Combination of Sequential Sampling Technique with GLR Control Charts for Monitoring Linear Profiles Based on the Random Explanatory Variables

Abstract

Control charts play a beneficial role in the manufacturing process by reduction of non-compatible products and improving the final quality. In line with these aims, several adaptive methods in which samples can be taken with variable sampling rates and intervals have been proposed in the area of statistical process control (SPC). In some SPC applications, it is important to monitor a relationship between the response and independent variables—this is called profile monitoring. This article proposes adaptive generalized likelihood ratio (GLR) control charts based on variable sampling interval (VSI) and sequential sampling (SS) techniques for monitoring simple linear profiles. Because in some real-life problems, it may be possible that the user cannot control the values of explanatory variables; thus, in this paper, we focus on such a scenario. The performance of the proposed method is compared under three different situations, i.e., the fixed sampling rate (FSR), VSI, and SS, based on average time to signal (ATS) criteria for phase II analysis. Since the SS approach uses a novel sampling procedure based on the statistic magnitude, it has a superior performance over other competing charts. Several simulation studies indicate the superiority as the SS approach yields lower ATS values when there are single-step changes in the intercept, slope, standard deviation of the error term, and explanatory variables. In addition, some other related sensitivity analysis indicates that other aspects of the proposed methods, such as computational time, comparison with other control charts, and consideration of fixed explanatory variables. Furthermore, the results are supported by a real-life illustrative example from the adhesive manufacturing industry.

1. Introduction

In some industrial processes, the intended quality characteristic is influenced by either one or more explanatory variables, and it is necessary to monitor the relationship between the primary quality characteristic (response variable) and independent (explanatory) variables. This approach, denoted as profile monitoring, has been extended in different areas of statistical process control (SPC) applications [1,2,3]. Most of the literature on profile monitoring deals with the analysis of linear profiles in which the relationship between the response and explanatory variables is assumed to be linear. In addition, most of these studies have been conducted in phase II, where the online monitoring of the process is the main purpose, and the underlying in-control model parameters are assumed to be known, to name but a few, see Kang and Albin [4], Kim, Mahmoud [5] and Zou, Tsung [6]. Monitoring of linear profiles in phase I could be found in Mahmoud and Woodall [7] and Mahmoud, Parker [8], Mahmood, Abbasi [9], Bandara, Abdel-Salam [10], and He, Song [11]. Since this paper focuses on phase II analysis, the other phase I studies are neglected for the sake of brevity.

Due to the simplicity of design and good performance for detecting a wide range of shifts, generalized likelihood ratio (GLR) control charts have been effectively implemented in monitoring quality characteristics. The change point model based on likelihood ratio tests (LRT) is used in the construction of the chart statistics [12,13,14]. In the case of profile monitoring, Mahmoud and Parker [8] extended this idea in phase I analysis, and after that, Xu and Wang [15] employed the GLR approach for monitoring linear profiles in phase II. Xu, Peng [16] detected sustained changes in the parameters of linear profiles by the development of a GLR control chart when there is an individual observation on each profile. A combination of the variable sampling interval (VSI) technique and GLR was applied by Hafez Darbani and Shadman [17] in monitoring generalized linear profiles. In addition, GLR control charts were extended to other profile types such as logistic [18], polynomial [19], Poisson [20,21], and geometric profiles for image data [22,23].

Many authors have shown that the performance of control charts can be improved by using adaptive approaches, in which at least one of the chart’s parameters, such as sampling size, interval, and so forth, is allowed to vary over time based on the past process data [24,25]. In the context of profile monitoring, Li and Wang [26] investigated the monitoring of linear profiles using an EWMA control chart with a VSI scheme. As a more efficient approach, Abdella and Yang [27] considered a T² control chart with variable sample size and sampling interval (VSSI) to monitor linear profiles. Similarly, De Magalhães, Fernandes [28] added the VSSI feature to the T² chart and investigated its statistical design. The idea of variable sample size (VSS) was implemented by Kazemzadeh and Amiri [29] in monitoring simple linear profiles in combination with a multivariate EWMA (MEWMA) chart. Wan and Zhu [30] considered monitoring profiles using the economic design of the VSI of $n{p}_x-\overline{X}$ control chart. An adaptive heuristic scheme for monitoring non-parametric profiles can also be found in Abbasi, Yeganeh [31].

Another useful approach to varying the sampling rate is sequential sampling (SS), in which the sample size is varied dynamically at each sampling time as a function of the data observed at the current sampling point [32]. This allows users to determine the number of samples at each sampling point as a result of chart statistic condition, and generally, it is suitable for situations where the time of taking an individual observation is so short, such that it could be ignored in comparison with the required time between two observations [33]. As the pioneer works in monitoring quality characteristics, Stoumbos and Reynolds [34] and Reynolds and Stoumbos [35] designed several SS charts based on a sequential probability ratio test (SPRT) at each sampling point. The SPRT charts are designed to detect one-sided shifts and are not appropriate for situations where the goal is to detect both increasing and decreasing shifts [36]. Considering this idea, Peng and Reynolds [37] utilized a GLR control chart with sequential sampling denoted by SS GLR to monitor the mean of a normal process. Simulation results have shown the proper detection ability of this chart in the presence of two-sided shifts. In a similar approach, Shahzad and Huang [38] applied the SS GLR approach in the process with geometric observations. To the best of the authors’ knowledge, there is no research that applies the SS charts in the profile monitoring area.

To bridge this knowledge gap, the SS GLR chart has been investigated for monitoring linear profiles in phase II in this study. Since one of the major applications of profile monitoring has been in system calibration [4,6], in which the explanatory variables could be fixed at some predefined levels, most of the previous research assumed that the values of the explanatory variable are fixed during the monitoring procedure. Although this assumption may be valid in a calibration application, it may not be possible to consider fixed design points in other industrial processes. For example, Ding and Tsung [39] monitored the quality of a specific type of glue based on the pressure and washing time as the explanatory variables, and due to the nature of these two variables, they were not able to consider fixed design points. Some other researchers, such as Shang, Tsung [40], Noorossana, Fatemi [41], Abbas, Abbasi [42], Malela-Majika, Shongwe [43] and Kumar, N. [44] provided some useful control charts based on the EWMA statistics with random design points; however, there is no research on the combination of VSI, SS, and generally adaptive methods with random explanatory variables in profile monitoring. Therefore, another contribution of this paper is the implementation and comparison of the VSI scheme in monitoring profiles with random predictors. To better analyze the adaptive methods, we investigated and compared the performance of GLR with a conventional T² control chart under FSR, VSI, and SS schemes. In addition, some comparisons with adaptive charts with fixed design points are provided. To sum up, the contributions of this paper could be summarized as follows:

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Combination of GLR and SS (denoted by SSGLR hereafter) as an adaptive approach in monitoring linear profiles.

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Considering the explanatory variable as random in the SSGLR.

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Adjustment of GLR with random explanatory variables under the VSI scheme (denoted by VSIGLR).

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Comparing GLR with T² control chart under adaptive scenarios.

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Comparing SSGLR under random and fixed explanatory variables.

The remainder of the paper is structured as follows. In Section 2, we introduce the SSGLR control chart statistic for monitoring simple linear profiles with random explanatory variables; then, we discuss the design and implementation of the FSR, VSI, and SS schemes in combination with the GLR control chart in Section 3. In Section 4, the mentioned methods are compared in terms of the average time to signal (ATS). In addition, some sensitivity analyses are performed in this section. A numerical example is provided in Section 5 to show how to implement the SSGLR control chart. Finally, the concluding remarks are presented in Section 6.

2. The SSGLR Control Chart with Random Explanatory Variables

In this section, we extend the GLR control chart proposed by Xu and Wang [15] for monitoring simple linear profiles in the case of sequential sampling and random explanatory variables. Suppose that the relationship between the response and the explanatory variable is adequately represented with a simple linear regression model, and the j^th process observation of t^th profile is collected based on the following model:

$$\begin{array}{c}{y}_{\mathit{tj} }{=\alpha }_{t }{+\beta }_t{x}_{\mathit{tj}}{+\epsilon }_{\mathit{tj}},\\ t=1,2, \dots ,\\ {j=1, \dots , n}_t,\end{array}$$

(1)

where ${\alpha }_t$ and ${\beta }_t$ are the intercept and slope of t^th profile, respectively, and ${\epsilon }_{\mathit{tj}}$ is a random error term that independently follows a normal distribution with mean 0 and variance of ${\sigma }_t^{2 }$. $y_{\mathit{tj}}$. is the j^th response observation of t^th profile, and the value of the explanatory variable for the j^th observation of t^th profile is indicated by x_tj, which is assumed to be independently and normally distributed with mean ${\mu }_t$ and variance ${\sigma }_{\mathit{tX}}^2$. Suppose at profile k, there are n_t paired observations for the previous k − 1 profiles as (x_tj,y_tj) in such a way that t = 1, …, k − 1 and j = 1, …, n_t (n_t > 2). Due to the nature of the SS scheme, there are l paired observations for the k^th (current) profile as ((x_kj,y_kj)|j = 1, …, l), where l is the decision variable of the SS scheme. Based on the change point model, the null hypothesis is defined as ‘there have not been any shifts in the in-control value of the parameters’ versus the alternative hypothesis, which is defined as ‘at least one parameter of the model (${\alpha }_t{, \beta }_t{, \sigma }_t^2{, \mu }_t{, \sigma }_{\mathit{tX}}^2$) has shifted from its in-control value at some unknown time between $\tau$ (called a “change point”) and τ + 1’. In other words, the alternative hypothesis can be formulated as follows:

$$H_1:\left\{\begin{array}{c}{ \alpha }_t{=\alpha }_0{ ,\beta }_t{=\beta }_0{ ,\sigma }_t^2{=\sigma }_0^{2 }{,\mu }_t{=\mu }_0{ ,\sigma }_{\mathit{tX}}^2{=\sigma }_{0X}^{2 } ,\\ t=1,2,\dots ,\tau ,\\ {\alpha }_t{=\alpha }_1{ ,\beta }_t{=\beta }_1{ ,\sigma }_t^2{=\sigma }_1^{2 }{,\mu }_t{=\mu }_1{,\sigma }_{\mathit{tX}}^2{=\sigma }_{1X}^{2 },\\ t=\tau +1,\tau +2 ,\dots ,k, \end{array}\right.$$

(2)

where ${\alpha }_0{, \beta }_0{, \sigma }_0^2{, \mu }_0$ and ${\sigma }_{0X}^{2 }$ are the in-control parameters of the model, which are assumed to be known in phase II, and ${\alpha }_1{, \beta }_1{, \sigma }_1^2{, \mu }_1$ and ${\sigma }_{1X}^{2 }$ denote the out-of-control parameters in which at least one of them would be different from its in-control parameter. It is obvious that the value of out-of-control parameter(s) along with the change points are unknown in phase II, so they should be estimated based on the out-of-control samples.

To develop a likelihood function in the case of a random explanatory variable, the basic principle of conditional probability can be used (see more details in Shang, Tsung [40]). Hence the joint likelihood of (x_tj,y_tj) for the j^th observation of t^th profile is written as:

$$L_{\mathit{tj}}\left(x_{\mathit{tj}}{,y}_{\mathit{tj}}\right)=f\left(x_{\mathit{tj}}{,y}_{\mathit{tj}}\right)=f\left(y_{\mathit{tj}}{|x}_{\mathit{tj}}\right)f\left(x_{\mathit{tj}}\right).$$

(3)

Under the null hypothesis, i.e., each of the parameters is not varied from its in-control value, the likelihood function at the current profile k is written as:

$$\begin{array}{l}L\left(\infty {,\alpha }_0{,\beta }_0{,\sigma }_0^2{,\mu }_0{,\sigma }_{0X}^{2 }{/y}_{\mathit{tj}}{,x}_{\mathit{tj}}{ ; t=1,2,\dots k ; j=1,\dots ,n}_t\right)=\\ [{\left({2\pi \sigma }_0^2\right)}^{\frac{-{{\displaystyle \sum }}_{t=1}^{k-1}{n}_t-l}{2}}{ e}^{-\frac{1}{{2\sigma }_0^{2 }}{\displaystyle \sum }_{t=1}^{k-1}{\displaystyle \sum }_{j=1}^{n_t}{\left(y_{\mathit{tj}}-{\alpha }_0-x_{\mathit{tj} }{\beta }_0\right)}^2}{\left({2\pi \sigma }_{0X}^2\right)}^{\frac{-{{\displaystyle \sum }}_{t=1}^{k-1}{n}_t-l}{2}}{ e}^{-\frac{1}{{2\sigma }_{0X}^2}{\displaystyle \sum }_{t=1}^{k-1}{\displaystyle \sum }_{j=1}^{n_t}{\left(x_{\mathit{tj} }-{\mu }_0\right)}^2} \\ e^{-\frac{1}{{2\sigma }_0^{2 }}{\displaystyle \sum }_{j=1}^l{\left(y_{\mathit{kj}}-{\alpha }_0-x_{\mathit{kj} }{\beta }_0\right)}^2}{ e}^{-\frac{1}{{2\sigma }_{0X}^2}{\displaystyle \sum }_{j=1}^l{\left(x_{\mathit{kj} }-{\mu }_0\right)}^2}].\end{array}$$

(4)

In order to calculate the likelihood function under the alternative hypothesis, the out-of-control parameters should be estimated. Using the maximum likelihood estimation (MLE) method and based on the out-of-control parameters, ${\alpha }_1$ and ${\beta }_1$ are estimated at the current profile (k^th profile) based on the k − τ previous samples as follows:

$$\begin{array}{c}{\widehat{\beta }}_{1\tau }=\frac{{{\displaystyle \sum }}_{t=\tau +1}^{k-1}{{\displaystyle \sum }}_{j=1}^{n_t}\left(y_{\mathit{tj}}-{\overline{y}}_{\tau }\right)\left(x_{\mathit{tj}}-{\overline{x}}_{\tau }\right)+{{\displaystyle \sum }}_{j=1}^l\left(y_{\mathit{kj}}-{\overline{y}}_{\tau }\right)\left(x_{\mathit{kj}}-{\overline{x}}_{\tau }\right)}{{{\displaystyle \sum }}_{t=\tau +1}^{k-1}{{\displaystyle \sum }}_{j=1}^{n_t}{\left(x_{\mathit{tj}}-{\overline{x}}_{\tau }\right)}^2+{{\displaystyle \sum }}_{j=1}^l{\left(x_{\mathit{kj}}-{\overline{x}}_{\tau }\right)}^2},\\ {\widehat{\alpha }}_{1\tau }={\overline{y}}_{\tau }-{\widehat{\beta }}_1{\overline{x}}_{\tau },\end{array}$$

(5)

where ${\widehat{\alpha }}_{1\tau }$ and ${\widehat{\beta }}_{1\tau }$ are the estimations of ${\alpha }_1$ and ${\beta }_1$, respectively, l is the sampling size of the k^th profile that is a decision variable and should be determined based on the SSGLR method (l > 2). The mean of the explanatory and response variables for out-of-control samples (from τ + 1 to k) is calculated as follows:

$$\begin{array}{c}\overline{x_{\tau }}=\frac{{{\displaystyle \sum }}_{t=\tau +1}^{k-1}{{\displaystyle \sum }}_{j=1}^{n_t}{x}_{\mathit{tj}}+{{\displaystyle \sum }}_{j=1}^l{x}_{\mathit{kj}}}{{{\displaystyle \sum }}_{t=\tau +1}^{k-1}{n}_t+l},\\ \overline{y_{\tau }}=\frac{{{\displaystyle \sum }}_{t=\tau +1}^{k-1}{{\displaystyle \sum }}_{j=1}^{n_t}{y}_{\mathit{tj}}+{{\displaystyle \sum }}_{j=1}^l{y}_{\mathit{kj}}}{{{\displaystyle \sum }}_{t=\tau +1}^{k-1}{n}_t+l}.\end{array}$$

(6)

Note that $\overline{x_{\tau }}$ is also the MLE of ${\mu }_1$ (${\widehat{\mu }}_{1\tau }=\overline{x_{\tau }}$). As it is obvious that the estimations occur in the τ time, we omitted the τ subscript index from ${\widehat{\alpha }}_{1\tau }$, ${\widehat{\beta }}_{1\tau }$, $\overline{x_{\tau }}$ and $\overline{y_{\tau }}$ for the ease of computations.

Considering the bias of MLE for ${\sigma }_1^{2 }$ and ${\sigma }_{1X}^2$, especially for a relatively large sample size, Kim, Mahmoud [5] suggested estimating these parameters using mean-squared estimators (MSE). The MSEs of ${\sigma }_1^{2 }$ and ${\sigma }_{1X}^2$ (${\widehat{\sigma }}_{1,MSE}^2$ and ${\widehat{\sigma }}_{1X,MSE}^2$) at the current profile k based on the k − τ previous samples are calculated as:

$$\begin{gathered} {\widehat{\sigma }}_{1,MSE\tau }^2=\frac{{{\displaystyle \sum }}_{t=\tau +1}^{k-1}{{\displaystyle \sum }}_{j=1}^{n_t}{\left(y_{\mathit{tj}}-{\widehat{\alpha }}_{1\tau }-x_{\mathit{tj} }{\widehat{\beta }}_{\tau }\right)}^2+{{\displaystyle \sum }}_{j=1}^l{\left(y_{\mathit{kj}}-{\widehat{\alpha }}_1-x_{\mathit{kj} }{\widehat{\beta }}_1\right)}^2}{{{\displaystyle \sum }}_{t=\tau +1}^{k-1}{n}_t+l-2}, \\ {\widehat{\sigma }}_{1X,MSE\tau }^2=\frac{{{\displaystyle \sum }}_{t=\tau +1}^{k-1}{{\displaystyle \sum }}_{j=1}^{n_t}{\left(x_{\mathit{tj}}-\overline{x}\right)}^2+{{\displaystyle \sum }}_{j=1}^l{\left(x_{\mathit{kj}}-\overline{x}\right)}^2}{{{\displaystyle \sum }}_{t=\tau +1}^{k-1}{n}_t+l-1}. \end{gathered}$$

(7)

Furthermore, since the major goal in many processes is to detect the increasing shift in the variance components (${\sigma }_1^{2 }$ and ${\sigma }_{1X}^2$, their estimations are revised as follows:

$$\begin{gathered} {\widehat{\sigma }}_{1\tau }^2=\mathit{Max}({\widehat{\sigma }}_{1,MSE\tau }^2,{\sigma }_0^2), \\ {\widehat{\sigma }}_{1X\tau }^2=\mathit{Max}({\widehat{\sigma }}_{1X,MSE\tau }^2,{\sigma }_{0X}^2). \end{gathered}$$

(8)

As mentioned, the τ subscript index is omitted from ${\widehat{\sigma }}_{1\tau }^2$ and ${\widehat{\sigma }}_{1X\tau }^2$ hereafter. Therefore, the likelihood function under the alternative hypothesis at the current profile k can be estimated based on the k − τ previous samples as:

$$\begin{array}{c}L\left(\tau ,{\widehat{\alpha }}_1,{\widehat{\beta }}_{1^{\prime }}{\widehat{\sigma }}_1^2,{\widehat{\mu }}_{1^{\prime }}{\widehat{\sigma }}_{1X}^2\mid y_{t{i}^{\prime }}{x}_{tj};t=1,2,\dots k,j=1,\dots ,n_t\right)=\hfill \\ {[\left(2\pi {\sigma }_0^2\right)}^{\frac{-{\sum }_{t=1}^{\tau }{n}_t}{2}}{e}^{-\frac{1}{2{\sigma }_0^2}{\sum }_{t=1}^{\tau }{\sum }_{j=1}^{n_t}{\left(y_{tj}-{\alpha }_0-x_{tj}{\beta }_0\right)}^2}\hfill \\ {\left(2\pi {\sigma }_{0X}^2\right)}^{\frac{-{\sum }_{t=1}^{\tau }{n}_t}{2}}{e}^{-\frac{1}{2{\sigma }_{0X}^2}{\sum }_{t=1}^{\tau }{\sum }_{j=1}^{n_t}{\left(x_{tj}-{\mu }_0\right)}^2}\hfill \\ {\left(2\pi {\widehat{\sigma }}_1^2\right)}^{\frac{-{\sum }_{t=\tau +1}^{k-1}{n}_t-l}{2}}{e}^{-\frac{1}{2{\widehat{\sigma }}_1^2}{\sum }_{t=\tau +1}^{k-1}{\sum }_{j=1}^{n_t}{\left(y_{tj}-{\widehat{\alpha }}_1-x_{tj}{\widehat{\beta }}_1\right)}^2}\hfill \\ {\left(2\pi {\widehat{\sigma }}_{1X}^2\right)}^{\frac{-{\sum }_{t=\tau +1}^{k-1}{n}_t-l}{2}}{e}^{-\frac{1}{2{\widehat{\sigma }}_{1X}^2}{\sum }_{t=\tau +1}^{k-1}{\sum }_{j=1}^{n_t}{\left(x_{tj}-{\widehat{\mu }}_1\right)}^2}\hfill \\ e^{-\frac{1}{2{\widehat{\sigma }}_1^2}{\Sigma }_{j=1}^l{\left(y_{kj}-{\widehat{\alpha }}_1-x_{kj}{\widehat{\beta }}_1\right)}^2}{e}^{-\frac{1}{2{\widehat{\sigma }}_{1X}^2}{\sum }_{j=1}^l{\left(x_{kj}-{\widehat{\mu }}_1\right)}^2}].\hfill \end{array}$$

(9)

In Equation (9), $\tau$ is an unknown change point (time) and it can be estimated as the time that the likelihood function is maximized. Finally, a log-likelihood ratio statistic for testing the hypothesis about the existence of a shift in the process parameters in the past k profiles is calculated as follows:

$$\begin{array}{c}{R}_{k,l}=Ln\frac{\underset{0\le \tau <k}{\overset{\mathit{max}}{}}L\left(\tau ,{\widehat{a}}_1,{\widehat{\beta }}_{1^{\prime }}{\widehat{\sigma }}_1^2,{\widehat{\mu }}_{1^{\prime }}{\widehat{\sigma }}_{1X}^2\mid y_{t{j}^{\prime }}{x}_{tj};t=1,2,\dots k j=1,\dots ,n_t\right)}{L\left(\infty ,{\alpha }_{0,},{\beta }_0,{\sigma }_0^2,{\mu }_0,{\sigma }_{0X}^2\mid y_{t{i}^{\prime }}{x}_{tj;}t=1,2,\dots k j=1,\dots ,n_t\right)}=\hfill \\ \underset{0\le \tau <k}{\overset{\mathit{max}}{}}[\frac{{\sum }_{t=\tau +1}^{k-1}{n}_t+l}{2}\mathit{Ln}\left(\frac{{\sigma }_0^2{\sigma }_{0X}^2}{{\widehat{\sigma }}_1^2{\widehat{\sigma }}_{1X}^2}\right)-\frac{1}{2{\widehat{\sigma }}_1^2}{\displaystyle \sum _{t=\tau +1}^{k-1}}{\displaystyle \sum _{j=1}^{n_t}}{\left(y_{tj}-{\widehat{\alpha }}_1-x_{tj}{\widehat{\beta }}_1\right)}^2\hfill \\ -\frac{1}{2{\widehat{\sigma }}_1^2}{\displaystyle \sum _{j=1}^l}{\left(y_{kj}-{\widehat{\alpha }}_1-x_{kj}{\widehat{\beta }}_1\right)}^2-\frac{1}{2{\widehat{\sigma }}_{1X}^2}{\displaystyle \sum _{t=\tau +1}^{k-1}}{\displaystyle \sum _{j=1}^{n_t}}{\left(x_{tj}-{\widehat{\mu }}_1\right)}^2-\hfill \\ \frac{1}{2{\widehat{\sigma }}_{1X}^2}{\displaystyle \sum _{j=1}^l}{\left(x_{kj}-{\widehat{\mu }}_1\right)}^2+\frac{1}{2{\sigma }_0^2}{\displaystyle \sum _{j=1}^l}{\left(y_{kj}-{\alpha }_0-x_{kj}{\beta }_0\right)}^2+\hfill \\ \frac{1}{2{\sigma }_0^2}{\displaystyle \sum _{t=\tau +1}^{k-1}}{\displaystyle \sum _{j=1}^{n_t}}{\left(y_{tj}-{\alpha }_0-x_{tj}{\beta }_0\right)}^2+\frac{1}{2{\sigma }_{0X}^2}{\displaystyle \sum _{t=\tau +1}^{k-1}}{\displaystyle \sum _{j=1}^{n_t}}{\left(x_{tj}-{\mu }_0\right)}^2+\frac{1}{2{\sigma }_{0X}^2}{\displaystyle \sum _{j=1}^l}{\left(x_{kj}-{\mu }_0\right)}^2].\hfill \end{array}$$

(10)

To alleviate the computational burden of R_k,l, a simple idea based on restricting the search to a window of the most recent profiles (window size) has been used in previous research [15,17]. Considering this idea (window of size m), the statistic of the SSGLR chart is modified as follows:

$$\begin{array}{c}{R}_{m,k,l}=\underset{max(0,k-m)\le \tau <k}{\overset{max}{}}[\frac{{\sum }_{t=\tau +1}^{k-1}{n}_t+l}{2}\mathit{Ln}\left(\frac{{\sigma }_0^2{\sigma }_{0X}^2}{{\widehat{\sigma }}_1^2{\widehat{\sigma }}_{1X}^2}\right)-\frac{1}{2{\widehat{\sigma }}_1^2}{\displaystyle \sum _{t=\tau +1}^{k-1}}{\displaystyle \sum _{j=1}^{n_t}}{\left(y_{tj}-{\widehat{\alpha }}_1-x_{tj}{\widehat{\beta }}_1\right)}^2\hfill \\ -\frac{1}{2{\widehat{\sigma }}_1^2}{\displaystyle \sum _{j=1}^l}{\left(y_{kj}-{\widehat{\alpha }}_1-x_{kj}{\widehat{\beta }}_1\right)}^2-\frac{1}{2{\widehat{\sigma }}_{1X}^2}{\displaystyle \sum _{t=\tau +1}^{k-1}}{\displaystyle \sum _{j=1}^{n_t}}{\left(x_{tj}-{\widehat{\mu }}_1\right)}^2\hfill \\ -\frac{1}{2{\widehat{\sigma }}_{1X}^2}{\displaystyle \sum _{j=1}^l}\left(x_{kj}\right.\hfill \\ {\left.-{\widehat{\mu }}_1\right)}^2+\frac{1}{2{\sigma }_0^2}{\displaystyle \sum _{j=1}^l}\left(y_{kj}\right.\hfill \\ {\left.-{\alpha }_0-x_{kj}{\beta }_0\right)}^2+\frac{1}{2{\sigma }_0^2}{\displaystyle \sum _{t=\tau +1}^{k-1}}{\displaystyle \sum _{j=1}^{n_t}}{\left(y_{tj}-{\alpha }_0-x_{tj}{\beta }_0\right)}^2+\frac{1}{2{\sigma }_{0X}^2}{\displaystyle \sum _{t=\tau +1}^{k-1}}{\displaystyle \sum _{j=1}^{n_t}}{\left(x_{tj}-{\mu }_0\right)}^2+\frac{1}{2{\sigma }_{0X}^2}{\displaystyle \sum _{j=1}^l}\left(x_{kj}\right.\hfill \\ \left.{\left.-{\mu }_0\right)}^2\right]]\hfill \end{array}$$

(11)

3. Combination of Adaptive Schemes with GLR Control Chart in Monitoring Linear Profiles

This section provides the designing and implementation of FSR, VSI, and SS schemes in combination with the GLR control chart denoted by FSRGLR, VSIGLR, and SSGLR in the next three subsections. It is noteworthy to mention that the proposed GLR statistics in Equation (11) are easily converted to Equation (12) for the FSR and VSI schemes in which the terms including l variable are omitted and n_t is replaced with n. For the sake of brevity, more details are not given here, and interested readers can refer to Xu, Wang [15] and Hafez Darbani and Shadman [17].

$$\begin{array}{c}{R}_{m,k}=\underset{max(0,k-m)\le \tau <k}{\overset{max}{}}[\frac{(k-\tau )n}{2}\mathit{Ln}\left(\frac{{\sigma }_0^2{\sigma }_{0X}^2}{{\widehat{\sigma }}_1^2{\widehat{\sigma }}_{1X}}\right)-\frac{1}{2{\widehat{\sigma }}_1^2}{\displaystyle \sum _{t=\tau +1}^k}{\displaystyle \sum _{j=1}^n}{\left(y_{tj}-{\widehat{\alpha }}_1-x_{tj}{\widehat{\beta }}_1\right)}^2\hfill \\ -\frac{1}{2{\widehat{\sigma }}_{1X}^2}{\displaystyle \sum _{t=\tau +1}^k}{\displaystyle \sum _{j=1}^n}{\left(x_{tj}-{\widehat{\mu }}_1\right)}^2+\frac{1}{2{\sigma }_0^2}{\displaystyle \sum _{t=\tau +1}^k}{\displaystyle \sum _{j=1}^n}{\left(y_{tj}-{\alpha }_0-x_{tj}{\beta }_0\right)}^2+\frac{1}{2{\sigma }_{0X}^2}{\displaystyle \sum _{t=\tau +1}^k}{\displaystyle \sum _{j=1}^n}{\left(x_{tj}-{\mu }_0\right)}^2]\hfill \end{array}$$

(12)

To compute the GLR statistic (R_m,k in FSRGLR and VSIGLR, or R_m,k,l in SSGLR), the samples taken from the process are stored until the m^th sample, and then (k > m) the new sample is replaced with the first one. By this procedure, the GLR statistic is the maximum value of the computed statistics based on possible values of τ. In fact, we have R_m,k,τ or R_m,k,l,τ for each possible value of change point and the maximum value among all of the creates R_m,k or R_m,k,l. Hereafter, we only write R_m,k for brevity.

The process terminates if the GLR statistic (R_m,k) plots beyond the upper control limit (UCL); otherwise, a new sample is taken from the process, and monitoring continues. In real industrial processes, some modifications should be done by process engineers to remove the assignable causes. Montgomery [32] proposed a comprehensive procedure called the Out-of-Control-Action Plan (OCAP) that can be used after an out-of-control signal. Interested readers are referred to Montgomery [32] for more details. Note that it is not necessary to consider the lower control limit (LCL) in the GLR control chart [15]. For better illustration, Figure 1 provides a schematic overview of the signaling procedure in the GLR control chart.

In phase II, control charts are usually compared based on the average run length (ARL) criteria when the number of required samples is important. In addition to comparing control charts, the control limits are adjusted based on the in-control value of ARL, denoted by ARL₀. On the other hand, the time of sampling may be more important in some situations in which ATS is used to compare and adjust control charts. Considering the nature of VSI and SS schemes, ATS criteria are selected in this paper. In addition, the standard deviation of time to signal (SDTS) is computed as a secondary criterion. It is expected to reach the lower out-of-control values for both criteria, denoted by ATS₁ and SDTS₁, by applying the VSI and SS schemes instead of FSR.

3.1. FSRGLR Control Chart

For the FSRGLR chart, the sample size and sampling interval are constant, and the sampling rate is defined as $\frac{n_0}{d_0}$. By assuming a sampling interval of 1 (d₀ = 1), it could be inferred that ATS₀ is equal to ARL in FSRGLR. Similar to other control charts in phase II [4,5,6], the UCL of FSRGLR, denoted by h, in this paper it is adjusted based on the predefined value of ATS₀ (ARL₀) in FSRGLR. For each sample, the GLR statistic (R_m,k) is computed by Equation (12), and a signal is triggered when R_m,k > h; otherwise, after a time interval of d₀ which is equal to 1, the next sample with size n₀ is taken. Pseudocode 1 illustrates the procedure of ATS₀ (ARL₀) and SDTS₀ (SDRL₀) computations in the FSRGLR control chart. The comments are indicated in green.

Pseudocode 1. The procedure of ATS0 (ARL0) and SDTS0 (SDRL0) computations in FSRGLR

Initialize n0, h, IterNum, m, and IC model. ATS = []; % ATS = ARL for t = 1:IterNum do  % loop for each iteration  RL = 0; Rm,k = 0;  while (Rm,k < h)   RL = RL + 1;   % in-control data generation   Generate normal error with size n0;   Generate explanatory variables with size n0;   Compute response variables from Equation (1);    % GLR statistic is computed by consideration of a window of m past data   Compute GLR statistic (Rm,k) from Equation (12);  end while  ATS = [ATS; RL]; % equal to ARL = [ARL; RL] end for ATS0 = mean(ATS); SDTS0 = std(ATS); % average and standard deviation of IterNum values

Pseudocode 1 conducts the Monte Carlo simulations by iterating the signaling procedure of the FSRGLR control chart a large number of times (IterNum times). To simplify the procedure, the GLR statistic computations are not shown in Pseudocode 1, but they are computed based on the steps proposed in Figure 1. It is noteworthy to mention that by changing the IC model parameters mentioned in Equation (1), one can easily obtain ATS₁ (ARL₁) and SDTS₁ (SDRL₁).

3.2. VSIGLR Control Chart

Note that there is one additional warning limit in the VSIGLR chart denoted by g [17]. Similar to FSRGLR, VSIGLR utilizes the fixed sample size (n₀), and the out-of-control signal is triggered in case R_m,k > h. The difference is that the sampling interval for the next sample is a function of the chart statistic (R_m,k) in the current sample. There are two possible sampling intervals in such a way that the short sampling interval (d₁) is used for the next sampling if g < R_m,k ≤ h and the next sample is taken with the long sampling interval (d₂) when R_m,k ≤ g. It is obvious that the short (long) sampling interval should be lower (greater) than the FSR sampling rate, i.e., d₂ > d₀ > d₁ or d₂ > 1 > d₁. By the process conditions and adjustments, one can consider proper values for d₁ and d₂.

There is no difference between the computation of h in FSRGLR and VSRGLR control charts (the obtained values for both are identical). The warning limit (g) is determined in such a way that the in-control average sampling rate would be equal to $\frac{n_0}{d_0}$. For this aim, the warning limit (g) is adjusted to have the same number of times for using each sampling interval. If the number of times for using each sampling interval is denoted by w₁ and w₂, it can easily be shown that $d_0=\frac{w_1{d}_1{+w}_2{d}_2}{w_1{+w}_2}=1$ [17,18]. Considering the known value of d₁ and d₂, the warning limit (g) is assigned to reach d₀ equal to 1. By this approach, ATS₀ in VSIGLR would be equal to ARL₀ in FSRGLR. This is discussed in more detail in empirical simulation studies in Section 4. Pseudocode 2 illustrates the procedure of ATS₀ and SDTS₀ computations in the VSIGLR control chart.

The difference between Pseudocodes 1 and 2 is in using different sampling intervals, which leads to different criteria, including ARL and ATS. As mentioned, by adjustment of g, the values of w₁ and w₂ should be obtained to reach $d_0=\frac{w_1{d}_1{+w}_2{d}_2}{w_1{+w}_2}=1$. It is noteworthy to mention that ARL criteria are not usually considered in VSI charts, and it is only computed in Pseudocode 2 for better illustration. Similar to FSRGLR, by changing the IC model parameters mentioned in Equation (1), one can easily obtain ATS₁ and SDTS₁.

Pseudocode 2. The procedure of ATS0 and SDTS0 computations in VSIGLR

Initialize n0, h, g, d1, d2, IterNum, m and IC model. ATS = []; ARL = []; w1 = w2 = 0; for t = 1:IterNum do  % loop for each iteration  RL = 0; Rm,k = 0;  dd1 = dd2 = 0; % variables for sampling interval time summation  while (Rm,k < h)   RL = RL + 1;   % in-control data generation   Generate normal error with size n0;   Generate explanatory variables with size n0;   Compute response variables from Equation (1);   % GLR statistic is computed by consideration of a window of m past data   Compute GLR statistic (Rm,k) from Equation (12);   if (Rm,k < g)    dd2 = d2 + dd2; w2 = 1 + w2; % take sample with long sampling interval   else    dd1 = d1 + dd1; w1 = 1 + w1; % take sample with short sampling interval   end if   end while  ATS = [ATS; (dd1+dd2)]; ARL = [ARL; RL]; end for ATS0 = mean(ATS); SDTS0 = std(ATS); % average of IterNum values ARL0 = mean(ARL); SDRL0 = std(ARL); % average of IterNum values

3.3. SSGLR Control Chart

The number of samples in each profile is not constant in the SS approach, so n_t is used instead of n (t = 1, 2, …, k − 1). In the current time (k^th profile), the number of samples, which is a decision variable, is shown with l (l > 2). Similar to VSIGLR, SSGLR has two limits (i.e., the control and warning limits) denoted by h and g, respectively.

To determine the value of l, the statistic of the SSGLR chart (R_m,k,l) is computed, and if it is lower than g, the sampling is stopped at the current profile as it is an in-control situation and n_t = n_k is set at l. On the other hand, the out-of-control signal is triggered when R_m,k,l > h, similar to FSRGLR and VSIGLR. When the chart statistic plots between warning and control limit (g < R_m,k,l ≤ h), the sampling procedure is iterated at the current profile (k^th profile) by adjustment l = l + 1.

The value of h is not equivalent to the obtained limit in FSRGLR and VSIGLR. To avoid arbitrary design for h and g, we defined three criteria for designing the SSGLR procedure. First, SSGLR should reach the predefined value of ATS₀. The average number of inspections (ANI) is defined as the expected number of observations taken from the start of the process to the signal time. Based on this criteria, the average sample number (ASN) is computed as $\frac{\mathit{ANI}}{\mathit{ARL}}$ [32,44]. Since the sample size at each sampling point is a random variable in the SSGLR chart, the in-control expected sample size (ASN₀) should be set to n₀ to achieve the same in-control average sampling rate as the FSRGLR and VSIGLR charts; however, several combinations for h and g could be found with these two criteria. Hence, reaching SDTS₀ equal to ATS₀ is the third criterion. Montgomery [32] stated that the ARL in the Shewhart chart followed the geometric distribution in which the mean and standard deviation are very close. Considering this idea, it is expected to reach nearly similar values for ATS₀ and SDTS₀ after the designing procedure. Pseudocode 3 illustrates the procedure of ATS₀ and SDTS₀ computations in the SSGLR control chart.

Pseudocode 3. The procedure of ATS0 and SDTS0 computations in SSGLR

Initialize n0, h, g, IterNum, m and IC model. ATS = []; ANI = []; for t = 1:IterNum do % loop for each iteration  TS = 0; Rm,k,l = 0; OC = 0;  Nt = []; % variable for storage of the number of inspected samples in each iteration   while (OC == 0) % first while for generation of new profiles   TS = TS + 1; NewSample = 0; StartSample = 0;   while (NewSample == 0) % second while for generation of new observation in each profile     if (StartSample == 0)     % in-control data generation with size n0 − 1     Generate normal error with size n0 − 1;     Generate explanatory variables with size n0 − 1;     Compute response variables from Equation (1);      l = no − 1;    else     Generate normal error with size 1;     Generate explanatory variables with size 1;     Compute response variables from Equation (1);     l = l + 1;    end if    % GLR statistic is computed by consideration of a window of m past data    Compute GLR statistic (Rm,k,l) from Equation (11);    StartSample = 1;    if (Rm,k,l < g)     % the procedure is terminated with in-control condition     NewSample = 1; Nt = [Nt; l];    elseif (Rm,k,l < h)     % it is not possible to decide about the process, so we take one more sample      NewSample = 0;    else % Rm,k,l > h     % the procedure is terminated with out-of-control condition     NewSample = 1; Nt = [Nt; l];     OC = 1; % exit from the first while     end if   end while % end first while    end while % end second while   % Nt is a vector of TS values  ATS = [ATS; TS]; ANI = [ANI; sum(Nt)]; end for ATS0 = mean(ATS); SDTS0 = std(ATS); % average of IterNum values ANI0 = mean(NI); SDNI0 = std(ANI); % average of IterNum values

Based on Pseudocode 3 the minimum sample size is considered as n₀ − 1 for the SSGLR. In each iteration of Pseudocode 3, the number of needed samples for a signal and the sample size in each sample are shown with TS and Nt, respectively. It is obvious that Nt is a vector of TS values in each iteration in such a way that the summation of Nt is equivalent to the number of samples taken in that iteration. As there are different sample sizes in comparison with FSRGLR and VSIGLR, the same warning and control limits may not be acquired in SSGLR. In the same manner as Pseudocodes 1 and 2, by changing the IC model parameters mentioned in Equation (1), one can easily obtain ATS₁ and SDTS₁.

4. Performance Comparison

This section provides several simulation studies to show the performance of the proposed methods. The first subsection describes the details of in-control parameters. The next subsection is about the designing procedure of each method under the in-control model. Section 4.3 gathers the ATS results of each model based on different shifts in the parameters. A comparison between GLR and other control charts is described in Section 4.4. Section 4.5 investigates the computational time of the proposed methods, and finally, the evaluation of the SS approach under fixed explanatory variables is illustrated in Section 4.6.

4.1. The Details of the In-Control Model

The in-control model was extracted from Kang and Albin [4], Kim, Mahmoud [5], Zou, Tsung [6], and Xu, Wang [15], in which it was assumed that ${\alpha }_0=3$, ${\beta }_0=2$ and ${\sigma }_0^2=1$. As these references considered fixed explanatory variables, we assumed standard normal distribution for the explanatory variable in the in-control condition (${\mu }_{X0}$= 0 and ${\sigma }_{X0}^2$= 1). Following Hafez Darbani and Shadman [17], other parameters entailing the sample size (n₀), sampling rate (d₀), short sampling interval (d₁), and long sampling interval (d₂) were adjusted at 4, 1, 0.1, and 1.9, respectively. The window size (m) was set at 400 based on the suggestion of Xu, Wang [15].

4.2. Designing of Each Control Chart under the In-Control Model

Considering 10,000 simulations, the value of h was obtained as 8.355 to reach ATS₀ equal to 200 for the FSRGLR and VSIGLR charts. To further discuss the performance of VSIGLR under different adjustments, some simulations under different values of d₁ and d₂ (d₂ > d₀ > d₁) were conducted. The results are presented in

Table 1. The adjustments of VSIGLR chart based on the ATS₀ = 200 and h = 8.355.

Scenario Number	g	d1	d2	In-Control Simulations			Out-of-Control Simulations
				w1	w2	d0	w1	w2	d0
1	5.91	0.01	1.1	1,767,090	156,100	1.00	428,520	118,230	0.86
2	3.599	0.1	1.9	1,044,732	969,942	1.02	175,800	323,400	0.73
3	2.8	0.33	3	505,104	1,431,502	1.03	104,953	430,942	0.85
4	2.15	0.55	5	216,365	1,965,601	0.99	54,638	493,019	0.99

, in which four designing scenarios are provided based on the h = 8.355 and ATS₀ = 200 in such a way that the warning limit was decreased in each designing scenario.

We simulated 10,000 in-control and out-of-control (with shift magnitude 0.2) profiles and then computed the number of times each interval had been selected (i.e., w₁ and w₂). Then, the average time for taking all samples was computed as $d_0=\frac{w_1{d}_1{+w}_2{d}_2}{w_1{+w}_2}$ for each situation. As the value of d₀ is assumed equal to 1 in the in-control condition, the warning limit (g) was assigned based on reaching d₀ = 1. The lowest d₀ in the out-of-control profiles is the best scenario. The reason is that lower ATS₁ values under the selected shift considering desired ATS₀ would be obtained. So, the highlighted situation in which w₁ $\approx$ w₂ was the best, and we selected it for the next simulations.

As the run length is a function of sample size in the SSGLR, the control limit (h) is not identical to FSRGLR and VSIGLR. Considering the three criteria, including ATS₀ = SDTS₀ = 200 and ANI₀ = n₀ × 200 = 800, the h and g were set at 8.74 and 4.24, respectively, in SSGLR (with m = 400).

4.3. Performance Evaluation in Term of ATS Criteria

To compare different methods, single-step shifts occurred at the initial time of the simulations to compute zero-state ATS. Considering m = 400 and similar adjustments for all three methods, five different shift types were employed in the in-control model. Three shifts were applied to the profile parameters; is, the new magnitudes of the intercept, slope, and standard deviation are equal to (${\alpha }_0{+\delta \sigma }_0$), (${\beta }_0{+\lambda \sigma }_0$) and (${\gamma \sigma }_0$), respectively. Two others were related to the explanatory variable in which the mean and standard deviation changed to (${\mu }_{X0}{+\phi \sigma }_{0X}$) and (${\kappa \sigma }_{0X}$).

Table 2. The ATS₁ values for the FSRGLR, VSIGLR, and SSGLR control charts under the individual shifts (m = 400 and ATS₀ = 200).

Control chart	δ
	0.10	0.20	0.30	0.40	0.50	0.60	0.80	1.00	1.20	2.00
FSRGLR	119.70	53.90	29.19	18.18	12.65	9.36	5.88	4.08	2.18	1.48
VSIGLR	96.10	39.99	21.94	13.97	9.79	7.19	4.47	3.09	1.62	1.14
SSGLR	85.04	35.41	20.38	11.88	8.69	6.61	4.13	2.95	1.73	1.21
Control chart	λ
	0.10	0.20	0.30	0.40	0.50	0.60	0.80	1.00	1.20	2.00
FSRGLR	120.92	54.01	29.41	18.31	12.92	9.64	6.04	4.29	2.46	1.75
VSIGLR	96.85	40.50	22.62	14.63	10.25	7.62	4.99	3.56	2.08	1.54
SSGLR	89.94	36.17	19.32	13.61	8.64	6.59	4.38	3.24	2.12	1.57
Control chart	γ
	1.10	1.15	1.20	1.25	1.30	1.40	1.60	1.80	2.20	2.60
FSRGLR	68.15	41.80	28.41	20.74	16.03	10.41	5.76	3.91	2.37	1.80
VSIGLR	51.12	31.41	21.09	15.60	11.87	8.04	4.62	3.20	2.03	1.58
SSGLR	46.75	30.44	21.07	14.48	11.46	6.67	4.24	2.84	2.07	1.66
Control chart	ϕ
	0.10	0.20	0.30	0.40	0.50	0.60	0.80	1.00	1.20	2.00
FSRGLR	120.32	53.33	28.81	18.14	12.66	9.24	5.84	4.07	2.19	1.47
VSIGLR	94.91	39.69	21.79	13.82	9.69	7.22	4.53	3.11	1.62	1.14
SSGLR	91.08	35.83	19.60	11.30	9.04	6.78	4.04	2.96	1.72	1.22
Control chart	κ
	1.10	1.15	1.20	1.25	1.30	1.40	1.60	1.80	2.20	2.60
FSRGLR	77.47	44.34	29.33	21.77	16.52	10.78	6.03	4.01	2.47	1.84
VSIGLR	53.66	32.66	22.05	16.36	12.76	8.51	4.88	3.34	2.12	1.64
SSGLR	39.72	25.91	16.36	12.11	9.33	6.95	3.98	2.89	2.11	1.76

reports the ATS₁ values under these five types of individual shifts for the FSRGLR, VSIGLR, and SSGLR control charts. The best result in each comparison is in boldface.

It can be inferred from Table 2 that the proposed SSGLR chart outperformed the VSIGLR and FSRGLR in nearly all the shifts based on ATS criteria except the last two largest shifts in which VSIGLR was the best technique. In the shifts in the standard deviation of the explanatory variable (the magnitude of the shifts is shown with ${\kappa \sigma }_{0X}$), more differences in the results were observed; for example, in κ = 1.1, the ATS₁ values were obtained as 39.72, 53.66, and 77.47 for SSGLR, VSIGLR, and FSRGLR, respectively. As another finding, it could be said that VSIGLR were completely able to reach lower ATS₁ than FSRGLR in all shifts in such a way that the magnitudes of differences were greater in the case of small shifts. As a novel future research direction, one can implement profile diagnosis methods for the profiles with random explanatory variables. Previous works such as Zou, Tsung [6], Zou, Tsung [45], Huwang, Wang [46], and Yeganeh and Shadman [47] were only able to identify the out-of-control profile parameters, but they were not able to detect the shifts in the explanatory variable.

In another study, the simultaneous shifts in the parameters were applied to the in-control model. There are several combinations for the simultaneous shifts, five of which are reported in

Table 3. The ATS₁ values for the FSRGLR, VSIGLR, and SSGLR control charts under the simultaneous shifts (m = 400 and ATS₀ = 200).

Control Chart	FSRGLR	VSIGLR	SSGLR	FSRGLR	VSIGLR	SSGLR	FSRGLR	VSIGLR	SSGLR
λ	δ
	0.10			0.20			0.40
0.02	112.52	89.80	88.03	52.42	39.53	35.74	18.38	12.62	12.63
0.08	98.31	74.58	67.23	47.53	38.25	30.51	16.53	12.38	12.02
γ	δ
	0.10			0.20			0.40
1.05	81.63	69.49	61.69	47.61	33.63	30.59	17.01	11.38	12.02
1.20	27.54	18.15	16.91	18.35	15.63	15.00	12.97	8.43	8.69
γ	λ
	0.05			0.10			0.40
1.05	100.42	87.83	82.63	80.42	58.32	61.55	18.45	12.43	11.63
1.20	26.74	21.74	18.42	26.30	19.15	16.04	13.52	8.05	8.59
ϕ	λ
	0.05			0.10			0.40
0.02	159.38	153.52	161.63	122.53	89.12	83.04	18.16	13.66	11.95
0.10	115.73	84.85	77.92	79.20	62.69	56.88	17.14	13.05	11.46
ϕ	γ
	1.05			1.10			1.20
0.02	117.39	100.04	101.75	71.24	44.62	47.35	29.17	18.62	18.21
0.10	80.39	64.51	61.94	51.75	39.75	38.79	25.19	22.71	17.96

. The results indicated that FSRGLR had the worst performance in all the shifts. Having compared VSIGLR and SSGLR, it is observed that SSGLR outperformed VSIGLR in most of the shifts, especially in small and moderate shifts. In some moderate and large shifts, VSIGLR was able to acquire the lowest ATS₁ values; as one example, ATS₁ values in the shift with δ = 0.4 and γ = 1.05 were 11.38 and 12.02 for VSIGLR and SSGLR, respectively. In one exception in small shifts, VSIGLR was the best in λ = 0.05 and φ = 0.02; however, SSGLR generally outperformed the other two competitors.

To better address the performance of each approach, ANI was also obtained for the SSGLR method. Note that ANI can easily be computed for FSR and VSI by multiplying the ATS values with n. For example, in λ = 0.2 (second part of Table 2), the ANI values were obtained as 162 (40.50 × 4) and 216.04 (54.01 × 4) for VSIGLR and FSRGLR, respectively. Based on Pseudocode 3, it was computed as 215.21 for the SSGLR control chart. With fewer required samples, VSIGLR had an obvious superior performance over the other two competitors. Similar results were also obtained for other shifts, but they are not reported for the sake of brevity.

4.4. Comparing Adaptive Approaches under T² Control Chart

To illustrate clearly the effect of adaptive approaches, they are applied here for the well-known T² control chart under the same in-control model with the parameters mentioned in Section 4.1. Implementation of the T² control chart under the random explanatory variables was defined in Noorossana, Fatemi [41] for FSR. We follow the same approach, but the formulas are not given for the sake of brevity. As the T² statistic follows chi-square distribution, the control limit of FSRT2 and VSIT2 would be χ²_(2,0.995), which is equal to 10.60 (type I error = α =$\frac{1}{{\mathit{ATS}}_0}$= 0.995). In the VSIT2, other parameters entailing the warning limit (g), sampling rate (d₀), short sampling interval (d₁) and long sampling interval (d₂) were adjusted at 1.39, 1, 0.1 and 1.9, respectively. For the SST2, the control limits did not follow the exact distribution and were computed by simulation as h = 12.97 and g = 4.49. For brevity, two individual shifts in the intercept and slope were added to the in-control model for comparison purposes. Figure 2 depicts the ATS₁ values when there are shifts in the intercept (left panels) and slope (right panels) of the in-control model.

Figure 2a,b compared the performance of the T² control chart under FSR, VSI, and SS schemes when there were shifts in the intercept (a) and slope (b). From the latter, it is expected that SST2 and VSIT2 will generate better results than the FSRT2 scheme. In addition, lower ATS₁ values are observed in SST2 than VSIT2 for small and moderate shifts, while the performance is nearly similar in large shifts. In the next two panels (Figure 2c,d), the VSIGLR was compared with VSIT2. Note that the results of VSIGLR were extracted from Table 2, and the VSIT2 results are identical to previous panels. It can be observed that VSIGLR is superior to VSIT2 in the small and moderate shifts, while the performance of VSIT2 is slightly better in the large shifts. For example, the ATS₁ values are 96.10 and 176.23 for VSIGLR and VSIT2 in case δ = 0.1 (intercept shift). On the other hand, these are 3.56 and 2.47 in the largest slope shift, i.e., λ = 1. Similar conclusions could be derived for the SS control charts in Figure 2e,f. Generally, SSGLR has a better detection ability than SST2 in small and moderate shifts; however, SST2 is able to reach slightly lower ATS₁ values for larger shifts. These findings could be justified due to the superiority of GLR over the T² control chart in the FSR condition reported in some references such as Xu, Wang [15], Hafez Darbani and Shadman [17], and Yao, Li [19].

4.5. Computational Time of Different Adaptive Approaches

The computational time of the three proposed methods in this paper is investigated in this subsection. Under the suggested in-control model in Section 4.1, the time of simulations for the shifts in the intercept was analyzed. As the time of computations is related to the ATS, it could be said that the same results will be obtained for the shifts with similar ATS values. Hence, we did not conduct the simulations for other shift types. The average time per one run of simulations (from a total of 10,000 runs) is gathered in

Table 4. The average time of simulations for each run of the FSRGLR, VSIGLR, and SSGLR control charts (m = 400 and ATS₀ = 200).

Control Chart	δ
	0.00	0.10	0.20	0.30	0.40	0.50	0.60	0.80	1.00
FSRGLR	0.4053	0.1340	0.0197	0.0051	0.0022	0.0012	0.0007	0.0004	0.0002
VSIGLR	0.4193	0.1053	0.0183	0.0048	0.0022	0.0010	0.0006	0.0003	0.0002
SSGLR	1.9042	0.0493	0.0473	0.0144	0.0066	0.0038	0.0023	0.0014	0.0010

(based on seconds). The simulations were done by MATLAB program on an AMD RYZEN7 5000 series with a 16.0 GB RAM computer.

In the in-control condition, the FSRGLR was the quickest method with a requirement of 0.4053 s for each run, and as the second method, VSIGLR was slightly slower with a 0.4193 s requirement. The difference between FSRGLR and VSIGLR computation time is related to additional comparing of statistics with the warning limit. The needed time of SSGLR (m = 400) in the in-control condition was much larger than the two other competitors in such a way that each run lasted 1.9042 s which was nearly five times greater than the two other methods. On the other hand, for the out-of-control shifts, VSIGLR is better than FSRGLR as it has lower ATS₁ values (see Table 2). The interesting finding here is that although SSGLR has the lowest ATS₁ values, it needed the most computational times in all shifts. It could be viewed as a negative aspect of this approach; however, this may not be very challenging nowadays due to the rapid extension of computer hardware.

As mentioned, we utilized the GLR control chart with m = 400. By reducing the value of m, the computational time of all methods will obviously be decreased in small shifts, but the performance may be changed. It could generally be expected that the ATS₁ results are decreased by increasing the value of m in small shifts due to feeding more information to the chart. Similar conclusions could be found in Hafez Darbani and Shadman [17]. We tested different values of m for the SSGLR, but it was decided to neglect reporting the results for two reasons. First, we were not able to reach a general rule for the performances since the ATS₁ values were not totally decreased by increasing m. Secondly, the aim of this study is not related to the reduction of computational times. To remedy this challenge, the development of a closed-form formula for the computation of the GLR statistic has also been suggested [16]. Reduction of the computational time of the SSGLR in different profile types is a proper potential future research direction.

4.6. Performance of SSGLR under the Fixed Explanatory Variables

In the previous simulations, the comparisons were reported based on random explanatory variables. It could be generally said that the nature of the SS approach dictates the random explanatory variables, as the next sample taken is not usually fixed. For this aim, we selected the random variables in this paper. However, this subsection provides some performance comparisons under the fixed explanatory variables. We followed the proposed GLR control chart and setups as done in Hafez Darbani and Shadman [17], in which the in-control model was the same as Kang and Albin [4], indeed the fixed explanatory variables were defined as 2:2:8. The GLR statistic based on the FSR and VSI approaches were extended in Hafez Darbani and Shadman [17], and we utilized their results. On the other hand, the combination of GLR and SS was proposed by Xu and Peng [16], but the setups were different as compared to the in-control model used in this paper. Therefore, we extended the GLR statistics based on our proposed in-control model (${\alpha }_0=3$, ${\beta }_0=2$ and ${\sigma }_0^2=1$). Based on the procedure in Xu, Peng [16], the first sample size was defined as equal to 3 (the values were 2, 4, and 6), and then, one sample was taken with the value 8. This sequence was repeated to reach an out-of-control signal. The developed method denoted by SSGLRF (F is due to fixed explanatory variables) was compared with the proposed control charts in Hafez Darbani and Shadman [17] denoted by FSRGLRF and VSIGLRF under the assumption of m = 400 and ATS₀ = 200. The control and warning limits were obtained as h = 6.764 and g = 2.576 (see Table 1 in Hafez Darbani and Shadman [17]). In addition, h and g were set at 7.84 and 2.03, respectively, for SSGLR.

Table 5. The ATS₁ values for the FSRGLRF, VSIGLRF, and SSGLRF control charts under the shifts in intercept, slope, and standard deviations when there are fixed explanatory variables (m = 400 and ATS₀ = 200).

Control chart	δ
	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80	0.90	1.00
FSRGLRF	110.00	47.20	25.50	16.00	11.10	8.30	6.27	5.20	4.28	3.60
VSIGLRF	89.10	36.50	19.60	12.30	8.60	6.40	5.18	3.90	4.01	2.70
SSGLRF	54.93	22.41	13.18	9.59	7.48	5.18	4.69	3.78	3.07	2.70
Control chart	λ
	0.025	0.0375	0.050	0.0625	0.075	0.100	0.125	0.150	0.200	0.250
FSRGLRF	78.40	45.40	29.40	20.70	15.30	9.60	6.70	5.00	3.10	2.30
VSIGLRF	61.20	34.70	22.60	16.25	11.97	7.48	5.24	3.87	2.43	1.69
SSGLRF	44.92	29.02	19.03	13.87	11.06	7.51	5.51	4.19	2.80	2.15
Control chart	γ
	1.10	1.15	1.20	1.25	1.30	1.40	1.60	1.80	2.20	2.60
FSRGLRF	57.80	35.50	24.20	17.80	13.60	9.00	5.10	3.50	2.20	1.70
VSIGLRF	44.9	26.8	18.4	13.83	10.81	7.16	4.10	2.93	1.90	1.50
SSGLRF	29.02	21.84	13.32	10.01	9.00	5.99	3.93	2.49	1.79	1.39

reports the ATS₁ values for the shifts in intercept, slope, and standard deviation when the explanatory variables are fixed.

In most of the shifts in Table 5, SSGLRF was the best method in all the shifts except for the intercept shifts with a range of λ > 0.075, in which VSIGLRF was the best. These results prove that the SS approach can perform well not only with random explanatory variables but also with fixed values when m = 400. However, its performance changes with other values of m (especially the lower values). The reason behind this finding is that the fixed samples are not able to provide new information from the process and only iterate some cycles during the monitoring procedure when there are lower m values. The simulations’ results for other values of m can be given to interested readers upon request.

5. Illustrative Example

Several examples and applications have been developed for the control charts in the literature. They are not only limited to industrial applications but are applicable to different areas, such as monitoring product comments in online shops [48], hydrological parameters [49], and so forth. Eventually, control charts were expanded for the aim of sustainable development [50], which is a very important issue [51,52].

Considering different applications, the aim of this section is to show the practical application of the proposed method. The example provided in Noorossana, Fatemi [41] was used to illustrate the implementation of the SSGLR control chart with FSR only. This example is related to the adhesive manufacturing industry, which is one of the most important products in different fields, such as chemical, industrial, and mechanical engineering. The adhesives market is classified into water-based adhesives, solvent-based adhesives, hot-melt adhesives, and others. On the basis of the end-user industry, the market is categorized into the packaging industry, construction industry, automotive industry, electrical & electronics, and others [53].

In these applications, tape thickness is a critical quality characteristic since the materials with a low thickness (in case locating of measurements below the lower specification limit (LSL)) may lead to customer complaints, whereas the production cost is augmented by thickness higher than the upper specification limit (USL). Therefore, monitoring tape thickness can bring several benefits for the related factories. It is noteworthy to mention that LSL and USL are acquired based on the production line requirements [32].

In the monitoring procedure, the operator measures the thickness of the tape at some randomly selected locations. Noorossana, Fatemi [41] assumed that there is a linear relationship between the tape thickness and randomly selected locations. They also deduced that the in-control profile is defined as:

$$\begin{gathered} y_{\mathit{tj}}{=x}_{\mathit{tj}}-{40+\epsilon }_{\mathit{tj}}, \\ t=1, 2, \dots , \\ {j=1, \dots , n}_t, \end{gathered}$$

(13)

where x_tj has a normal distribution with a mean of 80 and a standard deviation of 3; the error term is generated from the standard normal distribution. Since the explanatory variable is selected randomly, the SSGLR chart proposed in this paper can be used to monitor the process. Considering these settings, the values of h and g were set at 8.74 and 4.23, respectively, based on the ATS₀ and ASN₀ equal to 200 and 4 (n0).

The first five profiles are generated under the in-control condition (i.e., change point is 5) and after that, an artificial shift with magnitude 1 was added to the intercept (A₀ to A₀ + σ₀). The procedure started with l = n₀ − 1 = 3. The simulated profiles and the final sample size in each run are reported in

Table 6. The response and explanatory variables of the generated profiles in the illustrative example (the first five profiles are in-control).

Run Length (t)	nt	j											Variable
		1	2	3	4	5	6	7	8	9	10	11
1	3	41.57	44.71	35.45	-	-	-	-	-	-	-	-	ytj
		80.40	85.06	75.38	-	-	-	-	-	-	-	-	xtj
2	3	48.05	39.64	39.15	-	-	-	-	-	-	-	-	ytj
		88.54	79.82	78.82	-	-	-	-	-	-	-	-	xtj
3	5	51.12	41.29	41.00	40.59	36.16	-	-	-	-	-	-	ytj
		90.61	80.21	79.66	80.51	77.00	-	-	-	-	-	-	xtj
4	11	31.60	41.12	40.19	38.97	37.30	41.65	43.40	39.02	40.91	42.40	38.61	ytj
		70.35	81.60	78.38	79.38	76.91	80.89	84.07	79.38	81.83	81.71	78.36	xtj
5	3	41.72	40.15	39.49	-	-	-	-	-	-	-	-	ytj
		82.96	79.88	80.40	-	-	-	-	-	-	-	-	xtj
6	3	39.04	34.13	38.42	-	-	-	-	-	-	-	-	ytj
		78.67	74.53	79.47	-	-	-	-	-	-	-	-	xtj
7	9	41.89	44.91	40.18	39.00	36.78	43.97	42.72	44.62	33.14	-	-	ytj
		80.61	84.49	77.40	77.94	74.80	83.10	81.71	83.23	71.38	-	-	xtj

. The final sample size to reach a decision was indicated in the second column. It is obvious that SSGLR triggered an out-of-control signal at the seventh generated profile, i.e., it only needed two samples after the change point for out-of-control detection.

To better illustrate the SSGLR procedure, the values of GLR statistics in each run are shown in

Table 7. The values of GLR statistics in each run of simulation (yellow, green, and red cells are indicators of a requirement for a new sample, in-control, and out-control situations, respectively).

Run Length (t)	l − 2	Rm,k,l,t							Rm,k,l
1	1	0.21	-	-	-	-	-	-	0.21
2	1	1.79	1.53	-	-	-	-	-	1.79
3	1	5.36	5.34	4.71	-	-	-	-	5.36
3	2	4.85	4.73	3.73	-	-	-	-	4.85
3	3	3.96	3.71	2.56	-	-	-	-	3.96
4	1	6.46	6.15	5.26	3.45	-	-	-	6.46
4	2	5.74	5.38	4.38	3.02	-	-	-	5.74
4	3	5.63	5.25	4.34	3.58	-	-	-	5.63
4	4	5.42	5.02	4.14	3.13	-	-	-	5.42
4	5	5.48	5.07	3.96	2.85	-	-	-	5.48
4	6	4.91	4.48	3.36	2.48	-	-	-	4.91
4	7	4.51	4.08	2.88	2.32	-	-	-	4.51
4	8	4.38	3.93	2.75	1.95	-	-	-	4.38
4	9	4.10	3.65	2.53	1.93	-	-	-	4.10
5	1	3.33	2.92	1.77	2.01	1.17	-	-	3.33
6	1	2.74	2.37	1.41	1.83	1.08	1.19	-	2.74
7	1	3.52	3.08	2.16	2.07	0.57	1.20	4.72	4.72
7	2	3.66	3.23	2.39	2.49	0.90	1.75	4.65	4.65
7	3	5.01	4.59	3.82	4.27	2.66	3.76	6.81	6.81
7	4	5.05	4.62	3.83	3.86	2.30	3.53	7.01	7.01
7	5	5.08	4.65	3.89	3.78	2.38	3.73	7.44	7.44
7	6	5.30	4.86	4.11	3.62	2.61	4.30	8.33	8.33
7	7	7.82	7.40	6.63	6.14	4.83	7.15	10.84	10.84

. The decision about the process at each point is shown in the last column, with yellow, green, and red indicating the new sample, in-control, and out-of-control situations, respectively. The first column is the run length indicator, while the second column shows the procedure for the determination of the number of needed samples (we labeled it with l − 2). The other columns are the obtained statistics based on the GLR loops and, finally, the last column (R_m,k,l) is the maximum value of GLR statistics considering the previous samples, i.e., maximum of R_m,k,l,τ. For more details, see the proposed flowchart in Figure 1.

The procedure started with a sample size of three. In the first and second samples, SSGLR directly detected the process as in-control because the GLR statistics were lower than g = 4.23 (0.21 < 4.23 and 1.79 < 4.23). In the third run, the GLR statistic with sample size 3 was 5.36 > 4.23, so it was necessary to take another sample. Similarly, it was necessary to take another sample again, and the final sample size was acquired as five. Finally, the process was identified as out-of-control at the seventh run, with n₇ = 7 in such a way that the chart statistic was 10.84 > 8.74.

6. Concluding Remarks

In this paper, linear profiles are monitored using the GLR control chart under three adjustments (i.e., FSR, VSI, and SS approaches) with consideration of the random explanatory variables. To efficiently monitor, a novel framework with a combination of GLR and SS (i.e., SSGLR) approach was introduced when there is an IC simple linear profile model. The results showed that the SSGLR chart was able to reach the best detection ability in terms of ATS criteria in most of the shifts when compared to VSIGLR and FSRGLR charts. On the other hand, a comparison between FSRGLR and VSIGLR indicated that VSIGLR was completely superior to FSRGLR. Implementation of the proposed adaptive schemes under random explanatory variables was also developed on the T² control chart, and the results indicated a positive effect of adaptive schemes with the control chart. In another study, the performance of SSGLR under the fixed explanatory variables was evaluated, and the results showed that this approach was not optimal when the explanatory variables were fixed. Although FSRGLR was the best technique, it requires the most computational time in comparison with the other two competitors.

For future research, three directions are suggested here: (i) implementation of profile diagnosis techniques when there is a random explanatory variable, (ii) computation of steady-state ARL, and finally, (iii) reducing the computational time of adaptive GLR control charts especially SSGLR with random explanatory variables under different profile type.