Efficient homogeneously weighted dispersion control charts with an application to distillation process

Monitoring disturbances in process dispersion using control chart is mostly based on the assumption that the quality characteristic follows normal distribution, which is not the case in many real‐life situations. This paper proposes a set of new dispersion charts based on the homogeneously weighted moving average (HWMA) scheme, for efficient detection of shifts in process standard deviation (σ). These charts are based on a variety of σ estimators and are investigated for normal as well as heavy tailed symmetric and skewed distributions. The shift detection ability of the charts is evaluated using different run length characteristics, such as average run length (ARL), extra quadratic loss (EQL), and relative ARL measures. The performance of the proposed HWMA control charts is also compared with the existing EWMA dispersion charts, using different design parameters. Furthermore, an illustrative example is presented to monitor the vapor pressure in a distillation process.


INTRODUCTION
Statistical Process Control (SPC) is a tool-kit to enhance process performance by monitoring changes in process parameters. Walter A. Shewhart originally proposed the SPC concept in the 1920s. One of the most popular tools of SPC is control chart. The main aim of using control chart is to improve the quality of the process by detecting shifts occurring in either the process mean or/and the process variance as soon as possible to prevent them from occurring again.
There are two types of control charts, which are memory-type control charts and memory-less-type control charts. In recent years, the focus toward memory-type control charts has increased massively because of their high ability to detect small-to-moderate shifts. Page 1 and Roberts 2 proposed the CUSUM and the EWMA charts, for quick detection of small shifts in process parameters, respectively. These are the two most commonly used memory-type control charts for process monitoring because of their simplicity and good performance. These charts make use of current and previous sample information by accumulating small continual shifts and are helpful for reducing large financial loss, by quick detection of small shifts. The well-known Shewhart chart structures are only based on current sample information and hence are only suitable for handling large process shifts. More details about basic control chart designs can be found in Montgomery. 3 In process monitoring, the dispersion parameter is of more immediate concern. Indeed, there is an inverse relation between the improvement of the process and the process variance. Huwang et al 4 stated that monitoring shifts in process dispersion is more important compared to the monitoring of process mean. Many researchers, in recent years, devoted their studies on monitoring process dispersion using memory control charts. For instance, Abbas et al, [5][6][7] Abbasi and Miller, 8 Zhou et al, 9 Ahmad et al, 10 Castagliola et al, 11 Huang et al, 12 Haq, 13 Hossain et al, 14 Rajmanya and Ghute, 15 Osei-Aning and Abbasi, 16 Lee and Khoo, 17,18 Zaman et al, 19 and the references cited, therein.
The distributional assumption of most control chart structures for statistical process monitoring is that the quality characteristic follow the normal distribution; however, in practice, this assumption is mostly violated. Abbasi and Miller 20 indicated that control charts should be based on appropriate parent distribution of the process rather than just relying on normality assumption. Many authors have proposed EWMA control charts for monitoring process parameter when the assumption of normality is violated. Maravelakis et al 21 investigated robustness to nonnormality on EWMA dispersion charts based on individual measurements. Abbasi et al 22 investigated several EWMA dispersion charts for monitoring processes following normal and a variety of nonnormal distributions. Recently, Abbas 23 proposed a homogeneously weighted moving average (HWMA) control chart for monitoring small shifts in process mean. The HWMA chart assigns a specific weight to the current observation while the remaining weights are assigned equally to the past observations. This study presents the HWMA dispersion control chart based on a set of σ estimators. These estimators are sample range (R), standard deviation (SD), inter quartile range (Q), Downton's estimator (D), mean absolute deviation from median (MD), median absolute deviation (MAD), and two estimators based on interpoint distances, namely, S n and Q n . We have investigated processes that follow normal and nonnormal parent distributions. The distributions considered are the normal, two heavy-tailed symmetric and five skewed distributions. The distributions and the dispersion estimates investigated in this study are the same as the ones investigated by Abbasi et al 22  The remainder of the paper is organized as follows: The different dispersion estimates used in the study are given in Section 2. The design of the proposed HWMA dispersion chart is presented in Section 3 while Section 4 highlights the performance assessment and the steps taken in the simulation study. A comparison between the proposed HWMA and the EWMA dispersion control charts is given in Section 5 and an illustrative example is reported in Section 6. Finally Section 7 presents the conclusions of the study.

DISPERSION ESTIMATORS
This section describes a variety of σ estimators as described in the previous section. Suppose X is the quality characteristic of interest and 1 , 2 ,. . . , be a random sample of size n. Moreover, suppose be the ith order statistic (lowest to highest), be the sample mean, | | be the absolute value of X and̃be the sample median. The different dispersion estimates are briefly described below: (i) Sample range (R) based on order statistics: This is defined as (1) It is the most widely used dispersion estimator. It is the difference between the highest and lowest observations in the sample. R is too sensitive for extreme values and the normality requirement is not satisfied.
(i) Standard deviation (S) based on squared deviations from the mean: it is defined as Comparing S estimator with other estimators shows that S is the most efficient dispersion estimator for normal distribution but it is sensitive to outliers.
(i) Interquartile range (Q) based on order statistics: It is given as where 3 and 1 are the upper and lower quantiles. For large sample size, Q loses its efficiency while it is used as a robust measure of scale with a breakdown point of 25%. Abbasi and Miller 20 showed that for some nonnormal distributions, the Q chart performed better than R and S charts.
(i) Downton's Estimator (D) based on order statistics: It is defined as:

PROPOSED HWMA DISPERSION CHART
Let ( = 1, 2, …) represent a sequence of the observed values of an estimator T at time t, computed from a sample of size n. Here, T represents any of the dispersion estimators, mentioned in Section 2. The mean of the process is assumed to be stable as we are only interested in monitoring the process dispersion. Let the plotting statistic for HWMA dispersion charts be defined as: where is the smoothing parameter and it is selected to be between zero and one, that is, 0 < ≤ 1.̄− 1 is the average of previous (i − 1) dispersion statistics and is given as The initial value (̄0) is taken to be equal to the target mean of , that is, 0 (if target mean is unknown). Further, let Z defines relative dispersion statistic (ie, = ∕ ). Let ( ) = 2 and = 3 . 2 , and 3 are entirely dependent on sample size for a specific combination of T and parent distribution (see. . . . . . .).
Following Abbas, 23 we can show that: where E(T i ) = 2 and = 3 . For unknown process standard deviation, can be replaced by its estimator̂=̄2 .
Using these notations, the time varying upper control limit of the HWMA chart for the statistic is defined as: where is the width of the control limit and it is adjusted to achieve the desired in-control average run length (ARL 0 ). Furthermore, only the UCL is considered in this study as we are interested in detecting an upward shift in the process dispersion. Abbasi et al 22 reported the 2 and 3 values for different parent normal and nonnormal distributions, considering a variety of sample sizes.

PERFORMANCE EVALUATION OF HWMA CHART
The performance of control charts is usually evaluated using run length characteristics. The most common performance measures of control charts is the average run length (ARL). It is the average number of samples plotted before the detection of an out-of-control point. The in-control and the out-of-control average run lengths are denoted as ARL 0 and ARL 1 , respectively. For a comparison between control charts, the ARL 0 is fixed at a desired level and ARL 1 is compared, the chart with the lowest ARL 1 value has the best performance in detecting shifts. The distribution of ARL is mostly highly skewed; hence few researchers recommended to use some other run length measures such as median run length and standard deviation of the run length distribution. 22 So, some other performance measures have been proposed to evaluate the overall effectiveness of HWMA charts over the entire range of shifts. In this study, some useful measures to assess the performance of the proposed HWMA dispersion control charts are the average run length (ARL), standard deviation of the run length (SDRL), median run length (MDRL), extra quadratic loss (EQL), and the relative average run length (RARL).
Here, we used 10 4 Monte Carlo simulation runs developed in R language. Note that the results and report of study are based on subgroups size n = 5 and n = 7 using = 0.05 and = 0.25, respectively, for each HWMA dispersion chart with normal and nonnormal distributions. The main aim of this study is to investigate the performance of the HWMA charts and identify a chart (or group of charts) that is superior in the monitoring of dispersion for normal and nonnormal distribution processes. The nonnormal symmetric distributions are: logistic and Student's t. The skewed distributions are Weibull, chi-square, gamma and exponential, the extremely skewed distribution is log-normal. Furthermore, the density functions and the parameter values for different parent distributions used in this study are given below: Logistic ( , ), −∞ < ⟨∞, ⟩0 Student's t (k), > 0 Weibull( , ), > 0, > 0 Chi-square( ), > 0 Gamma( , ), > 0, > 0 Exponential( ), > 0 Log-normal ( , 2 ), −∞ < ⟨∞, ⟩0 Without loss of generality, we considered = 0 and = 1 for normal distribution, = 0 and = 1 for logistic distribution, k = 5 for Student's t-distribution, = 1.5 and = 1 for Weibull distribution, = 2 and = 1 for gamma distribution, = 1 for exponential distribution and = 0 and = 1 for the log-normal distribution. The extra quadratic loss (EQL) and relative average run length (RARL) measures are used to evaluate the overall shift detection ability of the proposed HWMA charts. These are briefly defined below:

4.1
Extra quadratic loss (EQL) and relative ARL (RARL) Reynolds Jr. and Stoumbos 27 and Wu et al 28 recommended the use of extra quadratic loss (EQL) and relative ARL (RARL) measures to evaluate the overall detection ability of control charts over the entire shift domain. EQL uses a weight 2 to compute the weighted average of ARL over the whole shift range. It is defined as: where ARL( ) is the ARL value of a specific chart at shift . A chart with the smallest EQL is considered better than competing charts.
RARL is defined as the ratio of the ARL value of a specific chart at a given shift and the ARL of benchmark chart. Mathematically, it is given by where ARL( ) is the ARL value of the chart for a specific shift and ARL benchmark is the ARL value of the benchmark chart, is mostly taken as a chart with the lowest EQL value.

PERFORMANCE COMPARISON
In this section, the ARL, SDRL, MDRL as well as the EQL and RARL performance measures of the proposed HWMA charts are compared for processes following normal and nonnormal distributions. Also, the performance of the proposed HWMA charts is compared with their respective EWMA dispersion charts. Table 1 reports the run length characteristics (ARL, MDRL, and SDRL) of the HWMA dispersion charts under the idea assumption of normality. To save space, only the ARL results are presented in Tables 2-9, for processes following other parent nonnormal distributions. In these tables, results are only reported for n = 7 and = 0.25. The control charts multipliers for each chart are given to achieve a desired ARL 0 set at 200 in the tables. The overall performance measures EQL and RARL for the different HWMA dispersion charts are reported in Table 9 (using n = 7 and = 0.25) and Table 10 (using n = 5 and = 0.05). To save space, we did not provide detailed results for other combinations of n and . The RARL comparison of the proposed charts is provided in Figure 1, considering using n = 5 and = 0.05. The control chart is said to be superior if its ARL 1 values are lower and have the minimum EQL compared to other competing charts. Furthermore, the sensitivity of the proposed HWMA chart for different smoothing constant and varying sample sizes n are evaluated for some special cases.

Proposed HWMA chart
Normal distribution: The ARL performance of a variety of HWMA dispersion charts is compared for normally distributed quality characteristics in Table 1 whereas EQL and RARL comparison is provided in Figure 1 and Tables

Effect of sample size (n)
We evaluated the performance of S H and D H charts for different levels of sample size n = 5, 7, 10, 12, and 15, when = 0.05. Table 11 reports the ARL values of the proposed S H chart for normally distributed quality characteristic whereas for D H chart, the ARL comparison is presented graphically in Figure 2. It can be observed from Table 11 and Figure 2 that the ARL 1 of the S H and the D H charts decrease with an increase in the sample size n. We observed this phenomenon for all dispersion HWMA charts that the detection ability of the charts improves as n increases.

Effect of smoothing parameter ( )
The performance of S H and MD H charts for varying levels of sample size = 0.03, 0.05, 0.1, and 0.25, when = 7 is evaluated, considering normal, logistic, and Gamma processes. Tables 13 and 14 present the ARL and EQL values of HWMA charts (together with EWMA charts). A graphical comparison of the ARL is also presented for MD H chart for Gamma distributed process in Figure 3. From Tables 12 and 13 and Figure 3, it is observed that the ARL 1 of the HWMA dispersion charts reduces with the decrease in . It indicates that the proposed HWMA dispersion charts are more sensitive in the detection of shifts for small values of . It is expected that for > 0.5, the HWMA dispersion chart will perform better for the detection of large shifts as compared to using small values of . As the purpose of the study is the efficient detection of small process shifts, hence we recommend the use of < 0.25 for the proposed HWMA dispersion charts.

Proposed HWMA charts versus EWMA charts
In this section, the performance of the proposed HWMA control charts is compared with the EWMA charts for monitoring shifts in process variability. Tables 12 and 13 Table 14 reports results of HWMA and EWMA charts based on R, S, D, MD, and QN dispersion estimators, considering all the parent distributions when n = 5 and when = 0.05. HWMA dispersion charts are better than the corresponding EWMA dispersion charts, among all distributions except for the extremely skewed distribution (log-normal) where the sensitivity of the EWMA chart is higher than that of the HWMA charts. For example, when = 0.05, the ARL values of the HWMA chart for logistic distribution when shift size = 1.5 for S H and D H estimators are 4.12 and 3.97, respectively, while that of EWMA chart are 6.69 and 6.57, respectively.

5.5
Proposed HWMA chart For n = 7 and = 0.25 (Table 1) • S H has the best performance as it has the lowest ARL, EQL, and RARL values among the other competing charts.
• D H is close competitor with S H ; they are the second best charts.
• R H and MD H charts lie between the best and worst chart performances.
• MAD H , Q H , SN H , and QN H have the worst performances in detecting shifts.
For n = 5 and = 0.05 (Table 10) • S H , D H , R H , and MD H charts have the best performance compared to other charts as their ARL values are the lowest for all shift sizes. • SN H and QN H charts lie between the best and worst chart performances.
• Q H has the highest ARL values and thus it is has the worst performance

Nonnormal symmetric distributions (logistic, and Student's t):
For n = 7 and = 0.25 (Tables 2 and 3) • MD H and D H charts for both parent distributions have the best performance in terms of their ARL values.  For n = 5 and = 0.05 (Table 10) • MD H chart is the best in terms of their EQL and RARL values.
• The R H , S H , SN H , and QN H lie between the superior and inferior charts.
• The MAD H charts for both distributions have the worst performance in terms of all the run length measures.

Skewed distributions (Weibull, chi-square, gamma, and exponential):
For n = 7 and = 0.25 (Tables 4-7) • The performances of MD H and D H charts are the best, as compared to all other competing charts.
• The S H , R H , and the Q H charts have almost the same performance among the four distributions but are more efficient for parent Weibull distribution, compared to the other distributions. • The MAD H , SN H , and QN H were the least efficient charts as their ARLs, EQLs, and RARLs are high.
For n = 5 and = 0.05 (Table 9) • The performances of MD H and D H charts are the best, as compared to all other competing charts.
• The S H , R H , and the Q H charts have almost the same performance among the four distributions but are more efficient for parent Weibull distribution, compared to the other distributions. • The MAD H , SN H , and QN H were the least efficient charts as their ARLs, EQLs, and RARLs are high. TA B L E 1 3 Average run length profiles of HWMA and EWMA mean deviation from sample median MD chart for normal, logistic, and gamma distribution with varying smoothing constant = 0.03, 0.05, 0.1, and 0.25 when n = 7 and ARL0 = 200

5.6
Effect of sample size (n) • It can be observed from Table 11 and Figure 2 that the ARL 1 of the S H and the D H charts decrease with an increase in the sample size n. • For all dispersion HWMA charts that the detection ability of the charts improves as n increases.

5.7
Effect of smoothing parameter ( ) • From Tables 12 and 13 and Figure 3, it is observed that the ARL 1 of the HWMA dispersion charts reduces with the decrease in . • The proposed HWMA dispersion charts are more sensitive in the detection of shifts for small values of .

Proposed HWMA charts versus EWMA charts
Tables 12 and 13 present the ARL values of the HWMA and EWMA charts using the S and MD estimators, considering normal, logistic, and Gamma distributions at varying levels of = 0.03, 0.05, 0.1, 0.25.

F I G U R E 3
Effect of on the ARL of MD H control chart for gamma distributed process when n = 7 • When the smoothing parameter is small (ie, = 0.03 or 0.05), the proposed HWMA charts are more sensitive than the EWMA dispersion charts. • As the smoothing parameter increase, the ARL of both HWMA and EWMA charts gets closer and closer to each other, with a slight advantage to HWMA dispersion charts. • It can be seen that the EQL values for HWMA charts in all cases are less than the EQL values of EWMA charts. It indicates that the overall performances of the HWMA charts are always better than their respective EWMA counterparts. Table 14 reports results of HWMA and EWMA charts based on R, S, D, MD, and QN dispersion estimators, considering all the parent distributions when n = 5 and when = 0.05.
• HWMA dispersion charts are better than the corresponding EWMA dispersion charts, among all distributions except for the extremely skewed distribution (log-normal) where the sensitivity of the EWMA chart is higher than that of the HWMA charts.

REAL DATA APPLICATION
This section provides the real-life application of the proposed charts for monitoring the vapor pressure of the distillation process. The dataset is obtained from https://openmv.net/info/distillation-tower. A brief explanation of the distillation process and the application of the proposed charts are provided below:

Distillation process
Distillation is an important process in chemical and industrial engineering. It separates the liquid components from a mixture based on their selective boiling and condensation, that is, vapor-and liquid-phase distribution. During this process, a distillation tower allows the liquids to distribute in multiple liquids by passing through different stages. In the first stage, time is a most important factor that allows the chemical mixtures to separate. However, distribution in later stages relies on the properties of each chemical, for example, vapor pressure, evaporation point, the properties of mixtures being separated, and the desired purity of the end distillate. In principle, these properties decide the fate of molecule to fall or rise depending on their heavy and light components followed by separation of different liquid components. A brief F I G U R E 4 A representation of distillation column (https://en.wikipedia.org/wiki/Distillation) explanation of the distillation process is shown in Figure 4 (https://en.wikipedia.org/wiki/Distillation). The schematic diagram of distillation unit represents a single feed that separates liquid mixtures into two produces: overhead product and bottom product. Usually, the bottom products are liquids (heavy components) while the overhead products can be either liquid or vapor or both (light components). 29,30 Furthermore, for necessary vaporization, the reboiler is used where it works on providing heat from the bottom and a vapor is produced where it flows upward through the distillation column. Thereon, the condenser is used for cooling and condensing the vapor from the top. A liquid is produced in reflux drum and it flows downward in the distillation column to be recycled back.

Quality control application
This section uses the vapor pressure variable, which is a measure of the tendency of a material to change into the vapor state. It is one of the chemical properties that decide the separation of liquids into different components in the distillation process and thus it is an important variable to be monitored. The following steps were taken to apply the proposed charts on this dataset: First, we find the distribution of the data to be used in constructing the upper control limits of the proposed HWMA charts. We used the fitdist function in fitdistrplus package in R and tested the data for four potential distributions which are normal, gamma, Weibull and lognormal. We identified the best-fitted distribution for this data as gamma distribution with scale and rate parameters 31.08 and 0.848, respectively. Moreover, we also used the descdist function to plot the Cullen and Frey Graph, given below in Figure 5; this graph plots the square of skewness against the kurtosis. It also indicated that gamma is the best-fitted distribution.
After distribution fitting, 50 samples size of n = 5 are drawn from the data and control limits are computed for a fixed ARL 0 = 200 using = 0.05. These samples were standardized to have mean and standard deviation equals 0 and 1, respectively. For illustration purpose, we used S H , D H, and MAD H control charts. The control chart coefficients for three charts are found to be L S = 1.350, L D = 1.361, and L MAD = 1.607.
To check the detection ability of the proposed HWMA charts, shift of = 3 was introduced in the standard deviation of last 10 samples. Figure 6 depicts the resulting control chart plots. As it can be seen in Figure 6, the S H chart detected 6 out-of-control points, 7 out-of-control points were detected by D H chart and 5 out-of-control points were detected by MAD H chart with one false alarm.

CONCLUSION
This paper proposed and investigated a set of HWMA charts for detecting shifts in process dispersion for normal and nonnormal parent distributions, considering a wide range of dispersion estimators.