978-1-5090-1897-0/16/$31.00 ©2016 IEEE
Study on the Impact of the NS in the Performance
of Meta-Heuristics in the TSP
A. S. Santos*, A. M. Madureira
**
, M. L. R. Varela
*
,
*
University of Minho / School of Engineering - Dept. of Production and Systems, Guimarães, Portugal
**
Polytechnic Institute of Porto / School of Engineering - GECAD Research Group, Porto, Portugal
abg@isep.ipp.pt, amd@isep.ipp.pt, leonilde@dps.uminho.pt
Abstract — Meta-heuristics have been applied for a long time to
the Travelling Salesman Problem (TSP) but information is still
lacking in the determination of the parameters with the best
performance. This paper examines the impact of the Simulated
Annealing (SA) and Discrete Artificial Bee Colony (DABC)
parameters in the TSP. One special consideration of this paper is
how the Neighborhood Structure (NS) interact with the other
parameters and impacts the performance of the meta-heuristics.
NS performance has been the topic of much research, with NS
proposed for the best-known problems, which seem to imply that
the NS influences the performance of meta-heuristics, more that
other parameters. Moreover, a comparative analysis of distinct
meta-heuristics is carried out to demonstrate a non-proportional
increase in the performance of the NS.
Keywords - Meta-heuristics, Simulated Annealing, Discrete
Artificial Bee Colony, Neighborhood Structures, TSP.
I.
I
NTRODUCTION
Complex Combinatorial Problems (COPs) such as the
Travelling Salesman Problem (TSP) is a classic combinatorial
optimization problem, which has been applied in, logistics,
transportation, networking and commercial domains [1-7]. It is
certainly one of the most well-known and widely studied
problem in the domain of COPs that can be defined as the
problem of finding a minimum distance tour of N cities,
starting and ending at the same city and visiting other city
exactly once [1]. In this paper, a Simulated Annealing (SA) and
Artificial Bee Colony algorithm (DABC) are used to approach
TSP and to determine how much the chosen NS impacts the
performance of either meta-heuristics.
Computational tests were performed to demonstrate that the
proposed meta-heuristics, for instance, the SA algorithm, and
the DABC algorithm are suitable for being used in the
Euclidian-TSP. Moreover, in this paper the authors intend to
show that NS should be treated as another parameter, that
influences the performance of the optimization techniques.
Therefore, in this paper four well-known NS are examined with
SA and DABC in the Symmetric Euclidian TSP. Furthermore,
a statistical analysis of the performance of the NS is presented.
The structure of this paper is the following: section II
overviews the developments in meta-heuristics, and presents in
more detail Simulated Annealing (SA) and Discrete Artificial
Bee Colony algorithm (DABC). Section III briefly describes
the neighbourhood structure (NS) examined in the
computational study. Section IV details the instances used in
the computational study, the parameters for both meta-
heuristics before the results are presented and explored and
finally the sections ends with the statistical study to
demonstrate how the NS influences the behaviour of the meta-
heuristics. Section V presents the conclusions.
II. M
ETA
-
HEURISTICS AND DECISION MAKING
Meta-heuristics are the main techniques to address complex
computational problems, such as, combinatorial optimization
problems, which have shown to be the approach techniques
with better performance [8]. They can be defined as an iterative
procedure inspired by nature that guides a subordinate heuristic
in the exploration of the solution space, [9,10]. Unlike the
Neighborhood Search methods that analyze solutions within
the solutions space belonging to a particular neighborhood,
meta-heuristics have mechanisms to move beyond the local
optima solutions, making them particularly interesting in
addressing Complex Combinatorial Problems (COPs). As an
approximation method, a meta-heuristic, cannot guarantee
optimal solutions, but usually outperform other less evolved
approximation optimization techniques. In general, meta-
heuristics parameters compromise between the technique
effectiveness and efficiency [9].
There are two distinct classes of meta-heuristics: the single
solution based meta-heuristics and population based meta-
heuristics. Meta-heuristics are then divided by the number of
solutions they use. Single solution meta-heuristics are inspired
by Neighborhood Search techniques, and the population-based
meta-heuristics are based on Genetic Algorithms (GA). Usually
the single solution based meta-heuristics have higher intensity
than population meta-heuristics, since population based meta-
heuristics simultaneously use several solutions to explore a
larger portion of the solution space, which makes them also
more time consuming [9,10].
A. Simulated Annealling
Simulated Annealing (SA) is one of the most efficient meta-
heuristic to approach complex optimization problems, through
adaptation of the procedure of neighborhood search techniques,
which would enable it to overtake local optimums [11,12]. It is
a particularly appealing meta-heuristic to address Complex
Combinatorial Problems (COPs), since it can find near-optimal
solutions, which, in a real-world environment are usually
suitable, without much computational effort [1,7]. Further we
discuss the parameters of SA to show how easily they can be
tuned to optimize the performance of the meta-heuristic in a
specific problem. Results in literature demonstrate how suitable
the SA meta-heuristics is in the Symmetric Euclidean-TSP
problem.
Simulated Annealing selects one solution between a finite
number of possible solutions, but unlike neighbourhood search
techniques it allows movements that worsen the current
solution. It can find near-optimal solutions for combinatorial
problems with low computational effort, and it is relatively
simple to implement and manipulate its parameters.
In the implemented software, SA provides alternative
solutions as well as the evolution path of the meta-heuristics
optimization procedure. Such implementation allows the user
to visualize the movement through the solution space with
advantages and disadvantages of the parameters. With such
information the decision maker understands how the
parameters impact the optimization technique. The motivation
for using SA is four fold. First, we can solve complex
combinatorial optimisation problems. Second, it can be easily
understood since its procedure is based on the annealing
process [11,12] and it does not include too many parameters.
Third, since it is a guided-random search algorithm [11,12], it
allows us to control its path through the solutions pace with the
parameters of the meta-heuristic [11,12]. Finally, the algorithm
is simple to implement and it does not require too much
computational time to find decent solutions, since the SA
search procedure is efficient and robust [11,12].
B. Discrete Artificial Bee Colony
Artificial Bee Colony (ABC) is one of the most recent
optimization techniques inspired by bee behaviour, the Queen
Bee Evolution (QBE), based on colony structure, Marriage in
Honey Bees Optimization (MBO) in the reproduction process,
and Bee Colony Optimization (BCO) or Virtual Bee Algorithm
(VBA), in search of food [20], which was developed by
Karaboga, in 2005 [13] and Pham et al., in 2005 [14] and was
inspired by the behaviour of a colony of bees in search for
food. Through this meta-heuristic the search within the solution
space is performed by three types of bees: worker bees,
opportunistic bees, and scout bees. ABC as been developed to
approach continuous problems, not discrete problems, such as,
scheduling problems. Moreover, recently there have been
proposed versions of the ABC for discrete problems, the
Discrete Artificial Bee Colony (DABC) procedure, including a
version proposed by Karaboga & Gorkemli to address the
Symmetric Euclidian TSP problem [15].
The DABC is similar to ABC, just changing the way the
bees move in the neighbourhood of a source of food, which is
carried out by manipulating the solutions, by transforming
them into neighbourhood solutions.
The DABC works through the interaction of three phases,
until the interruption criterion: the phase of the worker bees,
the phase of opportunistic bees and the phase of the scout bees.
Initially it is determined a food source (x
i
) for each working
bee, usually randomly or partially randomly with a heuristic
component. At the phase of working bees, each one will
explore a solution in the neighbourhood (v
i
) of its food supply,
and if it yields a superior performance than the current food
source, the new one should replace that food source. Next
occurs the phase of opportunistic bees, which wait for the
performance of each food source to determine which food
source they will explore. This means that the opportunistic bees
wait in the colony for the information provided by the worker
bees and probabilistically selects a food source. Once it chose a
food source, the opportunist bee will explore a solution in its
neighbourhood and if a solutions with a better performance
than the current source of food, then it replaces it. The phase of
scout bees, occurs once a food source has been abandoned. A
food source is abandoned once l iterations did occur without
improving the performance. Karaboga & Gorkemli in [15]
present values that indicate the number of iterations without
improvement that should occur before a food source can be
abandoned, knowing that the higher l conduct to a more
intensive search while the lower l conduct to a search with
higher diversity. Once a food source is abandoned, the working
bee will become a scout bee and move to a new food source.
III. N
EIGHBOURHOOD
S
TRUCTURES
One parameter common to all meta-heuristics, and even
other optimization techniques, is the neighborhood structure.
NS manipulate solutions to turn one solution into one another
in one basic movement. In SA, the NS determines how a
solution is turned into another, in each iteration, while in
DABC, the NS determines how worker and outlooker bees
move around their food source. Mechanisms that manipulate
several solutions, as the crossover mechanism in the Genetic
Algorithms, are also NS that cross two parental solutions into a
child solution. Mutation operators that manipulate, some or all,
of the created solutions, can be understood as part of the NS to
introduce diversity to the optimization procedure. There are
dozens of simple NS that are adequately broad-scoped to be
applied in a vast number of problems, other more advanced NS
are often developed to be used in a specific problem.
In the computational study, four NS will be reviewed. NS1,
2-swap or 2-Exchange, is a notorious neighbourhood structure
for scheduling problems, it selectes two random operations and
swaps their positions [16]. For example a execution sequence
of [1,2,3,4,5,6] where the 2
nd
and 4
th
operations were randomly
selected would become [1,4,3,2,5,6] as demostrated in figure 3.
Since TSP is a sequence problem, the NS1 can also be used.
NS2, or 2-opt, is a more tradional choice for TSP problems, it
selects two paths at random, disconnects and reconnects them
in another manner [6]. For example the tour [1,2,3,4,5,6],
where the path [1,2] and [4,5] were selected woud become
[1,4,3,2,5,6], as demostrated in figure 3. NS3 and NS4 combine
NS1 and NS2 into a more ample and diverse NS. NS3 uses
NS2 followed by NS1, or does a 2-opt movement followed by
2-swap, NS4 uses NS2 twice, or uses 2-opt twice.
Figure 1. NS1 and NS2
Studies about the performance of NS have examined how the
NS influences the behavior of local-search techniques. In. [17],
Ahuja et al presents an overview of well-known NS for the
Euclidian-TSP problem that demonstrate the need to balance the
size of the NS and the available time. In meta-heuristics, the
performance of NS has been the topic of much research, with
several NS proposed and studied for the best-known problems,
however this seems to infer that the NS is more important than
the other parameters. Since meta-heuristics are bestowed with
mechanisms to move from the local-optimums, the NS should be
presented as another parameter, which interacts with the other
parameters to determine the overall balance between the
intensity/diversity of the meta-heuristic search procedure.
IV. R
ESULTS
A
NALYSIS
SA and DABC, with the neighborhood structures (NS)
previously defined, were implemented in c in Microsoft Visual
Studio 2012. The computational tests were performed on a
MacBook Pro with a 3GHz Intel® Core i7 processor, 16GB of
1600MHz RAM and Windows 10 64-bit.
Both meta-heuristics were tested with all the implemented
NS in 5 academic benchmark instances of the Symmetric
Euclidian TSP [6], KroA100, KroB100, KroC100, KroD100
and KroE100, with 100 nodes each, which correspond to
9.33262
Ε
157 possible solutions. All the problems are available
in the TSPLIB, which also presents the optimal solutions for
each problem, for KroA100 is 21282, for KroB100 is 22141,
for KroC100 is 20749, for KroD100 is 21294 and for KroE100
is 22068, which allows a better comparison of the results of
both meta-heuristics with all the NS across the 5 instances,
since solutions can be normalized around the optimal solution
in each instance of the problem.
A. Parameterization
Meta-heuristics parameterization has concentrated the
attention of researchers, as an appropriate definition of the
parameters can improve the technique performance in a
specific application. However the parameterization can be time
intensive, often harder than the implementation of the meta-
heuristics [18]. Hunter et al. [19] presents a formal definition of
the parameterization, as: “Given a parameterised target
algorithm A, a set of problem instances I and a performance
metric c, the goal is to find parameter settings of A that
optimize c in I”.
In this paper the relation between the NS and the rest of the
parameters will be studied, instead of the overall performance
of the meta-heuristics. Moreover each meta-heuristics will be
tested with parameters that find solutions in less that 1 and 10
seconds, to examine how the performance of each NS varies
with the increase of the computational time, to examine the
evolution of the solutions. Since all the defined NS have
distinctive levels of intensity/diversity, it is expected that the
evolution of the solutions will depend on the characteristics of
each of the NS.
In SA the parameterization process is simpler than in other
meta-Heuristics, since with enough time SA performs well,
even without a detailed parameterization procedure [20, 21].
However, with limited computational time the parameters need
to be precisely determined, in order to conduct an efficient
search of the solution space. SA parameters include the Initial
Temperature, the Cooling Ratio, the Epoch Length, the
Neighborhood Structure (NS) and the Stoppage Condition.
Park & Kim in [20] examined the impact of each parameter in
SA and presented some parameterization rules: the initial
temperature (T
i
) should accept almost all movements, which
means the initial acceptance probability (P
i
) should be near 1;
the cooling ratio (
α
) should result in a slow decrease of the
temperature, which means that the cooling ration with a
proportional cooling function should be between [0.80;0.90];
finally, the epoch length (L) should be related to the instance
size, which means that it should be close to the dimension of
the problem or the number of neighborhood solutions.
Since this paper will study the performance of each NS
bellow 1 and 10 seconds of computational time, the Stoppage
Condition will be the number of iterations, while the NS will
be the one presented in the previous point. In table I are the rest
of the parameters for both 1 and 10 seconds. It is important to
notice that the overall performance of SA is not the main focus
of this paper; instead it will examine how the performance of
each NS varies, under different time limitations. It is expected
that when the computational time increases, the NS that
provides more diversity will result in a better performance.
In DABC the parameterization is more complex than SA, as
the parameters have a tremendous impact in the performance of
DABC, more pronounced than in SA, with less parameters.
DABC parameters include the Colony Size, the Limit, the
Neighborhood Structure (NS) and the Stoppage Condition.
Akay & Karaboga [22] examined the impact of each parameter
in ABC and presented some parameterization rules: the colony
size (L) does not need an increase of the population sizes to
solve large optimization problems since this parameter has an
enormous impact in the computational time; the limit (l) is
related to the colony size, larger colonies can have smaller
limits, since the lack of diversity from smaller limits are
balanced by the number of individuals in the population. In
[15] the authors presented some rules to determine the limit,
based on the number of bees and the dimension of the problem.
Other parameters considered in some implementations of ABC
and DABC, some authors have used a fixed number of scout
bees to increase the diversity of the Meta-Heuristic.
Like in SA, the stop criterion will be the number of
iterations, since the purpose is to examine the performance of
each NS bellow 1 and 10 seconds of computational time.
DABC requires more computational time per iteration than SA,
so there will be a difference in the number of iterations for SA
and DABC. In table I are the rest of the parameters for both 1
and 10 seconds. Once more, the focus on this paper is not the
overall performance of DABC; it will examine the performance
of each NS under different time limitations and compare how
the performance of each Meta-Heuristic is affected.
TABLE I. P
ARAMETERS OF
SA
AND
DABC
Parameter
1s 10s
SA
N. of Iterations 200000 2000000
Initial Temperature (T
i
) 2000 20000
Cooling Ratio (
α
)
0.95 0.95
Epoch Length (L) 500 500
DABC
N. of Iterations 10000 100000
Colony Size (L) 20 20
Limit (l) 1000 10000
SA and DABC parameters were determined so that the
procedure would conclude in less than 1 second, and then
adapted to conclude in less than 10 seconds, with an increased
of the number of iterations and the initial temperature in SA
and the limit in DABC. Some parameters were used for both 1
and 10 seconds and did not impact the run-time of either Meta-
Heuristic.
B. Computational Results
Both meta-heuristics were run five times for each problem
and achieved computational results close to the optimum. In
figure 4 the top row represents the computational result of SA,
and the bottom row represents the computational results of
DABC, for 1 and 10 seconds, with each neighborhood structure
results represented in a different color.
In SA with 1 second, NS2 found the best solutions (21459,
22552, 20872, 21508, 22122), close to the optimal solutions,
represented in blue. NS4 found the second best solutions
(24070, 24991, 24140, 24321, 24794) and NS3 (25038, 25730,
25526, 24587, 25662) the third best. NS3 and NS4 solutions
are similar, with a small lead in the performance of NS4. NS1
found the worst solutions (25527, 27718, 26510, 26357,
29442) and also appeared to have more variability across the
problems, for example, in KroA and KroE. In SA with 10
seconds, NS2 found the best solutions (21353, 22284, 20812,
21398, 22216). NS4 found the second best solutions, in KroB,
KroC, KroD and KroE, (21601, 22535, 20905, 21700, 22717),
NS3 the third best solutions in all problems but KroA (21357,
22888, 21007, 22128, 23078) and NS1 the worst solutions
(24031, 24566, 22888, 22814, 24311) in all problems.
All the NS performed better with the 10 seconds limit and,
other than NS2 that already found near optimal solutions with
the 1 second limit, all NS appeared to improve identically. NS1
and NS2 intense searches did not appear to be less competitive
than either NS3 or NS4, with their more diverse searches, even
with the increase of the available computational time.
In DABC with 1 second, NS2 found the best solutions,
(22225, 23650, 21981, 22552, 23161), but unlike SA the
difference to the other NS is more pronounced. NS4 found the
second best solution (31739, 32341, 31690, 32302, 31868),
followed by NS3 (34693, 35482, 34406, 35290, 36253) and
NS1 (36383, 35157, 37416, 36775, 36202). In DABC with 10
seconds, NS2 found the best solutions (21504, 22544, 21089,
21938, 22654) and NS4 found the second (22586, 24332,
22335, 22712, 23772), improving the solutions until they were
almost optimal. NS3 found the third best solutions (23989,
24814, 23624, 23793, 24834) and NS1 found the worst
solutions (31445, 32701, 33473, 32953, 34060), but this time
the difference in performance between NS1 and the other NS is
noticeable, this is evident in figure 1 that shows that NS1
solutions are almost 50% worse than the other NS.
Once more, all the NS improved their performance in the 10
seconds limit, nevertheless, in opposition to the SA, there
appears to be a difference between the improvement of the NS1
and NS2 more intense searches and the NS3 or NS4 more
diverse searches. NS3 and NS4 appeared to improve much
more that NS2 and specially NS1, which barely improved and
had the worst result from the computational test.
Overall SA performed better than the DABC, which is much
more noticeable in the 1 second experiment. In the 10 seconds
experiment the performance of both meta-heuristics improved,
but there appears to be a difference in improvement of NS1 and
NS2, and NS3 and NS4 in DABC, where NS1 and NS2 did not
improve as the NS with more diverse searches.
Figure 2. Results of SA and DABC
C. Statistical Results
In the computational trials, there appears to be a
difference in the evolution in the NS performance,
however to examine this difference in more detail the
computational results need to be normalized to compare
the results across all instances. Since the optimal solutions
KroA, KroB, KroC, KroD and KroE, are available in
TSPLIB, this document will use mean percent devotion
from the optimal solution, which is the best know metric
to compare different meta-heuristics [23]. Once the results
are normalized, the improvement of each of the NS
performance with the available time will be examined.
SA normalized results are presented in figure 3. In
KroA, the deviations for 1 second are 19.946%, 0.832%,
17.649% and 13.100% for NS1, NS2, NS3 and NS4, for
10 seconds the deviations are 12.917%, 0.334%, 0.352%
and 1.499%. In KroB, the deviations for 1 second are
25.189%, 1.856%, 16,210% and 12.872%, for 10 seconds
the deviations are 10.953%, 0.646%, 3.374% and 1.780%.
In KroC, the deviations for 1 second are 27.765%,
0.593%, 23.023% and 16.343%, for 10 the deviations are
10.309%, 0.304%, 1.243% and 0.752%. In KroD, the
deviations for 1 second are 23.777%, 1.005%, 15.464%
and 14.215%, for 10 the deviations are 7.138%, 0.488%,
3.917% and 1.907%. In KroE, the deviations for 1 second
are 33.415%, 0.245%, 16.286% and 12.353%, and for 10
seconds 10.164%, 0.671%, 4.577% and 2.941%.
Figure 3. Results from SA with 1 and 10 Seconds
DABC results are presented in figure 6. In KroA, the
deviations for 1 second are 70.957%, 4.431%, 63.016%
and 49.135%, for 10 seconds the deviations are 47.754%,
1.043%, 12.720% and 6.127%. In KroB, the deviations for
1 second are 58,787%, 6.815%, 60.255% and 46.068%,
for 10 seconds the deviations are 47.694%, 1.820%,
12.073% and 9.896%. In KroC, the deviations for 1
second are 80.327%, 5.908%, 65.820% and 52.730%, for
10 the deviations are 61.323%, 1.639%, 13.856% and
7.644%. In KroD, the deviations for 1 second are
72.701%, 5.908%, 65.727% and 51.695%, for 10 the
deviations are 54.753%, 3.024%, 11.736% and 6.659%. In
KroE, the deviations for 1 second are 64.047%, 4.953%,
64.279% and 44.408%, for 10 seconds the deviations
improve to 54.341%, 2.655%, 12.534% and 7.722%.
Figure 4. Results from DABC with 1 and 10 Seconds
Overall, the improvement of NS1 and NS2 appeared
less pronounced than the improvements of NS3 and NS4.
In SA the difference in the improvement of NS1 and NS2,
and NS3 and NS4 is not as prominent as in DABC. In SA,
NS1 and NS2 had a mean improvement of 8.069%, and
NS3 and NS4 an improvement of 13.517%, while in
DABC, the NS1 and NS2 improved 9.881% and NS3 and
NS4 improved 46.216%. However, to compare the
differences between the mean in improvement of NS1/2
and NS3/4 in SA and DABC, require the Wilcoxon Mann-
Whitney test, with the hypothesis:
H
:
/
/
H
:
/
/
The Wilcoxon test results, in table II, demonstrate that
there are statistical differences between the mean
improvement of NS1/2 and NS3/4 in DABC (0.000) but
not in SA (0.239).
TABLE II. W
ILCOXON
M
ANN
-W
HITNEY
T
EST
Parameter
SA DABC
Mann-Whitney U
34.000 0.000
Wilcoxon W 89.000 55.000
Z -1.210 -3.780
Asymp. Sig. (2-tailed) 0.226 0.000
Exact Sig. [2*(1-tailed Sig.)] 0.247 0.000
Exact Sig. (2-tailed) 0.239 0.000
Exact Sig. (1-tailed) 0.120 0.000
SA less impactful parameterization procedure can
explain the lack of variance in the performance of NS1/2
and NS3/4 with the increase of the available time. Park &
Kim in [25] state that SA do not require a detailed
parameterization procedure in order to perform well, if
there is enough computational time, which explain the SA
results in the 10s tests. In opposition, in DABC the results
demonstrate that NS3/4 improved more than NS1/NS2
with the increase of the available time, since DABC
requires a detail definition of the meta-heuristics
parameters.