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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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