Statistical quality control based on ranked set sampling ... - ThaiScience

Chiang Mai J. Sci. 2013; 40(3)

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Chiang Mai J. Sci. 2013; 40(3) : 485-498 http://it.science.cmu.ac.th/ejournal/ Contributed Paper

Statistical Quality Control Based on Ranked Set Sampling for Multiple Characteristics Adisak Pongpullponsak* and Peerawut Sontisamran Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand. *Author for correspondence; e-mail: [email protected] Received: 11 September 2012 Accepted: 18 July 2013

ABSTRACT Many quality control charts for mean have been developed from ranked set sampling (RSS), one of them, so-called ranked set sampling for multiple characteristics (RSSMC). This study is to compare a new chart based on RSSMC data with the control chart established from simple random sampling (SRS). The RSSMC control chart has better average run length (ARL) than the classical chart when a sustained shift in the process mean for error ranking characteristic by using other characteristics is added to help ranking data. To compare the RSSMC method with median ranked set sampling (MRSS), RSS, SRS, in term of out–of–control ARL performance, two characteristics of simulated data is used. It is found that the first data has error ranking, while the second data shows no error ranking when it is compared with the measuring data that is simulated. These data are subsequently used to build the corresponding control chart. All over the study, we use the simulated normal distribution data at the given mean and variance. Keywords: rank set sampling (RSS), quality control, multiple characteristics, average run length (ARL) 1. INTRODUCTION

Ranked set sampling (RSS) was introduced by McIntyre [1], who used the technique in estimation of the land for raising domesticated species without the requirement of transportation and harvesting. Later, Takahasi and Wakimoto [2] developed the theory and method of RSS for various problems in agricultural and forestry fields. In 1972, Dell and Clutter [3] studied the theory under the hypothesis of perfect and imperfect RSS.

Patil et al. [4] investigated the theory with the sample units of unlimited population size using different sampling approaches based on RSS. Subsequently, Bohn [5] modified RSS for nonparametric statistics before further developed for collecting of environmental samples which contained high variation within populations. For example, in sampling for forest condition survey, a wide range of tree sizes would be experienced extensively affecting the

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measurement of parameters such as height and width of the samples. Utilization of simple random sampling (SRS) or systematic sampling would consume high cost in recording the size or height of each sample. For this reason, RSS seems to be the best choice for solving differences of variables between populations. Malfunction of machines is one major problem faced in many industries, where inspection on the machines has high cost and it is difficult to perform as this may retard the production process. Accordingly, several statistical methods have been employed to enhance the production efficiency and reduce defective product numbers, thereby decreasing the cost. In quality control using quality control chart, SRS has been noted to be not suitable since it yields population with highly skewed distribution causing wide control margins, subsequently low efficiency of control. For this reason, RSS has been further developed to solve such a problem. This method shows higher sampling efficiency compared with the actual investigation with the same sample size as it gives narrower control margin which improves the process control, consumes low cost and short period in process. Halls and Dell [6] found that the efficiency of RSS is higher than SRS efficiency when the same sample size was examined. In 1997, Salazar and Sinha [7] proposed RSS and median ranked set sampling (MRSS), the newly modified sampling methods, for investigation of process. Few years later, Muttlak and Al-Sabah [8] developed new quality control charts for population mean using RSS and two of its modifications before compared the average run length (ARL) with that of the conventional control chart based on SRS method. Practically, in quality control by sample are required to be considered.


One example is evaluation of average dried weight of leaves by Ridout and Cobby [9]. Since it was very difficult to measure the precise dry weight of leaves, the spray deposits of water on both sides, the upper and lower surfaces of the leaves, were used in estimation. Consideration of spray deposits on one side may give high ranking error. Ridout and Cobby [9] employed the characteristic of interest in RSS from the products containing multiple characteristics in order to reduce error from ranking and increase efficiency of sampling. It was found that RSS is not appropriate for controlling production of goods with multiple characteristics. The aim of this study is to establish the quality control chart based on RSS for multiple characteristics and compare the efficiency with the previously mentioned quality control charts through ARL values. 2. SAMPLING METHODS

2.1 Ranked Set Sampling (RSS) RSS is the method facilitating sample selection for ranking by considering a variable which is related to a variable of interest or a variable that desires to measure. The step of RSS can be described as below. Step 1. Randomly select n sample units per set from the total n sets. Step 2. Allocate the selected samples into the sets by considering a variable related to a variable of interest. Step 3. Choose a sample unit for actual analysis starting from the smallest rank in the first set, then the smallest rank in the second set, continuing the procedure until the last set the largest rank is chosen. Step 4. Repeat steps 1 through 3 for r cycles until the desired sample size is obtained. As seen in Figure 1. Samples are randomly selected for 3 sets, where each


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set contains 3 sample units, then repeat this procedure for 4 times.

Figure 1. Sample units for RSS. Illustrating in Figure 1, each row contains ranked sample units, and only the sample units marked as @ is chosen for actual measurement. Therefore, from the total 36 sample units which are randomly selected from 4 cycles, only 12 units will be used for measuring a variable of interest. Furthermore, for some situations the selected samples could not be ranked due to the problem of making decision, another characteristic that is related to the characteristic of interest or consuming lower costs in process may be used in sample ranking. Given X(i:n)j is the ith order statistic from the set of size n in the jth cycle. Takahasi and Wakimoto [2] proposed an estimator of population mean for RSS as below; (1) which is an unbiased estimator of μ . The variance of rss,j is defined by var

(2)

2.2 Median Ranked Set Sampling (MRSS) MRSS was presented by Muttlak [10].

Using this method, the sample at the median of the sets is selected, if the set size n is odd. In case of the set size with even number, sample selection is from the (n/ 2)th order in the first half and the ((n+2)/ 2)th order in the second half of the set. The method of MRSS can be concluded as follows. Step 1. Select n sample units per set from the total n sample sets. Step 2. Allocate sample units into the set by using a variable related to a variable of interest in ranking. Steps 3. Choose the sample units for actual measurement by selecting the smallest rank in the ((n+2)/2)th order from the sample sizes with an odd number. For the sample sets with an even number, the smallest rank in the (n/2)th order of the first half and the smallest rank in the ((n+2)/2)th order in the second half are chosen. Step 4. Repeat steps 1 through 3 for r cycles until the desired sample size is obtained. As seen in Figure 2. Samples are

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Figure 2. Sample units for MRSS - case I. In Figure 2, each row contains ranked sample units, and only the sample units marked as @ is chosen for actual measurement. Therefore, from the total 36 sample units which are randomly selected for 4 cycles, only 12 units will be used for

measuring a variable of interest. As seen in Figure 3, samples are randomly selected for 4 sets, where each set contains 4 sample units, then repeat this procedure for 4 times.

Figure 3. Sample units for MRSS - case II. In the illustration above, each row contains ranked sample units, and only the sample units marked as @ is chosen for actual measurement. Therefore, from the total 36 sample units which are randomly selected for 4 cycles, only 12 units will be

used for measuring a variable of interest. For the sample sizes with an odd number, given X (i:n)j is the ( n/2) order statistic of the i th order from sample size n in the jth cycle.


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In case of an even number, X(i:n)j is the (n/2)th order statistic of the ith order from sample size n(i = 1,2,...,L where L = n/2), and the ((n+2)/2)th order statistic of the th order from sample size, where Muttlak [10] proposed an estimator of population mean for MRSS as below; (3) The variance of

mrss, j

is defined by (4)

2.3 Ranked Set Sampling for Multiple Characteristics (RSSMC) Generally, in RSS only one characteristic is used in ranking samples, where ranking errors may easily occur. Thus it is recommended to consider multiple characteristics in ranking to reduce errors. Ridout and Cobby [9] described the procedure for selection of characteristics used in ranking below: Step 1. Select characteristics of interest, then designate as i = 1, i = 2 and so on. Step 2. Let nij denote the number of samples ranked in the jth order for the ith characteristic, and Vi is the sample variance of the ith characteristic in the sample number nij . For a balanced

sampling, variance of the ith characteristic equals zero (Vi = 0). When no ranking error occurs, a new set of size n is established by calculating the Vi values from each characteristic in the sample set to estimate the values of V1 and V2 . Step 3. Select the sample that minimizes V =V1 +V2 from each sample set; in case of absence of ranking errors, select sample in the same way as MRSS; if there is more than one such sample, select one of them at random. Step 4. Repeat steps 1 through 3 for r cycles until the desired sample size is obtained. From the steps described above, the procedure of RSSMC can be performed as follows, giving the weight as the variable of interest or the variable to be actual measurement, and the height and width as variables that may relate to the weight. For this reason, we use the height and width as variables for ranking as shown below. Considering RSS for multiple characteristics, a selected sample is the one with the smallest variance or the one that is the median of the samples ranked by multiple characteristics. Muttlak and Al-Sabah [8] developed a control chart using the samples that are the median of MRSS data.

Table 1. A sample set from using the RSSMC method when the sample size n = 5a. Characteristic

Sample Sample Sample Sample Sample #1

#2

#3

#4

Sample Sample Sample Sample Sample

#5

#1

#2

#3

#4

#5

Height (A1)

63.17

72.61

65.99

69.28

79.62

V1

48.54

6.13

17.14

0.73

90.03

Width (A2)

19.96

16.99

12.12

14.82

16.99

V2

14.33

0.66

16.46

1.83

0.66

Weight

56.22

50.93

44.17 49.71

51.43

V

62.87

6.79

33.60

2.57

90.70

b

c

Note: a In each set, the data are obtained from the same method until we have n samples from n sets repeated for r rounds; b is the sample selected by calculating from Vi = (aij − i )2; c is a minimum value of V1 +V2 when A is the variable of characteristic used in ranking.

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Given X(i:mc)j is the order statistic that has the smallest sum variance value for sample size n in the jth cycle, where the RSS estimator for multiple characteristics could be calculated by (5) The variance of

rssmc, j

is defined by (6)

3. QUALITY CONTROL CHARTS

Statistical process control (SPC) is an effective method used to improve quality of an industry and production. The objective of SPC is to rapidly determine for any changes or problems in the process. SPC composes of two major tools involving in control of the production process, and sampling for acceptable and control chart. The control chart is a process control technique that has been widely employed. It requires only few samples in investigation of the process, making SPC becomes a high efficient approach in inspection of changes in the process that may affect to the product quality. Statistical quality control was established in 1924 after Shewhart [11] introduced a control chart for fractional nonconforming units. Subsequently, in 1950 Aroian and Levene [12] proposed the first example of establishing 3 parameters for consideration, which are sample size, control limit, and time between sampling. After that, Weiler [13] employed sample sizes in construction of a model used in estimation of the minimum mean before changes in process occur. By using his established model, time between sampling and probability of inspecting the impact when the process is out of control, so-called the average run length (ARL), could be estimated. In 1997,

Salazar and Sinha [7] developed an average control chart based on RSS when the population has normal distribution and the process has various shift values. They used the Monte Carlo model in calculating ARL when changes in process occur for average control charts based on both RSS and MRSS in cases of perfect and imperfect ranking. Later, Muttlak and Al-Sabah [8] modified RSS into more efficient methods; extreme ranked set sampling (ERSS), paired ranked set sampling (PRSS), and selected ranked set sampling (SRSS), when the population has normal distribution and the process has various shift values. They calculated various ARL values using computer simulations and concluded that every average control chart based on their established methods is more useful than that of Shewhart. The control chart used in this study is as follows. 3.1 Quality Control Chart Using SRS Let Xij when i = 1, 2,...,n and j = 1, 2,...,r, is the ith unit in the jth order of sample size n, and Xij ~ N (μ, σ 2). When the population mean and variance are μ and σ 2, respectively. The Shewhart control chart for (7) is defined by

(8)

when UCL, CL, and LCL are upper control limit, centerline, and lower control limit, respectively. After obtained the chart, the population mean j , j = 1, 2,...,r can be plotted into the upper control chart [14]. However, in actual measurement,


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the mean μ and variance σ 2 are unknown so both μ and σ are estimated from the collected data, where the unbiased estimator for μ is

(16) The estimate for suggested by Muttlak and Al-Sabah [8] will be as

(9) but

(17) when

(10) where

(18) and (19)

(11) is biased estimate for σ . We can use

/c4 as unbiased estimate for calculated from

where c4 is (12)

is the estimate for population mean of the

i th order statistic. Subsequently, the control chart can be constructed by using rss

and

as the equation below,

and the control chart for sample mean can be expressed as

(20)

(13)

3.3 Quality Control Chart Using MRSS

3.2 Quality Control Chart Using RSS

The MRSS mean mrss, j of the jth cycle can be plotted in the control chart based on MRSS proposed by Muttlak [10] as follows

The RSS mean rss,j at the jth cycle can be plotted in the control chart based on RSS proposed by Salazar and Sinha [7] as follows, (14)

(21)

where (22)

when (15) where the unbiased estimate for RSS [2] is

In practical, the values of μ and σ mrss are unknown so an unbiased estimator μ is calculated from MRSS data with normal distribution as shown below,

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(23) suggested by And the estimate for σ mrss Muttlak and Al-Sabah [8] is given by (24) where X(i:m)j is the estimator for population mean of the ith order statistic. Thus rss and σ^ mrss can be used to construct the control chart from MRSS as follows

where X(i:mc) j is the estimate for population mean of the ith order statistic. The control chart can be constructed using the and σ^ rssmc as follows

rssmc

(28)

(25)

4. RESULTS

4.1 Quality Control Chart Using RSSMC The control chart based on RSSMC and the RSSMC estimator was first proposed by Pongpullponsak and Sontisamran [15]. However, performance of the proposed estimator was unsatisfactory. For this reason, the aim of this study is to improve the estimator as could be concluded following. The RSSMC mean rssmc, j of the jth cycle which can be plotted on the control chart based on RSSMC is calculated by (26) where (27) In actual measurement, the μ and σ are unknown so the estimator for rssmc μ based on RSSMC data, when the distribution is normal, is given by and the estimator for

4.2 Control Limits Using SRS, RSS, MRSS and RSSMC Methods The study of limits of control chart by using SRS, RSS, MRSS and RSSMC shows how the constructing RSSMC can reduce the variation due to sampling method. In this study, we use the simulation data which has the normal distribution with mean μ and variance σ 2 by the statistical package R. After simulating data, we select the samples using SRS, RSS, MRSS and RSSMC and use them to construct the quality control chart. From using different sampling methods with the sample size n = 3, 4, the results show that (1) When the sample size is increased, the control limits from using SRS, RSS,MRSS, and RSSMCare narrowed which satisfy the principal of constructing control chart for mean. (2) From comparisons of all limits of control chart obtained from SRS, RSS,MRSS and RSSMC, both control charts based on SRS and MRSS have the maximum control chart width whereas the minimum width is seen only in the RSSMC control chart. Thus we can conclude that the centerline which is the other tool should be included in construction of a control chart. It is


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found that if the width of the control chart is large, the variation will be large too. We also observe that the control chart for RSSMC constructing in this

study has the minimum width which results in the minimum variation. The details are described as following in Table 2 and Figures 4-7.

Table2. Control limits of SRS, RSS, MRSS and RSSMC for n = 3, 4.

a

b

Figure 4. Quality control chart using SRS when sample size n=3 (a) and n=4 (b). a

b

Figure 5. Quality control chart using RSS when sample size n=3 (a) and n=4 (b). a

b

Figure 6. Quality control chart using MRSS when sample size n=3 (a) and n=4 (b).

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a

b

Figure 7. Quality control chart using RSSMC when sample size n=3 (a) and n=4 (b). 4.3 Comparisons of Efficiency of the Control Charts Constructed Using SRS, RSS, MRSS and RSSMC In constructing control chart, the range of average run length (ARL) is a criterion needed to be critically concerned. For this reason, to evaluate performance of the constructed control charts the ARL, where the process is under control with mean μ0 and standard deviation σ0, and the process may occasionally begin to be out of control e.g. shifting of the mean from μ 0 to μ 0 + − δσ0 /√ n = μ , is used in consideration. In the study, we assume that the process is following a normal distribution with mean μ 0 and variance μ 02 if the process is under control, and the shift on the process mean − is δ = (√ n / σ0)|μ – μ0|. If δ = 0 the process is under control and in this case when the point is outside the control limits, then it is a false alarm. In sample collection, sample sizes of each cycle for SRS, RSS, MRSS and RSSMC are n = 3, 4, 5, 6, 7, as described in sections 2.1 – 2.3. After constructing the control charts according to sections 3.1 - 3.3, and 4.1, the ARL for control limits of each chart is estimated by averaging the cycle times until the first out-of-control group is obtained. Normally, imperfect ranking will

occur if the variable of interest containing errors in ranking the units is ranked. To evaluate the constructed control chart performance, the data used in the study consist of 3 related variables with normal distribution. For each value of ARL we simulate 10,000 replications and the computer simulations are run for n=3, 4, 5, 6, 7 and δ = 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0. From the results shown in Table 3 and Figure 8, conclusion can be as follows. (1) In case that the process is under control, i.e. δ = 0, using MPRSS the number of false alarms will not be increased when compared with SRS, RSS and MRSS. In fact, there is small decrease in the ARL; for example, for n=3, ARL= 341.993 as compared to 346.090, 345.470 and 347.886 for SRS, RSS and MRSS respectively. (2) If the sample size increases, the ARL will decrease if δ > 0; for example if the sample size is 4 and δ = 1.0 the ARL is 15.944 as compared with 22.995 in the case of n=3. (3) The ARL value for the RSS will decrease much faster than the case of SRS if δ increases. This increase in ARL value will depend on the correlation between the variable of interest and the


concomitant variable that we use to estimate the rank of the variable of interest. (4) For the data with error ranking, the

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ARL for RSSMC is less than that of SRS because some variables used in ranking are related to a variable of interest without actual measurement.

Table 3. ARL value of each sampling method for n = 3, 4, 5, 6, 7 when δ is 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0.

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ARL for n = 3

ARL for n = 4

Table 3. (Continue).

Shift on the process mean

ARL for n = 5

ARL for n = 6




ARL for n = 7

Figure 8. Comparisons of ARL value of control chart using SRS, RSS, MRSS and RSSMC when n = 3, 4, 5, 6, 7.



5. CONCLUSIONS AND DISCUSSION

5.1 Conclusions Generally, for two groups of statistical process control (SPC), univariate statistical process control (USPC) and multivariate statistical process control (MSPC), RSS is proved to be more efficient when units are difficult and costly to measure, but will be easy and cheap to rank with respect to a variable of interest without actual measurement. In this study, we use the RSSMC in developing control charts for sample mean. When compared these control charts with the control charts mentioned above, we find that they have more efficiency and satisfy Muttlak and Al-Sabah’s [8] statistical quality control based on RSS. The following are some specific conclusions. (1) Quality control chart using RSSMC dominates the classical charts. If the process starts to get out of control, this chart reduces the number of ARLs substantially. (2) To overcome the problem of errors in ranking and/or to increase the efficiency of estimating the population mean, we suggest that using RSSMC instead of RSS and MRSS can reduce the errors in ranking. The RSSMC dominates all other methods in terms of reducing the ARL if the process starts to get out of control. Finally, we recommend using the RSSMC for construction of both USPC and MSPC, in case that the data have error ranking or unknown some characteristics related to a variable of interest without actual measurement, as it can reduce the ARL compared to the previously mentioned control charts. 5.2 Discussion Comparisons of the control chart for

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population mean developed in this study with the RSS, SRS, MRSS and ERSS control charts reveal that our constructed control chart is more efficient and satisfy Muttlak and Al-Sabah’s [8] statistical quality control based on RSS. In their report, they documented that MRSS and ERSS have more efficiency as the ARL values are less than those of the other methods when the shift on the process mean occurs because some data have error ranking. 6. Suggestion The results in the present study show that the ARL obtained from using RSSMC method is less than the ARL of other methods. However, when the variable used for ranking is highly related to a variable of interest without actual measurement, the ARL obtained from MRSS and RSS are not different from that of RSSMC. For the future research, we will optimize the cost and sample unit for RSS with multiple characteristics. REFERENCES [1] McIntyre G.A., A method for unbiased selective sampling, using, ranked sets, Aust. J. Agric. Res., 1952; 3: 385-390. [2] Takahasi K. and Wakimoto K., On the unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Stat. Math., 1968; 20: 1-31. [3] Dell T.R. and Clutter J.L., Ranked set sampling theory with order statistics background, Biom., 1972; 28: 545-553. [4] Patil G.P., Sinha A.K. and Taillie C., Ranked set sampling, A Handb. Stat., 1993; 1: 51-65. [5] Bohn L.L., A review of non-parametric ranked set sampling methodology,

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Comm. Stat. - Theory and Methods, 1996; 25: 2675-2685. [6] Halls L.K. and Dell T.R., Trial of ranked set sampling for forage yields, For. Sci., 1966; 12: 22-26. [7] Salazar R.D. and Sinha A.K., Control chart x-bar based on ranked set sampling, Comun. Tecica, 1997, No. 1-97-09 (PE/CIMAT). [8] Muttlak H.A. and Al-Sabah W.S., Statistical quality control based on ranked set sampling, J. Appl. Stat., 2003; 30: 1055-1078. [9] Ridout M.S. and Cobby J.M., Ranked set sampling with non-random selection of sets and errors in ranking, Appl. Stat., 1987; 36: 145-152. [10] Muttlak H.A., Median ranked set sampling, J. Appl. Stat. Sci., 1997; 6: 245-255.

[11] Shewhart W.A., Some applications of statistical methods to the analysis of physical and engineering data, Bell Technol. J., 1924; 3: 43-87. [12] Aroian L.A. and Levene H., The effectiveness of quality control charts. J. Am. Stat. Assoc., 1950; 45: 520-529. [13] Weiler H., On the most economical sample size for controlling the mean of a population, Ann. Math. Stat., 1952; 23: 247-254. [14] Montgomery D.C., Introduction to Statistical Quality Control, 5 thedn, New York, John Wiley & Sons, 2009. [15] Pongpullponsak, A. and Sontisamran P., Statistical quality control based on ranked set sampling for multiple characteristics, Proc. Int. Conf. Sustainable Greater Mekong Sub region, 2010; 26-27, August, Bangkok, Thailand.