Journal of Probability and Statistics
Volume 2019 (2019), Article ID 5269357, 11 pages
https://doi.org/10.1155/2019/5269357

Research Article

Generalized Performance Measures of Control Charts Based on Different Sampling Schemes

Rashid Mehmood,1,2 Muhammed Hisyam Lee,¹ Muhammad Riaz,³ and Iftikhar Ali²

¹Department of Mathematical Sciences, Universiti Teknologi Malaysia, Malaysia
²Department of Mathematics, University of Hafr Al Batin, Hafr Al Batin, Saudi Arabia
³Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Correspondence should be addressed to Muhammed Hisyam Lee; mhl@utm.my

Received 17 February 2019; Accepted 2 May 2019; Published 4 June 2019

Academic Editor: Aera Thavaneswaran

Copyright © 2019 Rashid Mehmood et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Different versions of control chart structure are available under various ranked set strategies. In these control charts, computation of performance measures was carried out through Monte Carlo simulation method (MCSM). In this article, we have defined a generalized structure of control charts under variant sampling strategies followed by derivation of their different performance measures. For the derivation of different performance measures, we have proposed pivotal quantity. For comparative analysis, we have presented results of generalized performance measures by involving numerical method (NM) as computation. We found that values of generalized performance measures based on NM are almost similar to values of performance measures based on MCSM. Also, NM is time efficient and can be considered as an alternative of MCSM.

1. Introduction

Shewhart (1931) proposed control chart under simple random sampling for monitoring the variations (natural or unnatural) in a process’s parameters. Afterwards, control charts became popular among researchers and practitioners of various fields. Therefore, different researchers and practitioners constructed improved forms of control charts in order to cope with different situations, for instance, when we have extra information (also termed as auxiliary information) associated with the variable of interest. Then, for such situation, a number of authors proposed and recommended auxiliary information based control charts (see [1–10]).

Salazar & Sinha [1] attempted Shewhart control chart based on ranked set sampling (RSS) for monitoring of location parameter. They utilized the auxiliary information for ranking the units of interest. Muttlak & Al-Sabah [2] extended the idea of Salazar and Sinha [1] by offering control charts under median ranked set sampling (MRSS) and extreme ranked set sampling (ERSS). Abujiya & Muttlak [3] contributed control chart under double ranked set sampling. All of the illustrated studies were designed for Shewhart control chart based on one point decision rule, that is, if any plotting statistic goes outside the control limits. Later on, Mehmood et al. [4] enhanced the performance of ranked set sampling based control charts for the detection of small-to-moderate shifts by incorporating more runs rules with the existing structures. It is worth mentioning that the common methodology used in the prescribed studies was the computation of performance measures through MCSM. One may think of NM such as quadratic and cubic polynomial (see [11–13]) as an alternate methodology for computing the values of performance measures. The most commonly used performance measures dealing with quality control charts are false alarm rate, power, and average run length (see [14]).

This study is projected to define a generalized design structure of control chart under different ranked set strategies along with derivation of generalized performance measures. For the said purpose, we will propose constant and pivotal quantity. It is important to mention that we are intended to show the significance of NM in terms of producing almost similar results as one can expect through MCSM. This practice may highlight more scope of NM to researchers especially those who have been working in the direction of quality control charts under variant ranked set schemes. Another purpose of introducing the NM in the place of MCSM is to prove it as time efficient methodology.

The Rest of the article is organized as follows: In Section 2 we will define generalized design structure of control chart under different ranked set strategies. Based on Section 2, we will propose pivotal quantity in Section 3. In Section 4, we will define performance measures under proposed constant and pivotal quantity. Section 5 will contain computation of performance measures under NM as well as MCSM, and results and discussion. Lastly, the study will end up with summary and conclusion.

2. Design Structure

In this section we define a generalized design structure of control chart under different ranked set strategies. The sampling strategies include simple random sampling (SRS), ranked set sampling (RSS), median ranked set sampling (MRSS), extreme ranked set sampling (ERSS), double ranked set sampling (DRSS), double median ranked set sampling (DMRSS), median double ranked set sampling (MDRSS), double extreme ranked set sampling (DERSS), and extreme double ranked set sampling (EDRSS). Now we define the structure as follows.

Let , and , denote the observation of sample collected under a sampling strategy as mentioned above with in-control known mean and standard deviation . The details of these sampling strategies can be seen in the article of Mehmood et al. [4]. Thus, generalized two sided control chart structure for monitoring the statistic is given below:

where represents the sampling strategy (as discussed earlier); indicates the sample size; is the amount of prefix false alarm rate of a process (see [14]); is the amount of correlation between study variable and concomitant variable which can attain different values, i.e., when case of perfect sampling strategy is considered and when case of imperfect sampling strategy is involved (see [4]); , are and quantiles of standard normal distribution, respectively. Furthermore, is a constant and given below:

where is the variance of order statistic in a sample of size from standard normal distribution under a ranked set scheme .

2.1. Construction of Constant

A brief procedure is illustrated for the construction of constant by considering the study of Mehmood et al. [15] as guidance. For readers concern, a brief procedure is illustrated as follows: generate 10⁵ random samples of size from standard normal distribution for the given choice of , , and followed by calculating the statistic () and standard deviation of statistic . The resulting values of are given in Table 1 at different choices of and for some choices of and . Likewise, one may approach other choices of and .

Table 1: values of at a given choice of sampling strategy and some choices of sample size .

SRS
RSS
MRSS
DRSS
DMRSS
MDRSS
ERSS
DERSS
EDRSS

0.75
5
1
0.800
0.778
0.755
0.696
0.718
0.825
0.778
0.775

6
1
0.778
0.755
0.735
0.688
0.701
0.82
0.775
0.778

7
1
0.762
0.748
0.725
0.68
0.695
0.802
0.767
0.762

8
1
0.764
0.738
0.716
0.675
0.687
0.805
0.759
0.769

9
1
0.753
0.729
0.71
0.671
0.684
0.794
0.753
0.753

10
1
0.762
0.721
0.707
0.671
0.686
0.794
0.748
0.742

1
5
1.000
0.599
0.537
0.472
0.286
0.368
0.645
0.552
0.540

6
1.000
0.563
0.494
0.429
0.254
0.328
0.642
0.536
0.554

7
1.000
0.527
0.460
0.394
0.211
0.290
0.606
0.507
0.507

8
1.000
0.498
0.431
0.365
0.192
0.267
0.611
0.492
0.522

9
1.000
0.478
0.407
0.345
0.166
0.242
0.581
0.472
0.488

10
1.000
0.458
0.389
0.324
0.153
0.226
0.587
0.465
0.502

3. Proposed Pivotal Quantity

Before defining the performance measures (see [14]) for structure (1), we have defined a pivotal quantity as follows:

where is standard normal variable. The theoretical justification for the relationship between and has been shown through Gaussian Kernel Density Plot (see Figures 1–6). The details of Gaussian Kernel Density Plot can be seen in the article of Sheether [16]. In short, Figures 1–6 showed that for a given choice of , , and the Gaussian Kernel Density Plots of standardized random variable (denoted as and random variable almost overlap with each other. This shows that probability distribution of random variable can be approximated by the random variable.

Figure 1: Kernel density plot of and at , , and .

Figure 2: Kernel density plot of and at , , and .

Figure 3: Kernel density plot of and at , , and .

Figure 4: Kernel density plot of and at , , and .

Figure 5: Kernel density plot of and at , , and .

Figure 6: Kernel density plot of and at , , and .

4. Derivation of Generalized Performance Measures

In this section we define performance measures (see [14]) for structure (1). The performance measures include false alarm rate, power, and average run length.

4.1. False Alarm Rate

False alarm rate (symbolically denoted as ) is a popular measure used for computing the probability that sample statistic goes outside the control limits (LCL, UCL) when actually the process is in-control. An expression for representing the definition of is given below:

and (5) can be written based on (3):

where is the area between two quantiles points to ) of standard normal curve (see [14]). Similarly is the area between two quantiles points ( to ).

4.2. Power

Power is defined as the probability that sample statistic goes outside the control limits (LCL, UCL) when actually a process is out-of-control. The out-of-control process implies that the in-control mean is shifted to new level , where denotes the amount of shift. An expression for representing the power is as follows:

4.3. Average Run Length

Average run length is defined as the average number of sample points that must be plotted on a control chart before an out-of-control signal is triggered. Moreover, average run length can be classified as in-control average run length and out-of-control average run length . is defined as the average number of sample points of an in-control process that must be plotted on the control chart before a plotting statistic breaches the control limits (LCL, UCL), whereas is defined as the average number of sample points of an out-of-control process that must be plotted on control chart before a plotting statistic breaches the control limits (LCL, UCL). For computing the and , one can use the results of (11).

5. Computation of Performance Measures and Discussion

In this section we have considered (8), (11), (12), and (13) to compute the false alarm rate, power, in-control average run length, and out-of-control average run length. For the said purpose, we have considered a prefix value of false alarm rate , various values of shifts (, and ), sampling strategies , sample size , and . Afterwards, for each choice of , , and we have solved (6), (9), (11), and (12) by using quadratic integration as numerical method (NM), and results are reported in Table 2. The details of quadratic and cubic polynomial integration can be seen in various articles (see [11–13]). Also, we have computed results of structure (1) through MCSM by taking into account all procedural details from the study of Mehmood et al. [4]. The number of simulations for implementing the MCSM is motivated from several existing studies such as Mehmood et al. [4] and Mehmood et al. [17]. Besides, we have recorded the time (see Table 3) utilized by both methods for obtaining the power . The time required for getting the is dependent on the choice of sampling strategy , sample size , and number of simulations . A method can prove to be time efficient if it consumes minimum time as compared to competitors given that all methods produced almost similar results.

(i)Table 2 shows that results obtained through both methods (MCSM and NM) are almost close to each other and particularly at large shifts results are similar. The following findings remain true for any choice of , , , and .(ii)Behavior of different sampling strategies remains consistent under the both methods as reported and discussed in the study of Mehmood et al. [4].(iii)NM has proved to be time efficient which reduced computation burden as compared to MCSM (see Table 3). In more detail, NM has consumed minutes to gain the intended results. In contrast, MCSM has utilized more time, depending on the choice of sampling strategy , sample size , and number of simulations .(iv)Structure (1) and equations (8), (11), (12), and (13) represent a generalized form of several studies such as Shewhart (1931) and Montgomery [14] when ; Salazar & Sinha [1] when ; Muttlak & Alsabah [2] when ; Abujiya & Muttlak [3] when and for varying choices of sampling strategy . Also, structure (1) represents the study of Mehmood et al. [4] under the one point decision rule.

Table 2: Performance measures through Monte Carlo simulation method (MCSM) and numerical method (NM) for varying choices of , , , and .

(a) and

Method
0.00
0.10
0.20
0.30
0.40
0.50

SRS
MCSM
0.0027
370.37
0.0032
312.50
0.0056
178.57
0.0104
96.15
0.0174
57.47
0.0299
33.44

NM
0.0027
370.37
0.0034
294.12
0.0056
178.57
0.0101
99.01
0.0177
56.50
0.0299
33.44

RSS
MCSM
0.0027
370.37
0.0049
204.08
0.0122
81.97
0.0303
33.00
0.0662
15.11
0.1289
7.76

NM
0.0027
370.37
0.0047
212.77
0.0122
81.97
0.0300
33.33
0.0659
15.17
0.1285
7.78

MRSS
MCSM
0.0027
370.37
0.0053
188.68
0.0148
67.57
0.0389
25.71
0.0904
11.06
0.1782
5.61

NM
0.0027
370.37
0.0052
192.31
0.0151
66.23
0.0392
25.51
0.0908
11.01
0.1787
5.60

ERSS
MCSM
0.0027
370.37
0.0046
217.39
0.0105
95.24
0.0252
39.68
0.0538
18.59
0.1031
9.70

NM
0.0027
370.37
0.0044
227.27
0.0107
93.46
0.0251
39.84
0.0534
18.73
0.1029
9.72

DRSS
MCSM
0.0027
370.37
0.0064
156.25
0.0205
48.78
0.0572
17.48
0.1335
7.49
0.2631
3.80

NM
0.0027
370.37
0.0063
158.73
0.0201
49.75
0.0571
17.51
0.1340
7.46
0.2634
3.80

DMRSS
MCSM
0.0027
370.37
0.0135
74.07
0.0763
13.11
0.2547
3.93
0.5499
1.82
0.8178
1.22

NM
0.0027
370.37
0.0133
75.19
0.0759
13.18
0.2552
3.92
0.5502
1.82
0.8181
1.22

MDRSS
MCSM
0.0027
370.37
0.0084
119.05
0.0376
26.60
0.1203
8.31
0.2837
3.52
0.5149
1.94

NM
0.0027
370.37
0.0085
117.65
0.0372
26.88
0.1199
8.34
0.2841
3.52
0.5154
1.94

DERSS
MCSM
0.0027
370.37
0.0057
175.44
0.0151
66.23
0.0375
26.67
0.0834
11.99
0.1646
6.08

NM
0.0027
370.37
0.0055
181.82
0.0148
67.57
0.0371
26.95
0.0837
11.95
0.1645
6.08

EDRSS
MCSM
0.0027
370.37
0.0055
181.82
0.0153
65.36
0.0402
24.88
0.0889
11.25
0.1748
5.72

NM
0.0027
370.37
0.0052
192.31
0.0150
66.67
0.0399
25.06
0.0892
11.21
0.1751
5.71

(b) and

Method
0.00
0.10
0.20
0.30
0.40
0.50

SRS
MCSM
0.0027
370.37
0.0037
270.27
0.0077
129.87
0.0161
62.11
0.0305
32.79
0.0558
17.92

NM
0.0027
370.37
0.0038
262.74
0.0076
130.86
0.0159
62.98
0.0309
32.41
0.0562
17.79

RSS
MCSM
0.0027
370.37
0.0074
135.14
0.0317
31.55
0.0981
10.19
0.2319
4.31
0.4362
2.29

NM
0.0027
370.37
0.0077
130.24
0.0314
31.84
0.0976
10.24
0.2323
4.30
0.4358
2.29

MRSS
MCSM
0.0027
370.37
0.0096
104.17
0.0461
21.69
0.1524
6.56
0.3536
2.83
0.6112
1.64

NM
0.0027
370.37
0.0097
103.36
0.0458
21.85
0.1519
6.58
0.3539
2.83
0.6109
1.64

ERSS
MCSM
0.0027
370.37
0.0058
172.41
0.0196
51.02
0.0532
18.79
0.1248
8.01
0.2456
4.07

NM
0.0027
370.37
0.0059
170.79
0.0191
52.44
0.0535
18.68
0.1252
7.99
0.2459
4.07

DRSS
MCSM
0.0027
370.37
0.0134
74.63
0.0729
13.72
0.2490
4.02
0.5386
1.86
0.8102
1.23

NM
0.0027
370.37
0.0131
76.41
0.0732
13.67
0.2489
4.02
0.5382
1.86
0.8098
1.23

DMRSS
MCSM
0.0027
370.37
0.0628
15.92
0.4792
2.09
0.9232
1.08
0.9980
1.00
1.0000
1.00

NM
0.0027
370.37
0.0631
15.86
0.4789
2.09
0.9229
1.08
0.9981
1.00
1.0000
1.00

MDRSS
MCSM
0.0027
370.37
0.0257
38.91
0.1882
5.31
0.5689
1.76
0.8908
1.12
0.9894
1.01

NM
0.0027
370.37
0.0261
38.31
0.1885
5.31
0.5693
1.76
0.8912
1.12
0.9890
1.01

DERSS
MCSM
0.0027
370.37
0.0081
123.46
0.0322
31.06
0.1017
9.83
0.2397
4.17
0.4497
2.22

NM
0.0027
370.37
0.0079
126.15
0.0322
31.09
0.1013
9.87
0.2401
4.16
0.4501
2.22

EDRSS
MCSM
0.0027
370.37
0.0075
133.33
0.0284
35.21
0.0838
11.93
0.1999
5.00
0.3849
2.60

NM
0.0027
370.37
0.0073
137.34
0.0280
35.73
0.0842
11.88
0.2005
4.99
0.3853
2.60

(c) and

Method
0.00
0.10
0.20
0.30
0.40
0.50

SRS
MCSM
0.0027
370.37
0.0041
243.90
0.0092
108.70
0.0200
50.00
0.0410
24.39
0.0782
12.79

NM
0.0027
370.37
0.0041
244.14
0.0091
109.97
0.0202
49.61
0.0414
24.17
0.0781
12.81

RSS
MCSM
0.0027
370.37
0.0108
92.59
0.0528
18.94
0.1769
5.65
0.4069
2.46
0.6759
1.48

NM
0.0027
370.37
0.0106
94.51
0.0528
18.93
0.1769
5.65
0.4066
2.46
0.6753
1.48

MRSS
MCSM
0.0027
370.37
0.0145
68.97
0.0854
11.71
0.2893
3.46
0.5999
1.67
0.8577
1.17

NM
0.0027
370.37
0.0145
69.09
0.0851
11.75
0.2888
3.46
0.6002
1.67
0.8576
1.17

ERSS
MCSM
0.0027
370.37
0.0072
138.89
0.0272
36.76
0.0828
12.08
0.1991
5.02
0.3800
2.63

NM
0.0027
370.37
0.0071
140.24
0.0273
36.57
0.0831
12.03
0.1995
5.01
0.3804
2.63

DRSS
MCSM
0.0027
370.37
0.0218
45.87
0.1459
6.85
0.4730
2.11
0.8204
1.22
0.9698
1.03

NM
0.0027
370.37
0.0215
46.42
0.1462
6.84
0.4728
2.12
0.8198
1.22
0.9701
1.03

DMRSS
MCSM
0.0027
370.37
0.1765
5.67
0.8721
1.15
0.9994
1.00
1.0000
1.00
1.0000
1.00

NM
0.0027
370.37
0.1765
5.67
0.8725
1.15
0.9993
1.00
1.0000
1.00
1.0000
1.00

MDRSS
MCSM
0.0027
370.37
0.0545
18.35
0.4220
2.37
0.8842
1.13
0.9955
1.00
1.0000
1.00

NM
0.0027
370.37
0.0548
18.25
0.4218
2.37
0.8846
1.13
0.9953
1.00
1.0000
1.00

DERSS
MCSM
0.0027
370.37
0.0106
94.34
0.0510
19.61
0.1673
5.98
0.3899
2.56
0.6566
1.52

NM
0.0027
370.37
0.0104
96.18
0.0506
19.77
0.1676
5.97
0.3904
2.56
0.6562
1.52

EDRSS
MCSM
0.0027
370.37
0.0093
107.53
0.0408
24.51
0.1336
7.49
0.3152
3.17
0.5599
1.79

NM
0.0027
370.37
0.0091
109.36
0.0410
24.39
0.1337
7.48
0.3158
3.17
0.5603
1.78

(d) and

Method
0.00
0.10
0.20
0.30
0.40
0.50

SRS
MCSM
0.0027
370.37
0.0033
303.03
0.0056
178.57
0.01
100
0.0176
56.82
0.0298
33.56

NM
0.0027
370.37
0.0034
294.12
0.0056
178.57
0.01
100
0.0177
56.5
0.0299
33.44

RSS
MCSM
0.0027
370.37
0.0038
263.16
0.0075
133.33
0.0154
64.94
0.0299
33.44
0.0545
18.35

NM
0.0027
370.37
0.0038
263.16
0.0075
133.33
0.0154
64.94
0.0299
33.44
0.0544
18.38

MRSS
MCSM
0.0027
370.37
0.0037
270.27
0.0077
129.87
0.0163
61.35
0.0321
31.15
0.059
16.95

NM
0.0027
370.37
0.0038
263.16
0.0078
128.21
0.0163
61.35
0.0321
31.15
0.059
16.95

ERSS
MCSM
0.0027
370.37
0.0038
263.16
0.0081
123.46
0.0174
57.47
0.0348
28.74
0.0643
15.55

NM
0.0027
370.37
0.0039
256.41
0.0082
121.95
0.0174
57.47
0.0348
28.74
0.0644
15.53

DRSS
MCSM
0.0027
370.37
0.004
250
0.0093
107.53
0.0209
47.85
0.0432
23.15
0.0817
12.24

NM
0.0027
370.37
0.0041
243.9
0.0093
107.53
0.0209
47.85
0.0432
23.15
0.0817
12.24

DMRSS
MCSM
0.0027
370.37
0.0041
243.9
0.0089
112.36
0.0195
51.28
0.0397
25.19
0.0744
13.44

NM
0.0027
370.37
0.0041
243.9
0.0089
112.36
0.0195
51.28
0.0397
25.19
0.0745
13.42

MDRSS
MCSM
0.0027
370.37
0.0037
270.27
0.0071
140.85
0.0143
69.93
0.0277
36.1
0.05
20

NM
0.0027
370.37
0.0037
270.27
0.0072
138.89
0.0144
69.44
0.0277
36.1
0.05
20

DERSS
MCSM
0.0027
370.37
0.0037
270.27
0.0078
128.21
0.0162
61.73
0.0321
31.15
0.058
17.24

NM
0.0027
370.37
0.0038
263.16
0.0078
128.21
0.0163
61.35
0.0321
31.15
0.059
16.95

EDRSS
MCSM
0.0027
370.37
0.0039
256.41
0.0079
126.58
0.0164
60.98
0.0325
30.77
0.0596
16.78

NM
0.0027
370.37
0.0039
256.41
0.0079
126.58
0.0165
60.61
0.0325
30.77
0.0597
16.75

Table 3: Time (expressed in minutes) consumed by the MCSM and NM for computing power at different choices of and .

MCSM
NM

5

6

SRS
7

8

9

10

5

RSS
6

MRSS
7

ERSS
8

9

10

DRSS
5

DMRSS
6

MDRSS
7

DERSS
8

EDRSS
9

10

6. Summary and Conclusions

In this study we have defined a generalized design structure of control charts and derived their different performance measures. In this regard, we have proposed constant and pivotal quantity. Afterwards, we have computed values of different generalized performance measures based on numerical method (NM). In conclusion, we have observed that NM has significant scope in terms of producing desired result as one can expect based on MCSM. Also, NM method may be considered as a time efficient methodology for handling performance measures of control chart under ranked set strategies.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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