Dynamic and Continuous Berth Allocation using Cuckoo Search

Optimization

Sheraz Aslam

, Michalis P. Michaelides

and Herodotos Herodotou

Department of Electrical Engineering, Computer Engineering and Informatics, Cyprus University of Technology, Cyprus

Keywords:

Berth Allocation Problem, Intelligent Sea Transportation, Cuckoo Search Algorithm, Metaheuristic

Optimization, Port Efﬁciency.

Abstract:

Over the last couple of decades, demand for seaborne containerized trade has increased signiﬁcantly and

it is expected to continue growing over the coming years. As an important node in the maritime industry,

a maritime container terminal (MCT) should be able to tackle the growing demand for sea trade. Due to

the increased number of ships that can arrive simultaneously at an MCT combined with inefﬁcient berth

allocation procedures, there are often undesirable situations when the ships have to stay in waiting queues

and delay both their berthing and departure. In order to improve port efﬁciency in terms of reducing the

total handling cost and late departures, this study investigates the dynamic and continuous berth allocation

problem (DC-BAP), where vessels are assigned dynamically as they arrive at their berth locations assuming

a continuous berth layout. First, the DC-BAP is formulated as a mixed-integer linear programming (MILP)

model. Since BAP is an NP-hard problem and cannot be solved by mathematical approaches in a reasonable

time, this study adopts the recently developed metaheuristic cuckoo search algorithm (CSA) to solve the DC-

BAP. For validating the performance of the proposed CSA method, we use a benchmark case study and a

genetic algorithm solution proposed in recent literature as well as compare our results against the optimal

MILP solution. From the simulation results, it becomes evident that the newly proposed algorithm has higher

efﬁciency over counterparts in terms of optimal berth allocation within reasonable computation time.

1 INTRODUCTION

The shipping industry covers 90% of the world

seaborne trade movements and 74% of the total goods

that are imported or exported in Europe travel with

ships (Aslam et al., 2020). According to Hsu et al.

(2017), 60% of the total sea transport is based on con-

tainers, which is also growing every year by 6.4%. So

the maritime container terminal (MCT) serves as an

important node in the shipping industry to deal with

increasing sea trade. A report presented in Barbosa

et al. (2019) stated that worldwide ports have han-

dled almost 701 million twenty-foot equivalent units

(TEUs) of containers in 2016. At the same time,

the throughput of container ports is also continuously

increasing, and the management of MCTs’ opera-

tions is becoming a challenging task. As a critical

and integral part of the global transportation network,

the MCTs serve the cost-efﬁcient delivery of various

https://orcid.org/0000-0003-4305-0908

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https://orcid.org/0000-0002-8717-1691

products in different markets. Linear shipping com-

panies use mega-ships in order to carry large contain-

ers up to 20,000 TEUs (De et al., 2020). Since the

MCTs have a huge importance in the maritime indus-

try, there is an exigent need to enhance the operational

efﬁciency of MCTs by mitigating the total turnaround

service times of vessels and achieving competitive

strategy along with customer satisfaction. Further-

more, port authorities always try to optimize MCT

operations by employing various strategies for the ef-

ﬁcient utilization of all the port resources.

MCT operations can be categorized into three

major operational areas, namely seaside, land-side,

and yard-side operations, as presented in Figure 1.

Among all MCT operations, the seaside operations

are the most important as they affect the overall per-

formance of MCTs. Inefﬁcient planning and im-

proper utilization of port resources may create sev-

eral issues, including congestion, long waiting times,

and late departures. For instance, 13, 647 vessels ar-

rived from Jan-Sep 2019 at Port of Shanghai, China,

from which almost 57% of vessels arrived late (more

Aslam, S., Michaelides, M. and Herodotou, H.

Dynamic and Continuous Berth Allocation using Cuckoo Search Optimization.

DOI: 10.5220/0010436600720081

In Proceedings of the 7th International Conference on Vehicle Technology and Intelligent Transport Systems (VEHITS 2021), pages 72-81

ISBN: 978-989-758-513-5

than 12 hours) (CargoSmart.ai, 2019). According to

a recent report presented in UNCTAD (2017), aver-

age waiting times for vessels from port-to-berth is 2.2,

2.4, and 2.7 hours in Malaysia, Dubai, and China,

respectively. In Michaelides et al. (2019), the effect

of time in port is considered by investigating the fac-

tors 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 imme-

diate beneﬁts from improving efﬁciency by assisting

all those involved in the port call process to engage

more easily to give shipping companies, port service

providers, and ship agents better information and de-

cision support systems to boost their efﬁciency and

that of their port (Lind et al., 2019). Hence, MCTs’

operators need to employ suitable strategies and ap-

proaches for proper utilization of the port resources

and to avoid the above-mentioned issues.

These challenges have motivated us to focus on

enhancing seaside operations. Thus, this study deals

with the berth allocation problem (BAP), which is a

well-known problem that aims to assign berthing po-

sitions to arriving vessels at the port in order to min-

imize or maximize the given objective function (e.g.,

minimize total waiting time, reduce late departures,

or maximize terminal performance). Before dealing

with the BAP, it is necessary to understand the prob-

lem environment. Based on the current literature,

there are two major factors affecting the BAP, i.e.,

the conﬁguration of quay/wharf and the arrival time

of ships. Quays can be conﬁgured in three differ-

ent ways: 1) continuous berthing layout, where arriv-

ing vessels can be moored at any location along the

wharf; 2) discrete berthing layout, where the wharf is

divided into a ﬁxed number of berths; and 3) hybrid

berthing layout, where we can have a mix of continu-

ous and discrete berthing layouts (Carlo et al., 2015).

In terms of vessel arrivals, there are two main types:

1) static arrivals where all the vessels are assumed to

be at the MCT and 2) dynamic arrivals meaning that

vessels are not at the MCT; however, the expected

time of arrival (ETA) is known for each vessel. This

study adopts the continuous berthing layout together

with dynamic vessel arrivals (i.e., DC-BAP).

Literature Review: Several approaches are reported

in the literature that deal with the BAP (Bierwirth

and Meisel, 2010, 2015). These approaches may pro-

vide exact solutions (Jos et al., 2019) or approxi-

mate (based on heuristic or metaheuristic) solutions

(Kavoosi et al., 2019; Dulebenets, 2017). However,

approximate approaches are more popular over exact

methods due to their efﬁciency in terms of compu-

tational complexity. The authors of Kavoosi et al.

Continuous berthing layout

Ship 1

Ship 2

Ship 3

Ship

Berth 1

Berth 2Berth m

Discrete berthing layout

Marshaling yard

Quay cranes

Gantry cranes

Waiting area

Sea side

Figure 1: An illustration of MCT with multiple berthing

positions assuming discrete and continuous berthing layout.

(2019) present a solution of the BAP by employing

evolutionary algorithms (EAs), particle swarm opti-

mization (PSO) and differential evolution (DE). An

EA-based solution is developed in Dulebenets (2017)

to deal with BAP, while the study presented in Xu

et al. (2018) proposed a simulated annealing (SA) al-

gorithm for the same problem. Jos et al. (2019) de-

veloped a mixed-integer linear programming (MILP)

model to deal with the BAP and a genetic algorithm

(GA) is developed in Hsu et al. (2019) to solve the

BAP. Another study (Chen and Huang, 2017) also de-

velops a GA-based approach to deal with the DC-

BAP to minimize penalty costs for late departures.

Mauri et al. (2016) also deal with the DC-BAP. The

objectives of this study are to minimize the total ser-

vice cost and the total ship stay time at port. To solve

the problem, a variant of neighborhood search method

called adaptive large neighborhood search (ALNS) al-

gorithm is developed. The algorithm works based on

the principle of destroy and recreate, where at each

iteration some solutions are destroyed and new ones

are generated in different ways to ﬁnd the best solu-

tion according to ﬁtness criteria.

The work presented in Frojan et al. (2015) deals

with an updated form of BAP, where multiple quays

and continuous berth layouts are considered. The

problem is ﬁrst formulated as an integer linear model

and then solved using GA. To conﬁrm the effec-

tiveness of the proposed method, several simulations

are performed with different sized datasets and the

results show the effectiveness against the compared

approaches. Han et al. (2015) solve the integrated

type of berth allocation where the BAP and the quay

crane (QC) allocation problem are considered simul-

taneously. They proposed a multi-objective particle

swarm optimization (PSO) algorithm, and a continu-

ous berth layout is considered for berth planning. The

primary objectives of this study are twofold: mini-

mizing the range of maximum and minimum QCs to

Dynamic and Continuous Berth Allocation using Cuckoo Search Optimization

save resources, and reducing the movements of QCs

to improve terminal’s efﬁciency. Another study pre-

sented in Cahyono et al. (2019) also addresses the in-

tegrated BAP and QC assignment problem with the

goals of minimizing the total handling and waiting

costs. A model predictive algorithm is developed to

achieve the desired objectives of this study. To vali-

date the proposed approach, several experiments are

conducted on real-time data collected from a con-

tainer terminal in Jakarta, Indonesia. In this study, a

novel cuckoo search algorithm (CSA) is proposed for

the ﬁrst time (according to the best of our knowledge)

to solve the dynamic and continuous berth allocation

problem (DC-BAP).

Contributions: This study investigates the DC-BAP

and aims to mitigate the total service cost of arriving

vessels, which includes the total handling cost as well

as the waiting times and late departures penalties. We

ﬁrst formulate DC-BAP as a mixed-integer linear pro-

gramming (MILP) model and solve it using the newly

developed cuckoo search algorithm (CSA). The sim-

ulation results are compared against two benchmarks

proposed in the recent literature: GA and an MILP

exact solution. From the provided simulation results,

it becomes evident that our proposed CSA shows ef-

ﬁcacy over counterparts.

Outline: The remainder of the paper is organized

as follows. Section 2 explains the investigated prob-

lem and provides its mathematical formulation. Our

newly proposed CSA method is described in detail in

Section 3 and simulation settings along with results

are presented in Section 4. Finally, Section 5 dis-

cusses future work and concludes the paper.

2 PROBLEM DESCRIPTION

This section ﬁrst describes in detail the DC-BAP con-

sidered in this work, followed by a mathematical

formulation as a mixed-integer linear programming

problem. Table 1 lists all abbreviations and notations

used in this section and throughout the paper.

In the dynamic and continuous berth allocation

problem, the MCT has one or more continuous

berthing layouts of known lengths that serve vessels

arriving at different points in time (i.e., in a dynamic

fashion). Let B = {1, 2, . . ., M } denote the set

of all possible berthing positions on the wharf of the

port. Typically, the BAP considers a particular time

period of vessel arrivals, such as the next 48 hours.

Hence, time is modeled as a set of time intervals

T = {1, 2, . . ., K } that can represent some time

duration of interest (e.g., an hour or a 15-minute in-

terval). Finally, let S = {1, 2, . . ., N } denote the set

Table 1: Nomenclature.

Name Explanation

Acronyms

BAP Berth allocation problem

BP Berthing position

CSA Cuckoo search algorithm

DC-BAP Dynamic and continuous BAP

ETA Estimated time of arrival

ETD Estimated time of departure

GA Genetic algorithm

HT Handling time

LoS Length of ship

MCT Maritime container terminal

PBP Preferred berthing position

QCs Quay cranes

WC Waiting cost

WT Waiting time

Notations

Berthing position of ship s

Berthing time of s

ETA

Estimated time of arrival of s

ET D

Estimated time of departure of s

Handling cost of s per time period

Handling time of s

Length of ship s

LDC

Late departure cost of s per time period

LDT

Late departure time of s

PBP

Preferred berthing position of s

W Length of wharf

Waiting cost of s per time period

W T

Waiting time of s

Indices

B = {1, 2, ...,M} Set of available berth positions

S = {1, 2,..., N} Set of ships

T = {1, 2, ..., K} Set of time periods

b Berthing position index

s Individual ship index

t Single time period index

of ships arriving at the terminal. For each ship, the es-

timated time of arrival (ETA), the preferred berthing

position (PBP), the ship’s length, and the estimated

(or requested) time of departure (ETD) are known in

advance.

In the ideal scenario, as soon as a vessel arrives at

the MCT, it should be moored at its preferred berthing

position. If the MCT cannot serve the vessel at the

time of arrival, the vessel must be towed to the wait-

ing area of the terminal, as shown in Figure 1. As a

result of the increased number of ships in the wait-

ing area, congestion and navigational challenges are

created at the seaside of the terminal. In this case,

the MCT incurs an extra waiting cost WC

against the

VEHITS 2021 - 7th International Conference on Vehicle Technology and Intelligent Transport Systems

ship s for the duration of s’ waiting time (e.g., calcu-

lated in EURO/hour).

Once the ships are moored at their assigned

berthing position, the quay cranes (QCs) start work-

ing in order to load/unload containers. Container han-

dling resources (e.g., number of QCs, gantry cranes)

are allocated to ships based on the handling rate that

is negotiated between the MCT operator and the ship-

ping company. The handling time for ship s at the

assigned berthing position is calculated based on the

total number of containers loaded on that ship and the

requested handling productivity. Note that this study

adopts a dataset for implementation with precom-

puted handling times for all arriving vessels. How-

ever, the handling productivity is reduced if the vessel

is assigned to a berth position other than its preferred

berthing position (PBP) (Bierwirth and Meisel, 2010,

2015). The PBP typically depends on vessel charac-

teristics such as the vessel length or vessel load as

well as port-related considerations such as the num-

ber of available quay cranes of the berthing area allo-

cated to a particular ship. Hence, the major cause of

handling productivity reduction is the increased load-

ing/unloading and transfer time of containers from the

assigned (suboptimal) berth to storage.

Finally, each ship s speciﬁes its own estimated (or

requested) time of departure ET D

and the MCT is

supposed to complete the tasks (loading/unloading) of

s before the ET D

, ∀ s ∈ S . Otherwise, the MCT is

liable to pay a late departure penalty cost LDC

for the

duration of the delay (e.g., calculated in EURO/hour)

to the shipping companies. Overall, the aim of the

MCT is to minimize the total waiting, handling, and

late departure costs for all arriving vessels at the port.

2.1 Mathematical Formulation

Before disclosing the mathematical formulation of

DC-BAP, we list the assumptions that are considered

in our work.

• The total number of arriving ships at the planning

horizon is known.

• Each berth position is able to handle only one ves-

sel at a particular time.

• A ship takes consecutive time intervals until load-

ing/unloading completion (i.e., no shifting).

• The ETA and ETD for each vessel are known and

will not change.

• Estimated processing time for each vessel is

known or can be easily computed.

• Each ship has a preferred berthing position and it

is known.

• All berths are idle at the start of the time horizon.

• The length of the wharf is known.

The total processing cost of a vessel s that is sched-

uled for berthing at position BP

at time BT

includes

a waiting cost, a handling cost, and late departure

penalty, expressed by the following function:

Cost(s, BP

, BT

) = W T

·WC

+ HT

· HC

· f (

− PBP

)

+ LDT

· LDC

(1)

The ﬁrst term in Equation (1), W T

·WC

, represents

the waiting cost when a vessel has to wait for berthing.

The waiting time W T

of vessel s is calculated as

the difference between the berthing time BT

and the

planned time of arrival ETA

W T

= BT

− ETA

, ∀ s ∈ S (2)

The second term in Equation (1) corresponds to the

total handling cost for loading/unloading containers.

The handling time HT

depends on the total volume

of containers to be loaded on the vessel, the number

of quay cranes available at this berth, and the aver-

age handling productivity of the cranes. Even though

we consider the handling time as input in this work,

we can easily extend our formulation to compute the

handling time. Without loss of generality, we also in-

troduce the term f (

− PBP

), which will penalize

the handling cost based on the absolute difference be-

tween the assigned berthing position BP

and the pre-

ferred berthing position PBP

. In this work, we assign

a ﬁx penalty when the BP

is different than the PBP

and zero otherwise.

The ﬁnal term in Equation (1), LDT

· LDC

, com-

putes the late departure penalty when a vessel departs

after its estimated time of departure. The delayed de-

parture time LDT

of vessel s (if any) is calculated as

the difference between the time s completes its oper-

ations and the estimated time of departure ET D

LDT

= max{BT

+ HT

− ET D

, 0}, ∀ s ∈ S (3)

The goal of the dynamic and continuous berth alloca-

tion problem is to ﬁnd the optimal berthing positions

and times for all vessels such that the total processing

cost is minimized, as shown by the following objec-

tive function:

minimize

∑

s ∈ S

∑

b ∈ B

∑

t ∈ T

sbt

· Cost (s, BP

, BT

) (4)

subject to the following set of constraints:

sbt

∈ {0, 1}, ∀ s ∈ S, b ∈ B, t ∈ T (5)

Dynamic and Continuous Berth Allocation using Cuckoo Search Optimization

The variable x

sbt

is 1 if vessel s is assigned to berthing

position b at berthing time t, and 0 otherwise.

∑

b ∈ B

∑

t ∈ T

sbt

= 1, ∀ s ∈ S, (6)

Constraint (6) ensures that each arrived ship at the

MCT will be assigned at a particular berthing posi-

tion only once during the planning time.

∑

6=s ∈ S

∑

b=BP

−L

+HT

∑

t=BT

−HT

= 0, ∀ s ∈ S,

(7)

Constraint (7) ensures that no two ships can share

(part of) the same berth during the handling times

of the two ships. For instance, suppose a ship s is

planned to be berthed at time 5h, has handling time

equal to 4h, utilizes berthing position 500m, and its

length is 300m. According to constraint (7), no other

ship can use berthing positions from 500m to 800m

(as length of ship s is 300m) in the time interval 5h

to 9h. In addition, a second ship s

with length 200m

and handling time 3h cannot use the berthing posi-

tions from 300m to 500m in the time interval 2h to

5h as it would overlap with ship s. Visually, this con-

straint ensures that any two rectangles (denoting the

time intervals and berthing positions allocated to ves-

sels) shown in Figure 3 can never overlap.

ETA

≤ BT s, ∀ s ∈ S. (8)

Constraint (8) warrants that the scheduled berthing

time BT

of ship s must always be later than or equal

to its planned time of arrival ETA

+ L

≤ W, ∀ s ∈ S, (9)

Constraint (9) guarantees that the berthing position

of ship s plus its length L

will always be less

than or equal to the total length W of the wharf.

3 PROPOSED METHODOLOGY

A metaheuristic-based cuckoo search algorithm

(CSA) was developed to solve the DC-BAP. In the

last decade, CSA has been applied in several do-

mains of computational intelligence and optimization,

where it demonstrated promising efﬁcacy. For in-

stance, CSA is implemented for task scheduling in

cloud computing (Agarwal and Srivastava, 2018), es-

timating solar panel model parameters (Chen and Yu,

2019; Kang et al., 2018), electricity load forecasting

(Dong et al., 2018), optimal placement of actuators

problem (Yang et al., 2018), various scheduling prob-

lems (Tein and Ramli, 2010; Chandrasekaran and Si-

mon, 2012; Aslam et al., 2017a,b), and constrained

optimization problems (Valian et al., 2013).

CSA is a swarm-based metaheuristic optimiza-

tion algorithm that was developed by Yang and Deb

(2009). The CSA emulates the breeding behavior of

some cuckoo species, which have a fascinating repro-

duction mechanism. In particular, some cuckoos lay

their eggs in nests of other birds (often nests of other

species’ nests), where they may discard eggs of other

birds in order to enhance the hatching ratio of their

own eggs. Then, the host birds take care of cuckoo

eggs as they presume that the eggs belong to them.

Nonetheless, sometimes the host birds distinguish be-

tween their own eggs and the alien eggs. Accordingly,

either the discovered alien eggs are thrown out of the

current nest or new nests are built in new locations.

Inspired by this particular mechanism of laying eggs

by the cuckoo birds, the following three standard rules

are adopted to employ CSA for optimization prob-

lems (Yang and Deb, 2009):

1. each cuckoo lays one egg at a time at a randomly

chosen nest;

2. the best nests with high-quality eggs will not be

removed and will be carried over to the next gen-

eration;

3. the quantity of host nests is ﬁxed and the egg

dumped by a cuckoo is discovered by a host bird

with a probability p

∈ (0, 1).

In this study, each nest denotes a solution set that

includes the berthing times and berthing positions

for all arriving vessels. An egg represents either a

berthing position or time, while a cuckoo egg rep-

resents a new (and better) berthing position or time.

The total number of host nests reﬂects the total search

space at each iteration of the algorithm. In this work,

100 host nests are considered and each nest contains

2N eggs, where N is the total number of vessels.

Hence, the total number of eggs in a nest is double the

total number of arriving vessels, as depicted in Figure

2. Overall, the high-level goal of the algorithm is to

use cuckoo eggs (better solutions) to replace not-so-

good eggs in the nests.

Algorithm 1 shows the procedure of the Cuckoo

Search Algorithm, which starts with a randomly dis-

tributed initial population of k=100 host nests over the

search space (line #1). In each iteration of the algo-

rithm, the reproduction step will be performed ﬁrst,

where new solutions are generated by replacing some

existing eggs with cuckoo eggs in randomly selected

nests (lines #3-8). The replacement follows the ra-

tionale that if a cuckoo egg is very similar to a host

VEHITS 2021 - 7th International Conference on Vehicle Technology and Intelligent Transport Systems

4 5 10 4 13 16 11 6 9 1 776 1425 963 1442 353 1015 443 1018 1040 92

4 6 10 6 13 17 11 6 10 2 776 1422 960 1431 370 1021 437 1010 1051 104

4 10 10 6 14 16 11 7 9 2 770 1416 948 1435 357 1022 436 999 1037 96

Nest i

Nest i+1

Nest k

. . .

Prospective generated berthing times for 10 vessels

Prospective generated berthing positions for 10 vessels

Figure 2: Example solution sets (nests) generated by cuckoo search.

Algorithm 1: Cuckoo Search Algorithm for DC-BAP.

1: X[1..k] = Generate initial population of host nests

2: for t = 1 to max number of iterations do

3: for i = 1 to k do

4: x

new

= X [i] + α ⊕ Levy(λ)

5: if (ﬁtness(x

new

) < ﬁtness(X [i])) then

6: X[i] = x

new

7: end if

8: end for

9: for i = 1 to k do

10: if (rand(0, 1) < p

) then

11: X[i] = Generate new host nest

12: end if

13: end for

14: x

best

= Find nest with lowest ﬁtness value in X

15: end for

egg, then this egg has lesser chances to be discovered.

Therefore, a random walk is performed through L

evy

ﬂights in order to generate new nests (i.e., new solu-

tions).

(t+1)

= X

(t)

+ α ⊕ Levy(λ), (10)

where t denotes the current iteration number, X

the

solution for nest i, and α (α > 0) the step size. The ⊕

operation denotes entry-wise multiplication. A ran-

dom walk in L

evy ﬂights is performed from a L

evy

distribution with a scale parameter λ (Sanajaoba and

Fernandez, 2016). The primary objective of perform-

ing random steps is to increase the possibility of ﬁnd-

ing the global solution instead of becoming stuck in

a local optimum. A new solution replaces a current

solution if its ﬁtness score is lower than the ﬁtness

score of the current solution. Note that the ﬁtness of

a possible solution is evaluated based on the objec-

tive function of the DC-BAP presented in Equation

(4) and accounts for the total processing cost (i.e., a

combination of handling, waiting, and late departure

penalty costs). Furthermore, some cuckoo eggs may

be discovered by host birds. These bad nests will be

abandoned and new ones are built so as to enhance

the exploration of the search space (lines #9-13). The

discovering probability p

in our work is set to 0.45

as reported in Yang and Deb (2009). Finally, we keep

track of the best solution (line #14). The above steps

repeat until either the total number of iterations is

reached (which equals 100 in this work) or there has

been no ﬁtness improvement for some iterations.

4 EXPERIMENTAL RESULTS

This section presents the experimental setup, exper-

imental data, and results of the experiments. In ad-

dition to the CSA algorithm, we have implemented

a state-of-the-art approach (i.e., GA) proposed in re-

cent literature (Salhi et al., 2019) as well as the ex-

act MILP approach. For our experiments, the prob-

lem dataset was taken from S¸ahin and Kuvvetli (2016)

and contains ten arriving vessels in a day along with

the estimated time of arrival, handling time, estimated

time of departure, preferred berthing position, and

length for each arriving vessel. The complete dataset

is shown in Table 2. The wharf is continuous with a

length of 2000m. The GA parameters of maximum

generations, population size, crossover rate, and mu-

tation rate are set to 500, 200, 0.5, and 0.4, respec-

tively (Salhi et al., 2019). The implemented algo-

rithms are coded in MATLAB 2019b on a Windows

10 PC with COREi7 processor and 8GB RAM.

Figure 3 presents the berth allocation solutions

that are generated by the three implemented algo-

rithms, namely, CSA, GA, and MILP. All three ap-

proaches allocate ships over available berthing posi-

tions and time slots based on the primary objective of

this study, which is to minimize total processing cost

as presented in Equation (4). In Figure 3, the vertical

Dynamic and Continuous Berth Allocation using Cuckoo Search Optimization

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (hour)

200

400

600

800

1000

1200

1400

1600

1800

2000

Wharf length = 2000m

(a)

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (hour)

200

400

600

800

1000

1200

1400

1600

1800

2000

Wharf length = 2000m

(b)

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (hour)

200

400

600

800

1000

1200

1400

1600

1800

2000

Wharf length = 2000m

(c)

Figure 3: Berth allocation solution generated by (a) CSA, (b) GA, and (c) MILP. Each rectangle denotes the time intervals

and berthing positions allocated to a particular vessel.

Table 2: Dataset used for simulations (S¸ahin and Kuvvetli,

2016).

Ship # ETA HT ETD PBP LoS

1 4 3 8 778 128

2 5 5 11 1416 113

3 10 5 15 957 334

4 4 2 8 1437 423

5 13 3 16 362 173

6 15 1 18 1015 391

7 11 3 15 434 338

8 6 2 9 1008 140

9 9 1 11 1043 302

10 2 1 3 102 194

axis shows the berthing positions, while the horizon-

tal axis presents time divided into hourly time inter-

vals. Each rectangle in this ﬁgure denotes the berthing

time periods and berthing positions allocated to an in-

coming vessel. The label within a rectangle shows the

vessel index. For instance, vessel 2, which arrives at

time period 5, is assigned to berthing time period 5

by MILP, time period 6 by CSA, and time period 7

by GA. Hence, with MILP, vessel 2 incurs no waiting

time, while with CSA, it will have to wait for 1 hour

as its preferred berthing position is allocated to vessel

4 (which is optimally assigned). Despite the 1 hour

berthing delay, vessel 2 will still achieve its requested

time of departure. On the other hand, with GA, ves-

sel 2 will have to wait for 2 hours to berth, which in

turn will cause 1 hour delay in its departure. All three

methods assign berthing position 1416, which corre-

sponds to the preferred berthing position of vessel 2.

The waiting time for each vessel for the three

methods is presented in Figure 4. It can be observed

from Figure 4 that no vessel needs to wait for its op-

timal berthing position (PBP) by employing MILP.

However, our newly proposed CSA provides a solu-

1 2 3 4 5 6 7 8 9 10

Ships

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

Time (hour)

Waiting time by CSA

Waiting time by GA

Waiting time by MILP

Figure 4: Waiting times when using CSA, GA, and MILP

for each vessel.

tion with a maximum waiting time of one time period

(hour) for any vessel, while with GA the waiting time

for any vessel is up to 2 hours. Furthermore, using

CSA, only three vessels need to wait for their optimal

berth position assignment; on the contrary, using GA,

ﬁve vessels need to wait for optimal assignment of

berthing position. From this ﬁgure, we can conclude

that the CSA outperforms GA in terms of reducing

waiting time.

Results presented in Figure 5 show the requested

departure times for all arriving vessels and the

planned departures given by the three implemented al-

gorithms, i.e., CSA, GA, and MILP. It can be clearly

seen from this ﬁgure that no ship departs late when

MILP is used, while only one vessel (vessel 5) is late

for one hour with CSA. On the contrary, when GA

is employed, two vessels depart late: vessel 2 is late

for one hour and vessel 5 for two hours. Once again,

we conclude that CSA exhibits higher performance in

terms of minimizing late departures compared to GA.

For comparison purposes, Table 3 lists computa-

VEHITS 2021 - 7th International Conference on Vehicle Technology and Intelligent Transport Systems

1 2 3 4 5 6 7 8 9 10

Ships

Time (hour)

Solution by CSA

Solution by GA

Solution by MILP

Requested departure time

Figure 5: Requested departure time and planned departure

times when using CSA, GA, and MILP for each vessel.

tional times along with total processing costs in Euro

for all tested methods. Furthermore, we also var-

ied the number of arriving vessels in order to investi-

gate the scalability of the proposed method. We have

tested the three methods on three datasets, where 10,

20, and 30 ships are considered, while all other pa-

rameters are the same, i.e., length of wharf, the vessel

arrival pattern, and berthing layout. It can be seen

from Table 3 that MILP gives the optimal solution in

terms of minimum processing cost in all cases. For

example, results from the 10 vessels show that MILP,

as expected, achieves the lowest cost with 275 Euro,

followed closely by CSA with 285 Euro, and GA with

a signiﬁcantly higher cost of 310 Euro. A similar

pattern is observed for the 20- and 30-vessel scenar-

ios; the solutions proposed by CSA are only slightly

costlier (3-5%) than the optimal MILP solution, while

the GA solution is much costlier (8-16%).

Table 3: Comparative analysis.

Method Total Cost (Euro) Comp. Time (sec)

# ships: 10 20 30 10 20 30

CSA 285 580 750 0.09 0.87 1.83

GA 310 635 785 0.08 0.55 1.05

MILP 275 550 725 28.35 47.22 85.02

In terms of computation times, CSA and GA ex-

hibit very similar behaviors, both in terms of ab-

solute numbers and scalability. Speciﬁcally, both

approaches are extremely efﬁcient with sub-second

times for the 10- and 20-vessel cases, and take less

than 2 seconds for the 3-vessel case. However, the

computation times for MILP are about two orders of

magnitude higher compared to CSA and GA. While

it only takes 1.83 seconds for CSA to ﬁnd a near-

optimal solution for the 30-vessel case, it takes 85

seconds for MILP to ﬁnd the optimal solution, which

is only 3% cheaper than the CSA solution. It is im-

portant to note that the above problem sizes are fairly

small. For larger, more realistic problem sizes, MILP

cannot be used as it has been reported to take over

100 hours of CPU time for real-world instances (Salhi

et al., 2019). Such times are certainly not accept-

able in the context of MCT operations. From the

above discussion, it can be observed that the CSA

provides near-optimal solutions in affordable compu-

tation times.

5 CONCLUSIONS AND FUTURE

WORK

In this paper, we have focused on the continuous berth

allocation problem with dynamic vessel arrivals. We

have developed a metaheuristic cuckoo search algo-

rithm (CSA) to solve the DC-BAP. Furthermore, two

benchmark schemes, a well-known metaheuristic GA

and an exact approach (MILP), have also been im-

plemented for comparison purposes. A benchmark

dataset is employed from recent literature for the ex-

periments. The results show that our proposed al-

gorithm has higher efﬁciency over GA in terms of

minimum processing cost for all arriving vessels.

The newly developed CSA outperforms GA by 8.1%,

8.7%, and 4.5% in terms of reduced cost consider-

ing 10, 20, and 30 vessels, respectively. Compared

to MILP, our proposed CSA provides a near-optimal

solution (<5% from the optimal) at a fraction of com-

putation time. Hence, we conclude that our proposed

CSA is an efﬁcient algorithm for near-optimal berth

allocation with minimum computational complexity.

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).

Dynamic and Continuous Berth Allocation using Cuckoo Search Optimization

REFERENCES

Agarwal, M. and Srivastava, G. M. S. (2018). A cuckoo

search algorithm-based task scheduling in cloud com-

puting. In Advances in Computer and Computational

Sciences, pages 293–299. Springer.

Aslam, S., Bukhsh, R., Khalid, A., Javaid, N., Ullah,

I., Fatima, I., and Hasan, Q. U. (2017a). An efﬁ-

cient home energy management scheme using cuckoo

search. In International Conference on P2P, Parallel,

Grid, Cloud and Internet Computing, pages 167–178.

Springer.

Aslam, S., Iqbal, Z., Javaid, N., Khan, Z. A., Aurangzeb, K.,

and Haider, S. I. (2017b). Towards efﬁcient energy

management of smart buildings exploiting heuristic

optimization with real time and critical peak pricing

schemes. Energies, 10(12):2065.

Aslam, S., Michaelides, M. P., and Herodotou, H. (2020).

Internet of ships: A survey on architectures, emerging

applications, and challenges. IEEE Internet of Things

Journal, 7:9714–9727.

Barbosa, F., Rampazzo, P. C. B., Yamakami, A., and Ca-

manho, A. S. (2019). The use of frontier techniques

to identify efﬁcient solutions for the berth allocation

problem solved with a hybrid evolutionary algorithm.

Computers & Operations Research, 107:43–60.

Bierwirth, C. and Meisel, F. (2010). A survey of berth al-

location and quay crane scheduling problems in con-

tainer terminals. European Journal of Operational Re-

search, 202(3):615–627.

Bierwirth, C. and Meisel, F. (2015). A follow-up survey of

berth allocation and quay crane scheduling problems

in container terminals. European Journal of Opera-

tional Research, 244(3):675–689.

Cahyono, R. T., Flonk, E. J., and Jayawardhana, B. (2019).

Discrete-event systems modeling and the model pre-

dictive allocation algorithm for integrated berth and

quay crane allocation. IEEE Transactions on Intelli-

gent Transportation Systems, 21(3):1321–1331.

CargoSmart.ai (2019). Vessels arrivals at China Ports.

https://www.cargosmart.ai/en/blog/vessels-arrive-at-

top-china-ports-with-shorter-delays-in-2019/.

Carlo, H. J., Vis, I. F., and Roodbergen, K. J. (2015).

Seaside operations in container terminals: literature

overview, trends, and research directions. Flexible

Services and Manufacturing Journal, 27(2-3):224–

262.

Chandrasekaran, K. and Simon, S. P. (2012). Multi-

objective scheduling problem: hybrid approach using

fuzzy assisted cuckoo search algorithm. Swarm and

Evolutionary Computation, 5:1–16.

Chen, L. and Huang, Y. (2017). A dynamic continu-

ous berth allocation method based on genetic algo-

rithm. In 2017 3rd IEEE International Conference on

Control Science and Systems Engineering (ICCSSE),

pages 770–773. IEEE.

Chen, X. and Yu, K. (2019). Hybridizing cuckoo search

algorithm with biogeography-based optimization for

estimating photovoltaic model parameters. Solar En-

ergy, 180:192–206.

De, A., Pratap, S., Kumar, A., and Tiwari, M. (2020). A hy-

brid dynamic berth allocation planning problem with

fuel costs considerations for container terminal port

using chemical reaction optimization approach. An-

nals of Operations Research, 290(1):783–811.

Dong, Y., Zhang, Z., and Hong, W.-C. (2018). A hybrid sea-

sonal mechanism with a chaotic cuckoo search algo-

rithm with a support vector regression model for elec-

tric load forecasting. Energies, 11(4):1009.

Dulebenets, M. A. (2017). Application of evolutionary

computation for berth scheduling at marine container

terminals: Parameter tuning versus parameter control.

IEEE Transactions on Intelligent Transportation Sys-

tems, 19(1):25–37.

Frojan, P., Correcher, J. F., Alvarez-Valdes, R., Koulouris,

G., and Tamarit, J. M. (2015). The continuous

berth allocation problem in a container terminal with

multiple quays. Expert Systems with Applications,

42(21):7356–7366.

Han, X., Gong, X., and Jo, J. (2015). A new continuous

berth allocation and quay crane assignment model in

container terminal. Computers & Industrial Engineer-

ing, 89:15–22.

Hsu, H.-P., Chiang, T.-L., Wang, C.-N., Fu, H.-P., and

Chou, C.-C. (2019). A Hybrid GA with Variable Quay

Crane Assignment for Solving Berth Allocation Prob-

lem and Quay Crane Assignment Problem Simultane-

ously. Sustainability, 11(7):2018–2038.

Hsu, H.-P., Wang, C.-N., Chou, C.-C., Lee, Y., and Wen, Y.-

F. (2017). Modeling and solving the three seaside op-

erational problems using an object-oriented and timed

predicate/transition net. Applied Sciences, 7(3):218.

Jos, B. C., Harimanikandan, M., Rajendran, C., and Ziegler,

H. (2019). Minimum cost berth allocation problem in

maritime logistics: new mixed integer programming

models. S

adhan

a, 44(6):149.

Kang, T., Yao, J., Jin, M., Yang, S., and Duong, T. (2018).

A novel improved cuckoo search algorithm for param-

eter estimation of photovoltaic (pv) models. Energies,

11(5):1060.

Kavoosi, M., Dulebenets, M. A., Abioye, O., Pasha, J.,

Theophilus, O., Wang, H., Kampmann, R., and Mik-

ijeljevi

c, M. (2019). Berth scheduling at marine con-

tainer terminals. Maritime Business Review, 5.

Lind, M., Michaelides, M., Ward, R., Herodotou, H., and

Watson, R. (2019). Boosting data-sharing to improve

short sea shipping performance: Evidence from limas-

sol port calls analysis. Technical Report 35, UNCTAD

Transport and Trade Facilitation Newsletter No. 82 -

Second Quarter.

Mauri, G. R., Ribeiro, G. M., Lorena, L. A. N., and Laporte,

G. (2016). An adaptive large neighborhood search for

the discrete and continuous berth allocation problem.

Computers & Operations Research, 70:140–154.

Michaelides, M. P., Herodotou, H., Lind, M., and Watson,

R. T. (2019). Port-2-port communication enhancing

short sea shipping performance: The case study of

cyprus and the eastern mediterranean. Sustainability,

11(7):1912.

VEHITS 2021 - 7th International Conference on Vehicle Technology and Intelligent Transport Systems

S¸ ahin, C. and Kuvvetli, Y. (2016). Differential evolution

based meta-heuristic algorithm for dynamic continu-

ous berth allocation problem. Applied Mathematical

Modelling, 40(23-24):10679–10688.

Salhi, A., Alsouﬁ, G., and Yang, X. (2019). An evolutionary

approach to a combined mixed integer programming

model of seaside operations as arise in container ports.

Annals of operations research, 272(1-2):69–98.

Sanajaoba, S. and Fernandez, E. (2016). Maiden applica-

tion of cuckoo search algorithm for optimal sizing of

a remote hybrid renewable energy system. Renewable

energy, 96:1–10.

Tein, L. H. and Ramli, R. (2010). Recent advancements

of nurse scheduling models and a potential path. In

Proc. 6th IMT-GT Conference on Mathematics, Statis-

tics and its Applications (ICMSA 2010), pages 395–

409.

UNCTAD (2017). UNCTAD: Ports in 2017.

https://unctad.org/system/ﬁles/ofﬁcial-document/rmt

2018ch4 en.pdf.

Valian, E., Tavakoli, S., Mohanna, S., and Haghi, A.

(2013). Improved cuckoo search for reliability op-

timization problems. Computers & Industrial Engi-

neering, 64(1):459–468.

Xu, Y., Xue, K., and Du, Y. (2018). Berth scheduling prob-

lem considering trafﬁc limitations in the navigation

channel. Sustainability, 10(12):4795–4816.

Yang, B., Miao, J., Fan, Z., Long, J., and Liu, X. (2018).

Modiﬁed cuckoo search algorithm for the optimal

placement of actuators problem. Applied Soft Com-

puting, 67:48–60.

Yang, X.-S. and Deb, S. (2009). Cuckoo Search via L

evy

Flights. In World Congress on Nature & Biologically

Inspired Computing (NaBIC), pages 210–214. IEEE.

Dynamic and Continuous Berth Allocation using Cuckoo Search Optimization