A Dynamic Ant Colony Optimization for Solving the Static Frequency
Assignment Problem
KHALED ALRAJHI
King Khaled Military Academy, Riyadh, SAUDI ARABIA
Abstract. This study proposes a dynamic ant colony optimization algorithm to solve the static frequency assignment problem. This
approach solves the static problem by modeling it as a dynamic problem through dividing this static problem into smaller sub-problems,
which are then solved in turn in a dynamic process. Several novel and existing techniques are used to improve the performance of this
algorithm. One of these techniques is applying the concept of a well-known graph colouring algorithm, namely recursive largest first
for each sub-problem. Furthermore, this study compares this algorithm using two visibility definitions. The first definition is based on
the number of feasible frequencies and the second one is based on the degree. Additionally, we compare this algorithm using two trail
definitions. The first one is between requests and frequencies. The second is between requests and requests. This study considers real
and randomly generated benchmark datasets of the static problem and our algorithm achieved competitive results comparing with other
ant colony optimization algorithms in the literature.
Keywords: dynamic ant colony optimization, graph colouring algorithm, frequency assignment problem.
Received: June 26, 2022. Revised: September 7, 2023. Accepted: October 11, 2023. Published: November 2, 2023.
1 Introduction
The frequency assignment problem (FAP) is related to
wireless communication networks, which are used in
many applications such as mobile phones, TV broadcast-
ing and Wi-Fi. The aim of the FAP is to assign frequen-
cies to wireless communication connections (also known
as requests) while satisfying a set of constraints, which
are usually related to prevention of a loss of signal qual-
ity. Note that the FAP is not a single problem. Rather,
there are variants of the FAP that are encountered in prac-
tice. The minimum order FAP (MO-FAP) is the first var-
iant of the FAP that was discussed in the literature, and
was brought to the attention of researchers by [1]. In the
MO-FAP, the aim is to assign frequencies to requests in
such a way that no interference occurs, and the number
of used frequencies is minimized. As the MO-FAP is NP-
complete [2], it is usually solved by meta-heuristics.
Many meta-heuristics have been proposed to solve the
MO-FAP including genetic algorithm (GA) [3], evolu-
tionary search (ES) [4], ant colony optimization (ACO)
[5], simulated annealing (SA) [6] and tabu search (TS)
[6, 7, 8, 9]. It can be seen from literature that there are
relatively few papers concerning the application of ACO
to solve the FAP. However, existing ACO algorithms in
the literature are unable to find a feasible solution in
some instances of the MO-FAP. Hence, this study inves-
tigates whether ACO can be improved to be an effective
solution method for the MO-FAP.
In this study, the dynamic ant colony optimization
(DACO) is mainly designed to solve MO-FAP by mod-
eling it as a dynamic problem through dividing this static
problem into smaller sub-problems, which are then
solved in turn in a dynamic process. Several novel and
existing techniques are used in this study to improve the
performance of DACO. One of these techniques is apply-
ing the concept of a well-known graph colouring algo-
rithm, namely Recursive Largest First (RLF), which was
proposed in [15]. RLF has not been used in ACO for the
static FAP in the literature. Furthermore, this study com-
pares DACO using two visibility definitions (see Section
5.3). The first definition is based on the number of feasi-
ble frequencies, which was previously used in ACO for
the graph colouring problem (GCP) [10]. The second one
is based on the degree, which was previously used in
ACO for the GCP [12]. Additionally, we compare
DACO using two trail definitions (see Section 5.4). The
first one is between requests and frequencies, which was
previously used in ACO for the static FAP [13]. Note that
ACO in [13] decreases the level of trail for bad solutions,
whereas we increase the level of trail for the unassigned
requests for all available frequencies in order to be more
attractive to be selected. This technique was previously
used in ACO for the examination scheduling problem
[11]. The second trail definition considered in this study
is between requests and requests, which was previously
used in ACO for the GCP [12].
This paper is organised as follows: the next section
gives an overview of the static MO-FAP. Section 3 pre-
sents the modeling the static MO-FAP as a dynamic
problem, section 4 shows the graph coloring model for
the static MO-FAP. Section 5 presents the main compo-
nents of our DACO algorithm for the static MO-FAP.
Results of this algorithm are given and discussed in Sec-
tion 6 before this study finishes with conclusions.
2 Overview of the Static MO-FAP
The main concept of the static MO-FAP is assigning a
frequency to each request while satisfying a set of con-
straints and minimizing the number of used frequencies.
International Journal of Applied Sciences & Development
DOI: 10.37394/232029.2023.2.18
Khaled Alrajhi
E-ISSN: 2945-0454
169
Volume 2, 2023
The static MO-FAP can be defined formally as follows:
given
a set of requests , where NR is the
number of requests,
a set of frequencies , where NF
is the number of frequencies,
a set of constraints related to the requests and frequencies
(described below),
the goal is to assign one frequency to each request so that
the given set of constraints are satisfied and the objective
function is minimized, where the objective function is
minimizing the number of used frequencies. Note that the
frequency that is assigned to requests is denoted
asthroughout of this study. The static MO-FAP has
four variants of constraints as follows:
1. Bidirectional Constraints: this type of constraint forms
a link between each pair of requests ,
where . In these constraints, the fre-
quencies and that are assigned to
and, respectively, should be distance apart. In
the datasets considered here, is always equal to a
constant value (238). These constraints can be written
as follows:
for
(1)
2. Interference Constraints: this type of constraint forms
a link between a pair of requests, where the pair
of frequenciesandthat is assigned to the pair
of requests and, respectively, should be more
than distance apart. These constraints can be
written as follows:
for
(2)
3. Domain Constraints: the available frequencies for
each request are denoted by the domain ,
where . Hence, the frequency which is
assigned to must belong to . For the datasets
considered in this study, there are 7 available domains.
4. Pre-assignment Constraints: for certain requests, the
frequencies have already been pre-assigned to given
values i.e. , where is given value.
3 Modeling the Static MO-FAP as
a Dynamic Problem
In this approach, the static MO-FAP is broken down into
smaller sub-problems, each of which is considered at a
specific time period. To achieve this, each request is
given an integer number between 0 and (where is a
positive integer) indicating the time period in which it
becomes known. In effect, the problem is divided into
smaller sub-problems, where n is
the number of sub-problems after the initial sub-problem
. Each sub-problemcontains a subset of requests
which become know at time period . The initial sub-
problem is solved first at time period. After that, the
next sub-problemis considered at time period and
the process continues until all the sub-problems are con-
sidered. In this study, we found that the number of sub-
problems does not impact on the performance of the ap-
proach for solving the static MO-FAP, so the number of
sub-problems is fixed at 21 (i.e. n = 20).
Based on the number of the requests known at time
period 0 (belonging to the initial sub-problem), 10 dif-
ferent versions of a dynamic problem are generated.
These versions are named using percentages which indi-
cate the number of requests known at time period 0.
These 10 versions are named 0%, 10%, 20%, 30%, 40%,
50%, 60%, 70%, 80%, 90% (note that 100% means all
the requests are known at time period 0 and so corre-
sponds to the static MO-FAP).
An example of how a static MO-FAP is modeled as a
dynamic problem is illustrated in Figure 1, where each
node represents a request, each edge a bidirectional or
interference constraint and each color a time period in
which a request becomes known for the first time.
After breaking the static MO-FAP into smaller sub-
problems, these sub-problems will be solved in turn.
Fig. 1. An example of modeling a static MO-FAP as a dy-
namic problem over 3 time periods.
4 Graph Coloring Model for the
Static MO-FAP
The graph coloring problem (GCP) can be viewed as an
underlying model of the static MO-FAP [16]. The GCP
involves allocating a color to each vertex such that no
adjacent vertices are in the same color class and the num-
ber of colors is minimized. The static MO-FAP can be
represented as a GCP by representing each request as a
International Journal of Applied Sciences & Development
DOI: 10.37394/232029.2023.2.18
Khaled Alrajhi
E-ISSN: 2945-0454
170
Volume 2, 2023
vertex and a bidirectional or an interference constraint as
an edge joining the corresponding vertices.
One useful concept of graph theory is the idea of
cliques. A clique in a graph can be defined as a set of
vertices in which each vertex is linked to all other verti-
ces. A maximum clique is the largest among all cliques
in a graph. Vertices in a clique have to be allocated to a
different color in a feasible coloring. Therefore, the size
of the maximum clique acts as a lower bound on the min-
imum number of colors.
As the requests belong to different domains, the graph
coloring model for each domain can be considered sepa-
rately and then a lower bound on the number of frequen-
cies that is required from each domain can be calculated.
An overall lower bound on the total number of frequen-
cies for a whole instance can also be calculated in a sim-
ilar way. A branch and bound algorithm is used to obtain
the set of all maximum cliques for each domain within
each sub-problem.
5 Overview of the Dynamic Ant
Colony Optimization
A key decision when designing DACO is how to choose
the solution space and cost function, request and fre-
quency selection, visibility definitions, trail definitions
and descent method.
5.1 Solution Space and Cost Function
The solution space of DACO is defined as the
set of all possible feasible assignments, that is,
satisfying all of the constraints. The corresponding
cost function is defined as the number of unassigned
requests.
5.2 Request and Frequency Selection
DACO selects a frequency greedily by selecting the
one which can be assigned feasibly to the most requests.
If there is more than one candidate frequency, then one
of them is randomly selected. After that, the frequency
is sequentially feasibly assigned to all possible re-
quests until no more can be feasibly assigned. The order
of selecting requests from among those that are feasible
for is based on probability given by Formula 3.
if
0 oth-
erwise
where is the set of frequencies which can be feasi-
bly assigned by an artificial ant to the request , The vis-
ibility of a request to be assigned a frequency
is defined in Section 3.3 and the trailis defined in
Section 3.4. The parameters control the relative
significance of the pheromone trail against the visi-
bility
After that, a different frequency is selected in the same
way and this process is repeated until all requests are fea-
sibly assigned, if possible. This process is inherited from
a well-known graph colouring algorithm, namely recur-
sive largest first. In fact, applying recursive largest first
aims to improve the performance of selecting frequencies
and requests to be assigned. In contrast, ACO for the
MO-FAP in the literature (see e.g. [13]) frequently se-
lects a request based on probability and then assign it to
a feasible frequency.
5.3 Visibility Definitions
The visibility gives some indication of the desirability
of choosing a request based on the experience of previous
ants. Hence, the visibility of a request acts as a greedy
heuristic. In this study, two types of visibility definition
are applied and compared. These two visibilities are de-
fined as follows:
i) Visibility of a request to be assigned a fre-
quency is based on the number of feasible frequencies
for (), which is given by Formula 4.
(4)
This definition prioritises those requests that have
fewer feasible frequencies. This type of visibility defini-
tion was previously used in ACO for the graph colouring
problem (GCP) [10].
ii) Visibility of a request to be assigned a fre-
quency is based on the degree of (), which is
defined as the numbers of unassigned requests that can-
not be assigned feasibly toand have a common inter-
ference constraint with. This visibility is given by For-
mula 5.
(5)
This visibility looks ahead and prioritises requests that
have more constraints in common with other requests
that cannot be assigned to the frequencies being consid-
ered currently. This visibility definition was previously
used in ACO for the GCP [12].
A request from among those that are feasible for the se-
lected frequencyis selected based on the probability
given by Formula 3 Here, assume that the trail and the
parameters and in Formula 3 are set to one. Then,
(3)
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the probability of selecting a request based on the two
visibility definitions would be calculated as follows:
i) The probability of selecting each request using the
first visibility definition is given in Table 1. Note that the
number of feasible frequencies of each request () is
invented and cannot be deduced from Figure 1.
Table 1. Requests selection based on probability using the first defi-
nition of visibility.
Σ
1
2
3
4
1
1/2
1/3
1/4
25/1
2
0.4
4
0.2
8
0.1
3
0.1
5
1
ii) The probability of selecting each request
using the second visibility definition is given in
Table 2. Note that the degree of each request
() can be deduced from Figure 1.
Table 2. Requests selection based on probability using
the second definition of visibility
Σ
3
0
1
1
+
1
4
1
2
2
9
0.4
4
0.1
1
0.2
2
0.2
2
1
In both cases, once the probabilities have
been calculated, one request is selected proba-
bilistically.
5.4 Trail Definitions
The purpose of the trail within DACO is to provide
information about previous construction solutions to in-
fluence future constructions. In this study, two different
trails are defined, where the initial values of these trails
are set to one. Moreover, evaporation and updating of
these trails are discussed. The definitions of these trails
are given as follows:
i) Trail between requests and frequencies (: the
key component of a solution is to decide to which fre-
quency each request is assigned. Therefore, the most ob-
vious trail definition is between each request and each
frequency, which is also previously used in ACO for the
static FAP [14]. The value of the trail indicates the qual-
ity of previous solutions when a request is assigned to a
frequency.
ii) Trail between requests and requests (): pre-
vious work on the graph colouring problem (GCP) in
[12] found that a trail between nodes and nodes was more
successful than a trail between nodes and colours. This is
because the important aspect of a graph colouring solu-
tion is not in which colour each node is placed, as the
colours are interchangeable. The important aspect is
which nodes are placed together in the same colour class.
When considering the static FAP, clearly the actual fre-
quency to which each request is assigned is important.
However, given the static FAP has the same underlying
model as the GCP, we decided to investigate whether a
trail based on which requests are assigned to the same
frequencies could be advantageous.
This trail measures the success of previous solutions
when requests are assigned to the same frequency us-
ing, which is the average trail between
the prospective request and all requests already assigned
to the candidate frequency, which is defined by For-
mula 6.
(6)
where is the set of requests already assigned to fre-
quencies .
5.4.1 Trail Evaporation
Both types of trail are evaporated after each genera-
tion by multiplying the trail by the evaporation parame-
ter, which will be determined experimentally The trail
evaporation can be defined by Formula 7.
.
(7) where the evaporation parameter is in the range [0,
1).
5.4.2 Trail Updates
The trails are updated using two reward functions,
namely Cost1 and Cost2, which are defined as follows:
- Cost1: counts the number of used frequencies in
the current solution. This is appropriate when a
solution is feasible.
- Cost2: counts the number of unassigned re-
quests in the current solution. This is appropri-
ate when a solution is infeasible.
The values of could have been updated using For-
mula 8.
(8)
where is the best minimum number of used fre-
quencies found so far in the search. Note that
can be equal to 0 when
and . Thus, we add 1 to the denominator of the
last term in Formula 8. A similar trail update function
was previously used in ACO for the GCP [12].
Similarly, the values of are updated using For-
mula 9.
(9)
Another problem of trail updates is that only requests
that have been assigned to frequencies are updated.
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Therefore, the trail values on any unassigned requests are
not increased, meaning such requests are likely to be se-
lected even later in the following construction processes.
As we would prefer to consider them earlier in the con-
struction process, the trail is increased on each unas-
signed request for all available frequencies. This idea
was previously used in ACO for the examination sched-
uling problem [11].
5.5 Descent Method
This method is executed only when no feasible solu-
tion can be found by all ants in a generation for a sub-
problem. In such generations, the descent method is exe-
cuted only for one ant which constructs the infeasible so-
lution with the minimum number of unassigned requests.
First, these requests are assigned to the frequencies
which lead to the least number of violations. Then, the
descent method aims to reduce the number of violations
with a fixed number of frequencies to find a feasible so-
lution, if possible
5.6 The DACO Algorithm Implementation
DACO solve each sub-problem through given number
of generations, each of which contains a given number of
ants, where each ant individually constructs a solution.
Each ant starts constructing a solution by selecting a fre-
quency to be assigned to all possible feasible requests.
The process is repeated until no frequencies can be se-
lected (see Section 5.2). After all ants in the current gen-
eration construct their solutions, if no feasible solution
can be found, then the descent method (see Section 5.5)
is used to attempt to achieve a feasible solution. Then,
the trail is evaporated and updated (see Section 5.4.1 and
5.4.2). After that, the next generation is executed by the
same process. The overall structure of the DACO algo-
rithm is illustrated in Figure 2.
.
Fig. 2. Overall structure of our DACO algorithm for each
sub-problem of the static MO-FAP
N
Y
N
Y
N
Y
Y
e
Initialize the phero-
mone trail and parame-
ters
Stop
Return the
best solu-
number of generations
= number of generations +
1
Is the
number of
genera-
tions > a
number of ants
= number of ants +
Is the
number
of ants >
Choose a re-
quest (Section 3.2)
N
Y
e
N
Choose a fre-
quency (Section 3.2)
Can
a re-
quest be
Can
a feasi-
ble so-
Descent
method (Section
Can a
frequency
be cho-
Assign the cho-
sen frequency to the
chosen request
Trail evapora-
tion (Section
Trail updates
(Section 5.4.2)
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6 Experiments and Results
This section presents and compared the performance
of DACO in three sections. The first section gives the
results of DACO for the static FAP. The second section
compares the performance of DACO with existing ACO
algorithms in the literature.
DACO is implemented in FORTRAN 95 and all ex-
periments were conducted on a 3.0 GHz Intel Core I3-
2120 Processor (2nd Generation) with 8GB RAM and a
1TB Hard Drive.
6.1 Results Comparison of the DACO
Algorithm
In this study, the number of generations of DACO is
100, where this number is selected based on experiments.
Moreover, the performance of DACO is compared based
on several options of the following components:
1. The number of ants,
2. The trail definition,
3. The visibility definition,
4. The parameters
Different values of the number of ants, two options of
the trail definition and two options for visibility defini-
tion are compared. For the parametersand, three
values of each parameter are tested. By considering all
these options, there are 756 versions of ACO to be com-
pared. Moreover, each version is tested on 10 instances
with 5 runs being performed on each instance. Therefore,
considering all the versions of ACO take excessive time.
Hence, the comparison is made for each component
while fixing the others; i.e. first, different numbers of
ants are compared while fixing the remaining compo-
nents. After selecting the best number of ants, the two
different trail definitions are compared. After that, two
definitions of the visibility are compared and finally, dif-
ferent values of the parameters (and) are compared
in the same way. Based on experiments, the best values
of the parameters and number of ants given in Table 3.
Table 3. The best values of the parameters and number of ants
based on experiments.
3
2
0.78
20
Moreover, the performance of DACO using is
better than using . The performance of DACO
using the two types of trail definitions is shown in Figure
3 (for the instances in which feasible solutions are
found).
Fig. 3. The performance of DACO using two types of trail
definitions.
It is found by the Wilcoxon signed-rank test at the 0.05
significance level that there is a significant difference be-
tween the performances of DACO using
and.
Moreover, the performance of DACO using the first
definition of visibility better than the second one.
6.2 Results Comparison with Existing ACO
Algorithms
The performance of our DACO is compared with exist-
ing ACO in the literature. To the best of my knowledge,
only one published research [13] applied ACO for the
MO-FAP using CELAR and GRAPH datasets. Table 4
shows the results in the form given in [13], i.e. in the
form of (y) where y is the number of violations. Note that
y is equal to 0 means a feasible solution is found.
Table 4. Results of DACO and existing ACO algorithm in the litera-
ture.
Instance
ACO [13]
Our DACO
CELAR 01
(0)
(0)
CELAR 02
(0)
(0)
CELAR 03
(0)
(0)
CELAR 04
(8)
(0)
CELAR 11
(2)
(1)
GRAPH 01
(0)
(0)
GRAPH 02
(0)
(0)
GRAPH 08
(0)
(0)
GRAPH 09
(0)
(0)
GRAPH 14
(0)
(0)
Table 4 shows that both of the algorithms struggled to
find a feasible solution for CELAR 11. Moreover, ACO
in [13] could not achieve a feasible solution for CELAR
04, whereas our DACO could. Overall, our DACO algo-
rithms performing better than ACO in [13].
0
5
10
15
20
25
Average numbers of used frequencies
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7 Conclusions
In this study, the DACO was introduced to solve the
static MO-FAP by modeling it as a dynamic problem
through dividing this static problem into smaller sub-
problems, which are then solved in turn in a dynamic pro-
cess. Several novel and existing techniques have been
used. One of the techniques was applied to improve the
performance of DACO is the recursive largest first. In
fact, this technique aims to improve the performance of
selecting frequencies and requests to be assigned. More-
over, DACO was compared using two trail definitions
and two visibility definitions. It was found that using the
trail between requests and frequencies led to better per-
formance than the other trail definition. Moreover, using
the visibility definition based on the number of feasible
frequencies resulted in better performance than another
visibility definition. Furthermore, several values for the
parameters were compared.
DACO is combined with a descent method to achieve
better results when no feasible solution can be found in a
generation. In such generations, the descent method is
executed for only one ant which constructs the infeasible
solution with the minimum number of unassigned re-
quests. Overall, our DACO algorithm performed better
than ACO in the literature.
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Conflict of Interest
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International Journal of Applied Sciences & Development
DOI: 10.37394/232029.2023.2.18
Khaled Alrajhi
E-ISSN: 2945-0454
176
Volume 2, 2023
