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wMathematical model of stochastic level that may be used within
the complex system of ferry-boat sea navigation
Sorin Baiculescu
Project Coordinator
National Company for Freight Railway Transport CFR MarfaS.A.
Ministry of Transport,38 Dinicu Golescu Av.,Sector1, Post code77113,Bucharest,Romania
Phone:00-4-021-6385588,E-mail:sorin_baiculescu@yahoo.com
Keywords:traffic,system,stochastic
Abstract
At present, within the complex system of ferry-boat sea navigation in Romania, there are specialized ships which navigate in the international waters of the Black Sea, along sea lines. In this paper there are several propositions of stochastic level mathematical models used to deter- mine the optimum time horizon and the minimum time required for these ships to reach the desti- nation harbors. These methods take into account some aleatory elements intervening in achieving a turnus, such as: weather conditions, possibilities to get into and out of straits, the situation of the navigable waters, the possibilities to get into and out of harbor berths, the strait-roads-berth-ship subsystem being considered a complex order expectation system, characterized by Poisson law.
The paper uses both the statistic data and a few methods of stochastic analysis and speci- fic operational research (Monte Carlo, Markov, expectation processes, prospective dynamic analysis, the decision theory, the games theory, the affecting theory), identifying the possibilities of an optimum feed-back. The obtained results are new in the field, having significance at theoretical and applying level, useful mainly in characterizing any complex system of ferry-boat navigation according to its local and general conditions.
1.Mathematical model of the complex system: Harbor A (Romania)-the Black Sea-the Bosphorus Strait-the Marmora Sea- Harbor B (Turkey)-Maritime Roads-Maritime Berth-Ferry Boat
General Considerations:
In the paper we consider the complex system consisting of Harbor A (Romania), the Black Sea, the Bosphorus Strait, the Marmora Sea, Harbor B (Turkey), accessible to the ferry-boat type ships, maritime roads, loading/unloading maritime berth, the ferry boat. It represents an expectation system, minimizing the stationing time implying reduced general serving time (minimum stop while waiting to go through the Bosphorus Strait), quick harbor operation, minimizing the expectation time in the harbor roads, reduction of the time required for harbor manoeuvres in order to take over the cargo, fuel supply (bunkering) and considering the weather conditions at the time (sea condition, wind intensity, mist, the navigation season, etc). We consider the complex system of the maritime line Harbor A (Romania)-Harbor B (Turkey). The respective mathematical model has the general form: (=((,U,(,X,Y,(,(,() (1) in which: T-the set of independent time variable values, affected by the following factors: the effective march of the ferry-boat in the Black Sea, the Bosphorus Strait and the Marmora Sea (both ways), transit expectation (both ways) of the Bosphorus Strait and its effective transiting (both ways), expectation in the roads to actually enter the landing berth (or operating berth) in Harbor B (one way) and Harbor A (return), steering expectation in Harbor B (both ways) and Harbor A (both ways), checking procedures in Harbor B (one way) and Harbor A (both ways), and performing the loading/unloading formalities in Harbor B (both ways) and Harbor A (both ways) cargo expectation (if necessary) in Harbor B (return) and Harbor A (one way), effective operation of the ferry-boat in Harbor A (both ways) and Harbor B (both ways), fuel supply (bunkering) (if necessary), impossibility to operate or handle the boat owing to unfavorable weather conditions in Harbor B (both ways) and Harbor A (both ways), auxiliary problems occurring in both harbors (both ways).
U-set of inlet variable values u described by function class (={(:T(U} (2) in which (={u(t)(t(T,u(t)(U}(3) represents the evolutin of the inlet variable accepted by the system and u represents the value at a certain time of function ( (cost)
(-function class of set of inlet variable values accepted by the system
X-set of state variable values marked x
Y-set of outlet variable values described by class (={(:T(Y}(4)
(-transition function of the considered system ((:TxTxXx((X ) (5), in which
x(t)=((t,(,x,() (6)
(-outlet function ((t)=((t , x (t)) (7) in which (:TxX(Y (8)
We mark with ({t1,t2}=((T ( ({t1,t2} (9) the restriction representing the inlet segment , (-the evolution of inlet variable accepted by the system
T=( ([ti,ti+1) (10) for i = 1,2,3,21, in which [ti,ti+1) represents the segments for the structure of time horizon implied by the main operations going on the analyzed ferry-boat line.
The following values are introduced:
[t1,t2)-time segment for ferry-boat marching (one way) in the Black Sea
[t2,t3)-time segment for expectation (one way)in order to sail through the Bosphorus Strait
[t3,t4)-time segment for passing through (one way) the Bosphorus Strait
[t4,t5)-time segment for ferry-boat marching (one way) in the Marmora Sea
[t5,t6)-time segment for waiting in Harbor B roads in order to enter landin berth and operating berth
[t6,t7)-timesegment for steering expectation to ensure inlet into the Bosphorus Strait and Harbor B
[t7,t8)-time segment for checking procedures and performing loading/unloading formalities of the ferry-boat in Harbor B
[t8,t9)- time segment for cargo waiting (if necessary) in Harbor B
[t9,t10)-time segment for ferry boat actual operation in Harbor B
[t10,t11)-time segment for fuel supply (bunkering) (if necessary)
[t11,t12)-time segment for impossiblity to operate the boat owing to unfavorable weather conditions in Harbor B area
[t12,t13)-time segment for auxiliary problems in Harbor B
[t13,t14)-time segment for ferry-boat marching (return) in the Marmora Sea
[t14,t15)-time segment for expectation (return) in order to pass through the Bosphorus Strait
[t15,t16)-time segment for passing through (return) the Bosphorus Strait
[t16,t17)-time segment for ferry-boat marching (return) in the Black Sea
[t17,t18) time segment for waiting in Harbor A roads in order to enter landing berth
[t18,t19)-time segment for checking procedures and performing formalities in Harbor A
[t19,t20)- time segment for steering expectation to ensure inlet into Harbor A
[t20,t21)-time segment for the impossibility to operate the boat owing to unfavorable weather conditions in Harbor A area
[t21,t22)-time segment for auxiliary problems in Harbor A
(1,2=u[t1,t2)-cost of ferry-boat marching (one way) in the Black Sea
(2,3=u[t2,t3)-cost of expectation (one way) in order to pass through the Bosphorus Strait
(3,4=u[t3,t4)-cost of passing (one way) through the Bosphorus Strait
(4,5=u[t4,t5)-cost of ferry-boat marching (one way) in the Marmora Sea
(5,6=u[t5,t6)-cost of waiting in Harbor B roads in order to enter landing berth and operating berth
(6,7=u[t6,t7)-cost of steering expectation to ensure inlet into the Bosphorus Strait and Harbor B
(7,8=u[t7,t8)-cost of checking procedures and performing loading/unloading formalities of the ferry-boat in Harbor B
(8,9=u[t8,t9)-cost of cargo expectation (if necessary) in Harbor B
(9,10=u[t9,t10)-cost of effective operation of ferry-boat in Harbor B
(10,11=u[t10,t11)-cost of fuel supply (bunckering) (if necessary)
(11,12=u[t11,t12)-cost of stationing owing to impossibility to operate the ferry-boat because of unfavorable weather conditions in Harbor B area
(12,13=u[t12,t13)-cost of ferry-boat stationing owing to auxiliary problems in Harbor B
(13,14=u[t13,t14)-cost of ferry-boat marching (return) in the Marmora Sea
(14,15=u[t14,t15)-cost of expectation (return) in order to pass through the Bosphorus Strait
(15,16=u[t15,t16)-cost of passing (return) through the Bosphorus Strait
(16,17=u[t16,t17)-cost of ferry-boat marching (return) in the Black Sea
(17,18=u[t17,t18)-cost of waiting in Port A roads to enter landing berth
(18,19=u[t18,t19)-cost of checking procedures and performing formalities in Harbor A
(19,20=u[t19,t20)-cost of steering to ensure Harbor A inlet
(20,21=u[t20,t21)-cost implied by the time segment for the impossibility to operate the boat owing to unfavorable weather conditions in Harbor A area
(21,22=u[t21,t22)-cost implied by the time segment for auxiliary problems in Harbor A
The function class of the set of inlet variable values accepted by the system will have the expression (={(i,i+1:T(U} (11) for i = 1,2,3,21.
According to (3) (i,i+1={u[ti,ti+1)(ti,ti+1(T,u[(ti,ti+1)](U} (12)
Under the conditions existing in the year 2003, the average values of time segment [ti,ti+1) recorded stochastic values, depending on the context of the situation, with effect on establishing the partial costs (I,I+1 and the general cost of the voyage. Reduction of the expectation time in harbors and exploitation of the ferry boat supposes the calculation of previous statistic elements (history), introduction of Poissona Law, taking into account stochasticity of certain elements intervening in the system (expectation system characterized by expectation discipline), optimum repartition of the boats to the berths (affecting theory), systematc optimization, the elements having implications in cost minimization.
Mathematical aspects of stochastic and determinist level that may be used in minimizing the main variables within the ferry-boat complex system.
The statistic elements regarding the ferry-boat exploitation activity within A-B maritime line give indications on the stochasticity and determinism of certain main elements, components of the system, such as: expectation time used to get permission to pass through the Bosphorus Strait, expectation time from boat arrival in the roads until its entering the landing berth/operating berth, the time used to expect the cargo and the time required for loading/unloading, etc. We consider the number of voyages .
It follows the average value of the time recorded in the period considered for expectation to pass through the Bosphorus Strait (t2,t3)med.=([t2,t3)i / n(13),(i-the serial number of the respective vo- yage, [t2,t3)i- the value of the time recorded for this operation(hours)during voyage number i) the average value of time recorded from boat
arrival at the roads until its entering the landing berth/operating berth [t5,t6)med.=([t5,t6)i / n (14) (for Harbor B) or the average value of the time recorded from boat arrival at the roads until its entering the landing berth/operating berth [t17,t18)med.=([t17,t18)i/n (14*) (for harbor A) (the component elements having the same significance as in (13) but with reference to the respective operation), the average value of the time used for expecting cargo in Harbor B [t8,t9)med. =([t8,t9)i / n (15), (the component elements having the same significance as in (14) but with reference to the respective operation), the average value of the time used for its loading/unloading in Harbor B [t9,t10)med.= ([t9,t10)i / n (16), (the component elements having the same significance as in (15) but with reference to the respective operation). The last two values may also be calculated for Harbor A (if necessary). These are the determining parameters extracted from the family of the characteristic parameters. The other elements are not decisive in the stochasticity of the time used and the costs for making the voyage, generally having invariable values. The empirical repartition curve for the cumulated expectation in order to sail through the Bosphorus Strait has the outline in the graph above. It follows that there are a number of 17 boats of similar capacity which may wait for passing through the Bosphorus Strait a cumulated time of about 3,000 minutes. For one such boat 176 minutes/operation are required, the equivalent of 3 hours and 33 minutes expectation. The number of situations in which the time duration for expectation to pass through the Bosphorus Strait is longer or shorter than the average time recorded in determined periods may be seen in the graph below. One can see that under the average of the expectation minutes there are a number of
situations, out of which three are cumulated around the value for interval [480, 540) minutes, and two situations around the value for intervals [540,600) miutes and [780,840) minutes respectively. The spline curve in the image indicates a number of maximum points with low frequency. Under the conditions in which the expectation duration for passing through the Bosphorus Straits were around the value of [480,540) minutes (or less) this interval may be considered optimum (8-9 hours).
Elements of the expectation theory are used, under the conditions of expectation disciplin and use of Poissons Law: Pn (t)=[((t)ne-(t]/n!, (17) (pentru expectations) and Qn= [(e-((n)] / n! (18) (for servings), n= 0,1,2,The algorithm which determines the optimum expectation duration (minimum) for passage is obtaind, taking into account the time for servings. In expressions (17) si (18)Pn (t)- the probability that in interval t should be n arrivals (servings), ,(t=a-the average number of arrivals in interval t,(-the average coefficient of arivals,(-the average number of servings in time unit. We note ( = [(/(] (19) the serving factor. The number and frequency of the statistically observed situations, the average and dispersion are evaluated in the calculations and .
Table 1
No.of observed cases
Frequency
Expectation period for a ship similar to a ferry-boat to pass through the Bosphorus Strait
(minutes) n mn(n-mn )2 fn110[60,120);[120,180);[240,300);[300,360);[720,780);[1020,1080);[1500,1560);[1980,2040);[2400,2460);[2640,3000)-0.3070.94222[540,600);[780,840)0.6930.96031[480,540)1.6932.866
Average m n={[(n f(n)]/N}=1.307 (20)
Dispersion sn2 ={[((n-mn)2 f(n)]/N}=0.366 (21)
N=13- total number of recorded frequencies
The hypothesis of the repartition of arrivals in expectation for passing through the Bosphorus Strait proves to be true according to Poissons Law.
It follows:
Pn=[(n e -(]/n! (22) . The component value of the repartition (2 is [(f(n)-NP(n))2/NPn] (23)
((=mn=1.307si n=1,2,3)
and repartition (2 has the expression :
(2=( [(f(n)-NP(n))2/NPn]=6.867 (24)
in which:Pn=P0*((n/n!) (25)
P0 =1/[((s/s!(1-(/s))+( (( k/k!)] ((26) k=0,1,s-1.
Table 2
No of observed cases
Frequency
Pn[(f(n)-NP(n))2/NPn]1100.3516.478220.2290.320310.1000.069 (2 = 6.867
Using test (2 we deduce that for a safety threshold p = 0.05and three freedom degrees, the value (02 = 7.81 (table of repartition square-hi in the statistic tables). As variable (2=6.867((02 has a low effective value, it follows that it is placed within the trust interval corresponding to the calculated safety coefficient - p = 0.95. The frequency observed for ferry-boat arrivals while expecting to pass through the Bosphorus Strait corresponds to Poissons Law. For serving at boat entrance in a free corridor for passing through the strait, it was checked, by the same method, that the duration of the operation also follows one of Poissons laws (rel.(18)). The average expectation time found statistically is considered the arithmetical mean between the recorded time of cumulated arrivals for one ferry-boat (3 hours and 33 minutes) and the median value of interval [480,540). Under the circumstances the average expectation time for boat enetering the free corridor to pass will be of 510 minutes. The average number of servings will be:
( =1440/510=2.82 (27)
The average number of arrivals on time unit is calculated by means of expression:
(={[(nf(n)]/(N/2)}=2.61(28)
and the serving factor:
(=(/( =0.92<1(stationary regime)
Cosidering that there are two free corridors for passing (s=2), the probability of not having to wait (rel.(26)) is:
P0 ={1/{[(0,922 )/(2!(1-0,92/2))]+1+0,92/1!}= 0.369 (29)
And the probability to have to wait is [(s /(s!(1-(/s))]P0 = 0.289 (30)
The average expectation time in line for ferry-boat type ships is:
tf* ={(s /[s s!( (1-((/s))2]}P0 =0.095 (31)
and the average number of units in expectation is: (*={(s /[s s!( (1-((/s))2]}P0=0.24 (32)
The apparent contradiction between these results (in hours) and the value recorded in exploitation {[480,540)min.}, is owing to the fact that through the Bosphorus Strait pass not only ferry-boat type ships in the same period. In the area there are also larger capacity ships in expectation. In addition there are problems arising from the way the Bosphorus Strait Administration organizes the passage for all the ships in the area. This operation is made according to certain criteria which are not always generated by the phenomena in expectation. A ferry-boat may pass through the strait (according to conclusions (31) and (32) ) waiting a minimum time of 0.0.95 in the area, only if it leaves the initial harbor so as to reach the Bosphorus area in the period when the traffic is not heavy (statistically between 12 a.m. and 4 a.m.) The boat should start the voyage 21 hours earlier, the respective voyage taking place in usual weather conditions. Cost reductions in this case are obvious. Boat entrance to the operating berth in effective conditions means using some expectation time in Harbor B roads. Statistically it is determined. Harbor B Administration makes the distribution of ferry-boats to the road, correlated with the distribution of the other ships existing in the harbor at the analyzed moment. (according to the vacant roads and order of arrival). For optimization elements from the affecting theory have been used.
The complex system ferry-boat-cybernetic system with multiple variables.
For the complex system considered in the paper as a cybernetic system with multiple variables the following configuration is identified:
I-in-put / E-out-put
State internal values S
7 6 5 4 3 2 1
Perturbating values P
1 1
2 2
3 3
4 4
5 5
6 6
7 7
Output values V
Execution values A
7 6 5 4 3 2 1
FEED-BACK APPLIED TO FERRY-BOAT Internal state
COMPLEX SYSTEM values S
(F)
(EXPLOITATION - MICROECONOMY)
1 1
2 2
3 3
4 4
5 5
6
Guiding values R 7 6
7
Fig.1 Configuration of ferry-boat complex system considered as a cybernetic system with multiple variables
For year one takes into account the values of year out-put.
The following matrix is introduced :
K-ferry-boat complex system
F-formator (exploitation and microeconomy)
A-formators effecting values
P-perturbating values
R-guiding values
x-state internal values (structural values) of the system
I- in-put values
E- out-put values
X(t)-state vector of the system
OI-in-put operator
OR-out-put operator
j-analyzed year
The calculation values use specific ecometric values:
OI
E= . I (33) general out-put values of the ferry-boat complex system
1 - OI . OR
OI
Ej= . Ij (34) out-put values of the year
1 - OI . OR
Ej-1= Ij (35) recurrent calculation relation between out-put value and . in-put
OI
Ij = ( )j-1Ij-1 (36) in-put values of the year
1 - OI . OR
Ej = ( OI )jEj-1 (37) out-put values of the year
1 - OI . OR
By mathematical processings a structure relation is obtained belonging to the ferry-boat complex system, characteristic of the year versus the year (rel.38) :
OI OI
Ij/Ej=[( )j-1 / ( )j ].[ Ij-1/Ej-1] (38)
1- OI . OR 1- OI . OR
Structural relations characteristic of year are obtained, according to the elements of year <2003>:
OI
Ij = I2003 (( )k (39) k =2003,,(j-1)
k 1 - OI.OR
Ej = E2003 (( OI )k (40) k = 2003,,j
k 1 - OI.OR
in which: (OI /(1 - OI.OR))k=(Ek 100)/100 (41)
j-1 OI j OI
Ij/Ej = [( ( )k / ( ( )k ].[ I2003 / E2003] (42)
k=2003 1 - OI . OR k=2003 1 - OI . OR
in which : (OI/(1 - OI.OR))j = (Ej ) / 100 (43)
100
( Ej ) / 100 parameter referring to the year previous to the calculation year of the structural .
100 relation , whose value is cosidered 100
Ij / Ej = (Ej-1 / 100 ) /(Ej / 100 ) / (Ij-1 / Ej-1) (44)
100 100
j-1 j
Ij / Ej =[( (Ek /100 ) /( (Ek /100 )] / (I2003 / E2003) (45)
k=2003 100 k=2003 100
( OI / (1 - OI.OR))j-1 - adjustment operator (46)
In the paper there are relations by which in-put value Ij attached to the ferry-boat complex system are expressed according to its in-put value Ij-1. An (exploitation or economic) parameter characteristic of a year may be expressed iteratively according to the same parameter corresponding to the previous year. The out-put value Ej is expressed according to the out-put value Ej-1. The obtained expressions are structural relations, applicable to any system indicator, according to the similar values of the preceding year or of the reference year. With the introduced notations {(i,I+1=u[ti,ti+1)}results specific of the ferry-boat complex system are obtained.
1.3 Differential equation system attached to the ferry-boat complex
We consider the ferry-boat complex system (, dimensionally finite, smooth, linear, characterized by previous state vector introduced in space RN, marked X=(x1,x2,xN).
The attached differential system has the general form :
dX(t)/dt=f(t,X(t),u(t)) (47)
in which vector u = (u1,u2,,um) (48) represents the check-up variable and the control is represented by measurable function u:[t0,T](Rm,u(t)=(u1(t),,um(t)) (49)
T([t0,T],f:[t0,T](RN(Rm(RN (50), f(t,x,u)=(f1(t,x,u),,fN(t,x,u) (51)
X:[t0,T](RN-system solution (47)),U-subset of space Rm, -set of measurable functions u:[t0,T](Rm,u(t)(U(t)-family of closed sets of space Rm
Functional cost will be represented by function:
(:AC([t0,T];RN)(((-(,+() (52)
((X,u)=( L(t,X(t),u(t))dt + l0(x(t0)) + l1(x(T)) (53) considered between integrating limits [t0,T]
AC([t0,T];RN)-space of absolutely continuous functions from [t0,T] in RN
L:[t0,T](RN(Rm(R
l0: RN(R si l1: RN(R given functions
L(t,X(t),u(t))-measurable function regardless u-measurable and x-continuous
Under the given conditions value ( with (i,i+1 i=1,2,3,,21 ( ferry-boat costs) may be equated.
Functional ( minimized on the function class X(AC([t0,T];RN) and checks (47).
The optimum pair (X*,u*) checks the equation:
((X*,u*)=min{((X,u);(X,u)(AC([t0,T];RN)(,(x,u)checks (47)} (48)
the problem of optimal control admitting at least one solution(x*,u*)(AC([t0,T];RN)(L1(0,T;Rm ) (49).
The mathematical model of stochastic level has completely fixed resources and enjoys the existence of the probabilistic mean. Repartition being a density function, an approximation of continuous probabilistic evolution follows , by means of a discrete probabilistic evolution. We obtained methods by which the fundamental values of the ferry-boat complex system dynamics may be minimized, therefore resulting optimization.
Bibliography:
1.G.Barles,Uniqueness of first order Hamilton-Jacobi equations and Hopf formula, J.Diff. Eqns.,1987
2.N.K.Bose,Multidimensional Systems Theory,Kluwer Print on Demand,2002
3.N.K.Bose,Multidimensional Systems Theory:Progress,Directions & Open Problems in Multidi-
dimensional Systems,D Reidel Pub Co,1986
4.A.S.Camara,A.S.Da Camara,Environmental Systems:A multidimensional Approach,Oxford
University Press,2002
5.G.B.Dantzig,M.A.H.Dempster,Markku Kallio,Large Scale Linear Programming, Procee- dings of a IIASA workshop,2-6 June 1980
6.R.C.Dorf,R.H.Bishop,Modern Control Systems,Prentice Hall,2000
7.A.Isidori,Nonlinear Control Systems,Springer Verlag,1997
8.R.E.Kalman,P.L.Falb,M.A.Arbib,Topics in Mathematical System Theory,Mc.Graw Hill,1969
9.H.K.Khalil,Nonlinear Systems,Prentice Hall,2001
10.L.A.Zadeh,C.A.Desoer,Linear System Theory,Mc.Graw Hill,1963
PAGE 1
PAGE 10
FERRY-BOAT COMPLEX SYSTEM
(K)
Ferry-boat subsystem
Maritime berth subsystem Harbor A (Romania)
The Black Sea subsystem
Bosphorus Strait subsystem
The Marmora Sea subsistem
Maritime Roads subsystem harborB (Turkey)
Maritime berth subsystem Harbor B(Turkey)
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