A review on the MCUSUM Charts in Detecting the Shifts of the Process with Comparison Study

In this paper, we compare the performance of different MCUSUM methods presented in the literature. First, we briefly introduce MCUSUM methods in multivariate normal distribution. In order to evaluate their performance, we present a comparative study with simulation. Furthermore, we compare the average out-of-control run length of MCUSUM methods under different scenarios of mean shifts, standard deviation shifts, and correlation shifts. The results of the simulation study show that MCUSUM methods have different efficiency in detecting process shifts and based on the required application, the appropriate MCUSUM chart should be selected .


Introduction
was the first who proposed the control techniques for Multivariate processes (Hotelling, 1947).
Multivariate control is divided into two stages.The first stage is the retrospective examination of process behavior.The second is prospective examination of process (Sullivan and Woodall, 1996).In the first stage, observations are analyzed to determine if the process was in control and the covariance, mean, and control limit are determined.In the second stage, a control chart is applied to control if the parameters determined in the first stage are correct. 1Corresponding author Email Address: fallahnezhad@yazd.ac.ir

International journal of innovation in Engineering
journal homepage: www.ijie.irJackson (1991) stated that any multivariate process control method must satisfy four conditions: they must provide a response to (1) whether the process is in control, (2) whether the specified probability of Type I error is preserved has it been or not and (3) whether they considered relationships between variables, and (4) answering the question, "If the process is out of control, what is the problem?" Must be available (Jackson, 1991).Bersimis et al. (2007) provide a literature review of multivariable process control chart techniques.They investigated multivariate extensions for a variety of univariate control charts, such as multivariate Shewharttype control charts, multivariate CUSUM control charts, and multivariate EWMA control charts.In addition, they explore unique methods for constructing multivariate control charts, based on multivariate statistical techniques such as principal component analysis (PCA) and partial least squares (PLS).MCUSUM control charts are divided into two categories.In the former, the direction of shift is known, while in the latter, the direction of change is assumed to be unknown (Bersimis et al., 2007).
A MCUSUM can be designed from CUSUMs based on two methods.One involves reducing each multivariate observation to a scalar then designing a CUSUM of the scalars.The second method is by accumulating the vectors before reducing it to a scalar, which is designing a MCUSUM directly from the observations (Crosier, 1988).Woodall and Ncube (1985) proposed a method for the bi-variate normal distributions, they showed that their MCUSUM method performs better than the Hotelling's T 2 method (Woodall and Ncube, 1985).Rasay et al. (2018) applied control charts as a condition monitoring technique, and inferences about the operating modes of the system are based on the information collected about the quality of the items produced (Rasay et al., 2018).
Akhavan Niaki and Fallah Nezhad (2009) proposed a new method in this paper to monitor the change of the overall mean and the classification of states of a multivariate quality control system Based on the Bayesian rule (Akhavan Niaki and Fallah Nezhad, 2009) .Jafarian-Namin et al. (2021) have examined the integration of triple components including statistical process monitoring (SPM), maintenance policy (MP) and economic production quantity (EPQ) (Jafarian-Namin et al., 2021).Rasay et al. (2019) considered a two-step affiliate process in which each step has a unique qualitative characteristic.According to the regression formula, the qualitative characteristic of the second stage is dependent on the first stage (Rasay et al., 2019).Today multivariate control charts are widely applied in industrial application.Thus selection of the appropriate multivariate control chart is very important in practice.
Industrial application of quality control methods are discussed in (Ghahremani, and Mohseni, 2021).
We compare the performance of different MCUSUM methods presented in the literature.For this purpose, we first briefly introduce MCUSUM methods in multivariate normal distribution.In order to evaluate their performance, we present a comparative study with simulation.Furthermore, we compare the average run lengths of in-and out-of-control MCUSUM methods under different scenarios of mean shifts, standard deviation shifts, and correlation shifts.The results of the simulation study show that MCUSUM methods have different efficiency in detecting process shifts and based on the required application, the appropriate MCUSUM chart should be selected.This paper is the result of a simulation analysis of the recent MCUSUM charts in the area of multivariate statistical process control.Section 2 describes the most significant multivariate cumulative sum (CUSUM) control charts.The simulation analysis is presented in Section 3. Finally, some concluding remarks are offered in Section 4.

MCUSUM Methods
Healy (1987) used the sequential probability ratio tests, in order to develop a MCUSUM chart (Healy, 1987).Let i x be the ith observation, that follows a multi normal distribution   00 , p N   with an in- control mean vector 0  and a known covariance matrix 0  .Let 1  be the out-of-control mean vector.
The CUSUM for detecting the out-of-control mean 1  may be written as  is the non-centrality parameter, and, When Si ≥ H then an out-of-control signal is observed.
where S0 ≥ 0 and k ≥ 0. When Si ≥ H then an out-of-control signal is observed, Crosier proposed the optimal value of k is the square root of the number of variables (Bersimis et al., 2007).
The second CUSUM proposed by Crosier is a CUSUM of vectors that is given by otherwise and   then an out-of-control signal is observed , h is selected to achieve an assumed in-control ARL by simulation.Crosier (1988) proposed that 1 0.5 ( ) k   and 0 0 S  (Crosier, 1988).


. The process is out of control if i S was more than a control limit H.
In 10000 independent replications, for an intended ARL0 of 320, we assumed  is calculated equal to 4 3 and the parameters of each MCUSUM method is determined.We pick the H parameters of the methods such that ARL0 of the methods becomes 320.For the comparison study, we estimate the ARL1 values of the MCUSUM methods by 10000 independent replications in each of the different scenarios of mean shifts.

34
The shifts are given as multiples of the process standard deviations and are shown in the first column of Table (1).The second up to the tenth column of Table (1 ) show the ARL1 values of the methods under consideration and their standard deviations.
Table 1: The results of ARL1 study for mean shifts (bi-variate normal) If we use the average of ranks as the performance criteria, we see from Table 2 that First MCUSUM method proposed by Crosier, (1988). is the best methods.Second MCUSUM method proposed by Crosier, (1988) is the second method in performance based on the assumed performance criteria.First MCUSUM method proposed by Pignatiello and Runger (1990) is preferred to the other remained control charts.In general, we see that first MCUSUM proposed by Pignatiello and Runger (1990) is the best chart in detecting the large mean shifts.The methods proposed by Healy, (1987) and Crosier, (1988) are the best charts for detecting medium mean shifts, and second MCUSUM proposed by Pignatiello and Runger (1990) is the best chart for detecting small mean shifts.From above analysis, it is concluded that hypothesis H0 that is the equality of performance of different method is rejected in  level 0.05.

Shifts in the Process Standard Deviation
The results of Table (3) show that the first MCUSUM method proposed by Crosier, (1988) is the best method in detecting the shifts of the standard deviation.Also first MCUSUM proposed by Pignatiello and Runger (1990) is the second best chart in detecting the standard deviation shifts.Since this method coincides with a similar procedure that is proposed by Healy, (1987) for controlling process dispersion, it was expected that this method denotes the good performance in detecting standard deviation shifts

Shifts in the Process Correlation
The results of Table (4) show that the MCUSUM proposed by Healy, (1987) is sensitive to the positive shifts in the correlation.As can be seen in Table (4), other methods are not sensitive to the shifts in the correlation.
number of subgroups since the most recent renewal (i.e.zero value) of the CUSUM chart, control chart with an upper control limit of H (H is investigated by simulation).If i MC exceeds H, then the process is out of control.
normal observations.If we define the quality characteristics to be X and Y random variables, assuming 0.5   , at stage i of the data gathering process

Table 2 :
Ranking of the different methods in detecting mean shifts

Table 3 :
The results of ARL1 study for standard deviation shifts (bi-variate normal)

Table 4 :
The results of ARL1 study for correlation shifts (bi-variate normal)