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Image Processing & Communication, vol. 17, no. 1-2, pp. 7-18
DOI: 10.2478/v10248-012-0011-5 7
SHORTEST PATH PROBLEM SOLVING BASED ON ANT COLONY
OPTIMIZATION METAHEURISTIC
MARIUSZ GŁ ˛ABOWSKI, BARTOSZ MUSZNICKI, PRZEMYSŁAW NOWAK, PIOTR
ZWIERZYKOWSKI
Pozna
´
n University of Technology, Faculty of Electronics and Telecommunications,
Chair of Communications and Computer Networks, Polanka 3, 60-965 Pozna
´
n, Poland
bartosz@musznicki.com, przemyslaw.nowak@inbox.com
Abstract. The Ant Colony Optimization
(ACO) metaheuristic is a versatile algorithmic
optimization approach based on the observation
of the behaviour of ants. As a result of numer-
ous analyses, ACO has been applied to solv-
ing various combinatorial problems. The ant
colony metaheuristic proves itself to be efficient
in solving N P-hard problems, often generat-
ing the best solution in the shortest amount of
time. However, not enough attention has been
paid to ACO as a means of solving problems
that have optimal solutions which can be found
using other methods.
The shortest path problem is undoubtedly one
of the aspects of great significance to navigation
and telecommunications. It is used, amongst
others, for determining the shortest route be-
tween two geographical locations, for routing
in packet networks, and to balance and opti-
mize network utilization. Thus, this article in-
troduces ShortestPathACO, an Ant Colony Op-
timization based algorithm designed to find the
shortest path in a graph. The algorithm consists
of several subproblems that are presented suc-
cessively. Each subproblem is discussed from
many points of view to enable researchers to
find the most suitable solutions to the problems
they investigate.
1 Introduction
This article focuses on a presentation of possibilities and
the usability of the application of the Ant Colony Opti-
mization metaheuristic for solving the Shortest Path prob-
lem. The following subsections present basic background
and introduce basic notions and parameters. Further sec-
tions discuss individual questions related to the algorithm
proposed by the authors. The article is concluded with a
summary of basic facts and conclusions.
1.1 Ant Colony Optimization algorithms
The ant colony algorithms were initially proposed by
Marco Dorigo, who in his doctoral dissertation of 1992 [1]
presented his first algorithm based on the behaviour of
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8 M. Gł ˛abowski, B. Musznicki, P. Nowak, P. Zwierzykowski
ants, called the Ant System (AS), in the application for
the classical Travelling Salesman problem. Arguably, the
most commonly used and the most successful research
line related to the ant colony algorithms concerns their ap-
plications to combinatorial optimization problems and is
called Ant Colony Optimization (ACO) [2]. The inspira-
tion for this algorithm was provided by the observation of
a colony of the Argentine ant Linepithema humile that are
capable of finding, subject to certain conditions being sat-
isfied, the shortest path from among a number of alterna-
tive routes between the nest and a source of food. Ants are
able to find food thanks to the phenomenon of stigmergy,
i.e. the exchange of information indirectly via the envi-
ronment by depositing pheromones, while the information
exchanged has a local scope; only an ant located where
the pheromones were left has a notion of them. During
their walk ants leave the so-called pheromone trail on the
ground that, though volatile and evaporating over time and
thus reducing its attractive strength, makes other ants, in-
cluding the ant that actually left the trail, follow the trail
to find the best way to the target place — food, or the way
back to the nest. The more "marked" the way is (i.e., has
a higher concentration of pheromones), the more likely it
is to be chosen by an ant running its errands. At the same
time, the trail is increasingly enhanced when a greater
number of ants chooses a given path, while in the case
of finding an even better way, the deposited pheromones
begin to evaporate. If, however, an ant does not find any
trail on its way, its choice of further path is purely random
— unless one of the paths of choice includes obstacles.
This mechanism can be described as a positive feedback
while on shorter routes and a negative feedback on longer
paths. The volatile nature of pheromones encourages ex-
ploration of new paths following a decrease in the inten-
sity of pheromone trails and, in this way, biases the choice
process of a given route. This skill of a colony to find out
and mark the best routes can be viewed as a process of
collective learning of ants.
Recently, ant algorithms have been gaining popularity
in such areas as combinatorial problems solving where
they can be applied in finding appropriate paths in a graph.
They constitute one of the numerical methods that have
been inspired by biology and have found application in
robotics and telecommunications, among others.
1.2 Problem of the shortest path
For the directed graph G = (V, E) where V is the set of
vertices and E is the set of edges we assign the cost
a
ij
to each of its edges (i, j) ∈ E (alternatively, this
cost can be also called the length). For the resulting
path (n
1
, n
2
, . . . , n
k
), its length can be expressed by For-
mula (1).
a
ij
=
k−1
X
i=1
a
n
i
n
i+1
(1)
A path is called the shortest path if it has the shortest
length from among all paths that begin and terminate in
given vertices. The shortest path problem involves find-
ing paths with shortest lengths between selected pairs of
vertices. The initial vertex will be designated as s, while
the end vertex as t.
A number of basic variants of the shortest path problem
can be distinguished:
• Finding the shortest path between a pair of vertices.
• Finding the shortest paths with one initial vertex.
• Finding the shortest paths with one end vertex.
• Finding the shortest paths between all pairs of ver-
tices.
This problem finds its application in a number of areas
such as routing in telecommunications networks, dynamic
programming or project management.
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Image Processing & Communication, vol. 17, no. 1-2, pp. 7-18 9
2 ShortestPathACO Algorithm
As a result of our study we present in this paper Shortest-
PathACO — a proposition of an algorithm to solve the
shortest path problem that is based on the metaheuris-
tic of ant colony. It is worthwhile to remark that in the
case of a problem that is so frequently studied and con-
sidered, such as the shortest path problem, it is difficult to
develop an algorithm that would outperform the already
existing solutions. An additional impediment is the fact
that the metaheuristic of ant colony has been developed
for solving N P-hard problems, whereby the results ob-
tained following its application provide approximations of
optimum results. With the above difficulties in mind, the
article aims at presenting an algorithm that would offer, at
least to a certain degree, a possibility of obtaining char-
acteristics (properties) comparable or better to those ob-
tained with the algorithms used hitherto that are capable
of calculating results in an accurate way.
The ShortestPathACO algorithm includes a number of
subproblems that will be discussed separately for con-
venience in the next sections. Each subproblem is ap-
proached from several different perspectives to allow re-
searchers to choose and adjust a method for a solution
of a specific subproblem. This kind of an approach has
the advantage of creating a procedure that would be best
suited for a solution of the whole problem in its entity.
The procedures presented in the following sections relate
to type of the widely adopted representation of ACO al-
gorithms [2].
3 Initiation of the algorithm
The ShortestPathACO algorithm uses the following pa-
rameters:
m — the number of ants
α — the parameter that defines the influence of pheromones on
the choice of the next vertex
β — parameter that determines the influence of remaining data
on the choice of the next vertex
ρ — parameter that determines the speed at which evaporation
of the pheromone trail occurs; takes on values from the
interval [0, 1]
τ
0
— initial level of pheromones on the edges
τ
min
, τ
max
— the minimum and maximum acceptable level of
pheromones on edges
s — initial vertex
t — end vertex (required when the shortest path between a pair
of vertices is to be calculated)
The number of ants m influences considerably the ac-
curacy of a solution obtained as a result of the operation
of the algorithm. At the same time, it increases time of its
operation. Hence, it is necessary to match their number
(the number of agents) with a corresponding and specific
case under investigation. This will make it possible to
negotiate a compromise between the accuracy of the al-
gorithm and the duration of its operation. The parameters
α and β allow us to modify a method for a selection of
the next vertex, which in turn influences considerably the
quality of the solution. The parameter ρ decides on the
speed at which the pheromone trail evaporates on indi-
vidual paths. The quicker it is evaporated, the more pos-
sibilities arise to increase exploration-oriented initiatives
on the part of ants and to find new solutions, whereas its
slower evaporation makes ant use solutions that have al-
ready been found. Parameters related to the pheromone
level enable to have it regulated (adjusted) accordingly,
which is then translated into a change in the characteris-
tics of the operation of ants.
Depending on a variant of the shortest path problem, the
initial vertex s and/or the end vertex t may be also re-
quired. Depending on a chosen method for finding paths,
discussed in the following section, a list storing vertex oc-
cupation time, i.e the amount of time needed by a given
ant to reach the vertex, and the current value of time may
turn out to be necessary as well. Whereas certain methods
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10 M. Gł ˛abowski, B. Musznicki, P. Nowak, P. Zwierzykowski
of choice of the next vertex require storing the vertices
and/or edges that have already been visited.
Function 3 shows a method to initiate the algorithm
that uses all the mentioned parameters. Additionally,
while determining the convergence, the auxiliary parame-
ters have been initiated.
4 Methods for finding paths
One of the more important elements of any algorithm that
is based on the use of the ant colony system paradigm are
the rules for the process of establishing paths leading from
the initial point to the end point. Subsections 4.1 and 4.2
show two methods for finding paths developed for the
ShortestPathACO algorithm. Item 4.3 presents the con-
clusions of the comparison of the main characteristics of
the two methods.
4.1 Finding paths for each ant one by one
In this method a choice of a subsequent vertex is made
iteratively for each ant one after another. In each itera-
tion the pheromone trail is evaporated. After reaching the
target vertex on a path that led the ant to this vertex, the
pheromone trail is reinforced by ∆τ . A further modifica-
tion of this method is also possible. The modification in-
volves finding, consecutively by each ant, the whole path
in each iteration, and then updating the pheromone trail
on these paths.
4.2 Finding paths according to the time list
Here, a time list is used for the purpose of recording times
(durations) in which an ant reaches a given vertex. This
list, in addition to putting order to ants grouping, has also
additional remarkable property — an ant that reaches the
target vertex earlier will be able to lay down pheromones
on the same path on which it arrived quicker. In this way it
is possible to increase the pheromone trail on the edges of
this path earlier than on the edges of longer paths, which
makes remaining ants more inclined in the following it-
erations to choose this path. Evaporation of pheromones
occurs with every transition to the next ant on the list. Up-
dating of the pheromone trail proceeds in turn in steps.
After reaching the end vertex, each ant recedes on the path
it came to reach the vertex and on each of its edges leaves
∆τ pheromones. This strategy makes it possible to mod-
ify the process of choice of the next vertex far more earlier
also for the ants that have not yet reached the end vertex.
This method introduces, however, certain redundancy re-
sulting from the use of the structure that sorts out ants
according to times (durations) of reaching the next vertex.
4.3 Observations related to the method
Depending on the character of a scenario to be solved and
its anticipated characteristics, it is necessary to choose an
appropriate method for finding paths and the parameters
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Image Processing & Communication, vol. 17, no. 1-2, pp. 7-18 11
that would make it possible to seek the best solution more
effectively and with better accuracy. After a compari-
son of the presented methods the following conclusions
should be highlighted:
• In search for paths step by step we favour paths that
consist of a lower number of edges because such
paths will be found after a lower number of itera-
tions, and what follows will enhance the pheromone
trail on them quicker. In such a case it is required
to apply an additional rule that concerns pheromone
trails left on the paths that have been found, e.g.
making this trail dependent on the ratio between the
length of a path and the length of the shortest path
found so far, or leaving pheromones only on selected
paths.
• The use of the time list induces that laying
pheromones requires as many iterations as the num-
ber of edges on the path. In the case of the fist
method, updating of pheromones takes place in the
same iteration as the reaching of the target vertex.
While comparing these two methods it may seem
that in the case of the time list iterations are unnec-
essarily dedicated to ants that get marched over the
path in opposite direction.
• The redundancy resulting from the usage of the time
list can be compensated by a stronger reinforcement
of short paths and better adjustment, with regard to
certain respects, of the state of pheromones to the
currently held information on the paths.
5 Choice of the next vertex
During the choice process of the next vertex the formula
presented in [3] can be used. The formula defines the tran-
sition probability of ant k that is in vertex i in its transition
to vertex j. Its general form is presented in (2).
p
k
ij
=
q
ij
P
{l:(i,l)∈E}
q
il
(2)
5.1 Parameters influencing the choice
The choice of the next vertex is considerably influenced
by the parameters α and β that regulate the reinforcement
of pheromones and additional data at the calculation of the
coefficient of a given edge q
ij
. Additional data is under-
stood to be all the information that is either available a pri-
ori or is gathered in the course of the performance of the
algorithm regardless on the pheromone trail. These can be
weights of edges or a marking whether a given edge has
been already visited. The edge coefficient can be calcu-
lated using many methods. Some exemplary methods are
presented in Equation (3).
q
ij
= τ
α
ij
η
β
(3a)
q
ij
= τ
ij
α + (1 − α)η (3b)
q
ij
=
(
τ
α
ij
(1 + β), for vi_nodes
j
= false
τ
α
ij
, elsewhere
(3c)
q
ij
=
(
τ
α
ij
(1 + β), for vi_edges
ij
= false
τ
α
ij
, elsewhere
(3d)
q
ij
=
τ
α
ij
(1 + β)
2
for
vi_nodes
j
= false
vi_edges
ij
= false
τ
α
ij
(1 + β) for
vi_nodes
j
= true
vi_edges
ij
= false
τ
α
ij
(1 + β) for
vi_nodes
j
= false
vi_edges
ij
= true
τ
α
ij
elsewhere
(3e)
The choice of a method for the calculation of the edge
coefficient is absolutely vital for the operation of the
whole algorithm. Depending on the applied formula, we
can reach a quicker convergence of the algorithm (not nec-
essarily for the most optimal solution), an increase in the
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12 M. Gł ˛abowski, B. Musznicki, P. Nowak, P. Zwierzykowski
exploration features of ants, or assertion as to the check of
all edges and vertices. It is also worthwhile to stress that
the coefficient should be concordant with and adjusted to
a given problem, i.e., an equation that would secure good
properties of the algorithm for one problem can be com-
pletely misleading for another one.
5.2 Optimization of the process of choice
One of the equations (3) is not enough to successfully se-
lect the next vertex for the ant. In order to increase the
chances of finding better paths for ants, a pseudo-random
generator can be used that will introduce randomness to
the process of edge selection. The pseudo-random gener-
ator is also necessary in case uniform (equal) probabili-
ties for a higher number of edges appear. When this is the
case, an edge to be followed by the ant should be selected
randomly, which will make it possible to avoid a situation
where the algorithm, when activated, chooses the same
edges again and again. Randomness of operations of ants
is very frequently a factor that favourably influences the
obtained results. Another idea worth mentioning here is
to apply an individual method for the selection of edge as
early as the initial stage of the operation of the algorithm.
This will make it possible to adjust the performance of the
algorithm to a specific problem and successfully prevents
the algorithm from being, for example, greedy, or to addi-
tionally reinforce randomness of ants in the choice of an
edge. The basic version of the method for selection of the
next vertex is presented with the help of the function 5.1.
This method uses the edge coefficient q
ij
only, because
calculations of the probability p
ij
in each of the paths is
not necessary in the process of the selection of the best
edge and requires additional computation.
5.3 Calculation of the edge coefficient
The method for calculation of the coefficient q
ij
is de-
picted by the function 5.2. The coefficient is calculated in
one of three ways depending on whether τ
ij
of the edge
under consideration is equal to 0 and whether ant k has al-
ready found 3 paths. If τ
ij
of a given path has not been yet
increased, we use one of the formulas from Equation (4).
Interestingly enough, this equation changes the meaning
of the parameter α adopted earlier — it is a purposeful op-
eration that aims at limiting the number of parameters of
the algorithm. Instead of introducing a new parameter, an
already available parameter is used, though incompatibly
with its original purpose. The parameter α is used here as
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Image Processing & Communication, vol. 17, no. 1-2, pp. 7-18 13
the power of the length of edge or its inverse, while the
parameter β retains its original application and influences
additional data — in this particular case it is the marking
whether a given vertex or edge have been already visited.
q
0
ij
= a
α
ij
(4a)
q
0
ij
=
1
a
ij
α
(4b)
q
0
ij
=
a
α
ij
(1 + β)
2
for
vi_nodes
j
= false
vi_edges
ij
= false
a
α
ij
(1 + β) for
vi_nodes
j
= true
vi_edges
ij
= false
a
α
ij
(1 + β) for
vi_nodes
j
= false
vi_edges
ij
= true
a
α
ij
elsewhere
(4c)
q
0
ij
=
1
a
ij
α
(1 + β)
2
for
vi_nodes
j
= false
vi_edges
ij
= false
1
a
ij
α
(1 + β) for
vi_nodes
j
= true
vi_edges
ij
= false
1
a
ij
α
(1 + β) for
vi_nodes
j
= false
vi_edges
ij
= true
1
a
ij
α
elsewhere
(4d)
If the pheromone trail has already been deposited on
the edge under consideration, then again we have a choice
of two options. In the case when an ant for which we
choose the next edge has not found yet three paths, we
return 0. This causes the algorithm to perform randomly
in the second stage of its operation, the first stage being
understood as the selection of edges on the basis of their
lengths or inverses of these lengths. This operation is nec-
essary if we are to solve a highly complex graph (i.e., one
that has many edges) or when the number of ants is rela-
tively small as compared to the size of a graph. Otherwise,
this stage can be omitted. It is only for an ant that is try-
ing to find the fourth or any successive number of path that
we use one of the basic formulas for the edge coefficient
presented in Equation (3).
6 Updating pheromone trail
The way of updating the pheromone trail is somehow re-
lated to a method for finding paths, but it is possible to
combine characteristic features of different methods to
obtain new strategies. These ways can be grouped into
the following categories:
in steps and progressive – pheromone trail is reinforced
with every passage of the ant from one vertex to an-
other during its search of a path leading to the end
vertex
in steps and backward (reverse) – reinforcement is in-
troduced during a change of a vertex by the ant, but
following a way back from the end vertex to the ini-
tial vertex (in an analogy with real life ants, it is tan-
tamount to a return of an ant to the nest)
overall – reinforcement is done in one iteration (the
whole path followed by an ant is reinforced in one
go)
selective – only a certain subset of the paths or one par-
ticular and defined path is reinforced, e.g. paths with
a weight not exceeding a given pre-defined thresh-
old, a path that is currently the best or the best path
in a given iteration
Depositing pheromones during the process of finding
the end vertex, i.e., concurrent enhancement of the vertex
during the relocation of ants along the edges of a graph, is
the most intuitive way of its upgrading. This process is the
one that resembles most the real behaviour of ants. Such
a strategy may turn out, however, to be of little effect as a
path has has not been found yet is reinforced, while it may
turn out that the path is far from being optimal. Due to the
fact that the pheromone trail deposited by ants influences
considerably decisions made by other ants on the move,
in this particular example the algorithm may terminate its
operation, while the first encountered path may be yielded
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14 M. Gł ˛abowski, B. Musznicki, P. Nowak, P. Zwierzykowski
as the result. This path, on account of the character of the
strategy for pheromone trail reinforcement, will be chosen
in successive iterations by an increasing number of ants
until the algorithm reaches convergence.
A good method to counteract risk of the occurrence of
the above situation is to reinforce the pheromone trail as
late as the moment at which the whole of a path is gener-
ated by an ant. At this point, we have precise and accurate
information on the quality of the path and we are in posi-
tion to appropriately apportion the amount of pheromones
to be deposited. Moreover, if we have knowledge on the
currently best solution we can make the pheromone trail
dependable on it, additionally increasing its amount de-
posited by an ant on its way back to the initial vertex.
The characteristic feature of this strategy is making the
pheromone trail dependant on the length of the path. This
is, however, done at the expense of the time (iteration)
spent on updating.
It may be so that the information on the state of
pheromones on individual edges of a graph has to be up-
dated very quickly and the earlier strategy is too slow
for this case. To retain some of its advantages, such as
the knowledge on the quality of the generated solution
(the weight of the path found by an ant), we can, after
reaching the end vertex, immediately update the whole
path instead of doing it in a series of steps. The state of
pheromones is quicker adjusted to the information cur-
rently held, whereas iterations related to a reconstruction
of the ant’s path is omitted. Such an approach, however,
levels off additional reinforcing of good solutions because
with the application of the time list the reaching time of
the end vertex and the return to the initial vertex with
short paths is decidedly shorter than with long paths, and
a quicker updating of the pheromone trail means a higher
probability of selection of given edges by other ants.
As it happens, however, all the above strategies may
bring no satisfactory results. When this is the case, it is
worthwhile then to consider reinforcing only a given sub-
set of paths obtained by ants. For example, we can rein-
force only
m
2
best paths in a given iteration, choose the
best path in this iteration, or reinforce only those paths
that are better than the best path at the initial stage of the
operation of the algorithm. The latter method, if not safe-
guarded by appropriate conditions, can eventually lead to
a too early convergence of the algorithm. Another thing
worth mentioning here is that only in the case of finding
whole paths for all ants one by one in each iteration, we
are in position to somehow compare them. With the ap-
plication of the time list of a path for all ants, they will
be available as late as when an ant that has chosen the
worst path finally reaches the end vertex, and at that time
the remaining ants will be on their way back to the initial
vertex or even involved in the next route. At this moment
it is already too late to update trails on the edges because
ants that are embarking on their successive routes have
information identical to that available at the beginning of
the operation of the algorithm or just few iterations ear-
lier. On the other hand, for those ants that have found
their paths to wait until the remaining ants find their routes
may turn out to be disadvantageous in view of the very
idea of the operation of the algorithm — ants should op-
erate independently and communicate only with the help
of pheromones. An iterative choice of the edge for each
ant one after another is followed by similar consequences,
with the difference that paths with a lower number of
edges are in this case more advantageous from the point
of view of the operation of the algorithm.
The above considerations show how complicated is the
task of adjusting (matching) a method for finding paths to
an appropriate way of updating edges. It is undoubtedly
necessary to apportion the required amount of time for ex-
periments that will illustrate which combinations perform
better or best for a problem under consideration. Obvi-
ously, it may turn out that finding a good combination is
not viable and the ant colony metaheuristic will not prove
to be of use in solving a given problem.
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Image Processing & Communication, vol. 17, no. 1-2, pp. 7-18 15
Updating of the pheromone trail can be also dependent
on the length of the performance of the algorithm and,
along with its increase, can undergo changes. A use of
a variant reinforcing policy in initial iterations may prove
to be a good solution with the case of a situation in which
the convergence of the algorithm ensues too quickly or the
algorithm generates sub-optimal solutions. At the same
time, it is also important to control transition periods and
do not let them be too long as this can lead to unnecessary
prolongation of the operating time of the algorithm or to
a situation in which the algorithm remains in unchanged
state.
∆τ = const (5a)
∆τ =
1
a
P
(5b)
∆τ =
C
a
P
(5c)
∆τ =
a
P
best
a
P
(5d)
where:
C = max
(i,j)∈E
a
ij
a
P
=
X
(i,j)∈P
a
ij
Equations (5) illustrate possible ways of the calculation
of the number of pheromones that are to be deposited on
a found path. C is the maximum cost a
ij
of the edge
(i, j) ∈ E, P denotes the path found by the ant, a
P
its
weight, and a
P
best
is the weight of the best path found so
far.
The choice of ∆τ is also very important and makes it
possible to control the operation of ants because it directly
influences decisions made by them. The better selected
∆τ, the quicker the algorithm reaches convergence and
the higher probability of avoiding invalid results.
6.1 Additional reinforcement of currently
optimal paths
If the algorithm returns non-optimal results it is worth-
while to attempt to improve its performance by reinforc-
ing additionally good solutions. One of the applicable
ways is to add a given number of pheromones to ∆τ that
has to be added to the edge of a path that has been found
by the ant. The number itself, just like ∆τ , can be fixed
or may depend on the length of a given path or on the
length of the best path hitherto found. Another way is
to multiply ∆τ by the number of ants m, which effects
in the edges of a given path to have a higher number of
pheromones, regardless of the number of paths found by
other ants. This can, however, lead to a situation in which
a reinforced path will not be optimal, which in turn can
result in a premature termination of the algorithm and the
algorithm returning invalid result. In general, just like in
the case of other methods, caution and appropriate expe-
rience are advisable.
6.2 Evaporation of pheromones
The mechanism for evaporation of pheromones itself is
easy and intuitive — the value of pheromones on each
of the edges is multiplied by an appropriate parameter.
This is presented with the help of the function 6.2. In
addition, pheromone values on edges cannot be lower than
the level determined by the parameter τ
min
. If it happens,
it is set on the value of this parameter. Such an approach
is drawn from MAX –MIN Ant System (MMAS) [4,
5]. A more difficult problem, however, is the selection of
frequency, the moment of evaporation of pheromones and
the value of the parameter ρ.
It is adopted that the most appropriate moment for
evaporation of pheromones is most frequently at the end
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16 M. Gł ˛abowski, B. Musznicki, P. Nowak, P. Zwierzykowski
of a current iteration, i.e., the moment when all remain-
ing actions have been completed. However, it should be
considered if this process should proceed after all these
actions have been completed by a single ant or the whole
of the colony. Combined with frequency of pheromone
evaporation, the following instances can be established:
1. evaporation after the step of an ant
2. evaporation after the step of all ants
3. evaporation after finding a path by ant
4. evaporation after finding paths by all ants
5. evaporation after a time change
6. evaporation per time unit
As it is easy observable, some instances depend on a
method for finding paths and can be applied only with
the application of one of them. The two earlier pre-
sented cases require the application of the time list, but
this method also includes applications of the remaining
instances.
Evaporation after each step of an ant can be applied with
both methods for finding paths presented in Section 4,
it may, however, turn out to be too frequent with these
modes of operation. The second and the forth case prove
better with finding paths for each ants one by one because
it is easier then to determine this particular moment with-
out additional calculations. On the other hand, the fifth
instance can be followed by evaporation that is too low,
which is in turn levelled off in the sixth instance.
The value of the parameter ρ makes it possible to con-
trol the speed at which pheromones evaporate. The higher
its value, the quicker pheromones are evaporated, while
for ρ = 0 evaporation does not take place at all. In the
case of some of the problems, a choice of this parameter
is absolutely vital for the operation of this algorithm. Its
too high value can result in a convergence to non-optimal
solutions, whereas its too low value to a complete lack of
convergence of the algorithm.
7 Termination of the algorithm
While taking into consideration the fact that the discussed
algorithm is an optimization algorithm and does not guar-
antee finding a solution that would be optimal, its termi-
nation can be made dependent on various and diverse fac-
tors.
The best condition for termination is its convergence,
which occurs when all ants from the colony find the same
solution. This situation takes place when pheromones that
indicate this solution have so high value that the succes-
sive iterations of the algorithm bring no further changes.
In the case of a great number of ants, reaching conver-
gence can take much time, hence a termination of the op-
eration of the algorithm is possible if a pre-defined frac-
tion of ants consecutively yields identical solution. This
provides a chance to shorten the operation of the algo-
rithm at the expense of an increase in getting a wrong (in-
valid) solution. Having this in mind, to terminate the op-
eration of the algorithm we have to select the percentage
of ants that has to reach the same solution appropriately.
In this way, the end task condition will be adjusted to the
solution under consideration.
In the case of improperly selected parameters of the al-
gorithm or when the algorithm is not suitable for a given
problem, it is worthwhile to introduce additional task end
conditions. Most frequently, it is enough to introduce a
limit/boundary to the iteration that the algorithm is to per-
form or to introduce a time limit for its performance. Both
methods provide an efficient mechanism to avoid unnec-
essary infinite repetition of the procedure that, in conse-
quence, may translate into lowering of the quality of ob-
tained solutions.
In practice, the best way is to attempt to apply skilfully
all of the methods discussed above to guarantee an accept-
able running time of the algorithm, with given required
credibility level of results. A selection of task end con-
ditions for the algorithm will make it possible to evaluate
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Image Processing & Communication, vol. 17, no. 1-2, pp. 7-18 17
its usefulness for a given problem and to proceed with op-
timization initiatives or to make a decision on finding a
necessary alternative method.
8 Conclusions
Algorithms based on the metaheuristic of ant colony do
not guarantee finding an optimal solution in all possible
cases. Accordingly, experimentation is particularly im-
portant to find and select parameters dedicated to each
of the problems under consideration. Individual elements
of the procedures applied in the process should be also
analysed with regard to their usability and purposefulness
of application. The construction of the presented Short-
estPathACO algorithm directly reflects this particular ap-
proach through the proposal and the discussion of vari-
ous variants of the execution of each of the elements of
the procedure. In this way, it is possible to improve the
method for the solution of the shortest path problem to
approach or reach optimal solutions. An evaluation of the
duration time and the quality of returned solutions will
provide information for making a decision on the imple-
mentation of a given scheme as being of optimum quality
or an alternative to more time-consuming procedures or
procedures with higher computational cost.
References
[1] M. Dorigo, Optimization, Learning and Natural
Algorithms, Ph.D. Thesis, Politecnico di Milano,
1992.
[2] M. Dorigo and T. Stützle, Ant Colony Optimization,
The MIT Press, Cambridge, 2004.
[3] M. Dorigo, V. Maniezzo and A. Colorni, The Ant
System: Optimization by a colony of cooperating
agents, in IEEE Transactions on Systems, Man, and
Cybernetics-Part B, 26(1):29–41, 1996.
[4] T. Stützle and H. H. Hoos, The MAX –MIN Ant
System and Local Search for the Traveling Sales-
man Problem, in Proceedings of IEEE International
Conference on Evolutionary Computation, pp. 309–
314, 1997.
[5] T. Stützle and H. H. Hoos, MAX –MIN Ant
System, in Future Generation Computer Systems,
16(8):889–914, 2000.
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