A Comparison of Shiryaev–Roberts and Cumulative Sum Procedures for Bivariate ZIP Process Monitoring

Abstract

The Zero-inflated Poisson (ZIP) model has been widely adopted in quality control for near-zero-defect processes with sporadic nonconformities, while prior research predominantly focused on univariate ZIP processes for single defect types, modern high-precision manufacturing environments often involve two or more correlated defect categories. This paper proposes a Shiryaev–Roberts (SR) control chart for detecting parameter shifts in bivariate ZIP processes and compares its performance with the existing cumulative sum (CUSUM) chart under both zero-state and steady-state scenarios. Through Monte Carlo simulations, we establish the upper control limits (h) for both schemes while maintaining an identical in-control average run length (ARL) to ensure fair comparisons. Numerical simulations demonstrate that the SR chart exhibits superior out-of-control ARL performance compared to the CUSUM chart under steady-state conditions for detecting small to moderate shifts. The proposed methodology is further validated through a case study in LED packaging, highlighting its practicality in advanced manufacturing quality assurance.

1. Introduction

Statistical Process Monitoring (SPM) encompasses a set of statistical methods for monitoring and controlling process behavior. These techniques are now ubiquitous, spanning applications from item-quality inspections in manufacturing to healthcare surveillance. Control schemes play a critical role across diverse fields—including manufacturing, healthcare, insurance, environmental monitoring, services, and network communication—by enabling process monitoring and change detection. High-quality processes (e.g., near zero-defect manufacturing) and health-related processes frequently generate count data with excessive zero counts [1,2]. When defects occur, they typically follow Poisson or negative binomial distributions [3]. The Zero-Inflated Poisson (ZIP) model, which extends the standard Poisson distribution, is employed to model the zero-heavy count data characteristic of high-quality processes.

Substantial research has been reported on developing appropriate control procedures that were used to monitor high-quality processes. Lambert [4] introduced the ZIP regression model, providing a robust framework for analyzing count data characterized by an excess of zero observations. Subsequently, the ZIP process was extensively studied by researchers [5,6,7]. Xie and Goh [2] and Xie et al. [8] introduced ZIP-based control schemes for monitoring near-zero-defect processes under random shocks. Conventional control procedures comprise Shewhart, cumulative sum (CUSUM), and exponentially weighted moving average (EWMA) charts. The Shewhart scheme relies solely on current sample information, rendering it memoryless, while maintaining effectiveness for detecting large shifts. Sim and Lim [9] developed a control scheme based on the combined use of the Shewhart $np$-chart and c-chart for monitoring ZIP-distributed data. Additionally, the performance of an upper one-sided Shewhart chart for monitoring ZIP processes with estimated parameters was investigated by Mamzeridou and Rakitzis [10].

Furthermore, the CUSUM and EWMA charts exhibit superior sensitivity to small and moderate shifts; while the CUSUM employs uniform weighting of past observations, the EWMA applies varying weights to past observations. Considerable research has been conducted on CUSUM charts and EWMA charts for the ZIP model. Chang and Gan [11] developed a CUSUM chart to monitor high-yield processes. He et al. [12] investigated two CUSUM schemes for monitoring ZIP processes: a combined scheme employing two separate charts to monitor increases in the two parameters of the ZIP process and a single chart designed to detect simultaneous shifts in both parameters. Then, He et al. [13] proposed a combined CUSUM scheme, comprising a conforming run length (RL) chart and a zero-truncated Poisson CUSUM chart, to detect increases in the parameters of the ZIP process.

Meanwhile, researchers have also explored the EWMA chart for the ZIP model. Fatahi et al. [14] introduced a ZIP-EWMA chart for monitoring rare health-related events, demonstrating its superiority over Shewhart-type charts. Aly et al. [15] demonstrated that their proposed adaptive EWMA chart for ZIP processes achieved superior performance over the standard ZIP-EWMA chart under conditions of increasing zero-inflation. An upper-sided ZIP-EWMA chart featuring a time-varying control limit was developed by Alevizakos and Koukouvinos [16], along with the introduction of a double EWMA (DEWMA) scheme for monitoring ZIP distributions (regarded as ZIP-DEWMA chart). Furthermore, Alevizakos and Koukouvinos [17] constructed a one-sided generally weighted moving average (GWMA) chart, termed the ZIP-GWMA chart, for detecting shifts in individual or both parameters of a ZIP process. Their study demonstrated that with parameters $q=0.95$ and $\alpha$ taking values within the interval from 0.5 to 0.7, the proposed chart outperformed the ZIP-EWMA, ZIP-DEWMA, and t-CUSUM charts when both parameters shifted. Hu and Liu [18] presented an upper-sided EWMA control chart based on a weighted score test statistic, which demonstrated its superiority over the existing ZIP control chart based on Pearson residuals for detecting small to moderate shifts. Lai et al. [19] proposed a residuals-based EWMA control chart with risk adjustments for the ZIP model to detect small shifts in zero-inflated data. Furthermore, Lai et al. [20] developed an EWMA control procedure based on the generalized likelihood ratio to better detect the risk-adjusted ZIP process. Their method was demonstrated to efficiently detect random shifts in both process parameters. Additionally, to monitor the variability of the random effects variance component in a ZIP mixed-effects model, Zhao et al. [21] derived a score test statistic based on the generalized Henderson’s joint likelihood function and subsequently constructed a risk-adjusted EWMA control chart. Then, Alevizakos et al. [22] presented a double GWMA (DGWMA) control chart for monitoring ZIP processes (regarded as ZIP-DGWMA chart) and revealed that the ZIP-DGWMA chart is more effective for shifts in the zero-inflated parameter p and small shifts in both of the zero-inflated and Poisson parameter.

Moreover, Chen et al. [23] developed a control procedure for monitoring attribute data using a generalized ZIP model. Subsequently, Mukherjee and Rakitzis [24] proposed progressive monitoring schemes based on Wald’s statistic, max-type statistic, and likelihood ratio statistic for ZIP processes. Their comparative analysis with He et al. [12] demonstrated that the Wald and likelihood ratio-based schemes achieved superior out-of-control (OOC) average run length (ARL) performance relative to the CUSUM approach. Then, Hassan and Aly [25] extended three of the existing multivariate EWMA, Hotelling’s T², and EWMA-R methods for phase II monitoring to the case of ZIP profiles. However, these studies focused exclusively on the scenario involving only a single type of defect.

In high-quality manufacturing environments, multiple defect types frequently arise from diverse equipment failure modes. This inherent complexity has driven recent research interest in multivariate zero-inflated models and their SPM applications. Li et al. [3] contributed to the development of multivariate ZIP models by proposing construction and parameter estimation methodologies. Based on the copula function approach, Fatahi et al. [26] utilized a copula approach to construct a joint distribution for correlated ZIP variables, thereby developing a bivariate ZIP (BZIP) control procedure for monitoring correlated rare events. Building on the multivariate framework of Li et al. [3], Pal and Gauri [27] extended this work by proposing a bivariate attribute control scheme for BZIP processes, with simulations confirming its effectiveness in detecting parameter shifts. However, the above-mentioned research is all based on copula function approaches. Following Li et al. [3], He et al. [28] developed a CUSUM chart procedure for monitoring BZIP processes, employing a p-CUSUM chart for shifts in the p-set parameters; a $\lambda$-CUSUM chart for shifts in the $\lambda$-set parameters; and a combined scheme for simultaneous monitoring of both parameter sets.

The Shiryaev–Roberts (SR) procedure, initially formulated by Shiryaev [29] for Brownian motion, utilizes a conditional average delay time criterion as its statistical foundation. Subsequent research has expanded its scope and compared its performance with other control schemes. For instance, a performance comparison between CUSUM and SR procedures for mean shift detection in Gaussian processes was conducted by Pollak and Siegmund [30] under conditions of an unknown in-control (IC) mean. Srivastava and Wu [31] evaluated the relative performance of EWMA, CUSUM, and SR control charts in monitoring shifts in the process mean. Kenett and Pollak [32] extended its application to non-homogeneous Poisson distribution surveillance. Methodological extensions include the simultaneous monitoring of process mean and variability by Zhang et al. [33], as well as adaptive SR procedures for mean and variance proposed by Zhang et al. [34] and Zhang et al. [35]. Performance evaluation has been advanced through numerical approximations of integral equations by Moustakides et al. [36]. Recently, Yu et al. [37] extended the study of Zhang et al. [33] and therefore developed a one-sided SR scheme for monitoring the scale parameter of the Weibull with Type-I censored data. Given that the shape parameter of the distribution directly controls the hazard rate, Yu et al. [38] discussed an SR type scheme for monitoring the shape parameter with a fixed scale parameter when using Type-II right-censored Weibull lifetime data.

Motivated by the SR procedure, we develop an SR-based monitoring scheme for the parameters of the BZIP model and compare the performance of the CUSUM chart and SR chart using the ARL. This study advances BZIP process monitoring through three key innovations: (1) Although the SR procedure has been successfully applied in various monitoring settings, no existing work has applied it to BZIP processes. This paper is the first to develop an SR-based monitoring framework specifically designed for the BZIP model. (2) Formulation of a joint monitoring framework enabling simultaneous tracking of all BZIP parameters, including the zero-inflation probability and the Poisson rate parameters. (3) While previous SR-based control charts have mainly been developed for continuous random variables, this paper is the first to extend the SR method to bivariate discrete random variables.

The remainder of this paper is structured as follows. Section 2 reviews fundamental concepts of the BZIP model and the relevant literature. Section 3 develops the proposed SR monitoring methodology for BZIP processes. A systematic comparison with the conventional CUSUM approach is presented in Section 4. Section 5 studies the effect of parameter estimation on the SR chart. Section 6 demonstrates practical implementation through a real-world case study. Concluding remarks and research perspectives are provided in Section 7.

2. Background and Existing Research

This section first provides a systematic overview of the BZIP model. It then reviews the CUSUM control chart method, as documented in the existing literature, for monitoring the parameters of this model.

2.1. Bivariate Zero-Inflated Poisson Distribution

The ZIP model, proposed by Cohen [39], is defined as a mixture of a Poisson distribution and a degenerate distribution at zero. It is suitable for analyzing data with an excess of zero observations. The model assumes that random shocks, which cause nonconformities in a product, occur with probability p. If a shock occurs, the number of defects Y follows a Poisson distribution with parameter $\lambda$; otherwise, $Y=0$. Therefore, based on Chen et al. [23], the probability mass function is given by the following equation:

$$P(Y=y)=\left\{\begin{array}{c}(1-p)+p{e}^{-\lambda }(y=0)\\ p{\displaystyle \frac{{\lambda }^y{e}^{-\lambda }}{y!}}(y=1,2,\cdots ),\end{array}\right.$$

where $0\le p\le 1$ and $\lambda >0$.

The BZIP model, formulated by Li et al. [3], is a mixture distribution comprising a bivariate Poisson component, two univariate Poisson components, and a point mass at $(0,0)$. Then

$$(Y_1,Y_2)\sim (0,0)withprobability{p}_{00},$$

(1)

$$\sim (Poisson\left({\lambda }_1\right),0)withprobability{p}_{10},$$

(2)

$$\sim (0,Poisson\left({\lambda }_2\right))withprobability{p}_{01},$$

(3)

$$\hspace{0.33em}\hspace{0.33em}\sim bivariatePoisson({\lambda }_{10,},{\lambda }_{20},{\lambda }_{00})withprobability{p}_{11},$$

(4)

where $0\le p_{00},p_{10},p_{01},p_{11}\le 1$, and $p_{00}+p_{10}+p_{01}+p_{11}=1$.

A bivariate Poisson distribution $(Y_1,Y_2)$ with parameters $({\lambda }_{10},{\lambda }_{20},{\lambda }_{00})$ can be represented as follows [40]:

$$Y_1=X_1+U,Y_2=X_2+U,$$

where $X_1$, $X_2$, and U are independent Poisson random variables with parameters ${\lambda }_{10}$, ${\lambda }_{20}$, and ${\lambda }_{00}$, respectively, and ${\lambda }_{10},{\lambda }_{20},{\lambda }_{00}>0$. As introduced by Marshall and Olkin [40], the covariance and non-negative correlation coefficient between $Y_1$ and $Y_2$ are

$$\begin{array}{cc}\hfill cov(Y_1,Y_2)& ={\lambda }_{00},\hfill \\ \hfill corr(Y_1,Y_2)& ={\displaystyle \frac{{\lambda }_{00}}{\sqrt{({\lambda }_{10}+{\lambda }_{00})({\lambda }_{20}+{\lambda }_{00})}}}.\hfill \end{array}$$

It follows that a non-zero ${\lambda }_{00}$ indicates correlation between the two count variables, thus precluding the representation of the BZIP data as independent univariate ZIP processes.

Li et al. [3] proposed the following probability distribution:

$$\begin{array}{cc}\hfill P(Y_1=0,Y_2=0)& =p_{00}+p_{10}{e}^{-{\lambda }_1}+p_{01}{e}^{-{\lambda }_2}+p_{11}{e}^{-\lambda },\hfill \\ \hfill P(Y_1=y_1,Y_2=0)& ={\displaystyle \frac{p_{10}{\lambda }_1^{y_1}{e}^{-{\lambda }_1}+p_{11}{\lambda }_{10}^{y_1}{e}^{-\lambda }}{y_1!}},\hfill \\ \hfill P(Y_1=0,Y_2=y_2)& ={\displaystyle \frac{p_{01}{\lambda }_2^{y_2}{e}^{-{\lambda }_2}+p_{11}{\lambda }_{20}^{y_2}{e}^{-\lambda }}{y_2!}},\hfill \\ \hfill P(Y_1=y_1,Y_2=y_2)& =p_{11}{\displaystyle \sum _{j=0}^{min(y_1,y_2)}}{\displaystyle \frac{{\lambda }_{10}^{y_1-j}{\lambda }_{20}^{y_2-j}{\lambda }_{00}^j{e}^{-\lambda }}{(y_1-j)!(y_2-j)!j!}},\hfill \end{array}$$

for $y_1,y_2=1,2,\cdots$, where $\lambda ={\lambda }_{10}+{\lambda }_{20}+{\lambda }_{00}$.

The parameters ${\lambda }_1$ and ${\lambda }_2$ in Equations (2) and (3) are defined as ${\lambda }_1={\lambda }_{10}+{\lambda }_{00}>0$ and ${\lambda }_2={\lambda }_{20}+{\lambda }_{00}>0$, following Li et al. [3].

According to Pal and Gauri [27], the marginal distributions of $Y_1$ and $Y_2$ are univariate ZIP. Their means and variances are given by

$$\begin{array}{cc}\hfill \mathrm{E}\left(Y_1\right)& =(p_{10}+p_{11}){\lambda }_1,\hfill \\ \hfill \mathrm{Var}\left(Y_1\right)& =(p_{10}+p_{11}){\lambda }_1[1+(1-p_{10}-p_{11}){\lambda }_1],\hfill \\ \hfill \mathrm{E}\left(Y_2\right)& =(p_{01}+p_{11}){\lambda }_2,\hfill \\ \hfill \mathrm{Var}\left(Y_2\right)& =(p_{01}+p_{11}){\lambda }_2[1+(1-p_{01}-p_{11}){\lambda }_2].\hfill \end{array}$$

It is evident that $\mathrm{E}\left(Y_1\right)<\mathrm{Var}\left(Y_1\right)$ and $\mathrm{E}\left(Y_2\right)<\mathrm{Var}\left(Y_2\right)$. Hence, the BZIP model is appropriate for overdispersed data.

2.2. CUSUM Chart Proposed by He et al. [28]

He et al. [28] defined shifted parameters for the BZIP model, including the shifted p-set parameters ($p_{001}$, $p_{101}$, $p_{011}$, $p_{111}$; summing to 1) and the shifted $\lambda$-set parameters (${\lambda }_{11}$, ${\lambda }_{21}$, ${\lambda }_{001}$). Since process deterioration is indicated by a decrease in $p_{00}$ or increases in the other parameters, the authors, motivated by this rationale, developed one-sided CUSUM procedures.

2.2.1. The p-CUSUM Procedure for Monitoring Shifts in the p-Set Parameters

The p-CUSUM procedure for monitoring BZIP-distributed sequences $Y_i=(y_{i1},y_{i2})$ ($i=1,2,\cdots$) is designed under the IC assumption of $\lambda$-set parameters, with its control statistic derived from the corresponding log-likelihood ratios. Then

$$S{P}_i=max(0,S{P}_{i-1}+L{P}_{ki}),$$

where $S{P}_0=0$, $k=1,2,3,4$ and $i=1,2,\cdots$. $L{P}_{ki}$ is defined as follows:

• $L{P}_{1i}$ = ln$(P_{001}/P_{000})$ for $y_{i1}=0,y_{i2}=0$ with

$$\begin{array}{cc}\hfill P_{001}& =p_{001}+p_{101}{e}^{-{\lambda }_1}+p_{011}{e}^{-{\lambda }_2}+p_{111}{e}^{-\lambda },\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill P_{000}& =p_{00}+p_{10}{e}^{-{\lambda }_1}+p_{01}{e}^{-{\lambda }_2}+p_{11}{e}^{-\lambda };\hfill \end{array}$$

(6)

• $L{P}_{2i}$ = ln$(P_{101}/P_{100})$ for $y_{i1}>0,y_{i2}=0$ with

$$\begin{array}{cc}\hfill P_{101}& ={\displaystyle \sum _{y_1=1}^{\infty }}{\displaystyle \frac{p_{101}{\lambda }_1^{y_1}{e}^{-{\lambda }_1}+p_{111}{\lambda }_{10}^{y_1}{e}^{-\lambda }}{y_1!}},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill P_{100}& ={\displaystyle \sum _{y_1=1}^{\infty }}{\displaystyle \frac{p_{10}{\lambda }_1^{y_1}{e}^{-{\lambda }_1}+p_{11}{\lambda }_{10}^{y_1}{e}^{-\lambda }}{y_1!}};\hfill \end{array}$$

(8)

• $L{P}_{3i}$ = ln$(P_{011}/P_{010})$ for $y_{i1}=0,y_{i2}>0$ with

$$\begin{array}{cc}\hfill P_{011}& ={\displaystyle \sum _{y_2=1}^{\infty }}{\displaystyle \frac{p_{011}{\lambda }_2^{y_2}{e}^{-{\lambda }_2}+p_{111}{\lambda }_{20}^{y_2}{e}^{-\lambda }}{y_2!}},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill P_{010}& ={\displaystyle \sum _{y_2=1}^{\infty }}{\displaystyle \frac{p_{01}{\lambda }_2^{y_2}{e}^{-{\lambda }_2}+p_{11}{\lambda }_{20}^{y_2}{e}^{-\lambda }}{y_2!}};\hfill \end{array}$$

(10)

• $L{P}_{4i}$ = ln$(P_{111}/P_{110})$ for $y_{i1}>0,y_{i2}>0$ with

$$\begin{array}{cc}\hfill P_{111}& ={\displaystyle \sum _{y_1=1}^{\infty }}{\displaystyle \sum _{y_2=1}^{\infty }}\left\{p_{111}{\displaystyle \sum _{j=0}^{min(y_1,y_2)}}{\displaystyle \frac{{\lambda }_{10}^{y_1-j}{\lambda }_{20}^{y_2-j}{\lambda }_{00}^j{e}^{-\lambda }}{(y_1-j)!(y_2-j)!j!}}\right\},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill P_{110}& ={\displaystyle \sum _{y_1=1}^{\infty }}{\displaystyle \sum _{y_2=1}^{\infty }}\left\{p_{11}{\displaystyle \sum _{j=0}^{min(y_1,y_2)}}{\displaystyle \frac{{\lambda }_{10}^{y_1-j}{\lambda }_{20}^{y_2-j}{\lambda }_{00}^j{e}^{-\lambda }}{(y_1-j)!(y_2-j)!j!}}\right\}.\hfill \end{array}$$

(12)

An OOC signal is triggered on the p-CUSUM chart when $S{P}_i>h_p$, where $h_p$ is the upper control limit (UCL) that is calibrated to achieve the desired IC performance.

2.2.2. The $\lambda$-CUSUM Procedure for Monitoring Shifts in the $\lambda$-Set Parameters

Under the analogous assumption of no shifts in the p-set parameters, He et al. [28] correspondingly developed a $\lambda$-CUSUM procedure using log-likelihood ratios to detect shifts in the $\lambda$-set parameters. The statistic is defined as follows:

$$S{L}_i=max(0,S{L}_{i-1}+L{L}_{ki}),$$

where $S{L}_0=0$, $k=1,2,3,4$ and $i=1,2,\cdots$. $L{L}_{ki}$ is defined as follows:

• $L{L}_{1i}$ = ln$(L_{001}/L_{000})$ for $y_{i1}=0,y_{i2}=0$ with

$$\begin{array}{cc}\hfill L_{001}& =p_{00}+p_{10}{e}^{-{\lambda }_{11}}+p_{01}{e}^{-{\lambda }_{21}}+p_{11}{e}^{-{\lambda }_a},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill L_{000}& =p_{00}+p_{10}{e}^{-{\lambda }_1}+p_{01}{e}^{-{\lambda }_2}+p_{11}{e}^{-\lambda };\hfill \end{array}$$

(14)

• $L{L}_{2i}$ = ln$(L_{101}/L_{100})$ for $y_{i1}>0,y_{i2}=0$ with

$$\begin{array}{cc}\hfill L_{101}& ={\displaystyle \frac{p_{10}{\lambda }_{11}^{y_{i1}}{e}^{-{\lambda }_{11}}+p_{11}{\lambda }_{101}^{y_{i1}}{e}^{-{\lambda }_a}}{y_{i1}!}},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill L_{100}& ={\displaystyle \frac{p_{10}{\lambda }_1^{y_{i1}}{e}^{-{\lambda }_1}+p_{11}{\lambda }_{10}^{y_{i1}}{e}^{-\lambda }}{y_{i1}!}};\hfill \end{array}$$

(16)

• $L{L}_{3i}$ = ln$(L_{011}/L_{010})$ for $y_{i1}=0,y_{i2}>0$ with

$$\begin{array}{cc}\hfill L_{011}& ={\displaystyle \frac{p_{01}{\lambda }_{21}^{y_{i2}}{e}^{-{\lambda }_{21}}+p_{11}{\lambda }_{201}^{y_{i2}}{e}^{-{\lambda }_a}}{y_{i2}!}},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill L_{010}& ={\displaystyle \frac{p_{01}{\lambda }_2^{y_{i2}}{e}^{-{\lambda }_2}+p_{11}{\lambda }_{20}^{y_{i2}}{e}^{-\lambda }}{y_{i2}!}};\hfill \end{array}$$

(18)

• $L{L}_{4i}$ = ln$(L_{111}/L_{110})$ for $y_{i1}>0,y_{i2}>0$ with

$$\begin{array}{cc}\hfill L_{111}& =p_{11}{\displaystyle \sum _{j=0}^{min(y_{i1},y_{i2})}}{\displaystyle \frac{{\lambda }_{101}^{y_{i1}-j}{\lambda }_{201}^{y_{i2}-j}{\lambda }_{001}^j{e}^{-{\lambda }_a}}{(y_{i1}-j)!(y_{i2}-j)!j!}},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill L_{110}& =p_{11}{\displaystyle \sum _{j=0}^{min(y_{i1},y_{i2})}}{\displaystyle \frac{{\lambda }_{10}^{y_{i1}-j}{\lambda }_{20}^{y_{i2}-j}{\lambda }_{00}^j{e}^{-\lambda }}{(y_{i1}-j)!(y_{i2}-j)!j!}};\hfill \end{array}$$

(20)

where the parameters are defined as: ${\lambda }_{101}={\lambda }_{11}-{\lambda }_{001}$, ${\lambda }_{201}={\lambda }_{21}-{\lambda }_{001}$, and ${\lambda }_a={\lambda }_{101}+{\lambda }_{201}+{\lambda }_{001}$.

According to He et al. [28], the $\lambda$-CUSUM chart triggers a signal when $S{L}_i>h_{\lambda }$, where $h_{\lambda }$ is the UCL that is determined by the desired IC performance.

Following He et al. [28], the p-set parameters ($p_{00}$, $p_{10}$, $p_{01}$, $p_{11}$) govern defect occurrence probabilities, while the $\lambda$-set parameters (${\lambda }_1$, ${\lambda }_2$, ${\lambda }_{00}$) determine defect counts in nonconforming products. A decrease in $p_{00}$ coupled with increases in $p_{10}$, $p_{01}$, and $p_{11}$ suggests a higher proportion of defective products. Similarly, an increase in any parameter of the $\lambda$-set corresponds to a higher number of defects per defective unit. Such parameter shifts reflect process deterioration and should be detected promptly.

3. The SR Control Procedure

This section develops an SR monitoring scheme for the p-set and $\lambda$-set parameters of the BZIP model. Following He et al. [28], the shifted p-set parameters are denoted as $p_{001}$, $p_{101}$, $p_{011}$, and $p_{111}$, satisfying $p_{001}+p_{101}+p_{011}+p_{111}=1$, while the shifted $\lambda$-set parameters are denoted as ${\lambda }_{11}$, ${\lambda }_{21}$, and ${\lambda }_{001}$. Since the decreases in $p_{00}$ and the increases in the other parameters indicate process deterioration, a one-sided SR procedure is developed accordingly.

3.1. The p-SR Procedure for Monitoring Shifts in the p-Set Parameters

For BZIP observations $Y_i=(y_{i1},y_{i2})$ ($i=1,2,\cdots$), the p-SR monitoring procedure is constructed under the IC assumption of $\lambda$-set parameters, with its monitoring statistic derived from the corresponding likelihood ratios. Consequently, following Moustakides et al. [36], the instantaneous likelihood ratio for each observation $Y_i$ is defined as ${\mathsf{\Lambda }}_i^{\left(p\right)}=g\left(Y_i\right)/f\left(Y_i\right)$, where $g\left(Y_i\right)$ and $f\left(Y_i\right)$ are the post-change and pre-change probability density functions, respectively. The SR-type plotting statistic is then given by

$$R{P}_i={\displaystyle \sum _{s=1}^i}{\displaystyle \prod _{j=s}^i}{\mathsf{\Lambda }}_j^{\left(p\right)},$$

where $R{P}_0=0$ and $i=1,2,\cdots$. To make the calculation more convenient, we compute the SR statistic with the following recursive expression:

$$R{P}_i=(1+R{P}_{i-1}){\mathsf{\Lambda }}_i^{\left(p\right)},$$

(21)

where ${\mathsf{\Lambda }}_i^{\left(p\right)}$ is computed as follows:

• ${\mathsf{\Lambda }}_i^{\left(p\right)}=P_{001}/P_{000}$ for $y_{i1}=0,y_{i2}=0$;
• ${\mathsf{\Lambda }}_i^{\left(p\right)}=P_{101}/P_{100}$ for $y_{i1}>0,y_{i2}=0$;
• ${\mathsf{\Lambda }}_i^{\left(p\right)}=P_{011}/P_{010}$ for $y_{i1}=0,y_{i2}>0$;
• ${\mathsf{\Lambda }}_i^{\left(p\right)}=P_{111}/P_{110}$ for $y_{i1}>0,y_{i2}>0$.

All of the parameters are the same with those of Equations (5)–(12).

The p-SR scheme produces an OOC signal when the statistic $R{P}_i>h_p$, where $h_p$ is determined to achieve the desired IC ARL.

3.2. The $\lambda$-SR Procedure for Monitoring Shifts in the $\lambda$-Set Parameters

Under the analogous assumption of stable p-set parameters, the $\lambda$-SR procedure is designed for monitoring $\lambda$-set parameters. Following Moustakides et al. [36], the instantaneous likelihood ratio ${\mathsf{\Lambda }}_i^{\left(\lambda \right)}=g\left(Y_i\right)/f\left(Y_i\right)$ is defined with $g\left(Y_i\right)$ and $f\left(Y_i\right)$ representing the post-change and pre-change probability density functions, respectively. The corresponding SR statistic is then derived as

$$R{L}_i={\displaystyle \sum _{s=1}^i}{\displaystyle \prod _{j=s}^i}{\mathsf{\Lambda }}_j^{\left(\lambda \right)},$$

where $R{L}_0=0$ and $i=1,2,\cdots$. And the SR statistic has the following recursive expression:

$$R{L}_i=(1+R{L}_{i-1}){\mathsf{\Lambda }}_i^{\left(\lambda \right)},$$

(22)

where ${\mathsf{\Lambda }}_i^{\left(\lambda \right)}$ is defined as follows:

• ${\mathsf{\Lambda }}_i^{\left(\lambda \right)}=L_{001}/L_{000}$ for $y_{i1}=0,y_{i2}=0$;
• ${\mathsf{\Lambda }}_i^{\left(\lambda \right)}=L_{101}/L_{100}$ for $y_{i1}>0,y_{i2}=0$;
• ${\mathsf{\Lambda }}_i^{\left(\lambda \right)}=L_{011}/L_{010}$ for $y_{i1}=0,y_{i2}>0$;
• ${\mathsf{\Lambda }}_i^{\left(\lambda \right)}=L_{111}/L_{110}$ for $y_{i1}>0,y_{i2}>0$.

where ${\lambda }_{101}={\lambda }_{11}-{\lambda }_{001},{\lambda }_{201}={\lambda }_{21}-{\lambda }_{001}$, with ${\lambda }_a={\lambda }_{101}+{\lambda }_{201}+{\lambda }_{001}$. And all of the parameters are the same as those of Equations (13)–(20).

The $\lambda$-SR procedure generates a signal when $R{L}_i>h_{\lambda }$, where the $h_{\lambda }$ is determined to ensure the desired IC performance.

3.3. The $p\lambda$-SR Procedure for Simultaneously Monitoring Shifts Both in the p-Set and $\lambda$-Set Parameters with a Combination of the p-SR and $\lambda$-SR Procedures

In this part, we employ a combined scheme using the p-SR and $\lambda$-SR charts to simultaneously monitor parameter shifts (referred to as the $p\lambda$-SR scheme). The $p\lambda$-SR chart signals when either the p-SR plotting statistic exceeds its control limit $h_p$ or the $\lambda$-SR plotting statistic exceeds its control limit $h_{\lambda }$. For comparison, we also implement a combined procedure based on the p-CUSUM and $\lambda$-CUSUM charts (denoted as $p\lambda$-CUSUM) to monitor parameter shifts simultaneously. Similarly, the $p\lambda$-CUSUM chart triggers a signal if either the p-CUSUM plotting statistic exceeds $h_p$ or the $\lambda$-CUSUM plotting statistic exceeds $h_{\lambda }$.

4. Simulation Studies

In this section, we compare the performance of the SR procedure with the CUSUM procedure. A common approach for evaluating the performance of a charting scheme is to examine its RL characteristics. In this part, we use the ARL to assess the performance of each control procedure. The ARL represents the average number of samples plotted on a control scheme before an OOC signal is triggered. In SPM, an effective control procedure should signal quickly when the process is OOC. Furthermore, it is desirable to have a large ARL when the process is IC (denoted as ARL₀), and a small ARL when the process is OOC (denoted as ARL₁).

We evaluate the performance of each control procedure using the ARL under both zero-state and steady-state conditions. The ZS-ARL characterizes a process that experiences a parameter shift immediately upon monitoring initiation, whereas the SS-ARL refers to a scenario where the process shifts to an OOC state only after an initial period of IC operation.

The performance of the designed p-SR and $\lambda$-SR schemes, which detect p-set and $\lambda$-set parameter shifts, respectively, is assessed through four representative simulation cases for the BZIP model based on Li et al. [3]:

(1)

$p_{00}$ = 0.89, $p_{10}$ = 0.03, $p_{01}$ = 0.03, $p_{11}$ = 0.05, ${\lambda }_1$ = 2.0, ${\lambda }_2$ = 2.0, ${\lambda }_{00}$ = 1.5;

(2)

$p_{00}$ = 0.745, $p_{10}$ = 0.10, $p_{01}$ = 0.075, $p_{11}$ = 0.08, ${\lambda }_1$ = 3.0, ${\lambda }_2$ = 4.0, ${\lambda }_{00}$ = 1.0;

(3)

$p_{00}$ = 0.63, $p_{10}$ = 0.135, $p_{01}$ = 0.135, $p_{11}$ = 0.10, ${\lambda }_1$ = 1.0, ${\lambda }_2$ = 1.0, ${\lambda }_{00}$ = 0.5;

(4)

$p_{00}$ = 0.54, $p_{10}$ = 0.10, $p_{01}$ = 0.20, $p_{11}$ = 0.16, ${\lambda }_1$ = 2.5, ${\lambda }_2$ = 1.5, ${\lambda }_{00}$ = 1.0.

It is clear that the parameters in the above four cases satisfy the constraints of the BZIP model described in Section 2.1.

When the parameters are unknown, estimation for the BZIP model can be performed via the method of moments or the maximum likelihood method. In the study of Li et al. [3], the moment estimators for the BZIP model were derived, and the directional grid search method proposed by Powell [41] was employed, using the moment estimates as initial values to obtain the maximum likelihood estimates.

To ensure comparability, our study adopts the framework of He et al. [28], evaluating three prespecified shift magnitudes in the p-set parameters—small, moderate, and large shifts. The corresponding shift sizes are specified as follows:

• shift magnitude 1: $p_{001}$ = $p_{00}-0.09$, $p_{101}$ = $p_{10}+0.03$, $p_{011}$ = $p_{01}+0.03$, and $p_{111}$ = $p_{11}+0.03$;
• shift magnitude 2: $p_{001}$ = $p_{00}-0.15$, $p_{101}$ = $p_{10}+0.05$, $p_{011}$ = $p_{01}+0.05$, and $p_{111}$ = $p_{11}+0.05$;
• shift magnitude 3: $p_{001}$ = $p_{00}-0.21$, $p_{101}$ = $p_{10}+0.07$, $p_{011}$ = $p_{01}+0.07$, and $p_{111}$ = $p_{11}+0.07$.

Correspondingly, we also consider three prespecified shift magnitudes in the $\lambda$-set parameters—small, moderate, and large shifts. The specific shift sizes are defined as follows:

• shift magnitude 1: ${\lambda }_{11}$ = ${\lambda }_1+0.5$, ${\lambda }_{21}$ = ${\lambda }_2+0.5$, and ${\lambda }_{001}$ = ${\lambda }_{00}+0.5$;
• shift magnitude 2: ${\lambda }_{11}$ = ${\lambda }_1+1.0$, ${\lambda }_{21}$ = ${\lambda }_2+1.0$, and ${\lambda }_{001}$ = ${\lambda }_{00}+1.0$;
• shift magnitude 3: ${\lambda }_{11}$ = ${\lambda }_1+2.0$, ${\lambda }_{21}$ = ${\lambda }_2+2.0$, and ${\lambda }_{001}$ = ${\lambda }_{00}+2.0$.

We have verified that, after applying the prespecified shift magnitudes, the shifted parameters continue to satisfy all constraints of the BZIP model for both the p-set and the $\lambda$-set parameters. In fact, the prespecified shift magnitudes depend solely on the practitioner’s knowledge and experience concerning the process being monitored.

In this section, we use Monte Carlo simulations to evaluate the performance of the proposed scheme. All simulations are run for 50,000 replications to estimate the target ARL values. Following He et al. [28], we set the sample size to $n=1$ for ease of comparison with their CUSUM chart. Other sample sizes are also feasible. For individual monitoring of shifts in the p-set or $\lambda$-set parameters, the ARL₀ is set to 400. For simultaneous monitoring of both parameter sets, the control limits of the p-SR and $\lambda$-SR schemes are adjusted separately to achieve a combined ARL₀ of 200. To simulate proper steady-state conditions, the process is run in the IC state for 50 consecutive samples using the SR chart’s control limits before introducing any parameter shift. If a signal is triggered during these initial 50 IC samples, that simulation run is discarded and a new set of 50 samples is generated. We compare the performance of the proposed SR chart with the CUSUM scheme of He et al. [28] under three scenarios: shifts only in the p-set parameters, shifts only in the $\lambda$-set parameters, and shifts in both parameter sets simultaneously.

The Monte Carlo method used to compute the control limits and the ARL₁ for the SR-based charts is described below. Algorithm 1 outlines the detailed steps for determining the control limits of the p-SR and $\lambda$-SR charts using the bisection method.

Algorithm 1 Computing the control limits of the p-SR and $\lambda$-SR charts via Monte Carlo simulation with bisection

Step 1.  Determine the values of the IC parameters (such as $p_{00}$, $p_{10}$, $p_{01}$, $p_{11}$, ${\lambda }_1$, ${\lambda }_2$ and ${\lambda }_{00}$), the prespecified shift magnitudes and the target ARL0. Set the number of replications to 50,000 and the sample size to $n=1$. Choose an initial search interval $[a,b]$ for the control limit such that ${\widehat{\mathrm{ARL}}}_0\left(a\right)>{\mathrm{ARL}}_0^{\mathrm{target}}$ and ${\widehat{\mathrm{ARL}}}_0\left(b\right)<{\mathrm{ARL}}_0^{\mathrm{target}}$, and set a tolerance $\epsilon$ and a maximum number of iterations K (e.g., $K=50$). Step 2.  Set the current control limit to $h_p\left(or{h}_{\lambda }\right)=c=(a+b)/2$. Generate a sample of size $n=1$ from the BZIP model with the corresponding IC parameters. Step 3.  Compute the plotting statistic $R{P}_i$ (or $R{L}_i$). If $R{P}_i\le h_p$ (or $R{L}_i\le h_{\lambda }$), return to Step 2. Otherwise, record the current sample number as the run length (RL) at which the OOC signal is triggered. Step 4.  Repeat Steps 2–3 a total of 50,000 times. The average of the 50,000 recorded RLs gives the estimated ${\widehat{\mathrm{ARL}}}_0\left(h_p\right)$ (or ${\widehat{\mathrm{ARL}}}_0\left(h_{\lambda }\right)$) for the current control limit $h_p$ (or $h_{\lambda }$). Then, update the search interval using the bisection method as follows:      (a) Let $c=(a+b)/2$ and set $h_p\left(or{h}_{\lambda }\right)=c$.      (b) Estimate ${\widehat{\mathrm{ARL}}}_0\left(c\right)$ by performing Steps 2–3 with $h_p\left(or{h}_{\lambda }\right)=c$.      (c) If $|{\widehat{\mathrm{ARL}}}_0\left(c\right)-{\mathrm{ARL}}_0^{\mathrm{target}}|<\epsilon$, stop and set the final control limit $h^{*}=c$.      (d) Otherwise, if $({\widehat{\mathrm{ARL}}}_0\left(a\right)-{\mathrm{ARL}}_0^{\mathrm{target}})·({\widehat{\mathrm{ARL}}}_0\left(c\right)-{\mathrm{ARL}}_0^{\mathrm{target}})<0$, then set $b=c$; else set $a=c$.      (e) Return to Step 2 with the updated interval and repeat until the convergence criterion in (c) is satisfied or the maximum number of iterations K is reached. The final value $h^{*}$ is taken as the upper control limit $h_p$ (for the p-SR chart) or $h_{\lambda }$ (for the $\lambda$-SR chart).

The procedure for the $p\lambda$-SR chart differs from that in Algorithm 1 and requires manually adjusting the control limits $h_p$ and $h_{\lambda }$ until the IC ARL₀, obtained via Monte Carlo simulation, reaches the target value within the desired accuracy. The specific steps are given in Algorithm 2.

Algorithm 2 Determining the control limits of the $p\lambda$-SR chart by Monte Carlo simulation

Step 1.  Specify the IC parameters, including $p_{00}$, $p_{10}$, $p_{01}$, $p_{11}$, ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_{00}$, the prespecified shift magnitudes, and the nominal ARL0. Use a sample size of $n=1$. Step 2.  Apply the procedure described in Algorithm 1 to compute the control limits $h_p$ and $h_{\lambda }$ for the p-SR and $\lambda$-SR charts, respectively. Set the ARL0 value to twice the nominal ARL0 during this computation. Step 3.  Substitute the obtained $h_p$ and $h_{\lambda }$ into the same simulation framework as in Algorithm 1. As stated in Section 3.3, the $p\lambda$-SR chart signals an alarm when the p-SR statistic exceeds $h_p$ or the $\lambda$-SR statistic exceeds $h_{\lambda }$. With these limits fixed, run the simulation to estimate the resulting IC ARL0. Step 4.  Compare the estimated ARL0 from Step 3 with the nominal ARL0. If the estimated value is smaller than the nominal value, increase both $h_p$ and $h_{\lambda }$. Otherwise, decrease both limits. Repeat this adjustment process until the estimated ARL0 matches the nominal value within the desired precision.

The OOC performance is quantified by ARL₁, which measures the detection speed after a process shift. The shift is obtained by moving the IC parameters to their OOC values. In this section, we study three types of parameter shifts in the BZIP model: shifts only in the p-set parameters, shifts only in the $\lambda$-set parameters, and simultaneous shifts in both parameter sets. In practice, the true shift sizes ($d_1$, $d_2$) and ($l_1$, $l_2$) are unknown in advance and will be defined later. Algorithm 3 provides the Monte Carlo procedure for computing the ARL₁ of the SR-based charts.

Algorithm 3 Obtaining the ARL1 of the p-SR, $\lambda$-SR and $p\lambda$-SR charts via Monte Carlo simulation

Step 1.  Specify the parameter values and the prespecified shift magnitudes, and obtain the corresponding control limits. Step 2.  Generate a single observation ($n=1$) under the true shift size $(d_1,d_2)$ or $(l_1,l_2)$, and compute the plotting statistic using Equation (21) or (22). Step 3.  If the control procedure signals, record the current run length as RL; otherwise, return to Step 2. Step 4.  After 50,000 replications of Steps 2–3, ARL1 is calculated by the average of RLs.

4.1. Individual Monitoring of p-Set Parameter Shifts

In this section, the p-SR procedure is dedicated solely to monitoring shifts in the p-set parameters of a BZIP process. The simulations with different shifted parameter combinations $p_{00}$, $p_{10}$, $p_{01}$, and $p_{11}$ are arranged as follows:

$$\left[\begin{array}{cccc}{p}_{00}& p_{10}& p_{01}& p_{11}\end{array}\right]=\left[\begin{array}{cccc}{p}_{00}& p_{10}& p_{01}& p_{11}\end{array}\right]+{△}_p,$$

where ${\mathsf{\Delta }}_p$ represents the shift of the p-set parameters from their IC values, with its value specified by each row of the matrix below:

$$\left[\begin{array}{cccc}0& 0& 0& 0\\ -d_1& d_1& 0& 0\\ -d_1& 0& d_1& 0\\ -d_1& 0& 0& d_1\\ -d_1& {\displaystyle \frac{d_1}{2}}& {\displaystyle \frac{d_1}{2}}& 0\\ -d_1& {\displaystyle \frac{d_1}{2}}& 0& {\displaystyle \frac{d_1}{2}}\\ -d_1& 0& {\displaystyle \frac{d_1}{2}}& {\displaystyle \frac{d_1}{2}}\\ -d_1& {\displaystyle \frac{d_1}{3}}& {\displaystyle \frac{d_1}{3}}& {\displaystyle \frac{d_1}{3}}\\ -d_1& -d_2& {\displaystyle \frac{d_1+d_2}{2}}& {\displaystyle \frac{d_1+d_2}{2}}\\ -d_1& {\displaystyle \frac{d_1+d_2}{2}}& -d_2& {\displaystyle \frac{d_1+d_2}{2}}\\ -d_1& {\displaystyle \frac{d_1+d_2}{2}}& {\displaystyle \frac{d_1+d_2}{2}}& -d_2\end{array}\right].$$

(23)

Similar to He et al. [28], the shift matrix (23) defines 11 parameter combinations for ARL evaluation. Combination 1 represents the IC state. Combinations 2–4 correspond to a downward shift $d_1$ in $p_{00}$ coupled with an upward shift $d_1$ in one other parameter. Combinations 5–7 feature a downward shift $d_1$ in $p_{00}$ with upward shifts of ${\displaystyle \frac{d_1}{2}}$ in two other parameters. Combination 8 exhibits a downward shift $d_1$ in $p_{00}$ and upward shifts ${\displaystyle \frac{d_1}{3}}$ in the other three parameters, consistent with the prespecified shift form. Combinations 9–11 consist of a downward shift $d_1$ in $p_{00}$, a downward shift $d_2$ in one other parameter, and upward shifts of ${\displaystyle \frac{d_1+d_2}{2}}$ in the remaining two. For each specified case, shift magnitude, and $(d_1,d_2)$, we are able to obtain a set of ARL values according to the shift matrix (23), where the first combination denotes the monitoring process IC state and the remaining 10 combinations all denote the OOC state. Accordingly, the ARL₁ for a true shift of $(d_1,d_2)$ is computed as the average ARL across the ten OOC combinations. These aggregated results are presented in the mean row of the tables, enabling a direct comparison of the control procedures’ detection performance.

It is worth noting that both the IC parameters and the OOC parameters after shifts must satisfy the constraint conditions of the BZIP model. We have already verified that the IC parameters for the four representative simulation cases satisfy these constraints. For the p-set parameters, as long as the sum of the true shift magnitudes is zero, the shifted parameters will also satisfy the BZIP model constraints. In the shift matrix (23), each row represents a specific pattern and magnitude of parameter shifts. Clearly, the sum of the shift magnitudes in each row is zero. For example, in the fifth row of matrix (23), the sum of the shift magnitudes is $-d_1+{\displaystyle \frac{d_1}{2}}+{\displaystyle \frac{d_1}{2}}+0=0$. Therefore, after applying these shifts, the shifted parameters still satisfy the constraint conditions of the BZIP model.

Table 1. The control limits of p-CUSUM and p-SR charts with ARL₀ = 400 for monitoring parameter shifts in the p-set parameters.

Case	Shift Magnitude	p-CUSUM		p-SR
		${\mathit{h}}_{\mathit{p}}$	ARL	${\mathit{h}}_{\mathit{p}}$	ARL
1	1	2.363	400.78	318.625	400.45
	2	2.936	399.04	290.500	399.97
	3	3.314	400.03	267.375	401.38
2	1	2.193	399.56	352.047	399.97
	2	2.863	399.61	332.625	400.00
	3	3.306	400.26	313.315	400.07
3	1	1.752	399.59	364.971	400.29
	2	2.385	399.74	350.508	400.81
	3	2.833	400.49	335.693	400.14
4	1	1.978	400.03	370.625	400.56
	2	2.656	400.08	348.047	399.82
	3	3.147	399.62	335.879	399.74

shows the control limits for monitoring shifts in the p-set parameters under different shift magnitudes for the previous four cases with ARL₀ = 400. Figure 1 displays the ZS-ARL₁ values for individually monitoring shifts in the p-set parameters under shift magnitude 1 for cases 1 to 4 with ARL₀ = 400. The results for other shift magnitudes are presented in Figure A1. The values of $(d_1,d_2)$ are set to $(0.006,0.001)$, $(0.012,0.002)$, $(0.06,0.003)$, $(0.09,0.004)$, $(0.15,0.005)$, $(0.21,0.006)$, and $(0.24,0.007)$, respectively. The horizontal axis (positions 1−7) corresponds to the same seven true shifts $(d_1,d_2)$ as previously defined. The blue solid line represents the ZS-ARL₁ values of the p-CUSUM chart, and the red dashed line represents those of the p-SR chart. As observed in Figure 1, under zero-state conditions, the p-SR method performs slightly better than the p-CUSUM method when the real shift $(d_1,d_2)$ is very small. For instance, in case 1 with shift magnitude 1, the ZS-ARL₁ value of the p-SR chart is slightly smaller than that of the p-CUSUM chart when $(d_1,d_2)=(0.006,0.001)$. However, the p-CUSUM scheme becomes much more effective as the actual shift size $(d_1,d_2)$ increases. Its ZS-ARL₁ values are significantly lower, meaning that it detects shifts faster than the p-SR scheme. Combining Figure 1 and Figure A1, we observe that the gap between the ZS-ARL₁ values of the two charts gradually narrows as the prespecified shift magnitude increases.

Table 2. The SS-ARL₁ values of individually monitoring the shifts in p-set parameters with ARL₀ = 400 for case 1.

No.	Shift Magnitude	(${\mathit{d}}_{\mathbf{1}}$,${\mathit{d}}_{\mathbf{2}}$)
		(0.006,0.001)			(0.012,0.002)			(0.06,0.003)			(0.09,0.004)			(0.15,0.005)			(0.21,0.006)			(0.24,0.007)
		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR
1	1	401.34	397.56		403.41	402.38		401.09	403.70		397.22	402.91		400.45	401.56		403.99	401.21		398.37	397.55
2		301.90	273.12		242.95	217.40		71.93	69.75		47.00	46.19		25.45	27.49		17.99	19.56		15.30	17.04
3		296.25	274.25		240.88	218.22		72.04	69.44		46.90	46.07		25.99	27.58		17.73	19.46		15.57	17.10
4		329.50	294.92		270.30	240.47		93.85	87.04		61.54	58.23		33.45	34.15		22.83	24.01		19.47	20.88
5		304.16	279.23		241.02	218.91		71.93	68.47		45.59	46.47		26.05	27.55		17.90	19.54		15.29	17.22
6		315.37	284.26		257.83	234.34		83.25	78.33		52.18	51.56		29.63	30.69		19.95	21.40		17.32	18.79
7		313.37	284.59		254.96	227.59		82.13	77.44		52.46	51.62		29.17	30.61		19.71	21.63		17.12	18.87
8		309.00	280.55		251.11	228.10		79.33	74.31		50.26	49.75		27.86	29.81		19.30	20.55		16.50	18.11
9		311.07	283.00		257.81	228.75		83.45	77.74		52.91	52.51		29.62	30.53		20.02	21.48		17.20	18.73
10		309.90	277.66		256.82	232.87		83.36	78.07		53.07	51.93		29.37	30.67		20.12	21.57		17.14	18.94
11		301.64	271.23		232.71	212.55		71.37	69.34		45.60	45.71		26.00	27.31		17.79	19.24		15.50	16.82
mean		309.21	280.28		250.64	225.92		79.26	74.99		50.75	50.00		28.26	29.64		19.33	20.85		16.64	18.25
1	2	401.05	398.46		397.99	401.80		399.74	400.42		401.66	400.01		401.75	403.72		401.88	398.75		397.74	398.25
2		306.19	292.74		251.33	233.51		77.84	72.34		49.18	46.53		26.71	26.09		17.63	17.78		15.20	15.19
3		309.57	291.86		248.32	241.12		77.06	72.43		49.10	46.44		26.56	26.25		17.49	17.89		15.18	15.48
4		329.26	308.20		277.79	261.40		101.63	89.07		63.93	58.44		33.35	32.10		21.35	21.67		18.77	18.70
5		306.36	289.68		248.53	229.88		78.02	72.84		49.50	45.98		26.09	26.15		17.75	17.92		15.09	15.46
6		319.83	296.74		262.90	246.73		89.82	79.45		55.06	51.21		29.33	28.69		19.63	19.70		16.68	16.86
7		318.00	299.38		270.53	244.70		87.25	79.88		55.47	51.24		30.21	28.41		19.53	19.35		16.50	16.90
8		309.29	301.92		262.07	243.45		82.54	76.96		52.91	49.92		28.37	27.58		18.93	19.07		16.10	16.13
9		316.68	302.60		264.80	253.48		89.61	80.67		55.60	51.42		29.13	28.71		19.59	19.59		16.79	16.86
10		319.90	304.72		264.20	243.47		88.87	80.30		55.53	51.07		29.26	28.46		19.49	19.71		16.85	16.89
11		303.03	285.13		242.44	233.41		76.44	70.35		48.11	45.92		26.17	25.69		17.69	17.77		15.11	15.26
mean		313.81	297.30		259.29	243.11		84.91	77.43		53.4	49.82		28.52	27.81		18.91	19.04		16.23	16.37
1	3	403.37	398.41		397.40	397.88		400.99	401.10		397.24	401.92		398.56	397.20		399.61	398.57		398.49	401.37
2		309.52	303.00		265.16	244.67		84.93	75.45		51.66	47.92		27.63	26.31		17.90	17.78		15.29	15.03
3		322.39	304.12		260.55	244.74		81.72	75.82		51.12	48.71		27.20	26.23		17.84	17.40		15.29	15.04
4		340.14	322.26		291.67	267.43		107.06	96.38		67.25	60.19		34.66	32.04		22.17	21.43		18.48	17.93
5		318.55	300.32		265.21	249.84		81.76	76.51		51.26	48.32		27.19	25.90		17.78	17.43		15.26	15.05
6		327.42	313.57		279.40	255.94		94.06	86.69		59.08	53.62		30.83	29.09		19.70	19.02		16.91	16.49
7		327.30	305.90		276.27	254.34		93.82	85.61		59.51	54.11		30.50	29.18		19.94	19.19		16.72	16.52
8		324.00	306.07		271.75	257.93		90.05	81.49		56.40	51.69		29.46	28.02		18.97	18.88		15.90	15.99
9		322.04	310.67		276.00	261.78		94.45	85.40		59.64	54.13		30.72	28.77		19.85	19.36		16.58	16.32
10		326.11	306.72		278.55	257.80		94.74	86.54		59.34	54.18		30.70	28.97		19.77	19.41		16.60	16.44
11		311.30	298.72		256.09	240.77		81.47	75.31		52.25	47.73		26.66	25.86		17.73	17.75		15.08	15.10
mean		322.88	307.14		272.07	253.52		90.41	82.52		56.75	52.06		29.56	28.04		19.16	18.76		16.21	15.99

presents the SS-ARL₁ values of individually monitoring the shifts in p-set parameters with different shift magnitudes for case 1, and the remaining three cases are presented in

Table A1. The SS-ARL₁ values of individually monitoring the shifts in p-set parameters with ARL₁ = 400 for case 2.

No.	Shift Magnitude	(${\mathit{d}}_{\mathbf{1}}$,${\mathit{d}}_{\mathbf{2}}$)
		(0.006,0.001)			(0.012,0.002)			(0.06,0.003)			(0.09,0.004)			(0.15,0.005)			(0.21,0.006)			(0.24,0.007)
		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR
1	1	397.31	397.61		403.90	399.78		397.90	399.42		402.79	398.03		403.05	398.23		400.73	399.86		403.80	397.12
2		328.86	298.54		275.86	257.97		116.40	107.91		75.95	75.59		42.71	45.89		28.77	32.61		24.81	28.45
3		317.22	292.41		263.10	241.47		91.79	89.72		60.42	61.34		33.10	36.62		22.76	26.22		19.56	23.01
4		317.73	289.66		269.37	244.92		94.96	92.55		60.94	62.75		34.18	37.69		22.84	26.97		19.83	23.22
5		318.10	290.43		271.67	250.12		101.76	98.53		68.03	68.10		37.40	40.79		25.23	28.80		21.91	25.47
6		326.87	293.46		271.56	250.36		103.63	98.83		68.67	69.04		38.61	41.06		25.78	29.19		22.10	25.60
7		311.76	288.75		262.56	241.23		93.46	90.43		60.17	62.48		33.96	37.18		22.89	26.47		19.62	22.99
8		318.97	295.03		268.05	249.78		101.05	97.35		65.59	65.56		36.22	39.85		24.40	28.01		21.15	24.66
9		309.39	288.82		256.38	241.84		94.18	90.77		59.29	61.81		33.45	37.36		23.12	26.38		19.64	22.99
10		319.83	293.14		273.25	252.41		105.64	99.62		68.37	68.74		38.28	41.18		25.87	29.41		22.19	25.51
11		319.92	298.28		278.15	252.51		103.80	98.23		68.43	67.82		37.68	40.46		25.55	29.08		21.98	25.21
mean		318.87	292.85		268.99	248.26		100.67	96.39		65.59	66.32		36.56	39.81		24.72	28.31		21.28	24.71
1	2	400.76	402.44		398.15	397.44		399.88	398.05		397.65	398.27		398.45	401.82		400.09	403.90		403.16	400.78
2		336.78	319.03		296.17	273.73		126.59	110.45		82.84	73.40		43.39	41.79		28.40	28.27		23.92	23.94
3		325.87	302.78		276.21	255.91		100.31	91.40		63.53	59.80		33.35	33.05		21.96	22.40		18.72	19.32
4		323.19	305.92		281.46	260.63		103.19	92.75		64.65	61.47		34.67	33.94		22.67	23.25		19.45	19.96
5		327.79	312.99		289.22	261.69		111.00	99.57		72.82	64.81		38.75	37.19		24.90	25.09		21.13	21.79
6		329.87	313.73		292.09	265.28		114.59	102.10		72.63	66.51		38.97	37.72		25.32	25.63		21.46	21.79
7		322.10	304.31		281.19	261.35		101.10	92.96		65.56	59.92		34.20	33.45		22.64	22.93		19.38	19.68
8		325.77	316.13		281.49	262.55		108.53	97.63		69.52	64.00		36.76	36.12		24.23	24.45		20.12	21.17
9		323.54	304.30		276.82	255.59		99.70	91.51		65.08	60.07		34.29	33.44		22.22	22.91		19.04	19.73
10		335.75	307.76		293.25	264.83		112.95	101.99		73.16	67.08		39.24	37.82		25.00	25.41		21.40	21.74
11		332.67	314.80		285.17	262.59		110.47	101.33		71.90	66.50		38.38	36.98		25.43	25.44		21.13	21.75
mean		328.33	310.17		285.31	262.41		108.84	98.17		70.17	64.35		37.20	36.15		24.28	24.58		20.58	21.09
1	3	398.19	402.11		399.11	398.19		398.30	397.96		399.86	400.17		397.17	399.61		398.12	402.05		399.92	400.59
2		338.37	334.73		306.03	293.80		135.71	121.37		90.85	78.06		46.23	42.53		28.69	27.96		23.94	23.43
3		336.55	321.25		285.40	270.65		108.23	98.98		67.90	62.22		35.13	32.92		22.56	22.31		18.69	18.49
4		331.69	319.14		285.58	276.65		109.03	100.72		70.67	64.61		36.57	34.31		22.87	22.57		19.20	19.09
5		336.41	323.97		288.39	281.77		120.17	107.62		78.32	70.54		39.46	37.90		25.60	24.67		21.16	20.50
6		337.97	325.84		290.42	280.17		121.59	110.73		77.77	71.68		40.99	38.26		25.75	24.50		21.10	21.02
7		329.71	315.13		280.25	276.71		109.26	98.51		68.69	63.57		35.38	34.28		22.68	22.30		19.17	18.89
8		335.77	316.62		286.96	281.25		117.32	104.22		75.77	68.74		39.22	36.29		24.21	24.13		20.33	20.39
9		331.87	310.85		288.42	269.10		108.80	98.61		68.84	63.83		34.90	34.09		22.49	21.96		19.18	18.62
10		343.06	332.59		293.00	285.54		122.70	111.38		78.48	71.42		40.92	38.39		25.91	24.77		21.24	21.23
11		329.95	331.47		295.94	282.47		119.64	108.95		76.95	70.64		40.45	38.11		25.44	24.21		21.35	20.95
mean		335.13	323.16		290.04	279.81		117.24	106.11		75.42	68.53		38.93	36.71		24.62	23.94		20.54	20.26

,

Table A2. The SS-ARL₁ values of individually monitoring the shifts in p-set parameters with ARL₀ = 400 for case 3.

No.	Shift Magnitude	(${\mathit{d}}_{\mathbf{1}}$,${\mathit{d}}_{\mathbf{2}}$)
		(0.006,0.001)			(0.012,0.002)			(0.06,0.003)			(0.09,0.004)			(0.15,0.005)			(0.21,0.006)			(0.24,0.007)
		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$ -CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR
1	1	397.98	398.76		397.58	399.87		403.77	398.93		402.33	399.98		398.94	401.23		397.25	398.81		398.41	403.15
2		325.03	316.27		299.72	280.81		147.97	145.45		104.18	110.14		62.00	72.35		42.64	53.72		36.94	47.16
3		329.40	313.62		297.13	278.20		145.62	149.40		104.47	111.82		61.97	72.30		43.09	53.52		36.90	47.12
4		318.38	295.33		279.27	261.73		111.67	118.85		75.22	84.76		43.51	54.93		30.22	39.97		26.33	35.16
5		329.59	313.96		302.00	279.72		146.54	146.40		105.39	110.54		62.68	71.69		42.99	53.38		36.61	47.32
6		322.72	309.11		285.15	273.69		125.29	131.17		87.51	97.55		50.82	63.07		35.67	45.55		30.58	40.07
7		324.75	307.55		288.23	271.03		129.11	131.41		89.26	97.68		51.83	62.33		35.58	46.02		30.87	40.45
8		323.27	308.61		287.65	276.39		135.34	134.96		91.35	100.83		54.86	65.56		37.86	47.64		32.70	42.55
9		326.57	304.07		279.04	270.85		126.34	130.30		87.54	96.86		51.75	61.95		35.03	45.48		30.64	40.46
10		324.51	302.66		286.37	267.68		126.18	129.34		86.81	97.10		51.75	61.96		35.09	45.25		30.90	40.26
11		332.01	316.28		302.73	282.22		148.65	147.89		107.92	112.15		62.97	73.34		43.73	53.96		37.37	48.10
mean		325.62	308.75		290.73	274.23		134.27	136.52		93.97	101.94		55.41	65.95		38.19	48.45		32.98	42.86
1	2	398.08	397.71		401.29	398.86		397.20	402.26		398.60	397.20		400.55	402.87		398.69	399.80		401.03	403.93
2		339.68	317.89		314.55	284.90		158.80	145.07		108.50	101.72		64.59	62.96		43.06	44.47		36.53	38.58
3		344.87	322.46		309.37	288.74		153.33	143.82		109.70	103.91		63.64	62.71		42.50	44.25		36.81	38.40
4		328.63	308.31		281.63	264.78		119.25	110.06		79.67	74.88		44.03	46.05		29.90	32.37		25.64	27.78
5		338.44	321.38		315.78	291.52		153.32	142.14		110.22	102.01		63.43	62.89		43.06	43.93		36.24	38.33
6		332.41	312.55		292.92	280.76		133.77	124.30		91.75	86.93		52.06	52.01		35.64	37.28		30.09	32.28
7		332.48	317.10		294.34	276.98		136.95	123.37		91.04	87.76		52.36	52.22		34.98	37.26		30.28	32.59
8		335.31	311.34		302.81	278.73		139.33	131.61		95.78	90.86		55.78	55.57		37.41	38.93		32.08	34.24
9		339.54	315.40		297.39	274.84		133.35	123.68		90.46	86.95		51.39	52.68		35.16	36.77		29.80	32.32
10		330.65	312.23		293.58	278.69		134.44	122.63		91.09	87.91		51.91	52.12		34.77	37.16		30.12	31.99
11		348.76	322.84		309.89	292.98		159.09	147.06		112.19	102.11		64.73	62.55		43.99	45.04		37.22	39.01
mean		337.08	316.15		301.23	281.29		142.16	131.38		98.04	92.51		56.39	56.18		38.05	39.75		32.48	34.55
1	3	401.76	400.06		402.28	403.41		402.79	400.66		401.48	402.26		399.90	403.97		400.91	399.25		402.37	402.94
2		348.41	333.06		319.61	294.85		164.90	148.16		116.34	105.64		67.87	61.76		45.07	42.07		37.29	36.10
3		352.94	327.40		316.21	297.43		164.81	149.27		115.65	105.54		66.74	61.11		43.93	42.49		37.31	36.35
4		333.66	319.51		292.92	274.05		124.27	112.92		83.14	76.70		45.72	43.77		29.57	30.01		25.58	25.83
5		343.43	331.38		314.20	300.88		163.95	150.32		116.12	103.45		68.80	61.86		44.27	42.11		37.29	35.94
6		336.87	325.06		309.98	288.05		143.35	128.16		98.22	88.42		55.28	52.03		35.78	35.43		30.36	30.06
7		337.09	321.48		304.50	287.36		140.73	128.41		97.54	89.32		55.65	51.32		35.71	35.07		30.18	30.00
8		348.66	329.66		307.09	294.32		148.26	136.00		103.76	94.67		58.60	54.21		37.99	36.82		31.72	32.06
9		344.14	327.03		307.27	282.16		142.62	127.67		97.90	89.56		54.31	51.10		35.94	35.19		30.57	30.20
10		341.55	326.97		297.57	289.92		140.44	129.22		98.18	88.51		54.43	51.07		35.33	34.60		30.29	30.08
11		352.55	333.49		324.72	305.79		166.13	147.89		120.71	106.23		67.18	62.22		45.02	42.95		37.77	36.77
mean		343.93	327.50		309.41	291.48		149.94	135.80		104.76	94.80		59.46	55.04		38.86	37.67		32.84	32.34

and

Table A3. The SS-ARL₁ values of individually monitoring the shifts in p-set parameters with ARL₀ = 400 for case 4.

No.	Shift Magnitude	(${\mathit{d}}_{\mathbf{1}}$,${\mathit{d}}_{\mathbf{2}}$)
		(0.006,0.001)			(0.012,0.002)			(0.06,0.003)			(0.09,0.004)			(0.15,0.005)			(0.21,0.006)			(0.24,0.007)
		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR		$\mathit{p}$-CUSUM	$\mathit{p}$-SR
1	1	401.65	403.50		402.66	397.72		398.83	401.98		399.19	402.84		400.28	402.07		398.56	401.16		398.23	402.91
2		312.73	300.70		267.10	252.23		98.51	102.34		64.79	70.79		36.19	43.47		25.04	31.53		21.83	27.59
3		337.22	320.03		305.49	287.70		158.31	149.35		110.25	109.32		64.27	70.43		43.22	50.53		36.83	43.82
4		325.87	301.41		280.97	266.22		114.44	113.76		76.26	81.39		42.27	49.28		28.90	35.73		24.92	31.28
5		332.53	311.36		284.95	265.62		120.62	121.35		81.80	86.23		46.64	53.82		32.07	38.58		27.29	33.77
6		311.97	298.46		270.01	255.46		105.57	108.55		69.58	75.57		39.35	46.56		26.90	33.70		23.10	29.44
7		331.78	319.09		293.05	274.91		130.17	130.45		89.08	92.88		52.33	57.64		35.15	41.66		30.14	36.64
8		318.43	308.24		279.39	265.36		119.14	118.35		80.99	85.05		45.29	52.46		30.66	37.87		26.58	33.11
9		333.42	311.23		292.01	276.26		133.93	130.91		93.11	95.24		52.55	58.51		35.17	42.45		30.58	37.22
10		316.26	295.80		269.39	252.86		104.10	106.75		68.52	74.55		39.31	46.49		26.75	33.20		22.94	28.96
11		322.84	308.53		289.29	270.68		121.02	122.00		83.55	87.36		46.88	54.12		31.73	39.12		27.56	34.19
mean		324.30	307.49		283.17	266.73		120.58	120.38		81.79	85.84		46.51	53.28		31.56	38.44		27.18	33.60
1	2	397.55	399.06		398.60	398.49		400.47	399.33		397.24	399.72		397.55	398.06		400.28	400.72		400.45	397.76
2		323.42	301.61		279.27	255.83		104.14	98.26		67.83	64.55		36.42	37.00		24.13	25.39		20.65	22.22
3		341.06	317.63		319.40	289.97		163.75	148.94		117.11	104.46		66.65	62.43		43.05	42.44		36.89	36.57
4		328.23	307.00		291.11	271.68		121.74	111.81		81.60	74.15		43.42	42.82		28.42	28.92		23.75	25.13
5		326.69	311.33		292.88	276.38		131.10	118.15		86.66	80.73		47.14	46.36		31.31	31.87		26.44	27.16
6		327.07	305.51		286.82	260.01		112.65	103.42		73.10	69.63		39.48	39.42		26.15	27.00		21.99	23.36
7		337.20	313.96		302.88	279.93		140.40	126.78		94.64	87.95		52.84	50.36		33.92	34.45		29.05	29.67
8		334.92	310.31		293.55	273.81		126.87	116.35		84.01	79.21		46.31	44.74		30.30	30.98		25.29	26.40
9		339.34	320.52		310.20	285.19		143.41	127.86		99.37	88.41		53.12	51.25		34.82	34.97		29.35	30.02
10		321.38	300.65		274.41	258.46		110.58	101.42		73.84	67.42		38.90	38.78		25.65	26.86		21.99	23.05
11		326.81	308.90		294.96	278.49		130.69	119.55		87.73	82.08		47.87	46.70		31.28	31.88		26.46	27.40
mean		330.61	309.74		294.55	272.97		128.53	117.25		86.59	79.86		47.22	45.99		30.90	31.48		26.19	27.10
1	3	397.93	401.95		402.84	397.37		403.39	402.00		402.42	398.49		401.57	400.81		399.76	401.70		399.75	398.25
2		332.56	315.76		289.22	276.06		113.68	106.25		73.90	68.66		38.04	37.21		24.72	24.57		20.45	21.04
3		354.99	338.21		327.06	312.37		179.64	157.83		127.14	110.24		71.58	64.17		45.47	41.90		37.13	35.25
4		342.78	325.40		304.22	289.02		132.37	119.43		86.78	77.29		45.62	41.95		28.25	27.76		23.72	23.39
5		343.26	326.30		305.71	288.72		144.14	126.70		94.26	85.56		50.10	46.78		32.58	31.07		26.49	26.38
6		331.67	321.20		294.53	278.49		122.02	111.15		78.51	72.29		41.21	39.19		26.41	25.95		22.24	22.25
7		354.09	331.24		313.57	302.14		156.07	138.73		104.51	92.33		56.02	51.27		35.35	33.80		29.04	28.22
8		333.43	321.87		307.53	289.81		137.67	123.71		91.05	82.61		48.60	45.38		30.55	29.42		25.43	25.57
9		349.13	339.45		314.77	305.70		154.41	141.41		106.18	93.72		56.15	52.11		35.18	34.20		29.60	28.90
10		333.81	322.55		289.64	271.45		117.47	109.38		78.72	71.70		40.71	38.81		25.92	25.41		21.72	21.84
11		342.76	325.92		302.86	291.31		144.42	127.90		93.24	84.85		50.50	47.25		31.95	31.17		26.59	26.08
mean		341.85	326.79		304.91	290.51		140.19	126.25		93.43	83.92		49.85	46.41		31.64	30.52		26.24	25.89

. Under steady-state conditions, simulation results indicate that, as shown in combination 8, the p-SR scheme exhibits superior detection performance compared to the p-CUSUM scheme when the actual shift is smaller than the prespecified shift magnitude. When the true shift equals the prespecified shift magnitude, the SS-ARL₁ of the p-SR procedure is slightly lower than that of the p-CUSUM procedure in most scenarios, though the two values are nearly equivalent. The only exception occurs under the shift magnitude 1 in cases 3 and 4, where the p-CUSUM chart yields a noticeably lower SS-ARL₁; however, the difference between the two is still relatively small. For example, in Table 2 under the shift magnitude 1 at row 8 with $(d_1,d_2)=(0.06,0.003)$, the SS-ARL₁ values for the p-CUSUM and p-SR charts are 79.33 and 74.31, respectively. When the actual shift exceeds the prespecified shift magnitude, the monitoring performance of the p-CUSUM chart surpasses that of the p-SR chart. For instance, in the same table under the shift magnitude 1 at row 8 with $(d_1,d_2)=(0.15,0.005)$, the SS-ARL₁ values are 27.86 for the p-CUSUM chart and 29.81 for the p-SR chart.

Furthermore, the SS-ARL₁ values in the mean row indicate that the p-SR procedure generally outperforms the p-CUSUM procedure when the true shift does not exceed the prespecified shift magnitude. This holds except for a few scenarios, including when $(d_1,d_2)=(0.09,0.004)$ in case 2, when $(d_1,d_2)=(0.06,0.003)$ and $(d_1,d_2)=(0.09,0.004)$ in case 3, and when $(d_1,d_2)=(0.09,0.004)$ in case 4. Even in these situations, the difference in SS-ARL₁ values between the procedures is negligible. For instance, in Table 2 under the shift magnitude 2 at the mean row with $(d_1,d_2)=(0.06,0.003)$, the SS-ARL₁ values for the p-CUSUM and p-SR charts are 84.91 and 77.43, respectively. For true shifts greater than the prespecified shift magnitude, the p-CUSUM procedure is more effective than its p-SR scheme. For example, in the same table, for the shift magnitude 2 at the mean row with $(d_1,d_2)=(0.21,0.006)$, the p-CUSUM and p-SR charts yield SS-ARL₁ values of 18.91 and 19.04, respectively.

4.2. Individual Monitoring of $\lambda$-Set Parameter Shifts

This subsection introduces the $\lambda$-SR procedure for individual monitoring of shifts in the $\lambda$-set parameters of a BZIP process. Its corresponding performance is evaluated through numerical simulations configured with the following parameter shifts:

$$\left[\begin{array}{ccc}{\lambda }_1& {\lambda }_2& {\lambda }_{00}\end{array}\right]=\left[\begin{array}{ccc}{\lambda }_1& {\lambda }_2& {\lambda }_{00}\end{array}\right]+{△}_{\lambda },$$

where ${\mathsf{\Delta }}_{\lambda }$ characterizes the shift of $\lambda$-set parameters from IC values and is given by the rows of the matrix:

$$\left[\begin{array}{ccc}0& 0& 0\\ l_1& 0& 0\\ 0& l_1& 0\\ l_1& l_1& 0\\ l_2& l_2& l_2\\ l_1+l_2& l_2& l_2\\ l_2& l_1+l_2& l_2\\ l_1+l_2& l_1+l_2& l_2\end{array}\right].$$

(24)

In matrix (24), the first combination represents this situation where the process is under IC state. The following two combinations denote the scenarios with an upward shift $l_1$ in parameters ${\lambda }_1$ and ${\lambda }_2$, respectively. The fourth combination represents the case with two upward shifts $l_1$ in both parameters ${\lambda }_1$ and ${\lambda }_2$. The fifth combination corresponds to the prespecified shift form, in which every parameter shifts upward by $l_2$. The sixth and seventh combinations denote the scenarios where there is an upward shift of $l_1+l_2$ in parameter ${\lambda }_1$ and ${\lambda }_2$, respectively, while the remaining parameters shift upward by $l_2$. The last combination represents the case where there are two upward shifts $l_1+l_2$ in parameters ${\lambda }_1$ and ${\lambda }_2$ and ${\lambda }_{00}$ shifts upwardly $l_2$. Similarly, for each specified case, shift magnitude, and $(l_1,l_2)$, we also are able to obtain a set of ARL values according to the shift matrix (24), where the first combination denotes the monitoring process IC state and the remaining 7 combinations all denote the OOC state. Consequently, the ARL₁ for a true shift $(l_1,l_2)$ corresponds to the mean ARL across the seven OOC combinations. These aggregated results are summarized in the mean row of the tables, providing a consolidated metric for comparative performance evaluation of the control procedures.

Similarly, both the IC and OOC $\lambda$-set parameters must satisfy the constraint conditions of the BZIP model. We have already verified that the IC parameters satisfy these constraints. Therefore, when the process is OOC, the shifted parameters will also satisfy the constraints as long as ${\lambda }_1>{\lambda }_{00}$, ${\lambda }_2>{\lambda }_{00}$, and ${\lambda }_{00}>0$. From the shift matrix (24), it can be seen that after applying the shift patterns to the IC $\lambda$-set parameters, these conditions remain satisfied. Therefore, the shifted parameters still satisfy the constraint conditions of the BZIP model.

The control limits of monitoring shifts in the $\lambda$-set parameter with different shift magnitudes for the previous four cases are shown in

Table 3. The control limits of $\lambda$-CUSUM and $\lambda$-SR charts with ARL₀ = 400 for monitoring parameter shifts in the $\lambda$-set parameters.

Case	Shift Magnitude	$\mathit{\lambda }$-CUSUM		$\mathit{\lambda }$-SR
		${\mathit{h}}_{\mathit{\lambda }}$	ARL	${\mathit{h}}_{\mathit{\lambda }}$	ARL
1	1	1.156	400.79	310.938	400.21
	2	1.681	399.96	251.250	399.40
	3	2.091	399.25	172.578	400.59
2	1	1.538	400.83	324.375	399.45
	2	2.209	399.03	270.452	399.86
	3	2.731	400.66	194.369	400.46
3	1	2.175	399.68	278.047	399.00
	2	2.778	399.66	212.300	399.55
	3	3.256	400.75	137.344	400.21
4	1	2.163	400.67	309.688	399.90
	2	2.872	400.90	242.813	400.63
	3	3.338	399.12	164.688	400.15

, where ARL₀ = 400. Figure 2 gives the ZS-ARL₁ values for cases 1 to 4 when individually monitoring $\lambda$-set parameter shifts under shift magnitude 1. The ZS-ARL₁ values for shift magnitudes 2 and 3 are provided in Figure A2. We set the values of $(l_1,l_2)$ to {(0.10,0.05), (0.25,0.10), (1.00,0.50), (3.00,0.75), (6.00,1.00), (15.00,2.00), (18.00,2.50)}, respectively. The horizontal axis (positions 1−7) corresponds to the preceding seven true shifts $(l_1,l_2)$. The blue solid line represents the ZS-ARL₁ values of the $\lambda$-CUSUM chart, and the red dashed line corresponds to the $\lambda$-SR chart. Figure 2 and Figure A2 together indicate that, under zero-state conditions, the ZS-ARL₁ values of the $\lambda$-CUSUM and $\lambda$-SR schemes are similar in most scenarios. In only a few scenarios do the ZS-ARL₁ values of the $\lambda$-CUSUM chart fall slightly below those of the $\lambda$-SR chart. The gap between the ZS-ARL₁ values of the two control methods decreases gradually as the prespecified shift magnitude increases.

Table 4. The SS-ARL₁ values of individually monitoring the shifts in $\lambda$-set parameters with ARL₀ = 400 for case 3.

No.	Shift Magnitude	(${\mathit{l}}_{\mathbf{1}}$,${\mathit{l}}_{\mathbf{2}}$)
		(0.10,0.05)			(0.25,0.10)			(1.00,0.50)			(3.00,0.75)			(6.00,1.00)			(15.00,2.00)			(18.00,2.50)
		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR
1	1	397.44	396.88		403.82	397.40		403.38	396.66		398.10	396.72		399.96	398.55		402.28	401.68		401.80	398.61
2		284.13	257.70		192.11	170.18		48.19	47.70		13.61	14.98		7.47	8.31		5.69	5.75		5.61	5.71
3		288.24	257.34		191.56	171.71		48.53	47.99		13.52	14.84		7.54	8.23		5.67	5.70		5.62	5.69
4		220.10	197.06		112.33	101.37		22.17	23.77		6.67	7.30		3.67	3.99		2.71	2.70		2.73	2.67
5		279.27	257.00		213.04	189.71		48.40	47.71		29.47	30.64		20.50	22.33		9.41	10.61		7.55	8.44
6		210.13	190.38		121.57	108.97		21.82	23.14		9.52	10.52		6.45	6.91		4.70	5.03		4.42	4.67
7		215.76	193.06		119.77	110.82		22.04	23.25		9.60	10.48		6.37	6.92		4.80	5.01		4.43	4.65
8		169.77	152.32		77.93	73.81		13.73	14.75		5.32	5.87		3.27	3.56		2.71	2.66		2.73	2.67
mean		238.20	214.98		146.90	132.37		32.13	32.62		12.53	13.52		7.90	8.61		5.10	5.35		4.73	4.93
1	2	404.23	402.04		397.55	401.30		403.17	400.32		402.14	397.11		404.77	403.27		404.83	397.59		401.60	403.60
2		305.36	286.82		206.99	190.63		51.79	48.24		12.84	12.77		7.14	7.25		5.64	5.63		5.69	5.61
3		308.61	282.96		207.23	189.41		50.99	47.32		12.95	12.93		7.16	7.18		5.58	5.63		5.57	5.69
4		244.22	223.37		128.28	115.64		22.76	22.02		6.07	6.13		3.37	3.50		2.71	2.72		2.66	2.70
5		294.99	274.65		231.09	210.26		52.95	48.69		30.90	29.13		21.20	20.39		9.06	9.09		7.06	7.09
6		238.49	219.30		132.14	120.86		22.11	21.52		8.96	9.07		5.99	6.09		4.68	4.68		4.32	4.22
7		235.02	213.39		135.36	119.52		22.14	21.34		8.88	9.04		5.95	6.14		4.60	4.69		4.21	4.23
8		186.47	171.78		88.47	81.08		13.15	13.17		4.91	5.03		3.11	3.16		2.70	2.70		2.71	2.70
mean		259.02	238.90		161.37	146.77		33.70	31.76		12.22	12.01		7.70	7.67		5.00	5.02		4.60	4.61
1	3	402.06	398.04		398.26	396.75		401.07	397.81		401.14	400.71		402.73	399.08		397.15	400.17		399.08	397.73
2		319.38	308.15		230.74	222.08		58.76	53.78		12.22	12.28		6.58	6.69		5.66	5.70		5.70	5.57
3		317.62	307.77		225.57	218.96		58.44	55.09		12.41	12.18		6.67	6.75		5.67	5.58		5.67	5.66
4		264.38	251.54		150.86	141.53		25.98	24.05		5.70	5.68		3.20	3.21		2.72	2.72		2.70	2.72
5		314.36	298.60		249.95	237.53		64.27	58.63		36.67	33.61		23.26	22.15		8.86	8.68		6.62	6.64
6		258.28	245.86		153.77	142.61		24.60	23.21		8.87	8.69		5.90	5.74		4.55	4.60		4.16	4.13
7		255.80	240.58		154.59	143.55		24.93	23.24		8.78	8.74		5.91	5.87		4.62	4.51		4.22	4.18
8		212.06	203.59		109.60	100.01		14.14	13.19		4.52	4.56		2.98	2.97		2.72	2.71		2.68	2.68
mean		277.41	265.16		182.15	172.32		38.73	35.88		12.74	12.25		7.79	7.63		4.97	4.93		4.54	4.51

presents SS-ARL₁ values from individually monitoring the shifts in $\lambda$-set parameters with different shift magnitudes for case 3, and the remaining three cases are presented in

Table A4. The SS-ARL₁ values of individually monitoring the shifts in $\lambda$-set parameters with ARL₀ = 400 for case 1.

No.	Shift Magnitude	(${\mathit{l}}_{\mathbf{1}}$,${\mathit{l}}_{\mathbf{2}}$)
		(0.10,0.05)			(0.25,0.10)			(1.00,0.50)			(3.00,0.75)			(6.00,1.00)			(15.00,2.00)			(18.00,2.50)
		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR
1	1	402.98	399.77		398.46	397.18		403.69	399.25		400.74	398.91		401.23	399.47		402.06	402.29		399.87	398.82
2		336.93	315.41		284.13	272.77		142.65	155.43		53.49	71.05		31.93	43.08		24.77	26.91		24.47	26.55
3		337.12	315.55		282.02	273.42		142.75	154.52		53.86	71.79		32.39	43.42		24.60	26.91		24.53	26.72
4		301.21	283.13		215.47	216.29		71.75	88.15		22.41	32.42		12.28	17.81		9.10	9.31		9.07	9.19
5		322.08	303.78		280.51	270.46		117.46	134.49		82.71	99.26		62.74	79.47		30.32	43.24		24.64	35.35
6		290.74	279.96		220.49	219.18		70.06	87.09		34.78	48.36		23.97	33.05		17.44	22.24		15.74	20.50
7		285.84	279.15		218.53	216.48		70.31	86.45		34.59	48.20		24.55	32.76		17.35	22.07		15.77	20.49
8		256.39	257.21		172.21	180.69		44.58	60.22		18.14	26.35		11.19	15.82		9.30	9.28		9.01	9.08
mean		304.33	290.60		239.05	235.61		94.22	109.48		42.85	56.78		28.44	37.92		18.98	22.85		17.60	21.13
1	2	402.65	397.58		397.46	399.69		403.99	398.85		401.87	397.75		402.47	397.73		401.92	403.87		398.43	397.76
2		345.15	317.31		295.45	267.50		151.07	138.02		53.91	55.43		31.30	33.21		24.64	24.81		24.74	25.09
3		347.52	316.06		298.43	268.95		147.90	140.05		53.75	55.69		30.61	33.07		24.76	24.92		24.67	24.87
4		312.79	284.79		230.80	209.12		72.83	73.43		21.55	23.46		11.87	12.73		9.02	9.24		8.92	9.01
5		334.33	311.78		287.97	269.98		123.88	115.83		85.52	81.44		63.47	63.14		29.17	31.56		23.48	25.49
6		301.73	273.07		232.01	206.80		72.67	70.03		34.35	35.83		23.74	24.86		17.18	18.29		15.59	16.53
7		302.86	274.39		226.05	212.37		69.78	70.31		33.72	35.91		23.88	25.22		17.50	17.85		15.58	16.19
8		280.76	244.98		183.66	167.97		44.50	45.72		16.99	18.71		10.77	11.60		9.08	9.12		9.11	9.03
mean		317.88	288.91		250.62	228.96		97.52	93.34		42.83	43.78		27.95	29.12		18.76	19.40		17.44	18.03
1	3	400.68	403.10		400.34	398.21		401.40	402.79		403.69	398.99		403.99	399.82		402.96	403.37		401.05	397.76
2		360.69	337.30		305.48	293.30		155.30	148.57		53.53	52.93		29.41	30.49		24.65	24.54		25.07	25.09
3		357.28	331.75		314.26	294.77		158.71	148.44		52.36	53.25		29.32	30.88		24.56	23.80		24.26	24.87
4		324.82	298.59		246.01	231.82		80.95	73.99		21.06	20.44		10.95	11.25		9.04	9.26		9.13	9.01
5		353.53	332.83		312.98	296.48		139.42	124.07		93.90	85.60		69.07	61.42		29.75	28.17		23.00	25.49
6		322.11	299.28		245.65	234.33		77.63	71.58		34.01	32.82		22.43	22.90		16.49	16.63		14.81	16.53
7		318.74	302.66		247.19	232.45		77.26	72.13		33.71	32.93		22.77	22.75		16.62	16.52		14.86	16.19
8		285.65	269.74		199.69	180.07		47.46	43.52		16.32	16.17		10.18	10.40		9.10	9.14		9.04	9.03
mean		331.83	310.31		267.32	251.89		105.25	97.47		43.56	42.02		27.73	27.16		18.60	18.29		17.17	18.03

,

Table A5. The SS-ARL₁ values of individually monitoring the shifts in $\lambda$-set parameters with ARL₀ = 400 for case 2.

No.	Shift Magnitude	(${\mathit{l}}_{\mathbf{1}}$,${\mathit{l}}_{\mathbf{2}}$)
		(0.10,0.05)			(0.25,0.10)			(1.00,0.50)			(3.00,0.75)			(6.00,1.00)			(15.00,2.00)			(18.00,2.50)
		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR
1	1	397.55	398.30		399.10	398.38		401.80	399.76		400.89	397.15		398.24	397.47		402.41	401.55		398.98	397.10
2		295.92	278.22		222.87	213.79		79.81	89.80		25.77	33.71		14.08	18.97		8.29	10.30		7.81	9.24
3		317.05	304.96		264.29	250.14		123.22	125.34		41.46	51.30		21.76	29.03		11.48	15.35		10.72	13.52
4		261.47	251.58		171.15	168.33		46.76	56.01		14.16	19.63		7.58	10.27		4.22	5.26		4.01	4.69
5		304.58	292.22		256.84	245.81		92.43	99.64		59.23	71.02		44.06	54.50		21.09	28.22		16.71	22.84
6		250.00	242.67		166.85	163.97		40.52	50.39		17.60	23.94		10.83	14.63		6.59	8.10		6.20	7.22
7		273.14	253.86		191.35	191.13		51.61	61.78		23.59	31.04		14.45	19.69		8.01	10.54		7.41	9.18
8		221.93	215.05		129.45	134.48		29.35	37.75		11.52	15.73		6.67	8.99		4.08	4.87		3.99	4.44
mean		274.87	262.65		200.40	195.38		66.24	74.39		27.62	35.20		17.06	22.30		9.11	11.81		8.12	10.16
1	2	397.76	402.85		397.88	398.39		397.58	403.06		398.61	402.43		401.75	401.13		398.79	400.57		397.87	399.50
2		314.23	285.59		239.55	223.26		84.79	79.50		24.76	25.55		13.13	13.96		7.81	8.17		7.69	7.98
3		333.64	313.85		284.51	258.43		131.81	119.38		40.35	40.44		20.25	21.55		10.84	11.64		10.38	10.92
4		279.87	262.36		183.18	167.28		47.30	46.06		13.20	14.07		7.03	7.43		4.13	4.20		4.00	4.10
5		322.64	301.13		273.61	252.48		98.70	90.66		63.58	59.91		44.99	43.58		20.14	20.89		15.64	16.69
6		276.48	244.80		182.36	167.70		41.41	41.25		16.49	17.56		10.02	10.74		6.44	6.63		6.05	6.24
7		287.75	261.33		208.10	186.55		53.53	51.03		22.72	23.69		13.42	14.42		7.7	8.14		7.06	7.30
8		241.78	220.68		145.44	129.09		28.37	29.09		10.46	11.31		6.19	6.57		3.99	4.04		3.95	3.92
mean		293.77	269.96		216.68	197.83		69.42	65.28		27.37	27.50		16.43	16.89		8.72	9.10		7.82	8.16
1	3	397.76	401.01		400.81	400.22		402.65	403.57		401.44	398.75		400.37	402.31		401.38	403.16		400.65	398.51
2		337.10	317.12		265.42	250.91		97.69	88.56		25.32	24.11		12.32	12.17		7.92	7.86		7.63	7.55
3		364.87	342.34		309.23	288.35		149.08	135.91		42.25	39.86		18.81	19.23		10.76	10.88		10.20	10.38
4		309.50	284.41		213.59	191.74		55.83	48.87		12.63	12.39		6.20	6.24		4.02	4.02		3.90	3.96
5		337.53	329.57		304.19	281.29		116.63	103.75		74.40	65.65		50.49	46.39		20.30	19.39		14.93	14.97
6		296.19	274.55		202.84	189.21		45.01	42.50		16.10	15.51		9.28	9.22		6.27	6.28		5.95	5.91
7		310.36	296.59		233.24	211.39		61.95	54.84		22.54	21.49		12.52	12.54		7.52	7.49		6.90	6.77
8		262.33	244.93		165.10	151.42		30.82	28.44		9.75	9.80		5.55	5.45		3.88	3.98		3.90	3.95
mean		316.84	298.50		241.94	223.47		79.57	71.84		29.00	26.97		16.45	15.89		8.67	8.56		7.63	7.64

and

Table A6. The SS-ARL₁ values of individually monitoring the shifts in $\lambda$-set parameters with ARL₀ = 400 for case 4.

No.	Shift Magnitude	(${\mathit{l}}_{\mathbf{1}}$,${\mathit{l}}_{\mathbf{2}}$)
		(0.10,0.05)			(0.25,0.10)			(1.00,0.50)			(3.00,0.75)			(6.00,1.00)			(15.00,2.00)			(18.00,2.50)
		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR		$\mathit{\lambda }$-CUSUM	$\mathit{\lambda }$-SR
1	1	399.23	403.30		399.88	403.93		399.43	402.34		397.19	402.97		397.69	398.41		397.07	401.12		398.35	402.48
2		335.51	309.64		273.56	246.07		109.54	98.60		30.68	32.57		15.31	17.07		7.66	8.62		7.28	7.72
3		259.80	239.38		158.91	150.41		38.43	40.02		11.72	13.45		6.63	7.62		4.37	4.59		4.31	4.49
4		228.10	212.45		120.57	115.46		25.41	27.44		7.29	8.53		3.90	4.45		2.27	2.39		2.20	2.30
5		281.55	262.62		224.86	201.87		54.22	55.09		32.85	34.96		23.56	25.58		10.64	12.24		8.35	9.72
6		248.33	229.87		166.30	150.76		33.11	34.60		15.30	17.00		9.70	10.91		5.42	6.08		4.90	5.42
7		200.43	188.28		109.83	100.61		19.71	21.84		8.44	9.63		5.29	6.05		3.68	4.00		3.50	3.74
8		178.24	165.24		86.42	82.71		15.23	17.39		5.83	6.75		3.42	3.93		2.20	2.30		2.21	2.21
mean		247.42	229.64		162.92	149.70		42.24	42.14		16.02	17.56		9.69	10.80		5.18	5.75		4.68	5.09
1	2	399.18	402.73		397.40	400.66		399.95	398.55		399.60	402.41		400.72	401.97		402.45	397.90		402.33	403.20
2		345.83	332.48		287.09	271.39		122.89	106.91		30.57	28.77		13.77	13.86		7.30	7.41		7.15	7.21
3		280.65	265.50		177.07	162.50		41.09	39.02		10.85	11.19		5.94	6.22		4.31	4.39		4.30	4.30
4		250.68	229.10		140.51	127.53		26.45	24.91		6.62	6.72		3.40	3.65		2.21	2.25		2.18	2.20
5		304.77	284.48		231.92	224.84		59.52	54.94		35.24	32.91		23.71	22.84		9.87	9.88		7.72	7.95
6		272.11	257.77		185.15	167.05		35.10	32.70		14.33	14.28		8.70	8.98		5.22	5.27		4.60	4.76
7		218.54	205.19		122.26	110.88		19.61	19.17		7.59	7.88		4.82	4.98		3.56	3.62		3.32	3.44
8		198.98	186.32		97.60	90.59		14.54	14.70		5.24	5.46		3.02	3.14		2.21	2.21		2.16	2.20
mean		267.37	251.55		177.37	164.97		45.60	41.76		15.78	15.32		9.05	9.10		4.95	5.00		4.49	4.58
1	3	397.31	400.44		398.66	399.00		398.55	397.14		398.63	397.26		399.49	398.65		397.53	401.54		400.99	398.62
2		359.65	347.69		311.26	296.05		142.05	129.55		31.05	28.71		12.51	12.46		7.21	7.15		7.00	7.03
3		308.92	284.34		205.17	197.18		49.12	46.68		10.69	10.59		5.76	5.76		4.25	4.30		4.27	4.25
4		274.88	266.62		166.44	158.63		30.17	28.01		6.12	6.15		3.15	3.10		2.17	2.18		2.18	2.17
5		319.60	311.31		269.34	248.46		73.56	68.52		40.93	38.56		27.19	24.87		9.63	9.35		7.18	7.22
6		292.60	283.02		207.29	197.17		40.99	37.82		14.82	14.19		8.33	8.23		4.99	4.96		4.32	4.41
7		246.26	238.14		143.15	135.27		21.91	20.29		7.23	7.17		4.60	4.53		3.47	3.44		3.25	3.28
8		227.58	216.92		120.24	111.64		15.34	14.68		4.85	4.75		2.79	2.78		2.16	2.17		2.21	2.21
mean		289.93	278.29		203.27	192.06		53.31	49.36		16.53	15.73		9.19	8.82		4.84	4.79		4.34	4.37

. Under steady-state conditions, simulation results based on the SS-ARL₁ values in row 5 indicate that the $\lambda$-SR chart generally triggers an alarm signal faster than the $\lambda$-CUSUM chart when the real shift does not exceed the prespecified shift magnitude, except for shift magnitude 1 in cases 1, 2, and 4, where the actual shift equals the prespecified value. Even in these excepted instances, however, the difference in SS-ARL₁ values between the two charts is small. For example, in Table 4 under shift magnitude 1 at row 5 with $(l_1,l_2)=(0.25,0.10)$, the SS-ARL₁ values for the $\lambda$-CUSUM and $\lambda$-SR schemes are 213.04 and 189.71, respectively. Conversely, the monitoring performance of the $\lambda$-CUSUM procedure is better than that of the $\lambda$-SR procedure for true shifts larger than the prespecified shift magnitude. This is illustrated in the same table for shift magnitude 1 at row 5 with $(l_1,l_2)=(6.00,1.00)$, where the SS-ARL₁ values are 20.50 for the $\lambda$-CUSUM chart and 22.33 for the $\lambda$-SR chart. Furthermore, the mean row shows that the $\lambda$-SR procedure triggers an alarm faster when the real shift is smaller than the prespecified shift magnitude. For instance, under shift magnitude 2 at the mean row with $(l_1,l_2)=(1.00,0.50)$ in the same table, the SS-ARL₁ values for the $\lambda$-CUSUM and $\lambda$-SR procedures are 33.70 and 31.76, respectively. When the true shift equals the prespecified magnitude, the SS-ARL₁ values of the $\lambda$-SR and $\lambda$-CUSUM procedures are nearly equivalent. However, the $\lambda$-CUSUM chart again shows better performance once the actual shift surpasses the prespecified magnitude. As a case in point, for shift magnitude 1 at the mean row with $(l_1,l_2)=(3.00,0.75)$ in Table 4, the SS-ARL₁ values are 12.53 for the $\lambda$-CUSUM chart and 13.52 for the $\lambda$-SR chart.

4.3. Simultaneously Monitoring Shifts Both in the p-Set and $\lambda$-Set Parameters with a Combination of the p-SR and $\lambda$-SR Procedures

In this section, we employ a combined p-SR and $\lambda$-SR scheme to simultaneously monitor the shifts in both the p-set and $\lambda$-set parameters. The performance of the control scheme is then assessed using Monte Carlo simulations under the following parameter shift configurations:

$$\left[\begin{array}{ccccccc}{p}_{00}& p_{10}& p_{01}& p_{11}& {\lambda }_1& {\lambda }_2& {\lambda }_{00}\end{array}\right]=\left[\begin{array}{ccccccc}{p}_{00}& p_{10}& p_{01}& p_{11}& {\lambda }_1& {\lambda }_2& {\lambda }_{00}\end{array}\right]+{▵}_{p\lambda },$$

where ${▵}_{p\lambda }$ is defined as the simultaneous shift of both parameter sets from their IC values. The specific values of ${▵}_{p\lambda }$ are given by each row of the following matrix:

$$\left[\begin{array}{ccccccc}-d_1& d_1& 0& 0& l_1& 0& 0\\ -d_1& 0& d_1& 0& 0& l_1& 0\\ -d_1& 0& 0& d_1& l_1& l_1& 0\\ -d_1& {\displaystyle \frac{d_1}{2}}& {\displaystyle \frac{d_1}{2}}& 0& l_2& l_2& l_2\\ -d_1& 0& {\displaystyle \frac{d_1}{2}}& {\displaystyle \frac{d_1}{2}}& l_1+l_2& l_2& l_2\\ -d_1& {\displaystyle \frac{d_1}{3}}& {\displaystyle \frac{d_1}{3}}& {\displaystyle \frac{d_1}{3}}& l_2& l_1+l_2& l_2\\ -d_1& -d_2& {\displaystyle \frac{d_1+d_2}{2}}& {\displaystyle \frac{d_1+d_2}{2}}& l_1+l_2& l_1+l_2& l_2\\ -d_1& {\displaystyle \frac{d_1}{3}}& {\displaystyle \frac{d_1}{3}}& {\displaystyle \frac{d_1}{3}}& l_2& l_2& l_2\\ -d_1& {\displaystyle \frac{d_1+d_2}{2}}& {\displaystyle \frac{d_1+d_2}{2}}& -d_2& l_2& l_1+l_2& l_2\end{array}\right].$$

(25)

In matrix (25), the first combination represents a case with a downward shift $d_1$ in $p_{00}$, an upward shift $d_1$ in $p_{10}$, and an upward shift $l_1$ in ${\lambda }_1$. Likewise, the second combination represents this situation where there is a downward shift $d_1$ in $p_{00}$, an upward shift $d_1$ in $p_{01}$, and an upward shift $l_1$ in ${\lambda }_2$. The third combination corresponds to the following scenario: a downward shift $d_1$ in $p_{00}$, an upward shift $d_1$ in $p_{11}$, and upward shifts $l_1$ in ${\lambda }_1$ and ${\lambda }_2$. The fourth combination represents a scenario with a downward shift $d_1$ in $p_{00}$, upward shifts ${\displaystyle \frac{d_1}{2}}$ in both $p_{10}$ and $p_{01}$, and upward shifts $l_2$ in all three $\lambda$-set parameters. The fifth combination represents this case that there is a downward shift $d_1$ in $p_{00}$, two upward shifts ${\displaystyle \frac{d_1}{2}}$ in parameters $p_{01}$ and $p_{11}$, an upward shift $l_1+l_2$ in ${\lambda }_1$ and two upward shifts $l_2$ in ${\lambda }_2$ and ${\lambda }_{00}$. The sixth combination represents the scenario where $p_{00}$ shifts downward by $d_1$ while the remaining three p-set parameters shift upward by ${\displaystyle \frac{d_1}{3}}$; and where ${\lambda }_2$ shifts upward by $l_1+l_2$ while the remaining two $\lambda$-set parameters shift upward by $l_2$. The seventh combination represents this situation with downward shifts $d_1$ and $d_2$ in $p_{00}$ and $p_{10}$, respectively; upward shifts ${\displaystyle \frac{d_1+d_2}{2}}$ in $p_{01}$ and $p_{11}$; upward shifts $l_1+l_2$ in ${\lambda }_1$ and ${\lambda }_2$; and an upward shift $l_2$ in ${\lambda }_{00}$. The eighth combination involves a downward shift of $d_1$ in $p_{00}$, with the remaining three p-set parameters each shifting upward by ${\displaystyle \frac{d_1}{3}}$. Alongside this, each $\lambda$-set parameter shifts upward by $l_2$, in accordance with the prespecified shift form. The last combination represents the case where there are downward shifts $d_1$ and $d_2$ in $p_{00}$ and $p_{11}$, respectively; upward shifts ${\displaystyle \frac{d_1+d_2}{2}}$ in $p_{10}$ and $p_{01}$; upward shifts $l_2$ in ${\lambda }_1$ and ${\lambda }_{00}$; and an upward shift $l_1+l_2$ in ${\lambda }_2$. Similarly, for each specified case, shift magnitude, and the true shift $\{(d_1,d_2),(l_1,l_2)\}$, nine OOC ARL values are obtained from the shift matrix (25). The average of these nine values is used to represent the ARL₁ under the true shift $(d_1,d_2),(l_1,l_2)$, and the results are presented in the mean row of the tables. Consequently, the performance of the control procedures is evaluated by comparing these ARL₁ values.

Based on the discussions in the preceding two subsections, when the p-set and $\lambda$-set parameters are shifted simultaneously according to the patterns and magnitudes in matrix (25), the resulting shifted parameters still satisfy the constraint conditions of the BZIP model.

The control limits of the $p\lambda$-CUSUM and $p\lambda$-SR charts for the four aforementioned cases with different shift magnitudes, where the ARL₀ = 200, are presented in

Table 5. The control limits of $p\lambda$-CUSUM chart with ARL₀ = 200 for simultaneously monitoring parameter shifts both in the p-set and $\lambda$-set parameters.

Case	Shift Magnitude	p-CUSUM	$\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-CUSUM
		${\mathit{h}}_{\mathit{p}}$	${\mathit{h}}_{\mathit{\lambda }}$	ARL
1	1	2.292	1.086	200.41
	2	2.878	1.590	200.02
	3	3.254	2.031	199.90
2	1	2.136	1.481	200.42
	2	2.785	2.166	199.53
	3	3.261	2.686	199.40
3	1	1.641	2.065	200.49
	2	2.285	2.682	198.92
	3	2.756	3.179	200.25
4	1	1.902	2.087	199.66
	2	2.599	2.825	200.61
	3	3.119	3.290	198.96

and

Table 6. The control limits of $p\lambda$-SR chart with ARL₀ = 200 for simultaneously monitoring parameter shifts both in the p-set and $\lambda$-set parameters.

Case	Shift Magnitude	p-SR	$\mathit{\lambda }$-SR	$\mathit{p}\mathit{\lambda }$-SR
		${\mathit{h}}_{\mathit{p}}$	${\mathit{h}}_{\mathit{\lambda }}$	ARL
1	1	253.625	245.938	199.46
	2	238.125	240.625	200.25
	3	251.375	156.578	199.44
2	1	293.047	265.375	200.07
	2	317.625	225.452	200.15
	3	299.895	180.949	199.93
3	1	290.971	204.047	199.47
	2	310.108	173.100	199.28
	3	315.703	117.344	199.53
4	1	304.625	244.688	200.41
	2	321.047	216.813	199.60
	3	322.879	151.688	199.55

, respectively. The ZS-ARL₁ values for simultaneously monitoring shifts in both the p-set and $\lambda$-set parameters under shift magnitude 1 are shown in Figure 3 for cases 1 to 4. The corresponding ZS-ARL₁ values for shift magnitudes 2 and 3 are given in Figure A3. The true shifts $\{(d_1,d_2)$, $(l_1,l_2)\}$ are set to the following values: $\left\{\right(0.006,0.001)$, $(0.10,0.05)\}$, $\left\{\right(0.012,0.002)$, $(0.25,0.10)\}$, $\left\{\right(0.06,0.003)$, $(0.50,0.25)\}$, $\left\{\right(0.09,0.004)$, $(1.00,0.50)\}$, $\left\{\right(0.15,0.005)$, $(2.00,1.00)\}$, $\left\{\right(0.21,0.006)$, $(2.50,2.00)\}$, and $\left\{\right(0.24,0.007)$, $(3.00,3.00)\}$. The horizontal axis (positions 1–7) corresponds to the seven true shifts $\{(d_1,d_2),(l_1,l_2)\}$ as defined above. The blue solid line represents the ZS-ARL₁ values of the $p\lambda$-CUSUM procedure, and the red dashed line represents those of the $p\lambda$-SR procedure. It can be seen from Figure 3 and Figure A3 that the ZS-ARL₁ values of the $p\lambda$-CUSUM chart are slightly lower than those of the $p\lambda$-SR chart under zero-state conditions. Therefore, the $p\lambda$-CUSUM chart demonstrates superior detection capability in most cases. As the prespecified shift magnitude increases, the difference between the ZS-ARL₁ values of the two control schemes gradually decreases. When the prespecified shift magnitude reaches shift magnitude 3, the monitoring performance of the two control charts is almost identical.

Table 7. The SS-ARL₁ values of simultaneously monitoring shifts both in the p-set and $\lambda$-set parameters with ARL₀ = 200 for case 4.

No.	Shift Magnitude	{(${\mathit{d}}_{\mathbf{1}}$,${\mathit{d}}_{\mathbf{2}}$), (${\mathit{l}}_{\mathbf{1}}$,${\mathit{l}}_{\mathbf{2}}$)}
		{(0.006,0.001), (0.10,0.05)}			{(0.012,0.002), (0.25,0.10)}			{(0.06,0.003), (0.50,0.25)}			{(0.09,0.004), (1.00,0.50)}			{(0.15,0.005), (2.00,1.00)}			{(0.21,0.006), (2.50,2.00)}			{(0.24,0.007), (3.00,3.00)}
		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR
1	1	150.98	132.90		121.96	108.92		59.70	58.83		36.20	38.44		18.02	20.02		12.39	13.79		9.82	11.08
2		136.11	119.46		95.02	86.05		44.66	44.46		22.57	24.06		9.91	11.03		6.93	7.88		5.55	6.26
3		123.33	109.01		77.68	71.21		35.56	36.42		17.39	18.62		7.52	8.38		5.47	6.09		4.35	4.85
4		141.13	125.99		115.00	101.91		54.06	52.96		30.21	32.20		14.78	16.46		7.01	7.79		4.53	5.10
5		129.55	117.14		96.81	85.38		43.17	42.23		22.32	23.60		10.23	11.19		5.21	5.86		3.62	3.96
6		115.83	102.89		72.30	67.11		30.08	31.68		14.75	16.28		6.74	7.63		4.07	4.54		2.98	3.34
7		107.93	94.66		61.12	57.92		24.82	25.76		11.38	12.45		4.97	5.57		3.12	3.48		2.34	2.62
8		140.71	125.42		112.34	102.01		53.44	52.00		29.61	31.51		14.27	15.81		6.71	7.50		4.37	4.88
9		116.61	104.57		72.79	66.85		31.11	32.31		15.01	16.29		6.90	7.74		4.17	4.68		3.01	3.39
mean		129.13	114.67		91.67	83.04		41.85	41.85		22.16	23.72		10.37	11.54		6.12	6.85		4.51	5.05
1	2	159.24	144.21		131.98	119.52		63.35	57.26		37.01	35.78		17.35	17.28		11.40	11.79		9.02	9.28
2		145.47	131.84		103.39	93.63		48.29	46.21		23.61	22.93		9.69	9.69		6.68	6.86		5.20	5.46
3		132.63	119.45		86.38	77.73		38.51	35.26		17.83	17.41		7.26	7.31		5.07	5.24		4.05	4.14
4		151.28	139.44		125.13	113.37		57.63	54.40		31.87	30.59		14.58	14.64		6.42	6.65		4.14	4.32
5		139.61	128.91		105.74	94.12		46.77	43.22		22.94	22.26		9.77	9.90		4.89	5.07		3.29	3.43
6		124.37	114.47		78.47	71.34		32.15	30.82		14.70	14.56		6.38	6.52		3.75	3.90		2.71	2.84
7		115.94	103.52		68.45	61.16		26.11	24.90		11.11	10.96		4.62	4.80		2.86	2.97		2.13	2.24
8		147.91	137.01		123.65	112.27		57.14	52.47		31.52	30.05		14.11	14.17		6.32	6.52		4.01	4.15
9		126.11	112.62		79.01	72.87		33.00	31.12		14.81	14.76		6.49	6.57		3.80	3.96		2.81	2.89
mean		138.06	125.72		100.25	90.67		44.77	41.74		22.82	22.14		10.03	10.10		5.69	5.88		4.15	4.30
1	3	168.25	156.33		143.36	131.40		67.18	63.06		39.87	36.83		17.27	16.81		11.22	10.93		8.36	8.46
2		154.18	144.06		115.87	105.83		56.65	51.84		26.35	25.12		9.87	9.82		6.58	6.60		5.12	5.07
3		141.93	136.07		98.57	89.30		44.89	40.18		20.26	19.02		7.31	7.13		5.03	5.06		3.87	3.90
4		158.87	148.64		133.73	122.51		64.92	60.23		34.99	32.84		15.14	14.75		6.36	6.39		3.91	3.85
5		150.48	138.72		114.48	107.70		53.69	49.21		26.31	24.19		10.20	10.25		4.73	4.73		3.10	3.09
6		134.35	126.76		92.80	82.92		36.06	34.10		15.24	15.23		6.17	6.23		3.46	3.52		2.56	2.55
7		130.73	115.44		79.31	71.23		30.34	28.07		11.67	11.03		4.36	4.34		2.62	2.65		1.96	2.00
8		161.76	149.96		132.98	122.57		62.66	58.02		35.46	33.26		15.20	14.71		6.18	6.21		3.72	3.74
9		133.68	124.81		89.39	81.98		36.71	34.49		15.62	15.33		6.16	6.25		3.50	3.57		2.59	2.59
mean		148.25	137.87		111.17	101.72		50.34	46.58		25.08	23.65		10.19	10.03		5.52	5.52		3.91	3.92

shows the SS-ARL₁ values for simultaneously monitoring shifts in both the p-set and $\lambda$-set parameters under different shift magnitudes for case 4, and the remaining three cases are presented in

Table A7. The SS-ARL₁ values of simultaneously monitoring shifts both in the p-set and $\lambda$-set parameters with ARL₀ = 200 for case 1.

No.	Shift Magnitude	{(${\mathit{d}}_{\mathbf{1}}$,${\mathit{d}}_{\mathbf{2}}$), (${\mathit{l}}_{\mathbf{1}}$,${\mathit{l}}_{\mathbf{2}}$)}
		{(0.006,0.001), (0.10,0.05)}			{(0.012,0.002), (0.25,0.10)}			{(0.06,0.003), (0.50,0.25)}			{(0.09,0.004), (1.00,0.50)}			{(0.15,0.005), (2.00,1.00)}			{(0.21,0.006), (2.50,2.00)}			{(0.24,0.007), (3.00,3.00)}
		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR
1	1	150.22	130.27		118.57	108.05		45.67	47.47		26.49	30.92		13.39	17.00		8.85	11.79		7.16	9.70
2		149.37	132.54		120.14	108.98		45.89	47.34		27.22	30.70		13.35	17.00		8.85	11.70		7.12	9.68
3		140.32	127.72		109.26	102.34		46.20	50.26		26.54	32.58		12.39	16.86		8.20	11.62		6.48	9.39
4		148.65	130.24		119.01	110.94		48.00	49.71		29.85	32.57		15.70	18.99		8.67	11.69		6.10	8.59
5		140.18	125.28		109.05	101.97		45.04	48.13		26.83	31.30		13.55	17.97		7.44	10.51		5.28	7.55
6		141.10	124.21		107.33	100.44		42.27	45.49		24.42	29.74		11.98	16.00		6.71	9.50		4.79	6.80
7		130.15	120.75		94.64	91.59		36.77	41.89		19.24	24.69		8.50	12.02		4.98	7.03		3.68	5.14
8		151.13	130.02		124.83	112.67		50.89	50.76		30.98	34.44		16.19	20.02		8.64	11.84		5.95	8.31
9		138.93	122.61		105.33	99.52		39.83	43.82		23.79	28.04		11.52	15.12		6.61	9.12		4.78	6.64
mean		143.34	127.07		112.02	104.05		44.51	47.21		26.15	30.55		12.95	16.77		7.66	10.53		5.70	7.98
1	2	152.66	136.46		122.23	110.62		46.42	45.03		27.24	27.73		12.70	13.94		8.43	9.31		6.83	7.58
2		153.32	137.19		122.33	112.42		46.95	44.88		26.77	28.10		12.95	13.89		8.50	9.37		6.75	7.61
3		146.54	132.96		113.07	103.56		48.78	47.92		26.90	28.81		12.32	13.88		8.25	9.26		6.51	7.34
4		151.40	136.71		125.76	114.51		49.81	47.01		29.61	30.46		15.64	16.06		8.35	9.33		5.72	6.51
5		148.30	133.46		113.41	103.06		45.84	45.22		27.06	27.41		13.43	14.71		7.34	8.29		5.17	5.92
6		143.91	130.58		109.98	103.31		43.16	42.70		24.64	25.41		11.77	12.84		6.41	7.43		4.71	5.32
7		138.45	124.70		96.71	90.99		37.75	37.79		19.20	20.79		8.29	9.39		4.74	5.47		3.57	4.02
8		156.53	141.37		128.79	118.67		52.04	49.37		31.68	31.48		16.05	16.93		8.34	9.46		5.76	6.55
9		143.44	129.26		107.68	100.78		41.85	40.61		23.42	24.18		11.07	12.14		6.24	7.21		4.57	5.19
mean		148.28	133.63		115.55	106.44		45.84	44.50		26.28	27.15		12.69	13.75		7.40	8.35		5.51	6.23
1	3	160.26	147.91		125.91	119.45		48.19	47.50		27.59	27.83		12.61	12.92		8.17	8.45		6.48	6.73
2		156.64	148.65		126.70	119.03		48.69	47.09		27.91	27.68		12.48	12.90		8.29	8.52		6.36	6.77
3		158.43	145.71		118.15	109.42		52.33	50.49		29.67	28.94		12.79	12.52		8.29	8.30		6.34	6.48
4		157.38	147.09		131.67	122.85		52.55	50.90		31.46	30.39		15.58	15.56		8.11	8.43		5.46	5.77
5		150.85	140.67		119.37	111.76		48.92	47.62		28.06	27.74		13.68	13.60		7.16	7.35		4.85	5.23
6		150.34	141.38		117.05	110.16		46.80	44.51		25.30	25.07		11.67	11.86		6.38	6.54		4.51	4.64
7		148.81	134.38		106.23	96.76		40.25	38.53		20.22	19.48		8.00	8.26		4.61	4.70		3.44	3.51
8		157.65	148.50		133.74	125.42		55.40	53.14		33.43	32.19		16.46	16.34		8.36	8.48		5.61	5.72
9		148.43	137.73		113.92	105.67		43.79	42.37		23.75	23.60		10.78	11.21		5.93	6.22		4.24	4.45
mean		154.31	143.56		121.42	113.39		48.55	46.91		27.49	26.99		12.67	12.80		7.26	7.44		5.25	5.48

,

Table A8. The SS-ARL₁ values of simultaneously monitoring shifts both in the p-set and $\lambda$-set parameters with ARL₀ = 200 for case 2.

No.	Shift Magnitude	{(${\mathit{d}}_{\mathbf{1}}$,${\mathit{d}}_{\mathbf{2}}$), (${\mathit{l}}_{\mathbf{1}}$,${\mathit{l}}_{\mathbf{2}}$)}
		{(0.006,0.001), (0.10,0.05)}			{(0.012,0.002), (0.25,0.10)}			{(0.06,0.003), (0.50,0.25)}			{(0.09,0.004), (1.00,0.50)}			{(0.15,0.005), (2.00,1.00)}			{(0.21,0.006), (2.50,2.00)}			{(0.24,0.007), (3.00,3.00)}
		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR
1	1	149.57	132.33		118.24	108.51		59.02	61.03		34.21	39.77		16.20	20.45		10.95	14.57		8.73	11.55
2		153.69	135.08		128.17	115.38		61.40	61.52		39.08	42.13		19.61	24.10		13.62	17.05		10.90	14.17
3		139.11	123.12		100.54	95.46		46.19	49.94		25.25	30.47		11.43	14.64		7.81	10.39		6.18	8.15
4		150.44	132.92		129.18	115.75		61.60	62.99		38.61	43.21		21.15	25.18		10.95	14.25		7.29	9.68
5		134.68	119.83		99.95	93.83		45.27	49.19		25.33	30.60		12.40	15.96		6.96	9.30		4.99	6.54
6		141.42	126.76		109.94	101.21		50.93	53.28		28.64	33.80		13.55	17.57		7.36	9.81		5.12	6.92
7		125.17	115.04		84.20	84.82		37.14	41.63		18.98	23.82		8.38	11.13		5.07	6.65		3.75	4.89
8		147.46	133.90		125.05	114.29		60.85	60.72		37.37	41.91		19.86	24.30		10.31	13.45		6.74	9.00
9		140.19	126.93		110.90	102.14		51.95	54.40		29.53	34.61		14.12	18.17		7.86	10.51		5.46	7.40
mean		142.41	127.32		111.80	103.49		52.71	54.96		30.78	35.59		15.19	19.06		8.99	11.77		6.57	8.70
1	2	155.12	137.17		124.71	112.00		62.78	57.17		34.87	33.49		15.74	15.93		10.60	11.03		8.34	8.73
2		160.69	141.09		133.39	116.03		63.74	58.87		38.89	37.20		18.89	19.33		12.59	13.37		10.29	10.64
3		145.60	126.47		106.99	93.30		49.54	45.93		25.77	25.30		11.12	11.40		7.49	7.94		5.87	6.16
4		154.64	139.17		134.81	118.00		64.54	60.00		40.31	38.81		20.27	20.69		10.17	10.85		6.77	7.11
5		142.32	122.17		106.32	92.92		47.12	44.67		25.07	24.87		11.72	12.15		6.52	6.98		4.63	4.98
6		145.96	130.35		116.26	102.85		52.27	49.15		28.45	28.33		13.13	13.42		6.90	7.24		4.81	5.09
7		131.54	117.43		92.13	80.17		38.74	37.14		18.73	18.91		7.88	8.34		4.67	4.97		3.48	3.73
8		157.67	138.80		131.95	117.51		63.89	59.97		38.76	37.84		19.81	19.98		9.68	10.22		6.35	6.75
9		146.93	130.75		117.20	102.59		54.87	50.21		29.60	28.83		13.68	13.74		7.24	7.71		5.03	5.33
mean		148.94	131.49		118.19	103.93		55.28	51.46		31.16	30.40		14.69	15.00		8.43	8.92		6.17	6.50
1	3	164.33	153.88		135.16	123.71		69.18	63.32		38.49	35.47		16.19	15.81		10.45	10.45		8.03	8.08
2		168.30	158.83		143.26	134.28		68.84	63.75		41.36	39.52		18.88	18.62		12.29	12.56		9.58	9.83
3		154.96	142.17		116.08	107.01		54.46	51.58		28.50	26.84		11.15	11.10		7.41	7.58		5.61	5.79
4		166.80	154.85		139.67	134.55		69.88	67.13		43.80	40.83		21.32	20.50		10.09	10.04		6.34	6.43
5		150.98	139.75		116.56	105.15		51.54	48.77		27.14	26.19		11.77	11.71		6.16	6.31		4.37	4.39
6		157.27	144.62		125.54	117.16		58.27	54.29		31.20	29.47		13.13	12.94		6.51	6.66		4.38	4.53
7		140.76	132.26		101.72	92.71		41.94	39.69		19.17	18.63		7.49	7.54		4.32	4.41		3.16	3.25
8		169.27	154.40		143.68	130.83		68.65	64.86		43.10	39.99		20.45	19.82		9.61	9.67		6.08	6.17
9		158.20	145.23		126.79	114.34		59.61	54.68		31.68	29.70		13.48	13.34		6.83	6.87		4.64	4.64
mean		158.99	147.33		127.61	117.75		60.26	56.45		33.83	31.85		14.87	14.60		8.19	8.28		5.80	5.90

and

Table A9. The SS-ARL₁ values of simultaneously monitoring shifts both in the p-set and $\lambda$-set parameters with ARL₀ = 200 for case 3.

No.	Shift Magnitude	{(${\mathit{d}}_{\mathbf{1}}$,${\mathit{d}}_{\mathbf{2}}$), (${\mathit{l}}_{\mathbf{1}}$,${\mathit{l}}_{\mathbf{2}}$)}
		{(0.006,0.001),(0.10,0.05)}			{(0.012,0.002),(0.25,0.10)}			{(0.06,0.003),(0.50,0.25)}			{(0.09,0.004),(1.00,0.50)}			{(0.15,0.005),(2.00,1.00)}			{(0.21,0.006),(2.50,2.00)}			{(0.24,0.007),(3.00,3.00)}
		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR		$\mathit{p}\mathit{\lambda }$-CUSUM	$\mathit{p}\mathit{\lambda }$-SR
1	1	140.33	120.98		97.06	89.06		44.91	45.39		22.45	23.81		9.65	10.54		6.72	7.13		5.35	5.57
2		140.13	121.37		96.70	89.10		44.24	44.74		22.91	24.08		9.74	10.30		6.73	7.13		5.26	5.64
3		112.74	99.38		65.08	62.26		27.43	29.11		13.80	14.42		6.13	6.53		4.38	4.62		3.56	3.72
4		137.58	121.48		107.23	97.42		48.67	47.50		26.09	27.10		12.06	13.03		5.63	5.94		3.71	3.89
5		114.74	100.40		71.82	66.53		29.46	30.13		14.70	15.52		7.02	7.46		4.01	4.26		2.95	3.03
6		112.05	100.86		71.25	67.57		28.66	30.20		14.26	15.00		6.56	7.00		3.85	4.04		2.74	2.89
7		93.47	85.86		51.45	49.64		20.36	21.31		9.52	10.12		4.32	4.54		2.69	2.83		2.15	2.23
8		135.14	121.36		106.08	96.46		46.10	46.06		24.68	26.27		11.74	12.42		5.40	5.77		3.55	3.79
9		116.58	102.08		72.40	67.50		31.14	30.76		14.72	15.67		6.75	7.02		3.88	4.08		2.84	2.97
mean		122.53	108.20		82.12	76.17		35.66	36.13		18.12	19.11		8.22	8.76		4.81	5.09		3.57	3.75
1	2	139.83	131.63		101.41	93.86		46.40	43.89		23.03	22.23		9.36	9.37		6.31	6.43		5.00	5.04
2		139.82	128.47		101.91	93.04		46.77	43.60		22.90	22.28		9.28	9.38		6.31	6.52		4.97	4.99
3		118.02	107.51		69.42	63.96		28.17	27.87		13.85	13.81		5.88	5.92		4.16	4.25		3.34	3.42
4		144.63	130.36		111.58	102.73		49.99	47.52		26.74	25.62		11.72	11.83		5.25	5.29		3.41	3.44
5		117.60	107.97		74.10	68.50		29.97	29.32		14.72	14.50		6.67	6.71		3.77	3.83		2.72	2.74
6		118.38	108.35		75.00	68.92		30.15	29.30		14.17	13.98		6.27	6.29		3.59	3.62		2.61	2.64
7		97.57	92.08		55.30	50.33		20.68	20.39		9.35	9.26		4.08	4.05		2.56	2.53		1.97	2.01
8		141.21	129.56		108.36	100.01		48.14	46.15		25.36	25.07		11.31	11.46		5.13	5.17		3.36	3.37
9		118.18	107.95		75.95	69.94		31.04	29.72		14.28	14.48		6.43	6.30		3.61	3.59		2.65	2.63
mean		126.14	115.99		85.89	79.03		36.81	35.31		18.26	17.91		7.89	7.92		4.52	4.58		3.34	3.37
1	3	153.29	137.75		108.75	100.62		50.86	47.12		24.16	23.23		9.65	9.25		6.16	6.17		4.83	4.75
2		149.24	141.85		109.14	99.67		51.58	46.31		24.40	23.24		9.37	9.26		6.21	6.05		4.79	4.76
3		126.47	112.92		77.28	69.36		31.57	29.92		15.14	14.49		6.17	5.92		4.23	4.09		3.28	3.29
4		149.07	139.14		122.01	113.09		55.30	51.24		28.68	26.99		12.42	12.21		5.16	5.15		3.25	3.30
5		125.55	113.66		82.42	76.09		33.34	31.54		15.59	15.37		6.88	6.72		3.71	3.73		2.61	2.62
6		128.11	116.63		81.34	75.26		33.06	30.70		15.29	14.61		6.27	6.25		3.43	3.45		2.48	2.43
7		107.57	98.13		61.37	55.04		22.75	21.49		9.86	9.29		3.87	3.84		2.40	2.44		1.91	1.93
8		153.06	137.45		119.40	108.21		53.24	48.49		27.57	26.33		12.24	11.73		5.02	5.05		3.15	3.17
9		124.07	118.51		83.20	75.50		34.08	31.38		15.54	14.85		6.23	6.19		3.44	3.43		2.52	2.45
mean		135.16	124.01		93.88	85.87		40.64	37.58		19.58	18.71		8.12	7.93		4.42	4.40		3.20	3.19

. Under steady-state conditions, from the SS-ARL₁ values in row 8, it can be seen that the $p\lambda$-SR procedure significantly outperforms the $p\lambda$-CUSUM procedure when the true shift is less than the prespecified shift magnitude. For instance, in Table 7, when the true shift is $\{(d_1,d_2)$, $(l_1,l_2)\}$ = $\left\{\right(0.012,0.002)$, $(0.25,0.10)\}$ at shift magnitude 1, the SS-ARL₁ values are 112.34 for the $p\lambda$-CUSUM scheme and 102.01 for the $p\lambda$-SR scheme. Conversely, the $p\lambda$-CUSUM chart demonstrates a marked performance advantage once the actual shift reaches or exceeds the prespecified magnitude. A case in point is shift magnitude 1 with $\{(d_1,d_2)$, $(l_1,l_2)\}$ = $\left\{\right(0.15,0.005)$, $(2.00,1.00)\}$ in the same table, where the SS-ARL₁ values are 14.27 for $p\lambda$-CUSUM and 15.81 for $p\lambda$-SR. This conclusion is further supported by the mean row. In most scenarios, the $p\lambda$-SR chart shows better monitoring performance than the $p\lambda$-CUSUM chart for actual shifts below the prespecified shift magnitude, with only occasional exceptions where their SS-ARL₁ values are nearly identical. To illustrate, in Table 7 under shift magnitude 2 with $\{(d_1,d_2)$, $(l_1,l_2)\}$ = $\left\{\right(0.06,0.003)$, $(0.50,0.25)\}$, the SS-ARL₁ values are 44.77 for $p\lambda$-CUSUM and 41.74 for $p\lambda$-SR. However, when the actual shift meets or exceeds the prespecified shift magnitude, the $p\lambda$-CUSUM scheme consistently outperforms the $p\lambda$-SR scheme, as seen for the same parameter set $\left\{\right(0.15,0.005)$, $(2.00,1.00)\}$ at shift magnitude 1, yielding SS-ARL₁ values of 10.37 and 11.54, respectively.

In practice, a control scheme with superior SS-ARL performance is more meaningful, since processes are usually initially in control but may shift to an OOC state after a change point. In this paper, we conclude that the p-SR, $\lambda$-SR, and $p\lambda$-SR charts outperform the p-CUSUM, $\lambda$-CUSUM, and $p\lambda$-CUSUM charts in terms of SS-ARL performance.

4.4. Run-Length Performance of the Proposed SR Procedures

Several run-length characteristics are presented in this section as performance measures for the SR control procedures. These include the ARL, the standard deviation of the run length (SDRL), and selected quantiles of the run-length distribution. The SDRL indicates the reliability of ARL estimation. Lower values reflect better control chart performance. The quantiles provide deeper insight into the run-length distribution. Specifically, we report the 5th, 25th, 50th, 75th and 95th percentiles, denoted as $Q_5$, $Q_{25}$, $Q_{50}$, $Q_{75}$ and $Q_{95}$, respectively.

In this part, we investigate the run length characteristics of the SR control procedures under both zero-state and steady-state conditions. Due to space limitations, we only present selected simulation results for case 1 with shift magnitude 2. The true shift forms and corresponding control limits for the p-SR, $\lambda$-SR and $p\lambda$-SR schemes are specified as combination 8 in matrix (23), combination 5 in matrix (24) and combination 8 in matrix (25), respectively. The corresponding control limits are given in Table 1, Table 3 and Table 6.

Table 8. ZS-ARL, SS-ARL, SDRL and run-length quantiles of the p-SR, $\lambda$-SR and $p\lambda$-SR procedures.

p-SR
Zero-state								Steady-state
$d_1$	ZS-ARL	SDRL	$Q_5$	$Q_{25}$	$Q_{50}$	$Q_{75}$	$Q_{95}$	$d_1$	SS-ARL	SDRL	$Q_5$	$Q_{25}$	$Q_{50}$	$Q_{75}$	$Q_{95}$
0.00	400.37	372.42	46	136	287	545	1160	0.00	398.35	366.90	46	135	286	543	1136
0.006	322.66	295.00	40	113	233	439	912	0.006	298.61	295.49	17	88	208	412	890
0.012	270.51	246.51	38	97	197	363	757	0.012	241.59	238.19	15	72	168	339	712
0.06	89.97	65.98	23	43	71	117	224	0.06	76.93	67.87	8	29	58	104	215
0.09	62.66	41.86	19	33	51	80	146	0.09	48.80	40.46	7	20	38	66	128
0.15	37.29	20.54	14	23	32	47	76	0.15	27.64	20.86	5	13	22	36	69
0.21	27.01	12.97	12	18	24	33	51	0.21	18.88	12.86	4	10	16	25	44
0.24	23.95	10.79	11	16	22	29	44	0.24	16.59	10.59	4	9	14	22	37
$\lambda$-SR
Zero-state								Steady-state
$l_2$	ZS-ARL	SDRL	$Q_5$	$Q_{25}$	$Q_{50}$	$Q_{75}$	$Q_{95}$	$l_2$	SS-ARL	SDRL	$Q_5$	$Q_{25}$	$Q_{50}$	$Q_{75}$	$Q_{95}$
0.00	397.45	335.33	72	161	295	529	1069	0.00	397.90	335.67	71	161	299	533	1054
0.05	343.58	282.47	65	144	260	456	904	0.05	309.35	289.49	28	102	221	425	886
0.10	303.31	250.87	59	131	231	392	793	0.10	266.40	245.36	25	94	195	360	750
0.50	145.90	98.96	40	76	119	187	341	0.50	115.07	95.97	13	46	89	156	307
0.75	107.67	69.26	32	60	91	137	243	0.75	81.09	65.63	11	35	64	109	208
1.00	85.02	50.17	27	49	73	108	181	1.00	63.09	49.69	8	27	51	85	161
2.00	47.15	25.64	15	29	42	61	95	2.00	31.53	23.30	4	14	26	43	77
2.50	38.70	21.13	13	23	35	49	79	2.50	25.37	19.11	3	11	21	35	63
$p\lambda$-SR
Zero-state								Steady-state
$d_1$, $l_2$	ZS-ARL	SDRL	$Q_5$	$Q_{25}$	$Q_{50}$	$Q_{75}$	$Q_{95}$	$d_1$, $l_2$	SS-ARL	SDRL	$Q_5$	$Q_{25}$	$Q_{50}$	$Q_{75}$	$Q_{95}$
0.00, 0.00	202.21	161.32	39	89	156	267	518	0.00, 0.00	199.31	159.41	37	88	156	261	516
0.006, 0.05	169.59	132.30	34	77	132	223	431	0.006, 0.05	138.68	127.46	11	46	100	194	391
0.012, 0.10	144.43	109.25	31	66	115	191	355	0.012, 0.10	115.54	107.38	10	40	83	158	329
0.06, 0.25	67.34	43.27	20	36	56	86	151	0.06, 0.25	49.94	42.51	6	19	38	68	133
0.09, 0.50	45.96	26.17	16	27	40	58	97	0.09, 0.50	31.45	25.02	4	13	25	43	81
0.15, 1.00	27.80	13.05	12	19	25	34	53	0.15, 1.00	16.99	12.32	3	8	14	23	41
0.21, 2.00	17.39	7.62	8	12	16	21	31	0.21, 2.00	9.41	6.73	2	4	8	13	23
0.24, 3.00	12.59	5.76	5	8	12	16	23	0.24, 3.00	6.55	4.73	1	3	5	9	16

reports the ZS-ARL, SS-ARL, SDRL and quantiles for all three schemes. Since the true shifts in combination 8 of matrix (23), combination 5 of matrix (24) and combination 8 of matrix (25) only contain $d_1$ and $l_2$, the table only includes these two values.

From Table 8, we observe that the SDRL values are all smaller than the corresponding ARL values. This indicates that the run lengths obtained by the SR schemes are relatively stable. For example, for the p-SR procedure, when the true shift magnitude $d_1=0.06$, the ZS-ARL and the corresponding SDRL are 89.97 and 65.98, respectively. Moreover, it can be seen from this table that the median run length, that is, $Q_{50}$, is significantly smaller than the corresponding ARL, suggesting that the run lengths are skewed to the right. For instance, for $\lambda$-SR procedure, when the true shift magnitude $l_2=0.50$, the SS-ARL and $Q_{50}$ are 115.07 and 89, respectively.

5. Sensitivity Analysis

In this section, we investigate the influence of biased parameter estimates and violation of the independence assumption on the monitoring performance of the proposed SR procedure.

5.1. Effect of Parameter Estimation for the SR Procedure with BZIP Model

We previously assumed that the IC parameters were known. In practice, however, they are usually unknown and must be estimated from suitable reference sample observations collected from an IC process. In this section, we study the effect of parameter estimation on the SR chart under the BZIP model. Since a comprehensive analysis over all possible models and parameters is not practical, we therefore select a few representative cases for illustration. The zero-state $p\lambda$-SR chart is used as an example. Specifically, we use case 4 with prespecified shift magnitude 2. The true shift form is taken as combination 8 in matrix (25). The nominal ${\mathrm{ARL}}_0$ is set to 200, and the corresponding control limits for the $p\lambda$-SR chart are $h_p=321.047$ and $h_{\lambda }=216.813$, as given in Table 6. We assume that the estimation errors are between $[-5\%,5\%]$. For convenience, we focus on four parameters: $p_{00}$, ${\lambda }_1$, ${\lambda }_2$ and ${\lambda }_{00}$. The estimation errors are introduced as follows: ${\widehat{p}}_{00}=p_{00}(1-{\eta }_{p_{00}})$, ${\widehat{\lambda }}_1={\lambda }_1(1-{\eta }_{{\lambda }_1})$, ${\widehat{\lambda }}_2={\lambda }_2(1-{\eta }_{{\lambda }_2})$ and ${\widehat{\lambda }}_{00}={\lambda }_{00}(1-{\eta }_{{\lambda }_{00}})$. Because the p-set and $\lambda$-set parameters must satisfy the constraints described in Section 2.1, an overestimation or underestimation of $p_{00}$ will also affect the estimates of $p_{10}$, $p_{01}$ and $p_{11}$. The specific estimated values are listed in

Table 9. Estimated parameter values after introducing estimation errors to $p_{00}$, ${\lambda }_1$, ${\lambda }_2$ and ${\lambda }_{00}$.

${\mathit{\eta }}_{{\mathit{p}}_{\mathbf{00}}}$	$\widehat{{\mathit{p}}_{\mathbf{00}}}$	$\widehat{{\mathit{p}}_{\mathbf{10}}}$	$\widehat{{\mathit{p}}_{\mathbf{01}}}$	$\widehat{{\mathit{p}}_{\mathbf{11}}}$	${\mathit{\eta }}_{{\mathit{\lambda }}_{\mathbf{1}}}$	$\widehat{{\mathit{\lambda }}_{\mathbf{1}}}$	${\mathit{\eta }}_{{\mathit{\lambda }}_{\mathbf{2}}}$	$\widehat{{\mathit{\lambda }}_{\mathbf{2}}}$	${\mathit{\eta }}_{{\mathit{\lambda }}_{\mathbf{00}}}$	$\widehat{{\mathit{\lambda }}_{\mathbf{00}}}$
0.05	0.513	0.109	0.209	0.169	0.05	2.375	0.05	1.425	0.05	0.95
0.00	0.540	0.100	0.200	0.160	0.00	2.500	0.00	1.500	0.00	1.00
−0.05	0.567	0.091	0.191	0.151	−0.05	2.625	−0.05	1.575	−0.05	1.05

.

Table 10. Effect of parameter estimation on the $p\lambda$-SR procedure for ARL₀ and ARL₁ under the zero-state. The control limits are obtained with an ARL₀ of 200 using case 4 and prespecified shift magnitude 2.

${\mathit{\eta }}_{{\mathit{p}}_{\mathbf{00}}}$	${\mathit{\eta }}_{{\mathit{\lambda }}_{\mathbf{1}}}$	${\mathit{\eta }}_{{\mathit{\lambda }}_{\mathbf{2}}}$	${\mathit{\eta }}_{{\mathit{\lambda }}_{00}}$	${\mathit{d}}_{\mathbf{1}},{\mathit{l}}_{\mathbf{2}}$
				${\mathit{d}}_{\mathbf{1}}=\mathbf{0.00}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{0.00}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.006}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{0.05}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.012}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{0.10}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.06}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{0.25}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.09}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{0.50}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.15}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{1.00}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.21}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{2.00}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.24}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{3.00}$
0.00	0.00	0.00	0.00	203.04	165.12	137.15	71.87	43.03	21.67	10.18	6.58
0.05	0.00	0.00	0.00	202.14	165.59	134.72	70.97	42.04	21.17	9.75	6.37
−0.05	0.00	0.00	0.00	203.23	167.59	137.99	72.07	43.87	22.57	10.45	6.79
0.00	0.05	0.00	0.00	203.40	164.08	136.61	71.86	43.07	21.76	10.07	6.51
0.00	−0.05	0.00	0.00	200.51	164.89	136.24	72.16	43.79	21.86	10.20	6.63
0.00	0.00	0.05	0.00	201.16	164.47	137.86	71.46	42.51	21.31	9.90	6.45
0.00	0.00	−0.05	0.00	201.01	164.70	137.06	71.26	43.77	22.09	10.22	6.68
0.00	0.00	0.00	0.05	202.93	165.44	136.43	71.76	43.36	21.61	10.06	6.57
0.00	0.00	0.00	−0.05	199.79	164.72	136.92	71.70	43.50	21.77	10.13	6.59
0.05	0.05	0.05	0.05	201.47	164.46	134.38	69.86	41.92	20.70	9.52	6.19
−0.05	−0.05	−0.05	−0.05	201.42	167.47	137.96	72.21	44.84	23.10	10.76	7.02

presents the ARL₀ and ARL₁ values under various estimation errors for the $p\lambda$-SR procedure. Since combination 8 in matrix (25) only involves the shift magnitudes $d_1$ and $l_2$, the table only includes these two shift magnitudes. The simulation results indicate that the SR scheme is robust to estimation bias. For example, in Table 10, when only ${\lambda }_1$ has estimation bias, the ARL₀ values are 203.40 and 200.51, respectively. When $d_1=0.09$ and $l_2=0.50$, the corresponding ARL₁ values are 43.07 and 43.79, respectively.

5.2. Effect of the Independence Assumption for the SR Procedure with BZIP Model

In the preceding discussion, we assumed that the observations collected at different time points are independent of each other. However, the monitoring performance of control charts based on model assumptions is more susceptible to deviations from those assumptions. It is therefore necessary to investigate the robustness of the proposed SR chart when the model is mis-specified. This section specifically investigates the influence of autocorrelation on the SR chart’s monitoring performance. For illustration, we study the performance of the $p\lambda$-SR chart under zero-state conditions using case 2 with prespecified shift magnitude 2, and the actual shift form is taken as combination 8 in matrix (25). The corresponding control limits for the $p\lambda$-SR chart are shown in Table 6 with ARL₀ = 200. Data are generated from a first-order bivariate zero-inflated negative binomial (BZINB(1)) autoregressive model. Let $\mathbf{X}$ and $\mathbf{R}$ be non-negative integer-valued random vectors, and let $\mathbf{A}$ be a $2\times 2$ diagonal matrix. The BINAR(1) process is expressed as follows:

$${\mathbf{Y}}_t=\mathbf{A}\circ {\mathbf{Y}}_{t-1}+{\mathbf{R}}_t=\left[\begin{array}{cc}{\alpha }_1& 0\\ 0& {\alpha }_2\end{array}\right]\circ \left[\begin{array}{c}{Y}_{1,t-1}\\ Y_{2,t-1}\end{array}\right]+\left[\begin{array}{c}{R}_{1t}\\ R_{2t}\end{array}\right],t\in Z.$$

The binomial thinning operator “∘” is defined as $\alpha \circ Y={\sum }_{i=1}^Y{X}_i$, where ${\left\{X_i\right\}}_{i=1}^Y$ is a sequence of independent and identically distributed Bernoulli random variables with $P(X_i=1)=1-P(X_i=0)=\alpha$, for $0\le \alpha \le 1$. The innovation ${\mathbf{R}}_t$ follows the BZIP model. Clearly, when ${\alpha }_1={\alpha }_2=0$, the BINAR(1) model degenerates into a sequence of independent observations.

We use the IC and OOC ARL to evaluate the effect of autocorrelation on the performance of the SR scheme. The simulation results are presented in

Table 11. Effect of autocorrelation on the performance of the $p\lambda$-SR procedure evaluated by ARL₀ and ARL₁ under the BINAR(1) process.

${\mathit{\alpha }}_{\mathbf{1}}$	${\mathit{\alpha }}_{\mathbf{2}}$	${\mathit{d}}_{\mathbf{1}},{\mathit{l}}_{\mathbf{2}}$
		${\mathit{d}}_{\mathbf{1}}=\mathbf{0.00}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{0.00}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.006}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{0.05}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.012}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{0.10}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.06}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{0.25}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.09}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{0.50}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.15}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{1.00}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.21}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{2.00}$	${\mathit{d}}_{\mathbf{1}}=\mathbf{0.24}$ ${\mathit{l}}_{\mathbf{2}}=\mathbf{3.00}$
0.0	0.0	198.66	169.57	148.43	81.09	56.70	33.50	18.39	12.33
0.1	0.1	78.40	70.80	66.87	44.52	35.86	26.01	18.32	13.99
0.2	0.2	45.84	43.92	41.44	30.88	26.37	20.55	15.98	13.08
0.3	0.3	32.88	31.81	30.60	24.16	21.31	17.27	14.04	11.84
0.4	0.4	25.84	24.97	23.98	20.08	18.09	15.21	12.55	10.67
0.5	0.5	21.02	20.30	19.75	17.09	15.49	13.36	11.25	9.56

. Since the true shift pattern in combination 8 of matrix (25) only involves the shift magnitudes $d_1$ and $l_2$, we only list these two values in the table. The simulation results show that the $p\lambda$-SR chart is adversely affected by violations of the independence assumption. For example, when $d_1=0.00$ and $l_2=0.00$, the ARL₀ values corresponding to ${\alpha }_1={\alpha }_2=0.1,0.2,0.3,0.4,0.5$ are $78.40,45.84,32.88,25.84,21.02$, respectively. These values are much smaller than the ARL₀ under independent observations. For the ARL₁, when $d_1=0.06$ and $l_2=0.25$, the ARL₁ values are $44.52,30.88,24.16,20.08,17.09$ as ${\alpha }_1$ and ${\alpha }_2$ increase from $0.1$ to $0.5$. These values are also much smaller than the ARL₁ of $81.09$ under independent observations. As the autoregressive parameters ${\alpha }_1$ and ${\alpha }_2$ increase, both the IC and OOC ARL values decrease gradually. This suggests that the $p\lambda$-SR chart becomes less reliable as the serial correlation becomes stronger.

6. An Illustrative Example

This section validates the proposed SR control procedure with an application from the light-emitting diode (LED) packaging industry [28]. The LED packaging process entails two sequential operations: component placement, performed by robotic systems to mount LEDs onto printed circuit boards (PCBs), and soldering, which establishes electrical connections with golden wires. The manufactured PCBs are characterized by two predominant defect types: LED mounting errors (Defect 1; denoted as D₁) and soldering errors (Defect 2; denoted as D₂). He et al. [28] collected count data from 2067 manufacturing batches. From this dataset, 100 IC samples were selected to estimate the process parameters using the method-of-moments estimation proposed by Li et al. [3]; the raw data for these samples are available in the original publication. The estimated parameters are as follows: $\widehat{p_{00}}$ = 0.6230, $\widehat{p_{10}}$ = 0.1016, $\widehat{p_{01}}$ = 0.2107, $\widehat{p_{11}}$ = 0.0647, $\widehat{{\lambda }_1}$ = 3.9077, $\widehat{{\lambda }_2}$ = 5.3014, and $\widehat{{\lambda }_{00}}$ = 2.6070.

Likewise, we also set the prespecified shift magnitudes as follows: $p_{001}$ = $\widehat{p_{00}}-0.15=0.4730$, $p_{101}$ = $\widehat{p_{10}}+0.05=0.1516$, $p_{011}$ = $\widehat{p_{01}}+0.05=0.2607$, $p_{111}$ = $\widehat{p_{11}}+0.05=0.1147$, ${\lambda }_{11}$ = $\widehat{{\lambda }_1}+1=4.9077$, ${\lambda }_{21}$ = $\widehat{{\lambda }_2}+1=6.3014$, and ${\lambda }_{001}$ = $\widehat{{\lambda }_{00}}+1=3.6070$. To independently monitor shifts in the p-set and $\lambda$-set parameters, the ARL₀ for both the p-CUSUM and p-SR charts is set to approximately 400. Simulations using the estimated parameters yielded the corresponding control limits: $h_p=2.863$ for the p-CUSUM chart and $h_p=334.995$ for the p-SR chart. Similarly, the ARL₀ for both the $\lambda$-CUSUM chart and the $\lambda$-SR chart is set to around 400. The control limits for the $\lambda$-CUSUM and $\lambda$-SR schemes are $h_{\lambda }=2.375$ and $h_{\lambda }=297.813$, respectively. Furthermore, to simultaneously monitor the p-set and $\lambda$-set parameters, the ARL₀ is set to approximately 200 for both the $p\lambda$-CUSUM and $p\lambda$-SR charts. Consequently, the adjusted control limits are $h_p$ = 2.812 and $h_{\lambda }$ = 2.324 for the $p\lambda$-CUSUM chart, and $h_p$ = 304.995 and $h_{\lambda }$ = 267.813 for the $p\lambda$-SR chart.

Since no OOC observations were collected from the actual process, we generated the Phase II OOC samples through Monte Carlo simulations. Therefore, this application serves only as an illustrative example. For phase II OOC samples, we generated 100 additional OOC samples corresponding to a shift in the p-set parameters, with the actual shift sizes defined as: $p_{00}={\widehat{p}}_{00}-0.09$, $p_{10}={\widehat{p}}_{10}+0.03$, $p_{01}={\widehat{p}}_{01}+0.03$, and $p_{11}={\widehat{p}}_{11}+0.03$ (see

Table 12. Generated 100 additional OOC samples with shift in the p-set parameters.

No.	D1	D2	No.	D1	D2	No.	D1	D2	No.	D1	D2	No.	D1	D2
1	5	0	21	0	0	41	0	10	61	0	0	81	0	5
2	0	0	22	0	0	42	5	8	62	0	0	82	0	0
3	2	0	23	0	9	43	0	0	63	7	7	83	0	0
4	0	7	24	0	0	44	1	6	64	0	0	84	0	0
5	0	0	25	0	5	45	0	0	65	5	0	85	0	0
6	0	4	26	0	3	46	0	6	66	7	8	86	2	0
7	0	7	27	0	5	47	0	0	67	0	2	87	0	7
8	0	4	28	0	0	48	0	0	68	0	4	88	0	0
9	1	2	29	0	0	49	4	4	69	5	0	89	7	9
10	0	5	30	0	0	50	0	0	70	0	0	90	6	9
11	0	0	31	0	4	51	1	0	71	0	0	91	0	0
12	4	0	32	0	5	52	0	0	72	1	6	92	0	0
13	0	0	33	0	3	53	0	0	73	1	0	93	4	5
14	0	0	34	0	0	54	0	8	74	0	0	94	0	0
15	0	0	35	0	6	55	0	0	75	0	0	95	0	6
16	0	0	36	0	6	56	0	4	76	0	6	96	0	3
17	0	0	37	0	8	57	0	0	77	0	0	97	0	8
18	0	7	38	0	0	58	0	0	78	6	0	98	0	0
19	0	0	39	0	0	59	0	4	79	0	0	99	0	0
20	0	0	40	0	0	60	0	0	80	4	0	100	0	0

). For monitoring the $\lambda$-set parameters, we also generated 100 additional OOC samples with the following shift sizes: ${\lambda }_1={\widehat{\lambda }}_1+0.5$, ${\lambda }_2={\widehat{\lambda }}_2+0.5$, ${\lambda }_{00}={\widehat{\lambda }}_{00}+0.5$ (see

Table 13. Generated 100 additional OOC samples with shift in the $\lambda$-set parameters.

No.	D1	D2	No.	D1	D2	No.	D1	D2	No.	D1	D2	No.	D1	D2
1	0	7	21	0	5	41	0	0	61	0	0	81	0	0
2	0	0	22	0	0	42	0	0	62	0	8	82	0	3
3	0	8	23	0	0	43	0	6	63	0	0	83	0	0
4	0	0	24	0	0	44	5	0	64	0	1	84	0	0
5	0	0	25	0	0	45	0	0	65	0	0	85	0	0
6	0	0	26	0	0	46	0	0	66	0	0	86	0	6
7	0	0	27	0	0	47	0	0	67	0	7	87	0	0
8	0	0	28	0	0	48	0	0	68	0	5	88	8	0
9	0	3	29	0	0	49	0	0	69	0	0	89	0	0
10	4	6	30	0	0	50	0	4	70	0	9	90	0	0
11	0	0	31	0	0	51	0	0	71	0	0	91	0	0
12	0	9	32	0	0	52	6	8	72	0	0	92	0	9
13	0	0	33	0	3	53	0	0	73	0	0	93	0	0
14	0	0	34	7	0	54	0	0	74	0	11	94	0	0
15	7	0	35	4	6	55	0	0	75	0	0	95	0	9
16	0	0	36	0	0	56	5	0	76	0	7	96	0	0
17	0	0	37	0	4	57	0	7	77	0	0	97	0	4
18	0	0	38	0	0	58	0	0	78	0	0	98	0	0
19	0	0	39	0	0	59	0	0	79	3	0	99	0	0
20	0	3	40	0	0	60	0	0	80	5	0	100	0	0

). In addition, we generated another 100 OOC samples involving simultaneous shifts in both the p-set and $\lambda$-set parameters, with the actual shift sizes given by: $p_{00}={\widehat{p}}_{00}-0.09$, $p_{10}={\widehat{p}}_{10}+0.03$, $p_{01}={\widehat{p}}_{01}+0.03$, $p_{11}={\widehat{p}}_{11}+0.03$, ${\lambda }_1={\widehat{\lambda }}_1+0.5$, ${\lambda }_2={\widehat{\lambda }}_2+0.5$, ${\lambda }_{00}={\widehat{\lambda }}_{00}+0.5$ (see

Table 14. Generated 100 additional OOC samples with shift both in the p-set and $\lambda$-set parameters.

No.	D1	D2	No.	D1	D2	No.	D1	D2	No.	D1	D2	No.	D1	D2
1	2	0	21	0	4	41	0	0	61	0	0	81	0	0
2	5	0	22	0	0	42	0	7	62	0	0	82	0	0
3	0	5	23	0	0	43	0	7	63	5	8	83	2	0
4	0	0	24	0	0	44	0	0	64	0	0	84	3	7
5	0	0	25	7	0	45	0	0	65	0	5	85	0	0
6	0	7	26	0	5	46	0	1	66	2	0	86	0	0
7	0	0	27	0	0	47	4	7	67	0	4	87	0	5
8	0	0	28	0	0	48	0	0	68	0	0	88	5	8
9	0	0	29	0	0	49	0	5	69	0	0	89	0	0
10	0	7	30	0	0	50	0	0	70	0	4	90	0	7
11	0	0	31	6	0	51	0	0	71	4	0	91	0	0
12	0	3	32	0	0	52	4	0	72	0	6	92	5	5
13	5	0	33	0	0	53	0	0	73	4	9	93	0	0
14	0	6	34	0	2	54	0	0	74	5	7	94	0	0
15	0	5	35	0	6	55	0	0	75	0	0	95	0	0
16	3	0	36	0	0	56	0	8	76	0	6	96	0	0
17	0	0	37	0	0	57	0	0	77	0	0	97	0	5
18	5	5	38	0	0	58	0	9	78	0	0	98	5	6
19	0	0	39	0	0	59	0	7	79	0	0	99	0	0
20	0	0	40	0	6	60	0	0	80	4	0	100	0	7

).

Figure 4 displays the plotting statistics for the p-CUSUM and p-SR procedures, along with the OOC samples from Table 12. The blue dotted lines represent the corresponding control limits $h_p$. As shown in Figure 4, the p-CUSUM scheme does not signal until the 90th observation, whereas the p-SR scheme signals at the 73th observation.

Figure 5 illustrates the $\lambda$-CUSUM and $\lambda$-SR plotting statistics for the OOC samples in Table 13. The blue dotted lines represent the corresponding control limits $h_{\lambda }$. The $\lambda$-CUSUM procedure signals at the 88th observation, while the $\lambda$-SR procedure signals earlier, at the 74th observation, indicating that the $\lambda$-SR scheme responds more quickly to small parameter shifts than the $\lambda$-CUSUM scheme.

Finally, Figure 6 presents the $p\lambda$-CUSUM and $p\lambda$-SR plotting statistics based on the OOC samples in Table 14. In each panel of Figure 6, the blue dashed line represents the control limit $h_p$ for monitoring shifts in the p-set parameters, and the red dashed line represents the control limit $h_{\lambda }$ for monitoring shifts in the $\lambda$-set parameters. From the figure, we observe that the $p\lambda$-CUSUM scheme signals at the 92th observation, and the $p\lambda$-SR scheme signals at the 88th observation. This result demonstrates that the $p\lambda$-SR scheme is more sensitive than the $p\lambda$-CUSUM scheme in detecting small shifts in both the p-set and $\lambda$-set parameters.

7. Conclusions

In modern high-quality manufacturing, multiple correlated defect types frequently arise simultaneously from complex production systems. Although univariate ZIP models have been extensively used for monitoring single defect categories, they are inherently inadequate when two or more defect types coexist. To address this shortcoming, we have developed an SR control chart to monitor the BZIP processes. Two distinct monitoring procedures are introduced: a p-SR chart that tracks the p-set parameters ($p_{00}$, $p_{10}$, $p_{01}$, $p_{11}$) and a $\lambda$-SR chart that monitors the $\lambda$-set parameters (${\lambda }_1$, ${\lambda }_2$, ${\lambda }_{00}$). A combined scheme, which integrates both charts, is also proposed to monitor simultaneous shifts in both parameter sets.

Through Monte Carlo simulations under the four parameter configurations, we compare the performance of the proposed procedure with that of the CUSUM scheme using the ARL. The simulation results show that the CUSUM chart performs better than the SR chart under zero-state conditions. Under steady-state conditions, the SR chart performs better when the true shift is smaller than the prespecified shift magnitude. The CUSUM chart performs better when the true shift is equal to or larger than the prespecified shift magnitude. Given that real manufacturing processes typically start in an IC state and only gradually shift towards OOC conditions during prolonged production, the steady-state advantage of the SR charts is of greater practical relevance for timely anomaly detection and quality assurance.

Finally, an LED manufacturing case study further validates the effectiveness of the proposed methods. Future work will extend the SR methodology to zero-inflated binomial and negative binomial models and develop multivariate control schemes capable of accommodating three or more defect types originating from distinct failure mechanisms. In addition, extending generalized likelihood ratio and adaptive monitoring schemes to the BZIP model is also a meaningful direction for future research.