CC BY
8
We assume that each component has a Weibull hazard
function and a constant repair rate. Components are
maintained preventively at periodic times.
Data is presented on Table 1.
First we present the nomenclature.
6;, pi - parameters of hazard function.
TTR - Mean Time to Repair (corrective maintenance).
TTP - Time of one preventive maintenance action.
PMC - Preventive maintenance cost.
CMC - Corrective maintenance cost.
t - time between two consecutive preventive
maintenance tasks.
With this preventive maintenance plan the availability achieved is about 90,30% and the life cycle cost is
122055,79.
The target for availability is 90%.
The objective function was slightly modified in order
to include the cost of down time.
MATLAB was used to optimise the objective function.
Table 2 shows the results. With this new preventive
maintenance policy we have a reduction of 5,5% in
Life Cycle Cost (LCC) and simultaneously the
availability A achieved (92,70%) is greater than the
existing one (90,30%).
With these results as initial conditions we have applied the tool "SOLVER" of Excel and we got a better solution (Table 3).
Table 3. Results of MatLab + Excel optimisation
5. Conclusion
This paper deals with a maintenance optimisation problem for a series system. First we have developed
an algorithm to determine the optimum frequency to perform preventive maintenance in systems exhibiting Weibull hazard function and constant repair rate, in order to ensure its availability. Based on this algorithm we have developed another one to optimise maintenance management of a series system based on preventive maintenance over the different system components. We assume that all components of the system still exhibit Weibull hazard function and constant repair rate and that preventive maintenance would bring the system to the as good as new condition. We define a cost function for maintenance tasks (preventive and corrective) for the system. The algorithm calculates the interval of time between preventive maintenance actions for each component, minimizing the costs, and in such a way that the total downtime, in a certain period of time, does not exceed a predetermined value. The maintenance interval of each component depends on factors such as failure rate, repair and maintenance times of each component in the system. In conclusion, the proposed analytical method is a feasible technique to optimise preventive maintenance scheduling of each component in a series system.
Currently we are developing a software package for the implementation of the algorithms presented in this paper.
References
[1] Andrews, J.D. & Moss, T.R. (1993). Reliability and Risk Assessment, Essex, Longman Scientific & Technical.
[2] Dekker, R. (1996). Applications of maintenance optimisation models: a review and analysis, Reliability Engineering and System Safety, 51, 229240.
[3] Duarte, J.C., Craveiro, J.T. & Trigo, T. (2006). Optimisation of the preventive maintenance plan of a series components system, International Journal of Pressure Vessels and Piping 83, 244-248.
[4] Duffuaa, S. A. (2000). Mathematical Models in Maintenance Planning and Scheduling. In M. Ben Daya et al (eds.), Maintenance, Modelling and Optimisation, Massachusetts, Kluwer Academic Publishers.
[5] Elsayed, E. A. (1996). Reliability Engineering, Massachusetts, Addison Wesley Longman, Inc.
[6] Garg, A. & Deshmukh, S.G. (2006). Maintenance management: literature review and directions. Journal of Quality in Maintenance Engineering. 12 No. 3, 205-238.
[7] Gertsbakh, I. (2000). Reliability theory: with applications to preventive maintenance, Berlin, Springer.
Table 1. Initial conditions
Compon bi TTR TTP PM CM ti
ents Cost Cost
1 4472,136 2 100 10 2000 4000 2000
2 1873,1716 2 50 40 2500 5000 1500
3 500,94 2 80 10 1000 2000 250
Table 2. Results of MatLab optimisation
t1 1600.2
t2 1246.8
MatLab t3 170.7535
Optimisation A - % 92.70
LCC 115345.22
D LCC- % -5.5
t1 1606.498
t2 1255.498
MatLab + Excel t3 175.4996
Optimisation A - % 93.02
LCC 113809.75
D LCC- % -6.8
[8] Henley; Ernest J. & Kumamoto, Hiromitsu (1992). Probabilistic risk assessment: reliability engineering, design, and analysis, New York, IEEE Press.
[9] McCormick, N. J. (1981). Reliability and Risk Analysis, San Diego, Academic Press Inc.
[10] Modarres, M., Kaminskiy, Mark and Krivtsov, Vasily (1999). Reliability and risk analysis, New York, Marcel Dekker, Inc.
[11] Nakagawa, T. (2005). Maintenance Theory of Reliability, London, Springer-Verlag
[12] Valdez-Flores, C. & Feldman, R. M. (1989). A Survey of Preventive Maintenance Models for Stochastically Deteriorating Single-Unit Systems, Naval Research Logistics, 36: 419-446.
[13] Vatn, J., Hokstad, P. & Bodsberg, L. (1996). An overall model for maintenance optimisation, Reliability Engineering and System Safety, 51, 241257.
Dziula Przemyslaw Jurdzinski Miroslaw Kolowrocki Krzysztof Soszynska Joanna
Maritime University, Gdynia, Poland
On multi-state safety analysis in shipping
Keywords
safety, multi-state system, operation process, complex system, shipping Abstract
A multi-state approach to defining basic notions of the system safety analysis is proposed. A system safety function and a system risk function are defined. A basic safety structure of a multi-state series system of components with degrading safety states is defined. For this system the multi-state safety function is determined. The proposed approach is applied to the evaluation of a safety function, a risk function and other safety characteristics of a ship system composed of a number of subsystems having an essential influence on the ship safety. Further, a semi-markov process for the considered system operation modelling is applied. The paper also offers a general approach to the solution of a practically important problem of linking the multi-state system safety model and its operation process model. Finally, the proposed general approach is applied to the preliminary evaluation of a safety function, a risk function and other safety characteristics of a ship system with varying in time its structure and safety characteristics of the subsystems it is composed of.
1. Introduction
Taking into account the importance of the safety and operating process effectiveness of technical systems it seems reasonable to expand the two-state approach to multi-state approach in their safety analysis [2]. The assumption that the systems are composed of multistate components with safety states degrading in time gives the possibility for more precise analysis and diagnosis of their safety and operational processes' effectiveness. This assumption allows us to distinguish a system safety critical state to exceed which is either dangerous for the environment or does not assure the necessary level of its operational process effectiveness. Then, an important system safety characteristic is the time to the moment of exceeding the system safety critical state and its distribution, which is called the system risk function. This distribution is strictly related to the system multi-state safety function that is a basic
characteristic of the multi-state system. Determining the multi-state safety function and the risk function of systems on the base of their components' safety functions is then the main research problem. Modelling of complicated systems operations' processes is difficult mainly because of large number of operations states and impossibility of precise describing of changes between these states. One of the useful approaches in modelling of these complicated processes is applying the semi-markov model [3]. Modelling of multi-state systems' safety and linking it with semi-markov model of these systems' operation processes is the main and practically important research problem of this paper. The paper is devoted to this research problem with reference to basic safety structures of technical systems [9], [10] and particularly to safety analysis of a ship series system [5] in variable operation conditions. This new approach to system safety investigation is based on the multi-
state system reliability analysis considered for instance in [1], [4], [6], [7], [8], [11] and on semi-markov processes modelling discussed for instance in [3].
2. Basic notions
In the multi-state safety analysis to define systems with degrading components we assume that:
- n is the number of system's components,
- Ei, i = 1,2,...,n, are components of a system,
- all components and a system under consideration have the safety state set {0,1,...,z}, z > 1,
- the safety state indexes are ordered, the state 0 is the worst and the state z is the best,
- T(u), i = 1,2,...,n, are independent random variables representing the lifetimes of components Ei in the safety state subset {u,u+1,...,z}, while they were in the state z at the moment t = 0,
- T(u) is a random variable representing the lifetime of a system in the safety state subset {u,u+1,...,z} while it was in the state z at the moment t = 0,
- the system and its components safety states degrade with time t,
- Ei(t) is a component Ei safety state at the moment t, t e< 0, ¥).
- S(t) is a system safety state at the moment t, t e< 0, ¥).
The above assumptions mean that the safety states of the system with degrading components may be changed in time only from better to worse. The way in which the components and the system safety states change is illustrated in Figure 1.
transitions
EZI:
worst state best state
Figure 1. Illustration of a system and components safety states changing
The basis of our further considerations is a system component safety function defined as follows.
Definition 1. A vector
si(t, ■ ) = [si(t,0), s,<t,1),..., s1(t,z)], t G< 0, ¥), (1) i = 1,2,..., n,
where
s(t,u) = P(E(t) > u | E(0) = z) = P(T(u) > t) (2)
for t e< 0,¥), u = 0,1,...,z, i = 1,2,...,n, is the probability that the component Ei is in the state subset {u,u +1,...,z} at the moment t, t e< 0, ¥), while it was in the state z at the moment t = 0, is called the multistate safety function of a component Ei.
Similarly, we can define a multi-state system safety function.
Definition 2. A vector
Sn(t, ■) = [Sn(t,0), Sn(t,1),..., Sn(t,z)], t e< 0, ¥), (3) where
Sn(t,u) = P(S(t) > u | S(0) = z) = P(T(u) > t) (4)
fort e< 0, ¥), u = 0,1,...,z, is the probability that the system is in the state subset {u,u +1,...,z} at the moment t, t e< 0, ¥), while it was in the state z at the moment t = 0, is called the multi-state safety function of a system.
Definition 3. A probability
r(t) = P(S(t) < r | S(0) = z) = P(T(r) £ t), (5)
t e< 0,¥),
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 multistate system.
Under this definition, considering (4) and (5), we have
r(t) = 1 - P(S(t) > r | S(0) = z) = 1 - Sn(t,r), (6)
t e< 0,¥),
and, if t is the moment when the risk exceeds a permitted level 5, 5 e< 0,1 >, then
t= r(5), (7)
where r -1(t), if it exists, is the inverse function of the risk function r(t) given by (6).
3. Basic system safety structures
The proposition of a multi-state approach to definition of basic notions, analysis and diagnosing of systems' safety allowed us to define the system safety function and the system risk function. It also allows us to define basic structures of the multi-state systems of components with degrading safety states. For these
basic systems it is possible to determine their safety functions. Further, as an example, we will consider a series system.
Definition 4. A multi-state system is called a series system if it is in the safety state subset {u, u +1,..., z} if and only if all its components are in this subset of safety states.
Corollary 1. The lifetime T(u) of a multi-state series system in the state subset {u, u +1,..., z} is given by
T(u) = min{Ti (u)} , u = 1,2,...,z.
1<i<n
The scheme of a series system is given in Figure 2.
Figure 2. The scheme of a series system
It is easy to work out the following result.
Corollary 2. The safety function of the multi-state series system is given by
sn(t, ■) = [1, Sn(t,1),..., sn(t,z)], t e< 0, ¥), (8)
where
Sn(t,u) = ns (t,u), t e< 0, ¥), u = 1,2,...,z. (9)
i=1
Corollary 3. If components of the multi-state series system have exponential safety functions, i.e., if
s(t, ■) = [1, s(t,1),..., s(t,z)], t e< 0, ¥),
where
si (t, u) = exp[-1i (u)t] for t e< 0, ¥), 1i (u) > 0, u = 1,2,...,z, i = 1,2,...,n,
then its safety function is given by
Sn(t, ■ ) = [1, Sn(t,1),..., Sn(t,z)], (10)
where
sn(t,u) = exp[-£ l. (u)t] for t e< 0, ¥), (11)
i=1
u = 1,2,...,z.
4. Basic system safety structures in variable operation conditions
We assume that the system during its operation process has v different operation states. Thus we can define Z(t), t e< 0,+¥>, as the process with discrete operation states from the set
Z = z2 , . . ., zv },
In practice a convenient assumption is that Z(t) is a semi-markov process [3] with its conditional lifetimes
9W at the operation state zb when its next operation state is zl, b, l = 1,2,..., v, b ^ l. In this case the
process Z(t) may be described by:
- the vector of probabilities of the process initial operation states [pb (0)]1 xv,
- the matrix of the probabilities of the process transitions between the operation states [pbl ]vxv , where pbb (t) = 0 for b = 1,2,..., v.
- the matrix of the conditional distribution functions [Hbl (t)]vxv of the process lifetimes 9bl, b ^ l, in the operation state zb when the next operation state is zl, where Hbl (t) = P(9u < t) for b, l = 1,2,...,v, b * l, and Hbb (t) = 0 for b = 1,2,..., v.
Under these assumptions, the lifetimes 9 bl mean values are given by
¥
Mbl = E[9bl ] =J tdHbl (t), b, l = 1,2,...,v, b * l. (12)
0
The unconditional distribution functions of the lifetimes 9 b of the process Z (t) at the operation states zb, b = 1,2,..., v, are given by
Hb (t) = ipwHw (t), b = 1,2,..., v.
l=1
The mean values E[9b ] of the unconditional lifetimes 9 b are given by
Mb = E[9b] = ipblMu , b = 1,2,..., v,
l=1
where Mbl are defined by (12).
Limit values of the transient probabilities at the operation states
pb (t) = P(Z(t) = zb) , t e< 0,+¥), b = 1,2,..., v, are given by
Pb = lim pb (t) =
P bMb
,M,
b = 1,2,..., v,
(13)
where the probabilities p b of the vector [p b ]1xv satisfy the system of equations
[p b ] =[p b ][ Pbl ]
I p, = 1.
l=1
We assume that the system is composed of n components E., i = 1,2,...,n, the changes of the process Z(t) operation states have an influence on the system components Ei safety and on the system safety structure as well. Thus, we denote the conditional safety function of the system component Ei while the system is at the operational state zb, b = 1,2,...,v, by
s(b)(t, ■)= [1, sf(t,1), s(b)(t, 2), ..., sf(t, z)], where
subset {u,u +1,...,z} is not less than t, while the process Z(t) is at the operation state zb.
In the case when the system operation time is large enough, the unconditional safety function of the system is given by
s„(t, ■)= [1 s„(t,1), s„(t,2), ..., s„(t,z)], t > 0, where
s„ (t, u) = P(T(u) > t) Pb su)
b=1
(14)
for t > 0, nb e{1,2,...,n}, u = 1,2,...,z, and T(u) is the unconditional lifetime of the system in the safety state subset {u,u +1,...,z}.
The mean values and variances of the system lifetimes in the safety state subset {u,u +1,...,z} are
m(u) = E[T(u)] pbm(b)(u), u = 1,2,...,z, (15)
b=1
where [2]
s(b)(t, u) = P(Tw(u) > t|Z(t) = zb)
for t e< 0, <x>), b = 1,2,...,v, u = 1,2,..., z, and the conditional safety function of the system while the system is at the operational state zb, b = 1,2,...,v, by
s nb' (t, ■)= [1, s lb}(t,1), s'n'b (t, 2), ..., s^t, z)],
Ub e {1,2,..., n},
<b)
where nb are numbers of components in the operation states zb and
slb^ u), = P(T(b)(u) > t|Z(t) = zb)
nb
for t e< 0, ¥), nb e {1,2,..., n}, b = 1,2,...,v, u = 1,2,... , z.
The safety function sj(b)(t, u) is the conditional probability that the component Ei lifetime Ti(b) (u) in the state subset {u,u +1,...,z} is not less than t, while the process Z(t) is at the operation state zb. Similarly, the safety function sU'5 (t, u) is the conditional
probability that the system lifetime T(b) (u) in the state
m(b)(u) = Js^(t,u)dt, nb e {1,2,..., n},
u = 1,2,..., z, and
■ (bV„M2 =
[c (b)(u)]2 = 2 J ts (t, u)dt - [m(b)(u)]2,
nb
(16)
(17)
u = 1,2,..., z,
for b = 1,2,...,v, and
¥
[c (u)]2 = 2 J ts n (t, u)dt - [m(u)]2, u = 1,2,..., z. (16)
The mean values of the system lifetimes in the particular safety states u, are [2]
m(u) = m(u) - m(u +1), u = 1,2,...,z -1, m (z) = m( z). (19)
5. Ship safety Model in constant operation conditions
We preliminarily assume that the ship is composed of a number of main technical subsystems having an essential influence on its safety. There are distinguished her following technical subsystems:
,=1
¥
S1 - a navigational subsystem, S 2 - a propulsion and controlling subsystem, S3 - a loading and unloading subsystem, S 4 - a hull subsystem, S5 - a protection and rescue subsystem, S 6 - an anchoring and mooring subsystem. According to Definition 1, we mark the safety functions of these subsystems respectively by vectors
s(t, ■) = [s(t,0), s(t,1),..., s(t,z)], t e< 0, ¥), (20) i = 1,2,.,6,
with co-ordinates
Si(t,u) = P(S(t) > u | Si(0) = z) = P(T(u) > t) (21)
fort e< 0, ¥), u = 0,1,...,z, i = 1,2,...,6, where Ti(u), i = 1,2,...,6, are independent random variables representing the lifetimes of subsystems Si in the safety state subset {u,u+1,...,z}, while they were in the state z at the moment t = 0 and Si(t) is a subsystem Si safety state at the moment t, t e< 0, ¥).
Further, assuming that the ship is in the safety state subset {u,u+1,...,z} if all its subsystems are in this subset of safety states and considering Definition 4, we conclude that the ship is a series system of subsystems Sj, S2, S3, S4, S5, S6 with a scheme presented in Figure 3.
S1 S2 S3 S4 S5 S6
Figure 3. The scheme of a structure of ship subsystems Therefore, the ship safety is defined by the vector
s6(t, ■)= [ S6 (t,0), s6 (t,1),..., S6(t, z)], (22)
t e< 0, ¥),
with co-ordinates s6 (t, u) = P(S(t) > u | S(0) = z) = P(T(u) > t) (23)
for t e< 0, ¥), u = 0,1,...,z, where T(u) is a random variable representing the lifetime of the ship in the safety state subset {u,u+1,...,z} while it was in the state z at the moment t = 0 and S(t) is the ship safety state at the moment t, t e < 0, ¥), according to Corollary 2, is given by the formula
s6(t,■)= [1, s6(t,1),..., s6(t,z)], t e< 0,¥X (24)
where
s6(t,u)= nSt(t,u),t e< 0,¥), u = 1,2,...,z. (25)
i =1
6. Ship operation process
Technical subsystems S1, S2, S3, S4, S5, S6 are
forming a general ship safety structure presented in Figure 3. However, the ship safety structure and the ship subsystems safety depend on her changing in time operation states.
Considering basic sea transportation processes the
following operation ship states have been specified:
z1 - loading of cargo,
z 2 - unloading of cargo,
z3 - leaving the port,
z 4 - entering the port,
z5 - navigation at restricted water areas,
z6 - navigation at open sea waters.
In this case the process Z(t) may be described by:
- the vector of probabilities of the initial operation states [pb (0)]1X6,
- the matrix of the probabilities of its transitions between the operation states [pbl ]6x6, where pbb (t) = 0 for b = 1,2,...,6,
- the matrix of the conditional distribution functions [Hbl (t)]6x6 of the lifetimes 9 bl, b * l, where Hbl (t) = P(9bl < t) for b, l = 1,2,...,6, b * l, and Hbb (t) = 0 for b = 1,2,...,6.
Under these assumptions, the lifetimes 9 bl mean values are given by
¥
Mbl = E[9bl ] =J tdHbl (t), b, l = 1,2,...,6, b * l. (26)
0
The unconditional distribution functions of the lifetimes 9 b of the process Z (t) at the operation states zb, b = 1,2,...,6, are given by
Hb (t) = i pblHbl (t), b = 1,2,...,6.
l=1
The mean values E[9b ] of the unconditional lifetimes 9 b are given by
Mb = E[9b ] = ^ pblMbl , b = 1,2,...,6, (27)
l=1
where Mbl are defined by (26).
Limit values of the transient probabilities at the operation states
Pb (t) = P(Z(t) = zb), t e< 0,+¥), b = 1,2,...,6,
are given by
Pb = lim Pb (t) =
P bMb Z P ,M,
,=1
b = 1,2.....6,
(28)
where the probabilities p b of the vector [p b ]1x6 satisfy the system of equations
[p b ] =[p b ][ Pbl ]
6
zp , = 1.
l=1
(29)
7. Safety model of ship in variable operation conditions
We assume as earlier that the ship is composed of n = 6 subsystems Si, i = 1,2,...,6, and that the changes of the process Z(t) of ship operation states have an influence on the system subsystems Si safety and on the ship safety structure as well. Thus, we denote the conditional safety function of the ship subsystem Si while the ship is at the operational state
zb, b = 1,2,...,6, by
sf(t, ■) = [1, s,(b)(t, 1), s(b)(t, 2), ..., s,(b)(t, z)]
(b).
(b)
where
s?\t, u) = P(Tw(u) > t|Z(t) = zb)
for t e< 0, ¥), b = 1,2,...,6, u = 1,2,..., z, and the conditional safety function of the ship while the ship is at the operational state zb, b = 1,2,...,6, by
sn?(t, ■)= [1, snb}(t,1), s^b(t,2), ..., snb}(t,z)],
<b)
where
snb)(t,u), = P(T(b)(u) > t|Z(t) = zb)
the state subset {u,u +1,...,z} is not less than t, while the process Z(t) is at the ship operation state zb.
Similarly, the safety function sU'5 (t, u) is the
nb
conditional probability that the ship lifetime T(b) (u)
in the state subset {u,u +1,...,z} is not less than t,
while the process Z(t) is at the ship operation state zb.
In the case when the ship operation time is large enough, the unconditional safety function of the system is given by
s6(t, ■)= [1, s6 (t,1), s6(t,2), ..., s6 (t,z)], t > 0,
where
s6(t, u) = P(T(u) > t) Pbsnb^ u)
'=1
(30)
for t > 0, nb e {1,2,3,4,5,6}, u = 1,2,...,z, and T(u) is the unconditional lifetime of the ship in the safety state subset {u,u +1,...,z}.
The mean values and variances of the ship lifetimes in the safety state subset {u,u +1,...,z} are
m(u) = E[T(u)} @ ZPbm(b)(u), u = 1,2,...,z, (31)
b=1
where
m (b)(u) = J s^(t, u)dt,
(32)
for b = 1,2,...,6, nb e {1,2,3,4,5,6}, u = 1,2,...,z, and
[c (u)]2 = D[T (u)] = 2 J ts 6(t, u)dt - [m(u)]2
0
u = 1,2,..., z,
(33)
The mean values of the system lifetimes in the particular safety states u, are
m(u) = m(u) - m(u +1), u = 1,2,...,z -1,
m (z) = m( z). (34)
nb
for t e< 0, ¥), b = 1,2,...,6, nb e {1,2,3,4,5,6}, u = 1,2,... , z.
The safety function si(b) (t, u) is the conditional
probability that the subsystem Si lifetime Ti(b) (u) in
8. Preliminary application of general safety model of ship in variable operation conditions
According to expert opinions [5] in the ship operation process, Z(t), t > 0, we distinguished seven operation
¥
states: z1, z2, z3, z4, z5, z6. On the basis of data coming from experts, the probabilities of transitions between the operation states are approximately given
by
[Pbl ]6x6 =
0.00 0.00 0.96 0.00 0.02 0.02
0.48 0.00 0.48 0.00 0.02 0.02
0.00 0.00 0.00 0.02 0.96 0.02
0.49 0.49 0.02 0.00 0.00 0.00
0.02 0.02 0.00 0.48 0.00 0.48
0.02 0.02 0.00 0.01 0.95 0.00
and the distributions of the ship conditional lifetimes in the operation states are exponential of the following forms:
Whereas, by (27), the unconditional mean lifetimes in the operation states are
M1 = E[q 1] = p13M13 + p15M15 + p16M16
= 0.96 ■ 2 + 0.02 -1 + 0.02 ■ 1 = 1.96, M 2 = E[9 2]
= p 21 M 21 + p 23M 23 + p 25M 25 + p26 M 26
= 0.48 ■ 2 + 0.48 ■ 2 + 0.02 ■ 1 + 0.02 ■ 1 = 1.96,
M 3 = E[9 3] = p34M 34 + p35M 35 + p36 M 36
= 0.02 ■ 0.04 + 0.96 ■ 0.04 + 0.02 ■ 0.08 = 0.0408,
H 13(f) = - exp[-0.5f], H15 (f) = 1- exp[-1.0f ], M 4 = £[0 4] = p 41M 41 + p42 M 42 + p43 M 43
H 16(f) = - exp[-1.0f], H21(f) = 1- exp[-0.5f ], = 0.49 • 0.08 + 0.49 • 0.08 + 0.02 • 0.04 = 0.0792,
H23 (f) = - exp[-0.5f], H25(f) = 1- exp[-1.0f ], M 5 = E[0 5]
H 26(f) = - exp[-1.0f], H34 (f) = 1- exp[ -25.0f ], = P51M 51 + P52M 52 + P54M 54 + P 56M 56
H35 (f) = - exp[-25.0f], H36 (f) = 1 - exp[-12.5f ], = 0.02 • 3 + 0.02 • 3 + 0.48 • 2 + 0.48 • 2 = 2.04,
H 51 (f) = - exp[-0.33f], H52(f) = 1 - exp[-0.33f ], M 6 = E[0 6]
H54 (f ) = - exp[-0.5f], H56 (f) = 1- exp[-0.5f ], = P61M 61 + P62M 62 + P64M 64 + P65M 65
H 61 (f ) = - exp[-0.2f], H62 (f) = 1- exp[-0.2f ], = 0.02 • 5 + 0.02 • 5 + 0.01 • 4 + 0.95 • 4 = 4.04.
H64 (f ) = - exp[-0.25f], H65 (f) = 1 - exp[-0.25f] Since from the system of equations
for f > 0. [P 1, P 2, P 3, P 4, P 5, P 6]
Hence, by (26), the conditional mean values of lifetimes in the operation states are
= [p ^ P 2 , P 3 , P 4 , P 5 , P 6][Pbl ]6x6 P 1 + P 2 + P 3 + P 4 + P 5 + P 6 = 1,
M13 = 2, M15 = 1, M16 = 1,
M21 = 2, M23 = 2, M25 = 1, M26 = 1,
M34 = 0.04, M35 = 0.04, M36 = 0.08,
M41 = 0.08, M42 = 0.08, M43 = 0.04,
M51 = 3, M52 = 3, M54 = 2, M56 = 2,
M61 = 5, M62 = 5, M64 = 4, M65 = 4.
we get
p j = 0.126, p 2 = 0.085, p 3 = 0.165,
p 4 = 0.155, p 5 = 0.312, p6 = 0.157,
then the limit values of the transient probabilities pb (t) at the operational states zb , according to (28), are given by
p1 = 0.145, p2 = 0.098, p3 = 0.004, p4 = 0.007,
p5 = 0.374, p6 = 0.372.
(35)
We assume that the ship subsystems , i = 1,2,...,6,
are its five-state components, i.e. z = 4, with the multistate safety functions
s,(b)(t, •) = [1, s,(b)(t,1), s?\t, 2), ..., s?\t, z)], b = 1,2,...,6, i = 1,2,...,6,
with exponential co-ordinates different in various ship operation states zb , b =1,2,...,6. At the operation states Zjand z2, i.e. at the cargo loading and un-loading state the ship is built of n = n2 = 4 subsystems S3, S4, S5 and S6 forming a series structure shown in Figure 4.
S3 — S4 — S5 — S6
Figure 4. The scheme of the ship structure at the operation states Zjand z2
We assume that the ship subsystems St, i = 3,4,5,6,
are its five-state components, i.e. z = 4, having the multi-state safety functions
s(b)(t, •)= [1, s(b)(t,1), s(b)(t,2), s(b)(t,3), s(b)(t,4) ], i = 3,4,5,6, b = 1,2,
with exponential co-ordinates, for b = 1,2, respectively given by:
- for the loading subsystem S3
s3(b)(t,1) = exp[-0.06t], s3b)(t,2) = exp[-0.07t], s3(b) (t,3) = exp[-0.08t], s3b) (t,4) = exp[-0.09t],
- for the hull subsystem S4
s4b)(t,1)= exp[-0.03t], s4b)(t,2) = exp[-0.04t],
s4b) (t,3) = exp[-0.06t], s4b) (t,4) = exp[-0.07t], for the protection and rescue subsystem S5
s5(b)(t,1)= exp[-0.10t], s5b)(t,2) = exp[-0.12t],
s5(b)(t,3) = exp[-0.15t], s5b)(t,4) = exp[-0.16t],
- for the anchor and mooring subsystem S6 ^6b\t,1)= exp[-0.06t], ^6b)(t,2) = exp[-0.08t], ^6b)(t,3) = exp[-0.10t], ^6b)(t,4) = exp[-0.12t].
Assuming that the ship is in the safety state subsets {u, u +1,..., z} , u = 1,2,3,4, if all its subsystems are in this safety state subset, according to Definition 1 and Definition 4, the considered system is a five-state series system. Thus, by Corollary 3, after applying (10)-(11), we have its conditional safety functions in the operation states z1 and z2 respectively for b = 1,2, given by
*T(t, •)
= [1, *ib)(t,1), sf b(t,2), s4b)(t,3), s4b)(t,4)], t > 0, b = 1,2,
where
S(b)(t,1) = exp[-(0.06 + 0.03 + 0.10 + 0.06)t]
= exp[-0.25t ],
sf)(t,2) = exp[-(0.07 + 0.04 + 0.12 +0.08)t]
= exp[-0.31t ],
SW(t,3)= exp[-(0.08 + 0.06 + 0.15 + 0.10)t]
= exp[-0.39t ],
sf)(t,4) = exp[-(0.09 + 0.07 + 0.16 + 0.12)t]
= exp[-0.44t] for t > 0, b = 1,2 .
The expected values and standard deviations of the ship conditional lifetimes in the safety state subsets calculated from the above result, according to (16)-(17), for b = 1,2, are:
m(b)(1) @4.00, m(b)(2) @ 3.26, m(b)(3) @2.56,
-(b)
m(b) (4) @ 2.27 years,
a (b)(1) @4.00, a (b)(2) @3.26, a (b)(3) @2.56
. (b)
(b)/
a (b)(4) @ 2.27 years,
- for the protection and rescue subsystem S5
and further, from (10), the ship conditional lifetimes in the particular safety states, for b = 1,2, are:
m (b)(1) @ 0.74, m (b)(2) @ 0.70, m (b)(3) @ 0.29,
m(b)(4) @ 2.27 years.
At the operation states z3 and z4 , i.e. at the leaving and entering state the ship is built of n3 = n4 = 5 subsystems S1, S2, S4, S5 and S6 forming a series structure shown in Figure 5.
Si
S?
S4
S5
Sfi
Figure 5. The scheme of the ship structure at the operation states z3 and z4
We assume that the ship subsystems St, i = 1,2,4,5,6,
are its five-state components, i.e. z = 4, having the multi-state safety functions
s(b)(f,•)= [1, s(b)(f,1),s(b)(f,2),s(b)(f,3),s(b)(f,4)], i = 1,2,4,5,6, b = 3,4,
with exponential co-ordinates, for b = 3,4, respectively given by:
- for the navigational subsystem S1
)(f,1) = exp[-0.15f], sib)(f,2)= exp[-0.20f],
5j(b) (i,3) = exp[-0.22t], 5j(b) (t,4) = exp[-0.25t],
- for the propulsion and controlling subsystem S 2
52b) (t,() = exp[-0.05t], 52b) (t,2) = exp[-0.06t],
s5(b) (t,1) = exp[-0.12t], s5b) (t,2) = exp[-0.14t], s5(b)(t,3) = exp[-0.16t], s5b)(t,4) = exp[-0.18t], - for the anchor and mooring subsystem S6 s6b)(t,1)= exp[-0.02t], s6b)(t,2) = exp[-0.04t],
s6b)(t,3) = exp[-0.06t], s6b)(t,4) = exp[-0.08t].
Assuming that the ship is in the safety state subsets {u, u +1,..., z} , u = 1,2,3,4, if all its subsystems are in this safety state subset, according to Definition 1 and Definition 4, the considered system is a five-state series system. Thus, by Corollary 3, after applying (10)-(11), we have its conditional safety functions in the operation states z3 and z4 respectively for b = 3,4, given by
s"5(b)(t, ■)
= [I s5b)(t,(), S5(b)(t,2), S5(b)(t,3), s5b)(t,4)L t > 0, b = 3,4,
where
s5(b)(t,() = exp[-(0.(5 + 0.05 + 0.04 +0.(2 + 0.02)t]
= exp[-0.38t],
i5(b)(t,2) = exp[-(0.20 + 0.06 + 0.05 +0.(4+ 0.04)t]
= exp[-0.49t ],
s(b)(t,3) = exp[-(0.22 + 0.07 + 0.07 + 0.(6 + 0.06)t]
= exp[-0.58t],
52b) (t,3) = exp[-0.07t], 52b) (t,4) = exp[-0.08t],
s5(b)(t,4) = exp[-(0.25 + 0.08 + 0.08 + 0.(8 + 0.08)t]
- for the hull subsystem S 4
54b) (t,() = exp[-0.04t], 54b) (t,2) = exp[-0.05t],
54b) (t,3) = exp[-0.07t], 54b) (t,4) = exp[-0.08t],
= exp[-0.67t] for t > 0, b = 3,4 .
The expected values and standard deviations of the ship conditional lifetimes in the safety state subsets calculated from the above result, according to (16)-(17), for b = 3,4, are:
m(b)(1) @ 2.63, mb) (2) @ 2.04, m(b)(3) @ 1.72, m(b) (4) @ 1.49 years,
a (b)(1) @ 2.63, a (b)(2) @ 2.04, a (b)(3) @ 1.72,
a (b)(4) @ 1.49 years,
and further, from (10), the ship conditional lifetimes in the particular safety states, for b = 1,2, are:
m(b)(1) @ 0.59, mb) (2) @ 0.32, m(b)(3) @ 0.23, m(b)(4) @ 1.49 years.
At the operation state z5 , i.e. at the navigation at restricted areas state the ship is built of n5 = 5 subsystems S1, S2 , S4 , S5 and S6 forming a series structure shown in Figure 6.
Si
S,
S4
S5
Sfi
Figure 6. The scheme of the ship structure at the operation state z 5
We assume that the ship subsystems St, i = 1,2,4,5,6,
are its five-state components, i.e. z = 4, having the multi-state safety functions
s(5) (t, •) = [1, s(5) (t,1), s(5) (t,2), s(5) (t,3), s(5) (t,4) ], i = 1,2,4,5,6,
with exponential co-ordinates respectively given by:
- for the navigational subsystem S1
s<5)(t,1) = exp[-0.18t], Sj(5)(t,2) = exp[-0.22t], s®(t,3) = exp[-0.24t], s<5)(t,4) = exp[-0.26t],
- for the propulsion and controlling subsystem S2 s25) (t,1) = exp[-0.06t], s25) (t,2) = exp[-0.07t],
s25) (t,3) = exp[-0.08t], s25) (t,4) = exp[-0.09t],
- for the hull subsystem S4
sf (t,1) = exp[-0.06t], sf (t,2) = exp[-0.08t],
545) (i,3) = exp[-0.09t], 545) (t,4) = exp[-0.10t],
- for the protection and rescue subsystem S5
555)(t,1)= exp[-0.14t], 555)(t,2)= exp[-0.15t],
sf> (t,3) = exp[-0.17t], s(55' (t,4) = exp[-0.20t],
- for the anchor and mooring subsystem S6
s65 (t,1) = exp[-0.02t], 565)(t,2)= exp[-0.03t],
s(6S) (t,3) = exp[-0.04t], s65) (t,4) = exp[-0.05t].
Assuming that the ship is in the safety state subsets {u, u +1,..., z} , u = 1,2,3,4, if all its subsystems are in this safety state subset, according to Definition 1 and Definition 4, the considered system is a five-state series system. Thus, by Corollary 3, after applying (10)-(11), we have its safety function given by
S55)(t, •)
= [1, s5(5) (t, 1), s5(5) (t, 2), s5(5) (t, 3), s5(5)(t, 4)], t > 0, where
s5(5)(t,1)= exp[-(0.18 + 0.06 + 0.06 + 0.14 + 0.02)t]
= exp[-0.46t ],
s5(5)(t,2)= exp[-(0.22 + 0.07 + 0.08 +0.15+ 0.03)t]
= exp[-0.55t],
s5(5)(t,3) = exp[-(0.24 + 0.08 + 0.09 + 0.17 + 0.04)t]
= exp[-0.62t ],
s5(5)(t,4)= exp[-(0.26 + 0.09 + 0.10 + 0.20 + 0.05)t] = exp[-0.70t] for t > 0.
The expected values and standard deviations of the ship lifetimes in the safety state subsets calculated from the above result, according to (16)-(17), are:
m
(()
(1) @ 2.17, m(6) (2) @ 1.82, m(6)(3) @ 1.61,
m(6)(4) @ 1.43 years,
c
(6)
(1) @2.17, c (6)(2) @ 1.82, c (6)(3) @ 1.61
c (6)(4) @ 1.43 years,
and further, from (10), the ship lifetimes in the particular safety states are:
m(6)(1) @0.35, m(6)(2) @0.21, m(6)(3) @0.18, m(6)(4) @ 1.43 years.
At the operation state z6, i.e. at the navigation at open sea state the ship is built of n6 = 4 subsystems S1, S2, S4, and S5 forming a series structure shown in Figure 7.
Si — S2 S4 — S5
Figure 7. The scheme of the ship structure at the operation state z 6
We assume that the ship subsystems St, i = 1,2,4,5,
are its five-state components, i.e. z = 4, having the multi-state safety functions
sf (t, •) = [1, sf (t,1), sf (t,2), sf (t,3), sf (t,4) ], i = 1,2,4,5,
with exponential co-ordinates respectively given by:
- for the navigational subsystem S1
si6)(t,1) = exp[-0.18t], Sj(6) (t,2) = exp[-0.22t], Sj(6) (t,3) = exp[-0.24t], s16) (t,4) = exp[-0.26t],
- for the propulsion and controlling subsystem S2
s26)(t,1)= exp[-0.06t], s26) (t,2) = exp[-0.07t], s26)(t,3) = exp[-0.08t], s26)(t,4)= exp[-0.09t],
- for the hull subsystem S4
sf(t,1)= exp[-0.05t], s46) (t,2) = exp[-0.06t],
s46)(t,3) = exp[-0.07t], s46) (t,4) = exp[-0.08t],
- for the protection and rescue subsystem S5 s56)(t,1) = exp[-0.15i], s(f)(t,2)= exp[-0.16t],
s56)(t,3) = exp[-0.18t], s56)(t,4)= exp[-0.22t].
Assuming that the ship is in the safety state subsets {u, u +1,..., z} , u = 1,2,3,4, if all its subsystems are in this safety state subset, according to Definition 1 and Definition 4, the considered system is a five-state series system. Thus, by Corollary 3, after applying (10)-(11), we have its safety function given by
s"i7)(t, •)
= [1, s47)(t,1), s47)(t,2), s47)(t,3), s47)(t,4)],t > 0, where
s46)(t,1) = exp[-(0.18 + 0.06 + 0.05 + 0.15)t]
= exp[-0.44t ], «46)(t, 2) = exp[-(0.22 + 0.07 + 0.06 +0.16)t]
= exp[-0.51t ], Sj6)(t,3) = exp[-(0.24 + 0.08 + 0.07 + 0.18)t] = exp[-0.57t ],
s46)(t,4) = exp[-(0.26 + 0.09 + 0.08 + 0.22)t]
= exp[-0.67t] for t > 0.
The expected values and standard deviations of the ship lifetimes in the safety state subsets calculated from the above result, according to (16)-(17), are:
m(6)(1) @ 2.27, m(6)(2) @ 1.96, m(6)(3) @ 1.75, m(6) (4) @ 1.49 years,
c (6)(1) @ 2.27, c (6)(2) @ 1.96, c (6)(3) @ 1.75, c (6)(4) @ 1.49 years,
and further, from (18), the ship lifetimes in the particular safety states are:
m(6)(1) @0.31, m(6)(2) @0.21, m(6)(3) @0.26, m(6)(4) @ 1.49 years.
In the case when the system operation time is large enough, the unconditional safety function of the ship is given by the vector
s6(t, •)
= [1, «6(t,1), S6(t,2), S6(t,3), *6(t,4)] t > 0, where, according to (14), the co-ordinates are
«6 (t, 1) = p sf (t, 1) + p2 sf (t, 1) + P3 «5(3) (t, 1)
+ p4 s(4)(t,1) + P( s(5)(t, 1) + P6 s46)(t,1) = 0.145 • exp[-0.25t] + 0.098 • exp[-0.25t] + 0.004 • exp[-0.38t] + 0.007 • exp[-0.38t] + 0.374 • exp[-0.46t] + 0.374 • exp[-0.44t ],
«6 (t, 2) = p, sf (t, 2) + p2 sf (t, 2) + P3 sf (t, 2)
+ p4s(4)(t,2) +p(sf(t,2) + p6 s46)(t,2) = 0.145 • exp[-0.31t] + 0.098 • exp[-0.31t] + 0.004 • exp[-0.49t] + 0.007 • exp[-0.49t] + 0.3.74 • exp[-0.55t] + 0.372 • exp[-0.51t],
s6 (t, 3) = px sf (t, 3) + p2 s42) (t, 3) + p3 s(3) (t, 3) + p4 s(4) (t, 3) + p5 s(5) (t, 3) + p6 s46) (t, 3) = 0.145 • exp[-0.39t] + 0.098 • exp[-0.39t] + 0.004 • exp[-0.58t] + 0.007 • exp[-0.58t] + 0.0374 • exp[-0.62t] + 0.372 • exp[-0.57t ],
s6 (t,4) = p, ,4) +p2 sf(t,4) + p3 sf(t,4) + p4 s(4) (t, 4) + p, sf (t, 4) + p6 sf (t, 4) = 0.145 • exp[-0.44t] + 0.098 • exp[-0.44t]
+ 0.004 • exp[-0.67t ] + 0.007 • exp[-0.67t ]
+ 0.374 • exp[-0.70t] + 0.372 • exp[-0.67t] for t > 0.
The mean values and variances of the system unconditional lifetimes in the safety state subsets, according to (31) and (33), respectively are
m(1) = pj m (1)(1) + p2 m (2)(1) + p3 m (3)(1) + p4 m (4)(1) + p( m (5)(1) + p6 m (6)(1),
@ 0.145 • 4.00 + 0.098 • 4.00 + 0.004 • 2.63 + 0.007 • 2.63 + 0.374 • 2.17 + 0.372 • 2.27 = 2.66.
[a (1)]2 @ 2[0.145 • [4.00]2 + 0.098 • [4.00]2
+ 0.004 • [2.63]2 + 0.007 • [2.63]2 + 0.374 • [2.17]2
+ 0.372 • [2.27]2] - [2.66]2 = [2.87]2, a(1) @ 2.87,
m(2) = p1 m(1) (2) + p 2 m (2)(2) + p3 m (3)(2)
+ p 4 m(4) (2) + p( m (5)(2) + p6 m (6)(2),
@ 0.145 • 3.26 + 0.098 • 3.26 + 0.004 • 2.04 + 0.007 • 2.04 + 0.374 • 1.82 + 0.372 • 1.96 = 2.22,
[a (2)]2 @ 2[0.145 • [3.26]2 + 0.098 • [3.26]2
+ 0.004 • [2.04]2 + 0.007 • [2.04]2 + 0.374 • [1.82]2
+ 0.372 • [1.96]2] - [2.22]2 = [2.38]2, a(2) @ 2.38,
m(3) = p1 m (1)(3) + p2 m (2)(3) + p3m (3)(3)
+ p4 m (4)(3) + p( m (5)(3) + p6 m (6)(3),
@ 0.145 • 2.56 + 0.098 • 2.56 + 0.004 • 1.72 + 0.007 • 1.72 + 0.374 • 1.61 + 0.372 • 1.75 = 1.89,
[a (3)]2 @ 2[0.145 • [2.56]2 + 0.098 • [2.56]2
+ 0.004 • [1.72]2 + 0.007 • [1.72]2 + 0.374 • [1.61]2
