Minimising Longest Path Length in Communication
Satellite Payloads via Metaheuristics
Apostolos Stathakis
Grégoire Danoy
Julien Schleich
Interdisciplinary Centre for
Security, Reliability and Trust
University of Luxembourg
Luxembourg
CSC Research Unit
University of Luxembourg
Luxembourg
CSC Research Unit
University of Luxembourg
Luxembourg
[email protected]
[email protected]
[email protected]
Pascal Bouvry
Gianluigi Morelli
CSC Research Unit
University of Luxembourg
Luxembourg
SES Engineering
Betzdorf
Luxembourg
[email protected]
[email protected]
ABSTRACT
1. INTRODUCTION
The payload system of a telecommunication satellite is in
charge of receiving and filtering the uplink signals, applying
frequency conversion, signal amplification and retransmitting those signals on the downlink. In order to fulfil these
missions, satellite payloads are composed of various components, such as amplifiers and channel filters, dedicated to the
retransmission of signals. These components are interconnected via a large set of switches in charge of relaying a signal
from a component to another. These sets of switches, also
known as switch matrices, can be configured, i.e., the position of each switch can be changed remotely. By doing so,
the topology of the potential paths changes, i.e., new links
are activated while some others are not accessible anymore.
This functionality allows the service provider to choose an
initial configuration for the routing of the signals, and it also
permits to reconfigure the payload depending on new communication needs and on mechanical or electronic failures.
Categories and Subject Descriptors
To meet the modern requirements, payloads are becomG.1.6 [Numerical Analysis]: Optimization; I.2.8 [Artificial
ing bigger, i.e., more switches, more channels and amplifiers.
Intelligence]: Problem Solving, Control Methods, and Search— Consequently, the current operational process consisting of
Heuristic methods; D.2.8 [Software engineering]: Metmanual configurations with experienced engineers is quickly
rics—Performance measures
reaching its limits. Indeed, in this context, the manual approach has become time consuming, difficult, and prone to
error. As a consequence, software-based solutions are needed
Keywords
to find optimal or near-optimal initial configurations and reMetaheuristics; Communication satellite; Payload optimisaconfigurations.
tion
Commercial software solutions already exist for the optimisation of satellite payloads configuration. However, they
suffer from various drawbacks from the point of view of the
satellite operator. First, their closed APIs do not allow efficient integrations in company workflows. Second, they do
not propose to incorporate custom satellite operator constraints or objectives and thus they are characterised by
their lack of flexibility.
Permission to make digital or hard copies of all or part of this work for
For all these reasons, in a previous work, Stathakis et al.
personal or classroom use is granted without fee provided that copies are
proposed to model real-world requirements with the Integer
not made or distributed for profit or commercial advantage and that copies
Linear Programming (ILP) methodology. This model was
bear this notice and the full citation on the first page. To copy otherwise, to
validated on real-world instances of satellite payloads with
republish, to post on servers or to redistribute to lists, requires prior specific
the help of the commercial solver CPLEX [1]. Although
permission and/or a fee.
the results were satisfying for small instances, this approach
GECCO’13, July 6-10, 2013, Amsterdam, The Netherlands.
The size and complexity of communication satellite payloads
have been increasing very quickly over the last years and
their configuration / reconfiguration have become very difficult problems. In this work, we propose to compare the efficiency of three well-known metaheuristic methods to solve
an initial configuration problem, which objective is to minimise the length of the longest channel path. Experiments
are conducted on real-world problem instances with realistic operational constraints (e.g., a maximum computation
time of 10 minutes) and Wilcoxon test is used to determine
with statistical confidence what technique is more suitable
and what are its limitations. The results of this work will
serve as an initial step in our research to design hybrid approaches to push even further the solving capabilities, i.e.,
tackling bigger payloads and more channels to activate.
Copyright 2013 ACM 978-1-4503-1963-8/13/07 ...$15.00.
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all feasible solutions in current and future communication
satellites payloads with a much larger number of switches,
channels and amplifiers might be time consuming or even
intractable. In addition, the set of all feasible solutions can
be very large and will contain a lot of solutions that would
never be chosen by the engineers due to their bad quality.
This would generate huge losses of time and as a consequence, it is more suitable to find the best solutions based
on well-defined objective functions.
In this vein, Stathakis et al. [9] proposed and validated
an Integer Linear Programming optimisation model (ILP)
for large payloads. This model allows the optimisation of
specific objectives like the number of switch changes or the
number of switches used in the new configuration. Besides,
it is flexible as it allows to deal with different failures or
switch types. In [10], the authors proposed and validated
an extension of their mathematical model focusing on the
minimisation of the number of channel interruptions for the
reconfiguration problem. However, the experimental results
obtained in these two papers [9, 10] showed that not all instances could be solved exactly within a realistic operational
time constraint of 10 minutes. This is mainly due to the intrinsic nature of the considered exact approach that returns
the optimal solution when found and nothing if it could not
make it on time.
To overcome this limitation, in this work, we propose to
apply and compare different population based metaheuristics in order to generate valid solutions within the same 10
minutes constraint. First, we develop a method to use metaheuristics for the telecommunication satellites payload configuration problem, that can be adapted for any objective
that interests the satellite operator, and in a second time we
evaluate their performances within a strict time limit focusing on minimising the length of the longest channel path.
did not scale for bigger instances with numerous channels to
activate.
As a consequence, in this paper we propose to relax the optimality constraint by solving various sizes of instances with
well-known metaheuristic methods: a simple Genetic Algorithm (sGA), a cellular GA (cGA) and the Particle Swarm
Optimisation (PSO) technique. Our main objective is to determine which method is the most suitable to efficiently solve
our problem within a 10 minutes operational constraint.
The remainder of this paper is organised as follows. The
next section provides a state of the art on commercial and
academic techniques to solve the considered problem. Section 3 proposes a formalisation of the payload optimisation problem and Section 4 provides details concerning the
methodology. Section 5 introduces and motivates the parameters of the experiments while Section 6 summarises and
analyses the obtained numerical results. Finally, the last
section concludes our work and provides some perspectives.
2.
RELATED WORKS
The satellites communications market is increasing and
rapidly evolving. Satellite operators aim, apart from providing the maximum possible capacity, to reach certain levels of
flexibility in satellites, in order to meet modern requirements
and adapt to market evolutions. Different levels of flexibility on communication satellites have been categorised in [4].
They concern among others flexibility at the orbit level (allowing the same mission to be achieved from different orbit
locations), flexibility at the frequency plan (allowing to modify number of channels and their bandwidth), and flexibility
at the routing level.
At this latter level, channel routing flexibility and redundancy are provided by large switch matrices used in todays satellites. As a consequence though, the increasing
size makes the manual management of payload configuration
and reconfiguration a difficult and time-consuming process
for engineers. Commercial software packages exist to deal
with this problem, like SmartRings [6] and TRECS [11].
Details concerning their algorithms and the models they use
however are not accessible due to commercial restrictions.
SmartRings uses a recursive search to compute all possible
payload configurations. The algorithm is controlled with the
use of constraint and optimisation parameters like the number of switches used or the number of interrupted paths. In
the concern of TRECS, the software finds all feasible solutions, while rejecting the non-feasible ones, and generates
the corresponding set of satellite commands for the engineers to reconfigure the payload. The main drawbacks of
these commercial tools though, is the lack of flexibility for
new potential operational constraints and objectives. Besides, they use a black-box solver that can not be changed
or customised based on the problem that has to be solved,
and most importantly they cannot be easily integrated in
companies workflows.
On the academic side, few works have been proposed for
the considered problem. In [7] a recursive algorithm was
proposed to perform a breadth-first-search (BFS) in order
to find all feasible paths that connect channels to amplifiers.
Experiments showed the efficiency of the proposed method
on a small switch network. For larger problems the BFS algorithm is limited due to its time complexity as every vertex
and every edge will be explored in the worst case.
However, since this operation is time critical, enumerating
3. PROBLEM DESCRIPTION
The general problem of payload reconfiguration can be
divided in three related subproblems. The first one is the
initial configuration problem, which consists in finding an
optimal payload configuration for connecting an initial set
of channels in an empty payload (i.e., without any preconnected channel path that carries service). The second
case is the reconfiguration problem, which occurs when there
exists a set of pre-connected channel paths carrying services
and some additional channels must be activated. The third
problem is the restoration problem, which arises when a set
of channels is already connected and one or more failures
occur in amplifiers or switches. In this case, the channels affected by the failures have to be rerouted through a different
path.
All the aforementioned subproblems have some common
operational objectives which are important for payload engineers. For example, the number of switch changes has
to be minimised, as changing the position of a switch may
cause its permanent failure. Interruptions on the established
channel paths have to be minimised since they impact the
quality of service provided to customers. Finally, the length
of the longest channel path is another important objective
that has to be minimised. Long paths imply high attenuation and may cause restrictions on future reconfiguration
process. In this work we deal with the initial configuration
problem and we focus on minimising the longest path length.
It is an objective of high interest for the satellite operators
1366
Input
switch matrix
Input
channels
Input
channels
Amplifiers
Initial Payload
Input
switch matrix
Amplifiers
Solution example
Figure 1: Simplified initial payload instance and example of a solution that connects channels 1 and 3 to
amplifiers 2 and 4
Position 1
Position 2
Position 3
Position 4
This section presents a detailed description of the three
metaheuristics in sections 4.1 to 4.3. The solution encoding
is then described in section 4.4 followed by details on the
fitness function and its evaluation in section 4.5.
4.1 Generational GA
Generational GAs (genGAs) are a type of panmictic algorithm, i.e., individuals are grouped into a single structureless population also referred to as panmixia. Individuals can
thus mate with any other individual in the population. The
genGA is a (μ, λ)-GA, where the newly generated individuals are placed in an auxiliary population which will replace
the current population when it is completely filled, i.e., when
the number of newly generated solutions is equal to the size
of this auxiliary population. In our case, the size of both the
auxiliary and the current population is the same (μ = λ).
Figure 2: Positions of the R-type switch
and as can be seen in the numerical results section is a difficult objective to be solved with an exact approach within
the strict operational time limit.
More precisely, given a set of channels to connect, the
initial payload configuration problem consists in finding the
positions of the switches that permit to establish the path
from each channel to its amplifier. A simplified payload
switch matrix example with 16 switches of R and C type is
shown in Fig.1. The input signals are crossing the switch
matrix and are guided for amplification to an appropriate
amplifier. The routing of signals in the payload is achieved
through reconfigurable switches organised in the switch matrix. Different types of switches may be used and each one
has different positions allowing different paths. The four
possible positions of an R-type switch are shown in Fig.2,
whereas the C-type switch has only 2 possible positions. For
example, one solution for connecting channel 1 to amplifier
2 and channel 3 to amplifier 4 is shown in Fig.1. In this case,
channel 1 follows a path of 7 switch crossings and channel
3 follows a path with 6 switch crossings, i.e., number of
switches used in the path.
4.
Algorithm 1 Pseudo-code of a canonical genGA
1: proc Evolve(genga)
// Parameters of the algorithm in
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
‘genga’
GenerateInitialPopulation(genga.pop);
Evaluation(genga.pop);
while ! StopCondition() do
for i←0 to genga.popSize do
parents ← Selection(genga.pop);
offspring ← Recombination(genga.Pc,parents);
offspring ← Mutation(genga.Pm,offspring);
Evaluation(offspring);
Add(auxiliary pop,offspring);
end for
genga.pop ← auxiliary pop;
end while
end proc Evolve
Algorithm 1 presents the pseudo-code of the generational
EA. The population is first randomly initialised (line 2).
Each generated individual is then evaluated using the fitness function defined for the tackled problem (line 3). The
genetic loop then starts in line 4 until some predefined termination condition is met, e.g., a number of fitness function
evaluations. “Parent” individuals are selected using some
stochastic selection operator to construct the mating pool
(line 6). Genetic variation is then ensured by the crossover
(also called recombination) and mutation operators, both
applied with some probability, which permit to visit other
METHODOLOGY
In order to overcome the limitations encountered with the
exact approach, in this work we propose to study the performance of three different population-based metaheuristics,
i.e., a panmictic GA, a structured GA and a swarm-based
optimisation algorithm.
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Selection
4.3 Particle Swarm Optimisation
Crossover
Particle swarm optimisation [8] is a population based metaheuristic based on the social behaviors observed in nature
(e.g., fish shoal). Solutions are represented as swarms that
evolve and look for a good solution to the problem without central authority. Each swarm evolution is conditioned
by the swarm’s best position (pbest) and its neighbors best
position (lbest).
A pseudo-code of a PSO is presented in Algorithm 3. The
swarm, i.e., population of particles, is initialized randomly
(line 2) and evaluated (line 3). Then each particle’s new
velocity v(t) is calculated (line 6) based on the two social
factors using the following equation:
Mutation
Replacement
Figure 3: cGA reproduction cycle with 5 × 5 population and L5 neighborhood.
v(t) = ω × v(t − 1) + c1 × ρ1 × (pbest − x(t − 1))
+ c2 × ρ2 × (lbest − x(t − 1)),
search space regions (line 7 and 8). The obtained offspring
is then evaluated and inserted into the auxiliary population
(line 9 and 10). The offspring population will become the
current one once full (line 12).
(1)
with ω the inertia weight, c1 and c2 the learning factors,
ρ1 and ρ2 two random variables in [0,1] . The position of
the particle is updated (line 7) using x(t) = x(t − 1) + v(t)
and its new fitness calculated (line 8). Each particle then
updates it lbest (line 9) and the lbest is updated once the
whole swarm has moved (line 11).
Originally introduced for continuous optimisation, PSO
variants were proposed to tackle combinatorial problems,
like the one considered in this work. In our case a particle is
associated with an n-dimensional binary vector (cf. section
4.4), we thus used a binary PSO. To update the position of
the particle a sigmoid function Sig(vid ) is used to transform
the velocities values vi into the [0, 1] interval, defined as:
4.2 Cellular GA
Contrary to panmictic GAs like the generational GA, the
cellular genetic algorithm (cGA) [3] uses a structured population. Individuals are spread in a two dimensional toroidal
mesh and are only allowed to interact with their neighbors.
An illustration of the cGA breeding loop is presented in Fig.
3 with individuals arranged on a 5X5 toroidal grid and the
neighborhood of the center individual, linear 5, is presented
as dashed lines.
A canonical cGA follows the pseudo-code presented in Algorithm 2. The population is usually structured in a regular
grid of d dimensions (d = 1, 2, 3), and a neighborhood is
defined on it. The algorithm iteratively considers as current
each individual in the grid (line 5), and individuals may only
interact with individuals belonging to their neighborhood
(line 6), so parents are chosen among the neighbors (line 7)
with a given criterion. Crossover and mutation operators
are applied to the individuals in lines 8 and 9, with probabilities Pc and Pm , respectively. Afterwards, the algorithm
computes the fitness value of the new offspring individual
(or individuals) (line 10), and inserts it (or one of them)
instead of the current individual in the population (line 11)
following a given replacement policy. This loop is repeated
until a termination condition is met (line 4).
Sig(vid ) =
1
1 + exp(−vid )
(2)
where vid is the velocity for particle i at dimension d. A
random number is generated in the same range and if the
generated number is less than Sig(vid ) the position of the
particle at the corresponding dimension is set to 1, otherwise
it is set to 0.
Algorithm 3 Pseudocode for a PSO
1: proc Evolve(pso)
2: GenerateInitialSwarm(swarm);
3: Evaluation(swarm);
4: while ! StopCondition() do
5:
for particle ← 1 to swarm.size do
6:
velocity ← UpdateVelocity(particle)
7:
particle ← UpdatePosition(particle)
8:
Evaluation(particle)
9:
UpdatepBest()
10:
end for
11:
UpdategBest()
12: end while
13: end proc Steps Up;
Algorithm 2 Pseudocode for a canonical cGA
1: proc Evolve(cga)
//Algorithm parameters in ‘cga’
2: GenerateInitialPopulation(cga.pop);
3: Evaluation(cga.pop);
4: while ! StopCondition() do
5:
for individual ← 1 to cga.popSize do
6:
n list←Get Neighborhood(cga,position(individual));
7:
parents←Selection(n list);
8:
offspring←Recombination(cga.Pc,parents);
9:
offspring←Mutation(cga.Pm,offspring);
10:
Evaluation(offspring);
11:
Add(position(individual),offspring,cga);
12:
end for
13: end while
14: end proc Steps Up;
4.4 Solution Encoding
When using such metaheuristics, the selection of solution
encoding is important as it drives the choice of the operators
and therefore the search space exploration. In the proposed
model of the payload configuration problem, a solution is the
set of each switch position. We use a binary encoding where
a switch position is encoded using two bits since a switch has
4 possible positions (see Fig. 2). The binary vector is thus
of size 2 ∗ n with n the number of switches in the payload.
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Payload switch matrix
Corresponding graph
1
1
Key
Switch node
2
2
Channel node
Amplifier node
3
3
Figure 5: Example of payload representation as a graph
Chromosome / Particle
00
10
10
vertex then the path is valid. A solution is thus valid if it exists such a connected component for each selected channel.
Due to the payload design, a valid connected component can
only represent a unique path from channel to amplifier. Its
length is thus equal to its total number of switches.
A simple example of the payload transformation into a
graph is presented in Figure 5. In the original payload
channel 1 is connected to amplifier 3, therefore in the corresponding graph it exists a connected component that contains channel 1 and amplifier 3 vertices. This path uses five
switches, its length is thus five.
11
sw1
sw3
sw2
sw4
5. EXPERIMENTAL SETUP
Figure 4: Solution encoding example.
In this section we provide details on the three metaheursitics configuration and on the payload problem instances
we have considered in our experiments.
A simple solution example with 4 switches is presented in
Fig. 4, where the first two bits (circled) encode the position
of switch 1 (sw1 ), i.e., “00” corresponds to position 1. Then
sw2 is in position 2 with “10”, etc.
5.1 Algorithms Parameters
Table 1 presents the parameters used for the GAs and the
PSO. All algorithms have a total of 49 individuals/particles
and a termination condition of 10 min, which is an operational constraint from the satellite operator. Two-point
crossover with probability pc =0.8, bit-flip mutation with
pm = chrom1length and an elitism strategy have been used.
The two parent individuals are selected using binary tournament for the genGA while one parent is the current individual for the cGA. The neighbourhood of the cGA is L5 (5
closest individuals measured in Manhattan distance). For
the PSO, the neighbourhood size is 2 and the inertia weight
linearly decreases in a range from 0.9 to 0.4. Finally the two
learning factors have an equal importance, i.e., c1 = c2 =
2. The three algorithms have been implemented using the
ParadisEO framework v1.3 [5].
The mathematical model used in the exact approach is
a variant of the ILP model presented in [9] adapted to the
minimisation of the length of the longest channel path. One
of the fastest non-commercial mixed integer programming
(MIP) solver, SCIP (Solving Constraint Integer Programs)
[2], was used.
4.5 Objective Function
The objective of this work is to find the best set of switch
positions that (1) creates a path between each selected channel and an amplifier and (2) minimises the size of the longest
created path. In order to assign a fitness value to the candidate solutions, we propose the following objective function:
lpl
(3)
100
where rc is the number of selected channels that have not
been connected to an amplifier and lpl is the longest path
length, i.e., the number of switches used in this path. The
real length of the path could not be used because this information is not made available and it cannot be deduced
from the payload schemas since they do not reflect the real
hardware design. Dividing lpl by 100 (the maximum switch
matrix size we consider is 100) permits to penalise unsatisfactory solutions proportionally to the number of channels
that remain to be connected. Using F we thus consider a
minimisation problem.
In order to evaluate the fitness F of a solution, we create
its corresponding payload graph. In this graph, each switch
is represented as four vertices, one per switch ports, and
edges between these vertices are created depending on the
switch position. Edges between switches are statically defined. Edges between the nodes of a switch are defined based
on its position. A unique partitioned graph is thus created
per possible combination of positions. If a connected component of the graph includes a channel vertex and an amplifier
F = rc +
5.2 Problem Instances Parameters
Experiments were performed on payload instances of 50
and 100 switches. For the small size payload, we have considered four different instance sizes from 8, 13, 18 and up to
the maximum of 23 channels to connect. The total number
of amplifiers is also 23. For the large size payload, we have
considered three instances with 25, 30 up to the maximum
of 35 channels to connect. The total number of amplifiers is
also 35. For each payload instance size, 30 different sets of
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Table 2: Comparison with exact method - hit-rate
Channels Exact
sGA
cGA
PSO
8
90 %
100 %
100 %
99.444 %
13
26.6 % 99.888 % 100 %
6.55 %
18
10 %
99.888 % 100 %
0%
23
0%
83.333 % 86.111 % 0 %
Population/Swarm size
Term. condition
49, 7 × 7 (cGA)
10 minutes
Selection
genGA, cGA
Neighborhood
Crossover
Mutation
Binary tournament (BT),
Current indiv. + BT (cGA)
L5 (cGA)
DPX, pc =0.8
bit flip, pm = chrom1length
Replacement strategy
Elitism
Replace if better (cGA)
1 individual
PSO
Table 1: Parameters used for the genGA, cGA and
PSO
Neighborhood size
Inertia weight
Learning factors
2
Linear decrease [0.9, 0.4]
c1 = c2 = 2.0
When considering only the best runs, we can notice that in
some cases, the metaheuristics have found all the optimal
solutions.
These results comforts us in our choice to use metaheuristics to cope with our problem, as we obtain a huge increase
of the hit-rate while the degradation of the fitness is almost
negligible compared to the exact approach.
channels to connect have been randomly generated. Indeed,
due to the payload design, two random sets of channels of
the same size will have a different complexity and solution
quality.
6.
6.2 Small Instances
The small instances consist of payloads composed of 50
interconnected switches. Intuitively, the complexity of the
problem increases with the number of channels to activate.
Indeed, if there are less channels to activate there should
be more flexibility in the payload matrix to reach amplifiers
without blocking other signals. Results displayed in Table 4
empirically confirm this trend. Indeed, all three techniques
are decreasing in their capability to find valid solutions (hitrate) in the given 10 minutes limits when the number of
channels to activate increases.
PSO is undoubtedly not adapted to this problem as its
hit-rate results decreases dramatically quick to finally reach
0 % for 23 channels to activate. On the contrary, the other
two approaches are performing well with slight performance
differences. Indeed, although cGA is able to find more valid
solutions than sGA, the difference between both methods
is quite small, i.e., 99.88 % (sGA) vs 100.0 % (cGA) for
18 channels and 83.0 % (sGA) vs 88.22 % (cGA) for 23
channels.
The fitness (see Table 4) results are similar to what can
be observed with the hit-rate. Indeed, cGA is always providing the best results, sometimes ex-æquo with the other
techniques. This is true for the average and the standard
deviation, and as a consequence comfort us in the idea that
cGA is the current best solution for the small instances.
Moreover, when the number of channels increases, PSO obtains very bad results very quickly. We also observe a strong
correlation between the complexity of the problem and the
fitness value. This is logical as the hit-rate is lower for more
complex problems and thus the fitness value is more often
higher than 1 thus mechanically increasing the average fitness value.
In order to obtain statistical confidence, we apply the
Wilcoxon test to all pairs of methods, i.e., sGA vs cGA,
sGA vs PSO and cGA vs PSO. Although cGA obtained better results than sGA, this was not detected as statistically
significant. As a consequence, we focused solely on representing ranking relations between the couple sGA, cGA on
one side and PSO on the other side. Wilcoxon test unanimously states that PSO proposes significantly worse results
(with 95% confidence) than both sGA and cGA for 13, 18
and 23 channels to activate, which is represented in Table 4
with a light grey background for PSO results.
NUMERICAL RESULTS
In this section, we review the performances of the different metaheuristics we used with the parameters detailed
in Section 5 to solve our problem. Statistical confidence in
our comparisons is assessed by performing the Wilcoxon test
[12]. In a first part, we compare the results obtained by the
exact approach to the chosen metaheuristics, i.e., sGA, cGA
and PSO. The second part reviews the results obtained for
the three tested methods on the small instances of payloads,
i.e., 50 switches. The third subsection compares the numerical results for the big instances and presents the limits
of all the different techniques. Finally, the last subsection
proposes a summary of the aggregated experience for our
problem.
6.1 Comparison with Exact Methods
The hit-rate of our exact approach, i.e., the percentage
of solved instances in 10 minutes, decreases extremely fast
with the number of channels to connect (see Table 2). These
results have been obtained for the small payload instances
(50 switches). Very encouraging results have been obtained
for sGA and cGA as their hit-rate remains very high. In the
case of metaheuristics, hit-rate stands for the percentage
of instances where all channels have been connected. PSO
exhibits very poor performances and thus cannot be considered a useful alternative to exact methods for this particular
problem. It seems that the exact solver was able to cut efficiently in the solutions space in order to find the optimal
solution very quickly while PSO’s exploration process got
stuck on invalid local optimum.
As previously mentioned, sGA and cGA are characterised
by very high hit-rate but the quality of their solutions needs
to be also checked with respect to those of the exact approach. To this end, we propose Table 3. This table is summarising results obtained only for the instances that have
been solved by the exact approach. For these particular
instances, Table 3 considers both the average of the 30 individual runs of each solved instance and the best runs of
each instance. The results of both sGA and cGA are quite
promising, as both the average and the best runs are very
close to the optimal values obtained by the exact approach.
1370
Avg
Best
Channels
8
13
18
8
13
18
#S
50
100
#Ch
8
13
18
23
25
30
35
Table 3: Comparison with exact
Exact
sGA
0.001892 ±0.000566 0.001938 ±0.000643
0.002500 ±0.000534 0.002600 ±0.000658
0.002660 ±0.000577 0.003160 ±0.000851
0.001892 ±0.000566 0.001923 ±0.000627
0.002500 ±0.000534 0.002500 ±0.000534
0.002660 ±0.000577 0.003000 ±0.001
method - fitness
cGA
0.001932 ±0.000634
0.002562 ±0.000603
0.003140 ±0.000954
0.001923 ±0.000627
0.002500 ±0.000534
0.003000 ±0.001
Table 4: Fitness and hit-rate results
sGA
cGA
fitness
hit-rate
fitness
hit-rate
0.0019 ±0.0005 100.0 % 0.0019 ±0.0005 100.0 %
0.0032 ±0.0006 100.0 % 0.0031 ±0.0006 100.0 %
0.0040 ±0.0333 99.88 % 0.0030 ±0.0008 100.0 %
0.1230 ±0.3221 88.22 %
0.1750 ±0.3754 83.0 %
0.068 ±0.2613
94.11 % 0.072 ±0.2627
93.66 %
0.332 ±0.6003
73.77 % 0.298 ±0.5932
77.33 %
3.666 % 2.130 ±1.0262
3.777 %
2.251 ±1.0475
6.3 Big Instances
PSO
0.001924 ±0.000618
0.048400 ±0.209
1.569000 ±0.670
0.001923 ±0.000627
0.002500 ±0.000534
0.337000 ±0.578
PSO
fitness
0.0019 ±0.0005
0.2830 ±0.4514
1.8830 ±0.8752
4.4070 ±0.7108
-
hit-rate
100.0 %
72.11 %
5.0 %
0.0 %
-
channels need to be activated, cGA is this time proposing
significantly better solutions than sGA (represented with a
dark grey background for cGA). Finally, for the biggest instance, i.e., 35 channels to activate, even if cGA provided
better results, the Wilcoxon test cannot state if a technique
is significantly better than the other, thus no specific background.
The big instances consist of payloads composed of 100
interconnected switches. For these instances, we chose to
focus on the most promising methods and thus discarded
PSO. As a consequence, we focus on comparing sGA with
cGA. Detailed results can be found in Table 4. The hit-rate
is again highly influenced by the complexity of the problem,
which is here mainly related to the number of channels to
activate. For the big instances, the hit-rate is relatively high
except for the extreme case of 35 channels where a bit less
than 4 % of the experiments were successful. This indicates
that we are very close to the maximum size of problem that
can be dealt within the 10 min constraint. In the bigger instances, cGA is not the best technique for all problem sizes.
Indeed, when 25 channels should be activated, sGA finds
valid solutions for 94.1 % of the payloads while cGA solves
only 93.66 % of them. However, when the problems become
more complex to solve, cGA obtains better results concerning the hit-rate, i.e., 77.33 % vs 73.77 % for 30 channels and
3.777 % vs 3.666 % for 35 channels.
This highly increasing complexity can also be observed in
the quality of the solutions via the fitness. Intuitively, the
easier the problem, the more time you have to explore the
solution space and thus, the higher is the quality of the final
best solution. On the contrary, it is highly probable that for
the biggest instances, for which less than 4 % of the payloads
have a valid solutions, the fitness results will be significantly
less good. sGA surprisingly provides better solutions than
cGA for the instances with 25 channels to activate, i.e., 0.068
vs 0.072. These very good results are however not repeated
for the more complex instances. Indeed, cGA obtains better
solutions for 30 resp. 35 switches to activate, i.e., 0.298 vs
0.332 resp. 2.130 vs 2.251.
We applied the Wilcoxon tests for all pairwise comparisons. For 25 channels to activate, cGA provides significantly
less good solutions than sGA. In Table 4, this is represented
with a light grey background for cGA results. When 30
6.4 Summary
These experiments allowed us to obtain valuable insights
on the suitability of the tested techniques for our problem.
We now know that PSO is not suitable for our problem with
the presented parameters settings. We observed that the
time-limit is not responsible for its poor results. Indeed,
preliminary analyses seem to suggest that PSO converges
very quickly and get stuck quite regularly in invalid solutions. It is thus not impossible that fine-tuning some of its
parameters, e.g., those in charge of the learning process, may
drastically change its performances. However, this was not
in the scope of this study and we tried to conduct it in the
fairest way possible by applying standard parameter values
to all of the methods.
Concerning the two other methods, cGA seems to be more
efficient than sGA, for both the average fitness and its standard deviation, and the hit-rate. This could be explained
by the fact that sGA may converge prematurely in some
local optimum whereas in the case of cGA though, this is
prevented thanks to the isolation by distance effect provided
by the use of a structured population. This is of course not
true for 100 switches and 25 channels to activate but that
is the only exception and even in this case the performance
difference between both method is not very important.
7. CONCLUSIONS AND PERSPECTIVES
In this work, we proposed a method to use metaheuristics for minimising the length of the longest channel path
in communication satellites payloads. We compared the ef-
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ficiency of three well-known metaheuristic methods , i.e., a
simple Genetic Algorithm (sGA), a cellular GA (cGA) and
the Particle Swarm Optimisation (PSO) technique, to solve
a specific payload problem, i.e., minimising the longest path
length. We proved the suitability of sGA and cGA compared
to the state of the art exact approach.
Our experimental results clearly disqualify PSO for its
very poor performances for both hit-rate and fitness, although both are correlated. cGA proposes the best solutions
in most of the considered cases and will thus be considered
for further experiments.
As a next step in our research, we plan in a first time
to enhance the performances under the time constraint by
fine-tuning the parameters of cGA and performing a sensibility analysis. If this is still not enough to perform on
the most challenging instances, we will propose hybridisation schemes for cGA, such as including a problem specific
local search mechanism. Finally, for the sake of flexibility,
we plan to work on multi-objective techniques in order to
propose panels of solutions to the deciders of the satellite
company.
Acknowledgement
A. Stathakis acknowledges the support of the National Research Fund of Luxembourg (FNR), with the AFR contract
no. 1346481. Experiments presented in this paper were
carried out using the HPC facility of the University of Luxembourg.
8.
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