Control charts using half-normal and half-exponential power distributions using repetitive sampling

This manuscript presents the development of an attribute control chart (ACC) designed to monitor the number of defective items in manufacturing processes. The charts are specifically tailored using time-truncated life test (TTLT) for two lifetime data distributions: the half-normal distribution (HND) and the half-exponential power distribution (HEPD) under a repetitive sampling scheme (RSS). To assess the effectiveness of the proposed control charts, both in-control (IC) and out-of-control (OOC) scenarios are considered by deriving the average run length (ARL). Various factors, including sample sizes, control coefficients, and truncated constants for shifted phases, are taken into account to evaluate the performance of the charts in terms of ARL. The behavior of ARLs is analyzed in the shifted process by introducing shifts in its parameters. The superiority of the HEPD-based chart is highlighted by comparing it with both the HND-based ACC and the ACC based on the Exponential distribution (ED) under TTLT using RSS. The results showcase the superior performance of the proposed HEPD-based chart, indicated by smaller ARL values. Additionally, the benefits of another proposed ACC using HND are compared with the ED-based ACC under RSS, further confirming the effectiveness of the HND-based approach through smaller ARLs Finally, the proposed control charts are evaluated through simulation testing and real-life implementation, emphasizing their practical applicability in real-world manufacturing settings.


Introduction of two life data distribution
This section provides a concise introduction about two important distributions: the Half-Normal Distribution (HND) and the Half-Exponential Power Distribution (HEPD).In Sect.2.1, we explore the application of the HND in the realm of statistical quality control.Furthermore, in Sect.2.2, we delve into the application of the HEPD within the field of statistical quality control.These discussions shed light on the significance and relevance of these distributions in the context of quality control practices.

Half Normal Distribution (HND)
The Half Normal Distribution (HND) is a widely employed statistical distribution commonly used for modeling life data.It was first developed and studied by Chou and Liu 31 , who extensively examined its properties and explored its applications in the field of quality control.Notably, the HND finds particular relevance when dealing with data related to fatigue, which refers to the structural deterioration that occurs when a material is consistently subjected to stress.In such cases, the HND serves as a valuable tool for analyzing and understanding the characteristics of fatigue-related data 32 .It is worth noting that the HND is a special case of the normal distribution, showcasing its significance and applicability in various statistical contexts.The Half Normal Distribution (HND) is a special case of Normal Distribution with mean 0 and its scale parameter α which is limited to the domain [0, ∞) .The probability distribution of half normally distributed variable t is given by here α is the scale parameter.The CDF denoted by F(t) is given by The erf is the error function defined as The hazard rate function, denoted as h(t) , for a continuous distribution is defined as the ratio of the prob- ability density function (PDF) to the survival function, given by Substituting the PDF and CDF of the HND into this formula will yield the hazard rate function for the HND and the graph of hazard rate function is also shown in Fig. 1 The positive trend in the hazard rate function graph for the half-normal distribution indicates an increasing failure rate with higher values of the random variable.This behavior portrays a growing likelihood of failure as the variable values progress, suggesting an escalating risk associated with larger values.Despite starting from zero, the hazard rate consistently rises as the variable increases, signifying a continuous elevation in the failure rate.The mean and variance of HND are given by (1) The versatile application of the Half Normal Distribution (HND) extends beyond quality control.It finds utility in various fields such as sports science, blowflies, physiology, stochastic frontier models, fiber buckling, and notably, reliability analysis.The significance of HND in Acceptance Sampling Plans can be observed in research studies conducted by Azam et al. 19,20 , Rao et al. 30 , Al-Omari et al. 33 and Lu et al. 34 .These studies highlight the diverse range of applications where the HND plays a crucial role, contributing to advancements and insights in various scientific disciplines.

Half Exponential Power Distribution (HEPD)
The Half-Exponential Power Distribution (HEPD) is a distribution that arises from truncating the Exponential Power Distribution, developed by Gui 35 .It serves as a generalized distribution that encompasses both the Half Normal Distribution (HND) and the Exponential Distribution (ED), specifically designed for non-negative variables.The HEPD finds extensive application in the fields of reliability analysis and quality control.In a study conducted by Gui 36 , a sampling plan for the HEPD utilizing the Time Truncated Life Test (TTLT) concept was developed.This plan provides guidelines for effective sampling and analyzing data from the HEPD distribution.Additionally, Gui and Xu 37 proposed a double acceptance sampling plan specifically tailored for the HEPD using TTLT.This plan aims to further enhance the efficiency and effectiveness of the sampling process in the context of the HEPD distribution.More recently, Naveed et al. 8 proposed the use of the HND and HEPD distributions in the development of an ACC.This further highlights the practical significance and broad applicability of the HEPD in reliability analysis and quality control applications.Overall, the HEPD is a crucial distribution that expands the range of applications for the HND and ED of non-negative variables.Its utilization in reliability analysis and quality control, as demonstrated by the aforementioned studies, underscores its practical importance in various fields.The probability distribution function (pdf) and cumulative distribution function (cdf) of HEPD are given by Here, the parameter α represents the scale parameter, while represents the shape parameter.The HEPD distribution exhibits different characteristics depending on the values of these parameters.Specifically, when the shape parameter is equal to 1, the HEPD is equivalent to the Exponential Distribution (ED).Similarly, when takes the value of 2, the HEPD transforms into the HND.The hazard rate function for the HEPD is given as we also display the graph of hazard rate function of HEPD in Fig. 2. The positive curve linear trend in the hazard rate function (HRF) of the half-exponential power distribution signifies a consistent and steady increase in the failure rate over time or across variable values, depicted by a curved linear pattern in the HRF graph.The mean of HEPD is as follows

Designing of the control chart using Half Normal Distribution
Here, we suppose that the failure time of manufacturing item denoted by t follows the HND.The probability that an item fails before time t 0 is given by replace the value of t 0 = qµ 0 where q indicates the truncated constant for HND and the value of α in term of µ using Eq. ( 6), then Eq. ( 12) can be written as The process is considered to be IC when µ = µ 0 then Eq. ( 13) becomes The inner and outer lower and upper control limits for proposed np charting structure using HND under RSS is as follows: The outer control limits for proposed chart are given as The inner control limits for proposed chart are given as Here W 1 andW 2 are the control coefficients with W 1 being greater than W 2 .These coefficients are determined based on the desired ARL for an in-control process.It is important to note the if W 1 = W 2 the proposed chart is converted to Naveed et al. 8 based on HND.
The working procedure for the presented chart is as follows: Step 1: Randomly select a sample of size n from each subgroup and conduct the TTLT with a duration of t 0 .Count the number of items that fail before time t 0 and denote it as X.
Step 2: Determine the control status based on the value of X .If X falls below the outer lower control limit LCL 1 or exceeds the outer upper control limit UCL 1 , the production process is considered out-of-control (OOC).If X falls within the range of the inner lower control limit LCL 2 and the upper control limit UCL 2 , the ongoing operation is considered in-control (IC).If X falls between LCL 1 and LCL 2 or between UCL 2 and UCL 1 , repeat Step 1.
It is important to note that the number of failure items X for the IC process follows the Binomial Distribu- tion (BD) with parameters n and P 0HND Here, P 0HND represents the probability of an item fails before time t 0 .In practical situations, the value of P 0HND is often unknown, To establish control limits that are applicable in real-world scenarios, a preliminary sample is taken from the IC process in order to estimate the value of P 0HND .This preliminary sample helps in determining the appropriate control limits for subsequent monitoring and control purposes.
The inner and outer control limits are then defined as follows: where X = X n denotes the mean failure time of items before time t 0 in a subgroup over a preliminary sample.The probability of declaring the running operation as out-of-control (OOC) under RSS when it is actually in control, is given as The probability of repetition for IC scenario is given as Hence the probability the ongoing process is declared to OOC when in fact it is IC using RSS is given as The effectiveness of the implemented CC is evaluated using the ARL.ARL quantifies the average number of subgroups observed before the process is declared as OOC.Hence the IC ARL using RSS is indicated by ARL 0HND orr 0HND and calculated as The in-control average sample size ASS 0HND of the proposed chart is given as Here, we consider a scenario where the scale parameter of the HND has been shifted due to an external source of variation from α 0 toα 1 = cα 0 .The value of c represents the magnitude of the introduced shift.The probability that an item failing before reaching the specified time t 0 represented by P 1HND is given as The probability of the process being in control (IC) when, in reality, it has undergone a switch due to a change in its scale parameter is calculated as The probability of repetition for OOC scenario is given as Hence the probability the ongoing process is declared to OOC when in fact it is OOC using RSS is given as ARL for shifted process under RSS denoted ARL 1HND orr 1HND is given as The OOC average sample size ASS 1HND of the proposed chart is given as To calculate the ARLs for the proposed control chart, the following algorithm is implemented: (1) Fix the value of IC ARL say(r 0HND ) and n (2) Determined the value of q, W 1 andW 2 based on the given sample size n for which ARL 0HND in Eq. ( 23) is nearer to r 0HND .(3) Utilize the values of q, candn obtained in step 2 to calculate the value of ARL 1HND and ASS 1HND using Eqs.(29-30) for different values of c.

Designing of the control chart using Half Exponential Power Distribution
Now, we suppose that the failure time of item follows the HEPD.The probability that an item fails before reaching time t 0 is given by where t 0 = q 1 µ 0 and q 1 represents the truncated constant for HEPD.Additionally, we substitute the value α in term of µ using Eq. ( 11), resulting in the following expression as The process is considered to be IC when µ = µ 0 (or α = α 0 and = 0 ) then Eq. ( 34) can be written as ( 28) The inner and outer lower and upper control limits for proposed np charting structure using HEPD under RSS is as follows.
The outer control limits for proposed chart are given as The inner control limits for proposed chart are given as here W 1 andW 2 are the control coefficients with W 1 being greater than W 2 .It is important to note the if W 1 = W 2 the proposed chart is converted to Naveed et al. 8 based on HEPD.The working procedure for the presented chart using HEPD is same as we have early discussed for the case of HND.For unknown value of P 0HEPD , the working inner and outer control limits are where X = X n denotes the mean failure time of items before time t 0 in a subgroup over a preliminary sample.The probability of declaring the process as OOC under RSS when it is in fact IC is given as The probability of repetition for IC scenario is given as Hence the probability the process is declared to OOC when in fact it is IC using RSS is given as Hence the IC ARL using HEPD under RSS is indicated by ARL 0HEPD orr 0HEPD and calculated as The in-control average sample size ASS 0HEPD of the proposed chart based on HEPD is given as Evaluation of the suggested CC using HEPD under RSS when its parameters are shifted In this section, we examine three different scenarios to evaluate the proposed chart.Each scenario involves a specific type of shift in the parameters of the distribution.These cases are as follow: www.nature.com/scientificreports/Shift in scale parameter.In this scenario, we explore the effects of a shift in the scale parameter of the distribution on the performance of the chart, aiming to assess its effectiveness.We consider a situation where the scale parameter, initially represented as α 0 undergoes a transformation to α 1 , where α 1 = eα 0 and e represents the magnitude of the introduced shift.The probability of an item failing before the specified time t 0 is denoted by P 1HEPD , which can be calculated as follows: Rewrite Eq. ( 35) Since, scale level is changed from α 1 = eα 0 , as a result the mean value of HEPD is also changed as µ 0 = µ 1 by substituting this information in above expression we have Now the probability that operation is IC for shifted process due to the change in scale parameter as follows: The probability of repetition for OOC scenario is given as Hence the probability the ongoing process is declared to OOC when in fact it is OOC using RSS is given as ARL for shifted process under RSS denoted ARL 1HEPD orr 1HEPD is given as The OOC average sample size ASS 1HEPD of the proposed chart is given as Shift in shape parameter.In this scenario, we consider a situation where the shape parameter of the distribution is shifted.We explore the impact of this adjustment on the probability of an object collapsing before a predefined time period, denoted as t 0 .Suppose that the shape parameter, initially represented by 0 has been modified to 1 , Where 1 = r 0 .Here, r signifies the extent of the introduced shift.Now, the probability of an object collapsing sooner than the predefined period t 0 designated by P 2HEPD is calculated as.
Rewrite Eq. ( 35) As shape parameter is shifted from 1 = r 0 , so its mean level is also changed as µ = µ 1 by substituting this information in above expression we have Now the probability that process is IC for shifted process due to the change in shape parameter as follows: Hence the probability the ongoing process is declared to OOC when in fact it is OOC using RSS is given as ARL for shifted process under RSS denoted ARL 2HEPD orr 2HEPD is given as The OOC average sample size ASS 2HEPD of the proposed chart is given as Shift in both parameters.In the last case, we examine a scenario where both the scale and shape parameters of the distribution undergo shifts.This combination of shifts presents a more complex situation, as both the scale and shape parameters of the process are changed simultaneously.We suppose that scale parameter is shifted as α 1 = qα 0 and shape parameter is shifted as 1 = w 0 .The probability of an item failing before the predeter- mined period t 0 designated by P 3HEPD , can now be calculated as: Rewrite Eq. ( 35) As both parameters are shifted as α 1 = eα 0 and 1 = r 0 , so its mean level is also changed as µ = µ 1 by sub- stituting this information in above expression we have Now the probability for IC when in fact it is OOC as follows.Now the probability that process is IC for shifted process due to the change in both parameters as follows: The probability of repetition for OOC scenario is given as Hence the probability the ongoing process is declared to OOC when in fact it is OOC using RSS is given as Vol.:(0123456789)The OOC average sample size ASS 3HEPD of the proposed chart is given as We will now proceed with the following calculations to create a Table of ARLs for the suggested CC using HEPD for all three cases.
(1) Set the values of the ARL (r 0HEPD ) , shape parameter, and sample size accordingly.
(2) Determine the values of q 1 , W 1 andW 2 based on the specified sample size n , so that ARL 0HEPD in Eq. ( 33) is approaches to r 0 .(3) Use the values of q 1 , W 1 andW 2 attain in step 2 to calculate the value of ARL 1HEPD , ARL 2HEPD , ARL 3HEPD , for different values of shifted constant.
Record the calculated ARL values in a table, along with the corresponding shifted constants.By following these steps, we can generate the Table of ARLs for the suggested CC using HEPD for all three cases.

Results discussion
In this section, we discuss the results obtained from two life data distributions.The values of ARL 1HND , ARL 1HEPD , ARL 2HEPD and ARL 3HEPD for various shifts in scale and shape parameters are given in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.These Tables display the ARLs of the proposed CC using HND and HEPD for different shifts in shape and scale parameters, as well as sample sizes, when the desired ARLs are set to 300, and 370.By analyzing these tables, we can identify certain patterns in the ARL values of the recommended control chart.
1.As we notice that larger shifts in the parameters result in a rapid decrease in the OOC ARL values for both distributions.Let's consider the case of HEPD, where the initial ARL value r oHEPD is set to 370, is 4, shifted constant e is 0.97, and the sample size is 30, the OOC ARL value is 65.45.However, if we decrease e to 0.70 while keeping the other parameters constant, the OOC ARL drops significantly to 1.86.The decreasing trend in ARL for larger shifted values using both HND and HEPD are also shown in Figs. 3 and 4.This demonstrates that larger shifts in the parameter have a notable impact on reducing the value of OOC ARL when the process goes out of control.2. Furthermore, we observe that increasing the value of shape parameter leads to a decrease in the value of OOC ARL.For instance, when r oHEPD is set to 370, is 2, shifted constant e is 0.97, and the sample size is 20, the OOC ARL value is 42.38.However, if we increase to 4 while maintaining the other parameters, the (65)

Advantages of recommended control chart
In this section, we conducted several comparisons between the proposed control charts (CC) and other existing charts, including the Shewhart type chart proposed by Naveed et al. 8 and the chart proposed by Aslam and Jun 1 .Additionally, we compared the suggested CCs with the Exponential Distribution (ED) CC under repetitive sampling.The evaluation of these control charts is based on their ARL values, which are presented in Table 11.Upon analyzing Table 11, it is evident that both RSS based control charts outperformed the Naveed et al. 8 chart.For example, when considering the proposed control chart based on the HND with the following parameters: r 0HND = 370, n = 20, q = 0.8602, W 1 = 1.2538W 2 = 0.974 and shifted constant c = 0.95 the value of OOC ARL for proposed control chart is 23.98.In comparison, the OOC ARL value for the chart proposed by Naveed et al. 8 was 250.99.The comparison of these two charts is also shown in Fig. 5. Similarly, for the proposed chart based on HEPD with parameters r 0HEPD = 370, n = 20, q = 0.8561334, W 1 = 1.0607,W 2 = 1.0064 and shift in  scale parameter e = 0.93 the value of ARL for proposed control chart is 13.82 and for Naveed et al. 8 it was 211.59 using HEPD.The comparison of these two charts is also shown in Fig. 6.Furthermore, we observed that the proposed HEPD-based CC performed better than the proposed HNDbased chart, and both of these proposed charts demonstrated superior performance compared to the ED based chart under RSS.For example, when r 0 = 370, n = 20, q = 0.8561334, W 1 = 1.0607,W 2 = 1.0064 and the shift in scale parameter e = 0.95 the value of ARL in the proposed control chart using HEPD is 21.27 and for HND the value of ARL is 23.98 using n = 20, q = 0.8602, W 1 = 1.2538 , W 2 = 0.974 and shifted constant c = 0.95.and in the cases of competitor exponential distribution based chart it was 52.22.
Additionally, we observed that the proposed control charts exhibited smaller ARL values compared to the charts proposed by Aslam and Jun 1 .For example, when utilizing the proposed control chart based on HEPD with parameters when r 0 = 370, n = 20, q = 0.8561334, W 1 = 1.0607,W 2 = 1.0064 and the shift in scale parameter e = 0.93 the value of ARL for the proposed control chart using HEPD is 13.82 and for HND the value of ARL is 15.60 using n = 20, q = 0.8602, W 1 = 1.2538 , W 2 = 0.974 and shifted constant c = 0.93.and for the cases of Aslam and Jun 1 it was 225.16.Overall, these comparisons demonstrate the superior performance of the proposed control charts in detecting variations and maintaining process control compared to the alternative approaches Table 4.The ARL 1HEPD and ASS 1HEPD values of proposed CC using HEPD when its scale parameter is shifted using r oHEPD = 300 and n = 30.Table 5.The ARL 1HEPD and ASS 1HEPD values of proposed CC using HEPD when its scale parameter is shifted using r oHEPD = 370 and n = 20.

Simulation study
In this section, we demonstrate the effectiveness of the proposed idea through the use of simulation data.We assume that the process is initially operating under stable conditions, and we generate 25 observations of subgroup size 20 from the HEPD using the RSS.The in-control scale parameter is set to α = 1 , and the shape parameter is set to = 2 .Next, we introduce a disruption to the stable process by altering its scale parameter due to external factors.The new scale parameter, denoted as α 1 , is defined as α 1 = eα 0 , where α 0 is the initial scale parameter and e = 0.95 is the shifted constant.We generate another set of 25 observations of subgroup size 20 from the HEPD using RSS, this time with α 1 = 0.80 and = 2 , representing the shifted process.We suppose that ARL 0HEP = 370 .The truncated time t 0 = (q 1 * µ) , calculated as using parameter α = 1, = 2 and r 0 = 370 , gives the value of q 1 from Table 5 as q 1 = 0.8602 , and value of µ using Eq. ( 11) against param- eter α = 1, = 2 calculated as µ = 0.7978 and truncation time t 0 = 0.8602 * 0.7978 = 0.6863.The num- ber of items which are failing before the time 0.6863 in each subgroup is designated by X and reported in Table 12.Using these values, we calculate the control limits from Eqs. (19-22) with W 1 =1.2538,W 2 =0.9740 as LCL 1 = 7, UCL 1 = 12, LCL 2 = 8, UCL 2 = 13.To visualize the data set of failed items, we plot the values in Table 6.The ARL 1HEPD and ASS 1HEPD values of proposed CC using HEPD when its scale parameter is shifted using r oHEPD = 370 and n = 30.
= Fig. 7.The plot shows that the process is disrupted at the 41-th item, which corresponds to the 16-th observation after the shift in the process.This result is consistent with the value reported in Table 5.We also plot the data set of competitor chart proposed by Naveed et al. 8 which shows the in-control process in Fig. 8. Therefore, it is evident that the suggested control chart efficiently detects the early shift in the process.Overall, this analysis demonstrates the capability of the proposed control chart in identifying process disruptions and detecting shifts in a timely manner using simulation data.

Real life examples
The proposed control chart is applied and evaluated using real-life data obtained from a study conducted by Gui 35 .The dataset consists of plasma ferritin cluster measurements of 202 athletes collected at the Australian Institute of Sport.This dataset has been previously analyzed by multiple researchers, including Naveed et al. 8 , Elal-Olivero et al. 27 , Azzalini and Valle 38 and Cook and Weisberg 39 .The data follows a HEPD with a mean of 76.88 plasma ferritin and a standard deviation of 47.50 plasma ferritin.The known scale parameter, α , is 97.1311, and the shape parameter, , is 2.5109.Assuming r 0HEPH = 300, n = 30 and q 1 = 0.49785 , we can obtain the value of P 0 using Eq. ( 36) as P 0 = 0.2913.The The inner and outer control limits using Eqs.(37-40)  are LCL 1 = 0, UCL 1 = 16, LCL 2 = 6, UCL 2 = 10 respectively, when W 1 = 3.185, W 2 = 0.844 .The truncated time t 0 = q 1 * µ 0 = 0.49785 * 76.88 = 38.27plasma ferritin.The following steps describe how the chart works:  www.nature.com/scientificreports/Stage 1: A sample of 30 items is randomly selected from the HEPD distribution with a scale parameter α = 97.1311and a shape parameter = 2.5109 .These items are then placed on a truncated time t 0 = 38.27plasma ferritin.The number of failed items (X) is counted during the test, and the results are plotted in Fig. 9.
Step 2: Declare the running process is under control if 6 ≤ X ≤ 10.
Step 3: The process is considered to be out if control if X > 16orX < 0.
Step 4: If the value of X lies in repetitive mode i.e. 0 ≤ X ≤ 6or10 ≤ X ≤ 16 then we take next values of X until we reach a decision of in-control or out of control process.

Concluding remarks
In this research paper, we introduced two attribute control charts specifically designed to detect early changes in process parameters.These charts are tailored for the HND and HEPD under the TTLT using repetitive sampling.We evaluated the performance of these charts by comparing them to existing competitor charts based on the ARL metric.The results of our study demonstrated that the proposed control charts exhibit superior performance in identifying minor process changes, as evidenced by their lower ARL values.This conclusion is supported by both simulation-based experiments and real-life datasets.The promising results obtained from these evaluations   highlight the effectiveness and relevance of our proposed control charting approach.In the context of these two proposed distributions, our analysis highlighted that the CC stemming from the HEPD exhibited superior performance in detecting smaller shifts within process parameter as compared to half normal distribution-based CC.Furthermore, we believe that this research opens up avenues for future investigations.In conclusion, our study introduces a valuable contribution to the field of control charts by presenting two attribute control charts specifically designed for the HND and HEPD under TTLT using repetitive sampling.The superior performance of these charts in detecting early process changes, as supported by simulations and real-life data, confirms their potential as effective tools for quality control.Investigating the control chart employing the cost model is a potential avenue for future research.Furthermore, exploring the control chart utilizing multiple dependent state repetitive sampling is another promising direction for future studies.

Figure 9 .
Figure 9. Graph of proposed CC using HEPD with real life data application.

Table 1 .
The ARL 1HND and ASS 1HND values of proposed CC using HND using r oHND = 300.

Table 2 .
The ARL 1HND and ASS 1HND values of proposed CC using HND using r oHND = 370.

Table 3 .
The ARL 1HEPD and ASS 1HEPD values of proposed CC using HEPD when its scale parameter is shifted using r oHEPD = 300 and n = 20.

Table 7 .
The ARL 2HEPD and ASS 2HEPD v values of proposed CC using HEPD when its shape parameter is shifted using r oHEPD = 300 and n = 20.

Table 8 .
The ARL 2HEPD and ASS 2HEPD values of proposed CC using HEPD when its shape parameter is shifted using r oHEPD = 370 and n = 20.

Table 9 .
The ARL 3HEPD and ASS 3HEPD values of proposed CC using HEPD when both parameters are shifted using r oHEPD = 300 and n = 20.

Table 10 .
The ARL 3HEPD and ASS 3HEPD values of proposed CC using HEPD when both parameters are shifted using r oHEPD = 370 and n = 20.

Table 11 .
Comparison of ARLs when r o = 370 and n = 20.