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32
A NEW CONTROL CHART FOR PROCESS DISPERSION
BASED ON RANKED SET SAMPLING
Chandrakant Gardi and Vikas Ghute*
Department of Statistics Punyashlok Ahilyadevi Holkar Solapur University, Solapur (MS), India chandrakant.gardi@gmail.com, vbghute_stats@rediffmail.com *Corresponding author
Abstract
In this paper, we propose a new control chart based on Downton's estimator (D) for monitoring the process dispersion using ranked set sampling design and runs rules. The performance of the proposed control chart is compared with the originally proposed D chart based on simple random sampling method when underlying process distribution is normal and non-normal. The average run length is used to evaluate the performance of the proposed control chart. It is observed that the proposed control chart is efficient in detecting shifts in process dispersion as compared with the chart using simple random sampling method. The performance of the proposed chart is further investigated using runs rules. The efficiency of the designed runs rules RSS-D chart is compared with its existing counterparts and is found to be superior.
Keywords: Control chart, Average run length, Process dispersion, Ranked set sampling, Runs rules, Downton estimator.
1. Introduction
In statistical process control (SPC), control charts are the most popular tools used to monitor the quality characteristics in an industrial process. Shewhart R chart and S chart are the commonly used control charts for monitoring the process dispersion. The R chart is based on the sample range (R) whereas S chart is based on sample standard deviation (S). These charts are based on subgroups of sample size n using simple random sampling (SRS) method. Both R and S charts are easy to implement and are effective in the detection of large shifts, but are less sensitive in the detection of small and moderate shifts in the process dispersion (Montgomery[1]).Several alternatives have been proposed in the literature to improve the performance of Shewhart type control charts. One recent approach is the construction of control charts based on ranked set sampling (RSS) scheme. The RSS scheme was first suggested by McIntyre [2]. This scheme ensures increased precision through ranking of observations. He pointed out that this sampling scheme works superior to standard simple random sampling (SRS) for estimation of the population mean. Takahasi and Wakimoto [3] laid the necessary mathematical formulation for this sampling scheme. Several researchers have studied applications of ranked set sampling in different streams. Kvam and Samaniego [4] showed that ranked set sampling may occur naturally in survival analysis. Kvam and Samaniego [5] studied the applications of this sampling scheme in reliability. Recently, RSS scheme has got considerable attention in the construction of control charts. Salazar and Sinha [6] first suggested control charts for monitoring process mean using RSS scheme. It was shown that the RSS based control charts perform better than the charts based on SRS. Muttlak and Al-Sabah [7]
developed control charts for process mean based on different RSS schemes and showed that the proposed RSS based control charts perform better than the classical SRS charts. Al-Naseer and Al-Rawwash [8] developed the mean control chart using robust RSS method. Al-Omari and Al-Naseer [9] suggested a new quality control chart for mean using robust extreme ranked set sampling (RERSS) method. It was shown that RERSS chart performs better than all other charts based on SRS and RSS methods. Al-Omari and Haq [10] developed Shewhart-type control charts to improve the monitoring the process mean by using the double quartile-ranked set sampling, quartile double ranked set sampling and double quartile extreme ranked set sampling methods. Al-Rawwash et al. [11] developed control charts of the sample mean based on several sampling techniques including RSS as well as some of its recent modifications assuming that the underlying distribution is normal
2
with mean j and variance & . Mehmood et al. [12] discussed the details of different ranked set strategies and their applications in an industrial process using control charts. Abbasi [13] investigated the performance of the mean chart based on various sampling schemes under normal and non-normal process distributions. Tahir et al. [14] designed and investigated dispersion control charts based on different estimates of process dispersion under different RSS strategies for normal and non-normal processes.
Most of the papers published so far on the RSS based control charts have concentrated on monitoring the process location and few control charts has been developed for monitoring the process dispersion. Motivated by the attractive features of RSS scheme, in this paper, we propose a D chart based on RSS scheme (denoted as RSS-D chart) for monitoring process dispersion. It is shown that the proposed RSS-D chart performs uniformly better than the D chart without RSS scheme for detecting increasing shifts in the process dispersion. Although the RSS scheme improves the performance of the D chart in detecting shifts in process dispersion, further improvement can be achieved if runs rules scheme is implemented to the RSS-D chart.In case of SRS, a number of authors have defined runs rules schemes for different control charts to improve the detecting ability for small and moderate shifts in process parameters. Klein [15] proposed two control charts based on runs rule as efficient alternatives to Shewhart control chart. These control charts were named as 2-out-of-2 runs rule control chart and 2-out-of-3 runs rule control chart. In 2-out-of-2 runs rule control chart, if two successive points fall beyond a special control limit, then process is declared to be out of control. In 2-out-of-3 runs rule control chart, if two points among successive three points fall beyond a special control limit, then process is declared to be out of control. This special limit is determined in such a way that the in-control ARL equals with specified value ARL0. This chart has been proven to perform better than Shewhart chart.Antzoulakos and Rakitzis [16] have studied the control charts with supplementary runs rules for monitoring the process standard deviation. In general, the use of runs rules increases a chart's sensitivity but also produces more false alarms. To keep the same rate of false alarm, control limits have to be adjusted (Cheng and Chen [17]). Although runs rules have been widely used in practice to increase the sensitivity of control charts, to our best knowledge, little research has been done on the RSS based control charts.
The purpose of this paper is to improve the performance of the D chart proposed by Abbasi and Miller [18] to detect shifts in process dispersion by using RSS scheme and further to implement runs rule scheme to proposed RSS-D chart. The rest of the paper is organized as follows: In section2, we outline the process of constructing the D control chart based on SRS (denoted as SRS-D chart) for monitoring the process dispersion. The description of ranked set sampling scheme used in this study is given in section 3. The charting structure of the proposed RSS-D chart is presented in section 4. RSS based D-chart with runs rules is discussed in section 5.The performance evaluation of proposed RSS-D chart and RSS-D chart with runs rules and its comparative study with SRS-D control chart is given in section 6. In section 7, an illustrative example based on simulated data sets is provided for a better understanding about implementation of the proposed control chart. A real life application of the proposed control chart is illustrated in section 8. Conclusions are given in the final section 9.
2. D Chart for Process Dispersion Based on SRS
LetX1,X2, ...,Xn be a simple random sample of size n from a normal process whose mean is fi and standard deviation is a. Further, letX(1),X(2), .~,X(n) be the order statistics corresponding to this sample. Downton [19] proposed following statistic as an estimator for process standard deviation a .
which is an unbiased estimator of a for normally distributed process data. Abbasi and Miller [18] proposed the control chart based on D to monitor the process variability. It was shown that for normally distributed process, the D chart is equally efficient to the Shewhart S chart for detecting shifts in the process standard deviation. Since the distribution of D is not symmetric at least for small to moderate values of n, they have used probability limits instead of three-sigma control limits in the construction of D chart. The probability limits of D chart are computed by using quantile points of the distribution of = The upper control limit (UCL) of the D chart is given as
UCL = Z1-aD with P(Z> Z1-a) = 1-a (2)
where a is the specified probability of making Type-I error, Za is ath quantile point of the distribution of Z. The process dispersion is then monitored by plotting values of the D statistic on the chart with UCL given by Eq. (2). If D < UCL, the process is regarded as in-control, otherwise there is an increase in process standard deviation and process is deemed out-of-control.
To design the D chart based on simple random sample (SRS), let X1,X1,...,Xnbe independent and identically distributed^random sample of sizen, where n0and are the in-control mean and variance, respectively. As the concern of this paper is monitoring process dispersion, let us assume that process standard deviation has a shift of size5. Hence, the new standard deviation for the process becomes a1 = Sa0, where c1is out-of-control value of process standard deviation. For positive shift (5 > 1), that is, for increase in a, k+a0is the upper control limit and process is declared to be out-of-control if D > k+a0. For negative shift (5 < 1), that is for decrease ina, k-a0 is the lower control limit and process is declared to be out-of-control if D < k-a0. The values of k+ and k- are found for fixed in-control ARL value based on simulations.
A commonly used performance measure for a control chart is its ARL. For a given shift in process parameter, the ARL is the average number of points plotted on a control chart until an out-of-control signal is obtained. In case of Shewhart control chart, which signals as soon as a point falls beyond the control limits, ARL = 1/-p, wherep is the probability that a point exceeds the control limits. For positive shift S,
p = Pr(D > k+a0/a = Sa0)
= Pr(z>*i) = 1~F(i)
In-control ARL is the reciprocal of the detecting power p.
ARL(S) = ~ = —U (3)
p 1-F{-)
For negative shift S,
p = Pr(D < k a0/a = Sa0)
= Pr(z<T)
where F(.) is distribution function of Z.
In-control ARL is the reciprocal of the detecting power p.
ARL(S) = -- —
i
i
(4)
3. Ranked Set Sampling Scheme
The mechanism of ranked set sampling (RSS) is as follows. Let X be the characteristic under study. In order to get a RSS sample of sizen, total of n2 units are drawn from the population. These n2 units are randomly arranged in n sets each of sizen. Let^^i = 1,2,... ,n}be the kth set, where= 1,2,.,n . The observations in the kth set [Xik,i = 1,2, ..,n} are ranked either by visual magnitude of characteristic X or by using some other variable, which is significantly correlated withX. If the ranking of Xis exact, then the sampling scheme is called RSS with perfect ranking, and if ranking is not exact, then the sampling scheme is called RSS with imperfect sampling. In the present study, we have assumed perfect RSS scheme. Let kthordered set be [X[iik], i = 1,2,., n}, where X[ik], i = 1,2, ...,n; k = 1,2, ...,n is the ith order statistic in kthset. These sets are called the ranked sets. Then ith, i = 1,2, ...,n order statistic is picked from ith, i = 1,2, ...,n ranked set. That is, first order statistics from first ranked set, second order statistics from second ranked set and so on. Thus, a final selected sample of size n is [X[1t1],X[22], ...,X[nn]}, which is termed as a ranked set sample (RSS).
In this section, a working mechanism of RSS-D chart is described. A subgroup sample from the underlying process is obtained using RSS scheme of sampling. It is assumed that ranking is perfect. Using this scheme, we generate subgroups of size n and repeat them for n cycles from the distribution that is under study. Downton statistic based on RSS sample{X[11],X[2 2], .,X[nn]} of size n is given as
known, simulation approach is used to compute ARL values of the proposed RSS-D chart. The control limits for RSS-D chart are found outusing extensive simulation. 50000 ranked set samples of size n were simulated and quantile points of D statistic based on RSS scheme were realized. Using these quantile points, the values of fc+and k-are evaluated using equations (3) and (4) respectively for positive and negative shifts in process standard deviation a. The estimated ARL values of the proposed RSS-D chart along with standard error of the chart are found with 50000 simulations for each shift of magnitude 5in process standard deviations. The ARL performance of the RSS-D chart under both normal and non-normal process distributions is presented in Section 6.
In this section, we study the performance of the proposed RSS-D chart using the following runs rules. Note that we would usually use only one of these rules at a time. Each rule would have different control limits.
The process is declared to be out of control when
(i) Two consecutive points plot outside the control limits (2-of-2-rule).
(ii)Any two of three successive points plot outside the control limit (2-of-3-rule)
The ARL performance of the RSS-D chart using these runs rules has been evaluated under both normal and non-normal process distributions using a simulation study in Section 6.
4. RSS Based D Chart for Process Dispersion
5. RSS-D Chart with Runs Rule
6. Performance Evaluation and Comparison 6.1.Performance evaluation under normality
This section evaluates and compares the performance of RSS-D chart and RSS-D chart using runs rules and D chart with simple random sampling. To compare the performance of these charts under normality, in-control observations are generated from N(p.0,o2) distribution. Without loss of generality, we take n0 = 0 and o^ = 1. For out-of-control process, observations are generated from N(p.0, of) distribution. Without RSS scheme, the values of the control chart statistic D for the D chart are computed using Eq. (1) for subgroups of size n. With the RSS scheme, the values of the control chart statistic D for the RSS- D chart are computed using Eq. (5) based on RSS sample of size n. For increase in standard deviation (that is, a1 > a0), where a1 = Sa0, the magnitude of shift in process standard deviation considered are 5 = 1.0,1.1,1.2,1.3,1.4,1.5 and 2.0, while for decrease in process standard deviation (that is, a1 < a0 ),the magnitude of shift in process standard deviation considered are 5 = 1.0,0.9,0.8,0.7,0.6,0.5 and 0.1. All charts have calibrated in-control ARL of 200 and subgroup size of n = 5,8 and 10 are considered. The values of k+ and k- are obtained by designing the chart to have an in-control ARL of 200. The out-of-control ARL values of the charts are found with 50000 simulations for each of shift of magnitude 5 in the process standard deviation a.
ARL values for positive shifts and negative shifts in process standard deviation with different sample sizes asn = 5,8,10 are given in Table 1 and Table 2 respectively.
Table 1: ARL comparison for process dispersion with positive shift under normal distribution.
Sample Shift SRS-D RSS-D RSS-D RSS-D
n S (2-out-of-2 ) (2-out-of-3)
1.0 200 200.27(1.816) 200.39 (1.795) 200.25 (1.817)
1.1 66.65 62.43 (0.561) 60.35 (0.540) 55.36 (0.491)
1.2 29.67 25.90 (0.230) 25.76 (0.223) 23.22 (0.197)
1.3 15.56 13.48 (0.117) 14.52 (0.119) 12.78 (0.102)
5 1.4 9.59 8.35 (0.071) 9.50 (0.074) 8.26 (0.061)
1.5 6.49 5.66 (0.048) 6.87 (0.050) 6.11 (0.042)
2.0 2.31 2.05 (0.014) 3.22 (0.016) 2.98 (0.013)
k+ 2.068 1.9985 1.5377 1.60286
1.0 200 199.65(1.822) 200.03 (1.779) 200.85(1.823)
1.1 54.00 45.14 (0.407) 40.46 (0.358) 37.27 (0.326)
1.2 20.62 15.75 (0.138) 14.63 (0.122) 13.14 (0.105)
ß 1.3 10.40 7.47 (0.064) 7.61 (0.057) 6.80 (0.048)
8 1.4 6.05 4.33 (0.035) 4.91 (0.033) 4.44 (0.027)
1.5 4.13 2.92 (0.022) 3.69 (0.021) 3.46 (0.018)
2.0 1.55 1.27 (0.005) 2.24 (0.006) 2.19 (0.004)
k+ 1.78 1.6296 1.33955 1.43986
1.0 200 199.55(1.852) 200.03 (1.809) 199.49 (1.792)
1.1 48.33 36.44 (0.333) 32.07 (0.282) 29.49 (0.252)
1.2 17.45 11.56 (0.101) 10.67 (0.085) 9.58 (0.074)
10 1.3 8.27 5.20 (0.042) 5.39 (0.037) 5.02 (0.032)
10 1.4 4.86 3.02 (0.023) 3.66 (0.021) 3.39 (0.017)
1.5 3.28 2.08 (0.014) 2.88 (0.013) 2.74 (0.010)
2.0 1.34 1.10 (0.003) 2.06 (0.003) 2.05 (0.002)
k+ 1.676 1.508285 1.27474 1.307509
Table 2: ARL comparison for process dispersion with negative shift under normal distribution.
Sample n Shift S SRS-D RSS-D RSS-D (2-out-of-2 ) RSS-D (2-out-of-3)
1.0 201 200.64 (1.851) 200.77 (1.816) 200.98 (1.852)
0.9 135.62 176.49 (1.592) 118.89 (1.055) 116.73 (1.067)
0.8 86.12 152.17 (1.386) 62.21 (0.548) 62.80 (0.564)
5 0.7 51.93 128.76(1.173) 30.37 (0.265) 30.30 (0.265)
0.6 29.34 99.68 (0.917) 14.24 (0.118) 14.27 (0.117)
0.5 15.42 72.75 (0.667) 6.58 (0.047) 6.53 (0.048)
0.1 1.00 1.42 (0.007) 2.00(0.00) 2.00(0.00)
k- 0.240 0.1184 0.483308 0.483291
1.0 200 200.47(1.824) 199.59(1.786) 200.79(1.832)
0.9 102.77 101.45(0.911) 61.40(0.548) 62.08(0.562)
0.8 51.32 47.47(0.425) 20.01(0.169) 19.89(0.167)
8 0.7 23.99 20.34(0.182) 7.24(0.0543) 7.22(0.054)
0.6 10.72 8.29(0.071) 3.37(0.018) 3.40(0.019)
0.5 4.63 3.20(0.024) 2.22(0.006) 2.23(0.006)
0.1 1.00 1.00(0.00) 2.00(0.00) 2.00(0.00)
k- 0.388 0.4425 0.674418 0.67443
1.0 201.00 200.82(1.818) 201.01(1.859) 201.59 (1.823)
0.9 91.63 72.29(0.658) 43.57(0.390) 43.54(0.379)
0.8 39.72 24.76(0.220) 11.52(0.093) 11.81(0.095)
10 0.7 16.37 8.50(0.072) 4.19(0.026) 4.23(0.026)
0.6 6.77 3.06(0.023) 2.39(0.008) 2.40(0.008)
0.5 2.81 1.43(0.007) 2.02(0.001) 2.02(0.002)
0.1 1.00 1.00(0.00) 2.00(0.00) 2.00(0.00)
k- 0.449 0.549 0.7365543 0.736105
Following are the findings from Table 1 and Table 2.
• It is observed that with an increase in sample size n, the out-of-control ARL values of all the charts under investigation decreases. This shows that large sample sizes help control chart statistic to quickly detect the shifts in process dispersion of the production process.
• For a fixed value of sample size n, with an increase or decrease in the shift of process dispersion, the out-of-control ARL values of all the charts under investigation decreases. The same trend is observed in standard errors of the estimated ARL values.
• For increase in process dispersion, the proposed RSS-D chart consistently produces smaller out-of-control ARL values than SRS-D chart for the entire range of shifts in the process dispersion.
• The RSS-D chart with runs rule shows a better performance than SRS-D chart for smaller shifts (5 < 1.2) and smaller sample size (n = 5), while for larger sample size (n = 8 or 10), chart shows better performance for small to moderate shifts.
• The RSS-D chart with runs rule shows better performance than RSS-D chart only for small shifts in process dispersion.
• For decrease in process dispersion, for smaller sample size (n = 5), the out-of-control ARL values of the proposed RSS-D chart are greater than that of SRS-D chart. As sample size increases, the RSS-D chart becomes more efficient than the SRS-D chart.
• RSS-D chart with runs rule shows better performance than RSS-D chart and SRS-D chart for decreasing shifts in process dispersion.
• The performance of RSS-D chart with 2-out-of-2 and 2-out-of-3 runs rule is similar.
6.2. Performance evaluation under non-normality
In this section, we study the performance of the RSS-D and RSS-D chart by implementing runs rules schemes under non-normal distributions and it is compared with the D chart based on SRS scheme. In order to study the performance of these charts under non-normality, we considered heavy-tailed symmetric and skewed distributions. Specifically, we simulated observations in the heavy-tailed symmetric case from double exponential distribution and in the skewed case from gamma distribution. The probability density functions of these non-normal distributions are given as
Double exponential distribution: The double exponential distribution, denoted by Double Exponential (p, A) has the probability density function,
f(x;^,A) = <x < w,A > 0,-m < ^ < rn (6)
2A
where ^ and A are respectively the location and scale parameters. The mean and variance of a double exponential distribution are n and 2A2 respectively.
Gamma distribution: The gamma distribution, denoted by Gamma(a.p) has the probability density function,
f(x;a,p)=^xae-Px,x>0; a,p > 0 (7)
where a and p are respectively the shape and scale parameters. The mean and variance of a gamma distribution are a/p and a/p2 respectively.
To study the performance of the RSS-Dcontrol chart under non-normality, we have considered the process data from double exponential distribution with location parameter ^ = 0 and scale parameter A = 1 and gamma distribution with shape parameter a = 2 and scale parameter p = 1. Simulation approach is used to compute ARL values of these charts. The ARL values of the charts are found with 50000 simulations for each shift of magnitude 8 in the process standard deviation. The standard errors of the estimated ARL values of these charts are given (in parentheses) along with ARL values. The value offc+isdetermined based on 100 iterations of 50,000 simulations from the underlying process with no shift, such that in-control ARL is 200. In this study we consider the case of increase in standard deviation only.
Tables 3 and 4 present the ARL values of the charts under investigation to detect increase in process standard deviation for different magnitudes for in-control ARL of 200 and sample sizes n = 5,8 and 10, when underlying process distributions are gamma and double exponential.
Table 3: ARL comparison for process dispersion with positive shift under gamma distribution.
Sample Shift SRS-D RSS-D RSS-D RSS-D
n S (2-out-of-2 ) (2-out-of-3)
1.0 200.23 (0.897) 200.61 (0.904) 199.50 (1.808) 200.26 (1.807)
1.1 93.56 (0.415) 92.95 (0.41) 77.59 (0.703) 74.34 (0.668)
1.2 50.63 (0.226) 48.38 (0.215) 37.54 (0.327) 34.74 (0.300)
5 1.3 30.75 (0.135) 28.33 (0.126) 20.92 (0.178) 19.58 (0.163)
1.4 20.19 (0.089) 17.99 (0.078) 13.36 (0.11) 12.38 (0.098)
1.5 14.04 (0.060) 12.17 (0.052) 9.32 (0.073) 8.65 (0.063)
2.0 4.46 (0.018) 3.533 (0.013) 3.59 (0.02) 3.41 (0.017)
k+ 2.452 2.4051 1.635 1.73
1.0 200.32 (0.897) 201.30 (0.897) 201.5 (1.814) 200.46 (2.786)
1.1 80.14 (0.357) 74.60 (0.330) 58.25 (0.517) 55.35 (0.788)
1.2 38.73 (0.171) 22.76 (0.099) 23.21 (0.198) 21.53 (0.280)
ß 1.3 21.50 (0.094) 13.52 (0.058) 11.69 (0.095) 10.85 (0.084)
8 1.4 13.31 (0.057) 8.81 (0.037) 7.16 (0.054) 6.76 (0.047)
1.5 9.00 (0.038) 6.27 (0.026) 4.977 (0.033) 4.71 (0.029)
2.0 2.75 (0.010) 1.82 (0.005) 2.364 (0.008) 2.34 (0.006)
k+ 2.0315 1.8919 1.411 1.468
1.0 201.12 (0.892) 199.84 (0.887) 200.5 (1.814) 201.10 (1.809)
1.1 74.03 (0.329) 65.79 (0.293) 47.66 (0.421) 46.55 (0.405 )
1.2 33.71 (0.149) 26.84 (0.119) 17.43 (0.148) 16.57 (0.134)
1f) 1.3 17.92 (0.078) 12.85 (0.055) 8.418 (0.064) 7.98 (0.058)
10 1.4 10.88 (0.046) 7.18 (0.029) 5.129 (0.035) 4.97 (0.031)
1.5 7.11 (0.030) 4.49 (0.018) 3.708 (0.021) 3.65 (0.019)
2.0 2.21 (0.007) 1.39 (0.003) 2.137 (0.005) 2.12 (0.003)
k+ 1.885 1.717 1.33 1.378
Table 4: ARL comparison for process dispersion with positive shift under double exponential distribution.
Sample Shift D RSS-D RSS-D RSS-D
n S (2-out-of-2 ) (2-out-of-3)
1.0 200.61 (0.889) 199.86 (0.891) 200.00 (1.825) 200.00 (1.801)
1.1 94.25 (0.422) 91.15 (0.407) 83.25 (0.745) 79.02 (0.709)
1.2 51.29 (0.228) 48.36 (0.214) 41.50 (0.370) 38.04 (0.333)
R 1.3 31.22 (0.137) 28.46 (0.124) 24.08 (0.206) 21.82 (0.182)
5 1.4 20.65 (0.089) 18.37 (0.079) 15.44 (0.129) 14.10 (0.115)
1.5 14.54 (0.063) 12.64 (0.054) 11.20 (0.090) 9.88 (0.076)
2.0 4.75 (0.019) 3.85 (0.014) 4.25 (0.026) 3.91 (0.021)
k+ 2.521 2.5075 1.745 1.798
1.0 200.95 (0.893) 200.36 (0.901) 200.31 (1.778) 200.45 (1.817)
1.1 83.00 (0.369) 75.99 (0.336) 62.75 (0.557) 60.27 (0.525)
1.2 40.38 (0.178) 34.99 (0.154) 26.41 (0.227) 24.44 (0.208)
ß 1.3 22.65 (0.099) 18.41 (0.080) 13.94 (0.115) 12.95 (0.102)
8 1.4 14.22 (0.061) 10.96 (0.046) 8.69 (0.068) 7.99 (0.058)
1.5 9.65 (0.04) 7.20 (0.030) 6.088 (0.043) 5.67 (0.038)
2.0 2.95 (0.01) 2.11 (0.007) 2.67 (0.011) 2.59 (0.009)
k+ 2.095 1.975 1.457 1.523
1.0 199.84 (0.895) 199.48 (0.894) 200.40 (1.808) 200.33 (1.813)
1.1 75.13 (0.332) 67.76 (0.301) 53.10 (0.476) 50.23 (0.439)
1.2 34.70 (0.153) 28.44 (0.130) 20.59 (0.179) 19.13 (0.161)
1.3 18.79 (0.082) 14.20 (0.060) 10.51 (0.085) 9.65 (0.073)
10 10 1.4 11.49 (0.049) 8.19 (0.034) 6.33 (0.046) 6.00 (0.040)
1.5 7.66 (0.032) 5.26 (0.021) 4.50 (0.029) 4.25 (0.025)
2.0 2.38 (0.008) 1.61 (0.004) 2.28 (0.007) 2.26 (0.005)
k+ 1.938 1.792 1.371 1.424
Following are the important findings based on Table 3 and Table 4.
• When underlying process distribution is gamma, the ARL values of RSS-D chart and RSS-D charts with runs rules are smaller than that of the SRS-Dchart for all sample sizes n and entire range of shifts considered. The RSS-D charts with runs rules show a better performance than RSS-D and SRS-D charts. The RSS-D chart with 2-out-of-3 runs rule is very effective for detecting shifts than 2-out-of-2 chart.
• When underlying process distribution is double exponential, we see that the ARL values of RSS-D chart and RSS-D charts with runs rules are smaller than that of the SRS-D chart for all sample sizes n and entire range of shifts considered. The RSS-D charts with runs rules show a better performance than RSS-D and SRS-D charts. The RSS-D chart with 2-out-of-3 runs rule is very effective for detecting shifts than 2-out-of-2 chart.
• The ARL performance of RSS-D chart and SRS-D chart under non-normal process distributions is higher than that for the normal process distribution with sample sizes n = 5,8 and 10
Chandrakant Gardi, Vikas Ghute RT&A, No 2 (73) RSS BASED DISPERSION CONTROL CHART_Volume 18, June 2023
• In general, under skewed and heavy tailed process distributions, the RSS-D chart is more efficient than the SRS-D chart for monitoring shifts in process dispersion.
7. Illustrative Example
In this section, we provide an example to illustrate the proposed chart. For this purpose, two datasets are generated which will be later referred as dataset 1 and dataset 2.
Dataset 1 contains 40 simple random samples each of size n = 5, out of which first 10 samples are generated from N(0,1)referring to in-control situation and remaining 30 samples are generated from N(0,1.2) referring to a shift in a process standard deviation. The D-statistics of these 40 samples are computed and are presented in Table 5. The graphical display of the control chart applied to data set 1 with UCL=1.676 is given in Figure 1.
Table 5: D-values for simulated data with SRS.
In-control Process
Out of control process with Shift (5) =1.2
No. D-value No. D-value No. D-value No. D-value
1 1.04679 11 1.285956 21 0.884477 31 1.100131
2 1.165552 12 1.513122 22 1.113954 32 1.534435
3 1.468238 13 0.722581 23 0.887508 33 1.196817
4 0.4986 14 1.658094 24 1.431795 34 1.590498
5 0.574144 15 1.40786 25 1.294181 35 1.173697
6 1.231162 16 1.263608 26 1.52249 36 0.677852
7 0.84646 17 1.518884 27 1.379621 37 0.991026
8 0.728805 18 0.896195 28 1.349834 38 1.412045
9 0.767625 19 1.431129 29 1.029293 39 1.136405
10 0.920877 20 1.032453 30 1.981427 40 1.138973
UCL— 1.676 |
fifty
0 10 20 30 40
sample number
Figure 1: SRS-D control chart.
From Figure 1, we can see that D control chart gives first out of control signal at point no. 30, which is the only out of control signal.
Dataset 2 consists of 40 ranked set samples each of sizen = 10 of which first 10 samples are generated from N(0,1) referring to in-control situation and remaining 30 RSS samples are generated from N(0,1.2) referring to a shift in a process standard deviation. The D-statistics based on these 40 ranked samples are computed and are presented in Table 6. The graphical display of the control chart applied to data set 2 with UCL=1.5082 is given in Figure 2.
Table 6: D-values for simulated data with RSS
In-control Process
Out of control process with Shift (5) =1.2
No. D-value No. D-value No. D-value No. D-value
1 0.76489 11 0.955223 21 1.658042 31 1.41809
2 1.11167 12 1.340428 22 1.200549 32 1.12013
3 0.928814 13 1.440937 23 1.057834 33 1.22928
4 1.029337 14 0.856949 24 1.827535 34 1.1807
5 1.046913 15 1.241251 25 1.452098 35 1.476113
6 1.197435 16 1.410242 26 0.963128 36 1.039491
7 1.258457 17 0.76041 27 1.096964 37 0.93336
8 1.291416 18 1.293803 28 0.947557 38 1.461749
9 1.262475 19 1.189676 29 1.917295 39 1.562749
10 0.871265 20 1.502025 30 1.601436 40 0.895917
sample number
Figure 2: RSS-D control chart
From Figure 2, we can see that RSS D chart gives first out of control signal at point no.21 and total out of control signals given are05. These signals are at point numbers 21, 24, 29, 30, 39.
From the above two figures, it is clear that RSS D chart signals earlier than D chart, also giving more number of signals than D chart. The above example clearly indicates that the proposed RSS-D chart performs better than D chart without RSS.
8. A Real Life Application
The use of RSS and other sampling schemes based on ranking of observations in SPC has increased in the recent years. As pointed out by Nawaz and Han [20], many a times, the actual measurement process has the constraints on the time and cost. Therefore, it is advisable to adopt such sampling designs which provide greater efficiency with relatively small sample sizes. Instead of taking physical measurements, sometimes it is easier to rank the sampling units on the basis of size, weight and volume by visual inspection or judgment. For example, in an automatic bottle filling plant, the actual quantification of filled volume of liquid in bottles, is costly and time consuming. Instead, it is easier and inexpensive torank the bottles on the basis of the amount of filled volume of liquid in bottles by visual inspection before actual measurement.
Abujiya and Muttlak [21] constructed control charts for mean using double ranked set
sampling. To demonstrate the performance of proposed chart, they used a real dataset, which was originally taken by Muttlak and Al-Sabah [22]. The dataset consisted of 54 subgroups, each of size three. The subgroup data were collected using the SRS and RSS schemes. The data were taken from a Pepsi Cola production company in Khobar, Saudi Arabia. The quality characteristic understudy was considered to be the unfilled part of the bottle, which was measured with the help of an instrument. By visual inspection of levels of the soft drinks, the bottles in each set were ranked. Then the samples were realized with newly proposed sampling techniques. Haq and Munir [23] also used the same dataset to demonstrate the performance of improved CUSUM charts for monitoring process mean based on RSS.The detection abilities of the CUSUMcharts with RSS and ordered RSS schemes were compared.
Nawaz and Han [20] considered a real dataset taken from Tufekci [24] for monitoring process location by using new RSS-based memory control charts. Since the original data was not collected from thequality control prospective, therefore, the available data was considered as a population and RSS samples were drawn from it to demonstrate the process monitoring.
In this section, we consider dataset on inside diameter measurements (mm) for automobile engine piston rings, given in Montgomery [1]. The dataset consists of total 40 subgroups each of size 5 collected using SRS method. Based on this dataset, we estimated the population mean and population standard deviation (a) as^ = 74.00361 and 8 = 0.010144. To compare the performance of proposed RSS-D control chart with SRS-D chart, 8 RSS subgroups are formed using given 40 subgroups assuming that the ranking is perfect. To have fair comparison, out of available 40 SRS samples, 8 equidistant subgroups are selected. Using the estimated values of parameters, 22 SRS and 22 RSS samples are generated with a shift of size 8 = 1.3 in process dispersion. These values are tabulated in Table VII. The graphical display of the control charts applied to these data sets with UCL=2.068 for SRS and UCL= 1.9985 for RSS is given in Figure 3.
Table 7: D-values for SRS and RSS samples of Inside Diameter Measurements (mm) for Automobile Engine
Piston Rings.
D values for SRS Samples
In-control Process Out of control process with Shift (5) = =1.3
No. D-value No. D-value No. D-value No. D-value
1 1.624979 9 1.373168 17 0.613449 25 2.546284
2 0.90859 10 1.278336 18 1.419153 26 2.014164
3 0.297039 11 0.993755 19 1.312875 27 1.862361
4 0.803753 12 0.417964 20 0.786504 28 1.006743
5 1.310467 13 0.778037 21 2.358553 29 0.645239
6 1.799708 14 1.593989 22 1.268609 30 1.791942
7 1.118265 15 1.286196 23 1.984099
8 1.45025 16 0.627962 24 0.736714
D values for RSS Samples
In-control Process Out of control process with Shift (5) =1.3
No. D-value No. D-value No. D-value No. D-value
1 1.048373 9 0.749542 17 0.50747 25 2.456585
2 1.048373 10 1.469715 18 2.005007 26 1.333404
3 0.873645 11 1.827723 19 1.612167 27 1.005756
4 1.467723 12 1.485399 20 0.765135 28 1.214323
5 1.415304 13 1.57744 21 1.68864 29 0.801106
6 0.995955 14 1.691586 22 1.294557 30 1.63209
7 1.57256 15 2.131071 23 1.493243
8 1.555087 16 1.51146 24 1.607816
SRS-D chart
UCL=2.068 ^ /\
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Sample number
RSS-D chart
A UCL=1.9985 /\
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Sample number
Figure 3: SRS-D and RSS-D control chart for a real dataset.
From Figure 3, we can see that the SRS D chart gives first out of control signal at point no 21 and total out of control signals given are02, which are at point numbers 21,25. Whereas, RSS D chart gives first out of control signal at point no. 15 and total out of control signals given are 03. These signals are at point numbers 15, 18, 25.
9. Conclusions
In this paper, a control chart based on Downton's estimator D is developed using ranked set sampling scheme for monitoring the process dispersion of normally and non-normally distributed processes. The performance of the proposed chart is investigated in terms of ARL and is compared with originally proposed D chart without RSS for monitoring the process dispersion. The proposed chart is found to be performing well for entire range of shifts in process dispersion under both normal and non-normal process distributions. The efficiency of proposed chart is further enhanced using two runs rules schemes. The chart with runs rules is found to be performing well for small and moderate shifts under normal process and performing uniformly better for entire range of shifts in dispersion under heavy tailed and skewed process distributions.
References
[1] Montgomery, D. C. Introduction to Statistical Quality Control. John Wiley & Sons 2009; 6th edn. Inc. New York.
[2] McIntyre, G. A. (1952). A method for unbiased selective sampling using ranked sets.Australian Journal of Agricultural Research, 3: 385-390.
[3] Takahasi, K. and Wakimoto, K. (1968). On the unbiased estimates of the population mean based on thesample stratified by means of ordering. Annals of Institute of Statistical Mathematics, 20: 1-31.
[4] Kvam, P. H. and Samaniego, F. J. (1993). On maximum likelihood estimation based on ranked set sampling with application to reliability. In: A. Basu (Ed.) Advanced in Reliability, 215-229.
[5] Kvam, P. H. and Samaniego, F. J. (1994). Non-parametric maximum likelihood estimation based on ranked set samples. Journal of American Statistical Association, 89: 526-537.
[6] Salazer, R. D. and Sinha, A. K.(1997). Control chart X based on ranked set sampling. Communication Tecica, No. 1-97-09 (PE/CIMAT).
[7]Muttlak, H. and Al-Sabah, W. (2003).Statistical quality control based on ranked set sampling. Journal of Applied Statistics, 30 (9):1055-1078.
[8] Al-Naseer, A. D. and Al-Rawwash, M. A.(2007). A control chart based on ranked data. Journal of Applied Science, 7(14): 1936-1939.
[9] Al-Omari, A. I. and Al-Naseer, A. D. (2011). Statistical quality control limits for the sample mean chart using robust extreme ranked set sampling. Economic Quality Control 2011; 26(1): 73-89.
[10] Al-Omari, A. I. and Haq, A. (2012). Improved quality control charts for monitoring the process mean using double ranked set sampling methods. Journal of Applied Statistics, 39(4): 745763.
[11] Al-Rawwash, M., Al-Naseer, A. D., Al-Omari, M. I. (2014). Quality control charts using different ranked set sampling schemes. Mathematical Methods in Science and Mechanics, 173-177.
[12] Mehmood, R., Riaz, M., Does, R. J. M. M. (2014). On the application of different ranked set sampling schemes. Quality Engineering, 26: 370-378.
[13] Abbasi, S. A. (2019). Location charts based on ranked set sampling for normal and nonnormal processes. Quality and Reliability Engineering International, 35 (6): 1603-1620.
[14] Tahir, A., Zaman, B., Atir, A., Riaz M., Abbasi, S. A.(2019). On improved dispersion control charts under ranked set schemes for normal and non-normal processes. Quality and Reliability Engineering International, Special Issue: 1-29.
[15] Klein, M. (2000). Two alternatives to the Shewhart X control chart. Journal of Quality Technology, 32(4): 427-431.
[16] Antzoulakos, D. L and Rakitzis, A. C. (2010). Runs rules schemes for monitoring process variability. Journal of Applied Statistics, 37 (97): 1231-1247.
[17] Cheng, C. S. and Chen, P. W. (2011). An ARL unbiased design of time-between-events control charts with runs rules. Journal of Statistical Computation and Simulation, 81 (7): 857-871.
[18] Abbasi, S. A. and Miller, A. (2011). D chart: An efficient alternative to monitor process dispersion.Proceedings of the World Congress on Engineering and Computer Science (WCECS 2011), 2 San Francisco, USA.
[19] Downton, F. (1966). Linear estimates with polynomial coefficients. Biometrika, 53: 129-141.
[20] Nawaz, T. and Han, D. (2020). Monitoring the process location by using new ranked set sampling-based memory control charts. Quality Technology & Quantitative Management, 17(3), 255284.
[21] Abujiya, M. A. and Muttlak, H. (2004). Quality control chart for the mean using double ranked set sampling. Journal of Applied Statistics, 31(10): 1185-1201.
[22] Muttlak, H. and Al-Sabah, W. (2003). Statistical quality control based on ranked set sampling. Journal of Applied Statistics, 30(9): 1055-1078.
[23] Haq, A. and Munir, W. (2018). Improved CUSUM charts for monitoring process mean. Journal of Statistical Computation and Simulation,88(9): 1684-1701.
[24] Tüfekci, P. (2014). Prediction of full load electrical power output of a base load operated combined cycle power plant using machine learning methods. International Journal of Electrical Power & Energy Systems 60: 126-140.
