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RELIABILITY, RISK AND AVAILABILITY BASED OPTIMIZATION OF COMPLEX TECHNICAL SYSTEMS OPERATION PROCESSES
PART 1
THEORETICAL BACKGROUNDS
K. Kolowrocki, J. Soszynska. •
Gdynia Maritime University, Gdynia, Poland e-mail: katmatkk@am.gdynia.pl, joannas@am.gdynia.pl
ABSTRACT
A convenient new tool for solving the problem of reliability and availability evaluation and optimization of complex technical systems is presented. Linking a semi-markov modeling of the system operation processes with a multi-state approach to system reliability and availability analysis is proposed to construct the joint general model of reliability and availability of complex technical systems in variable operation conditions. This joint model and a linear programming is proposed to complex technical systems reliability and availability evaluation and optimization respectively.
1 INTRODUCTION
Most real technical systems are very complex and it is difficult to analyze their reliability and availability. Large numbers of components and subsystems and their operating complexity cause that the evaluation and optimization of their reliability and availability is complicated. The complexity of the systems' operation processes and their influence on changing in time the systems' structures and their components' reliability characteristics is often very difficult to fix and to analyze. Usually the system environment and infrastructure have either an explicit or an implicit strong influence on the system operation process. As a rule some of the initiating environment events and infrastructure conditions define a set of different operation states of the technical system. A convenient tool for solving this problem is a semi-markov modeling of the system operation processes linked with a multi-state approach for the system reliability and availability analysis and a linear programming for the system reliability and availability optimization.
2 MODELLING SYSTEM OPERATION PROCESS
We assume that the system during its operation process is taking v, re N. different operation states. Further, we define the system operation process Z(t), ie<0,+°°>, with discrete operation states from the set of states Z = {zl,z2,...,zv}. Moreover, we assume that the system operation process Z(t) is semi-markov (Grabski 2002) with the conditional sojourn times 0w at the operation states zh when its next operation state is zu h. I = 1.2..... v. b^l. Under these assumptions, the system operation process may be described by [1], [2], [6] the vector of probabilities of the system operation process Z(t) initial operation states [ph (0)]|XI, the matrix of probabilities of the system
operation process Z(f) transitions between the operation states [pbl ]1XV and the matrix of conditional distribution functions of the system operation process Z(t) conditional sojourn times Ohl in the operation states [Hbl(t)]mvor equivalently by the matrix of corresponding conditional density functions [MOLv
From the formula for total probability it follows that the unconditional distribution functions of the sojourn times 9b,b = l,2,...,v, of the system operation process Z(t) at the operation states zh. b =1,2,..., v, are given by (Kolowrocki, Soszynska 2008)
Hb(f) = XpbIHbI(t), b = l,2,...,v.
1=1
(1)
Hence, the mean values E[0b ] of the unconditional sojourn times 0b, b =1,2,..., v, are given by
Mb=E[0b] = ^pblMbl,b = 1,2,..., v, i=i
(2)
where Mbl are defined by the formula
Mu = E[0bI] = ]tdHbl(t) = ]thbl(t)dt, b,l = \,2,...,v,b*l.
(3)
The limit values of the transient probabilities pb (t) at the particular operation states are given by (Grabski 2002, Kolowrocki, Soszynska 2008)
7TbMb
Pb = --, b= 1,2,-, V,
i=i
(4)
where Mb, b =1,2,..., v, are given by (2), while the stationary probabilities nh of the vector [7tb\x satisfy the system of equations
K] = K][Jpw]
2>,=1.
/=i
(5)
3 RELIABILITY, RISK AND AVAILABILITY OF MULTI-STATE SYSTEMS IN VARIABLE OPERAION CONDITIONS
In the multi-state reliability analysis to define systems with degrading (ageing) components we assume that:
- n is the number of system components,
- Eu i = 1,2,...,«, are components of a system,
- all components and a system under consideration have the state set {0,l,...,z}, z>\,
- the state indexes are ordered, the state 0 is the worst and the state z is the best,
- Ti(a), 7=1,2,...,«, are independent random variables representing the lifetimes of components Ei in the state subset {//,//+l,...,z}, while they were in the state r at the moment t= 0,
- T(u) is a random variable representing the lifetime of a system in the state subset {//,//+l,...,z} while it was in the state z at the moment t= 0,
- the system state degrades with time t without repair,
- ei(t) is a component Ei state at the moment t, te< 0,°°), given that it was in the state z at the moment t= 0,
- s(t) is a system state at the moment t, te< 0,°°), given that it was in the state z at the moment t
= 0.
The above assumptions mean that the states of the system with degrading components may be changed in time only from better to worse (Kolowrocki 2004, Kolowrock 2007). Under these assumptions, a vector
= te / = 1,2,...,«,
where
Ri{t,u)=P{ei{t) > u | el(Q) = z) = P(Tl(ii) >t),te u = 0,1,...
is the probability that the component Et is in the state subset {u.u + L...z} at the moment i, te< 0,°°), while it was in the state z at the moment t = 0, is called the multi-state reliability function of a component Ei. Similarly, a vector
Rn{t,-) = [Rn(t,0), Rn(t,\),..„ Rn{Kz)l te
where
Rn(t,u) = P(s(t) > u | s(0) = z) = P{T{u) >t),ts (-,-), w = 0,1,(6)
is the probability that the system is in the state subset {u,u z} at the moment t, te< 0,»), while it was in the state z at the moment t = 0, is called the multi-state reliability function of a system.
A probability
r(t) = P(s(t) < r | 5(0) = z) = P(T(r) <t),te »),
that the system is in the subset of states worse than the critical state r, r e {1 ,...,z} while it was in the state z at the moment t = 0 is called a risk function of the multi-state system or, in short, a risk (Kolowrocki 2004).
Under this definition, from (6), we have
r(t) = 1- P(s(t) > r|s(0) = z) = 1- Rn{ij\ te H»). (7)
and if ris the moment when the risk exceeds a permitted level S, then
T=r-\8), (8)
where r (!), if it exists, is the inverse function of the risk function r(t).
Further, we assume that the changes of the process Z(t) states have an influence on the system multi-state components E, reliability and the system reliability structure as well. Thus, we denote the conditional reliability function of the system multi-state component E, while the system is at the operational state zb, b = 1,2,...,v, by (Kolowrocki et all 2008, Limnios, Oprisan 2001, Ross 2007, Soszynska 2006a,b)
[R^t, -)](6) = [1, [i?,a,l)](6),..., [R,{t,z)n i = 1,2,...,n, where for te<0,°°), u = 1,2,..., z, b = 1,2,...,v,
[R,(t,i0r=P(T^(i0>t\Z(t) = zb)
and the conditional reliability function of the system while the system is at the operational state zb,
b = 1,2,..., v, by
[R.(t,-)]ib)=[i, [Rm(t,z)]w],
where for te<0,°°), u =1,2,..., z, b = 1,2,..., v.
[R,XtM(b) =P(T(b\i<)>t\Z{t) = zb\
and Tib)(u) is the system conditional lifetime at the operational state zh. dependent on the components conditional lifetimes at the operational state zb.
The reliability function is the conditional probability that the component £\ lifetime
T^b){u) in the state subset {u,u + l,...,z} is greater than i, while the process Z(l) is at the operation state zh. Similarly, the reliability function [Rn (l,u)]a > is the conditional probability that the system lifetime T(b)(u) in the state subset {u,u + l,...,z} is greater than t, while the process Z(t) is at the operation state zb. In the case when the system operation time 0 is large enough, the unconditional reliability function of the system
Rn{t,■)=[!, Rn{t,!),..., Rn{t,z)l (9)
where
Rn{t,u)=P{T{u)>t), for te<0,°°), u= 1,2,..., z,
and T{u) is the unconditional lifetime of the system in the system reliability state subsets is given by
R„(t,u) = ipb[R„(t,«)l<b) for t> 0, u= 1,2,...,=, (10)
and the mean value of the system unconditional lifetime in the system reliability state subsets is
i
= "= 1,2,...,-, (11)
6=i
where
M«) = ][R„(t,»)](b)dt, u= 1,2,...,z, (12)
0
and pb are given by (4) and the variance of the system unconditional lifetime in the system reliability state subsets is
a\u) = 2]t Rn(t,u)dt-[¿1(11)]2, u= 1,2,...,z. (13)
0
Additionally, according to (3.19) (Blokus et all 2008), we get the following formulae for mean values of the unconditional lifetime of the system in particular reliability states
Ji(u) = ju(u)~ ju(u +1), u = 0,1,..., z — 1, Ji(z) = ju(z). (14)
The main characteristics of multi-state renewal system with ignored time of renovation related to their operation process can be approximately determined by using results of the research report (Blokus et all 2008) formulated in the form of the following theorem. Theorem 3.1
If components of the multi-state renewal system with ignored time of renovation at the operational states have exponential reliability functions and the system reliability critical state is r,
re{l,2,...,z}, then:
i) the distribution of the time SN(r) until the Nth exceeding of reliability critical state r of this system, for sufficiently large TV, has approximately normal distribution N{N/.i{r),4Na{r)), i.e.,
F(f, r) = P(SN (r) <t) = Fm>1) t e Ho, oo), r e {1,2,..., z},
VNa(r)
ii) the expected value and the variance of the time SN{r) until the Nth exceeding the reliability critical state r of this system take respectively forms
D[SN(r)] = iV[cr(r)]2,re {1,2,..., =},
iii) the distribution of the number N{t,r) of exceeding the reliability critical state r of this system up to the moment t,t> 0, for sufficiently large i, is approximately of the form
Nu(r)-t (N + i)u(r)-t
P(N(t,r) = N) = FM(01)( - FN(01)(--), N = 0,1,2,..., re {1,2,..., z},
o-(r)
——
Mr) 1
Kr)
iv) the expected value and the variance of the number N{t,r) of exceeding the reliability critical state r of this system at the moment t,t> 0, for sufficiently large i, approximately take respectively forms
H(t,r) = ^— D{t,r) = —t—[a(r)]2, re {1,2,...,z}, M(r) [//(*■)]
where and ¿i(r) and a(r) are given by (11 )-(13) for u = r.
The main characteristics of multi-state renewal system with non-ignored time of renovation related to their operation process can be approximately determined by using results of the research report [1] formulated in the form of the following theorem. Theorem 3.2
If components of the multi-state renewal system with non-ignored time of renovation at the operational states have exponential reliability functions, the system reliability critical state is r, re {1,2,..., z}, and the successive times of system's renovations are independent and have an identical distribution function with the expected value /uQ (r) and the variance cr02 (r), then:
i) the distribution function of the time SN(r) until the Nth system's renovation, for sufficiently large N, has approximately normal distribution
N(N(ji(r) + ju0(r)),^N(a2(r) + <r»)), i.e.,
= (A° = t-N(u(r)+u (r))
F (t,r) = P(SN(r)<t) = FN((i (—=== ; ^ JJ), = 1,2,..., r e {1,2,...,z},
- V#(a2(r) + a;(r))
ii) the expected value and the variance of the time SN(r) until the Nth system's renovation take respectively forms
E[S„(r)] = N(ju(r) + Mo (r)), D[S„(r)] = N(<j2 (r) + a; (r)), r e {1,2,..., zj,
iii) the distribution function of the time SN(r) until the Nth exceeding the reliability critical state r of this system takes form
V#(a2(r) + a02(r))-a;(r)
re {1,2,...,z},
iv) the expected value and the variance of the time SN (r) until the Nth exceeding the reliability critical state r of this system take respectively forms
E[Sn (r)] = Nfi(r) + (N - l)Mo (r), D[Sn (r)] = Na2 (r) + (N - 1)<to2 (r), re {1,2,..., z}, v) the distribution of the number N(t,r) of system's renovations up to the moment t,t> 0, is of the
N(u(r)+u (r))-t (N + l)(u(r) +li (r))-t
P(N(t,r) = N)=Fmoi)( | =) -Fm0l)( "-AM =),
(CT2(r) + CTl(r)) J , ' , (cr2(r) + cr2(r))
V //(r) + //0 (r) ° //(r) + //0(r)
W = l,2,..., r e {1,2,..., z},
vi) the expected value and the variance of the number N(t,r) of system's renovations up to the moment t,t> 0, take respectively forms
H(t,r)= * D(t,r)= 1 (<x2(r) + <V(r)), r e {1,2,..., zj,
vii) the distribution of the number N(t,r) of exceeding the reliability critical state r of this system up to the moment t,t> 0, is of the form
N(u(r) + u (r)) -t- u, (r) (N +1 )(//(r) + u (r)) -t-jun (r)
N = 1,2,..., re{l,2,...,z},
t + ^ir) (<x2(r) + <x2(r)) //(r) + //0(r) ° \
t + ^ir) (a-(r) + a-(r)) //(r) + //0(r)
viii) the expected value and the variance of the number N{t,r) of exceeding the reliability critical state r of this system up to the moment t,t> 0, for sufficiently large i, are approximately respectively given by
H(Lr) = ■ D(t,r)= t+/*{r) ,(a-(r) + a;(r)), r e {1,2,...,z},
ix) the availability coefficient of the system at the moment t is given by the formula
A{t,r) = —^— t >0, re {1,2,...,zj,
x) the availability coefficient of the system in the time interval <t,t + v), r > 0, is given by the formula
A(t,r,r) = ——^——]Rn(t,r)dt, t> 0, r > 0, re {1,2,...,z}, n(r) + n0(r) r
where R„{t,r) is given by the formula (10) and /u(r)and a(r) are given by (11)-(13) for u = r.
4 OPTIMIZATION OF A SYSTEM OPERATIONPROCESS 4.1 Optimal transient probabilities maximizing system lifetime
Considering the equation (10), it is natural to assume that the system operation process has a significant influence on the system reliability. This influence is also clearly expressed in the equation (11) for the mean values of the system unconditional lifetimes in the reliability state subsets.
From linear equation (11), we can see that the mean value of the system unconditional lifetime ju(u), u = 1,2,..., z, is determined by the limit transient probabilities pb, b = 1,2,..., v, of the system
operation states given by (4) and the mean values /ub{u), b = 1,2,..., v, u = 1,2,..., z, of the system conditional lifetimes in the reliability state subsets {u,u z}, u = 1,2,..., r, given by (3.6). Therefore, the system lifetime optimization approach based on the linear programming can be proposed. Namely, we may look for the corresponding optimal values pb, b= 1,2,..., v, of the transient probabilities pb, b = 1,2,..., v, in the system operation states to maximize the mean value ju(u) of the unconditional system lifetimes in the reliability state subsets {u,u +1,..., zj, u = 1,2,..., z, under the assumption that the mean values /ub{u), b= 1,2,..., v, u=\,2,...,z, of the system conditional lifetimes in the reliability state subsets are fixed. As a special case of the above formulate system lifetime optimization problem, if r, r =1,2,..., z, is a system critical reliability state, then we want to find the optimal values pb, b= 1,2,..., v, of the transient probabilities pb, b =1,2,..., v, in the system operation states to maximize the mean value ju(r) of the unconditional system lifetimes in the reliability state subset {r,r +1zj, r = 1,2,..., z, under the assumption that the mean values jub(r), b= 1,2,..., v, r = 1,2,..., r, of the system conditional lifetimes in this
reliability state subset are fixed. More exactly, we formulate the optimization problem as a linear programming model with the objective function of the following linear form
r
M(r) = HpbMb(r) (15)
for a fixed r e {1,2,..., zj and with the following bound constraints
IPb=l (16)
6=1
pb<pb<pb, b = 1,2,..., v, (17)
where
Hb{r), fj.b{r)> 0, b = 1,2,..., v, are fixed mean values of the system conditional lifetimes in the reliability state subset {r,r + l,...,z}
Pt, 0<pb<l and pb, 0<A<1 pb<pb,b = \,2,...,v,
are lower and upper bounds of the unknown transient probabilities pb, b = 1,2,..., r. respectively. Now, we can obtain the optimal solution of the formulated by (15)-(18) the linear programming problem, i.e. we can find the optimal values pb of the limit transient probabilities pb, b = 1,2,..., v, that maximize the objective function (15). First, we arrange the system conditional lifetime mean values mib)(r), b = 1,2,..., v, in non-increasing order
Mh (r) ^ Mb, (r)>...> Mbv ir), where bi e {1,2,..., v} for / = 1,2,..., v.
Next, we substitute
xi = Pb , x, = pbi, x, = pb. for i = 1,2,..., v (19)
and we maximize with respect to x, , / = 1,2,..., r. the linear form (15) that after this transformation takes the form
M(r) = ±x,Mbi(r) (20)
/=i
for a fixed r e {1,2,..., z) with the following bound constraints
ix,=l, (21)
7 = 1
X, <x, <x„ / = 1,2,..., V, (22)
where
Mbi(rl Mbi(r">- ' = 1,2,..., v,
are fixed mean values of the system conditional lifetimes in the reliability state subset {r,r + l,...,z} arranged in non-increasing order and
x, , 0 < x, < 1 and x,, 0 < x, < 1, x, <x,, i = 1,2,..., v, (23)
are lower and upper bounds of the unknown limit transient probabilities x, , i = 1,2,..., v,
respectively.
We define
x = T,Xn y = l~x (24)
7 = 1
x° = 0, x° = 0 and X1 =ix„ xJ=ix, for / = 1,2,..., v. (25)
7=1 /=1
Next, we find the largest value I e {0,1,...,r} such that
x'-x'<y (26)
and we fix the optimal solution that maximize (20) in the following way:
i) if I = 0, the optimal solution is
xl=y + xl and x,. = x,. for i = 2,3,..., v; (27)
ii) if 0 < I < v, the optimal solution is
X,. = x, for / 1.2.....I. x/+1 = y-x1 +x' +x/+1
x, = x, for ; = I + 2,1 + 3,..., V- (28)
iii) if I = v, the optimal solution is
x, = x, for i = 1,2,..., v. (29)
Finally, after making the inverse to (19) substitution, we get the optimal limit transient probabilities
pbj = x, for ; = 1,2,..., v, (30)
that maximize the system mean lifetime given by the linear form (15) giving its optimal value in the following form
A(r) = ipbMr) (31)
4=1
for a fixed r e {1,2,..., z}.
From the above, replacing r by u, u = 1,2,..., z. we obtain the corresponding optimal solutions for the mean values of the system unconditional lifetimes in the reliability state subsets {u,u +1,..., zj of the form
/"00 = £PbMb00 for u = 1,2,..., z,
4=1
and by (13) the corresponding values of the variances of the system unconditional lifetimes in the system reliability state subsets is
cr{ii) = l\i Rn(t,u)dt-[U(u)]2, u = 1,2,..., z, (33)
where ju(u) is given by (32) and R„(t,it), according to (9)-(10), is the coordinate of the corresponding optimal unconditional multistate reliability function of the system
R(t,) =[1, J?,(U),..„ RM(t,z)], (34)
given by
R„(t,u) =±pb[Rn(t,u)]lb) for t> 0, u 1.2.....r. (35)
4=1
and by (14) the optimal solutions for the mean values of the system unconditional lifetimes in the particular reliability states are of the form
Jl(u) = ju(u)~ju(u+Y), u = 0,1,...,z-1, ju(z) = ju(z). (36)
Moreover, considering (7) and (8), the corresponding optimal system risk function and the moment when the risk exceeds a permitted level S, respectively are given by
r(i)=l - R (t,r), te K»), (37)
t = r'\S),
(38)
where r~'(t), if it exists, is the inverse function of the risk function r{t).
Finally, replacing ju(r) by ju(r) and a(r) by a(r) in the expressions for the renewal systems characteristics pointed in Theorem 1 and Theorem 2, we get their corresponding optimal values.
4.2. Optimal sojourn times in operation states maximizing system lifetime
Replacing in (4) limit transient probabilities pb in operational states by their optimal values pb found in the previous section and the mean values Mb of the unconditional sojourn times in operational states by their corresponding unknown optimal values Mb we get the system of equations
7tMu
pb = ^r, 1.2.....V. (39)
After simple transformations the above system takes the form
(pl -1 )7llMl + pxn2M2 + ... + pl7lvMv =0
p27T1M1 + (p2 -1 )k2M2 +... + p27TvMv = o
pv7lxM
1 + pv7T2M2 + ... + (pv - \)nvMv = 0,
(40)
where Mb are unknown variables we want to find, pb are optimal limit transient probabilities determined by (30) and nb are probabilities determined by (2).
Since the above system is homogeneous then it has nonzero solutions when the determinant of the system equations main matrix is equal to zero, i.e. if its rank is less than v. Moreover, in this case the solutions are ambiguous. Anyway, if we fix the optimal values Mb of the mean values Mb of the unconditional sojourn times in operational states, for instance by arbitrary fixing one or a few of them, then it is also possible to look for the optimal values Mbl of the mean values Mu of the conditional sojourn times in operational states using the following system of equations
obtained from (2) by replacing Mb by Mb and Mu by Mu, were pbl are known probabilities of the
system operation process transitions between the operation states.
5 REFERENCES
Blokus-Roszkowska, A., Guze, S., Kolowrocki K, Kwiatuszewska-Sarnecka, B., Soszynska J. 2008. Models of safety, reliability and availability evaluation of complex technical systems related to their operation processes. WP 4 - Task 4.1 - English - 31.05.2008. Poland-Singapore Joint Project. MSHE Decision No. 63/N-Singapore/2007/0.
Grabski, F. 2002. Semi-Markov Models of Systems Reliability and Operations. Warsaw: Systems Research Institute, Polish Academy of Sciences.
Kolowrocki, K. 2004. Reliability of Large Systems, Elsevier.
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Kolowrocki, K. 2007b. Reliability modelling of complex systems - Part 2. International Journal of Gnedenko e-Forum "Reliability: Theory & Application", Vol. 2, No 3-4, 128-139.
Kolowrocki, K. & Soszynska, J. 2008. A general model of technical systems operation processes related to their environment and infrastructure. WP 2 - Task 2.1 - English - 31.05.2008. Poland-Singapore Joint Project. MSHE Decision No. 63/N-Singapore/2007/0.
Kolowrocki, K, Soszynska, J., Baranowski, Z., Golik, P. 2008. Preliminary modeling, statistical identification and evaluation of reliability, risk, availability and safety of port, shipyard and ship technical systems in constant and variable operation conditions. WP4 - Task 4.2. Preliminary reliability, risk and availability analysis and evaluation of a port oil transportation system in constant and variable operation conditions. WP4 - Sub-Task 4.2.1 - English - 30.11.2008. Poland-Singapore Joint Project. MSHE Decision No. 63/N-Singapore/2007/0.
Limnios, N., Oprisan, G. 2001. Semi-Markov Processes and Reliability. Birkhauser, Boston.
Ross, S. M. 2007. Introduction to Probability Models. Elsevier, San Diego, 2007.
Soszynska, J. 2006a. Reliability of large series-parallel system in variable operation conditions. International Journal of Automation and Computing. Vol. 3, No 2: 199-206.
Soszynska, J. 2006b. Reliability evaluation of a port oil transportation system in variable operation conditions. International Journal of Pressure Vessels and Piping. Vol. 83, Issue 4: 304-310.
Zpb№u=Mb, b = 1,2,...,v.
(41)
/=i
