8.3. Формы функции принадлежности лингвистических

2’2012
.
8.3.
1.
:
.,
:
T
x = fT x 1, x 2 , x 3 = 0,
,
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,
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-
,
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-
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,
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gaus
-
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[4].
,
(x)
f ( m, )
1,
x = x2 ;
x x4
xB x4
x2 < x < x3 ;
0,
x
x
e
f ( m, )
e
.
.
x x1
x A x1
x1 < x < x 2 ;
1,
x2
x x4
xB x4
x3 < x < x 4 ;
0,
x
( x m )2
a 2
x3 ;
x4.
.
-
x
.
[3]
,
,
[ 0 ,1 ]
(x)
R
.
,
x1 ;
x m
lapl ( x )
.
.
x3 .
:
,
,
x1 < x < x 2 ;
:
-
,
x1 ;
x x1
x A x1
–
x = fT x 1, x 2 , x 3 , x 4 = 0,
T
,
x
13
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2.
:
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,
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,
3.
(
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[0, 1].
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4.
:
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,
2.
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,
1.
(x)
f ( x 1 , x1 , x 3 , x 4 )
-
:
1
f1 ( x )
x2
x1
x
x
x3 ;
x2 ;
f2 ( x )
0
x3
x
x4 ;
[2].
[3, 5, 1, 8, 9].
0
f1(x) –
;
.
(1)
.
f2(x) –
.
.
-
.
1.
:
(1)
x
x1
(x)
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,
-
x2
x2
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,
x1
-
,
x2 .
x
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.
-
–
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,
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.
,
,
/
,
.
(x2 = x3).
2.
,
.
x1
-
,
x
x2 .
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-
,
,
/
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3.
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3.
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. 1.
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.1
.
f( x)
1,
( sup( ( x ))
(1).
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1 ).
-
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4.
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1
2’2012
. 2).
,
,
circ=lambda a,
b: lambda x: abs(1 – ((x – a) / abs(b – a))**2)**0,5.
,
-
.
:
quad=lambda a, b: lambda x: 1 – ((x – a) / abs(b –
– a))**2.
[7]
-
,
,
.
-
,
,
,
lin
:
(x)
c
x
a
f ( a, b,c , x )
b
;
b
(3)
c
2
. 2.
(
,
)
quad
(x)
f ( a, b,c, x )
a–
,
1
–
(4)
) –
;
b–
(1),
,
,
,
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b
b
,
(
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(
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–
x
a
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-
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(
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–
),
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-
–
,
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.
,
lin
c>1
:
( x1 , x 2 )
f ( x , x1 , x 2 )
x
x2
x1
,
x1
(2)
, c < 1,
«
-
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,
-
»
.
c=0
.
-
,
;
,
-
(
–
):
lins = lambda a, b: lambda x: ((x – b) / (a – b)).
,
,
[200, 350]
, 500
,
lins(350, 500).
.
.
,
:
,
.
(
–
),
.
-
,
,
50
,
:
lins(200, 50).
[3]:
. 3.
(
)
2
c =
,
.3
-
.
,
-
,
.
3.
.
.
-
–
–
.
.
-
,
,
,
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-
. 5.
4.
.
-
,
:
(x)
.
f ( x , x 1 , x 2 , x 3 , x 4 , c l , c r , fl , fr )
x2
x1
x3
1
fl ( x 2 , x 1 , c l , x )
f r ( x 3 , x 4 , cr , x )
x
x
x
x3 ;
x2 ;
x4;
5.
(5)
0,5-
.
0
,
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,
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-
,
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,
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,
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-
.
0,5.
.
,
,
. 4.
,
:
c
lin
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-
,
.
.
,
,
x =
lin
x
a
b
b
log c b 0.5
a b
,
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,
:
. 4).
-
.
quad
1.
–
x =1
-
,
.
x
a
(6),
b
b
2
log
1
:
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x1
x4.
-
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,
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-
(8)
;
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-
2 0.5
0,5-
.
2.
c b
a b
-
x2
-
x3.
(7)
;
(5)
-
(
(6)
a b
-
,
x b
= 0,5 ;
x =
a b
c = log x b 0.5 ;
,
.
-
.
5
:
1,0,
0,0
-
0,5.
3
2’2012
2
2
x a
;
ln0,5
=
(10)
2
x
gauss
a ln0,5
2
c a
x =e
;
,
(11)
[3]:
:
x
lapl
x =e
a ln0,5
c a
(12)
;
:
cauc
1
x =
1+
. 5.
.
x
gauss
x =e
a
c2
:
,
2
;
(13)
:
log
x =
ln3
;
a
a
2
2
1+ e
(9)
x
c
a
a
2
.
(14)
.6
:
.
. 6.
, (e)
4
x
c
, (f)
. (a)
, (g)
, (b)
, (h)
, (c)
, (j)
, (d)
-
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:
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.,
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8.3. MEMBERSHIP FUNCTIONS OF
THE ECONOMIC FACTORS'S
LINGUISTIC VARIABLES
-
M.V. Koroteev, Department of Information Systems in
Economics Lab. Assistant
.
,
.
(
):
:
Volgograd state technical university (VSTU),
Volgograd, Russia
(7);
(8);
:
This article is viewing new method of estimating the risk
level of investment projects based on a well-known
mathematical approaches in addition with fuzzy logic in
order to formalize economic uncertainty. This method can
be widely used in making manager's decision, projects
ranking and estimation.
(11);
(12);
(
,
«
-
.
.,
(13);
(14).
)
1. Collan M., Fuller R., Mezei J. A fuzzy pay-off method for real
option variation // Journal of applied mathematics and decision sciences. 2009. Vol. 2009.
2. Dubois D., Prade H. Fuzzy numbers: an overview // Analysis
of fuzzy information. 1987. Vol. 1. Pp. 3-39.
3. Mitaim S., Kosko B. The shape of fuzzy sets in adaptive function approximation // IEEE Transactions on fuzzy systems.
2001. Vol. 9. 4.
4. Sugeno M., Yasukawa T. A fuzzy-logic-based approach to
qualitative modeling // IEEE Transactions on fuzzy systems.
1993. Vol. 1. 1.
5. Rutkovskaya D., Pilinsky M., Rutkovsky M. Neural networks,
genetic algorithms and fuzzy systems: Translator: I.D. Rudinsky. Telecom, 2004. 452 p.
6. Weisstein E.W. Wigner's semicircle law. From MathWorld-A
Wolfram Web Resource. http://mathworld.wolfram.com/ WignersSemicircleLaw.html
7. Zadeh L.A. The concept of a linguistic variable and its application to approximate reasoning // Information sciences.
1975. Vol. 8. Pp. 199-249.
8. Zadeh L.A. Fuzzy sets // Information and control. 1965. Vol.
8. Pp. 338-353.
9. Zaychenko Y.P., Zayec I.O., Kamocky O.V., Pavluk O.V. Study of
different types of membership of function parameters of fuzzy
forecasting models in fuzzy arguments group method.
;
;
;
;
;
;
Literature
1. M. Sugeno, T. Yasukawa. A fuzzy-logic-based approach to
qualitative modeling. // IEEE Transactions on fuzzy systems.
Vol. 1, 1. 1993.
2. Dubois, D. and Prade, H., «Fuzzy Numbers: An Overview», //
Analysis of Fuzzy Information 1:3-39, CRCPress, BocaRaton,
1987.
3. Rutkovskaya D., Pilinsky M., Rutkovsky M. Neural networks,
genetic algorithms and fuzzy systems: Translator: I.D. Rudinsky. Telecom, 2004. 452 p ISBN 5-93517-103-1.
4. S. Mitaim, B. Kosko. The shape of fuzzy sets in adaptive function
approximation. // IEEE Transactions on fuzzy systems. Vol. 9,
4. 2001.
5. Weisstein, Eric W. «Wigner's Semicircle Law». From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/ WignersSemicircleLaw.html
6. L.A. Zadeh. The concept of a linguistic variable and its application to approximate reasoning. // Information sciences 8, pp
199-249. 1975.
7. Y.P. Zaychenko, I.O. Zayec, O.V. Kamocky, O.V. Pavluk. Study
of different types of membership of function parameters of fuzzy
forecasting models in fuzzy arguments group method.
8. L.A. Zadeh. Fuzzy sets. // Information and control, 8(3), pp.
338-353. 1965.
9. M. Collan, R. Fuller, J. Mezei. A fuzzy pay-off method for real
option variation. // Journal of applied mathematics and decision sciences, Volume 2009.
Keywords
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;
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;
Investment project; uncertainty; risk; fuzzy logic; membership function; fuzzy arithmetic; mathematical modeling;
net present value; internal rate of return; fuzzy relation.
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