International Journal of Civil Engineering and Technology (IJCIET)
Volume 10, Issue 1, January 2019, pp.509–519, Article ID: IJCIET_10_01_048
Available online at http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=1
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
©IAEME Publication
Scopus Indexed
CONTROL CHARTS FOR MULTIPLE
DEPENDENT STATE REPETITIVE SAMPLING
PLAN USING FUZZY POISSON DISTRIBUTION
Sreeja M Krishnan and O.S.Deepa
Department of Mathematics
Amrita School of Engineering, Coimbatore
Amrita Vishwa Vidyapeetham, India
ABSTRACT
Multiple Dependent State Repetitive Sampling(MDSRS) plan is a combination of
Multiple deferred state sampling plan as well as repetitive sampling plan.This paper
deals with multiple dependent repetitive state sampling plan for certain attribute
control chart with respect to Poisson, gamma Poisson, fuzzy Poisson andfuzzy
Binomial distributions. The average run length for various distributions in MDSRS
are tabulated. Graphical illustrations are also made. The average run lengths are
compared for smaller shifts in the process using control charts for different parameter
values. The proposed method will be much useful in industry during monitoring of
manufacturing process. An example of earthquake data set from UCI respiratory is
considered and the average run length is computed based on fuzzy Poisson
distribution.
Key words: Poisson distribution, Gamma Poisson, Fuzzy Poisson and Fuzzy
Binomial distributions, Multiple Dependent Repetitive State Sampling Plan, Control
charts.
Cite this Article: Sreeja M Krishnan and O.S.Deepa, Control Charts For Multiple
Dependent State Repetitive Sampling Plan Using fuzzy Poisson Distribution,
International Journal of Civil Engineering and Technology (IJCIET), 10 (1), 2019, pp.
509–519.
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1. INTRODUCTION
The quality of the products in the share market plays an important role for the customers.
Different statistical techniques are used to increase the quality of the product. It was Walter. A
.Shewhart who invented control chart and is known as Shewhart charts. The control chart is a
graph which changes depending on time. Through historical data one can find average line,
upper line and lower line of the graph. Control chart is one among the seven basic tools of the
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Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson
Distribution
quality control. This is mainly used for manufacturing purposes, laboratory purposes,
industrial purposes as well as education purposes.
If the number of defectives taken from a sample is less than the acceptance number c, then
the lot is accepted in case of single sampling plan. But a repetitive sampling can have repeated
samplings so that taking decision from the first sampling is impossible. If decisions are not
taken from the first sampling, Multiple deferred state sampling (MDS) will be useful. The
decision from the process state of MDS can be made when the preceding “i” subgroups can be
concluded in control or not in control. There is another sampling called multiple dependent
state repetitive sampling (MDSRS) which is the combination of MDS Sampling as well as
repetitive sampling. If the manufactured process has shifts then by using the MDSRS, shifts
can be detected.
The work in this paper is carried with multiple dependent state repetitive sampling
(MDSRS) for

Poisson Distribution

Gamma Poisson Distribution

Fuzzy Poisson Distribution

Fuzzy Binomial Distribution

Application of Fuzzy Poisson Distribution in earth quake data set of UCI respiratory
Operating Procedure
The steps involved in multiple dependent state sampling plan for p charts of any distributions
is based on sample size n and the number of defectives d.
1. Let LCL1,UCL1,LCL2,UCL2be the four control limits constructed during the process .
If LCL1  LCL2  d  UCL1  UCL2 then the process is in control and ifd  UCL and d 
LCL then the process is out of control.
2. For a given i , if the preceding “i” subgroups is concluded as in-control , then the
entire process is in control.
Let k1 and k2 be two control coefficients and the four control limits of MDRS can be
defined as:
UCL1    k1 
LCL1  max( 0,   k1  )
UCL2    k 2 
LCL2  max( 0,   k 2  )
where   np is the parameter for Poisson
The MDSRS sampling has more advantage than the MDS sampling because in MDS
sampling the declaration of the process to be in-control is done once the preceeding “i”
subgroups are in –control. But the MDSRS will repeat until a proper declaration is made even
if preceeding “i” subgroups do not have a conclusion.
ALGORITHM FOR THE CONTROL COEFFICIENTS (K1,K2and ARL)
Consider the values of ARL, r0,p0 and Iand determine the values of k1 and k2 for which ARL0  r0 .
Based on k1 and k2,determine the values of ARL1. For MDS plan the probability of
considering the process to be in control is given by
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P
01
in,1
 UCL2 e  d
 
 d  LCL 1 d!
2

UCL1
  LCL2 e  d
e   d
 
 
  d  LCL 1 d!
d!
d  UCL2 1
11
 
  UCL2 e  d
 
  d  LCL 1 d!
2
 




where   np0
For repeated sampling:
UCL1
UCL2
 LCL2 e   d
e   d  
e   d 

P rep 


1

 1 d! d  UCL
 1 d!   d  LCL
 1 d! 
 d  LCL
11
2
2


 
For MDSRS plan the probability of considering the process to be under control if the
P 0 in,1
process actually under control is given by P 01in 
1  P 0 rep
01
For the process to be in control, the average run length is ARL01 
1
1  P 01in
ARL FOR SHIFTED PROCESS
For the shifted process, p1  p0  cp0
The probability of considering in-control for MDSRS when the process shifts is given as
 UCL2 e   d
P11in,1   
 d  LCL 1 d!
2

UCL1
  LCL2 e   d
e   d
 
 
  d  LCL 1 d!
d!
d  UCL2 1
11
 
  UCL2 e   d
 
  d  LCL 1 d!
2
 




Where   np1
For repeated sampling:
For shifted process, the probability of repeated sampling given as
UCL1
 LCL2 e   d
e   d
P11rep   
 
 d  LCL 1 d!
d!
d  UCL2 1
11

UCL2
 d
 
  1   e 
  d  LCL 1 d!
2
 
For shifted process to be in control, the MDSRS is P11in 
For the shifted process : ARL11 




P11in,1
1 P11rep
1
1  P11in
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Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson
Distribution
Table 1 Different values of Average Run Length values for r=30 and i=2 :
p0
k1
k2
n
c
0
0.01
0.03
0.05
0.07
0.1
0.13
0.15
0.17
0.2
0.25
0.5
0.7
0.8
1
0.1
2.8127
2.04671
90
30.7622
30.4300
19.7642
21.5419
21.4846
24.0670
26.2857
25.9144
31.2035
23.1923
27.3560
34.0999
29.9500
36.7936
28.7078
0.2
0.3
2.8244
2.9504
2.1727
2.1651
80
81
ARL
37.4334 47.4013
36.9533 58.0619
38.6264 43.9958
42.2963 53.2946
37.0645 64.7105
40.2828 47.7972
46.2045 46.4837
35.6754 53.5851
40.0402 55.9675
43.2538 52.2996
38.8027 54.2359
41.5200 59.6298
49.0899 63.9907
45.0309 66.4436
49.9821 65.4073
0.4
2.9304
2.2021
83
58.6402
65.2745
60.9321
64.0138
55.8716
59.4761
61.8043
61.6230
61.7673
67.3314
63.6404
67.8871
73.7792
74.3189
80.4005
Table 2 Different values of Average Run Length values for r=3 and i=2 :
k1
k2
n
c
0
0.01
0.03
0.05
0.07
0.1
0.13
0.15
0.5
0.7
0.8
1
p0
0.1
0.2
0.3
0.4
3.530558 3.318249 3.067156 3.034194
0.596302 0.595697 1.036403 1.975097
83
84
78
98
ARL
2.9795 3.6451 8.4067 48.930
2.9878 3.6503 8.4280 46.0873
3.0017 3.6706 8.2350 46.2865
3.0220 3.6195 8.2921 45.7233
2.2203 3.6425 9.6036 43.8623
2.2273 4.4717 8.2685 48.3511
2.9801 3.6205 8.1831 46.3517
2.9908 3.6241 9.3360 53.3124
2.9323 4.3149 11.3786 55.3986
3.7390 4.2943 10.8866 58.3943
3.7030 5.1143 10.7910 65.8548
3.6398 5.0114 11.7217 64.4092
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Example:
The Average Run Length is increasing from 0.1 to 0.4 in both Table 1 and Table 2 when c is
0.01 and for various values of p0 is 0.1,0.2,0.3,0.4.Also for different values of c, the average
run length increases from 0.1 to 0.4.
Gamma Poisson Distribution
For MDS plan the probability of considering the process to be in control is given by
P 02 in,1 
( m  d  1)! np 



d
d  LCL2 1 !( m  1 p )! m  np 
UCL2
d
m
 m 

 
 m  np 
d
m
UCL1
 LCL2 ( m  d  1)!  np  d  m  m
(m  d  1)!  np   m  


 
  

 


 d  LCL1 1 d !(m  1 p )!  m  np   m  np 

d  UCL2 1 d !( m  1 p )!  m  np   m  np 


( m  d  1)! np 


* 
d
d  LCL2 1 !( m  1 p )! m  np 
UCL2
d
 m 


 m  np 
at
m
For repeated sampling:
P
02
rep
d
m
UCL1
 LCL2 m  d  1!  np  d  m  m

m  d  1!  np   m  


 
  

 


 d 

LCL1 1 d ! ( m  1 p )!  m  np   m  np 
d  UCL2 1 d ! ( m  1 p )! m  np   m  np 


d
m
UCL2






(
m

d

1
)!
np
m

 
 
 1  
 d  LCL2 1 d!(m  1 p)! m  np   m  np  


at
For MDSRS plan the probability of considering the process to be under control if the
process actually under control is given by
P 02 in 
P 02 in,1
1 P 02 rep
Hence ARL01 
1
1  P 01in
For shifted process, the process to be in control is P12 in 
P12 in,1 
( m  d  1)! np 


d  LCL2 1 d !( m  1 p )! m  np 
UCL2

d
P12 in,1
where
1  P12 rep
m
 m 

 
 m  np 
d
m
UCL1
 LCL2 ( m  d  1)!  np  d  m  m
(m  d  1)!  np   m  












 
 
 d 
d !(m  1 p )!  m  np   m  np 
d  UCL2 1 d !( m  1 p )!  m  np   m  np 
 LCL1 1

( m  d  1)! np 


d  LCL2 1 d !( m  1 p )! m  np 
UCL2
*

d
 m 


 m  np 
m
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Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson
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For repeated sampling:
P
12
rep
d
m
UCL1
 LCL2 m  d  1!  np  d  m  m

m  d  1!  np   m  


 
  

 


 d 

d!(m  1 p)!  m  np   m  np 
d  UCL2 1 d ! ( m  1 p )! m  np   m  np 
 LCL1 1

d
m
UCL2

(m  d  1)! np   m  


 

 1 
 d  LCL2 1 d!(m  1 p)! m  np   m  np  


at
Table 3 Different values of Average run length for Gamma Poisson when m=1
p0
k1
k2
n
c
0
0.01
0.03
0.05
0.07
0.1
0.13
0.15
0.17
0.2
0.1
2.812753
2.046714
90
2.3922
2.6831
2.5993
2.5233
2.4541
2.3611
2.4103
2.3545
2.3029
2.2324
0.2
0.3
2.824495 2.950435
2.172771 2.165179
80
81
ARL
1.6607
1.5845
1.8239
1.5757
1.7972
1.559
1.7721
1.5432
1.7486
1.5883
1.7595
1.5462
1.7274
1.5406
1.7075
1.5263
1.6886
1.5365
1.6996
1.5165
Figure 1
0.4
2.930443
2.20219
83
1.4692
1.4626
1.45
1.4555
1.4436
1.427
1.4115
1.4171
1.4075
1.3938
Figure2
The graph for table 3 with m=1 and p=0.3 The graph for table 3 with m=2 and p=0.4
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Table 4 Different values of Average run length for Gamma Poisson whenm=2:
p0
k1
k2
n
c
0
0.01
0.03
0.05
0.07
0.1
0.13
0.15
0.17
0.2
0.25
0.3
0.4
0.5
0.7
0.8
1
0.1
2.812753
2.046714
90
3.6859
4.2413
4.0628
3.9015
3.7551
3.5594
3.745
3.6209
3.5068
3.3518
3.1295
3.1831
2.8427
2.7557
2.4859
1.6753
2.3829
0.2
0.3
2.824495 2.950435
2.172771 2.165179
80
81
ARL
2.1995
2.0632
2.5354
2.0411
2.4762
2.0078
2.4211
1.9739
2.3697
2.1032
2.4023
1.9883
2.3319
1.9939
2.2886
1.9619
2.2478
1.9857
2.2775
1.941
2.1859
1.8746
2.1059
1.8367
2.1656
1.7961
2.0902
1.7441
1.9241
1.6716
2.0194
1.6392
1.7828
1.5871
0.4
2.930443
2.20219
83
1.8377
1.8236
1.7966
1.8084
1.7833
1.7484
1.7163
1.7279
1.708
1.6801
1.6654
1.6263
1.6056
1.5914
1.5265
1.5056
1.446
Table 5 Different values of Average run length for Gamma Poisson whenm=1:
p0
k1
k2
n
c
0
0.01
0.03
0.05
0.07
0.1
0.1
3.530558
0.596302
83
1.5414
1.5238
1.5121
1.4774
1.5562
1.5148
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0.2
0.3
3.318249 3.067156
0.595697 1.036403
84
78
ARL
1.2356
1.3041
1.23
1.299
1.2196
1.289
1.2697
1.2949
1.2581
1.2852
1.2423
1.2718
515
0.4
3.034194
1.975097
98
1.3657
1.3607
1.3656
1.356
1.3468
1.334
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Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson
Distribution
Table 6 Different values of Average run length for Gamma Poisson whenm=2:
p0
k1
k2
n
c
0
0.01
0.03
0.05
0.07
0.1
0.1
3.530558
0.596302
83
2.0589
2.0006
1.9619
1.8697
2.1659
2.022
0.2
0.3
3.318249 3.067156
0.595697 1.036403
84
78
ARL
1.5035
1.5916
1.4881
1.5788
1.4597
1.5544
1.5876
1.5848
1.5541
1.56
1.5101
1.5264
0.4
3.034194
1.975097
98
1.6457
1.635
1.6449
1.6246
1.6055
1.5789
Fuzzy Poisson Distribution:
The fuzzy Poisson distribution is given by
~
~
 e   d

 e   d

U
PK [ ]  min 
|   [ ] and PK [ ]  max 
|   [ ]
 A d!

 A d!

L
~
Where [ ]  [1 [ ], 2 [ ]]
 [nK  na2 , nK  na3  n(a3  a 2 ) K
~
~
p a  P( A)[ ]  [ PKL [ ], PKU [ ]]
ARL for Poisson process,(in-control):
 UCL2 e   d
P 03in,1   
 d  LCL 1 d!
2

UCL1
  LCL2 e   d
e   d
 
 
  d  LCL 1 d!
d!
d  UCL2 1
11
 
  UCL2 e   d
 
  d  LCL 1 d!
2
 




Where   np0
For repeated sampling:
UCL1
UCL2
 LCL2 e  d
e   d  
e   d 
P 03 rep   
 
 1 
 d  LCL 1 d!
d!   d  LCL2 1 d! 
d  UCL2 1
11

The UCL and LCL values are MDSRS when the process is in-control is :
P 03in 
P 03in,1
1 P 03 rep
For the process is in-control then:
1
1  P 03 in
Then to find ARL1i.e., the shifted process when the process is in control is
ARL03 
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 UCL2 e   d
P in,1   
 d  LCL 1 d!
2

13
UCL1
  LCL2 e   d
e   d
 
 
  d  LCL 1 d!
d!
d  UCL2 1
11
 
  UCL2 e   d
 
  d  LCL 1 d!
2
 




Where   np1
For repeated sampling:
UCL1
UCL2
 LCL2 e  d
e   d  
e   d
P13 rep   
 
 1 
 d  LCL 1 d!
d!   d  LCL2 1 d!
d  UCL2 1
11

For the process to be in-control in case of shifted processis:
13
P
in




P13in,1

1 P13 rep
Hence ARL13 
1
1  P13in
Table 7 Different values of Average run length for Fuzzy Poisson Distribution
p0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.4
c
0.1
0.11
0.111
0.23
0.24
0.243
0.38
0.6
n
90
90
90
80
80
80
81
83
k1
2.812753
2.812753
2.812753
2.824495
2.824495
2.824495
2.950435
2.930443
k2
2.046714
2.046714
2.046714
2.172771
2.172771
2.172771
2.165179
2.20219
ARL,L
20.70716
26.21117
26.32208
31.93199
37.65789
38.59999
45.20291
48.38394
ARL,U
21.27237
26.31821
26.34777
32.54151
37.98218
38.78242
46.02123
48.98715
Example:
Consider the value of p0= 0.1 , c = 0.1 and n = 90 then the ARL lower limit value is 20.70 and
the ARL upper limit value is 21.27and is represented as (20.70,21.27).
Fuzzy Binomial Distribution:
For MDS plan the probability of considering the process to be in control is given by
UCL1
  UCL2  n  d
 UCL2  n  d
  LCL2  n  d

n  d
nd
nd
nd
nd
P 04 in,1      p0 1  p0        p0 1  p0      p0 1  p0        p0 1  p0  
 d  LCL 1  d 
  d  LCL 1  d 



d  UCL2 1  d 
2
11

 

  d  LCL2 1  d 
or repeated sampling:
P
04
rep
UCL1
UCL2
 LCL2  n  d
 

n  d
n  d
nd
nd
nd



  p0 1  p0      p0 1  p0 

 1     p0 1  p0  


 d  LCL 1  d 
  d  LCL 1  d 
d  UCL2 1  d 
11
2


 
For the MDSRS, the process to be in control is P 04 in 
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517
P 04 in,1
1 P 04 rep
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Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson
Distribution
The average run length for the process to be in control is ARL04 
1
1  P 04 in
For the shifted process, p1  p 0  cp 0
Then to find ARL1
UCL1
  UCL2  n  d
 UCL2  n  d
  LCL2  n  d

n  d
nd
nd
nd
nd
P14 in,1      p1 1  p1        p1 1  p1      p1 1  p1        p1 1  p1  
 d  LCL 1  d 
  d  LCL 1  d 

  d  LCL 1  d 
d  UCL2 1  d 
2
2
11

 

 
For repeated sampling:
14
P
rep
UCL1
UCL2
 LCL2  n  d
 

n  d
n  d
nd
nd
n d



  p1 1  p1      p1 1  p1 

 1     p1 1  p1  


 d  LCL 1  d 
  d  LCL 1  d 
d  UCL2 1  d 
11
2


 
For shifted process, the process to be in-controlis
P14 in 
Hence
P14 in,1
1  P14 rep
ARL14 
1
1  P14 in
Table 6 The Average Run Length for the Fuzzy Binomial:
p0
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
c
0.1
0.12
0.13
0.14
0.15
0.02
0.03
0.04
0.05
0.06
n
90
90
90
90
90
32
32
32
32
32
k1
2.812753
2.812753
2.812753
2.812753
2.812753
2.824495
2.824495
2.824495
2.824495
2.824495
k2
2.046714
2.046714
2.046714
2.046714
2.046714
2.172771
2.172771
2.172771
2.172771
2.172771
ARL,L
5.81629
5.816294
3.837679
3.837679
3.837679
2.539819
2.539819
2.539819
2.539819
2.539819
ARL,U
36.30648
36.30806
30.93731
30.93731
30.93731
5.887213
5.887213
5.887213
5.887213
5.887213
Example:
Consider the second row when the value of p = 0.1 and c= 0.12 and n = 90 then the ARL
lower limit value is 5.816 and the ARL upper limit value is 36.308 and can be represented as
(5.816,36.308).
EARTHQUAKE DATASET
Procedure
100 data was selected from UCI respiratory dataset. A target was fixed for each criteria
(genergy, gpuls, gdenergy, gdpuls). After fixing the target, the data was again classified into
10 sub data of the 100 data. By checking the values against the target, the defects were
estimated from each sub data. Fuzzy Poisson is then applied to the data with the n as 100 and
the p value as the total defects by n. By this procedure the average run length of lower limit
and upper limit of the data set can be obtained.
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Table 7 The average run length for the fuzzy Poisson distribution
p0
0.27
0.27
0.27
c
k1
k2
ARL,l ARL,u
0.37 2.930443 2.20219 61.62233 61.32813
0.378 2.930443 2.20219 62.7359 62.07409
0.38 2.988402 2.319525 69.66456 68.89455
CONCLUSION
The average run length increases for various c values in case of Poisson, Gamma Poisson and Fuzzy
Poisson distributions and decreases for Fuzzy Binomial distributions. The efficiency of various
distributions are compared. It is seen that the average run length increases very slowly for Gamma
Poisson distribution and drastically for Poisson distribution and Fuzzy Poisson distribution. Compared
to the proposed distributions and existing distribution for multiple deferred state repetitive sampling
plan, the average run length using Poisson distribution and fuzzy Poisson distribution are found to be
better. This method can be applied to industry for better monitoring of the process.
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