A Weibull process monitoring with AEWMA control chart: an application to breaking strength of the fibrous composite

In recent times, there has been a growing focus among researchers on memory-based control charts. The Exponentially Weighted Moving Average (EWMA) and Cumulative Sum (CUSUM) charts and the adaptive control charting approaches got the attention. Control charts are commonly employed to oversee processes, assuming the monitored variable follows a normal distribution. However, it's worth noting that this assumption does not hold true in many real-world situations. The use of the algebraic expression for normalization, which can be used for all kinds of skewed distributions with a closed-form distribution function, using the proposed continuous function to adapt a smoothing constant, motivates this study. In the present manuscript, we design an EWMA statistic-based adaptive control chart to monitor the irregular variations in the mean of two parametric Weibull distribution and use Hasting approximation for normalization. The adaptive control charts are used to update the smoothing constant according to the estimated shift. Here we use the proposed continuous function to adapt the smoothing constant. The average run length and standard deviation of run length are calculated under different parameter settings. The effectiveness of the proposed chart is argued in terms of ARLs over the considered EWMA chart through Monte-Carlo (MC) simulation method. The proposed chart is examined, followed by a real data set to demonstrate the design and application procedures.


Design of the proposed control charts for Weibull distribution
In this segment, we design new EWMA and AEWMA-I CCs for observing irregular changes in the mean of a Weibull process.Let U denote a Weibull random variable.We note U ∼ W(η, θ) , where η and θ respectively, denote the scale and the shape parameters ( η, θ > 0).The following cumulative distribution function can char- acterize this distribution: The Exponential distribution is a reduced model of Weibull distribution for θ is equal to 1. Assuming the random sample of size n is taken from the sequence {U t } , which follows a two-parameter Weibull distribution at time t as {U 1t , U 2t , . . ., U nt , } ; U it gives the ith observation in the tth sample with i = 1, 2, . . ., n, and t ≥ 1.

Hastings's approximation to normal
The algebraic approximation proposed by 25 converts the random variable of skewed distribution to the standard normal variate Z using its distribution function. where: now, For the in-control process, V t is given as, i.e., V t ∼ N(0, 1).

The EWMA control chart for Weibull distribution
The plotting statistic E t of the EWMA CC suggested by 3 based on the sequence {V t } is where E 0 = 0 is the initial value and ∈ (0, 1] .The mean and variance of E t is The asymptotic variance for t → ∞ is therefore, the control limits of the EWMA CC are as follows where H is the EWMA CC's coefficient of control limit for the given in-control ARL.The value of H (H > 0) is a threshold for a given n and λ.In order to guarantee the in-control ARL optimum sensitivity of the EWMA CC statistic |E_t | at a predetermined fixed level (let's say ARL 0 ), the value of H is calculated.where H is the coefficient of control limit of the EWMA CC for specified in-control ARL.The MC method is used for simulation study.
To get the mean and the standard deviation of RL of the EWMA chart, the sample sizes n = 3, 4, and 5 are used.
We have fixed in-control ARL (ARL 0 = 370) with φ = 0.15 and 0.20 for some fixed values of shift sizes.The RL profiles of the EWMA CC are presented in Tables 1 and 2.

Design of the proposed AEWMA control chart for Weibull distribution
Practically speaking, the magnitude of shift size is not known ahead of time.So here we consider the estimator proposed by 26 to estimate the magnitude of shift size as (2) (3) x = ln 1 (5) Var  www.nature.com/scientificreports/where φ ∈ (0, 1] , δ * 0 = 0 .Consider ⌣ δ t = δ * * t when estimating the magnitude of shift size.Using the sequence {V t } , the plotting statistic of the AEWMA-I CC for observing the process mean is given by where F 0 = 0 and g ⌣ δ t ∈ (0, 1] such that Drawing inspiration from binary logistic regression.Where the response function is constrained within the range of 0 to 1, we conducted an empirical exploration.This is done by experimenting with different functions (such as logarithmic, exponential, and others) along with different constants.Our aim was to fine-tune the traditional MEWMA scheme to achieve near-optimal performance in detecting shifts within the predefined ranges.Since the design of the function plays a crucial role in managing the false alarm rate, it's important to note that the standard deviation of the run-length is closely tied to this false alarm rate.This is why our proposed AMEWMA exhibits a more compact run-length profile when compared to competing CCs.
The constants used in the function are suggested with some values.The value of the EWMA plotting statistic F t is determined from the proposed continuous function g ⌣ δ t such that the AEWMA-I CC becomes optimum in the quick recognition of the shift in the mean of the process.The working of the AEWMA-I CC is like that of the AEWMA CC proposed by 22 .

Decision rule
In a one-sided AEWMA-I chart, if the plotting statistic |F t | > L, then the process prompted the out-of-control signal.Similarly, the process prompted the out-of-control signal if the proposed statistic F t > L or F t < -L in the case of a two-sided AEWMA-I chart.
For a specified n and φ , the value of L (L > 0) is a threshold.The value of L is determined such that the in-control ARL optimal sensitivity of the AEWMA-I CC statistic |F t | is ensured at some chosen fixed level (say ARL 0 ).For all specified parametric combinations of n and φ the value of L is determined separately.The parametric combinations of φ , n with some given specified δ overwhelmingly affect the optimum in-control RL threshold performance.The L value is utilized to fix the in-control ARL as the ARL 0 ; threshold by selecting the recommended adaptive functional methodology.
The out-of-control ARL is impacted by the value of Weibull distribution parameters, sample size, and design parameters φ, L. The trouble in concentrating out-of-control ARL is that there are two parameters in the Weibull distribution.We consider the shift level of the shape parameter θ 1 /θ 0 = 1.This is sensible since, in applied appli- cations, the scale parameter will undoubtedly change as a result of assignable causes, while the shape parameter is more related to the regular properties of the system and is somewhat stable.When keeping the shape θ as a constant, the value of out-of-control ARL of the AEWMA-I chart with transformed Weibull data relies upon the shift level of the scale parameter η 1 /η 0 .

Algorithm: To work out the RL profiles of the AEWMA-I CC for Weibul process utilizing the MC technique is described in the steps given below,
Step 1: Fix the values of parameters, sample size (n), and a smoothing constant (φ).
Step 2: Generate an item (or subgroup of the desired size) using simple random sampling at time t and measure its quality characteristic U t using (1) with specified Weibull distribution parameters.
Step 3: Normalize the Weibull-distributed data using Hastings's approximation given in ( 2), ( 3) and calculate Step 4: Estimate mean shift δ(say ⌣ δ t ) by using δ * * t and by using this estimated ⌣ δ t , the proposed continuous function g ⌣ δ t will get the value of the smoothing constant for the AEWMA-I CC and then calculate the plot- ting statistic F t ("Design of the proposed AEWMA CC for Weibull distribution" contains the details of the AEWMA-I CC design).
For in-control process i. Decide the in-control ARL 0 (say 370).ii.Select the value of L (threshold).This is done so that we get a fixed level of ARL 0 reaches some chosen fixed level (say 370).We have considered the 50,000 replicates.iii.A similar practice is done with different parametric combinations, at some given specified δ, before the use of the chart at phase II.
For out-of-control process i.If |F t | > named as out-of-control, becomes RL; note the iteration number as RL.Else, recurrence steps 2-4.ii.Continue choosing the random sampling units from MC simulation till the accomplishment of 50,000 replications.iii.The desired average and standard deviation of RLs are computed.iv.The RL profiles at several quantified given δ are evaluated through respective simulations.
Hence, the RL profiles stated in Appendix Tables 2 and 3 have been widely produced for the performance assessment of this study.

Performance comparison
Performance assessment of a CC employing its RL abilities, ARL, and SDRL is quite common in SPC.On the same lines, we use RL abilities as a demonstration standard in this study.For a given ARL 0 , smoothing constant φ and the magnitude of the shift (δ), any CC is supposed to be good enough than its competitor CCs if its out- of-control ARLs are significantly lesser.The sensitivity of the CC increases as the value of the EWMA parameter φ decreases.In practice, φ is set within the interval [0.05 ≤ φ ≤ 0.25] with φ = 0.15, and 0.20 being popular choices.A rule of thumb is to use the small values φ of to detect smaller shifts.(Montgomery, 2009).
The AEWMA-I CC has been evaluated with EWMA CC.The ARL 0 = 370 with sample size n = 3, 4, 5.The φ = 0.15, 0.20 are taken.The 50,000 iterations are done to calculate the RL attributes of CCs..The revelations connected with the outcomes are examined under: i. From Table 2, when ARL 0 = 370 and φ = 0.15, the increase in sample size from 3 to 5 and increase in shape parameter from 0.5 to 3 resulted in a significant decrease in ARL and SDRL of AEWMA-I as compared to EWMA on all scale shifts from 1.1 to 4.5.For instance, with θ = 0.5, η 1 /η 0 = 1.1 , the ARLs are 247.77and 175.63 while at shift 4.5 the ARLs are 4.58 and 3.54 for EWMA and AEWMA-I CCs respectively.ii.Similarly, same result has been observed in Appendix Table 4, when ARL 0 = 370 and φ = 0.20.
It very well may be seen from Appendix Tables 3 and 4 that the AEWMA-I chart performs uniformly and extensively better than the EWMA CC in all cases, which shows the predominance of the AEWMA-I chart over its counterpart.

Illustrative example
In this section, a typical practice that has been trailed by various researchers to clarify the execution and working of the CCs with the assistance of real datasets.To clarify the operation and implementation of the suggested CC, we take a look at a real-world dataset.Observing the breaking strength of the fibrous composite is vital in the ventures that ensure the security of material utilized in the aero industry and construction of bridges.
For this purpose, the real-life dataset used by 26,27 is considered.It is a dataset related to carbon fibers' breaking strengths in manufacturing fibrous composite materials.This is obtained from an investigation by the U.S. Armed Force Materials Technology Laboratory in Watertown, Massachusetts.We assume the data set incorporates 20 samples with a sample size n = 5 is the Weibull process with scale parameter ( η = 2.9437) and shape parameter ( θ = 2.7929).The Anderson-Darling test is applied to check the goodness of fit that results in the p-value of 0.8306, so the data fits well to the Weibull distribution.Draw 15 random-size samples (n = 5) without replacement and consider them in-control samples.Now contaminate the data by adding 1 in each observation, draw 10 random samples of size n = 5 without replacement, and consider them as out-of-control samples.The parameter estimation is done by the maximum likelihood estimation (MLE) method.
We apply two competing CCs on this data with an ARL 0 = 370.The parameter choices considered for the CCs are EWMA with ( φ = 0.15, LCL = -0.7971,UCL = 0.7971); and the AEWMA-I with ( φ = 0.15, L = 0.1685).The resultant values CCs are given in Appendix Table 6 and demonstrated in Figs. 1 and 2 following the design given in "Design of the proposed control chart for Weibull distribution".2. It is evident from Figures 1 and 2 that during the first 15 samples, both CCs stay constant.Nevertheless, both CCs start to show out-of-control signals in the process mean after that point.Interestingly, the AEWMA-I chart exhibits a faster out-of-control signal than its corresponding chart.An out-of-control signal is triggered at the 17th and 19th samples by the AEWMA-I chart, respectively.

Main findings
The extensive simulation results are obtained to evaluate the performance of the proposal,the results of the proposed chart are discussed as: i. i.It is qualified to rapidly identify small shifts to monitor mean in the process in which the under-study variable follows the Weibull distribution.For instance, from Table 2, at scale shift, 10%, n = 5, θ = 0.5, the ARL of the AEWMA-I chart is 175.63 with φ = 0.20, and at similar parameters, the ARL of EWMA is

Conclusion
The adaptive CCs have procured great thought as they are not simply more fragile than the non-adaptive CCs.They are valuable in providing better security when the shift in the process is dependent upon existing in some range.In the current work, we have suggested the AEWMA-I CC to monitor irregular variations in the process's mean, which follows the Weibull distribution.First, the data ought to be normalized utilizing Hasting transformation.The MC simulation method is used for RL profile calculation.During the fair assessment of the RL profiles, it was observed that the AEWMA-I CC with transformed Weibull data performs better than the EWMA chart in recognizing the changes in scale parameter when the shape parameter is fixed.The numerical example based on industrial data is given to delineate using the AEWMA-I CC.Based on the current research, AEWMA charts that simultaneously observe the mean and variance or screen shift in the process variance can be created.
It may be possible to extend the current work to create new AEWMA charts for additional processes that are not generally/normally distributed.Additionally, new AEWMA charts for additional non-normal bivariate and multivariate distributions could be planned using the current work as a basis.In the current study, we presented AEWMA-I CC for monitoring the process means which follow the Weibull distribution.For further study, we can monitor the process dispersion CCs such as variance and Coefficient-of-variation CCs by considering nonnormal processes. https://doi.org/10.1038/s41598-023-47159-9