Mixed Exponentially Weighted Moving Average—Moving Average Control Chart with Application to Combined Cycle Power Plant

Abstract

Statistical process control (SPC) consists of various tools for effective monitoring of the production processes and services to ensure their stable and satisfactory performance. A control chart is an important tool of SPC for detecting the process shifts that may undermine the quality of the products or services. In the literature, a mixed exponentially weighted moving average–moving average (EWMA–MA) control chart for monitoring the process location is proposed to enhance the overall shift detection ability of the EWMA control chart. It is noted that the moving averages terms were considered as independent irrespective of their order. Consequently, the covariance terms are ignored while deriving the variance expression of the monitoring statistic. However, the successive moving averages of span w might not be independent since each term includes w − 1 preceding samples’ information. In this study, the variance expression of the mixed EWMA-MA charting statistic is derived by considering the dependency among the sequential moving averages. The control limits of the mixed EWMA-MA control chart are revised and the run-length profile is studied by using Monte Carlo simulations. The performance of the mixed EWMA-MA chart is compared with the existing counterparts and its robustness under various process distributions is studied. In the end, a real-life example is provided to demonstrate its application by using the data from a combined cycle power plant.

1. Introduction

Statistical process control (SPC) is a technique of quality control that employs statistical methods for monitoring sequential processes (production or services) to ensure their stable and satisfactory performance. The concept of a control chart was first introduced by Shewhart [1] as an effective tool of SPC to identify the process shifts. Shewhart-type charts are very sensitive in detecting large shifts because their charting statistic is based on the most recent sample only. Due to this reason, the Shewhart-type control charts are categorized as memory-less control charts. The cumulative sum (CUSUM) control chart introduced by Page [2] and the exponentially weighted moving average (EWMA) control chart proposed by Roberts [3] are memory-type control charts as their charting statistics incorporates the current as well as previous samples information. These charts are very efficient in detecting small to moderate process shifts. Roberts [3,4] discussed another control chart called the moving average (MA) control chart which is more effective than the Shewhart-type charts in detecting small to moderate process shifts. For further details see Montgomery [5].

To improve the shift detection ability of the control charts, several mixed control charting schemes have been introduced in SPC literature. For example, Lucas [6] proposed the combined Shewhart-CUSUM control chart which integrates the large and small shift detection abilities of Shewhart and CUSUM schemes, respectively, and protects against a range of shifts. Similarly, Lucas and Saccucci [7] have combined the features of Shewhart and EWMA control charting schemes for improved process monitoring. Shamma and Shamma [8] developed a double EWMA (DEWMA) control chart for improved performance to detect small to moderate shifts in the process location. Khoo and Wong [9] proposed a double moving average (DMA) control chart that improves the small to moderate shift detection ability of the MA control chart for a normally distributed process. Haq [10] proposed a new hybrid exponentially weighted moving average control chart with variable control limits for effective monitoring of smaller shifts in the process mean. For improved small shift detection ability, Abbas et al. [11] incorporated the EWMA statistic into the CUSUM charting statistic and named it as a mixed EWMA-CUSUM control chart, but it is not effective when quick detection of large shifts is of interest. Zaman et al. [12] proposed a reversed of EWMA-CUSUM and named it as mixed CUSUM-EWMA control chart for quick detection of smaller shifts in the process. Ali and Haq [13] proposed a mixed GWMA-CUSUM control chart for effective monitoring of small shifts in the process mean. It includes the CUSUM and EWMA-CUSUM charts as its special cases. Alevizakos et al. [14] provided the correct variance of the double moving average control chart that was initially developed by Khoo and Wong [9] for the monitoring of normally distributed quality characteristics. Alevizakos et al. [15] proposed a triple exponentially weighted moving average control chart by combining the three EWMA statistics to detect small process shifts in the location parameter. For more details regarding the mixed control charting schemes, see Rasheed et al. [16], Engmann and Han [17], Raza et al. [18], and Han et al. [19].

Recently, Sukparungsee et al. [20] proposed a mixed EWMA-MA control chart which is in line with Khan et al. [21] to monitor the process location. They considered the MA terms independent of each other and ignored the covariances between them while obtaining the variance of the EWMA-MA statistic. Thus, their results are of limited use and can lead to misleading conclusions in practice because the successive MA statistics of span w might be dependent as each MA term includes w − 1 preceding samples’ information. To be more precise, MA_i and MA_j (∀ i < j) terms of span w are dependent when j – i < w, and otherwise they are independent. Sukparungsee et al. [20] studied the performance of the EWMA-MA chart to detect only a specified shift. However, in practice the shift size is usually unknown and there is need for the assessment of the overall performance of the control chart for a range of shifts. Moreover, the comparison of the mixed EWMA-MA control charts with other existing combined charting structures is not considered by Sukparungsee et al. [20].

Keeping in view the shortcomings of the existing mixed EWMA-MA control chart, an in-depth study is carried out to gain insights into design structure, performance, and application of the EWMA-MA control chart. To be more specific, this study presents:

• The revised variance of the mixed EWMA-MA statistic along with control limits by considering both the independent and dependent behavior of MA terms which enhances its applicability;
• The run-length profile and the associated performance metrics such as average run length, standard deviation of run length, and median run length using Monte Carlo simulations;
• The use of average extra quadratic loss and relative mean index for a more thorough assessment of the EWMA-MA control chart to detect a range of shifts;
• A comprehensive table of the design parameters of the EWMA-MA chart to aid practitioners in implementation;
• An extensive simulation-based comparative study to assess the performance of the EWMA-MA chart against several competing traditional and combined/mixed control charts;
• The robustness of the mixed EWMA-MA control chart for various continuous symmetrical and skewed process distributions;
• The practical advantage of using the EWMA-MA chart over its various competitors by using a real dataset from a combined cycle power plant where the ambient pressure is monitored that can affect the performance of gas turbine.

The rest of the paper is organized as: Section 2 describes the charting structure of the existing mixed EWMA-MA control chart. Section 3 provides the revised version of the EWMA-MA control chart. The run-length profile of the mixed EWMA-MA control chart is discussed in Section 4 for various settings of the design parameters. The robustness of the EWMA-MA control chart to non-normality is also assessed in this Section for various process distributions. A comprehensive comparative study is conducted in Section 5 to evaluate the performance of the EWMA-MA control chart. Moreover, an example of real-life application is also provided in this Section. Finally, Section 6 concludes the paper.

2. The Existing Mixed EWMA-MA Control Chart

The EWMA-MA control chart was developed by Sukparungsee et al. [20], which integrates the moving averages into the EWMA statistic for monitoring the process location. Suppose that there are k independent and identically distributed random samples of the critical quality characteristic (X) each of size n ≥ 1 and ${\stackrel{-}{X}}_i$, for i = 1, 2, …, k, be the average of the ith sample, where ${\stackrel{-}{X}}_i=\frac{X_{i1}+X_{i2}+\cdots +X_{in}}{n}.$ The MA_i for a span w at time i is defined as:

$$M{A}_i=\{\begin{array}{cc}\frac{{{\displaystyle \sum }}_{j=1}^i{\stackrel{-}{X}}_j}{i},& for i<w\\ \frac{{{\displaystyle \sum }}_{j=i-w+1}^i{\stackrel{-}{X}}_j}{w},& for i\ge w\end{array}$$

(1)

The monitoring statistic for the mixed EWMA-MA control chart is:

$$Z_i=\lambda M{A}_i+\left(1-\lambda \right)Z_{i-1}, i=1,2,\dots$$

(2)

where λ is the smoothing parameter with 0 < λ < 1, and Z₀ is the initial value which is equal to the targeted mean, i.e., Z₀ = μ₀. The mean and variance of the existing EWMA-MA control chart are:

$$E\left(Z_i\right)={\mu }_0$$

(3)

$$Var\left(Z_i\right)=\left(\frac{{\sigma }^2}{w}\right)\left(\frac{\lambda }{2-\lambda }\right)$$

(4)

The control limits of the existing EWMA-MA chart are defined as:

$$\begin{array}{c}UCL={\mu }_0+K\sqrt{\left(\frac{{\sigma }^2}{w}\right)\left(\frac{\lambda }{2-\lambda }\right)}\\ LCL={\mu }_0-K\sqrt{\left(\frac{{\sigma }^2}{w}\right)\left(\frac{\lambda }{2-\lambda }\right)}\end{array}\}$$

(5)

where K determines the width of the control limits. The Z_is statistics are plotted against the control limits and if any Z_i falls beyond the control limits, the process is out-of-control (OOC). Otherwise, the process is in-control (IC).

3. The Revised Mixed EWMA-MA Control Chart

Sukparungsee et al. [20] considered the MA terms as independent of each other in the variance expression of the existing EWMA-MA charting statistic. This approach can lead to misleading conclusions in practice because the successive MA statistics of span w might be dependent as each MA term includes w − 1 preceding samples’ information. For instance, MA_i and MA_j (∀ i < j) terms of span w will be dependent when j – i < w because in this situation MA_i and MA_j consists of at least one common $\overline{X}$ term, otherwise these are independent. Consequently, the control limits and results of the existing EWMA-MA chart are dubious. In this paper, the variance of the mixed EWMA-MA charting statistic has been modified by considering the dependence structure of the MA terms (hereafter, the term EWMA-MA control chart will refer to the revised mixed EWMA-MA control chart). The mean of the EWMA-MA control chart remains the same as given in Equation (3). The EWMA-MA statistic given in Equation (2) can be expanded as follows:

$$Z_i=\lambda {\displaystyle \sum }_{j=0}^{i-1}{\left(1-\lambda \right)}^{j}M{A}_{i-j}+{\left(1-\lambda \right)}^i{Z}_0$$

(6)

The revised variance expression of the EWMA-MA control chart is given below:

$$Var\left(Z_i\right)={\lambda }^2{\displaystyle \sum }_{j=0}^{i-1}{\left(1-\lambda \right)}^{2j}Var\left(M{A}_{i-j}\right)+ 2{\lambda }^2{\displaystyle \sum }_{1\le j_1<j_2\le i}{\left(1-\lambda \right)}^{i-j_1}{\left(1-\lambda \right)}^{i-j_2} Cov\left(M{A}_{j_1},M{A}_{j_2}\right)$$

$$Var\left(Z_i\right)={\lambda }^2{\displaystyle \sum }_{j=0}^{i-1}{\left(1-\lambda \right)}^{2j}Var\left(M{A}_{i-j}\right)+2{\lambda }^2{\displaystyle \sum }_{j_1=1}^{i-1}{\displaystyle \sum }_{j_2=j_1+1}^i{\left(1-\lambda \right)}^{2i-j_1-j_2} Cov\left(M{A}_{j_1},M{A}_{j_2}\right)$$

(7)

where

$$Var\left(M{A}_i\right)=\{\begin{array}{cc}\frac{{\sigma }^2}{ni},& for i<w\\ \frac{{\sigma }^2}{nw},& for i\ge w\end{array}$$

(8)

and

$$Cov({MA}_{j_1},{MA}_{j_2})=\{\begin{array}{cc}\frac{{\sigma }^2}{n{j}_2},& for j_1,j_2<w\\ \left(\frac{j_1-j_2+w}{j_{1}w}\right)\left(\frac{{\sigma }^2}{n}\right),& for j_1<w, j_2\ge w, j_2-j_1<w\\ \left(\frac{j_1-j_2+w}{w^2}\right)\left(\frac{{\sigma }^2}{n}\right),& for j_1,j_2\ge w, j_2-j_1<w\\ 0,& for j_2-j_1\ge w\end{array}$$

(9)

The revised control limits of the EWMA-MA chart are given by:

$$\begin{array}{c}UCL={\mu }_0+L\sqrt{Var\left(Z_i\right)}\\ LCL={\mu }_0-L\sqrt{Var\left(Z_i\right)}\end{array}\}$$

(10)

where L is the coefficient of control limits for the EWMA-MA control chart. The EWMA-MA triggers an OOC signal at time i if Z_i falls outside the control limits. Otherwise, the process is IC.

4. Performance Evaluation

There are many measures to evaluate the performance of a chart. The frequently used performance measures are average run length (ARL), the standard deviation of run length (SDRL), and median run length (MRL). The IC and OOC ARLs are represented by ARL₀ and ARL₁, respectively. The ARL is computed as:

$$ARL=\frac{{\sum }_{t=1}^{N}R{L}_t}{N}$$

(11)

where RL is the number of points that falls within the control limits before an OOC signal is produced. The SDRL is used to estimate the spread of RL distribution which is formulated as:

$$SDRL=\sqrt{\frac{{\sum }_{t=1}^N{\left(R{L}_t\right)}^2}{N}-AR{L}^2}$$

(12)

The MRL is the central value of run-length distribution that provides ease of interpretation as 50% of the run lengths are always greater than it.

$$MRL=Median \left(RL\right)$$

(13)

The average extra quadratic loss (AEQL) and relative mean index (RMI) are usually used to compare the overall performance of a control chart to detect a range of shifts in the process parameters. The formula for computing AEQL is given as:

$$AEQL=\frac{1}{{\delta }_{max}-{\delta }_{min}}{\displaystyle \sum }_{\delta =0}^{{\delta }_{max}}{\delta }^{2}ARL\left(\delta \right),$$

(14)

where δ_max and δ_min are the maximum and minimum magnitude of shifts to be used. A lower AEQL value of a control chart indicates its overall superior shift detection ability as compared to other competing charts. The RMI is defined as:

$$RMI=\frac{1}{N}{\displaystyle \sum }_{i=1}^N\left\{\frac{AR{L}_{({\delta }_i)}-AR{L}_{\left({\delta }_i\right)}^{*}}{AR{L}_{\left({\delta }_i\right)}^{*}}\right\}$$

(15)

where the total number of shifts is denoted by N; $AR{L}_{({\delta }_i)}$ is the ARL value of the competing control chart for a specific shift δ_i; and $AR{L}_{\left({\delta }_i\right)}^{*}$ represents the control chart that has the minimum ARL at the specific shift δ_i. The control chart that has the smallest value of RMI outperforms the other charts [22,23].

There are different methods for estimating the run-length characteristics such as Monte Carlo simulation, Markov chain, and integral equation. The Monte Carlo simulation is used in this study to estimate the RL characteristics of the EWMA-MA control chart because it is more flexible and efficient [24]. The simulation code is developed in R software and computations are based on 10,000 iterations. The design parameter (L) value of the EWMA-MA chart is determined for a specific combination of λ and w at a desired ARL₀. The IC and OOC run-length profiles are computed by using the following algorithm.

4.1. Algorithm for Computing RL Characteristics

4.1.1. Computation of the IC RL Characteristics

• Step 1: select the smoothing parameter λ and span w for calculation of the charting statistic Z_i
• Step 2: choose the value of the limit coefficient L to get the desired ARL₀ (such as 200, 370, and 500) under fixed values of λ and w;
• Step 3: generate a random sample of size n from a specific distribution such as the normal distribution with mean μ₀ = 0 and variance σ² = 1 as a realization of the IC process state;
• Step 4: calculate MA_i using Equation (1), and compute the monitoring statistic Z_i.
• Step 5: Calculate the control limits of the EWMA-MA control chart and check whether the monitoring statistic Z_i falls within the limits or not. If Z_i ≤ LCL or Z_i ≥ UCL, record the sample number as it will be the RL. Otherwise, go to Step-3;
• Step 6: repeat Steps 3 to 5 m (say 100,000) times which results in m run lengths;
• Step 7: Calculate the average of m run lengths, i.e., ARL₀. If the calculated ARL₀ is equal to the desired ARL₀, go to Step 8. Otherwise, go to Step 2 to adjust the value of L;
• Step 8: Calculate the other run-length characteristics such as SDRL and MRL.

4.1.2. Computation of the OOC RL Characteristics

• Step 9: Generate the test sample of size n from the OOC process under a specific distribution by introducing a process shift (δ ≠ 0) in the mean. For example, the sample can be taken from a normal distribution with a mean ${\mu }_1={\mu }_0+\delta \frac{{\sigma }_0}{\sqrt{n}}$ and variance σ² = 1;
• Step 10: compute the OOC RL by repeating Steps 4 to 6 under the shifted process and then calculate ARL₁, SDRL₁, and MRL₁;
• Step 11: After computing ARL₁, SDRL₁, and MRL₁ for all the shifts considered in the study, the average extra quadratic loss (AEQL) is computed for estimating the overall performance of a control chart by using Equation (14).

Using the above simulation algorithm, the run-length characteristics are calculated for different shifts (δ), sensitivity parameters (λ = 0.05, 0.25, and 0.75), and span (w = 2, 3, 4, 5, 8, and 10).

Table 1. The limit coefficient, L, values of the EWMA-MA control chart for various combinations of λ and w.

ARL0	λ	w
		2	3	4	5	8	10
200	0.05	2.145	2.100	2.062	2.031	1.973	1.928
	0.10	2.368	2.315	2.274	2.235	2.161	2.118
	0.25	2.595	2.531	2.487	2.445	2.358	2.315
	0.50	2.713	2.660	2.617	2.582	2.525	2.499
	0.75	2.768	2.730	2.700	2.679	2.644	2.639
370	0.05	2.426	2.379	2.335	2.310	2.237	2.205
	0.10	2.620	2.567	2.526	2.490	2.410	2.367
	0.25	2.819	2.760	2.717	2.676	2.592	2.552
	0.50	2.925	2.869	2.826	2.798	2.730	2.713
	0.75	2.966	2.933	2.903	2.882	2.849	2.838
500	0.05	2.547	2.503	2.465	2.435	2.367	2.330
	0.10	2.738	2.683	2.645	2.603	2.528	2.487
	0.25	2.923	2.868	2.821	2.784	2.698	2.660
	0.50	3.017	2.974	2.929	2.893	2.828	2.809
	0.75	3.059	3.027	2.996	2.976	2.945	2.932

presents the choices of the limit coefficient L to achieve the desired ARL₀ ≅ 200, 370, and 500, while

Table 2. The run-length characteristics of the EWMA-MA control chart for λ = 0.05 and n = 1 under the normal distribution.

Shift (δ)	w = 2			w = 3			w = 4			w = 5			w = 8			w = 10
	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL
0	372.5	361.4	261	371.8	366.3	259	370.6	364.5	259	369.1	363.5	257	368.5	363.5	252	368.0	363.3	252
0.10	215.9	200.9	154	217.9	204.9	156	214.4	201.4	154	214.3	204.7	154	208.3	204.7	148	211.0	209.2	147
0.25	67.4	58.9	52	67.1	58.8	51	66.8	58.8	51	66.8	58.8	51	64.3	55.8	50	64.7	57.9	48
0.50	23.6	16.9	21	23.7	16.8	21	22.3	16.4	19	22.0	16.4	19	21.9	15.8	18	21.0	15.7	17
0.75	13.0	8.3	13	12.6	8.0	12	12.1	7.7	11	11.6	7.7	10	11.0	7.7	10	11.0	7.7	9
1.00	8.6	5.8	8	8.2	5.3	8	8.0	4.8	8	7.6	4.7	7	7.3	4.5	7	6.9	4.5	6
1.50	4.3	3.2	4	4.3	3.1	4	4.2	2.7	4	4.2	2.7	4	4.0	2.5	4	3.8	2.3	3
2.00	2.6	1.8	2	2.6	1.7	2	2.6	1.7	2	2.6	1.7	2	2.5	1.5	2	2.4	1.4	2
2.50	1.8	1.0	1	1.8	1.0	1	1.8	1.1	1	1.8	1.1	1	1.8	1.1	1	1.8	0.9	1
3.00	1.5	0.6	1	1.4	0.6	1	1.4	0.7	1	1.4	0.7	1	1.3	0.7	1	1.4	0.6	1
EQL	23.9			23.6			23.2			22.9			22.7			22.1

,

Table 3. The run-length characteristics of the EWMA-MA control chart for λ = 0.25 and n = 1 under the normal distribution.

Shift (δ)	w = 2			w = 3			w = 4			w = 5			w = 8			w = 10
	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL
0	370.9	366.5	261	370.8	366.5	261	370.7	366.4	260	370.7	365.9	260	371.6	360.3	260	370.2	360.0	259
0.10	285.7	278.5	240	284.4	276.5	238	284.1	276.0	237	283.6	275.0	236	282.1	272.5	232	281.5	271.7	230
0.25	127.8	120.5	115	127.5	119.5	114	126.6	119.0	113	124.7	118.2	112	123.7	117.4	110	122.3	116.4	109
0.50	36.6	33.8	27	35.4	31.5	26	35.2	31.3	25	35.1	31.2	25	32.3	28.4	24	31.4	27.1	23
0.75	16.1	12.8	12	15.5	11.7	12	15.4	11.7	12	15.3	11.6	12	14.8	10.3	12	14.7	9.9	13
1.00	9.4	7.0	7.5	9.1	6.2	8	9.1	6.1	8	9.1	5.7	8	8.9	5.6	8	8.8	5.5	8
1.50	4.4	2.6	4	4.6	2.6	5	4.7	2.6	5	4.6	2.7	4	4.5	2.7	4	4.5	2.7	4
2.00	2.9	1.5	3	3.0	1.7	3	2.9	1.7	3	2.9	1.7	3	2.8	1.7	2	2.8	1.7	2
2.50	2.0	1.2	2	2.1	1.2	2	2.0	1.2	2	2.0	1.1	2	1.9	1.1	2	1.9	1.1	2
3.00	1.6	0.8	1	1.5	0.8	1	1.5	0.8	1	1.5	0.8	1	1.5	0.7	1	1.4	0.7	1
EQL	28.9			28.7			28.6			28.5			28.0			27.3

and

Table 4. The run-length characteristics of the EWMA-MA control chart for λ = 0.75 and n = 1 under the normal distribution.

Shift (δ)	w = 2			w = 3			w = 4			w = 5			w = 8			w = 10
	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL	ARL	SDRL	MRL
0	370.2	367.8	255	370.0	367.0	255	370.1	365.6	244	370.0	360.9	237	369.7	359.1	206	369.5	333.0	167
0.10	331.5	329.3	231	323.2	327.4	222	316.0	325.9	209	304.4	321.3	193	285.9	320.5	148	266.5	305.6	114
0.25	203.5	199.5	142	184.9	195.5	125	165.0	183.3	107.5	150.5	170.0	97	119.1	148.6	62	100.0	144.0	39
0.50	79.2	79.6	55	62.2	65.6	42	52.7	57.8	34	43.0	50.2	26	30.6	40.0	16	23.6	34.5	6
0.75	33.5	33.2	23	25.5	25.7	17	19.9	21.8	13	16.6	19.3	10	10.6	13.8	4	8.7	11.9	3
1.00	16.3	15.9	12	11.9	12.2	8	9.5	10.0	6	7.8	8.6	5	5.2	6.5	2	4.3	5.6	2
1.50	5.6	4.9	4	4.2	3.8	3	3.4	3.2	2	2.9	2.8	2	2.1	2.1	1	1.9	1.8	1
2.00	2.8	2.1	2	2.2	1.7	2	1.9	1.4	1	1.7	1.2	1	1.4	0.9	1	1.3	0.7	1
2.50	1.8	1.1	1	1.5	0.9	1	1.3	0.7	1	1.2	0.6	1	1.1	0.4	1	1.1	0.4	1
3.00	1.4	0.7	1	1.2	0.5	1	1.1	0.4	1	1.1	0.3	1	1	0.2	1	1	0.2	1
EQL	39.9			31.5			28.3			23.6			18.4			16.5

depict the run-length characteristics of the EWMA-MA control chart under different choices of design parameters. The main findings based on the results given in Table 1, Table 2, Table 3 and Table 4 are summarized as follows:

• It is observed that the values of limit coefficient L decrease by increasing the values of span (w) for the fixed value of the smoothing parameter. On the other hand, the value of limit coefficient L increases with λ for the fixed value of w;
• For a fixed value of λ, the values of ARL₁, SDRL₁, and MRL₁ tends to decrease with the increase in the magnitude of shift (δ) and w;
• It is also observed that the obtained values of AEQL decrease as the values of w increase;
• Furthermore, the ARL₁, SDRL₁, MRL₁, and AEQL values tend to increase with λ by keeping other design parameters at a fixed level.

4.2. Robustness of the Mixed EWMA-MA Chart to Non-Normality

Generally, the control charting structures assumes the underlying process follows some specific distribution such as the normal distribution. If the IC run-length distribution of a control chart remains approximately the same when the fundamental distributional assumption is violated, it is considered robust [5]. The behavior of the IC RL profile of a control chart under different process distributions provides useful insight for practical implementation.

In this article, the IC run-length characteristics such as ARL₀, SDL₀, and MRL₀ of the EWMA-MA control chart are computed for various process distributions such as the normal distribution, N(0,1); the gamma distribution with shape parameters α = 0.5, 1, 2, 5 and the scale parameter β = 1; the Weibull distributions with shape parameters β = 0.5, 2, 3.5, 10 and scale parameter η = 1; the Student’s t_v distribution with v = 4, 10, and 20 degrees of freedom; the logistic distribution with location and scale parameters as (2,1), (6,2), and (9,4), respectively; the Laplace distribution with zero as location parameter, and 2, 1, and $\frac{1}{\sqrt{2}}$ as shape parameters, respectively. These distributions are considered to study the behavior of the IC run-length characteristics under symmetric and asymmetric process distributions. In

Table 5. In-control run-length characteristics of the EWMA-MA control chart at w = 5 and n = 5 under various probability distributions.

Parameter (s)	(λ, L)
	(0.05, 2.310)		(0.10, 2.490)		(0.25, 2.676)		(0.50, 2.798)		(0.75, 2.882)
	ARL0	SDRL0	ARL0	SDRL0	ARL0	SDRL0	ARL0	SDRL0	ARL0	SDRL0
Normal distribution
Normal (0, 1)	370.00	358.3	370.4	366.9	370.6	367.1	370.3	367.3	370.7	368.3
Gamma distribution
Gamma (5, 1)	374.2	370.4	368.3	365.4	364.8	362.9	363.1	361.9	354.0	349.6
Gamma (2, 1)	371.2	370.5	363.5	362.5	347.8	347.4	340.4	335.6	324.4	323.7
Gamma (1, 1)	371.6	370.1	361.3	359.8	332.0	330.7	308.1	301.9	304.6	299.8
Gamma (0.5, 1)	365.5	360.4	353.9	351.6	303.0	301.5	260.7	257.9	257.4	254.1
Weibull distribution
Weibull (0.5, 1)	366.2	350.1	294.2	285.5	200.4	198.5	180.8	173.9	177.9	168.5
Weibull (2, 1)	369.8	366.5	368.9	366.9	366.9	365.2	365.5	361.6	360.7	357.1
Weibull (3.5, 1)	370.3	366.7	369.3	362.2	368.2	366.7	365.7	362.2	361.8	357.5
Weibull (10, 1)	378.1	368.3	375.3	372.2	374.6	370.2	367.1	364.5	362.7	358.8
t-distribution
t (20)	374.8	369.9	364.7	363.3	363.3	362.1	360.9	358.9	361.1	358.0
t (10)	370.7	362.7	360.5	367.0	361.0	359.4	354.7	352.6	352.9	350.2
t (4)	358.5	357.4	337.6	336.6	293.9	290.8	285.6	281.0	281.9	275.9
Logistic distribution
Logis (2, 1)	380.3	375.8	365.9	363.9	362.1	357.9	355.8	353.6	350.2	346.9
Logis (6, 2)	364.5	361.7	360.7	358.2	352.4	349.2	349.9	345.2	347.1	345.2
Logis (9, 4)	373.9	370.2	365.1	360.5	357.3	355.5	353.4	350.9	348.0	345.7
Laplace distribution
Laplace (0, 2)	367.4	364.0	360.0	355.6	331.1	327.9	319.6	309.2	316.7	301.3
Laplace (0, 1)	362.8	356.8	356.5	352.1	330.7	325.1	318.2	309.1	307.1	298.4
Laplace (0, $\frac{1}{\sqrt{2}}$)	359.5.8	350.5	352.7	348.2	329.8	324.8	309.4	299.7	298.9	291.5

, assuming a pre-fixed ARL₀ = 370, the attained IC run-length characteristics of the EWMA-MA chart are presented for various choices of the sensitivity parameter λ under a fixed value of span w = 5 and sample size n = 5. The following points summarize the results:

• The mixed EWMA-MA chart exhibited a somewhat robust behavior for the smoothing parameter λ = 0.05 as the ARL₀ under the gamma, student’s t, logistic, and Laplace distributions remain within 10% of the normal scheme;
• For λ = 0.10, the ARL₀ values of the mixed EWMA-MA chart under gamma, student’s t, Weibull, Logistic, and Laplace distributions remain slightly lower than the pre-defined value of 370, especially when the shape of the distribution becomes extremely skewed such as the Weibull (0.5,1);
• Like λ = 0.10, the ARL₀ of the EWMA-MA chart for λ = 0.25, 0.50, and 0.75 remains lower than the chosen value of 370 under non-normal process distribution which results in an increased false alarms rate;
• In general, the mixed EWMA-MA control chart is found to be quite resistant to non-normality when the process follows the symmetrical distributions.

5. Comparative Study

The performance of the EWMA-MA chart is compared with some existing control charts which include conventional MA, EWMA, CUSUM, mixed EWMA-CUSUM, combined Shewhart-EWMA, and combined Shewhart-CUSUM control charts. The comparison is made using ARL, SDRL, and MDRL to identify the shift of a specific magnitude. Furthermore, the AEQL and RMI are used to evaluate the overall performance of the EWMA-MA chart and several existing control charts. For a rational comparison, the control charts are constructed under a fixed ARL₀ = 370 and a sample of size n = 5 with their respective design parameters. The results are provided in

Table 6. Run-length characteristics of the MA, EWMA, CUSUM, mixed EWMA-CUSUM, combined Shewhart-EWMA, combined Shewhart-CUSUM, and EWMA-MA control charts for n = 1 at ARL₀ = 370.

Control Chart	RL Characteristic	Shift (δ)											AEQL	RMI
		0.0	0.05	0.10	0.25	0.50	0.75	1.0	1.50	2.0	2.50	3.0
MA at w = 5, L1 = 2.88	ARL	368.6	348	306.7	161.4	51.3	20.9	11.2	4.6	2.8	2	1.6	32.8	0.47
	SDRL	373.5	348.4	302.8	162	49.4	19	9.2	3	1.5	0.9	0.7
	MRL	252	236	214	110	37	15	8	4	2	2	1
EWMA at λ = 0.05, L2 = 2.492	ARL	369.8	316.1	222.6	73.1	26.8	15.3	10.7	6.8	5	4	3.4	41.5	0.50
	SDRL	352.7	299.2	207.8	58	15.3	6.9	4.1	2	1.2	0.9	0.7
	MRL	258	227	158	56	23	14	10	7	5	4	3
CUSUM k = 0.50, h = 4.77	ARL	368.9	348.4	280.6	121.8	34.9	16.2	9.9	5.5	3.9	3	2.5	36.1	0.46
	SDRL	363.7	346.2	277.3	115.5	29	10.6	5.3	2.2	1.3	0.8	0.6
	MRL	258	243.5	194	86	26	13	9	5	4	3	2
Mixed EWMA-CUSUM λ = 0.05, k = 0.50, h = 45.54	ARL	370.6	303.6	194.9	74	37.3	26.5	21.4	15.8	12.9	11.1	9.7	98.9	2.03
	SDRL	327.5	257.8	153.8	39.5	11.5	6	3.9	2.1	1.4	1.1	0.9
	MRL	269	227	148	64	35	26	21	16	13	11	10
Combined Shewhart-EWMA λ = 0.05, L2 = 2.91, L3 = 3.11	ARL	371.2	337.5	264.7	101.1	33.2	18	12	7.1	4.5	3	2.1	37.3	0.48
	SDRL	365.7	334	258.3	85.4	20.7	8.8	5.3	3.1	2.3	1.8	1.3
	MRL	262	238	185	76	29	17	12	7	5	3	2
Combined Shewhart-CUSUM k = 0.50, h = 5.77, L3 = 3.11	ARL	370.9	350	310.7	153.1	43.1	18.8	11.4	6	3.9	2.7	2	36.7	0.55
	SDRL	369.7	347.7	306.3	146.7	35	12.1	6.2	2.8	1.9	1.4	1.1
	MRL	259	241	215	109	33	16	10	6	4	3	2
Mixed EWMA-MA λ = 0.05, w = 5, L = 2.311	ARL	370.4	307.3	208.9	65.9	22.5	12.4	7.9	4.2	2.6	1.7	1.3	23.2	0.008
	SDRL	360.9	304.2	201.2	57.9	16.2	7.8	4.9	2.8	1.7	1.1	0.7
	MRL	256	218	148.5	51	19	11	7	4	2	1	1

. The following are some interesting findings regarding the EWMA-MA control chart.

It is found that the EWMA-MA chart has a minimum ARL₁ values as compared to competing charts except for shifts of magnitude δ < 0.25 for which the mixed EWMA-CUSUM control chart has slightly better performance. However, the mixed EWMA-CUSUM has the worst performance for moderate to large process shifts. Additionally, the results from Table 6 indicate that the EWMA-MA chart has lower AEQL and RMI values than its competitors which highlights its overall superior shift detection ability. To further support the findings, an application of the EWMA-MA and its competing control charts is considered in the following subsection.

Application

The application of control charts for monitoring the wind turbines generators (WTG) was considered by Cambron [25] where four different parameters, including electrical energy produced, tower vibration, nacelle yaw, and gearbox temperature, were monitored for possible abnormal behavior by plotting the difference of a specific WTG and the average of all remaining WTGs using the EWMA control chart. In another application of control charts in monitoring and controlling the energy performance in compressed air generation and use for enhanced energy management, Benedetti [26] employed CUSUM control charts to identify optimal operating conditions and identification of changes in consumption patterns and degradation of the energy performance due to faults.

In this paper, for practical implementation of the EWMA-MA chart, a real dataset from a combined cycle power plant (CCPP) originally collected by Tüfekci [27] is considered. The CCPP generates electricity by combining gas and steam turbines in a cycle. The electricity is generated by gas and steam turbines and then transmitted from one turbine to another. These elements are connected in a cycle where four different types of input variables can influence the overall energy output, which are atmospheric temperature (AT), relative humidity (RH), exhaust vacuum (EV), and ambient pressure (AP) (see Tüfekci [27] for more details). To conserve space, in this study only the AP is used as a variable of interest to demonstrate the application of the EWMA-MA control chart. However, it can be applied to other variables in a similar fashion. AP can affect the gas turbine’s performance because at lower AP the efficiency of the gas turbine decreases which affects the output of the power plant. The original dataset contains 9568 data points collected from CCPP. For convenience, the first 600 observations are taken as a realization of the IC process. The mean of AP is found to be 1013.55, while the standard deviation is 5.822. The normality of the dataset is tested through Anderson-Darling (A = 0.0745, p = 0.475) and Kolmogorov–Smirnov (D = 0.0258, p = 0.8196) tests which indicate that data follows the normal distribution as p-value > 0.05. The histogram of the data shown in Figure 1 also indicates that the AP approximately follows a normal distribution.

Considering the first 600 observations as a population, 50 samples of size 5 each are drawn where the first 20 samples are taken assuming the IC process state, and the next 30 samples are drawn after introducing the shift in the process mean such that ${\mu }_1={\mu }_0+0.5\frac{\sigma }{\sqrt{n}}$ to imply the OOC process state. To monitor the process, the existing MA, EWMA, CUSUM, mixed EWMA-CUSUM, combined Shewhart-EWMA, combined Shewhart-CUSUM, and the EWMA-MA control charts are set up for a pre-fixed ARL₀ = 370 and n = 5.

The existing MA control chart is constructed using w = 5 and displayed in Figure 2. It is observed that the MA chart fails to identify the shift in the process mean indicating the IC process state. The EWMA control chart is set up by using λ = 0.05 and shown in Figure 3. The EWMA control chart detects the shift at sample number 47. The existing CUSUM control chart is constructed using the design parameters k = 0.50 with h = 4.77 and shown in Figure 4. The CUSUM chart triggers an OOC signal at sample number 48. The existing mixed EWMA-CUSUM chart is set up by using the design parameters λ = 0.05, k = 0.50, and h = 45.54. Figure 5 depicts the mixed EWMA-CUSUM control chart that fails to detect the shift and declares the process to be IC. The combined Shewhart-EWMA control chart is set up by using L = 3.11, λ = 0.05, and K = 2.91. It detects the process shift at sample number 48 as shown in Figure 6. The combined Shewhart-CUSUM control chart is computed using L = 3.11, k = 0.5, and h = 5.75 and displayed in Figure 7. The chart fails to detect the shift in the process mean. Finally, the EWMA-MA control chart is set up with design parameters λ = 0.05, w = 5, and L = 2.311 as shown in Figure 8. It triggers an OOC signal at point 41 which is the 21st sample after the introduction of a shift in the process mean. These results clearly show that the EWMA-MA control chart has a superior mean shift detection ability as compared to other existing control charts, which further confirms the simulated results.

6. Summary and Conclusions

In the SPC literature several mixed/combined control charting strategies were introduced to enhance the sensitivity of the control charts to quickly detect a range of shifts in the process parameters. Recently, Sukparungsee et al. [20] developed a mixed EWMA-MA control chart for quick detection of small to moderate shifts in the process mean. Their charting strategy assumed successive moving averages as independent and excluded the covariance terms from the variance expression of the EWMA-MA statistic. This approach is inappropriate as the MA terms are dependent and have at least one common $\overline{X}$ term when their distance is less than the span w. Thus, their strategy may result in misleading decisions about the process state. In this study, the revised mixed EWMA-MA control chart is presented, and a comprehensive comparison of its performance is made with several competing combined control charting structures by using various run-length characteristics that were lacking in Sukparungsee et al. [20]. The comparison established the better shift detection ability of the revised mixed EWMA-MA control chart. Moreover, a robustness study is carried out to assess the effects of non-normality on the in-control run-length distribution of the EWMA-MA control chart. The results of the robustness study showed that the revised EWMA-MA control is somewhat resistant to non-normality, especially for smaller values of the smoothing parameter λ. Finally, to demonstrate the practicability of the EWMA-MA chart, an example from real-life data from a combined cycle power plant was used to monitor the ambient pressure that can affect the efficiency of gas turbine. The results further confirm the superiority of the EWMA-MA chart over the existing individual and combined charting schemes.