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Vol. 1 Issue IV, Nov ember 2013 ISSN: 2321-9653
INTERNATIONAL JOURNAL FOR RESEARCH IN AP PLIED SCIENCE AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T)
Evaluating Failure of a Refrigeration cycle using Triangular Intuitionstic Fuzzy Approach Neeraj Lata Dept.of Mathematics, TMU, Moradabad, (U.P.) sirohimaths@gmail.com
Abstract: In real life systems, the information may be inaccurate or might have linguistic representation. In such cases the estimation of precise values of probability becomes very difficult. In order to handle this situation, triangular fuzzy approach is used to evaluate the failure rate status. In this paper we introduced triangular fuzzy fault tree analysis for evaluating failure range of the refrigeration cycle system.
Key Words: triangular Intuitionstic fuzzy approach, fuzzy fault tree, failure rate, refrigeration cycle etc. real number as a membership grade and in such cases it may be useful to identify meaningful lower and upper bounds for
1. INTRODUCTION The theory of fuzzy sets (FSs), proposed by Zadeh (1965) [6]
the membership grade. In 1986, Atanassov [7] introduced
has gained successful applications in various fields. However, Intuitionistic fuzzy sets (IFSs) which have been found to be the membership function of the fuzzy set is a single value very useful to deal with uncertainty information. between zero and one, which combines the favouring evidence and the opposing evidence. Due to fuzzy boundaries,
The concept of the IFSs is a generalization of that of the FSs.
this single value for the membership grade is the result of the
IFS's are being studied and used in different fields of science.
combined effect of evidences in favour and against the
Among the research works on these sets we can mention
inclusion of the element in the set the utility of the application
Atanassov [2,3,4]; Atanassov and Gargov [1]; Szmidt and
of fuzzy sets depends on the capability of the user to construct
Kacprzk (2001) [7] proposed the definition of Intuitionistic
appropriate membership functions, which are often very
Fuzzy numbers (IFN) and studied the perturbation of IFN.
precise. In many contexts it is difficult to assign a particular
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Vol. 1 Issue IV, Nov ember 2013 ISSN: 2321-9653
INTERNATIONAL JOURNAL FOR RESEARCH IN AP PLIED SCIENCE AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T) ~ A ( xi , A ( xi )) : xi X
2. FUZZY SETS
A set is a well-defined collection of objects. It has a sharp
Where A~ ( x ) 0, 1
boundary to distinguish which element of the universe of discourse belongs to the set. In real life applications situations, it is not possible to distinct these elements by such a sharp layer due to the uncertainty involved. A fuzzy set is a set that consists of the elements having varying degrees of belongingness in the sets. So these situations may be better explained by the fuzzy sets, the set which contains all the elements of the universe but with different degrees of
2.1 Definition of Intuitionistic Fuzzy Sets (IFSs):Fuzzy set theory was first introduced by Zadeh in 1965 [6]. Let X be universe of discourse defined by X = {x1, x2,...,xn}. The grade of membership of an element xi ∈ X in a fuzzy set is represented by real value between 0 and 1. It does indicate
the evidence for xi ∈X, but does not indicate the evidence
against xi ∈X. Atanassov in 1984 [4] presented the concept of IFS, and pointed out that this single value combines the
membership. A crisp set A may be defined over a universe X and may
evidence for xi ∈X and the evidence against xi ∈ X. An IFS
( ) and a
in X is characterized by a membership function be characterized by its characteristic function
A as
A ( x, A ( x)) Where
Intuitionistic fuzzy set
A : X {0,1} is defined by if x A if x A
function
called
membership
( ) >:
∈ }
→ [0, 1]
define
respectively,
Where the functions
function
A~ : X [0, 1] to characterize a fuzzy set defined over the universe X as follows.
of E is an object having the form
( ),
= {< ,
In analogy to the characteristic function, we can a
( ).
2.2 Intuitionistic Fuzzy Set: - Let E be a fixed set. An
1 A ( x) 0
define
non membership function
:
:
→ [0 , 1] the
and
degree
of
membership and the degree of non-membership of the element ∈
to the set A, which is a subset of E and for every
∈ , 0≤
( )+
( ) ≤ 1.
When the universe of discourse E is discrete, an IFS written as
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can be
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Vol. 1 Issue IV, Nov ember 2013 ISSN: 2321-9653
INTERNATIONAL JOURNAL FOR RESEARCH IN AP PLIED SCIENCE AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T) =
[
( ), 1 −
( )]/ , ∀
Fig.2 Membership and non-membership functions
∈
of TIFN 2.4 Arithmetic operations on IFNs:The arithmetic operations denoted generally by *, of two IFNs is a mapping of an input subset of R x R (with elements =( ,
)) onto an output subset of R (with elements
denoted by y). Let
resultant of operations then:
be two IFNs, and (
∗
) the
Fig.1 Membership and non-membership functions of ∗
2.3 Triangular Intuitionistic Fuzzy Numbers (TIFN):The TIFN
is an Intuitionistic Fuzzy number ( ) is an
=
Intuitionistic Fuzzy set in R with five real numbers ( ,
,
, ′,
′′ )
triangular functions
( )=
ℎ( ′≤ ,
0,
,
≤ ≤
≤
ℎ
≤
≤
′′
) and two
With
2 2
≤
≤
2 2
( ) ∨ = ∧ =
∗ ∗
, [ ( )∧ [ ( )∨
∗
( )], ( )]
,∀ ,
( ) =∧ =
∗
∗
( ) =∨ =
, ∗ [
∈ [
( )∧
( )∨
3. NUMERICAL COMPUTATIONS
Here we have a simple fault tree structure of failure of refrigeration cycle system. The failure of the system depends ( )=
on different factors like –electric supply ,hot cooling factor ,condenser failure , compressor failure etc .There are two major factors one is failure power supply and the second one be heat capacity exceeds. For both of these there are two sub factors .The following notation has been used to get failures
′′
′ ≤ the ≤ failure of refrigeration system =′ , Represents
, ≤ ≤ ′′ = Represents the failure of electric supply 1, ℎ
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INTERNATIONAL JOURNAL FOR RESEARCH IN AP PLIED SCIENCE AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T) = Represents the failure of cooling system.
= (0.03,0.03,0.05; 0.010.03,0.07)
= (0.02,0.04,0.06; 0.03,0.04,0.05)
= Represents the failure of the condenser
= Represents the failure of the compressor
= Represents the excess of commodity
= (0.03,0.04,0.07; 0.02,0.04,0.05)
Step 1
Failure of electric supply =
= Represents the failure of refrigerant.
×
= (0.0006,0.0009,0.002; 0.0001,0.0009,0.0035)
Step 2 failure of cooling system=1− (1−
)(1−
)
= 1−(0.98,0.96,0.94, ; 0.97,0.96,0.95) (0.97,0.96,0.93;0.98,0.96,0.95)
Failure of refrigeration system
=1−(0.9506,0.9216,0.8742; 0.9506,0.9216,0.9025)
AND
= (0.0494,0.0784,0.1258; 0.0494,0.0784,0.0975) Now failure of the refrigeration system
No Electric Supply
Failure of cooling system
= 1− (1−
4. Conclusion:AND
OR
)(1−
)
= (0.05,0.08,0.12; 0.04,0.08,0.10)
In this present paper we have discussed failure rate of a
refrigeration cycle using triangular Intuitionistic fuzzy sets. Intuitionistic fuzzy fault tree analysis is efficient and simple to
Failure of condenser
Failure of compressor
Failure of the failure of system of all fields. A new TIFN Excess of computingrefrigerant commodity fault tree analysis model is proposed in this paper that modifies the fuzzy set arithmetic operations for implementing
Fig.3 Fuzzy fault tree of refrigerant system Let the reliability of events are = (0.02,0.03,0.04; 0.01,0.03,0.05)
fault tree analysis. Results of TIFN fault tree are more flexible than the fuzzy fault tree analysis because it have more sharp boundary.
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INTERNATIONAL JOURNAL FOR RESEARCH IN AP PLIED SCIENCE AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T) REFERENCES
1. Atanassov K.T., Gargov G., Interval-valued Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31 (3) (1989) 343349.
2. Atanassov K.T., Intuitionistic Fuzzy Sets, PhysicaVerlag, Heidelberg, f J/Kevi York, (1999).
3. Atanassov K.T., Intuitionistic fuzzy sets. Fuzzy Sets and Systems, Vol 20, No. 1, (1986) pp. 87-96.
4. Atanassov K.T., More on Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 33 (1) (1989) 37-46.
5. Atanassov K.T., Two theorems for Intuitionistic fuzzy sets., Fuzzy Sets and Systems, (2000) 110: 267-269.
6. Zadeh L.A., Fuzzy sets. Information Control, 8 (1965), pp. 338-353.
7. Szmidt E., Kacprzyk J., Entropy for Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 118 (2001) 467-477.
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