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Abstract

The precision of process monitoring often encounters challenges in determining the exact shift size. Therefore, combined control charts have gained considerable attention because of their excellent speed to detect simultaneously small-to-moderate and large-size shifts. The effectiveness of the applied quality control methods strongly depends on the performance of the measurement system. Measurement error presence contributes significantly negatively toward the performance of the usual control charting schemes. This article proposes novel two-sided combined Shewhart-Cumulative EWMA-sum (Shewhart-CUESUM) control charts designed to efficiently monitor the mean of normally distributed processes. In addition, to address measurement errors, the M-Shewhart-CUESUM chart is proposed, incorporating an additive measurement error model. Evaluation of the charts through Monte-Carlo simulations, considering metrics such as average run length (ARL), extra quadratic loss, relative ARL, and performance comparison index. It is found that the combined Shewhart-CUESUM outperforms than CUESUM chart. The results show that the presence of measurement errors can significantly diminish the charts’ performance, which can be mitigated by utilizing a multiple measurements scheme. Among the different well-established combined charts examined, the M-Shewhart-CUESUM chart shows considerably more sensitive to detecting simultaneously detect small and large size shifts. To employ simulated datasets to illustrate the impact of measurement errors and demonstrate the implications of the proposed charts on process mean shifts.

Keywords

Combined chart measurement error statistical process control multiple measurements Monte-Carlo simulations

Introduction

The Statistical Process Control (SPC) toolkit encompasses a wide range of tools that are widely employed in contemporary industrial settings to effectively monitor and control manufacturing as well as service processes. It represents a robust assortment of analytical techniques aimed at attaining process stability and enhancing capability through the reduction of variability.¹ Among the vital components of SPC, the univariate control chart was originally devised by Dr. Walter A. Shewhart² in 1924, standing as an indispensable tool in this domain.

In general, control charts can be classified into two categories: memoryless and memory-type control charts. Memoryless charts, such as Shewhart-type charts, excel at quickly detecting large shifts. However, they exhibit less sensitivity toward small and moderate shifts in the process parameters, such as location and dispersion. To address this limitation, memory-type charts were introduced as enhancements to the memoryless monitoring chart. The Cumulative Sum (CUSUM) chart and the Exponentially Weighted Moving Average (EWMA) chart were developed to improve the detection capability for small shifts.^3,4 These memory-type charts utilize in cooperation with previous and recent realizations, resulting in an increased sensitivity toward small and moderate shifts in the process parameters. Subsequently, further enhancements were proposed for these two charts. Crosier’s CUSUM (or CCUSUM) chart, introduced by Crosier,⁵ and the Generally Weighted Moving Average (GWMA) chart, presented by Sheu and Lin,⁶ are notable examples. Furthermore, Haq⁷ recently suggested the Cumulative EWMA-SUM (CUESUM) chart for monitoring the mean of normally distributed processes. Through detailed comparisons, it has been observed that the CUESUM chart exhibits greater sensitivity to small-size shifts compared to the CUSUM, EWMA, CCUSUM, double EWMA, and mixed EWMA-CUSUM charts.

To overcome the limitations of memory-less and memory-type control charts, Lucas⁸ proposed a combined chart called the Shewhart-CUSUM chart. This chart was developed with the aim of monitoring both small-to-moderate and large shifts in process parameter(s). He found that the Shewhart-CUSUM chart exhibited superior performance in terms of average run length (ARL) compared to both the standard CUSUM and Shewhart chart. Specifically, Lucas observed that the CUSUM chart worked greatly for controlling the tiny and small shifts, but its efficiency decreased when faced with moderate-to-large shifts. On the other hand, the Shewhart chart demonstrated better efficiency than the Shewhart-CUSUM chart when the shift size was substantial.

Building upon a similar methodology, Lucas⁹ introduced the Shewhart-EWMA chart as an enhancement to the detection capability of the EWMA chart. Their study demonstrated that the Shewhart-EWMA chart achieved higher sensitivity compared to using the EWMA chart independently. Additionally, they recommended employing Shewhart limits that are larger than those utilized in a standard Shewhart chart to avoid a substantial decrease in the ARL under in-control conditions. Furthermore, it is essential to be cautious while employing the Shewhart-EWMA chart when dealing with processes that produce occasional outliers. To enhance the detection capability for large shifts, Sheu and Lin⁶ proposed the Shewhart-GWMA chart in more recent times. They suggested that if detecting small shifts in the process mean takes precedence over minimizing false alarms and associated time and cost, choosing the Shewhart-GWMA chart might be a more favorable choice. On the similar lines, for monitoring the process variation, Abujiya et al.¹⁰ proposes new combined Shewhart–CUSUM S charts based on the extreme variations of the ranked set sampling technique.

Furthermore, Munir et al.¹¹ constructed the Shewhart-CCUSUM chart, building upon the work of Lucas.⁸ Their study demonstrated that the Shewhart-CCUSUM chart showed slight efficiency in detecting small shifts compared to the Shewhart-CUSUM chart while maintaining similar properties to the latter. Accurately predicting the precise magnitude of shifts from the target value in practical process settings can be challenging. Consequently, combined charts are recognized as powerful SPC tools that excel in monitoring both small-to-moderate and moderate-to-large process shifts effectively.

These charts offer the advantage of simplicity in implementation by incorporating additional Shewhart limits into memory-type charts, without requiring complex charting structures. Considering the benefits of combined charts, our objective is to propose the Combined Shewhart-CUESUM chart. This chart aims to provide efficient monitoring of both small-to-moderate and large-size shifts in the mean of a normally distributed process. By leveraging the strengths of Shewhart and CUESUM techniques, this proposed chart offers an effective solution for process control in practical industrial scenarios.

In SPC, a foundational assumption is that observations from the underlying process are ideally recorded through a flawless measurement system. However, in practical SPC applications, this assumption is routinely challenged by various factors influencing the characteristics of interest during measurement. These factors, including environmental conditions (e.g. temperature and light), human elements (e.g. improper operation and error collection), and nuances within the measurement system itself, are inherent in measurement processes and introduce errors into true observations (cf., Maleki et al.¹²; Bennett¹³).

Measurement error, defined as the disparity between a characteristic’s recorded and true value, is omnipresent. For instance, precise measurement of the exact volume of liquid in bottles on a production line is unattainable, and errors often arise during the generation and measurement of peak areas in mass-spectrometry analyses within analytical laboratories. Analogously, analog blood pressure machines in medical settings may not consistently provide accurate readings (Riaz¹⁴). Such variations contribute to an increased observed process variability, potentially leading to misguided deductions and decisions. Consequently, the need for an accurate measuring system, characterized by an observed mean close to the true value and high precision, is emphasized (International Organization for Standardization¹⁵).

Concerning control charts, their statistical performance is compromised with heightened measurement variation for a given process variation (Linna and Woodall¹⁶). Inaccurate measurements elevate the occurrence of false alarms in in-control situations and significantly diminish the charts’ ability to detect out-of-control (OC) situations.

In practical SPC applications, variations stemming from the measurement system can result in erroneous deductions and decisions. Therefore, a capable measurement system – precise, accurate, and stable – is essential to facilitate process improvements. Nevertheless, acknowledging the inherent errors in all measurements is crucial, as many of these errors can be substantial enough to lead to misleading conclusions about the measured phenomenon. The impact of imprecise measurements on different control charts in SPC is notable, introducing additional process variability and elevating the occurrence of false alarms in in-control situations. Moreover, inaccurate measurements significantly impede control chart performance during out-of-control situations. As the variance of measurement errors increases, so does the uncertainty related to observed process variability, ultimately deteriorating the statistical performance of control charts, especially when the variability due to measurement errors surpasses the inherent process variability. Addressing and minimizing measurement errors are therefore crucial for preserving the integrity and effectiveness of SPC methodologies.

Research interest in the impact of measurement errors on control chart performance has been examined by Abraham.¹⁷ Kanazuka¹⁸ specifically studied the power of control charts when measurement errors are present, focusing on the process average and variance. Linna and Woodall¹⁶ investigated the effect of measurement errors on $\bar{X}$ and S2 charts using a linear covariate model. Their investigation revealed that measurement errors can diminish the ability to detect shifts in the underlying process variable, resulting in reduced statistical power. Consequently, it may be advantageous to obtain multiple measurements for each item in a subgroup, as suggested by Linna and Woodall.¹⁶ Recently, Hu et al.¹⁹ delved into the statistical design of synthetic charts considering the presence of measurement errors. Maleki et al.¹² introduced a conceptual classification scheme that employed the content analysis method to analyze and categorize research exploring the impact of measurement errors on various aspects of statistical process monitoring.

The Shewhart median chart and the adaptive EWMA $\bar{X}$ chart with measurement error were examined by Tang et al.²⁰ Tran et al.²¹ investigated the performance of coefficient of variation charts in the presence of measurement errors. Lee et al.²² focused on the two-sided Shewhart coefficient of variation chart with measurement error, utilizing an error model with a linear covariate model to detect both increases and decreases in the coefficient of variation in short-run processes.

Nguyen et al.²³ examined the impact of measurement error on the performance of EWMA control charts specifically designed for monitoring the ratio of two variables that follow a normal distribution. Thanwane et al.²⁴ assessed the performance of the homogenously weighted moving average $\bar{X}$ monitoring chart in the presence of measurement error. In their studies, they utilized a linear covariate error model, considering both constant and linearly increasing variances. Kahraman and Öztürk²⁵ investigated the prediction of surface roughness measurement uncertainty in grinding operations, particularly beneficial for precision grinding wheel design accuracy assignment. This deterministic tool systematically estimates and controls both repeatable and non-repeatable errors of grinding tools. For the first time, Monte Carlo simulation techniques were applied to experimental data and grinding process modeling, allowing for a probabilistic assessment of the influences of stochastic uncertainties in predictors that deterministic approaches couldn’t capture. Researchers discovered that measurement error has a detrimental effect on the performance of control charts. For more information, please see Öztürk and Kahraman,²⁶ Li et al.,²⁷ Munir et al.,¹¹ and Liu et al.,²⁸ a few cited therein.

Nevertheless, there have been no prior investigations conducted to establish the Combined Shewhart-CEUSUM chart. It is evident that the impact of measurement errors on this chart has not been examined. The objective of this paper is to address this research gap by concentrating on the development of a more effective combined chart while considering the decision risks arising from measurement errors for the mean of processes that follow a normal distribution. Furthermore, investigate the utilization of a multiple measurement strategy as a means to mitigate the introduced variability caused by measurement errors.

The subsequent sections of this paper are organized as follows: Section 2 offers a concise review of existing CUSUM and combined control charts. Section 3 introduces the new combined Shewhart-CUESUM and investigates the performance of this chart in the presence of measurement error. The results of Monte Carlo simulations of different control charts are presented in Section 4. Section 5 presents an illustrative example utilizing simulated datasets, while Section 6 concludes the paper with final remarks.

A review of the existing quality control charts

This section offers a concise overview of existing control charts known for their effectiveness in quickly identifying both small and large size shifts in the mean of a process that follows a normal distribution.

Let ${X_{t}; t \geq 1}$ represent the primary quality of interest variable in the manufacturing process, following a normal distribution $N (μ, σ^{2})$ at time $t \geq 1$ . A random sample of size $n : (X_{t, 1}, X_{t, 2}, \dots, X_{t, n})$ is continually obtained from the process. The sample mean ${\bar{X}}_{t} = X_{t, 1} + X_{t, 2} + \dots + X_{t, n}$ /n is then calculated at time t to monitor variations in the process mean μ. In relation to the ${X_{t}}, {X_{t}; t \geq 1}$ , ${\bar{X}}_{t}$ represents a sequence of independent and identically distributed random variables following a normal distribution with mean $μ_{\bar{X}} = μ$ and standard deviation $σ_{\bar{X}} = σ / \sqrt{n}$ .

Suppose the process to be in control when the process mean $μ = μ_{0}$ and has shifted to an out-of-control state $μ = μ_{1} = μ_{0} + Δ \times \frac{σ}{\sqrt{n}}$ . Here $Δ = \frac{(μ_{1} - μ_{0}) \sqrt{n}}{σ}$ is the standardized shift leading to the new out-of-control mean.

The CUESUM chart

In a recent study, Haq⁷ introduced a novel control chart called the Cumulative EWMA-sum (CUESUM) chart, designed to efficiently monitor small-size shifts in the process mean of normally distributed processes. Haq explored the concept of an EWMA statistic based on the unshrunk CUSUM and utilized it to develop a new Crosier CUSUM chart for process mean monitoring. By incorporating additional memory through the EWMA statistic, the CUESUM chart demonstrated superior performance compared to other competitive counterparts, particularly for small to moderate-sized shifts.

Like the CCUSUM chart, the CUESUM chart follows a control charting structure defined by the following equations:

\begin{matrix} D_{t} = E_{t} \times \max (0, 1 - \frac{K \times λ}{| F_{t} |}) \\ E_{t} = λ F_{t} + (1 - λ) E_{t - 1} \end{matrix}}

(1)

Where $F_{t} = {\bar{X}}_{t} + D_{t - 1}$ with initial values of $D_{0}$ and $E_{0}$ set to zero. The CUESUM chart represents the values of $D_{t}$ plotted against two control limits: an upper limit ( $h$ ) and a lower limit ( $- h$ ), which are symmetric. When the absolute value of $D_{t}$ exceeds the limit ( $| D_{t} | > h$ ), an out-of-control signal is triggered, indicating a shift in the process mean. Specifically, $D_{t} > h$ suggests an upper-side shift in mean, whereas $D_{t} < - h$ suggests a downside shift in mean. It is noting that the CUESUM chart simplifies to the CCUSUM chart when $λ$ is equal to 1. The in-control ARL of the CUESUM chart, given a specific $(λ, k)$ value, is fixed by the limit $h$ . For further details on this chart, see Haq.⁷

The Shewhart-CUSUM chart

Lucas⁸ combined the features of Shewhart and CUSUM charts and proposed the combined Shewhart-CUSUM chart for efficiently detecting a shift in the process mean. This chart’s functions similar to the classical Shewhart (large shifts) and CUSUM (small and moderate shifts) charts.

By using the order of ${{\bar{X}}_{t}}$ , the Shewhart-CUSUM chart incorporates both upward and downward CUSUM statistics, denoted as $A_{t}^{+}$ and $A_{t}^{-}$ , respectively. These statistics are used in conjunction with the Shewhart control chart. The chart can be represented as follows:

\begin{matrix} A_{t}^{+} = \max [0, A_{t - 1}^{+} + ({\bar{X}}_{t} - μ_{0}) - K] \\ A_{t}^{-} = \max [0, A_{t - 1}^{-} - ({\bar{X}}_{t} - μ_{0}) - K] \end{matrix}},

(2)

$K$ represents the sensitive parametric value and defined as $K = k σ / \sqrt{n}$ of the Shewhart-CUSUM chart. A frequently adopted approach in the CUSUM parametric setting is to set the value of $k$ as half the magnitude of the shift (δ) in the process mean that one aims to detect, which can be represented as $k = δ / 2$ .

The Shewhart’s chart upper control limit (UCL) and lower control limit (LCL) of the Shewhart chart can be determined as follows:

UCL = μ_{0} + L_{1} \frac{σ}{\sqrt{n}} and LCL = μ_{0} - L_{1} \frac{σ}{\sqrt{n}} .

(3)

The two-sided Shewhart-CUSUM chart determines an out-of-control process if the sample mean ${\bar{X}}_{t}$ beyond the CL defined in equation (2) in any side (upper/lower), or if either $A_{t}^{+}$ or $A_{t}^{-}$ cross the decision value $H = h σ / \sqrt{n}$ . The constant h and L₁ are arbitrarily selected to achieve the desired in-control ARL. For further information, please refer to Lucas.⁸

The Shewhart-Crosier CUSUM chart

Munir et al.¹¹ devised a novel two-sided Shewhart-CCUSUM chart capable of effectively monitoring irregular changes in the process mean, accounting for scenarios with and without measurement error. The construction of this chart relies on the utilization of the following statistical measures.

\begin{matrix} B_{t} = 0, if C_{t} \leq K \\ B_{t} = ({\bar{X}}_{t} - μ_{0} + B_{t - 1}) (1 - K / C_{t}), if C_{t} > K \end{matrix}},

(4)

Where the term $C_{t} = | {\bar{X}}_{t} - μ_{0} + B_{t - 1} |$ , with $B_{0}$ initially set as 0. The reference parameter $K$ ( $= k σ / \sqrt{n}$ ; k > 0) and control limits H( $= \frac{h σ}{\sqrt{n}}; h > 0$ ), LCL, and UCL in the Shewhart-CCUSUM chart share similarities with those in the Shewhart-CUSUM chart mentioned in Section 2.1.1. The two-sided Shewhart-CCUSUM chart raises an out-of-control signal when $B_{t}$ exceeds $H$ or is less than $- H$ , and/or when ${\bar{X}}_{t}$ falls below LCL or exceeds UCL. Unlike the Shewhart-CUSUM charting-statistics $A_{t}^{+}$ and $A_{t}^{-}$ , the CUSUM charting-statistic $B_{t}$ is updated using ${\bar{Y}}_{t}$ , and then adjusted toward zero, enhancing the chart’s sensitivity to detect small shifts. For more details on the Shewhart-CCUSUM chart, readers can refer to Munir et al.¹¹

Proposed combined control charts

In this section, at first hand, the new Combined Shewhart-CUESUM for efficiently detecting the irregular changes in process mean, is developed. The Combined Shewhart-CUESUM, referred to Shewhart-CUESUM, also proposed in the presence of measurement error.

The Shewhart-CUESUM chart

The Combined Shewhart-CUESUM control chart amalgamates the strengths of the Shewhart chart, which is particularly effective in targeting large size shifts, with the sensitivity of CUESUM in detecting small-to-moderate size shifts in the process mean. The overarching objective is to integrate these dual strengths into a single chart, thereby empowering it with a comprehensive capability to handle both large and small-to-moderate size shifts in the process mean. Building upon Lucas⁸ concept of enhancing detection capabilities for small-sized shifts using the CUESUM chart, further advanced the methodology by integrating the Shewhart chart with the CUESUM chart. As a result, proposed a new combined control chart called the Shewhart-CUESUM chart. This innovative approach enables efficient and simultaneous monitoring of process mean shifts, encompassing both small-to-moderate shifts as well as large shifts in normally distributed processes.

Using the order of ${{\bar{X}}_{t}}$ based on equation (11) plotting-statistics { $P_{t} and {\bar{X}}_{t}}$ . The combined SCE mean CC is given bellows.

\begin{matrix} P_{t} = N_{t} \times \max (0, 1 - \frac{K \times λ}{| M_{t} |}) \\ N_{t} = λ M_{t} + (1 - λ) N_{t - 1} \end{matrix}}

(5)

Where $M_{t} = {\bar{X}}_{t} + P_{t - 1}$ and $P_{0} = N_{0} = 0$ . $K = k σ_{X} / \sqrt{n} (> 0)$ and $H = h σ_{X} / \sqrt{n} (> 0)$ are the sensitive parameter and threshold of Shewhart-CUESUM chart. To reach pre-defined ARL₀, the control limits h and L are determined. Whenever $P_{t} > H$ or $P_{t} < - H$ and/or ${\bar{X}}_{t}$ >UCL, UCL/LCL are defined in equation (3), then two-sided Shewhart-CUESUM chart indicates an out-of-control condition when a signal is triggered. The UCL(LCL) of the Shewhart in combined Shewhart-CUESUM chart, targeting the large size shifts while H(-H) are the control limits of the CUESUM chart in combined Shewhart-CUESUM, detecting the small-to-large size shifts. These combining features make the Shewhart-CUESUM chart more sensitive against process mean shifts. It is important to note that Shewhart-CUESUM chart simplifies toward the Shewhart-CCUSUM chart when the parameter λ is set to 1.

Measurement error-based combined Shewhart-CUESUM chart

In this subsection, development of the combined Shewhart-CUESUM chart to effectively monitor the process mean when measurement errors are present, called the M-Shewhart-CUESUM chart.

Let us consider an ongoing manufacturing process where successive observations of the quality characteristic of interest, $X$ , are generated at different time points $t = 1, 2, . . .$ , denoted as ${X_{t, 1}, X_{t, 2}, \dots, X_{t, n}}$ . It is assumed that the $X_{t, j}$ ’s are normally distributed and independent random variables that is, $X_{i, j} ~ N (μ_{0}, σ)$ , with mean $μ_{0}$ and the standard-deviation $σ$ . Linna and Woodall¹⁶ proposed the assumption that the quality characteristic $X_{t, j}$ is not directly observable, but instead can only be evaluated based on a set of $m \geq 1$ measurement operations, denoted as ${Y_{t, j, 1}, Y_{t, j, 2}, \dots, Y_{t, j, m}}$ , where each $Y_{i, j, k}$ corresponds to a linearly covariate error model.

Y_{t, j, k} = A + B X_{t, j} + ε_{t, j, k},

(6)

In the given model, A and B represent known constants, while $ε_{t, j, k}$ denotes the normal random error term resulting from measurement inaccuracy. The error term ε follows a normal distribution $ε ~ N (0, σ_{M})$ , and it is independent of $X_{t, j}$ . The measurement error variance ${σ_{M}}^{2}$ can either be a constant that is unrelated to the nominal mean $μ_{0}$ or, as mentioned by Linna and Woodall,¹⁶ it can have a linear dependence on $μ = μ_{0} + Δ σ$ , such as ${σ_{M}}^{2} = C + D μ$ , where C and D are constants. It is important to note that measurement error might also incorporate a constant misrepresentation caused by factors like poor calibration. In this discussion, here considered the case where the measurement error variance is constant and independent of the nominal mean, with the error having a mean of zero. Similar methods can be applied to the case of a linearly increasing measurement error.

Given that the variable $Y_{t, j, k}$ is subject to measurement imperfections, it is customary to employ a strategy of multiple measurements (referred to as an m-measurements strategy, where $m \geq 1$ ) to mitigate the impact of measurement error. This approach intends to reduce the overall effect of measurement errors by obtaining several measurements for each item in a sample. Linna and Woodall¹⁶ proposed this remedial scheme, emphasizing the importance of averaging multiple measurements per item to decrease the variance caused by measurement errors compared to a single measurement approach. However, it is crucial for practitioners to strike a balance between the additional costs and time associated with multiple measurements and the acceptable level of measurement errors. Careful consideration should be given to selecting an appropriate value of m that ensures a reasonable trade-off between these factors.

At each time point $t = 1, 2, . . .$ , and for each $j = 1, 2, . . ., n$ , and $k = 1, 2, . . ., m$ , with have a total of $m \times n$ observations denoted as $Y_{t, j, k}$ , and the sample mean ${\bar{Y}}_{t}$ is calculated as follows:

{\bar{Y}}_{t} = \frac{1}{mn} \sum_{j = 1}^{n} \sum_{k = 1}^{m} Y_{t, j, k} = A + \frac{1}{n} (B \sum_{j = 1}^{n} X_{t, j} + \frac{1}{m} \sum_{j = 1}^{n} \sum_{k = 1}^{m} ε_{t, j, k}) .

(7)

It can be demonstrated that mean $E ({\bar{Y}}_{t})$ and variance $V ({\bar{Y}}_{t})$ of $Y_{t}$ are equivalent to:

E ({\bar{Y}}_{t}) = A + B (μ_{0} + Δ σ),

(8)

V ({\bar{Y}}_{t}) = (B^{2} σ^{2} + \frac{σ_{M}^{2}}{m}),

(9)

or equivalently

V ({\bar{Y}}_{t}) = \frac{B^{2} σ^{2}}{n {C^{2}}_{(n, m, γ)}},

(10)

where

C^{2} = \frac{m}{m + {(σ_{M} / σ_{0})}^{2}} = \frac{m}{m + γ^{2}} .

The ratio γ, where γ= $\frac{σ_{M}}{σ_{0}}$ (with γ > 0), represents the relationship between the variability of the measurement system ( $σ_{M}$ ) and the variability of the process ( $σ_{0}$ ).

Using the order of ${{\bar{Y}}_{t}}$ based on equation (7) plotting-statistics { $Q_{t} and {\bar{Y}}_{t}}$ . The M-Shewhart-CUESUM CC are given bellows.

\begin{matrix} Q_{t} = R_{t} \times \max (0, 1 - \frac{K \times λ}{| S_{t} |}) \\ R_{t} = λ M_{t} + (1 - λ) R_{t - 1} \end{matrix}}

(11)

The control limits for the Shewhart chart can be calculated as follows:

\frac{LCL}{UCL} = μ_{Y} \mp L \frac{σ_{Y}}{\sqrt{n}},

(12)

Where $S_{t} = {\bar{Y}}_{t} + Q_{t - 1}$ and $Q_{0} = R_{0} = 0$ . K $(= \frac{k σ_{Y}}{\sqrt{n}}; > 0)$ is the sensitive parameter and H $(= \frac{h σ_{Y}}{\sqrt{n}}; > 0)$ is the threshold for initiating the signals by the Shewhart-CUESUM chart. To achieve the pre-fixed in-control ARL, the values of $k, h$ , and $L,$ are determined. Whenever $Q_{t} > H$ or $Q_{t} < - H$ and/or ${\bar{Y}}_{t} < LCL$ or ${\bar{Y}}_{t}$ >UCL, UCL and LCL defined in equation (12), the two-sided M-Shewhart-CUESUM chart triggers an out-of-control signal in the process mean. In this M-Shewhart-CUESUM chart setting, this chart is a special case of M-Shewhart-CCUSUM chart when $λ$ set to be 1.

Run-length profiles and performance comparison

There are various performance measures to evaluate the control chart’s sensitivity which is based on run length distributions. These measures include Average Run Length (ARL), Standard Deviation of Run Length (SDRL), Extra Quadratic Loss (EQL), Relative Average Run Length (RARL), and Performance Comparison Index (PCI).

The performance analysis of a control chart is typically based on its ARL. A run refers to a consecutive sequence of samples in which the process remains under control. The ARL is defined as the reciprocal of the probability that the statistic value, used for monitoring the process, exceeds the control limits. It represents the number of samples needed to detect a shift, whether the process is in control or out of control. The ARL calculation assumes that the process is initially in control. For a control chart, a higher ARL indicates better performance when the process is in control, while a lower ARL is desirable when the process is out of control. ARL serves as a comparative measure among different charts for a specific shift size. The ARL can be determined using either the Markov Chain approach or Monte Carlo simulation. In this study, the Monte Carlo simulation method is employed. Further insights and details on ARL analysis can be found in the works of Lucas⁸ and Lucas and Saccucci.⁹

In SPC, performance measures like ARL and SDRL are commonly employed to assess the performance profiles of control charts. They are used to evaluate the sensitivity of a control chart in detecting a specific size of shift. However, when the shift magnitude is unknown and can reasonably be assumed to fall within a certain range, it is more appropriate to consider measures that provide an overall assessment of performance across the range. In the SPC literature, three well-established performance measures are used for this purpose: EQL, RARL, and PCI. The mathematical expressions for these measures are as follows:

EQL = \frac{1}{s + 1} \sum_{i = 0}^{s} {δ_{i}}^{2} ARL (δ_{i})

(13)

RARL = \frac{1}{s + 1} \sum_{i = 0}^{s} \frac{AR L_{c} (δ_{i})}{AR L_{opt} (δ_{i})}

(14)

PCI = EQL / EQ L_{opt} .

(15)

Here, define $δ_{i}$ as $δ_{i}$ = $a + i (b - a) / s$ , where s is a given number. For computing these measures, it is considered the two-sided optimal conventional CUSUM chart. The overall performance of a control chart can be assessed based on the value of the EQL. A lower EQL value indicates higher overall sensitivity of the control chart, signifying better performance. For the optimal chart, the RARL value is equal to one. Consequently, the nearer the RARL value is to 1, the more effective the given control chart will be compared to the optimal control chart. PCI is defined as the ratio between the EQL of a specific chart and the EQL of the optimal chart under the same conditions. This index facilitates performance comparison by enabling ranking based on EQL values.

The performance evaluation of the two-sided Shewhart-CUESUM and CUESUM charts is carried out when the in-control ARL is set at 370. For both charts, here considered the commonly used value of the reference parameter as $k = 1 / 2$ while also for Shewhart-CUESUM chart under varying values of $λ \in (0.1, 0.25, 0.5, 0.75, 0.9)$ in order to explore the detection abilities without measurement error.

The performance of the combined Shewhart-CUESUM, Shewhart-CUSUM, and Shewhart-CCUSUM charts under the presence of measurement error are investigated. Then evaluated their performance in terms of ARL and SDRL. To conduct the investigation, by using different values for the number of measurement replications $m \in {1, 2, 3, 4}$ , ratios of measurement system variance to process variance $γ \in {0.0, 0.1, \dots, 1.0}$ , and shift sizes Δ $\in {0.25, 0.50, \dots, 3.00}$ . Additionally, considering different values for the parameter $λ \in (0.10, 0.25, 0.5, 0.75, 0.9)$ , while setting $k = 0.5$ . These parameter values were selected to encompass a wide range of possible shifts, accuracy errors, and measuring times for each item in a sample. A γ value of 0 indicates no measurement error, while a γ value of 1 represents a significant accuracy error. In simulations to compute the out-of-control ARL, the chart’s parameters are selected to ensure the desired in-control ARL when no measurement error exists. Following are the main comparative analyses presented in the subsections.

CUESUM versus Shewhart-CUESUM chart

By analyzing the performance presented in Table 1 and Figure 1, a comparison was made between the Shewhart-CUESUM and CUESUM charts in terms of their IRARL, EQL, and PCI. The results indicate that the Shewhart-CUSUM chart reveals greater sensitivity than the CUESUM chart in detecting shifts of different magnitudes in the process mean, thereby demonstrating its overall effectiveness. For instance, the CUESUM chart, at λ = 0.5, the values of (IRARL, EQL, PCI) were observed to be (1.67, 27.02, 1.79), while for the Shewhart-CUESUM chart, they were (1.27, 18.79, 1.25) respectively. Moreover, it is noteworthy that as the value of λ increases, the sensitivity of the Shewhart-CUESUM chart also increases, as evident from the pairs (λ, IRARL) with values (0.1, 1.52), (0.5, 1.27), and (0.9, 1.20).

Table 1.

Run-length performance comparisons of two-sided Combined Shewhart-CUESUM and CUESUM charts when ARL₀=500.

Δ	RL	λ = 0.10		λ = 0.25		λ = 0.50		λ = 0.75		λ = 0.90
Δ	RL	SCE	CE	SCE	CE	SCE	CE	SCE	CE	SCE	CE
0.25	ARL	79.24	75.08	83.39	79.21	98.25	93.50	119.39	113.27	136.11	126.97
0.25	SDRL	46.46	44.19	62.31	59.06	84.36	79.94	109.63	104.02	128.21	119.64
0.50	ARL	34.59	32.93	30.63	29.18	30.34	29.15	33.14	31.80	36.38	34.38
0.50	SDRL	14.16	13.31	15.61	14.91	18.82	18.21	24.32	23.27	28.74	27.07
0.75	ARL	21.89	20.93	18.19	17.45	16.54	15.92	16.28	15.65	16.69	15.98
0.75	SDRL	7.45	6.70	7.25	6.86	7.88	7.60	9.17	8.77	10.40	9.90
1.00	ARL	15.91	15.43	12.94	12.47	11.24	10.90	10.49	10.16	10.33	9.99
1.00	SDRL	4.98	4.18	4.55	4.07	4.53	4.25	4.88	4.64	5.25	5.02
1.25	ARL	12.39	12.24	10.01	9.76	8.52	8.31	7.69	7.52	7.41	7.23
1.25	SDRL	3.89	2.93	3.31	2.76	3.10	2.79	3.14	2.96	3.26	3.08
1.50	ARL	9.98	10.17	8.07	8.07	6.80	6.76	6.06	5.99	5.78	5.68
1.50	SDRL	3.33	2.19	2.69	2.05	2.38	2.02	2.30	2.07	2.33	2.13
1.75	ARL	8.19	8.73	6.68	6.88	5.63	5.70	4.97	5.00	4.71	4.69
1.75	SDRL	3.07	1.74	2.37	1.59	2.00	1.55	1.84	1.57	1.81	1.59
2.00	ARL	6.76	7.67	5.61	6.02	4.75	4.96	4.20	4.29	3.96	4.01
2.00	SDRL	2.88	1.42	2.19	1.30	1.78	1.24	1.57	1.24	1.50	1.25
2.25	ARL	5.54	6.84	4.70	5.36	4.03	4.40	3.59	3.78	3.39	3.52
2.25	SDRL	2.70	1.19	2.05	1.08	1.64	1.03	1.40	1.01	1.31	1.02
2.50	ARL	4.52	6.17	3.95	4.85	3.45	3.95	3.09	3.40	2.94	3.14
2.50	SDRL	2.50	1.02	1.92	0.93	1.52	0.88	1.28	0.86	1.19	0.85
2.75	ARL	3.65	5.65	3.27	4.43	2.94	3.61	2.67	3.08	2.55	2.85
2.75	SDRL	2.22	0.89	1.76	0.81	1.41	0.76	1.18	0.75	1.09	0.74
3.00	ARL	2.94	5.21	2.74	4.09	2.51	3.33	2.33	2.83	2.24	2.60
3.00	SDRL	1.91	0.79	1.58	0.72	1.29	0.67	1.08	0.67	1.00	0.65
3.25	ARL	2.40	4.84	2.29	3.80	2.15	3.09	2.03	2.62	1.97	2.41
3.25	SDRL	1.59	0.72	1.38	0.66	1.16	0.60	0.99	0.60	0.92	0.57
3.50	ARL	1.97	4.52	1.91	3.55	1.85	2.89	1.78	2.44	1.73	2.26
3.50	SDRL	1.29	0.65	1.15	0.60	1.01	0.56	0.89	0.55	0.83	0.49
3.75	ARL	1.67	4.25	1.64	3.33	1.61	2.72	1.57	2.29	1.54	2.14
3.75	SDRL	1.02	0.59	0.94	0.54	0.85	0.55	0.78	0.48	0.73	0.42
4.00	ARL	1.44	4.02	1.44	3.16	1.42	2.55	1.41	2.17	1.38	2.04
4.00	SDRL	0.79	0.55	0.76	0.48	0.71	0.53	0.66	0.41	0.63	0.37
4.25	ARL	1.29	3.81	1.29	3.01	1.28	2.40	1.28	2.09	1.26	1.97
4.25	SDRL	0.61	0.53	0.60	0.43	0.57	0.50	0.55	0.34	0.52	0.35
4.50	ARL	1.19	3.61	1.19	2.89	1.19	2.27	1.18	2.02	1.18	1.90
4.50	SDRL	0.47	0.53	0.47	0.43	0.46	0.45	0.45	0.29	0.42	0.36
4.75	ARL	1.12	3.42	1.12	2.76	1.12	2.17	1.12	1.97	1.11	1.83
4.75	SDRL	0.37	0.51	0.37	0.46	0.36	0.37	0.36	0.28	0.34	0.40
5.00	ARL	1.07	3.26	1.07	2.62	1.07	2.09	1.07	1.92	1.07	1.75
5.00	SDRL	0.28	0.45	0.28	0.49	0.28	0.29	0.28	0.30	0.27	0.44
IRARL		1.52	2.52	1.37	2.01	1.27	1.67	1.21	1.49	1.20	1.42
EQL		22.08	41.68	20.19	33.13	18.79	27.02	17.93	23.82	17.57	22.37
PCI		1.46	2.76	1.34	2.20	1.25	1.79	1.19	1.58	1.17	1.48

CE: cumulative EWMA-sum; SCE: Shewhart-Cumulative-EWMA-sum.

Figure 1.

Comparisons of the ARL curves of the two-sided Shewhart-CUESUM (SCE) and CUESUM (CE) chart when $λ = 0.1, λ = 0.25, λ = 0.50, λ = 0.75$ and $λ = 0.90$ .

By examining Figure 1, it becomes evident that for shift sizes $δ < 2$ , the detection ability of the Shewhart-CUESUM chart is comparable to that of the CUESUM chart across different values of λ. This observation similar to Locus⁹ studied in that in combined Shewhart-CUSUM chart, the CUSUM chart performs more or less same as to single CUSUM but effective in detecting moderate-to-large size shifts and when comparing overall performance. However, as the magnitude of the shift increases (δ > 2), the Shewhart-CUESUM chart consistently exhibits higher sensitivity compared to the CUESUM chart. Additionally, when considering the SDRL values presented in Table 1, certain trends can be observed. Consequently, when aiming to simultaneously detect small-to-moderate as well as large-size shifts, the Shewhart-CUESUM chart emerges as the preferable choice for effective process controlling.

Shewhart-CUESUM versus Shewhart-CUSUM and Shewhart-CCUSUM charts

Table 2 presents a comparison between Shewhart-CUSUM, Shewhart-CCUSUM, and Shewhart-CUESUM charts as to ARL and SDRL. Evaluation of the detection sensitivity performance of the Shewhart-CUESUM chart by examining it under varying magnitudes of λ. Results indicate that irrespective of whether λ is small or large, the Shewhart-CUESUM chart consistently demonstrated greater sensitivity compared to the existing Shewhart-CUSUM and Shewhart-CCUSUM charts in detecting small-to-moderate shift sizes (δ ≤ 2). For example, when considering a shift size (δ) of 0.50, the out-of-control ARL values for the Shewhart-CUSUM, Shewhart-CCUSUM, and Shewhart-CUESUM charts with λ = 0.10 are observed to be 80.87, 79.86, and 30.97, respectively. It is important to note that when the target is to detect small shifts, selecting smaller values of λ (less than 0.2) is advisable, while larger values of λ are more suitable for detecting large shifts. In the case of larger values of λ, the Shewhart-CUESUM chart maintains its superior sensitivity in detecting small-to-moderate size shifts compared to the Shewhart-CUSUM and Shewhart-CCUSUM charts, while still performing similarly to them. It is important to highlight that the Shewhart-CUESUM chart represents a specific instance of the Shewhart-CCUSUM chart, occurring when λ = 1, consistent with Haq’s⁷ explanation of how the CUESUM control chart transforms into the CCUSUM chart under these conditions.

Table 2.

Run-length performance comparisons of two-sided Combined control charts.

δ	RL	SC	SCC	SCE (λ = 0.10)	SCE (λ = 0.25)	SCE (λ = 0.50)	SCE (λ = 0.75)	SCE (λ = 0.90)
0.25	ARL	256.15	255.41	69.96	73.05	85.04	101.32	112.60
0.25	SDRL	255.36	253.26	42.31	55.27	72.88	92.58	105.72
0.50	ARL	80.87	79.86	30.97	27.72	27.66	29.89	32.44
0.50	SDRL	69.59	68.51	13.15	14.60	17.64	22.11	25.91
0.75	ARL	30.38	29.97	19.72	16.57	15.26	15.02	15.31
0.75	SDRL	19.49	19.26	6.90	6.84	7.52	8.69	9.62
1.00	ARL	17.13	16.90	14.35	11.81	10.40	9.77	9.62
1.00	SDRL	9.44	9.29	4.61	4.26	4.30	4.65	5.00
1.25	ARL	11.54	11.35	11.21	9.13	7.90	7.21	6.96
1.25	SDRL	6.06	5.96	3.55	3.05	2.93	3.02	3.13
1.50	ARL	8.35	8.25	9.10	7.40	6.32	5.69	5.42
1.50	SDRL	4.50	4.43	3.01	2.47	2.22	2.19	2.22
1.75	ARL	6.31	6.24	7.51	6.16	5.25	4.68	4.44
1.75	SDRL	3.63	3.55	2.74	2.14	1.84	1.74	1.71
2.00	ARL	4.85	4.81	6.23	5.19	4.45	3.95	3.74
2.00	SDRL	2.98	2.94	2.55	1.96	1.62	1.47	1.41
2.25	ARL	3.81	3.77	5.17	4.40	3.80	3.40	3.21
2.25	SDRL	2.47	2.43	2.41	1.84	1.50	1.30	1.23
2.50	ARL	2.99	2.98	4.27	3.72	3.27	2.95	2.79
2.50	SDRL	2.03	2.01	2.24	1.73	1.39	1.18	1.11
2.75	ARL	2.41	2.40	3.50	3.14	2.82	2.57	2.44
2.75	SDRL	1.65	1.62	2.03	1.60	1.30	1.10	1.01
3.00	ARL	1.97	1.97	2.86	2.64	2.42	2.26	2.15
3.00	SDRL	1.31	1.29	1.78	1.45	1.19	1.01	0.94

SC: Shewhart-CUSUM; SCC: Crosier-Shewhart-Cumulative-EWMA-sum.

By utilizing the computed ARL values, overall performance was evaluated of the existing and proposed charts in terms of IRARL, EQL, and PCI. These results are presented in Figure 2. It is evident from the figure that the Shewhart-CUESUM chart outperforms the existing counterparts in terms of efficiency for monitoring the process mean. Additionally, when considering different values of λ, the Shewhart-CUESUM chart exhibits higher sensitivity, particularly at larger values of λ.

Figure 2.

IRARL, EQL, and PCI performance of the combined charts.

Effect of measurement error on combined chart

Tables 3 to 6 demonstrate that as the magnitude of measurement error (γ) increases from 0 (representing a scenario without any measurement error, i.e. perfect measurement) up to 1 (representing the worst-case scenario with significant measurement error), the ARL₁ values of the Combined Shewhart-CUSUM, Shewhart-CCUSUM, and M-Shewhart-CUESUM charts (for different values of $m : 1, 2, 3, and 4$ ) exhibit an upward trend. The ARL₁ values are directly proportional to the level of γ, indicating that higher measurement error leads to higher ARL₁ values. For instance, from Table 3, considering the M-Shewhart-CUESUM chart with $λ = 0.5$ , the (γ, ARL₁) values are observed as (0.00, 85.04), (0.40, 137.42), (0.70, 173.59), and so on, reaching (1.0, 203.66) for δ = 0.25 with $m = 1$ .

Table 3.

ARL₁ profile of the Two-sided combined charts, with ARLo = 370 and m = 1.

Δ	Chart	ϒ
Δ	Chart	0.00	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80	0.90	1.00
0.25	SC	256.15	272.00	284.79	292.81	297.83	303.19	308.95	312.15	315.81	319.64	319.48
	SCC	255.41	264.36	271.65	278.20	286.24	291.26	296.93	299.60	303.60	307.07	311.76
	SCE(λ = 0.10)	69.96	78.30	86.65	95.20	103.35	112.69	120.21	129.31	137.69	145.75	153.23
	SCE(λ = 0.25)	73.05	83.98	94.65	105.37	116.19	127.06	137.83	147.51	157.17	166.27	175.53
	SCE(λ = 0.50)	85.04	98.20	111.02	124.65	137.42	149.79	162.02	173.59	182.92	192.74	203.66
	SCE(λ = 0.75)	101.32	116.84	132.31	146.95	160.92	175.17	187.45	197.69	209.02	219.20	228.43
	SCE(λ = 0.90)	112.60	128.97	145.66	159.93	174.87	188.54	200.69	211.13	222.50	231.12	243.16
0.50	SC	80.87	96.76	114.74	132.49	150.95	166.50	179.81	194.08	204.86	212.96	220.30
	SCC	79.86	90.15	100.00	109.97	119.25	129.01	137.81	146.37	154.22	161.98	170.54
	SCE(λ = 0.10)	30.97	34.59	38.29	41.90	45.78	49.64	53.60	57.37	61.47	65.56	69.85
	SCE(λ = 0.25)	27.72	31.47	35.51	39.57	44.05	48.56	53.38	58.11	62.87	68.39	73.54
	SCE(λ = 0.50)	27.66	32.19	36.99	42.22	47.84	53.61	59.50	65.90	71.87	78.38	84.90
	SCE(λ = 0.75)	29.89	35.65	41.89	48.24	55.43	62.57	69.86	77.39	85.49	93.25	101.01
	SCE(λ = 0.90)	32.44	39.06	45.91	53.37	61.49	69.30	77.63	86.65	95.30	103.54	112.38
0.75	SC	30.38	35.67	41.66	48.22	55.58	63.60	71.99	81.56	91.21	101.15	112.08
	SCC	29.97	33.52	36.99	40.36	44.31	48.22	52.19	55.99	60.51	64.96	69.10
	SCE(λ = 0.10)	19.72	21.89	24.16	26.38	28.68	30.97	33.48	35.83	38.17	40.73	43.36
	SCE(λ = 0.25)	16.57	18.66	20.79	22.99	25.32	27.72	30.15	32.75	35.37	38.19	40.93
	SCE(λ = 0.50)	15.26	17.44	19.76	22.19	24.80	27.72	30.70	33.75	37.23	40.47	44.30
	SCE(λ = 0.75)	15.02	17.57	20.27	23.34	26.58	30.08	33.85	37.66	41.89	46.04	50.49
	SCE(λ = 0.90)	15.31	18.15	21.22	24.64	28.54	32.47	36.59	41.19	45.86	50.80	56.05
1.00	SC	17.13	19.71	22.52	25.48	28.83	32.47	36.27	40.52	45.19	49.69	54.82
	SCC	16.90	18.51	20.07	21.76	23.42	25.11	26.93	28.69	30.53	32.37	34.13
	SCE(λ = 0.10)	14.35	15.97	17.57	19.21	20.81	22.42	24.14	25.82	27.59	29.23	31.04
	SCE(λ = 0.25)	11.81	13.17	14.63	16.06	17.61	19.20	20.73	22.45	24.23	25.92	27.73
	SCE(λ = 0.50)	10.40	11.78	13.21	14.74	16.26	18.00	19.74	21.55	23.59	25.62	27.72
	SCE(λ = 0.75)	9.77	11.20	12.78	14.47	16.27	18.25	20.30	22.61	24.92	27.43	30.06
	SCE(λ = 0.90)	9.62	11.15	12.83	14.71	16.71	18.90	21.25	23.88	26.47	29.50	32.52
1.25	SC	11.54	13.08	14.78	16.63	18.57	20.74	22.98	25.43	27.93	30.67	33.42
	SCC	11.35	12.38	13.35	14.29	15.35	16.27	17.31	18.37	19.36	20.38	21.41
	SCE(λ = 0.10)	11.21	12.49	13.75	15.03	16.26	17.56	18.83	20.12	21.45	22.77	24.12
	SCE(λ = 0.25)	9.13	10.18	11.24	12.34	13.48	14.60	15.75	16.99	18.22	19.46	20.77
	SCE(λ = 0.50)	7.90	8.86	9.90	10.95	12.04	13.24	14.45	15.64	17.01	18.34	19.73
	SCE(λ = 0.75)	7.21	8.17	9.21	10.34	11.53	12.74	14.12	15.52	17.05	18.63	20.36
	SCE(λ = 0.90)	6.96	7.94	9.04	10.24	11.48	12.84	14.28	15.89	17.65	19.39	21.27
1.50	SC	8.35	9.47	10.65	11.89	13.27	14.71	16.17	17.80	19.47	21.29	23.22
	SCC	8.25	8.97	9.69	10.38	11.10	11.81	12.45	13.16	13.81	14.52	15.23
	SCE(λ = 0.10)	9.10	10.16	11.23	12.32	13.31	14.37	15.41	16.48	17.55	18.63	19.73
	SCE(λ = 0.25)	7.40	8.28	9.13	10.02	10.91	11.81	12.70	13.68	14.62	15.57	16.59
	SCE(λ = 0.50)	6.32	7.10	7.89	8.69	9.53	10.40	11.31	12.20	13.20	14.17	15.22
	SCE(λ = 0.75)	5.69	6.42	7.21	7.99	8.86	9.76	10.72	11.71	12.76	13.84	15.07
	SCE(λ = 0.90)	5.42	6.15	6.95	7.78	8.68	9.61	10.63	11.72	12.83	14.08	15.32
1.75	SC	6.31	7.14	8.01	8.96	9.93	10.96	12.05	13.21	14.40	15.73	17.09
	SCC	6.24	6.81	7.37	7.92	8.47	8.96	9.52	10.00	10.51	11.02	11.54
	SCE(λ = 0.10)	7.51	8.47	9.38	10.32	11.22	12.12	13.02	13.93	14.79	15.73	16.65
	SCE(λ = 0.25)	6.16	6.91	7.65	8.39	9.14	9.89	10.67	11.41	12.22	12.98	13.82
	SCE(λ = 0.50)	5.25	5.90	6.54	7.22	7.90	8.58	9.30	10.05	10.80	11.57	12.36
	SCE(λ = 0.75)	4.68	5.29	5.90	6.53	7.19	7.87	8.61	9.39	10.16	11.02	11.85
	SCE(λ = 0.90)	4.44	5.01	5.63	6.28	6.95	7.65	8.41	9.20	10.04	10.92	11.83
2.00	SC	4.85	5.49	6.16	6.89	7.62	8.43	9.25	10.10	11.05	11.96	12.99
	SCC	4.81	5.29	5.74	6.22	6.66	7.10	7.52	7.91	8.34	8.76	9.17
	SCE(λ = 0.10)	6.23	7.13	7.99	8.83	9.62	10.42	11.23	12.01	12.80	13.58	14.40
	SCE(λ = 0.25)	5.19	5.87	6.53	7.18	7.83	8.49	9.13	9.78	10.45	11.12	11.78
	SCE(λ = 0.50)	4.45	5.01	5.57	6.13	6.72	7.29	7.90	8.50	9.12	9.75	10.40
	SCE(λ = 0.75)	3.95	4.46	4.97	5.50	6.05	6.60	7.19	7.79	8.43	9.10	9.76
	SCE(λ = 0.90)	3.74	4.22	4.72	5.24	5.77	6.34	6.93	7.56	8.22	8.90	9.66
2.25	SC	3.81	4.30	4.81	5.38	5.98	6.58	7.18	7.91	8.59	9.29	10.10
	SCC	3.77	4.19	4.56	4.94	5.32	5.69	6.06	6.40	6.75	7.11	7.43
	SCE(λ = 0.10)	5.17	6.03	6.83	7.60	8.37	9.08	9.81	10.51	11.23	11.93	12.61
	SCE(λ = 0.25)	4.40	5.03	5.65	6.24	6.82	7.40	7.98	8.54	9.14	9.71	10.34
	SCE(λ = 0.50)	3.80	4.32	4.83	5.32	5.83	6.33	6.83	7.35	7.88	8.43	8.97
	SCE(λ = 0.75)	3.40	3.85	4.29	4.75	5.21	5.69	6.17	6.70	7.19	7.73	8.29
	SCE(λ = 0.90)	3.21	3.64	4.06	4.49	4.96	5.43	5.92	6.40	6.95	7.49	8.06
2.50	SC	2.99	3.38	3.78	4.22	4.69	5.18	5.66	6.19	6.74	7.33	7.93
	SCC	2.98	3.31	3.65	3.97	4.31	4.61	4.93	5.23	5.53	5.82	6.13
	SCE(λ = 0.10)	4.27	5.09	5.87	6.61	7.31	7.99	8.65	9.31	9.95	10.57	11.23
	SCE(λ = 0.25)	3.72	4.34	4.90	5.47	6.00	6.52	7.06	7.58	8.10	8.62	9.14
	SCE(λ = 0.50)	3.27	3.75	4.22	4.67	5.13	5.57	6.03	6.49	6.93	7.41	7.89
	SCE(λ = 0.75)	2.95	3.36	3.76	4.16	4.56	4.98	5.39	5.83	6.27	6.72	7.20
	SCE(λ = 0.90)	2.79	3.17	3.55	3.93	4.32	4.73	5.13	5.57	6.01	6.47	6.94
2.75	SC	2.41	2.71	3.04	3.37	3.74	4.11	4.51	4.93	5.37	5.80	6.26
	SCC	2.40	2.68	2.96	3.22	3.51	3.78	4.06	4.31	4.59	4.83	5.10
	SCE(λ = 0.10)	3.50	4.27	5.02	5.72	6.40	7.06	7.68	8.30	8.89	9.48	10.08
	SCE(λ = 0.25)	3.14	3.72	4.26	4.81	5.31	5.82	6.30	6.77	7.24	7.74	8.19
	SCE(λ = 0.50)	2.82	3.28	3.71	4.14	4.55	4.95	5.37	5.77	6.19	6.60	7.02
	SCE(λ = 0.75)	2.57	2.95	3.32	3.68	4.05	4.42	4.79	5.17	5.54	5.95	6.36
	SCE(λ = 0.90)	2.44	2.80	3.13	3.48	3.83	4.18	4.54	4.90	5.30	5.68	6.07
3.00	SC	1.97	2.20	2.46	2.73	2.99	3.29	3.63	3.93	4.29	4.64	5.00
	SCC	1.97	2.20	2.42	2.66	2.89	3.12	3.35	3.58	3.82	4.04	4.27
	SCE(λ = 0.10)	2.86	3.57	4.28	4.95	5.60	6.24	6.84	7.43	8.00	8.55	9.09
	SCE(λ = 0.25)	2.64	3.18	3.72	4.23	4.73	5.19	5.66	6.10	6.54	6.98	7.41
	SCE(λ = 0.50)	2.42	2.86	3.27	3.67	4.06	4.45	4.82	5.20	5.57	5.95	6.32
	SCE(λ = 0.75)	2.26	2.61	2.95	3.29	3.62	3.96	4.30	4.63	4.98	5.33	5.68
	SCE(λ = 0.90)	2.15	2.47	2.79	3.11	3.42	3.74	4.05	4.39	4.72	5.06	5.42

SC: Shewhart-CUSUM; SCC: Crosier’s CUSUM; SCE: Shewhart-Cumulative-EWMA-sum.

Table 4.

ARL₁ profile of the Two-sided combined charts, with ARLo = 370 and m = 2.

Δ	Chart	ϒ
Δ	Chart	0.00	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80	0.90	1.00
0.25	SC	257.40	268.96	276.05	283.75	288.99	293.77	296.12	298.92	302.80	304.92	305.49
	SCC	256.94	260.06	264.56	269.35	274.27	276.35	278.51	282.21	286.17	287.82	291.66
	SCE(λ = 0.10)	69.96	75.87	82.59	88.91	94.87	101.50	107.15	113.61	119.82	126.08	131.97
	SCE(λ = 0.25)	73.08	81.42	88.64	96.85	105.07	112.99	120.95	127.88	135.63	143.88	150.34
	SCE(λ = 0.50)	84.44	94.22	105.17	114.54	124.64	132.91	142.63	151.74	159.96	168.68	176.72
	SCE(λ = 0.75)	101.28	113.22	124.78	134.93	146.33	156.86	166.69	177.18	186.14	193.36	201.74
	SCE(λ = 0.90)	112.65	124.33	136.53	148.37	159.55	170.27	181.04	190.30	198.91	207.17	216.44
0.50	SC	80.48	91.25	103.71	115.25	127.14	139.05	150.52	161.20	168.94	179.57	186.79
	SCC	80.13	85.23	90.24	95.39	100.20	105.51	110.30	114.51	119.58	124.05	129.20
	SCE(λ = 0.10)	31.05	33.67	36.42	39.07	41.81	44.56	47.37	50.17	53.13	55.84	58.70
	SCE(λ = 0.25)	27.79	30.47	33.36	36.47	39.53	42.80	45.92	49.17	52.50	56.03	59.89
	SCE(λ = 0.50)	27.71	31.02	34.61	38.21	41.88	46.02	50.33	54.44	58.93	62.85	67.38
	SCE(λ = 0.75)	30.10	34.20	38.66	43.34	47.93	53.06	58.53	63.77	69.45	74.58	80.63
	SCE(λ = 0.90)	32.55	37.19	42.51	47.63	53.27	58.80	65.08	70.64	76.70	83.27	89.14
0.75	SC	30.38	33.79	37.70	41.66	46.06	50.74	55.47	60.76	66.39	71.94	77.46
	SCC	30.13	31.73	33.41	35.03	36.75	38.58	40.53	42.21	44.16	46.27	48.16
	SCE(λ = 0.10)	19.69	21.34	23.00	24.62	26.32	27.96	29.64	31.39	33.08	34.87	36.60
	SCE(λ = 0.25)	16.59	18.11	19.68	21.27	22.87	24.60	26.30	28.12	29.93	31.75	33.65
	SCE(λ = 0.50)	15.22	16.83	18.57	20.34	22.09	24.06	26.11	28.18	30.17	32.45	34.92
	SCE(λ = 0.75)	15.06	16.90	18.82	20.99	23.22	25.51	28.07	30.66	33.45	36.20	39.12
	SCE(λ = 0.90)	15.39	17.41	19.68	22.06	24.65	27.27	30.10	33.10	36.02	39.47	42.91
1.00	SC	17.12	18.86	20.69	22.65	24.72	26.77	29.13	31.58	34.12	36.80	39.62
	SCC	16.94	17.73	18.49	19.26	20.07	20.92	21.76	22.57	23.44	24.28	25.03
	SCE(λ = 0.10)	14.40	15.54	16.75	17.93	19.12	20.26	21.48	22.73	23.94	25.12	26.38
	SCE(λ = 0.25)	11.81	12.83	13.89	14.93	16.01	17.15	18.30	19.40	20.59	21.67	22.95
	SCE(λ = 0.50)	10.40	11.42	12.44	13.55	14.65	15.74	17.04	18.25	19.52	20.87	22.26
	SCE(λ = 0.75)	9.76	10.85	11.94	13.12	14.35	15.72	17.13	18.53	20.04	21.66	23.26
	SCE(λ = 0.90)	9.64	10.77	11.98	13.23	14.60	16.16	17.66	19.29	20.97	22.75	24.63
1.25	SC	11.50	12.58	13.67	14.84	16.10	17.45	18.82	20.24	21.80	23.39	25.05
	SCC	11.36	11.87	12.33	12.82	13.36	13.85	14.34	14.84	15.36	15.77	16.33
	SCE(λ = 0.10)	11.22	12.17	13.09	14.03	14.97	15.87	16.84	17.76	18.68	19.62	20.55
	SCE(λ = 0.25)	9.13	9.93	10.72	11.51	12.31	13.16	13.95	14.77	15.67	16.52	17.35
	SCE(λ = 0.50)	7.89	8.62	9.35	10.11	10.92	11.70	12.54	13.40	14.26	15.15	16.04
	SCE(λ = 0.75)	7.18	7.92	8.68	9.47	10.27	11.14	12.05	12.94	13.92	14.92	15.99
	SCE(λ = 0.90)	6.93	7.70	8.48	9.30	10.17	11.10	12.05	13.00	14.10	15.25	16.44
1.50	SC	8.36	9.06	9.83	10.64	11.46	12.34	13.29	14.25	15.29	16.34	17.40
	SCC	8.23	8.64	8.95	9.32	9.72	10.03	10.39	10.75	11.08	11.43	11.77
	SCE(λ = 0.10)	9.09	9.89	10.69	11.46	12.21	13.01	13.77	14.53	15.32	16.06	16.85
	SCE(λ = 0.25)	7.41	8.05	8.69	9.32	9.96	10.63	11.30	11.92	12.60	13.27	14.00
	SCE(λ = 0.50)	6.34	6.90	7.48	8.06	8.67	9.28	9.89	10.55	11.21	11.85	12.55
	SCE(λ = 0.75)	5.70	6.23	6.80	7.38	7.95	8.58	9.24	9.91	10.59	11.32	12.09
	SCE(λ = 0.90)	5.42	5.97	6.53	7.13	7.76	8.39	9.08	9.79	10.49	11.26	12.05
1.75	SC	6.31	6.84	7.35	7.93	8.50	9.22	9.82	10.49	11.22	11.94	12.71
	SCC	6.24	6.54	6.82	7.09	7.37	7.65	7.90	8.17	8.44	8.70	9.00
	SCE(λ = 0.10)	7.50	8.22	8.93	9.60	10.27	10.96	11.62	12.25	12.90	13.57	14.20
	SCE(λ = 0.25)	6.16	6.71	7.26	7.82	8.37	8.90	9.46	9.99	10.53	11.10	11.65
	SCE(λ = 0.50)	5.26	5.73	6.19	6.69	7.18	7.68	8.18	8.69	9.21	9.76	10.27
	SCE(λ = 0.75)	4.68	5.12	5.56	6.03	6.50	6.98	7.48	8.00	8.53	9.04	9.63
	SCE(λ = 0.90)	4.43	4.88	5.31	5.77	6.24	6.72	7.24	7.75	8.33	8.86	9.47
2.00	SC	4.86	5.24	5.64	6.05	6.52	6.96	7.42	7.94	8.41	8.98	9.49
	SCC	4.81	5.08	5.31	5.52	5.76	6.00	6.21	6.42	6.66	6.87	7.10
	SCE(λ = 0.10)	6.24	6.93	7.55	8.17	8.79	9.39	9.98	10.54	11.11	11.69	12.26
	SCE(λ = 0.25)	5.19	5.70	6.20	6.67	7.17	7.65	8.12	8.58	9.06	9.53	10.01
	SCE(λ = 0.50)	4.44	4.87	5.29	5.70	6.12	6.54	6.95	7.39	7.82	8.25	8.70
	SCE(λ = 0.75)	3.96	4.33	4.70	5.09	5.49	5.87	6.28	6.71	7.11	7.56	7.99
	SCE(λ = 0.90)	3.74	4.09	4.47	4.84	5.22	5.62	6.02	6.43	6.88	7.32	7.76
2.25	SC	3.78	4.07	4.37	4.69	5.01	5.35	5.68	6.07	6.44	6.81	7.21
	SCC	3.76	3.97	4.17	4.37	4.55	4.75	4.95	5.15	5.33	5.52	5.70
	SCE(λ = 0.10)	5.19	5.83	6.42	7.02	7.60	8.14	8.67	9.21	9.71	10.23	10.74
	SCE(λ = 0.25)	4.40	4.88	5.33	5.78	6.23	6.66	7.07	7.49	7.92	8.32	8.75
	SCE(λ = 0.50)	3.81	4.20	4.57	4.94	5.31	5.67	6.05	6.40	6.76	7.16	7.52
	SCE(λ = 0.75)	3.40	3.73	4.07	4.40	4.73	5.06	5.40	5.75	6.11	6.48	6.83
	SCE(λ = 0.90)	3.21	3.53	3.85	4.16	4.48	4.80	5.13	5.49	5.85	6.21	6.58
2.50	SC	2.99	3.20	3.42	3.66	3.90	4.15	4.42	4.69	4.97	5.25	5.52
	SCC	2.97	3.15	3.33	3.48	3.66	3.81	3.97	4.15	4.31	4.47	4.63
	SCE(λ = 0.10)	4.27	4.88	5.50	6.03	6.57	7.10	7.60	8.10	8.58	9.05	9.51
	SCE(λ = 0.25)	3.72	4.18	4.61	5.03	5.44	5.85	6.24	6.62	6.98	7.38	7.75
	SCE(λ = 0.50)	3.27	3.63	3.98	4.33	4.65	4.98	5.32	5.64	5.97	6.29	6.63
	SCE(λ = 0.75)	2.95	3.25	3.56	3.85	4.15	4.44	4.73	5.04	5.35	5.65	5.97
	SCE(λ = 0.90)	2.79	3.08	3.35	3.63	3.92	4.20	4.48	4.79	5.08	5.40	5.69
2.75	SC	2.41	2.56	2.72	2.90	3.07	3.26	3.45	3.68	3.87	4.08	4.28
	SCC	2.39	2.53	2.67	2.81	2.96	3.09	3.23	3.37	3.50	3.66	3.78
	SCE(λ = 0.10)	3.51	4.07	4.64	5.19	5.70	6.21	6.67	7.14	7.59	8.06	8.47
	SCE(λ = 0.25)	3.14	3.57	3.99	4.40	4.79	5.16	5.53	5.88	6.23	6.58	6.93
	SCE(λ = 0.50)	2.81	3.15	3.49	3.79	4.12	4.43	4.73	5.02	5.31	5.61	5.92
	SCE(λ = 0.75)	2.58	2.85	3.13	3.40	3.68	3.94	4.19	4.46	4.74	5.02	5.29
	SCE(λ = 0.90)	2.44	2.71	2.96	3.22	3.46	3.72	3.98	4.23	4.50	4.75	5.02
3.00	SC	1.97	2.09	2.22	2.34	2.49	2.63	2.76	2.92	3.07	3.22	3.39
	SCC	1.96	2.08	2.19	2.31	2.43	2.53	2.65	2.77	2.88	3.01	3.14
	SCE(λ = 0.10)	2.86	3.40	3.90	4.43	4.93	5.41	5.88	6.33	6.77	7.20	7.60
	SCE(λ = 0.25)	2.65	3.05	3.44	3.84	4.21	4.57	4.93	5.26	5.60	5.91	6.24
	SCE(λ = 0.50)	2.42	2.74	3.06	3.36	3.67	3.94	4.23	4.50	4.78	5.05	5.31
	SCE(λ = 0.75)	2.26	2.52	2.78	3.02	3.28	3.51	3.77	4.01	4.25	4.49	4.75
	SCE(λ = 0.90)	2.14	2.39	2.64	2.86	3.09	3.33	3.55	3.78	4.02	4.26	4.50

SC: Shewhart-CUSUM; SCC: Crosier’s CUSUM; SCE: Shewhart-Cumulative-EWMA-sum.

Table 5.

ARL₁ profile of the Two-sided combined charts, with ARLo = 370 and m = 3.

Δ	Chart	ϒ
Δ	Chart	0.00	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80	0.90	1.00
0.25	SC	258.32	267.15	275.34	279.76	285.42	290.60	292.83	293.50	296.87	295.89	299.50
	SCC	255.78	257.69	263.05	264.06	267.27	270.15	273.64	276.66	276.65	281.36	283.85
	SCE(λ = 0.10)	69.66	75.37	80.61	86.32	91.97	96.96	102.40	107.77	112.71	118.26	123.76
	SCE(λ = 0.25)	73.14	80.32	87.48	94.33	101.15	108.08	114.43	121.65	127.82	134.60	141.26
	SCE(λ = 0.50)	85.30	93.66	102.30	111.04	119.26	127.66	135.98	142.89	151.02	158.75	164.62
	SCE(λ = 0.75)	101.19	112.13	121.50	131.70	141.55	150.16	158.25	167.89	175.18	183.56	190.91
	SCE(λ = 0.90)	112.48	123.38	134.48	144.12	153.74	163.52	172.70	181.36	190.22	197.23	206.23
0.50	SC	80.77	90.16	98.84	109.03	119.34	129.33	138.29	147.73	156.34	163.64	170.80
	SCC	80.37	83.14	86.66	89.83	93.55	96.50	100.25	103.53	106.64	109.61	113.04
	SCE(λ = 0.10)	31.03	33.33	35.73	38.07	40.54	42.93	45.25	47.64	50.01	52.55	54.78
	SCE(λ = 0.25)	27.67	30.19	32.69	35.35	37.95	40.67	43.31	46.20	48.96	51.77	54.74
	SCE(λ = 0.50)	27.74	30.67	33.62	36.84	40.11	43.38	46.82	50.39	54.21	57.66	61.66
	SCE(λ = 0.75)	30.02	33.77	37.77	41.67	45.75	50.15	54.30	58.92	63.39	67.93	72.58
	SCE(λ = 0.90)	32.39	36.54	41.15	45.62	50.58	55.25	60.04	65.46	70.20	75.82	80.67
0.75	SC	30.38	33.20	36.36	39.52	42.96	46.54	50.52	54.38	58.74	63.00	67.53
	SCC	30.02	31.14	32.30	33.46	34.52	35.74	36.94	37.98	39.23	40.50	41.65
	SCE(λ = 0.10)	19.72	21.16	22.60	24.08	25.44	26.93	28.33	29.79	31.27	32.73	34.18
	SCE(λ = 0.25)	16.60	17.96	19.28	20.71	22.08	23.56	24.95	26.36	27.92	29.45	30.99
	SCE(λ = 0.50)	15.25	16.65	18.10	19.64	21.21	22.81	24.59	26.13	28.01	29.93	31.74
	SCE(λ = 0.75)	15.07	16.67	18.42	20.23	22.14	24.04	26.05	28.24	30.38	32.58	34.96
	SCE(λ = 0.90)	15.38	17.13	19.13	21.10	23.41	25.56	27.92	30.36	32.69	35.44	38.05
1.00	SC	17.19	18.53	20.06	21.66	23.31	25.05	26.84	28.80	30.85	32.80	35.02
	SCC	16.94	17.45	17.96	18.50	19.07	19.57	20.16	20.60	21.19	21.73	22.36
	SCE(λ = 0.10)	14.39	15.39	16.43	17.51	18.52	19.54	20.55	21.62	22.63	23.65	24.75
	SCE(λ = 0.25)	11.80	12.72	13.65	14.57	15.47	16.44	17.34	18.33	19.34	20.30	21.41
	SCE(λ = 0.50)	10.41	11.28	12.21	13.16	14.10	15.07	16.08	17.11	18.11	19.26	20.37
	SCE(λ = 0.75)	9.77	10.70	11.68	12.73	13.76	14.89	15.99	17.16	18.43	19.71	20.97
	SCE(λ = 0.90)	9.63	10.62	11.70	12.77	13.90	15.15	16.42	17.81	19.23	20.52	22.09
1.25	SC	11.51	12.35	13.31	14.22	15.32	16.35	17.44	18.60	19.76	21.04	22.18
	SCC	11.34	11.71	12.02	12.34	12.68	13.00	13.35	13.71	14.01	14.31	14.67
	SCE(λ = 0.10)	11.20	12.05	12.89	13.67	14.50	15.29	16.10	16.91	17.65	18.45	19.28
	SCE(λ = 0.25)	9.14	9.84	10.53	11.21	11.93	12.61	13.31	14.01	14.74	15.43	16.20
	SCE(λ = 0.50)	7.89	8.53	9.18	9.85	10.49	11.16	11.86	12.59	13.31	14.04	14.82
	SCE(λ = 0.75)	7.19	7.83	8.50	9.15	9.88	10.59	11.32	12.12	12.93	13.75	14.60
	SCE(λ = 0.90)	6.94	7.60	8.28	8.99	9.71	10.49	11.33	12.11	12.96	13.90	14.90
1.50	SC	8.35	8.93	9.58	10.16	10.87	11.58	12.30	13.04	13.85	14.58	15.46
	SCC	8.23	8.48	8.72	8.96	9.24	9.46	9.71	9.92	10.15	10.39	10.63
	SCE(λ = 0.10)	9.09	9.81	10.50	11.18	11.85	12.52	13.17	13.83	14.47	15.13	15.78
	SCE(λ = 0.25)	7.39	7.97	8.54	9.10	9.64	10.21	10.78	11.35	11.89	12.47	13.03
	SCE(λ = 0.50)	6.34	6.84	7.35	7.85	8.35	8.91	9.42	9.95	10.47	11.05	11.63
	SCE(λ = 0.75)	5.68	6.17	6.66	7.16	7.66	8.18	8.74	9.29	9.84	10.45	11.03
	SCE(λ = 0.90)	5.42	5.89	6.40	6.91	7.43	7.98	8.52	9.12	9.71	10.33	10.97
1.75	SC	6.30	6.72	7.17	7.61	8.07	8.56	9.06	9.58	10.12	10.68	11.23
	SCC	6.23	6.45	6.62	6.83	7.00	7.20	7.38	7.58	7.76	7.92	8.09
	SCE(λ = 0.10)	7.50	8.14	8.77	9.37	9.93	10.54	11.09	11.64	12.20	12.77	13.31
	SCE(λ = 0.25)	6.16	6.66	7.14	7.61	8.10	8.55	9.02	9.50	9.97	10.45	10.90
	SCE(λ = 0.50)	5.26	5.68	6.10	6.50	6.95	7.35	7.79	8.21	8.65	9.09	9.55
	SCE(λ = 0.75)	4.68	5.07	5.46	5.86	6.26	6.69	7.09	7.53	7.96	8.40	8.87
	SCE(λ = 0.90)	4.43	4.82	5.20	5.61	6.00	6.44	6.85	7.28	7.74	8.18	8.66
2.00	SC	4.86	5.14	5.45	5.80	6.10	6.45	6.80	7.16	7.55	7.91	8.30
	SCC	4.81	4.98	5.13	5.31	5.46	5.61	5.76	5.93	6.06	6.23	6.36
	SCE(λ = 0.10)	6.24	6.83	7.42	7.96	8.49	9.01	9.51	10.01	10.50	11.01	11.47
	SCE(λ = 0.25)	5.19	5.65	6.08	6.52	6.93	7.34	7.75	8.14	8.57	8.95	9.34
	SCE(λ = 0.50)	4.44	4.83	5.18	5.55	5.92	6.26	6.63	6.98	7.36	7.71	8.09
	SCE(λ = 0.75)	3.96	4.29	4.61	4.95	5.29	5.64	5.97	6.32	6.64	7.04	7.40
	SCE(λ = 0.90)	3.74	4.06	4.37	4.70	5.03	5.36	5.71	6.08	6.42	6.76	7.15
2.25	SC	3.78	4.00	4.23	4.44	4.68	4.94	5.19	5.45	5.69	5.95	6.23
	SCC	3.76	3.92	4.03	4.17	4.29	4.44	4.56	4.70	4.81	4.96	5.07
	SCE(λ = 0.10)	5.18	5.76	6.29	6.82	7.31	7.78	8.25	8.70	9.18	9.61	10.04
	SCE(λ = 0.25)	4.39	4.82	5.23	5.63	6.01	6.38	6.76	7.11	7.46	7.85	8.17
	SCE(λ = 0.50)	3.81	4.14	4.48	4.80	5.11	5.44	5.77	6.07	6.39	6.69	7.03
	SCE(λ = 0.75)	3.41	3.70	3.99	4.29	4.56	4.85	5.12	5.43	5.74	6.03	6.33
	SCE(λ = 0.90)	3.22	3.49	3.77	4.05	4.32	4.60	4.88	5.18	5.47	5.76	6.07
2.50	SC	3.00	3.14	3.29	3.48	3.64	3.82	3.98	4.17	4.36	4.55	4.73
	SCC	2.98	3.08	3.21	3.32	3.43	3.55	3.64	3.76	3.87	3.98	4.09
	SCE(λ = 0.10)	4.26	4.82	5.35	5.82	6.31	6.77	7.21	7.62	8.06	8.48	8.86
	SCE(λ = 0.25)	3.73	4.13	4.51	4.89	5.24	5.60	5.93	6.26	6.58	6.91	7.24
	SCE(λ = 0.50)	3.28	3.60	3.89	4.20	4.49	4.79	5.06	5.34	5.62	5.89	6.18
	SCE(λ = 0.75)	2.95	3.22	3.48	3.74	4.00	4.25	4.50	4.76	5.02	5.28	5.54
	SCE(λ = 0.90)	2.80	3.05	3.29	3.53	3.78	4.02	4.26	4.50	4.77	5.01	5.28
2.75	SC	2.41	2.52	2.63	2.75	2.86	3.00	3.12	3.25	3.39	3.52	3.66
	SCC	2.40	2.49	2.57	2.69	2.76	2.86	2.95	3.04	3.14	3.23	3.33
	SCE(λ = 0.10)	3.50	4.02	4.52	4.98	5.44	5.88	6.31	6.71	7.13	7.50	7.88
	SCE(λ = 0.25)	3.13	3.52	3.89	4.26	4.59	4.93	5.23	5.56	5.86	6.16	6.45
	SCE(λ = 0.50)	2.81	3.12	3.41	3.70	3.97	4.23	4.50	4.75	5.00	5.25	5.50
	SCE(λ = 0.75)	2.58	2.83	3.07	3.30	3.53	3.75	4.00	4.23	4.45	4.68	4.91
	SCE(λ = 0.90)	2.44	2.67	2.90	3.11	3.34	3.55	3.78	3.99	4.21	4.44	4.67
3.00	SC	1.97	2.05	2.13	2.22	2.30	2.40	2.48	2.58	2.69	2.79	2.89
	SCC	1.97	2.04	2.11	2.19	2.26	2.35	2.42	2.50	2.57	2.65	2.74
	SCE(λ = 0.10)	2.86	3.33	3.79	4.24	4.68	5.12	5.53	5.91	6.30	6.67	7.04
	SCE(λ = 0.25)	2.64	3.00	3.36	3.69	4.02	4.34	4.64	4.95	5.23	5.52	5.80
	SCE(λ = 0.50)	2.42	2.70	2.99	3.25	3.51	3.76	4.01	4.26	4.48	4.72	4.95
	SCE(λ = 0.75)	2.25	2.49	2.71	2.94	3.16	3.36	3.57	3.78	3.99	4.20	4.41
	SCE(λ = 0.90)	2.16	2.37	2.58	2.78	2.98	3.17	3.37	3.57	3.76	3.97	4.17

SC: Shewhart-CUSUM; SCC: Crosier’s CUSUM; SCE: Shewhart-Cumulative-EWMA-sum.

Table 6.

ARL₁ profile of the Two-sided combined charts, with ARLo = 370 and m = 4.

Δ	Chart	ϒ
Δ	Chart	0.00	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80	0.90	1.00
0.25	SC	255.60	265.06	272.43	279.48	281.72	286.44	288.27	292.23	292.74	295.02	296.93
	SCC	255.68	257.19	261.43	263.37	264.54	266.33	271.17	271.71	274.26	273.74	278.53
	SCE(λ = 0.10)	69.74	75.05	79.88	84.83	90.16	95.43	100.02	104.99	110.32	114.66	119.46
	SCE(λ = 0.25)	73.22	79.47	86.79	92.58	98.93	105.06	111.17	117.50	123.29	129.27	135.30
	SCE(λ = 0.50)	85.01	92.98	100.67	109.09	116.44	124.46	132.09	139.08	146.49	153.38	159.72
	SCE(λ = 0.75)	101.23	111.07	120.94	129.47	138.75	146.05	155.22	163.03	170.02	177.27	185.65
	SCE(λ = 0.90)	112.15	122.79	132.25	141.92	151.55	159.26	168.29	177.54	183.43	191.15	197.71
0.50	SC	80.76	89.40	96.77	106.63	115.06	123.70	132.07	139.75	148.30	155.03	162.27
	SCC	80.16	82.22	84.95	87.55	90.28	92.53	95.21	97.14	100.64	102.62	104.92
	SCE(λ = 0.10)	30.94	33.16	35.41	37.62	39.73	41.82	44.07	46.15	48.43	50.66	52.92
	SCE(λ = 0.25)	27.77	30.03	32.34	34.68	37.08	39.67	42.23	44.67	47.12	49.56	52.22
	SCE(λ = 0.50)	27.80	30.41	33.26	36.18	39.29	42.34	45.01	48.41	51.61	55.19	58.42
	SCE(λ = 0.75)	30.07	33.59	37.07	40.68	44.45	48.25	52.26	56.34	60.51	64.56	68.92
	SCE(λ = 0.90)	32.46	36.28	40.33	44.65	49.09	53.36	58.06	62.73	67.13	71.87	76.41
0.75	SC	30.26	33.08	35.60	38.68	41.56	44.59	47.85	51.43	54.86	58.64	62.23
	SCC	30.19	30.85	31.71	32.53	33.39	34.18	35.21	36.09	36.87	37.63	38.50
	SCE(λ = 0.10)	19.75	21.06	22.45	23.77	24.98	26.44	27.74	28.97	30.30	31.62	32.95
	SCE(λ = 0.25)	16.55	17.89	19.14	20.33	21.70	22.98	24.35	25.64	26.98	28.28	29.75
	SCE(λ = 0.50)	15.23	16.59	17.93	19.31	20.66	22.21	23.66	25.25	26.90	28.40	30.14
	SCE(λ = 0.75)	15.05	16.58	18.13	19.87	21.47	23.17	25.11	26.91	28.91	30.95	32.96
	SCE(λ = 0.90)	15.39	17.06	18.83	20.70	22.67	24.65	26.69	28.96	31.27	33.55	35.93
1.00	SC	17.12	18.47	19.79	21.16	22.55	24.16	25.78	27.33	29.23	31.00	32.74
	SCC	16.94	17.28	17.71	18.13	18.51	18.84	19.34	19.67	20.10	20.57	20.95
	SCE(λ = 0.10)	14.37	15.33	16.32	17.27	18.21	19.17	20.11	21.03	21.96	22.89	23.78
	SCE(λ = 0.25)	11.79	12.63	13.49	14.35	15.21	16.10	16.96	17.79	18.66	19.58	20.50
	SCE(λ = 0.50)	10.38	11.22	12.07	12.94	13.79	14.69	15.60	16.53	17.47	18.39	19.41
	SCE(λ = 0.75)	9.76	10.64	11.55	12.48	13.43	14.41	15.48	16.56	17.60	18.72	19.92
	SCE(λ = 0.90)	9.60	10.59	11.50	12.54	13.62	14.71	15.77	16.95	18.27	19.59	20.77
1.25	SC	11.47	12.30	13.10	13.99	14.83	15.80	16.69	17.66	18.76	19.82	20.85
	SCC	11.41	11.59	11.85	12.09	12.40	12.58	12.86	13.09	13.34	13.56	13.82
	SCE(λ = 0.10)	11.20	12.00	12.76	13.51	14.24	14.98	15.72	16.47	17.16	17.90	18.60
	SCE(λ = 0.25)	9.14	9.77	10.42	11.07	11.71	12.34	12.97	13.64	14.27	14.93	15.56
	SCE(λ = 0.50)	7.87	8.49	9.09	9.71	10.30	10.92	11.59	12.20	12.84	13.48	14.13
	SCE(λ = 0.75)	7.17	7.79	8.42	9.05	9.65	10.33	10.98	11.64	12.36	13.15	13.87
	SCE(λ = 0.90)	6.95	7.56	8.18	8.84	9.49	10.23	10.93	11.65	12.42	13.25	14.03
1.50	SC	8.36	8.87	9.44	10.02	10.57	11.22	11.78	12.46	13.13	13.76	14.46
	SCC	8.24	8.44	8.62	8.79	8.97	9.13	9.33	9.53	9.70	9.82	10.02
	SCE(λ = 0.10)	9.09	9.76	10.39	11.05	11.63	12.27	12.87	13.46	14.09	14.64	15.21
	SCE(λ = 0.25)	7.40	7.95	8.47	9.00	9.50	10.00	10.53	11.03	11.53	12.03	12.55
	SCE(λ = 0.50)	6.34	6.80	7.28	7.74	8.22	8.69	9.17	9.66	10.13	10.64	11.10
	SCE(λ = 0.75)	5.68	6.13	6.59	7.06	7.53	7.99	8.47	8.99	9.46	10.00	10.56
	SCE(λ = 0.90)	5.42	5.86	6.35	6.79	7.28	7.77	8.26	8.79	9.32	9.88	10.43
1.75	SC	6.31	6.65	7.06	7.41	7.84	8.24	8.67	9.10	9.53	9.98	10.44
	SCC	6.23	6.39	6.52	6.65	6.81	6.95	7.08	7.24	7.38	7.53	7.64
	SCE(λ = 0.10)	7.51	8.11	8.68	9.23	9.80	10.30	10.82	11.35	11.87	12.35	12.86
	SCE(λ = 0.25)	6.15	6.63	7.09	7.51	7.95	8.38	8.80	9.22	9.63	10.07	10.50
	SCE(λ = 0.50)	5.25	5.64	6.04	6.43	6.82	7.22	7.60	7.98	8.36	8.75	9.15
	SCE(λ = 0.75)	4.68	5.04	5.41	5.76	6.13	6.51	6.90	7.28	7.67	8.06	8.48
	SCE(λ = 0.90)	4.44	4.79	5.16	5.51	5.89	6.26	6.65	7.02	7.42	7.85	8.27
2.00	SC	4.85	5.11	5.36	5.65	5.90	6.21	6.49	6.81	7.06	7.36	7.71
	SCC	4.82	4.94	5.05	5.16	5.29	5.41	5.55	5.64	5.76	5.86	5.96
	SCE(λ = 0.10)	6.25	6.81	7.33	7.85	8.34	8.81	9.29	9.74	10.20	10.63	11.07
	SCE(λ = 0.25)	5.20	5.61	6.02	6.42	6.81	7.19	7.55	7.92	8.29	8.64	9.01
	SCE(λ = 0.50)	4.44	4.80	5.14	5.48	5.81	6.13	6.45	6.79	7.11	7.45	7.77
	SCE(λ = 0.75)	3.96	4.27	4.58	4.89	5.19	5.50	5.80	6.12	6.44	6.76	7.08
	SCE(λ = 0.90)	3.73	4.03	4.33	4.63	4.93	5.24	5.53	5.86	6.19	6.50	6.83
2.25	SC	3.78	3.96	4.15	4.34	4.53	4.73	4.92	5.13	5.32	5.53	5.74
	SCC	3.76	3.87	3.97	4.07	4.16	4.26	4.37	4.47	4.57	4.67	4.76
	SCE(λ = 0.10)	5.19	5.72	6.22	6.71	7.16	7.60	8.04	8.46	8.88	9.25	9.68
	SCE(λ = 0.25)	4.40	4.80	5.18	5.54	5.89	6.24	6.57	6.90	7.22	7.55	7.87
	SCE(λ = 0.50)	3.80	4.13	4.44	4.72	5.03	5.32	5.62	5.90	6.18	6.46	6.75
	SCE(λ = 0.75)	3.40	3.67	3.94	4.22	4.47	4.74	5.00	5.27	5.53	5.79	6.09
	SCE(λ = 0.90)	3.21	3.47	3.73	3.98	4.25	4.49	4.75	5.02	5.27	5.54	5.81
2.50	SC	3.00	3.11	3.24	3.37	3.50	3.65	3.77	3.91	4.08	4.19	4.35
	SCC	2.99	3.07	3.15	3.24	3.33	3.40	3.49	3.56	3.65	3.74	3.81
	SCE(λ = 0.10)	4.27	4.78	5.26	5.73	6.20	6.61	6.99	7.41	7.80	8.15	8.54
	SCE(λ = 0.25)	3.72	4.09	4.46	4.81	5.14	5.46	5.77	6.08	6.38	6.66	6.96
	SCE(λ = 0.50)	3.26	3.58	3.86	4.14	4.40	4.67	4.93	5.18	5.44	5.70	5.95
	SCE(λ = 0.75)	2.95	3.20	3.45	3.69	3.91	4.15	4.39	4.63	4.84	5.07	5.32
	SCE(λ = 0.90)	2.79	3.03	3.25	3.48	3.70	3.93	4.15	4.37	4.60	4.82	5.06
2.75	SC	2.40	2.50	2.58	2.68	2.77	2.85	2.96	3.06	3.16	3.26	3.35
	SCC	2.40	2.46	2.53	2.60	2.68	2.75	2.82	2.88	2.95	3.01	3.10
	SCE(λ = 0.10)	3.51	3.98	4.45	4.88	5.32	5.73	6.11	6.48	6.86	7.24	7.56
	SCE(λ = 0.25)	3.14	3.50	3.85	4.17	4.49	4.80	5.09	5.39	5.66	5.93	6.20
	SCE(λ = 0.50)	2.81	3.09	3.37	3.63	3.88	4.13	4.37	4.61	4.84	5.07	5.31
	SCE(λ = 0.75)	2.57	2.81	3.03	3.25	3.47	3.68	3.89	4.09	4.31	4.52	4.72
	SCE(λ = 0.90)	2.44	2.66	2.87	3.08	3.27	3.48	3.67	3.87	4.07	4.26	4.48
3.00	SC	1.96	2.04	2.10	2.16	2.24	2.31	2.35	2.44	2.51	2.58	2.65
	SCC	1.97	2.02	2.08	2.13	2.18	2.25	2.31	2.35	2.41	2.48	2.54
	SCE(λ = 0.10)	2.86	3.30	3.73	4.15	4.56	4.96	5.34	5.70	6.06	6.40	6.72
	SCE(λ = 0.25)	2.64	2.99	3.31	3.63	3.93	4.23	4.51	4.79	5.05	5.32	5.56
	SCE(λ = 0.50)	2.42	2.68	2.95	3.20	3.44	3.66	3.90	4.11	4.33	4.54	4.74
	SCE(λ = 0.75)	2.26	2.48	2.68	2.89	3.09	3.28	3.48	3.67	3.85	4.05	4.23
	SCE(λ = 0.90)	2.15	2.35	2.55	2.74	2.92	3.10	3.29	3.46	3.64	3.82	4.00

SC: Shewhart-CUSUM; SCC: Crosier’s CUSUM; SCE: Shewhart-Cumulative-EWMA-sum.

On the similar lines of Linna and Woodall,¹⁶ Munir et al.,¹¹ Nguyen et al.²³ to follow their suggestions and utilization of the multiple measurement scheme was employed as a means to mitigate the impact of measurement error and consequently reduce the ARL₁ values for both the proposed and existing charts. This effect can be observed in Tables 3 to 6. Specifically, considering the M-Shewhart-CUESUM chart with $λ = 0.1$ , the corresponding values for ( $m$ , ARL₁) are as follows: (1, 112.69), (2, 101.50), (3, 96.96), and (4, 95.43) for $γ = 0.5$ at $δ = 0.25$ . These results indicate that as the value of m increases from 1 to 4, there is a decrease in ARL₁. This trend is also observed across all the control charts, demonstrating that the effectiveness of a chart is directly proportional to the number of measurements ( $m$ ).

When evaluating the effectiveness of the combined control charts under the influence of measurement error, the M-Shewhart-CUESUM chart demonstrates significantly superior performance compared to the Shewhart-CUSUM and Shewhart-CCUSUM charts. For instance, when considering a shift size (δ) of 0.50 with a measurement replication of $m = 4$ , ARL₁ values obtained for Shewhart-CUSUM, Shewhart-CCUSUM, and Shewhart-CUESUM charts are 162.27, 104.92, and 52.92, respectively. This substantial difference in ARL₁ values highlights the superior performance of the Shewhart-CUESUM chart compared to the other charts.

Illustrative examples (applications of study)

In this section, it utilized simulated datasets to illustrate the implementation of both the combined charts without measurement errors and the ones that incorporate measurement errors in the detection of shifts in the process mean.

The Shewhart-CUESUM and CUESUM chart applied to simulated dataset

Assuming a normally distributed underlying process { $X_{t}$ } for $t \geq 1$ , process considered to be in control for $t \leq t_{0}$ , characterized by parameters ( $μ_{0} = 0; σ = 1$ ). However, due to a significant upward shift with a magnitude of δ = 2.0, the process goes out of control, resulting in a new mean $μ_{1} = 2.0$ . To monitor this process, first draw 70 samples ( $t_{0}$ = 30) from the in-control process and 30 samples ( $t_{0}$ = 20) from the out-of-control process, each with a sample size of $n = 5$ . For employing two monitoring techniques, the two-sided Shewhart-CUSUM chart and the CUSUM chart, using different sets of process parameters. For the two-sided Shewhart-CUSUM chart, we use $λ = 0.1, h = 1.473$ , and $L = 3$ . On the other hand, for the CUSUM chart, we use $λ = 0.1, h = 1.374$ , with $k = 0.5$ . Figure 3 illustrates the displays of these monitoring charts.

Figure 3.

The existing and proposed charts for the simulated dataset.

The observed control charts clearly indicate that the underlying process remains in the in-control state for the initial 30 samples. However, in the last 20 samples, the process becomes out-of-control, as evidenced by the occurrence of out-of-control signals on these control charts. The Shewhart-CUESUM chart and the CUESUM chart both first detect out-of-control signals at the 35th and 37th samples, respectively. It is worth noting that the Shewhart-CUESUM chart exhibits earlier signal detection compared to the CUESUM chart, suggesting that the former is more sensitive than the latter.

The combined charts applied to measurement error-based simulated data

Assuming a normal distribution for the underlying process ${Y_{t}}$ with $t > 1$ , we establish that this process stays in a statistical control state for $t \leq t_{0}$ , for the process parameters { $μ = 0, σ = 1, and γ = 0.0}$ . The first 30 observations are obtained from a statistically controlled process, whereas the following 20 observations are obtained against a process that experiences an upward shift after a certain time point, denoted as $t > t_{0}$ , with $δ = 0.25$ . The combined charts are applied to these datasets.

To examine different scenarios of measurement errors, we consider three situations: no errors (γ = 0), error variation accounting for 40% of the process variation (γ = 0.4), and error variation accounting for 80% of the process variation (γ = 0.8). The charts’ parameters of the Shewhart-CUSUM, Shewhart-CCUSUM, and M-Shewhart-CUESUM charts are as follows: $(h = 9.9, L = 3), (h . 9.7, L = 3)$ , and $(λ = 0.1, h λ = 74, L = 3)$ , respectively.

These parameters of the studied charts are to achieve the ARL₀ of 370 at the process is in-control state. Figures 4 to 6 display the control charts for the aforementioned scenarios.

Figure 4.

The Shewhart-CUSUM control chart applied to the simulated dataset when ϒ = 0.0 (a), ϒ = 0.4 (b), and ϒ = 0.8 (c).

Figure 5.

The Shewhart-CCUSUM control chart applied to the simulated dataset when ϒ = 0.0 (a), ϒ = 0.4 (b), and ϒ = 0.8 (c).

Figure 6.

The Shewhart-CUESUM control chart applied to the simulated dataset when ϒ = 0.0 (a), ϒ = 0.4 (b), and ϒ = 0.8 (c).

The figures demonstrate the decline in statistical power for the three charts as the magnitude of measurement errors increases.

In the absence of measurement error denoted by (A) in Figures 4 to 6, the Shewhart-CUSUM, Shewhart-CCUSUM, and M-Shewhart-CUESUM charts detect the first out-of-control signal at sample numbers 37, 36, and 33, respectively. However, by (B) with 40% variation in measurement errors, the out-of-control signals appear at sample numbers 42, 42, and 35, respectively. Similarly, by (C) with 80% variation in measurement errors, the out-of-control signals appear at sample numbers 47, 46, and 37 respectively. These results support the conclusions drawn in Section 4.2, emphasizing the substantial influence of measurement errors on the effectiveness of combined control charts.

Conclusion

This paper proposed the novel Combined Shewhart-Cumulative EWMA-sum (Shewhart-CUESUM) chart for efficient detecting different kind of shifts in mean of a normally distributed process. The performance of the Shewhart-CUESUM chart in the presence of measurement errors is investigated. Through Monte-Carlo simulations, comprehensive run length profiles were computed, including ARL, SDRL, EQL, IRARL, and PCI. Findings indicate that the Shewhart-CUESUM chart outperforms the CUESUM, Combined Shewhart-CUSUM, and Shewhart-CCUSUM charts overall. Despite being influenced by measurement errors, the M-Shewhart-CUESUM chart remains significantly more sensitive than other charts. Utilizing multiple measurements per item helps mitigate the impact of measurement inaccuracy. However, when measurement errors are unavoidable, we recommend using the measurement error-based Shewhart-CUESUM chart as a preferred approach.

For recommendation research work, a Shewhart-CUSUM chart can be designed to simultaneously address violations of the normality and perfect measurement data assumptions. This can be done following the work of Rahlm.²⁹ Additionally, this study could be extended to consider the combined effects of measurement error and autocorrelation, as done by Costa and Castagliola.³⁰

Footnotes

Acknowledgements

The authors extend their appreciation to King Saud University for funding this work through the Researchers Supporting Project number (RSPD2024 R803), King Saud University, Riyadh, Saudi Arabia.

Author’s contribution

Tahir Munir: conceptualization, methodology, software, writing - original draft preparation. Fahad M. Alqahtani: conceptualization; methodology, reviewing, and editing, revision. Afaf Alrashidi: Formal analysis, writing - review & editing. Abdu R Rahman: visualization, reviewing, and editing. Salman Arif Cheema: writing - original draft, idea-building discussion. Yi Li: supervision, validation, finalizing.

Funding

This research was supported by Researchers Supporting Project number (RSPD2024 R803), King Saud University, Riyadh, Saudi Arabia.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

ORCID iD

Tahir Munir

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Novel combined Shewhart-CUmulative EWMA-SUM mean charts without- and with measurement error

Abstract

Keywords

Introduction

A review of the existing quality control charts

The CUESUM chart

The Shewhart-CUSUM chart

The Shewhart-Crosier CUSUM chart

Proposed combined control charts

The Shewhart-CUESUM chart

Measurement error-based combined Shewhart-CUESUM chart

Run-length profiles and performance comparison

CUESUM versus Shewhart-CUESUM chart

Shewhart-CUESUM versus Shewhart-CUSUM and Shewhart-CCUSUM charts

Effect of measurement error on combined chart

Illustrative examples (applications of study)

The Shewhart-CUESUM and CUESUM chart applied to simulated dataset

The combined charts applied to measurement error-based simulated data

Conclusion

Footnotes

Acknowledgements

Author’s contribution

Funding

Declaration of conflicting interests

ORCID iD

Data availability statement

References