Evaluation Heuristics for Tug Fleet Optimisation Algorithms
A Computational Simulation Study of a Receding Horizon Genetic Algorithm
Robin T. Bye and Hans Georg Schaathun
Faculty of Engineering and Natural Sciences, Aalesund University College, Postboks 1517, NO-6025
˚
Alesund, Norway
Keywords:
Receding Horizon Control, Genetic Algorithm, Dynamic Optimisation, Algorithm Evaluation, Modelling,
Computational Simulation.
Abstract:
A fleet of tugs along the northern Norwegian coast must be dynamically positioned to minimise the risk of
oil tanker drifting accidents. We have previously presented a receding horizon genetic algorithm (RHGA) for
solving this tug fleet optimisation (TFO) problem. Here, we first present an overview of the TFO problem, the
basics of the RHGA, and a set of potential cost functions with which the RHGA can be configured. The set
of these RHGA configurations are effectively equivalent to a set of different TFO algorithms that each can be
used for dynamic tug fleet positioning. In order to compare the merit of TFO algorithms that solve the TFO
problem as defined here, we propose two evaluation heuristics and test them by means of a computational
simulation study. Finally, we discuss our results and directions forward.
1 INTRODUCTION
Several thousand ships transit along the northern Nor-
wegian coastline every year, with the latest figures of
2013 including 1,584 so-called “risky transports,” of
which 298 were ships with oil or other petroleum-
related cargo on board (Vardø VTS, 2014b). With the
recent increase in traffic through the Northwest Pas-
sage and the projected increase in oil exploration in
the High North (Havforskningsinstituttet, 2010), the
Norwegian coastline is increasingly exposed to the
risk of incidents with potentially high impact on the
environment. Indeed, in 2013 alone, the Vard Vessel
Traffic Service (VTS) registered 286 operational inci-
dents, including 186 incidents of drifting vessels, 29
of grounding, 36 of pollution, 10 of fire, and 7 ship-
wrecks (Vardø VTS, 2014a).
The Vard VTS is located at the northeasternmost
point of Norway and is run by the Norwegian Coastal
Administration (NCA) (see Figure 1). Among other
duties, the VTS constantly monitors ship movements,
maintains dialogue with ships, and manages the tug
fleet of Norway.
As noted above, there is an incident of a drift-
ing vessel occurring about every second day on av-
erage. A number of these vessels are high-risk ships
such as oil tankers, which if allowed to drift aground
can cause serious damage to the environment due to
spillage of oil and fuel. In a measure to avoid such
incidents, the VTS is constantly instructing its pa-
Figure 1: Northern Norwegian coastline and the Vard VTS
(shown with its call signal NOR VTS). Solid line is the ge-
ographical baseline; stapled line is the border of the Norwe-
gian Territorial Waters (NTW); thick pink line is the traffic
corridor for the Traffic Separation Scheme (TSS). Adapted
from (Vardø VTS, 2011).
trolling fleet of tugs to move to new positions in a
manner such that if an oil tanker loses manoeuvrabil-
ity, e.g., because of engine or propulsion problems or
steering failure, tugs should be sufficiently close that
it can intercept the drifting oil tanker before it runs
aground (Eide et al., 2007a).
A set of risk-based decision support tools based
on dynamical risk models have been developed pre-
viously (Eide et al., 2007a; Eide et al., 2007b). The
models incorporate a number of factors such as wind,
waves, currents, geography, types of ships in tran-
sit, and potential environmental impact should drift
grounding occur. Whilst such tools can aid the human
270
T. Bye R. and Georg Schaathun H..
Evaluation Heuristics for Tug Fleet Optimisation Algorithms - A Computational Simulation Study of a Receding Horizon Genetic Algorithm.
DOI: 10.5220/0005217802700282
In Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pages 270-282
ISBN: 978-989-758-075-8
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
operators at the VTS in making informed decisions
about tug fleet positioning, they do not tell explicitly
where tugs should move; instead, they give the oper-
ators a real-time risk map divided into zones of low,
medium, and high risk.
The number of oil tanker transits is expected to
rise significantly in coming years (Havforskningsin-
stituttet, 2010), therefore, the problem of command-
ing tugs to “good” positions may become unmanage-
able for human operators. Motivated by this chal-
lenge, our Dynamic Resource Allocation with Mar-
itime Application (DRAMA) research group at the
Aalesund University College (AAUC) has over the
last few years developed and refined a receding hori-
zon genetic algorithm (RHGA) (Bye et al., 2010; Bye,
2012; Bye and Schaathun, 2014). The algorithm it-
eratively plans individual movement trajectories for
the fleet of tugs such that the net collective behaviour
of the tugs is optimised, that is, it employs a genetic
algorithm (GA) in order to minimise cost functions
that have been specifically designed to reduce the risk
of drift grounding accidents. We have also investi-
gated using mixed integer programming (MIP) for the
optimisation component of the algorithm (Assimizele
et al., 2013).
In our most recent work (Bye and Schaathun,
2014), we identified a flaw in the cost function we
had employed previously, and suggested a number of
other cost functions that could be used instead. A
challenge, however, is the problem of comparing and
evaluating the merit of different cost functions, or in
general, of different TFO algorithms. This challenge
is the focus of the work we present here.
In the following sections, we proceed by present-
ing a model of what we have coined as the tug fleet
optimisation (TFO) problem, before introducing our
RHGA and suggesting a set of possible cost functions
that can be used in the algorithm. Next, we propose a
two new and objective evaluation heuristics designed
for making comparisons of TFO algorithms. Finally,
we test the method on our RHGA with the set of cost
functions in a simulation study and discuss the viabil-
ity of our approach as well as future work.
2 METHOD
2.1 A Model of the TFO Problem
We employ a 1D model of the TFO problem and adopt
most of the principles and assumptions in our earlier
work (Bye et al., 2010; Bye, 2012; Assimizele et al.,
2013; Bye and Schaathun, 2014).
Oil tankers are required by law to follow a pre-
defined corridor, or lane, parallel to the coastline, de-
picted as the pink TSS in Figure 1. In topological
space, the corridor constitutes a curve, which is lo-
cally homeomorphic to a straight line. This means
that the curve can be deformed into a straight line by a
continuous, invertible mapping, and vice versa. Con-
sequently, for model simplicity, we assume that N
o
oil
tankers move in one dimension only along a straight
line of motion z.
To the inside of the corridor, a fleet of tugs patrol
the coastal waters. Ignoring a rugged coastline with
islands, peninsulas and shoals, and by the same topo-
logical argument above, we may assume that N
p
tugs
are patrolling along a line of motion y parallel to z,
e.g., the geographical baseline depicted in Figure 1.
We do appreciate, however, that this approximation
is only locally correct, since the curvature of y and
z will make the outermost line longer than the inner-
most line. We also realise that is may be possible that
better protection is achieved by allowing the tugs to
move freely in a 2D space confined to the area be-
tween the coastline and the oil tanker corridor rather
than being confined to a 1D line of motion. Indeed,
our DRAMA research group is currently investigating
using 2D probabilistic models and exact and heuristic
optimisation techniques such as MIP, stochastic pro-
gramming, and tabu search, thus extending our previ-
ous work (Assimizele et al., 2013).
We have been informed by the NCA that up until
the end of 2013, three tugs have operated in the area
depicted in Figure 1, and that the stretch of coastline
they protect is about 1,500 km. The number of tugs
have since the beginning of 2014 been reduced to two.
Consequently, we give the patrol line y a length of
1,500 km, and are mainly interested in fleets of two
or three tugs. For simplicity, we model y with “hard”
borders to the north and to the south, outside of which
the tugs will ignore drifting ships. In reality, ship traf-
fic that may result in drift grounding outside the bor-
ders of the patrol line may still be rescued by an NCA
tug, especially to the south, which is Norwegian terri-
tory.
Fundamental to our modelling approach is the ex-
istence and availability of real-time ship traffic in-
formation such as the direction and speed of the oil
tankers. This information is readily available by the
automatic identification system (AIS) that all ships
above 300 gross tonnage are required to use on in-
ternational voyages due to a regulation by the Inter-
national Maritime Organization (IMO).
In addition, we require accurate simulation mod-
els that can predict the future positions of oil tankers
along z and the corresponding potential drift tra-
EvaluationHeuristicsforTugFleetOptimisationAlgorithms-AComputationalSimulationStudyofaRecedingHorizon
GeneticAlgorithm
271
jectories, given real-time and predicted information
about the tanker movements and the environment the
tankers are travelling through. Developing such mod-
els is outside the scope of the research presented here.
Instead, we are concerned with the planning and con-
trol of a fleet of tugs given that these models and re-
sulting information is readily available.
We note, however, that due to the relatively slow
dynamics of oil tankers, which cannot easily and
quickly change speed or direction, obtaining reason-
ably accurate predicted future positions of the tankers
can be done simply by using dead reckoning or linear
extrapolation or by more advanced techniques such as
a Kalman filter.
Drift trajectory models, on the other hand, repre-
sents a more complex problem, and depend on the
ever-changing dynamics of the ocean, including cur-
rents, waves, and wind, as well the size and shape of
the oil tankers. Nevertheless, although we do not inte-
grate any such models here, they do exist and are cur-
rently an active focus of research (e.g., see (Sørg
˚
ard
and Vada, 1998; Hackett et al., 2006; Breivik and
Allen, 2008; Breivik et al., 2011)).
For any oil tanker moving along the line z, there is
a small probability that an incident may occur at the
position z(t), resulting in the tanker starting to drift
at t = t
d
. Naturally, most of the time, nothing will
happen, and the tanker will continue sailing along z.
Employing a discrete-time model with a sampling pe-
riod of t
s
= 1 hour, we assume that we can estimate
the future tanker positions at discrete points in time,
limited to a prediction horizon T
h
hours into the fu-
ture. For each of the oil tankers, this results in a
set of future tanker positions given by
{
ˆz(t|t
d
)
}
for
t = t
d
+ 1,t
d
+ 2, . . . ,t
d
+ T
h
.
Furthermore, we assume that we can determine,
for example through Monte Carlo simulations, the
most likely hypothetical predicted drift trajectories
that emanate from each predicted tanker position
ˆz(t|t
d
). Such trajectories would depend on a number
of actual and forecast conditions in the area, such as
ocean currents and wind speed and direction, and may
or may not intersect the patrol line y after an estimated
drift duration
ˆ
∆ into the future.
According to (Eide et al., 2007a), situations of
“fast drift” can have drift durations as fast as 8–12
hours, whereas more typical drift durations are in
the range 16–24 hours. In previous work, in order
to be conservative rather than optimistic, we there-
fore either set the estimated drift duration
ˆ
∆ to be
8 hours for all oil tankers (Assimizele et al., 2013),
or to be drawn randomly for each oil tanker such that
ˆ
∆ ∈ {8, 9 . . . , 12} hours (Bye et al., 2010; Bye, 2012;
Bye and Schaathun, 2014).
It should also be kept in mind that there will in-
evitably be a detection delay δ between the time when
an oil tanker begins drifting at the drift time t
d
1
and
the time when the VTS centre detects, or is notified
of, the incident at time t
a
some hours later, which we
call the alarm time. The detection delay is thus given
by δ = t
a
−t
d
.
If we examine all the future predicted positions for
all the oil tankers as well as all the corresponding drift
trajectories, we obtain a distribution of cross points
located at points where future drift trajectories will
intersect the patrol line y. A cross point of the cth oil
tanker’s drift trajectory at time t can be defined as the
position y
c
t
. Assuming a drift duration
ˆ
∆, a drift tra-
jectory starting on z(t) at t = t
d
will have a cross point
on y at t = t
d
+
ˆ
∆. Assuming the same drift duration
for all drift trajectories and considering the prediction
horizon T
h
, there is a predicted set of cross points for
the cth oil tanker given by
{y
c
t
} =
n
y
c
t
d
+
ˆ
∆
, y
c
t
d
+1+
ˆ
∆
, . . . ,y
c
t
d
+T
h
o
. (1)
Moreover, we define a patrol point as the pth tug’s
position on y at time t as y
p
t
.
Based on the predicted future distribution of cross
points, we define the TFO problem as the problem
of calculating patrol trajectories (sequences of patrol
points) that start at t = t
d
and have some duration T
h
,
along y for each of the patrolling tugs such that the
risk of an oil tanker in drift not being reached and
prevented from grounding is minimised.
Figure 2 shows a graphical summary of the TFO
problem as presented above, exemplified by two pa-
trolling tugs and three oil tankers.
Oil tanker line
yo
Patrol tugs line
yp
Coast line
Oil tanker
Oil tanker
Oil tanker
Patrol tug
Patrol tug
D
r
i
f
t
t
r
a
j
e
c
t
o
r
y
D
r
i
f
t
t
r
a
j
e
c
t
o
r
y
D
r
i
f
t
t
r
a
j
e
c
t
o
r
y
Cross
point
Cross
point
Cross
point
t = td
t = td
t = td + 1
t = td + 2
t = td + 3
t = td
t = td
t = td + drift time
t = td
t = td + 2 + drift time
t = td+3 + drift time
t = td + 1 + drift time
t = td + drift time
t = td + drift time
Figure 2: TFO problem: Where should the tugs move?
1
Note that t
d
also is used as the start time for planning
patrol trajectories for the tugs to follow.
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
272
2.2 The RHGA
The TFO algorithm that we study in this paper is
the RHGA (Bye et al., 2010; Bye, 2012; Bye and
Schaathun, 2014). The algorithm consists of two
main components: receding horizon control (RHC)
and a genetic algorithm (GA). The GA is a search
heuristic for solving search and optimisation prob-
lems and is inspired by elements in natural evolu-
tion, such as inheritance, mutation, selection, and
crossover. It has been attributed to (Holland, 1975),
with subsequent popularisation by (Goldberg, 1989),
and is currently a very popular optimisation tool
across many different disciplines, including opera-
tions research. The GA we have implemented in our
RHGA is based on work by (Haupt and Haupt, 2004).
The optimisation problem must be defined as a
cost function such that, when evaluated for a set of
candidate solutions, the GA is able to distinguish
good solutions from bad ones. Specifically, for the
TFO problem, the cost function must be designed
such that its solution is a set of future position tra-
jectories, or collective movement plan, for the fleet of
tugs that minimises the risk of drift grounding acci-
dents to happen.
At any given point in time, the GA can incorpo-
rate real-time information about the current situation,
as well as a prediction of the future, to calculate an op-
timal set of patrolling tug trajectories. However, due
to the dynamic nature of the environment and the pa-
rameters involved, the solution will quickly become
outdated. We therefore require some feedback mech-
anism in the algorithm that can update the solution
with changes in ocean conditions such as wind, cur-
rent, and waves, as well as speed and direction of oil
tankers. The mechanism we adopt is the principle of
RHC.
From control theory, it is known that RHC, which
is also called model predictive control (MPC), is one
of very few control methods able to handle constraints
in the design phase of a controller and not via post
hoc modifications (e.g., see (Goodwin et al., 2001;
Maciejowski, 2002; Rossiter, 2004)). For the TFO
problem, one such constraint is the maximum speed
of tugs, which is constrained by factors such as ship
design and weather conditions. This maximum speed
will necessarily limit the number of reachable cross
points. Using RHC it is possible to constantly incor-
porate such constraints in the planning of tug patrol
trajectories, even as conditions change.
In our RHGA, the GA component plans a set of
tug trajectories starting at t
d
and with a prespecified
duration, namely the prediction horizon T
h
introduced
previously. However, the tugs only execute the very
first time step of their trajectories. In the mean time,
with a start time of t
d
+ 1, another set of of tug trajec-
tories is planned, based on new and predicted infor-
mation available. This new solution replaces the old
one but again only the first portion is implemented.
This process repeats as a sequence of planning steps,
thus creating a feedback loop where updated informa-
tion is fed back to the GA. Effectively, the prediction
horizon keeps being shifted into the future, and this
has led to the term receding horizon control.
A thorough presentation of our simulator frame-
work and algorithm implementation is not possible
within the the scope limitations of this paper. Whilst
the RHGA was implemented in Matlab in earlier ver-
sions (Bye et al., 2010; Bye, 2012), we recently
rewrote the entire code base in the advanced, purely-
functional programming language Haskell (Bye and
Schaathun, 2014), and have used this framework for
the work presented here. For further details about var-
ious aspects of our implementation, we refer to (Bye
et al., 2010; Bye, 2012; Bye and Schaathun, 2014).
2.3 Cost Functions
Determining suitable cost functions for a TFO algo-
rithm is a key design challenge for the algorithm to be
successful. Below, we will present three possible cost
functions, each of which can be configured to yield
different properties by means of parameterisation.
2.3.1 Cost Function f
1
For the cost function we used in our earliest work
(Bye et al., 2010; Bye, 2012; Assimizele et al., 2013),
we employed a metric defined as the sum of the dis-
tances between all cross points and the nearest patrol
points, based on the argument that if an oil tanker in
drift can be saved by the nearest tug, then it is not rel-
evant if the other tugs are also able to save the tanker,
and if an oil tanker in drift cannot be saved by the
nearest tug, then it cannot be saved by the other tugs
either. This argument assumes that the tugs all have
the same maximum speed. In this case, this metric
is equivalent to minimum rescue time, since distances
will be directly proportional to rescue times. If in-
stead the tugs do not have identical maximum speeds,
one can easily define rescue time as distance divided
by maximum tug speed and add up the minimum res-
cue times for each cross point.
Recently, we decided to examine some other met-
rics, namely squaring the distances and also incorpo-
rating a safe zone (Bye and Schaathun, 2014). The
effect of squaring the distance from a cross point to
a patrol point is that cross points further away will
EvaluationHeuristicsforTugFleetOptimisationAlgorithms-AComputationalSimulationStudyofaRecedingHorizon
GeneticAlgorithm
273
be penalised more in the cost function. It makes in-
tuitive sense that higher costs should be awarded to
the cross points of tankers that are less likely to be
saved. Similarly, tankers with cross points that are
very close to the position of one or several tugs are
very likely to be saved and can thus be ignored in the
cost function. We incorporate these situations by the
inclusion of a “power” parameter e and raise the dis-
tance to the power e = 2 for squaring, whereas we use
a safe region parameter r to impose no penalty in the
cost function for close cross points.
Combining our original cost function with the re-
cent modifications, we obtain the cost function f
1
given by
f
1
(t) =
t
d
+T
h
∑
t=t
d
∑
o∈O
max
0, min
p∈P
y
c
t
− y
p
t
e
− r
(2)
for N
o
oil tankers o ∈ O = {o
1
, . . . ,o
N
o
}
, N
p
patrol
tugs p ∈ P = {p
1
, . . . , p
N
p
}
, e ∈ {1, 2}, and r chosen
as some distance that can confidently be reached by a
tug to enable drift interception and hookup to the ship.
A reasonable and conservative choice for r could for
instance be half the expected distance a tug can travel
from an alarm is received until the first hypothetical
cross points occur.
In terms of minimising this cost function for non-
zero r, a challenge will be that of flat cost surface re-
gions for cross points within the safe range, which
makes it more difficult for the GA to find an optimal
solution.
Note that letting e = 1 and r = 0 yields the cost
function used in our earlier work (Bye et al., 2010;
Bye, 2012; Assimizele et al., 2013).
2.3.2 Cost Function f
2
In (Bye and Schaathun, 2014), we identified a prob-
lem in cost function f
1
, namely the lack of taking
the detection delay δ = t
a
− t
d
into account. That is,
there will be a delay from the time t
d
when an oil
tanker starts drifting until the VTS and its tugs are
being alarmed at the time t
a
some hours later. Cost
function f
1
actually implicitly assumes that the VTS
will be notified immediately when a ships starts drift-
ing; an assumption that is clearly far too optimistic.
Instead, we should assume that oil tankers typically
have drifted for some time before the tugs are be-
ing alarmed, and consequently, we must define a new,
and shorter, drift-from-alarm (DFA) time
ˆ
∆
a
=
ˆ
∆ − δ,
which is the drift time from the tugs receive an alarm
at t
a
until the drifting tanker crosses the patrol line at a
cross point. We will keep our somewhat arbitrary, but
realistic, choice of δ = 3 hours presented previously
(Bye and Schaathun, 2014) in this paper.
In (Bye and Schaathun, 2014), we also discov-
ered a serious flaw with cost function f
1
in that it
implicitly assumes that tugs will continue to execute
their original plans even after receiving an alarm, be-
cause it compares cross points and patrol points at the
same times into the future. Instead, the tugs should of
course abandon their original plans immediately upon
an alarm about a drifting tanker, and make every effort
to intercept it before it runs aground. Therefore, the
cost function should compare the positions of the tugs
(patrol points) when they receive the alarm at time t
a
,
and the hypothetical future positions where drifting
tankers will cross the patrol line some
ˆ
∆
a
hours later,
where
ˆ
∆
a
is the total drift time
ˆ
∆ (8–12 hours) less the
detection delay δ (3 hours), leaving
ˆ
∆
a
in the range
5–9 hours.
To address the issues raised above regarding the
original cost function f
1
, we propose a modified cost
function f
2
given by
f
2
(t) =
t
a
+T
h
∑
t=t
a
∑
o∈O
max
0, min
p∈P
y
c
t+
ˆ
∆
a
− y
p
t
e
− r
.
(3)
Compared with f
1
, we observe that the cost evaluation
starts at the notification time of alarm t
a
, and not at the
time of start of drift t
d
, and that in the distance term,
we measure the distance between each cross point at
some future cross time t +
ˆ
∆
a
and the position of the
nearest tug at the alarm time t, and do this for the
current alarm time t = t
a
and future potential alarm
times t = t
a
+ 1,t
a
+ 2, . . . ,+t
a
+ T
h
.
2.3.3 Cost Function f
3
In contrast with f
1
and f
2
above, another and proba-
bly more realistic cost funtion f
3
is simply the number
of unsalvageable tankers. That is, from a pragmatic
point of view, we merely want to consider whether a
tug can reach a drifting tanker in time to prevent it
from grounding, and the distance is otherwise imma-
terial.
We may use the safe range r for counting the num-
ber of unreachable cross points and let this number
constitute a measure for unsalvageable tankers. If
cross points are outside the safe range, we add 1 to
the accumulated cost, otherwise we add 0. The cost
function f
3
can then be described by
f
3
(t) =
t
a
+T
h
∑
t=t
a
∑
o∈O
g
min
p∈P
y
c
t+
ˆ
∆
a
− y
p
t
− r
, (4)
g(x) =
(
1, x > 0 (outside r),
0, x ≤ 0 (inside r),
(5)
where g(x) is the Heaviside unit step function.
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
274
Note that cost function f
3
accumulates a binary
penalty (1 or 0) and relies on a safe region r only,
which makes it much more difficult to optimise for
the GA than f
1
and f
2
due to numerous plateaus of
flat cost surface regions in the cost landscape that the
GA searches through.
2.4 Algorithm Evaluation
How can we compare the performance of different
TFO algorithms, or in our case, the performance
of the RHGA configured with various different cost
functions? By definition, the metrics of different cost
functions are not generally directly comparable, and
it is not always possible to use a cost function which
directly reflects the real cost of the solution. In the
TFO problem, there are many random elements with-
out well-understood probability models. Incorporat-
ing these elements in the cost function would make it
too complex to be practical.
2.4.1 Simulation Framework
The solution is a Monte Carlo simulation as shown in
Figure 3, where the complete optimisation algorithm
with cost function can be tested against a large num-
ber of (pseudo) random scenarios.
RHGA
PRNG
Event
PRNG
Cost
Function
Evaluation
Heuristic
Problem
Solution
Outcome
Result
Figure 3: Simulation model. A pseudo-random number
generator (PRNG) generates simulation scenarios in which
the RHGA uses some cost function Cost 1 to determine a
solution (where tugs should move). The PRNG then gener-
ates an event of drift, which depending on the current posi-
tion of the tugs may be critical or not. The outcome (saving
or not saving the drifting ship) is then quantified as Cost 2
by an evaluation heuristic.
We have two pseudo-random algorithms (PRNG);
one to generate the problem as observed by the opti-
misation algorithm (RHGA), and one to generate the
situation, or event, where the solution is to be exe-
cuted. For example, the positions of the tankers is
known a priori, and is part of the problem. A tanker
starting to drift is an event which is only known after
the RHGA has provided the solution. Given the so-
lution and the event, we can evaluate the cost of the
result, e.g., of a grounding accident or a successful
rescue.
The evaluation heuristic may look similar to the
cost function, but there is a critical difference. The
evaluation heuristic evaluates the cost of a particular
event. The cost function has to evaluate a solution
without the knowledge of which event will occur.
There are several stochastic processes governing
the outcome of an event in the model. These pro-
cesses can be internalised either in the Monte Carlo
simulation or integrated analytically in the evaluation
heuristic. We can illustrate this with an example. The
event in the simulation model can be subdivided into
two stages:
1. Oil tanker o starts drifting.
2. Oil tanker o grounds.
If the first event occurs, one ore more patrol tugs will
attempt to rescue the drifting tanker. This rescue op-
eration may or may not succeed, depending largely
on the maximum tug speed as determined by weather
conditions. If it does not, the second event occurs.
One possibility is to stop the Monte Carlo estima-
tion after the first event, and let the cost be equal to
the conditional probability of the second event, that
is, the probability of a failed rescue operation. The
second possibility is to simulate the entire rescue op-
eration. If it succeeds the cost is zero, otherwise it is
the cost of the grounding accident, which may depend
on the type of cargo, geographical location, weather
conditions, and so on.
For the purpose of this work, we evaluate the re-
sult when tanker o starts drifting. We do not simu-
late the rescue operation. The steps of the evaluation
method can be summarised as follows:
1. randomly generate a deterministic and repro-
ducible simulation scenario;
2. run the RHGA (or another TFO algorithm) for a
given number of planning steps;
3. considering each oil tanker separately, assume
each tanker begins drifting and count the number
of salvageable tankers;
4. for the same simulation scenario, repeat (2) and
(3) with a different cost function configuration in
the RHGA (or a different TFO algorithm); and
5. repeat steps (1)–(4) for a number of different sim-
ulation scenarios and find the accumulated evalu-
ation cost for each RHGA configuration (or TFO
algorithm).
Note that instead of evaluating one random event, we
evaluate one event for each tanker o, where o starts to
drift. This is possible because the number of tankers
is small, and it lets us evaluate a larger number of sce-
narios with little extra time. We propose two candi-
date evaluation heuristics h
1
and h
2
.
EvaluationHeuristicsforTugFleetOptimisationAlgorithms-AComputationalSimulationStudyofaRecedingHorizon
GeneticAlgorithm
275
2.4.2 Evaluation Heuristic h
1
The first heuristic is similar to cost function f
3
count-
ing the number of salvagable tankers at some alarm
time t
a
. We will simply assume that each patrol tug p
can save any ship with cross points inside the safe re-
gion r = v
p
max
ˆ
∆
a
away, where
ˆ
∆
a
is the DFA time and
v
p
max
is the pth tug’s maximum speed, that is, within
the maximal reach of a tug upon a drift alarm. In a
more realistic model, the heuristic should probably be
weather dependent and direction dependent, e.g., go-
ing against the wind is slower than going with it. It
should also include hookup times.
A simulation scenario in this case is simply a set
of pre-determined oil tanker movements and the re-
sulting hypothetical drift trajectories and cross points
for a pre-specified duration. For testing purposes, we
can generate a number of such scenarios offline and
use them as input data for testing TFO algorithms. In
a real-world application, the actual scenario is what
that is happening right now, and future oil tanker po-
sitions, drift trajectories, and cross points would have
to be predicted in real-time.
To sum up, we define the evaluation heuristic h
1
as
h
1
(t
a
) =
∑
o∈O
g
min
p∈P
y
c
t
a
+
ˆ
∆
a
− y
p
t
a
− r
, (6)
r = v
p
max
ˆ
∆
a
(7)
g(x) =
(
1, x > 0 (outside r),
0, x ≤ 0 (inside r).
(8)
Other possible objective measures exist, e.g., we
could sum up the total fuel consumption and use it as a
component of an overall objective measure if that is of
interest. Furthermore, the cost of measuring the num-
ber of salvagable tankers does not need to be discrete
(yes/no) but could instead have a continuous proba-
bility distribution attached to it. We could then sum
these probabilities to find an evaluation cost for the
TFO algorithm.
2.4.3 Evaluation Heuristic h
2
The evaluation heuristic h
1
does not discriminate be-
tween cross points far away from the nearest tug and
cross points that are much closer, as long as they are
all inside the maximal reach from any tug as given by
r = v
p
max
ˆ
∆
a
. However, it is clear that due to varying
and non-optimal weather conditions, the maximum
speed of each tug may be much lower than during
ordinary operation. Moreover, h
1
does not take into
account that there will be a hookup time when the tug
attaches itself to the drifting ship. In an attempt to
address these issues, we suggest the following evalu-
ation heuristic h
2
given by
h
2
(t
a
) =
∑
o∈O
max
0, min
p∈P
y
c
t
a
+
ˆ
∆
a
− y
p
t
a
− r
2
,
(9)
r = v
p
min
ˆ
∆
a
, (10)
where the safe region has been reduced to the area
reachable for any tug with some minimum speed v
p
min
,
which we assume the tug will always be able to main-
tain. Inside the safe region, there is zero cost for
cross points of salvageable tankers, whereas outside,
the cost increases with the square of the distance to
cross points of unsalvageable tankers. Squaring en-
sures that we punish larger distances more.
3 SIMULATION STUDY
3.1 Basic Parameters
Until the end of 2013, the NCA have been using N
p
=
3 tugs for patrolling the northern Norwegian coast.
However, since the beginning of 2014, the number of
tugs have been reduced to N
p
= 2. In previous papers,
we have consistently assumed three tugs in the fleet.
Using our evaluation heuristics, we are able to exam-
ine by means of simulations whether there are any
differences in coastal protection (as we have defined
it by our evaluation heuristics) from this reduction in
fleet size. For completeness, and in order to test our
propositioned evaluation heuristics, we will also ex-
amine tug fleets of four, five, or six tugs, as well as
the case of a single patrol tug, hence N
p
∈ {1, . . . , 6}.
The stretch of coastline patrolled by the tug ves-
sels that we have termed the patrol line y is about
1, 500 km long. Hence, we define our patrol zone
Y as unidimensional along y in the continuous inter-
val Y = [−750, 750] km, and constrain cross points y
c
and patrol points y
p
to lie in Y , or y
c
, y
p
∈ Y . For im-
plementation purposes, we also define a tanker zone
Z in the same interval much further away from land
but underline that the (simulated) VTS will still ob-
serve ship traffic outside of this zone. The reason is
of course that a tanker outside the tanker zone Z may
still drift and ground inside the patrol zone Y .
For simplicity, we will assume a patrol zone with
“hard” borders, where cross points outside the patrol
zone are ignored, but as argued in Section 2.1, we do
realise that in reality, the VTS will also consider po-
tential cross points outside of such borders.
Based on historical traffic data, we have previ-
ously assumed that the typical average number of
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
276
tankers sailing along z and being watched by the Vard
VTS is N
o
= 6. From communication with the NCA,
the actual number can be less on certain days but also
higher, especially if we add high risk ships other than
oil tankers that may also be watched carefully. In-
deed, as mentioned in the introduction, there were
close to 1600 risky transports in the area in 2013
alone, of which about 300 were carrying petroleum-
related cargo (Vardø VTS, 2014b). Additionally, it is
desirable to obtain results comparable with our previ-
ous work. Thus we keep this number of oil tankers
unchanged in this study.
Under normal conditions, we assume that the pa-
trol tugs are limited to a maximum speed of v
p
max
=
20 km/h, whereas the speed of each oil tanker v
o
is
randomly drawn from a uniform distribution such that
v
o
∈ [20, 30] km/h. Note that compared with previ-
ous work, we have reduced the maximum speed of
the tugs from 30 km/h to 20 km/h, thus making it
more difficult for tugs to cover potential cross points.
These speeds are in line with the literature (e.g., (Det
Norske Veritas, 2009), (Eide et al., 2007a)). We also
assume that even in very bad weather conditions, the
tugs are able to maintain at least a minimum speed of
v
p
min
= 5 km/h.
Drift trajectories are set to be perpendicular (east-
bound) onto the south-north patrol line y, with as-
sociated drift times drawn randomly from the inter-
val {8, 9, . . . , 12} hours and cross points generated
from extrapolating the predicted future positions of
oil tankers and their resulting drift trajectories.
The parameter settings are summarised in Table 1.
3.2 Simulation Scenarios
A simulation scenario consists of simulation-
generated tanker movements along z as well as hy-
pothetical drift trajectories with corresponding cross
points on y. The scenario acts as an input to a TFO
algorithm such as the RHGA and is completely inde-
pendent of what the RHGA calculates and how the
tugs move.
We initialise a scenario by placing N
o
oil tankers
at random positions and with random speeds along
z, headed in either the southbound or the northbound
direction. Next, we sample each of the tankers’ posi-
tions, speeds, and directions at every simulation step
t
s
= 1 h from the start of the simulation at t
i
= 0 h
to the final simulation time at t
f
= 24 h. For any
simulation time t
d
in
{
t
i
,t
i
+t
s
, . . . ,t
f
}
, we suppose
that we have precise real-time information about the
speed and direction of each oil tanker, as provided by
AIS. We also assume that we have an accurate model
that, given this real-time actual information, is able
to predict future positions and speeds of the tankers
at future times t
d
+t
s
,t
d
+ 2t
s
, . . . ,t
d
+ T
h
, where T
h
is
the prediction horizon. Finally, we assume that we
have another accurate model that is able to predict hy-
pothetical drift trajectories and cross points for each
tanker if it starts drifting at time t
d
and also at the fu-
ture times just listed.
Note that since the scenario consisting of oil
tanker movements, drift trajectories, and cross points
is independent of how the fleet of tugs move, we can
replay the same scenario as an input to other TFO al-
gorithms (or variations of the RHGA) in order to eval-
uate and compare the algorithms.
3.3 TFO by the RHGA
For all scenarios, the N
p
tugs are initialised at simula-
tion time t
i
= 0 by being uniformly positioned along
the coast at stationary base stations in a manner such
that they can cover as much of the patrol line y as pos-
sible. For example, since we have defined y as a line
constrained to [−750, 750] km, a single tug will be
placed at y
p
t
i
= 0, a fleet of two tugs will be placed at
y
p
t
i
= {−375, 375}, a fleet of three tugs will be placed
at y
p
t
i
= {−500, 0, 500}, and so on. From these ini-
tial positions, tugs will begin to actively pursue good
positions for reducing the risk of drift grounding ac-
cidents depending on how the scenario plays out and
how the TFO algorithm will control them.
At any simulation time t
d
, the GA component in
the RHGA uses the predicted distribution of potential
cross points to calculate a plan, which consists of a
position trajectory for each of the tugs in the fleet. The
plan consists of future desired positions for each tug at
times {t
d
+ 1, t
d
+ 2, .. . ,t
d
+ T
h
}. The plan is optimal
(or close to optimal) in the sense that it minimises (or
tries to minimise) a cost function.
Using RHC, we let the tugs execute only the first
step of this plan and move the tugs from their posi-
tions at t = t
d
to future positions at t = t
d
+ 1. At
t = t
d
+ 1, the GA plans a new set of desired trajec-
tories from t = t
d
+ 2 to t = t
d
+ T
h
+ 1, but again,
we let the tugs execute only the first step from t =
t
d
+ 1 to t = t
d
+ 2. This process repeats until the
final simulation time t
f
= 24, at which we plan for
t = t
f
,t
f
+ 1, . . . ,t
f
+ T
h
, and again, and finally, let the
tugs execute only the first step from t = t
f
to t = t
f
+1.
We have then completed one simulation of this
particular scenario using one particular TFO algo-
rithm, in our case, the RHGA employing a particu-
lar configuration of one of the cost functions f
1
– f
3
.
The end-of-simulation positions of tugs and tankers
and their cross points are then be used as input to the
evaluation heuristics.
EvaluationHeuristicsforTugFleetOptimisationAlgorithms-AComputationalSimulationStudyofaRecedingHorizon
GeneticAlgorithm
277
Table 1: Simulation parameters, settings, and units.
Parameters Settings Units
Patrol zone (south-north line) Y = [−750, 750] km
Tanker zone (south-north line) Z = [−750, 750] km
Number of oil tankers N
o
= 6 -
Set of oil tankers O = {1, 2 . . . , N
o
} -
Number of tugs N
p
= {1, . . . ,6} -
Set of tugs P = {1, 2 . . . , N
p
} -
Initial tug positions (base stations) Uniformly distributed km
Random initial tanker positions y
o
∈ Z, ∀o ∈ O km
Maximum speed of tugs v
p
max
= 20, ∀p ∈ P km/h
Minimum speed of tugs v
p
min
= 5, ∀p ∈ P km/h
Random speed of oil tankers v
o
∈ [20, 30], ∀o ∈ O km/h
Initial simulation time t
i
= 0 h
Simulation step t
s
= 1 h
Final simulation time t
f
= 24 h
Prediction horizon T
h
= 24 h
Time of start of drift t
d
∈
{
t
i
,t
i
+ 1, . . . , t
f
}
h
Detection delay δ = 3 h
Alarm time t
a
= t
d
+ δ ∈
{
t
i
+ δ,t
i
+ δ + 1, . . .,t
f
+ δ
}
h
Drift direction Eastbound -
Estimated drift times
ˆ
∆ ∈ {8, 9, . .. , 12} h
Drift-from-alarm (DFA) times
ˆ
∆
a
=
ˆ
∆ −δ ∈ {5, 6, . . . , 9} h
Static strategy y
p
t
= y
p
t
i
, ∀t km
Cost functions F = { f
1
, f
2
, f
3
} -
Distance power e = {1, 2}, in f
1
, f
2
-
Safe region r =
{0, 50, 100}, in f
1
, f
2
{50, 100}, in f
3
v
p
max
ˆ
∆
a
= [100, 180], in h
1
v
p
min
ˆ
∆
a
= [25, 45], in h
2
km
TFO algorithms Configurations of RHGA( f
i
, e, r, N
p
) -
Number of RHGA( f
i
, e, r, N
p
) configurations N
conf
= 15 -
Number of scenarios N
sc
= 1600 -
Total number of simulations N
sim
= N
conf
× N
sc
× dimN
p
= 144, 000 -
3.4 GA Description and Settings
The GA we employ in this study is based on the con-
tinuous GA presented in (Haupt and Haupt, 2004)
and has been presented in detail in our previous work
(Bye, 2012; Bye et al., 2010). We initialise the
GA with a population size of chromosomes that are
randomly generated. At every iteration of the GA,
N
keep
= 10 chromosomes are selected from the popu-
lation by roulette wheel selection, with low cost chro-
mosomes having a greater chance of being picked.
These chromosomes survive from one generation to
the next and are also used for mating to generate new
offspring that replace the chromosomes that were not
picked. Mating is performed by a combination of
an extrapolation method and a single crossover point
to obtain new offspring variable values bracketed by
the parents variable values (?, see)for details]haupt04.
After mating, the new population of chromosomes is
ranked and the N
elite
= 10 best chromosomes are cat-
egorised at elite chromosomes and are not allowed
to mutate. Of the remaining non-elite chromosomes,
each has a mutation rate µ = 0.1 probability of being
mutated. After mutation has taken place, the GA re-
peats the process for a total of N
iter
= 200 iterations,
after which the best solution obtained is used for mov-
ing the tugs one step ahead as per the RHC strategy
presented above.
We chose these GA parameter settings by man-
ually evaluating a number of test runs, where we
were able to find suitable settings that ensured satis-
factory run times for all RHGA configurations while
at the same time obtaining satisfactory minimisation
of cost functions. That is, only negligible improve-
ments were attainable from tuning the GA to other,
often more time-consuming settings, e.g., increasing
the population size or number of iterations.
Our choice of GA settings are summarised in Ta-
ble 2.
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
278
Table 2: GA settings.
Parameters Settings
Population size N
pop
= 50
Chromosomes kept for mating N
keep
= 10
Elite chromosomes N
elite
= 10
Mutation rate µ = 0.1
Number of iterations N
iter
= 200
3.5 RHGA Configurations
The three cost functions f
1
, f
2
, and f
3
are param-
eterised by a distance power e and a safe region r,
whereas the evaluation heuristics h
1
and h
2
are param-
eterised by a safe region r, which in turn is a function
of v
p
max
or v
p
min
for h
1
or h
2
, respectively.
We decided to implement and evaluate 14 differ-
ent configurations of the cost functions for the RHGA
and evaluate each configuration using both the eval-
uation heuristics. In addition, we wanted to evaluate
a static strategy, in which tugs are stationary at base
stations and do not move until notified about a drift-
ing ship. For convenience in our data processing, we
have labelled the static strategy as configuration #0
and defined its configuration as f
0
, with e and r both
set to zero. For the same reason, we have set e = 0
for f
3
, where e is not applicable. Each configuration
can be thought of as a unique TFO algorithm. Indeed,
our approach generalises to the evaluation of any TFO
algorithm able to calculate tug fleet control decisions
based on the parameters, settings, and input scenarios
that we have described above.
The 15 configurations are summarised in Table 3.
Table 3: RHGA configurations.
Cost function f
i
Power e Safe region r #
0 0 0 1
1
1
0 2
50 3
100 4
2
0 5
50 6
100 7
2
1
0 8
1 50 9
100 10
2
0 11
50 12
100 13
3 0
50 14
100 15
4 RESULTS
We randomly generated N
sim
= 1600 unique simula-
tion scenarios and tested the performance of tug fleet
optimisation for each of the 15 RHGA configurations
given in Table 3 when faced with N
p
= {1, . . . , 6} tugs
to control, yielding a grand total of 144,000 simula-
tions. For each configuration, that is, each combina-
tion of cost function f
i
, distance power e, and safe
region r, denoted as f
i
(e, r), the sample mean, stan-
dard deviation, coefficient of variance (relative stan-
dard deviation), standard error (standard deviation of
the sample mean), and relative standard error for the
evaluation heuristics h
1
and h
2
were calculated for
N
p
= {1, . . . , 6} tugs, respectively. Both heuristics
were evaluated at the end of each simulated scenario
at t = t
f
. In the sections below, the results of the sam-
ple means as well as comparisons of the active control
configurations of the RHGA versus the static strategy
are our main concern and will be presented graph-
ically, whereas the other statistics will be presented
briefly in text.
4.1 Evaluation Heuristic h
1
The evaluation heuristic h
1
is a measure of the number
of unsalvageable tankers. Figure 4 shows the max-
imum (worst performance) and minimum (best per-
formance) sample mean
¯
h
1
of cost functions f
1
– f
3
over all configurations (combinations of power e and
safe region r), as well as the static strategy, evaluated
for 1–6 tugs. Unsurprisingly, the size of the tug fleet
static
max(f
1
)
max(f
2
)
max(f
3
)
min(f
1
)
min(f
2
)
min(f
3
)
h
1
0
1
2
3
4
Number of tugs
1 2 3 4 5 6
Figure 4: Maximum and minimum mean h
1
evaluated for
1–6 tugs and cost functions f
1
– f
3
and the static strategy.
EvaluationHeuristicsforTugFleetOptimisationAlgorithms-AComputationalSimulationStudyofaRecedingHorizon
GeneticAlgorithm
279
strongly affects h
1
. With a single tug,
¯
h
1
was in the
range [4.15–4.87], and then decreased with the num-
ber of tugs to the range [0.034–0.30] for six tugs.
For all configurations with 1–3 tugs, the standard
deviation showed no trend and was in the range [0.99–
1.25], whereas it was decreasing with the number of
tugs for configurations with 4–6 tugs and ranged from
0.19 to 0.91. The standard error of the mean was in
the range [0.005–0.032] for all configurations, with
typically smaller values for smaller means. The rel-
ative standard error (found by dividing by the mean)
was small for all configurations, increased with num-
ber of tugs (and thus smaller means), and was in the
range [0.0051–0.13].
As expected, all RHGA configurations of f
1
– f
3
outperform the static strategy for the same number of
tugs. In addition, when the RHGA is configured with
a tug fleet with one tug less than the static strategy,
the following observations are made: With a single
tug, no RHGA configuration is able to outperform the
static strategy with two tugs; with two tugs, the best
configurations of f
2
and f
3
outperform the static strat-
egy with three tugs; and with 3–5 tugs, all configura-
tions of f
2
and the best configuration of f
3
outperform
the static strategy with 4–6 tugs, respectively.
Comparing the RHGA configurations with respect
to the number of tugs, we observe the following: For
a single tug, f
3
has a better respective minimum and
maximum performance than the other cost functions;
for two tugs, f
2
and f
3
have approximately equal
minimum and maximum performance; for 3–6 tugs,
f
2
has a better minimum and maximum performance
than the other cost functions. Also, for any number of
tugs, the best configuration of f
3
is with safe region
r = 100 and the worst is with r = 50. For any number
of tugs, the best configuration of f
3
is not much worse
than that of f
2
while the worst configuration of f
3
is
clearly worse than that of both f
1
and f
2
for 4–6 tugs.
Finally, we note that when compared with f
2
, f
1
has a similar trend and relationship between maxi-
mum and minimum
¯
h
1
with increasing number of tugs
but consistently with worse performance.
4.1.1 Comparison with Static Strategy
Figure 5 shows the normalised mean of h
1
for all con-
figurations of cost functions f
1
(left), f
2
(middle), and
f
3
(right) evaluated for 1–6 tugs and normalised by di-
viding the results with those of the static strategy.
For both cost functions f
1
and f
2
and 1–3 tugs,
a power setting of e = 1 has better performance than
e = 2, whereas there is a slight overall performance
improvement for e = 2 for 4–6 tugs. The overall best
safe region setting is r = 50.
f
1
(1,0)
f
1
(1,50)
f
1
(1,100)
f
1
(2,0)
f
1
(2,50)
f
1
(2,100)
f
2
(1,0)
f
2
(1,50)
f
2
(1,100)
f
2
(2,0)
f
2
(2,50)
f
2
(2,100)
f
3
(0,50)
f
3
(0,100)
h
1
normalised
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Number of tugs
1 2 3 4 5 6
Figure 5: Mean h
1
for cost functions f
1
(left), f
2
(middle),
and f
3
(right), normalised by the static strategy.
For f
3
, a safe region of r = 100 performs well,
whilst r = 50 performs badly, especially with increas-
ing number of tugs.
Finally, only f
2
(all configurations) and f
3
(r =
100) is able to steadily improve its normalised perfor-
mance with increasing number of tugs, reaching an
improvement of 73–88% compared to the static strat-
egy for any configuration with a tug fleet of six tugs.
4.2 Evaluation Heuristic h
2
Figure 6 shows the same results as that of Figure 4 but
for the evaluation heuristic h
2
, which is a measure of
the sum of squared distances to cross points of unsal-
vageable tankers.
As for h
1
, the size of the tug fleet strongly affects
this evaluation heuristic. With a single tug,
¯
h
2
was in
the range [8.5–9.7]·10
5
, and then decreased with the
number of tugs to the range [3.7–9.5]·10
3
for six tugs.
Generally, for any configuration, the standard de-
viation decreased with the number of tugs, ranging
from 3.3 · 10
3
to 5.1 · 10
5
. The standard error of the
mean was in the range from 82 to 9.3· 10
3
for all con-
figurations, with typically smaller values for smaller
means. The relative standard error was small for all
configurations and in the range [0.0091–0.026].
Due to the large difference in magnitude of
¯
h
2
(due
to the square term in the heuristic) depending on the
number of tugs, we have plotted
¯
h
2
on a logarithmic
scale to enhance readability. The results are similar to
those for evaluation heuristic h
1
, with the same rela-
tionships between the various cost functions and con-
figurations. The exception is f
3
when employed with
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
280
static
max(f
1
)
max(f
2
)
max(f
3
)
min(f
1
)
min(f
2
)
min(f
3
)
h
2
(log scale)
10
4
10
5
10
6
Number of tugs
1 2 3 4 5 6
Figure 6: Maximum and minimum mean h
2
evaluated for
1–6 tugs and cost functions f
1
– f
3
and the static strategy.
1–4 tugs, for which its relative performance compared
to the other cost functions is worse than when evalu-
ated with h
1
.
4.2.1 Comparison with Static Strategy
Figure 7 shows the same results as that of Figure 5 but
for the evaluation heuristic h
2
.
f
1
(1,0)
f
1
(1,50)
f
1
(1,100)
f
1
(2,0)
f
1
(2,50)
f
1
(2,100)
f
2
(1,0)
f
2
(1,50)
f
2
(1,100)
f
2
(2,0)
f
2
(2,50)
f
2
(2,100)
f
3
(0,50)
f
3
(0,100)
h
1
normalised
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Number of tugs
1 2 3 4 5 6
Figure 7: Mean h
2
for cost functions f
1
(left), f
2
(middle),
and f
3
(right), normalised by the static strategy.
For both cost functions f
1
and f
2
and any number
of tugs, with the exception of a safe region of r = 50, a
squared power setting of e = 2 has better performance
than e = 1, especially for 4–6 tugs.
For f
3
, a safe region of r = 50 performs better for
1–2 tugs, whilst letting r = 100 is better for 3–6 tugs.
Finally, only f
2
(all configurations except
f
2
(1, 100)) is able to steadily improve its normalised
performance with an increasing number of tugs,
reaching an improvement of 65–69% compared to the
static strategy for any configuration with a tug fleet of
six tugs.
5 DISCUSSION
Both evaluation heuristics are able to quantify the per-
formance of TFO algorithms designed to solve the
TFO problem as defined in this paper. The small
standard error for both heuristics is obtained by em-
ploying a large number of simulation scenarios (total
1,600) and means that the uncertainty in the means of
both h
1
and h
2
is small. This is important in order to
reliably measure the performance of TFO algorithms.
The general effect of increasing the number of
oil tankers is that a static strategy will become in-
creasingly suitable, whereas a dynamic scheme such
as the RHGA configurations tested here will become
less important. Thus, it is very impressive that cost
function f
2
is able to increase its performance rela-
tive to the static strategy as measured by both eval-
uation heuristics, even with five or six tugs, and for
all its configurations. The evaluation heuristics show
that the RHGA configured with cost function f
2
has
the best overall performance, with most configura-
tions outperforming the other cost functions. The best
choice of safe region for f
2
was r = 50, whereas the
best power setting was e = 1 for h
1
and e = 2 for h
2
,
for for 1–3 or 1–4 tugs, respectively. With these con-
figurations, f
2
was able to outperform the static strat-
egy even with one less tug.
Cost function f
3
also performs well if r = 100,
and is comparable with f
2
when evaluated by h
1
and
to a lesser extent when evaluated by h
2
. However, if
r = 50, f
3
is the worst of all the RHGA configurations.
The similarities between cost function f
3
and h
1
in
measuring the number of tankers are salvageable or
not is probably what makes f
3
perform better for h
1
than for h
2
.
Cost function f
1
is similar to but consistently
worse than f
2
and and the best configuration of f
3
and
should be rejected. We propose that this is a direct re-
sult of its flaw that we have documented previously
(Bye and Schaathun, 2014).
EvaluationHeuristicsforTugFleetOptimisationAlgorithms-AComputationalSimulationStudyofaRecedingHorizon
GeneticAlgorithm
281
5.1 Future Work
The main hurdle before our RHGA can be used in
real-world systems is to test and verify it under real-
istic conditions. This includes considering historical
data of oil tanker traffic, realistic estimates of the vari-
able maximum tug speeds attainable under various
conditions, realistic drift trajectories and cross point
distributions, downtime of tugs due to secondary mis-
sions or change of crew, and so on. It may also be
necessary to extend the algorithm to 2D, in particu-
lar high risk scenarios where oil tankers enter or leave
port and therefore are much closer to land than when
sailing along the TSS corridor. Although challenging,
we do welcome the prospect of TFO algorithms be-
ing adopted as decision-support tools for VTS centres
around the world.
ACKNOWLEDGEMENTS
The DRAMA research group is grateful for the sup-
port provided by Regionalt Forskningsfond Midt-
Norge and the Research Council of Norway through
the project Dynamic Resource Allocation with Mar-
itime Application (DRAMA), grant no. ES504913.
REFERENCES
Assimizele, B., Oppen, J., and Bye, R. T. (2013). A sustain-
able model for optimal dynamic allocation of patrol
tugs to oil tankers. In Proceedings of the 27th Euro-
pean Conference on Modelling and Simulation, pages
801–807.
Breivik, Ø. and Allen, A. (2008). An operational search and
rescue model for the Norwegian Sea and the North
Sea. Journal of Marine Systems, 69(1–2):99–113.
Breivik, Ø., Allen, A., Maisondieu, C., and Roth, J. (2011).
Wind-induced drift of objects at sea: The leeway field
method. Applied Ocean Research, 33(2):100–109.
Bye, R. T. (2012). A receding horizon genetic algorithm for
dynamic resource allocation: A case study on optimal
positioning of tugs. Series: Studies in Computational
Intelligence, 399:131–147.
Bye, R. T. and Schaathun, H. G. (2014). An improved re-
ceding horizon genetic algorithm for the tug fleet op-
timisation problem. In Proceedings of the 28th Euro-
pean Conference on Modelling and Simulation, pages
682–690.
Bye, R. T., van Albada, S. B., and Yndestad, H. (2010).
A receding horizon genetic algorithm for dynamic
multi-target assignment and tracking: A case study
on the optimal positioning of tug vessels along the
northern Norwegian coast. In Proceedings of the In-
ternational Conference on Evolutionary Computation
(ICEC 2010) – part of the International Joint Con-
ference on Computational Intelligence (IJCCI 2010),
pages 114–125.
Det Norske Veritas (2009). Rapport Nr. 2009-1016. Re-
visjon Nr. 01. Tiltaksanalyse — Fartsgrenser for skip
som opererer i norske farvann. Technical report,
Sjøfartsdirektoratet.
Eide, M. S., Endresen, Ø., Breivik, Ø., Brude, O. W.,
Ellingsen, I. H., Røang, K., Hauge, J., and Brett, P. O.
(2007a). Prevention of oil spill from shipping by mod-
elling of dynamic risk. Marine Pollution Bulletin,
54:1619–1633.
Eide, M. S., Endresen, Ø., Brett, P. O., Ervik, J. L., and
Røang, K. (2007b). Intelligent ship traffic monitoring
for oil spill prevention: Risk based decision support
building on AIS. Marine Pollution Bulletin, 54:145–
148.
Goldberg, D. E. (1989). Genetic Algorithms in Search, Op-
timization, and Machine Learning. Addison-Wesley
Professional.
Goodwin, G. C., Graebe, S. F., and Salgado, M. E. (2001).
Control System Design. Prentice Hall, New Jersey.
Hackett, B., Breivik, Ø., and Wettre, C. (2006). Forecast-
ing the Drift of Objects and Substances in the Ocean.
In Ocean Weather Forecasting: An Integrated View of
Oceanography, pages 507–523. Springer.
Haupt, R. L. and Haupt, S. E. (2004). Practical Genetic
Algorithms. Wiley, 2nd edition.
Havforskningsinstituttet (2010). Fisken og havet, sær-
nummer 1a-2010: Det faglige grunnlaget for opp-
dateringen av forvaltningsplanen for Barentshavet og
havomr
˚
adene utenfor Lofoten. Technical report, Insti-
tute of Marine Research (Havforskningsinstituttet).
Holland, J. H. (1975). Adaptation in natural and artificial
systems: An introductory analysis with applications to
biology, control, and artificial intelligence. University
of Michigan Press, Oxford, England.
Maciejowski, J. M. (2002). Predictive Control with Con-
straints. Prentice Hall, first edition.
Rossiter, J. A. (2004). Model-based Predictive Control.
CRC Press.
Sørg
˚
ard, E. and Vada, T. (1998). Observations and mod-
elling of drifting ships. Technical report, Det Norske
Veritas Research, Høvik Norway.
Vardø VTS (2011). The Vardø Vessel Traffic Ser-
vice – For increased safety at sea. Information
pamphlet accessed at http://kystverket.no/Om-
Kystverket/Brosjyrer-skjema-og-andre-
publikasjonar/Brosjyrer2/Brosjyre-om-Vardo-VTS/.
Vardø VTS (2014a). Annual report on petroleum incidents.
Technical report, Norwegian Coastal Administration.
Vardø VTS (2014b). Annual report on petroleum transport
statistics. Technical report, Norwegian Coastal Ad-
ministration.
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
282
