Interactive and Non-Interactive Hybrid Immigrants
Schemes for Ant Algorithms in Dynamic
Environments
Michalis Mavrovouniotis
Centre for Computational Intelligence (CCI),
School of Computer Science and Informatics,
De Montfort University, The Gateway,
Leicester, LE1 9BH, U.K.
Email: mmavrovouniotis@dmu.ac.uk
Shengxiang Yang
Centre for Computational Intelligence (CCI),
School of Computer Science and Informatics,
De Montfort University, The Gateway,
Leicester, LE1 9BH, U.K.
Email: syang@dmu.ac.uk
Abstract—Dynamic optimization problems (DOPs) have been
a major challenge for ant colony optimization (ACO) algorithms.
The integration of ACO algorithms with immigrants schemes
showed promising results on different DOPs. Each type of immi-
grants scheme aims to address a DOP with specific characteristics.
For example, random and elitism-based immigrants perform
well on severely and slightly changing environments, respectively.
In this paper, two hybrid immigrants, i.e., non-interactive and
interactive, schemes are proposed to combine the merits of
the aforementioned immigrants schemes. The experiments on a
series of dynamic travelling salesman problems showed that the
hybridization of immigrants further improves the performance
of ACO algorithms.
I. INTRODUCTION
Ant colony optimization (ACO) algorithms are inspired
from nature, i.e., the foraging behaviour of real ant colonies
[2], [3]. ACO has been initially developed for the travelling
salesman problem (TSP) and later on has been applied on
several other applications [4], [7], [17]. Most of the optimiza-
tion problems addressed so far by ACO assume a stationary
environment. However, the environment in many real-world
applications changes over time. The difference between sta-
tionary and dynamic optimization problems is that the aim of
the former problems is to locate the static global optimum
efficiently whereas the aim of the latter problems is to track
the moving global optimum efficiently [10].
Addressing dynamic optimization problems (DOPs) is a
challenging task to ACO algorithms, and generally to all
optimization algorithms. Once the algorithm converges to an
optimum, then it is difficult to escape from it in order to
track the new optimum when a dynamic change occurs. The
pheromone trails, generated with ACO algorithms, of the
previous environment may bias the population of ants into the
optimum of the previous environment. A straightforward way
to address this issue is to consider every dynamic change as
the arrival of a new problem instance that needs to be solved
from scratch by re-initializing all the pheromone trails with an
equal amount. However, such strategy may be computational
expensive and requires the detection of a dynamic change.
In case the changing environments have similarities, the re-
optimization time may be improved by transferring knowledge
from previous environments [1], [10], [12].
Over the years, several strategies have been proposed to en-
hance the performance of ACO algorithms in DOPs including,
maintain diversity strategies [5], [8], [12], increase diversity
via immigrants [11], [13], [14], memory-based schemes [9]
multi-colony schemes [16]. Among the approaches immigrants
schemes have proved that they are straightforward to apply
and have good performance on difference DOPs. The policy
of immigrants schemes in ACO is to replace a small portion of
ants from the population with other randomly generated ants in
order to deposit pheromone [11]. Another type of immigrants
schemes, is when elitism-based immigrants are generated using
the best ant from the previous environment as the base to
generate ants and replace other ants in the population [14].
Random and elitism-based immigrants have shown good
performance on DOPs with severely and slightly changing
environments, respectively [13], [14]. Since different immi-
grants schemes are suitable for different DOPs, then hybrid
immigrants may combine the merits of two or more immigrants
schemes [18], [19]. In this paper, we propose two types
of hybrid immigrants. The first one generates both types of
immigrants on each iteration, whereas the second one each
type of immigrant is triggered depending on the current of
status of the algorithm.
Based on the dynamic benchmark generator proposed in
[15], a series of dynamic test cases are constructed from several
stationary TSP instances and experiments are systematically
carried out for several ACO algorithms, including the proposed
ones. The experiments show that the hybrid immigrants fur-
ther improves the performance of ACO algorithms in many
dynamic test cases. The rest of the paper is organized as
follows. Section II describes the dynamic TSP generated by the
benchmark generator. Section III reviews existing integrations
of immigrants schemes with ACO. Section IV describes the
proposed hybrid immigrants schemes and section V gives the
experimental results. Section VI concludes this paper with
discussion on future work.
II. D
YNAMIC TRAVELLING SALESMAN PROBLEM
A. Description of the Problem
The TSP can be described as follows: given a collection of
cities, the objective is to find the shortest path that starts from
1542
2014 IEEE Congress on Evolutionary Computation (CEC)
July 6-11, 2014, Beijing, China
978-1-4799-1488-3/14/$31.00 ©2014 IEEE
one city and visits each of the other cities once before returning
to the starting city. The TSP is classified to the 𝒩𝒫-complete
combinatorial optimization problems. Usually, the problem is
represented by a fully connected weighted graph 𝐺 =(𝑁,𝐴),
where 𝑁 = {1,...,𝑛} is a set of nodes and 𝐴 = {(𝑖, 𝑗):
𝑖 ∕= 𝑗} is a set of arcs. The collection of cities is represented
by the set 𝑁 and the connections between them by the set 𝐴.
Each connection (𝑖, 𝑗) is associated with a non-negative value
𝑑
𝑖𝑗
which represents the distance between cities 𝑖 and 𝑗.
Formally, the TSP can be described as follows:
𝑓(𝑥)=𝑚𝑖𝑛
𝑛
𝑖=1
𝑛
𝑗=1
𝑑
𝑖𝑗
𝜓
𝑖𝑗
, (1)
subject to:
𝜓
𝑖𝑗
=
1, if (𝑖, 𝑗) is used in the tour,
0, otherwise,
(2)
where 𝜓
𝑖𝑗
∈{0, 1}, 𝑛 is the number of cities, and 𝑑
𝑖𝑗
is the
distance between cities 𝑖 and 𝑗.
B. Generate Dynamic Test Environments
In order to generate dynamic TSPs (DTSPs), the dy-
namic benchmark generator for permutation-encoded problems
(DBGP) [15] is used, which can convert any static permutation-
encoded benchmark problem instance to a dynamic envi-
ronment. The main advantage of DBGP compared to other
benchmark generators is that in case the optimum of the
benchmark problem instance is known, then it will remain
known during the environmental changes. DBGP shifts the
population of the algorithm to search to a new location in the
fitness landscape. Other existing DTSP benchmark generators,
e.g., the DTSP with traffic factors [14] and the DTSP with
exchangeable cities [9], modify the fitness landscape and the
optimum value changes in every dynamic change.
Considering the TSP description, each city 𝑖 ∈ 𝑁 has a
location defined by (𝑥, 𝑦) and each arc (𝑖, 𝑗)∈𝐴 is associated
with a non-negative distance 𝑑
𝑖𝑗
. Usually, the distance matrix
of a problem instance is defined as D =(𝑑
𝑖𝑗
)
𝑛×𝑛
.DBGP
generates the dynamic case as follows. Every 𝑓 iterations a
random vector
𝑉 (𝑇 ) is generated that contains exactly 𝑚 ×𝑛
cities, where 𝑇 = ⌈𝑡/𝑓 ⌉ is the index of the period of change,
𝑡 is the iteration count of the algorithm, 𝑓 determines the
frequency of change, 𝑛 is the size of the problem instance,
and 𝑚 determines the magnitude of change. More precisely,
𝑚 ∈ [0.0, 1.0] defines the degree of change, in which only
the first 𝑚 ×𝑛 of
𝑉 (𝑇 ) cities are swapped. Then a randomly
re-ordered vector
𝑈(𝑇 ) is generated that contains the cities of
𝑉 (𝑇 ). Therefore, exactly 𝑚×𝑛 pairwise swaps are performed
in D using the two random vectors (
𝑉 (𝑇 )⊗
𝑈(𝑇 )), where “⊗”
denotes the swap operator.
III. ACO
IN DYNAMIC ENVIRONMENTS
Conventional ACO algorithms consist of a population of 𝜇
ants where they construct solutions and share their information
via their pheromone trails. ACO was initially established for
the TSP [3]. More precisely, every ant 𝑘 is able to construct
a complete TSP solution by using a probabilistic rule to move
from city 𝑖 to city 𝑗 as follows:
𝑝
𝑘
𝑖𝑗
=
[𝜏
𝑖𝑗
]
𝛼
[𝜂
𝑖𝑗
]
𝛽
𝑙∈𝒩
𝑘
𝑖
[𝜏
𝑖𝑙
]
𝛼
[𝜂
𝑖𝑙
]
𝛽
, if 𝑗 ∈𝒩
𝑘
𝑖
, (3)
where 𝜏
𝑖𝑗
is the existing pheromone trail between cities 𝑖 and
𝑗, 𝜂
𝑖𝑗
=1/𝑑
𝑖𝑗
is the heuristic information available a priori,
where 𝑑
𝑖𝑗
is the distance between cities 𝑖 and 𝑗 and 𝒩
𝑘
𝑖
denotes
the set of unvisited cities for ant 𝑘 incident to city 𝑖.
The pheromone trails are updated after all ants construct
solutions, where each ant deposits pheromone according to
their solution quality. In this way the next solution construction
is more likely to guide ants to promising areas because of the
distribution of the pheromone trails. Additionally, pheromone
evaporation is performed where small amount of pheromone
is deducted from all trails.
This process continuous until the algorithm eventually
converges. For the DTSP, premature convergence is often
dangerous because when a dynamic change occurs it is difficult
for the population to escape from the old optimum in order
to adapt to the newly generated one. This is due to the high
concentration of pheromone trails around the old optimum that
bias the ants to still construct solutions for the previous envi-
ronment rather than exploring for the new one. The pheromone
evaporation may be able to eliminate the high intensity of
pheromone trails in case they are not frequently used from
the ants.
Several other strategies were proposed to eliminate the
high intensity of pheromone trails in ACO and enhance their
performance in different DOPs such as adaptive pheromone
evaporation [12], immigrant schemes [14], population-based
ACO [9], pheromone re-initialization strategies [8] and multi
caste ACO [16]. The integration of ACO with immigrants
schemes proved that enhances the performance on different
DOPs such as the DTSP with traffic factors [14], [16] and the
DTSP with exchangeable cities [9], [11]. The basic principle
of immigrants schemes in ACO is to introduce new ants
(immigrants) every iteration to replace other ants from the
constructed population and, thus, distribute the pheromone
trails into several areas in the search space rather than to a
single area (usually near the global optimum).
IV. H
YBRID IMMIGRANTS SCHEMES
A. Algorithms’ Framework
Immigrants schemes mainly differ on the way immigrant
ants are generated. Random immigrants ACO (RIACO) and
elitism-based immigrants ACO (EIACO) were proposed for
DTSPs [14]. Random immigrants are generated randomly to
represent random TSP tours, whereas elitism-based immigrants
are generated by swapping cities from the best solution of the
previous environment.
The solution construction of RIACO and EIACO is defined
as in Eq. (3). However, the pheromone update of RIACO
and EIACO differs from conventional ACO algorithms in the
following ways: a) only the 𝑘
𝑠
best ants are allowed to deposit
pheromone in addition with immigrant ants; b) pheromone
evaporation is not available; and c) the pheromone trails exist
only for a single iteration.
1543
Every 𝑡 iteration, the pheromone table is associated with
a short-term memory of 𝑘
𝑠
size, denoted 𝑘
𝑠ℎ𝑜𝑟𝑡
(𝑡) in this
paper, and any change to 𝑘
𝑠ℎ𝑜𝑟𝑡
(𝑡 +1) causes an update to the
pheromone table. Within RIACO and EIACO the 𝑘
𝑠
best ants
from the current iteration 𝑡 replace the ants in 𝑘
𝑠ℎ𝑜𝑟𝑡
(𝑡 − 1).
Furthermore, for RIACO a set 𝑆
𝑟𝑖
of 𝑛
𝑟𝑖
random immigrants
are generated to replace the worst ants in 𝑘
𝑠ℎ𝑜𝑟𝑡
(𝑡).For
EIACO, the best ant from the previous environment is used as
the base to generate a set 𝑆
𝑒𝑖
of 𝑛
𝑒𝑖
elitism-based immigrants
to replace the worst ants in 𝑘
𝑠ℎ𝑜𝑟𝑡
(𝑡). For the DTSP an elitism-
based immigrant is generate by swapping cities with a 𝑝
𝑖
𝑚
probability (usually 𝑝
𝑖
𝑚
=0.01).
When the worst ants are replaced by immigrant ants, the
pheromone trails of each 𝑘-th worst ant are removed, as
follows:
𝜏
𝑖𝑗
← 𝜏
𝑖𝑗
− Δ𝜏
𝑘
𝑖𝑗
, ∀ (𝑖, 𝑗) ∈ 𝑇
𝑘
, (4)
where 𝑇
𝑘
represents the tour of ant 𝑘 and Δ𝜏
𝑘
𝑖𝑗
=(𝜏
𝑚𝑎𝑥
−
𝜏
0
)/𝑘
𝑠
, where 𝜏
𝑚𝑎𝑥
and 𝜏
0
denote the maximum and initial
pheromone values, respectively. Furthermore, the pheromone
trails of each 𝑘-th immigrant ant are added, as follows:
𝜏
𝑖𝑗
← 𝜏
𝑖𝑗
+Δ𝜏
𝑘
𝑖𝑗
, ∀ (𝑖, 𝑗) ∈ 𝑇
𝑘
, (5)
where Δ𝜏
𝑘
𝑖𝑗
and 𝑇
𝑘
are as defined in Eq. (4).
B. Non-Interactive Hybrid Immigrants
RIACO and EIACO were proposed to address severely
and slightly changing DTSPs, respectively. This is natural
because in case the environment changes slightly EIACO is
more suitable due to the knowledge transferred via elitism-
based immigrants. On the other hand, RIACO is suitable in
more severely changing environments.
However, there is a high risk with RIACO to disturb the
optimization process of ACO because of too much random-
ization and with EIACO to trap the optimization process of
ACO into a local optimum because of too much knowledge
transferred. In order to limit both risks and further improve
the performance of ACO algorithms in DOPs, RIACO are
hybridized together with EIACO, to form hybrid immigrants
ACO (HIACO-I). In [11] and [13] hybrid immigrants showed
good offline performance when applied on the DTSP with
exchangeable cities and with traffic factors, respectively. In
this paper HIACO-I is applied on the DTSPs constructed from
the DBGP.
Every iteration 𝑡 within HIACO-I a set 𝑆
ℎ𝑖
(𝑡) of 𝑛
𝑟𝑖
random immigrants and 𝑛
𝑒𝑖
elitism-based immigrants are
generated. In this way, the half of the generated immigrants
will be random immigrants and the other half elitism-based
immigrants. HIACO-I follows the solution construction defined
in Eq. (3) and the same pheromone policy with RIACO and
EIACO defined in Eqs 4 and 5.
C. Interactive Hybrid Immigrants
Within HIACO-I, the two types of immigrants, random
and elitism-based, are generated without the consideration
of the status of algorithm. For example, when the ACO
algorithm reaches stagnation behaviour
1
or when too much
1
A term use to define that behaviour when all ants construct the same
solution from early stages
knowledge in transferred, random immigrants may be more
suitable than elitism-based immigrants, since more diversity is
needed to help escape from local optima. Therefore, another
type of hybrid immigrants in which random and elitism-based
immigrants are triggered interactively can be proposed to
combine the merits of RIACO and EIACO, denoted HIACO-II
in this paper. The basic idea is to generate only elitism-based
immigrants every iteration and when the algorithm reaches
stagnation behaviour to generate only random immigrants.
The 𝜆-branching factor [6] is an efficient way to define
whether ACO has reached stagnation behaviour. The idea of
𝜆-branching is described as follows: If for a given city 𝑖 ∈ 𝑉 ,
the concentration of pheromone trails on almost all the incident
arcs becomes very small but is large for a few others, then the
freedom of exploring other paths from city 𝑖 is very limited.
Therefore, if this situation arise simultaneously for all cities
of graph 𝐺, the search space that is searched by ants becomes
relatively small.
The average
¯
𝜆(𝑡) branching factor at iteration 𝑡 is defined
as follows:
¯
𝜆(𝑡)=
1
2𝑛
𝑛
𝑖=1
𝜆
𝑖
, (6)
where 𝑛 is the number of cities and 𝜆
𝑖
is the 𝜆-branching
factor for city 𝑖, which is defined as follows:
𝜆
𝑖
=
𝑑
𝑗=1
𝐿
𝑖𝑗
, (7)
where 𝑑 is the number of available arcs incident to city 𝑖 and
𝐿
𝑖𝑗
is defined as follows:
𝐿
𝑖𝑗
=
1, if (𝜏
𝑖
𝑚𝑖𝑛
+ 𝜆(𝜏
𝑖
𝑚𝑎𝑥
− 𝜏
𝑖
𝑚𝑖𝑛
)) ≤ 𝜏
𝑖𝑗
,
0, otherwise,
(8)
where 𝜆 is a constant parameter (𝜆 =0.05 by default [6]),
𝜏
𝑖
𝑚𝑖𝑛
and 𝜏
𝑖
𝑚𝑎𝑥
are the minimum and maximum pheromone
trail values on the arcs incident to city 𝑖, respectively. A value
of
¯
𝜆(𝑡) close to 1 indicates stagnation behaviour.
According to the behaviour of the algorithm in terms of
searching and the rules described above, either random or
elitism-based immigrants are triggered as follows:
𝑆
ℎ𝑖
(𝑡)=
𝑆
𝑒𝑖
, if
¯
𝜆(𝑡) >𝜖,
𝑆
𝑟𝑖
, otherwise,
(9)
where
¯
𝜆(𝑡) is defined in Eq. (6) that indicates the status of
the algorithm and 𝜖 is a threshold value. When 𝜖 =1, which
is the default value, indicates stagnation behaviour whereas
a higher value 𝜖>1 indicates exploration. Hence according
to the rules defined in Eq. (9) whenever HIACO-II reaches
stagnation behaviour random immigrants are triggered to move
the population into different areas in the search space.
V. E
XPERIMENTAL STUDY
A. Experimental Setup
In the experiments, we investigate the performance of
HIACO-I and HIACO-II compared with standard ACO
1544
TABLE I. E XPERIMENTAL RESULTS REGARDING THE OFFLINE PERFORMANCE OF ACO ALGORITHMS
Algorithms
& DTSPs
kroA100(Optimum=21282) kroA150(Optimum=26524) kroA200(Optimum=29368)
𝑓 =10,
𝑚 ⇒
0.1 0.25 0.5 0.75 0.1 0.25 0.5 0.75 0.1 0.25 0.5 0.75
SACO 25103.7 26944.7 28749.8 29397.2 31977.0 34683.4 37484.0 38419.8 36939.2 39901.5 43211.3 44379.3
RIACO 24327.5 25843.7 26600.2 26680.7 30871.2 32544.6 33267.3 33414.2 34854.8 36746.6 37446.0 37538.6
EIACO 22992.1 24696.1 25388.5 25440.9 29426.7 31220.2 31824.9 31967.9 33170.7 35228.0 35853.7 35938.9
HIACO-I 23588.1 25395.7 26139.3 26209.6 30222.2 32069.3 32723.2 32874.0 34187.7 36211.0 36894.8 36949.8
HIACO-II 23260.9 24812.1 25144.6 25292.9 29583.8 31350.8 31570.8 31714.6 33192.2 35238.3 35648.0 35719.8
𝑓 = 100,
𝑚 ⇒
0.1 0.25 0.5 0.75 0.1 0.25 0.5 0.75 0.1 0.25 0.5 0.75
SACO 23347.9 23956.2 24187.8 24286.0 29737.6 30422.8 30931.8 30973.5 33895.2 34508.6 35103.3 35195.1
RIACO 22002.0 22465.4 22779.2 22777.8 28015.7 28441.1 28703.2 28689.7 30782.4 31557.3 32057.9 32028.0
EIACO 22096.6 22494.5 22834.0 22874.8 27865.6 28410.3 28567.5 28622.1 30749.1 31347.1 31716.1 31805.9
HIACO-I 21921.6 22288.7 22675.2 22694.4 27708.1 28263.7 28504.5 28497.6 30603.2 31262.6 31710.9 31803.1
HIACO-II 21917.4 22345.8 22756.3 22785.6 27935.5 28378.3 28595.3 28544.7 30620.8 31285.6 31684.2 31766.8
(SACO), RIACO and EIACO. All common algorithmic param-
eters were set as follows: 𝛼 =1, 𝛽 =5and the population
size 𝜇 =50. 𝑘
𝑠
was set to 6 and the number of immigrants
was set to 𝑛
𝑟𝑖
= 𝑛
𝑒𝑖
= 𝑛
ℎ𝑖
=2for RIACO, EIACO, HIACO-
I and HIACO-II. 𝜌 was set to 0.5 for SACO and 𝜖 was set to
1 for HIACO-II.
DTSPs are generated from three stationary benchmark
instances obtained from TSPLIB
2
using the DBGP generator
with 𝑓 set to 10 and 100 indicating quickly and slowly chang-
ing environments, respectively, and 𝑚 set to 0.1, 0.25, 0.5
and 0.75, indicating slowly, to medium, to severely changing
environments, respectively. Totally, a series of 8 DTSPs are
constructed from each stationary instance.
For each ACO algorithm on a DTSP, 30 independent runs
were executed on the same set of random seed numbers. For
each run 1000 iterations were allowed and the best so far
ant after a dynamic change was recorded every iteration. The
overall offline performance [10] of an ACO on a DTSP is
defined as follows:
¯
𝑃
𝑂𝐹𝐹
=
1
𝐸
𝐸
𝑖=1
1
𝑅
𝑅
𝑗=1
𝑃
∗
𝑖𝑗
, (10)
where 𝑅 is the number of runs, 𝐸 is the number of iterations,
and 𝑃
∗
𝑖𝑗
defines the tour cost of the best ant of iteration 𝑖 of
run 𝑗. Moreover, the population diversity [13] was recorded as
follows:
¯
𝑇
𝐷𝐼 𝑉
=
1
𝐸
𝐸
𝑖=1
1
𝑅
𝑅
𝑗=1
𝐷𝐼𝑉
𝑖𝑗
, (11)
where 𝐷𝐼𝑉
𝑖𝑗
is the diversity of the population in iteration 𝑖
of run 𝑗. For the DTSPs, 𝐷𝐼𝑉
𝑖𝑗
can be calculated as follows:
𝐷𝐼𝑉
𝑖𝑗
=
1
𝜇(𝜇 − 1)
𝜇
𝑝=1
𝜇
𝑞∕=𝑝
𝑀(𝑝, 𝑞), (12)
2
http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/
where 𝜇 is the size of population and 𝑀 (𝑝, 𝑞) is the similarity
metric between ants 𝑝 and 𝑞 defined as:
𝑀(𝑝, 𝑞)=1−
𝑐
𝐸
𝑝𝑞
𝑛
, (13)
where 𝑐
𝐸
is defined as the number of common edges (arcs)
between the solutions of ants 𝑝 and 𝑞, and 𝑛 is the number
of cities. A value 𝑀(𝑝, 𝑞) closer to 0 means that the ants are
more similar.
B. Basic Experimental Results and Analysis
The experimental results regarding the offline performance
of the investigated ACO algorithms for each dynamic test case
are presented in Table I. The corresponding statistical results
are presented in Table II, where Kruskal–Wallis tests were
applied followed by posthoc paired comparisons using Mann–
Whitney tests with the Bonferroni correction. In Table II,
the results are shown as “+”, “−” and “∼” when the first
algorithm is significantly better than the second one, when
the second algorithm is significantly better than the first one
and when the two algorithms are not significantly different,
respectively.
Moreover, the dynamic behaviour of ACO algorithms in
terms of offline performance of the first 10 environmental
changes with 𝑓 =10and 𝑚 =0.1 for the first 5 environmental
changes with 𝑓 = 100 and 𝑚 =0.75 is presented in Fig. 1
and Fig. 2, respectively. The corresponding dynamic behaviour
in terms of population diversity is presented in Fig. 3 and
Fig. 4. From the experimental results, several observations
can be made by comparing the behaviour of the investigated
algorithms.
First, RIACO significantly outperforms SACO in all dy-
namic test cases; see the comparisons of RIACO ⇔ SACO in
Table II. This supports our claim that the conventional SACO
biases the population towards non-promising areas due to the
high concentration of pheromone trails generated from previ-
ous environments. On the other hand, RIACO eliminates the
pheromone trails of the previous environments directly because
of the different pheromone update policy. Furthermore, RIACO
significantly outperforms or it is comparable with EIACO
1545
TABLE II. EXPERIMENTAL RESULTS REGARDING THE OFFLINE PERFORMANCE OF ACO ALGORITHMS
Algorithms & DTSPs kroA100 kroA150 kroA200
𝑓 =10, 𝑚 ⇒ 0.1 0.25 0.5 0.75 0.1 0.25 0.5 0.75 0.1 0.25 0.5 0.75
RIACO ⇔ SACO ++ ++ ++ ++ ++ ++
EIACO ⇔ SACO ++ ++ ++ ++ ++ ++
EIACO ⇔ RIACO ++ ++ ++ ++ ++ ++
HIACO-I ⇔ SACO ++ ++ ++ ++ ++ ++
HIACO-I ⇔ RIACO ++ ++ ++ ++ ++ ++
HIACO-I ⇔ EIACO −− −− −− −− −− −−
HIACO-II ⇔ SACO ++ ++ ++ ++ ++ ++
HIACO-II ⇔ RIACO ++ ++ ++ ++ ++ ++
HIACO-II ⇔ EIACO −− ++ −− ++ ∼∼ ++
HIACO-II ⇔ HIACO-I ++ ++ ++ ++ ++ ++
𝑓 = 100, 𝑚 ⇒ 0.1 0.25 0.5 0.75 0.1 0.25 0.5 0.75 0.1 0.25 0.5 0.75
RIACO ⇔ SACO ++ ++ ++ ++ ++ ++
EIACO ⇔ SACO ++ ++ ++ ++ ++ ++
EIACO ⇔ RIACO ∼∼ −− + ∼ + ∼ ∼ +++
HIACO-I ⇔ SACO ++ ++ ++ ++ ++ ++
HIACO-I ⇔ RIACO ++ ++ ++ ++ ++ ++
HIACO-I ⇔ EIACO ∼ +++++ ∼ + ++ ∼∼
HIACO-II ⇔ SACO ++ ++ ++ ++ ++ ++
HIACO-II ⇔ RIACO ++ ∼∼ ∼∼ ++ ++ ++
HIACO-II ⇔ EIACO ∼∼ ++ ∼∼ ∼∼ + ∼∼∼
HIACO-II ⇔ HIACO-I ∼∼ −− −− −∼ ∼∼ ∼∼
in some dynamic test cases (i.e., kroA100) with 𝑓 = 100
and 𝑚 =0.5 and 𝑚 =0.75 which can be observed from
Fig. 2. However, RIACO usually performs significantly worse
than EIACO in most DTSPs because random immigrants may
disturb the optimization process. This can be observed from
Fig. 3 and Fig. 4 where RIACO maintains the highest diversity
among other ACO algorithms that follow the same pheromone
update policy.
Second, EIACO significantly outperforms SACO in all
dynamic test cases and RIACO in most dynamic test cases
(except DTSPs with 𝑚 =0.5 and 𝑚 =0.75); see the
comparisons of EIACO ⇔ SACO and EIACO ⇔ RIACO,
respectively, in Table II. This supports our claim when the
changing environments are similar then the knowledge trans-
ferred via elitism-based immigrants in EIACO may guide the
population towards promising areas which can be observed
from Fig. 1. Moreover, it can be observed from Fig. 3 and
Fig. 4 that EIACO maintains the lowest diversity than its
competitors.
Third, HIACO-I significantly outperforms SACO and RI-
ACO in most dynamic test cases; see the comparisons of
HIACO-I in Table II. When comparing the performance of
HIACO-I over EIACO it can be observed that HIACO-I
significantly outperforms EIACO on DTSPs with 𝑓 = 100,
whereas EIACO significantly outperforms HIACO-I on DTSPs
with 𝑓 =10. Fig. 2 shows that HIACO-I often performs
better than its competitors. This supports our claims that the
hybridization may further improve the performance of ACO
algorithms since it may keep a good balance of the diversity
generated. This can be observed from Fig. 3 and Fig. 4 where
HIACO-I maintains lower level of diversity than RIACO but
higher level of diversity than EIACO.
Finally, HIACO-II significantly outperforms SACO and
RIACO in most dynamic test cases; see the comparison of
HIACO-II in Table II. When comparing the performance of
HIACO-II over EIACO it can be observed that HIACO-II
significantly outperforms EIACO on DTSPs with 𝑓 =10and
𝑚 =0.5 and 𝑚 =0.75, whereas EIACO is comparable with
HIACO-II on most DTSPs with 𝑓 = 100. When comparing the
performance of the two hybrid algorithms it can be seen that
HIACO-II significantly outperforms HIACO-I in DTSPs with
𝑓 =10whereas HIACO-I outperforms HIACO-II in DTSPs
with 𝑓 = 100; see the comparisons of HIACO-II ⇔ HIACO-I
in Table II.
C. Experimental Results of the 𝜖 Parameter in HIACO-II
The performance of HIACO-II is furthermore investigated
with different 𝜖 parameters defined in Eq. (9). This parameter
defines the threshold to trigger random immigrants. For exam-
ple, when 𝜖 =1.0 then
random immigrants are triggered when
HIA
CO-II reaches stagnation behaviour. When 𝜖>1.0, then
random immigrants are triggered when HIACO-II reaches near
to stagnation behaviour. The offline performance of HIACO-
II with different 𝜖 values is given in Fig. 5 for DTSPs with
𝑓 = 100
3
.
In the basic experiments 𝜖 was set to 1.0 that indicates
stagnation behaviour. However, it can be clearly observed from
Fig. 5 that when 𝜖 =1.1 the performance of HIACO-II is
improved on most DTSPs. This shows that once the population
reaches stagnation behaviour, then it is difficult to escape from
it even if only random immigrants are generated. However,
3
The corresponding results for DTSPs with 𝑓 =10have similar observa-
tions and thus they are not presented
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