CC BY
10
-¥< g(a)< aç í x - ar ö Fè J <a< 1 0
aç £ g (a ) < b œ x - ^ö œ x - arö FV oM o^ x - oF-1 (a)- aç 2(bç- aç)
£ g (a)< cç œ x - crö œ x - ö FV „T^H o?ö x -oF-1 (a) + cç- 2bç 2 ( cç- b)
g (a ) ^ cç 0 <a<F^x cçJ 1
Table 3. Integration range with respect to a
degenerates to a crisp number, and is maximum when the fuzzy variable is an equi-possible one, i.e., all values have the same possibility. In order to address such a requirement, Li and Liu [6] provided a new definition based on credibility measure.
Definition 6.1.(Fuzzy Entropy) Let x be a continuous fuzzy variable defined on a credibility space (©,2®, Cr ), then the fuzzy entropy, H[x], is defined by
where Z = g (a) = x - cO-1 (a) .
Then we obtain the average chance measure for the
event {X (w ,9) < x}
F
%
s
( x v / x -oF 1(a) - a
ch{X(w,0) £ x}= J 1 V.
œ x-^ö 2bz - a )
o
F
| x-bç ! f| x-cç
è s J x-oF-1(a) + cç - 2b è s
+ J -y ' -Z-da + J 1x (47)
of
2(cç - bJ
which leads to the average chance distribution
Y(x ) =
x - a.
ç
œ œ x - a ö
2(bç - aJ
F
V V ° 0
-F
œ x - bz öö
v s 00
x + cç - 2bç œ œ x - bz ö
2(cç - bJ
F
V v s 0
-F
œx -cz öö
+ F
V s 0
o
2(b - aç)
V s 00
x-aç o
Juf(u)du
x
o
2(cç - bJ
Juf(u)du
z x-cz
o
(48)
6. Fuzzy repair effect estimation under fuzzy maximum entropy principle
Entropy is a measure of uncertainty. The entropy of De Luca and Termini [1] characterizes uncertainty resulting primarily from the linguistic vagueness rather than resulting from information deficiency, and vanishes when the fuzzy variable takes all the values with membership degree 1. However, we hope that the degree of uncertainty is 0 when the fuzzy variable
H [X ] = J S (Cr ({0 : X (0 ) = u}))du
-¥
where
S (t) = -t ln t - (1 -1)ln(1 -1)
(49)
(50)
For convenience, we name S (t) as entropy density at point t.
The maximum entropy principle provides a route such that it is possible to select the parameter(s) l that maximizes the value of entropy function and satisfies certain given constraints for specifying a membership function with a given form. However, what we aim at is not obtaining parameters from the theoretical entropy function rather we must determine the parameters based on observations of the fuzzy variable, say, x . In other words, we need to develop a criterion to obtain data-assimilated membership function. Therefore, we suggest an empirical fuzzy entropy function for parameter searching since the optimal value of the data-dependent object function has to reflect the constraints specified by observational data implicitly. The data assimilated object function is the average of entropy densities evaluated at {z1,z2,L ,zn}respectively, i.e.,
J [- A, L2 ] =1 S S (Cr {Z (0 ) = z, ; (S )}) (51)
n i=1
where a finite interval [-Lj, L2], L2 > L1 > 0 is defined for the domain of the entropy. Note that with the finiteness of empirical entropy, J[-L1, L2 ] ® H[Z; l ] asymptotically with parameter constrained by the data structure and Z e [-L1, L2], L2 > L1 > 0 which guarantees the theoretical entropy H [Z] exists and finite in general. Then, we can estimate the parameter (az, , cz) of
the membership of fuzzy composite error in terms of maximum entropy principle. Furthermore, we can isolate a few repair as bad-as-old regime and thus
x-
o
+
o
repair effect is zero for estimation parameter o for specifying e i, the translation error because under triangular membership assumption, the empirical membership can be defined and satisfies the asymptotical requirements.
7. Conclusion
In this paper, we argue that a differential equation motivated regression model will result in a regression model with random fuzzy error terms and thus complete our mission for solidifying a rigorous mathematical foundation for the grey modelling on system repair effects proposed by Guo [3], [4]. The maximum entropy principle facilitates a way for fuzzy parameter estimation. However, the average change distribution is also providing a way for parameter data-assimilation.
Acknowledgements
This work was supported partially by South Africa Natural Research Foundation Grant FA2006042800024. We are grateful to Professor B.D. Liu for his constant encouragements of applying his fuzzy credibility measure theory. The error frequency in Figure 1 is provided by my MSc student, Mr Li Xiang.
References
[1] De Luca, A. & Termini, S. (1972). A Definition of Non-probabilistic Entropy in the Setting of Fuzzy Sets Theory. Information and Control 20, 301-312.
[2] Deng, J. L. (1985). Grey Systems (Social L Economical). The Publishing House of Defence Industry, Beijing (in Chinese).
[3] Guo, R. (2005a). Repairable System Modelling Via Grey Differential Equations. Journal of Grey System, 8 (1), 69-91.
[4] Guo, R. (2005b). A Repairable System Modelling by Combining Grey System Theory with Interval-Valued Fuzzy Set Theory. International Journal of Reliability, Quality and Safety Engineering, 12 (3), 241-266.
[5] Liu, B. (2004). Uncertainty Theory: An Introduction to Its Axiomatic Foundations. Springer-Verlag Heidelberg, Berlin.
[6] Li, P. & Liu, B. Entropy of Credibility Distributions for Fuzzy Variables. IEEEE Transactions on Fuzzy Systems,( to be published).
Guze Sambor
Maritime University, Gdynia, Poland
Numerical approach to reliability evaluation of two-state consecutive "k out of n: F" systems
Keywords
two-state system, consecutive " k out of « : F" system, reliability, algorithm Abstract
An approach to reliability analysis of two-state systems is introduced and basic reliability characteristics for such systems are defined. Further, a two-state consecutive " k out of n: F" system composed of two-state components is defined and the recurrent formulae for its reliability function are proposed. The algorithm for numerical approach to reliability evaluation is given. Moreover, the application of the proposed reliability characteristics and formulae to reliability evaluation of the system of pump stations composed of two-state components is illustrated.
1. Introduction
The assumption that the systems are composed of two-state components gives the possibility for basic analysis and diagnosis of their reliability. This assumption allows us to distinguish two states of system reliability. The system works when its reliability state is equal to 1 and is failed when its reliability state is equal to 0. In the stationary case the system reliability is the independent of time probability that the system is in the reliability state 1. The main results determining the stationary reliability and the algorithms for numerical approach to this reliability evaluation for consecutive "k out of n: F" systems are given for instance in [1], [5]-[6], An exemplary technical consecutive "k out of n: F" system can be found in [3]. There is considered the ordered sequence of n relay stations /',,, /',,, • ,En,, which have to reroute a signal from a source station E(l to a target station En+j. A range of each station is equal to k. It means, when the station En i = 0,1,...,«, is operating, it sends a signal directly to a station EM,...,Emm(j+t . The failed station does not send any
signal. The probability of efficiency of the stations E(l and En+1 is equal to 1. The signal from E(l to En+1
cannot be sent, if at least k consecutive stations out of /',,, /',,,• , En, are damaged.
The paper is devoted to extension of these stationary results to the non-stationary case and applying them in transmitting then for two-state consecutive "" k out of n: F" systems with dependent of time reliability functions of system components ([3]). Then, the reliability function, the lifetime mean value and the lifetime standard deviation are basic characteristics of the system.
2. Reliability of two-state consecutive "k out of n: F" systems
In the non-stationary case of two-state reliability analysis of consecutive "k out of n: F" systems we assume that ([3]):
- «is the number of system components,
- E-, i = 1,2are components of a system,
- Tj are independent random variables representing the lifetimes of components En i = 1.2.......
- Ri(t) = P(Ti >t),t g<0,cc\ is a reliability function of component E1, / = 1,2
- ^(0 = 1-R, (0 = P{T, < t),t g< 0,oo), is the
distribution function (unreliability function) of component Ei, i = 1,2,...,n.
Definition 1. A two-state system is called a two-state consecutive " k out of n: F" system if it is failed if and only if at least its k neighbouring components out of n its components arranged in a sequence of E1, E2, ..., En, are failed.
The following auxiliary theorem is proved in [3], [6].
Lemma 1. The stationary reliability of the two-state consecutive " k out of n: F" system composed of components with independent failures is given by the following recurrent formula
Rkn = <
1
n
1 -Ek
j=i
Pn Rk ,n-1
k-1
+ ^ pn-i Rk, n—i-1 ¿=1
for n < k, for n = k,
(1)
• n
j=n-i+1
for n > k,
where
- pt is a stationary reliability coefficient of component Ei, i = 1,2,...,n,
- qi is a stationary unreliability coefficient of component Ei, i = 1,2,...,n,
- Rkn is the stationary reliability of consecutive "k out of n: F" system.
After assumption that:
- Tkn is a random variable representing the lifetime of a consecutive "k out of n: F" system,
- Rkn(t) = P(Tkn > t), t î< 0, ¥), is the reliability function of consecutive "k out of n: F" system,
- Fk n (t) = 1 — Rkn (t) = P(Tkn £ t), t î< 0, ¥), is the distribution function of consecutive "k out of n: F" system,
we can formulate the following result.
Lemma 2. The reliability function of the two-state
consecutive " k out of n : F" system composed of
components with independent failures is given by the following recurrent formula
Rk.n (t ) =
1
1—Ef (t )
j=1
Rn (t)Rk,n—1(t)
k—1
for n < k, for n = k,
(2)
+ ^ Rn_t (t ) R i =1
• nn Fj (t)
k,n—i—1
(t)
for n > k,
j=n—i+1
for t e< 0, ¥).
Motivation. When we assume in formula (1) that
pt (t) = R (t), q (t) = F (t) for t e< 0,¥), i = 1,2,..., n,
we get formula (2).
From the above theorem, as a particular case for the system composed of components with identical reliability functions, we immediately get the following corollary.
Corollary 1. If components of the two-state consecutive "k out of n : F" system are independent and have identical reliability functions, i.e.
R (t) = R(t), Ft (t) = F(t) for t e< 0,¥), i = 1,2,..., n,
then the reliability function of this system is given by
Rk.n (t ) =
1
1 — [ F (t)]n
R(t ) Rk ,n—1 (t )
k—1
+ R(t ) Z F1 (t )
i=1
• R
for n < k, for n = k,
(3)
k, n—i
-1(t )
for n > k,
for t e< 0, œ).
In further considerations we will used the following reliability characteristics:
- the mean value of the system lifetime,
¥
E[TKn ] = J Rk,n(t)dt, (4)
- the second order ordinary moment of the system lifetime,
E[Tk2n ] = 2J t Rk, n(t)dt,
(5)
- the standard deviation of the system lifetime,
s=V D[Tkn ],
where
D[T,n)] = E[Tk2n ] - (E[Tkn ])2.
(6)
(7)
3. Algorithm for reliability evaluation of a two-state consecutive „k out of n: F" system
For numerical approach to evaluation of the reliability characteristics, given by (3)-(6), we use the trapezium rule of numerical integration. In particular situation, for 10 = 0, step h, we have
E[Tkn ] = J Rk,n(t)dt
h E Rk,n (t 0 + ih) + Rkn (t 0 + (i +1) • h)l (8)
2 i=0
E[T2n ] = 2J t Rk, n(t)dt
n-1 t
= h£{(t 0 + ih) • Rk ,n (t 0 + ih)
i=0
+ (t0 + (i +1) • h) • Rk n (t0 + (i +1) • h)}.
(9)
2. If k > n then Rk,n (t) = 1
3. else if k = n Rk,n (t) = 1 -[F(t)]n
4. else
5. for i = 0 to t do
6. {
7. for j = 1 to k - 1 do
8. temp := temp + [F (i)]1 • Rk ,n-1 -1 (i);
9. Rk ,n (i) = R(i) • Rk ,n-:(i) + temp;
10. }
where
Necessary in (7)-(8) values of function Rknn(t) are calculated from (2) using the following algorithm.
- k is a length of the sequence of consecutive components,
- n is a number of all components in sequence,
- t is an end of the time interval,
- F(t) is a distribution function of components,
- R(t) is a reliability function of components.
Example implementation of Algorithm 1 and formulas (3), (8)-(9) in the D programming language is given in Appendix.
4. Application
From Corollary 1, in a particular case, substituting k = 3 in (3), we get:
- for n = 1
R31 (t) = 1 for t e< 0,¥), (10)
- for n = 2
R3 2(t) = 1, for t e< 0,¥), (11)
- for n = 3
R3 3 (t) = 1 - F3(t) for t e< 0,¥), (12)
- for n > 4
RXn (t) = R(t) RXn-1 (t) + R(t)F(t) RXn-2 (t) + R(t)[F(t)]2 R3 n-3 (t) for t e< 0, ¥), (13)
U.i
Algorithm 1.
1. Given: t, k, n, F(t), R(t);
Example 1. Let us consider the pump stations system with n = 20 pump stationsEj,E2,...,E20. We assume that this system fails when at least 3 consecutive pump stations are down. Thus, the
considered pump stations system is a two-state consecutive "3 out of 20: F" system, and according to (9)-(12), its the reliability function is given by
R320(t)=R(t) Rug(t)
+ R(t)F(t)Rus(t)
+ R(t)[F(t)]2RU7(t) (14)
for t e< 0, oo).
In the particular case when the lifetimes '/,', of the pump stations /v, 7 = 1,2,* ,20 have exponential distributions of the form
F(t) = l-e-°mt for f > 0,
i.e. if the reliability functions of the pump stations /v, / = 1,2,* ,20 are given by
R(t) = e-°mt for t > 0,
considering (9)-(12), (13) we get the following recurrent formula for the reliability Rr ,n (!) of pump stations system
Ru(t) = 1 for fe<0,oo), (15)
R32(t) = \ for fe<0,oo), (16)
R33(t) = l- [l-e-0'01']3 for fe<0,oo), (17)
R3 Jt)=e-omt R3 n^)
+ e""u[]-e""u]Ryn2U)
+ e-001' [1 -e-001']2 R3^3(t) for t e< 0,oo), (18) 77 = 4,5,...,20.
The values of reliability function of the system of pump stations given by (14), calculated by the computer programme based on the formulae (10)-(18) and Algorithm 1, are presented in Table 1 and illustrated in Figure 1.
Table 1. The values of the two-state reliability function of the pump stations system for X = 0.01
t R 2 (>) 21 RX20(t)
0.0 1.0000 0.0000
5.0 0.9980 9.9800
10.0 0.9859 19.7189
15.0 0.9583 28.7499
20.0 0.9137 36.5474
25.0 0.8535 42.6743
30.0 0.7811 46.8657
35.0 0.7008 49.0561
40.0 0.6170 49.3614
45.0 0.5337 48.0347
50.0 0.4541 45.4117
55.0 0.3805 41.8584
60.0 0.3144 37.7282
65.0 0.2564 33.3331
70.0 0.2066 28.9274
75.0 0.1647 24.7024
80.0 0.1299 20.7893
85.0 0.1016 17.2662
90.0 0.0787 14.1688
95.0 0.0605 11.5004
100.0 0.0462 9.2416
105.0 0.0350 7.3588
110.0 0.0264 5.8107
115.0 0.0198 4.5531
120.0 0.0148 3.5426
125.0 0.0109 2.7385
130.0 0.0081 2.1044
135.0 0.0060 1.6082
140.0 0.0044 1.2229
145.0 0.0032 0.9255
150.0 0.0023 0.6974
155.0 0.0017 0.5235
160.0 0.0012 0.3916
165.0 0.0009 0.2918
170.0 0.0006 0.2168
175.0 0.0004 0.1607
180.0 0.0003 0.1188
185.0 0.0002 0.0876
190.0 0.0002 0.0644
195.0 0.0001 0.0473
200.0 0.0000 0.0347
205.0 0.0000 0.0253
210.0 0.0000 0.0185
215.0 0.0000 0.0135
220.0 0.0000 0.0098
225.0 0.0000 0.0072
230.0 0.0000 0.0052
235.0 0.0000 0.0038
240.0 0.0000 0.0027
245.0 0.0000 0.0020
