)   Figure 9(d). The first out-of-control signal is detected at the 16 th sample. All of-out-of control are marked with B ++ , starting from the 16 th , because of the shift in both the mean vector and the covariance matrix. In addition, the 20 th , 28 th , 32 nd , and 33 rd samples are marked with M ++ since the out-of-controls are caused only by the shift in the mean vector. Figure 10 displays the conventional Max-Mchart applied to the simulation data. Figure 10   The C i statistics are compared with UCL = 3.595, and the first out-of-control signal is detected at the 16 th sample since the statistic C 16 exceeds the UCL = 3.595. The out-of-control signal is caused only by the change in the covariance matrix. Therefore, the outof-control signals are marked with V ++ . However, not all samples in Phase II for Data set 3 are detected as out-ofcontrol signals. The 29 th sample is in-control because C 29 is less than UCL = 3.595. The monitoring result for Data set 4 is displayed in Figure 10(d). The first out-of-control signal is detected at the 16 th sample. All out-of-control signals are marked with B ++ , starting from the 16 th sample, because of the shift in both the mean vector and the covariance matrix. We can conclude from this simulation scenario that the proposed Max-half-Mchart has more consistent results with the Hotelling T 2 and GV control charts in detecting shift both mean vector and covariance matrix than the conventional Max-Mxchart.

B. APPLICATION TO REAL DATA
The proposed Max-Half-Mchart is applied to Woven Poly Propylene (WPP) i.e., plastic bag production [17]. The data were measured in a day in three shifts (morning, evening, and night). Data measurements carried out include data from 2 October until 30 October insisted 54 observations and has two quality characteristics. The variables used in this paper refer to the characteristics of quality Woven Poly Propylene products that as identified by the company; these variables include the length of Woven Poly Propylene (WPP) and weight of Woven Poly Propylene [17]. In Figure 1, the Hotelling T 2 control chart detects seven out-of-control signals at the 1 st , 4 th , 9 th , 31 st , 33 rd , 35 th , 42 nd observations. However, in the GV control chart, no out-of-control signals were detected.
In Figure 11, the proposed Max-Half-Mchart detected seven out-of-control signals; these are the same out-ofcontrol signals found by the Hotelling T 2 control chart. These seven out-of-control signals are marked with M ++ because the out-of-control signals are caused only by the change in the mean vector. Meanwhile, the conventional Max-Mchart detected the nine out-of-control signals exhibited in Figure 3. Six out-of-control observations are marked with M ++ , including the 4 th , 31 st , 33 rd , 35 th , and 42 nd observations. The other three out-of-control signals are marked with V ++ at the 6 th , 16 th , 20 th and 46 th observations because they are caused only by the change in the covariance matrix. These results show that the conventional Max-Mchart does not consistently monitor Woven Poly Propylene product with the GV control chart since the GV control chart does not