The Measurement Errors and Their Effects on the Cumulative Sum Schemes for Monitoring the Ratio of Two Correlated Normal Variables

Abstract

Monitoring the ratio of two correlated normal random variables is often used in many industrial manufacturing processes. At the same time, measurement errors inevitably exist in most processes, which have different effects on the performance of various charting schemes. This paper comprehensively analyses the impacts of measurement errors on the detection ability of the cumulative sum (CUSUM) charting schemes for the ratio of two correlated normal variables. A thorough numerical assessment is performed using the Monte Carlo simulation, and the results indicate that the measurement errors negatively impact the performance of the CUSUM scheme for the ratio of two correlated normal variables. Increasing the number of measurements per set is not a lucrative approach for minimizing the negative impact of measurement errors on the performance of the CUSUM charting scheme when monitoring the ratio of two correlated normal variables. We consider a food formulation as an example that illustrates the quality control problems involving the ratio of two correlated normal variables in an industry with a measurement error. The results are presented, along with some suggestions for further study.

1. Introduction

Statistical process monitoring (SPM) and control play a significant role in inspecting and ensuring the quality of products or services and maintaining efficient designs. The SPM techniques include many practical tools employed to reduce fluctuations in the product’s quality characteristics and stable process. Among these practical tools, the process monitoring scheme is well-known and widely used for monitoring processes in various domains such as production lines, medical and healthcare systems, and economic conditions (see Montgomery [1], Qiu [2]). While Phase-I charting tools analyze historical data retrospectively, Phase-II charting schemes are employed for online or sequential monitoring. An SPM scheme can also be categorized as a memoryless or memory-type scheme. The Shewhart scheme proposed by Shewhart [3] is classified as a memoryless scheme because it only uses current information to detect the shifts occurring during the process. The CUSUM scheme, proposed by Page [4], and the exponentially weighted moving average (EWMA) scheme, introduced by Roberts [5], are both classified as memory-based schemes. These memory-type SPM schemes are sensitive to small to moderate and persistent shifts because they combine past and present data during the detection process.

The problems involving the ratio of two correlated normal variables are often essential in practical applications, including product quality monitoring in production plants. For example, in the food and drug sectors, the ratio of two main ingredients is directly related to food taste and the efficacy and safety of drugs; see Celano and Castagliola [6]. In the statistical literature on distribution theory, the ratio of one normal variable to another has a long history. Spisak [7] initially discussed the SPM schemes for the ratio of two random variables. Oksoy et al. [8] suggested a series of guiding methods for applying Shewhart control charts in the online monitoring process of glass processing production. The Shewhart-type scheme for individual observations monitoring the ratio of two normal variables was proposed recently by Celano et al. [9]. Celano and Castagliola [6] extended the Shewhart schemes to the case of $n\ge 1$ sample units. Celano and Castagliola [10] developed a Phase-II Synthetic scheme for monitoring the ratio of two normal variables, which is superior to the Shewhart scheme in terms of statistical sensitivity. A study using run rules through the incorporation of two one-sided limits for monitoring the ratio of one normal variable to another was proposed by Tran et al. [11]. Tran et al. [12,13] proposed the CUSUM and EWMA schemes for the ratio of normal variables to improve the sensitivity of these schemes for small to moderate shifts, respectively. Further, Tran and Knoth [14] proposed a steady-state ARL-unbiased EWMA scheme for monitoring the ratio of one normal variable to another. Nguyen et al. [15] and Nguyen et al. [16] combined the variable sampling interval (VSI) with the EWMA and CUSUM schemes, respectively, to monitor the ratio of the two normal variables. A two-sided run sum scheme for the ratio of one normal variable to another was made by Abubakar et al. [17].

Most control charts are designed based on the idealized assumption of accurate measurements. In practice, measurement errors exist in almost all processes with varying severity. Statistical properties of specific process characteristics may be misleading if one ignores the presence of measurement errors in designing SPM schemes when it has a dominating presence. In a bottle-filling plant, for example, it is not easy for a quality control engineer to obtain an accurate estimate of the volume of the liquid inside the bottle. Likewise, analog or digital blood pressure machines often give inaccurate readings in medical applications. The performance of various control charts will be affected by the presence of measurement errors as the process fluctuation increases. For example, the measurement error reduces the performance of control charts to detect the Out-Of-Control (OOC) process. In recent years, some scholars have noticed and analyzed the influence of measurement errors on the performance of control charts (see Linna and Woodall [18], Cocchi and Scagliarini [19], Maravelakis [20], Abbasi [21], Maleki et al. [22], Amiri et al. [23], Liu et al. [24], Sabahno et al. [25], Shongwe and Malela-Majika [26], Thanwane et al. [27], Maleki et al. [28], and Munir et al. [29]).

Additionally, some scholars have studied the performance of a charting scheme used to monitor the ratio of one normal variable to another in the presence of measurement errors. Tran et al. [30] investigated the effect of measurement errors on the Shewhart scheme assuming that the measurement error follows a linear covariate error model. Nguyen and Tran [31] investigated the effect of measurement errors on the two one-sided Shewhart schemes for the ratio of two normal variables under a similar supposition. The performance of the EWMA scheme in monitoring the ratio in the presence of measurement errors is examined by Nguyen et al. [32], who expanded the linear covariate error model used in earlier studies to a more generic situation.

In order to guarantee the stability and reliability of control charts, it is essential to identify and carefully manage the impact of measurement errors. Control charts can only accurately reflect the actual state of the process by means of such careful consideration, providing a strong basis for well-informed decision-making. This paper presents an innovative CUSUM scheme tailored to monitor the ratio of two correlated normal variables in the presence of measurement errors, addressing a crucial limitation of the conventional approach that assumes error-free measurements. Recognizing the ubiquitous nature of measurement errors in industrial production settings and their detrimental impact on the effectiveness of control charts, this study endeavors to enhance the practicality and robustness of the CUSUM methodology for ratio monitoring.

The main contributions of this paper are as follows: (1) An innovative design of a CUSUM scheme for monitoring the ratio of two correlated normal variables tailored to accommodate measurement errors. (2) A detailed examination of the effects of measurement errors on the CUSUM scheme for monitoring the ratio of two correlated normal variables. (3) Increasing the number of measurements per set is found not to reduce the effects of measurement errors on the CUSUM scheme for monitoring the ratio of two correlated normal variables.

The remaining portion of this paper is organized as follows: The ratio distribution for the normal random variables is reviewed in Section 2. In Section 3, the linear covariate error model with the Gaussian error component for the sample ratio is presented. The CUSUM scheme for the ratio of two normal variables impacted by measurement errors is created and put into practice in Section 4. Section 5 discusses the CUSUM scheme’s performance for the ratio of two correlated normal variables with measurement errors. In Section 6, an example is provided to show how the discussed technique is implemented. Section 7 offers some recommendations and conclusions.

2. The Review of the Ratio Distribution

This section briefly reviews the background of the distribution of the ratio of two correlated normal variables. Consequently, the joint distribution of the two variables is assumed to be bivariate normal $\left(BVN\right)$. A bivariate normal random vector is $\mathbf{Y}={(U,V)}^T\sim BVN\left({\mathit{\mu }}_{\mathit{Y}},{\mathbf{\Sigma }}_{\mathit{Y}}\right)$. Here,

$${\mathit{\mu }}_{\mathbf{Y}}=\left(\begin{array}{c}{\mu }_U\hfill \\ {\mu }_V\hfill \end{array}\right),$$

(1)

where ${\mu }_U$ is the mean of U and ${\mu }_V$ is the mean of V. Further,

$${\mathbf{\Sigma }}_{\mathbf{Y}}=\left(\begin{array}{cc}{\sigma }_U^2& {\rho }_{UV}{\sigma }_U{\sigma }_V\\ {\rho }_{UV}{\sigma }_U{\sigma }_V& {\sigma }_V^2\end{array}\right),$$

(2)

where ${\sigma }_U$ is the standard deviation of U, ${\sigma }_V$ is the standard deviation of V, and ${\rho }_{UV}$ is the correlation coefficient between U and V.

The ratio of U to V is expressed as $Z=\frac{U}{V}$. ${\eta }_U=\frac{{\sigma }_U}{{\mu }_U}$ is the coefficient of variation (CV) of U. Similarly, ${\eta }_V=\frac{{\sigma }_V}{{\mu }_V}$ is the CV of V. $\omega =\frac{{\sigma }_U}{{\sigma }_V}$ is the standard deviation ratio.

There are many studies involving the distribution of Z; for example, Geary [33], Hayya et al. [34], Cedilnik et al. [35], and Pham-Gia et al. [36]. In this paper, according to Tran et al. [30], “In fact, for stable and predictable processes with normally distributed quality parameters, the population dispersion should be significantly smaller than the mean in order to limit the number of nonconforming units vs. the specification interval and to obtain sufficiently large capability index values. For this reason, in these processes, it is very frequent that the coefficient of variation takes small values within the range (0, 0.2]”. To this end, we assume ${\eta }_U$ and ${\eta }_V$ are in the range (0, 0.2]. The Z distribution is approximated as follows:

$$F_Z\left(z\mid {\eta }_U,{\eta }_V,\omega ,{\rho }_{UV}\right)\simeq \Phi \left(\frac{A}{B}\right),$$

(3)

where $\Phi (.)$ is the cumulative distribution function (CDF) of $N(0,1)$, and A and B are functions in terms of ${\eta }_U,{\eta }_V,z,\omega$, and ${\rho }_{UV}$ as follows:

$$A=\frac{z}{{\eta }_V}-\frac{\omega }{{\eta }_U},$$

(4)

$$B=\sqrt{{\omega }^2-2{\rho }_{UV}\omega z+z^2}.$$

(5)

The probability density function (PDF) of Z is approximated as follows:

$$f_Z\left(z\mid {\eta }_U,{\eta }_V,\omega ,{\rho }_{UV}\right)\simeq \left(\frac{1}{B{\eta }_V}-\frac{(z-{\rho }_{UV}\omega )A}{B^3}\right)\times \varphi \left(\frac{A}{B}\right),$$

(6)

where $\varphi (.)$ is the PDF of $N(0,1)$.

An approximate expression of the inverse distribution function $F_Z^{-1}\left(p\right|{\eta }_U,{\eta }_V,\omega ,{\rho }_{UV})$ can be obtained by solving the equation of $F_Z\left(z\mid {\eta }_U,{\eta }_V,\omega ,{\rho }_{UV}\right)=p$ with respect to z.

$$F_Z^{-1}\left(p\right|{\eta }_U,{\eta }_V,\omega ,{\rho }_{UV})=\left\{\begin{array}{cc}\frac{-C_2-\sqrt{C_2^2-4C_1{C}_3}}{2C_1}& p\in (0,0.5],\\ \frac{-C_2+\sqrt{C_2^2-4C_1{C}_3}}{2C_1}& p\in [0.5,1),\end{array}\right.$$

(7)

where $C_1,C_2$, $C_3$ can be expressed as follows:

$$\begin{array}{cc}& C_1=\frac{1}{{\eta }_V^2}-{\left({\Phi }^{-1}\left(p\right)\right)}^2,\hfill \\ & C_2=2\omega \left({\rho }_{UV}{\left({\Phi }^{-1}\left(p\right)\right)}^2-\frac{1}{{\eta }_U{\eta }_V}\right),\hfill \\ & C_3={\omega }^2\left(\frac{1}{{\eta }_U^2}-{\left({\Phi }^{-1}\left(p\right)\right)}^2\right),\hfill \end{array}$$

where ${\Phi }^{-1}(.)$ is the inverse distribution function of $N(0,1)$.

3. The Linear Covariate Dependent Error Model for the Sample Ratios

In this section, we will summarize the main conclusions. We consider the linear covariate error model using the sample ratios.

In the production process, the true quality characteristic is $\mathbf{Y}={(U,V)}^T\sim BVN\left({\mathit{\mu }}_{\mathbf{Y}},{\mathbf{\Sigma }}_{\mathbf{Y}}\right)$, and ${\mathit{\mu }}_{\mathbf{Y}}$ and ${\mathbf{\Sigma }}_{\mathit{Y}}$ are defined by Equations (1) and (2). At time i ($i=1,2,\dots$), we have n independent samples, which are referred to as $\{{\mathbf{Y}}_{i,1},{\mathbf{Y}}_{i,2},\dots ,{\mathbf{Y}}_{i,n}\}$. ${\mathbf{Y}}_{i,j}={(U_{i,j},V_{i,j})}^T$ is the jth sample ($j=1,\dots ,n$) at time i.

In general, because of measurement errors in practice, the true value ${\mathbf{Y}}_{i,j}$ is not easy to measure. We observe a set of multiple measurements of the jth sample at the time i; say $\{{\mathbf{Y}}_{i,j,1}^{*},{\mathbf{Y}}_{i,j,2}^{*},\dots ,{\mathbf{Y}}_{i,j,m}^{*}\}$. m is the number of observations in each set of current configurations. In this paper, we discuss employing multiple measurements as a strategy for reducing measurement errors.

For the model of the linear covariate error, see Linna et al. [37]. According to them, the observation is related to the true value ${\mathbf{Y}}_{i,j}$, and is represented as

$${\mathbf{Y}}_{i,j,k}^{*}=\mathbf{K}+{\mathbf{SY}}_{i,j}+{\epsilon }_{i,j,k},k=1,2,\dots ,m,$$

(8)

where ${\mathbf{K}}_{2\times 1}$ is a vector of constants, and $\mathbf{K}={(k_U,k_V)}^T$. $\mathbf{S}=\left(\begin{array}{cc}s& 0\\ 0& s\end{array}\right)$; here, s is a constant. $\epsilon \sim N(\mathbf{0},{\mathbf{\Sigma }}_{\mathit{\epsilon }})$ and $\mathbf{Y}$ are independent of each other.

$${\mathbf{\Sigma }}_{\mathit{\epsilon }}=\left(\begin{array}{cc}{\mathbf{\sigma }}_{{\epsilon }_U}^2& {\rho }_{\epsilon }{{\sigma }_{\epsilon }}_U{{\sigma }_{\epsilon }}_V\\ {\rho }_{\epsilon }{{\sigma }_{\epsilon }}_U{{\sigma }_{\epsilon }}_V& {\sigma }_{{\epsilon }_V}^2\end{array}\right),$$

(9)

where ${\sigma }_{{\epsilon }_U}^2$ is the variance of the measurement error for U, ${\sigma }_{{\epsilon }_V}^2$ is the variance of the measurement error for V, and ${\rho }_{\epsilon }$ is the correlation coefficient between the measurement error for U and V.

The mean of observations $\{{\mathbf{Y}}_{i,j,1}^{*},{\mathbf{Y}}_{i,j,2}^{*},\dots ,{\mathbf{Y}}_{i,j,m}^{*}\}$ is ${\overline{\mathbf{Y}}}_{i,j}^{*}={({\overline{U}}_{i,j}^{*},{\overline{V}}_{i,j}^{*})}^T$. ${\overline{\mathbf{Y}}}_{i,j}^{*}$ is usually regarded as the representative value of ${\mathbf{Y}}_{i,j}$. ${\overline{\mathbf{Y}}}_{i,j}^{*}\sim BVN({\mathit{\mu }}_{{\mathbf{Y}}^{*}},{\mathbf{\Sigma }}_{{\mathbf{Y}}^{*}})$, where

$${\overline{\mathbf{Y}}}_{i,j}^{*}=\mathbf{K}+{\mathbf{SY}}_{i,j}+\frac{1}{m}\sum _{t=1}^m{\epsilon }_{i,j,t}.$$

(10)

The mean vector and the covariance matrix of ${\overline{\mathbf{Y}}}_{i,j}^{*}$ are denoted by (11) and (12), respectively.

$${\mathit{\mu }}_{{\mathbf{Y}}^{*}}=\mathbf{K}+\mathbf{S}{\mathit{\mu }}_{\mathbf{Y}},$$

(11)

$$\begin{array}{cc}\hfill {\mathbf{\Sigma }}_{{\mathbf{Y}}^{*}}& =\mathbf{S}{\mathbf{\Sigma }}_{\mathbf{Y}}{\mathbf{S}}^T+\frac{1}{m}{\mathbf{\Sigma }}_{\epsilon }\hfill \\ & =s^2{\mathbf{\Sigma }}_{\mathbf{Y}}+\frac{1}{m}{\mathbf{\Sigma }}_{\epsilon }.\hfill \end{array}$$

(12)

Let the means of ${\overline{U}}_{i,j}^{*}$ and ${\overline{V}}_{i,j}^{*}$ be ${\mu }_{U^{*}}$ and ${\mu }_{V^{*}}$, the variances of ${\overline{U}}_{i,j}^{*}$ and ${\overline{V}}_{i,j}^{*}$ be ${\sigma }_{U^{*}}^2$ and ${\sigma }_{V^{*}}^2$, and the CVs of ${\overline{U}}_{i,j}^{*}$ and ${\overline{V}}_{i,j}^{*}$ be ${\eta }_{U^{*}}$ and ${\eta }_{V^{*}}$. Finally, let ${\rho }_{UV}^{*}$ be the correlation coefficient between ${\overline{U}}_{i,j}^{*}$ and ${\overline{V}}_{i,j}^{*}$, where

$${\mu }_{U^{*}}=k_U+s{\mu }_U,$$

(13)

$${\mu }_{V^{*}}=k_V+s{\mu }_V,$$

(14)

$${\sigma }_{U^{*}}^2=s^2{\sigma }_U^2+\frac{{\sigma }_{{\epsilon }_U}^2}{m},$$

(15)

$${\sigma }_{V^{*}}^2=s^2{\sigma }_V^2+\frac{{\sigma }_{{\epsilon }_V}^2}{m},$$

(16)

$${\rho }_{UV}^{*}=\frac{s^2{\rho }_{UV}{\sigma }_U{\sigma }_V+{\rho }_{\epsilon }\frac{{\sigma \epsilon }_U{\sigma \epsilon }_V}{m}}{{\sigma }_{U^{*}}{\sigma }_{V^{*}}}.$$

(17)

When the process is in control (IC), the mean vector of ${(U_{i,j},V_{i,j})}^T$ is denoted as ${\mu }_{\mathbf{0},\mathit{Y}}={({\mu }_{0,U},{\mu }_{0,V})}^T$, and the ratio is $z_0=\frac{{\mu }_{0,U}}{{\mu }_{0,V}}$. The correlation coefficient between $U_{i,j}$ and $V_{i,j}$ is ${\rho }_{UV}={\rho }_0$.

When the process is OOC, $z_1=\tau z_0$, we denote the shift size as $\tau$. ${\rho }_{UV}$ is changed from ${\rho }_0$ to ${\rho }_1$, and the mean vector is from ${\mathit{\mu }}_{\mathbf{0},\mathit{Y}}={({\mu }_{0,U},{\mu }_{0,V})}^T$ to ${\mathit{\mu }}_{\mathbf{1},\mathit{Y}}={({\mu }_{0,U}+{\delta }_U{\sigma }_U,{\mu }_{0,V}+{\delta }_V{\sigma }_V)}^T$, where ${\delta }_U$ is the mean shift of $U_{i,j}$, and ${\delta }_V$ is the mean shift of $V_{i,j}$.

$$\begin{array}{cc}\hfill z_1& =\frac{{\mu }_{0,U}+{\delta }_U{\sigma }_U}{{\mu }_{0,V}+{\delta }_V{\sigma }_V}\hfill \\ & =\tau z_0\hfill \\ & =\tau \times \frac{{\mu }_{0,U}}{{\mu }_{0,V}},\hfill \end{array}$$

(18)

and further,

$$1+{\delta }_U{\eta }_U=\tau \left(1+{\delta }_V{\eta }_V\right).$$

(19)

Therefore, (13) and (14) can be rewritten as

$${\mu }_{U^{*}}=k_U+s({\mu }_{0,U}+{\delta }_U{\sigma }_U),$$

(20)

$${\mu }_{V^{*}}=k_V+s({\mu }_{0,V}+{\delta }_V{\sigma }_V).$$

(21)

Because of the measurement errors, the CVs of ${\overline{U}}_{i,j}^{*}$ and ${\overline{V}}_{i,j}^{*}$ are, respectively, expressed as

$$\begin{array}{c}\hfill {\eta }_{U^{*}}=\frac{\sqrt{s^2{\sigma }_U^2+\frac{{\sigma }_{{\epsilon }_U}^2}{m}}}{k_U+s\left({\mu }_{0,U}+{\delta }_U{\sigma }_U\right)},\end{array}$$

(22)

$$\begin{array}{c}\hfill {\eta }_{V^{*}}=\frac{\sqrt{s^2{\sigma }_V^2+\frac{{\sigma }_{{\epsilon }_V}^2}{m}}}{k_V+s\left({\mu }_{0,V}+{\delta }_V{\sigma }_V\right)}.\end{array}$$

(23)

Following Nguyen et al. [32], dividing the numerator by ${\sigma }_U$, and the denominator of (22) by ${\mu }_{0,U}$, making necessary adjustments and combining with (19), (22) can be expressed as

$$\begin{array}{cc}\hfill {\eta }_{U^{*}}& =\frac{\sqrt{s^2+\frac{{\zeta }_U^2}{m}}}{{\theta }_U+s\left(1+{\delta }_U{\eta }_U\right)}\times {\eta }_U\hfill \\ & =\frac{\sqrt{s^2+\frac{{\zeta }_U^2}{m}}}{{\theta }_U+s\tau \left(1+{\delta }_V{\eta }_V\right)}\times {\eta }_U,\hfill \end{array}$$

(24)

and similarly,

$${\eta }_{V^{*}}=\frac{\sqrt{s^2+\frac{{\zeta }_V^2}{m}}}{{\theta }_V+s\left(1+{\delta }_V{\eta }_V\right)}\times {\eta }_V,$$

(25)

$${\rho }_{UV}^{*}=\frac{s^2{\rho }_{UV}+{\rho }_{\epsilon }\frac{{\zeta }_U{\zeta }_V}{m}}{\sqrt{s^2+\frac{{\zeta }_U^2}{m}}\sqrt{s^2+\frac{{\zeta }_V^2}{m}}},$$

(26)

$${\omega }^{*}=\sqrt{\frac{s^2+\frac{{\zeta }_U^2}{m}}{s^2+\frac{{\zeta }_V^2}{m}}}\times \omega ,$$

(27)

where ${\zeta }_U=\frac{{\sigma \epsilon }_U}{{\sigma }_U},{\zeta }_V=\frac{{\sigma \epsilon }_V}{{\sigma }_V}$, ${\theta }_U=\frac{k_U}{{\mu }_{0,U}},{\theta }_V=\frac{k_V}{{\mu }_{0,V}}$, ${\eta }_U=\frac{{\sigma }_U}{{\mu }_{0,U}}$, and ${\eta }_V=\frac{{\sigma }_V}{{\mu }_{0,V}}$.

When the process is IC, the OOC ratios with measurement errors are represented as follows:

$$z_0^{*}=\frac{{\theta }_U+s}{{\theta }_V+s}\times z_0.$$

(28)

$$z_1^{*}=\frac{{\theta }_U+s\tau \left(1+{\delta }_V{\eta }_V\right)}{{\theta }_V+s\left(1+{\delta }_V{\eta }_V\right)}\times z_0.$$

(29)

4. The Design of the CUSUM Scheme for the Ratio of Two Correlated Normal Variables with Measurement Errors

In this section, we discuss the CUSUM scheme for the ratio of two correlated normal variables with measurement errors based on the model introduced in Section 3. The mean of the observed value $\{{\mathbf{Y}}_{i,j,1}^{*},{\mathbf{Y}}_{i,j,2}^{*},\dots ,{\mathbf{Y}}_{i,j,m}^{*}\}$ is ${\overline{\mathbf{Y}}}_{i,j}^{*}={({\overline{U}}_{i,j}^{*},{\overline{V}}_{i,j}^{*})}^T$, and the monitored statistic with measurement errors at each sampling period is

$${\widehat{Z}}_i^{*}=\frac{{\widehat{\mu }}_{U_i^{*}}}{{\widehat{\mu }}_{V_i^{*}}}=\frac{{\overline{\overline{U}}}_i^{*}}{{\overline{\overline{V}}}_i^{*}}=\frac{{\sum }_{j=1}^n{\overline{U}}_{i,j}^{*}/n}{{\sum }_{j=1}^n{\overline{V}}_{i,j}^{*}/n},$$

(30)

where ${\overline{\overline{U}}}_i^{*}\sim N\left({\mu }_{U^{*}},\frac{{\sigma }_{U^{*}}}{\sqrt{n}}\right)$ and ${\overline{\overline{V}}}_i^{*}\sim N\left({\mu }_{V^{*}},\frac{{\sigma }_{V^{*}}}{\sqrt{n}}\right)$; hence, the CVs ${\eta }_{{\overline{\overline{U}}}^{*}}$ and ${\eta }_{{\overline{\overline{V}}}^{*}}$ are given by

$${\eta }_{{\overline{\overline{U}}}^{*}}=\frac{{\sigma }_{U^{*}}/\sqrt{n}}{{\mu }_{U^{*}}}=\frac{{\eta }_{U^{*}}}{\sqrt{n}},$$

(31)

$${\eta }_{{\overline{\overline{V}}}^{*}}=\frac{{\sigma }_{V^{*}}/\sqrt{n}}{{\mu }_{V^{*}}}=\frac{{\eta }_{V^{*}}}{\sqrt{n}},$$

(32)

and the standard deviation ratio ${\omega }_{{\widehat{Z}}^{*}}$ is given by

$${\omega }_{{\widehat{Z}}^{*}}=\frac{{\sigma }_{U^{*}}/\sqrt{n}}{{\sigma }_{V^{*}}/\sqrt{n}}=\frac{{\sigma }_{U^{*}}}{{\sigma }_{V^{*}}}={\omega }^{*}.$$

(33)

The CDF of ${\widehat{Z}}_i^{*}$ is expressed as $F_Z\left(z\mid {\eta }_{{\overline{\overline{U}}}^{*}},{\eta }_{{\overline{\overline{V}}}^{*}},{\omega }^{*},{\rho }_{UV}^{*}\right)$, and the PDF of ${\widehat{Z}}_i^{*}$ is abbreviated as $f_Z\left(z\mid {\eta }_{{\overline{\overline{U}}}^{*}},{\eta }_{{\overline{\overline{V}}}^{*}},{\omega }^{*},{\rho }_{UV}^{*}\right).$

The downward CUSUM scheme detects the decrease in the ratio; its monitoring statistic is

$$D_i^{*-}=\max (0,D_{i-1}^{*-}-({\widehat{Z}}_i^{*}-z_0^{*})-{\lambda }^{-}{z}_0^{*}),$$

(34)

with $D_0^{*-}=0$, and $LC{L}^{-}=H^{-}\times z_0^{*}(H^{-}>0)$.

On the other hand, the upward CUSUM scheme detects the increase in the ratio; its monitoring statistic is

$$D_i^{*+}=\max (0,D_{i-1}^{*+}+({\widehat{Z}}_i^{*}-z_0^{*})-{\lambda }^{+}{z}_0^{*}),$$

(35)

with $D_0^{*+}=0$, and $UC{L}^{+}=H^{+}\times z_0^{*}(H^{+}>0)$.

The average run length (ARL) is the average set of samples extracted from the control chart when it detects a production problem warning. When the process is IC, the larger the ARL value, the better. When the process is OOC, the smaller the ARL value, the better. In the literature research and practical application, we generally fix the ARL of IC, denoted as ARL₀, and try to minimize the ARL under OOC, denoted as ARL₁. Three methods can be used to calculate the ARL values, the Markov Chain, integral equation, and Monte Carlo simulation. The Markov Chain is widely used to compute the ARL of a control chart without running a large number of simulations. Brook and Evans [38] detailed the use of the Markov Chain to calculate ARL.

In general, we need to know the definite value of the shift when calculating ARL, which is actually very difficult to obtain. References Tran et al. [30], Nguyen and Tran [31], Nguyen et al. [32], and Yeong et al. [39] have applied the calculation formula of the expected average run length, denoted as EARL. Knowing the definite magnitude of the shift in advance is not necessary for calculating EARL. However, the possible range of likely shifts should be known. It has a Bayesian flavor, and when the typical density of the shift parameter is unknown, often a uniform prior is used for simplicity. The EARL may be expressed as follows:

$$\mathrm{EARL}={\int }_{\Omega }\mathrm{ARL}\times f_{\tau }\left(\tau \right)\mathrm{d}\tau ,$$

(36)

where $f_{\tau }\left(\tau \right)$ is the PDF of the random shift size $\tau$ in $\Omega$, and ARL is calculated by the Markov Chain method. See Appendix A for further details. From the previous literature on statistical process control, we know that $\tau$ follows a uniform distribution in [a, b], that is, $f_{\tau }\left(\tau \right)=\frac{1}{b-a},\tau \in [a,b]$. It has been described in several research articles, see, for example, Tran et al. [13].

The optimal design of the control chart discussed in this paper is composed of the optimal pairings $({\lambda }^{*-},H^{*-})$ for the downward CUSUM and $({\lambda }^{*+},H^{*+})$ for the upward CUSUM.

In general, determining the optimal pairings typically involves two steps:

Step 1.

Seek possible pairings $({\lambda }^{-},H^{-})$ or $({\lambda }^{+},H^{+})$ that satisfy the condition where ARL is equal to ARL₀.

Step 2.

Among these possible pairings, the optimal pairing $({\lambda }^{*-},H^{*-})$ or $({\lambda }^{*+},H^{*+})$ is the one that achieves the minimum value of EARL when the process is OOC.

That is,

$$({\lambda }^{*-},H^{*-})=\underset{\left({\lambda }^{-},H^{-}\right)}{argmin}\mathrm{EARL}\left(n,{\lambda }^{-},H^{-},{\eta }_{U^{*}},{\eta }_{V^{*}},{\rho }_{UV}^{*},{\omega }^{*},{\rho }_1\right),$$

and the constraint is as follows:

$$\mathrm{ARL}\left(n,{\lambda }^{-},H^{-},{\eta }_{U^{*}},{\eta }_{V^{*}},{\rho }_{UV}^{*},{\omega }^{*},{\rho }_1={\rho }_0,\tau =1\right)={\mathrm{ARL}}_0.$$

Similarly,

$$({\lambda }^{*+},H^{*+})=\underset{\left({\lambda }^{+},H^{+}\right)}{argmin}\mathrm{EARL}\left(n,{\lambda }^{+},H^{+},{\eta }_{U^{*}},{\eta }_{V^{*}},{\rho }_{UV}^{*},{\omega }^{*},{\rho }_1\right),$$

and the constraint is as follows:

$$\mathrm{ARL}\left(n,{\lambda }^{+},H^{+},{\eta }_{U^{*}},{\eta }_{V^{*}},{\rho }_{UV}^{*},{\omega }^{*},{\rho }_1={\rho }_0,\tau =1\right)={\mathrm{ARL}}_0.$$

5. The Performance of the CUSUM Scheme for Monitoring the Ratio of Two Correlated Normal Variables with Measurement Errors

In this section, by observing the efficiency of the discussed schemes, we summarize the influence law of measurement errors on the charting scheme.

We consider two different ranges of shift size $\tau$ over the [0.9, 1) (downward CUSUM scheme) and (1, 1.1] (upward CUSUM scheme), and we set the target of ARL₀ $=200$, $z_0=1,{\delta }_V=1$.

Referring to Celano and Castagliola [6], we set ${\rho }_0\in \{-0.8,$$-0.4,0,0.4,0.8\}$ to account for varying strengths of correlation between U and V, which may be strong, weak, or absent. To ensure compliance with the lower bound for acceptable measurement system precision, we assign ${\zeta }_U={\zeta }_V=0.28$. “The accuracy error depends on the gauge calibration and it is usually equal to some percentage points of the true measure”. In this context, we assume ${\theta }_U={\theta }_V=0.05$ to investigate the influence of measurement system accuracy. Additionally, we consider the case where ${\rho }_{\epsilon }=0.4$, indicating that precision errors are positively correlated. For comprehensive discussions on parameter selection, please refer to Tran et al. [30].

Table 1. The optimal pairings of the CUSUM scheme (${\lambda }^{*-},H^{*-}$) in the upper row and $({\lambda }^{*+},H^{*+})$ in the lower row.

${\mathit{\eta }}_{\mathit{U}}$	${\mathit{\eta }}_{\mathit{V}}$	${\mathit{\rho }}_0$	$\mathit{n}=1$	$\mathit{n}=5$	$\mathit{n}=15$
0.2	0.2	−0.8	(0.0100, 2.0748)	(0.0147, 0.9223)	(0.0116, 0.5163)
			(0.0635, 3.8486)	(0.0261, 1.1544)	(0.0176, 0.5441)
		−0.4	(0.0060, 1.9933)	(0.0151, 0.7953)	(0.0137, 0.4066)
			(0.0547, 3.1892)	(0.0231, 0.9919)	(0.0157, 0.4723)
		0	(0.0088, 1.7129)	(0.0162, 0.6350)	(0.0124, 0.3405)
			(0.0446, 2.5376)	(0.0201, 0.8058)	(0.0145, 0.3765)
		0.4	(0.0122, 1.3282)	(0.0132, 0.5082)	(0.0113, 0.2519)
			(0.0337, 1.8295)	(0.0176, 0.5733)	(0.0128, 0.2704)
		0.8	(0.0138, 0.7780)	(0.0115, 0.2780)	(0.0111, 0.1192)
			(0.0217, 0.9559)	(0.0140, 0.2899)	(0.0104, 0.1362)
0.01	0.01	−0.8	(0.0105, 0.0549)	(0.0074, 0.0161)	(0.0078, 0.0038)
			(0.0107, 0.0571)	(0.0072, 0.0172)	(0.0076, 0.0043)
		−0.4	(0.0107, 0.0428)	(0.0094, 0.0093)	(0.0075, 0.0027)
			(0.0105, 0.0461)	(0.0095, 0.0097)	(0.0077, 0.0028)
		0	(0.0097, 0.0345)	(0.0097, 0.0057)	(0.0070, 0.0016)
			(0.0102, 0.0347)	(0.0073, 0.0092)	(0.0070, 0.0024)
		0.4	(0.0107, 0.0194)	(0.0086, 0.0033)	(0.0054, 0.0014)
			(0.0103, 0.0211)	(0.0058, 0.0069)	(0.0049, 0.0018)
		0.8	(0.0110, 0.0057)	(0.0056, 0.0018)	(0.0040, 0.0001)
			(0.0105, 0.0070)	(0.0056, 0.0019)	(0.0041, 0.0001)

presents the optimal pairing $({\lambda }^{*-},H^{*-})$ and $({\lambda }^{*+},H^{*+})$ of the CUSUM scheme for monitoring the ratio of two correlated normal variables with measurement errors, as exemplified by the parameters set of ${\eta }_U\in \{0.01,0.2\}$, ${\eta }_V\in \{0.01,0.2\}$, $n\in \{1,5,15\}$, ${\rho }_0\in \{-0.8,$$-0.4,0,0.4,0.8\}$, ${\zeta }_U={\zeta }_V=0.28$, ${\theta }_U={\theta }_V=0.05$, $m=1,s=1$.

5.1. The Performance of the CUSUM Scheme with Precision Errors

In order to evaluate the effect of precision errors (${\zeta }_U$ and ${\zeta }_V$) on the CUSUM scheme, that is, ${\theta }_U={\theta }_V=0$ (no effect of accuracy errors), the ranges of ${\zeta }_U$, ${\zeta }_V$, and n are ${\zeta }_U\in \{0.0,0.1,0.2,\dots ,1.0\}$, ${\zeta }_V\in \{0.0,0.1,0.2,\dots ,1.0\}$, and $n\in \{1,15\}$. When ${\eta }_U={\eta }_V=0.2$, ${\rho }_{\epsilon }=0,m=1,s=1$, the EARL values are presented in

Table 2. The effect of precision errors (${\zeta }_U$ and ${\zeta }_V$) on the CUSUM scheme ($\tau \in [0.9,1)$ in the upper row, and $\tau \in (1,1.1]$ in the lower row.

${\mathit{\zeta }}_{\mathit{U}}={\mathit{\zeta }}_{\mathit{V}}$		0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
${\rho }_0={\rho }_1=-0.8$
$n=1$		82.14	82.29	82.71	83.42	84.36	85.58	86.98	88.56	90.29	92.13	94.10
		88.92	89.10	89.64	90.54	91.78	93.35	95.24	97.45	99.98	102.82	106.00
$n=15$		31.82	31.89	32.09	32.43	32.88	33.45	34.12	34.88	35.72	36.68	37.66
		33.00	33.08	33.30	33.67	34.16	34.77	35.49	36.31	37.23	38.20	39.27
${\rho }_0=-0.4,{\rho }_1=-0.8$
$n=1$		52.02	52.31	53.16	54.57	56.56	58.97	61.63	64.66	67.82	71.17	74.61
		54.95	55.22	56.03	57.35	59.16	61.41	64.07	67.11	70.49	74.22	78.30
$n=15$		21.05	21.16	21.52	22.02	22.79	23.78	24.73	25.97	27.24	28.61	30.09
		21.97	22.01	22.34	22.82	23.57	24.53	25.54	26.72	28.02	29.41	30.79

. It is obvious that the value of EARL increases as ${\zeta }_U$ and ${\zeta }_V$ increase, regardless of whether ${\rho }_0$ and ${\rho }_1$ are equal or not. Therefore, we may conclude that precision errors degrade the performance of the CUSUM scheme. In the calculation results, roughly, we can conclude that the precision errors have no obvious effect on the CUSUM scheme when ${\zeta }_U\le 0.4$ and ${\zeta }_V\le 0.4$; the effect is pronounced when ${\zeta }_U>0.4$ and ${\zeta }_V>0.4$. As an example, take the case of $n=15$, when ${\rho }_0={\rho }_1=-0.8,\tau \in (1,1.1]$, we have $\mathrm{EARL}=33.08$ (${\zeta }_U={\zeta }_V=0.1$), $\mathrm{EARL}=34.16$ (${\zeta }_U={\zeta }_V=0.4$), and $\mathrm{EARL}=39.27$ (${\zeta }_U={\zeta }_V=1.0$); when ${\rho }_0=-0.4,{\rho }_1=-0.8,\tau \in [0.9,1)$, we have $\mathrm{EARL}=21.16$ (${\zeta }_U={\zeta }_V=0.1$), $\mathrm{EARL}=22.79$ (${\zeta }_U={\zeta }_V=0.4$), and $\mathrm{EARL}=30.09$ (${\zeta }_U={\zeta }_V=1.0$).

5.2. The Performance of the CUSUM Scheme with Accuracy Errors

We investigate the effect of accuracy errors (${\theta }_U$ and ${\theta }_V$) on the CUSUM scheme without precision errors. The values of EARL with ${\theta }_U={\theta }_V$, $s=1,m=1,{\rho }_{\epsilon }=0$, and $n\in \{1,15\}$ are shown in

Table 3. The effect of ${\theta }_U={\theta }_V$ on the CUSUM scheme ($\tau \in [0.9,1)$ in the upper row, and $\tau \in (1,1.1]$ in the lower row.

${\mathit{\theta }}_{\mathit{U}}={\mathit{\theta }}_{\mathit{V}}$		0.000	0.005	0.010	0.015	0.020	0.025	0.030	0.035	0.040	0.045	0.050
${\rho }_0={\rho }_1=-0.8$
$n=1$		82.14	82.13	82.12	82.11	82.10	82.10	82.09	82.08	82.07	82.06	82.05
		88.92	88.86	88.81	88.75	88.69	88.64	88.59	88.53	88.48	88.43	88.38
$n=15$		31.82	31.82	31.82	31.83	31.83	31.83	31.84	31.84	31.84	31.85	31.85
		33.00	33.00	33.00	32.99	32.99	32.99	32.99	32.98	32.98	32.98	32.99
${\rho }_0=-0.4,{\rho }_1=-0.8$
$n=1$		52.02	52.01	52.00	52.01	51.99	51.97	51.96	51.94	51.93	51.91	51.89
		54.94	54.91	54.89	54.86	54.84	54.81	54.79	54.76	54.74	54.72	54.70
$n=15$		21.05	21.04	21.05	21.05	21.05	21.05	21.04	21.04	21.05	21.05	21.05
		21.97	21.96	21.95	21.94	21.92	21.91	21.90	21.90	21.89	21.88	21.88

. We can see that for the same value of ${\theta }_U$ and ${\theta }_V$, the change in the values of EARL is slight as ${\theta }_U$ and ${\theta }_V$ increase when ${\rho }_0={\rho }_1=-0.8$ and ${\rho }_0=-0.4$, ${\rho }_1=-0.8$. However, when ${\theta }_U$ and ${\theta }_V$ are changed in the opposite direction, the change in the values of EARL is obvious. For example, in

Table 4. The effect of ${\theta }_U\ne {\theta }_V$ on the CUSUM scheme ($\tau \in [0.9,1)$ in the upper row, and $\tau \in (1,1.1]$ in the lower row.

${\mathit{\theta }}_{\mathit{U}}$	0.005	0.05
${\mathit{\theta }}_{\mathit{V}}$	0.05	0.005
${\rho }_0={\rho }_1=-0.8$
$n=1$	86.78	90.05
	93.50	95.78
$n=15$	36.83	41.18
	41.00	38.94
${\rho }_0=-0.4,{\rho }_1=-0.8$
$n=1$	54.95	54.91
	57.11	58.90
$n=15$	23.20	27.93
	24.02	24.73

, with $n=1$, ${\rho }_0={\rho }_1=-0.8$, and $\tau \in [0.9,1)$, we have $\mathrm{EARL}=86.78$ when ${\theta }_U=0.005,{\theta }_V=0.05$ and $\mathrm{EARL}=90.05$ when ${\theta }_U=0.05,{\theta }_V=0.005$.

5.3. The Performance of the CUSUM Scheme When Changing ${\rho }_{\epsilon }$

The effects of ${\rho }_{\epsilon }$ on the CUSUM scheme are shown in Figure 1. The range of ${\rho }_{\epsilon }$ is $\{-1.0,-0.8,-0.6,\dots ,0.6,0.8,1.0\}$. Figure 1 shows the values of EARL with ${\zeta }_U={\zeta }_V=0.28$, ${\theta }_U={\theta }_V=0.05,m=1,s=1$. It distinctly illustrates a negative association between ${\rho }_{ϵ}$ and EARL. When the ${\rho }_{ϵ}$ is large, the EARL values are low. For example, in the condition of ${\rho }_0=-0.4,{\rho }_1=-0.8$, and $\tau \in [0.9,1),n=1$, $\mathrm{EARL}=55.86$ when ${\rho }_{\epsilon }=-0.8$, and $\mathrm{EARL}=52.86$ when ${\rho }_{\epsilon }=0.8$. In the case of ${\rho }_0=-0.4,{\rho }_1=-0.8$, and $\tau \in (1,1.1],n=1$, $\mathrm{EARL}=58.46$ when ${\rho }_{\epsilon }=-0.8$, and $\mathrm{EARL}=55.57$ when ${\rho }_{\epsilon }=0.8$.

5.4. The Performance of the CUSUM Scheme When Increasing m

In addition, Linna and Woodall [18] proposed the method of increasing the number of measurements per set to reduce the effect of measurement errors. Nguyen and Tran [31] and Nguyen et al. [32] applied this method. Based on these, we increase m from 1 to 20 and detect changes in EARL at the same time. When ${\zeta }_U={\zeta }_V=0.28,{\theta }_U={\theta }_V=0.05, s=1,{\rho }_{\epsilon }=0.4$, the change in EARL with the increase in m is shown in Figure 2. The values of EARL change indistinctively with the increase in m. For example, in the condition of ${\rho }_0={\rho }_1=-0.8$, and $n=1,\tau \in [0.9,1)$, $\mathrm{EARL}=82.74$ when $m=1$, and $\mathrm{EARL}=82.12$ when $m=10$. Compared with $m=10,15$, the change in EARL is slight when $m=20$. In the case of ${\rho }_0=-0.4,{\rho }_1=-0.8$, and $n=15,\tau \in (1,1.1]$, $\mathrm{EARL}=22.64$ when $m=1$, and $\mathrm{EARL}=22.12$ when $m=10$. It follows that increasing the measurement times of each item does not obviously change the efficiency of the discussed scheme.

5.5. The Performance of the CUSUM Scheme When Changing s

Finally, we study the effect of changing the value of s on the CUSUM scheme for monitoring the ratio of two correlated normal variables with measurement errors. Given ${\zeta }_U={\zeta }_V=0.28$, ${\theta }_U={\theta }_V=0.05$, $m=1$, and ${\rho }_{\epsilon }=0.4$, the changes in EARL for $s\in \{0.8,0.85,\dots ,1.15,1.2\}$ are listed in Figure 3. As can be seen from Figure 3, when n is 1 and 15, respectively, the value of EARL does not change obviously with the increase in s.

6. An Illustrative Example

This section illustrates the application of the CUSUM scheme for the ratio of two correlated normal variables with measurement errors in an example. We apply the requirements for a muesli recipe, and the objective of monitoring is to recognize and manage systemic issues that could lead to a continuous deterioration in product quality.

Tran et al. [30] considered that the muesli recipe consists of sunflower oil, seeds (pumpkin, flaxseed, sesame), oatmeal, etc. The product must be formulated with an equal weight of pumpkin seeds and flaxseeds to meet the desired nutritional standards of the product and to have a good taste.

In the production process, we take five boxes of mixed cereal at 30-minute intervals and separate the “pumpkin seeds” and “flaxseeds” from each box. The weight of the pumpkin seeds is denoted by U, and the weight of the flaxseeds is denoted by V. In this way, we can obtain the average of the two seeds $\overline{U}$ and $\overline{V}$. The ratio $Z=\frac{\overline{U}}{\overline{V}}$ is monitored to determine whether the process is stable.

Table 5. The data of the example with $n=5$.

Sample Number	${\mathit{U}}_{\mathit{i},\mathit{j}}^{*}$ 1					${\overline{\mathit{U}}}_{\mathit{i}}^{*}$	${\widehat{\mathit{Z}}}_{\mathit{i}}^{*}=\frac{{\overline{\mathit{U}}}_{\mathit{i}}^{*}}{{\overline{\mathit{V}}}_{\mathit{i}}^{*}}$	${\mathit{D}}_{\mathit{i}}^{*+}$
	${\mathit{V}}_{\mathit{i},\mathit{j}}^{*}$ 2					${\overline{\mathit{V}}}_{\mathit{i}}^{*}$
1	26.052	25.687	26.497	25.747	26.603	26.117	1.001	0
		26.025	25.741	26.214	26.005	26.441	26.085
2	26.195	27.540	26.277	24.990	25.122	26.025	0.995	0
		26.230	26.834	26.191	25.767	25.809	26.166
3	26.922	25.997	26.234	26.216	25.866	26.249	0.996	0
		26.819	26.202	26.359	26.263	26.100	26.349
4	26.073	25.703	25.793	25.214	25.810	25.719	0.993	0
		25.997	26.453	25.995	25.503	25.600	25.910
5	26.520	26.587	25.341	26.738	26.230	26.283	1.003	0
		26.505	26.338	25.957	26.098	26.467	26.273
6	26.476	27.304	26.377	26.874	25.679	26.542	1.007	0.004
		26.442	26.661	26.355	26.416	25.959	26.367
7	25.687	27.056	25.827	25.499	25.593	25.932	0.991	0
		25.741	26.717	26.355	25.957	26.030	26.160
8	26.638	26.890	27.098	25.743	26.147	26.503	1.006	0.003
		26.447	26.486	26.543	26.255	25.949	26.336
9	27.212	25.930	25.827	26.884	26.026	26.376	0.998	0
		26.780	26.491	26.371	26.587	25.936	26.433
10	26.216	26.630	26.194	26.082	26.052	26.235	1.003	0
		26.325	26.205	26.210	26.061	26.025	26.165
11	27.120	27.047	27.131	26.864	27.238	27.100	1.026	0.023
		26.409	26.367	26.603	26.066	26.590	26.407
12	26.461	26.434	26.844	27.098	26.262	26.620	1.004	0.024
		26.614	26.296	26.465	26.782	26.438	26.519
13	25.898	25.524	25.992	26.357	26.606	26.075	0.997	0.018
		26.176	25.989	25.828	26.482	26.255	26.146
14	25.944	26.484	25.896	26.313	26.713	26.270	1.002	0.017
		25.848	26.344	25.881	26.343	26.548	26.193
15	25.867	26.168	26.393	25.360	26.945	26.147	0.993	0.007
		26.236	26.144	26.507	26.079	26.689	26.331

¹ Sample weight of pumpkin seeds (gr). ² Sample weight of flaxseeds (gr).

presents a set of simulated samples. Until the sample point 10, the process is IC. There is a 1% upward shift occurring between sample point 10 and sample point 11. In practice, it is required that the process should signal when the upward shift of the ratio reaches 1%.

The choice of parameters and assumptions is consistent with the previous literature, ${\eta }_U=0.02,{\eta }_V=0.01,{\rho }_0=0.8,$${\theta }_U={\theta }_V=0.05,{\zeta }_U={\zeta }_V=0.28,{\rho }_{\epsilon }=0.5$. The $UCL$ of the CUSUM scheme with measurement errors is 0.0211 when ${\lambda }^{+}=0.0030$, and ARL₀ $=200$. From Figure 4, it is evident that sampling points 11 and 12 are where the OOC signal is observed. The OOC signal is observed at sampling point 11 in Tran et al. [30]. This demonstrates the applicability of the proposed charting scheme to quality control issues.

7. Conclusions and Suggestions for Future Research

In this paper, we study the ratio of two correlated normal variables with measurement errors monitored by the CUSUM scheme. The optimal pairings are identified in the event of an uncertain shift using the shift interval. The effects of measurement errors on the CUSUM scheme for monitoring the ratio of two correlated normal variables are examined deeply. Moreover, it is discovered that this effect cannot be reduced by increasing the number of measurements per set. The applicability of this charting scheme to quality control issues is demonstrated with the assistance of an example. Finding ways to reduce the effects of measurement errors on the ratio control charts is the focus of future research. Further, reducing the negative performance on the ratio control chart through a combination of adaptive strategies like variable sampling intervals, variable sampling sizes or with run rules, or adding the auxiliary information could be considered for future research.