Robust Surveillance Schemes Based on Proportional Hazard Model for Monitoring Reliability Data

Abstract

Product reliability is a crucial component of the industrial production process. Several statistical process control techniques have been successfully employed in industrial manufacturing processes to observe changes in reliability-related quality variables. These methods, however, are only applicable to single-stage processes. In reality, manufacturing processes consist of several stages, and the quality variable of the previous stages influences the quality of the present stage. This interdependence between the stages of a multistage process is an important characteristic that must be taken into account in process monitoring. In addition, sometimes datasets contain outliers and consequently, the analysis produces biased results. This study discusses the issue of monitoring reliability data with outliers. To this end, a proportional hazard model has been assumed to model the relationship between the significant quality variables of a two-stage dependent manufacturing process. Robust regression technique known as the M-estimation has been implemented to lessen the effect of outliers present in the dataset corresponding to reliability-related quality characteristics in the second stage of the process assuming Nadarajah and Haghighi distribution. The three monitoring approaches, namely, one lower-sided cumulative sum and two one-sided exponentially weighted moving average control charts have been designed to effectively monitor the two-stage dependent process. Using Monte Carlo simulations, the efficiency of the suggested monitoring schemes has been examined. Finally, two real-world examples of the proposed control approaches are provided in the study.

1. Introduction

Reliability and survival analyses are both specific areas of statistics that are designed to study particular kinds of time-to-event random variables. Reliability analysis involves the techniques for assessing and analyzing the performance of products. These days, products enter the market with the warranty that they will perform smoothly for their expected operating time. This guarantee relies on the reliability of the product [1]. On the other hand, in survival analysis, statistical methods are employed to analyze data where the outcome variable of interest is the time to occurrence of an event, for example, a person’s demise, disease, or some other individual experience [2]. Other widely used statistical techniques might provide insight into how long something might take to occur. Take regression analysis as an example. However, regression analysis generally works with both positive and negative values of the variables, whereas the survival time variable is always positive following a skewed distribution. In addition, linear regression is unable to account for censoring, which occurs when the survival data are incomplete for several reasons. The key advantage of survival analysis is attributed to the analysis of time to event and censoring data [3].

Presently, manufacturing processes consist of several stages, each of which has an impact on the stage that comes after it. Such processes with multiple stages are referred to as multistage processes, e.g., semiconductor manufacturing systems, health care systems, and telecommunication systems. In these processes, the performance of a product is assessed by output quality variables [4]. An essential feature that must be taken into consideration in monitoring a multistage process is the interdependence between stages of the process. This feature of the multistage process is called the cascade property [5]. Therefore, to implement the statistical process control (SPC) schemes in multistage processes, it is crucial to examine how the attributes of stages interact. In the past two decades, the use of SPC to multistage processes has increased extensively [4]. For this reason, we require a model for the modeling and analysis of such data that has three key components. First, the dependent variable is the waiting time before the occurrence of a specified event. Second, the observations are censored, i.e., the event has not yet happened while the data are being collected for certain units. Lastly, there are other variables whose impact we want to examine on the waiting time [6].

To effectively monitor a process while accounting for the impacts of significant additional variables, many researchers have suggested numerous model-based techniques. In a multivariable process, Hawkins [7] proposed model-based regression adjustments for each variable, where a change in one variable might influence dependent stage variables. Sulek et al. [8] compared the performance of the cause-selecting control chart to the classical Shewhart chart and employed it to monitor a two-stage service process. Asadzadeh et al. [9] provided a thorough analysis of cause-selecting charts (CSCs) in multistage processes. They showed that for dependent phase processes, cause-selecting charts outperformed classical Shewhart charts. Cox regression (or proportional hazards regression) is a method for investigating the effect of several variables on the time a specified event happens. In a proportional hazards model, the unique effect of a unit increase in a covariate is multiplicative with respect to the hazard rate. Let $X_i=(X_{i1},\cdots ,X_{ip})$ be the realized values of the covariates for subject i. The hazard function for the Cox proportional hazards model has the form $h\left(t\right|X_i)=h_0\left(t\right)exp({\beta }_1{X}_{i1}+\cdots +{\beta }_p{X}_{ip})$. This expression gives the hazard function at time t for subject i with a covariate vector (explanatory variables) $X_i$. Please note that between subjects, the baseline hazard does not depend on i. Zheng et al. [10] discussed a novel hybrid repair-replacement model in the proportional hazards model with a stochastically increasing Markovian covariate process. By taking into account the Cox regression model, Biswas and Kalbfleisch [11] suggested a risk-adjusted cumulative sum (CUSUM) chart. To examine a two-stage dependent process with reliability data, Nabeel et al. [12] suggested EWMA and CUSUM control charts assuming generalized exponential distribution for the second stage variable. They mixed proportional hazard (PH) modeling with robust regression techniques to deal with the issues of outliers in reliability data and the cascade property of the multistage process. A risk-adjusted survival time CUSUM chart was studied by Sego et al. [13] using the accelerated failure time (AFT) model. The authors compared the suggested risk-adjusted survival time CUSUM with the risk-adjusted Bernoulli CUSUM chart whereas survival times were modeled using log-logistic and Weibull distributions. Olteanu [14] studied the CUSUM chart for censored data by employing the likelihood-ratio test for non-normal distribution. Regression-adjusted control techniques based on the AFT model were proposed by Asadzadeh and Aghaie [15] and Goodarzi et al. [16] to monitor a multistage process with reliability data by assuming different distributions, including Weibull and log-normal corresponding to the output variable. For a phase-I investigation, Zhang et al. [17] developed a risk-adjusted Shewhart chart by taking into account the likelihood-ratio test statistic. To monitor the Weibull mean and variance concurrently, Wang and Cheng [18] developed an exponentially weighted moving average (EWMA) chart by utilizing the likelihood-ratio test and inverse error function. To lessen the negative effects of outliers present in the historical data, Asadzadeh and Baghaei [19] presented control schemes based on the AFT model to examine a Weibull-distributed outgoing quality characteristic with reliability data in a two-stage dependent process. A new risk-adjusted EWMA control chart was developed by Ding et al. [20] to track ongoing surgical outcomes assuming log-logistic distribution. Recently, Ali et al. [21] proposed memory-type control charts for censored reliability data following generalized exponential distribution assuming a frailty model.

To evaluate a therapy process, Kazemi et al. [22] designed a risk-adjusted multivariate Tukey’s CUSUM control chart. Asif and Noor-ul Amin [23] applied the idea of an adaptive EWMA control chart in their study and discussed a novel adaptive risk-adjusted EWMA control chart by utilizing the AFT regression. Vasconcelos et al. [24] developed a control chart with Weibull-distributed quality characteristics for monitoring a process mean. They used transformation to make use of a gamma distribution to analytically determine control limits to monitor the process mean. The PH model has been incorporated with robust regression approaches to model reliability data with outliers. To analyze a two-stage dependent process, EWMA and CUSUM charts are suggested by Nabeel et al. [25] using logistic distribution. Park and Lee [26] considered each stage of a multistage process as a profile structure and suggested a multivariate multiple linear regression model implemented in each stage using the orthogonal design.

The use of the least squares method is not suggested when there are outliers or extreme values in a dataset. Therefore, a robust parameter estimation approach is required, where the estimates of the parameters are less impacted by outliers in the dataset. Generally, a robust regression method is suggested as an alternative to least squares regression in such cases. In robust regression estimation, weights are assigned to the observation according to how well-behaved they are. M-estimation, S-estimation, and MM-estimation are three popular robust estimation techniques [27]. The robust M-estimation technique is an extension of the maximum likelihood approach [28], whereas S-estimation and MM-estimation are extensions of the M-estimation [29].

Numerous researchers have presented various robust regression approaches [30,31,32,33]. When designing a control strategy for data including outliers, Jearkpaporn et al. [34] considered the generalized linear model. Shu et al. [35] studied Shewhart and EWMA control charts to study the performance of the run lengths by changing the parameters with their estimates. In the presence of outliers in the data, Ampanthong and Suwattee [36] estimated the coefficients of a multiple regression model using weights based on a postulated influence function. Alma [37] and Mutlu and Sazak [38] compared linear regression estimates with robust regression approaches such as M-estimators, MM-estimators, and S-estimators. Onur and Cetin [39] compared S-estimators with other robust estimators and conventional least squares estimators. For the robust regression parameter estimation, Susanti et al. [28] proposed algorithms for MM-estimators, M-estimators, and S-estimators. In the presence of outliers in the phase-1 data, Kumar and Jaiswal [40] developed an estimator that is a function of the sample median and used it to generate the phase-II control limits. Maleki et al. [41] developed and compared a multivariate robust $T^2$ control chart assuming an M-estimator and traditional estimators. A multivariate robust control chart is proposed as an alternative to the traditional Hotelling $T^2$ technique. The robust shrinkage estimators developed by Cabana et al. [42] and further investigated by Cabana et al. [43] formed the basis for the analysis of the chart [29].

The previous studies did not consider a situation where reliability data may have mode at zero and yet allows for increasing, decreasing, and constant hazard rates as the Weibull distribution. Thus, the goal of this paper is to develop a monitoring procedure for a two-stage dependent manufacturing process with reliability data that may have the mode at zero with different hazard shapes. The robust estimation method has been applied to the survival model to account for the impact of outliers in the historical data. This paper is structured as follows: Section 2 discusses the model and its assumptions. In Section 3, the proposed control schemes are covered in detail. Section 4 analyzes the effectiveness of suggested monitoring approaches. Section 5 provides examples of two real-world applications of the suggested monitoring techniques. Section 6 concludes the study with future recommendations.

2. Model and Assumptions of the Process

Consider a manufacturing process with two interdependent stages, where the first stage’s quality variable is represented by “X”, which follows a normal distribution, i.e., $X\sim N(\mu ,\sigma )$. The second stage’s reliability-related quality variable is represented by “T”, and it follows the Nadarajah and Haghighi (NH) distribution [44], i.e., $T\sim NH(\nu ,s)$. The probability density function of NH distribution is given as

$$f\left(t\right)=s\nu {(1+st)}^{\nu -1}exp(1-{(1+st)}^{\nu }).$$

(1)

Here, the scale parameter is denoted by $s>0$ and the shape parameter is denoted by $\nu >0$. The reason for considering NH distribution is its mode is at zero and yet allows for increasing, decreasing, and constant hazard rates as the Weibull distribution. Additionally, the cumulative distribution function is closed form as does the Weibull distribution, and exponential distribution is a special case of NH distribution. The impact of the quality variable from the first stage on the outgoing variable from the second stage must be taken into consideration to monitor the aforementioned process with reliability data. Generally, survival analysis regression models are used to model reliability data while incorporating the impact of significant variables [12]. The most commonly used survival regression model is the Cox PH model which examines how predictors and the time to event are related using a hazard function. It is based on the assumption that the predictors have a multiplicative effect on the risk and that effect is constant across time. A PH model is a parametric model that is used as an alternative to the Cox model. It adopts the form of the Cox model but assumes a parametric distribution for the baseline hazard function [45].

In this study, the hazard and survival functions of the reliability-related variable in the second stage are constructed using the PH model. The parametric PH model is based on the assumption that the baseline hazard function follows a parametric distribution [46]. Assuming the covariates have a multiplicative impact on the hazard function, the PH model analyzes the link between a group of significant covariates and the reliability-related variable [45]. The PH model’s hazard and survival functions are as follows.

$$\begin{array}{c}h\left(t\right|x)=h_0\left(t\right)\psi (x,\beta )\end{array}$$

(2)

$$\begin{array}{c}s\left(t\right|x)=s_0{\left(t\right)}^{\psi (x,\beta )}\end{array}$$

(3)

The positive function $\psi (x,\beta )$ can take on several functional forms. In this analysis, the exponential form $exp(-\beta x)$ has been considered for $\psi (x,\beta )$ [47]. As one significant variable has been taken into account in this investigation, the hazard and survival functions in Equations (2) and (3), respectively, may be expressed as

$$\begin{array}{c}h\left(t\right|x)=h_0\left(t\right)exp(-\beta x)\end{array}$$

(4)

$$\begin{array}{c}s\left(t\right|x)=s_0{\left(t\right)}^{exp(-\beta x)}\end{array}$$

(5)

where t is the outgoing quality variable in the second stage of reliability data, $\beta$ is a regression model parameter, and x stands for the incoming quality variable in the first stage. The functions $h_0\left(t\right)$ and $s_0\left(t\right)$, respectively, indicate the baseline hazards and baseline survival functions of the ith observation. The baseline hazard and survival functions for the NH distribution are

$$\begin{array}{c}{h}_0\left(t\right)=s\nu {(1+st)}^{\nu -1}\end{array}$$

(6)

$$\begin{array}{c}{s}_0\left(t\right)=exp(1-{(1+st)}^{\nu }).\end{array}$$

(7)

R-estimators, L-estimators, and M-estimators are a few robust estimators that may be used to minimize the impact of outliers on the parameter estimations of the model. In this analysis, M-estimators are employed due to their high breakpoint and effectiveness. Additionally, they offer the most accurate estimates when the data associated with the response variable contains outliers. In this method, the residuals function $\rho$ is minimized and weight is given to each observation. As a result, the effect of outliers on the model’s parameter estimates reduces. The two most popular M-estimators are the Tukey bi-square estimator and the Huber estimator [12]. The following are the weight and objective functions for the Huber estimators.

$$\begin{array}{ccc}\hfill \rho \left(r\right)& =& \left\{\begin{array}{cc}\frac{1}{2}{r}^2,& \mathrm{for}\left|r\right|\le k\\ k\left|r\right|-\frac{1}{2}{k}^2,& \mathrm{for}\left|r\right|>k\end{array}\right.\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill W\left(r\right)& =& \left\{\begin{array}{cc}1,& \mathrm{for}\left|r\right|\le k\\ \frac{k}{\left|r\right|},& \mathrm{for}\left|r\right|>k\end{array}\right.\hfill \end{array}$$

(9)

Likewise, the Tukey bi-square estimator’s weight and objective functions are given as follows.

$$\begin{array}{ccc}\hfill \rho (r)& =& \left\{\begin{array}{cc}\frac{k^2}{6}(1-{(1-{(\frac{r}{k})}^2)}^3),& \mathrm{for}\left|r\right|\le k\\ \frac{k^2}{6},& \mathrm{for}\left|r\right|>k\end{array}\right.\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill W\left(r\right)& =& \left\{\begin{array}{cc}{(1-{(\frac{r}{k})}^2)}^2,& \mathrm{for}\left|r\right|\le k\\ 0,& \mathrm{for}\left|r\right|>k\end{array}\right.\hfill \end{array}$$

(11)

where “k” refers to the turning constant used in Huber and Tukey bi-square estimators. The Huber and Tukey bi-square estimators are $95\%$ efficient when the errors are normal and the turning constants are $k=1.345\sigma$ and $k=4.685\sigma$, respectively [27]. The function $\rho$ is the objective function, $w\left(r\right)$ is the weight function, and “r” stands for the Cox-Snell residual [27], which are obtained by taking the natural logarithm of the survival function for each observation. The Cox-Snell residuals are defined as $r_i=-log\left(s_i\right)$ where $s_i$ is the ith observation’s survival function. Irrespective of the kind of survival function, the Cox-Snell residuals have an exponential distribution with a unit mean $(\lambda =1)$ [48]. Mathematically, $g\left(r\right)=exp(-r)$.

3. Process Monitoring Procedures

To identify downward mean shifts in the outgoing quality variable, we present three process monitoring strategies in this section, i.e., two one-sided EWMA control charts and one lower-sided CUSUM control chart are developed. However, to detect both upward and downward shifts, two-sided control charts with two control limits can also be taken into account.

The charting statistic of a one-sided CUSUM control chart [49] is defined as

$$C_j=min(0,C_{j-1}-w_j)$$

(12)

where $C_0=0$ and $w_j$ is the CUSUM score defined by $w_j=log\left\{\frac{f_1\left(t\right|x)}{f_0\left(t\right|x)}\right\}$, where the in-control (IC) and out-of-control (OOC) NH distributions of the reliability-related variable t are denoted by $f_0\left(t\right)$ and $f_1\left(t\right)$, respectively. The weights can be re-expressed as

$$w_j=log\left\{\frac{h_1\left(t\right|x)s_1\left(t\right|x)}{h_0\left(t\right|x)s_0\left(t\right|x)}\right\}.$$

(13)

Following the substitution of the NH distribution’s hazard and survival functions in Equation (13), we obtain,

$$w_j=log\left\{\frac{\nu \alpha s{(1+\beta )}^{\nu -1}exp\left(\beta x\right)({(exp(1-{(1+\alpha st)}^{\nu }))}^{exp\left(\beta x\right)}}{\nu s{(1+\beta )}^{\nu -1}exp\left(\beta x\right)\left(\right(exp{(1-{(1+st)}^{\nu })}^{exp\left(\beta x\right)}}\right\}.$$

(14)

where $\alpha$ is the pre-specified shift in the reliability-related variable’s parameter, and the CUSUM control chart has been designed to identify shifts. A signal is generated by the CUSUM control chart when $(C_j<h_1)$, where $h_1$ is the lower control boundary to reach the predetermined average run length (ARL) denoted by $AR{L}_0$.

The second monitoring plan is the EWMA control chart. According to [50], the typical EWMA statistic is

$$Q_j=\lambda t_j+(1-\lambda )Q_{j-1}$$

(15)

where $\lambda$ is a smoothing parameter, ranging from zero to one. The EWMA statistic has an initial value of $Q_0={\mu }_t$ at $j=1$, here ${\mu }_t$ is the IC mean of the NH distribution and can be computed numerically using IC parameters. When $(Q_j<h_2)$, the one-sided EWMA produces an OOC signal. Here, $h_2$ is fixed to achieve the desired $AR{L}_0$. Similarly, the EWMA statistic with a reflecting barrier is defined as

$$Q_j=min(\lambda t_j+(1-\lambda )Q_{j-1},{\mu }_t).$$

(16)

If the EWMA statistic drops below the lower control limit (LCL) $h_3$. i.e., $Q_j<h_3$, the chart generates a signal.

4. Performance Evaluation and Comparison

This section compares and evaluates the suggested control schemes to determine a more effective monitoring strategy for detecting downward changes in the response variable to deal with process deterioration. To this end, the model’s parameters are first estimated in both robust and non-robust scenarios and then the performance evaluation criteria are used.

4.1. Estimation of Model Parameters

The IC parameters in non-robust, robust Huber, and Tukey bi-square have been determined via the maximum likelihood estimation in this study. The NH distribution has been assumed for the quality variable in the second stage with a shape parameter of $\nu =4$ and a scale parameter of $s=1$. The distribution of the quality variable in the first stage is assumed to be normal with mean $\mu =5$ and standard deviation $\sigma =1$. Additionally, it is assumed that the regression coefficient for the PH model is $\beta =-0.1$. The estimates for the three scenarios have been obtained by averaging 10,000 simulation runs and the results are listed in

Table 1. Estimation of the Parameters in the Non-Robust and Robust Scenarios.

Scenario	$\mathit{\beta }$	S.E ($\mathit{\beta }$)
Non-robust	$-0.4229$	$4.7875\times {10}^{-5}$
Robust (Huber)	$-0.1607$	$4.1878\times {10}^{-6}$
Robust (Tukey bi-square)	$-0.09660$	$3.2780\times {10}^{-6}$

.

4.2. Performance Comparison Criteria

The performance of the control charts is usually assessed using the ARL criterion, which is computed assuming a pre-fixed shift in a process. Contrary to this, the efficacy of the control charts over a range of shifts can be measured using the extra quadratic loss (EQL) and performance comparison index (PCI) measures. In comparison to other control schemes, the control chart with a lower EQL will be more effective. Similarly, among all control charts, the chart with a smaller PCI value is considered best. Therefore, this study uses measures such as the ARL, the standard deviation of run length (SDRL), EQL, and PCI to compare the performance of the control charts. The EQL and PCI are defined as

$$\begin{array}{c}\mathrm{EQL}=\frac{1}{{\delta }_{max}-{\delta }_{min}}\underset{{\delta }_{min}}{\overset{{\delta }_{max}}{\int }}{\delta }^2\mathrm{ARL}\left(\delta \right)d\delta \end{array}$$

(17)

$$\begin{array}{c}\mathrm{PCI}=\frac{\mathrm{EQL}}{{\mathrm{EQL}}_{Benchmark}}\end{array}$$

(18)

where ${\delta }_{max}$ and ${\delta }_{min}$ stand for the upper and lower bounds of a shift in a process. The ARL($\delta$) represents the ARL value at $\delta$ shift. Since there is no closed form expression for the $\mathrm{ARL}\left(\delta \right)$ for CUSUM and EWMA charts, we computed $\mathrm{EQL}=\frac{{\sum }_{{\delta }_{min}}^{{\delta }_{max}}{\delta }^2\mathrm{ARL}\left(\delta \right)}{{\delta }_{max}-{\delta }_{min}}$.

4.3. Performance Evaluation

To identify which control scheme works best at detecting decreasing mean shifts of sizes $2.5\%$, $5\%$, $10\%$, $15\%$, $20\%$, $25\%$ and $30\%$, this section analyzes the performance of the charts. The statistical software R language [51] has been used to carry out the simulation-based investigations. To calculate the zero-state ARL and quartiles of the run length distribution, the simulation runs were repeated 10,000 times. All the proposed control charts have had their LCLs set so that the ${\mathrm{ARL}}_0\approx 200$. For two one-sided EWMA control charts, the smoothing parameter values have been set at $0.05,0.1$, and $0.2$. A shift denoted by $\alpha$ is introduced in the scale parameter of NH distribution. This is done because the scale parameter has a direct relationship with the mean of the NH distribution [44].

Table 2. Composition of the CUSUM and EWMA control charts in non-robust and robust scenarios, SD = standard deviation.

$\mathit{\alpha }$			1	0.975	0.95	0.9	0.85	0.8	0.75	0.7	EQL	PCI
Non-robust scenario
CUSUM		ARL	200.7443	177.465	152.5909	112.2530	82.2053	59.6388	45.1340	34.7939	1.6864	1.4611
		SD	165.6702	145.2413	123.1018	87.2894	61.6965	43.0580	30.8276	23.0219
		$Q_1$	55.0000	77.0000	66.0000	51.0000	39.0000	47.0000	24.0000	19.0000
		$Q_2$	200.7443	177.465	152.5909	112.2530	82.2053	59.6388	45.1340	34.7939
		$Q_3$	265.0000	233.0000	201.0000	146.0000	108.0000	77.0000	57.0000	44.0000
EWMA	$\lambda =0.05$	ARL	199.5043	154.6325	120.8129	78.8784	54.3385	40.2960	30.6520	24.5845	1.1542	1.0000
		SD	190.8764	147.6743	112.9557	68.9558	43.4726	29.0869	19.7133	13.9121
		$Q_1$	62.0000	50.0000	41.0000	30.0000	24.0000	20.0000	17.0000	15.0000
		$Q_2$	199.5043	154.6325	120.8129	78.8784	54.3385	40.2960	30.6520	24.5845
		$Q_3$	274.0000	211.0000	162.0000	106.0000	71.0000	52.0000	39.0000	31.0000
	$\lambda =0.1$	ARL	199.7100	160.7357	133.0093	90.9987	62.6067	45.7976	34.7260	26.8265	1.3080	1.1333
		SD	193.6052	151.2406	125.7737	81.8185	53.6595	36.2904	25.9265	18.4091
		$Q_1$	62.0000	52.0000	44.0000	33.0000	24.0000	20.0000	16.0000	14.0000
		$Q_2$	199.7100	160.7357	133.0093	90.9987	62.6067	45.7976	34.7260	26.8265
		$Q_3$	277.0000	222.0000	180.0000	124.0000	84.0000	60.0000	45.0000	34.0000
	$\lambda =0.2$	ARL	200.8715	165.2947	140.6019	103.8111	76.1626	55.9641	42.2472	32.2309	1.5709	1.3610
		SD	197.5640	158.6682	134.8404	96.8044	69.7239	50.0309	36.3282	25.7809
		$Q_1$	59.0000	53.0000	45.0000	34.0000	26.0000	21.0000	17.0000	14.0000
		$Q_2$	200.8715	165.2947	140.6019	103.8111	76.1626	55.9641	42.2472	32.2309
		$Q_3$	277.0000	226.0000	194.000	144.0000	104.0000	75.0000	56.0000	43.0000
$\alpha$			1	0.975	0.95	0.9	0.85	0.8	0.75	0.7	EQL	PCI
EWMA	$\lambda =0.05$	ARL	199.9123	165.6395	137.7184	94.5257	68.1334	50.2377	38.2298	31.0543	1.4317	1.2404
with reflecting barrier		SD	186.8441	150.7021	121.5976	80.4225	52.8940	35.6086	24.0687	17.5112
		$Q_1$	68.0000	59.0000	51.0000	38.7500	31.0000	25.0000	21.0000	19.0000
		$Q_2$	199.9123	165.6395	137.7184	94.5257	68.1334	50.2377	38.2298	31.0543
		$Q_3$	271.0000	223.0000	185.0000	124.0000	88.0000	64.0000	48.0000	38.0000
	$\lambda =0.1$	ARL	201.5071	171.3319	145.3346	103.6928	77.4990	56.5839	42.6646	33.1553	1.5933	1.3804
		SD	194.0158	160.1877	133.9086	93.1039	67.3644	46.6611	32,2181	23.2945
		$Q_1$	65.0000	57.0000	50.0000	38.0000	30.0000	24.0000	20.0000	17.0000
		$Q_2$	201.5071	171.3319	145.3346	103.6928	77.4990	56.5839	42.6646	33.1553
		$Q_3$	273.2000	233.0000	198.0000	139.0000	103.0000	74.0000	55.0000	42.0000
	$\lambda =0.2$	ARL	199.7246	175.8626	152.8292	117.6264	89.2111	67.3432	51.6096	39.6198	1.7372	1.5051
		SD	189.7030	170.0812	145.6179	110.8450	81.4139	60.3279	44.5509	33.04623
		$Q_1$	64.0000	55.0000	49.3800	39.0000	31.0000	25.0000	20.0000	16.0000
		$Q_2$	199.7246	175.8626	152.8292	117.6264	89.2111	67.3432	51.6096	39.6198
		$Q_3$	273.2000	241.0000	209.0000	161.0000	122.0000	91.0000	69.0000	52.0000
Robust scenario based on Huber function
CUSUM		ARL	201.1451	169.6328	140.8195	99.6323	70.5558	50.1916	36.8214	28.0148	1.4163	1.2546
		SD	166.2194	137.3228	111.9262	76.8709	52.6802	36.1056	24.9941	18.0536
		$Q_1$	84.0000	74.0000	63.0000	46.0000	34.0000	25.0000	19.0000	15.0000
		$Q_2$	201.1451	169.6328	140.8195	99.6323	70.5558	50.1916	36.8214	28.0148
		$Q_3$	265.0000	221.0000	185.2000	130.0000	90.0000	64.0000	47.0000	35.0000
EWMA	$\lambda =0.05$	ARL	200.8337	150.9337	119.7106	78.6085	53.5001	39.2193	29.8674	23.6825	1.1289	1.0000
		SD	195.4095	144.8701	110.8733	69.4721	42.4874	28.0109	18.8696	13.1857
		$Q_1$	61.0000	49.0000	41.0000	30.0000	24.0000	20.0000	16.0000	14.0000
		$Q_2$	200.8337	150.9337	119.7106	78.6085	53.5001	39.2193	29.8674	23.6825
		$Q_3$	279.0000	207.0000	162.2000	105.0000	70.0000	50.0000	38.0000	29.0000
	$\lambda =0.1$	ARL	199.4049	162.2354	133.0822	88.3851	62.2124	44.9408	33.8370	25.9593	1.2827	1.1362
		SD	195.3274	144.8592	125.1256	81.2541	54.0456	36.1379	25.3104	17.5833
		$Q_1$	61.0000	51.0000	44.0000	31.0000	24.0000	20.0000	16.0000	14.0000
		$Q_2$	199.4049	162.2354	133.0822	88.3851	62.2124	44.9408	33.8370	25.9593
		$Q_3$	274.0000	221.0000	183.2000	117.0000	82.0000	59.0000	44.0000	33.0000
	$\lambda =0.2$	ARL	200.1266	167.2870	145.7537	103.5113	76.8572	55.9620	42.1337	32.0882	1.5746	1.3948
		SD	193.2887	161.6693	140.2150	96.2823	69.6910	49.9447	35.3759	25.6641
		$Q_1$	61.0000	52.0000	46.0000	34.0000	27.0000	21.0000	17.0000	14.0000
		$Q_2$	200.1266	167.2870	145.7537	103.5113	76.8572	55.9620	42.1337	32.0882
		$Q_3$	280.0000	232.0000	199.0000	141.0000	105.0000	75.0000	56.0000	42.0000
EWMA	$\lambda =0.05$	ARL	198.9547	164.6895	134.0704	95.0891	67.1230	49.5734	37.8907	30.2922	1.4138	1.2524
with reflecting barrier		SD	185.7313	152.2588	120.2918	80.8726	52.1491	35.2607	24.0434	16.7665
		$Q_1$	68.0000	58.0000	49.7500	38.0000	31.0000	25.0000	21.0000	18.0000
		$Q_2$	198.9547	164.6895	134.0704	95.0891	67.1230	49.5734	37.8907	30.2922
		$Q_3$	268.0000	222.0000	180.0000	127.0000	87.0000	64.0000	47.0000	38.0000
	$\lambda =0.1$	ARL	199.5805	169.4613	143.1575	104.5654	74.7270	55.0920	41.8321	31.9012	1.5547	1.3772
		SD	191.5354	159.4234	132.4548	95.0874	63.4440	44.6270	31.6745	22.2395
		$Q_1$	64.0000	56.0000	49.0000	38.0000	29.0000	23.0000	20.0000	16.0000
		$Q_2$	199.4805	169.4613	143.1575	104.5654	74.7270	55.0920	41.8321	31.9012
		$Q_3$	269.0000	231.2000	193.0000	140.0000	100.0000	73.0000	54.0000	41.0000
	$\lambda =0.2$	ARL	199.2833	171.7889	155.2576	117.8723	88.2755	67.2280	51.2123	39.2312	1.8648	1.6519
		SD	191.5813	165.2328	150.6022	111.0126	80.4561	61.0228	44.7726	32.4167
		$Q_1$	63.0000	55.0000	50.0000	39.0000	31.0000	24.0000	19.0000	16.0000
		$Q_2$	199.2833	171.7889	155.2576	117.8723	88.2755	67.2280	51.2123	39.2312
		$Q_3$	273.0000	233.0000	210.0000	159.0000	119.0000	90.0000	69.0000	52.0000
Robust scenario based on Tukey bi-square function
CUSUM		ARL	198.6599	167.2256	139.5164	95.0839	65.1233	46.5104	34.6734	25.9419	1.3268	1.1730
		SD	161.8994	135.0049	112.5885	74.1585	48.3893	33.1497	23.8619	16.7490
		$Q_1$	85.0000	73.0000	60.0000	43.0000	31.0000	23.0000	18.0000	14.0000
		$Q_2$	198.6599	167.2256	139.5164	95.0839	65.1233	46.5104	34.6734	25.9419
		$Q_3$	262.0000	219.0000	183.0000	124.0000	83.0000	60.0000	44.0000	33.0000
EWMA	$\lambda =0.05$	ARL	199.6212	152.3438	120.0416	78.8739	53.3664	39.3274	29.9474	23.8362	1.1311	1.0000
		SD	193.9231	144.6941	113.4220	68.4978	42.2735	27.9959	18.8809	13.3006
		$Q_1$	59.0000	48.0000	40.0000	30.0000	23.0000	20.0000	16.0000	14.0000
		$Q_2$	199.6212	152.3438	120.0416	78.8739	53.3664	39.3274	29.9474	23.8362
		$Q_3$	277.0000	212.0000	163.0000	106.0000	70.0000	50.0000	38.0000	30.0000
	$\lambda =0.1$	ARL	199.9025	160.4655	130.8211	90.0447	62.7029	45.5106	33.6244	26.2518	1.5871	1.4031
		SD	190.4568	154.3083	125.9188	81.6334	54.4819	36.8057	24.8628	17.4120
		$Q_1$	61.0000	50.0000	42.0000	32.0000	24.0000	19.0000	16.0000	14.0000
		$Q_2$	199.9025	160.4655	130.8211	90.0447	62.7029	45.5106	33.6244	26.2518
		$Q_3$	274.0000	221.0000	174.0000	122.0000	83.0000	60.0000	44.0000	33.0000
	$\lambda =0.2$	ARL	200.2844	166.9600	142.8648	103.5326	76.1354	54.9989	41.9356	31.6754	1.5603	1.3795
		SD	197.3578	162.6991	135.2637	98.5446	68.8390	48.8380	36.0554	25.4658
		$Q_1$	61.0000	52.0000	46.0000	34.0000	27.0000	21.0000	17.0000	14.0000
		$Q_2$	200.2844	166.9600	142.8648	103.5326	76.1354	54.9989	41.9356	31.6754
		$Q_3$	273.0000	231.0000	197.2000	139.0000	103.0000	74.0000	56.0000	41.0000
EWMA	$\lambda =0.05$	ARL	198.2270	165.7226	138.2534	95.9348	67.3676	49.7240	37.8654	30.1437	1.4179	1.2536
with reflecting barrier		SD	185.2698	152.0990	121.4520	81.2702	51.6793	35.3376	23.9713	16.8474
		$Q_1$	67.0000	58.0000	51.0000	39.0000	30.0000	25.0000	21.0000	18.0000
		$Q_2$	198.2270	165.7226	138.2534	95.9348	67.3676	49.7240	37.8654	30.1437
		$Q_3$	269.0000	223.0000	185.0000	126.0000	89.0000	63.0000	47.0000	37.0000
	$\lambda =0.1$	ARL	199.5478	170.5499	147.4620	104.0158	76.3843	55.9152	42.0644	32.8232	1.5763	1.3936
		SD	192.9706	159.1970	135.8105	92.2968	65.9376	45.4057	31.3486	22.9317
		$Q_1$	64.0000	57.0000	49.0000	38.0000	29.0000	24.0000	20.0000	17.0000
		$Q_2$	199.5478	170.5499	147.4620	104.0158	76.3843	55.9152	42.0644	32.8232
		$Q_3$	273.0000	232.0000	202.0000	141.0000	103.0000	74.0000	55.0000	42.0000
	$\lambda =0.2$	ARL	199.2022	176.5519	151.4087	116.0709	87.8592	67.6121	51.3464	38.9756	1.8609	1.6452
		SD	191.4593	167.6931	143.9507	110.1907	80.0792	61.5244	44.3267	32.1573
		$Q_1$	61.0000	56.0000	48.0000	38.0000	31.0000	24.0000	19.0000	16.0000
		$Q_2$	199.2022	176.5519	151.4087	116.0709	87.8592	67.6121	51.3464	38.9756
		$Q_3$	276.0000	242.0000	209.0000	155.0000	118.0000	91.0000	69.0000	51.0000

lists the results for the proposed charts.

The findings listed in Table 2 clearly show that the Tukey bi-square CUSUM is the best when compared to the Huber-CUSUM and non-robust CUSUM. For instance, the Huber-CUSUM and Tukey bi-square CUSUM both produce OOC signals at 71 and 65 sample numbers, respectively, whereas non-robust CUSUM produces a signal at 82 sample point for a $15\%$ shift size. These findings also reveal the dependency of the CUSUM control chart on the model parameters as can be seen from Figure 1. In addition, if ARL and SDRL values of the CUSUM control chart in each of the three scenarios are closely examined, one can notice that these values decline as the shift size increased. This suggests that the CUSUM control chart will detect an OOC signal more quickly and the dispersion of the distribution of run duration will be smaller. For non-robust CUSUM, for example, the SDRL value at a shift of size $10\%$ is $87.2894$, whereas, with a shift of size $15\%$, it is $61.6965$. Likewise, the SDRL value for Huber-CUSUM is $52.6802$ for a $15\%$ shift and $36.1056$ for a $20\%$ shift. This demonstrates how the values of SDRL fall as the shift size rises, resulting in a narrower spread in the run length distribution. Finally, we observe that the quartiles of the run length distribution in both non-robust and robust cases decrease as the shift’s size becomes larger. For instance, the quartiles for the CUSUM control chart in the Huber robust scenario for a $10\%$ shift are Q${}_1=46.00$, $Q_2=99.6323$, and $Q_3=130$, while $Q_1=63.00$, $Q_2=140.8195$, and $Q_3=185.2$, respectively, for a $5\%$ shift.

Concentrating on the one-sided EWMA control charts without a reflecting barrier, it is evident that the EWMA control charts with a certain smoothing parameter do not depend on the model’s parameters. For instance, the ARL values for the EWMA control chart without a reflecting barrier are $120.8129$, $119.7106$, and $120.0416$ in three cases with smoothing parameter $0.05$ and shift of size $5\%$. This indicates that the performance of one-sided EWMA control is almost the same for three scenarios with a particular smoothing parameter. Moreover, one can observe from Table 2 that the values of ARL and SDRL for the one-sided EWMA control chart decline as the shift size increases. For instance, the values of ARL are $53.3664$, $39.3274$, and $29.9474$, respectively, for the Tukey bi-square EWMA control chart with smoothing parameter of $0.05$ and shift of sizes $15\%$, $20\%$, and $25\%$, respectively. Similarly, the SDRL values for $15\%$, $20\%$, and $25\%$ shifts with smoothing parameter $0.05$ are $42.2735$, $27.9959$, and $18.8809$, respectively. It is also important to notice that the quartiles of the run length distribution decrease as the shift size grows larger for the EWMA control chart, i.e., consider the one-sided EWMA control chart with a smoothing parameter of $0.1$ in a Tukey bi-square scenario. One can see that $Q_1=24.0$, $Q_2=62.7029$, and $Q_3=83.0$ at a $15\%$ shift while with a $20\%$ shift, $Q_1=19.00$, $Q_2=45.5106$, and $Q_3=60.00$.

Furthermore, if we examine the one-sided EWMA control chart without a reflecting barrier in three cases, we observe that the EWMA chart with a smoothing value of $0.05$ shows a higher performance as compared to the EWMA charts with smoothing parameters of $0.1$ and $0.2$. In other words, as the smoothing parameter value increases, the performance of the one-sided EWMA control chart decreases. To provide an example, consider the EQL values of the Huber one-sided EWMA control charts without a reflecting barrier for smoothing parameters of $0.05$, $0.1$, and $0.2$. The values are $1.1289$, $1.2827$, and $1.5746$, respectively. Thus, increasing the smoothing parameter’s value over $0.05$ results in increased values of the EQL. This implies that the performance of the one-sided EWMA control chart without a reflecting barrier deteriorates as the value of the smoothing parameter increases. Figure 2 depicts the results for easier comprehension.

A thorough examination of the robust and non-robust scenarios clearly shows that the one-sided EWMA control chart with a smoothing value of $0.05$ and without a reflecting barrier outperforms the lower-sided CUSUM control in both scenarios. In the non-robust case, the PCI value for the lower-sided CUSUM is $1.4611$ while $1.0$ for the one-sided EWMA control chart without the reflecting barrier. Similarly, the one-sided EWMA control chart without a reflecting barrier with a smoothing parameter of $0.05$ outperforms the lower-sided CUSUM control chart in the robust scenario based on the Huber and Tukey bi-square functions. To have a clear idea. The PCI values for the lower-sided CUSUM and one-sided EWMA without the reflecting barrier are $1.2546$ and $1.00$ for Huber and $1.1730$ and $1.00$ for Tukey bi-square. This demonstrates that the lower-sided CUSUM control chart is outweighed by the one-sided EWMA without the reflecting barrier, as can be confirmed from Figure 3.

Focusing on the one-sided EWMA control chart with a reflecting barrier, one can see from Table 2 that the OOC ARL and EQL values become large for a large smoothing parameter, resulting in a reduction in the control chart’s performance. In comparison to the other proposed control approaches, it is noticed that the EWMA control chart with a reflecting barrier has the highest OOC ARL values, confirming its poor performance.

The quality of a multistage process can be classified as the total and specific quality. The total quality refers to the combined quality of the current stage and previous stages while specific quality is the quality of the current stage only. As the current study aims to monitor the changes occurring in the mean of the second stage quality, we report here the results of sensitivity analysis to assess the effect of the first stage variable. For the performance evaluation of the control charts, we used normal distribution with $\mu =3$ and $\sigma =1$ for the first stage process. To examine the performance by changing the parameters of the normal distribution, we report results for the bi-square CUSUM for different values of mean and the standard deviation of the normal distribution. From

Table 3. Sensitivity Analysis of Tukey bisquare CUSUM with different parameters of the first stage process.

${\mathit{\alpha }}^{\prime }$			1	0.975	0.95	0.9	0.85	0.8	0.75	0.7
CUSUM		ARL	198.6599	167.2256	139.5164	95.0839	65.1233	46.5104	34.6734	25.9419
$X\sim N(3,1)$		S.D	161.8994	135.0049	112.5885	74.1585	48.3893	33.1497	23.8619	16.7490
		Q1	85.0000	73.0000	60.0000	43.0000	31.0000	23.0000	18.0000	14.0000
		Q3	262.0000	219.0000	183.0000	124.0000	83.0000	60.0000	44.0000	33.0000
CUSUM		ARL	199.4822	169.7044	137.4374	93.1159	63.2452	44.0059	32.9774	24.5984
$X\sim N(1.5,3)$		S.D	163.4584	139.6599	109.3847	71.2261	46.5578	30.4992	22.2035	15.8303
		Q1	85.0000	72.0000	59.0000	42.0000	30.0000	22.0000	17.0000	14.0000
		Q3	259.0000	222.0000	181.0000	121.0000	82.0000	57.0000	42.0000	31.0000
CUSUM		ARL	199.5094	170.5114	141.2416	97.189	66.9093	48.2992	35.8364	27.1819
$X\sim N(4,1.5)$		S.D	158.0576	140.6099	115.2269	76.3812	48.7513	34.5305	23.9865	17.4026
		Q1	85.0000	71.0000	63.0000	44.0000	32.0000	24.0000	19.0000	15.0000
		Q3	265.0000	223.0000	185.0000	126.0000	87.0000	62.0000	45.0000	34.0000

, it can be noticed that the CUSUM surveillance scheme is not much affected by changing the first stage mean and standard deviation as the ARL values are almost the same with minor changes due to random data generation. This is because the effect of the previous step is eliminated by modeling the quality characteristics of both stages by the PH model. Thus, the results may not be different for other combinations of parameters. The reason for this conclusion is the performance assessment by the ARL which is a scale-free performance measure [52]. However, the situation will be different in the case of average time to signal (ATS) because, unlike the ARL, the ATS metric is dependent on the data scale.

5. Illustrative Examples

To assess the efficacy of suggested control techniques, two real-life examples of the concrete manufacturing process in civil engineering are discussed in this section. Concrete is an important material in civil engineering and its compressive strength in its hardened state is a crucial property. The proportion of ingredients used in concrete affects its strength, and thus it is important to find the right proportions for structural requirements. The process of manufacturing concrete involves measuring and mixing dry ingredients to create a consistent mixture, adding water to achieve a uniform texture, transferring the wet mixture to molds, and compacting it. The molds are then covered and left undisturbed for 24 h, after which the specimen is removed from the mold and submerged in fresh clean water. To determine the compressive strength of concrete, a specimen is removed from water after 7 or 28 days and is subjected to a compression test using a machine. Load is gradually applied until the specimen breaks, and the maximum load at which it breaks is recorded. The compressive strength is then calculated by dividing the maximum load by the cross-sectional area of the specimen. Three readings are taken, and the average is used to represent the batch [53].

Example 1.

For our analysis, we first consider the concrete compressive strength dataset which has been extracted from the UCI repository Lab [54]. There are no missing values in the dataset. The dataset comprises 1030 values, each with eight components for the input vector and one output value (compressive strength). The concrete samples manufactured with Portland cement and cured under normal circumstances were assessed. These specimens were all converted into 15 cm cylinders to test the compressive strength of concrete [55]. Since the proportion of ingredients in the first stage affects the compressive strength of concrete, we have considered the amount of cement (kg/m${}^3$) as the input variable for the first stage and the concrete compressive strength, expressed in MPa (Mega Pascal), is taken as the outgoing quality variable for the second stage.

To determine the effect of input variables on the output variable, 350 observations are taken and the first 150 points are considered in phase I to estimate the parameters of the model while the last 200 observations have been used in phase II to assess the performance of the suggested control charts. We first use a simple linear regression model in a non-robust scenario to obtain the IC estimates of the PH model. The estimates of the parameters are $\widehat{\beta }=-0.00426,\widehat{\nu }=232.6338$ and $\widehat{s}=0.7006$. Then, the IC estimates of the PH model are calculated using the robust Tukey bi-square function. The resulting estimates are $\widehat{\beta }=-0.0040,\widehat{\nu }=232.6338$ and $\widehat{s}=0.7006$. In both non-robust and Tukey bi-square robust scenarios, the lower-sided CUSUM control chart and the one-sided EWMA control chart with smoothing parameter = $0.05$ are employed to identify a decreasing shift of size $25\%$. The one-sided EWMA statistic, CUSUM score, and lower-sided CUSUM statistic are then computed and plotted on the control charts as shown in Figure 4 and Figure 5.

Figure 4 shows that an OOC signal is triggered at the 201st data point for a non-robust one-sided EWMA control chart and the 263rd data point for a non-robust lower-sided CUSUM control chart. Therefore, corrective measures must be implemented in both circumstances. When compared to the lower-sided CUSUM control chart, the one-sided EWMA with a smoothing value of $0.05$ demonstrated superior performance. From Figure 5, the Tukey bi-square lower-sided CUSUM control chart triggers a signal at the 263rd data point whereas the Tukey bi-square one-sided EWMA control chart gives an OOC signal at the 201st data point with smoothing parameter $0.05$. Thus, the EWMA chart is preferable to the lower-sided CUSUM control chart in both non-robust and Tukey bi-square robust cases.

Another important property of concrete is workability which refers to its ease and homogeneity in mixing, placing, consolidating, and finishing [56]. The workability of concrete is greatly influenced by its consistency, which is related to the flow behavior of fresh concrete. Although consistency may not be able to evaluate workability directly, it can serve as an indicator of concrete workability. Measuring the consistency of a concrete mix is typically done using the slump test [57]. To evaluate the flowability of fresh concrete, the slump-cone test is employed to gauge its consistency. This test involves measuring the slump by determining the drop from the highest point of the slumped concrete and assessing the slump flow by measuring its diameter [55].

Example 2.

The second dataset taken into account is the concrete slump test dataset. Initially, the dataset contained 78 data points, but over several years, an additional 25 data points were obtained [54]. Different combinations of mixed proportions were tested to gather the data. The database consists of 103 records, each containing seven components of the input vector and three output values. The consistency of fresh concrete depends upon the optimum quantity of water and cement in the initial stage [58]. The amount of cement expressed in $kg/m^3$ is considered the input variable in the first stage, while for the second stage, the concrete slump flow (from 20 to 70 cm) is considered the output variable. It is calculated by measuring the diameter of the slumped fresh concrete. The bottom diameter of the slump cone, which is 20 cm, represents the minimum slump flow [59]. In this study, 100 observations are taken, of which 50 observations for phase I and the remaining 50 data points for phase-II analysis. A simple linear regression model has been used to estimate the PH model’s parameters in the non-robust scenario. The estimates are $\widehat{\beta }=-0.00214,\widehat{\nu }=79.4092$, and $\widehat{s}=1.9286$. Similarly, in the Tukey bi-square instance, the IC estimates for the PH model are $\widehat{\beta }=-0.0021,\widehat{\nu }=79.4092$ and $\widehat{s}=1.9286$. In both non-robust and robust Tukey bi-square situations, the CUSUM score, CUSUM statistic, and EWMA statistic are computed.

The one-sided EWMA control charts and the lower-sided CUSUM control charts are designed to find a $15\%$ decreasing shift. Figure 6 and Figure 7 show the resulting charts. From Figure 6. the non-robust one-sided EWMA control chart generates an OOC signal at the 78th observation, whereas the non-robust CUSUM control chart gives an OOC signal at the 87th observation. Similarly, from Figure 7, the Tukey bi-square robust EWMA control chart triggers an OOC signal at the 78th position, whereas the Tukey bi-square robust CUSUM control chart generates an OOC signal at the 87th position. These analyses demonstrate that in non-robust and Tukey bi-square cases, the one-sided EWMA control chart with smoothing parameter $0.05$ outperforms the lower-sided CUSUM chart.

6. Conclusions

To effectively monitor a multistage process in the presence of cascade property, three monitoring schemes are discussed in this paper. The incoming quality characteristic in the first stage has a normal distribution, whereas the outgoing reliability-related quality characteristic in the second stage follows an NH distribution. To account for the impacts of significant variables, the PH model is employed to model reliability data. Moreover, two M-estimators, i.e., the Tukey bi-square estimator and Huber estimator, have been implemented to remove the detrimental impact of outliers on the regression model’s coefficients. This is done by obtaining the Cox-Snell residuals. Then, using these Cox-Snell residuals, the weights for the Huber and Tukey bi-square functions are determined. Next, these weights are allocated to the outgoing quality variable in the second stage. The unbiased estimates of the regression model’s parameters are then computed. Next, using Monte Carlo simulations, the proposed monitoring schemes, namely one-sided EWMA and CUSUM control charts are analyzed based on ARL, SDRL, EQL, and PCI criteria. The results show that in both robust and non-robust settings, the one-sided EWMA control chart with a smoothing parameter $0.05$ performs better than the lower-sided CUSUM control chart. In addition, the one-sided EWMA control chart with a smoothing value of $0.05$ performs better than the one-sided EWMA control charts with smoothing values of $0.1$ and $0.2$, respectively. However, the performance of the one-sided EWMA control chart degrades as the smoothing parameter value increases. Moreover, the Tukey bi-square lower-sided CUSUM control chart performs better than the lower-sided CUSUM in no-robust and Huber robust cases.

The analysis also shows that the one-sided EWMA control chart with a reflecting barrier performs almost the same as the one-sided EWMA control chart without a reflecting barrier. Finally, the findings from the two real-world examples confirm the results of the simulation study, i.e., in both non-robust and Tukey bi-square robust circumstances, the one-sided EWMA control chart with no reflecting barrier and a smoothing parameter of $0.05$ is better than the lower-sided CUSUM control chart.

For future studies, the suggested control strategies can be extended to dependent processes with more than two stages. Future research may also benefit from using discrete distributions such as the Poisson distribution and discrete beta distribution, etc., for outgoing variables to examine the dependency structure in multistage processes.