International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 2
Number 2
June 2008
173
Construction of the Ship’s Technical Failure
Model to Assess its Navigational Safety
L. Gucma & R. Gralak
Maritime University of Szczecin, Poland
ABSTRACT: The ship technical failures contribution in overall number of navigational accidents are
significantly smaller than those caused by human factor but in safety analysis they cannot be neglected. The
paper presents methodology of modeling the technical failures of ships with respect of most important ship
systems such as main engine, power generators and steering gear. The repair time is also taken into account.
The data for simulation was obtained from analysis of ships statistical data of polish owners. The model could
be used mostly in assessment with projecting phase of ship appliances, simulating owner’s economical
analysis or generating random events in marine simulators.
1 INTRODUCTION
Construction of ships can be designed using the
probabilistic methods of reliability analysis. In virtue
of its complicate level the ship technical appliances
such as engines, auxiliaries or other in the engine-
room are sensitive to environmental and human
factor influence. The interrelation between events
causing failures can be examined by application of
computer simulation method and reliability analysis.
Assuming the definition of reliability as feature of
device which allows of functioning without failure
in specific conditions and given time, we determine
its numerical and functional measures, which usually
are:
− expected value of working time until failure
u
T
,
− reliability function R(t),
− failure rate function λ(t).
These are measures of probabilistic nature,
because based on the assessment of probability for
occurrence of an event of failure preventing from
functioning, what is meant by providing function
intended for the technical device [1].
2 SIMULATION MODEL’S RESEARCH
POPULATION
Simulation model of ship failures to assess its
safety is an mathematical algorithm estimating, what
is the probability of damage occurrence or ship
failure, having the influence navigational safety.
The calculations are based on statistical distributions
properties, which reflect (on the level of suitable
confidence degree) the course of ships’ elements
damages in reality.
Rate of ship damages depends on many factors,
among others:
– the age of the unit,
– the type of the ship,
– tonnage,
– type of navigation,
– navigation basin,
– others.
Obtaining such detailed data to compile a
numerous and reliable research population is very
problematic, or even unavailable. Population on
which the model is based, is Polska Żegluga Morska
fleet and data from annual reports from ship repair
yards and producers of engineering ship devices.
174
Statistical data (table 2) on ship damages cover the 6
year-time period (1999-2005) and concern all the
company’s ships. Currently PŻM owns in total 77
ships, with total deadweight 2,1 mln dwt. These are
bulk carriers in groups: coaster (4 400 dwt),
numerous group of handy-size and panamaxes
(73 500 dwt). Except bulk carriers PŻM owns 4
sulfur cariers and ferries m/f Polonia and m/f Gryf
managed by Unity Line Szczecin.
Process of given system damage probability
estimation, finds its reflection in properties of
exponential distribution.
3 MODEL’S INPUT DATA
Input data and random variable of the damage and
ship’s failure model, have been prepared basing on
the statistical samples of PŻM fleet, Ship Repair
Yard and publications connected with the life-span
and use of ships’ systems.
Input variable is the amount of annual utilization
periods of the ship. By annual utilization period we
understand an average number of running days of the
ship’s systems per year (tab. 1). It fluctuates around
208 for ships with deadweight 1000-100000 DWT
(90% of PŻM fleet, being the research population).
Research population is the data with average number
of damages and failure of ships, during the
utilization period and average or limited repair times
(tab. 2).
Table 1. Annual average utilization time of ship. Source: [3]
DWT
Engine working time
Annual exploitation
time of the ship
Number of calls
to port
Number of days in
the port
Engine working time
Number of days
at sea
Time of demurrage
Max days at sea
(~5 days of demurrage)
[h]
[day]
-
[day]
[day]
[day]
[day]
[day]
[tons]
A
B
C
D=C x
1,5
E=A/
24
G=B - D
H=360 -
B
I=360 - D
< 5,000
4000
240
100
150
167
90
120
210
5,000 –
100,000
5000
270
60
90
208
180
90
270
> 100,000
6000
300
35
53
250
247
60
307
Average of
world fleet
5840
300
70
105
243
210
60
255
Table 2. Number of PŻM fleet damages/failures. Based on
PŻM data
Damage/failure
Annual number of damages/failures
1999
2000
2001
2002
2003
2004
2005
Damage of machinery
appliances :
[%] of
overall PŻM ships
-
main engine
10.1
2.2
18.5
23.0
17.1
21.3
5.3
-
auxiliares
7.9
6.6
7.4
10.8
9.2
8.0
6.7
-
other appliances
1)
1.1
0.0
2.5
5.4
5.3
4.0
2.7
Damage of steering:
[%] of overall PŻM ships
-
rudder
2.2
2.2
1.2
4.1
1.3
1.3
4.0
-
installation of main shaft
and propeller
6.7
5.5
7.4
5.4
3.9
5.3
2.7
By damage we understand the faultiness state of
the system, which is eliminated just and only by the
ship crew. By the failure of the ship, we mean the
faultiness state which occur during the utilization of
the ship, which can not be eliminated by the means
of ship’s crew, what results in necessity of repair in
shipyard or by service (tab. 3).
It has been accepted that every failure is preceded
by the inspection (minimum 30 minutes) of
unsuccessful trial to repair the damage.
Table 3. Number of events on one unit in the 1-year-period of
utilization time and approximated times of repair. Based on
PŻM and “Gryfia” Repair Yard data
*
– for remaining devices we account among others: boilers,
steam devices, cooling installation, fuel installation and other
secondary systems, excluding the main engine and auxiliary
engines. For damages we account also damages caused by fires.
Model does not take into consideration ageing
processes and renewal of particular systems’
elements, due to the lack of access to this type of
data.
Damage/failure
Average
number of
failures
Time of
repair
Average
number of
damages
Time of
repair
[failures/per
utiliz. period]
min
[h]
max
[h]
[damages
/ per
utiliz. period]
min
[h]
max
[h]
Damage of
machinery
appliances :
- main engine
0.139
168
2160
12.83
0.5
72
- auxiliares
0.081
60
336
5.3
84
- other
machinery
appliances *
0.030
4
250
4.24
110
Damage of
steering:
- ruder or
steering
instalation
0.023
264
496
1.1
0.5
69
- installation of
main shaft and
propeller
0.053
24
1465
-
-
-
175
4 THE FAULT-TREE MODEL’S ALGORITHM
Simulation model in the form of statistical fault-tree
simulates the probabilities of systems’ damages and
ships failures (fig. 1) in chosen amount of annual
periods of use. The above model has been worked
out basing on the random variable of X = t
exponential distribution simulation, describing the
course of probabilities for occurrence of an event,
after time t, described with the dependency:
)exp(1)( xxF
λ
−−=
(1)
Rn
tx
−
==
1
1
ln
1
λ
(2)
where: Rn = random variable of the uniform distri-
bution, containing in the range
]1,0[∈Rn
;
λ
= number of failures for one ship in 1-year period
of utilization (~208 days).
Simulation of damage of failure occurrence
proceeds according to the dependency:
)()( tPtP
LDD
>
(3)
where: P
D
(t) = probability of damage occurs;
P
LD
(t) = probability obtained from the random
variable of uniform distribution
]1,0[∈Rn
.
We assume that every failure is preceded by the
damage, which is impossible to remove in the scope
of abilities of ship crew members. Time needed to
remove the damage is simulated on the basis of
random variable of uniform distribution, assuming,
that it has a random character (according to the ship
repair yard employees) and that the minimal time of
repair equals to 30 minutes.
Simulation of the failure occurrence proceeds
according to the dependency:
)()( tPtP
LFF
>
(4)
where: P
F
(t) = probability of failure occurs;
P
LF
(t) = probability obtained from the random
variable of uniform distribution
]1,0[∈Rn
Results of computer simulation of the statistical
event-tree using the Monte Carlo method, are
obtained on the basis of dependency:
)()...()()(
21
tPtPtPtP
DnDDD
=
(5)
)()...()()(
21
tPtPtPtP
FnFFF
=
(6)
where:
)(tP
n
= probability of last damage or failure
that occurred in simulation and:
DnDDD
tttt +++= ...
21
(7)
FnFFF
tttt +++= ...
21
(8)
where: t
n
= time needed to repair the damage or
failure.
Fig. 1. Algorithm of the fault-tree model
5 RESULTS OF SIMULATION
From the results of simulation and from the analysis
of gathered data it comes out, that together with the
increase in number of ship utilization periods,
susceptibility for given system failure increases as
well. It is due to the fact, that every ship device is
subjected to the process of wearing out, intensity of
which depends on utilization factors like, among
others, the professionalism of the crew, accuracy
of design and workmanship, the environmental
conditions in which it works.
From the performed simulations, basing on the
experts’ knowledge and the research materials, it
results that the process of ships systems damages in
its great measure has the random character. It is
impossible to explicitly state and translate into
tabular manner, the ratio of number of specific types
of damages occurrence to the beforehand assumed
age of the unit. It happens that the relatively new
ships suffer from damages, which have never
occurred in old and utilized units.
176
0
0.2
0.4
0.6
0.8
1
Annual utilization period
Probability
P
D
(t)
0.25
0.22
0.67
0.47
0.9
0.99
0.96
0.99
1
1
1
1
1
1
P
F
(t)
0
0.1
0
0
0.33
0.06
0.7
0.22
0.14
0.91
0.58
0.04
0.01
0.19
0.1
0.25
0.5
0.75
1
2
3
4
5
6
7
8
9
10
Fig. 2. Example results of safety model simulation for ship –
probability of damages or ship systems failure occurrence for
the number of utilization periods from 0.1 to 10
0
500
1000
1500
2000
2500
3000
3500
Annual utilization period
Time of repair [h]
t
D
61.9
129
108
36.8
160
64.9
241
224
184
277
205
184
147
139
t
F
0
369
0
0
1188
2396
1118
1142
2158
589
1790
1766
3145
3375
0.1
0.25
0.5
0.75
1
2
3
4
5
6
7
8
9
10
Fig. 3. Example results of safety model simulation for ship –
time of repair of damages or ship systems failures for the
number of utilization periods from 0.1 to 10
Similar distribution has the characteristic of
probability density for damage occurrence in relation
to characteristic of failure occurrence. Intensity of
both events is a random process and depends on
numerous factors (quality of service, physical and
chemical phenomena), which one can not, in any
way, consider in this type of model.
Fig. 4. Probability of system damage occurrence after the time t
for the number of utilization periods from 1 to 10
Fig. 5. Probability of system failure occurrence after time t for
the number of utilization periods from 1 to 10
6 CONCLUSION
Using the properties of exponential distribution,
statistical event-tree method and Monte Carlo
method, as well as considering abovementioned real
statistical data, I have created the analytical model
simulating the probability of damage or failure on
the ship in the assumed time. It determines the
chance of event occurrence after chosen amount of
1-year utilization periods. Using the properties of
exponential distribution and the mentioned Monte
Carlo method, it was possible to simulate the times,
which are required to bring the state of system
efficiency back, for the systems which were the
subject of failure in the mentioned model of
probabilities intensity.
Calculations and simulation analysis which were
carried out, create the picture of event probability
rate on the specific units. Such simulations allow to
state, how does the damage and failure distribution
develop, for the given type of ship (in the model,
data concerning bulk carriers of tonnage from 10 to
100 thousand DWT was used) and on this basis
conclude which systems on the ship requires close
attention in the aspect of increasing the safety and
reliability of service. Moreover they can be also used
by different types of institutions responsible for
navigation safety, procedure planning, planning of
recommendations and orders applied to utilization
and design of solutions for the marine industry.
REFERENCES
Hann M., 2005: On the Possibility of Applying Reliability
Theory for the Practice of the Ship’s Structural Design, II
West-Pomeranian Science Congress, Maritime University
of Szczecin, Szczecin.
Borgoń J. & Jaźwiński J. & Klimaszewski S. & Żmudziński Z.
& Żurek J., 1998: Simulation methods for flights security
research, Science Publishinghouse SKON, Warszawa.
Corbett J., 2004: Considering alternative input parameters in an
activity-based ship fuel consumption and emissions model.
Reply to comment by Øyvind Endresen et al. on
‘‘Updated emissions from ocean shipping’’,
http://www.ocean.udel.edu/cms/jcorbett/CorbettKoehler200
4.pdf.