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Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28) Spring 2011, pp 9-22
ISSN: 2008-594x
Quadrature Rules for Fuzzy Integrals
T. Allahviranloo1*, A. Borhanifar2, S. Hajihosseinlou2, Y. Nejatbakhsh1
1Department
of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran
2Department
Received: 3 April 2010
Accepted: 22 September 2010
Abstract
In this paper the Newton Cot's Quadrature methods with positive coefficient for fuzzy
integrations are discussed and then the concluded Eqs, is translated to an n  n fully fuzzy
liner systems (FFLS) . For this intent we discusses about the (FFLS) using embedding
approach to find its positive fuzzy number solutions. We investigate an n  n (FFLS) and
replace the original n  n (FFLS) by an 2n  2n parametric linear system and finally,
numerical example is used to illustrate the mentioned approach.
Keywords: Newton Cot's Methods, Fuzzy Integral, Fuzzy Numbers, Fully Fuzzy Linear Systems.
1 Introduction
Considering approximation theory, the integration problem plays major role in
various areas such as mathematics, physics, statistics, engineering and social sciences.
Since in many applications at least some of the system's parameters and measurements
are represented by fuzzy rather than crisp numbers, it is important to develop fuzzy
integration and solve them. The concept of fuzzy numbers and arithmetic operations
with these numbers were first introduced and investigated by Zadeh [1] and others. The
topic of fuzzy integration is discussed by [7]. The structure of this paper is organized as
follows.
In Section 2, we bring some basic definitions and results on fuzzy numbers, fuzzy
integrations and (FFLS) . In Section 3, we introduce the integration formulas of Newton
Cot's methods for fuzzy integration with convergence theorem. In section 4, our
procedure for finding positive solutions of (FFLS) is introduced. The proposed
algorithms are illustrated by solving an example in Section 5 and conclusions are drawn
in Section 6.
*Corresponding author
E-mail address: [email protected]
9
A. Allahviranloo, et al, Quadrature Rules for Fuzzy Integrals
2 Preliminaries
We represent an arbitrary fuzzy number by an ordered pair of functions
(u (r ), u (r )), 0  r  1 which satisfy the following requirements:
 u ( r ) is a bounded left continuous nondecreasing function over [0, 1].
 u ( r ) is a bounded left continuous nondecreasing function over [0, 1].
 u ( r ) and u ( r ) are right continuous in 0.
 u (r )  u (r ) , 0  r  1.
A crisp number  is simply represented by u (r )  u (r )   , 0  r  1 . The set of all
the fuzzy numbers is denoted by E 1 . A fuzzy number is the triangular fuzzy number
u  a1 , a2 , a3  which
 x  a1
a a ,
 2 1
a x
u x    3
,
a

a
3
2

 0,


a1  x  a2
a2  x  a3
otherwise .
It's parametric form is:
u  r   a2  a1  r  a1 ,
u  r    a3  a2  r  a3 .
Lemma 1. [6] Let v ,w  E 1 and s be real number. Then for 0  r  1
u=v if and only if u  r   v  r  and u  r   v  r  ,
v  w  v  r   w  r  ,v  r   w  r   ,
v w  v  r  w  r  ,v  r  w  r   ,

v .w  min v  r  .w  r  ,v  r  .w  r  ,v  r  w  r  ,v  r  w  r  ,
max v  r  .w  r  ,v  r  .w  r  ,v  r  w  r  ,v  r  w  r 
sv  s v  r  ,v  r   .

E 1 with addition and multiplication as defined by lemma 1 is a convex cone which is
then embedded isomorphically and isometrically in to a Banach space.
Definition 1 [4] For arbitrary fuzzy numbers u  u ,u  and v  v ,v


 the quantity
D u ,v   sup max  u  r  v  r  , u  r  v  r  
0 r 1
is the distance between u and v . It is shown that (E 1 , D ) is a complete metric space.
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Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28), Spring 2011, pp 9-22
Definition 2 A function f : R 1  E 1 is called a fuzzy function. If for arbitrary fixed
t 0  R 1 and   0 , there exist   0 such that
t  t 0    D  f t  , f  t 0    
f is said to be continuous.
Throughout this work we also consider fuzzy functions which are defined only over a
finite interval a, b  . We now follow Goetschel and Voxman [3] and define the integral
of a fuzzy function using the Riemann integral concept.
Definition 3 Let f : a, b   E 1 . For each partitions p  t 0 ,t1 ,···.t n  of a, b  and for
arbitrary i : t i 1  i  t i , 1  i  n , let
n
R   f i t i  t i 1 
i 1
The definite integral of f (t ) over a, b  is

b
a
f (t ) dt  lim R ,
max1i n t i  t i 1  0
provided that this limit exists in the metric D .
If the fuzzy function f (t ) is continuous in the metric D , its definite integral exists [3].
Further more,
  f (t ; r )dt    f t ; r dt ,
b
b
a
a
(1)
 b

 a f (t ; r ) dt    f t ; r  dt .

 a
b
It should be noted that the fuzzy integral also can be defined using the Lebesgue-type
approach [5, 7]. More details about properties of the fuzzy integral are given in [3, 7].
Now we are conceder the definition of FFLS.
Definition 4 The n  n linear system of equations
 a11x 1  a12 x 2 
a x  a x 
 21 1 22 2


an 1x 1  an 2 x 2 
 a1n x n  b1 ,
 a2 n x n  b 2 ,
(2)
 ann x n  bn .
where the elements , aijr , of the coefficient matrix A , 1  i , j  n and the elements,
bi ,of the vector b are fuzzy numbers is called a FFLS .
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A. Allahviranloo, et al, Quadrature Rules for Fuzzy Integrals
Definition 5 For any FFLS AX  b and for all r  [0,1], the n  n interval linear system
of equations
 a11r x 1r  a12r x 2r 
 r r
r
r
a21x 1  a22 x 2 


anr1x 1r  anr 2 x 2r 
 a1rn x nr  b1r ,
 a2rn x nr  b 2r ,
(3)
r
 ann
x nr  b nr ,
where aijr , x rj and
bir , 1  i , j  n , r  [0,1] are
r-cut sets of fuzzy numbers
aij , x j ,bi , 1  i , j  n , is called r-cut system of linear system and represented by
Ar Xr  br ,0  r  1 . Any r- cut system is called interval linear system.
Definition 6 A fuzzy number vector (x 1 , x 2 ,..., x n )t given by 1  i  n , 0  r 1 is called a
solution of FFLS , if
n
n
j 1
j 1
 aij x j  r    aij x j  r   bi (r ),
n
n
j 1
j 1
 aij x j  r    aij x j  r   bi (r )
(4)
Now, we define positive fuzzy number solution of FFLS as follows:
Definition 7 A fuzzy number solution vector (x 1 , x 2 ,..., x n )t of FFLS is called positive
fuzzy number solution if for all i , (i  1, 2,..., n ) x i  0.
Necessary and sufficient condition for the existence of a positive fuzzy number solution
of FFLS is:
Theorem: If FFLS AX  b has a fuzzy number solution, then FFLS AX  b has
positive fuzzy number solution if and only if 0  cut system of linear system represented
by A 0 X0  b 0 has positive solution.
proof : Let (x 1 , x 2 ,..., x n )t is fuzzy number solution of AX  b. AX  b has positive
solution, if and only if x i  0, i  1, 2,
, n , if and only if
supp x i  {x | x  and x i (x )  0}  [x i (0), x i (0)]
are positive intervals if and only if A 0 X0  b 0 has positive interval solution.
1 Newton cot's methods
Let f be a fuzzy function. For any natural number n, the Newton Cot's formulas

b
a
n
f (x ) dx  h   i f i  E , f i  f  x i  , h 
i 0
provide approximate values for

b
a
n
f (x ) dx  w i f i  E
i 0

b
a
b a
n
(5)
f (x ) dx . If we take h i  w i for i  0,1,..., n
, f i  f x i 
12
(6)
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28), Spring 2011, pp 9-22
which wi , i = 0; 1; :::; n are the weights of integration. The parametric form of (5) is as
follows:
b

n
f  x ; r  dx  w i  r f  x i ; r   E  f ; r 
a
i 0
b
n
a
i 0
(7)
 f  x ; r  dx  w  r f  x ; r   E f ; r 
i
i
In the Newton cot's methods integration points are definite. Thus for a method with
n + 1 point, n + 1 values w 0 ,w 1 ,...,w n are unknown. For determine these values we
position E  0 for f (x )  x j , j  0,1,..., n and we obtain the n + 1 equations who give
me the weights. Thus by the definition (1) we have
n
b

D   f  x  dx , w i f i   0
i 0
a

,
f i  f (x i )
and by the properties of the Hausdorff distance
n
b
 n
b

D   x j dx ,w i x ij    D   x j dx ,w i x ij   0
i 0
a
 i 0  a

hence for all j  0,1,..., n that is enough
b

D   x j dx ,w i x ij   0
a

(8)
b
n
a
i 0
and we take I   x j dx and R  w i x ij . Therefore from the relation (8) we have
n
b
D   x j dx ,w i x ij
i 0
a

  sup max I  R , I  R  0
 r 0,1


Where
b
I  x
a
j
 r  dx
b
,
I  x
j
 r  dx ,
a
And
13
(9)
A. Allahviranloo, et al, Quadrature Rules for Fuzzy Integrals
n
n
R  w i x ij  r  ,
R  w i x ij  r ,
i 0
i 0
are The parametric form of I and R. Then from the (6) we can write
b

sup max   x
r 0,1

a
b
n
n


a
b
Therefore

 r  dx  w i x ij  r  ,  x j  r  dx  w i x ij  r    0
i 0
i 0
j
b
n
x
 r  dx  w i x i  r 
j
j
x
and
i 0
a
n
j
a
 r  dx  w i x ij  r 
tend to zero.
i 0
Hence
b
 x  r  dx
j
n
 w i x ij  r 
b
x
(10)
i 0
a
n
j
 r  dx  w i x ij  r 
(11)
i 0
a
Thus from the (8) we have
w 0x
0
0
 r  w
b
0
1 1
x
 r  ···w n x  r    x 0  r  dx
0
n
a
b
w 0 x 01  r   w 1x 11  r  ···w n x n1  r    x 1  r  dx
a
.
.
.
w 0x
n
0
 r  w
(12)
.
.
.
b
n
1 1
x
 r  ···w n x  r    x n  r  dx
n
n
a
also
b
w 0 x 00  r   w 1x 10  r  ···w n x n0  r    x 0  r  dx
a
b
w 0 x 01  r   w 1x 11  r  ···w n x n1  r    x 1  r  dx
a
.
.
.
w 0x
n
0
 r  w
.
.
.
b
n
1 1
x
 r  ···w n x  r    x n  r  dx
n
n
a
14
(13)
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28), Spring 2011, pp 9-22
2 Positive Solutions of FFLS
In this section, we are going to find positive fuzzy number solutions of FFLS. We
suppose that aij the elements of A are in the three forms:
1) aij  0 2) aij  0
3) aij (r )  0 and aij (r )  0.
Let AX  b be FFLS . Consider i-th equation of this system:
n
a x
j 1
ij
j
 bi ,
i  1, 2,..., n
(14)
Since we suppose that elements of coefficient matrix are in three forms, wemay define
three n  n matrix of fuzzy numbers A1 , A 2 , A3 as follows:
aij

(A1 )ij  
0

if aij  0
aij

(A3 )ij  
0

if aij (r )  0 and aij (r )  0
aij

(A 2 )ij  
0

if O .w
if aij  0
if O .w
(15)
if O .w
It is clear that
A=A1 +A2 +A3 ,
(16)
And
AX=A1X+A2 X+A3X.
(17)
Hence the i-th equation of system AX  b is transformed to this equation:
n
n
n
n
j 1
j 1
j 1
j 1
bi   aij x j  a1ij x j   a2ij x j   a3ij x j ,
If we use definition 4, and if x  (x 1 ,
following equations must be true:
i  1,..., n
(18)
, x n )t is fuzzy number solution of AX  b ,
n
n
n
n
j 1
j 1
j 1
j 1
bi  r    aij x j  r   a1ij x j  r    a2ij x j  r    a3ij x j  r  ,
15
i  1,..., n
(19)
A. Allahviranloo, et al, Quadrature Rules for Fuzzy Integrals
And
n
n
n
n
j 1
j 1
j 1
j 1
bi  r    aij x j  r   a1ij x j  r    a2ij x j  r    a3ij x j  r  ,
i  1,..., n
(20)
Now, let AX  b has a positive fuzzy number solution, since z j  0, (1  j  n ) and by
applying Lemma 1 we can write:
a1ij x j (r )  a1ij (r ).x j (r )
a1ij x j (r )  a1ij (r ).x j (r )
a2ij x j (r )  a2ij (r ).x j (r )
a2ij x j (r )  a2ij (r ).x j (r )
a3ij x j (r )  a3ij (r ).x j (r )
(21)
a3ij x j (r )  a3ij (r ).x j (r )
Now, if we replace above expressions in (19) and (20), they can be rewritten as:
 a x  r   a  r  x  r    a  r   a  r  x  r  b  r  ,
n
n
j 1
ij
j
j 1
n
1ij
j
j 1
2ij
3ij
j
i
i  1,..., n
(22)
And
 a x  r   a  r  x  r    a  r   a  r  x  r  b  r  ,
n
n
j 1
ij
j
j 1
n
1ij
j
j 1
2ij
3ij
j
i
i  1,..., n
(23)
Hence i-th equation of AX  b (14) is transformed to two parametric equations (22)
and (23). Now we illustrate these equations in matrix forms.
If C l l  1, 2,3, 4 are parametric n  n matrices by elements
(C 1 )ij  a1ij (r , )
(C 2 )ij  a2ij (r )  a3ij (r ),
(24)
(C 3 )ij  a1ij (r )  a3ij (r ),
(C 4 )ij  a2ij (r ).
and if X 1 , X 2 , B1 and B 2 are parametric n-vectors by elements
X 1  (z 1 (r ),..., z n (r ))t ,
X 2  (z 1 (r ),..., z n (r ))t ,
(25)
B1  (b1 (r ),..., bn (r ))t ,
B 2  (b1 (r ),..., b n ( r ))t ,
the FFLS AX  b can be represented in matrix form as:
16
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28), Spring 2011, pp 9-22
 C 1 C 2  X 1   B1 

    
C 4 C 3   X 2   B 2 
(26)
where this matrix of coefficients is represented in 2n  2n. If we solve this system, its
solution is positive solution of FFLS .
Note that before solving FFLS AX  b we must have information about its solution
such as: 1- Does this system has a positive solution? and 2- Is x j positive fuzzy
number? Hence, we must solve 0-cut system of AX  b to find FFLS AX  b solution
supports. After solving this interval system we can find our questions' answers. If that
interval system has positive interval solution, we can transform FFLS AX  b to
2n  2n parametric linear system as form (26). The algorithm for solving FFLS and to
find its positive solution is illustrated as follows:
Positive Solution of FFLS 0s Algorithm Suppose AX  b is a FFLS:
ˆ ,vertbˆ ). If this system
 1. Solve A0 X0  b0 system and find its hull  (vertA
has positive solution then go to 2 else go to 5.
 2. Transform AX  b to (26) system using A0 X0  b0 solutions.
 3. Solve 2n  2n parametric system (26).
 4. If the solutions of (26) can define fuzzy numbers, this solution is a positive
fuzzy number solution of FFLS AX  b go to 6.
 5. This system has a non positive solution or this system does not have any
solution and this algorithm can not solve it.
 6. End.
If at least one of the obtained solutions are not fuzzy number then we have the
following remark.



Remark 1 [8] Let X  x i  r  , x i  r  ,1  i  n denote the unique solution of Ax=b.



The fuzzy number vector u  u i  r  ,u i  r  ,1  i  n defined by


u i  r   min x i  r  , x i  r  , x i 1

(27)

u i  r   max x i  r  , x i  r  , x i 1
is called the fuzzy solution of Ax=b. The use of x(1) in Eq. (27) is meant to eliminate
the possibility of fuzzy numbers whose associated triangle possess an angle greater than
90 .
If  x i  r  , x i  r   ,1  i  n are all fuzzy numbers then u i  r   x i  r  , u i  r   x i  r  ,1  i  n
U is called a strong fuzzy solution. Otherwise, U is a weak fuzzy solution.
2 Numerical Example
17
A. Allahviranloo, et al, Quadrature Rules for Fuzzy Integrals
1
1
5

1 3
3
Let f : 1, 2  E 1 , with x   x  , x , x   , 1   ,1,  and 2   , 2,  are
2
2
2

2 2
2
x

1
positive fuzzy numbers and
. Then we want to evaluate the fuzzy integral

2
1
f (x )dx . First, we have
1
1
x r   x  r  ,
2
2
1
1
1 r   r  ,
2
2
1
3
2 r   r  ,
2
2
1
5
3 r   r  ,
2
2
1
1
x r   x  r  ;
2
2
1
3
1 r    r  ;
2
2
1
5
2r    r  ;
2
2
1
7
3 r    r  .
2
2
1
1
1 1
3
1
If f  x   1 then f  x ; r    r  ,  r   where f  x ; r   r  ,
2
2
2 2
2
2
1
3
f  x ; r    r  . Then we get
2
2
3
3
1
1
1 f  x ; r  dx  1  2 r  2  dx  r  1;
3

1
3
 1
f  x ; r  dx     r   dx  3  r .
2
2
1
3
Therefore from the Newton Cot's formulas we have
1
1 1
3
1
1 1
3
 r  1,3  r   w 0 ,w 0  r  ,  r    w 1 ,w 1  r  ,  r  
2 2
2
2 2
2
2
2






1 1
3
1
 w 2 ,w 2  r  ,  r  
2 2
2
2
1
1
1
1
1
1
 r  , x  r   where f  x ; r   x  r  ,
2
2
2
2
2
2

If f  x   x then f  x ; r    x

1
1
f  x ; r   x  r  . Therefore from the Newton Cot's formulas we have
2
2
 r  3,5  r   w 0 ,w 0  


1
1 1
3
3 1
5
1
r  ,  r    w 1 ,w 1  r  ,  r  
2 2
2
2 2
2
2
2


5 1
7
1
 w 2 ,w 2  r  ,  r  
2 2
2
2
If f  x   x 2 then, from Definition 1, we have
18
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28), Spring 2011, pp 9-22
1
1
1
1
1
1

f  x ; r    x 2  r 2   r  1 x  r  , x 2  r 2  1  r  x  r  
4
2
4
4
2
4

Where
1
1
1
1
1
1
f  x ; r   x 2  r 2   r  1 x  r  , f  x ; r   x 2  r 2  1  r  x  r  .
4
2
4
4
2
4
Then we get
1
1
1 2
31
 2 1 2
1 f  x ; r  dx 1  x  4 r   r  1 x  2 r  4  dx  2 r  3r  6 ;
3
3

 f  x ; r  dx   x
3
3
1
1
2
1
1
1
1
1
79
 r 2  1  r  x  r   dx  r 2  r  .
4
2
4
2
2
6
Therefore from the Newton Cot's formulas we have


31 1 2 1
79 
1 1 2 3
9
1 2
1 2 1
 r  3r  , r  r    w 0 ,w 0  r  r  , r  r  
6 2
2
6 
2
4 4
2
4
2
4




3
9 1
5
25 
3
25 1
7
49 
1
1
 w 1 ,w 1  r 2  r  , r 2  r    w 2 ,w 2  r 2  r  , r 2  r  
2
4 4
2
4 
2
4 4
2
4 
4
4
hence conceder the full fuzzy linear system


1 1
3
1 1
3
1 1
3
1
1
1
 r  , r  
 r  , r  
 r  , r  


2 2
2
2 2
2
2 2
2
2
2
2




1 1
3
3 1
5
5 1
7
1
1
1


 r  , r  
 r  , r  
 r  , r  
2 2
2
2 2
2
2 2
2
2
2
2


 1 2 1
1 1 2 3
9 1 2 3
9 1 2 5
25   1 2 3
25 1 2 7
49  
 r  r  , r  r    r  r  , r  r    r  r  , r  r   
2
4 4
2
4 4
2
4 4
2
4  4
2
4 4
2
4 
 4




 

   
  12 r
 w 0  r  ,w 0  r 

 w  r  ,w  r 
1
 1

 w 2  r  ,w 2  r 

We solve this system by our algorithm as follows:
It's 0-cut is
19





31 1
1
79 
2
 3r  , r 2  r   
6 2
2
6  
 r  1,3  r 
 r  3,5  r 
A. Allahviranloo, et al, Quadrature Rules for Fuzzy Integrals
 1 3   1 3 
1 3 
 ,  
 2 , 2   2 , 2 




 2 2   w 0  0  ,w 0  0 


 1 3   3 5 
5 7 
 ,   , 
 ,    w 1  0  ,w 1  0 
2 2 
 2 2   2 2 

 1 9   9 25   25 49    w 2  0  ,w 2  0 
 ,   ,   ,  
 4 4   4 4   4 4  







  1,3 


    3,5 
  316 , 796 
Its vertex hull solution:




1 1
4 4
w 0  0   w 0  0  ,w 0  0    ,  , w 1  0   w 1  0  ,w 1  0    ,  ,
3 3
3 3


1 1
w 2  0   w 2  0  ,w 2  0    , 
3 3
This interval system has positive solution. And C l , l  1, 2,3, 4 are defined as:
1
1

r

2
2

1
1
C1  
r

2
2

 1 r 2  1 r  1
2
4
4
1
1
r
2
2
1
3
r
2
2
1 2 3
9
r  r
4
2
4
1
1

r

2
2

1
5
,
r

2
2

1 2 3
25 
r  r 
4
2
4 
1
3

 2r 2

1
3
C3    r 

2
2

 1 r 2  3 r  9
2
4
4
1
3
 r
2
2
1
5
 r
2
2
1 2 5
25
r  r
4
2
4
1
3 
 r
2
2 

1
7 
 r
,
2
2 

1 2 7
49
r  r  
4
2
4 
0 0 0


C2 C4  0 0 0
0 0 0


And matrix of coefficients is
20
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28), Spring 2011, pp 9-22
 C1

C 4
1
 1
 2r 2

 1r1
 2
2
1
1
2
 r  r1
C2  4
2
4

C3  
0



0



0

1
1
r
2
2
1
3
r
2
2
1 2 3
9
r  r
4
2
4
1
1
r
2
2
1
5
r
2
2
1 2 3
25
r  r
4
2
4
0
0
0
0
0
0
0
0
0
0
0
0
1
3
 r
2
2
1
3
 r
2
2
1 2 3
9
r  r
4
2
4
1
3
 r
2
2
1
5
 r
2
2
1 2 5
25
r  r
4
2
4




0



0

1
3 
 r

2
2 
1
7 
 r

2
2 
1 2 7
49 
r  r 
4
2
4
0
Where it is 6  6 parametric linear system and its solutions are:
w 0 r  
2
,
6  3r
w 0 r  
4  27 r
,
12
w 1 r  
8  6r
,
6  3r
w 1 r  
8  27r
,
6
w 2 r  
2
,
6  3r
w 2 r  
4  27 r
.
12
Since w 0  r  ,w 2  r  are not fuzzy number, by consideration of Remark 1, we have the
following fuzzy weak solutions.
2
 4  27 r
w 0  r   w 0  r  ,w 0  r   
,
6  3r
 12



;

 8  6r 8  27 r 
w 1  r   w 1  r  ,w 1  r   
,
;
6 
 6  3r


2
 4  27 r
w 2  r   w 2  r  ,w 2  r   
,
6  3r
 12



.

Where these solutions are the positive weights of integration.
3 Conclusion
In this paper at first the Newton Cot's method to approximate fuzzy integration was
introduced and then for finding positive weights we considered a new method to solve
fully fuzzy linear system of equations.
21
A. Allahviranloo, et al, Quadrature Rules for Fuzzy Integrals
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22