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Maritime University of Szczecin

Akademia Morska w Szczecinie

2010, 20(92) pp. 128–133 2010, 20(92) s. 128–133

A discrete approach to assess the risks to the ship routes

Dyskretne podejście do oceny ryzyka dla tras statku

Leszek Smolarek, Sambor Guze

Gdynia Maritime University, Faculty of Navigation Akademia Morska w Gdyni, Wydział Nawigacyjny

81-345 Gdynia, ul. Morska 81-87, leszsmol@am.gdynia.pl, sambor@am.gdynia.pl

Key words: hazard situation, operational model, safety of navigation Abstract

In the paper the hazard estimation model of assessment of risk on ship routes in terms of supporting the decision depending on the traffic stream parameters has been proposed. It has been obtained by applying of semi-Markov model of safety and partial models based on consecutive “k out of n: G” systems. Furthermore, the considerations on the aspects of the selection methodology for assessment of risk to a ship route have been presented.

Słowa kluczowe: sytuacja zagrożenia, model operacyjny, bezpieczeństwo nawigacji Abstrakt

W pracy przedstawiono model określania ryzyka tras morskich w kategoriach wspomagania decyzji w zależ-ności od parametrów natężenia ruchu. Zostało to uzyskane przez zastosowanie semi-Markowskiego modelu bezpieczeństwa i częściowych modeli opartych na systemach progowych typu kolejnych “k z n” zdatnych. Ponadto, zaprezentowano rozważania dotyczące aspektów wyboru metodologii do oceny ryzyka trasy statku.

Introduction

The safety considered in the three directions: the passengers, the cargo and the environment involved in the process of transport, is one of the most important criteria for the evaluation of the process. In the maritime transport the most important factors making up the safety include: the technical efficiency of the ship, the qualifications of the people in charge of the ship and the conditions under which the transport process takes a place. In hazard situations it is useful to have methods and criterions to assess the safety of traffic. Then, the evaluation of the activities what lead to settle the hazard situation and the evaluation of quality control and assessment in terms of traffic safety ([1, 2, 3, 4]) should be considered. This assessment can help to develop the best control or the best manoeuvres for given hazard situation [5, 6, 7, 8].

Two basic measures – the CPA and TCPA have been introduced for better assess the hazard

situation at sea [3, 9, 10]. These measures help to assess distances in sea traffic.

The ship domain analysis is the another approach to determined the measures of risk [5, 7, 11, 12]. It is very important thing for safety on sea, to make fast and reliable decisions. It depends on a human factor [4, 7].

The paper is devoted to use cellular approach [13] and consecutive k out of n systems [14, 15, 16, 17] to the safety assessment in maritime traffic.

Basic notations

For operational process we assume that:

Xij – two dimensional binary random variable

representing random state of component eij,

which is equal to 1, when component eij is

free and is equal to 0, in the other case, i = 1, 2, ..., n, j = 1, 2, ..., li;

pij = P(Xij = 1) – probability of event that

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qij = 1 – pij = P(Xij = 0) – the probability that

component eij is busy, for i = 1, 2, ..., n,

j = 1, 2, ..., li; l

k,n

A – the event, when k consecutive blocks out of n are free; ) 1 ( _,   l n k l k,n P X

P – the probability of event l

k,n A ; ) 0 ( 1 _, _, ,   _kl_n  _kl_n  l n k P P X

P – the probability that

the event l k,n

A is not true;

rij – importance coefficient of component eij, for

i = 1, 2, ..., n, j = 1, 2, ..., li.

Model of operational process

The following definitions are used to describe the model of operational process [13].

Definition 1. The set of cells of the water belts designating specified length of geometric shape is called the operational block.

Definition 2. Block Yj, j = 1, 2, .., n is

opera-tionally efficient, if and only if every cell in block is free.

 

ij i

j X

Y min , i = 1, 2, ..., l

Definition 3. The waterway or the belt of water-ways with in front of a vessel, partitioned to n equal blocks is the system.

Definition 4. The system is called homogeneous when for any j = 1, 2, ..., n, blocks Yj have the same

distribution function.

Definition 5. The system is called non-homo-geneous, when there is at least one pair of random variables Xij, with different distribution functions.

Definition 6. The n dimensional binary vector is called the system state, and its value is equal to 1, when the j-th block is operationally efficient and equal to 0 in the other hand.

Fig. 1. The general concept of the system in operational model Rys. 1. Ogólna koncepcja systemu w modelu operacyjnym

Further, some classes of models are considered. They are described by different distributions of

random variables Xij, i = 1, 2, ..., l, j = 1, 2, ..., n,

i.e. by the following types of distributions.

Class 1. Homogeneous system, with pk = p for

k  N.

Class 2. The probabilities pk are given by

for-mula: k k m k m p        1 ) , ( , where m  ,k N (1)

for-mula:









                               n c k c n k c c n c k n k c n k c p_k for 1 1 for 1 1 , , 2 (2) where c1, k N.

for-mula:





                              nb k b n k b b nb k na n k na k n k a n k b a p_k 1 1 ) , , , ( 3 (3) for a, b < 1, k  N.

Generally, the following theorem is true for the operational model.

Theorem 1. The probability, that at least k con-secutive blocks of water way out of n is free, is given by the following formula:

 





            k i l j ij l j k j l n k i k p p A P 1 1 1 1, , 1 1 , k  N (4)

In particular situations, we can consider the fol-lowing two cases.

Case 1. The one belt model of operational system Let we assume that l = 1, i = 2, j = 1, 2, ..., n, i.e. we consider the path size of n in front of a ship (see figure 2).

Fig. 2. The ship at the main waterway – one-belt model Rys. 2. Statek na torze głównym – model z jednym pasem

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Case 2. The three belt model of operational system Under these assumptions, from (4), we get the following auxiliary theorem.

Theorem 2. Probability of event is given by for-mula:

 



   k r r k k,n q p A P 1 1 1 ₍₅₎ k  N.

Thus, for each class of models and according to (1–3), we get the following equations:

in case of Class 1

 

k k,n qp A P 1  _{, k  N} ₍₆₎ in case of Class 2

 



                        k r r k k,n m m A P 1 1 1 ₁ 1 1 _{, m, k  N (7)}

 







              k r k r k k,n m m A P 1 1 1 1 1 _{, m, k  N} ₍₈₎ in case of Class 3

 









                                                  _                           



  k r k r k,n n c k c n r c c c n k c c n c k n r c n k c A P 1 1 2 2 1 1 1 1 1 1 1 1 1 k  N, c < 1 (9) in case of Class 4

 

 

   

 









 

                                              _                                    



   k nb r nb na r na r k,n nb k b n r b b b n k b b nb k na n r n k na k n r a n k a A P 1 1 1 1 1 1 1 1 1 1 1 3 2 3 2 1 k  N, a < 1, b < 1 (10)

Case 2. The three belt model of operational system Let we assume that l = 3, i = 1, 2, 3, j = 1, 2, ..., n, i.e. we consider n blocks with length equal to 3 in front of a ship (see figure 3).

Fig. 3. Three-belt model for a ship on main waterway Fig. 3. Model z trzema pasami dla statku na trasie głównej

For this case, under (4), the following theorem is true.

Theorem 3. The probability, that k consecutive blocks of waterway out of n is free, is given by the following formula:

 





           k i j ij j k j n k p p A P 1 3 1 3 1 1, 3 , 1 , k  N (11)

Under assumptions, that all blocks of the water-way are the same because of the probabilities of being free, i.e. they are homogeneous, and in addi-tion, we consider the blocks with regular length equal to 3 and pij = rijp then we get the following

theorem.

Theorem 4. The probability, that at least k con-secutive regular and homogeneous blocks out of n is free, we get the following formula:

 



            k i i i i j k j n k p p r rr r A P 1 1 2 3 3 1 1, 3 3 3 , 1 , k  N (12)

Under assumption that l = 3 and according to the formulae (6–12), and taking into account the impor-tance coefficient for each block of water way, the following equations are given by:

 for Class 1:

  





       k i i i i k k k k,n p r r r p r r r A P 1 1 2 3 3 3 , 1 2 , 1 1 , 1 3 3 ₁ k  N (13)  for Class 2:

 



    _                        k i i i i k k k k,n r r r m r r r m A P 1 1 2 3 3 3 , 1 2 , 1 1 , 1 3 3 1 1 1 1 1 m, k  N (14)

 for Class 3 formula (15);  for Class 4 formula (16).

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Safety model

For further analysis it is necessary to use the fol-lowing definitions [13].

Definition 6. Safety is the state, in which a ship can continue to move in an unimpeded manner (planned) without a need to change the ship course.

Definition 7. Safety risk is the state, in which a ship has to perform the unplanned change the course.

Definition 8. Danger is the state, in which a ship has to perform rapid manoeuvres to change the course.

For each value of vessel’s speed, three numbers nB, nZB, nNB determining the limit values for safety,

safety risk and dangers of the system.

Then the set of states the safety of the system is divided into subsets:



i B



B s i n

S  :  – the set of components, which are in the safe states;



i ZB B



ZB s n i n

S  :   – the set of components, which are in the safety risk state;



i ZB



NB s i n

S  :  – the set of components, which are in the danger state.

Due to the safety interest in a concern only inci-dent events si  SB, si  SZB, si  SNB, determine the

safety states of the system.

Suppose P(SB), P(SZB), P(SNB) mean the

prob-ability of being in the respective states of the

system safety model of one-belt case. Additionally, let P(SB), P(SZB), P(SNB) mean the probability of

being in the respective states of the system safety model of three-belt case.

Then the probability, that the vessel is in the safe state, equals to the probability that at least nB

consecutive sections of the waterway out of n be-fore it is free. Thus

 for the case of the one-belt model:

 

SB PnBn

P  _, (17)

 for the case of the three-belt model:

 

3 ,n n B PB S P  (18)

Next, the probability, that the vessel is in the risk state, equals to the probability that at least nZB

and at most nB consecutive sections of the

water-way out of n before it is free. So taking into account previous models we get:

 

SZB Pn_ZBn Pn_Bn

P  ,  , (19)

 

3 , 3 ,n n n n ZB PZB PB S P   (20)

And finally the probability, that the vessel is in the risk state, equals to the probability that at most nZB consecutive sections of the waterway out of n

 









) 15 ( 1 0 , , 1 1 1 1 1 1 1 , 1 1 1 1 1 1 1 1 2 3 3 3 1 2 1 1 1 3 1 1 2 3 3 2 3 1 2 1 1 1 3 2 3                                                 _                            _                                              



        c N k n c k r r r c n k c c r r r c n k c c n c k r r r n k c r r r n k c A P k i i, i, i, , k , k , k k i i, i, i, , k , k , k k,n

 

 

   

 









 

  ) 16 ( 1 0 , , 1 1 1 1 , 1 1 , 1 1 1 1 3 2 1 3 3 1 2 1 1 1 3 3 2 1 3 3 1 2 1 1 1 3 1 1 2 3 9 6 3 1 2 1 1 1 9 6 3                                           _                                                                                    



            b a N k nb k r r r b n k b b r r r b n k b b nb k na r r r n k r r r n k na k r r r n k a r r r n k a A P k nb i i, i, i, , k , k , k nb na i i, i, i, , k , k , k na i i, i, i, , k , k , k k,n

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before it is free. So taking into account previous equations we get:

 

SNB PnZBn

P 1 _, (21)

 

3 , 1 _n _n NB PZB S P   (22) Applications

On the following graphs there is shown a com-parison of the different classes of operational mo-dels in case for random variable. For example, after assuming k = 3 while n = 10, and k = 4 while n = 10, then we obtain for case 2, the following graphs probability that at least 3 blocks out of 10 and at least 4 blocks out of 10 before the unit are not occupated.

Fig. 4. Comparison of the models – three-belt case, the models as functions of a) parameter m and b) parameter c

Rys. 4. Porównanie modeli – przypadek z trzema pasami, modele jako funkcje a) parametru m, b) parametru c

In the case of safety model, we assume that nB = 4, nZB = 3 and nNB = 2. Thus and according to

(16–21) we get the following equations:

 

SB P n

P  ₄_, (22)

 

3 , 4 n B P S P  (23)

 for the case of the one-belt model

 

SZB P n P n

P  ₃_,  ₄_, (24)

 

3 , 4 3 , 3n n ZB P P S P   (25)

 

SNB Pn

P 1 ₃_, (26)

 

3 , 3 1 _n NB P S P   (27)

Let us consider the model of Class 2. Thus and according to (22–27) we get following graphs of safety model.

Fig. 5. Graphs of states’ probabilities as a functions of para-meter m

Rys. 5. Wykresy prawdopodobieństw stanów bezpieczeństwa jako funkcji parametru m

0,993 0,994 0,995 0,996 0,997 0,998 0,999 1 1,001 0 5 10 15 20 m 25

Probability of non-occupancy Class 2

0 0,0001 0,0002 0,0003 0,0004 0,0005 0,0006 0,0007 0,0008 0 0,2 0,4 0,6 0,8 1 c 1,2 Probability of non-occupancy 0 0,001 0,002 0,003 0,004 0,005 0,006 0,007 0,008 0 10 20 30 40 50 P ro ba bil it y m 0,998 0,9985 0,999 0,9995 1 0 10 20 30 40 m 50 P(SNB) P(SZB) P(SB)

 

3 ,n k A P k = 4, n = 10 k = 3, n = 10 a)

 

3 ,n k A P b) Class 3, k = 4, n = 10 Class 4, k = 4, n = 10 Class 3, k = 3, n = 10 Class 4, k = 3, n = 10

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Conclusions

The four classes of models depending on diffe-rent parameters for operational process have been proposed. The two particular cases of models: one-belt model and the three-one-belt model have been described. Next, the comparison of the class of the three belt model has been shown on exemplary graphs.

Further, the safety model, basis on operation process has been defined and exemplary evaluation has been given on graphs under consideration of Class 2 of the models.

The safety model and operational process model can be more complicated after assuming the semi- -Markov model of operational process instead of the Markov model.

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