A variable sampling interval one-sided CUSUM control chart for monitoring the multivariate coefficient of variation

2.1 Basic properties of the sample MCV squared

Denote by $\left\{{\mathit{X}}_1,{\mathit{X}}_2\cdots ,{\mathit{X}}_{\mathit{n}}\right\}$ a random sample of size $n$ . Each ${\mathit{X}}_{\mathit{i}}$ is a p-dimensional vector from a multivariate normal distribution with mean parameter $\mathit{\mu }$ and covariance parameter $\mathrm{\Sigma }$ , i.e., ${\mathit{X}}_{\mathit{i}}\sim {\mathrm{N}}_{\mathit{p}}\left(\mathit{\mu },\mathrm{\Sigma }\right)$ , $i=1,2,3,\cdots ,n$ . The population MCV is then derived as

Then the sample MCV is

To monitor the MCV, the distribution of the MCV $\gamma$ or the MCV squared ( ${\widehat{\gamma }}^2$ ) is of interest. It has been derived that the cumulative distribution function (c.d.f.) of ${\widehat{\gamma }}^2$ is

2.2 The existing one-sided VSI EWMA MCV charts

Considering that the Shewhart chart focuses only on the latest sample information, a one-sided MCV chart based on the EWMA scheme was proposed in Giner-Bosch et al. (2019). The VSI feature is then incorporated by Ayyoub et al. (2022) to further enhance the EWMA MCV chart.. In both studies, only upward shifts were considered, while downward shifts were neglected. Motivated by this, Nguyen et al. (2021) suggested two one-sided VSI EWMA MCV charts with a “restart state” to overcome the “inertia problem” in the previous VSI EWMA scheme. The construction methods of their charts are described as follows.

The charting statistic for the upward EWMA MCV chart is

Furthermore, the upper warning limit (UWL) and lower warning limit (LWL) are defined in Nguyen et al. (2021) as

2.3 The existing one-sided CUSUM MCV charts

To make use of the advantage of the CUSUM control chart in detecting small shifts in univariate CV, Hu et al. (2023) have suggested two one-sided MCV charts based on the CUSUM scheme. The charting methods are as follows.

The charting statistic for the upward CUSUM chart is

The charting statistic for the downward CUSUM chart is

3.1 The proposed one-sided VSI CUSUM MCV charts

The above CUSUM MCV charts proposed in Hu et al. (2023) belongs to the non-adaptive type charts, which adopt a fixed sampling interval (FSI) strategy. To be consistent with the VSI EWMA MCV charts reviewed in Section 2.2, the VSI CUSUM MCV control charts are constructed by adapting the fixed sampling interval in the original FSI chart into the variable version $h$ , while the UCL and the LCL remains the same. The VSI $h$ of the proposed VSI CUSUM MCV control charts changes at both $h_S$ and $h_L$ levels for different values of the current statistic $C_t^{+}$ or $C_t^{-}$ , where $h_S<h_L$ . To implement the VSI CUSUM MCV chart, it is necessary to introduce new warning limits (UWL and LWL), which are defined as follows:

• When the monitored statistic $C_t^{+}$ or $C_t^{-}$ falls into the safe region, the process is declared as IC and a long sampling interval $h_L$ is adopted to draw the next sample.

• When the monitored statistic $C_t^{+}$ or $C_t^{-}$ falls into the warning region, the process is IC but with a high risk of being OOC. Hence, the next sample interval it set to be short, say $h_S$ , to detect potential assignable causes more quickly.

• When the monitored statistic $C_t^{+}$ or $C_t^{-}$ falls into the OOC region, the process is declared as OOC and prompt actions should be taken to locate and remove assignable causes quickly.

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Fig. 1

The region division of the upward VSI CUSUM MCV control chart.

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3.2 Performance measures

Traditionally, in order to evaluate the performance of FSI type charts, criteria based on the run length distribution is widely used. Among these criteria, the average run length (ARL) is the most commonly adopted. For VSI type charts, the time between two successive samples varies. Therefore, ARL is not applicable. To evaluate the properties of VSI type control charts,. the ATS and standard deviation of time to signal (SDTS) are suggested. ATS indicates the expected time to the appearance of an OOC signal since the process monitoring is started, and SDTS measures the variability of the time to signal. A well-performed control chart is expected to falsely alarm at a low rate for an IC process and to trigger a signal as fast as possible for an OOC process. Therefore, a large IC ATS ( ${\mathit{ATS}}_0$ ) and a small OOC ATS ( ${\mathit{ATS}}_1$ ) are considered to be the indication of excellent chart performance.

For FSI type charts, the ATS is represented as

For VSI type charts, the ATS is computed as

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Fig. 2

Sub-division of the IC range.

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By definition, $H_j$ , $j=0,1,\cdots ,s$ , denotes the midpoint of the $j^{\mathrm{t}\mathrm{h}}$ subinterval $\left(H_j-\delta \right.,\left.H_j+\delta \right]$ . The width of each subinterval is $2\delta$ and decreases with $s$ since $\delta =UCL/2s$ . When $s$ is sufficiently large, the subinterval becomes so narrow that all the values in the $j^{\mathrm{t}\mathrm{h}}$ subinterval can be approximated as $H_j$ . When $j=0,$ $H_0=0$ indicates the proposed charts to “restart”. The transition probability $Q_{i,j},i,j=0,1,\cdots ,s$ from state $i$ to state $j$ is provided as follows.

• For the upward chart,

• For the downward chart,

From Eqs. (17) and (21), the expected sampling interval $E\left(h\right)$ is obtained as:

The denominator in Eq. (23) is the formula for ARL as in Brook and Evans (1972). To make fair comparison, the same IC values of ${\mathit{ATS}}_0$ and $E_0\left(h\right)$ are imposed for both FSI and VSI types of charts. Then, the OOC metrics ${\mathit{ATS}}_1$ and $E_1\left(h\right)$ are evaluated for the comparison purpose. Without the loss of generality, we assumed that $h_F=1$ time unit, which leads to ${\mathit{ATS}}_0^{\mathit{FSI}}={\mathit{ARL}}_0$ by plugging $h_F$ into Eq. (16).

3.3 Optimization algorithm

This section provides an optimization algorithm for minimizing ${\mathit{ATS}}_1$ and ${\mathit{EATS}}_1$ of the VSI CUSUM MCV charts. For the known shift size case, Section 3.3.1 presents an optimization model to determine the chart parameters that minimizes the ${\mathit{ATS}}_1$ of the proposed control charts. For the unknown shift size case, the EATS measure is proposed and the corresponding optimization program is designed in Section 3.3.2.

4.1 Optimal triples of the VSI CUSUM MCV charts

The optimal triples $\left(K_D^{\ast },H_D^{\ast },h_L^{\ast }\right)$ when $\tau =\left\{0.5,0.75,0.9\right\}$ , and $\left(K_U^{\ast },H_U^{\ast },h_L^{\ast }\right)$ when $\tau =\left\{1.1,1.25,1.5\right\}$ are presented in the Table 1 for $n=\left\{10,15\right\}$ , $p=2$ , ${\gamma }_0=\left\{0.1,0.2,0.3,0.5\right\}$ and $W=\left\{0.1,0.6,0.9\right\}$ . For the sake of brevity, similar tables for $p=\left\{3,4,5\right\}$ are presented in Table A1–A3 in the supplementary material. From these tables, the following conclusions are drawn.

• In general, given ${\gamma }_0$ , $n$ , $p$ and $W_U\left(W_D\right)$ , the values of $\left(K_D^{\ast },H_D^{\ast },h_L^{\ast }\right)$ and $\left(K_U^{\ast },H_U^{\ast },h_L^{\ast }\right)$ vary with ${\mathrm{\Delta }}_{\tau }=\left|\tau -1\right|$ . Specifically, $K_U^{\ast }\left(K_D^{\ast }\right)$ decrease and $H_U^{\ast }\left(H_D^{\ast }\right)$ , $h_L^{\ast }$ increase as ${\mathrm{\Delta }}_{\tau }$ decreases. Considering the setting $n=10$ , $p=2$ , ${\gamma }_0=0.1$ and $W_U=0.1$ , the optimal triples $\left(K_U^{\ast },H_U^{\ast },h_L^{\ast }\right)$ for the VSI CUSUM MCV control chart are $(K_U^{\ast }=0.872,H_U^{\ast }=3.773,h_L^{\ast }=1.25)$ if ${\mathrm{\Delta }}_{\tau }=0.5$ and these values change to $(K_U^{\ast }=0.191,H_U^{\ast }=8.588,h_L^{\ast }=2.83)$ if ${\mathrm{\Delta }}_{\tau }=0.1$ (Table 1).

• In general, given $\tau$ , ${\gamma }_0$ , $p$ and $W_U\left(W_D\right)$ , the values of $\left(K_D^{\ast },H_D^{\ast },h_L^{\ast }\right)$ and $\left(K_U^{\ast },H_U^{\ast },h_L^{\ast }\right)$ vary with $n$ . Specifically, the values of $H_U^{\ast }\left(H_D^{\ast }\right)$ and $h_L^{\ast }$ increase and the values of $K_U^{\ast }\left(K_D^{\ast }\right)$ decrease as $n$ decreases. In the case that $\tau =1.1$ , $p=2$ , ${\gamma }_0=0.1$ and $W_U=0.1$ , the optimal triples $\left(K_U^{\ast },H_U^{\ast },h_L^{\ast }\right)$ for the VSI CUSUM MCV control chart are $(K_U^{\ast }=0.191,H_U^{\ast }=8.588,h_L^{\ast }=2.83)$ if $n=10$ and these values change to $(K_U^{\ast }=0.241,H_U^{\ast }=7.516,h_L^{\ast }=2.46)$ if $n=15$ (Table 1).

• In general, given $\tau$ , ${\gamma }_0$ , $n$ and $W_U\left(W_D\right)$ , the values of $\left(K_D^{\ast },H_D^{\ast },h_L^{\ast }\right)$ and $\left(K_U^{\ast },H_U^{\ast },h_L^{\ast }\right)$ vary with $p$ . Specifically, the values of $H_U^{\ast }\left(H_D^{\ast }\right)$ and $h_L^{\ast }$ decrease and the values of $K_U^{\ast }\left(K_D^{\ast }\right)$ increase as $p$ decreases. For instance, when $n=10$ , $\tau =1.1$ , ${\gamma }_0=0.1$ and $W_U=0.1$ , the optimal triples $\left(K_U^{\ast },H_U^{\ast },h_L^{\ast }\right)$ for the VSI CUSUM MCV control chart are $(K_U^{\ast }=0.191,H_U^{\ast }=8.588,h_L^{\ast }=2.83)$ if $p=2$ and these values are $(K_U^{\ast }=0.150,H_U^{\ast }=9.665,h_L^{\ast }=3.16)$ if $p=5$ (Table 1 and Table A1 in the supplementary material).

• In general, given $\tau$ , ${\gamma }_0$ , $n$ and $p$ , the values of $\left(K_D^{\ast },H_D^{\ast },h_L^{\ast }\right)$ and $\left(K_U^{\ast },H_U^{\ast },h_L^{\ast }\right)$ vary with $W$ . Specifically, the values of $K_U^{\ast }\left(K_D^{\ast }\right)$ and $h_L^{\ast }$ decrease and the values of $H_U^{\ast }\left(H_D^{\ast }\right)$ increase as $W$ increases. For instance, when $n=10$ , $\tau =1.1$ , ${\gamma }_0=0.1$ and $p=2$ , the optimal triples $\left(K_U^{\ast },H_U^{\ast },h_L^{\ast }\right)$ for the VSI CUSUM MCV control chart are $(K_U^{\ast }=0.191,H_U^{\ast }=8.588,h_L^{\ast }=2.83)$ if $W=0.1$ and $(K_U^{\ast }=0.162,H_U^{\ast }=9.231,h_L^{\ast }=2.05)$ if $W=0.9$ .

• In general, given $\tau$ , $p$ , $n$ and $W_U\left(W_D\right)$ , the IC ${\gamma }_0$ has a slight effect on the values of $\left(K_D^{\ast },H_D^{\ast },h_L^{\ast }\right)$ and $\left(K_U^{\ast },H_U^{\ast },h_L^{\ast }\right)$ . For instance, when $n=10$ , $\tau =1.1$ , $W=0.1$ and $p=2$ , the optimal triples $\left(K_U^{\ast },H_U^{\ast },h_L^{\ast }\right)$ for the VSI CUSUM MCV control chart are $(K_U^{\ast }=0.191,H_U^{\ast }=8.588,h_L^{\ast }=2.83)$ if ${\gamma }_0=0.1$ and $(K_U^{\ast }=0.142,H_U^{\ast }=10.116,h_L^{\ast }=3.03)$ if ${\gamma }_0=0.5$ . That is, an increase of ${\gamma }_0$ leads to a slight increase in $H_U^{\ast }\left(H_D^{\ast }\right)$ and $h_L^{\ast }$ and a slight decrease in $K_U^{\ast }\left(K_D^{\ast }\right)$ .

4.2 OOC performance of the VSI CUSUM MCV charts

Table 2 presents the ${\mathit{ATS}}_1$ values of the three competing charts for ${\gamma }_0=\left\{0.1,0.2,0.3,0.5\right\}$ , $p=2$ , $\tau =\left\{0.5,0.75,0.9,1.1,1.25,1.5\right\}$ , $W_U\left(W_D\right)=\left\{0.1,0.6,0.9\right\}$ and $n=\left\{10,15\right\}$ . For the sake of brevity, similar tables for $p=\left\{3,4,5\right\}$ are presented in Tables B1–B3 in the supplementary material. Conclusions are presented as follows (the SDRL values are also commented on in the paper even if they are not shown in these tables):

• Given the values of $\tau$ , ${\gamma }_0$ , $p$ and $W_U\left(W_D\right)$ , performance of the VSI CUSUM MCV charts is largely influenced by the sample size $n$ . For instance, when $\tau =1.1$ , $p=2$ , ${\gamma }_0=0.1$ and $W_U=0.1$ , the $({\mathit{ATS}}_1,{\mathit{SDTS}}_1)$ values of the upward VSI CUSUM MCV chart for $n=\left\{10,15\right\}$ are (16.68, 13.45) and (11.58, 8.95), respectively (see Table 2).

• Given the values of $n$ , $\tau$ , $p$ and $W_U\left(W_D\right)$ , an increase in the value of the IC ${\gamma }_0$ negatively impact the performance of the VSI CUSUM MCV charts. For instance, with $n=10$ , $\tau =1.1$ , $W_U=0.1$ and $p=2$ , the ${\mathit{ATS}}_1$ value of the VSI CUSUM MCV chart increases from 16.68 to 17.35 when the value of ${\gamma }_0$ increases from 0.1 up to 0.2 (see Table 2).

• Given the values of $\tau$ , ${\gamma }_0$ , $n$ and $W_U\left(W_D\right)$ , the detection efficiency of the VSI CUSUM MCV charts is enhanced when the dimension $p$ has an increasing trend. In the scenario that $n=10$ , $\tau =1.1$ , ${\gamma }_0=0.1$ and $W_U=0.1$ , we have ${\mathit{ATS}}_1=16.68$ and ${\mathit{SDTS}}_1=13.45$ if $p=2$ , and we have ${\mathit{ATS}}_1=18.43$ and ${\mathit{SDTS}}_1=15.03$ if $p=3$ (see Table 2 and Table B1 in the supplementary material).

• Given the values of $\tau$ , ${\gamma }_0$ , $n$ and $p$ , the warning limit coefficient $W_U\left(W_D\right)$ slightly affected the OOC performance of the VSI CUSUM MCV charts. For instance, when $n=10$ , $\tau =1.1$ , ${\gamma }_0=0.1$ and $p=2$ , we have ${\mathit{ATS}}_1=16.68$ and ${\mathit{SDTS}}_1=13.45$ if $W_U=0.1$ , and we have ${\mathit{ATS}}_1=17.34$ and ${\mathit{SDTS}}_1=14.09$ if $W_U=0.9$ (see Table 2).

4.3 Comparisons with some existing MCV control charts

Control charts for comparison include the VSI CUSUM MCV charts, the corresponding FSI CUSUM MCV control charts proposed by Hu et al. (2023) and VSI EWMA MCV charts designed by Nguyen et al. (2021). The results from numerical experiments show the superiority of the VSI CUSUM MCV charts over the competing ones.