Abstract
The normality assumption is a significant part of the development of control charts. This underlying assumption of normality most likely does not hold true in real scenarios. One of such designs usually devised to observe the target parameter of the Maxwell quality characteristics is the -control chart. In general, quality practitioners preferably have to observe the scale parameter rather than in examined processes. The contemporary -control chart is relying on the -statistic which does not hold the unbiasedness property with respective to parameter of the Maxwell probability model. In view of this, implementation of the -chart is not an appropriate design in monitoring a real parameter of the underlying Maxwell data. To accommodate the monitoring of the parameter of the Maxwell model, a novel design of the -chart is mainly proposed in this work. To support a statistical understanding of the -chart, power function, characteristic function, and the average run length have been essentially established. The parameters of the -chart are determined from the results of the sampling distribution of the derived statistic. Analytical findings are further applied to determine the performance of the study proposal with its existing counterpart. Substantially, the better performance of the proposed technique has been observed because of statistical power used as a performance measure. Eventually, the computational plan of the -chart is considered both for the simulated and real datasets with the aim of illustrating the theory of the proposed design.
1. Introduction
There are two kinds of variations in all production processes, namely, normal and unnatural variations. Normal variations spontaneously cause harmless disparities in all systems, whereas unnatural variations are unique and trigger discrepancies because of such process defects [1]. The abnormal changes are especially important because their presence may harm the quality of the final product [2]. Regulation of the statistical process control (SPC) provides a valuable range of methods for managing irregular process variations [3]. One of the most powerful methods for process monitoring in SPC is the control chart approach. Control charts in the industry are now widely recognized and applied to learn, monitor, and improve the process sequentially [4]. The traditional control chat designs are based on the notion of the normal or approximately normal distribution of the quality characteristics under study. The assumption of normality in the process population in many cases is, however, not appropriate. The distribution may be skewed for many real datasets [5]. Accelerated life testing observations, measurements obtained from chemical operations, semiconductor analysis, and cutting wear processes are few examples of the positively skewed data [6]. If the quality variable is distorted in its distribution, analyzing the process with typical control charts can be misleading [7]. When the skewness increases due to population variability, implementation of the conventional Shewhart charts under asymmetrical distributions increases the risk of type-I error [8]. Several researchers have investigated quality monitoring in the context of positively skewed data [9]. The Shewhart control chart has been analyzed by Raza et al. [10] under the type-I censoring Rayleigh distributed data. The quality monitoring of the compressing strength of concrete for the log-normal model has been studied by Aichouni et al. [11]. Derya and Canan developed the design for the distorted distributions, namely, Gamma, log-normal, and Weibull [12]. Guo and Wang proposed a Weibull-based control chart using the type-II censoring data [13]. Designs of control chart rooting in mixture distribution models have been also suggested [14, 15]. Other related studies on this subject include Yang and Xie [16], Chan and Cui [17], Santiago and Smith [18], Chang and Bai [19], Aslam et al. [20], and Yang et al. [21].
In literature, the -control chart rooted on the Maxwell model has been proposed for monitoring the target parameter (i.e., square quantity of the monitoring quality) of the Maxwell process [22]; however, the·scale parameter of the Maxwell model is occasionally desired to be estimated rather than .
A new design called the -control chart that allows monitoring of the parameter of the Maxwell process is presented in this study. The structure of the existing -control is definitely not relying on an unbiased statistic of so unable to observe the target parameter of the Maxwell quality characteristics.
Discussion is structured in the following sequence. Section 2 discusses the current model of the -chart. A novel design of the -chart is discussed in Section 3. The performance study of the proposed design is conducted in Section 4. Construction of the suggested -chart using simulated data is presented in Section 5. Section 6 offers a comparative analysis. In Section 7, the implementation of the proposed chart with an illustration of real-world data is given. Eventually, the results of the study are concluded in Section 8.
2. The Existing Approach of the -Chart
The theoretical model employed in the design of the -chart is rooted on the assumption that the data generating process supports the Maxwell model with a single scale parameter [22]. The quality of interest is said to assume the single parameter Maxwell probability model if it takes the cumulative distribution function (CDF) and probability density function (PDF) as described below:where .
Assuming the scale parameter equal to 0.75, the graphical overview of the CDF and PDF of the Maxwell distribution is depicted in Figure 1.
It is clearly shown in Figure 1 that the Maxwell distribution is skewed particularly for a smaller value of . The scale parameter also plays a significant role in describing the shape of the curve. The PDF curve becomes symmetric for the greater value of , while CDF progressively reaches to unity for increased values of . In certain real-life cases, the value of the characterised Maxwell distribution parameter is not known. To find the unknown quantity , a variety of estimation methods may be used [23]. An estimate of using the maximum likelihood approach has been given by Krishna et al. [24]:
The -chart is now derived on the random variable as defined below:where the random variable assumes the Maxwell distribution, and the description refers to the defined fixed realization .
The distribution underlying random variable is the Gamma distribution having parameters and . The -chart parameters based on the estimator defined in (3) are given bywhere is an unbiased statistic of the target parameter and the multiplier is a percentile of the Gamma probability model at given value of i.e., probability of incorrect out-of-control signals.
Note that the specified statistic used to estimate the quantity is not in the same measuring unit as the actual quality variable is observed. Moreover, the resulting estimator is not unbiased with respect to parameter . A new proposal of the -chart for monitoring the stability of target parameter of the Maxwell process has been suggested in the next section.
3. Statistical Development of the -Chart
The following -statistic has been considered in the development of the -chart:
Using the following theorems, the distribution of -statistic can be determined.
Theorem 1. The ratio of the random variable is a Gamma distribution having parameters and 2 if the underlying distribution of the random variable is Maxwell with a single scale parameter , i.e.,
Proof. Suppose that the random variable has Maxwell distribution with a real parameter ; subsequently, the PDF is given byThe transformation is a one-one transformation from to with the inverse solution of is given byConsequently,where is the Jacobian of the transformation.
Thus, the distribution of random variable can be immediately written asSolving (10) gives the density of :Thus, .
Theorem 2. If are independent identical distributed Gamma random variables each having the PDF as defined in (11), thenwhere .
Proof. Since each with independent and identical distributed property, therefore using the reproductive property of Gamma distribution [25], we can write:and hence, it is proved.
Theorem 3. The transformation follows the chi distribution (-dist) with parameter if .
Proof. Given that , thus PDF of is defined byThe transformation is a one-one transformation from to with the value of inverse solution and Jacobian:Thus, the new random variable gets the PDF asFurther simplification provideswhich is the PDF of -dist with parameter, and hence, it is proved.
Now, the mean and variance of can be achieved asConsequently, mean and variance of the estimator would be set by the following relation:This providesThe relationship defined in (20) yields the mean and variance of the -estimator as follows:where the function depends on . Besides, the -statistic is not an unbiased estimator of the target parameter , as indicated in (21). The unknown parameter of the underlying Maxwell quality characteristic can therefore be estimated by an unbiased estimator :The percentiles of -statistic are essential for probability limits of the -chart. As the random variable follows -dist with degree of freedom , the -percentile of the statistic can be written asHere, denotes the CDF of the -dist.
Thus,Further simplification of (24) implies thatThus, the of the -chart at a given level of significance is provided bywhere and .
In addition, the based on the estimated value of is given bywhere and .
The coefficient pairs and are the quantiles of the -dist with df multiplied by the constants and , respectively. These coefficient values at given the false signal probability and various values are calculated and given in Table 1.
A general model based on -sigma limits may also be developed for any chart. To this end, two initial moments of the -statistic are required. Utilizing properties (21) of the -estimator, the -sigma limits are given bywhere is a known quantity, is the quantile point of the -dist at specified α, and the customary constants are given bywhere factor is calculated in Table 2 for fixed and .
If the in-control is not provided, the unbiased estimator specified in (22) can be used to estimate it. Let us assume that initially samples are present, and let be an estimator, computed for the sample; then, the samples’ average isThus, the model of the -control chart could be defined asTypically, constants and are given byThe resulting constants , , , and are listed in Table 3 for defined and sample size .
4. Performance Measures
Several statistical metrics such as the power curve (PC), operating characteristic curve (OCC), and run-length (RL) distributional properties are readily available to find out the performance of a control chart. The OCC and PC are employed to identify the control chart’s ability to respond against an infrequent change in the quality process observed, while the metric reflects the expected number of points plotted prior to a certain change is identified for the process operating under any potential cause. As the approach to the control chart is identical to the strategy of hypothesis testing, thus the process conditions can be set as (i.e., the process is in-control) (i.e., the process is shifted), where with a shift constant . Let the process be undergone some special cause that led to a change from to in the control value. In relation to type-II error, the ability of the -control to not detect this shift can thus be defined aswhere and represent upper and lower probability bounds of the design.
Simplifying (33) leads towhere is the -dist function with df.
Subsequently, out-of-control ARL and power function can be obtained as
For , (35) turns into (in-control average run). In order to illustrate the implementation of (35), let us assume that , and we are desired to know the probability of identifying a shift of the amount in parameter of the Maxwell distributed quality at a specified value of taken after the shift has occurred. Thus, the efficiency of the proposed chart for detecting this change is summarized in the form of power curves at equal to 0.0027 in Figure 2.
From examining power curves in Figure 2, we note that the change of various amounts is efficiently detected by the suggested -chart. In addition, Figure 2 shows that the probability of identifying the change of a particular amount increases with the rise in the sample size. As an illustration, a fairly greater change of amount 4 in accordance with sample size equals to 2, 4, and 8, and the chances of detection are 46%, 90%, and 95%, respectively.
Along with power curves, a set of plots for the proposed chart have been considered. In Figure 3, the curves for the same sample sizes are presented.
To demonstrate the use of Figure 3, if we desire to use a sample size of to detect a shift of amount , then the average number of samples essential to determine the shift would be . Note, also, that if sample size increases to 30, is decreased to about 1.
5. Simulation Analysis
Simulated data has been employed in this section to establish the parameters of the -chart. The Maxwell model with parameter has been used for the selection of twenty-five preliminary sample subgroups, each with 4 observations. The -statistic has now been computed from each subgroup observations. Consequently, from all 25 batches, 25 values have been obtained. The factors and are obtained from Table 3 for the fixed probability and . The parameters of the -chart are thus determined as
Figure 4 illustrates the graphical display of the values along with computed parameters of the -chart. All the plotted points in Figure 4 represent a nonrandom pattern within the -sigma limits constructed, which explicitly implies that the underlying process is stable. Similarly from Table 1, coefficients and are used to establish the . The thus obtained at chosen sample and false alarm risk are given below:
These with calculated values of estimator are depicted in Figure 5.
Figure 5 indicates again that there is no sign of an out-of-control scenario and for the current process are slightly wider than -sigma limits. We deliberately implemented various magnitudes of shifts in data to determine the sensitivity of the -chart against different shifts i.e., how the proposed chart would identify these shifts. After batch 15, a shift of upward amount has been incorporated, and the rest 10 subgroups with 5 observations each are generated from the Maxwell probability model with parameter of scale equal to 2, 2.5, 3, and 3.5, respectively. The -control chart with conventional -sigma limits have been employed on all 25 subgroups. For each subgroup, the -statistic has been determined and graphical displayed in Figure 6.
(a)
(b)
(c)
(d)
Figure 6 demonstrates that the last 10 points are all above the central line and in one direction, clearly showing that the observed process is in an out-of-control situation. Additionally, all points of the last 10 subgroups are not above the upper control limit, as shown in Figure 6(a), while in Figures 6(b)–6(d), all points are above the designed upper limit. Certainly, the rationale is the quantity of shift that is comparatively larger in latter cases. The proposed chart effectively identified the shift, thus demonstrating an efficient approach to detecting a shift of any amount.
6. Comparison Study
This section presents a comparative study to determine the efficacy of the proposed design compared to the alternative counterpart of the -control. For this comparison, many performance metrics may be used, but PC construction is commonly employed in numerous studies [26, 27]. Therefore, the efficiency of the design proposed is assessed in the form of the power function. We have constructed power curves for the proposed chart using Monte Carlo simulations under with sample sizes and , and false alarm risk is performed. Different values of the constant have been considered in the construction of power curves. In Figure 7, the probabilities of identifying a dramatic change in the parameter of the Maxwell distribution are graphed.
(a)
(b)
It is evident from Figure 7 that the power is very close to the fixed value when the process is under control. If the process goes in an out-of-control situation, with the rise of , power increases for varied values of . The proposed control chart, however, is evidently observed in Figure 7 to be more efficient and responsive to detect the change of a certain size at a specified value of . For example, a shift of size has been detected by the proposed design with a probability 0.821 for a sample size which is greater than the respective probability of 0.397 of the existing design. In addition, the proposed chart more effectively detects the shift with increased sample size.
7. Real Application
The real dataset on carbon fiber strengths has been employed to illustrate the computational process of the -control chart. The strength data was originally reported as an application example by Badar and Priest [28]. Data show the strengths of single fibers trialed under stress at various gauge levels. Here, we consider 60 measurements of the strength data tested at gauge length 20 mm. To acquire a good fit, original data is slightly modified by subtracting 1.2 from each value to shift the origin of the dataset. This modified data is then employed in further analysis. Initially, we have used the probability plotting as a graphical tool to check if the strength data are consistent with a hypothesized Maxwell distribution based on a subjective visual analysis of the given data. The general process is straightforward and can be carried out easily using R software. The probability plot for our sample dataset is shown in Figure 8.
In Figure 8, a straight line has been drawn through plotted points as the best fit line assuming the Maxwell model. Figure 8 shows that all plotted points are covered up by this straight line, and the Maxwell model reasonably portrays the given dataset. Following the graphical approach, the data are further analyzed with formally goodness of fit Kalmogorov–Simirnov (KS) test. The value of the KS test for the analyzed dataset is 0.670 which indicates that the Maxwell model may be appropriate for given data. This dataset has now been used for the proposed chart for the Maxwell scale parameter monitoring. Ignoring the last three observations in the modified dataset, all 60 observations are classified simply in 12 subgroups each of size 5, as shown in Table 4.
These subgroups are further considered in the construction of -sigma limits and for the -control chart. Now, -coefficients for and are and . Thus, the -sigma limits using (31) are computed as
These computed statistics in the proposed chart are displayed in Figure 9.
Figure 9 shows that the proposed chart has not indicated any unstable state i.e., no single sample lies outside the designed limits of the chart and plotted statistic displays a random pattern. Similarly, constants and for and sample size have been utilized from Table 1 to find the of the -chart. This provides the parameters of the proposed chart in terms of as and . The chart with is depicted in Figure 10. The constructed in Figure 10 again justifies that the quality characteristic under investigation is in the in-control state.
8. Conclusions
The Maxwell distribution properties are described, and the -control chart as a novel design for monitoring the quality characteristic, which is best defined by the Maxwell model, has been proposed in this work. The properties and power function have been derived to find out the performance of the suggested control. Numerical results obtained in the simulated environment demonstrate that, with the increased process shift in addition to increment of sample size, -control chart performs efficiently for identifying the shift. The -chart has been compared to the contemporary design of the -chart. The power curves produced in this comparison have shown that the suggested chart is more effective and powerful than the competitor -chart in identifying the infrequent changes in the target parameter of the Maxwell model. On the contrary, the distribution also indicates that the proposed chart is more responsive with a substantial shift in the monitoring parameter. A simulated date from the Maxwell distribution has been applied to exhibit the computational methodology of the proposed chart. Finally, the implementation procedure has also been demonstrated using real example data on fiber’s stress strength.
Data Availability
The data used to support the findings of the study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.