397
1 INTRODUCTION
Ro‐Ro and Ro‐Pax shipping lines schedules are
plannedinadvanceandRo‐RoandRo‐Paxshipstry
tokeeparrivaltimestoportsasaccurateaspossible.
AccurateRo‐RoandRo‐Paxshipsarrivaltimetoany
port is important for the all involved
parties in this
business: passengers, shipping lines operators,
transport units (road and railway), port itself and
terminals. In the same time different type of
difficulties occur during Ro‐Ro and Ro‐Pax ships
voyage at sea between ports, such as: storm
conditions, fog situations, when it is necessary to
decrease
ship’sspeed,shipsqueueattheentranceof
theportsandetc.takesadditionalwaitingtimeforthe
Ro‐Ro and Ro‐Pax ships during the entrance to the
portsordeparturesfromtheports[2].
Evaluation methods of Ro‐Ro and Ro‐Pax ships
arrival time into the port depends
on the weather
conditions,shipsqueueneartheportentrance,Ro‐Ro
shipsfailuresandsoonbasedonstatisticaldataandit
is possible used mathematical methods, which can
help forallRo‐Ro and Ro‐Pax shipping participants
and can assist in better planning of ports and
terminals
activity, passengers and transport units
informationregardingpossibledelays[5,11,13].
2 ANALYSISOFRO‐ROSHIPSARRIVALTIMETO
PORT
Ro‐Ro, Ro‐Pax and container ships works according
to the stable schedule and try to keep maximum
possibleaccuracyofthearrivaltime,whichdepends
onthe departure time from
the port of departure. It
requeststoadoptship’sspeedandincase,ifRo‐Roor
Ro‐Paxvesselshavesomereservetime,itispossible
tofulfillthearrivaltimeschedule[7].
Inthesametime,duringRo‐RoandRo‐Paxships
voyages sometimes situations occur, like
storm
conditions,waitinginqueuetoentertheport,ifthere
arewaitingpassengerships,aswellasanyproblems
in the port approach and inside navigational
Accuracy Evaluation of Ro-Ro and Ro-Pax Ships Arrival
to the Ports
V.Paulauskas&D.Paulauskas
KlaipedaUniversity,Klaipeda,Lithuania
ABSTRACT:Ro‐RoandRo‐Paxshipsworksonbasisofregularscheduleandtrykeepthisscheduleasaccurate
as possible. In same time difficulties during Ro‐Ro and Ro‐Pax ships voyages like bad weather conditions,
waiting time to entry in to port, shipʹs
failures and others, sometime request more time as it was planned
initially.Ro‐RoandRo‐PaxshipsarrivalsintoportevaluationisimportantnotonlyjustfortheRo‐RoandRo‐
Paxshippinglinesplanningbutaswellitisveryimportantforpassengers,transportunits(trucks)
andport
terminals. In this Article are analyzed delay times of Ro‐Ro and Ro‐Pax ships arrival to port on basis of
statisticaldataandusedmathematicalmodelsforcalculationofthearrivaltimepass.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 14
Number 2
June 2020
DOI:10.12716/1001.14.02.17
398
channels, etc. It makes very difficult to keep the
schedules [12]. Analysis of Ro‐Ro and Ro‐Pax
shippinglinesinSouthpartoftheBalticseashowed
that about 30 % of Ro‐Ro and Ro‐Pax ships arrivals
are delayed up to 20 minutes, about 20 % of
ships
arrivalsaredelayedupto40minutesandabout10%
ofshipsarrivalsaredelayedonehourormore.Some
statisticaldata,astheexample,ofonemonthofRo‐Ro
and Ro‐Pax vessels arrival delays on Ro‐Ro line
Klaipeda – Kiel and Klaipeda – Karlshamn
are
presentedonFigure1.
Figure1. One month of Ro‐Pax ships delay in minutes at
Klaipeda port on Ro‐Pax lines Klaipeda – Kiel (KL‐KIEL)
andKlaipeda–Karlshamn(KL‐KARLS)
Mentioned Ro‐Pax ships “Victoria Seaways”
(Klaipeda – Kiel line) and “Optima Seaways
(Klaipeda – Karlshamn line) speed distribution
during the voyages are presented on Figure 2 and
Figure3.
Figure2. Ro‐Pax ship’s “Victoria Seaways” speed
distribution during the voyage Klaipeda – Kiel” (AIS
information)
Figur3.Ro‐Paxship’s“OptimaSeaways”speeddistribution
during the voyage Klaipeda – Karlshamn(AIS
information)
Ro‐Pax ships arrival time delay increase during
autumnand winter periods and more precise to the
schedule are in summer time, but it very much
dependsoftheotherfactors,likeholidays,weekends
aswellasspaceutilizationof theRo‐Paxships,port
terminalscapacities.
3 THEORETICALBASISFOR
THERO‐ROAND
RO‐PAXSHIPSARRIVALSACTUALTIME
EVALUATIONANDCASESTUDYOFTHERO‐
PAXSHIPSARRIVALTIMEDELAY
EVALUATION
Ro‐Ro and Ro‐Pax ships arrival evaluation is
importantfortheportandterminalactivityplanning.
For Ro‐Ro ships arrival evaluation could be used
several
methods, for example maximal distribution
method or dispersion method [4, 8]. On basis of
maximaldistributionmethoditispossibletoreceive
shipsdelaytimepass,butdensityinthispasswillbe
different.
3.1 Dispersionandmaximaldistributionmethodsforthe
Ro‐RoandRo‐Paxshipsarrivaltimedelay
calculation
Dispersion method presents delay time pass and
dependsontheprobabilitywiththesamedensityin
passwidth.
Average circle error (e) in dispersion method is
comparablewithdispersion(
y
)[11]:
y
e
(1)
Dispersioncanbecalculatedasfollows:
22
1
()
1
ytiyi
i
Tm
n
, (2)
where:
i
n ‐number of the statistical points (days);
yi
m
‐ mathematical hope of the Ro‐Pax ships of the
arrivaltimedelaycanbecalculatedasfollows:
1
yi ti
i
mT
n
,(3)
On the basis of Ro‐Pax ships arrival delays,
accuracy (arrival time pass) could be calculated as
follows:
(max) ( )arrival arrival scedule delay
TT Te
, (4)
(min) ( )arrival arrival scedule delay
TT Te
. (5)
Ro‐Paxshipsveryrarearrivebeforethescheduled
time.Incase,ifRo‐Paxshiparrivaltimeisbeforethe
scheduledtime, captains always decrease speed and
save some fuel. That means formula (5) just
theoretically can be used, practically just in
emergency situations, for example in case
of
399
requestedurgentmedicalassistanceforpassengersor
crewmembers,Ro‐Paxshipcanentryinportearlier,
because in usual conditions port terminals are not
ready for mooring of ships and provide cargo
handlingorpassengerembankmentoperations.
In some cases it is possible to use maximal
distributionmethodfor
theRo‐Paxshipsarrivaldelay
calculation. Main dependence could be expressed as
follows[14]:
max min
'
AV n
TT kPT
(6)
where:
AV
T ‐ average Ro‐Pax ship’sarrival time, can
be taken as
yi
m
;
n
k
‐coefficient depends on the
numberofdata,incase,ifnumberofdataare3–this
coefficientwillbe‐0,55;incasenumberofdata4‐
thiscoefficientwillbe0,47andsimilardependsofthe
datanumber:5–0,43;6–0,395;7–0,37;8–
0,351;9–
0,337;10–0,329;11–0,325;12–0,322andsoon,but
minimum of this coefficient could be about 0,315 in
case number of data will be more as 15.
'
P
‐
probability dependence coefficient (in case of this
coefficient1–probabilitywillbe68,3%,incaseof
thecoefficient2–probabilitywillbe95,3%,incaseof
thiscoefficient3–probabilitywillbe99,7%);
max min
T
‐differencebetweenminimumandmaximumresults,
that means between earliest and latest Ro‐Pax ships
arrivaltime.
As the case study of Ro‐Pax ships arrival time
delay evaluation were taken Ro‐Pax shipping lines
Klaipeda–KielandKlaipeda–Karlshamn.BothRo‐
Paxshippinglinesaredaily,just
sailingtime to one
directionontherouteKlaipeda–Kieltakesabout20
hoursandtimeinporttakesabout4hoursandonthe
routeKlaipeda–Karlshamnsailingtimetakesabout
14 hours and shipʹs stay at the port takes about 10
hours. Analysis covers few
months during summer
timeingoodweatherconditionsandinwintertime,
whenweatherconditionsweremorecomplicated.
Inwinter andsummer time wereexcludedstorm
days,whenwindwasmorethan20m/sandRo‐Pax
ships cannot entry into the port or leave port
according to the schedule. In
total were taken more
than 100 arrivals and received next results:
mathematical hope of the Ro‐Pax ships on line
Klaipeda–Kieltoportarrivaldelaywas17minutes,
andon line Klaipeda –Karlshamn‐was21 minutes
anddispersionsoraveragecircleerrorreceived+/‐8
minutesfor
theRo‐PaxlineKlaipeda–Kieland+/‐6
minutesfortheRo‐PaxlineKlaipedaKarlshamn.
According to maximal distribution method was
taken probability 68,3 %, coefficient depends on the
number of data was taken 0,315, and difference
betweenminimumandmaximumresultsfortheRo‐
PaxlineKlaipeda
–Kielwasreceived62minutes,for
theRo‐PaxlineKlaipeda‐Karlshamnwasreceived51
minutes.
Average Ro‐Pax ship’s arrival delay time for the
Ro‐PaxlineKlaipeda–Kielwasreceived17minutes
incomparisonwithtimetableandfortheRo‐Paxline
Klaipeda – Karlshamn
was received 12 minutes in
comparison with the schedule. Finally according to
maximaldistributionmethodRo‐Paxshipsdelaytime
for the line Klaipeda –Kiel was 36 minutes and for
theRo‐Paxline Klaipeda –Karlshamn–28 minutes.
Differences between evaluation methods based on
different density, because in
dispersion method
density of the results is similar in all received time
pass and in maximal distribution method density is
differentonreceivedtimepass.
Next important problem, which is necessary to
solve: sustainable information system, which has be
useful for ports, terminals and clients, that means
accurateinformationfor
passengersandtruckdrivers,
because it links with city limitations to use some
streetsinrushhoursfortheheavytransportsdriving
toandfromportterminals.
3.2 Graphtheorypossibilitiesoptimizefreightand
passengerstransportrichRo‐Roterminals
ThisproblemfortherichRo‐Roterminalinport via
cities during rush hours can be solved on basis of
graph theory. For developing an optimal streets
network,which arenotbusyduring rush hoursand
couldbeusedfortheheavytransport.Theapplication
ofgraph theorymethodis used,wherethemodelis
buildinthatincorporatesa
setofvertices,whichare
representing permit streets cross places and a set of
edges,whichrepresentsthedistancesbetweenpermit
streetscrossingpoints.Theoptimalpermitstreetsfor
heavytransportsduringrushhoursnetworkmodeled
asagraphisexpressedasfollows[6,8]:
(, )GVE
, (7)
where:V‐thesetofvertices;E‐thesetofedges.
As an example, the permit streets network could
becreatedinKlaipedacity streetsnetwork,whichis
notbusyduringrushhours,asshownonFigure4.
Figure4. Not busy streets network in rush hours as the
graphtree
Forthegraphtree,presentedonfigure4,thesets
of vertices and the set of edges can be expressed as
follows[1,4,9]:
123456
,,,,,Vvvvvvv
(8)
12 23 24 25 56
( , )( , )( , )( , )( , )E vvvvvvvvvv
(9)
400
The all‐vertex incidence matrix of a non‐empty
andlooplessdirectedgraphforthepresentedgraph
treeGis[3,6]:
ij
A
a
, (10)
1 if v is the initial vertex of e
-1 if v is the terminal vertex of e
0 otherwise.
ij
ij i j
where a
In this study case for permit streets for heavy
transportsnetworkincidencematrixcanbeexplained
asfollows[6]:
123456
2
3
4
5
6
,,,,,vvvvvv
v
v
A
v
v
v
(11)
For the graph tree covering city streets network,
whichisexplainedonFigure4,mentionedmatrixin
formula(11)canbecomputedasfollows:
010000
101110
010000
010000
010001
000010
A
(12)
Finallyfortheoptimumdistancesbetweenpermit
streets cross points for the freight transport in rush
hoursoroptimalpriceinnetworkcouldbeusednext
optimizationformula[3,6]:
:
f
ER
, (13)
Anditisnecessaryfindgraphtree
'
()TVE
price
oroptimaldistanceF(T)like,
'
() ( )
xy E
F
Tfxy
, (14)
where:
()
f
xy ‐minimumpriceoroptimaldistance.
In study case the edges
exyE
as minimum
priceoroptimaldistancecouldbefindasfollows[3,
4,6]:
() min ( )
xy E
f
efxy
, (15)
Based on the suggestion of graph theory it is
possible argues that it is possible to design permit
streetstoorfromterminalsforthefreighttransportin
rushhoursnetwork, andtoconsider additionallyon
Ro‐Pax ship’s entry to port delay time evaluation,
sinceitis
possibletoidentifythesetsofverticesand
thesetofedgesweights.Inconsiderationofweights
of the sets of vertices and edges it is possible to
improve or identify the optimal permitted streets
locationnetworkforheavytransportsandpassengers
carsduringrushhours.
Themainquestion,howit
ispossibletoimplement
oratleasttakeinaccountresults,whicharereceived
intheArticle. Firstofall,ifRo‐Pax shipshavemain
engine power reserve, it is possible to increase the
main engine power and compensate Ro‐Pax ship’s
arrivaltimedelaytotheport,but
inthesametimeit
willincreasefuelconsumptionandshipssailingcosts.
ItispossibletoimplementthissystemonRo‐Paxline
Klaipeda – Karlshamn, because on this line Ro‐Pax
shipsoftendonotusedfullenginepower,orschedule
is changed slightly, that means departure time does
notchangetoomuch.
On Ro‐Pax line like Klaipeda – Kiel ship’s used
mainenginepowerisclosetothefull(fullseaspeed),
soitisnecessarytousefullmainenginepowerandas
experimental results show, to compensate ship’s
arrival to port about 36 minutes, additional
fuel
consumptionisaround3,5tons.
Possibility to keep Ro‐Pax ships schedule and
decreasearrivaltotheportdelayscouldbeincluded
in tickets for passengers and trucks price and more
accuratecalculationforpossiblelossesordecreaseof
profitability.
4 CONCLUSIONS
1 Ro‐Pax ships entry to the port
delay time
evaluation methods presented in Article could
assistfortheRo‐Paxshippingcompaniesformore
accurate preparation of schedules to avoid
problems for the passengers and heavy road
transport.
2 Moreaccurateinformationondelaytimeofentry
into the port of Ro‐Pax ships could evaluate
requested
additional sources and align services
tariffs.
3 Possibility use graph theory methods for the
preparing optimal ways to reach terminals and
avoidconflictswithcitiesauthoritiescanoptimize
all the transport process especially in big port
cities.
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