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Rajesh Dangwal, Kapil Naithani RT&A, No 2 (53) DEFUZZIFICATION OF HEALTHCARE RELATED PROBLEM..._Volume 14, June 2019
Defuzzification of Healthcare Related Problem by Using (X, p) Intervalued Trapezoidal Fuzzy Numbers
and Their Functional
Dr. Rajesh Dangwal
Professor, B.G.R. Campus H.N.B. Garhwal
University, Srinagar, Uttarakhand, E-Mail: dangwalrajeshhnbgu@gmail.com
Kapil Naithani
Research Scholar, B.G.R. Campus H.N.B. Garhwal University, Srinagar, Uttarakhand, E-Mail: kapilpintu88@gmail.com
Abstract
In healthcare related problems there exist some medical error due to involvement of human errors and technologies error. In general the available data is not sufficient to assess the clinical process. This study uses level (A,p) intervalued trapezoidal fuzzy numbers and their functional to evaluate the estimate reliability of system in the fuzzy sense, In this paper we uses fault tree diagram of the mixed system (series and parallel system) and we change their crisp probability to trapezoidal fuzzy number and a new approach of functional of fuzzy number has developed and Computed results have been compared with results obtained from other existing techniques.
Key Words: Healthcare, Fuzzy sets, inter-valued trapezoidal fuzzy numbers, Functional of fuzzy numbers, Fault tree analysis, Defuzzification.
I. Introduction
In the traditional set theory, an element either belong to the set or not, that is the answer become yes or no rather than more or less. Fuzzy set theory provide means of uncertainty. Probability theory also a primary tool for representing uncertainty but they are random. And all the uncertainty are not random. Fuzzy set theory is marvellous tool for modelling such kind of uncertainty associated with vagueness which are not random.
Healthcare is a series of process for a patient to receive medication..A Joint Health Commission report indicates that medical errors result in the death of between 44,000 and 98,000 patients every year and concludes that healthcare is a high risk [5] There are several examples where reliability analysis methods such as root cause analysis (RCA), failure mode and effect analysis (FMEA), fault tree analysis (FTA) and event tree analysis (ETA) have been applied for patient safety risk modelling in healthcare [6,7-10]. Fault tree analysis has been extensively used as a powerful technique in health related risk analysis from both qualitative and quantitative
perspectives [8-10]. . Some of the suggested healthcare areas where FTA can be used are equipment failures and malfunctions, material faults, human errors, environment-related risks, management deficiencies, communication and measurement errors, etc
II. Fuzzy Sets
In real world there exist fuzzyness i.e vagueness , uncertainty. Fuzzy sets were introduced independently by Lotfi A. Jadeh and Dieter Klaua in 1965 as an extension of classical notion of a set .In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent conditions that is an element either belong to set or not, fuzzy set theory permits the gradual assessment of the membership of elements in a set this is described with the aid of a membership function valued belong to the unit interval [0,1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special case of membership functions of fuzzy sets, if the latter only take values 0 or 1. Classical bivalent sets are usually called crisp sets it is used in a wide range of domains in which information is incomplete or imprecise
A fuzzy set is defined by a membership function from the universal set to the interval [0,1], as given below;
j (x): X ^ [0,1] (1)
where ja (x) gives the degree of belongingness of x in the set A. A fuzzy set A can be expressed as follows:
A = {{x,Ma (x)): x e X} (2)
Fuzziness can be found in many areas of daily life such as in engineering, medicine, manufacturing and others. In all areas in which human judgements, evaluation and decision are important. These are the areas of decision making reasoning, learning and so on.
III. Fuzzification and Inter-Valued Trapezoidal Fuzzy Number
In various situations, the exact values of any parameters of a system are not known due to unavailability of data and complete knowledge about the system, and thus the uncertainty arise. In order to quantify uncertainty, Fuzzification of parameter's value or collected data are done by system experts. In the process of fuzzification, crisp value is transformed into fuzzy value with the help of fuzzy membership functions. For analysing safety and healthcare related problems, inter-valued trapezoidal fuzzy membership functions or more simply inter-valued trapezoidal fuzzy number (IVTPFNs) are often utilized to provide more precise descriptions and to obtain more accurate solutions [27]. In this paper, IVTPFNs are used for quantifying data uncertainty associated with basic events. Mathematically, An interval-valued fuzzy set A(i — v fuzzzyset) on
R is derived by A = ^(x,[f2L (x),M~u (x)])/ x e r} 0 < j (x) < fi~u (x) < 1V x e R, It is
denoted by (x) = [f (x) v-AU (x),J x e R or A = [AL, AU ]
The i-v fuzzy set A indicates that, when the membership grade of x belongs to the interval [ jf~L (x) fu~j (x),J the largest grade is jf~U (x) and the smallest grade is f~L (x)
l (x) = <
X( x a)
b - a
X
X(d - x )
d - c
0
a < x < b b < x < c c < x < d otherwise
(3)
Therefore, A L = ( a, b, c, d ,A) a < b < c < d Let
(x) =
p(.x -e) e < x < b
b - e
p b < x < c
p(f - x) c<x<f
(f -c)
0 otherwise
(4)
Therefore AU = (e,b,c, f ,p),e<b<c< f, Consider the case in which 0<A< p<1 and e < a < b < c < d < f, from (3) and (4) we obtain A=[AL, AU ] [(a,b,c, d;A), (e,b, c , f;p)] , Which is called the level (A,p) i - v fuzzy number.
e a b
c d f A
Fig1: i-v trapezoidal fuzzy number A
IV. Fault Tree Analysis and Fuzzy probability
Fault-tree analysis (FTA) is a top-down. Deductive failure analysis in which an undesired state of a system is analyzed using Boolean logic to combine a series of lower level basic events it is to evaluate probability of an accident resulting from sequences and combinations of faults and failure events. A fault tree describes an accident-model and interprets the relations between components.
Thus, the fault tree is useful for understanding logically the mode of occurrence of an accident. Furthermore, given the failure probabilities of system components, the probability of the top event can be calculated. In fault tree diagram there are two gates are used one is "AND" and another is "OR" AND (conjunction) means the failure probability is depend on all those event which are associated with AND gate and in OR gate the event associated with OR gate they are work independently to failure next event.
The fuzzy failure probability can be calculated by following arithmetic operation on fuzzy numbers, here when we take probability in fuzzy sense the FTA is called FFTA( fuzzy fault tree analysis)
Table1:
Row. Approach Gate Operation Equation
No.
(1) Traditional OR
FTA AND
(2) Traditional OR
FFTA AND
Conjunction Por = 1 — [(1 — ) x (1 — q2) x... x (1 — qn)]
Intersection Pand = ql x q2 x.... x qn
Conjunction Por = T0[(T0~ ) ® (T©~2) ®.... ® (T©~ )]
Intersection Pand = ® ~2 ®......®
V. Defuzzification
Defuzzification is the process of producing a quantifiable result in crisp logic. Given fuzzy sets and corresponding memberships degree it is the process to transformed fuzzy numbers to crisp value there are many rule to transform a number of variable into a fuzzy set and then defuzzified. Defuzzification is interpreting the membership degrees of the fuzzy sets into a specific decision or real value the simplest method to defuzzification is choose to set of highest membership function another a common and useful defuzzification technique is centre of gravity. For the interval valued trapezoidal fuzzy number we will take the mean of the COG of upper and lower trapezoidal fuzzy numbers. Also we use the signed distance method and bisector of area method to evaluate the defuzzified the fuzzy numbers.
VI. Steps of the methodology
Step1. Construction of fault-tree
Construct fault-tree for some healthcare related problems (e.g. patient transfer without infection control, etc.) by using fault-tree logical symbols.
Step2. Obtain fundamental events failure probabilities in the form of level (A, p) interval-valued trapezoidal fuzzy numbers
Possible failure probability of each fundamental event is obtained by aggregating experts knowledge and experience, and represented in terms of level (k,p) interval-valued trapezoidal fuzzy numbers.
Step3. Computation of system top event fuzzy failure probability (qT)
Using fault-tree diagram and possible failure of fundamental events represented in terms (A,p) interval-valued trapezoidal fuzzy numbers, the system top event fuzzy failure probability (qT) can be computed by using operations given in table 4. Also, the defuzzified value of system top event can be easily computed using various defuzzification methods as COG, bisector of area, middle of maxima.
Step4.Compute system top event fuzzy reliability
Compute system top event fuzzy reliability which is equal to one minus the fuzzy failure probability of the top event.
Step5. Find the most and least influential fundamental events of the problem.
Tanaka et. al. V —index will be extended for level (A, p) interval-valued trapezoidal fuzzy numbers, the most and least influential fundamental events of the considered problems will be evaluated by finding max{V(qT, qTj)v ¿} and min{7((|r, qTj)v ¿} values respectively for the system, where qT. is the system top event fuzzy failure probability after eliminated ith basic event.
Step6. Analyze the results and give suggestions based on it for improving the efficiency of considered healthcare related problems.
Index V, measure the difference between q~ and q~ , and defined as
V( qT, qTi ) =
(q1LTl - q\ ) + (q\ - qLTh ) + (qLT3 - q% ) + (qLT4 - qLT,4 ) +
> 0 (5)
(qUT5 -qUTk ) + (qUT2 - qUn2 ) + (qUT3 -qUTh ) + (qUT6 - qUT,6 ) V( qT, qTi ) indicates the extent of improvement in eliminating the failure of the ith component.
If V( qT, qT ) > V( qT, qT ) then preventing failure of i-th component is more effective than j-th component
VII. Example
Fault tree analysis of patient transfer without infection control precaution here is an example of study extent and execution of redundant process during in patient transfer to radiology, and their impact on errors during the transfer process; in which there are given some basic events which are fundamental event to reliable whole system. There were four ways to communicate the required infection control precautions to a porter: (1) verbal handover at Radiology; (2) written handover at Radiology using a transfer form; (3) verbal handover at the ward during patient collection; and (4) verification of the transfer form by the ward nurse fig 2 depicts a fault tree for the events leading to inadequate infections control precautions during transfers. The basic events are given as follow.
T=Inadequate infections control precautions A= Patient was infectious
B= patient infectious status was not communicated through verbal handover at radiology C= Patient infectious status was communicated through verbal handover at radiology, but error still occurred
D= Patient infectious status was not communicated through written handover at radiology F= Patient infectious status was not communicated through verbal handover at ward G= Patient infectious status was not communicated through verbal handover at ward, but error still occurred
H= Patient infectious status was not communicated through written handover at ward. I= Patient infectious status was not communicated through written handover at ward, but error still occurred
The fault tree diagram of "Patient transfer without infection control precaution" is follow, here the (^1 Shape show "AND" and Q for "OR" gate
We express and gate by Intersection n and OR gate by conjunction (union) u . The fault tree can be expressed as T = A n J = A n (k n L)
= A n[(M n N) 0 (O n
= A n[(B u C)n(D u E)n(F u G)n(H u /)] (6)
The mathematical formula of failure probability o top event T is given by q~ = q2 x q~
= q~ x q~ x q~
=q-Ä xhüx q~ )xfex q~)]
= q-i *1{l-(l - q~X1 - q~ M-(i - q-6 X1 - q~ M-(l - q~X1 - q~ M-(i - q~ )(i - q~)}] (7)
Using this formula the failure probability q~ by various method can be calculated and the reliability of system calculated by l - q~.
TABLE 2: Crisp values and their corresponding trapezoidal fuzzy numbers S.No. Failure Crisp value IVTPFN _Probability_
1
4
5
q~A
q~
q~
q~ q~
0.27
0.93
0
0.52 0.22 0.74
0.23,0. 0.21,0.
' 0.89,0 v 0.87,0
^0.00,0. 0.00,0.
0.48,0. v 0.46,0.
' 0.18,0 v 0.16,0 ^0.70,0 0.68,0.
25,0.29,0.31 25,0.29,0.33
91,0.95,0.97 91,0.95.0.99
00,0.00,0.00 00,0.00,0.00
50,0.54,0.56 50,0.54,0.58 20,0.24.0.26 20,0.24,0.28 72,0.76,0.78 72,0.76,0.80
0.8 1.0
0.8 1.0
0.8 1.0 0.8 10 0.8 1.0 y 0.8^ 1.0
2
3
6
S.No.
Failure Probability
Crisp value
IVTPFN
7
8
0 0.56
0.30
0.00,0.00,0.00,0.00: 0.8 0.00,0.00,0.00,0.00:1.0 ^0.52,0.54,0.58,0.60: 0.8^ v0.50,0.54,0.58,0.62:1.0y ^0.26,0.28,0.32,0.34: 0.8^
v
0.24,0.28,0.32,0.36:1.0
y
TABLE 3: Fuzzy operation of two intwervalued fuzzy numbers
OPERATION
TRAPEZOIDAL FUZZY INTERVALUED NUMBERS
MULTIPLICATION
COMPLEMENT
a1, b1, c1, d1 :À ^b , c , fi: p
, b , , : ^ V e2 , b2 , C2 , ./2 : p
a1 a2 ,b1b2, c1c2, d1d2 : ^
Veie2, b1b2 , C1C2 , f1f2 : p y
1-
a,b,c,d :Ä e, b, c, f : p
y
1 - d,1 - c,1 - b,1- a :Ä 1 - f ,1 - c,1 - b,1 - a : p
9
x
VIII. Failure probability by various method
VIII(I). Traditional Method: The fuzzy failure probability of top event by traditional method taking crisp value of failure probability is given by 0.080441705 and reliability is 0.919558295 VIII(II). Max Min Method:-It is for crisp value of failure probability, it is introduced by huang.et.al.,in this method we take maximum for union and minimum for intersection. In given example the fuzzy failure probability can be calculated as follow from equation (5)
qM = max(qB, qc) = max(0.93,0.00) = 0.93 qw = max(qD, q£) = max(0.52,0.22) = 0.52 % = maxq, qG) = max(0.74,0.00) = 0.74 qp = max(qff, q ) = max(0.56,0.30) = 0.56 qr = min(qM, qw) = min(0.93,0.52) = 0.52 qL = min(q0, qp) = min(0.74,0.56) = 0.56 q7 = min(qr, qL) = min(0.52,0.56) = 0.52 qr = minq, q7) = min(0.27,0.52) = 0.27
That is fuzzy failure probability of top event is 0.27 and fuzzy reliability of top event is 0.73
VIII(III). TANAKA ET AL METHOD: The fuzzy failure probability of top event of fault tree by tanaka. et. al method from table 2 and is given by (0.05271925680.06682515840.098663675,10.11793798488 08
v0.04186685620.06682515840.09866367510.13964485711.0 ,
( 0.88206201530.90133632490.93317484160.9472807432 0.8^ 0.86035514290.9013363249,0.93317484163.9581331438} 1.0
and the fuzzy reliability
of top event is
0.1
Failure Probability
0.8
0.6
0.4
0.2
0
0.05
0.15
Fig3: Failure Probability Of Top Event In Trapezoidal Fuzzy Number
Fig4: Fuzzy Reliability Of Top Event In Trapezoidal Fuzzy Numbers
VIV. DEFUZZIFICATION METHODS
Defuzzificztion is a process when we transformed fuzzy numbers to discrete value which is most probable value among all the probabilities there are some important defuzzification methods, which are following.
VIV(I) . MEAN OF COG METHOD
COG method is one of the most defuzzification method. In this method we take mean value of lower and upper trapezoidal fuzzy numbers, the upper left and right and similarly lower left and right membership function are
i
fuU (x) = P
JL (x) =A
x — e ,b — e. x — aN b — a.
J u U (x) = P
\
J L (x) =A
x — f
V C — f J f x — d
c — d
_ I (x)dx
Now centre of gravity is given by x — ^-
I jUA (x)dx
v _ |xU .Mau (x)dx
J^ U ( x)dx
r J xL l (x)dx
xL -A-
JvA L (x)dx
MEAN OF COG =1 (xU + xL ) 2 '
1
x — — 2
J x.p| x—e Jdx + J xpdx + J x.pj^x—f \dx
J p fzf \dx+J pdx+J p f—f\dx
J x.pj x—— Jdx + J xAdx + J x.A| x—— Jdx
; — d
J pf x—— Jdx + J AAdx + J A x—d jdx
(8)
1
x — — 6
c2 + f2 — f2 — e2 — fc — ef I I d2 — f2 — a2 + cd — af |
c + f — e — f
c + d — a — f
(9)
In this method the fuzzy failure probability obtained by tanaka. et. al. of top event can be calculated and is equal to 0.0504603867 and fuzzy reliability of top event is 0.949539613
P
VIV(II). BISECTOR OF AREA METHOD:
e a b
g h c d f
FIG 5: Bisector Of Area
A
in this method we evaluate the point where the area are bisected let 'g" and "h" be the point where the upper and lower trapezoidal region is bisected for upper trapezoidal area
1 [(g - b)+ (g - e)]x p =1 [(f - g )+(c - g )]xp
2g - (b + e) = (f + c) - 2g 4g = (f + c) + (b + e)
g =1 (f + c + b + e) Similarly for lower trapezoidal area h = 1 (d + c + b + a)
(10) (11)
Therefore mean of the bisector of upper and lower trapezoidal fuzzy number will be given by
1 (g + h) =1 (a + e + d + f + 2b + 2c) (12)
2 8
In this method the fuzzy failure top event is = 0.0853933277 and the fuzzy reliability of top event is 0.914606672
f
d
f
a
c
f
VIV(III). MIDDLE OF MAXIMA: -
In this method we get the value of x which is middle of maximum membership function which is
given by xMOM =1 (b + c) (13)
By this method the fuzzy failure probability of top event is =0.0827444168 and the fuzzy reliability is =0.917255583
VIV(IV) FUNCTIONAL OF FUZZY NUMBERS:
It is defined as a function of function of x that is if we give the membership value to membership function .it is defined as follow jA (x) : X ^ [0,1] jf (x)): j (x) ^ [0,1]
A = K^ ja (x)), J (JA (x))): x e X}
Let ju(jua (x)) = —1— and 0 < —1— < 1 for any non negative value of x then the fuctional form
1
x + 1
of fuzzy number as following
x +1
TABLE 4
S.N Failure Crisp o. Probabil value
_iy_
Functional value of IVTPFN ((x, J (x)), ff(f (x)))
q ~
qb
3 qc
q~ q~ q~
q~
q~ q~
0.27
0.93
( 0.23 v 0.21
( 0.89 0.87
0.8 : 0.813,0.25 : 0.8 : 0.8,0.29: 0.8 : 0.775,0.31:0.8: 0.763A 1.0 : 0.826,0.25 :1.0: 0.8,0.29:1.0 : 0.775,0.33 : 1.0 : 0.751 ,
0.8 : 0.529,0.91: 0.8 : 0.523,0.95: 0.8 : 0.512,0.97: 0.8.0.507^ 1.0 : 0.534,0.91: 1.0: 0.523,0.95: 1.0 : 0.512,0.99:1.0 : 0.502
0.52
0.22
0.74
0.56
0.30
(0.00: 0.8 : 1,0.00 : 0.8 : 1,0.00 : 0.8 : 1,0.00: 0.8 : 1 v0.00: 1.0 : 1,0.00 : 1.0 : 1,0.00: 1.0 : 1,0.00 : 1.0 : 1 ,
(0.48: 0.8 : 0.675,0.50: 0.8 : 0.666,0.54: 0.8 : 0.649,0.56: 0.8 : 0.641^ v0.46: 1.0 : 0.684,0.50: 1.0 : 0.666,0.54: 1.0 : 0.649,0.58: 1.0 : 0.632 (0.18: 0.8: 0.847,0.20: 0.8: 0.833,0.24: 0.8: 0.806.0.26: 0.8: 0.793^ v0.16:1.0: 0.862,0.20:1.0: 0.833,0.24:1.0: 0.806,0.28:1.0: 0.781 (0.70: 0.8: 0.588,0.72: 0.8: 0.581,0.76: 0.8: 0.568,0.78: 0.8: 0.561^ v0.68:1.0: 0.595,0.72:1.0: 0.581,0.76:1.0: 0.568,0.80:1.0: 0.555 ^ (0.00: 0.8: 1,0.00: 0.8: 1,0.00: 0.8 : 1,0.00: 0.8: v0.00:1.0: 1,0.00:1.0: 1,0.00:1.0: 1,0.00:1.0: 1 , r0.52: 0.8: 0.657,0.54: 0.8: 0.649,0.58: 0.8 : 0.632,0.60: 0.8 : 0.625 0.50:1.0 : 0.666,0.54: 1.0 : 0.649,0.58: 1.0: 0.632,0.62:1.0 : 0.612
(0.26: 0.8 : 0.793,0.28: 0.8 : 0.781,0.32: 0.8: 0.757,0.34: 0.8 : 0.746^ 0.24:1.0: 0.806,0.28:1.0: 0.781,0.32:1.0 : 0.757,0.36:1.0: 0.735
1
2
0
4
5
6
0
7
8
9
The fuzzy functional failure probability of top event is given by from tanaka.et.al. method from table 2 and is given by
(0.05271925681 0.8 : 0.949,0.0668251584: 0.8 : 0.937,0.0986636751 0.8 : 0.910,0.11793798481 0.8 : 0.988^
I 0.0418668562 1.0 : 0.959,0.0668251584:1.0 : 0.937,0.09866367511.0 : 0.910,0.13964485711.0 : 0.877 J
and reliability of top event given by 0.8820620152 0.8: 0.531,0.9013363249 0.8: 0.525,0.933174841(5 0.8: 0.517,0.9472807432! 0.8: 0.513 0.860355142991.0:0.537,0.901336324991.0: 0.525,0.93317484161.0: 0.517,0.958133143811.0: 0.510
FUNCIONAL OF INTERVALUED TRAPEZOIDAL FUZZY NUMBER
E
^ 0.9
0.8 1
X: 0.09834 X: 0.06683 Y: 1 Y: 1 Z: 0.9105 09866
Z: 0.9374 06683 Y : 0.8 ! y: 0.8 Z: 0.9102
membership value
0.05
0.15
failure probability
Fig6: Graph Of Functional Value Of Failure Probability F Top Event
FUNCTIONAL VALUE OF RELIABILITY OF TOP EVENT
0.54 v
0.5
X: 0.9013 X Y: 1 Y
Z: 0.5259 '.9011 Z j y : 0.7919
Z: 0.526
0.9334 0.9899 0.5172 9332
Z: 0.5173
0.2
membership of reliability
0.88
0.9
Reliability
Fig7: Graph Of Functional Value Of Reliabilty Of Top Event
For defuzzification of functional of fuzzy numbers we are using the centre of gravity method(centre of volume)
Now to determine the centre of gravity we can use the formula . _ J x fA (x)f(fA (x))dx
(xX"("a (x))dx
(14)
We evaluate the defuzzified value of each lower and upper trapezoidal fuzzy numbers and then we take the mean value of both defuzzification value.
= 1
0
0.56
0.52
X =
X = -
ix.pl'X—e ).—1— dx + fx.p.—1— dx + fx.p i V b - e ) x +1 { x +1 i
f r \
f - x
f - c
1
x +1
dx
1 S ) x71dx + ! x71dx +1 p
\
x =-
f - x f - c
x +1
dx
fx.l(x—a .—1— dx + fx.l.—1— dx + fx.l| a V b - a ! x +1 b x +1 c
d - x d - c
x + 1
dx
b b|
x-a b-a
— dx + |-dx + f
+1 x +1
]
|
d - x d - c
x+1
dx
1 x
2
* 1 I u , z* x = — Ix + x
f + c - b - e e +1, --+-in 2 b - e +11-inf c +11- f + 1lni ve + 1 ) Vb + 1j f - c V y+11] .c + 1 )
" f +1. --in J - c f f+1 V c +1 , 1-e+1in b-e f b +11 V e +1 ) + in f c +11 V b +1 )
x =
d + c - b - a a +1, f b +1 1
+-ln|
- ■ i)
b - a v a +1
ln
c + 1 b +1
d +1 J d +
!
d - c V c +1
d+1 inf d+11-a+1 mi b+1 V mi c+1
d - c V c +1 ! b - a V a +1 ! V b +1
(15)
(16)
Defuzzified value of failure probability of top event of upper trapezoidal fuzzy number is 0.083528098 and defuzzified value of failure probability of lower trapezoidal fuzzy number is 0.083528098.Similarly defuzzified value of reliabilty of lower trapezoidal fuzzy number is 0.916471902 and defuzzified value of reliability of lower trapezoidal fuzzy number is is 0.912979297 .and the mean value of both values is 0.9147255995
In traditional method the fuzzy failure probability can be calculated as follow
= 0.27 x [{1 - (1 - 0.93)(1 - 0.00)}x {l - (l - 0.52)(l - 0.22)}x {l - (l - 0.74)(l - 0.00)}x {l - (l - 0.56)(l - 0.30)}] =0.080441705
1
1
1
a
!
And the fuzzy reliability of top event is 0.919558295
The difference of reliability value from traditional method and the tanaka et al method is 0.004832695 to the left of trapezoidal fuzzy numbers.
Table 5.Ranking of basic event of Example using failure difference
Eliminated event
4t1
V fe, ~TU )
Rank
A(i = 1) f0.00,0.00,0.00,0.001 V0.00,0.00,0.00,0.00J 0.6831466219 1
B(i = 2) f0.00,0.00,0.00,0.001 V0.00,0.00,0.00,0.00J 0.6831466219 1
C(i = 3) f0.05271925680.06682515840.09866367510.11793798481 0.00 6
V0.0418668562,0.0668251584)0.09866367510.1396448571lJ
D(i = 4) f0.01624908(60.0219098880.03589945730.04488272261 0.4389908241 2
V0.012022069220.0219098880.03589945730.0553832294)
Rajesh Dangwal, Kapil Naithani DEFUZZIFICATION OF HEALTHCARE RELATED PROBLEM... RT&A, No 2 (53) Volume 14, June 2019
Eliminated event ~Tk V fev ~T, ) Rank
E(i = 5) '0.04333089(60.0547747220.08077377890.0966704794^ v0.034563449,0.0547747220.08077377890.1147221028 0.1227626968 5
F(i = 6) '0.00,0.00,0.00,0.00^ v0.00,0.00,0.00,0.00J 0.6831466219 1
f) II f0.05271925680.06682515840.098663675,0.117937984!^ 0.00 6
^0.04186685620.06682515840.098663675 J0.139644857lj
H (i = 8) f0.02175715360.0279770440.04419425540.054482212 ^ 0.3615252583 3
^0.01661383180.027977040.044 1 9425540.0664272576
I (i = 9) f0.0435143026D.0539557220.08010208780.09614509631 0.1263569586 4
[0.03461214960.0539557220.08010208780.1144024992
Conclusion
It is seen that when the membership function of fuzzy failure probability is also in fuzzy sense than reliability of top event can also go to another value. Here the error outcome from trapezoidal fuzzy failure probability to left side, and using equation 5 and table 5, the value of index is analyzed that the most critical basic events are A B and F whereas least critical basic events is C and G. The order of all critical basic events are given below in decreasing manner (A, B, F) > D > H > I > E > C > G.
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