An enhanced nonparametric quality control chart with application related to industrial process

In various practical situations, the information about the process distribution is sometimes partially or completely unavailable. In these instances, practitioners prefer to use nonparametric charts as they don’t restrict the assumption of normality or specific distribution. In this current article, a nonparametric double homogeneously weighted moving average control chart based on the Wilcoxon signed-rank statistic is developed for monitoring the location parameter of the process. The run-length profiles of the newly developed chart are obtained by using Monte Carlo simulations. Comparisons are made based on various performance metrics of run-length distribution among proposed and existing nonparametric counterparts charts. The extra quadratic loss is used to evaluate the overall performance of the proposed and existing charts. The newly developed scheme showed comparatively better results than its existing counterparts. For practical implementation of the suggested scheme, the real-world dataset related to the inside diameter of the automobile piston rings is also used.


Wilcoxon signed-rank test
Suppose V is the quality characteristic and its median is the target value ( α o ), and V it represents the ith observation in the tth sample of size n > 1 for i = 1, 2, 3, • • • , n and t = 1, 2, 3, • • • .Let R + it measures the absolute rank differences |V it − α o |, i = 1, 2, 3, • • • , n , then the WSR statistics following Graham et al. 25 is described as: Here, WR t is the sum of I it R + it .It is revealed that that the WSR statistic is linearly associated to the Mann-Whitney test statistic W + n through WR = ((cf.Gibbons and Chakraborti 33 ).Based on the relationship between WRandW + n , the WSR statistic has a mean and variance as follows: Also in WSR test, it is assumed that the distribution of differences between paired observations is symmetric.This implies that there should be an equal number of positive and negative differences, and they should be evenly spread out.Violation of this assumption can undermine the validity of the WSR test results.

Nonparametric SEWMA signed-rank (NPSR-E) control chart
Graham et al. 25 developed an NPSR-E control chart by accumulating the statistic WR t of Shewhart-type SR chart of Bakir 22 .The charting statistic of the NPSR-E control chart is written as based on (1): where the initial value of the statistic given in ( 2) is taken as zero (ES o = 0)and 0 < β ≤ 1 is the smoothing parameter of the NPSR-E chart.The upper and lower time-varying limits of the NPSR-E control chart by using the mean and variance of (2) are as follows: The "steady-state" control limits when t → ∞ are given below: If a single point of ES t statistic is plotted outside the control limits then the process is considered out of control (OOC). (1) µ = 0 and σ 2 = n(n + 1)(2n + 1) 6 (2) The NPSR-H control chart was introduced by Raza et al. 19 for monitoring the process location and exhibit early shift detection.The design structure of NPSR-H chart that is based on (1) is described below: where the average of first (t − 1) samples of WR statistics and the initial value of WR o is set equals to zero.For obtaining the parameters of the plotting statistic H t , given in (5) can also be written as below: For obtaining the mean of this chart applying expectation on (6): We will get the variance of H t as follows: The control limits corresponding to the NPSR-H chart are given below: where T denotes the width of the control limits based on the IC ARL and Var(H t ) is defined in (7).In (7), if the no of samples approaches to infinity then the variance will be β 2 n(n+1)(2n+1)

6
. Therefore, the control limits of the NPSR-H chart based on asymptotic variance are given as below (cf.Raza et al. 19 ):

Nonparametric SEWMA sign (NPSN-E) control chart
Yang et al. 11 suggested the NPSN-E for monitoring of changes in process location.The plotting statistics of NPSN-E chart is defined as: where M t = n i=1 I i and The mean and variance of the NPSN-E chart are: If the time is infinite then the asymptotic variance NPSN-E chart, σ 2 0 = β 2−β np 1 − p .Therefore, the asymptotic control limits of NPSN-E chart are: where β and L are set according to the desire ARL o .

Parametric SEWMA control chart
Robert 3 developed SEWMA control chart for detecting of the small shifts in process location.Robert 3 defined the plotting statistic of SEWMA chart as follows: where β lies in the range 0 < β ≤ 1 .The intial value of EWMA 0 = µ o .The IC mean and asymptotic variance of the SEWMA chart are: The control limits constructed on the basis of the aforementioned mean and variance for the SEWMA chart is given as: (5)

Parametric SHWMA control chart
Abbas 7 proposed a SHWMA chart that allocates equal weights to the previous observation, unlike the EWMA chart.The plotting statistic of HWMA chart is given as: where is the mean of the previous t − 1 samples and the starting value of Z 0 = µ o .The control limits of SHWMA chart based on the statistic ( 14) are defined as: where L is the coefficient of the control limits and its value depends on the ARL o .

Design structure of the proposed NPSR-DH control chart
The newly proposed NPSR-DH control chart is based on the WSR test, an alternative of one sample t-test for distribution-free data which means the assumption of normality or particular distribution is not required.The plotting statistic of the NPSR-DH chart based on the statistic of the WSR test is as follows: The mean of the previous (t − 1) statistics of WR is WR t−1 = t−1 i=1 WR i t−1 and the initial value of WR o = 0 .If β sets equal to one in (14) then it turns to Shewhart type WSR chart of Bakir 22 .The mean and variance of DH t can be obtained as follows: and The limits of the NPSR-DH chart based on the aforementioned E(DH t ) and Var(DH t ) are described as: (13) where T indicates the coefficient of control limit to determine its width according to the pre-defined IC ARL i.e., ARL o .The statistic DH t is plotted against the limits given in (17).If any value of DH t is plotted beyond the limits given in ( 17), the process is seemed to be OOC, else, it is considered to be IC.

Performance evaluation
In literature, several statistical measures are available to judge the performance of control schemes.Some of them are for a single value of shift while others for a range of shifts.The ARL is the one used more often.There are two types of ARL that is ARL o and ARL 1 are used to assess the performance of the control chart.The ARL o is the  expected number of samples before an OOC point is detected when the process is IC while ARL 1 is the expected number of samples before an OOC signal is received when the process is shifted to an OOC state.A chart is considered more effective as compared to other charts if it has a minimum value of ARL 1 for the same amount of shift i.e., δ .We have also assessed the performance of the NPSR-DH chart by using the other measures of the RL such as (standard deviation of the RL (SDRL), a median of the RL (MDRL), and some percentile points of the RL ( PRL 25 , PRL 50 &PRL 75 )) because of the skewed behavior of the RL distribution (cf.Naveed et al. 34 , Bataineh et al. 35 and Shafqat et al. 36 ).The RL measures are calculated through Monte Carlo simulations in R language.The ARL o of the newly developed NPSR-DH chart depends on the values of n, Tandβ, i.e., ARL o = f (n, β, T).The computational algorithm of the NPSR-DH chart in the form of the flow chart is presented in Fig. 1.
The values of T of the NPSR-DH chart for several choices of β and sample size (n) for ARL 0 ≈ 370&500 are reported in Table 1.It is noted that the values of are increased as β increases when ARL 0 ≈ 370&500 (cf.Table 1).It is also noted that for fixed β the value of T also increases as n is increased (for instance β = 0.20, n = 10, T = 2.31, β = 0.20, n = 15, T = 2.37 , andβ = 0.20, n = 20, T = 2.4 , when ARL O = 500 (cf.Table 1).

Normal and non-normal environments
Both the normal and non-normal environments are used to evaluate the performance of the proposed chart.The following distributions are employed to assess the performance of the proposed NPSR-DH chart: (a) A bellshaped, symmetrical normal distribution denoted by N(0,1), with mean and variance; (b) Platykurtic-shaped symmetrical Student's t-distribution with 4 degrees of freedom and a heavy tail; (c) Laplace distribution, also known as the double exponential distribution, denoted as Lap(0, 1/ √ 2 ); (d) Logistic distribution represented as Log(0, √ 3/π); and (e)Contaminated normal distribution with 5% level of contamination CN(0.05) used to assess the behavior of the proposed chart in the presence of outliers.All the distributions under consideration are adjusted with zero mean and unit standard deviation for valid comparisons.The density functions these distributions are detailed in Table 2.
(1) Standard normal; , where; V ∈ R, T 0 = 0andσ 2 = u u+2 and u = 4 (3) Logistic; , where; , where;V ∈ R, T 0 = 0andσ 2 = 0.95 + 0.05σ  www.nature.com/scientificreports/consideration (cf.Table 2).Furthermore, the distribution of the ARL o is positively skewed for all distributions i.e., ARL o > PRL 50 (cf.Table 2).The ARL 1 performance of the NPSR-DH chart is relatively better in the case of a Laplace(0,1/ √ 2 ) distribution against other distributions.For instance when δ = 0.1, 0.25 , the corresponding ARL 1 values for Lap(0,1/ √ 2 ), N(0,1), t(4), t(8), Log (0, √ 3/π) , and CN(0.05) distributions are (16.95, 4.83), (24.94, 6.57), (18.31, 4.95), (21.97, 5.85), (22.10, 5.87), and (20.69, 5.51), respectively (cf.Table 2).Moreover, the ARL 1 performance of the NPSR-DH chart for the Log (0, √ 3/π) distribution is comparatively worst against all other distributions (cf.Table 3).A measure called a percentage decrease in ARL(PD ARL ) is also used for comparison purpose and mathemati- cally, it is defined as:  www.nature.com/scientificreports/A control chart with a larger PD ARL value is considered to be efficient.The PD ARL is just used to compare the performance of the control charts for specific shifts and not use for assessing the overall performance of the control charts.So to evaluate the overall performance of the proposed and existing charts, a well-known overall performance measure called the extra quadratic loss ( EQL ) suggested by Zhang and Wu 37 is also utilized in this study.The EQL is the weighted average of ARL 1 values over a range of shifts (δ) by taking into account δ 2 as weight.Mathematically, the EQL is defined as: A control chart can be considered best among others if it shows minimum value of EQL provided that all control charts have same ARL 0 values.
The distributional comparisons between proposed and existing charts are given below:

Industrial application of the NPSR-DH chart
In this section, we have presented the real-life industrial application of the NPSR-DH chart.In this industrial process the "inside diameter of the automobile engine piston rings" is considered as the variable of interest (cf.Montgomery 1 ).The piston ring is a cap with a spring-like property and it is positioned on the outside diameter of an automobile engine.The main advantage of the piston ring is that its stops the leakage of the boiling gases from the combustion slot.The size of the piston ring diameter of an engine is generally considered to be within the range of 74.000 mm ± 0.05 mm.The pictorial description of the automobile engine piston rings is presented in Fig. 3 (cf.Zafar et al. 38 ).Also, the values of the design parameters of the proposed NPSR-DH and existing charts for the piston ring data set is reported in Table 9 when ARL o ≈ 500.
The NPSR-E and NPSR-H charts issue the OOC signals at the 14th sample (cf. Figure 4a and Fig. 4b).But, the proposed NPSR-DH chart issue the OOC signal at the 13th (cf.Fig. 4c).So, the proposed NPSR-DH chart has an enhanced shift detection capability than to the existing counterparts and these results also coincide with the results given in Section "Performance evaluation".

Conclusion and recommendations
Control charts are magnificent statistical monitoring techniques that are commonly applied in industrial and non-industrial processes.For producing and manufacturing high-quality products these industries required highly sensitive monitoring devices which trace deteriorations in the processes effectively.In most of the ongoing processes assumption of normality is hard to meet which leads to parametric monitoring structures invalid comparisons.In this article, a new NPSR-DH scheme has designed based on Wilcoxon signed rank statistic to address small shifts in the process location efficiently.The proposed design proved in control robust for all distributions and more effective for heavy tailed and skewed distributions.The EQL values indicate that the proposed NPSR-DH chart shows superiority against the existing counterpart's charts selected for this study.
The scope of the charting structure explored in this article may also be extended for individual observations based on univariate and multivariate versions.Moreover, the Bayesian aspects of the proposed design structure may also be the potential for future research.

Figure 1 .
Figure 1.The flow chart based compuatational algorithm of the NPSR_DH chart.

Figure 3 .
Figure 3. Pictorial description of inside diameter of the automobile piston rings.
1 L is the control limits coefficient, which depends on β and ARL o . where

Table 5 .
The OOC run length characteristics of the proposed and existing charts for β = 0.05andn = 10atARL o ≈ 500 under t(4).
Vol:.(1234567890) Scientific Reports | (2024) 14:13561 | https://doi.org/10.1038/s41598-024-64084-7www.nature.com/scientificreports/Distributional comparisons based on ARL 1 values A comprehensive comparison of the newly developed control chart versus existing counterparts is done, in this section.Also, the plotting statistic and control limits of the existing and proposed charts are reported in Table 1.The efficiency comparisons of the NPSR-DH chart with the SEWMA chart, SHWMA chart, NPSN-E chart, NPSR-E chart, and NPSR-H chart are made on the basis of ARL, SDRL, and MDRL values for all selected distributions, and results are reported in Tables 4, 5, 6, 7, 8.

Table 8 .
The OOC run length characteristics of the proposed and existing charts for β = 0.05andn = 10atARL o ≈ 500 under CN with 5% contamination.

Table 9 .
Values of design parameters of the existing and proposed charts.