The three Statistical Control Charts using Ranked Set Sampling

Mu’azu Ramat Abujiya Department of Mathematical Sciences

Faculty of Sciences, Universiti Teknologi Malaysia Skudai 81310, Johor, Malaysia

abujiya@gmail.com

Muhammad Hisyam Lee Department of Mathematical Sciences

Faculty of Sciences, Universiti Teknologi Malaysia Skudai 81310, Johor, Malaysia

mhl@utm.my

Abstract—This article investigated the performance of the three common statistical control charts, the Shewhart chart, cumulative sum (CUSUM) chart, and exponentially weighted moving average (EWMA) chart for location using ranked set sampling (RSS) instead of the traditional simple random sampling (SRS). Considering a normal population, a Monte Carlo simulation is carried out for several shift values for each of the control chart. The average run length (ARL) showed that the control charts based on RSS data are superior to their corresponding SRS counterparts with no significant difference between CUSUM and EWMA charts. There is an interesting increase in the sensitivity of RSS based Shewhart chart relative to other charts.

Keywords— Average run length, CUSUM chart, EWMA chart, ranked set sampling, Shewhart control chart.

I. INTRODUCTION Control charts are graphical techniques for continuous

monitoring of the quality of a manufacturing process. Their primary objective is to distinguish between chance and assignable causes of process variation. The chance cause is part of a stable system and is usually very small in magnitude, while an assignable cause is due to factors that are not part of the process. When a process operates in the presence of assignable cause, it is not stable and is out-of-control. Control charts can help quickly detect the formation of assignable causes of process disturbances so that investigation and corrective measure is taken before many nonconforming units are produced. In general, control charts are effective tools in eliminating process variability as well as estimating the parameters of the production process [1].

A control chart consists of three horizontal lines, the upper control limit (UCL), the centerline (CL) and the lower control limit (LCL). A process is considered stable, i.e., in-control when the plotting points falls within the control limits. A point outside the control limits indicates an out-of-control signal and requires corrective action to bring the process back in-control and improve the quality of the process. The three types of control charts widely used in practice include Shewhart control chart, cumulative sum (CUSUM) chart and exponentially weighted moving average (EWMA) chart [2].

The performance of these control charts are often compared in terms of their average run length (ARL) properties. ARL

represents the average number of samples plotted on a control chart until an out-of-control sample is observed. It measures how quickly a chart responds to process disturbances. Generally, the ARL for an in-control process should be high, and low, when the process mean shifts to an unsatisfactory level. Statistically, the Shewhart charts are slow in detecting small shifts in the process but handles large shifts perfectly, while CUSUM and EWMA charts are very good with small shifts, [3] and [4].

Several authors have studied the ARL performance of these control charts, but most of the reports in the literature are based on simple random sampling (SRS) which is considerably less effective in estimating the population mean as compared to ranked set sampling (RSS) with the same subgroup size. This sampling technique has proven to be very effective in situations where measurements of quality characteristics of interest are difficult or expensive, but could readily be ordered by visual inspection or some cheap method not requiring actual measurement [5] and [6].

There are, however, some recent researches that used RSS scheme to improve the efficiency of the control charts in detecting changes in process characteristics. For example, [7] used ranked sampling with equal and unequal allocation to develop Shewhart charts, [8] used several modifications of ranked sampling, [9] used double ranked sampling, [10] used robust ranked sampling and very recently, [11] and [12] used the scheme to develop combined Shewhart-EWMA and combined Shewhart-CUSUM control charts respectively.

While previous studies has shown the statistical significance of RSS based control charts for mean, no attempt have been made to compare the performance of the three commonly used control charts for the same subgroup size with same pair of shifts using RSS. Therefore, this paper investigates the performance of the Shewhart , CUSUM and EWMA charts using RSS.

Using Monte Carlo simulation, we compute the ARL values for the RSS based Shewhart , CUSUM and EWMA control charts. Comparisons among the newly developed control charts are made and in addition, we compare the results with the classical control charts using SRS. We also give a real life example to demonstrate the simplicity of the scheme.

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978-1-4673-5814-9/13/$31.00 ©2013 IEEE

II. CONTROL CHARTS BASED ON RSS

A. Ranked Set Sampling The RSS scheme consists of drawing n random samples,

each of size n from target population, and ranking the units within each set with respect to a variable of interest. The total n measured observations are then obtained by taking the smallest observation from the first set, second smallest from the second set and continues until the largest unit is been selected from nth set. The procedure may be repeated m times until nm units have been measured. These nm observations are the ranked set samples.

Suppose : denote the ith order statistic from the ith sample of size n in the jth cycle, then the unbiased estimator for the population mean based on RSS is

∑ : (1)

with 1, 2, 3, … , and the variance of is given by

∑ : (2)

where : : : is the population variance of the ith order statistic, [6]. Furthermore, [13] showed that regardless of the presence of errors in ranking, the RSS scheme may still produce improved precision in the estimator of population mean than the random sampling.

B. The Shewhart Control Chart Suppose that a process follows a normal distribution , , with a known mean and a known variance :

of ith order statistic for RSS then, equation (1) can be plotted on the RSS based Shewhart control limits as follows [7]

3 3 (3)

where 1⁄ ∑ : and is calculated using the

known results from the tables of order statistics for standard normal distribution [14].

Although, the Shewhart control chart is still widely in use, it is only sensitive to large process shifts. This is because it uses only the information about the last sample observation and ignores all the information in the entire sequence of points, thereby making it less effective in detecting small process shifts. Other control charts, such as the CUSUM and EWMA charts are good complementary tools to the Shewhart control chart when small process shifts are of interest [1].

C. The CUSUM Control Chart The CUSUM control chart proposed by [15], uses

information contained in sequence of points. There are two types of CUSUM charts, the two-sided (V-mask) and two one-sided (tabular) CUSUM. The later plots more closely like

Shewhart and EWMA charts and is chosen for use. Assuming a process follows a normal distribution with a known population mean and variance : , the CUSUM charts based on RSS are given by

0, 0, (4)

where the starting values are 0. The statistics and are the one-sided upper and lower RSS CUSUMs, respectively. When either or exceeds the predetermined decision interval h, 0, the process is considered to be out of control. For a good choice of chart’s parameters h and k to archive a specific ARL, see [16] for detail.

D. The EWMA Control Chart Like Shewhart and CUSUM control charts, the EWMA control chart introduced by [17] is easy to implement and interpret. For RSS data, it’s based on the statistic

: 1 : (5)

where λ is a smoothing constant with 0 1 , and is the starting value of : when the process is in-control. A small value of λ increases the chart’s sensitivity to small shifts and a large value of λ increases the chart’s sensitivity to large shifts. The variance of : is given by

: 2⁄ 1 1 . (6)

Thus, the EWMA control limits based RSS are defined as follows

: : (7)

where L is the width of the control limits. The term 11 approaches one, as j gets larger. When 1, the EWMA chart behaves like the Shewhart chart. See [18] for choices of the λ and L.

III. AVERAGE RUN LENGTH COMPARISON The ARL assumes that the process is in a state of statistical

control with mean and standard deviation , and at some point in time the process may start to get out of control with a shift in mean from to /√ , [1]. Suppose the process follows a normal distribution with mean and variance when the process is in-control, the shift on the process mean is given by – / . If 0, the process is in state of statistical control and any point outside the control limits is a false alarm. In this case, the ARLs should be close to the target value.

To compare the ARL values using RSS data, we used Monte Carlo simulation. For a better comparison, we set the in-

control ARL values of the three control charts based on RSS at ARL 370. For each mean shift, we simulated 500,000 iterations. Using the standard Shewhart chart three-sigma, the CUSUM’s reference value and decision interval were set at 0.50 and 4.7749 respectively, [16]. We also set EWMA’s chart parameters at 0.12 and 2.75, [18].

A. Perfect Ranking Perfect ranking is when ranking of the units in each

subgroup can be done with respect to variable of interest without errors in ranking. The simulation was carried out for a normal process with 0 and 1 using various shift values δ, when the subgroup sizes are 3 and 5. Control limits were opened up little bit, where necessary, to archive the target false alarm rate of ARL 370. The results are presented in Table I. Note that the ARL values for the classical SRS chart reported in Table I are independent of the subgroup size [3], [7]. To measure the overall performance of the control charts, we implore the Average Ratio of ARL (ARARL) given by d (8)

where and are the minimum and maximum mean shifts respectively. is generated from the best chart and the results obtained are displayed at the bottom of Tables I.

TABLE I. ARL COMPARISON OF THE THREE CONTROL CHARTS USING RSS WHEN ARL0 = 370.

RSS SRS

Shewhart CUSUM EWMA Shewh. CUSUM EWMA

n = 3 n = 5 n = 3 n = 5 n = 3 n = 5 any sample size

0.00 370.3 370.1 370.1 370.0 370.0 370.2 370.3 370.0 370.4

0.25 228.2 195.1 71.7 50.5 56.3 41.3 287.9 124.1 98.0

0.50 97.4 67.8 18.9 13.5 17.1 12.7 156.9 35.4 29.7

0.75 41.6 26.0 9.4 7.1 9.2 7.1 81.9 16.2 14.8

1.00 19.6 11.4 6.2 4.8 6.3 4.9 44.4 9.9 9.6

1.50 5.8 3.3 3.7 3.0 3.9 3.2 15.1 5.5 5.6

2.00 2.5 1.6 2.7 2.3 2.9 2.4 6.4 3.9 4.0

2.50 1.5 1.1 2.2 1.9 2.3 2.0 3.3 3.0 3.2

2.66 1.3 1.1 2.1 1.8 2.2 1.9 2.7 2.8 2.93.00 1.2 1.0 1.9 1.6 2.0 1.8 2.0 2.5 2.6

4.00 1.0 1.0 1.4 1.1 1.6 1.2 1.2 2.0 2.1

ARARAL 2.21 1.51 1.19 0.96 1.23 1.00 1.79 1.76 2.21

Table I shows that the Shewhart, CUSUM and EWMA control charts based on RSS gives better ARL performance as compared to their corresponding control charts for mean using SRS when a process is out of control, 0. Observe that as the sample size n increases in RSS based control charts, the ARL values decreases. In other words, the ARL values are inversely proportional to sample size n. For small shift in the

process, 0.25 0.75, the ARL values of EWMA indicates that the chart is doing better than both CUSUM and Shewhart. The CUSUM on the other hand dominates the EWMA chart when moderate shift is of interest. The overall performance in terms of ARARL shows no significant difference between the CUSUM and EWMA control charts when 0.25 4. However, beyond this region, the overall performance of CUSUM dominates both the EWMA and Shewhart control charts (cf. Table I).

A very interesting finding in Table I is that, the RSS based Shewhart chart provide reasonably fast detection of shifts in process mean when 3. Note that the classical Shewhart chart is better in handling large process shifts than CUSUM and EWMA only when 2.66, [3]. For example, in RSS based control charts, if 2.00 and n varies from 3 to 5 , the Shewhart ARL values of are 2.5, 1.6 as compared to CUSUM’s 2.7, 2.3 and EWMA’s 2.9, 2.4 . While in classical SRS charts, if 2.00, the Shewhart ARL value is 6.4 which is not better than the CUSUM’s and EWMA’s 3.9 and 4.0, respectively.

B. Imperfect Ranking In reality, ranking of the units in a subgroup with respect to

main variable of interest may not be possible but could be carried out by its concomitant variable . This is called imperfect ranking. Since ranking is carried out on concomitant variable, the accuracy of the process depends on the correlation

between and . Perfect ranking and SRS are special cases of imperfect ranking with 1 and 0 respectively.

Let , denote a bivariate normal random vector and suppose the regression of on is linear. The unbiased estimator for the mean of the variable of interest with judgment ranking based on concomitant variable using imperfect ranked set sampling (IRSS) is given by

∑ : (9) 1, 2, 3, … , and the variance of is

1 ∑ : (10)

where is the variance of the variable of interest and : is the variance of the ith order statistic [19].

The Shewhart, CUSUM and EWMA chart statistics in equations (3), (4) and (6) are based on perfect ranked sampling. To accommodate imperfect ranking, we replace in equations (3), (4) and (6) with

1 ∑ : . (11)

Using equation (11), simulations were carried for each combination of 2, 3, 4, 5, 6; 0.0, 0.25, 0.5, 0.75, 1.0,1.5, 2.0, 2.5, 3.0, 4.0 and 0.0, 0.25, 0.5, 0.75. The results are displayed in Tables II – VI, and the following remarks can be made.

• If the process is in-control 0, the ARL values for the three control charts are approximately the same and sufficient enough to compare the control charts.

• When the ranking is perfect or almost perfect, the EWMA control chart appears to be more robust to RSS data than the CUSUM and Shewhart charts. For example, if 1, the in-control ARL value of EWMA is 370.1 as compared to the 366.9 of CUSUM and 344.5 of Shewhart charts.

TABLE II. ARL VALUES FOR THE THREE CONTROL CHARTS USING RSS WHEN 2.

Control Charts

0.00 0.25 0.50 0.75 1.00 1.50 2.00 2.50 3.00 4.00

Shewhart 3 sigma

0.00 370.8 282.1 155.7 80.9 43.8 15.0 6.3 3.2 2.0 1.20.25 369.3 279.7 152.7 79.5 42.8 14.5 6.1 3.1 2.0 1.20.50 370.6 273.1 146.3 75.3 39.7 13.2 5.6 2.9 1.8 1.10.75 366.0 264.5 135.7 66.5 34.5 11.2 4.7 2.5 1.6 1.11.00 344.5 240.7 114.4 54.3 27.1 8.5 3.6 2.0 1.4 1.0

CUSUM 0.50 4.7749

0.00 370.1 123.9 35.3 16.2 9.9 5.5 3.9 3.0 2.5 2.00.25 370.8 122.3 34.6 15.9 9.8 5.5 3.8 3.0 2.5 1.90.50 369.8 116.5 32.5 15.0 9.3 5.2 3.7 2.9 2.4 1.90.75 369.9 106.0 29.0 13.6 8.5 4.9 3.4 2.7 2.3 1.81.00 366.9 90.8 24.2 11.6 7.4 4.3 3.1 2.5 2.1 1.7

EWMA 0.12 2.75

0.00 371.9 95.9 29.5 14.9 9.6 5.6 4.0 3.2 2.6 2.10.25 371.2 94.9 29.2 14.6 9.5 5.5 4.0 3.1 2.6 2.00.50 372.7 89.8 27.6 14.0 9.1 5.3 3.8 3.0 2.5 2.00.75 372.7 82.6 24.8 12.8 8.4 5.0 3.6 2.9 2.4 1.91.00 370.1 70.8 21.3 11.1 7.4 4.5 3.3 2.6 2.2 1.8

TABLE III. ARL VALUES FOR THE THREE CONTROL CHARTS USING RSS WHEN 3.

Control Charts

0.00 0.25 0.50 0.75 1.00 1.50 2.00 2.50 3.00 4.00

Shewhart 3 sigma

0.00 369.2 281.5 154.5 81.3 44.0 15.0 6.3 3.2 2.0 1.20.25 369.7 279.1 152.3 78.7 42.3 14.3 6.0 3.1 1.9 1.20.50 370.9 271.2 142.4 71.9 37.7 12.4 5.2 2.7 1.7 1.10.75 367.5 256.0 123.5 59.0 29.7 9.4 3.9 2.1 1.4 1.01.00 343.3 216.7 92.2 39.9 18.9 5.7 2.5 1.5 1.1 1.0

CUSUM 0.50 4.7749

0.00 368.7 124.0 35.4 16.2 9.9 5.5 3.9 3.0 2.5 2.00.25 368.3 121.1 34.2 15.7 9.7 5.4 3.8 3.0 2.4 1.90.50 368.3 112.2 31.2 14.5 9.0 5.1 3.6 2.8 2.3 1.90.75 370.5 96.6 25.9 12.3 7.8 4.5 3.2 2.6 2.2 1.71.00 363.3 71.9 18.7 9.4 6.2 3.7 2.7 2.2 1.9 1.4

EWMA 0.12 2.75

0.00 371.9 96.3 29.5 14.9 9.6 5.6 4.0 3.2 2.6 2.10.25 371.0 94.1 28.8 14.5 9.4 5.5 3.9 3.1 2.6 2.00.50 372.7 87.1 26.5 13.5 8.8 5.2 3.7 3.0 2.5 2.00.75 372.6 75.3 22.6 11.7 7.7 4.6 3.4 2.7 2.3 1.91.00 369.2 56.4 17.0 9.1 6.2 3.8 2.8 2.3 2.0 1.6

• The ARL values are independent of the sample size for the three control charts when the process is in-control 0, and inversely proportional to sample size n as the process gets out of control, 0.

• As the sample size increase and 0, the three control charts based on RSS offer much faster detection of shifts in the process mean than their corresponding classical charts using SRS.

TABLE IV. ARL VALUES FOR THE THREE CONTROL CHARTS USING RSS WHEN 4.

Control Charts

0.00 0.25 0.50 0.75 1.00 1.50 2.00 2.50 3.00 4.00

Shewhart3 sigma

0.00 365.8 280.8 155.2 81.3 44.0 14.9 6.3 3.2 2.0 1.20.25 374.0 277.3 152.0 78.5 42.1 14.2 6.0 3.1 1.9 1.20.50 369.8 270.2 140.1 69.7 36.4 11.9 5.0 2.6 1.7 1.10.75 369.0 248.2 116.9 54.2 26.9 8.4 3.5 1.9 1.3 1.01.00 346.7 199.2 76.7 31.0 14.1 4.2 1.9 1.3 1.1 1.0

CUSUM 0.50 4.77490.00 370.4 124.2 35.3 16.2 9.9 5.5 3.9 3.0 2.5 2.00.25 370.4 120.5 34.0 15.7 9.6 5.4 3.8 2.9 2.4 1.90.50 369.9 109.5 30.3 14.1 8.8 5.0 3.5 2.8 2.3 1.90.75 370.4 90.4 24.1 11.5 7.4 4.3 3.1 2.5 2.1 1.71.00 368.0 59.4 15.6 8.0 5.4 3.3 2.4 2.0 1.8 1.2

EWMA 0.12 2.75

0.00 371.9 96.2 29.5 14.9 9.6 5.6 4.0 3.2 2.6 2.10.25 372.9 93.5 28.6 14.5 9.4 5.5 3.9 3.1 2.6 2.00.50 371.5 85.2 25.9 13.2 8.6 5.1 3.7 2.9 2.4 2.00.75 371.8 70.5 21.1 11.0 7.3 4.4 3.2 2.6 2.2 1.81.00 369.8 47.4 14.4 7.9 5.5 3.4 2.6 2.1 1.9 1.4

TABLE V. ARL VALUES FOR THE THREE CONTROL CHARTS USING RSS WHEN 5.

Control Charts

0.00 0.25 0.50 0.75 1.00 1.50 2.00 2.50 3.00 4.00

Shewhart3 sigma

0.00 366.2 280.6 154.0 81.1 43.8 15.0 6.3 3.2 2.0 1.20.25 366.4 280.7 151.3 78.3 41.7 14.1 5.9 3.1 1.9 1.20.50 368.7 267.2 137.2 68.3 35.4 11.6 4.8 2.5 1.6 1.10.75 365.2 243.1 111.7 50.9 25.0 7.7 3.2 1.8 1.3 1.01.00 343.5 184.0 65.2 25.0 11.0 3.3 1.6 1.1 1.0 1.0

CUSUM 0.50 4.77490.00 369.8 124.3 35.3 16.2 9.9 5.5 3.9 3.0 2.5 2.00.25 369.5 120.0 33.9 15.6 9.6 5.4 3.8 2.9 2.4 1.90.50 370.1 108.1 29.6 13.9 8.7 4.9 3.5 2.7 2.3 1.80.75 370.2 86.0 22.8 11.0 7.1 4.2 3.0 2.4 2.0 1.61.00 367.4 50.6 13.5 7.1 4.8 3.0 2.3 1.9 1.6 1.1

EWMA 0.12 2.75

0.00 370.6 96.1 29.7 14.9 9.6 5.6 4.0 3.2 2.6 2.10.25 371.4 93.0 28.5 14.4 9.3 5.5 3.9 3.1 2.6 2.00.50 371.0 83.6 25.3 13.0 8.5 5.0 3.6 2.9 2.4 2.00.75 372.9 67.2 20.2 10.6 7.1 4.3 3.2 2.5 2.2 1.81.00 369.4 40.9 12.7 7.1 5.0 3.2 2.4 2.0 1.8 1.2

• When the process starts to deviates frstate with small shifts, 0 0.5, appears to dominate all other charts further springs out of control with chart appears to be more effective than

• Although imperfect ranking using conreduces efficiency, the control charts athe classical charts using SRS most0.5.

• The classical Shewhart chart detecting large shift in the procesCUSUM and EWMA counterparts (cf

TABLE VI. ARL VALUES FOR THE THREE CONTROWHEN 6.

Control Charts

0.00 0.25 0.50 0.75 1.00 1.5

Shewhart 3 sigma

0.00 368.6 278.9 156.7 81.0 43.9 150.25 372.2 279.9 150.3 78.2 41.8 140.50 368.0 269.4 135.9 67.2 34.9 110.75 370.8 244.3 107.6 48.4 23.7 71.00 349.6 173.7 56.0 20.7 8.9 2

CUSUM 0.50 4.7749

0.00 369.8 123.8 35.3 16.2 9.9 50.25 370.2 120.0 33.8 15.5 9.6 50.50 370.8 106.9 29.3 13.7 8.6 40.75 371.2 83.1 21.9 10.6 6.9 41.00 368.9 44.2 12.0 6.4 4.4 2

EWMA 0.12 2.75

0.00 372.0 96.0 29.6 14.9 9.6 50.25 373.2 92.9 28.4 14.3 9.3 50.50 372.6 82.9 25.1 12.9 8.4 50.75 372.3 64.9 19.4 10.3 6.9 41.00 369.9 36.1 11.4 6.5 4.6 2

IV. EXAMPLE OF APPLICAT

The fill volume of a soft drink bottle is ancharacteristic. Thus, we used a real data secollected from production line of Pepsicompany Al-Khobar, Saudi Arabia, to illustof the three statistical control charts. The vdrink bottle based on perfect ranked set samwas obtained by measuring the unfilled parbottle when the production line was in a stdata covers fifty-four samples each of subAfter calculating the mean, standard devcharacteristics, the three control charts were the behavior of the process (cf. Figures 1 – 3)

A. Comments on the Control Charts Base on real data used in this example

cannot be generalized, the RSS based Shewappears to have fewer fluctuations and ther

from the in-control the EWMA chart

but as the process 0.5, the CUSUM n EWMA.

ncomitant variable, are still better than t especially when

0, is better at ss mean than its f. Tables II – VI).

OL CHARTS USING RSS

50 2.00 2.50 3.00 4.005.0 6.3 3.2 2.0 1.24.0 5.9 3.0 1.9 1.21.4 4.7 2.5 1.6 1.17.2 3.1 1.7 1.3 1.02.7 1.4 1.1 1.0 1.0

5.5 3.9 3.0 2.5 2.05.4 3.8 2.9 2.4 1.94.9 3.5 2.7 2.3 1.84.1 2.9 2.3 2.0 1.62.8 2.1 1.8 1.5 1.0

5.6 4.0 3.2 2.6 2.15.5 3.9 3.1 2.6 2.05.0 3.6 2.9 2.4 2.04.2 3.1 2.5 2.1 1.82.9 2.2 1.9 1.7 1.1

TION n important quality et obtained by [8], i-Cola production trate the simplicity volume of the soft mpling techniques

rt of the soft drink table process. The bgroup size three. viation and chart constructed to see

).

, which of course whart control chart refore more stable

than SRS chart counterpart (CUSUM and EWMA control cBoth appears to be havingcorresponding SRS charts (cf. better job in the estimation of p

Fig. 1. Shewhart control chart using S

Fig. 2. CUSUM control chart using S

Fig. 3. EWMA control chart using SR

cf. Figure 1). The RSS based charts are not doing badly either. g less fluctuation than their Figures 2 and 3) there by doing opulation mean.

SRS and RSS when 3

RS and RSS when 3

RS and RSS when 3

V. CONCLUSION The ranked set sampling procedure has proven to be very effective where measurements of quality characteristics are difficult or expensive but could readily be ranked with respect to the variable of interest by visual inspections or through concomitant variable. This study evaluated the average run lengths performance of the three commonly used control charts to monitor the process mean using ranked set sampling. We found that RSS has substantially reduced the average run lengths of the three control charts and this will play significant

role in reducing production of nonconforming units. Using RSS, the EWMA chart is more sensitive to detecting very small shifts in process mean closely followed by CUSUM. The Shewhart chart using RSS can detect early large shifts.

ACKNOWLEDGMENT The first author gratefully acknowledges the help of Dr.

Mohammad Riaz of King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia.

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