An enhanced EWMA chart with variable sampling interval scheme for monitoring the exponential process with estimated parameter

Control charts have been used to monitor product manufacturing processes for decades. The exponential distribution is commonly used to fit data in research related to healthcare and product lifetime. This study proposes an exponentially weighted moving average control chart with a variable sampling interval scheme to monitor the exponential process, denoted as a VSIEWMA-exp chart. The performance measures are investigated using the Markov chain method. In addition, an algorithm to obtain the optimal parameters of the model is proposed. We compared the proposed control chart with other competitors, and the results showed that our proposed method outperformed other competitors. Finally, an illustrative example with the data concerning urinary tract infections is presented.


Structural design of the proposed control chart
Let X = {X 1 , X 2 , . . .} be a random variable following an exponential distribution with the scale parameter η , denoted as X i ∼ exp(η) , where i = 1, 2, • • • .The probability density function (pdf) of the exponential distribu- tion is This study uses the two-sided chart to design the proposed control chart.The null and alternative hypotheses are presented as follows: where η 0 and η 1 represent the scale parameter of the exponential distribution for the in-control and out-of-control states.Let δ = η 1 /η 0 , which represents the magnitude of the shift.0 < δ < 1 and δ > 1 represent downward and upward shifts, respectively.Let Y = X 1/3.6 ; the advantage of such transformation 30 is to make the data asymptoti- cally symmetric, thereby using symmetric control limits.Many studies have adopted this statistic 16,31,32 .It is evident that Y follows a Weibull distribution with a scale parameter of η 1/3.6 0 and a shape parameter of 3.6, denoted by Y ∼ Weibull η 1/3.6 0 , 3.6 .Next, the EWMA statistic is shown as follows: where is the smoothing parameter with a range of (0, 1] .This study employs the variable sampling interval (VSI) scheme, where h 1 and h 2 represent the longer and shorter sampling intervals, respectively.When samples are within the central region ( CR ), indicating a low risk of process shift, h 1 is utilized; conversely, when samples fall within the warning region ( WR ), indicating a higher shift risk, h 2 is employed, as shown in Fig. 1.Based on the control chart theory, the definitions of LCL , UCL , LWL , and UWL are as follows: and (1) η , x > 0, η > 0.
where µ 0 (Z) and σ 0 (Z) represent the mean and standard deviation of the statistic Z when the process is in- control, and K and W are model parameters to be optimized in section "Investigation of performance measures for the proposed scheme".Equation ( 2) can be equivalently written as a moving average of the current and past observations: where the initial value Z 0 is often taken to be the target value or the process mean.Then, if the Y i are independent and have a common standard deviation σ 0 (Y ) , we have Hence, Eqs.(3), (4), ( 5) and ( 6) are easily derived accordingly.and where µ 0 (Y ) and σ 0 (Y ) are equal to and η 0 = m j=1 X j /m is the estimated process parameter, where m denotes the number of samples for the in-control state.Note that as i increases, the term ( 1 − (1 − ) 2i ) converges to unity.Thus, these limits converge toward constant levels given as ( Step 1: Collect m in-control samples and estimate scale parameter η 0 .
Step 2: Calculate the control and warning limits based on the optimal model parameters obtained from the proposed optimization model later.
Step 3: Let i = i + 1 , draw a random sample X i with the sampling interval h 1 , and transform it to X i 1/3.6 .Then, calculate the statistic Z i .
Step 4: If Z i ∈ CR , go to Step 3. If Z i ∈ WR , go to Step 5. Otherwise, go to Step 6.
Step 5: Let i = i + 1 , draw a random sample X i with the sampling interval h 2 , and calculate the statistic Z i .Then, go to Step 4.
Step 6: If Z i ∈ AR , stop the process and eliminate assignable causes.

Optimization algorithm design for model enhancement
In this section, we propose an optimization algorithm to adjust the model parameters ( , K, W, h 1 , h 2 ) .Our goal is to enhance detection efficiency across various shift levels while ensuring the in-control average performance.
When the process parameter is known, Yeong et al. 28 proposed an optimization algorithm for optimizing model parameters.Inspired by Yeong et al. 28 , we propose an optimization algorithm for the scenario where the process parameter η 0 is unknown.Additionally, the sampling intervals ( h 1 and h 2 ) are not predetermined like Yeong et al. 28 , but obtained through model optimization.The optimal model parameters ( * , K * , W * , h 1 * , h 2 * ) are obtained as follows: The performance measures AATS 0 and AATS 1 correspond to in-control and out-of-control states, respec- tively, calculated using Eq.(20).AASI 0 represents the in-control average sampling interval, computed as AASI 0 = AASI(δ = 1) = CASI δ = 1| η 0 f (γ )dγ , where CASI δ = 1| η 0 = p 1 h 1 + p 2 h 2 denotes the con- ditional average sampling interval for the in-control state, with p 1 and p 2 representing the probabilities of using long and short sampling intervals ( h 1 and h 2 ).τ is the specified value of AATS 0 , set at 370.4 in this study, and h 0 is the given value of average sampling interval ( h 2 < h 0 < h 1 ).Without loss of generality, we set h 0 = 1.
Here are the steps for the model optimization algorithm we provide, please refer to the supplementary file for the corresponding R code.
Step 8: Terminate the loop and obtain the optimal model parameters * , K * , W * , h 1 * , h 2 * corresponding to the smallest AATS 1 .

Comparison of proposed and existing schemes
We present boxplots of CATS 0 for different values of m ( m = 50, 200 ) in Fig. 2. "unadjusted" refers to CATS 0 is calculated using model parameters based on the assumption of known η 0 , while "adjusted" indicates CATS 0 calculated using adjusted model parameters optimized through the optimization model detailed in section "Optimization algorithm design for model enhancement".Notably, "unadjusted" yields CATS 0 values mostly below 370.4,indicating a higher false alarm rate.Conversely, the "adjusted" scenario shows improved CATS 0 Subject to the constraints values.Therefore, the effect of the estimated parameter on CATS 0 was mitigated when using the optimal model parameters.

Optimal model parameters
(3) The AATS 1 values of the proposed VSIEWMA-exp chart are consistently lower than the AATS 0 value, demonstrating unbiasedness.However, both Shewhart-exp and VSIShewhart-exp charts exhibit bias, as evident from the AATS 1 curve in Fig. 3.
As we know, the mean of the transformed exponential data ( X 1/3.6 ) is u 0 = η 1/3.6 0 Ŵ(1 + 1/3.6) , and the standard deviation is σ 0 = η 1/3.6 0 Ŵ(1 + 2/3.6) − Ŵ 2 (1 + 1/3.6) .Based on the normality assumption with mean and standard deviation u 0 and σ 0 , we developed an EWMA-type control chart with a VSI scheme (VSIEWMA-nor).In the case of m = +∞ , we have obtained the optimal model parameters for the VSIEWMA- nor chart at different shift levels δ , and calculated the AATS 0 of the transformed data accordingly, denoted as AATS 0 _N , and display it in Table 5.We also provide the AATS 0 of the VSIEWMA-exp control chart, calculated based on the optimal model parameters from Table 4, denoted as AATS 0 _E , and display it in Table 5.It is observed that the AATS 0 _N is not equal to 370.4.Because the transformed data only approximates a normal distribution, not a normal distribution, it results in AATS 0 _N greater than 370.However, the AATS 0 of our proposed VSIEWMA-exp chart is 370.4,as the transformed data follows a Weibull distribution.Therefore, utilizing the proposed control chart is more reliable.

Implementation of the proposed schemes
In this section, a dataset of urinary tract infections (UTIs) is considered to demonstrate our proposed control chart, which is presented in Table 6 by Santiago et al. 31 .The purpose of this data is to monitor the changes in the infection rate of UTI, so the days in between discharge of males in nosocomial UTIs in patients is recorded.For more details, please refer to Santiago et al. 31 .It can be observed from Fig. 4 that this dataset exhibits significant characteristics of an exponential distribution.Moreover, the p-value of the K-S (Kolmogorov-Smirnov) test is 0.8112, indicating that this dataset follows the exponential distribution.Firstly, we use in-control Phase I data to estimate η 0 .Subsequently, we apply the optimization algorithm to obtain optimal model parameters for calculating the control and warning limits.Figure 5 displays the plotted points for the Phase I data.It can be observed that none of the control charts show any false alarms.It is consistent with Santiago et al. 31 , indicating that they can be utilized to monitor Phase II data.
Figure 6 shows the detection results that the process is out-of-control.Regarding detection capability, Shewhart-exp and VSIShewhart-exp control charts failed to detect shifts.The FSIEWMA-exp chart identifies only one out-of-control sample, while the proposed VSIEWMA-exp chart detects five samples.Regarding detection efficiency, only the first example falls within the central region, while the others all fall within the warning

Figure 1 .
Figure 1.The VSIEWMA-exp chart with action and warning control limits.

Figure 2 .
Figure 2. The distribution of CATS 0 for adjusted and unadjusted model parameters.( ATS 0 = 370.4).The green and blue boxplots correspond to scenarios where m = 50 and m = 200 , respectively.

Figure 3 .
Figure 3. Curves of AATS 1 .The vertical axis is logarithmic for the sake of comparison.

Figure 6 .
Figure 6.The VSIEWMA-exp control chart for phase II dataset.

Table 5 .
Performance of the transformed data based on VSIEWMA-nor chart.