International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 3

Number 4

December 2009

449

1 INTRODUCTION

Loss of the propulsion function by a ship is one of

the most serious categories of hazardous events

1

in

shipping. In specific external conditions it may lead

to a loss of ship together with people aboard. The

loss of propulsive power may be an effect of the

propulsion system (PS) failures or of errors commit-

ted by the crew in the system operation process. In

the safety engineering language we say that the pro-

pulsion loss probability depends on the reliability of

the PS and of its operators. Determination of that

probability is in practice confronted with difficulties

connected with shortage of data on that reliability.

This pertains particularly to the cases of estimation

in connection with decisions taken in the ship opera-

tion. In such cases we have to rely on subjective es-

timations made by persons with practical knowledge

in the field of interest, i.e. experts. The experts, on

the other hand, prefer to formulate their opinions in

the linguistic categories, in other words in the lan-

guage of fuzzy sets. The author's experience tells also

that in the expert investigations it is difficult to

maintain proper correlation between the system data

and the system component data. The paper presents

a method of the subjective estimation of propulsion

loss probability by a ship, based on the numerical-

fuzzy expert judgments. The method is supposed to

ensure that proper correlation. It is adjusted to the

1

Hazardous event is defined as an event bringing about dam-

age to human beings as well as to the natural and/or technical

environment. It is also called "accident" or “initiating event”.

knowledge of experts from ships’ machinery crews

and to their capability of expressing that knowledge.

The method presented has been developed with

an intention of using it in the decision taking proce-

dures in risk prediction during the seagoing ship op-

eration, in the shortage of objective reliability data

situations.

2 DEFINITION OF THE SHIP PROPULSION

LOSS AS A HAZARDOUS EVENT

The propulsion hazard is connected with the loss by

the PS system of its capability of performing the as-

signed function, i.e. generating the driving force of a

defined value and direction. It appears as an effect of

a catastrophic failure

**

of the PS. Such failure may

cause immediate (ICF) or delayed (DCF) stoppage

of a ship. In the latter case the stoppage is connected

with renewal, which may be carried out at any se-

lected moment. It is obvious that only the former

case of the forced stoppage creates a risk of damage

or even loss of ship - it is a hazardous event.

We will relate the probability of ICF to an arbi-

trary time interval determined by the analyst. For in-

stance, it may be duration of one trip, time interval

between the ship class renewal surveys or one year,

as it is usually assumed in risk analyses. Such an ap-

**

Catastrophic failure is defined as loss of the capability of

performing by the object of its assigned function.

Estimation of the Probability of Propulsion

Loss by a Seagoing Ship Based on Expert

Opinions

A. Brandowski & W. Frackowiak

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: The event of the loss of propulsion function has been defined as hazardous event to a seagoing

ship. It has been formalized. The procedure of acquisition of expert opinions on frequency of the event occur-

rence has been described. It may be considered to be of a numerical-fuzzy character. The fuzzy part was

transferred to the numerical form by the pair comparison method. An example of the ship propulsion system

comprising a low speed internal combustion engine and a solid propeller illustrates the method presented. It

may be used wherever a hazard analysis has to be performed of a system involving human and technical as-

pects and there is a shortage of objective data on the investigated object.

450

proach is useful in the ship operation risk manage-

ment process.

The ICF type failure consequences may be divid-

ed into casualties and incidents (IMO 1997). In gen-

eral, the ship casualties are non-repairable at sea by

means of the ship own resources and may have very

serious consequences, with the ship towing at the

best and the loss of ship at the worst. The problem of

consequences is not the subject of this paper.

The ICF type failure frequency depends mainly

on the type of PS and the ship operation mode (liner

trade, tramping etc.). On the other hand, the conse-

quences are strongly dependent on the ship size and

type and the environmental conditions, first of all the

water region, season, time of day, atmospheric and

sea conditions. They are also dependent on the navi-

gational decisions and on the type and fastening of

cargo in the holds and on deck. In general, these are

the factors connected with the type of shipping car-

ried out and the shipping routes the ship operates on.

3 FORMAL MODEL OF ICF EVENT

We assume the following:

− We are interested only in the "active" phase of

ship operation, when it is in the shipping traffic.

We shall exclude from the model the periods of

stays in ship repair yards or in other places con-

nected with renewals of the ship equipment.

− The investigated PS system may be only in the

active usage or stand-by usage state. The ICF

type PS failures may occur only in the former

state.

− A formal model of the ICF type PS failures is the

homogeneous Poisson process (HPP). This as-

sumption is justified by the expert elicited data,

which indicate that this type of failures occur fair-

ly often, several times a year, but their conse-

quences in general mean only a certain loss of

operation time. More serious consequences, caus-

ing longer breaks in the normal PS system opera-

tion, occur seldom. The exponential distribution

of time between failures, taken place in the HPP

stream model, is characteristic of a normal opera-

tion of many system classes, including also the

ship systems (Gniedienko B.W. & Bielajew J.K.

& Solowiew A.D. 1965, Modarres M., Kaminskiy

M. & Krivtsov V. 1999). It is appropriate in the

case when the modeled object failures and the

operator errors are fully random abrupt failures

and not gradual failures caused by the ageing

processes and/or wear of elements. This corre-

sponds with the situation when scrupulously per-

formed inspections and renewals prevent the lat-

ter type of failure from occurring.

− Experts are asked only about two numerical val-

ues: number of ICF type failures N(t) during time

period t = 1 year (8760 hours), and the time at sea

percentage share κ 100% during their seamanship

period - this is within their capability of answer-

ing.

− The opinions on the failures of PS system com-

ponents are elicited in the linguistic form.

The seagoing ship system active usage time t(a) is

strongly correlated with the specific ship operational

state times, mainly with the "at sea" state including

"sailing", "maneuvers" and "anchoring". The follow-

ing approximation may be adopted for the system,

also for the PS:

,

)()(

ttt

ma

κ

==

(1)

where t

(a)

= active usage time; t

(m)

= time at sea; t =

calendar time of the system observation;

tt

m

/

)(

=

κ

= time at sea factor (

∈

κ

〉〈 1,0

).

In view of these assumptions, the ICF type PS fail-

ures may occur only in the system active usage state,

i.e. for the PS system in the t

(m)

time, although their

observed yearly numbers are determined by experts

in relation to the calendar time t. The model ICF

probability has the vector form:

( )

{ }

{ }

==

==

−

Kke

k

t

tPtP

t

ka

a

a

,...,2,1:

!

)(

)(

)(

κλ

κλ

κ

(2)

where P{t

(a)

} = the vector of probabilities of ICF

type event occurrence within time interval

),0 t〈

;

jj

J

j

j

a

ttN

κλ

∑

≈

=1

)(

)(

= intensity function of

HPP (ROCOF) (and at the same time the failure rate

of the exponential distributions of time between fail-

ures in that process, [1/h];

=

j

N

annual number of

the ICF type events elicited by j-th expert, [1/y];

j

κ

= time at sea factor elicited by j-th expert; t

j

= cal-

endar time of observation by j-th expert [h];

J = number of experts; K = the maximum number of

possible ICF type failures in the time interval

),0 t〈

;

t = the time of probability prediction.

The

)(

a

λ

formula is based on the theorem on the

asymptotic behaviour of the renewal process (Gnie-

dienko B.V., Bielajev J.K. & Soloviev A.D. 1965):

,

1)]([

lim

λ

==

∞→

o

t

Tt

tNE

(3)

where

=

o

T

mean time between failures.

The number of ICF type events in the

),0 t〈

peri-

od may be 0,1,2,…or K with well-defined probabili-

ties. The maximum of these probabilities is the as-

451

sumed measure of the probability of ICF type event

occurrence:

( )

{ }

t

ka

Kk

a

a

e

k

t

tP

κλ

κλ

)(

!

)(

max

)(

},...,2,1{

max

−

∈

(4)

The λ and

κ

parameters determined from the

elicited opinions may be adjusted as new operation

process data arrive on the investigated system fail-

ures.

Expressions (2) and (4) allow to estimate the

probabilities of ICF type hazardous events in the de-

termined time interval t. Another problem is estima-

tion of the risk of consequences of these events, i.e.

damage to or total loss of the ship and connected

human, environmental and financial losses. This is a

separate problem not discussed in this paper.

4 DATA ACQUISITION

The PS will be further treated as a system consisting

of subsystems and those consisting of the sets of de-

vices.

Experts are asked to treat the objects of their

opinions as anthrop-technical systems, i.e. composed

of technical and human (operators’ functions) ele-

ments. They elicit their opinions in three layers in

such a way that proper correlation is maintained be-

tween data of the system and data of the system

components. In layer 0 opinions are expressed in

numbers, in layers I and II - in linguistic terms. For

layers I and II separate linguistic variables (LV) and

linguistic term-sets (LT-S) have been defined (Pie-

gat A. 1999).

Layer 0 – includes PS as a whole.

Estimated are the annual numbers of type ICF type

failures of PS N(t) and the percentage share of time

at sea

%100

κ

in the time of expert’s observation.

Layer I – includes decomposition of PS to a subsys-

tem level.

− LV = share of the number of subsystem failures

in the number of type ICF failures of PS.

− LT-S = A1-very small/none, B1-small, C1-

medium, D1-large, E1-very large.

Layer II – includes the decomposition of subsys-

tems to the sets of devices (set of devices is a part of

subsystem forming a certain functional entity whose

catastrophic failure causes catastrophic failure of the

subsystem - e.g. it may be a set of pumps of the

cooling fresh water subsystem).

− LV = share of the number of failures of the sets of

devices in the number of catastrophic failures of

the respective PS subsystem.

− LT-S = A2-very small/none, B2-small, C2-

medium, D2-large, E2-very large.

The structure of data acquisition procedure pre-

sented here implies a series form of the reliability

structures of subsystems (layer I) and sets of devices

(layer II). Elements of those structures should be so

defined that their catastrophic failures cause equally

catastrophic failures of the PS system and subsystem

respectively. The division into subsystems and sets

of devices should be complete and disjunctive.

The data acquisition procedure presented here

takes into account the expert potential abilities. It

seems that their knowledge should be more precise

in the case of a large operationally important system,

as the PS is, and less precise as regards individual

components of the system.

5 ALGORITHM OF EXPERT OPINION

PROCESSING

In layer 0 the experts elicit annual numbers of the

ICF type failures, which, in their opinion, might

have occurred during 1 year in the investigated PS

type:

JjtN

j

,...,2,1)( =

(5)

and shares of the time at sea in the calendar time of

ship operation:

Jj

j

,...,2,1%

100 =

κ

(6)

where j = experts index; J = number of experts.

These sets of values are subjected to selection due

to possible errors made by the experts. In this case a

statistical test of the distance from the mean value

may be useful, as in general we do not have at our

disposal any objective field data to be treated as a

reference set.

If the data lot size after selection appears insuffi-

cient, it may be increased by the bootstrap method

(Efron & Tibshirani 1993).

From the data (5) and (6), parameters

)(

a

λ

and

κ

of expression (2) and (4) are determined. Number of

opinions J may be changed after the selection.

In layer I experts elicit the linguistic values of

subsystem shares in the number of ICF type failures

of the investigated PS type (they choose LV value

from the {A1, B1, C1, D1, E1} set). The data are

subjected to selection.

The elicited data with linguistic values are com-

pared in pairs - estimation of each subsystem is

compared with estimation of each subsystem. The

452

linguistic estimations are transformed into numerical

estimations according to the following pattern:

,21⇒=− BSLT

,31⇒=− CSLT

,41⇒=− DSLT

.51⇒=− ESLT

Numerical estimates of each subsystem are sub-

tracted from estimates of each subsystem. In this

way the difference values are obtained, which may

have the following values: -4,-3,-2,-1, 0, 1, 2, 3, 4.

Those differences are transferred into preference es-

timates (as given in Table 1) in accordance with the

following pattern:

4

⇒

9, absolute preference,

3

⇒

7, clear preference,

2

⇒

5, significant preference,

1

⇒

3, weak preference,

0

⇒

1, equivalence,

-1

⇒

1/3, inverse of weak preference,

-2

⇒

1/5, inverse of significant preference,

-3

⇒

1/7, inverse of clear preference,

-4

⇒

1/9, inverse of absolute preference.

From these differences, by the pair comparison

method, a matrix of estimates is constructed. The es-

timates depend on the "distance" of the linguistic

values LT-S of a given variable LV. For instance,

preference A1 in relation to E1 has the value 9 as-

signed, in relation to D1 a value 7, in relation to C1

a value 5. in relation to B1 a value 3 and in relation

to A1 a value 1. The inverses of those preferences

have the values, respectively: 1/9, 1/7, 1/5, 1/3 and

1. The matrix of estimates is approximated by the

matrix of weight quotients of the sought arrange-

ment. The recommended processing method is the

logarithmic least squares method. The result is a

vector of normalized arrangements of the subsystem

shares (Saaty 1980, Kwiesielewicz 2002)

***

:

],,...,,...,[

21 Ii

ppppp =

(7)

where

i

p

= share of the i-th subsystem as a cause

of an ICF type PS failure; I = number of subsystems.

Now we can determine in a simple way the inten-

sity functions of individual subsystems arising from

catastrophic failures:

.,...,2,1,

)(

)(

Iip

i

a

a

i

==

λλ

(8)

***

The Saaty method, criticised in scientific circles, is widely

applied in the decision-taking problems.

Table 1. Expert preference estimates acc. to Saaty (1980)

__________________________________________________

Estimate Preference

__________________________________________________

1 Equivalence

3 Weak preference

5 Significant preferencje

7 Strong preference

9 Absolute preferencje

Inverse of Inverse of the above described

the above numbers preference

__________________________________________________

In layer II experts elicit the linguistic values of

subset shares in the number of catastrophic subsys-

tem failures (they choose LV value from the {A2,

B2, C2, D2, E2} set). As in the case of subsystems,

the expert opinions are processed to the form of

normalized vectors of the arrangements of set

shares:

KkIi

ppppp

iKikiii

,...,2,1 ,...,2,1

],...,,...,,[

21

==

=

(9)

where p

i

= vector of the shares of i-th subsystem sets

as causes of catastrophic failures of that subsystem;

=

ik

p

share of the k-th set of i-th subsystem; K =

number of sets in a given subsystem.

Then, the intensity functions of sets contained in

individual subsystems arising from catastrophic fail-

ures are determined:

.,...,2,1,,...,2,1

)()(

KkIip

ik

a

i

a

ik

===

λλ

(10)

6 EXAMPLE

The example discusses investigation of a PS consist-

ing of a low speed piston combustion engine driving

a solid propeller, installed in a container carrier ship.

Experts were marine engineers with long experience

(50 persons). Special questionnaire was prepared for

them containing definition of the investigated object,

schematic diagrams of subsystems and sets, precise-

ly formulated questions and tables for answers. It

was clearly stated in the questionnaire that an ICF

type failure may be caused by a device failures or by

a crew actions. Out of 50 opinions elicited by ex-

perts, 3 were estimated as very unlikely (2 elicited

numbers of the ICF events in a year were extremely

underestimated and one was overestimated). They

were eliminated and the remaining 47 opinions were

further processed.

Figs. 1 and 2 present statistical estimates of the

expert opinion data (5) and (6).

453

Figure 1. Box and whiskers plot of ICF yearly numbers

Figure 2. Box and whiskers plot of time at sea share

Table 2. Basic results of propulsion system investigation

___________________________________________________

Averaged

5,2)1( =yN

expert elicited

1325,1)]1([ =yN

σ

data

%95745,83100 =

κ

%24406,7]100[ =

κσ

___________________________________________________

Risk model

ht 411720

47

1

=

∑

Parameters

hE

a

10439922,3

)(

−=

λ

0,83957=

κ

___________________________________________________

Table 2 contains averaged basic data elicited by

47 experts in relation to the PS as a whole and the

model parameters of ICF type event probability

(equation (2)) determined from these data.

From the Table 2 data the probabilities of deter-

mined numbers of ICF type event occurrences in 1

year were calculated. Fig.3 diagram presents results

of those calculations. The numbers of probable ICF

events in 1 year are equal 1, 2, …, 5. The maximum

probability is 0.2565, which stands for 2 ICF type

events during 1 year, and the probability that such

event will not occur amounts to 0.0821.

Figure 3. Distribution of ICF event numbers’ probability

Table 3 contains the subsystem intensity function

(ROCOF) data calculated from equation (8). The

main PS risk "participants" are main engine and the

electrical subsystem and the least meaningful is the

propeller with shaft line. This is in agreement with

the experience of each shipbuilder and marine engi-

neer.

Table 3. Intensity functions of the subsystems

___________________________________________________

No

Subsystem

i

p

5)(

10

−a

λ

___________________________________________________

1 Fuel oil subsystem 0,1330 4,5203

2 Sea water cooling subsystem 0,0437 1,4852

3 Low temperature fresh water 0,0395 1,3426

cooling subsystem

4 High temperature fresh water 0,0620 2,1074

cooling subsystem

5 Starting air subsystem 0,0853 2,9006

6 Lubrication oil subsystem 0,0687 2,3352

7 Cylinder lubrication oil 0,0446 1,5147

subsystem

8 Electrical subsystem 0,1876 6,3770

9 Main engine 0,1987 6,7536

10 Remote control subsystem 0,1122 3,8146

11 Propeller + shaft line 0,0247 0,8410

___________________________________________________

Table 4 contains the fuel supply subsystem inten-

sity function (ROCOF) data calculated from equa-

tion (10).

Table 4. Intensity functions of the fuel oil subsystem sets

___________________________________________________

No Set

ik

p

6)(

10

−a

ik

λ

___________________________________________________

1 Fuel oil service tanks 0,0488 2,2062

2 Fuel oil supply pumps 0,1672 7,5572

3 Fuel oil circulating pumps 0,1833 8,2840

4 Fuel oil heaters 0,0944 4,2666

5 Filters 0,1540 6,9599

6 Viscosity control arrangement 0,2352 10,6323

7 Piping + heating up steam 0,1172 5,2965

Arrangement

___________________________________________________

0,0821

0,2052

0,2565

0,2124

0,1336

0,0668

0

0,05

0,1

0,15

0,2

0,25

0,3

0 1 2 3 4 5 6

Annual numbers of ICF events N(1y) [1/y]

Probability of ICF yearly

numbers

454

7 SUMMARY

The paper presents a method of subjective estima-

tion of the hazard connected with losing by a seago-

ing ship of the propulsion function capability. The

estimation is based on opinions elicited by experts -

experienced marine engineers. The method is illus-

trated by an example of such estimation in the case

of a propulsion system with a low speed piston com-

bustion engine and a solid propeller installed in a

container carrier.

The given in section 6 do not raise any objec-

tions. The authors do not have at his disposal suffi-

cient objective data to evaluate precisely the adequa-

cy of those data. It has to be taken into account that

results of a subjective character may, by virtue of the

fact, bear greater errors than the objective results

achieved from investigations in real operational

conditions.

The presented method may be used in the proce-

dures of the ship propulsion risk prediction. It allows

to investigate the impact of the PS system compo-

nent reliability on the probability values of ICF type

event. It may also be used with other types of ship

systems and not only to ship systems, particularly in

the situations of hazardous event probability estima-

tions with insufficient objective data at hand.

In this place the authors thank Prof. Antoni Pod-

siadlo and Dr. Hoang Nguyen for their cooperation,

particularly in the scope of the acquisition of expert

opinions and their processing.

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