Application of the probabilistic-fuzzy method of assessing the risk of a ship manoeuvre in a restricted area L. ~ucrna', Z. pietrzykowski2 Maritime University of Szczecin ul. Waly Chrobrego 1/2 70-500 Szczecin, Poland Abstract The paper presents possible applications of a new method of assessing navigational risk of a ship manoeuvring in a fairway. The method, based on the probabilistic analysis of risk and elements of fuzzy logic, allows to evaluate the risk taking into account such factors as the danger to the safety and the intuition that the safety of a ship manoeuvre is threatened. 1. Introduction In order to evaluate navigational risk in restricted areas (fairways, harbour entrances, harbour basins, turning basins etc.) the most frequently used are methods of quantitative risk analysis based on probabilistic methods. One approach in such cases utilises data from simulated passages conducted by navigators. From the data one can estimate parameters of the density distribution function of the random variable of the distance from danger. The use of the density function enables to determine the probability that a ship will move out of the safe boundaries of the fairway, in which case a ship may run aground or strike a port structure. Such an event results in the system status being changed to navigation system failure, or the probability of grounding. One shortcoming of the probabilistic method is that it is impossible to assess a situation when safety is threatened, that is a situation when a ship is
manoeuvring close to the fairway boundary. This state, if no or inadequate measures are taken, can result in navigation system failure. 2. The probabilistic method of assessing the risk of ship manoeuvre in a restricted area The probabilistic method of evaluating the risk consists in determining the probability of a collision of a ship in motion, understood as ship's deviation outside a designated manoeuvring area. Depending on the area shape, a collision may mean hitting a wharf, port structure or grounding. The determination of collision probability was based on simulated studies consisting of a series of trials in which navigators handle a ship through a fairway. In the trials, the fairway in question was divided into sections defined by lines perpendicular to the fairway centre line. The trials of reliable size, guaranteeing a preset level of confidence, were followed by the estimation of parameters of distributions of the function of density of random variable probability: maximum distance of ship's points to the starboard side and port side from the fairway centre line, separately for particular sections. Two random variables are determined in each of the sections: maximum distances of ship's extreme points to the starboard and port side from the fairway centre line. When the phenomenon model (type of distribution) is determined, its parameters are estimated. In order to calculate the probability of grounding, the following relations are used: density function of the random variable X of the maximum distance of ship's extreme points to the starboard side from the fairway centre line in the i-th section, density function of the random variable X of the maximum distance of ship's extreme points to the port side from the fairway centre line in the i-th section, distance frqm the center of the waterway to the waterway border (half of the waterway width). Thus obtained vectors, with the order equal to the number of the section, represent a probability of a ship's deviation from the fairway (collision occurrence) for particular fairway sections (see Fig. 1).
Port side Starboard side Figure 1. Calculation of the navigation system failure. 3. The probabilistie-fuzzy method The approach presented above does not account for dangerous situations, in which a ship comes too close to the fairway boundary, as such situations may lead to situations of navigation system failure. This refers to areas difficult for navigation in which, although not many collisions have occurred, dangerous situations are observed. The probability of a fuzzy event refers to a situation when there co-exist uncertainties of fuzzy and probabilistic type. Such a case takes place when an attempt is made to determine a probability of a dangerous situation for a ship moving in a restricted area, i.e. a fairway. A dangerous situation is meant to be a system state which, if no or inadequate countermeasures are taken, may change to the state of navigation system failure. The term: dangerous situation is an example of a linguistic variable described by the membership function of the fuzzy set 'dangerous situation'. which roughly corresponds to the fuzzy set 'dangerous distance'. An event understood as an occurrence of such situation is an example of a fuzzy event. Hence the determination of a probability that a dangerous situation will occur comes down to the determination of a probability of a fuzzy event. A fuzzy event A in the space R" is called a fuzzy set A [2], 131: with the membership function i(i(x) measurable according to Borel. The probability of a fuzzy event A is then defined as the Lebesques-Stieljes integer:
For n=l and when the probability can be described by the density function g(x) PI: the probability of a fuzzy event A can be written as follows: where: m P@) = JP, (x)g(x)d(x) X=- pa(x) - function of membership to a fuzzy set, g(x) - function of density of random variable X. 4. Interpretation of the function for the determination of a probability of a dangerous situation The function used for calculation of the probability of a dangerous event, defined by the relation (6), is a product of two functions: 1. the density function of a random variable of ship's extreme points occurring in the fairway, 2. the membership function of a fuzzy set 'dangerous event'. The former contains information on a ship's position in the fairway in a given series of simulated passages. Its mean represents the expected value of the ship's position. This function allows to determine the width of ship's swept paths. Intervals of confidence are usually used for this purpose. The latter function is determined by an expert method and represents a level (degree) of danger in the fairway depending on the distance to a danger (fairway boundary), where the value "0" denotes a safe situation, and "l" denotes a navigational accident. The function for calculating the probability of a dangerous situation computed according to the relation (6) is determined in the (-m +m) interval and may assume values ranging from 0 to 1. The function depends on: parameters of the density function of a random variable of the occurrence of ship's extreme points in the fairway, mainly on the mean i of the standard deviation; function of membership to a fuzzy set 'dangerous situation' assuming the values between 0 and 1; particularly on such parameters of the function as the point at which this function assumes a value larger than 0 and on the rate at which its value increases.
Analysing the relation (6) one may find out that when the membership function of the fuzzy set 'dangerous situation' reaches the value of 0, there is no danger. Whilst in the case when the function of membership equals 1, the function for determining the probability of a dangerous situation assumes the values of the function of the density of random variable of extreme points occurring in the fairway. In order to interpret the function for determining the probability that a dangerous situation occurs, for the sake of simplification the same function of membership to a fuzzy set 'dangerous situation' was taken (Fig. 2). \ 0 I I -100-80 40-40 -20 G 20 40 60 80 100 X lml Figure 2. An example of the function of membership to the fuzzy set 'dangerous situation'. Three examples of the density functions of random variable of ship's extreme points occurring in the fairway were assumed; their standard deviation o=8m and the mean values were as follows: 1. m,=-13m 2. m2= -25m 3. m3= -35m Their corresponding functions for determining the probability of a dangerous situation are shown in Figs. 3, 4 and 5. X lml Figure 3. A function for determining the probability of a dangerous situation for ol=8m and m,=-13m.
0 06 m 1 004 003 - - X Iml Figure 4. A function for determining the probability of a dangerous situation for ol=8m and ml=-25m. 002 - g,(n 001 - $X) g,(x) -700-80 60-40 -20 0 20 40 60 80 100 X Imi Figure 5. A function for determining the probability of a dangerous situation for ol=8m and m,=-35m. l The probability that a dangerous situation occurs and the probability that a ship moves beyond the fairway boundaries are shown in Table 1. Table 1. The probability of a dangerous situation and that of a navigational accident for a standard deviation. Situation Probability of a dangerous situation l (ml=-13m) 0.0055 2 (m2=-25m) 0.0941 3 (m3=-35m) 0.3568 Probability that a ship moves beyond the fairway boundaries (accident) 1.8. 10.1~ 2.1. 10-l0 2.9. 10" It will be noted that, as the mean shifts towards the danger (boundaries of the fairway), for the constant membership function, the probability of a dangerous situation clearly increases. The point in which the function for determining the probability of a dangerous situation reaches its maximum value is more difficult to be interpreted. It should be noticed, however, that this point shifts from the mean of ship's extreme points towards danger depending on the angle at which the membership function is
inclined. The point indicates that in a given location of the fairway there occurred the largest number of dangerous situations at the same level of danger. The influence of standard deviation of the density function of the random variable of ship's extreme points occurring in the fairway are presented for two examples of such functions having the same mean value m=-13m and these two standard deviations: 1. 01=8m 2, 02=20m The resultant function for determining the probability of a dangerous situation is shown in Fig. 3 (the case is identical to the one analysed before) and Fig. 6. X Iml Figure 6. Function for determining the probability of a dangerous situation for ol=20m and m]=-13m. The probability that a dangerous situation occurs and the probability that a ship crosses the fairway boundaries are shown in Table 2. Table 2. The probability of a dangerous situation and that of a navigational accident for a constant mean of the densitv function Probability of a Probability that a ship moves outside Situation dangerous situation the fairway boundaries (accident) l (ol=grn) 0.0055 1.8. 10-l5 It can be noted that as the standard deviation increases, the probability of a dangerous situation rises visibly, whilst the point in which this function reaches its maximum shifts towards the danger. In summary, the function determining the probability of a dangerous situation in the fairway expresses that probability quantitatively. Its maximum indicates points in the fairway where the largest number of dangerous situations occurred for the same level of danger. 5. Research In order to present the proposed probabilistic-fuzzy method of assessing a ship manoeuvre risk and to compare the method with the probabilistic method commonly used, simulated and expert research was conducted.
Simulation research The research was camed out on a fairway bend, 150 m wide at the bottom, with the curve radius of 600m (Fig. 7). The ship used for research was a loaded tanker with 40000 DWT capacity, length overall L=196m, beam B=28m and draft T=llm. The tanker was equipped with the right-hand turn propeller and a conventional rudder. 1400 - -I 1400 1600 1800 2000 2200 2400 26W 2BW 1WO 3200 MOO 3600 Figure 7. The area of the bend divided into sections. In the studies a ship-handling simulator working in real time was used for the experiments. Ships captains and pilots were invited to participate. The navigators were told to carry out a series of simulated passages, handling the ship through a part of the fairway. In the manoeuvres the rudder was mainly operated. The engine was only used in particular situations, and the ship was moving at full manoeuvring speed. After a series of simulated trials, whose reliable size guaranteed a preset confidence level, parameters of two density functions of the random variable were estimated: maximum distance of ship's points to the starboard and port side from the fairway centre line. The fairway was divided into sections with limits running perpendicular to its centre line. With the distance of 25 metres between sections their total number amounted to n=90. Two random variables were identified in each section. It was assumed that a normal distribution well describes the distribution of ship's points to the starboard and port side of the fairway centre line, which had been confirmed by relevant statistical tests [l]. Previously estimated parameters of the distributions (mean and variance) were used in order to determine the probability of grounding (navigation system failure). For the whole stretch of the fairway (all sections) the grounding probability vectors were defined for the starboard and port sides of the fairway. Expert research The expert research was aimed at determining the membership function of the set 'dangerous situation' for the starboard and port sides of the fairway in each of
he sections, paip(x) and pail(x), respectively. The basic criterion chosen for assessing a situation was the distance of the extreme port and starboard points of the stem from the fairway centre line. Experienced navigators who participated in the expert studies were those who had taken part in the simulated research. Data from the questionnaires utilising the method of psychological scaling [2] was used to specify degrees of membership, found as dangerous by the navigators, to the set 'dangerous situation'. The form of the membership function of the set 'dangerous situation' for both fairway sides, as it is known in the literature [5], was modified: p,,, (X) = 1 - exp(-((x -.X, )/ ~ 1 ) ) for ~ X > X,, P,.,, (X) = 0 otherwise pa, (X) = 1 - exp(-((x - X, )l XI)') for X < x0 paii (X) = 0 otherwise (7) (8) then its parameters (xo i xl) were estimated. The determination of the density function of the random variable X and the membership function of the set 'dangerous situation' enabled calculating the vectors of the probability of a dangerous situation. 6. The results Three sections of the examined fairway were chosen: 1) section 30 - a straight section just before entering the bend; 2) section 55 - fairway bend; 3) section 75 - a straight segment just after leaving the bend. Fig. 8 presents density functions of the random variable X for the port and starboard sides of the fairway. X Iml Figure 8. Density functions of the random variable X for the port and starboard sides of the fairway. On this basis the respective vectors of navigation system failure were determined:
Pail = [O.OOOO, 0.0073,0.0304]; P,,, = [O.OOOO, 0.0169,0.0039] The functions of membership to the set 'dangerous situation' for the chosen sections of the fairway are shown in Fig. 9. X [m1 Figure 9. Functions of membership to the set 'dangerous situation' for the chosen sections of the fairway. The corresponding functions used for the determination of the probability of a fuzzy event 'dangerous situation' are illustrated in Fig. 10. 10.~ 'r -,, 6 -, 58, < ' 1 - T5 U m L 4-3 ' -,, - / X Iml Figure 10. Functions for the determination of the probability of a fuzzy event 'dangerous situation'. They provided a basis to determine the vectors of probability of a dangerous situations; their respective values were as follows: PAII = [0.0037, 0.1053, 0.2103]; PAIp = [0.0673, 0.1937, 0.05971 It may be intuitively stated that the probability of a dangerous situation p(a) is higher than the probability that a collision P, occurs. The statement is derived when we take into account situations when navigational safety was threatened but no collision occurred (e.g. thanks to a proper response of the navigator). Analysing Fig. 9 presenting three examples of membership functions, we may notice that the experts found that the most dangerous situations took place in section 30. This may be due to the fact that the navigators' task was to pass through the bend in the fairway. This manoeuvre has to be started at a strictly specified point. Any delay or false start of the turning manoeuvre may make a safe manoeuvre impossible to be carried out.
On the starboard side of the fairway a dangerous situation occurs as early as when the 20 metre wide area from the fairway centre line is crossed, while on the port side going beyond the 25 metre zone creates a dangerous situation. The function reflecting the starboard side of the fairway shows a clearly slower rise as compared with the corresponding function for the port side. In this context the low value of the probability of determining a dangerous situation on the port side seems interesting if compared with the starboard side (Fig. 9). Inter alia, it results from the fact that, in spite of the port side being safer in the navigators' opinion, (turn is facilitated by the turning moment of the channel effect) ship's positions were more frequently found in the area on the starboard side of the fairway due to a natural tendency of a navigator to accelerate the manoeuvre and proceed along the shortest trajectory. Probabilities of a dangerous situation in further sections (55 and 75) are much higher which may be caused by large scattering of ship's positions in the fairway. The safety level in these sections is considered by navigators as higher, which is confirmed by the membership function (Fig. 9). It should be noted that, despite the navigator's classification of a given situation as dangerous, he is not capable of manoeuvring the ship so as to avoid the danger and, consequently, the probability of a dangerous situation will be high. 7. Summary The presented method has a practical implication as it allows for a quantitative determination of the probability of a dangerous situation. The paper presents an application of the method in determining occurrences of dangerous situations in which a manoeuvring ship may find itself in the fairway bend. Such assessment of the manoeuvre safety may further be used in the optirnisation of the fairway safe shapes or the assessment of the safety of conducting a given manoeuvre. The paper also presents a practical interpretation of the presented method and the dependence of the function for determining a dangerous situation on the parameters of the membership function of the fuzzy set 'dangerous situation' and on the density function of the random variable of ship's extreme points occurring in the fairway. References 1. Iribarren, J.R. Determining the horizontal Dimensions of Ship Manoeuvring Areas. PIANC Bulletin No 100. Bruxelles 1999. 2. Kacprzyk J., Fuzzy sets in system analysis, PWN Warszawa 1986, (in Polish) 3. Driankov, D., Hellendorn, H., Reinfrank, M., An Introduction to Fuzzy Control, Springer Berlin, 1993 4. Rommelfanger, H., Fuzzy Decision Support Systems, Springer Berlin, 1994 (in German) 5. James, C., Modelling the Decision Process in Computer Simulation of Ship Navigation, Journal of Navigation Vol. 39, 1986