ARITHMETIC OPERATIONS ON Z-NUMBERS DESCRIBED BY PAIR OF TRAPEZOIDAL MEMBERSHIP FUNCTIONS
F.N.Iskandarova
Tashkent University of Information Technologies [email protected]
S.N.Iskandarova
Samarkand Branch of Tashkent University of Information Technologies, Street Shoxrux Mirzo 47A, Samarqand, Uzbekistan
[email protected] ABSTRACT
This paper considers implementation of arithmetic operations on Z-numbers described by pairs of trapezoidal membership functions. Trapezoidal numbers allow good approximation for a wide range of fuzzy numbers and are quite frequently used in fuzzy systems. Trapezoidal representation of fuzzy numbers is quite effective and efficient in preserving the systems from the loss of information after series of operations combining fuzzy numbers. We suggest an operational approach to implement arithmetic operations on Z-numbers the components of which are trapezoidal fuzzy numbers.
1. INTRODUCTION Trapezoidal Fuzzy Model in Representing Z-Numbers
Z-numbers are suggested by Professor Lotfi Zadeh as reliable data structures to store the information at presence of various forms of uncertainty [1,2]. A variable X described by a Z- number has two components (A, B), the first of which specifies the constraints on the values of the variable and the second one is the measure of reliability of the first component. Both components are usually fuzzy numbers.
There are very few researches on practical computation with Z-numbers. The paper [3]
considers operations on discrete fuzzy numbers. In our research we consider Z-numbers (A,B) where A and B are fuzzy numbers with trapezoidal membership functions.
The following trapezoidal model is suitable for representing botha trapezoidal membership function and a distribution function of probability measure (abcd):
. ,
0
, ,
, ,
, ,
, ,
0
), , , , , ( ) (
d x
d x c c
d x h d
c x b h
b x a a
b a h x
a x
x h d c b a T
x
When representing a normal MF, we have (h =1):
. ,
0
, ,
, ,
1
, ,
, ,
0
), 1 , , , , ( ) (
d x
d x c c
d x d
c x b
b x a a
b a x
a x
x d c b a T
x
For a distribution function of probability measure of random variable (ad, abcd):
. ,
0
, ) ,
( ) (
2
, ) ,
( ) (
2
, ) ,
( ) (
2
, ,
0
) , ( ) ( , 2 , , , )
(
d x
d x c c
d x d b a d c
c x b b
a d c
b x a a
b a x b a d c
a x
b x a d d c
c b a T x P
Trapezoidal functions can also be described by other parameter sets, when more convenient:
), 1 , , , , ( ) 1 , , , ,
(
A A A A A A1 A2 A3A
T a b c d T a
), 1 , , , , ( ) 1 , , , ,
(
B B B B B B1 B2 B3B
T a b c d T a
) , 2
( , 2 , , ) ,
( ) (
, 2 , , , )
(
3 2 1
3 2
1
p p p
p p p p p
p p p p p p
p T a
b a d
d c c b a T x P where:
1 A A
,
A
b a
2 A A
,
A
c b
3 A A
,
A
d c
1 B B
,
B
b a
2 B B
,
B
c b
3 B B
,
B
d c
1 p p
,
p
b a
2 p p
,
p
c b
3 p p
,
p
d c
2. IMPLEMENTING AN ARITHMETIC OPERATION ON Z-NUMBERS
Y X Z *
).
, (
* ) , ( )
,
( A
ZB
Z A
XB
X A
YB
Y
The computation of the variable part ( A ) is quite easy:
Y X
Z
A A
A *
where * is any of +, −, ×, /.
For trapezoidal fuzzy numbers (approximation for “× ” and “ / ”):
), , , ,
(
X X X XX
T a b c d
A
), , , ,
(
Y Y Y YY
T a b c d
A
), , , ,
(
Z Z Z ZZ
T a b c d
A
), ,
, ,
(
X Y X Y X Y X YY
X
A T a a b b c c d d
A
), ,
, ,
(
X Y X Y X Y X YY
X
A T a d b c c b d a
A
)), ,
, ,
max(
), ,
, ,
max(
), ,
, ,
min(
), ,
, ,
(min(
Y X Y X Y X Y X
Y X Y X Y X Y X
Y X Y X Y X Y X
Y X Y X Y X Y X Y
X
d d
a d
d a
a a
c c b c c b b b
c c b c c b b b
d d
a d
d a
a a
T A A
. 0 ,
, , ), / 1 , / 1 , / 1 , / 1 ( ) , , , (
/
Y
X X X X
Y Y Y Y Y Y Y Y
X
A T a b c d T d c b a a b c d
A
Let’s consider the computation of the second part ( B ).
If the probability density functions where known, the resultant probability functions for arithmetic operations could have been computed using the convolutions [4]:
For summation ( + ):
R
. ) ( ) ( )
( v P v u P u du
P
Z X YFor subtraction ( − ):
R
. ) ( ) ( )
( v P v u P u du
P
Z X YFor multiplication (× ):
R
. ) ( ) / 1 (
)
( P v u P u du
v u
P
Z X Y .4For division ( / ):
R
. ) ( ) ( )
( v u P vu P u du
P
Z X YAssume the distributions X p and Y p are trapezoidal functions. Then, pZXY(v) can be replaced by the parameterized function, which can be computed numerically:
wrt, in, [ , ,] with, acc, . ,, , , , , , , , , , ,
.,
u h d c b a T u v h d c b a T range
u Integrate
v d c b a d c b a p
Y Y Y Y Y X
X X X X Y
Y Y Y X X X X
p p p p p p
p p p p p
p p p p p p p Z
Then:
, ) ( ) ( )
( ) ( sup
) (
,
u p u du u p u du
p B X B Y
p Z
p X Y
pY X
Z
where pZ is the function defined above, subject to compatibility constraints (COG below stands for “Center Of Gravity”):
X X
A COGp
COG
Y Y
A COGp
COG
Centroid for a trapezoidal function can be computed as [5]:
) . ( ) (
) (
) (
3
1
2 2 2 2) ), , , , ,
(
b a d c
b ab a d
cd COG
T abcdh xc
Hence the compatibility constraint:
.
3 2 1 3 2
1 2
, 2 , , , )
1 , , , , (
p p p p p p P A
A A
A T a
d c b a
T
COG
COG
Then:
) , , , , , ,
( A A A A p1 p2 p3
p S a b c d
a , where S(...) is the 7-argument function found from solving the above equation, is a dependent variable.
2 .
3 3
6 3
2 3
(...) 1
3 2 1
3 2 3
1 2
1 2
3 2
2 2
1 )
1 , , , ,
(
p p p
p p p
p p
p p
p p
d c b a T
p
S COG
A A A Aa
For our case, the above formula for (pZ) can be rewritten as:
, max
2 , , 2
, , , ,
1 , , , , ,
, 2 ,
, 2 , , , ,
1 , , , , ,
max ,
, , , , ,
3 2 1
3 2 1
3 2 1
3 2 1
3 2 1
3 2 1 3
2 1 3 2 1
, ,
, , ,
*
*
*
*
*
*
pY pY pY
pX pX pX
Y Y Y
Y Y Y Y
Y Y Y Y
X X X
X X X X
X X X X
Y Y Y X X Z X
u a
T u d c b a T u wrt Integrate
u a
T u d
c b a T u wrt Integrate
v
p p p
p p p p A
A A A
p p p
p p p p A
A A A
p p p p p p p
subject to:
ForZXY (summation):
, ,, , , *, , *, , *, , * , , , * , * , * , ,
.*
*
*
*
*
*
3 2 1 1
3 2 1
3 2 1 3 2 1
u h a
T u v h a
T u wrt Integrate
v
Y Y Y Y Y
Y X X X X X
Y Y Y X X Z X
p p p p p p
p p p p p
p p p p p p p
For Z X Y (subtraction):
u h a
T u v h a
T u wrt Integrate
v
Y Y Y Y Y
Y X X X X X
Y Y Y X X Z X
p p p p p p
p p p p p
p p p p p p p
, , , , , ,
, , , , , ,
, , , , , ,
*
*
*
*
*
*
*
*
*
*
*
*
*
3 2 1 1
3 2 1
3 2 1 3 2
1
For Z XY (multiplication):
u h a
T u v h a
uT u wrt Integrate
v
Y Y Y Y Y
Y X X X X X
Y Y Y X X Z X
p p p p p p
p p p p p
p p p p p p p
, , , , , /
, , , , , 1 ,
,
, , , , , ,
*
*
*
*
*
*
*
*
*
*
*
*
*
3 2 1 1
3 2 1
3 2 1 3 2
1
For Z X /Y (division):
u h a
T vu h a
T u u wrt Integrate
v
Y Y Y Y Y
Y X X X X X
Y Y Y X X Z X
p p p p p p
p p p p p
p p p p p p p
, , , , , ,
, , , , , ,
, , , , , ,
*
*
*
*
*
*
*
*
*
*
*
*
*
3 2 1 1
3 2 1
3 2 1 3 2
1
Then:
), , , , , , ,
( X X X X X1 X2 X3
X A A A A p p p
p S a b c d
a
).
, , , , , ,
( Y Y Y Y Y1 Y2 Y3
X A A A A p p p
p S a b c d
a
subject to:
. ) ( )
(u p u du
w A Z
Z
The computation of (w)
BZ
for any given w in our case can be done using the algorithm described below.
For a value w find such values of parameters
apX,bpX,cpX,dpX,apY,bpY,cpY,dpY
that minimize the expression:
pY pY pY pY
pX pX pX pX
Y Y Y Y X X X X Z
Z Z Z
d c b a
d c b a
p p p p p p p p Z A
A A
A b c d u p a b c d a b c d u w
a T u wrt Integrate
, , ,
, , , ,
2
min
, , , , , , , , ,
1 , , , , ,
and maximize the expression:
pY pY pY pY
pX pX pX pX Y Y Y Y Y Y X X X X X X Z
d c b a
d c b p a
p p p p p p p p p p p
p b a c b d c b a c b d c
, , ,
, , ,
, min
, ,
, ,
,
The corresponding value for ( ) Z B w μ then would be:
, , , , ,
.)
( Z X X X X X X Y Y Y Y Y Y
Z p p p p p p p p p p p p p
B w b a c b d c b a c b d c
3. NOTE ON PROGRAM IMPLEMENTATION
All of the algorithms described above are implemented using an evolutionary optimization method. We use the DE algorithm [6,7] with constraints as one of the best one for numerical optimization.
Substitution of variables to ensure account of constraints for trapezoids (e.g. abcd) and effective DE optimization (almost the same domain of variation for all optimization variables) is done as follows:
Optimization variables: ti, i1,...,8. t1
apX
1 2
1 t t
t
bpX
2 3 1 2
1 t t t t
t
cpX
3 4 2 3 1 2
1 t t t t t t
t
dpX t5
apY
5 6
5 t t
t
bpY
6 7 5 6
5 t t t t
t
cpY
7 8 6 7 5 6
5 t t t t t t
t
dpY
4. CONCLUSION
We presented a set of algorithms that can be used to implement of arithmetic operations over Z-numbers. All underlying optimization is to be done by one of leading evolutionary optimization method, a version of DE with constraints, implemented by the author.
REFERENCES
1. Zadeh LA (2011) A Note on Z-numbers. Information Sciences 181, 2923-2932.
2. Zadeh LA (2011) The Concept of a Z-Number—A New Direction in Uncertain Computation (http://www.cs.berkeley.edu/ ~zadeh/presentations%202010 /IRI%202011- The%20concept%20of%20a%20Z-number-A%20New% 20Direction% 20Aug%203%20Las%
20Vegas.pdf).
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Information Sciences (accepted)
4. Williamson RC (1989) A thesis submitted for the degree of Doctor of Philosophy. Department of Electrical Engineering University of Queensland, August, 1989 (http://users.cecs.anu.edu.au/ ~williams/papers/ thesis300dpi.pdf).
5. Wang YM, Yang JB, Xu DL, Chin KS (2006) On the Centroids of Fuzzy Numbers. Fuzzy Sets and Systems 157, 919–926.
6. Storn R, Price K (1997) Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 11: 341–359.
doi:10.1023/A:1008202821328.
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