
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

38575656
58675
10.1038/s41598-024-58675-7
Article
An enhanced EWMA chart with variable sampling interval scheme for monitoring the exponential process with estimated parameter
Bai Yajie
Chiang Jyun-You
Liu Wen Wenwen_LIU@foxmail.com

Mou Zhengcheng
https://ror.org/04ewct822 grid.443347.3 0000 0004 1761 2353 School of Statistics, Southwestern University of Finance and Economics, Chengdu, 611130 China
4 4 2024
4 4 2024
2024
14 795821 11 2023
2 4 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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.

Keywords

Exponential process
Estimated parameter
Exponentially weighted moving average
Variable sampling interval
Markov chain method
Optimization algorithm design
Subject terms

Engineering
Mathematics and computing
http://dx.doi.org/10.13039/501100012226 Fundamental Research Funds for the Central Universities JBK2201058 Chiang Jyun-You issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

The control chart is one of the important tools in statistical process control, mainly used to detect process shifts in the manufacturing process. Shewhart1 first proposed a control chart, which effectively detects large shifts. Subsequently, the memory control charts, such as the CUSUM2,3 and EWMA4,5 control charts, were developed to detect moderate and small shifts. Most recent studies have designed the charts based on the two assumptions. The first assumption is that the quality characteristic follows a normal distribution. The second assumption is that the in-control process parameter is known. When the normal distribution assumption is violated, it may lead to a high false alarm rate for the in-control process6. Therefore, many non-normal control charts have been proposed7–9. Additionally, in real applications, process parameters are often unknown and need to be estimated. Considering the impact of parameter estimation on control charts, some studies have focused on designing control charts with estimated parameters10,11.

In practical applications, the exponential distribution is usually used to fit non-normal distribution data, such as lifetimes or failure times of product, disease infection rate, etc.12,13. Consequently, many studies have been conducted on control charts for the exponential processes. For example, Xie et al.12 and Zhang et al.14 developed the Shewhart-type control charts for the exponential process. Besides, the CUSUM and EWMA-typed charts have been employed to monitor the exponential process15,16. However, most studies on monitoring the exponential processes focused on assuming the process parameter is known. In practice, the process parameter needs to be estimated, so the control chart with an estimated parameter should be designed. In addition, due to the asymmetry of the exponential distribution, the performance measure of the two-sided control chart is biased. The unbiasedness refers to the values of the in-control performance measure being consistently greater than that for the out-of-control state17. To avoid this bias, one-sided charts are used to monitor the exponential process18. However, as we know, a two-sided chart can display both upward and downward shifts on the same chart. Therefore, the two-sided chart is also necessary.

The control charts mentioned in the above literature are all standard fixed-parameter control charts. When sample size, sampling intervals, or other control chart components vary, the chart is referred to as the adaptive control chart. The adaptive control charts are superior to the standard fixed-parameter control charts in monitoring small and medium shifts19. Sabahno et al.20 categorized adaptive charts into four types: VSS (Variable Sample Sizes)21, VSI (Variable Sampling Intervals)22,23, VSSI (Variable Sample Sizes and Sampling Intervals)24, and VP (Variable Parameters, if all of the chart parameters are allowed to vary)25. In this paper, we consider using VSI scheme to construct the proposed control chart for the following reasons: (1) Aykroyd et al.26 highlighted the VSI scheme as a recent research hotspot through bibliometric analysis. (2) While much VSI research are based on the normal distributions27,28, some studies have extended its applicability to the non-normal distributions24,29. However, no studies have yet applied the VSI scheme to monitoring the exponential processes. (3) Similar to studies by Liu et al.16, Santiago et al.31, Aslam et al.32, etc., this research employs transformed the exponential data with a sample size of 1 as the quality characteristic to be monitored, rendering the VSI scheme suitable for use.

To sum up, the exponential distribution is essential in fitting skewed data. The process parameter is unknown and needs to be estimated in practice. Additionally, the performance measure is usually biased when the two-sided chart monitors skewed distributed data. Therefore, it is necessary to design an efficient control chart with unbiased properties and parameter estimation to monitor the exponential process. The main contributions of this study are as follows:Designed a two-sided VSI EWMA control chart to monitor the exponential process with unknown parameter.

Derived the transition probability matrix of the proposed VSIEWMA-exp control chart, enabling the Markov chain method to be used to calculate the performance measures of the control chart.

Taking inspiration from Yeong et al.28, we propose an optimization algorithm for unknown parameters. This algorithm aims to achieve optimal out-of-control detection efficiency at different shift levels while ensuring average in-control performance. Moreover, the performance measure of the optimized VSIEWMA-exp chart is unbiased.

The remainder of this paper is organized as follows: Section "Structural design of the proposed control chart" introduces the proposed VSIEWMA-exp chart control chart. In Section "Investigation of performance measures for the proposed scheme", performance measures are investigated using the Markov chain method. Section "Optimization algorithm design for model enhancement" introduces the optimization algorithm for adjusting model parameters. Additionally, a numerical comparison is presented in Section "Comparison of proposed and existing schemes". A real data on urinary tract infections is used to demonstrate the proposed control chart in Section "Implementation of the proposed schemes". Finally, the concluding remarks are given in Section "Summary remarks".

Structural design of the proposed control chart

Let X={X1,X2,⋯} be a random variable following an exponential distribution with the scale parameter η, denoted as Xi∼expη, where i=1,2,⋯. The probability density function (pdf) of the exponential distribution is1 fx=1ηe-xη,x>0,η>0.

This study uses the two-sided chart to design the proposed control chart. The null and alternative hypotheses are presented as follows:H0:η1=η0;H1:η1≠η0,

where η0 and η1 represent the scale parameter of the exponential distribution for the in-control and out-of-control states. Let δ=η1/η0, which represents the magnitude of the shift. 0<δ<1 and δ>1 represent downward and upward shifts, respectively. Let Y=X1/3.6; the advantage of such transformation30 is to make the data asymptotically symmetric, thereby using symmetric control limits. Many studies have adopted this statistic16,31,32. It is evident that Y follows a Weibull distribution with a scale parameter of η01/3.6 and a shape parameter of 3.6, denoted by Y∼Weibullη01/3.6,3.6. Next, the EWMA statistic is shown as follows:2 Zi=λYi+1-λZi-1,i=1,2,⋯,

where λ is the smoothing parameter with a range of (0,1]. This study employs the variable sampling interval (VSI) scheme, where h1 and h2 represent the longer and shorter sampling intervals, respectively. When samples are within the central region (CR), indicating a low risk of process shift, h1 is utilized; conversely, when samples fall within the warning region (WR), indicating a higher shift risk, h2 is employed, as shown in Fig. 1. Based on the control chart theory, the definitions of LCL, UCL, LWL, and UWL are as follows:LCL=μ0Z-Kσ0Z,

UCL=μ0Z+Kσ0Z,

LWL=μ0Z-Wσ0Z,

andUWL=μ0Z+Wσ0Z,

where μ0Z and σ0Z 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: Zi=λ∑j=0i-11-λjYi-j+1-λiZ0, where the initial value Z0 is often taken to be the target value or the process mean. Then, if the Yi are independent and have a common standard deviation σ0Y, we haveFigure 1 The VSIEWMA-exp chart with action and warning control limits.

μ0Z=μ0Y,

σ0Z=λ2-λ1-1-λ2iσ0Y.

Hence, Eqs. (3), (4), (5) and (6) are easily derived accordingly.3 LCLi=μ0Y-Kλ2-λ1-1-λ2iσ0Y,

4 UCLi=μ0Y+Kλ2-λ1-1-λ2iσ0Y,

5 LWLi=μ0Y-Wλ2-λ1-1-λ2iσ0Y,

and6 UWLi=μ0Y+Wλ2-λ1-1-λ2iσ0Y,

where μ0Y and σ0Y are equal to7 μ0Y=η^013.6Γ1+13.6,

and8 σ0Y=η^013.6Γ1+23.6-Γ21+13.6.

η^0=∑j=1mXj/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-λ2i) converges to unity. Thus, these limits converge toward constant levels given as9 LCL=η^013.6lcl,

10 UCL=η^013.6ucl,

11 LWL=η^013.6lwl,

and12 UWL=η^013.6uwl,

where13 lcl=Γ1+13.6-Kλ2-λΓ1+23.6-Γ21+13.6,

14 ucl=Γ1+13.6+Kλ2-λΓ1+23.6-Γ21+13.6,

15 lwl=Γ1+13.6-Wλ2-λΓ1+23.6-Γ21+13.6,

and16 uwl=Γ1+13.6+Wλ2-λΓ1+23.6-Γ21+13.6,

Next, the VSIEWMA-exp control chart works as follows:

Step 1: Collect m in-control samples and estimate scale parameter η0.

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 Xi with the sampling interval h1, and transform it to Xi1/3.6. Then, calculate the statistic Zi.

Step 4: If Zi∈CR, go to Step 3. If Zi∈WR, go to Step 5. Otherwise, go to Step 6.

Step 5: Let i=i+1, draw a random sample Xi with the sampling interval h2, and calculate the statistic Zi. Then, go to Step 4.

Step 6: If Zi∈AR, stop the process and eliminate assignable causes.

Investigation of performance measures for the proposed scheme

In this section, we evaluate the performance of the proposed chart using the conditional average time to signal (CATS), which depends on the estimated process parameter η^0. CATS is computed utilizing the Markov chain method33.

The in-control region LCL,UCL is divided into 2N+1 discrete subintervals. A larger N indicates a more accurate result. However, as N increases, the computation time also grows, and typically, 2N+1 is taken to be 50 or greater. Saccucci et al.34 consider 2N+1=83 to be sufficient. The width of the subinterval is d=(UCL-LCL)/(2N+1). Lj=LCL+(j-1)(UCL-LCL)/2N+1 and Uj=LCL+j(UCL-LCL)/2N+1 indicate the lower and upper bounds of the jth discrete subinterval, where j=1,...,2N+1. Let lj=lcl+(j-1)(ucl-lcl)/2N+1 and uj=lcl+j(ucl-lcl)/2N+1, then, we have Lj=η^01/3.6lj and Uj=η^01/3.6uj. Mj=Lj+Uj-Lj/2 represents the midpoint of the jth discrete subinterval. It can be rewritten as Mj=η^01/3.6lj+uj-lj/2=η^01/3.6mj. Naturally, the transition probability is equal to17 P^kjδ|η^0=PLj<Zi<Uj|Zi-1=Mk=PLj<λYi+1-λZi-1<Uj|Zi-1=Mk=Pη^0/η0δ13.6lj-1-λmkλ<Xiη113.6<η^0/η0δ13.6uj-1-λmkλ=FWBγδ13.6uj-1-λmkλ;1,3.6-FWBγδ13.6lj-1-λmkλ;1,3.6,

where FWB·;1,3.6 represents the Weibull distribution with the scale parameter of 1 and the shape parameter of 3.6. Here, γ=η^0/η0 follows a Gamma distribution, denoted by γ∼Gamma(m,1/m), where 1/m and m represent the corresponding scale and shape parameters, respectively. Then, the transition probability matrix of this Markov chain is given as follows:P^δ|η^0=Q^r^0′1

18 =p^0,0p^0,1⋯p^0,2N1-p^0,0-p^0,1-⋯-p^0,2Np^1,0p^1,1⋯p^1,2N1-p^1,0-p^1,1-⋯-p^1,2N⋮⋮⋮⋮⋮p^2N,0p^2N,1⋯p^2N,2N1-p^2N,0-p^2N,1-⋯-p^2N,2N00⋯01.

The submatrix Q^ is a (2N+1)×(2N+1) matrix of transition probabilities for the transient states, while r^ is a (2N+1)×1 vector satisfying r^=(1-Q^)1, where 1=(1,1,⋯,1)′. Inspired by Saccucci et al.22, we compute CATS as follows:19 CATSsδ|η^0=b′I-Q^δ|η^0-1h,s=0,δ=11,δ≠1,

where b is a (2N+1)×1 vector of initial probability, defined as b=1Zi∈[Lj,Uj]0Zi∉[Lj,Uj].

I is the identity matrix. h=(h1∗,h2∗,⋯h2N+1∗) is a (2N+1)×1 vector, where each element satisfies the following conditions: for Mk∈CR, hi∗=h1, and for Mk∈WR, hi∗=h2.

As we know, the CATS varies with η^0. Therefore, the unconditional measure AATS is required to assess the chart’s average performance. The AATS is calculated as follows:20 AATSsδ=∫CATSsδ|η^0fγdγ,s=0,δ=11,δ≠1.

Optimization algorithm design for model enhancement

In this section, we propose an optimization algorithm to adjust the model parameters (λ,K,W,h1,h2). 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 η0 is unknown. Additionally, the sampling intervals (h1 and h2) are not predetermined like Yeong et al.28, but obtained through model optimization. The optimal model parameters (λ∗,K∗,W∗,h1∗,h2∗) are obtained as follows:21 λ∗,K∗,W∗,h1∗,h2∗=argminλ,K,W,h1,h2,mAATS1λ,K,W,h1,h2,δ,mSubjecttotheconstraintsAATS0=τAASI0=h0,

The performance measures AATS0 and AATS1 correspond to in-control and out-of-control states, respectively, calculated using Eq. (20). AASI0 represents the in-control average sampling interval, computed as AASI0=AASIδ=1=∫CASIδ=1|η^0fγdγ, where CASIδ=1|η^0=p1h1+p2h2 denotes the conditional average sampling interval for the in-control state, with p1 and p2 representing the probabilities of using long and short sampling intervals (h1 and h2). τ is the specified value of AATS0, set at 370.4 in this study, and h0 is the given value of average sampling interval (h2<h0<h1). Without loss of generality, we set h0=1.

Here are the steps for the model optimization algorithm we provide, please refer to the supplementary file for the corresponding R code.

Step 1: Specify δ and m.

Step 2: Set h2=0.1, h1=h0+0.1.

Step 3: Set λ=0.03. Solve for K and W based on the constraints AATS0=τ and AASI0=h0.

Step 4: Compute AATS1 using Eq. (20).

Step 5: Increment λ by 0.01 while maintaining h1 and h2. Repeat Steps 3–4 until λ=1.

Step 6: Increment h1 by 0.1 while maintaining h2. Repeat Steps 3–5 until h1=2.5.

Step 7: Increment h2 by 0.1. Repeat Steps 3–6 until h2=0.9.

Step 8: Terminate the loop and obtain the optimal model parameters λ∗,K∗,W∗,h1∗,h2∗ corresponding to the smallest AATS1.

Comparison of proposed and existing schemes

We present boxplots of CATS0 for different values of m (m=50,200) in Fig. 2. “unadjusted” refers to CATS0 is calculated using model parameters based on the assumption of known η0, while “adjusted” indicates CATS0 calculated using adjusted model parameters optimized through the optimization model detailed in section "Optimization algorithm design for model enhancement". Notably, “unadjusted” yields CATS0 values mostly below 370.4, indicating a higher false alarm rate. Conversely, the “adjusted” scenario shows improved CATS0 values. Therefore, the effect of the estimated parameter on CATS0 was mitigated when using the optimal model parameters.Figure 2 The distribution of CATS0 for adjusted and unadjusted model parameters. (ATS0=370.4). The green and blue boxplots correspond to scenarios where m=50 and m=200, respectively.

To establish the superiority of the VSIEWMA-exp chart in shift detection, we conduct a numerical comparison with the three existing charts (Shewhart-exp, VSIShewhart-exp, and FSIEWMA-exp). The former two are Shewhart-type control charts using FSI and VSI schemes, respectively, while the latter is an EWMA-type control chart using the FSI scheme. Set m=50,200,500,+∞, N=100 and δ={0.9,0.8,0.6,0.5,0.4,0.2,0.1,1.1,1.25,1.5,2,2.5,5,10}, where {0.9,0.8,0.6,0.5,0.4,0.2,0.1} and {1.1,1.25,1.5,2,2.5,5,10} represent the downward and upward shifts, respectively. From Tables 1, 2, 3 and 4, we can conclude the following results:As m increases, the optimal model parameters and the AATS1 are both gradually converge between the two cases of unknown η0 (m<+∞) and known η0 (m=+∞). Additionally, when the shift level is large, the difference in AATS1 becomes small. This indicates that when m and shift levels are significant, the impact of parameter estimation on control chart performance is relatively small.

Across various combinations of (m,δ), the proposed VSIEWMA-exp chart consistently exhibits the smallest AATS1 values among competitive charts, suggesting superior sensitivity in detecting process shift.

The AATS1 values of the proposed VSIEWMA-exp chart are consistently lower than the AATS0 value, demonstrating unbiasedness. However, both Shewhart-exp and VSIShewhart-exp charts exhibit bias, as evident from the AATS1 curve in Fig. 3.

Table 1 The values of AATS1 with optimal model parameters for m=50 (in-control AATS0=370.4).

Optimal model parameters	m=50	
δ	λ∗	K∗	W∗	(h1∗,h2∗)	VSIEWMA-exp	FSIEWMA-exp	VSIShewhart-exp	Shewhart-exp	
0.9	0.03	2.5628	0.5071	(2.5, 0.1)	325.02	332.56	500.59	501.24	
0.8	0.03	2.5628	0.5071	(2.5, 0.1)	196.11	215.89	625.87	627.43	
0.6	0.03	2.5622	0.5573	(2.3, 0.1)	32.90	51.61	687.91	691.95	
0.5	0.06	2.7403	0.6218	(2.1, 0.1)	15.97	29.69	595.70	608.81	
0.4	0.11	2.8526	0.6227	(2.1, 0.1)	9.18	18.99	446.41	492.93	
0.2	0.26	2.9212	0.8070	(1.7, 0.1)	3.77	8.37	120.53	246.98	
0.1	0.40	2.9064	0.9542	(1.5, 0.1)	2.39	5.20	23.10	123.74	
1.1	1.00	2.7387	0.6218	(2.1,0.1)	258.51	262.85	258.51	262.85	
1.25	0.95	2.7428	0.6218	(2.1, 0.1)	148.36	155.15	148.65	155.50	
1.5	0.05	2.7014	0.5088	(2.5, 0.1)	44.04	57.18	65.30	71.73	
2	0.10	2.8368	0.7017	(1.9, 0.1)	12.33	18.50	20.87	24.70	
2.5	0.18	2.9071	0.9542	(1.5, 0.1)	7.07	10.49	10.44	12.87	
5	0.55	2.8629	1.3668	(1.2, 0.1)	2.64	3.33	2.91	3.54	
10	0.79	2.7801	1.6589	(1.1, 0.1)	1.64	1.85	1.67	1.87	

Table 2 The values of AATS1 with optimal model parameters for m=200 (in-control AATS0=370.4).

Optimal model parameters	m=200	
δ	λ∗	K∗	W∗	(h1∗,h2∗)	VSIEWMA-exp	FSIEWMA-exp	VSIShewhart-exp	Shewhart-exp	
0.9	0.03	2.3798	0.6108	(2.1, 0.1)	277.02	287.15	530.16	530.47	
0.8	0.03	2.3798	0.6108	(2.1, 0.1)	112.99	135.20	686.16	689.93	
0.6	0.05	2.5519	0.5537	(2.3, 0.1)	22.52	37.76	721.56	723.88	
0.5	0.08	2.6828	0.5846	(2.2, 0.1)	13.08	24.65	611.24	619.94	
0.4	0.12	2.7690	0.6547	(2.0, 0.1)	8.22	16.75	451.37	497.85	
0.2	0.28	2.8681	0.8673	(1.6, 0.1)	3.61	7.85	118.12	249.22	
0.1	0.42	2.8672	1.0467	(1.4, 0.1)	2.35	4.98	22.21	124.86	
1.1	0.94	2.7481	0.6545	(2.0, 0.1)	243.62	249.11	244.08	249.61	
1.25	0.03	2.3798	0.6108	(2.1, 0.1)	91.36	105.63	133.53	141.41	
1.5	0.05	2.5519	0.5537	(2.3, 0.1)	29.78	40.54	57.98	64.82	
2	0.13	2.7852	0.7446	(1.8, 0.1)	10.90	16.18	19.26	23.12	
2.5	0.19	2.8398	0.8670	(1.6, 0.1)	6.66	9.70	9.94	12.33	
5	0.55	2.8413	1.3561	(1.2, 0.1)	2.59	3.27	2.85	3.50	
10	0.81	2.7690	1.6458	(1.1, 0.1)	1.63	1.84	1.67	1.87	

Table 3 The values of AATS1 with optimal model parameters for m=500 (in-control AATS0=370.4).

Optimal model parameters	m=500	
δ	λ∗	K∗	W∗	(h1∗,h2∗)	VSIEWMA-exp	FSIEWMA-exp	VSIShewhart-exp	Shewhart-exp	
0.9	0.03	2.3133	0.5472	(2.3, 0.1)	249.56	260.16	538.91	539.24	
0.8	0.03	2.3133	0.5472	(2.3, 0.1)	93.12	114.42	704.57	705.43	
0.6	0.06	2.5604	0.5822	(2.2, 0.1)	21.37	35.47	731.06	733.43	
0.5	0.09	2.6751	0.5838	(2.2, 0.1)	12.65	23.58	617.23	625.20	
0.4	0.13	2.7567	0.6538	(2.0, 0.1)	8.06	16.22	455.11	501.61	
0.2	0.28	2.8545	0.8659	(1.6, 0.1)	3.61	7.72	118.37	251.08	
0.1	0.42	2.8588	1.0451	(1.4, 0.1)	2.33	4.92	22.17	125.79	
1.1	0.03	2.3178	0.6439	(2.0, 0.1)	227.73	235.42	240.80	246.54	
1.25	0.03	2.3178	0.6439	(2.0, 0.1)	78.08	91.33	130.63	138.63	
1.5	0.06	2.5604	0.5822	(2.2, 0.1)	27.86	37.87	56.77	63.61	
2	0.14	2.7724	0.7435	(1.8, 0.1)	10.65	15.71	19.01	22.85	
2.5	0.19	2.8204	0.8656	(1.6, 0.1)	6.55	9.53	9.85	12.25	
5	0.55	2.8361	1.3539	(1.2, 0.1)	2.58	3.26	2.85	3.50	
10	0.81	2.7685	1.6432	(1.1, 0.1)	1.63	1.84	1.67	1.87	

Table 4 The values of AATS1 with optimal model parameters for m=+∞ (in-control AATS0=370.4).

Optimal model parameters	m=+∞	
δ	λ∗	K∗	W∗	(h1∗,h2∗)	VSIEWMA-exp	FSIEWMA-exp	VSIShewhart-exp	Shewhart-exp	
0.9	0.03	2.2510	0.5730	(2.2, 0.1)	220.45	229.92	545.63	545.98	
0.8	0.03	2.2510	0.5730	(2.2, 0.1)	82.16	101.19	712.67	719.95	
0.6	0.06	2.5209	0.6497	(2.0, 0.1)	20.58	33.73	738.54	740.96	
0.5	0.09	2.6426	0.6519	(2.0, 0.1)	12.41	22.75	622.47	629.83	
0.4	0.14	2.7494	0.6531	(2.0, 0.1)	7.95	15.80	458.53	505.08	
0.2	0.28	2.8439	0.8650	(1.6, 0.1)	3.58	7.61	118.77	252.81	
0.1	0.42	2.8525	1.0440	(1.4, 0.1)	2.32	4.88	22.18	126.66	
1.1	0.03	2.2604	0.7824	(1.7, 0.1)	200.69	207.70	238.58	244.45	
1.25	0.03	2.2532	0.6406	(2.0, 0.1)	70.86	82.26	128.77	136.85	
1.5	0.06	2.5209	0.6497	(2.0, 0.1)	26.65	36.00	56.01	62.86	
2	0.16	2.7756	0.7428	(1.8, 0.1)	10.53	15.36	18.85	22.69	
2.5	0.19	2.8049	0.8647	(1.6, 0.1)	6.47	9.41	9.80	12.19	
5	0.54	2.8345	1.3525	(1.2, 0.1)	2.57	3.26	2.85	3.50	
10	0.81	2.7683	1.6415	(1.1, 0.1)	1.63	1.84	1.67	1.87	

Figure 3 Curves of AATS1. The vertical axis is logarithmic for the sake of comparison.

As we know, the mean of the transformed exponential data (X1/3.6) is u0=η^01/3.6Γ1+1/3.6, and the standard deviation is σ0=η^01/3.6Γ1+2/3.6-Γ21+1/3.6. Based on the normality assumption with mean and standard deviation u0 and σ0, 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 AATS0 of the transformed data accordingly, denoted as AATS0_N, and display it in Table 5. We also provide the AATS0 of the VSIEWMA-exp control chart, calculated based on the optimal model parameters from Table 4, denoted as AATS0_E, and display it in Table 5. It is observed that the AATS0_N is not equal to 370.4. Because the transformed data only approximates a normal distribution, not a normal distribution, it results in AATS0_N greater than 370. However, the AATS0 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.Table 5 Performance of the transformed data based on VSIEWMA-nor chart.

δ	λ∗	K∗	W∗	(h1∗,h2∗)	AATS0_N	AATS0_E	
0.9	0.03	2.2699	0.5822	(2.1, 0.1)	370.25	370.4	
0.8	0.03	2.2699	0.5822	(2.1, 0.1)	370.25	370.4	
0.6	0.05	2.4705	0.5575	(2.2, 0.1)	372.65	370.4	
0.5	0.09	2.6637	0.6280	(2.0, 0.1)	380.61	370.4	
0.4	0.13	2.7649	0.6295	(2.0, 0.1)	391.58	370.4	
0.2	0.27	2.9106	0.8378	(1.6, 0.1)	449.20	370.4	
0.1	0.37	2.9497	1.0154	(1.4, 0.1)	509.98	370.4	
1.1	0.03	2.2699	0.5822	(2.1, 0.1)	370.25	370.4	
1.25	0.03	2.2699	0.5822	(2.1, 0.1)	370.25	370.4	
1.5	0.06	2.5368	0.6255	(2.0, 0.1)	374.37	370.4	
2	0.14	2.7826	0.7169	(1.8, 0.1)	394.79	370.4	
2.5	0.22	2.8774	0.7726	(1.7, 0.1)	425.29	370.4	
5	0.44	2.9684	1.3277	(1.2, 0.1)	567.42	370.4	
10	0.68	2.9942	1.6330	(1.1, 0.1)	867.50	370.4	

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.Table 6 The urinary tract infections (UTIs) dataset.

Phase I	Phase II	
i	Samples	i	Samples	i	Samples	i	Samples	i	Samples	
1	0.57014	19	0.12014	37	0.27083	1	0.38231	19	0.30506	
2	0.07431	20	0.11458	38	0.04514	2	0.44270	20	0.25665	
3	0.15278	21	0.00347	39	0.13542	3	0.28831			
4	0.14583	22	0.12014	40	0.08681	4	0.18758			
5	0.13889	23	0.04861	41	0.40347	5	0.09674			
6	0.14931	24	0.02778	42	0.12639	6	0.20090			
7	0.03333	25	0.32639	43	0.18403	7	0.57208			
8	0.08681	26	0.64931	44	0.70833	8	0.49713			
9	0.33681	27	0.14931	45	0.15625	9	0.13790			
10	0.03819	28	0.01389	46	0.24653	10	0.31507			
11	0.24653	29	0.03819	47	0.04514	11	0.74995			
12	0.29514	30	0.46806	48	0.01736	12	0.09923			
13	0.11944	31	0.22222	49	1.08889	13	0.49534			
14	0.05208	32	0.29514	50	0.05208	14	0.68715			
15	0.12500	33	0.53472	51	0.02778	15	0.24903			
16	0.25000	34	0.15139	52	0.03472	16	0.22786			
17	0.40069	35	0.52569	53	0.23611	17	0.13581			
18	0.02500	36	0.07986	54	0.35972	18	0.63413			

Figure 4 Histogram of Urinary Tract Infections rate.

Firstly, we use in-control Phase I data to estimate η0. 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 5 The VSIEWMA-exp control chart for phase I dataset.

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 region. This suggests that a shorter sampling interval is employed after the first example. Consequently, the proposed VSIEWMA-exp chart takes less time to trigger an out-of-control signal than the FSIEWMA-exp chart.Figure 6 The VSIEWMA-exp control chart for phase II dataset.

Summary remarks

The purpose of this study is to propose a VSIEWMA-exp control chart. The theoretical method is to use the VSI scheme to construct an EWMA-type control chart for the exponential distribution and consider the situation of unknown parameter. In addition, we use the Markov chain approach to propose two performance measures for the proposed control chart, CATS and AATS. We also provide an optimization algorithm for model parameters. In Section "Comparison of proposed and existing schemes", we compared our proposed control chart with the other three control charts, and the results showed that our proposed control chart outperformed other competitors in most cases. Finally, the proposed control chart is demonstrated using the UTI data as an example, and the results showed that the proposed control chart has high efficiency in monitoring the Phase II data in this case.

It is worth noting that this study only used the VSI scheme. Future research can consider monitoring statistical data X¯ based on subgroup sample size. At this point, X¯ follows a gamma distribution, and EWMA control charts designed by VSS or VSI can be considered.

Supplementary Information

Supplementary Information.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-58675-7.

Acknowledgements

Dr Chiang’s work was supported by the Fundamental Research Funds for the Central Universities No. JBK2201058.

Author contributions

Y. J. B. conceived the study and wrote the manuscript. J.-Y. C. provided methodology utilized in conducting the study. W. L. carried out the experiments and provided R codes. Z. C. M. contributed to the data analysis.

Data availability

All data generated or analyzed during this study are included in this published article.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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