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978-1-5386-7942-5/19/$31.00 ©2019 IEEE
Solving Traveling Salesman Problem using Firefly
algorithm and K-means Clustering
Ameera Jaradat
Department of Computer Sciences
Yarmouk University
Irbid-Jordan
ameera@yu.edu.jo
Bara’ah Matalkeh
Department of Computer Sciences
Yarmouk University
Irbid-Jordan
bfmatalkah@hotmail.com
Waed Diabat
Department of Computer Sciences
Yarmouk University
Irbid-Jordan
aldiabatw@gmail.com
Abstract—Traveling Salesman Problem (TSP) is a
significant combinatorial optimization problem. TSP is NP-
hard problem which involves finding the minimum tour length
among a given set of nodes and return to the starting node
knowing that each node must be visited once. This paper
proposes a solution to TSP using Firefly Algorithm (FA) and k-
means clustering. The proposed approach comprises three
major steps: cluster the nodes, find the optimal path in each
cluster, and reconnect the clusters. The first step uses k-means
clustering to divide the nodes into sub-problems, the second
step utilizes FA to find the optimal path in each cluster, finally
reconnect all clusters and return the path between them. The
experimental results show that the proposed approach
provides better results when compared to other algorithms
from the literature.
Keywords—traveling salesman problem,firefly algorithm,k-
means clustering, NP-hard problem
I. INTRODUCTION
Travelling Salesman Problem (TSP) is a classical
combinatorial optimization problem, easy to describe but
hard to solve [1]. TSP consists of salesman and N cities, the
goal of this problem is to visit each city by the salesman once
with minimum tour length. The challenge in TSP is to find
the route that has a minimum distance.
TSP has many applications in vehicle routing and
communication networks [2]. Therefore, finding a solution to
the TSP is of a great importance. TSP is classified as a hard
problem, since finding the shortest route over a set of n cities
requires the total comparisons of (n-1)! Permutation vectors
[2]. Researchers have attempted to apply variety of heuristic
and Meta heuristic approaches. Meta heuristic attempts to
find optimal solutions through iteratively enhancing a
candidate solution taking in respect a measure of solution
quality [2, 3]. Although optimal solution is not guaranteed
by these heuristics, a near optimal results can be produced.
The goal of TSP is to find the minimum route among all
provided cities, visiting each city exactly once and returning
to the point of departure. TSP seems to be a simple problem,
but it is one of the most significant classical optimization
problems and it has been verified difficult to solve [1, 2].
This paper proposes a new approach to solve the TSP. First,
the approach assembles the nodes into a number of clusters.
Then, the firefly algorithm is applied to find the minimum
rout length in each cluster. Finally, the clusters are grouped
together through one global route.
Clustering involves partitioning or grouping a given set
of data into disjoint clusters. K-means is a simple
unsupervised learning algorithm that was developed to the
problem of clustering [6]. K-means has been widely
implemented in clustering applications due its effectiveness
in producing good results [8, 9].
Firefly algorithm is heuristic practices swarm
optimization algorithm that was formulated by Xin-She
Yang in 2008 [4]. The firefly algorithm stimulates the
flashing activities of fireflies and the phenomenon of
bioluminescent communication in solving problems. Firefly
algorithm has proven to be effective in solving many
optimization problems [5, 10].
TSP, classified NP complete, has attracted many
researchers of this field to find optimal tour. Recently,
researchers approaches the problem by developing heuristic
algorithms [8, 9].
Saloni Gupta and Poonam Panwar [7] provided a solution
to TSP using Genetic Algorithm (GA). The proposed
approach calculates the distances between all cities to be
visited and form a matrix of these distances. Then, the initial
city is selected randomly, which is followed by sequence of
genetic steps to extract the best path.
Li-Zhuang et al. [8] combined Genetic Algorithm (GA)
with k-mean clustering method to solve the TSP. The
approach involves dividing the cities into several groups by
using k-mean clustering. Then, apply Genetic Algorithm on
each group to find the sub-path.
Tanasanee Phienthrakul [9] proposed clustering
techniques (Gaussian mixer and k-means) to divide the cities
to smaller problems by grouping the nearest nodes with each
other. Then the optimal path is found in each cluster by
applying GA and Ant Colony Optimization (ACO). Finally,
clusters are connected to find the optimal path.
The main goal of this work is to propose a new approach
for solving travelling salesman problem (TSP). The approach
divides the graph into groups of nodes. Then, it uses FA
algorithm to generate best routs for each group. Later, these
groups are connected together in one global rout.
The rest of this paper is organized as follows: Section II
illustrates the firefly algorithm. Section III describes the k-
means clustering technique. The proposed method in Section
IV. Experimental results in Section V. Finally conclusion in
Section VI.
II. FIREFLY ALGORITHM
Firefly algorithm is a meta-heuristic algorithm based on
flashing patterns and behavior of fireflies, which was
introduced by Xin-She Yang in 2008 [4].Yang applied the
following assumptions regarding fireflies [4, 5]:
586
•Any Firefly will be attracted to all other fireflies as
fireflies are unisexual.
•The brighter firefly will attract the less bright firefly.
However, the light intensity decreases as the distance
between the fireflies increases.
•The firefly will move randomly if there is no brighter
firefly.
Next, we refer to the Pseudo code of the firefly algorithm
that was proposed by Yang, X. S [4] for more detailed
description of the firefly optimization. Fig. 1 summarizes the
basic steps of the firefly algorithm.
1. Data input and initialization:
Generate an initial population of fireflies
, and import the array of dataset. Each firefly generates an
initial solution randomly.
The initial value of the light intensity parameter is
initialized with initial path length. Firefly selects the next
city to fly to, its selection probability is associated with the
light intensity and path lengths between cities. Firefly will
choose a shorter path, and a city with higher intensity.
(Gamma) is the absorption coefficient and its value = 1, and
is the distance between two positions which is calculated
using Eq. (2).
2. Light Intensity and Attractiveness:
The light intensity decreases when a firefly moves, and
consequently its attractiveness will change. Attractiveness β
of a firefly is determined by its brightness which is
associated with the encoded objective function, light
intensity decreases with the distance from its source, varying
with the distance between firefly and firefly and light
that is absorbed in the media.
Light intensity varies according to the inverse square
law , is the intensity at the source.
(1)
3. Distance and Movement:
The distance between any two fireflies and at and ,
respectively, is the Cartesian distance:
(2)
The firefly with the best fitness value is selected to
participate in the next phase of optimization.
In case of equal brightness, the Firefly move randomly.
The firefly modifies its position using the current position.
The movement of a firefly attracted to another more
attractive (brighter) firefly is determined by:
(3)
!
"#$%
Fig.1. Steps of the Firefly Algorithm
III. K-MEANS CLUSTERING
K-means is considered among the simplest unsupervised
algorithms to group a given set of patterns into disjoint
clusters [6].The algorithm categorizes the data set into
&
groups of similarity. Similarity is measured using the
Euclidean distance [6, 8].
The main steps of k-mean algorithm can be summarized
as follows: first, define centers, one for each cluster or
&
partition. Next, associate each point in the given data set to
the nearest center. Then recalculate new centroid of
&
clusters resulting from the previous step. Now, apply a new
binding between the data set points and the nearest new
centers. This step will be repeated until centers do not move
any more [6]. Fig.2 illustrates the algorithmic steps of k-
means clustering.
Fig.2. Flowchart of the k-mean clustering method.
IV. PROPOSED METHODOLOGY
In this work, we are proposing a new heuristic approach
587
that combines K-mean clustering algorithm with firefly
algorithm to solve TSP.
Let be a weighted graph of nodes representing the
'()
cities and edges with weights representing the distances
between these cities. The proposed algorithm can be
summarized in three basic steps:
•Apply k-means clustering on nodes, this step will
divide the nodes into k clusters.
•Apply Firefly algorithm to find the minimal tour in
each cluster.
•Connect the clusters with each other and locate the
final solution.
Next, we provide a detailed explanation of each step of
the proposed algorithm through an illustrative
example.
A. K-means Clustering on Nodes
This paper uses k-means clustering because it is a
distance-based clustering algorithm. The first step of this
method has grouped the nodes in the TSP using k-means
clustering technique. The number of clusters is defined as:
(4)
& *
Where is the number clusters, is the number of nodes
&*
for TSP. Initially, the centroids are selected randomly from
TSP nodes. Each node in the TSP is assigned to the nearest
cluster based on the minimum distance between it and the
centroids then recalculate the centroid of clusters, as shown
in Fig. 3 This will be repeated until the values of centroids
are not changed.
Fig. 3. The application of K-means clustering.
Fig. 1 illustrates the result of applying k-mean clustering
on the input graph of (17) nodes, where according to
&+
Eq. (4).
B. Finding Optimal Path in Each Cluster
The optimal path in each cluster will be found after all of
the nodes in the TSP are clustered. This paper uses firefly
algorithm to find the shortest path in the clusters. FA starts
by generating an initial population for the fireflies randomly,
where each firefly represents one solution of TSP.
The firefly that has less bright will move towards the
brightest firefly based on the brightness of fireflies, after that
update the light intensity and find the current best solution.
This will continue looping until the maximum iteration is
reached. Fig. 4 illustrates the result of applying firefly
algorithm to find the minimum tour in each cluster.
Fig. 4. Finding the optimal path in each cluster.
C. Connect the Clusters
After finding the optimal path in each cluster, the clusters
will be connected with each other. At this point, we have k-
clusters each contains local minimal tour. The goal is to
reconnect these clusters by connecting the endpoints of each
tour to produce one global minimal tour.
To connect the tour of the k-clusters we apply the following
procedure:
1. Calculate the distances between the centroids of
each cluster.
2. Select the two clusters with the minimum distance
between their centroids.
3. Combine the selected clusters in one larger cluster.
4. Concatenate the tours of these two clusters.
5. Repeat step 2 until the global minimum tour is
generated.
Fig. 5. Cluster reconnection.
Fig. 5 illustrates the concatenation of cluster 2 and cluster
3 as they have a minimum distance between their centroids.
Then Fig. 6 shows the final concatenation that leads to the
global minimum tour.
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Fig. 6. Final path of Traveling Salesman Problem.
V. EXPERIMENTAL RESULTS
In this paper, we combine K-mean clustering with Firefly
to generate a solution to the travelling salesman problem
using java eclipse on a platform with the specifications of
Intel CORE i3 and 4GB RAM.
To evaluate the proposed methods we tested the
algorithm on variety of TSP datasets (eil51, eil76, pr76 and
ulysses16) from TSPLIB [11]. These datasets in two-
dimensional Euclidean distance. The experiment was
repeated 20 times to produce the average and minimum tour
length on each dataset. The experimental results were
compared against other proposed algorithm that also use
clustering approach.
Table 1 shows the resulting tour length upon applying
our proposed approach (k-mean clustering and firefly). It
also compares these results to the results of applying other
algorithm from the literature on the same data set [7, 8, 9].
Results showed that combining k-mean clustering with
firefly produces either better or competitive tour in most of
the cases. The proposed algorithm scores better results for
the data sets (eil76, pr76 and ulysses16). On the other hand,
Ant colony algorithm seems to produces better results for the
data set (eil51), however, the difference is very small.
TABLE I. COMPARISON BETWEEN CLUSTERING ANT COLONY,
FIREFLY AND CLUSTERING FIREFLY ALGORITHMS
Datasets Optimal
Path Algorithm Minimum Average
Ant Colony
using Clustering 484 –
Genetic
Algorithm using
Clustering
484 -
Firefly
Algorithm 435.60 451.37
eil51 426
Firefly using
Clustering 485.31 531.95
Ant Colony
using Clustering 624 –
Genetic
Algorithm using
Clustering
624
Firefly
Algorithm ––
eil76 538
Firefly using
Clustering 613.02 633.15
Datasets Optimal
Path Algorithm Minimum Average
Ant Colony
using Clustering 125464 –
Genetic
Algorithm using
Clustering
125243 -
Firefly
Algorithm ––
pr76 108159
Firefly using
Clustering 112386.74 119931.63
Ant Colony
using Clustering 7400 7808
Genetic
Algorithm using
Clustering
7399 7400
Firefly
Algorithm 7399 7408
ulysses16 6859
Firefly using
Clustering 7084 7326
In average, firefly algorithm using k-means clustering
technique can minimize the length of TSP tour in
comparison with clustering ant colony algorithm and firefly
algorithm
VI. CONCLUSION
This paper has proposed Firefly algorithm with clustering
technique to solve TSP. The proposed method involves three
major steps: First, the nodes are divided into sup problems
using k-means clustering method. Then, the firefly algorithm
is applied in each cluster to find the optimal path. Finally, the
local best paths of all clusters are connected to form one
global path that traverse all nodes. The experimental results
prove the effectiveness of the firefly algorithm with k-means
clustering in providing a competitive solution to the TSP.
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