A Heuristic Threshold Policy for Fault Detection in rejection since its publication in 1763, so that Bayes’
mode of reasoning, which was finally buried on so
uni-variate Statistical Quality Control
many occasions, has recently risen again with
Environments
astonishing vigor [1].
1, *
Mohammad Saber Fallah Nezhad
This research focuses on sequential methods based
1
Industrial Engineering Department,
on stochastic dynamic programming techniques for
Yazd University, Yazd, Iran.
process quality control. For the statistics literature,
SDP was studied mostly from Markov's decision
process perspective. Traditional SPC methods
provide a group of statistical tests for a general
hypothesis, in which the mean value for the quality
characteristic of a process, or a process mean, for
Abstract
In this research, the decision on belief (DOB) short, is consistent with its target level. A variety of
approach was employed to analyze and classify the graphical tools have been developed for monitoring a
states of uni-variate quality control systems. The process mean by Shewhart charts [2], CUSUM charts
concept of DOB and its application in decision [3], and EWMA charts [4].
making problems were introduced, and then a The process mean is desired to be maintained at its
methodology for modeling a statistical quality target level consistently; however, random process
control problem by DOB approach was discussed. errors, or random “shocks”, can shift the process
For this iterative approach, the belief for a system mean to an unknown level. A control chart is
being out-of-control was updated by taking new required to detect this shift as soon as possible. At
observations on a given quality characteristic. This the same time, it should not signal too many false
can be performed by using Bayesian rule and prior alarms when the process mean is on the target. These
beliefs. If the beliefs are more than a specific criteria are usually defined in terms of the Average
threshold, then the system will be classified as an Run Length of the control chart for the in-control
out-of-control condition. Finally, a numerical operation and out-of-control operation of the process,
example and simulation study were provided for i.e., in-control ARL and out-of-control ARL,
evaluating the performance of the proposed method. respectively [5].
For the quality control of a manufacturing process,
Keywords: Statistical Quality Control, Bayesian one essential task is to detect any possible abnormal
Rule, Decision on Belief.
change in the process mean and remove it. However,
a control chart alone does not explicitly provide a
process adjustment scheme even when the process
mean is deemed to be off-target. Consider a
manufacturing process, in which detecting off-target
state is very important and a control charting method
is not sufficient. We present a stochastic dynamic
programming approach for theses types of processes.
This approach, named Decision on Belief (DOB),
was firstly presented by Eshragh and Modarres [6].
They applied sequential analysis and stochastic
dynamic programming to find the best underlying
*
Corresponding Author Email:
probability distribution for the observed data, also
[email protected]
Eshragh and Niaki applied the DOB concept as a
Phone: +989122140128 / Fax: +983518210699
decision-making tool in Response Surface
methodology [7 & 8]. Fallahnezhad and Niaki [9]
applied the concept in the problem for determining
the best binomial distribution. Also they applied this
approach to acceptance sampling plans and
production systems [10, 11].
Sequential analysis has been used in control charts
by some authors [12, 13]. Wu et al proposed a
1. Introduction
scenario for continuously improving the X & S
The value of Bayes’ theorem, as a basis for statistical
control charts during charts usages. It uses the
inference, has swung between acceptance and
information collected from the out-of-control cases
in a manufacturing process to update charting
parameters [12].
Bayesian inference is one of the best tools in
Sequential Analysis. Some researchers have applied
Bayesian analysis in Quality control [14, 15]. Chun
and Rinks [14] assumed that the proportion defective
is a random variable that follows a Beta distribution
and regarding Bayesian inference, they derived
Bayes producer's and consumer's risks. Fallahnezhad
and Niaki [16] proposed a new monitoring design for
uni-variate statistical quality control charts based on
an updating method. Marcellus [17] proposed a
Bayesian statistical process control and compared it
with CUSUM charts. He showed the advantage for
turning from Shewhart chart or CUSUM chart into
Bayesian monitoring in situations where the required
information about the process structure is obtained.
The rest of the paper is organized as follows: DOB
modeling for SPC is presented in Section 2. Section
3 provides the belief and approach of its
improvement. A Stochastic Dynamic Programming
Approach is discussed in section 4. The numerical
demonstration on the proposed methodology
application is provided in section 5. Section 6
introduces a simulation experiment and Section 7
provides a conclusion.
making process exits. First, the decision space can be
divided into two candidates; an in-control or out-ofcontrol production process. Second, the problem
solution is one of the candidates for either an incontrol or out-of-control process. Finally, a belief is
assigned to each candidate so that the belief shows
the probability for the process being in or out-ofcontrol condition. Based upon the belief updated for
any iteration, we may decide if the process is in outof-control condition.
3. Learning: the beliefs and approach for its
improvement
For simplicity, only one single observation on the
quality characteristic of interest in any iteration for
data gathering process was assumed. At iteration k
for data gathering process, Ok  ( x1 , x2 ,..., xk ) was
defined as the observation vector where
x1 , x2 ,..., xk resemble observations for previous
iterations 1, 2, …, k. The decision-making process
after any iteration is in a stochastic space such that
we can never say surely that the production process
is in out-of-control state. After taking a new
observation, xk , the probability of being in an out-ofcontrol
state
is
defined
as B( xk , Ok 1 )  Pr{Out  of  control xk , Ok 1} .
At this iteration, we want to improve the belief of
being in out-of-control state based on observation
2. DOB modeling for SPC problems
In a uni-variate quality control environment, the vector Ok 1 and new observation xk . If we define
collected observations for a quality characteristic of a B(O )  B( x , O )
k 1
k 1
k  2 as the prior belief of an outproduct contain so much information about
production process that if we limit ourselves to apply of-control state, in order to update the posterior
a control charting method, we will not be using most belief B( xk , Ok 1 ) , since we may assume that the
of them. In fact, the main aim of a control charting observations are taken independently in any iteration,
method is to detect undesired variation quickly then we will have
occurred in the process. However, applying both Pr{x Out  of  control, O }  Pr{x Out  of  control}
k
k 1
k
sequential analysis concept and Bayesian rule, at any
iteration in which some observations on the quality
characteristic are available, based upon the With this feature, using Bayesian rule the updated
observations at hand we may calculate the belief for belief is:
the process being out-of-control. Regarding these B (x k ,O k 1 )  Pr{Out  of  control x k ,O k 1} 
beliefs and a stopping rule, we may find and specify Pr{Out  of  control , x k O k 1}
a decision interval on these beliefs and when the
Pr{x k O k 1}
updated belief in any iteration is not within this
interval, an out-of-control signal is observed. The  Pr{Out  of  control O k 1}Pr{x k Out  of  control ,O k 1}
Pr{x k O k 1}
control interval is determined based on the value for
type-one error.
(1)
In these problems, first, all probable solution spaces As candidates are independent, we can write
will be divided into several candidates (the solution equation (1) as
is one of the candidates), then a belief will be
assigned to each candidate considering our
experiences and finally, the beliefs are updated and
the optimal decision is selected based on the current
situation. In a SPC problem, a similar decision-
2
B (x k ,O k 1 ) 
Thus B  (O k ) is determined by the following
equation,
B  (O k 1 ) 1    x k  

B (O k )  
B (O k 1 ) 1    x k    0.5(1  B  (O k 1 ))
Pr{Out  of  control O k 1}Pr{x k Out  of  control }
Pr{Out  of  control O k 1}Pr{x k Out  of  control }  Pr{In  control O k 1}Pr{x k In  control }

B (O k 1 ) Pr{x k Out  of  control }
B (O k 1 ) Pr{x k Out  of  control }  (1  B (O k 1 )) Pr{x k In  control }
(2)
Assuming the quality characteristic of interest
follows a normal distribution with mean  and
(6)
4. A Stochastic Dynamic Programming Approach
variance  2 , we use equation (2) to calculate both We present a decision making approach in terms of
beliefs for occurring positive or negative shifts in the Stochastic Dynamic Programming approach.
process mean  .
Presented approach is like an optimal stopping
problem.
Suppose n stages for decision making is remained
Positive shifts in the process mean

The values of B (Ok ) , showing the probability of and two decisions are available.
1. A positive shift is occurred in the process mean
occurring a positive shift in the process mean, will be 2. No positive shift is occurred in the process mean
calculated
applying
equation
(2) Decision making framework is as follows:
recursively. Pr{x k In  control } is defined by the 1.
Gather a new observation.
2.
Calculate the posterior Beliefs in terms of prior
following equation,
Beliefs.
Pr{x k In  control }  0.5
3.
Order the current Beliefs as an ascending form
For   0 , the probability that the observations in and choose the maximum.
iteration
k
is
out-of-control, 4.
Determine the value of the least acceptable
Pr{xk Out  of  control } , is calculated using belief ( d   n  is the least acceptable belief for
equation (3).
detecting the positive shift and d   n  is the least
Pr{x k Out  of  control }    x k 
acceptable belief for detecting the negative shift)
(3)
5.
If the maximum Belief in step 3 was more than
where  (.) is the cumulative probability distribution the least acceptable belief, d   n  , select the belief
function for the normal distribution with mean  candidate with maximum value as a solution else go
and variance  2 . Above probabilities are not exact to step 1.
probabilities and they are a kind of belief function to In terms of above algorithm, the belief with
ascertain good properties for B  (Ok ) thus B  (Ok ) maximum value is chosen and if this belief was more
than a control threshold like d   n  , the candidate
is determined by the following equation,
B  (O k ) 
B  (O k 1 )  x k 
of that Belief will be selected as optimal candidate
else the sampling process is continued. The objective
of this model is to determine the optimal values
of d   n  . The result of this process is the optimal
B (O k 1 )  x k   0.5(1  B (O k 1 ))
(4)


 Negative shifts in the process mean
The values of B  (Ok ) , showing the probability of
negative shift in the process mean, will also be
calculated using equation (2) recursively. In this
case, Pr{x k In  control } is defined by the
strategy with n decision making stages that maximize
the probability of correct selection.
Suppose new observation x k is gathered. (k is the
number of gathered observations so far).
V n , d   n  is defined as the probability of correct
following equation,
selection when
n decision making stages are
remained and we follow d   n  strategy explained

Pr{x k In  control }  0.5

Pr{xk Out  of  control} , is calculated using above also V ( n ) denotes the maximum value of
V  n , d   n   thus,
equation (5).
Pr{x k Out  of  control }  1    x k 
V (n )  Max
V  n ,d   n 

(5)
d (n )
3




CS is defined as the event of correct selection. S1 is
 B  (O k ) Pr B  (O k )  d   n  
defined as selecting the out-of-control condition

(positive shift) as an optimal solution and S2 is
(1  B  (O k )) Pr 1  B  (O k )  d   n 
defined as selecting the in-control condition as an

  Pr{CS NS }
optimal decision and NS is defined as not selecting V  n   max
0.5d   n  1 
any candidate in this stage.
1  Pr B  (O k )  d   n   
Hence, using the total probability law, it is concluded


that:






 Pr 1  B (O k )  d

V  n , d   n    Ma x{Pr{CS }}  Pr{CS S 1}Pr{S 1} 

 n 





 B  (O ) Pr B  (O )  d   n  

k
k





 max
1  B (O k )  Pr 1  B (O k )  d  n  
0.5d   n  1 

1  Pr B  (O k )  d   n   
 V (n  1) 




  Pr 1  B (O k )  d  n   

 

Pr{CS S 2 }Pr{S 2 }  Pr{CS NS }Pr{NS }
(7)
Pr{CS S 1} denotes the probability of correct
selection when candidate S1 is selected as the optimal
candidate and this probability equals to its
belief, B  (Ok ) , and with the same discussion, it is
concluded that Pr{CS S 2 } =1- B  (Ok )
(8)


1.
Pr{S1} is the probability of selecting out In terms of above equation, V n , d   n  is defined
of control candidate ( positive shift) as the solution as follows:
thus following the decision making strategy, we


should have B  (Ok )  max( B  (Ok ),1  B  (Ok ))


and
B  (Ok )  d   n 
that
is
equivalent
to
following,
Pr{S1} = Pr B  (O k )  d   n  , d   n   0.5,1


With the same reasoning, it is concluded that,
Pr{S 2 } = Pr 1  B  (O k )  d   n  ,
d
2.

 B  (O ) Pr B  (O )  d   n  

k
k







V  n , d  n    1  B (O k )  Pr (1  B (O k ))  d  n  



1  Pr B  (O k )  d   n    
 V n  1 

 



 Pr 1  B (O k )  d  n   

 

(9)
 n  0.5,1
Pr{CS NS } denotes the probability of
correct selection when none of candidates has been
selected and it means that the maximum value of the
beliefs is less than d   n  and the process of
decision making continues to latter stage. As a result,
in terms of Dynamic Programming Approach, the
probability of this event equals to maximum of
probability of correct selection in latter stage(n-1),
V (n  1) , but since taking observations has cost,
then the value of this probability in current time is
less than actual value of it and by using the
discounting factor  , it equals V (n  1)
3.
Since the entire solution space is
partitioned,
it
is
concluded
that
Pr{NS}  1  (Pr{S1}  Pr{S2})
By the above preliminaries, the function V ( n ) is
determined as follows:
4












Calculation method forV n , d   n  :

First the value of Pr B  (O k )  d   n  will be
B  ( gr ,O k ) and B  (sm ,O k ) are defined as follows: determined:
Pr B  (O k )  d   n  
B  ( gr ,O k )  max B  (O k ),1  B  (O k )


  x k  B  O k 1 

B  (sm ,O k )  min B  (O k ),1  B  (O k )
Pr 

d
n





   x k  B O k 1   1  B O k 1   0.5

Now equation (9) is rewritten as follows:
V  n ,d   n  
 Pr   x   h d   n   0.5  1  0.5h d   n  

 B ( gr ,O )  V  n  1 
Pr B ( gr ,O )  d  n  
 B (sm ,O )  V  n  1 
Pr B (sm ,O )  d  n   V  n  1

k

k

Since   x k

k
thus it follows a uniform Distribution function in
interval [0, 1], thus the above equality is concluded.
With the same reasoning, it is concluded that:

k


(12)
is a cumulative distribution function
Pr 1  B  (O k )  d   n  

k
Pr 1  d   n   B  (O k )
(10)
There are three conditions:
1. B  ( gr ,O k )  V (n  1) :
 0.5h 1  d   n  
In this condition, both B  ( gr ,O k )  V (n  1) and
B (sm ,O k )  V (n  1)
(13)
Now
equation
(8)
can
be
written as follows:
are negative, thus to
maximizeV n , d   n  ,
it



is
concluded
that
V (n ) 








B O k  1  0.5h d  n   
don’t select any candidate in this condition and


sampling process continues.




max
1  B O k   0.5h 1  d  n  


2. B  (sm ,O k )  V (n  1) :


0.5d  n  1


1  0.5 1  h d   n     
In this condition, both B ( gr ,O k )  V (n  1) and





V
n

1




B (sm ,O k )  V (n  1) are positive, thus to
 0.5h 1  d   n  



 

maximize V  n , d   n   , it is concluded that
d   n   1 and since B  ( gr ,Ok )  d   n   1 we




d   n   0.5 and since B  ( gr ,Ok )  d   n   0.5 ,
(14)
we select the belief candidate B  ( gr ,O k ) as the And equation (10) can be written as follows:
solution.
3. B  (sm ,O k )  V (n  1)  B  ( gr ,O k ) :
V n , d   n   B  O k   V  n  1 1  h d   n  0.5 
In this condition, one of the probabilities in equation


(10) has positive coefficient and one has negative 1  B O k   V  n  1 0.5h 1  d  n   V  n  1


coefficient, to maximize V n , d

 n  ,

h d

 n  


(15)

d   n  1  B  (O k 1 ) 
1  d   n   B  (O k 1 )


optimality
methods will be applied.
Definition: h d   n  is defined as follows:


 
Since V
*
V
 n   0.5Max
d  n  1 

 n , d  n  

thus it is
sufficient to maximize the real value function
V  n , d   n   , therefore; we should find the
function value in points where the first order
deviation equals zero,
(11)
5
 
V  n , d   n  
d   n 
no positive shift is occurred in the process mean and
decision making stops.
0
 B O   V  n  1 


k
1  B O   V  n  1 

1  d  n  



6. If Max B  O k  ,1  B  O k
2
then data is not sufficient for detecting the positive
shift and go to stage 2.
7. If Min  B  O k  ,1  B  O k    V  n  1 then
d  n 
2
k
1
 d  n  

B

O k   V  n  1 
1  B O   V  n  1 

   V  n 1 ,
if
Max  B  O k  ,1  B  O k
   B  O k 
it
is
concluded that a negative shift is occurred the
process mean and decision making stops and if
1
Max  B  O k  ,1  B  O k
k
(16)
   1  B  O k  ,
then
no negative shift is occurred in the process mean and
The optimal threshold d  n  is determined by the decision making stops.
8. If Max B  O k  ,1  B  O k   V  n 1 ,
above equation. Since the optimal value of d   n 
should be in the interval [0.5, 1] thus it is concluded then data is not sufficient for detecting the negative
shift and go to stage 2.
that the optimal value of d   n  will be determined





O k  ,1  B O k   
,
V  n  1  Min  B  O k  ,1  B  O k  
then determine the value for d   n  (least acceptable
as follows:
9. If
d  n  






1
Max 
, 0.5


B  O k   V  n  1 

 


1


1  B  O k   V  n  1 
Max  B
belief for detecting the positive shift) by the
following equation:






1
d   n   Max 
, 0.5 
(17)

 B  O k   V  n  1   1 
 


The above method is presented for detecting the
1  B  O k   V  n  1 


positive shifts for the process mean and can be
(18)
adapted for detecting the negative shifts with the
same reasoning.
The general decision making algorithm is 10. If
summarized as follows:
Max B  O k  ,1  B  O k   V  n  1 
,
1. Set
k=0
and
the
initial




Min
B
O
,1

B
O




k
k
beliefs B O0   0.5, B O0   0.5 .

2. Gather
an
observation
and then determine the value for d  n  (least
set k  k  1, n  n  1 .
acceptable belief for detecting the negative shift) by
3. If n  0 , then no shift is occurred in the process the following equation:
mean and decision making stops.
4. Update
the
values
for
the
beliefs




B Ok  , B Ok  by equation (2).





5. If Min  B  O k  ,1  B  O k
if
   V  n  1 , then d   n   Max 

Max  B  O k  ,1  B  O k    B  O k  , it is
 


concluded that a positive shift is occurred in the
process mean and decision making stops, also if
Max  B  O k  ,1  B  O k
   1  B  O k  ,
1
 B O   V  n  1 

k
1  B O   V  n  1 

k



, 0.5 


1


(19)
then 11. If B  O  d  n , then a positive shift is
 k
 
occurred
6
and
decision
making
stops,
and
if 1  B  O k   d   n  , then no positive shift is B  (O 2 )  0.67
occurred and decision making stops, else go to stage B  (O )  0.33
2
2.
Step
3:
Order the beliefs.
12. If B  Ok   d   n  , then a negative shift is B  ( gr ,O )  B  (O )  1  B  (O )  B  (sm ,O )
2
2
2
2
occurred and decision making stops, and If




B ( gr ,O 2 )  1  B (O 2 )  B (O 2 )  B (sm ,O 2 )
1  B  Ok   d   n  , then no negative shift is
occurred and decision making stops, else go to stage Step 4:
2.
The approximate value of V  n  1 based on the Since

14

discount factor  in the stochastic dynamic B ( gr ,O 2 )   V (0)  0.620  B (sm ,O 2 ) and
programming approach is  nV  0 .
B  ( gr ,O 2 )   14V (0)  0.620  B  (sm ,O 2 ) ,
thus we are in the condition 3 of decision making
process, hence first, the values of d  14  and
5. Numerical example
A numerical example is provided to detect the
positive and negative shift of mean for the process.
The assumption for a quality characteristic in a
process follows the standard normal distribution and
a shift   1 for the process mean. It was assumed
that sampling had a cost. Only 15 observations could
be sampled (n=15), also the values of   0.97 and
V (0)  0.95 were selected as the parameter values.
The initial value for the out-of-control or in-control
beliefs is equal to 0.5.
d  14  are determined by equation (16):
d  (14) 
1





14
  B O 2    V (0)  1 


1  B  O 2    14V (0)
max 
  0.7
1


, 0.5


0.67  0.62






1

B (O 0 )  0.5, B (O 0 )  0.5
0.33  0.62


To show the process of decision making on Beliefs,

random numbers of normal distribution with d (14) 
parameters   0,   1,   1 were generated.






1
1. First observation:
max 
 0.29, 0.5   0.5

14
Step 1: Number x 1  0.25 generation.
  B O 2    V (0)  1


14
Step 2: Posterior belief calculation.


1

B
O


V
(0)


2


B  (O1 )  0.59
B  (O1 )  0.41
Step 5:
Step 3: Order the belief.
Since
B  ( gr ,O 1 )  B  (O1 )  1  B  (O1 )  B  (sm ,O 1 )



B  ( gr ,O 2 )  B  (O 2 )  0.67  d  14  0.7 and

B ( gr ,O 1 )  1  B (O1 )  B (O1 )  B (sm ,O 1 )
Step
4:
Since B  ( gr ,O 1 ), B  ( gr ,O 1 ) are
B  ( gr ,O 2 )  d  14  0.5 , then it is concluded
less
that no negative shift is occurred in the process mean
than  15V (0)  0.6 , it is in the condition 1 of but additional observations are needed for decision
making about the positive shifts.
decision making method. Hence, d  15  1 .
2.
Take the third observation:
Step 5: Since B  ( gr ,O 1 ), B  ( gr ,O 1 ) are less Step 1: x  0.26
3
than d  15  1 , then there is not any selection. Step 2: Posterior beliefs calculation.
Therefore, another observation will be selected.
B  (O 3 )  0.75
1.
Second observation:
Step 3: the beliefs are ordered.
Step 1: x 2  0.24
B  ( gr ,O 3 )  B  (O 3 )  1  B  (O 3 )  B  (sm ,O 3 )
Step 2: Posterior beliefs calculation.
7
Step 4:
second type error and first type error. Also as V (0)
decreases, the probability of first type error increases
Since
but the probability of second type error decreases in
B  ( gr ,O 3 )   13V (0)  0.639  B  (sm ,O 3 ) ,
small value of shifts and increases in large values of
thus we are in the condition 3 of decision making shifts therefore the optimal value of V (0) should be
process, hence first, the value of d  13 is selected based on out of control value of mean that
should be detected.
determined by equation (16):
d  13 
1





13
  B O 3    V (0)  1 
 1  B  O 3    13V (0)

Max 
  0.64
1


, 0.5


0.75  0.639
 

1
0.25  0.639


Step 5:
Since
B  ( gr ,O 3 )  B  (O 3 )  0.75  d  (13)  0.64 ,
thus it is concluded that a positive shift is occurred in
the process mean.
6. Simulation Experiment
For the simulation methodology, the standard normal
observations were generated. Then, the proposed
procedure was implemented for the simulated
gathered data in different iterations.
Table 1 shows the estimated probabilities for
detecting a positive shift for the proposed
methodology. Five different parameter sets with
10,000 independent replications of the process were
selected.
Table (1): The Probabilities of detecting a positive
shift in simulation experiments
Table (1) shows that the overall values of
probabilities are favorably large. Further, as the
magnitudes of the mean shifts increase, the
probabilities become larger. Also, when the value of
 increases, the probability of first type error,
denoting the probability of detecting a positive shift
while no shift is occurred in the process mean,
decreases and the probability of second type error
decreases in most of the cases . Also as the number
of decision making stage (n) decreases, the
probability of first type error decreases but the
probability of second type error increases
simultaneously therefore the optimal value of n
should be selected based on the tradeoff between
8
Table (2): The ARL values in simulation
experiments
Table (2) shows that the overall values of ARL are
favorably small. Further, as the magnitudes of the
mean shifts increase, the ARL values become
smaller. Also, when the value of  increases, both
values of ARL0 and ARL1 increases. Also as the
number of decision making stage (n) decreases, both
values of ARL0 and ARL1 decreases. Also as V (0)
decreases,
both
values
of
ARL0
and
ARL1 decreases. Therefore it is concluded that
optimal parameter should be selected based on the
required application and performance.
6.
Conclusions
In this paper, the concept of decision on belief
(DOB) approach is employed to analyze and classify
the states of uni-variate quality control systems. In
this iterative approach, we tried to update the beliefs
of a system being out-of-control by taking new
observations on the given quality characteristic.
Decision making is based on comparing the values of
beliefs with a control threshold. The value for control
threshold is determined by solving a stochastic
dynamic programming problem. The optimal values
of control thresholds are determined in order to
maximize the probability of selecting correct design.
A numerical example is presented to illustrate how
the proposed procedure can be applied to design a
process control method. Also, in simulation
experiment, the performance of proposed method for
detecting the positive shifts of the process mean is
evaluated and it denotes that the proposed method
performance for detecting the shifts of the process
mean is satisfactory. Also, small values of ARL in
the proposed method indicate an advantage for
proposed method.
9
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