A Nonparametric HEWMA-p Control Chart for Variance in Monitoring Processes

Control charts are considered as powerful tools in detecting any shift in a process. Usually, the Shewhart control chart is used when data follows the symmetrical property of a normal distribution. In practice, the data from the industry may follow a non-symmetrical distribution or an unknown distribution. The average run length (ARL) is a significant measure to assess the performance of the control chart. The ARL may mislead when the statistic is computed from an asymmetric distribution. To handle this issue, in this paper, an ARL-unbiased hybrid exponentially weighted moving average proportion (HEWMA-p) chart is proposed for monitoring the process variance for a non-normal distribution or an unknown distribution. The efficiency of the proposed chart is compared with the existing chart in terms of ARLs. The proposed chart is more efficient than the existing chart in terms of ARLs. A real example is given for the illustration of the proposed chart in the industry.


Introduction
The aim of quality refers to the quality of those product characteristics that will appeal to potential customers.It takes into account what it would cost to produce the product and what the customers are willing to pay for the product.It can be thought of as the perspective for accomplishing manufactured quality.Once the manufacturing process has started, the process does not always produce a unit in conformity with what was proposed.This may be due to causes of defects arising in materials, parts, subassemblies, assemblies, and in the final product.Due to defective or nonconforming resources, parts, assemblies, and finished products that are discarded or reworked during the manufacturing process result in increased cost and customer dissatisfaction.The waste of time and effort in manufacturing the defective or nonconforming product, the delays in delivery, and other associated costs attributable to a poorly manufactured product are the consequence of manufactured quality.If a company wishes to produce higher quality products, it usually needs higher costs for manufactured products.However, the aim should always be to offer customers good quality at a low cost.Therefore, quality is also part of the corporate approach.Understanding quality concepts leads to correct implementation and management of product quality, which adds benefits to the entire production endeavor.If an industry understands and applies quality control principles in their manufactured products, it will produce well finished products, and reduction of the costs of the products may be possible.
The control charts are effectively used for the monitoring of the process.The Shewhart control chart is designed under the assumption that the data coming from the industry follows a normal distribution.This chart is more effective in detecting a relatively large shift in a process.The Shewhart control chart cannot be applied for a monitoring process when the industrial data follow the non-normal distribution or unknown distribution.Several authors focused on designing control charts for monitoring process mean, including for example [1][2][3].Some authors designed control charts for monitoring the process variance; see for example [4,5].More details on control charts can be seen in [6][7][8][9][10][11][12][13].
As mentioned by [14] practitioners are often not statisticians and may have problems in implementing control charts based on non-parametric approaches.Keeping in mind this issue, several authors, including for example, references  focused on designing control charts for monitoring the process mean that were easier to apply as compared to existing charts.Later on, Yang et al. [42] worked on the extension of the chart designed in [43] to monitor the process variance using a simple arcsin transformed symmetric exponentially weighted moving average (EWMA) statistic.Yang et al. [42] further extended the work of [5] by mixing the arcsin transformed EWMA and simple EWMA statistic.Yang et al. [5] designed the arcsin transformed EWMA to monitor process variance.The average run length (ARL) is used to assess the performance of a control chart.Smaller ARL means more efficient and faster detection of a shift in the process.According to [5] "For a monitoring statistic with an asymmetrical distribution, the control chart leads to a biased ARL.That is, the in-control ARL may be smaller than any out-of-control ARL, thus taking longer to detect shifts in the parameter than to trigger a false alarm".Lowry et al. [44] proposed a chart to tackle this issue.
Haq [45] proposed a control chart using two EWMA statistics and called it a hybrid EWMA (HEWMA) chart.He claimed that their proposed chart performed better than the usual EWMA chart.The HEWMA statistic consists of two EWMA statistics and two smoothing constants.The control chart based on HEWMA statistic has the ability to detect the shift in the process earlier than the EWMA-based control chart.The operational process of the HEWMA based control chart is the same as the EWMA control chart.Several authors worked on these statistics in the literature.There are various variable charts for joint monitoring; see for example, Nyau et al. [11], who designed the multivariate EWMA chart for the median run-length.Riaz et al. [12] proposed a mixed Tukey EWMA-CUSUM (cumulative sum) control chart and Osei-Aning et al. [13] worked on the mixed EWMA-CUSUM and mixed CUSUM-EWMA.Haq [46] presented a discussion on the HEWMA control chart.Noor-ul-Amin et al. [47] worked on the HEWMA chart for the regression estimator.Several authors proposed the attribute control chart using EWMA and HEWMA statistics.Aslam et al. [48] designed the mixed EWMA control chart.Haq [49] worked on the nonparametric EWMA chart.Riaz et al. [50] designed the nonparametric double EWMA control chart.Aslam et al. [51] worked on the HEWMA-CUSUM chart for the Weibull distribution.Aslam et al. [52] worked on the HEWMA chart for the COM-Poisson distribution.
Yang et al. [14] designed an ARL unbiased EWMA-p chart.According to the best of our knowledge, there is no work on designing a EWMA-p chart using a hybrid EWMA.The proposed methodology presents an approach to evaluate the performance using the combination of a hybrid EWMA control chart with weighted moving average proportion (EWMA-p) control chart.The hybrid EWMA and EWMA-p charts are chosen since it has been shown that these charts are efficient in detecting small but possibly detrimental shifts in the process.Aslam et al. [52] also pointed out that in a general manufacturing process, an exponentially weighted moving average EWMA control chart is more efficient in detecting small process shifts.The control chart based on the EWMA method consists of an exponential weight factor applied to the data, which gives current or recent past observations more weight than older data values.The combination of a hybrid EWMA chart and EWMA-p charts will be explored to determine the best conditions, i.e., the appropriate values of control variables for monitoring concrete strength.In this paper, we will present the enhanced hybrid exponential weighted moving average proportion (enhanced HEWMA-p) chart.The structure of the proposed chart will be presented and its efficacy will be compared with [14].

Design of the Enhanced HEWMA-p Chart
In this section, we will present some equations taken from [14] and present the operational procedure of the proposed control chart.
Let a random sample of size n be drawn from process X whose distribution is unknown with a variance σ 2 to practitioners.Yang et al. [14] suggested to select an even sample size n for convenience.Assuming that these samples are independently distributed with known variance, let Then, where Therefore, when the process is in control, V is distributed as a binomial with parameters 0.5n and p v0 , where the value of p v0 depends on the distribution of X i .Let us define p v0 = P(Y * j > σ 2 ).The null hypothesis is that the process is in control state at p v0 .The alternative hypothesis is that the process has been shifted at p v1 .According to [14], the statistic of V t /0.5n has the mean of p v0 and the variance of p v0 (1 − p v0 )/0.5n.
We define the following two EWMA statistics: where λ 1 ∈ [0, 1] and λ 2 ∈ [0, 1] are smoothing constants, and HEW MA p t is the statistic of enhanced HEWMA-p at t.The proposed control chart is stated as follows: Step 1: Select a random sample of size n(X 1 , . . ., X n ) from the process at time t.Compute V t using (3) and HEW MA p t using (5).
Step 2: The process is declared to be as out-of-control if HEW MA p t ≥ UCL or HEW MA p t ≤ LCL and to be in-control if LCL < HEW MA p t < UCL, here LCL and UCL show the lower control limit and upper control limit.
The proposed control chart is the extension of the chart proposed by [14].The proposed chart reduces to [14] chart when λ 1 = λ 2 = λ or λ 1 = 1 or λ 2 = 1.The proposed chart becomes the Shewhart chart when λ 1 = 1 and λ 2 = 1.It is assumed that the starting value of HEW MA p t is the mean of p v0 , i.e., HEW MA p t = p v0 for an in control process.By following [45], the mean and variance of statistic HEW MA p t is given by E HEW MA p t = p v0 (6) and Thus, the asymptotic variance of HEW MA p t is given as As suggested by [14] "the new variance chart may be constructed based on the distribution of the monitoring statistic V t /0.5n, which is an asymmetric distribution having similar defects to those of the corresponding Shewhart p chart".Therefore, monitoring the process variance is the same as the monitoring process proportion p v0 , as the proportion p v0 of statistic V t /0.5n may not be same.The control limits of the proposed control chart are given as where k 1 > k 2 are control limit coefficients.

The Average Run Length of Enhanced HEWMA-p Control Chart
The proposed enhanced hybrid exponential weighted moving average proportion (enhanced HEWMA-p) control chart performance measure can be used as the average run length (ARL).In this paper, we have limited our study to non-normal distributions with finite variance.The control limits of the enhanced HEWMA-p control chart are determined by setting the in-control ARL (ARL v0 ) to be a specified value, usually 370.The ARL represents the expected number of samples until a control chart signals.The proposed control chart comprises of two control coefficients, k 1 and k 2 , which are obtained by considering the desired in-control ARL.Once the coefficients k 1 and k 2 are determined, the control limits of the enhanced HEWMA-p control chart are obtained and the out-of-control ARLs (ARL v1 ) can be obtained according to various values of shift in proportion, p v1 = c p v0 , c = 1, and 0 < p v1 ≤ 1.We use the following Monte Carlo simulation procedure to compute control coefficients k 1 and k 2 , and to calculate the out-of-control ARL (ARL v1 ) under a specified n, p v0 , λ 1 , λ 2 and ARL v0 values.
Step 2. Evaluation of proposed control chart coefficients k 1 and k 2 for in-control process 2.1.Generate 10,000 possible values of control chart coefficients k 1 and k 2 .2.2.When the process is in-control, from a binomial distribution with the in-control parameters 0.5n and p v0 .a random sample of size 2000 is generated, i.e., V t ∼ binomial(0.5n,p v0 ) at time t.2.3.The enhanced HEWMA-p statistic HEWMA-p is computed for each subgroup of size 2000.2.4.The proposed statistic HEWMA-p is plotted and in-control if LCL ≤ HEW MA p ≤ UCL; go to step 2.5 and the run length for out of control process is noted.2.5.Repeat 10,000 times steps 2.2 through 2.3, to compute run lengths.If the average of these run lengths (ARLs) is equal to the specified ARL v0 note the corresponding values of k 1 and k 2 , and move to step 3, otherwise select other possible values of k 1 and k 2 , and repeat the procedure from steps 2.2.
Step 3 Evaluation of ARL v1 for proposed control chart when the process is shifted 3.1.Let the out-of-control proportion, p v1 , be a proportion of the in-control proportion, p v0 .That is, p v1 = c p v0 , c = 1, and 0 < p v1 ≤ 1, where c is the amount of shift in the process proportion, p v0 .3.2.From binomial distribution, with the in-control parameters, 0.5n and p v1 , a random sample of size 2000 is generated, i.e., V t ∼ binomial (0.5n, p v1 ) at time t.3.3.The Enhanced HEWMA-p Statistic HEWMA-p is Computed for Each Subgroup of Size 2000.3.4.Using the Values of k 1 and k 2 , the proposed statistic HEWMA-p is plotted and in-control if LCL < HEW MA p t < UCL; go to step 3.5 and the run length for out of control process is noted.3.5.Repeat 10,000 times steps 3.2 through 3.3, to compute run lengths.The average of run length (ARL v1 ) and standard error of run length (SERL v1 ) for each specified amount of shift is computed.
From Tables 2 and 4 we observe the following trend in ARL v1 1.If n is increased, there is a decrease in ARL v1 and SERL v1 values, as we expected.For example, for 0.5n = 4 and p v1 = 0.05 from Table 2 we have ARL v1 = 107.03and SERL v1 = 0.8830, whereas if 0.5n = 15, we have ARL v1 = 18.12 and SERL v1 = 0.0921.We also observed a similar trend from Table 4.
2. The ARL v1 and SERL v1 values decrease when p v1 is far away from p v0 .
The R codes for this study are given in the Appendix A.

Comparative Study
Now, we discuss the performance of the proposed control chart with the existing control charts proposed by [4,14] for λ = 0.2.The proposed chart reduces to [14] chart when λ 1 = λ 2 = λ = 0.2 (for example).We present the values of ARL v1 for the proposed control chart as well as control charts given by [4,14] in Table 5 when in-control ARL v0 ≈ 370.
From Table 5, we observe that the proposed control chart has smaller values of ARL v1 as compared to the existing two control charts.For example, when 0.5n = 6, λ 1 = λ 2 = λ = 0.2, p v0 = 0.3, p v1 = 0.4 the proposed control chart gives ARL v1 is 27.30, the ARL v1 from the two existing control charts are 31.36and 34.20, respectively.Thus, the proposed control chart performs better than the existing control charts.
Table 5 ARL v1 s comparison between the chart proposed by [4,14] for λ = 0.2 and enhanced HEWMA-p control chart for λ 1 = 0.2, λ 2 = 0.2. Figure 1 depicts the ARL v1 profile comparison at p v0 = 0.1 and p v1 = 0.2 for different values of n under HEWMA-p chart and two existing charts.From Figure 1, we noticed that ARL v1 values of enhanced HEWMA-p control chart are smaller than in the two existing control charts.Hence, our proposed enhanced HEWMA-p control chart performed well as compared with existing charts.From Table 5, we observe that the proposed control chart has smaller values of ARLv1 as compared to the existing two control charts.For example, when 0.5n = 6,  1 =  2 =  = 0.2, pv0 = 0.3, pv1 = 0.4 the proposed control chart gives ARLv1 is 27.30, the ARLv1 from the two existing control charts are 31.36and 34.20, respectively.Thus, the proposed control chart performs better than the existing control charts.
Table 5. ARLv1s comparison between the chart proposed by [4] and [14] for  = 0.2 and enhanced HEWMA-p control chart for 1 1 depicts the ARLv1 profile comparison at pv0 = 0.1 and pv1 = 0.2 for different values of n under HEWMA-p chart and two existing charts.From Figure 1, we noticed that ARLv1 values of enhanced HEWMA-p control chart are smaller than in the two existing control charts.Hence, our proposed enhanced HEWMA-p control chart performed well as compared with existing charts.

Example
In this section, we present an example given by [14].The service time of a bank branch in Taiwan is used to illustrate the application of the proposed enhanced HEWMA-p control chart to monitor the variability of service time.According to [14] "From the historical data, the in-control data of service times (unit: minutes) is a non-normal/unknown distribution with variance 27.805.Reference [14] illustrated that the resulting in-control probability that the service time is larger than the in-control variance is  0 = (  * > 27.805) = 0.31".To construct the enhanced HEWMA-p control chart, we also use the same value of  0 .The upper and lower control limits of the enhanced HEWMA-p control chart with  1 = 0.2,  2 = 0.2 when in-control ARLv0 ≈ 370 are UCL = 0.4454 and LCL = 0.1963.

Example
In this section, we present an example given by [14].The service time of a bank branch in Taiwan is used to illustrate the application of the proposed enhanced HEWMA-p control chart to monitor the variability of service time.According to [14] "From the historical data, the in-control data of service times (unit: minutes) is a non-normal/unknown distribution with variance 27.805.Reference [14] illustrated that the resulting in-control probability that the service time is larger than the in-control variance is p v0 = P(Y * j > 27.805) = 0.31".To construct the enhanced HEWMA-p control chart, we also use the same value of p v0 .The upper and lower control limits of the enhanced HEWMA-p control chart with λ 1 = 0.2, λ 2 = 0.2 when in-control ARL v0 ≈ 370 are UCL = 0.4454 and LCL = 0.1963.
Ten new samples of size 10 each from new automatic service system of the bank branch under study were considered [14] and listed in Table 6.To illustrate the out-of-control detection ability, for each sample in Table 6, the statistic, V t and the monitoring statistic HEW MA p t = λ 1 EW MA p t + (1 − λ 1 ) HEW MA p t−1 where EW MA p t = λ 2 V t /0.5n + (1 − λ 2 )EW MA p t−1 at time t, t = 1, 2, . . . ,10, were computed.The corresponding enhanced HEWMA-p control chart detected out-of-control variance signals from the third sample onward (samples 3-10 on the enhanced HEWMA-p control chart) (Figure 2).By comparing Figure 2 with the chart in [14], it can be seen that the existing chart indicated a shift at the 4th sample.Therefore, the proposed chart was more efficient in detecting a shift in the process as compared to existing chart of Yang and Arnold [14].The same performance was also shown by the results in Tables 2 and 4. For this study, we can conclude that the proposed chart shows better performance than the existing two charts.Ten new samples of size 10 each from new automatic service system of the bank branch under study were considered [14] and listed in Table 6.To illustrate the out-of-control detection ability, for each sample in Table 6, the statistic, Vt and the monitoring statistic    =  1    + (1 −  1 )   −1 where EWM   =  2   0.5 ⁄ + (1 −  2 )  −1 at time t, t = 1, 2,…, 10, were computed.The corresponding enhanced HEWMA-p control chart detected out-of-control variance signals from the third sample onward (samples 3-10 on the enhanced HEWMA-p control chart) (Figure 2).By comparing Figure 2 with the chart in [14], it can be seen that the existing chart indicated a shift at the 4 th sample.Therefore, the proposed chart was more efficient in detecting a shift in the process as compared to existing chart of Yang and Arnold [14].The same performance was also shown by the results in Tables 2 and 4. For this study, we can conclude that the proposed chart shows better performance than the existing two charts.

Concluding Remarks
In this paper, an enhanced hybrid EWMA-p chart is proposed for monitoring the process variance.A simulation procedure is presented for calculating its average run lengths (ARLs).Some tables are presented for practical use.The simulation study supports that the proposed chart is more efficient in detecting a shift in the process.A real example is presented for illustration purposes.The proposed control chart can be used in the industry for the monitoring of processes when the distribution is unknown in practice.The limitation of the proposed chart is that it can be used for only a fixed sample size.The variable sample size enhanced hybrid EWMA-p chart will be considered as our future research.In addition, the proposed control chart for a variable sample size can be considered as future research.The proposed chart using autocorrelation can be considered as future research.

Nomenclature:
ARL: Average run length

Concluding Remarks
In this paper, an enhanced hybrid EWMA-p chart is proposed for monitoring the process variance.A simulation procedure is presented for calculating its average run lengths (ARLs).Some tables are presented for practical use.The simulation study supports that the proposed chart is more efficient in detecting a shift in the process.A real example is presented for illustration purposes.The proposed control chart can be used in the industry for the monitoring of processes when the distribution is unknown in practice.The limitation of the proposed chart is that it can be used for only a fixed sample size.The variable sample size enhanced hybrid EWMA-p chart will be considered as our future research.In addition, the proposed control chart for a variable sample size can be considered as future research.The proposed chart using autocorrelation can be considered as future research.

Figure 1 .
Figure 1.The ARL v1 profile comparison at p v0 = 0.1 and p v1 = 0.2 for different values of n under HEWMA-p chart and Yang and Arnold [4,14].

Figure 2 .
Figure 2. The enhanced HEWMA-p chart for the example.

Figure 2 .
Figure 2. The enhanced HEWMA-p for the example.
First row ARL and second row SERL (standard error of run length).
First row ARL and second row SERL (standard error of run length).

Table 5 .
The comparison of the proposed chart with existing charts.

Table 6 .
The new service times from 10 counters in a bank branch.EWMA is exponentially weighted moving average.

Table 6 .
The new service times from 10 counters in a bank branch.EWMA is exponentially weighted moving average.