Computational Intelligence - February 2016 - 56

been fully considered previously, leaving
an important gap in the current research.
This paper attempts to close this gap by
proposing an EA to solve the FSP by
considering the above factors. The proposed algorithm will be tested on two
case studies of container ports 2. The
motivation for choosing ports as the case
studies is due to their vital role in international supply chains: 90% of the world's
non-bulk trade goods were carried by
containers as of 2007 [2].
Specifically, the outcome of this
research will help answering the following questions for the first time: 1) How
to determine the optimal/robust number of vehicles in static/uncertain
ESTTs, especially CTs? 2) How to analyze the impact of uncertainty on the
optimal number of vehicles? 3) What is
the impact of the buffer size on the
optimal/robust fleet size?
The novelty of this research can be
summarized as follows: First, an EA is
proposed to solve the FSP in ESTTs,
with better performance than existing
state-of-the-art methods. Second, a new
formulation for the FSP is developed so
that EA components can be built upon.
Third, the following EA elements are
developed: a representation, a local
search, two operators and an adaptive
learning mechanism. Fourth, uncertainty
in the FSP in CTs is taken into account
and solved. Two high-fidelity simulation
models are also developed to serve as the
benchmark for the EA in the uncertain
cases. Finally, a set of test cases is developed using realistic data from European
CTs to resolve the issue of the lack of
benchmarks.
The rest of this paper is structured as
follows. Sec. II describes the FSP in container ports. Sec. III discusses the gap of
knowledge in research on the FSP in
ESTTs. Sec. IV describes the proposed
EA for the static case and its different
components. In Sec. V, a combination of
the proposed EA with a Monte Carlo
(MC) simulation to determine the
robust number of vehicles under uncertainty is described. The test cases are
2
These two container ports have committed to consider the result of this research to improve their
operations.

Pi at a time s i and deliver it to a
delivery point D i, and then can still
travel to a pickup point P j to pick
up job j within the time window
of job j. Mathematically, i and j
a re c o m p a t i b l e i f s i + t Pi Di +
t Di Pj e [a j, b j], where t PD is the travel
time from point P to point D and
[a j, b j] is the time window of job j.
3) Problem modelling: The FSP was modelled in [3] as a graph where each
node represents one of the possible
pickup times for a job. This graph has
a source node from which all other
nodes originate and a sink at which
all other nodes terminate. Nodes that
are compatible, i.e. nodes whose jobs
can be done by the same vehicle, are
connected by arcs. A set of connected
arcs going from the source to the sink
is called a path. Each path represents
the sequence of jobs to be done by
one vehicle. The total number of
paths represents the total number of
vehicles. The objective is to find the
minimum number of paths which
start from the source node and end at
the sink node subject to the following constraints: 1) each job can start
only once (this means that among all
the possible pickup time nodes of a
job, only one should be included in
one of the paths); 2) each job cannot
be done by more than one vehicle
(this means that each node in the
graph cannot be included in more
than one path).
Fig. 1 shows an example of using a
graph to model the solution of one simple FSP with three jobs and two vehicles.
Job 1 has two possible pickup times, represented by nodes j 11 and j 12. Job 2 has

given in Sec.VI. The experimental results
in the static case are presented in Sec.VII.
Experiment results in the uncertain case
are described in Sec. VIII. Finally, the
conclusion is provided in Sec. IX.
II. Terminologies and
Problem Descriptions
A. Basic Terminologies and an
Integer Programming Model
for the FSP

Here we explain some detailed concepts
of the FSP, as adopted from [3].
1) Jobs and time windows: A job is defined
as the process of moving a good from
one PDP to another by a vehicle. For
each job a time window [a i, b i] is
associated where a i is the release time
of job i from a PDP and b i is the latest time to start job i. The value of b i
is calculated based on the release time
of subsequent jobs of job i and the
size of the buffer. For example, with a
buffer of size n (i.e. the capacity of n
goods) the due time for job i is calculated as: b i = a i + n. This means that
job i should start before the release
time of job i + n to have at least one
available slot for job i + n in the buffer. To determine the exact pickup
time of each job from the buffer, each
time window needs to be discretized
into multiple intervals, each with a
duration of d. The pickup time of a
job is set at the beginning of one of
the intervals.
2) Compatible jobs: Two jobs are compatible if they can be done consecutively by one vehicle. Jobs i and j
are compatible if one vehicle can
pick up job i from a pickup point

: A Node in a Path

j11

Veh.
1

: A Node Not in a Path

j12

s : The Source Node

j31
s
Veh.
2

j32
j21

t : The Sink Node
: An Arc in the Path
of Vehicle 1
: An Arc in the Path
of Vehicle 2
: An Arc Not in a Path

Figure 1 The solution for an FSP problem with three jobs.

IEEE ComputatIonal IntEllIgEnCE magazInE | FEbruary 2016

Table of Contents for the Digital Edition of Computational Intelligence - February 2016