515
1 INTRODUCTION
Given that, the Vessel Turnaround Time (VTT) is one
of the key performance measures used by
international shipping lines to determine which
transshipment hub ports to use, this research looks
into the factors that influence container vessel
turnaround times. It was observed that a number of
factors influence vessel turnaround time; as a result,
crucial parameters under the control of the terminal
operator were analyzed for a more in-depth analysis
using the stochastic model. VTT refers to the amount
of time a ship spends in port from arrival to
departure. (Daganzo & Goodchild, 2005).
Despite the fact that it is offered as a separate time
measure, VTT is a summation of several sub activities
such as waiting for a berth, maneuvering time,
mooring/unmooring time, idle time, container
handling time, and other time components until the
vessel exits port limits (Moon, 2018).
Independent Variables (Xi) are used to describe the
influential components, while VTT is a Dependent
Variable (Y). Because ships are built to sail, the more
time they spend on the water, the more money they
make. As a result, ship-owners and shipping lines
expect faster port operations, which will lead to
shorter VTTs and more load trips per year.
In the year 2000, an average-sized container vessel
spent around 60% of its time at berths, with a daily
cost of $65000, according to Ghotb, Kia, and Shayan
(2000).
Container terminals are specified operational
entities specializing in the handling and storage of
containerized cargo activities, according to Ting
(2018), where containers can be unloaded, loaded,
received, delivered, stored, and assisted movement
Stochastic Model to Estimate the Waiting Time for
Container Vessel Turnaround Times
A. Elentably, K. Fisher, S. Holger, A. Alghanmi & S. Alhrbi
King Abdul-Aziz University, Jeddah, Saudi Arabia
ABSTRACT: By developing possible solutions for Saudi ports to limit the increase in damage to the marine
ecosystem, the random system for estimating the waiting time of ships in Saudi ports has been developed as a
model to guide the application of the multiple benefits to the beneficiaries such as ship owners, shipping
companies and port authorities so that it is applied to create multiple economic and environmental savings. An
imperative optimization model for solving container slot allocation problems for time-sensitive commodities
under the dynamics of port congestion pricing. The proposed new pricing mechanism has proven to be effective
when compared to a generic slot allocation model that does not take into account shipping time limits and port
congestion, with results showing that the proposed pricing scheme can significantly improve ship companies'
revenues and improve customer satiation. In terms of reducing carbon emissions from the ship's stay for a
longer period at the docks.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 16
Number 3
September 2022
DOI: 10.12716/1001.16.03.14
516
across different modes of transportation (vessels,
barges, trucks and railways etc.). To establish the key
predictors of container port technical efficiency in
Niavis & Tsekeris, the researchers used data from a
Tobit Regression and a Truncated Regression (TR), as
well as Parametric Bootstrapping Models (PBM)
(2012).
The study, which is based on previous research,
finds that port performance improves as port size
grows, meaning that larger ports outperform smaller
ports. According to Alemán et al. (2016)
Efficiency Analysis (EA) research, port efficiency
in growing regions is increasing, with time series data
indicating a 10% increase from 51% in 2000 to 61% in
2010. According to the study (Alemán et al., 2016),
improvements in liner connectivity, private sector
participation, government sector corruption
reduction, and the development of multi-modal links
have all influenced port efficiency levels in rising
regions. Cariou and Oliveira (2015) , used a TR and a
PBM to investigate how competition affects container
port efficiency at various levels of impact.
In Sanchez, Tovar, and Wilmsmeier, the
researchers used non-parametric Data Envelopment
Analysis to study the effects of dynamic economic
circumstances on container terminal efficiency and
productivity (2013). Between 2005 and 2011, the study
looked at 20 container terminals in ten countries
across Latin America, the Caribbean, and Spain.
Waiting time, container handling time, and overall
ship turnaround time are the primary metrics used to
evaluate container terminal performance
(Budipriyanto, et al., 2017).
The availability and assignment of a suitable berth
for an arriving vessel can have a significant impact on
the measure above. He (2016) employed a Mixed
Integer Programming model for berth allocation and
QGC assignment, which provides optimal solutions
for time and energy savings in overall berthing
expenses.
2 STOCHASTIC APPROACH USING
DISTRIBUTIONS IN CONTAINER VESSEL
Recommend Erlang random variables for two crane
kinds, however Choi and Yun (2000) proposes normal
random variables (quay, yard). In terms of crane cycle
time, Koh et al. (1994) recommend using a Weibull
random variable; Bugaric and Petrovic (2007) propose
normal random variables for a bulk cargo terminal
and report the estimated parameters. Traditional
numerical discretization approaches, on the other
hand, are often used to ordinary or partial differential
equations stated as underlying physical or
mathematical problems (Khowaja, Saeed, and Alvi
2004).
3 METHODOLOGY
In Saudi Arabia there are total three Ports namely:
Dammam, Jeddah and Jubail. Among these three
ports there is no large TEUs with a capacity of more
than 4 million. The two ports namely, Dammam and
Jeddah comes under Medium (0.5-4 million TEUs)
and Jubail port is Small (less than 0.5 million TEUs).
East Asian ports dominate the top 50 ranking ports.
Yokohama port (Japan) is the top-ranked container
port in the CPPI 2020, followed by King Abdullah
port (Jeddah, Saudi Arabia) in second place.
Regardless of the approach, these two ports occupy
the same two locations.
In order to explore within the limited scope of this
work, the following assumptions are established for
assessing terminal data for our study:
1. Waiting time (c), time gap between anchor in and
anchor out
2. Loaded and empty containers (μ) can be handled at
the same speed
3. Containers on decks and hatches (β) can be handled
at the same speed
4. Crane operators maintain the same handling speed
from start to finish (d)
Figure 1. Flow chart of VVT to assess the expected waiting
time
4 MODEL TO ESTIMATE THE EXPECTED
WAITING TIME
The existence or absence of a random variable is
referred to as "stochastic." Stochastic processes have
several essential hypotheses about time and state
variables. Stochastic models will help us identify and
predict the hidden components. The Shifted
Exponential Distribution can be used to simulate
lifetime data with growing, decreasing, and upside-
down bathtub shaped disintegration rates (SED). In
SED, only the mean changes, while the variance stays
the same. The loaded and empty containers,
containers on decks and hatches and crane operators
are the parameters of the SED.
( )
( ) ( ) ( )
00
xd
i k k
P X Y g x H x dx g x e
−
−
= =
On simplifications we get,
k
*
1d
g
β
−
=
(1)
( ) ( )
S t P T t= =
The survival function S(t) which is
the probability that VVT survives for a time t.
It is known from renewal process that
517
( ) ( ) ( ) ( ) ( )
( )
*
1
00
1
**
1
1
11
1
k
k i k k
kk
k
k
k
d
P T t F t P X y F t F t g
dd
g F t g
+
==
−
=
−
= = − =
−−
=−
(2)
( ) ( )
P T t L t = =
the distribution function of life time
(t)
Using convolution theorem for Laplace transforms,
F0(t)=1 and on simplification, it can be shown that,
( ) ( )
1
**
1
11
1 1
k
k
k
dd
L t g F t g
−
=
−−
= − −
(3)
Let the random variable with waiting time (c),
follows exponential with parameter,
( )
*
c
fs
cs
=
+
, on
simplification we get,
( )
( )
( )
* * *
*
* * *
11
1 1
11
1
dd
g f S g c
ls
dd
g f S c s g c
−−
−−
==
−−
− + −
(4)
After first and second derivatives on simplification
we get the estimated waiting time of VVT through
SED as seen in equation (6),
( )
1
1
d
ET
cd
+−
=
−
5 NUMERICAL ILLUSTRATION
The stochastic character of the variables involved in
equations (6), modeling as a management tool enables
for the modeling of intricate processes that would
otherwise be difficult to investigate analytically. The
model is used in simulation to calculate the Estimated
waiting time E(T) as a function of various parameters,
allowing for the evaluation of technical and economic
options without the need for large capital
investments. When μ,β,d is 0.3, 0.5, or 0.7, Table 1
illustrates fixed VVT for varied waiting times, also
described in figure 1. When one parameter VVT is
clear and the other two parameters VVT have waiting
time clearance, how the expected waiting time
behaves is illustrated in Table 2, Figure 2. VVT
clearance for two parameters is clear and for One
parameter needs waiting time clearance is shown in
Table 3 and Figure 3.
Table 1. Fixed VVT for different waiting time period
_______________________________________________
c 0.3 0.5 0.7
_______________________________________________
0.1 11.29 15 26.33
0.2 5.64 7.5 13.17
0.3 3.76 5 8.78
0.4 2.82 3.75 6.58
0.5 2.26 3 5.27
0.6 1.88 2.5 4.39
0.7 1.61 2.14 3.76
0.8 1.41 1.87 3.29
0.9 1.25 1.67 2.93
1.0 1.23 1.5 2.63
_______________________________________________
15
10
0
4
8
5
0
12
10
0
20
0.3
0.5
0.7
Figure 1. Fixed VVT for different waiting time period
Table 2. VVT clearance for one parameter at different
waiting time period
_______________________________________________
c d=0 β=0 μ=0
_______________________________________________
0.1 12.5 10 10
0.2 6.25 5 5
0.3 4.17 3.33 3.33
0.4 3.13 2.5 2.5
0.5 2.5 2 2
0.6 2.08 1.67 1.67
0.7 1.79 1.43 1.43
0.8 1.56 1.25 1.25
0.9 1.39 1.11 1.11
1.0 1.25 1 1
_______________________________________________
1001010.10.01
99
90
80
70
60
50
40
30
20
10
5
3
2
1
Data
Percent
3.662 10 0.723 0.229
2.929 10 0.723 0.229
2.929 10 0.723 0.229
Mean N AD P
d
ß
µ
Variable
Probability Plot of d, ß, µ
Exponential - 95% CI
Figure 2. VVT clearance for one parameter at different
waiting time period
Table 3: VVT clearance for two parameter at different
waiting time period
_______________________________________________
c μ,β=0 d,β=0 d,μ=0
_______________________________________________
0.1 10 10 10
0.2 5 5 5
0.3 3.33 3.33 3.33
0.4 2.5 2.5 2.5
0.5 2 2 2
0.6 1.67 1.67 1.67
0.7 1.43 1.43 1.43
0.8 1.25 1.25 1.25
0.9 1.11 1.11 1.11
1.0 1 1 1
_______________________________________________
518
1010.10.01
99
90
80
70
60
50
40
30
20
10
5
3
2
1
Data
Percent
2.929 10 0.723 0.229
2.929 10 0.723 0.229
2.929 10 0.723 0.229
Mean N AD P
d
ß
µ
Variable
Probability Plot of d, ß, µ
Exponential - 95% CI
Figure 3. VVT clearance for two parameter at different
waiting time period
6 DISCUSSION
A general description is given of the issue of ship
scheduling, where the waiting time of ships in ports is
uncertain, as well as the response time. As noted in
the Introduction section of the manuscript, marine
station operators frequently encounter unpleasant and
unpredictable events that cause port congestion,
further affecting ships' waiting times and port
handling times. Scheduling and managing good
container terminal operations can reduce waiting
times for container ships. Port wait times are not
easily predictable for liner companies due to a few
common and uncontrollable factors of terminal
operations. This negatively affects the marine
ecosystem and the simulation results indicate a
significant reduction in container vessel waiting
times, which may be beneficial for key operations
functions and container terminal design. And we
showed that it can be used to analyze production time
trade-offs for alternative mooring policy, different
types of vehicles, and changing vessel capabilities.
After presenting the results of scenarios with both
ships of current size and large ships, we discuss how
our analytical model can be applied to assign mooring
to live operations. In this case, the AIS data is used to
derive the times that ships stay in port initially, after
which a prediction model is made to predict the
length of time a ship will stay at a particular berth.
We developed upper and lower limits of
theoretical estimates of throughput times from our
model, and provided an extension for the case where
the final processing time depends on the quantity of
loaded/unloaded containers.
Instead of simply summarizing completion times
for all new vessels scheduled during the given time
frame, ships already docked in that time frame
according to the terminal's operations data are
scheduled again by the model in that time frame.
Next, vessel arrival data, vessel dimensions, TEU
loads, and vessel handling times must be entered into
the system.
As ships are restricted by schedule in/after port
arrival times. The right-hand side of Equation (2)
represents the total ship turnover time, estimated as
the sum of the ships' total sailing time, the total
expected waiting time for ships in port, and the total
expected handling time for ships in port.
Average waiting times for ships at terminals were
9.1, 8.6 and 8.1 hours for larger, medium and smaller
vessels. An important fact that caught the attention of
station planners was the analysis of RCP indicators by
classes of ships. Although port authorities favor larger
vessels, there is also a need to focus on smaller vessels
in order to improve the terminal's overall capacity. An
important aspect regarding berthing decisions such as
port expansion is that due to the additional berths to
be built outside the peninsula forming spaces, berths
are expected to have different handling times,
different types of machinery used, and varying
distances from container storage yards. As waiting
times increase at ports of discharge, container
shippers will have a higher risk of delays on time-
sensitive goods and will be charged for delays in
delivery. Transporting the goods in this way may
result in additional charges for storage, demurrage
and detention, in the event that the truck driver is not
able to reach the port terminal on time, due to delays
due to weather or strikes during transport.
Therefore a deterministic optimization model is
proposed to solve the container slot allocation
problems for time-sensitive commodities under the
dynamics of port congestion pricing. The proposed
new pricing mechanism has proven to be effective
when compared to a generic slot allocation model that
does not take into account shipping time limits and
port congestion, with results showing that the
proposed pricing scheme can significantly improve
ship companies' revenues and improve customer
satiation. In terms of reducing carbon emissions from
the ship's stay for a longer period at the docks.
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