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.

1. http://www.iaeme.com/IJCIET/index.asp 509 editor@iaeme.com
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.
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=1
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

2. Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson
Distribution
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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:
),0max(
),0max(
22
22
11
11




kLCL
kUCL
kLCL
kUCL




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 00 rARL  .
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

3. Sreeja M Krishnan and O.S.Deepa
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

























  






 2
2
1
2
2
11
2
2 1111
1,
01
!!!!
UCL
LCLd
dUCL
UCLd
dLCL
LCLd
dUCL
LCLd
d
in
d
e
d
e
d
e
d
e
P
 
where 0np
For repeated sampling:

















  




 2
2
1
2
2
11 111
01
!
1
!!
UCL
LCLd
dUCL
UCLd
dLCL
LCLd
d
rep
d
e
d
e
d
e
P
 
For MDSRS plan the probability of considering the process to be under control if the
process actually under control is given by
rep
in
in
P
P
P 0
1,
0
01
1

For the process to be in control, the average run length is
inP
ARL 0101
1
1


ARL FOR SHIFTED PROCESS
For the shifted process, 001 cppp 
The probability of considering in-control for MDSRS when the process shifts is given as


























  






 2
2
1
2
2
11
2
2 1111
1,
11
!!!!
UCL
LCLd
dUCL
UCLd
dLCL
LCLd
dUCL
LCLd
d
in
d
e
d
e
d
e
d
e
P
 
Where 1np
For repeated sampling:
For shifted process, the probability of repeated sampling given as

















  




 2
2
1
2
2
11 111
11
!
1
!!
UCL
LCLd
dUCL
UCLd
dLCL
LCLd
d
rep
d
e
d
e
d
e
P
 
For shifted process to be in control, the MDSRS is
rep
in
in
P
P
P 11
1,
11
11
1

For the shifted process :
inP
ARL 1111
1
1



4. Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson
Distribution
http://www.iaeme.com/IJCIET/index.asp 512 editor@iaeme.com
Table 1 Different values of Average Run Length values for r=30 and i=2 :
p0
0.1 0.2 0.3 0.4
k1 2.8127 2.8244 2.9504 2.9304
k2 2.04671 2.1727 2.1651 2.2021
n 90 80 81 83
c ARL
0 30.7622 37.4334 47.4013 58.6402
0.01 30.4300 36.9533 58.0619 65.2745
0.03 19.7642 38.6264 43.9958 60.9321
0.05 21.5419 42.2963 53.2946 64.0138
0.07 21.4846 37.0645 64.7105 55.8716
0.1 24.0670 40.2828 47.7972 59.4761
0.13 26.2857 46.2045 46.4837 61.8043
0.15 25.9144 35.6754 53.5851 61.6230
0.17 31.2035 40.0402 55.9675 61.7673
0.2 23.1923 43.2538 52.2996 67.3314
0.25 27.3560 38.8027 54.2359 63.6404
0.5 34.0999 41.5200 59.6298 67.8871
0.7 29.9500 49.0899 63.9907 73.7792
0.8 36.7936 45.0309 66.4436 74.3189
1 28.7078 49.9821 65.4073 80.4005
Table 2 Different values of Average Run Length values for r=3 and i=2 :
p0
0.1 0.2 0.3 0.4
k1 3.5305583.3182493.0671563.034194
k2 0.5963020.5956971.0364031.975097
n 83 84 78 98
c ARL
0 2.9795 3.6451 8.4067 48.930
0.01 2.9878 3.6503 8.4280 46.0873
0.03 3.0017 3.6706 8.2350 46.2865
0.05 3.0220 3.6195 8.2921 45.7233
0.07 2.2203 3.6425 9.6036 43.8623
0.1 2.2273 4.4717 8.2685 48.3511
0.13 2.9801 3.6205 8.1831 46.3517
0.15 2.9908 3.6241 9.3360 53.3124
0.5 2.9323 4.3149 11.3786 55.3986
0.7 3.7390 4.2943 10.8866 58.3943
0.8 3.7030 5.1143 10.7910 65.8548
1 3.6398 5.0114 11.7217 64.4092

5. Sreeja M Krishnan and O.S.Deepa
http://www.iaeme.com/IJCIET/index.asp 513 editor@iaeme.com
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
mdUCL
LCLd
LCL
LCLd
UCL
UCLd
mdmd
mdUCL
LCLd
in
npm
m
npm
np
pmd
dm
npm
m
npm
np
pmd
dm
npm
m
npm
np
pmd
dm
npm
m
npm
np
pmd
dm
P



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
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 

2
2
2
1
1
2
2
2
1
1 1
1
1,
02
)!1(!
)!1(
*
)!1(!
)!1(
)!1(!
)!1(
)!1(!
)!1(
at
For repeated sampling:
   







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


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
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

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
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
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

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
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

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



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














 

 
mdUCL
LCLd
mdLCL
LCLd
UCL
UCLd
md
rep
npm
m
npm
np
pmd
dm
npm
m
npm
np
pmd
dm
npm
m
npm
np
pmd
dm
P
2
2
2
1
1
2
1
1 1
02
)!1(!
)!1(
1
)!1(!
!1
)!1(!
!1
at
For MDSRS plan the probability of considering the process to be under control if the
process actually under control is given by
rep
in
in
P
P
P 02
1,
02
02
1

Hence
inP
ARL 0101
1
1


For shifted process, the process to be in control is
rep
in
in
P
P
P 12
1,
12
12
1
 where
mdUCL
LCLd
LCL
LCLd
UCL
UCLd
mdmd
mdUCL
LCLd
in
npm
m
npm
np
pmd
dm
npm
m
npm
np
pmd
dm
npm
m
npm
np
pmd
dm
npm
m
npm
np
pmd
dm
P



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


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
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
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

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
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

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
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
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

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






 


 

2
2
2
1
1
2
2
2
1
1 1
1
1,
12
)!1(!
)!1(
*
)!1(!
)!1(
)!1(!
)!1(
)!1(!
)!1(
at

6. Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson
Distribution
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For repeated sampling:
   





























































 

 
mdUCL
LCLd
mdLCL
LCLd
UCL
UCLd
md
rep
npm
m
npm
np
pmd
dm
npm
m
npm
np
pmd
dm
npm
m
npm
np
pmd
dm
P
2
2
2
1
1
2
1
1 1
12
)!1(!
)!1(
1
)!1(!
!1
)!1(!
!1
at
Table 3 Different values of Average run length for Gamma Poisson when m=1
p0
0.1 0.2 0.3 0.4
k1 2.812753 2.824495 2.950435 2.930443
k2 2.046714 2.172771 2.165179 2.20219
n 90 80 81 83
c ARL
0 2.3922 1.6607 1.5845 1.4692
0.01 2.6831 1.8239 1.5757 1.4626
0.03 2.5993 1.7972 1.559 1.45
0.05 2.5233 1.7721 1.5432 1.4555
0.07 2.4541 1.7486 1.5883 1.4436
0.1 2.3611 1.7595 1.5462 1.427
0.13 2.4103 1.7274 1.5406 1.4115
0.15 2.3545 1.7075 1.5263 1.4171
0.17 2.3029 1.6886 1.5365 1.4075
0.2 2.2324 1.6996 1.5165 1.3938
Figure 1 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

7. Sreeja M Krishnan and O.S.Deepa
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Table 4 Different values of Average run length for Gamma Poisson whenm=2:
p0
0.1 0.2 0.3 0.4
k1 2.812753 2.824495 2.950435 2.930443
k2 2.046714 2.172771 2.165179 2.20219
n 90 80 81 83
c ARL
0 3.6859 2.1995 2.0632 1.8377
0.01 4.2413 2.5354 2.0411 1.8236
0.03 4.0628 2.4762 2.0078 1.7966
0.05 3.9015 2.4211 1.9739 1.8084
0.07 3.7551 2.3697 2.1032 1.7833
0.1 3.5594 2.4023 1.9883 1.7484
0.13 3.745 2.3319 1.9939 1.7163
0.15 3.6209 2.2886 1.9619 1.7279
0.17 3.5068 2.2478 1.9857 1.708
0.2 3.3518 2.2775 1.941 1.6801
0.25 3.1295 2.1859 1.8746 1.6654
0.3 3.1831 2.1059 1.8367 1.6263
0.4 2.8427 2.1656 1.7961 1.6056
0.5 2.7557 2.0902 1.7441 1.5914
0.7 2.4859 1.9241 1.6716 1.5265
0.8 1.6753 2.0194 1.6392 1.5056
1 2.3829 1.7828 1.5871 1.446
Table 5 Different values of Average run length for Gamma Poisson whenm=1:
p0
0.1 0.2 0.3 0.4
k1 3.530558 3.318249 3.067156 3.034194
k2 0.596302 0.595697 1.036403 1.975097
n 83 84 78 98
c ARL
0 1.5414 1.2356 1.3041 1.3657
0.01 1.5238 1.23 1.299 1.3607
0.03 1.5121 1.2196 1.289 1.3656
0.05 1.4774 1.2697 1.2949 1.356
0.07 1.5562 1.2581 1.2852 1.3468
0.1 1.5148 1.2423 1.2718 1.334

8. Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson
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Table 6 Different values of Average run length for Gamma Poisson whenm=2:
p0
0.1 0.2 0.3 0.4
k1 3.530558 3.318249 3.067156 3.034194
k2 0.596302 0.595697 1.036403 1.975097
n 83 84 78 98
c ARL
0 2.0589 1.5035 1.5916 1.6457
0.01 2.0006 1.4881 1.5788 1.635
0.03 1.9619 1.4597 1.5544 1.6449
0.05 1.8697 1.5876 1.5848 1.6246
0.07 2.1659 1.5541 1.56 1.6055
0.1 2.022 1.5101 1.5264 1.5789
Fuzzy Poisson Distribution:
The fuzzy Poisson distribution is given by

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
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
 

A
d
U
K
d
e
P ][|
!
max][
~




Where
KaannanKnanK )(,[
]][],[[][
2332
21
~




]][],[[])[(
~~
 U
K
L
Ka PPAPp 
ARL for Poisson process,(in-control):


























  






 2
2
1
2
2
11
2
2 1111
1,
03
!!!!
UCL
LCLd
dUCL
UCLd
dLCL
LCLd
dUCL
LCLd
d
in
d
e
d
e
d
e
d
e
P
 
Where 0np
For repeated sampling:

















  




 2
2
1
2
2
11 111
03
!
1
!!
UCL
LCLd
dUCL
UCLd
dLCL
LCLd
d
rep
d
e
d
e
d
e
P
 
The UCL and LCL values are MDSRS when the process is in-control is :
rep
in
in
P
P
P 03
1,
03
03
1

For the process is in-control then:
inP
ARL 0303
1
1


Then to find ARL1i.e., the shifted process when the process is in control is

9. Sreeja M Krishnan and O.S.Deepa
http://www.iaeme.com/IJCIET/index.asp 517 editor@iaeme.com


























  






 2
2
1
2
2
11
2
2 1111
1,
13
!!!!
UCL
LCLd
dUCL
UCLd
dLCL
LCLd
dUCL
LCLd
d
in
d
e
d
e
d
e
d
e
P
 
Where 1np
For repeated sampling:

















  




 2
2
1
2
2
11 111
13
!
1
!!
UCL
LCLd
dUCL
UCLd
dLCL
LCLd
d
rep
d
e
d
e
d
e
P
 
For the process to be in-control in case of shifted processis:
rep
in
in
P
P
P 13
1,
13
13
1

Hence
inP
ARL 1313
1
1


Table 7 Different values of Average run length for Fuzzy Poisson Distribution
p0 c n k1 k2 ARL,L ARL,U
0.1 0.1 90 2.812753 2.046714 20.70716 21.27237
0.1 0.11 90 2.812753 2.046714 26.21117 26.31821
0.1 0.111 90 2.812753 2.046714 26.32208 26.34777
0.2 0.23 80 2.824495 2.172771 31.93199 32.54151
0.2 0.24 80 2.824495 2.172771 37.65789 37.98218
0.2 0.243 80 2.824495 2.172771 38.59999 38.78242
0.3 0.38 81 2.950435 2.165179 45.20291 46.02123
0.4 0.6 83 2.930443 2.20219 48.38394 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
        

















































  







2
2
1
2
2
11
2
2 1
00
1
00
1
00
1
001,
04
1111
UCL
LCLd
dnd
UCL
UCLd
dnd
LCL
LCLd
dnd
UCL
LCLd
dnd
in pp
d
n
pp
d
n
pp
d
n
pp
d
n
P
or repeated sampling:
      


































  





2
2
1
2
2
11 1
00
1
00
1
00
04
1111
UCL
LCLd
dnd
UCL
UCLd
dnd
LCL
LCLd
dnd
rep pp
d
n
pp
d
n
pp
d
n
P
For the MDSRS, the process to be in control is
rep
in
in
P
P
P 04
1,
04
04
1


10. Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson
Distribution
http://www.iaeme.com/IJCIET/index.asp 518 editor@iaeme.com
The average run length for the process to be in control is
inP
ARL 0404
1
1


For the shifted process, 001 cppp 
Then to find ARL1
        

















































  







2
2
1
2
2
11
2
2 1
11
1
11
1
11
1
111,
14
1111
UCL
LCLd
dnd
UCL
UCLd
dnd
LCL
LCLd
dnd
UCL
LCLd
dnd
in pp
d
n
pp
d
n
pp
d
n
pp
d
n
P
For repeated sampling:
      


































  





2
2
1
2
2
11 1
11
1
11
1
11
14
1111
UCL
LCLd
dnd
UCL
UCLd
dnd
LCL
LCLd
dnd
rep pp
d
n
pp
d
n
pp
d
n
P
For shifted process, the process to be in-controlis
rep
in
in
P
P
P 14
1,
14
14
1

Hence inP
ARL 1414
1
1


Table 6 The Average Run Length for the Fuzzy Binomial:
p0 c n k1 k2 ARL,L ARL,U
0.1 0.1 90 2.812753 2.046714 5.81629 36.30648
0.1 0.12 90 2.812753 2.046714 5.816294 36.30806
0.1 0.13 90 2.812753 2.046714 3.837679 30.93731
0.1 0.14 90 2.812753 2.046714 3.837679 30.93731
0.1 0.15 90 2.812753 2.046714 3.837679 30.93731
0.2 0.02 32 2.824495 2.172771 2.539819 5.887213
0.2 0.03 32 2.824495 2.172771 2.539819 5.887213
0.2 0.04 32 2.824495 2.172771 2.539819 5.887213
0.2 0.05 32 2.824495 2.172771 2.539819 5.887213
0.2 0.06 32 2.824495 2.172771 2.539819 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.

11. Sreeja M Krishnan and O.S.Deepa
http://www.iaeme.com/IJCIET/index.asp 519 editor@iaeme.com
Table 7 The average run length for the fuzzy Poisson distribution
p0 c k1 k2 ARL,l ARL,u
0.27 0.37 2.930443 2.20219 61.6223361.32813
0.27 0.378 2.930443 2.20219 62.7359 62.07409
0.27 0.38 2.9884022.31952569.6645668.89455
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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