Optimizing Multi-Quay Berth Allocation using the Cuckoo Search
Algorithm
Sheraz Aslam
a
, Michalis P. Michaelides
b
and Herodotos Herodotou
c
Department of Electrical Engineering, Computer Engineering and Informatics, Cyprus University of Technology, Cyprus
Keywords:
Multi-Quay Container Terminals, Multi-Quay Berth Allocation Problem, Cuckoo Search Algorithm, Port
Efficiency, Intelligent Maritime Transportation.
Abstract:
Proper utilization of port resources and efficient berth planning play a crucial role in minimizing port conges-
tion and overall handling costs. Therefore, this study focuses on efficient berth planning in maritime container
terminals composed of multiple quays. In particular, this study addresses the Multi-Quay Berth Allocation
Problem (MQ-BAP), where a continuous berthing layout is considered along with dynamic ship arrivals and
practical constraints such as safety time windows and safety distances between ships. Since MQ-BAP is an
NP-hard problem, this study proposes a metaheuristic-based approach, the Cuckoo Search Algorithm (CSA)
for solving the problem. A comparative study is also performed using real data instances collected from the
Port of Limassol, Cyprus, against a genetic algorithm solution proposed in the recent literature, as well as
the optimal exact solution implemented using MILP. The results of the experiments show the effectiveness of
our proposed CSA approach in handling real-world berth allocation in ports with multiple quays while also
considering practical constraints.
1 INTRODUCTION
Maritime transport accounts for 90% of the world’s
seaborne trade and 74% of all goods imported in, or
exported from Europe are carried by ships (Aslam
et al., 2020). According to a recent report on maritime
transport conducted by the United Nations Confer-
ence on Trade and Development (UNCTAD) in 2021
(UNCTAD, 2021), total global containerized trade
has increased by 45.45%, with Twenty-Foot Equiv-
alent Units (TEUs) totaling 110 million in 2010 and
rising to 160 million TEUs in 2021. The container
traffic has also increased in 2021, even though it was
reduced in 2020 compared to 2019 due to the pan-
demic situation.
Maritime Container Terminals (MCTs) play a crit-
ical role in meeting the growing demand for seaborne
trade. To deal with the growing demand for MCTs,
there is a need to optimize their operations, benefit-
ing from current technologies and optimization-based
approaches. Following this practical need, the devel-
opment of novel and efficient methods for optimiz-
a
https://orcid.org/0000-0003-4305-0908
b
https://orcid.org/0000-0002-0549-704X
c
https://orcid.org/0000-0002-8717-1691
ing terminal operations has attracted immense atten-
tion from academia and industry (Lind et al., 2020,
2021; STEAM, 2022; STM, 2022). Michaelides et al.
(2019) have investigated the factors influencing the
various waiting times at the Port of Limassol, Cyprus,
both from a quantitative and a qualitative perspective.
For shipping, and particularly for short sea shipping,
there are obvious and immediate benefits from im-
proving efficiency by supporting all actors involved
in the port call process to engage more easily, to give
shipping companies, port service providers, and ship
agents better information and decision support sys-
tems to boost their efficiency and that of their port
(Lind et al., 2019). Hence, MCTs’ operators need to
employ suitable strategies and approaches for proper
utilization of the port resources and to avoid the afore-
mentioned issues.
As MCTs handle huge volumes of containers, they
are constantly challenged to increase their produc-
tivity by introducing numerous software and hard-
ware innovations, e.g., in terminal design, cargo han-
dling equipment, automated/efficient berthing oper-
ations, and operations research. MCTs can be di-
vided into three main areas, namely seaside, mar-
shaling yard, and landside areas, with each area hav-
ing its own set of terminal operations (Hsu and Chi-
124
Aslam, S., Michaelides, M. and Herodotou, H.
Optimizing Multi-Quay Berth Allocation using the Cuckoo Search Algorithm.
DOI: 10.5220/0011081200003191
In Proceedings of the 8th International Conference on Vehicle Technology and Intelligent Transport Systems (VEHITS 2022), pages 124-133
ISBN: 978-989-758-573-9; ISSN: 2184-495X
Copyright
c
2022 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
ang, 2019). The main problems in the seaside oper-
ations include the Berth Allocation Problem (BAP),
the Quay Crane Assignment Problem (QCAP), and
the Quay Crane Scheduling Problem (QCSP) (Aslam
et al., 2021). In this study, we address the first deci-
sion problem, namely the allocation of berths to ar-
riving ships, known as BAP. On the fleet side, the
berth allocation plan determines both the berthing po-
sition and the berthing time for arriving vessels to re-
duce costs, waiting times, handling times, and depar-
ture delays. On the port side, on the other hand, an
efficient berth allocation plan determines how many
ships can be handled in a planning period, with the
goal of maximizing profit and making appropriate use
of port resources.
The BAP has attracted a lot of attention
from the research community and several ap-
proaches/techniques have been developed to opti-
mize seaside operations. For example, in Dulebenets
(2020), an evolutionary algorithm is proposed to deal
with BAP, where the main objective is to reduce the
weighted total service cost. Another study also deals
with BAP, while additionally considering uncertain-
ties in the operational time of ships (Xiang and Liu,
2021). In particular, an exact approach is developed
and K-means clustering is used to model the uncer-
tainty. The proposed method performs well; how-
ever, it is only suitable for small data instances. A
hybrid heuristic-based Genetic Algorithm (GA) is de-
veloped in Bacalhau et al. (2021) to solve the BAP
to avoid the issue of high computation time in ex-
act approaches. The authors combine Dynamic Pro-
gramming (DP) with the standard GA to solve large-
scale problems that minimize service cost. Guo et al.
(2021) deal with BAP, taking into account the uncer-
tainty in handling times of vessels. They develop
an optimization method to solve the BAP, combin-
ing particle swarm optimization and machine learn-
ing. Here, machine learning is used to check the re-
lationship between handling time and weather condi-
tions to model the uncertainty in handling time. They
conduct several experiments and the results confirm
the effectiveness of the proposed method compared
to its counterparts. A study presented in Cheimanoff
et al. (2021) also addresses continuous and dynamic
BAP, in which a metaheuristic-based reduced vari-
able neighborhood search method is developed to op-
timally allocate berths to arriving vessels. The study
also considers tidal constraints for optimal berth as-
signment, and the main objective of this study is to
reduce the total service time of vessels. In addition, a
machine learning algorithm is used to tune the param-
eters of the metaheuristic. The proposed method has
been tested on several datasets (small and large scale)
Time (hour)
s3
s1
s8
s9
s6
Time (hour)
s4
s2
s5
s10
s7
Quay length (m)
Quay 1 Quay 2
Quay length (m)
Figure 1: An illustration of two continuous berthing quays
with 10 arriving ships.
and the results were compared with the exact method.
The comparative study shows the effectiveness of the
metaheuristic approach over other methods proposed
in the literature.
Most of the aforementioned studies deal only with
the allocation of berths at a single quay, assuming
that it forms one straight line in which vessels can
be berthed according to their length and the positions
of other vessels. However, this assumption is not re-
alistic for several ports around the globe, which con-
sist of multiple separate line segments or quays for
berthing (Frojan et al., 2015). For example, the Port
of Limassol in Cyprus has seven continuous berthing
quays. Considering multiple quays adds a new dimen-
sion to the BAP; the problem of assigning vessels to
quays in addition to assigning berthing positions and
times for each separate quay. This requires a multi-
ple space-time representation, as can be seen in Fig-
ure 1. Very few studies have dealt with the Multi-
Quay BAP (MQ-BAP). One study presented in Fro-
jan et al. (2015) proposes a set of priority rules and
uses GA to address the MQ-BAP. However, in their
problem formulation, the total length of the quay is
evenly distributed among the number of quays, while
the evaluation is only performed over random data.
Another study proposes a set of heuristics based on
general variable neighborhood search (Krimi et al.,
2020). Even though this approach is shown to be very
efficient in solving this problem, in certain cases it
proposes solutions that are far (up to 40%) from the
optimal.
The motivation of this study is to develop a Multi-
Quay BAP model and solution that can be applied
in a real port, such as the Port of Limassol, Cyprus.
As such, we consider additional practical constraints
of the port, including the preferred berthing quay of
each arriving ship, a safety time interval between con-
secutive vessels entering the port, as well as a safety
time and distance between vessels berthed at a par-
ticular quay. First, the MQ-BAP is modeled as a
mixed-integer linear model and then solved by the
Optimizing Multi-Quay Berth Allocation using the Cuckoo Search Algorithm
125
metaheuristic-based cuckoo search algorithm (CSA).
Furthermore, we also implement the exact method us-
ing MILP and a popular heuristic method using GA
for comparison. We conduct experiments using real
data from one week of operations at the Port of Li-
massol and the results confirm the effectiveness of our
proposed method.
The remaining paper is organized as follows.
Problem definition and formulation are presented in
Section 2. Next, Section 3 presents the details of the
proposed CSA method and Section 4 discloses simu-
lation setting, data instances, and simulation results.
Finally, Section 5 concludes this study.
2 PROBLEM DESCRIPTION
In this section, we first introduce the MQ-BAP along
with assumptions, and then we formulate the problem
as a mixed-integer linear problem.
2.1 Problem Definition
In contrast to existing studies, and to make the prob-
lem more practical, this work considers MCT with
multiple quays (having continuous berthing layouts)
to berth arriving vessels, i.e., Q = {1, ...|Q|}. A con-
tinuous quay q Q consists of a section of the berth
line at the MCT and arriving ships can be moored at
any point along the berth line, representing the avail-
able berthing positions at q, i.e., B
q
= {1, ...|B
q
|}.
The set T indicates the set of time intervals consid-
ered for planning, i.e., T = {1, 2, 3, ...|T |}. There is
a set of arriving ships S = {1, 2, 3, ...|S|} and each
ship s S has multiple known characteristics, includ-
ing Length of Ship (LoS), Expected Time of Arrival
(ETA), Expected Time of Departure (ETD), Handling
Time (HT), and Preferred Berthing Quay (PBQ). Ta-
ble 1 presents the mathematical notations that are
used in this section.
The objective of this study is to determine the
berthing quay, berthing position, and berthing time
for all arriving ships in S in order to minimize the
total cost associated with the berthing process. The
cost against ship s includes handling cost C
h
s
, waiting
cost C
w
s
, penalty cost due to late departures C
ld
s
, and
penalty cost due to allocation of ships to non-optimal
berthing quay C
noq
s
. The handling cost includes the
cost of loading and unloading of containers and de-
pends on the handling time T
h
s
of ship s. The wait-
ing cost is calculated based on the waiting time T
w
s
,
which is the difference between the estimated arrival
time T
ea
s
and the berthing time T
b
s
. The late departure
penalty cost C
ld
s
depends on the late departure time
T
ld
s
, which is defined as the difference between the
operations finishing time T
f
s
and the requested depar-
ture time T
rd
s
of each ship. The last penalty cost C
noq
s
is added if the ship s is not moored to its preferred
berthing quay (PBQ) but rather it is moored to an al-
ternative berthing quay (ABQ), since more resources
are needed to move containers over a longer distance.
ABQs are introduced in this study to constraint the
algorithm to select only appropriate quays (i.e., either
the preferred or alternative quays) for berthing a ves-
sel since in practical scenarios, not all quays can han-
dle all vessels (e.g., due to lack of proper equipment).
2.2 Assumptions
The problem under consideration and the solution are
based on the following assumptions:
The number of incoming ships in the planning pe-
riod is known;
When a vessel commences operations at a partic-
ular quay, it cannot be interrupted until the opera-
tions are completed;
Berths from any quay become available immedi-
ately after a ship completes its operations;
The number and length of the quays are known;
The ETA along with ETD for all arriving ships are
known;
The handling time for each vessel is known;
Each vessel has a PBQ and an optional list of
ABQs that are known in advance;
All berths are assumed to be free at the beginning
of the time horizon (time = 0);
All penalty costs for all vessels are known;
2.3 Mathematical Formulation
The total handling cost of a ship s that is planned
for berthing at position B
s
of a particular quay Q
s
at
time T
b
s
includes a waiting cost, a handling cost, and
a penalty for late departure, presented as:
Cost(s, Q
s
, B
s
, T
b
s
) = T
w
s
·C
w
s
+ T
h
s
·C
h
s
· f (Q
s
, B
s
, PBQ
s
, ABQ
s
, C
noq
s
)
+ T
ld
s
·C
ld
s
(1)
The first term in Equation (1), T
w
s
·C
w
s
, shows the wait-
ing cost when a ship s has to wait for mooring. The
waiting time T
w
s
of ship s is calculated as the differ-
ence between the ETA T
ea
s
and berthing time T
b
s
,
VEHITS 2022 - 8th International Conference on Vehicle Technology and Intelligent Transport Systems
126
Table 1: Mathematical Notations.
Name Explanation
ABQ
s
Alternative berthing quays for ship s
B
s
Berthing position of ship s at quay Q
s
C
h
s
Handling cost per time unit (hour) for ship s
C
w
s
Waiting cost per time unit (hour) for ship s
C
noq
s
Penalty cost against non-optimal berthing
quay (fixed cost) for ship s
C
ld
s
Penalty cost against late departure per time
unit (hour) for ship s
L
s
Length of ship s
L
q
Length of quay q
PBQ
s
Preferred berthing quay of ship s
Q
s
Berthing quay of ship s
SD Safety distance (in meters) between two
ships during berthing
SE Safety entrance time between two ships
ST Safety time between two ships during
berthing on the same quay
T
b
s
Berthing time of ship s
T
ea
s
Expected/estimated arrival time of ship s
T
f
s
Finish time of operations (loading and/or
unloading) of ship s
T
h
s
Handling time of ship s
T
ld
s
Late departure time of ship s
T
rd
s
Requested departure time of ship s
T
w
s
Waiting time of ship s
s S = {1, 2, ..., |S|} Individual ship
q Q = {1, 2, ..., |Q|} Berthing quay
b B
q
= {1, 2, ..., |B
q
|} Berthing position in quay q
t T = {1, 2, ..., |T |} Single time period
T
w
s
= T
b
s
T
ea
s
, s S (2)
The second term in Equation (1), T
h
s
· C
h
s
·
f (Q
s
, B
s
, PBQ
s
, ABQ
s
, C
noq
s
) represents the total pro-
cessing cost of ship s that was incurred by unloading
and loading containers from/to ship s, where T
h
s
and
C
h
s
denote handling time for ship s and handling cost
per unit time, respectively.
Without loss of generality, this work
also introduces the penalty function
f (Q
s
, B
s
, PBQ
s
, ABQ
s
, C
noq
s
), which penalizes the
handling cost due to non-optimal quay assignment,
as presented in Equation (3). The penalty cost equals
0 if ship s is moored at its preferred berthing quay
PBQ
s
and C
noq
s
if s is moored at one of the alternative
berthing quays ABQ
s
. Otherwise, the penalty is set
to infinite to ensure the algorithm will never perform
such an assignment.
f (Q
s
, B
s
,PBQ
s
, ABQ
s
, C
noq
s
) =
0 , if Q
s
= PBQ
s
C
noq
s
, else if Q
s
ABQ
s
, otherwise
(3)
The final term T
ld
s
·C
ld
s
in Equation (1) calculates the
late departure penalty cost against ship s when it de-
parts after a requested departure time. The delay in
departure time T
ld
s
of ship s is computed as the differ-
ence between the time of operations completion T
f
s
and the requested time of departure T
rd
s
.
T
ld
s
= max{T
f
s
T
rd
s
, 0}, s S (4)
where, T
f
s
can be calculated as,
T
f
s
= T
b
s
+ T
h
s
, s S (5)
The primary objective of the MQ-BAP is to allo-
cate optimal quays and berthing positions along with
berthing times to arriving ships such that the total pro-
cessing cost (that includes waiting cost, handling cost,
and various penalties) can be minimized, as presented
by the following objective function:
minimize
s S
q Q
b B
q
t T
Cost(s, q, b, t) · x
sqbt
(6)
subject to the following constraints.
x
sqbt
{0, 1}, s S, q Q, b B
q
, t T (7)
The variable x
sqbt
is 1 if ship s is assigned to berthing
position b of quay q at berthing time slot t, and 0 oth-
erwise.
q Q
b B
q
t T
x
sqbt
= 1, s S, (8)
Constraint (8) states that each ship is assigned to a
particular quay q and berthing position b only once
(at time t) during the planning period T .
T
b
s
T
ea
s
, s S. (9)
Constraint (9) defines that the proposed berthing time
T
b
s
for particular ship s must be equal or greater than
its expected arrival time T
ea
s
.
B
s
+ L
s
L
Q
s
, s S, (10)
Constraint (10) ensures that the summation of the pro-
posed berthing position B
s
and the ship length L
s
must
always be equal to or less than the quay length L
Q
s
.
Optimizing Multi-Quay Berth Allocation using the Cuckoo Search Algorithm
127
j̸=s S
B
s
+L
s
+SD
b=B
s
L
j
SD+1
T
b
s
+T
h
s
+ST
t=T
b
s
T
h
j
ST +1
x
jqbt
= 0,
s, j S, q Q
s
(11)
Constraint (11) avoids scheduling overlapping of two
or more ships. For example, suppose ship s is sched-
uled to berth at time 6h, has a processing time of 5h,
uses berth position 600m, and its length is 300m. As
per constraint (11), no other vessel can use position
from 600m to 900m (as length of ship s is 300m) in
the time interval 6h to 11h. Furthermore, another ves-
sel j with a length of 200m and a processing time of
4h cannot be berthed at positions from 401m to 900m
(ignoring the safety distance) in the time interval 3h to
11h (ignoring the safety time) because it would over-
lap with previous vessel s. Furthermore, constraint
(11) is also responsible for keeping the safety distance
SD and safety time ST between two ships in order to
avoid any dangers during berthing. Visually, this con-
straint ensures that any two rectangles (denoting the
time intervals and berthing positions allocated to ves-
sels) shown in Figure 1 can never overlap.
T
b
s
T
b
j
SE s ̸= j S, (12)
Finally, constraint (12) ensures a minimum safety en-
trance time period SE between sequential berthings,
regardless of berthing quay, since at the Port of Li-
massol, like many other ports around the world, there
is a single port entrance that all ships must sequen-
tially enter to berth (see the red triangle in Figure 2).
3 PROPOSED METHOD
In this study, we employ the Cuckoo Search Algo-
rithm (CSA) to address MQ-BAP. CSA is a meta-
heuristic optimization algorithm developed by (Yang
and Deb, 2009). The CSA is inspired by the breed-
ing mechanism of some cuckoo species, which are
fascinating because of their beautiful sounds and ag-
gressive reproduction mechanism. Some cuckoos lay
their eggs in communal nests of other species, where
they try to remove the eggs of other birds in or-
der to improve the hatching probability of their own
eggs. Then, other birds, probably from other species,
known as host birds take care of cuckoo eggs. How-
ever, if the host birds realize that some eggs do not
belong to them, then the cuckoo eggs are disposed
of or current nests are destroyed and built elsewhere.
In particular, some cuckoo species (e.g., new world
brood-parasitic Tapera) specialize in the mimicry of
the pattern or color of eggs and they lay their eggs in
nests of relevant species in order to reduce the prob-
ability of their eggs being thrown or destroyed (Gan-
domi et al., 2013). Overall, the CSA works based on
the behavior of cuckoos for laying eggs and adopts
three idealized rules (Yang and Deb, 2009):
1. each cuckoo bird dumps only one egg at a time in
a random nest;
2. the best nests having high quality eggs are kept
and used for the next generation;
3. the number of host nests is fixed and the egg laid
by a cuckoo is detected by a host bird with proba-
bility p
α
(0, 1).
The mapping of CSA to MQ-BAP is as follows. A
single nest shows a set of solutions containing the
mooring positions and mooring times of all arriving
ships. An egg in a nest denotes a berthing time or
berthing position in a berthing quay for an arriving
ship, whereas, a cuckoo egg shows a novel (or better)
solution (i.e., a berthing time or position in a quay).
The total search space of the problem at each iteration
is reflected by the total number of host nests, which is
fixed (100 host nests are assumed in this study). In
addition, each nest includes 2N eggs, where N shows
the number of ships that have arrived at a given time.
Therefore, the total number of eggs in a nest is twice
the total number of ships arriving. This is because
we need two solutions for each ship (i.e., one is the
berthing time and the other is the berthing position).
The overall goal of the algorithm is to use cuckoo
eggs (better solutions) to replace the not-so-good eggs
in the various nests.
4 SIMULATION RESULTS
In this section, we present the simulation setup along
with the simulation results. We propose and imple-
ment metaheuristic-based CSA for the berth alloca-
tion problem when considering multiple quays (MQ-
BAP). We also implement the GA, a popular meta-
heuristic approach proposed in Salhi et al. (2019), and
the exact solution using MILP for comparison pur-
poses. Simulations are performed on a PC with Core
i7 using MATLAB 2018b. Regarding the dataset,
we employ real-world data collected from the Port of
Limassol, Cyprus, which contains information about
each arriving ship, i.e., ETA, ETD, LoS, processing
time of ship, and preferred berthing quay (as the Port
of Limassol has multiple quays). The data are col-
lected directly from a port calls database and are con-
verted into a more suitable format for processing. For
example, time is discretized into fixed-time intervals
for easier processing by the scheduling algorithms. In
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128
Table 2: Example data for 10 ships (out of the 28 ships) in our dataset that arrived at the Port of Limassol, Cyprus).
Ship # ETA (day\time) HT (min.) ETD (day\time) PBQ LoS (meters)
1 1\04:00 919 1\22:30 Container/ Ro-Ro Quay 194
2 1\05:30 1490 2\06:50 East Quay 139
3 1\14:00 1285 2\12:50 West Quay 84
4 1\15:00 5700 5\14:03 East Quay 89
5 1\17:00 5970 5\21:00 West Quay 190
6 2\04:30 470 2\13:50 Container/ Ro-Ro Quay 159
7 2\05:00 168 2\09:30 Container Quay 196
8 2\08:00 440 2\15:55 North Quay 155
9 3\04:00 905 3\20:50 Container/ Ro-Ro Quay 175
10 3\03:30 1331 4\06:15 Container Quay 277
Figure 2: A satellite view of the Port of Limassol, Cyprus illustrating its seven berthing quays (taken from ais.cut.ac.cy
(2022)).
the future, we plan to implement a graphical user in-
terface for visualizing the output of the algorithm to
the berth planner in a user-friendly manner. Table 2
shows a sample of the collected data for 10 arriving
ships. The full dataset contains a total of 28 ships that
arrived at the port during a period of one week. We
implement the proposed and compared methods on
the same dataset and the results are presented later in
this section. As for cost calculations, we assume the
handling cost is 10 euro/hour, the waiting cost is 20
euro/hour, and the late departure cost is 20 euro/hour.
A satellite view of the Port of Limassol is shown in
Figure 2. There are a total of seven quays in the Port
of Limassol but only five appear in the dataset (i.e.,
only the ones that handle commercial traffic), namely
Container/ Ro-Ro Quay, Container Quay, East Quay,
West Quay, and North Quay, and are all of different
lengths, as shown in Table 3. Note that all arriving
ships have a preferred berthing quay for performing
loading and unloading operations, but currently, the
dataset does not include a preferred berthing position
or alternative berthing quays.
Figure 3 shows the berth allocation plan obtained
by the 3 implemented solutions: (a) CSA, (b) GA,
and (c) MILP. The berthing plan is developed accord-
ing to the objective function of this study (given in
Equation (6)) and considering all constraints. In this
figure, the horizontal axis shows the berthing time di-
vided into 30-minute intervals, and the vertical axis
shows the berthing positions of the arriving vessels.
Optimizing Multi-Quay Berth Allocation using the Cuckoo Search Algorithm
129
(a) Solution by CSA
(b) Solution by GA
(c) Solution by MILP
Figure 3: Berth allocation solutions over one week for five quays at the Port of Limassol, Cyprus.
VEHITS 2022 - 8th International Conference on Vehicle Technology and Intelligent Transport Systems
130
Table 3: Quays at the Port of Limassol, Cyprus.
Quay # Quay name Length (m)
1 Container/ Ro-Ro Quay 450
2 Container Quay 800
3 East Quay 480
4 West Quay 770
5 North Quay 430
5 10 15 20 25
Ships
0
1
2
3
4
5
6
7
8
9
Time (30-min interval)
Waiting time by CSA
Waiting time by GA
Waiting time by MILP
Figure 4: Waiting time for all arriving ships when using
all three algorithms, i.e., CSA, GA, and MILP. The only
induced waiting times are reported when using GA.
Each rectangle demonstrates the berthing positions
and berthing time intervals assigned to each ship.
Furthermore, the label next to the rectangle corre-
sponds to the vessel index. For example, ship 8 ar-
rives at 8:00 on the second day and the PBQ is ‘North
Quay’. It can be observed that the CSA (see Figure
3a) efficiently assigns a berth (without any waiting
time) to ship 8 at its PBQ. However, if we look at
the berth allocation for ship 8 for GA (see Figure 3b),
we can observe that ship 8 has to wait for an opti-
mal berthing and the waiting time is 5 time slots (2.5
hours). However, despite the delay of 5 time slots
in berthing, ship 8 can reach the desired departure
time. Figure 3c shows the optimal berth allocation
plan computed using the exact method, i.e., MILP.
Figure 4 clearly shows that MILP and CSA always
provide an optimal solution with regards to minimiz-
ing waiting times because there is no waiting time
before berthing. However, using GA, 14 out of the
28 ships have to wait, with a maximum waiting time
of 9 slots (4.5 hours) before they can berth. As for
the departure time, we can see from Figure 5 that no
ship departs late when using any of the three algo-
rithms, i.e., CSA, GA, and MILP. Even with GA, sev-
eral ships are late in docking, but they can still achieve
the desired departure time. In this case, penalty cost
for waiting time is calculated, and therefore the to-
5 10 15 20 25
Ships
0
50
100
150
200
250
300
350
Time (30-min interval)
Solution by CSA
Solution by GA
Solution by MILP
Requested departure time
Figure 5: Requested and proposed departure times when
using all three algorithms, i.e., CSA, GA, and MILP.
tal handling cost (shown in Equation 1) using GA is
higher than our proposed CSA and MILP methods.
Note that the total handling cost is the sum of the total
waiting cost, the handling cost (including the penalty
for non-optimal quay assignment), and the penalty for
late departure.
Table 4 shows the total handling cost for the 28
vessels and the computational time for all three algo-
rithms. From this table, it can be seen that the total
costs of the CSA and MILP methods are the same,
which shows that CSA achieves an optimal solution
for all arriving ships for this particular dataset. How-
ever, for GA, the total cost is 11.2% higher because
some of the ships have to wait for a long time before
berthing. As for the computation time, MILP takes
910.76 seconds and, as expected, its computation time
is orders of magnitude higher than the other two algo-
rithms. However, our proposed CSA method and GA
solve the same problem in near real time, with only
4.49 and 1.79 seconds, respectively. Even though GA
achieves the minimum computation time (1.79 sec-
onds) to handle a week’s worth of data, it fails to pro-
vide an optimal solution.
It is worth noting that for larger problems, the
exact method (MILP) cannot be used because it re-
portedly requires over 100 hours CPU time for large
datasets (Aslam et al., 2021). Such times are cer-
tainly not acceptable in the context of MCT oper-
ations. From the above discussion, it appears that
CSA can provide an optimal or near-optimal solution
within an acceptable computation time.
Table 4: Comparison of proposed and benchmark methods.
Total Cost (Euro) Comp. Time (Sec)
Method: CSA GA MILP CSA GA MILP
1-week 5385 6025 5385 4.49 1.79 910.76
Optimizing Multi-Quay Berth Allocation using the Cuckoo Search Algorithm
131
5 CONCLUSION
Over the past decade, international maritime trade has
increased dramatically. In order to serve the growing
number of vessels arriving at the terminal for loading
and unloading, MCTs need to increase their efficiency
by using various technologies and methods. This
study focuses on the Multi-Quay BAP and proposes
a metaheuristic-based CSA method to solve the prob-
lem. A continuous berthing layout is considered and
the ships arrive dynamically. The problem is first for-
mulated as a mixed-integer linear problem and then
solved by CSA. In addition, two benchmark meth-
ods (i.e., GA and MILP) are also implemented in this
study for comparison. To confirm the performance of
our proposed method, simulations are conducted us-
ing real data from the Port of Limassol, Cyprus. The
data contains 28 ships arriving in a period of one week
and intending to dock at five different quays. The re-
sults of the experiments confirm the benefits of the
proposed method with 11.2% lower cost than GA and
200x lower computation time than MILP.
In the future, we plan to examine the perfor-
mance of the proposed CSA method on larger real-
world datasets, containing several vessels and span-
ning longer planning time periods (days to weeks).
We also plan to extend the modeling to incorporate
a hybrid berthing layout that includes both discrete
and continuous berthing layouts. Finally, we plan to
investigate the application of the CSA in solving the
berth allocation problem combined with the related
quay crane assignment and scheduling problems.
ACKNOWLEDGEMENTS
This work was supported by the European Re-
gional Development Fund and the Republic of Cyprus
through the Cyprus Research and Innovation Founda-
tion (STEAM Project: INTEGRATED/0916/0063).
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