Neutrosophic multivariate EWMA control chart

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Introduction
One of the methods which are used to discover process shifts is the control chart.This technique is extensively utilized in industry to observe the manufacturing process.The multivariate charts that are commonly used in the manufacturing process include  Hotelling (Chong et al., 2019;Tiryaki & Aydin, 2022), MCUSUM (Xie et al., 2021;Imran et al., 2022), and MEWMA (Ajadi et al., 2021).In the service industry, one of the multivariate charts that is extensively applied is the MEWMA chart, which is designed to investigate the process mean vector and is more efficient for detecting small shifts (Montgomery, 2020).The MEWMA control chart involves information on current and previous observations and is an expansion of the EWMA univariate chart.While the traditional MEWMA chart is appropriate when the observations are correct, certain, or precise, in practice, the results of observations or measurements of quality characteristics often produce ambiguous data.In this situation, the use of conventional charts is not appropriate.When there is ambiguity or randomness in the observed characteristics, the traditional charts are not appropriate to use.In case the observations or parameters studied are vague or have ambiguity, control charts based instead on fuzzy logic are a suitable choice for monitoring the process.Zadeh discusses how the fuzzy-logic method has been widely used in the unpredictability of a situation (Zadeh, 2005).Senturk and Erginel (2009) state that "Human judgments, evaluations, and decisions are included in observations, and a numerical random variable of a manufacturing process consists of the variability due to the measurement devices, environmental conditions, or human subjectivity".The variables cause ambiguity in the measurement system.As a result, fuzzy charts have generally been used in situations where there is uncertainty.According to Khademi and Amirzadeh, "fuzzy data lies in the present manufacturing and service process" (Khademi & Amirzadeh, 2014), so many researchers have concentrated on such control charts, such as (Khan et al., 2018;Rowlands & Wang, 2000).Several actual examples of the implementation of fuzzy control charts involve the manufacture of synthetic buttons by a clothing business in Turkey, the creation of piston rings for automobile engines through a forging process to ensure statistical control of the rings' interior diameter, and a food industry operation in Pakistan that fills cooking oil containers.Smarandache explained that the fuzzy approach is a part of neutrosophic analysis (Smarandache, 2010).Several fundamental works on neutrosophic statistics are cited in (Smarandache, 2014).Neutrosophic statistics are applicable when the samples or parameters are fuzzy, inexact, indecisive, or vague, whereas the classical statistics method assumes that all data observations are determined, precise, and certain.For the analysis of imprecise data, neutrosophic statistics are more powerful than conventional statistics.
Several authors focused on the development of a control chart based on neutrosophic sets.They performed a rock study based on neutrosophic statistics (NS) and they present the efficacy of these statistics (Chen, Ye, & Du, 2017;Chen, Ye, Du, et al., 2017).Another researcher suggested the  chart based on the neutrosophic exponentially weighted moving average (NEWMA) to monitor quality under an uncertain environment (Aslam et al., 2019).Aslam et al. developed variable charts and neutrosophic-based attributes. Aslam and Khan (2019) suggested the  chart by using neutrosophic statistics.Using the NS, both variable and attribute charts were proposed by Aslam, employing resampling (Aslam, 2019).This chart contains two statistics and two pairs of control limits ( ,  )  ( ,  ).However, so far, the development of neutrosophic control charts has been limited.Several researchers have developed such control charts based on univariate cases, such as NEWMA (Aslam et al., 2019), neutrosophic control charts S (Z.Khan, Gulistan, Chammam, et al., 2020), and neutrosophic mean deviation (Khan, Gulistan, Hashim, et al., 2020).Recently, for univariate control charts, researchers have suggested monitoring scheme with X-Bar chart by employing the Neutrosophic-Based Generalized Multiple Dependent State Sampling (Khan et al., 2022), Neutrosophic Maxwell Distribution based chart (Shah et al., 2023), and Moving average control chart with neutrosophic statistics (Aslam et al., 2023).Meanwhile, cases of multivariate quality characteristics require handling with multivariate neutrosophic control charts.When used to test the mean of two populations, neutrosophic  Hotelling statistics are more sensitive compared to the conventional  Hotelling statistics.A neutrosophic  Hotelling chart has been developed based on the neutrosophic  Hotelling statistics, and used in investigating abnormality in the glass manufacturing process (Wibawati et al., 2022), while other researchers also proposed the Multivariate T 2 Chart for Neutrosophic Data applied to the chemical sector (Saritha & Varadharajan, 2023).Neutrosophic control charts, on average, are more sensitive than conventional control charts.Meanwhile, Hotelling's  control chart is ineffective for small process shifts, whereas the MEWMA control chart can handle this problem.
According to the background, this research focuses on developing a neutrosophic-based multivariate EWMA chart that can overcome the problem of ambiguous data.The proposed chart is named as NMEWMA and its ability to observe process shifts is investigated using the Average Run Length (ARL) and Standard Deviation Run Length (SDRL).The suggested chart is used to track data from the glass manufacturing process.

Materials and Methods
The Multivariate EWMA (MEWMA) chart was first suggested by Lowry et al.MEWMA charts perform well for monitoring a small shift in the mean process (Lowry et al., 1992).Let   ,  = 1,2,3, . . ., , , be a p×1 random vector and follow a pvariate normal distribution,  ∼  (, Σ).The MEWMA control chart statistics are stated as follows: where 0 <  ≤ 1, The samples are stated as control if, for all i,  < ; if at least one observation falls beyond the control limit, the process is called out-of-control.MEWMA control limits for some smoothing parameters ( ) and the number of quality characteristics () are presented in Table 1, which contains the ARL performance for MEWMA for various values of  for quality characteristics () equal to 2, 4, 6, 10, and 15.The control limit (H) is estimated to produce an in-control  = 200 (Montgomery, 2020).The design of the suggested chart is developed according to the traditional MEWMA chart through the following steps: Let  ∈  ,  be a neutrosophic vector random variable with a sample size  ∈  ,  , where  and  are the minimum and maximum observations, respectively.The neutrosophic vector random variable can be written as (Aslam & Arif, 2020): By adopting the traditional MEWMA, we define the vectors of NMEWMA: where initially  =  ,  ∈  ,  and 0 <  ≤ 1 is the smoothing constant, and the NMEWMA statistics are defined as: The process sends an out-of-control (OOC) if

Performance of Neutrosophic MEWMA (NMEWMA) Control Chart
The neutrosophic MEWMA control chart's performance was investigated by employing the ARLs and SDRLs, and using various process shifts, parameter smoothing (λ), the number of characteristics (p), and correlation of the quality characteristics (r).The NMEWMA's performance is compared to that of the classical MEWMA.In this simulation, several scenarios are given for parameter smoothing (λ), including 0.1, 0.2 and 0.3.The suggested chart is made up of two charts, NMEWMA lower ( ) and NMEWMA upper ( ).The value of ARL and SDRL is obtained from the first out-of-control signal from  or  for each level shift as the run length (RL) for 10,000 replications.This section compares the proposed chart's performance with MEWMA chart.The comparison of various shifts  = (0.1, 0,1) ; (0.2,0.2) ; . . ., (2,2)′ with r = 0.25; p = 2 and λ = 0.3 is shown in Fig. 1 and Table A (Appendix).Fig. 1 shows that the ARLs produced by NMEWMA are smaller than MEWMA for small shifts (0.1-0.6).The value of  equals to 200, and the value of  NMEWMA is close to 200 for high correlation ( = 0.9).This chart indicates that, for larger shifts, NMEWMA is quicker to detect an out-of-control process.The other schemes of quality characteristics correlation (r) are shown in Fig. 2.There are three schemes of correlation,  = 0.25,  = 0.5, and  = 0.9, representing small, medium, and high correlation, respectively.Based on Fig. 2, Table A, and Table D, among the scheme of correlations, the performance of NMEWMA is more sensitive than the MEWMA charts.For both NMEWMA and MEWMA, the higher the correlation in the chart, the more sensitive it is.The suggested chart's ability in detecting shift is also compared with the MEWMA chart for various values of parameter smoothing (λ) and the number of quality characteristics (p), as shown in Fig. 3 and Fig. 4. The efficiency of the NMEWMA and MEWMA control charts is similar, as illustrated in Fig. 3, Table B, and Table E, where the smaller λ, the more sensitive is the performance of NMEWMA and MEWMA for small-to-medium process mean shifts.This is because the control chart's ARL shrinks at small smoothing parameters/weights.For the larger shifts, both of these charts have the same performance.According to Fig. 4, Fig. 5, and Table C, it is visible that MEWMA and NMEWMA have the same performance for the various numbers of quality characteristics.

Application of Neutrosophic MEWMA (NMEWMA) Control Chart
The NMEWMA chart was used on simulated data to assess how well the control chart detects out-of-control observations on modest and large process mean shifts.The data was generated by following a normal multivariate distribution of 50 samples with a number of quality characteristics equal to 4. In this simulation, the data is separated into two parts.The first 80% of the initial sample had a normal multivariate distribution with a neutrosophic mean vector  and a covariance matrix  .The next 20% of samples had a normal multivariate distribution with a mean vector  =  +  , where  0. These samples were classified as either in-control or out-of-control.In this simulation, the in-control sample is generated from a normal multivariate distribution with parameter  =  .Meanwhile, the out-of-control sample is generated from a normal multivariate distribution with parameter  , using  = 0.75 for small process shifts and  = 3 for large process shifts, respectively.The smoothing parameter/weight λ is equal to 0.1.Based on the synthetic data, the results of the NMEWMA chart for small process shifts are depicted in Fig. 6.The statistical value of the neutrosophic MEWMA (NMEWMA) is obtained (shown in Table 3), and the control limit  = 12.73.Based on the performance evaluation, when  equals 4, the NMEWMA is more sensitive for the parameter smoothing.In this case,  equals 0.1, so in this case study, we use  = 0.1.Fig. 8 shows the findings of the neutrosophic MEWMA (NMEWMA) control chart.Fig. 8 shows the MEWMA neutrosophic control chart (NMEWMA), indicating that the mean of the glass production process has not been controlled.The NMEWMA control chart produces 1 point that is out of the  control limit, namely the upper NMEWMA control chart.The comparison of the neutrosophic MEWMA control chart with the conventional MEWMA control chart is shown in Fig. 9.By using NMEWMA, the second observation is detected as an out-of-control observation, while, by using MEWMA, the second observation is still detected as in-control.This is because NMEWMA involves two control charts with two control limits, meaning that this chart is more sensitive.

Conclusions
This work develops a new Neutrosophic Multivariate EWMA (NMEWMA) control chart.The performance of the proposed NMEWMA chart was assessed using the ARL and SDRL.The performance of the proposed NMEWMA chart is measured using various process shifts (), parameter smoothing correlation of quality characteristics ().Based on ARL and SDRL, NMEWMA is more sensitive than MEWMA.Control charts are used for both simulation and real data.Based on the reallife application and simulation studies, the proposed NMEWMA chart shows better performance when compared to the conventional chart.The Neutrosophic MEWMA for the subgroup will be investigated in other performance analyses and also for different uncertainty levels for future research.

Fig. 6 .
Fig. 6.MEWMA (a) NMEWMA lower, (b) upper, (c) for small shifts Fig. 6 shows that both the MEWMA and the NMEWMA control charts indicate that the mean process has been controlled.This is because, in the resulting control chart, there are no out-of-control points.Therefore, based on these simulation data, both the MEWMA control chart and the NMEWMA control chart are insensitive in identifying small process mean changes.The results of utilizing the NMEWMA chart for the simulated observation for large process shifts are shown in Fig. 7.The neutrosophic MEWMA (NMEWMA) control chart application uses data from the Quality Assurance division for glass production quality characteristics in 2022.Four quality characteristics are used: zebra left ( ), zebra right ( ), cutter line ( ), and edge distortion ( ).The zebra's left and right targets are 60 mm, the cutter line is 115 mm, and the edge distortion is 40 mm.Table 2 displays data from the four quality characteristics.