On designing a sequential based EWMA structure for efficient process monitoring

Control chart is a popular technique that is widely used in statistical process control to identify any possible deviations from a stable state of a process. Shewhart charts are famous for identifying larger shifts, while cumulative sum and exponentially weighted moving average control charts are well known for identifying smaller shifts in process parameters. This study examines the performance of a sequential-based EWMA (namely ) chart for observing the location of a normally distributed process. The performance of is oberved by using several overall run length properties (like average, median and standard deviation). The comparative study reveals that the overall performance of the proposed design is better than the existing counterparts. Moreover, the superiority depends on the choice of the design parameters of the proposed chart. A real-life data set from a steel rod manufacturing industry were considered to show the real-life illustration of the proposed design.


Introduction
Statistical process control (SPC) is a mechanism that helps in monitoring the stability of an ongoing process to produce quality products. It comprises extremely powerful tools such as pareto chart, cause and effect diagram, fishbone diagram, flow diagram and control chart (cf. Montgomery [1]). SPC is vital for the reputation of a manufacturing industry that relies on the quality of products in a global market competition. Within the SPC toolkit, control charts are the most important tools used for observing and detecting the special cause variations and ultimately enhancing the performance of the product (cf. Shewhart [2]). In order to monitor any ongoing process, the use of apropriate control charting techniques is essential to ensure the stability of the process and to provide several benefits such as reduction in cost and time and increase in efficiency of the industrial process. Shewhart charts are well known for the detection of large shifts in process parameters while exponentially weighted moving average (EWMA) (cf. Roberts [3]) and cumulative sum (CUSUM) charts (cf. Page [4]) are mostly used for the detection of small to moderate shifts. In this study, we introduce an efficient sequential-based EWMA chart which is more sensitive to shifts in process location. Several authors worked to enhance the performance of classical EWMA chart (cf. Yang et al. [5]). The aim of this article is to design an EWMA chart based on a sequential testing algorithm to enhance the performance of the traditional EWMA chart and to increase its efficiency over the classical setup.
In the existing literature, there are mainly two forms of charts where sequential process monitoring is used. That is, some SPC practitioners adopt sequential sampling in their study, while others implement SPRT control charts for online monitoring. Both approaches have an advantageous mechanism for quality control charts for the monitoring of process parameters. According to Matheson [6], the probability of identifying the shift using sequential sampling is more than or equivalent to the likelihood of identifying the shift using a random sampling technique in-control charts as it needs fewer number of samples at a smaller cost. Green [7] highlighted that there is little advantage to sequential sampling if the cost of sample collection is similar to the cost of sample analysis. That is, if sequential sampling is to be more economical than many used approaches, less time should be needed to take a sample in the field than to analyse it in the laboratory. Sequential sampling is appropriate when the time required to gather at least one observation is a very small interval. In this way, it is likely to take numerous samples successively at a sampling point. A useful discussion in this context may be seen in Green [8] and the references therein. As a natural extension of the ideas from sequential analysis, sequential sampling methods have been widely used in the area of acceptance sampling but not so much in process monitoring (cf. Stoumbos and Reynolds [9,10]).
The sequential probability ratio test (SPRT) chart is useful where testing is very costly or difficult. Peng and Reynolds [11] applied sequential sampling using gen-eralized likelihood ratio (GLR) charts. Steiner at el. [12] designed the exact expression for the Wald-SPRT and CUSUM design. Various authors also worked on single directional process monitoring using sequential sampling (SS). Reynolds [13][14][15] and Reynolds et al. [16,17] designed sequential sampling-based control charts which were very effective for detecting changes in a single direction in the process, with some tuning parameters specified. Reynolds and Arnold [18] constructed a one-sided EWMA design for VSI following a sequential sampling algorithm. Zhang et al. [19] designed an EWMA chart under a sequential sampling scheme for single directional process monitoring, and this design was more sensitive to single direction shift detection in the process. Stoumbos and Reynolds [20] designed SPRT chart for process location which was more efficient and highly effective with administrative benefits. Ou at el. [21] designed an SPRT location chart using several sampling intervals. Ou at el. [22] further enhanced the performance of SPRT location chart. Xu and Jeske [23] used SPRT technique to deal with the INAR model. Godase and Mahadik [24] introduced an SPRT dispersion chart.
Keeping in view of several dynamic aspects of SPRT and sequential sampling, we intend to design a sequential algorithm-based EWMA chart, named SEWMA Ò . This scheme uses two pairs of control limits as follows in SPRT. The novelty of this article is to propose a more flexible EWMA control chart for monitoring the process location by employing an SPRT decision algorithm. The rest of this paper is organized as follows: Section 2 provides the design structure of the proposed sequential EWMA chart. Section 3 demonstrates procedural flow for the computation of run length measures. The performance evaluation and comparative analysis are provided in Section 4. A real-life application concerning the monitoring of breaking strength of steel rods is presented in Section 5. Finally, Section 6 lists the main finding and recommendations of the study.

Design structure of the proposed SEWMA Ò chart
This section provides the design structure of the proposed chart. For the rest of the current article, we will denote the proposed chart as a SEWMA Ò chart. In the SPRT technique, observations are taken in groups of one or more observations. If the single observation falls in an indecision zone then another observation is taken until a decision is achieved. For the sake of simplicity, we have used individual observations for this study; however, subgroups of different sizes may also be considered on similar lines (cf. Wald [25]).
Let X 1 , X 2 , X 3 . . . . . . . . . ..,X n be a set of independent individual observations, selected from a specific probability distribution (say normal distribution). An individual observation may be classified into three nonoverlapping regions categorized as Region I: the acceptance region (where the process is considered to be in-control); Region II: the resampling region (where the process needs to be resampled); Region III: the rejection region (where the process is considered to be out-of-control).
For a single observation, let α be the probability of falling in the rejection region and γ be the probability of falling in the indecision region. Then under a resampling setup, the respective probabilities for the three regions are given as shown in Figure 1.
Let us denote R s as the resampling region. Now based on observed sample values X js we may define a plotting statistic, say Z j , of the SEWMA Ò chart as follows: where E[Z 0 ] = μ 0 and 0 < λ g ≤ 1. Also for compactness, we can alternatively denote I j as follows: where N j is the number of data points sequentially sampled in the jth monitoring batch, k is just the counter for the sequential sampling, and Y j,k is the resampled point in the kth sequence of the jth monitoring batch. Or alternatively, we can denote I j as We note that the probability of a sample Z j ∈ R s is given by P(Z j ∈ R s ) = γ . Samples will be drawn at each stage j until Z j / ∈ R s . Moreover, for simplicity of derivations and for finding the first two centred moments of Z j , we will use the transformation λ * g = 1 − λ g (check Appendix for details). It is to be mentioned that λ g is the smoothing or the weighting factor (0 < λ g ≤ 1) and it may take varying values depending on how frequently the monitoring statistic is falling in the indecision region. In other words, it may be used to cater to the sensitivity of the shifts that might occur in a process. The initial value of SEWMA Ò statistic is usually set to be Z 0 = μ 0 . Moreover, X and Y are independently distributed random variables that follow the normal distribution with μ 0 as mean and variance σ 2 0 . The mean and variance of SEWMA Ò statistic are (3) It is to be mentioned that the abovementioned expressions are asymptotic versions of mean and variance of Z j (cf. Appendix). The details regarding the derivations of these expressions can be seen in the Appendix.
The control limits of the proposed SEWMA Ò control chart are in the form of two pairs of upper and lower control limits because of the three regions (as indicated in Figure 1). Using the expressions given in Equation (3) and the splitted regions in Figure 1, the control limits of the proposed SEWMA Ò chart are defined as where LCL 1 , UCL 1 determine the out-of-control and LCL 2 , UCL 2 determine the in-control regions, respectively, and accordingly the inner region (i.e. the area between UCL 1 & UCL 2 and LCL 1 & LCL 2 ) corresponds to the indecision region where we need to resample (cf. Figure 1). It is to be mentioned that when the monitoring statistic falls in the indecision region, we continue resampling and keep updating the plotting statistic (using the new sample information) until it falls in one of the decisive regions (i.e. Regions I or III). The quantities L 1 and L 2 are the control limit coefficients that are used to adjust the in-and out-of-control limits to achieve the desired in-control average run length (denoted by ARL 0 ). These coefficients must hold the condition that L 1 ≥ L 2 . Note that for L 1 = L 2 , the performance of SEWMA Ò chart is exactly the same as the usual EWMA chart.

i. Percentile Adjustment for Indecision zone
In the sequential sampling setup, it is important to adjust the indecision zone in order to achieve the desirable properties corresponding to a prespecified ARL 0 . In this study, we adjust the width of the indecision region, according to the following relation where L is the control limit coefficient (at a prespecified ARL 0 ) for the SEWMA Ò chart, which matches its performance with the EWMA chart (i.e. L 1 = L 2 = L), and Ò is the factor used to adjust the width of the decisive zone, 0≤ Ò ≤. For Ò = 0, L 1 = L 2 = L, the performance of the SEWMA Ò chart is exactly the same as the proposed SEWMA Ò chart.

Decision criteria
Using Equations (3)-(6), we compute the two pairs of control limits for a fixed ARL 0 . For each new monitoring sample, we calculate the plotting statistic Z j using Equation (1) sequentially. If the plotting statistic Z j falls in one of the decisive regions I or III, then we make the decision accordingly. If Z j falls in the indecision region, we continue resampling to update the plotting statistic (using the new sample information), until it falls in one of the decisive regions (i.e. Regions I or III). Figure 2 provides an exemplary charting display for the proposed SEWMA Ò chart. It is to be noted that: (i) the sensitivity of the regions may be controlled by the design parameters, namely λ g , Ò, L 1 and L 2 ; (ii) for the sake of simplicity, we have used λ g = λ for each occasion where we needed to resample; (iii) classical EWMA chart is a special case of the proposed chart when L 1 = L 2 = L.

Performance evaluation techniques
To assess the performance of the SEWMA Ò chart, we have used different run length properties, such as the average (ARL), the median (MRL) and the standard deviation of the run length (SDRL) distribution. ARL is the frequently used measure to evaluate the performance of control charts. ARL can be categorized into in-control (IC) average run length (ARL 0 ) and out-of-control ARL (ARL 1 ), defined as ARL 0 : Average number of samples plotted on a control chart until the first point goes outside the limits, when the process is in-control (IC). Similarly, MRL and SDRL can be defined based on an in-control or out-of-control state of the process.
To examine the detection ability of the SEWMA Ò chart, shifts are introduced in the in-control (IC) process location say, μ 0 and the OOC process mean level is defined as μ 1 = μ 0 + δσ 0 , where δ refers to the amount of shift.
These run length properties may be calculated by using the following expressions (cf. Chakraborti [26], Khaliq et al. [27])

Algorithm for SEWMA Ò chart construction
In this subsection, we discuss how to construct the SEWMA Ò chart via an algorithm. The following are the necessary steps of the algorithm to achieve this work.
(i) Draw a random observation from a normal distribution with in-control mean μ 0 and standard deviation σ 0 . Without loss of generality, we used μ 0 = 0 and σ 0 = 1. Find the monitoring statistic for the SEWMA Ò chart (using Equation (1)). (ii) Select L 1 = L 2 = L and plot the monitoring statistic (using the criterion mentioned in (2)) against the control limits (given in Equation (4)), to achieve the desired IC-ARL 0 . (iii) Decide L 2 using Equation (5) after selecting a level of p. For a given specified L 2, L 1 is then chosen to achieve the desired ARL 0 .
We now discuss the simulation procedure to assess the performance of the SEWMA Ò chart.

Algorithm for RL properties
Monte Carlo simulations are used to obtain the RL properties of the SEWMA Ò chart, following Quesenberry [28] and Khaliq at el. [29]. The steps taken in the simulation study are as follows.
(i) With the sequential charting algorithm described in the previous subsection, construct the SEWMA Ò chart. (ii) For getting out-of-control RL, shifts are introduced in the process mean level by plotting the monitoring statistic of the shifted samples against the control limits (using the criterion mentioned in Equation (2)). The OOC run length is computed as the number of samples plotted on the control chart before the first point goes beyond LCL2 or UCL2. (iii) The process is repeated 100,000 times to record RL values of the proposed design. (iv) Using the RL's from step (iii) above, the ARL, MRL and SDRL are computed as the mean, median and standard deviations of these RLs respectively.

Performance evaluation and comparisons
In this section, we evaluate the performance of the proposed SEWMA Ò chart at different levels of design parameters λ, Ò, and ARL 0 , L 1 and L 2. The performance of the proposed SEWMA Ò chart is compared with some existing charts for efficient detection of shifts in the process mean level. Tables 1-3

SEWMA Ò performance analysis
This section is about the Run Length (RL) performance of SEWMA Ò chart by varying Ò and λ. Several performance measures were used to observe the performance of the proposed design, i.e. average, median and standard deviation (cf. ARL, MRL and SDRL). Tables 1 and  2 represent the ARL 1 , MRL and SDRL performance of SEWMA Ò versus EWMA chart by varying λ and Ò. Here, λ is the smoothing or weighting factor that has a similar effect on the performance of the EWMA chart. The Ò has a significant impact on the performance of the SEWMA Ò chart. Here, Ò is control limits adjusting factor (percentile) and it guarantees the better ARL 1 performance of SEWMA Ò chart. Several choices of Ò are 2%, 4%, 6%, 8%, 10% and 12%. It may be observed from the tables that we have smaller ARL 1 , MRL and SDRL values from SEWMA Ò than classical EWMA.

i. ARL 1 Performance Analysis
Although both designs performed equally well based on the ARL under the in-control situation, our proposed design has better ARL 1 performance in the out-of control situations than the classical EWMA chart as it has smaller ARL 1 for all choices of Ò. As an illustration, consider  Tables 1 and 2. That is, our proposed design has a smaller variation in RL than the classical EWMA design.

General Comments
The proposed SEWMA Ò chart has better ARL, MRL and SDRL performance than the classical EWMA design. The proposed design is more efficient than the classical one from small, moderate and larger shifts. In particular, SEWMA Ò is more efficient than the EWMA chart when Ò > 0.02 as may be seen in Figures 3 and 4. It is to be noted that the SEWMA Ò reduces to the EWMA chart when Ò = 0.

Comparative analysis of SEWMA Ò versus EWMA, CUSUM and FIR-EWMA
This section is designed to show the comparative performance of SEWMA Ò versus classical EWMA, CUSUM and FIR-EWMA chart using several ARL 0 values. Table  3 demonstrates the ARL performance of SEWMA Ò (Ò = 10% and 12.5%) against classical EWMA and CUSUM design at ARL 0 = 168, 400 and 500. Moreover, Table 3 also presents the ARL performance of SEWMA Ò (Ò = 10% and 12.5%) versus FIR-EWMA at ARL 0 . The findings are as discussed below.

General Comments
The proposed design exhibits better performance than EWMA for all choices of Ò > 0. As Ò increases, the performance of the SEWMA Ò chart becomes increasingly better than EWMA, especially for higher values of smoothing parameter λ (cf. Table 3). The proposed SEWMA Ò chart outperforms the CUSUM chart at all levels of shift δ. The performance of the SEWMA Ò chart (compared with CUSUM chart) gets better with increase in p. Table 3 also provides the comparison of SEWMA Ò chart (Ò = 10% and 12.5%) versus FIR-EWMA at λ = . 25 and ARL 0 = 500 [36]. It is noticed that as Ò increases, SEWMA Ò chart is more sensitive for the detection of shifts as compared with FIR-EWMA chart (cf. last row of Table 3). Overall, the proposed SEWMA Ò chart performs It is to be mentioned that the current study is designed under normality; however, one may extend this for other distributional environments such as Maxwell, Weibull, Burr, Power, etc. (cf. Al Mutairi and Volodin [31], Basheer [32], Haq et al. [33], Hossain et al. [34,35]).

A real application of SEWMA Ò to steel rod manufacturing industry
This section presents a real-life application of the proposed technique in a production line that relates to an ongoing monitoring process of steel rod manufacturing. The steel industry is a booming industry and it plays an essential role in the development of a country and helps strengthen the economy. Most of the steel manufactured by an aluminium plant or a steel mill is moulded through extrusion or rolling into long continuous strips of numerous shape and size. Several testing techniques are commercially available and can be customized to meet specific inspection needs. Whether manual, automated or phased array, ultrasonic  testing can help manufacturers of bars, tubing and related metal products to assure product quality and customer satisfaction. But without proper SPC techniques, it will be time consuming and costly. To reduce the cost, it is necessary to use such techniques which provide a timely identification of any deviations from the stable state of a process. Several authors designed efficient SPC techniques to identify the unusual variations present during several processes. Lyu and Chen [36] designed a multivariate SPC technique to detect defects which often occur during manufacturing process. Pérez-González et al. [37] applied SPC to get the proper decision from the entire emergency and security system. In several steel manufacturing industries, quality control techniques are implemented to enhance process monitoring of designing and manufacturing steel rods. Christyanti [38], Naeem et al. [39] and the references therein may be seen for further details related to the topic. They used different statistical approaches like DOE (design of experiment) andX chart for the identification of special causes. They suggested that 6-sigma procedures really helps to identify the special causes during process monitoring. Figure 5 provides a graphical representation of the monitoring process in steel rod manufacturing, where one can potentially use the proposed SEWMA Ò chart.
The data we have used here is related to the steel manufacturing industry. The breaking strength of the rods is our quality characteristic of interest that needs monitoring over time for a better quality of the products related to the process output. A sample of 90 steel rods measurements is collected from the aforementioned process, and their breaking strength is measured. Through distribution fitting tests, we identified that the breaking strengths of the steel rods follow a normal distribution. We have constructed SEWMA Ò and the classical EWMA charts for this dataset using λ = .05 and Ò = 0.1. Figure 6 presents these comparative charts where UCL and LCL refer to limits of the classical EWMA, while the other two pairs (LCL 1 , UCL 1 ) and (LCL 2 , UCL 2 ) correspond to the SEWMA Ò chart. It is observed that the proposed design is not only indicating the shift earlier than the classical EWMA, but it is also giving a larger number of out-of-control signals (this is exactly in accordance with the findings in Section 4).
These earlier and larger detections by SEWMA Ò design has many practical implications for process monitoring. It may be used as a tool needed to reduce the amount of labour and to perform cost-effective experimentations. The potential factors causing out-ofcontrol signals might include changes in these factors: mechanical properties of metal, hardness level, yield strength, intensity of temperature, the grain size, atmospheric pressure, the quality of the protective surface film, the presence of certain agents reducing any corrosive effects, the presence of surface cracks or discontinuities and other environmental condition. These kinds of abnormalities may be identified by using appropriate SPC techniques such as the newly proposed SEWMA Ò chart of this study.

Concluding remarks
This study introduces a memory control chart, namely SEWMA Ò chart, for an efficient monitoring of shifts in the process location parameter. The performance of the proposed design is assessed by using several RL properties (such as ARL, MRL and SDRL) at varying values of design parameters λ g , Ò, L 1 and L 2 . The results advocate that the proposed design provides enhanced RL properties as compared with the classical EWMA, CUSUM and FIR-EWMA charts at several p values. As p increases, the proposed design offers more sensitivity towards shifts in the process mean relative to the competing charts. A real-life application concerning the breaking strength of steel rods highlights the significance of our study.
The current work can be extended to study the performance of CUSUM, mixed / combined Shewhart, EWMA-CUSUM charts both in univariate and multivariate directions for location and dispersion parameters. The scope may further be extended to cover mixed and sequential sampling techniques in SPC for efficient process monitoring of non-normally distributed process.
For derivations of some results in this appendix, we will rely on the following Theorem, which we present without proof, on the moments of the random sum of random variables from Klugman et al. (2014) Theorem 1: ( [40], Equation (9.9)). Let the random sum, S = N j=1 X j where X j is iid from any distribution f (x; μ, σ 2 ) and the size random variable, N, has a distribution h(n; m, π). Then, the first two moments of S are as follows: Definition 1: Let us denote R s as the resampling region. With 0 < λ g ≤ 1 and for j = 1, 2, 3, . . . , the sequential EWMA random variable Z j is defined as follows Z j = λ g I j + (1 − λ g )Z j−1 For compactness, we can alternatively denote I j as follows: Or alternatively, we can denote I j as We note that the probability of a sample Z j ∈ R s is given by P(Z j ∈ R s ) = γ . We note that the random sample Y j,k will be drawn sequentially at each stage j until Z j / ∈ R s . As such, the random variable N j measures the sample size needed until Z j is outside of the resampling region. Thus, N j is a geometrically distributed random variable. We also note that the random sample X j comes from a normal distribution with a mean of μ 0 and variance of σ 2 0 . The sequential random sample Y j,k also comes from a normal distribution with a mean of μ 0 and variance of σ 2 0 . For simplicity of derivations and results for the first two centred moments of Z j , we will use the transformation λ * g = 1 − λ g .
A) Derivation of the Mean of Z j .
due to Theorem 1 (by Klugman et al.) 41 where N 1 ∼ geom(1−γ ) and Y 1,k is iid with μ 0 . Thus due to Theorem 1 (by Klugman et al.) 41 where N 2 ∼ geom(1−γ ) and Y 2,k is iid with μ 0 . Thus The following summarizes the results, we have thus far: Based on these, we now let Then by induction, we have due to Theorem 1 (by Klugman et al.) 41 where N j ∼ geom(1−γ ) and Y j,k is iid with μ 0 . Thus, Since 0 < λ * g < 1, we can also reexpress the above as follows:

B) Asymptotic behaviour of the mean of Z j .
Since 0 < λ * g < 1, as j grows large, we have the following: C) Derivation of the variance of Z j .

D) Asymptotic behaviour of the variance of Z j .
Since 0 < λ * g ≤ 1, as j grows large we have the following: