Novel Mixed EWMA Dual-Crosier CUSUM Mean Charts without and with Auxiliary Information

Research Institute of Educational Science, Hunan University, Changsha, China Department of Statistics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan Department of Mathematics, Women University of Azad Jammu and Kashmir, Bagh, Pakistan School of Management, Nanjing University of Posts and Telecommunications, Nanjing, China Statistics Program, Department of Mathematics, Statistics, and Physics, College of Arts and Science, Qatar University, Doha 2713, Qatar Statistical Consulting Unit, College of Arts and Science, Qatar University, Doha 2713, Qatar


Introduction
Statistical process monitoring (SPM) provides many techniques to help practitioners monitor the manufacturing or industrial processes for assignable causes of variation. Control charts are a widely used SPM technique to detect the process shifts by assignable causes and also to help to establish the process capability. Shewhart and Van Nostrand [1] introduced a memoryless control chart used to monitor large shifts in a process parameter. Regardless of the wide prominence and simplicity of the Shewhart charts, these charts are ineffective in industries or manufacturing processes where small shifts occur. erefore, the memory-type control charts, including the cumulative sum (CUSUM) by Page [2] and the exponentially weighted moving average (EWMA) control chart by Roberts [3], are widely used to deal with the sooner detection of small to moderate shifts.
In SPM, it is a common practice to enhance the sensitivity of the control chart with the induction of an auxiliary information-based (AIB) estimator of the underlying process parameter(s). Abbas et al. [4] enhanced the detection ability of the classical EWMA chart and suggested the AIB-EWMA chart. e AIB-EWMA mean chart significantly and uniformly outperforms the classical EWMA mean chart. Likewise, Haq [5] and Abbasi and Haq [6] suggested adaptive EWMA and adaptive CUSUM charts for the process mean when the shift is not known in advance. Similarly, Haq and Khoo [7] suggested a multivariate process mean chart using auxiliary information. In order to further increase the sensitivity of the mixed EWMA-CUSUM chart, Riaz et al. [8] suggested an AIB mixed version of the EWMA-CUSUM mean chart. Similarly, Aslam et al. [9] introduced mixed control chart for attribute data. Likewise, Aslam [10] suggested mixed EWMA-CUSUM and mixed accelerated hybrid censored control chart for Weibull distribution. Also, Aslam et al. [11] suggested mixed chart using EWMA statistic. On similar lines, Haq and Bibi [12] suggested the dual CUSUM charts with the help of auxiliary information to improve the process mean monitoring. Similarly, Anwar et al. [13] suggested AIB-mixed EWMA-CUSUM and AIB-mixed CUSUM-EWMA charts for efficient monitoring of process location. Also, Anwar et al. [14] proposed an AIB-modified EWMA chart for improved process location monitoring. Anwar et al. (2021a) suggested an AIB double homogeneously weighted moving average control chart for the process mean on the same lines. It has been found that these charts are superior to the non-AIB-based chart for detecting various process mean shifts. For more related works, we refer to Aslam et al. [15], Lee [16], Abid et al. [17], Khan et al. [18], Haq et al. [19], Hussain et al. [20], Hussain et al. [20], Anwar et al. [21], and the references cited therein.
Existing control charts such as CUSUM, AIB-CUSUM, EWMA, and AIB-EWMA charts are designed to detect a particular shift level efficiently by selecting an appropriate reference value. However, the shift size is rarely known and can only be assumed to fall within a certain range. e charts with reference values designed for a prespecified shift usually do not perform well for the whole shift range. Hence, it is important to design a control chart for detecting a range of known or unknown shift levels. Because we rarely know the exact shift size before it is detected, several techniques have been developed to handle the issue of detecting unknown shifts within a certain range. For example, Zhao et al. [22] proposed a method by combining two different CUSUM charts, namely, the dual CUSUM (DC) charts which are used to detect a range of shifts. Hany and Mahmoud [23] provided Crosier CUSUM (CC) control chart that performs better than the classical CUSUM chart. Similarly, Haq and Munir [24] suggested proposed CC and Shewhart-CUSUM (SC) charts for the process mean under ranked set sampling (RSS) scheme. Moreover, the false initial response features were attached to improve the control chart's detection ability. e RSS-based CC and SC charts are very efficient for monitoring small and large shifts in the process mean compared to the existing counterparts based on a simple random sampling scheme. Later, Abbas et al. [25] suggested a new charting scheme called mixed EWMA dual-CUSUM (EDC) chart to monitor the process mean. e EDC chart provided a good overall detection over the DC and dual-Crosier CUSUM (DCC) charts when detecting a range of mean shift sizes. After that, Song et al. [26] presented an efficient approach of designing distribution-free exponentially weighted moving average schemes with dynamic fast initial response for joint monitoring of location and scale. Furthermore, Saha et al. [27] advocated two CUSUM schemes for simultaneous monitoring of unknown parameters of a shifted exponential process and its application in monitoring of call durations in telemarketing. Afterward, Haq and Syed [28] recently introduced CUSUM and dual CUSUM charts for the enhanced process mean monitoring. Also, Haridy et al. [29] instigated an improved design of exponentially weighted moving average scheme for monitoring attributes. For more related works, readers can refer to Raji et al. [30], Haq and Bibi [31], Haq and Bibi [31], Javaid et al. [32], Aslam and Anwar [33], Rasheed, et al. [34], and the references therein.
As mentioned before that the CC, EDC, and DCC charts are very efficient for the detection of different shift sizes when the process parameters are univariate and free from any extra information like auxiliary information. Sometimes, the practitioners are interested in monitoring processes where the quality characteristic is accessed along with auxiliary information variables. In this case, the CC, EDC, and DCC charts are inefficient. Moreover, the performance of the EDC chart is better than CC and DCC charts. Taking motivation from these facts, we propose an EWMA-DCC chart, called EDCC chart, for monitoring variations in the process mean, considering shifts lying within an interval. In addition, by using the AIB mean estimator, we propose AIBbased EDC (AIB-EDC) and AIB-based EDCC (AIB-EDCC) charts for efficiently monitoring the process mean shifts within an interval. e performance of the proposed charts is evaluated with the help of the average run length (ARL), the standard deviation of run-length (SDRL) and median run length (MRL), extraquadratic loss (EQL), and integral relative ARL (IRARL). e findings revealed that the proposed AIB-EDC and AIB-EDCC charts are more sensitive in detecting process mean shifts than their competitors used in this study. e proposed AIB-EDC and AIB-EDCC charts also outperform in terms of overall performance. A similar trend is observed when these benchmark CUSUM charts are compared with the proposed charts with FIR features. e remainder of this paper is organized as follows. In Section 2, the DC, DCC, and EDC charts are presented. Section 3 presents the details of the proposed EDCC, AIB-EDCC, and AIB-EDC charts with and without FIR features. en, the RL performance of different control charts and some design guidelines of the charts are provided in Section 4. Section 5 contains a real data example to illustrate the application of our proposed charts. Some conclusions and discussions are given in Section 6.

Existing CUSUM Charts
In this section, a brief review of some existing CUSUM location charts is provided. Let us assume X t,1 , . . . , X t,n , t � 1, 2, . . ., represents a sample of size n ≥ 1 from normally distributed quality characteristics having mean μ 0 and standard deviation σ 0 , i.e., X t ∼ N(μ 0 + δσ 0 , σ 0 ), where δ represents the shift in the process mean level. e process stays in control if δ � 0 and considered out-of-control for δ ≠ 0. Also, X t � n j�1 X t,j is the sample mean and μ X t � μ 0 + δσ 0 and σ X t � σ 0 / � n √ , respectively. Without loss of generality, the process mean μ 0 and σ 0 are assumed to be 0 and 1, respectively, and the sample size n � 1.

Classical CUSUM Chart.
e usual CUSUM X chart is defined as where ) is the reference value. An out-ofcontrol signal is detected when C + t (for the upper-sided chart) or C − t (for the lower-sided chart) exceeds the decision interval h, where h is determined to obtain a desired incontrol ARL. When the mean shift size δ is known, it has become well known that the upper-sided or lower-sided CUSUM X chart can be optimally designed using K � δ/2.

DC Chart.
Zhao et al. [22] suggested using DC chart to detect different levels of mean shift. e DC chart outperforms the CUSUM chart when detecting mean shifts in different ranges. e chart consists of plotting two upper (A + 1,t , A − 1,t ) and two lower (A + 2,t , A − 2,t ) CUSUM statistics, respectively, based on the sample mean X t at time, given by where A + i,0 � A − i,0 � 0 and k i are the reference parameters for i � 1, 2.
e two-sided DC chart gives an out-of-control signal whenever . e values of h i are selected so that the in-control ARL of the DC chart reaches the desired level. On similar lines, the one-sided DC chart triggers an out-of-control signal whenever only A + i,t > h i for the upper-sided chart to detect an upward shift or A − i,t > h i for the lower-sided chart to detect a downward shift in the process mean, respectively. In order to increase the sensitivities of the DC chart against the initial process shifts, it is possible to incorporate the FIR features into this chart. On the lines of Lucas and Crosier [35], the FIR features are applied to the DC chart by modifying As our interest lies in detecting shifts that are expected to lie with an interval, for all the control charts studied in this paper, we consider k 1 � (3a + b)/8 and k 2 � (a + 3b)/8 with k 1 h 1 � k 2 h 2 and k 1 + h 1 > k 2 + h 2 , which were also recommended by Zhao et al. [22].

EDC Chart.
Abbas et al. [25] proposed a more sensitive EDC control chart for detecting small shifts in the process mean. Similar to the DC chart, the plotting statistics where Now, the mean and variance of statistic are given as where λ ∈ (0, 1] is the smoothing parameter, exceeds h i , the lower-sided EDC triggers an out-of-control signal. For more details on the EDC chart, readers can refer to Abbas et al. [25]. Similar to the FIR-DC chart, the FIR-EDC chart with the 50% head start E + i,0 � E − i,0 � h i /2, for i � 1, 2, are also suggested to protect the initial problems.

DCC Chart.
e CC chart has been suggested by Crosier [36] for monitoring the process mean. e CC chart is more sensitive than the classical DC chart. Using the sequence X t , the plotting-statistic of the DCC mean chart is given as follows: Here, k i > 0 with i � 1, 2 are the reference parameters of the DCC chart. e DCC chart works with two different plotting statistics, D 1,t and D 2,t . An out-of-control signal is initiated by the two-sided DCC chart whenever D i,t > h i or D i,t < − h i , where h i , i � 1, 2 are the decision intervals. For the one-sided charts, the upper-sided chart triggers an out-of-control signal when D i,t > h i and the lower-sided chart gives an outof-control signal when D i,t < − h i . In addition, with fixed k 1 and k 2 for a given mean shift interval, the values of either h 1 or h 2 are determined to get the desired in-control ARL for the DCC chart. To increase the sensitivities of the DCC chart, the FIR feature is applied by adjusting the DCC chart with the 50% head start to protect the initial problems, i.e.,

Proposed Control Chart
is section proposes three new CUSUM control charts to efficiently monitor a wide range of shifts in the process mean level.
ese include an EWMA-DCC (EDCC) chart, the auxiliary information-based EDC (AIB-EDC) chart, and the auxiliary information-based EDCC (AIB-EDCC) chart. Both the one-sided and two-sided versions of these charts are described as follows.

EDCC Chart.
In order to enhance the detection ability of the DCC chart, the mixture charting structure of EWMA with DCC charts are integrated into a single control chart. e plotting statistics of two CUSUM statistics, say G 1,t and G 2,t , based on M t , are given by and M t is defined in (11). Here, k i , i � 1, 2, are the reference values of the EDCC chart. e EDCC chart works with two different plotting statistics, i.e., G 1,t and G 2,t . An out-ofcontrol signal is initiated by the two-sided EDCC chart whenever G i,t > h i or G i,t < − h i , where h i , i � 1, 2 are the decision intervals. For the one-sided charts, the upper-sided chart triggers an out-of-control signal when G i,t > h i and the lower-sided chart gives an out-of-control signal when G i,t < − h i . e values of h i are selected so that the in-control ARL of the EDCC chart reaches the desired level. For the EDCC chart, an FIR feature is attached to guard against the initial/startup problems, i.e., G i,0 � h i /2, for i � 1, 2. is FIR feature enables the EDCC chart to respond to an off-target process at the startup quickly.

AIB-EDC Chart.
Suppose there exists an auxiliary characteristic Y which is correlated with the quality characteristic of interest X. e observations of X and Y are obtained in paired form, i.e., (X t , Y t ) for t ≥ 1, from the bivariate normal distribution, i.e., representing the variances and ρ is a correlation coefficient between X and Y. A bivariate random sample of size n is taken from the process (X t , Y t ) at the time t, denoted by (X tj , Y tj ), for t � 1, 2, . . .. Let Y t � (1/n) n j�1 Y tj , denotes the sample mean of the auxiliary characteristic Y. Using these notations, the regression estimator of the process mean is given by (cf. Riaz [37] and Abbas et al. [4]): e mean and variance of the statistic Q t in (13) are given as (13) and (14) imply that Q t is also an unbiased estimator of μ X t and σ 2 Q t < σ 2 X t . Using the sequence Q t based on the regression estimator in (12), the upper (E + 1,t , E + 2,t ) and two lower (E − 1,t , E − 2,t ) plotting statistics of the AIB-EDC mean chart are given by where . k i and h i are the reference parameters and decision intervals, respectively, of the AIB-EDC chart. e value of h i is selected to obtain the in-control ARL of the AIB-EDC chart at a desired level. e two-sided and one-sided AIB-EDC charts work similarly to EDC charts presented in Section 2.3. e statistic M t in (15) and (16) is defined as, where λ ∈ (0, 1]. Now the mean and variance of M t are given as: With the time t increasing, the factor (1 − (1 − λ) 2t ) tends to unity, and ultimately the variance of M t will converge to σ 2 Q t λ/(2 − λ). An FIR feature with the AIB-EDC chart to increase its sensitivity against the initial problems or shifts in the process mean, called the FIR-AIB-EDC chart, is also suggested. e FIR-AIB-EDC chart is suggested with the 50% head start, i.e.

AIB-EDCC Chart.
e auxiliary information based EDCC, namely the AIB-EDCC chart, is based on two plotting statistics (G 1,t and G 2,t ), defined below: and M t is defined in (17). k i represents the reference value and h i is the decision interval. e two-sided and one-sided AIB-EDCC charts work similarly to the EDCC charts presented in Section 3.1. An FIR feature can also be attached to the AIB-EDCC chart by setting G i,0 � h i /2 for i � 1, 2, called the FIR-AIB-EDCC chart.

Performance Comparison
A detailed comparative study of the proposed control chart with some of the existing control charts is provided in this section. e performance of a control chart is generally evaluated in terms of its run length properties, including the ARL, MRL and SDRL. ese measures are used to evaluate the sensitivity of a control chart for a specific shift. For an incontrol process, the in-control ARL (ARL 0 ) should be large enough to avoid frequent false alarms. While for an out-ofcontrol process, the out-of-control ARL (ARL 1 ) should be as small as possible to detect the out-of-control signals quickly. e same implies to the MRL and SDRL. e smaller the SDRL of a control chart for a given shift, the better performance of a control chart at a specific shift will be. Many approaches are frequently used to approximate the run length characteristics of a control chart, including the integral equations, Markov chain, and the Monte Carlo simulation. In this paper, the Monte Carlo simulation method is used to obtain estimates of the run length characteristics due to the complexity of the charting statistics.
In order to study the overall performance of a control chart for a range of shift sizes, an alternative performance measure IRARL that describes the effectiveness of a control chart over the whole process shift domain (a ≤ δ ≤ b) is considered in our study to investigate the run length performance of the control charts, see Zhao et al. [22] and Wu et al. [38]. e IRARL can be formulated as follows: Here, the traditional optimal one-sided CUSUM X chart is considered as the benchmark control chart for a specific shift. Also, the ARL c (δ) represents the ARL 1 of the compared control chart under the mean shift δ and ARL opt (δ) can be defined as the ARL 1 of the traditional CUSUM X chart in Section 2.1 with k � δ/2 for various values of δ in the range [a, b]. In addition, F(δ) is the cumulative distribution function (CDF) of shift δ. In practice, if we have no prior information about the mean shift, the CDF of uniform distribution U[a, b] can be used, see Zhao et al. [22] and Tanaka et al. [39]. IRARL values are obtained by applying the Simpsons method using the integral function available in R "Bolstad" statistical package.
Based on 10 5 replications under each simulation, these control charts' run length characteristics are computed and reported in Tables 1-3. e main findings are listed in the following points.
As might be anticipated that, having fixed the ARL 0 , the out-of-control ARL/MRL/SDRL of the proposed charts decrease as a function of δ, i.e., as the value of δ increases, the out-of-control ARL/MRL/SDRL tends to decrease and vice versa. For example, in Table 1, when λ � 0.5, the ARL/MRL/ SDRL values of the AIB-EDCC chart decrease from 44.37/ 35.00/32.58 down to 7.90/7.00/2.69 with δ increasing from 0.25 up to 1. Overall, the proposed one-sided EDC and EDCC charts with AIB outperform the exiting control charts for different shifts. For example, from Table 1, it can be noted that when λ � 0.5, the IRARL is 0.86 for the AIB-EDC and AIB-EDCC charts, which are both smaller than the IRARL values of the DC (1.04), DCC (0.96), and EDC (1.03) charts. On the other hand, the AIB-EDCC control chart is very efficient than the AIB-EDC chart for a larger value of ρ. For example, at δ � 0.25 and ρ � 0.75, the ARL value AIB-EDC chart is 33.26, whereas the ARL value AIB-EDCC chart is 33.15.
In addition, it can also be noted that the EDC and EDCC charts without and with AIB outperform the DC and DCC charts for a specific small shift in the shift ranges. For example, when δ � 0.25 in Table 3 Table 3).
From Table 4, when δ increasing from 1 up to 3 at λ � 0.5, the ARL/MRL/SDRL values of the AIB-EDCC decrease from 6.45/6/3.09 to 1.91/2/0.51. Likewise, for the overall performance measures of the control charts, the proposed AIB-EDC and AIB-EDCC charts have minimum values of EQL, i.e., 8.77 and 8.77, respectively. is implies that the proposed AIB-EDC and AIB-EDCC charts are more efficient in terms of overall performance measures (see Table 4). Furthermore, the proposed charts are also presented at another range of 1 to 5. From Table 5, when λ � 0.5, the AIB-EDCC ARL/MRL/SDRL values decrease from Mathematical Problems in Engineering        For example, at λ � 0.05, 0.25, 0.50, the AIB-EDCC chart (with ρ � 0.75 and δ � 1) provides ARL of 5.91, 5.09, and 4.37, respectively (see Table 5).

Mathematical Problems in Engineering
It can be noted that, from Table 6, the FIR AIB-EDC and ABI-EDCC charts are better than other FIR charts in the detection of a whole range of shifts. For example, when λ � 0.5, the IRARL � 0.83 of the FIR AIB-EDC and FIR AIB-EDCC charts are smaller than IRARL � 0.92 and IRARL � 0.90 of the FIR DC and DCC charts. In addition, for comparison of Tables 3 and 6, it is noted that the FIRbased charts in Table 6 surpass the charts without FIR features in Table 3 to detect shifts in a whole range. For example, when λ � 0.5, the IRARL of the FIR-DC, FIR-DCC, FIR-EDC, FIR-EDCC, FIR AIB-EDC, and FIR-EDCC charts are 0.92, 0.90, 0.95, 0.95, 0.83, and 0.83, respectively (see the last row in Table 6). ese values are all smaller than the corresponding IRARL values in the last row of Table 3. e two-sided charts are presented in Table 7; it can be concluded that two-sided AIB-EDC and AIB-EDCC charts also perform better than the charts without AIB and DC and DCC charts for the detection of a whole range of shifts. For example, at λ � 0.5, the IRARL are 1. 19 Table 7). Compared with the one-sided charts, it can be noted that all the one-sided charts perform better than the corresponding two-sided charts. For example, when λ � 0.5, all the ARL/ MRL/SDRL values of the one-sided DC, DCC, EDC, EDCC, and AIB-EDC charts for a specific shift δ are smaller than the ones of the corresponding two-sided charts and the IRARL of one-sided charts are also smaller than the IRARL of twosided charts.

Illustrative Example
is section considers an illustrative example to demonstrate the implementation of the proposed charts in real-life scenarios by using the film thickness dataset. e film thickness is one of the prime quality characteristics for coating purposes. e film is a coating material used for protection over reflection to absorption, permeation, transparency, flexibility, transmission, and electrical polarization. Film material is used in many products and technologies such as packaging, sensors, semiconductors, photovoltaics, displays, and biological and medical devices and equipment. e deepness of the coating applied is known as the film thickness, and generally, it can be classified into two major categories based on the time at which it has been applied: wet film thickness (WFT) and dry film thickness (DFT). e thickness of liquid-based coating like wet paints is known as wet film thickness. Initially, when the coating is applied, it has the wettest thickness but gradually decreases with the evaporation of solvents from the film coating. In order to acquire the particular level of dry film thickness, it is essential to determine the quantity of the material applied while providing the best possible protection against damage and corrosion. After the coating has dried, its level of thickness is measured on top of the layer, which refers to dry film thickness (DFT). e number of layers could be more than one.
DFT is measured only when the coating dries completely. e process which is adopted for the application of coating determines the thickness of the coating. By following the approved coating parameters, a proper level of thickness can be achieved. A manufacturing process produces a polymer film that is in the shape of a rectangle. As a quality control measure for each film produced, the thickness (microns) is measured in four specified locations: bottom left, bottom right, top left, and top right. According to our data of a plastic film thickness after being cut, we can measure the thickness by using the Gauge model 4000/F. Gauge model 4000/F can measure the thickness in the range of "0.001 mm" to "1 mm." In addition, including the depth of the jaw "35 mm," and this model is special for measuring like polyethylene bags and thickness of the plastic film and many more things. Here, we consider two variables for the brevity of discussion: X (study variable) top thickness and Y (auxiliary variable) as the bottom thickness measurements from this process. One hundred sixty samples for these variables are used, and the correlation between these variables is 0.75. e data are taken from https://openmv.net/info/film-thickness. We standardized both variables before being used for control charting. For illustration purposes, we used four charts, i.e., AIB-EDC, AIB-EDCC, DC, and DCC charts. For a fair comparison, the control limits are adjusted so that all the control charts have the same ARL 0 � 300. Specifically, we used (k 1 � 0.22, k 2 � 0.41, h 1 � 8.5727, and h 2 � 4.6000) for the one-sided DC (k 1 � 0.22, k 2 � 0.41, h 1 � 7.7566, and h 2 � 4.1621) for the DCC (k 1 � 0.22, k 2 � 0.41, h 1 � 48.4500, and h 2 � 26.0000) for AIB-EDC and (k 1 � 0.22, k 2 � 0.41, h 1 � 48.2000, and h 2 � 25.8600) AIBEDCC charts, considering λ � 0.05 and ρ � 0.75. ese control charts are designed to efficiently detect the mean shift δ in the range [0. 25,1]. To check the detection ability of the four charts, shift δ � 0.5 is introduced in the last 30 samples, and the resulting control charts are presented in Figure 1. We can observe from Figure 1 that the proposed AIB-EDC and AIB-EDCC charts detect 18 out-of-control signals, whereas DC and DCC charts detect only 3 and 11 signals, respectively.
is superiority of the proposed charts over the existing charts is in accordance with the findings of the simulation study.

Conclusions and Recommendations
e Crosier-CUSUM (CC) is the advanced version of the CUSUM chart used to monitor the process when the shift is not known in advance. Similarly, the mixed EWMA-double CUSUM (EDC) is very efficient for the monitoring of process mean. is paper integrated double CC (DCC) and EWMA chart structures and proposed a new mixed EWMA-DCC (EDCC) charting scheme for monitoring the process mean. In addition, the performance of AIB EDC and mixed EWMA-DCC (EDCC) charts is also investigated for efficient process monitoring. Monte Carlo simulations are used to compute the run-length characteristics of the proposed charts. e ARL, SDRL, MRL, EQL, and IRARL are used as performance measures of the proposed control charts. e proposed one-sided or two-sided EDC and EDCC charts with AIB outperform the existing control charts for different shifts. To guard against the initial/startup problems in the process, FIR features are attached to different control charts, and the results show that the FIR AIB-EDC and AIB-EDCC charts are better than other FIR charts in the detection of a whole range of shifts. e current study can be extended to nonnormal, nonparametric, neutrosophic, and multivariate scenarios.

Data Availability
e data for the study can be provided from the corresponding author on reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.