Current research Journal of Biological Sciences 4(2): 198-201, 2012
ISSN: 2041-0778
© Maxwell Scientific Organization, 2012
Submitted: November 21, 2011
Accepted: January 04, 2012
Published: March 10, 2012
On the Overflow Alert System Based on Fuzzy Models
R. Saneifard and H. Ahmadianrad
Department of Applied Mathematics, Urmia Branch, Islamic Azad University, Oroumieh, Iran
Abstract: An overflow alert system is a non structural measure for overflow mitigation. This article illustrates
an ordered impressionistic fuzzy analysis that has the capability to simulate the unknown relations between a
set of meteorological and hydrological parameters. In this article, the researchers first define ordered
impressionistic fuzzy sets and establish some results on them.
Key words: Defuzzification, fuzzy sets, initial universe set, overflow alert, soft set, universe set
Bishu, 1998; Zimmermann, 1991; Saneifard, 2009a). The
location important parameters are termed as weighted
indices. If the weighted indices are unity, then ordered
impressionistic fuzzy sets coincide with impressionistic
fuzzy sets. In this article we define ordered
impressionistic fuzzy sets and establish some results of
them.
INTRODUCTION
An efficient overflow alert system may significantly
improve public safety and mitigate damages caused by
inundation. Overflow forecasting is undoubtedly a
challenging field of operational hydrology, and over the
year lots of analytical works have accumulated in related
areas. Although conceptual models allow a deep
understanding of the hydrological processes, their
calibration requires the collection of a great amount of
information regarding the physical properties of the
watershed under study, which may be expensive and very
time consuming. Since overflow alert systems do not aim
at providing an explicit knowledge of the rainfall process,
black box models are largely used besides the traditional
physically-based ones. Over the last decade, fuzzy
technology has been increasingly used in hydrological
forecasting. A overflow alert system is a non structural
measure for overflow mitigation. Several parameters are
responsible for overflow related disasters. A quick
responding overflow alarm system is required for
effective overflow mitigation measures. Atmospheric
parameters that affect overflowing are rainfall, wind
speed, wind direction, relative humidity and surface
pressure. In our daily life, we frequently deal with vague
or imprecise information. Information available is
sometimes vague, sometimes inexact or sometimes
inefficient. Out of the several higher order fuzzy sets,
impressionistic fuzzy sets by (Abel and Sostak, 2007;
Sezer, 2008) has been found to be highly useful in dealing
with vagueness. impressionistic fuzzy sets is described by
two functions: a membership function and a non
membership function. The importance of membership
degree varies in different situations. The importance of
wind speed may be different for different locations. So we
need a generalization of impressionistic fuzzy sets, which
is called ordered impressionistic fuzzy sets (Kim and
PRELIMINARIES
We start this section with a simple definition of a
fuzzy set and then point out some definitions which are
used in this paper (Tanaka and Ishibuchi, 1992; Ma et al.,
2002; Tanaka et al., 1982; Saneifard, 2009b;
DeWnition: Let U be an initial universe set and E be a set
of parameters. Let P(U) denote the power set of U and A
dE. A pair (F , A) is called a soft set over U, where F is a
mapping given by F:A÷P(U).
DeWnition: Let the initial universe U= {L1, L2, L3, L4, L5}
be the five selected locations in West Azerbaijan, and E
= {P1, P2, P3, P4} be the atmospheric parameters, where
P1, P2, P3, and P4 are wind speed, wind direction, relative
humidity and surface pressure respectively. Suppose that
F(P1) = {L1, L2, L4}, F(P2) = {L3,L5}, F(P3) = {L1, L2,
L3}, and F(P4) = {L2, L3, L5}.
Each approximation has two parts:
C
C
A predicate p
An approximate value set
Consider F(P1), here predicate name is wind speed and
value set is{L1, L2, L4}.
Definition: Let U be an initial universe set and E be a set
of parameters. Let P(U) denote the set of all fuzzy sets of
U and AdE. A pair (F, A) is called a fuzzy soft set over U,
where F is a mapping given by F : A÷P(U).
Corresponding Author: R. Saneifard, Department of Applied Mathematics, Urmia Branch, Islamic Azad University, Oroumieh,
Iran
198
Curr. Res. J. Bio. Sci., 4(2): 198-201, 2012
Definition: (Yen et al., 1999). Let E be a fixed set and
AdE. Impressionistic fuzzy set in E is an object having
the form A = {(x, : A(x), <A(x))*x , E}, where the
function :A : E ÷[0,1] and < A : E÷ [0,1] define the
degree of membership and non membership respectively
of the element x to the set A. Also 0 # :A (x) + <A (x) # 1
and B A (x) = 1-:A (x) – <A (x) is called the in deterministic
part for x. Clearly 0 # B A (x) # 1.
C
C
The application of fuzzy modeling normally includes
three procedures, i.e., fuzzification, logic decision and
defuzzification. Fuzzification involves the identification
of the input variables and the control variables, the
division of both input and the control variables into
different domains, and choosing a membership and non
membership function. Logic decision involves process
design, and the determination of output which may retain
a fuzzy or crisp nature. If the output produced is of a
fuzzy nature it will require defuzzification which will
produce a crisp output. Interpretation of real world
situations may be based on either the output of the logic
decision or the defuzzified crisp output.
is possible only if t = u = 1. Therefore the weighted
indices of an ordered super impressionistic fuzzy set are
unity.
Definition: For two ordered impressionistic fuzzy sets Ap,q
and Br,s, Ap,q is said to dominate Br,s
if
t ,u , p , q
Tdk
}
)1
t ,u ,
⎫
x ∈ P⎬
⎭
is called impressionistic fuzzy set.
C
C
C
t ,u , p ,q
(S
t ,u , B p ,q
)
A p ,q =
{ x , ( µ ( x ) ) , (v ( x ) )
q
xi ∈ E
}
Br ,s =
{ x , ( µ ( x)) , ( µ ( x))
q
xi ∈ E
}
Definition: If Ap,q and Br,s are two ordered impressionistic
fuzzy sets of the fuzzy set E, then:
C
)
A p ,q ≥ Tdk
Similarity measures between ordered impressionistic
fuzzy sets: In this study, we introduce the similarity
measures of ordered super impressionistic fuzzy sets. Let
P(U) be the set of all ordered impressionistic fuzzy sets
of X. Let Ap,q , P(U) and Br,s , P(U) be two ordered
impressionistic fuzzy sets where:
ordered in deterministic part for x. Clearly 0 # π Ap ,q (x) #
) (
(S
The ordered super impressionistic fuzzy set St ,u clearly
dominates all over the ordered super impressionistic
fuzzy set of E.
where, the functions :pA : E÷[0,1] and <pA : E÷[0,1]
define the degree of membership and non membership
respectively of the element x to the set A. p and q are
called weighted indices of the set A. Also 0 #(: A(x))p
+(<A(x))q #1, π Ap ,q (x) = 1-(: A(x))p +(<A(x))q is called the
(
⎫
⎪
x ∈ E ⎬.
⎪⎭
Definition: Ordered impressionistic fuzzy set St,u of E is
said to be the ordered super impressionistic fuzzy set of
E if (:S (x))t = 1, (<S (x))u = 0 and π S t ,u ( x ) = 0, ∀x ∈ E . This
Definition: Let E be a fixed set and AdE. For p, q , N, an
ordered impressionistic fuzzy set in E is an object having
p
q
the form:
Ap ,q = x , ( µ A ( x )) , (ν A ( x )) x ∈ E
1
⎧
A1,1 = ⎨ x , µ A ( x ) , ν A ( x )
⎩
Ap ,q * Br ,s
⎧ x , ( µ ( x )) p . ( µ ( x )) r , (ν ( x )) q
⎪
A
B
A
= ⎨
s
q
s
⎪⎩ + (ν B ( x )) − (ν A ( x)) . (ν B ( x ))
⎫
⎪
x ∈ E ⎬.
⎪⎭
Definition: Let U be an initial universe set and E be a set
of parameters. Let Pp,q(U) denote the set of all ordered
impressionistic fuzzy sets of U and AdE. A pair (Fp,q , A)
is called an ordered impressionistic fuzzy set over U,
where Fp,q is a mapping given by Fp,q : A ÷ Pp,q (U).
METHODOLOGY AND FORMULATION
1. If p = q 1, then:
⎧ x , ( µ ( x )) p + ( µ ( x ) ) r −
⎪
A
B
A p ,q + Br ,s = ⎨
p
r
q
s
⎪⎩ ( µ A ( x )) . ( µ B ( x )) , (ν A ( x )) . (ν B ( x ))
C
Definition: (Ruoning, 1997). Let U be an initial universe
set and E be a set of parameters. Let P(U) denote the set
of all impressionistic fuzzy sets of U and AdE. A pair
(F, A) is called an impressionistic fuzzy set over U, where
F is a mapping given by F: A ÷ P(U).
{
A p ,q
⎫
⎧ x , max(( µ ( x )) p ,( µ ( x )) r ),
A
B
⎪
⎪
∪ Br ,s ⎨
x
∈
E
⎬.
q
r
⎪⎭
⎪⎩ min(( µ A ( x )) ,( µ B ( x )) )
Ap,qdBr,s iff œx0E& l(µA(x))p#(µB(x))r and (vA(x))q
$(µA(x))sm.
Ap,q = Br, s iff œx0E& l(µA(x))p = (µB(x))r and (vA(x))q
= (µA(x))sm
Ap,q = {(x, (vA(x))q, µA(x))p >*x0E}
Ap,q1Br,s = {#x, min((µA(x))p , µB(x))r ), max((vA(x)q,
(vB(x)s)$*x0E}
Let
Td k be
p
i
A
A
p
i
B
B
(1)
(2)
a mapping such that Td k : ( P(U )) 2 → [0,1], k = 1,2.
Definition: The similarity measure between two ordered
impressionistic fuzzy sets is defined as:
199
Curr. Res. J. Bio. Sci., 4(2): 198-201, 2012
(
)
Tdp1 ,q ,r ,s A p ,q , Br ,s = 1 −
1 n ⎡ ( p ,r )
∑ ⎢ M (i )
2n i = 1 ⎣
p+ r
2
+ N ( q ,s ) (i )
q+ s
2
w⎤
+ I ( p ,q ,r ,s) (i ) 4 ⎥
⎦
p ,q
Td2
(3)
where
(A
p ,q , B p ,q
p+ q ⎤
n ⎡
p
q
∑ ⎢ M p (i ) + N q (i ) + I ( p,q ) (i ) 2 ⎥
i = 1⎢⎣
⎥⎦
p+ q ⎤
n ⎡
p
q
( p ,q )
p
q
∑ ⎢ E (i ) + F (i ) + G
(i ) 2 ⎥
i = 1⎢⎣
⎥⎦
) = 1−
E p (i ) = ( µ A ( xi )) + ( µ B ( xi ))
p
M ( p ,r ) (i ) = ( µ A ( xi )) − ( µ B ( xi ))
p
N ( q ,s ) (i ) = (ν A ( xi )) − (ν B ( xi ))
q
F q (i ) = (v A ( xi )) + (v B ( xi ))
r
p
s
Remark: 0 ≤ Td
And n is the number of attributes of the system and w= p
+ q +r + s. If r = P and s = q, then the above formula
becomes:
T
(
1
−
2n
)
⎡
∑ ⎢⎣ M
p
p
q
(i ) + N q (i ) + I ( p ,q ) (i )
⎤
⎥
⎦
(A
p ,q
)
, Br ,s ≤ 1
)
(
(
) if and only if A p,q= Br,s, i.e.,
(: A(x)) = (: B(x)) and (< A(x))q = (< B(x))s for any xi , E
(4)
r
(
)
p ,q ,r , s
A p,q , Br , s = 0 and
Remark: If Td k
(
)
Td k p ,q ,t ,u Ap ,q , Ct ,u = 0, Ct ,u ∈ P(U ),
where
M p (i ) = ( µ A ( xi )) − ( µ B ( xi ))
p
N q (i ) = (v A ( xi )) − (v B ( xi ))
q
(
Similarity measure algorithm: Reliable overflow
prediction cannot be done by subjecting available data to
conventional methods of analysis. We therefore turn to
ordered impressionistic fuzzy sets and develop a simple
but effective model which has been designed in such a
way as to produce reliable output in the prediction of
overflow possibility. The inputs are basic parameters
related to overflow occurrence and fuzzy membership and
non membership degrees were assigned to each
parameters. The model processes the ordered
impressionistic fuzzy set constructed from collected data
and identifies the most overflow prone location.
q
Definition: Another form of similarity measure between
two ordered impressionistic fuzzy sets is defined as:
p ,q ,r ,s
(A
p ,q , Br ,s
) = 1−
p+r
q+s
w
n ⎡
⎤
∑ ⎢ M ( p ,r ) (i ) 2 + N ( q ,s) (i ) 2 + I ( p ,q ,r ,s) (i ) 4 ⎥
i =1 ⎣
⎦
,
p+ r
q+s
w
n ⎡
⎤
( p ,r )
(
,
)
(
,
,
,
)
q
s
p
q
r
s
∑⎢E
(i ) 2 + F
(i ) 2 + G
(i ) 4 ⎥
i =1 ⎣
⎦
C
C
C
(5)
Selection of a desired number of parameters (Pj)
Selection of a desired number of locations (Li)
Constructing ordered impressionistic fuzzy set:
⎧
⎛
⎞ ⎛
p j , ⎜ µ Li ( p j )⎟ , ⎜ ν
⎝
⎠ ⎝
p
,
q
⎩⎪
p
⎪
Lip ,q ⎨
where
E(p, r)(i) = (µA(xi))p+(µB(xi))r
F(q, s)(i) = (vA(xi))q+(µB(xi))s
G(p, q, r, s)(i) =
)
πA
p ,q
(x ) + π
i
Br , s
then
Td k r ,s,t ,u Br ,s , Ct ,u = 0 , for k = 1, 2.
p
I p,q (i ) = π A p ,q ( xi ) − π B p ,q ( xi )
Td2
)
Td k p ,q ,r , s A p,q , Br ,s = 1
p
i =1
p ,q ,r , s
k
(
Remark:
p+ q
2
q
Remark: Td k p ,q ,r ,s A p ,q , Br ,s = Td k p ,q ,r ,s , Br ,s , A p ,q
= A p ,q , B p , q = − 1
n
p
G p ,q (i ) = π A p ,q ( xi ) + π B p ,q ( xi )
I ( p ,q ,r ,s ) (i ) = π A p ,q ( xi ) − π B r ,s ( xi )
p ,q
d1
(6)
( x)
C
Calculating π L p ,q ( Pj )
C
Calculating Td k
C
If r = p and s = q, then the above formula becomes:
200
i
t ,u , p ,q
Find L for which:
(S
⎞
( p j )⎟
Lip ,q
⎠
i
t ,u , L p ,q
q
⎫⎪
pj ∈ E⎬
⎭⎪
) for k = 1, 2
Curr. Res. J. Bio. Sci., 4(2): 198-201, 2012
(
)
(
Td tk,u , p ,q St ,u , Li p,q = max i Td tk,u, p ,q St ,u , Li p,q
C
Kim, B. and R.R. Bishu, 1998. Evaluation of fuzzy linear
regression models by comparing membership
functions. Fuzzy Set. Syst., 100: 343-352.
Ma, M., M. Kandel and M. Friedman, 2002. A New
Approach for Defuzzification. Fuzzy Set. Syst. 128:
351-356.
Ruoning, X., 1997. S-curve regression model in fuzzy
environments. Fuzzy Set. Syst., 90: 317-326.
Saneifard, R., 2009a. Ranking L-R fuzzy numbers with
weighted averaging based on levels. Int. J. Indus.
Math., 2: 163-173.
Saneifard, R., 2009b. A method for defuzzification by
weighted distance. Int. J. Indus. Math., 3: 209-217.
Sezer, S., 2008. OIIF model of overflow alert. Fuzzy Set.
Syst., 24: 279-300.
Tanaka, H., S. Uejima and K. Asay, 1982. Linear
regression analysis with fuzzy model. IEEE Trans.
Syst. Man Cybernet., 12: 903-907.
Tanaka, H. and H. Ishibuchi, 1992. Possibilistic
Regression Analysis Based on Linear Programming.
In: Kacprzyk, J., (Ed.), Fuzzy Regression Analysis.
Omnitech Press, Heidelberg, pp: 47-60.
Yen, K.K., S. Ghosray and G. Roig, 1999. A Linear
Regression Model Using Triangular Fuzzy Number
Coefficient. Fuzzy Set. Syst., 106: 167-177.
Zimmermann, H.J., 1991. Fuzzy Sets Theory and its
Applications. Kluwer Academic Press, Dordrecht.
(8)
If Li is not unique, choose that one corresponding to
which π L ip,q ( Pj ) =
C
)
m
∑ π L ip,q ( Pj )
i =1
is the greatest
i
The optimal solution is L .
CONCLUSION
This study, it is hoped, may go a long way in
exploring the possibility of using fuzzy technology to
model real time overflow prediction. There are varieties
of uncertainties in rainfall and overflow prediction, and it
is difficult to treat these uncertainties by using traditional
deterministic methods. In this study it has been
demonstrated that impressionistic model has its potential
usage in overflow prediction. The impressionistic model
presented for overflow alarm system has furnished very
promising results. This model is applied for five selected
area of West Azerbaijan, Iran. Two local parameters were
also considered. A critical discussion on the results of
proposed model has been conducted. It provides a new
way that helps disaster management studies to cope with
fatal and rapid changes in highly sensitive parameters.
REFERENCES
Abel, M. and A. Sostak, 2007. A fuzzy soft flood alarm.
Comp. Stat. Data Anal., 42: 47-72.
201