International Journal of Applied Science and Engineering
2006. 4, 1: 71-82
Fuzzy Reliability of a Marine Power Plant Using Interval
Valued Vague Sets
Amit Kumara, Shiv Prasad Yadava*, and Surendra Kumarb
a
Department of Mathematics,
Indian Institute of Technology Roorkee, Roorkee, India
b
Department of Electrical Engineering,
Indian Institute of Technology Roorkee, Roorkee, India
Abstract: In conventional reliability analysis, the failure probabilities of the components of a
system are treated
as exact values. It is often difficult to obtain data for failure probabilities
under changing environmental conditions. Hence fuzzy sets are used to analyze the fuzzy system
reliability, where a fuzzy number represents the reliability of each component. Chen [14] analyzed the fuzzy system reliability using vague set theory. The values of the membership and
non-membership of an element, in a vague set, are represented by a real number in [0, 1]. A specialist is always uncertain about the values of the membership and non-membership of an element in a set. Hence, it is better to represent the values of the membership and non-membership
of an element in a set by intervals of possible real numbers instead of real numbers. In this paper
a new method has been developed for analyzing the fuzzy system reliability of a series and parallel system using interval valued trapezoidal vague sets, where the reliability of each component
of each system is represented by an interval valued trapezoidal vague set defined in the universe
of discourse [0, 1].The developed method has been used to analyze the fuzzy reliability of a marine power plant. The major advantage of the proposed approach (the concept of interval valued
vague sets) over the existing approaches [7, 8, 14] is that the proposed approach separates the
positive and negative evidence for the membership of an element in a set. Also in the proposed
approach the values of the membership and non- membership of an element in a set are intervals
instead of a single real number.
Keywords: fuzzy system reliability; vague set theory; fuzzy fault tree; interval valued vague
sets.
1. Introduction
One of the important engineering tasks in
design and development of a technical system
is reliable engineering. It is well known that
the conventional reliability analysis, using the
probabilities, has been found to be inadequate
to handle uncertainty of failure data and modeling. To overcome this problem, the concept
*
of fuzzy approach has been used in the
evaluation of the reliability of a system.
Fuzzy set theory was first introduced by
Zadeh [1] in 1965. Singer [2] presented a
fuzzy set approach for fault tree and reliability
analysis. Cai et al. [3, 4, 5] gave a different
insight by introducing the possibility assumption and fuzzy state assumption to replace the
probability and binary state assumptions.
Corresponding author; e-mail:[email protected]
Accepted for Publication: October 31, 2005
© 2006 Chaoyang University of Technology, ISSN 1727-2394
Int. J. Appl. Sci. Eng., 2006. 4, 1
71
Amit Kumar, Shiv Prasad Yadav, and Surendra Kumar
Mon et al. [7] presented a method for the
fuzzy system reliability analysis for components with different membership functions via
non-linear programming techniques. Chen [8]
presented a method for fuzzy system reliability analysis using fuzzy number arithmetic
operations. The collection of papers by Onisawa and Kacprzyk [9] presents many other
different approaches for the fuzzy reliability.
Cai [11] presented an introduction to system
failure engineering.
Chen [14] presented a new method for analyzing the fuzzy system reliability based on
vague sets.
In this paper, the concept of vague sets is
extended by idea of interval valued vague sets.
Also we have introduced: some definitions
related to interval valued vague set, definition
of interval valued trapezoidal vague set and
arithmetic operations between two interval
valued trapezoidal vague sets. Further, a new
method has been developed for analyzing the
fuzzy system reliability of a series and parallel system using interval valued trapezoidal
vague sets, where the reliability of each component of each system is represented by an
interval valued trapezoidal vague set defined
in the universe of discourse [0,1].The developed method has been used to analyze the
fuzzy reliability of a marine power plant. The
proposed method can model and analyze the
fuzzy system reliability in a more flexible and
intelligent manner in comparison to the
method given by Chen [14].
This paper is organized as follows. Sections
2, presents some definitions. Section 3, presents the arithmetic operations between two
interval valued trapezoidal vague sets. Section
4, presents the fuzzy reliability calculations of
a series and parallel system using interval
valued trapezoidal vague sets. Section 5, presents the case study of the system. Section 6,
presents the algorithm for analyzing fuzzy
reliability. Section 7 presents the assumed
data. Sections 8 presents the results .The conclusions are discussed in section 9.
2. Some definitions
In this section, some definitions are reviewed and some other definitions are introduced.
2.1. Definitions reviewed
We briefly review some definitions [6, 12,
14] here.
2.1.1 Definition
An interval valued fuzzy set F(over a basic set X ) is specified by a function
TF~ : X  D([0,1]), where D([0,1]) is the set
of all intervals within [0, 1], i.e. for all
x X , TF~ ( x) is an interval [ 1 , 2 ],
0 1 2 1.
2.1.2 Definition
~
A vague set V , in a basic set X , is characterized by a truth membership function
tV~ , tV~ : X  [0,1], and a false membership function f V~ , f V~ : X  [0, 1]. If the generic element of X i
sde
not
e
dby‘xi ’t
he
n
the lower bound on the membership grade of
xi derived from evidence for xi is denoted
by tV~ ( xi ) and the lower bound on the negation of xi is denoted by f V~ ( xi ) , tV~ ( xi ) and
f V~ ( xi ) both associate a real number in the
interval [0,1] with each point in X ,where
~
tV~ ( xi ) f V~ ( xi ) 1. A vague set V in the
universe of discourse X is shown in Fig. 1.
~
When X is continuous, a vague set V can
be written as
~
tV~ (xi ),1 fV~ (xi )/ xi , xi X
V 
X
~
When X is discrete a vague set V can be
written as
~ n
V 
tV~ ( xi ),1 fV~ ( xi )
/ xi , xi X .
i
1
72
Int. J . Appl. Sci. Eng., 2006. 4, 1
Fuzzy Reliability of a Marine Power Plant
Using Interval Valued Vague Sets
tV~ ( X ),1 f V~ ( X )
1  f V~ ( X )
1  f V~ ( x i )
t V( X )
t V~ ( x i )
0
X
xi
Figure 1. A vague set.
2.1.3 Definition
Let A = [a1, a2] and B = [b1, b2] be two arbitrary intervals then the minimum of A and B is
repr
e
s
e
nt
e
dby“
MI
N[
A,B]
”a
ndi
sde
f
i
ne
d
by MIN ([a1, a2]; [b1, b2]) = [min (a1, b1), min
(a2, b2)]
2.1.4 Definition
The complement of an interval A = [a1, a2]
is denoted by A and is defined by
A = [1-a2, 1-a1].
2.2. Definitions introduced
The definition of interval valued vague set
and definitions related to interval valued
vague set are introduced here.
2.2.1 Definition
~
An interval valued vague set V over a basic set X is defined as an object of the form
~
V [ xi ;TV~ ( xi );1 FV~ ( xi )] 
, xi X
where
TV~ : X  D[0,1] and FV~ : X  D[0,1]
a
r
ec
a
l
l
e
d“
Tr
ut
hme
mbe
r
s
hi
pf
unc
t
i
on”a
nd
“
Fa
l
s
eme
mbe
rs
hi
pf
unc
t
i
on”r
e
s
pe
c
t
i
ve
l
ya
nd
where D ([0,1]) is the set of all intervals
within [0, 1].
2.2.2 Definition
~
An interval valued vague set V is said to be
convex if and only if x1 and x 2 X
TV~ 
x1 
1 
x 2 Min
TV~ 
x1 
, TV~ ( x 2 )

1 FV~ 
x1 
1 
x2 Min
1 FV~ 
x1 
,1 FV~ 
x2 
where [0,1]
2.2.3 Definition
An interval valued vague number is an interval valued vague subset in the universe of
discourse X that is both convex and normal.
2.2.4 Definition
An interval valued trapezoidal vague set
~
V (over a basic set X ) can be represented by
~
V  
( x , y , z , w ); [ 1 , 2 ]; [1 , 2 ] 

where
0  1  2 1 2 1
as shown in Fig. 2
Int. J. Appl. Sci. Eng., 2006. 4, 1
73
Amit Kumar, Shiv Prasad Yadav, and Surendra Kumar
defined by
TV~ ( X ),1 FV~ ( X )
~
V1 
( x1 , y1 , z1 , w1 );[ 11 , 12 ];[11 ,12 ]
~
V2 
( x 2 , y 2 , z 2 , w2 );[ 21 , 22 ];[21 ,22 ]

1
2
1
1 FV~ ( X )
2
1
TV~ ( X )
where
0 11 21 12 22 11 21 12 22 1
The
arithmetic
operations
~
~
V1 and V2 are defined as follows
X
0 x
y
z
w
Figure 2. Interval valued trapezoidal vague set
~
V.
3. Arithmetic operations between two interval valued trapezoidal vague sets
Certain arithmetic operations between two
interval valued trapezoidal vague sets are developed here.
Let us consider two interval valued trape~
~
zoidal vague sets, V1 and V2 , as shown in Fig. 3
(x1 x2 , y1 y2 , z1 z 2 , w1 w2 );

~ ~
V1 V2 
[min(11, 21 ), min(12 , 22 )]; 




[min(11,21 ), min(12 ,22 )]


(x1 w2 , y1 z2 , z1 y2 , w1 x2 );

~ ~

[min(11, 21), min(12 , 22 )]; 
( V1 ΘV2 ) 
>


[min(11,21), min(12 ,22 )]


( x1 x2 , y1 y2 , z1 z2 , w1 w2 );

~ ~

V1 V2 
[min(11, 21 ), min(12 , 22 )]; 



[min(11,21 ), min(12 ,22 )]


~
V1
TV~1 ( X ),1 FV~1 ( X )
TV~2 ( X ),1 FV~2 ( X )
4.
1
22
12
22
12
21
11
X
x1 y1
z1
w1 x2 y2
z2 w2
Figure 3. Interval valued trapezoidal vague sets
~
~
V1 and V2 .
74
( x1 / w 2 , y 1 / z 2 , z 1 / y 2 , w1 / x 2 ); 

~

V 2 
[min( 11 , 21 ), min( 12 , 22 )]; 



[min( 11 ,21 ), min(12 , 22 )]


Fuzzy reliability calculations of a series
and parallel system using interval valued trapezoidal vague sets
In this section, a new method has been developed for the fuzzy reliability calculation of
a series and parallel system using interval
valued trapezoidal vague sets, where the reliability of each component of each system is
represented by an interval valued trapezoidal
vague set defined on the universe of discourse
[0, 1].
21
11
0
between
Int. J . Appl. Sci. Eng., 2006. 4, 1
4.1. Series system
Let us consider a series system consisting
of‘
n’c
ompone
nt
sa
ss
howni
nFi
g
.4. The
~
fuzzy reliability R of the series system
shown below can be evaluated as follows:
Fuzzy Reliability of a Marine Power Plant
Using Interval Valued Vague Sets
~
Rn
~
R2
~
R1
Figure 4. Series system.
n
RR
i
i
1
n
n
n
n
i
1
i
1
i
1
i
1
tors G1 and G2 one located at the stern and the
other at bow. Each generator is connected to
its respective micro switch board-1 and micro
switch board-2. The distributive switch board
receives the supply from the switch boards
through cables C1 and C2 and respective junction boxes D and E .The two micro switch
boards are interconnected through a long cable C3 and the junction boxes A and B. The
schematic diagram is shown in Fig. 6.
[( x i , y i , z i wi );
n
n
n
n
i
1
i
1
i
1
i
1
~
R1
[min(i1), min( i 2 )];[min(i1), min(i 2 )]] 
~
R2
where
R
[( x i , y i , z i , wi );[ i1, i 2 ];[i1,i 2 ] 
i
~
Rn
represents the reliability of the ith component.
Figure 5. Parallel system.
4.2. Parallel system
Let us consider a parallel system consisting
of‘
n’c
omp
one
nt
sa
ss
howni
nFi
g
.5. The
~
fuzzy reliability R of the parallel system can
be evaluated as follows:
G1
+
G2
+
n
~
~
R 1 Θ (1 Θ Ri )
i
1
n
n
i
1
i
1
[(1 (1 x i ),1 (1 y i ),
n
i
1
A
D
C3
B
E
n
[min( i1 ), min( i 2 )];
i
1
n
MSB −2
n
1 (1 z i ),1 (1 w i ));
i
1
n
MSB−1
i
1
n
[min( ' i1 ), min( ' i 2 )]] 
i
1
i
1
5. Case Study
A marine power plant [13] has two genera-
C1
C2
DSB
Figure 6. Marine power plant.
Let us assume that basic components subjected to failure are
(a) Generators G1 and G2.
(b) Micro switch board-1 (MSB-1) and Micro
Int. J. Appl. Sci. Eng., 2006. 4, 1
75
Amit Kumar, Shiv Prasad Yadav, and Surendra Kumar
switch board-2 (MSB-2).
(c) Interconnecting cable C3 and junction
boxes A and B, all are treated as one unit.
(d) Junction boxes D and E.
(e) Distributive switch board (DSB).
5.1 Notations
~
V1 
x1 , y1 , z1 , t1 
;[11, 12 ];['11 , '12 ]
,
represents the unreliability of generator G1.
~
V2 
x2 , y2 , z2 , t2 ;[21, 22];['21 , '22 ],
represents the unreliability of generator G2.
~
V3 
x3 , y3 , z 3 , t 3 
; [ 31 , 32 ];[ '31 , '32 ],
represents the unreliability of micro switch
board-1.
represents the reliability of the event that there
is no power supply to micro switch board-1.
~
F3 
X 3 , Y3 , Z 3 , T3 
; [31 ,32 ]; [' 31 ,' 32 ]

represents the reliability of the event that there
is no power supply to the junction box D.
~
F4 
X 4 , Y4 , Z 4 , T4 
; [41 ,42 ]; [' 41 ,' 42 ]

represents the reliability of the event that there
is no power supply from the junction box D.
~
F5 
X 5 , Y5 , Z 5 , T5 
; [51 ,52 ]; [' 51 ,' 52 ]

represents the reliability of the event that there
is no power supply through the junction boxes
D and E.
~
X 6 , Y6 , Z 6 , T6 
F6 
; [61 ,62 ]; [' 61 ,' 62 ]
~
V4 
x 4 , y 4 , z 4 , t 4 
; [ 41 , 42 ];[ ' 41 , ' 42 ],
represents the unreliability of micro switch
board-2.
represents the reliability of the event that there
is no power supply to micro switch board-2.
~
F7 
X 7 , Y7 , Z 7 , T7 
; [71 ,72 ]; [' 71 ,' 72 ]

~
x5 , y5 , z5 , t5 
V5 
;[51 , 52 ];['51 , '52 ],
represents the reliability of the event that there
is no power supply to the junction box E.
represents the unreliability of the junction
boxes A and B.
~
X 8 , Y8 , Z 8 , T8 
F8 
;[81 ,82 ];['81 ,'82 ]
~
x6 , y6 , z6 , t6 
V6 
;[61 , 62 ];[' 61 , ' 62 ],
represents the unreliability of the junction box
D.
~
V7 
x7 , y 7 , z 7 , t 7 
; [ 71 , 72 ]; [ ' 71 , ' 72 ]
,
represents unreliability of the junction box E.
~
x8 , y8 , z8 , t8 
V8 
; [ 81 , 82 ]; [ '81 , '82 ],
represents the unreliability of the distributive
switch board.
~
X 1 , Y1 , Z 1 , T1 
F1 
; [11 ,12 ]; ['11 ,'12 ]
,
represents the reliability of the event that there
is no power supply through the junction boxes
A and B.
~
X 2 , Y2 , Z 2 , T 2 
F2 
; [21 ,22 ]; [' 21 ,' 22 ]

76
Int. J . Appl. Sci. Eng., 2006. 4, 1
represents the reliability of the event that there
is no power supply from the junction box E.
~
X 9 ,Y9 , Z9 ,T9 
F9 
;[91,92 ];['91 ,'92 ]
represents the reliability of the event that no
power is
coming to distributive switch
board.
~
F , represents the fuzzy unreliability of the
system.
~
R , represents the fuzzy reliability of the system.
5.2. Fault tree
The fault tree for the marine power plant
(Fig. 6) is shown in Fig. 7.
Fuzzy Reliability of a Marine Power Plant
Using Interval Valued Vague Sets
Total Failure of
the System
No power
coming to D.S.B
D.S.B is burnt
No power from
junction box D
No power from
junction box E
No power to
junction box D
No power to
junction box E
D is burnt
E is burnt
M.S.B-1 is
burnt
M.S.B-2 is
burnt
No power to
M.S.B-1
No power to
M.S.B-2
G1 is burnt
G2 is burnt
No supply
through A and B
No supply
through A and B
M.S.B-2 is
burnt
G2 is burnt
G1 is burnt
A and B
damaged
M.S.B-1 is
burnt
A and B
damaged
Figure 7. Fault tree of marine power plant.
Int. J. Appl. Sci. Eng., 2006. 4, 1
77
Amit Kumar, Shiv Prasad Yadav, and Surendra Kumar
5.3. Fuzzy fault tree
Using fault tree, the fuzzy fault tree (Fig. 8)
is constructed for evaluating the fuzzy reliability of the above marine power plant. It is
based on fuzzy logic and can handle fuzzy
arithmetic with the application of fuzzy set
theory.
6. Proposed algorithm
With the help of fuzzy fault tree the following algorithm is proposed for analyzing
the fuzzy reliability of the above marine
power plant.
Step 1
~
~
~
~
F1 1Θ(1ΘV2 ) (1ΘV5 ) (1ΘV4 )

X 1, Y1, Z 1, T1 ;[11,12 ];['11,'12 ]


where
X 1 1 (1 x 2 )(1 x 5 )(1 x 4 ),
Y1 1 (1 y 2 )(1 y 5 )(1 y 4 ),
Z 1 1 (1 z 2 )(1 z 5 )(1 z 4 ),
T1 1 (1 t 2 )(1 t 5 )(1 t 4 ),
11 min( 21, 41, 51 ),
12 min( 22 , 42 , 52 ),
'11 min(' 21, ' 41, ' 51 ),
'12 min( ' 22 , ' 42 , ' 52 )
Step 2
2 1 m in ( 1 1 , 2 1 , 4 1 , 5 1 ),
2 2 m in ( 1 2 , 2 2 , 4 2 , 5 2 ),
' 2 1 m in ( ' 1 1 , ' 2 1 , ' 4 1 , ' 5 1 ),
' 2 2 m in ( ' 1 2 , ' 2 2 , ' 4 2 , ' 5 2 )
Step 3
~
~
~
F3 1Θ(1 ΘV3 ) (1 ΘF2 )

X 3 , Y3 , Z 3 , T3 ;[31,32 ];[' 31,' 32 ]


where
X 3 1(1x3)(1X 2 ), Y3 1(1y3)(1Y2 ),
Z 3 
1 (1z 3)(1Z 2 ), T3 
1 (1t 3)(1T2 ),
31 min (11, 21, 31, 41, 51),
32 min (12, 22, 32, 42 , 52 ),
' 31 min ('11, ' 21, ' 31, ' 41, ' 51),
32 min ('12, ' 22, ' 32, ' 42, ' 52 )
Step 4
~
~
~
F4 1 Θ(1ΘV6 ) (1ΘF3 )


( X 4 , Y4, Z 4 , T4 );[41,42 ];[' 41,' 42 ]


where
X 4 1(1x 6 )(1X 3 ),
Y4 1(1y 6 )(1Y3),
Z 4 1(1z 6 )(1Z 3),
T4 1(1t 6 )(1T3 ),
where
41 min(11, 21, 31, 41, 51, 61),
42 min(12 , 22 , 32 , 42 , 52 , 62 ),
' 41 min('11, ' 21, ' 31, ' 41, ' 51, ' 61),
' 42 min('12 , ' 22 , ' 32 , ' 42 , ' 52, ' 62 )
X 2 x1 X 1 , Y2 y1 Y1,
Step 5
 
F
2 V1 F1

X 2 ,Y2 , Z 2 , T2 ;[21,22 ];[' 21,' 22 ]


Z 2 z1 Z 1, T2 t1 T1
78
Int. J . Appl. Sci. Eng., 2006. 4, 1
~
~
~
~
F5 =1Θ(1ΘV1 ) (1ΘV5 ) (1ΘV3 )
Fuzzy Reliability of a Marine Power Plant
Using Interval Valued Vague Sets

( X 5 , Y5 , Z 5 , T5 );[51,52 ];[' 51,' 52 ]
~
F
where
X 5 1 (1 x1)(1 x 5 )(1 x 3 ),
OR
~
F
~
V
Y5 1 (1 y1)(1 y 5 )(1 y 3 ),
8
9
Z 5 1 (1 z1)(1 z 5 )(1 z 3 ),
AND
T5 1 (1 t1)(1 t 5 )(1 t 3 ),
~
F
 5 1  m i n ( 1 1 , 3 1 , 5 1 ) ,
 5 2  m i n ( 1 2 , 3 2 , 5 2 ) ,
 ' 5 1  m in ( '1 1 , ' 3 1 , ' 5 1 ) ,
~
F
4
8
OR
~
V
 ' 5 2  m in ( '1 2 , ' 3 2 , ' 5 2 )
OR
~ ~
VOR F
~
F
6
7
3
7
OR
OR
Step 6
~
V

F
V
6
2 F5


X 6,Y6, Z6,T6 ;[61,62];[' 61,' 62]


X
6
~
V
~
F
~
F
4
2
3
6
AND
AND
~
V
x 2 X 5 , Y 6  y 2 Y 5 ,
~
F
1
~
V
1
Z 6 z 2 Z 5 , T 6 t 2 T 5 ,
2
61 m in( 11 , 21 , 31 , 51 ),
62 m in( 12 , 22 , 32 , 52 ),
' 61 m in( ' 11 , ' 21 , ' 31 , ' 51 ),
' 62 m in( ' 12 , ' 22 , ' 32 , ' 52 )
~
F
5
OR
~
V
2
~
V
5
~
V
4
OR
Step 7
~
V
~
~
~
F7 1 Θ(
1ΘV4 ) (1 ΘF6 )
1
Figure 8.

X 7 ,Y7 , Z 7 , T7 ;[71,72 ];[' 71,' 72 ]


where
X 7 1(1x 4 )(1X 6 ), Y7 1(1y 4 )(1Y6 ),
Z 7 1(1z 4 )(1Z 6 ), T7 1(1t 4 )(1T6 )
71 min(11, 21, 31, 41, 51),
72 min(12, 22 , 32 , 42, 52 ),
' 71 min('11, ' 21, ' 31, ' 41, ' 51),
' 72 min('12 , ' 22 , ' 32 , ' 42 , ' 52 )
~
V
5
~
V
3
Fuzzy fault tree.
Step 8
~
~
~
F8 1 Θ(1ΘV7 ) ( 1ΘF7 )

X 8 , Y8 , Z 8 , T8 ;[81,82 ];[' 81,' 82 ]


where
Int. J. Appl. Sci. Eng., 2006. 4, 1
79
Amit Kumar, Shiv Prasad Yadav, and Surendra Kumar
X 8 1(1x 7 )(1X 7 ), Y8 1(1y 7 )(1Y7 ),
Z 8 1(1z 7 )(1Z 7 ), T8 1(1t 7 )(1T7 )
81 min(11, 21, 31, 41, 51, 71),
82 min(12 , 22 , 32 , 42 , 52 , 72 ),
' 81 min('11, ' 21, ' 31, ' 41, ' 51, ' 71),
' 82 min('12 , ' 22 , ' 32 , ' 42 , ' 52 , ' 72 )
Step 9
 
F
9 F4 F8


X 9,Y9, Z9,T9 ;[91,92];[' 91,' 92]


where
X 9 X 4 X 8, Y9 Y4 Y8,
Z9 Z 4 Z8, T9 T4 T8,
91 min(11, 21, 31, 41, 51, 61, 71),
92 min(12, 22, 32, 42, 52, 62, 72),
' 91 min('11, ' 21, ' 31, ' 41, ' 51, ' 61, ' 71),
' 92 min('12, ' 22, ' 32, ' 42, ' 52, ' 62, ' 72)
Step 10
~
~
~
F 1Θ(1ΘV8 ) (1ΘF9 )

1 (1 x 8 )(1 X 9 ),1 (1 y 8 )(1 Y9 ),

1 (1 z 8 )(1 Z 9 ),1 (1 t 8 )(1 T9 ) 
;
[min(11, 21, 31, 41, 51, 61, 71, 81 );
min(12 , 22 , 32 , 42 , 52 , 62 , 72 , 82 )];
[min('11, ' 21, ' 31, ' 41,
' 51, ' 61, ' 71, ' 81 );
min ('12 , ' 22 , ' 32 , ' 42 ,
' 52 , ' 62 , ' 72 , ' 82 )]
analyzing the fuzzy reliability of the above
marine power plant:
~
V1 < [(0.003, 0.006, 0.013,
0.92]; [0.94, 0.98]]>
~
V2 < [(0.006, 0.008, 0.015,
0.89]; [0.92, 0.97]]>
~
V3 < [(0.011, 0.022, 0.033,
0.91]; [0.93, 0.96]]>
~
V4 < [(0.012, 0.024,0 .040,
0.91]; [0.94, 0.98]]>
~
V5 < [(0.021, 0.032, 0.053,
0.92]; [0.93, 0.98]]>
~
V6 < [(0.041, 0.062, 0.093,
0.90]; [0.93, 0.99]]>
~
V7 < [(0.052, 0.083, 0.140,
0.90]; [0.92, 0.98]]>
~
V8 < [(0.111, 0.132, 0.163,
0.92]; [0.94, 0.97]]>
0.026); [0. 85,
0.028); [0. 85,
0.046); [0. 82,
0.059); [0. 84,
0.084); [0. 87,
0.134); [0. 81,
0.175); [0. 85,
0.184); [0. 86,
8. Results
~
The fuzzy unreliability F and the fuzzy re~
~
liability, R 1 Θ F , of the above marine
power plant has been computed using the
above data and the proposed algorithm and
obtained as the following interval valued
trapezoidal vague sets
~
F < [(0.114, 0.140, 0.181, 0.217); [0. 81,
0.89]; [0.92, 0.96]]>
and
~
R < [(0.783, 0.819, 0.860, 0.886); [0. 81,
0.89]; [0.92, 0.96]]>
7. Data
The following data in terms of interval valued trapezoidal vague sets are assumed for
80
Int. J . Appl. Sci. Eng., 2006. 4, 1
Interval valued trapezoidal vague sets repre~
~
senting F and R are shown in Figs. 9 and
10, respectively.
Fuzzy Reliability of a Marine Power Plant
Using Interval Valued Vague Sets
sets and vague sets, is that interval valued
vague sets separate the positive and negative
evidence for the membership of an element in
a set and also the values of membership and
non membership of an element in a set are intervals instead of a single real number. Interval valued trapezoidal vague sets are used
which is very effective in representing the
failure possibility of the basic events under
fuzzy environment.
TF~ (r) ,
1 F F~ ( r )
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.12
0.14
0.16
0.18
0.2
0.22
Unreliability ( r )
Lines from top to bottom:
Represents supremum value of
the interval 1 FF~ (ri ) for each
ri r
TR~ (r ) ,
1 FR~ ( r )
1
0.9
0.8
0.7
0.6
Represents infimum value of the
interval 1 FF~ (ri ) for each
ri r
0.5
0.4
0.3
0.2
Represents supremum value of
the interval TF~ (ri ) for
each ri
r
Represents infimum value of the
interval TF~ (ri ) for each ri r
Figure 9. An interval valued trapezoidal vague set
~
representing F .
9. Conclusion
In this paper, a new method has been developed for analyzing the fuzzy system reliability of a series and parallel system using
interval valued trapezoidal vague sets, where
the reliability of each component of each system is represented by an interval valued
trapezoidal vague set defined in the universe
of discourse [0, 1]. The developed method has
been used to analyze the fuzzy reliability of a
marine power plant. The major advantage of
using interval valued vague sets, over fuzzy
0.1
0
0.77
0.79
0.81
0.83
0.85
0.87
0.89
Reliability ( r )
Lines from top to bottom:
Represents supremum value of
the interval 1 FR~ (ri ) for each
ri r
Represents infimum value of the
interval 1 FR~ (ri ) for each
ri r
Represents supremum value of
the interval TR~ (ri ) for
each ri
r
Represents infimum value of the
interval TR~ (r ) for each ri r
Figure 10. An interval valued trapezoidal
~
vague set representing R .
Int. J. Appl. Sci. Eng., 2006. 4, 1
81
Amit Kumar, Shiv Prasad Yadav, and Surendra Kumar
Acknowledgement
The authors are thankful to the reviewers
for their critical and fruitful comments. The
authors also acknowledge the financial support given by the Council of Scientific and
Industrial Research, Govt. of India, INDIA
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