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Performance index to quantify reliability using fuzzy subset theory
Microelectron Reliab.. Vol. 21, No. 4, pp. 543-549, 198I. Printed in Great Britain.
0026-2714/81/040543-07$02.00/0 © 1981 Pergamon Press Ltd.
PERFORMANCE INDEX TO QUANTIFY RELIABILITY USING FUZZY SUBSET THEORY K. B. MISRA Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302, W.B., India and ARUNA SHARMA Electrical Engineering Department, University of Roorkee, Roorkee 247672. U.P.. India
(Receivedfor publication 11 September 1980) Abstract--The system performance is generally assessed through reliability. There are gaps in the quantification through probability. An alternate concept of performance index is proposed in the paper which is expected to permit the consideration of any number of graded states of a system or a subsystem states and laws to compute the performance index. The theory of fuzzy sets permits the mathematical models to be flexible and close to human thinking, assessment or interpretations. This helps considerably in decision making process as it brings human thinking very close to engineering system performance. probabilities associated with these random events are designated "reliability" and "unreliability" of the Very recently, H o t o y a m a [ 4 ] published a paper, in system. which he had considered a situation where a system In reality, the situation is not that simple. The could have three states instead of two discrete states transition from one state to the other state is not that viz., good and bad. The third state Hotoyama con- sharp or abrupt. In fact, since the performance assesssidered was designated as a "degraded" state. Thus it ment is closely related to human assessment of the was obviously an extension of usual binary algebra situation, it is subject to subjective assessment of the to take into consideration the multi-state structure performance. Thereby the transition from one state to function associated with the system. However, the the other is not that precise or sharp. There may be paper made use of decomposition technique in order situations where one could classify the performance to compute reliability of a system based on two- of system or of an equipment not in two precise state models. categories but many more, by saying the performance The purpose of the present paper is to draw is poor, satisfactory, good, very good or excellent. In attention to a more versatile theory which permits the this way there could be many more states possible consideration of system performance assessment where transition from one to the other is not very involving a n y number of states. Besides it is also sharp or alternatively the states are fuzzy states. Undoubtedly, that two-state model of reliability being intended to bring to notice the gap that exists at present in reliability of the theory of probability for used at present is a boundary states model in which reliability assessment. An alternate approach has no information is available on the intermediate states or, alternatively, performance is not defined for interbeen suggested in this paper. The IEC publication 271, 1974 provides the follow- mediate states. Further the mathematical tools are ing definition of reliability. only aids to human decision making process. The It is defined as "the ability of an item to perform a final decision making is done by human beings based on some mathematical calculations and interrequired function under stated conditions for a stated pretations. It will be more logical and proper if period of time". Obviously, the ability of an item to perform a mathematical tools or aids reflect the thinking function may be quantified in many ways, one of process of human beings and are close to it. The gap that exists in the mathematical models at them being the probability of successful operation which has been widely accepted as the definition of present is due to the fact that the models are not reliability. But the moment we quantify the per- consistent with the thinking of persons who handle formance in terms of probability, all related theories the engineering systems or eventually, based on their of probability and probability calculus become decision making capability, control them. This explains why people are not accepting the "statistical applicable to solve the related performance models. In usual two-state models in reliability studies, we risk" calculations, realistic or practical as they are, based on "randomness" of events and have been consider the system to exist either in "good" or "bad" state. These two states constitute the two basic or leaning on "probability" computation using primary events. The probability of "success event" statistical data. The "perceived risk", on the other constitutes the reliability of the system. These two hand, is based on human judgement and is events are also considered to be random and the "possibilistic" in nature. A very interesting difference 543 I. I N T R O D U C T I O N
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K.B. MISRAand ARUNASHARMA
of these two types of risks has been brought out by Carsten Bee [5]. In fact, in the recent past these considerations have generated such strong opinions or feelings that they have virtually threatened various Governments in Western Europe. The difference between "statistical risk" and "perceived risk" is the same as the difference between "probability" and "possibility". An interesting article on the Theory of Possibility has been published by Zadeh[3], who has very clearly brought out the various aspects of the two situations. The theory of fuzzy sets as proposed by Zadeh [2] allows consideration of more than two states of a system. It also permits us to develop mathematical tools which are close to our thinking process and form the basis of decision making capability. The authors of this paper intend to provide the basic mathematical tools which allow these considerations of performance assessment of a system in terms of "performance index" rather than through reliability which is quantified in terms of "probability". The involvement of the theory of fuzzy subset opens up a vast vista of mathematical theories, in order to ease the consideration of system performance assessment through "performance index". Also, the theory of fuzzy sets allows the users of engineering systems to work out their own "operators" to carry out the analysis associated with the assessment, in place of usual "union and intersection" law of probability. In short the theory of fuzzy sets offers a very wide range of mathematical tools to bring the models close to human thinking, interpretations and decision making. 2. N O M E N C L A T U R E
ordinary set or subset fuzzy subset xl is an element of fuzzy subset A grade of membership of x in A (membership characteristic function) c is a subset (inclusion) U union t'3 intersection complement of A pseudo-complement of A O empty set O(s) structure function of the system S qJ(S/ps) performance function of the system S with /as as the membership characteristic function W(A), W(B) performance function of the components A~B~...connected in a system represents the summation of fuzzy variables states of components i ai X ( a t , a 2 ..... a,) states of components in the system Path vector or a vector X such that O(X) > 0 Route vector Path set Ca (X) of a path vector Minimal path set path set of a minimal path vector or minimal route set A A xiEA ~lA(X)
Cut vector, or antiroute vector Cut set or antiroute set Minimal cut vector Minimal cut set PI,P2 ..... Pp Kt, K2,...,Kt<
a vector X such that Q(X) = 0 Co(X) of a cut vector a cut vector X such that Y > X implies O(Y) > 0 cut set of a minimal cut vector the minimal route set of Q the minimal antiroute set of Q.
2.1. Fuzzy subset The theory of fuzzy subsets is an advancement in precision of the classical maths and the pervasive imprecision of the real world. A set, characterized by a membership function ranging between zero and one, is defined as a fuzzy subset. Thus the usual set theory is a particular case of the theory of fuzzy subsets, which permits only two grades of membership: unity and zero. Generally in the physical world the boundaries of the objects are not defined precisely. For example, the performance of a system may be satisfactory, good, excellent, bad, poor etc. indicating that the membership grades of the system performance are of importance. If A represents a subset of set E, then mathematically the fuzzy subset A is defined as follows: i f x is an element of E then x e A then A = {(xl/0,0 ), (x2/0,3), (X3/0,5), (X4/0,8), (Xs/1,0)} is a fuzzy subset with different membership functions of x~ in a set A. This explains element x 1 is not present in A because pA(Xl)= 0,0 here, element x 2 is present with #A(X2) = 0,3 and so on. The fuzzy subset theory brings out the following interesting information. 1. Fuzzyness and randomness are entirely different properties. 2. Fuzzyness governs human thinking and decisions. 3. To deal with fuzzyness effectively old concepts have to be modified for the study of cybernetics including human assessment of engineering system performance. 2.2. Possibility distribution If F be a fuzzy subset of universe of discourse U~ and Pe is the characteristic function, X is a variable taking values in U; R(X) is the fuzzy restriction F on variable X; then X is F represents R ( X ) = F possibility distribution rcx
represent
the
7rx = R(X). The possibility distribution function 7tx associated with X is numerically equal to the grade of membership of F, i.e.
2.3. Difference between the fi~zzyness and randomness The difference between the fuzzyness and random-
Performance index to quantify reliability using fuzzy subset theory ness can be explained by membership grade function, by considering degree of compatibility (or possibility) rather than probability. Let us consider an example of the number of failures of an equipment in a system in a given time. The probability distribution (Py) and the possibility distribution (#y) for failure of equipments are given below: No. of ~ilures
1
2
3
4
5
6
7
pf
1 0.7
1 0.2
0.8 0.1
0.5 0
0.2 0
0 0
0 0
545
Further the probability theory also takes into consideration the following properties, which are not valid for fuzzy subset theory: a.d=0
and
a+d=l.
The corresponding expression for the above two are aAi:#0;
except for
a=0
or
a= 1
aVi~l;
except for
a=0
or
a = 1.
4. N O T I O N OF PERFORMANCE FUNCTION'
PI
This explains that (1) the possibility information is more basic than probability information. Possibility is also an important property; (2) the membership function for possibility is not same for the probability. Reliability index I means that the performance is up to the mark, and system can fail, that is the system cannot perform with 100~'0 reliability but from possibility aspect it can do so; (3) the lower value of possibility implies lower value of probability and higher value of possibility may have high value of probability but low value of probability does not mean low value of possibility but may have high value of possibility.
As illustrated previously that performance index plays an important role in describing the system performance, this gives interesting connections between the theory of fuzzy variables and the theory of functions of structure. In certain problems the operation of an equipment not only considers whether it is operating or not operating, but some intermediate states are used governed by certain performance levels. For example: (1) (2) (3) (4) (5)
functions perfectly functions very satisfactorily functions fairly well functions rather poorly does not function.
If x~E E; E = {x 1, x 2. . . . . x,}; A is fuzzy subset then ~tA(X3~[0, 1] represents the fuzzy variable. If a level of system for series system and for parallel system is given by a functions Ws and We respectively. Ws(xl,x2 . . . . . x,,) = xl A x2 A ... A x,,,
3. SOME OF T H E AXIOMS OF FUZZY SETS
and If ~/A(X) is the membership function of element x in the fuzzy subset A and ~LB(X)in the fuzzy subset B, then ifa =/ZA(X), b =/~n(x)...a, b . . . ~ M = 1-0,1], the following operations are performed on the quantities a, h , . . .
a A b = min(a, b) a V b = max(a, b) fi =l-a a~b=(aAb) V(aAli)
kI'/p(Xl,X 2 . . . . . Xn) = X 1 V X 2 V . . . V x n.
Then Ws and We are called the performance functions for series and parallel systems respectively. Considering the performance index (Reliability) of a parallel system containing components A and B in parallel. If a, b are the state variables, a, b e {0, 1} then the structure function is given algebraically as: O(a,b) = a S b - a . b
a A b = b A a] commutativity aVb=bV aS
and the performance index = W(S) = WIA,~8~ = max(A, B).
(a A b) A c = a A (b A c)~ associativity (aVb) Vc aV(bVc) J
This shows that redundancy does not improve the performance index.
aAa=a} idempotence aVa a
c)}
a A (b V c) = (a A b) V (a A aV (bAc) (aVb) A (aVc)
distributivity
aA0=0 aV0=a aAl=a aVl=l
(~)
=
a
aAb=fivli} _ aVb iAli
DeMorgan's theorems.
5. PERFORMANCE F U N C T I O N AND RELIABILITY DETERMINATION
Consider a system S, and A and B are the components of this system. If A and B have fuzzy states as a t and b~ as follows: at, b ~ = 1 represents perfectly good or excellent performance of the components A and B a2, b 2 = 0,8 represent good performance a3, b 3 = 0,6 represent fairly good performance a4, b4 = 0,4 represents poor performance as, b5 = 0,0 represents components have failed.
546
K.B. MISRAand ARUNASHARMA
In
Out 0
O
Fig. 1. Series system. The performance of the system S depends on the performance of the components and the mode of operation of the components. This can be represented on the similar lines as s~ = s2 = s3 = s4 = ss =
1,0 0,8 0,6 0,4 0,0
= = = = =
excellent system performance good system performance satisfactory system performance poor system performance system has failed.
The X, Y and S represent the set of states of components A and B and system X = {(al/1,0), (a2/0,8), (a3/0,6), (a4/0,4), (as/0,0) } Y = {(b,/l,O), (b2/0,8), (b3/0,6), (b4/0,4), (bs/O,O)} S = {(sl/1,0), (s2/0,8), (s3/0,6), (s4/0,4), (Ss/0,0)lj. The system performance for different arrangements can be calculated by making use of different operators. Here conjunction and disjunction operators are used. 5.1. Series system If two components A and B are connected in series mode of operation. The performance function of this system depends on the individual performance of the connected components: the structure function of this system is given as
Q(s) = @(A)n Q(B) and the performance function is given as qJ(S) = q'(.4 ~ t~)The performance of the system in this case depends on the worst performance component, obviously the reliability also depends on less reliable component. Hence the performance index of the system can be expressed as W(S) = minlX, YI ---- [{(al/1,0), (a2/0,8), (a3/0,6), (a4/0,4), (a5/0,0)] n {(b J1,0), (b2/0,8), (b3/0,6), (b,/0,4),
(bdo,o)}] = [minl(a~/1,0)(bt/1,0), (al/1,0)(b2/0,8), (at/1,0)(b3/0,6), (a ~/1,0)(b,,/0,4), (al/1,0)(bs/0,0)l, mini (a2/0,8)(bt/1,0),
(a 2/O,8 ) (b2/0,8 ), (a z/O,8 ) (b3/0,6 ), (a2/0,8)(b4/0,4), (a2/O,8)(bs/O,O)l, minl(a3/0,6), (hi/i,0), (a3/O,6)(b2/0,8), (a3/0,6) (b,,/0,4), (a3/0,6)(bs/0,0)l, min I(a,,/0,4)(b~/1,0), (a4/O,4)(b2/0,8), (a4/0,4) (b3/0,6)(a4/0,4) (b,,/0,4), (a,/0,4)(b5/0,0)[, mini (as/0,0)(bl/1,0), (as/O,O)(b2/O,8), (as/0,0) (b3/0,6), (as/0,0)(b4/0,4), (as/0,0)(b5/0,0)l-].
'•
i,o
,,o
,,o
o,e
o,6
0,4
o,o o,o
\\\', o,s
o,s
\\\\ \\\\ 0,4 0,0
0,4 0,0
o,s \0,6\ o,4 ,,,\'\\
o,o
\\\\
~\\
o,o
,\\\
0,4
0,4
0,4
0,0
0,0
0,0
0,0
0,0
Fig. 2. The performance fuzzy graph for 2 components series system. This expression can be represented graphically using fuzzy logic graph as shown in Fig. 2. If the system performance is 0,6, this means that either of the component has 0,6 as performance index. The possible combination for this performance index is shown in Fig. 2. The marked portion represent the possible combinations. ~(s3/0,6 ) = [min(al/l,0)(b3/0,6), min(a,/0,8)(b3/0,6), (a3/0,6)(b3/0,6), min(bt/l,0)(a3/0,6), min(bj0,8)(a3/0,6)]. Algebraically the performance index is given as • (S) = T(A). q~(B). 5.2. Parallel s3,stem The system is working iff at least one component is good, it failed iff all the components are bad. Consider A and B components are connected in parallel mode, then the system structure function is written as: (Z)(S) = (Z)(A) u (Z)(B) and the performance function is • (S) = "PH u 8). The performance of the system depends on the performance of best component i.e. the component
In L ~
A ~
Fig. 3. Parallel system.
Out
Performance index to quantify reliability using fuzzy subset theory
547
®(s) = [((Z)(A)n ®(B))~ ((Z)(A)n (Z)(c)) ho
0,8
0,4
0,0
u ((Z)(B) ~ (Z)(C))] and the pertormance function is
1,0
\\\\\\\',
~P(S)= [~P((A~ B) w (An C) u (B ~ C))] \,
o~
0,6
0,6
~
0,4
,~\\\ \\\',\\\\
Using the same method (as followed for parallel and for series systems) for graphical representation of the system, it can be represented as in Figs. 5-7.
\0,8 \
\
,\\\\
= max[min(X, Y), rain(X, Z), min(Y, Z)].
~0 \0,4\ ,\\ \\\\
qo
y • 1,0
0,8
0,6
0,4
0,0
Fig. 4. Performance graph for parallel system.
ho
i,o
0,8
0,6
0,4
o,o
having highest performance. So the performance function is expressed as
0,8
0,8
0,8
0,6
0,4
0,0
qqS) = maxlX, YI
0,6
0,6
0,6
0,6
0,4
0,0
0,4
0,4
0,4
0,4
0,4
0,0
o,o
o,o
o,o
o,o
o,o
o,o
1,0
0,8
0,6
0,4
0,0
1,0
1,0
0,8
0,6
0,4
0,0
0,8
0,8
0,8
0,6
0,4
0,0
0,6
0,6
0,6
0,6
0,4
0,0
0,4
0,4
0,4
0,4
0,4
0,0
0,0 0,0
0,0
0,0
0,0
0,0
= [maxl(a,/l,O)(az/O,8)(a3/O,6)(a4/0,4) (%/0,0) w (bl/1,O), (b2/O,8)(b3/O,6)(b.jO,4)(bUO.O)l]
= D(al/1,O) u (bl/l,O), (al/1,O) w (b2/0,8), (al/1,O) u (b3/0,6), (al/1,O) u (bJO,4), (at/l,0) u (bs/0,0)l, I(a2/0,8) w (b~/1,0), (a2/0,8) L9 (b2/0,8), (a2/0,8) ~ (b3/0,6), (a2/0,8) ~ (b4/0,4), (a2/0,8)w (bs/O,O)l, L(a3/O,6) u (bl/1,O), (a3/0,6) w (b2/0,8), (a3/0,6) w (b3/0,6), (a3/0,6) w (b4/0,4), (a3/0,6) w (bs/0,0) I, [(a4/0,4)u (bl/i,0), (a4/0,4) w (bz/0,8), (a4/0,4) w (b3/0,6), (a4/0,4) w (b4/0,4), (a4/0,0) w (bs/0,0)l, 1(%/0,0) w (bl/0,0), (%/0,0) u (bz/0,8), (as/0,0) u (a3/0,6), (as/0,0) u (b4/0,4), (as/O,O) w (bs/O,O)/-J. So the perfoi'mance depends on the maximum value of the network component. Graphical representation of the function is given in Fig. 4. If the system performance is 1,0 then either component A or B's performance is excellent. The possible state combination is shown shaded in the fuzzy logic graph.
Z~
z~
~,0
0,8
0,6
0,4
0,0
I,o
~,0
0,8
0,6
0,4
0,0
0,8
0,8
0,8
0,6
O,4
o,o
If we use algebraic operation, then the performance index is given as hv(S) = qqA)$q~(B).
0,6
o,6
o#
0,6
o,4
o,o
5.3. Two out of three components
0,4
0,4
0,4
0,4
0,4
0,0
o,o
o,o
o,o
o,o
o,o
ud(S/1,O) = I(a~/l,O w b,/1,O), (al/1,O w b3/0,6), (al/l,O w bs/O,O), (bt/1,O w b3/0,6), (bt/l,0 w %/0,0)1
(al/1,O w (al/1,O w (bl/1,O u (b,/1,O u
b2/0,8), b4/0,4), a2/0,8), b4/0,4),
Three components are connected in parallel mode in a system. The system is working, if, any two components are working. Let the component C also has the Z as a set of states and C~ states, i = 1 to 5. The structure function is
o,o
Figs. 5, 6, 7. Fuzzy performance graph of two out of three system.
548
K.B. MISRAand ARUNASHARMA
Voter
J
Out
Fig. 8. Majority voting system.
J#
0,8
0,6
0,4
0,0
i,o
i,o
0,9
o,8
o,7
0,5
0,8
0,9
0,8
0,7
0,6
O,4
0,6
0,8
0,7
0,6
0,5
O,3
0,4
0,7
0,6
0,5
0,4
0,2
0,0
0,5
0,4
0,3
0,2
o,o
For finding the performance index of the system firstly the minimum value of the three combinations is found out using series system method and then by using parallel system operation maximum value is found. 5.4. Majority voting system Majority voting is considered as a special case of (K, n) i.e. K out of n system. In this n must be an odd number, so that majority average value is considered. The figure shows this system. Assumption: voting equipment is working perfectly whether components are working or not working. The structure function for this system is
O(S) = ½{(O(A)$ ~)(B)) u (®(A)$ ~)(C)) u (O(B)$ O(C))}
Z~
1,0 0,8 0,6 0,4
0,0
1,0
1,0
0,9
0,8
0,7
0,5
0,8
0,9
0,8
0,7
0,6
o,4
0,6
0,8
0,7
0,6
O,5
0,3
0,4
07
0,6
0,5
0,4
0,2
o,o
o,5
o,4
o,3
o,2
o,o
Z~
1,0
O,8
0,6
0,4
W(S) = (sl/1,0 , s2/0,8 , s3/0,6 , s J0,5, s5/0,4, s6/0,0). This can also be represented graphically in the same manner as explained in (2,3) system. This is given in Figs. 9-11. If components A and B are in working state and the component C is functioning with less performance index, then the performance index of the system is represented as the average value of the performance index of the components operating in the system.
0,0
5.5. Non-series parallel system A non-series parallel system is one which does not follow the simple series or parallel system's laws. For example consider a bridge network.
1,0
1,0
0,9
0,8
0,7
0,5
o,8
o,9
o,8
0,7
o,6
0,4
0,6
0,8
0,7
o,6
0,5
O, :5
0,4
0,7 0,6
0,5
0,4
0,2
0,0
0,5
0,-3 0,2
O,0
0,4
which is explained as the system is working if the average sum of performance of components is greater than 1/2. The system performance level for this system is defined as:
:igs. 9, t0, lI. Fuzzy performance graph for majority voting system.
Fig. 12. Bridge network.
Performance index to quantify reliability using fuzzy subset theory
4r3qxq4x3O
549
The antiroutes are KI, K2, K 3 and K 4 K l= K2 = K3 = K4 =
1-(1-xl)(1-x2) 1-(1-x4)(1-Xs) 1-(1-xl)(1-x3)(1-Xs) 1-(1-x2)(1-x3)(1-x4). 6. CONCLUSION
Fig. 13. A gen SP to a simple parallel arrangement.
The approach suggested in this paper is more realistic than the one which makes use of probability calculus in assessing the performance of the system.
Fig. 14. A gen SP to a simple series arrangement. A bridge network is an example of a generalized series-parallel system which is not series-parallel. Figure 12 represents a bridge network with xl, x2, x3, x4 and xs as the states of the components. The structure function O (x) is given as G ( X ) = (xl n x , ) u ( x l n x3 n Xs) u (x2 n x3 c~ x4) u (x2 n xs).
The maximal simple routes are PI, P2, P3 and P , el
= XIX3X5
P 2 ~- X2X3X4
P3 = XlX,lP4 ~ x2xs" This can be represented by a parallel structure (Fig. 13). The same arrangement can be represented by a series-parallel function for this is:
~ ( x ) = (xl u x2) c~ (x~ u x 3 u x s ) n (x~ u x3 u x , ) r~ ( x , u Xs).
The system performance can be judged through linguistic variables such as good, very good, etc. and thus the h u m a n thinking. Its interpretations are understandable and would help in making decisions promptly and effectively. The concept of performance index based on the fuzzy subset theory is intended to open up many versatile theories in the area of system performance assessment. REFERENCES
1. List of basic term, definitions and related mathematics for reliability, International Electrotechnical Commission IEC Pub. 271 (1974). 2. L. A. Zadeh, Fuzzy sets, lnformat. Cont. 8, 338-353 (1965). 3. L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets syst. I, 3-28 (1978). 4. Y. Hatoyama, Reliability analysis of 3-state systems, IEEE Trans. Reliab. R-28, 386-393 (1979). 5. Carsten BOe, Some views on risk and risk acceptance, Proceedings, 5th Symposium on Reliability Technology, University of Bradford, pp. 173-182 (1978).
0026-2714/81/040543-07$02.00/0 © 1981 Pergamon Press Ltd.
PERFORMANCE INDEX TO QUANTIFY RELIABILITY USING FUZZY SUBSET THEORY K. B. MISRA Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302, W.B., India and ARUNA SHARMA Electrical Engineering Department, University of Roorkee, Roorkee 247672. U.P.. India
(Receivedfor publication 11 September 1980) Abstract--The system performance is generally assessed through reliability. There are gaps in the quantification through probability. An alternate concept of performance index is proposed in the paper which is expected to permit the consideration of any number of graded states of a system or a subsystem states and laws to compute the performance index. The theory of fuzzy sets permits the mathematical models to be flexible and close to human thinking, assessment or interpretations. This helps considerably in decision making process as it brings human thinking very close to engineering system performance. probabilities associated with these random events are designated "reliability" and "unreliability" of the Very recently, H o t o y a m a [ 4 ] published a paper, in system. which he had considered a situation where a system In reality, the situation is not that simple. The could have three states instead of two discrete states transition from one state to the other state is not that viz., good and bad. The third state Hotoyama con- sharp or abrupt. In fact, since the performance assesssidered was designated as a "degraded" state. Thus it ment is closely related to human assessment of the was obviously an extension of usual binary algebra situation, it is subject to subjective assessment of the to take into consideration the multi-state structure performance. Thereby the transition from one state to function associated with the system. However, the the other is not that precise or sharp. There may be paper made use of decomposition technique in order situations where one could classify the performance to compute reliability of a system based on two- of system or of an equipment not in two precise state models. categories but many more, by saying the performance The purpose of the present paper is to draw is poor, satisfactory, good, very good or excellent. In attention to a more versatile theory which permits the this way there could be many more states possible consideration of system performance assessment where transition from one to the other is not very involving a n y number of states. Besides it is also sharp or alternatively the states are fuzzy states. Undoubtedly, that two-state model of reliability being intended to bring to notice the gap that exists at present in reliability of the theory of probability for used at present is a boundary states model in which reliability assessment. An alternate approach has no information is available on the intermediate states or, alternatively, performance is not defined for interbeen suggested in this paper. The IEC publication 271, 1974 provides the follow- mediate states. Further the mathematical tools are ing definition of reliability. only aids to human decision making process. The It is defined as "the ability of an item to perform a final decision making is done by human beings based on some mathematical calculations and interrequired function under stated conditions for a stated pretations. It will be more logical and proper if period of time". Obviously, the ability of an item to perform a mathematical tools or aids reflect the thinking function may be quantified in many ways, one of process of human beings and are close to it. The gap that exists in the mathematical models at them being the probability of successful operation which has been widely accepted as the definition of present is due to the fact that the models are not reliability. But the moment we quantify the per- consistent with the thinking of persons who handle formance in terms of probability, all related theories the engineering systems or eventually, based on their of probability and probability calculus become decision making capability, control them. This explains why people are not accepting the "statistical applicable to solve the related performance models. In usual two-state models in reliability studies, we risk" calculations, realistic or practical as they are, based on "randomness" of events and have been consider the system to exist either in "good" or "bad" state. These two states constitute the two basic or leaning on "probability" computation using primary events. The probability of "success event" statistical data. The "perceived risk", on the other constitutes the reliability of the system. These two hand, is based on human judgement and is events are also considered to be random and the "possibilistic" in nature. A very interesting difference 543 I. I N T R O D U C T I O N
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of these two types of risks has been brought out by Carsten Bee [5]. In fact, in the recent past these considerations have generated such strong opinions or feelings that they have virtually threatened various Governments in Western Europe. The difference between "statistical risk" and "perceived risk" is the same as the difference between "probability" and "possibility". An interesting article on the Theory of Possibility has been published by Zadeh[3], who has very clearly brought out the various aspects of the two situations. The theory of fuzzy sets as proposed by Zadeh [2] allows consideration of more than two states of a system. It also permits us to develop mathematical tools which are close to our thinking process and form the basis of decision making capability. The authors of this paper intend to provide the basic mathematical tools which allow these considerations of performance assessment of a system in terms of "performance index" rather than through reliability which is quantified in terms of "probability". The involvement of the theory of fuzzy subset opens up a vast vista of mathematical theories, in order to ease the consideration of system performance assessment through "performance index". Also, the theory of fuzzy sets allows the users of engineering systems to work out their own "operators" to carry out the analysis associated with the assessment, in place of usual "union and intersection" law of probability. In short the theory of fuzzy sets offers a very wide range of mathematical tools to bring the models close to human thinking, interpretations and decision making. 2. N O M E N C L A T U R E
ordinary set or subset fuzzy subset xl is an element of fuzzy subset A grade of membership of x in A (membership characteristic function) c is a subset (inclusion) U union t'3 intersection complement of A pseudo-complement of A O empty set O(s) structure function of the system S qJ(S/ps) performance function of the system S with /as as the membership characteristic function W(A), W(B) performance function of the components A~B~...connected in a system represents the summation of fuzzy variables states of components i ai X ( a t , a 2 ..... a,) states of components in the system Path vector or a vector X such that O(X) > 0 Route vector Path set Ca (X) of a path vector Minimal path set path set of a minimal path vector or minimal route set A A xiEA ~lA(X)
Cut vector, or antiroute vector Cut set or antiroute set Minimal cut vector Minimal cut set PI,P2 ..... Pp Kt, K2,...,Kt<
a vector X such that Q(X) = 0 Co(X) of a cut vector a cut vector X such that Y > X implies O(Y) > 0 cut set of a minimal cut vector the minimal route set of Q the minimal antiroute set of Q.
2.1. Fuzzy subset The theory of fuzzy subsets is an advancement in precision of the classical maths and the pervasive imprecision of the real world. A set, characterized by a membership function ranging between zero and one, is defined as a fuzzy subset. Thus the usual set theory is a particular case of the theory of fuzzy subsets, which permits only two grades of membership: unity and zero. Generally in the physical world the boundaries of the objects are not defined precisely. For example, the performance of a system may be satisfactory, good, excellent, bad, poor etc. indicating that the membership grades of the system performance are of importance. If A represents a subset of set E, then mathematically the fuzzy subset A is defined as follows: i f x is an element of E then x e A then A = {(xl/0,0 ), (x2/0,3), (X3/0,5), (X4/0,8), (Xs/1,0)} is a fuzzy subset with different membership functions of x~ in a set A. This explains element x 1 is not present in A because pA(Xl)= 0,0 here, element x 2 is present with #A(X2) = 0,3 and so on. The fuzzy subset theory brings out the following interesting information. 1. Fuzzyness and randomness are entirely different properties. 2. Fuzzyness governs human thinking and decisions. 3. To deal with fuzzyness effectively old concepts have to be modified for the study of cybernetics including human assessment of engineering system performance. 2.2. Possibility distribution If F be a fuzzy subset of universe of discourse U~ and Pe is the characteristic function, X is a variable taking values in U; R(X) is the fuzzy restriction F on variable X; then X is F represents R ( X ) = F possibility distribution rcx
represent
the
7rx = R(X). The possibility distribution function 7tx associated with X is numerically equal to the grade of membership of F, i.e.
2.3. Difference between the fi~zzyness and randomness The difference between the fuzzyness and random-
Performance index to quantify reliability using fuzzy subset theory ness can be explained by membership grade function, by considering degree of compatibility (or possibility) rather than probability. Let us consider an example of the number of failures of an equipment in a system in a given time. The probability distribution (Py) and the possibility distribution (#y) for failure of equipments are given below: No. of ~ilures
1
2
3
4
5
6
7
pf
1 0.7
1 0.2
0.8 0.1
0.5 0
0.2 0
0 0
0 0
545
Further the probability theory also takes into consideration the following properties, which are not valid for fuzzy subset theory: a.d=0
and
a+d=l.
The corresponding expression for the above two are aAi:#0;
except for
a=0
or
a= 1
aVi~l;
except for
a=0
or
a = 1.
4. N O T I O N OF PERFORMANCE FUNCTION'
PI
This explains that (1) the possibility information is more basic than probability information. Possibility is also an important property; (2) the membership function for possibility is not same for the probability. Reliability index I means that the performance is up to the mark, and system can fail, that is the system cannot perform with 100~'0 reliability but from possibility aspect it can do so; (3) the lower value of possibility implies lower value of probability and higher value of possibility may have high value of probability but low value of probability does not mean low value of possibility but may have high value of possibility.
As illustrated previously that performance index plays an important role in describing the system performance, this gives interesting connections between the theory of fuzzy variables and the theory of functions of structure. In certain problems the operation of an equipment not only considers whether it is operating or not operating, but some intermediate states are used governed by certain performance levels. For example: (1) (2) (3) (4) (5)
functions perfectly functions very satisfactorily functions fairly well functions rather poorly does not function.
If x~E E; E = {x 1, x 2. . . . . x,}; A is fuzzy subset then ~tA(X3~[0, 1] represents the fuzzy variable. If a level of system for series system and for parallel system is given by a functions Ws and We respectively. Ws(xl,x2 . . . . . x,,) = xl A x2 A ... A x,,,
3. SOME OF T H E AXIOMS OF FUZZY SETS
and If ~/A(X) is the membership function of element x in the fuzzy subset A and ~LB(X)in the fuzzy subset B, then ifa =/ZA(X), b =/~n(x)...a, b . . . ~ M = 1-0,1], the following operations are performed on the quantities a, h , . . .
a A b = min(a, b) a V b = max(a, b) fi =l-a a~b=(aAb) V(aAli)
kI'/p(Xl,X 2 . . . . . Xn) = X 1 V X 2 V . . . V x n.
Then Ws and We are called the performance functions for series and parallel systems respectively. Considering the performance index (Reliability) of a parallel system containing components A and B in parallel. If a, b are the state variables, a, b e {0, 1} then the structure function is given algebraically as: O(a,b) = a S b - a . b
a A b = b A a] commutativity aVb=bV aS
and the performance index = W(S) = WIA,~8~ = max(A, B).
(a A b) A c = a A (b A c)~ associativity (aVb) Vc aV(bVc) J
This shows that redundancy does not improve the performance index.
aAa=a} idempotence aVa a
c)}
a A (b V c) = (a A b) V (a A aV (bAc) (aVb) A (aVc)
distributivity
aA0=0 aV0=a aAl=a aVl=l
(~)
=
a
aAb=fivli} _ aVb iAli
DeMorgan's theorems.
5. PERFORMANCE F U N C T I O N AND RELIABILITY DETERMINATION
Consider a system S, and A and B are the components of this system. If A and B have fuzzy states as a t and b~ as follows: at, b ~ = 1 represents perfectly good or excellent performance of the components A and B a2, b 2 = 0,8 represent good performance a3, b 3 = 0,6 represent fairly good performance a4, b4 = 0,4 represents poor performance as, b5 = 0,0 represents components have failed.
546
K.B. MISRAand ARUNASHARMA
In
Out 0
O
Fig. 1. Series system. The performance of the system S depends on the performance of the components and the mode of operation of the components. This can be represented on the similar lines as s~ = s2 = s3 = s4 = ss =
1,0 0,8 0,6 0,4 0,0
= = = = =
excellent system performance good system performance satisfactory system performance poor system performance system has failed.
The X, Y and S represent the set of states of components A and B and system X = {(al/1,0), (a2/0,8), (a3/0,6), (a4/0,4), (as/0,0) } Y = {(b,/l,O), (b2/0,8), (b3/0,6), (b4/0,4), (bs/O,O)} S = {(sl/1,0), (s2/0,8), (s3/0,6), (s4/0,4), (Ss/0,0)lj. The system performance for different arrangements can be calculated by making use of different operators. Here conjunction and disjunction operators are used. 5.1. Series system If two components A and B are connected in series mode of operation. The performance function of this system depends on the individual performance of the connected components: the structure function of this system is given as
Q(s) = @(A)n Q(B) and the performance function is given as qJ(S) = q'(.4 ~ t~)The performance of the system in this case depends on the worst performance component, obviously the reliability also depends on less reliable component. Hence the performance index of the system can be expressed as W(S) = minlX, YI ---- [{(al/1,0), (a2/0,8), (a3/0,6), (a4/0,4), (a5/0,0)] n {(b J1,0), (b2/0,8), (b3/0,6), (b,/0,4),
(bdo,o)}] = [minl(a~/1,0)(bt/1,0), (al/1,0)(b2/0,8), (at/1,0)(b3/0,6), (a ~/1,0)(b,,/0,4), (al/1,0)(bs/0,0)l, mini (a2/0,8)(bt/1,0),
(a 2/O,8 ) (b2/0,8 ), (a z/O,8 ) (b3/0,6 ), (a2/0,8)(b4/0,4), (a2/O,8)(bs/O,O)l, minl(a3/0,6), (hi/i,0), (a3/O,6)(b2/0,8), (a3/0,6) (b,,/0,4), (a3/0,6)(bs/0,0)l, min I(a,,/0,4)(b~/1,0), (a4/O,4)(b2/0,8), (a4/0,4) (b3/0,6)(a4/0,4) (b,,/0,4), (a,/0,4)(b5/0,0)[, mini (as/0,0)(bl/1,0), (as/O,O)(b2/O,8), (as/0,0) (b3/0,6), (as/0,0)(b4/0,4), (as/0,0)(b5/0,0)l-].
'•
i,o
,,o
,,o
o,e
o,6
0,4
o,o o,o
\\\', o,s
o,s
\\\\ \\\\ 0,4 0,0
0,4 0,0
o,s \0,6\ o,4 ,,,\'\\
o,o
\\\\
~\\
o,o
,\\\
0,4
0,4
0,4
0,0
0,0
0,0
0,0
0,0
Fig. 2. The performance fuzzy graph for 2 components series system. This expression can be represented graphically using fuzzy logic graph as shown in Fig. 2. If the system performance is 0,6, this means that either of the component has 0,6 as performance index. The possible combination for this performance index is shown in Fig. 2. The marked portion represent the possible combinations. ~(s3/0,6 ) = [min(al/l,0)(b3/0,6), min(a,/0,8)(b3/0,6), (a3/0,6)(b3/0,6), min(bt/l,0)(a3/0,6), min(bj0,8)(a3/0,6)]. Algebraically the performance index is given as • (S) = T(A). q~(B). 5.2. Parallel s3,stem The system is working iff at least one component is good, it failed iff all the components are bad. Consider A and B components are connected in parallel mode, then the system structure function is written as: (Z)(S) = (Z)(A) u (Z)(B) and the performance function is • (S) = "PH u 8). The performance of the system depends on the performance of best component i.e. the component
In L ~
A ~
Fig. 3. Parallel system.
Out
Performance index to quantify reliability using fuzzy subset theory
547
®(s) = [((Z)(A)n ®(B))~ ((Z)(A)n (Z)(c)) ho
0,8
0,4
0,0
u ((Z)(B) ~ (Z)(C))] and the pertormance function is
1,0
\\\\\\\',
~P(S)= [~P((A~ B) w (An C) u (B ~ C))] \,
o~
0,6
0,6
~
0,4
,~\\\ \\\',\\\\
Using the same method (as followed for parallel and for series systems) for graphical representation of the system, it can be represented as in Figs. 5-7.
\0,8 \
\
,\\\\
= max[min(X, Y), rain(X, Z), min(Y, Z)].
~0 \0,4\ ,\\ \\\\
qo
y • 1,0
0,8
0,6
0,4
0,0
Fig. 4. Performance graph for parallel system.
ho
i,o
0,8
0,6
0,4
o,o
having highest performance. So the performance function is expressed as
0,8
0,8
0,8
0,6
0,4
0,0
qqS) = maxlX, YI
0,6
0,6
0,6
0,6
0,4
0,0
0,4
0,4
0,4
0,4
0,4
0,0
o,o
o,o
o,o
o,o
o,o
o,o
1,0
0,8
0,6
0,4
0,0
1,0
1,0
0,8
0,6
0,4
0,0
0,8
0,8
0,8
0,6
0,4
0,0
0,6
0,6
0,6
0,6
0,4
0,0
0,4
0,4
0,4
0,4
0,4
0,0
0,0 0,0
0,0
0,0
0,0
0,0
= [maxl(a,/l,O)(az/O,8)(a3/O,6)(a4/0,4) (%/0,0) w (bl/1,O), (b2/O,8)(b3/O,6)(b.jO,4)(bUO.O)l]
= D(al/1,O) u (bl/l,O), (al/1,O) w (b2/0,8), (al/1,O) u (b3/0,6), (al/1,O) u (bJO,4), (at/l,0) u (bs/0,0)l, I(a2/0,8) w (b~/1,0), (a2/0,8) L9 (b2/0,8), (a2/0,8) ~ (b3/0,6), (a2/0,8) ~ (b4/0,4), (a2/0,8)w (bs/O,O)l, L(a3/O,6) u (bl/1,O), (a3/0,6) w (b2/0,8), (a3/0,6) w (b3/0,6), (a3/0,6) w (b4/0,4), (a3/0,6) w (bs/0,0) I, [(a4/0,4)u (bl/i,0), (a4/0,4) w (bz/0,8), (a4/0,4) w (b3/0,6), (a4/0,4) w (b4/0,4), (a4/0,0) w (bs/0,0)l, 1(%/0,0) w (bl/0,0), (%/0,0) u (bz/0,8), (as/0,0) u (a3/0,6), (as/0,0) u (b4/0,4), (as/O,O) w (bs/O,O)/-J. So the perfoi'mance depends on the maximum value of the network component. Graphical representation of the function is given in Fig. 4. If the system performance is 1,0 then either component A or B's performance is excellent. The possible state combination is shown shaded in the fuzzy logic graph.
Z~
z~
~,0
0,8
0,6
0,4
0,0
I,o
~,0
0,8
0,6
0,4
0,0
0,8
0,8
0,8
0,6
O,4
o,o
If we use algebraic operation, then the performance index is given as hv(S) = qqA)$q~(B).
0,6
o,6
o#
0,6
o,4
o,o
5.3. Two out of three components
0,4
0,4
0,4
0,4
0,4
0,0
o,o
o,o
o,o
o,o
o,o
ud(S/1,O) = I(a~/l,O w b,/1,O), (al/1,O w b3/0,6), (al/l,O w bs/O,O), (bt/1,O w b3/0,6), (bt/l,0 w %/0,0)1
(al/1,O w (al/1,O w (bl/1,O u (b,/1,O u
b2/0,8), b4/0,4), a2/0,8), b4/0,4),
Three components are connected in parallel mode in a system. The system is working, if, any two components are working. Let the component C also has the Z as a set of states and C~ states, i = 1 to 5. The structure function is
o,o
Figs. 5, 6, 7. Fuzzy performance graph of two out of three system.
548
K.B. MISRAand ARUNASHARMA
Voter
J
Out
Fig. 8. Majority voting system.
J#
0,8
0,6
0,4
0,0
i,o
i,o
0,9
o,8
o,7
0,5
0,8
0,9
0,8
0,7
0,6
O,4
0,6
0,8
0,7
0,6
0,5
O,3
0,4
0,7
0,6
0,5
0,4
0,2
0,0
0,5
0,4
0,3
0,2
o,o
For finding the performance index of the system firstly the minimum value of the three combinations is found out using series system method and then by using parallel system operation maximum value is found. 5.4. Majority voting system Majority voting is considered as a special case of (K, n) i.e. K out of n system. In this n must be an odd number, so that majority average value is considered. The figure shows this system. Assumption: voting equipment is working perfectly whether components are working or not working. The structure function for this system is
O(S) = ½{(O(A)$ ~)(B)) u (®(A)$ ~)(C)) u (O(B)$ O(C))}
Z~
1,0 0,8 0,6 0,4
0,0
1,0
1,0
0,9
0,8
0,7
0,5
0,8
0,9
0,8
0,7
0,6
o,4
0,6
0,8
0,7
0,6
O,5
0,3
0,4
07
0,6
0,5
0,4
0,2
o,o
o,5
o,4
o,3
o,2
o,o
Z~
1,0
O,8
0,6
0,4
W(S) = (sl/1,0 , s2/0,8 , s3/0,6 , s J0,5, s5/0,4, s6/0,0). This can also be represented graphically in the same manner as explained in (2,3) system. This is given in Figs. 9-11. If components A and B are in working state and the component C is functioning with less performance index, then the performance index of the system is represented as the average value of the performance index of the components operating in the system.
0,0
5.5. Non-series parallel system A non-series parallel system is one which does not follow the simple series or parallel system's laws. For example consider a bridge network.
1,0
1,0
0,9
0,8
0,7
0,5
o,8
o,9
o,8
0,7
o,6
0,4
0,6
0,8
0,7
o,6
0,5
O, :5
0,4
0,7 0,6
0,5
0,4
0,2
0,0
0,5
0,-3 0,2
O,0
0,4
which is explained as the system is working if the average sum of performance of components is greater than 1/2. The system performance level for this system is defined as:
:igs. 9, t0, lI. Fuzzy performance graph for majority voting system.
Fig. 12. Bridge network.
Performance index to quantify reliability using fuzzy subset theory
4r3qxq4x3O
549
The antiroutes are KI, K2, K 3 and K 4 K l= K2 = K3 = K4 =
1-(1-xl)(1-x2) 1-(1-x4)(1-Xs) 1-(1-xl)(1-x3)(1-Xs) 1-(1-x2)(1-x3)(1-x4). 6. CONCLUSION
Fig. 13. A gen SP to a simple parallel arrangement.
The approach suggested in this paper is more realistic than the one which makes use of probability calculus in assessing the performance of the system.
Fig. 14. A gen SP to a simple series arrangement. A bridge network is an example of a generalized series-parallel system which is not series-parallel. Figure 12 represents a bridge network with xl, x2, x3, x4 and xs as the states of the components. The structure function O (x) is given as G ( X ) = (xl n x , ) u ( x l n x3 n Xs) u (x2 n x3 c~ x4) u (x2 n xs).
The maximal simple routes are PI, P2, P3 and P , el
= XIX3X5
P 2 ~- X2X3X4
P3 = XlX,lP4 ~ x2xs" This can be represented by a parallel structure (Fig. 13). The same arrangement can be represented by a series-parallel function for this is:
~ ( x ) = (xl u x2) c~ (x~ u x 3 u x s ) n (x~ u x3 u x , ) r~ ( x , u Xs).
The system performance can be judged through linguistic variables such as good, very good, etc. and thus the h u m a n thinking. Its interpretations are understandable and would help in making decisions promptly and effectively. The concept of performance index based on the fuzzy subset theory is intended to open up many versatile theories in the area of system performance assessment. REFERENCES
1. List of basic term, definitions and related mathematics for reliability, International Electrotechnical Commission IEC Pub. 271 (1974). 2. L. A. Zadeh, Fuzzy sets, lnformat. Cont. 8, 338-353 (1965). 3. L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets syst. I, 3-28 (1978). 4. Y. Hatoyama, Reliability analysis of 3-state systems, IEEE Trans. Reliab. R-28, 386-393 (1979). 5. Carsten BOe, Some views on risk and risk acceptance, Proceedings, 5th Symposium on Reliability Technology, University of Bradford, pp. 173-182 (1978).