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1 INTRODUCTION

The global societies and economies depend on the

safe and reliable transport of cargo, goods, and

people. Modern transport systems are becoming more

and more complex. Apart from their structural

complexity, technical systems may be characterized

by the complexity of the operation process. The

system can operate differently at different stages of its

operation. It also happens that the operating process

of a technical system consists of a series of separate

time intervals in which various tasks (processes) are

performed. According to the literature ([6], [9], [19]),

such these systems are defined as complex multiple-

phased mission systems (MPMS). This type of

systems is considered in many practical applications

such as aerospace, nuclear power, airborne weapon

systems, and distributed computing systems. The

system mission involves multiple, consecutive, and

non-overlapping phases of operation ([8], [9], [16],

[20], [23] - [24]). During each phase, the system must

accomplish a specified task and be subject to different

stresses and environmental conditions and reliability

requirements ([8], [9], [20], [23] - [24]). This type of

systems is characterized by the following properties

[16]:

− task carried out in the phase may differ from the

tasks in the remaining phases,

− performance and reliability requirements may

vary between phases,

− during some phases, the system may be subjected

to powerful environmental influences, which may

significantly increase the intensity of damage,

− the structure of the system may change as a

function of time, depending on the functional and

reliability requirements formulated for the phase

that is currently being performed,

− the correct execution of tasks in each phase can

bring other effects for the system than those

obtained in other phases.

There are two approaches to considering the

operation process in the analysis and modelling of

MPMS. One, the so-called synthetic model, which

cover the entire system's operation process. The

second approach is models in which individual

phases are considered separately. However, it must

be remembered that building a synthetic model is not

A Multiphase Process Approach to the Analysis of the

Reliability and Safety in Maritime Transport Systems

D. Chybowska

Maritime University of Szczecin, Szczecin, Poland

S. Guze

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: The main aim of the paper is to develop an original approach to the analysis of the reliability and

safety of maritime transport systems and the processes. It is done with accordance to multiphase process

approach to reliability and safety analysis. To achieve this goal, the basic knowledge of the reliability theory of

technical systems was discussed. The multiphase systems and damage trees were also characterized. The next

steps of the developed solution were discussed using the s / v Ramdas sea disaster. Finally, a summary of the

obtained results is presented.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 14

Number 4

December 2020

DOI: 10.12716/1001.14.04.25

986

easy. On the other hand, when every phase is

considered separately, there is a need to consider the

phases' relationship. In modelling the reliability of

systems with dependencies, several results can be

adapted to MPMS (e.g. [2], [4] - [6]).

According to data, the most effective mode to

move the large quantities of cargo is maritime

transport. However, this mode of transport is also

one of the more vulnerable to changes in hydro-

meteorological conditions and human errors. The

combination of several unfavourable factors can lead

to a sea disaster. It can result in material losses,

environmental pollution and, worst of all, human

losses. Due to these consequences, it is so important

to conduct research into methods to improve

maritime safety [11], [13], [17]. One of the ways is to

analyze maritime accidents and draw appropriate

conclusions from them.

Thus, the article's main goal is to develop an

original approach to the analysis of the reliability and

safety of maritime transport systems and the

processes taking place in them. The result of the

assumed goal is a stepwise method of dealing with

the accident data collection, which leads to the

estimation of the probability of a marine accident.

The content of the article is divided as follows.

Section 2 contains basic notations and well-known

results for reliability modelling of technical systems

and failure trees. In contrast, Section 3 contains the

main result of the work presented in the example of a

selected sea disaster. The work ends with conclusions

and tips for further work.

2 BASICS NOTATIONS AND METHODS

The well-known information about the reliability of

technical systems and the analysis of fault trees

presented in this section constitute the basis for

implementing the above-mentioned objective.

2.1 Basics on systems reliability

When analyzing the reliability of technical systems,

we can choose between two basic approaches: two-

state, multi-state. The first assumes that the system is

working or not. On the other hand, the multi-state

approach presents a fuller and closer to the analysed

system's real picture. In this case, concerning the

system and its elements, we assume that it is an

ordered set of their states, where 0 is considered the

worst, and z - the best, in terms of reliability. It can

also be assumed that the two-state approach is a

special case of a multi-state, assuming that z = 1. Then

the previously indicated set will contain only two

states 0 and 1. In theory, it presented, inter alia, in the

works [4], [14] it is taken into account that the system

and its components deteriorate during their operation

unless they are repaired.

With these assumptions, vector functions of

element and system reliability were defined in the

literature relating to multi-state complex technical

systems' reliability theory. According to them [14]:

− reliability function of components

1,2,...,in=

is described by the vector:

( , ) [ ( ,0), ( ,1),..., ( , )]

i i i i

R t R t R t R t z=

, (1)

where

( , ) ( ( ) | (0) )

i i i

R t u P e t u e z=  =

(2)

is the probability of the event that the component

1,2,...,in=

, at time t, is in one of the states of

the subset

{ , 1,..., },u u z+

while at time t = 0 it was

in the best state - z;

− system’s reliability function is given as the vector:

( , ) [ ( ,0), ( ,1),..., ( , )]t t t t z=R R R R

, (3)

where

( , ) ( ( ) | (0) )t u P s t u s z=  =R

, (4)

is the probability of the event that the system at

time t, is in one of the states of the subset while at

time t = 0 it was in the best state - z.

Probability that the component

1,2,...,in=

in state u if at time t = 0 it was in state z is calculate as

the following vector:

( , ) [ ( ,0), ( ,1),..., ( , )]

i i i i

p t p t p t p t z=

, (5)

where:

( , ) ( ( ) | (0) ),

i i i

p t u P e t u e z= = =

(6)

for

0, ),t  +

1,2,...,uz=

On the other hand, the probability that the system

is in the state u under the condition that at time t = 0 it

was in the state z is given by the vector:

( , ) [ ( ,0), ( ,1),..., ( , )]p t p t p t p t z=

, (7)

where:

( , ) ( ( ) | (0) )p t u P s t u s z= = =

, (8)

for

0, ),t  +

1,2,...,uz=

The introduction of the above concepts allows

defining the reliability structures of multi-state,

ageing technical systems and their components and

the parameters characterizing these systems. In this

paper we consider only two of them, series and

parallel.

Definition 1

A multi-state technical system is called a serial

system if the expression

T(u) =

min{ ( )}



1,2,...,uz=

(9)

describes the time T(u) of its stay in a subset of

reliability states

{ , 1,..., },u u z+

987

According to above definition, a system is in a

reliability state subset if every component of this

system is in this subset.

Definition 2

A multi-state technical system is called a parallel

system if the expression

T(u) =

max{ ( )}



1,2,...,uz=

. (10)

describes the time T(u) of its stay in a subset of

reliability states .

From definition 2, it follows that the system is in a

subset of reliability states

{ , 1,..., },u u z+

if at least one

of its elements resides in it.

Regarding the reliability structures defined above,

the coordinates of the reliability function vector are as

follows [14]:

− for multi-state series system:

( , )tuR

( , )

R t u



0, ),t  +

1,2,...,uz=

, (11)

− for multi-state parallel system:

( , )tuR

1−

( , )

F t u



0, ),t  +

1,2,...,uz=

, (12)

Furthermore, if we assume, that u=1 in definitions

1, 2 and in formulae (1) – (12), the reliability of the

two-state technical system is described.

Extremely popular method in the reliability

analysis of a technical system is reliability block

diagram (RBD). One of the advantages of this method

is possibility to determine the critical component

form reliability point of view. This method also

indirectly determines the necessity to indicate the

components essential for ensuring the continuity of

the system operation in the described structure. The

figures 1 and 2 presents the exemplary RBD for series

and parallel systems, respectively.

E1 E2 En

Figure 1. The exemplary series system

Figure 2. The exemplary parallel system

Identification of the most important components

into the system can be provided with following

significance measures:

− Birnbaum’s measure of component importance:

()



(13)

− criticality importance measure:

( ) (1 ( )) (1 ( ))

( ) ( )

( ) (1 ( )) (1 ( ))

S C C

C S S

R t R t R t

I t I t

R t R t R t

 − −

=  = 

 − −

(14)

− reliability importance measure:

()

NSF t

(15)

where

()

is a system reliability and

()

describes component reliability,

()

NSF t

is number

of system failures caused by component

()

NSF t

is a total number of system failure,

1,2,...,in=

0, )t  +

2.2 Fault Tree Analysis basic concepts

The Fault Tree Analysis (FTA) is the technique which

is based on the probability theory and Boolean logic

[21], [22]. The first element allows estimating of the

system failure probability. Also, it is possible to

describe it as the function of time. On the other hand,

the Boolean logic’s principles reduce the fault tree

structure and show the combination of events that

lead to the system’s failure (top event). There are

some logic gates. Two most used are AND logic and

OR logic gates. The figure 3 presents their symbols.

1 2

(a)

(b)

Figure 3. The exemplary logic gates in FTA: (a) AND; (b)

OR.

When the failure probability is describing as a

function of time, this method can estimate the

function of the system’s risk level. This is one of the

basic characteristics, apart from the technical system's

reliability, which allows rationalizing the safety of the

system operation. In many publications, the risk

function describes combination (product, not

multiplication!) of the occurrence probability of an

event and its effects ([15], [21], [22]).

To understand how calculate the top event

probability I the FT, we assume that

and are

the probabilities of the events 1 – 3 given in figure

3. The event number 3 is the top one. Then the

formula for occurrence probability of this event is

given as follows:

988

− AND gate

3 1 2

,p p p=

(16)

− OR gate:

3 1 2

1 [(1 )(1 )]p p p= − − −

(17)

When we look at formulas 16-17, a certain

observation arises. In combining reliability and risk

analysis using reliability block diagrams and failure

trees, the use of the serial structure determines the

AND gate. For a parallel structure, use an OR gate.

2.3 Introduction to Multiple-Phased Mission Systems

Generally, the multiple-phased systems (MPS) are

defined as systems whose operational process can be

divided into several disjoint periods, called phases

[25]. Considering the results given in [9], the phased

mission system (PMS) is defined as a system whose

relevant configuration (block diagram or fault-tree)

changes during consecutive time periods (phases). In

other words, it is a sequence of tasks performed to

achieve the purpose of the system. A different subset

of the system may need running to perform each task.

During the phase operation, component or subsystem

can fail at any time. If these elements of the particular

phase's system are not critical, it will not affect the

system performance. But considering the domino

effect, if a critical component of one phase is failed in

the previous one, it can lead to this phase and whole

mission failure. In this case, the transition from one

phase to the next in sequence is a critical event.

Different features (e. g. reliability, unreliability) may

be of interest during various phases, according to the

specific task being performed in that phase and the

user's need. Each phase is identified by phase

number, time interval, system configuration, the task

to be performed, a parameter of interest and

maintenance policy.

3 APPLICATION AND DISCUSSION

Now, we introduce some new approach to disaster

analysis. This method bases on the mixture of the

tools introduced in Section 2 and approach to disaster

process analysis defined in [12].

We show the procedure on the example of one of

the thirty analyzed sea disasters, described in [1].

The first step, after collecting data from reports, is

categorization into the following five phases [12]:

− phase I – latent phase (LPh);

− phase II – initiating phase (IPh);

− phase III – escalating phase (EPh);

− phase IV – critical phase (CPh);

− phase V – energy release (EnPh).

Let us consider the disaster of s/v Ramdas on 17

July 1947 off India's coast in the Arabian Sea, several

nautical miles from Bombay's port. The course of this

catastrophe was as follows.

As the elementary event in LPh, we distinguish a

problem with the lack of forecasts concerning bad

hydrometeorological conditions on the planned ship's

route.

Thus, the ship is in storm conditions. On the other

hand, the crew tried to change the ship's course in

vain. This classifies as the Iph.

Due to the above, two high waves hit the

starboard. This is the single element of EPh.

To protect themselves from the waves, passengers

go to the port board. As a result, the ship increases tilt

in that direction. This is the critical phase. Next, the

ship loses stability within 2 minutes and is sinking.

As a result of this catastrophe, 669 people died. And

according to [11] it is energy release phase.

This short description of the disaster helps to

classify individual events into larger, thematically

coherent classes of events. Particularly, we define

them as follows:

− KI – Knowledge incomplete / No knowledge;

− EHC – Extreme hydrometeorological conditions;

− INM – Inefficient navigational maneuver;

− NPF – Negative physical factors;

− CSC – Change of stability characteristics;

− ST – Ship tilt;

− LS – Loss of stability;

− LL – Loss of life;

− SS – Ship sinking.

Based on above shortcuts and taking into account

the results in [11] there is possibility to build the

block diagram of this disaster (see figure 5).

Energy relase phase

Critical phase

Escalating phaseInitiating phase

Latent phase

KI EHC

ST LS

NODE 1 NODE 2

INM NPF

CSC

DISA

STER

NODE 3 NODE 3

Figure 5. Block diagram of s/v Ramdas disaster

The nodes 1 – 4 presented in figure 5 can be

treated as barriers between stages of a modeled

catastrophe.

Next step in the proposed procedure is to build a

fault tree of disaster. This should be done based on a

block diagram that is built from the collected data.

The fault tree for s/v Ramdas disaster is in figure 6.

Phase I Phase IIIPhase II Phase IV Phase V

SHIP

DISASTER

KI EHC INM NPF CSC ST LS

LL SS

Figure 6. Fault Tree for s/v Ramdas disaster

989

Considering the FT and formulae (16) – (17) the

probability of the top event can be calculated. The

defined classes of events make up a discrete set.

Therefore, the discrete approach to the probability of

events should be used. In our case, the probability of

event describes following formula:

()

XXX

(18)

where

- i-th event from class XX or XXX (defined above);

XXX

- number of all incidents belonging to class XX

or XXX in the total disaster population;

- total number of events in disaster population.

The value of top event’s probability is equal to

0.0128.

Next step is to calculate the importance, criticality,

and improvement potential measures. When the

discrete values are given, then the formulae (13) – (14)

have to be redrawn with (18) as follows:

− discrete Birnbaum’s measure of component

importance:

()



, (19)

− discrete criticality importance measure:

()

( ) ( )

()

DC DB

C i C i

I e I e

=

, (20)

where

- i-th event in particular phase Ph,

- an intermediate event corresponding to the

implementation of phase F of the disaster.

Furthermore, the expression defines the

improvement potential as follows:

( ) ( ) ( )

i C i i

IP e I e P e=

, (21)

where

()

and

()

are given in (19), (18),

respectively.

Based on formulae (18) – (21), the numerical

results for s/v Ramdas disaster are given in table 1.

Table 1. Importance of events in particular phases of the

disaster.

_______________________________________________

Event Birnbaum’s Criticality Improvement

importance measure potential

measure

_______________________________________________

Latent phase

KI 1,0000 1,0000 0,0148

Initiating phase

INM 0,0544 1,0000 0,0017

EHC 0,0320 1,0000 0,0017

Escalating phase

NPF 1,0000 1,0000 0,0444

Critical phase

CSC 0,0496 1,0000 0,0018

ST 0,0353 1,0000 0,0018

Energy release phase

LS 0,0645 1,0000 0,0014

SS 0,0217 0,5289 0,0008

LL 0,0216 0,4545 0,0007

_______________________________________________

After determining these measures, for each phase

of a disaster, it is possible to determine the ranking of

the importance of events, including the above-

mentioned definitions and the resulting impact of a

given event on the occurrence of a given disaster

phase. Finally, also, it is possible to calculate the

impact of the entire disaster.

4 CONCLUSION

In the paper, the concept of tool to post-disaster

analysis has been introduced. This tool has based on

multiple-phase system concepts, reliability theory

and fault tree analysis. It has been performed as the

mixture of these elements. Thus, some well-known

concepts, definitions and notations in reliability

theory and fault tree analysis have been described.

Presented way of thinking allows finding

information on the probability that a catastrophe

would not happen. The importance measures of

events in a particular phase can help improve

maritime transport safety.

Further work will concern preparing a simulation

tool to conduct these analyses for a specific class of

sea disasters.

REFERENCES

[1] Australian Associated Press 669 Die in Ship Disaster

Available online:

https://trove.nla.gov.au/newspaper/article/26402001

(accessed on Feb 26, 2020).

[2] B. Babiarz, A. Blokus, “Dependency of technological

lines in reliability analysis of heat production,” ENER-

GY, vol. 211, art. no. 118593, pp. 1-17, 2020.

[3] Z.W. Birnbaum, “On the importance of different com-

ponents in a multicomponent system,” in Multivariate

Analysis, P. R. Krishnaiah, Ed: Academic Press, 1969,

vol. 11.

[4] A. Blokus, “Multistate System Reliability with Depend-

encies,” 1st edn. Elsevier Academic Press, United

Kingdom, 2020.

[5] A. Blokus, K. Kołowrocki, “Influence of component de-

pendency on system reliability,” in: Zamojski W., Ma-

zurkiewicz J., Sugier J., Walkowiak T., Kacprzyk J.

(eds.) Theory and Applications of Dependable Com-

puter Systems. DepCoS-RELCOMEX 2020. Advances in

Intelligent Systems and Computing, vol. 1173, pp.

105−114. Springer, Cham, 2020.

[6] A. Bondavalli, S. Chiaradonna, F. Di Giandomenico, I.

Mura, “Dependability Modeling and Evaluation of

Multiple-Phased Systems Using DEEM,” IEEE Transac-

tions on Reliability, 53:4, 2004.

[7] L. Chybowski, K. Gawdzińska, “On the Present State-of-

the-Art of a Component Importance Analysis for Com-

plex Technical Systems,” Advances in Intelligent Sys-

tems and Computing, Volume 445, Springer Interna-

tional Publishing, pp. 691-700, 2016.

[8] J. B. Dugan, “Automated analysis of phased-mission re-

liability,” IEEE Transactions on Reliability, 40:1, 1991.

[9] J. D. Esary, H. Ziehms, “Reliability analysis of phased

missions,” In: Barlow RE, Fussell JB, Singpurwalla ND,

editors. Reliability and fault tree analysis: Theoretical

and applied aspects of system reliability and safety

assessment. Philadelphia, PA, SIAM, pp. 213–236, 1975.

[10] J. F. Espiritu, D. W. Coit, U. Prakash, "Component

criticality importance measures for the power industry,"

990

Electric Power Systems Research, Vol. 77, Issues 5–6,pp.

407 - 420, 2007.

[11] K. Formela, T. Neumann and A. Weintrit, “Overview

of Definitions of Maritime Safety, Safety at Sea,

Navigational Safety and Safety in General,” TransNav,

the International Journal on Marine Navigation and

Safety of Sea Transportation, Vol. 13, No. 2,

doi:10.12716/1001.13.02.03, 2019, pp. 285-290.

[12] J. Håvold, “The RMS Titanic disaster seen in the light

of risk and safety theories and models,“ In Safety,

Reliability and Risk Analysis; CRC Press, pp. 475–482,

2013.

[13] C. Hetherington, R. Flin, K. Mearns, “ Safety in

Shipping: The human element, ” Journal of Safety

Research, vol. 37, 401‐411, 2006.

[14] K. Kołowrocki, J. Soszynska-Budny, “Reliability and

Safety of Complex Technical Systems and Processes,

Modeling – Identification – Prediction – Optimization,”

Springer-Verlag, 2011.

[15] W. Kuo, X. Zhu, “Importance measures in reliability,

risk, and optimization. Principles and application,” John

Wiley & Sons, Ltd., 2012.

[16] T. Nowakowski, S. Werbińska, “On problems of

multicomponent system maintenance modelling, “Int. J.

Autom. Comput. 6, 364, 2009.

[17] Z. Pietrzykowski, “Assessment of Navigational Safety

in Vessel Traffic in an Open Area,” TransNav, the

International Journal on Marine Navigation and Safety

of Sea Transportation, Vol. 1, No. 1, pp. 85-88, 2007.

[18] M. Rausand, A. Høyland, “System Reliability Theory:

Models, Statistical methods, and Applications (2nd

ed.),” Wiley, Hoboken, 2004.

[19] J. Reason, “Human error,” Cambrige University Press,

New York, 1990.

[20] A.K. Somani, J. A. Ritcey, S. H. L. Au,

“Computationally efficient phased-mission reliability

analysis for systems with variable configurations,” IEEE

Transactions on Reliability, 41(4), pp. 504–511, 1992.

[21] M. Stewart, R. Melchers, “Probabilistic Risk

Assessment of Engineering Systems,” Chapman Hall,

London, 1997.

[22] W. Vesely, M. Stematelators, J. Dugan, J. Fragola, J.

Minarick, J. Railsback, “Fault Tree. Handbook with

Aerospace Applications,” NASA Research Center,

Washington, 2002.

[23] L. Xing, “Reliability evaluation of phased-mission

systems with imperfect fault cover age and common-

cause failures,” IEEE Transactions on Reliability, 56 (1),

pp. 58–68, 2007.

[24] L. Xing, J. B. Dugan, “Analysis of generalized phased

mission system reliability, performance and

sensitivity,” IEEE Transactions Reliability, 51, pp. 199-

211, 2002.

[25] W. Yueqin and R. Zhanyong, "Mission reliability

analysis of multiple-phased systems based on Bayesian

network," 2014 Prognostics and System Health

Management Conference (PHM-2014 Hunan),

Zhangiiaijie, pp. 504-508, 2014, doi:

10.1109/PHM.2014.6988224.