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A Heuristic Approach for Cluster TSP
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DOI: 10.1007/978-3-030-34152-7_4
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Book Title Recent Advances in Intelligent Information Systems and Applied Mathematics
Series Title
Chapter Title A Heuristic Approach for Cluster TSP
Copyright Year 2020
Copyright HolderName Springer Nature Switzerland AG
Author Family Name Manna
Particle
Given Name Apurba
Prefix
Suffix
Role
Division Department of Computer Science
Organization P. K. College
Address Contai, Purba Medinipur, 721404, W.B., India
Email apurba.manna2008@gmail.com
Author Family Name Maity
Particle
Given Name Samir
Prefix
Suffix
Role
Division
Organization OM Group, Indian Institute of Management
Address Calcutta, India
Email samirm@iimcal.ac.in
Corresponding Author Family Name Roy
Particle
Given Name Arindam
Prefix
Suffix
Role
Division Department of Computer Science
Organization P. K. College
Address Contai, Purba Medinipur, 721404, W.B., India
Email royarindamroy@yahoo.com
Abstract This investigation took an attempt to solve the cluster traveling salesman problem (CTSP) by the heuristic
approach. In this problem, nodes are clustered with given a set of vertices (nodes). Given the set of vertices
is divided into a prespecified number of clusters. The size of each cluster is also pre-specified. The main
aim is to find the least cost Hamiltonian tour based on the given vertices. Here vertices of each cluster
visited contiguously, and the clusters are visited in a specific order. Standard GA is used to find a
Hamiltonian path for each cluster. The performance of the algorithm has been examined against two
3. existing algorithms for some symmetric TSPLIB instances of various sizes. The computational results
show the proposed algorithm works well among the studied metaheuristics regarding the best result and
computational time.
Keywords Cluster TSP - GA - Heuristic
4. A Heuristic Approach for Cluster TSP
Apurba Manna1
, Samir Maity2
, and Arindam Roy1(B)
1
Department of Computer Science, P. K. College, Contai,
Purba Medinipur 721404, W.B., India
apurba.manna2008@gmail.com, royarindamroy@yahoo.com
2
OM Group, Indian Institute of Management, Calcutta, India
samirm@iimcal.ac.in
Abstract. This investigation took an attempt to solve the cluster trav-
eling salesman problem (CTSP) by the heuristic approach. In this prob-
lem, nodes are clustered with given a set of vertices (nodes). Given the
set of vertices is divided into a prespeciïŹed number of clusters. The size
of each cluster is also pre-speciïŹed. The main aim is to ïŹnd the least cost
Hamiltonian tour based on the given vertices. Here vertices of each clus-
ter visited contiguously, and the clusters are visited in a speciïŹc order.
Standard GA is used to ïŹnd a Hamiltonian path for each cluster. The
performance of the algorithm has been examined against two existing
algorithms for some symmetric TSPLIB instances of various sizes. The
computational results show the proposed algorithm works well among
the studied metaheuristics regarding the best result and computational
time.
Keywords: Cluster TSP · GA · Heuristic
1 Introduction
Traveling salesman problem (TSP) has many diïŹerent variations. The clustered
traveling salesman problem (CTSP) is one of them. At ïŹrst, CTSP was proposed
by Chisman [4]. DiïŹerent approaches are taken by various researcher during last
decades to solve cluster traveling salesman problem (CTSP). Few of them are
New Hybrid Heuristic approach by Mestria [11], using Neighborhood Random
Local Search a heuristic approach by Mestria [10], another approach is based on
with d-relaxed priority rule by Phuong et al. [12], a Metaheuristic approach by
Zhang et al. [15], applying the Lin-Kernighan-Helsgaun Algorithm by Helsgaun
[5], etc. CTSP is deïŹned as follows: consider a complete undirected graph G.
Where, G = (V, E). Here V = set of vertices and E = set of edges. If the number
of node is N, then V = {v1, v2, v3, · · · , vN } and it is divided into K prespeci-
ïŹed clusters. The prespeciïŹed clusters are {C1, C2, C3, · · · , Ck}. A cost matrix
COST = [cij] is present. This matrix represents the travel costs, distances, or
travel times which is deïŹned on the edge set E = {(vi, vj) : vi, vj â V, i = j}.
Till now, diïŹerent variants of CTSP is available based on diïŹerent conditions.
c
Springer Nature Switzerland AG 2020
O. Castillo et al. (Eds.): ICITAM 2019, SCI 863, pp. 1â10, 2020.
https://doi.org/10.1007/978-3-030-34152-7_4
Author
Proof
5. 2 A. Manna et al.
Suppose the number of clusters is two then it is treated as TSP with backhauls
(TSPB) [8]. In the case of free CTSP, the eïŹective number of cluster is deter-
mined dynamically, not determined by prespeciïŹed order. The routing between
clusters is also an important part of this paper. In the case of free CTSP, it is
determined simultaneously. If all variations of CTSP are colligation of classical
TSP, they are all NP-hard. In real life, CTSP is important, and it also has a
huge application like vehicle routing [3], warehouse routing [7], integrated circuit
testing [6], production planning [6], etc. Chisman [4] ïŹrst proposed that CTSP
can be represented as a TSP by adding or subtracting a big impulsive constant
I to or from the cost of every inter-cluster edge. So, at the end of conversion,
a speciïŹc algorithm for the TSP also apply to solve the problem precisely. The
use of the heuristic procedure is practical in CTSP when the number of nodes
is large or very large. Most common heuristic algorithms are approximate algo-
rithms, artiïŹcial neural network, tabu search, genetic algorithm (GA) and so on.
To solve TSP and its variation, Genetic Algorithm (GA) is treated as best. Now
our proposed algorithm Heuristic Approach is a variation of GA to ïŹnd the opti-
mal solution of given problem. The eïŹectiveness of our proposed algorithm has
been compared against lexisearch algorithm (LSA) [1] and hybrid GA(HGA)
[2] for few symmetric TSPLIB [13] instances. At last, we have taken a set of
solutions of large size TSPLIB [13] instances and compared with Hybrid GA
(HGA).
The proposed algorithm have following key features:
âą Cluster creation
âą Genetic Algorithm (GA)
âą Probabilistic selection
âą Cyclic crossover
âą Random crossing point
âą Random mutation
âą Routing between clusters
âą Test on TSPLIB instances
The present paper is prepared as follows: Sect. 1, a short introduction is pro-
duced. In Sect. 2, required mathematical pre-requisite. In Sect. 3, the proposed
algorithm is presented. In Sect. 4, a numerical tests are ïŹnished. Again in Sect. 5,
a brief discussion is given. Finally, in Sect. 6, a conclusion with future scope is
studied.
2 Classical Definition of CTSP
The CTSP is outlined on a loop-free undirected graph G. Where, G = (V, E).
Here V = set of vertex and E = set of edge. If the number of node is N, then,
V = {v1, v2, v3, · · · , vN } and it is divided into K cluster. Here, K is pre-speciïŹed.
The pre-speciïŹed clusters are {C1, C2, C3, · · · , Ck}. A cost matrix COST = [cij]
between ith
and jth
node is present. This matrix represents the travel costs,
which is deïŹned on the edge set E = {(vi, vj) : vi, vj â V, i = j}. There is a
Author
Proof
6. A Heuristic Approach for Cluster TSP 3
decision variable xij, xij = 1 iïŹ a tour completed between vi to vj, otherwise,
xij = 0. The framing of CTSP can be represented as follows:
Minimize Z =
i=j
c(i, j)xij
subject to
N
i=1
xij = 1 for j = 1, 2, ..., N
N
j=1
xij = 1 for i = 1, 2, ..., N
iâvk
jâvk
xij = |vk|, â|vk| â V, |vk| â„ 1, k = 1, 2, 3, · · · , m
where xij â {0, 1}, i, j = 1, 2, · · · , N
â«
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
âŹ
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
âȘ
â
(1)
Then the above CTSP reduces to
determine a complete tour (x1, x2, ..., xN , x1)
to minimize Z =
Nâ1
i=1
c(xi, xi+1) + c(xN , x1)
where xi = xj, i, j = 1, 2..., N.
â«
âȘ
âȘ
âŹ
âȘ
âȘ
â
(2)
along with sub tour elimination criteria
N
iâS
N
jâS
xij †|S| â 1, âS â Q (3)
3 Proposed Heuristic Based Genetic Algorithm
A well-known heuristic based GA is used for solve the CTSP. Using GA or any
other heuristic method we get a better solution for small size TSP or medium
size TSP very easily. But when the size of TSP has increased then complexity
increases in a parallel way. To overcome this problem, TSP transformed to CTSP,
which is a variation of TSP. The proposed algorithm performs in three steps.
First, all nodes are divided into the pre-speciïŹed number of cluster. The number
of nodes in each cluster may be the same or not. Second, each cluster is optimized
using GA. Third, reconstruct a Hamiltonian cycle using all optimized cluster.
All optimized cluster contains a Hamiltonian path, not cycle.
3.1 Cluster Creation
The number of clusters is pre-speciïŹed. At ïŹrst, we ensure the size of each cluster.
Then, selected nodes are inserted into each speciïŹed cluster. It is clear that every
cluster must contain a unique set of nodes. That is after the optimization of each
cluster, generate a diïŹerent and unique Hamiltonian path.
Author
Proof
7. 4 A. Manna et al.
Algorithm:
1. Begin.
2. State the number of cluster.
3. Generate a random number(r) between (0 to Nâ1).
4. Calculate the cost(c) from node r to each node.
5. Select a node in a cluster depend on minimum cost(c).
6. Ignore the node selected in step 5.
7. Calculate the cost(c) from node r to each remain node.
8. Repeat steps 5 to 7 until all nodes are distributed based on previous cluster
size of each cluster.
9. End.
3.2 Genetic Algorithm(GA)
Proposed algorithm have the concept of the generation of the cluster. Here given
nodes are divided between pre-speciïŹed clusters based on subsection 3.1. Initially,
each cluster contains a number of nodes. Based on these nodes initial population
is created randomly. Each cluster strictly follows this step, and strictly GA is
applied to each cluster to produce a Hamiltonian path. i.e GA is applied to
optimize each cluster. So our proposed GA is as follows.
Genetic Algorithm is a well-known randomized search method. There is
a natural rule that, survival of the ïŹttest among the species based on their gene
architecture of the chromosomes. Gene structure constructed based on random
change on it and it is evolved from one iteration(generation) to next. Every
iteration with the following three operations.
(a) Selection: It is a stochastic process which simulates the quotation
survival -of-fittest. An objective function took a vital role and based on it few
chromosomes are copied from a predeïŹned population of the chromosome. All
selected chromosomes are used for the next operation. Our proposed algorithm
uses the Boltzmanâs probabilistic selection process [9].
(b) Crossover: It is known as a binary operator. It works with a pair of
parent chromosome. Parents are selected with a signiïŹcant probability, and as
a result, two new oïŹspring chromosomes are prepared. Its importance in GA
is very much. The proposed algorithm uses Cyclic Crossover [14] as a crossover
operator.
(c) Mutation: It is known as a unary operator. It is applied to every chro-
mosome with a small probability. The mutation also important part to diversify
the GA search space. The proposed algorithm uses random mutation as a muta-
tion operator.
GA starts with a randomly generated initial population and repeat the
above three operations until the stopping criterion is satisïŹed. Crossover creates
a new opportunity over GA generating new oïŹspring chromosomes. An example
of a successful heuristic algorithm to solve a classical TSP and its variations is
GA. It never gives the guarantee about the optimal solution, but it can ïŹnd a
near-optimal solution in a concise time.
Author
Proof
8. A Heuristic Approach for Cluster TSP 5
3.3 Inter Cluster Re-linking
We aim to ïŹnd a Hamiltonian cycle. Optimized each cluster have a Hamiltonian
path. To produce a Hamiltonian cycle we have maintained the following steps.
1. Store the number of cluster
2. Store each Hamiltonian path of each cluster
3. Calculate possible combinations of given clusters
4. Arrange the cluster sequence based on combination sequence
5. Merge each combination and prepare a ïŹnal path
6. Calculate the cost of each combination
7. The Least cost combination is treated as best result of our proposed algorithm
3.4 Proposed Algorithm
1. Start
2. Input the number of cluster.
3. DeïŹne the size of each cluster.
4. To determine the nodes for each cluster, do following steps:
(A) Generate a random number(r) between (0 to N-1).
(B) Calculate the cost(c) from node r to each node.
(C) Select a node in a cluster depend on minimum cost(c).
(D) Ignore the node selected in step (C).
(E) Calculate the cost(c) from node r to each remain node.
(F) Repeat steps (C) to (E) until all nodes are distributed based on previous
cluster size of each cluster.
5. After creation of each cluster with its respective nodes, a randomly generated
population is prepared on the basis of stored nodes of each cluster.
6. Proposed GA is applied to each cluster to generate a Hamiltonian path
based on the speciïŹed nodes of each cluster.
7. Prepare possible combinations of given clusters.
8. Calculate objective function value of each combination(path).
9. Find minimum cost(objective function value) among all combinations, this
will be the best solution of our proposed algorithm.
10. Stop
4 Numerical Tests
Proposed algorithm is guided by few parameters, namely, crossover probability
(pc), mutation probability (pm) and population size (pv) and also termination
condition. Proper functioning of GA depends on a proper selection of these
parameters. Table 1 shows the comparison of performance between proposed
Heuristic based GA (HbGA), LSA [1] and HGA [2] also.
Table 2 shows a comparative study between HGA and HbGA based on sym-
metric TSPLIB instances. Taken TSPLIB instances are larger than TSPLIB
instances of Table 1.
Author
Proof
11. 8 A. Manna et al.
Table 3. (continued)
Cluster pc pm popsize result cpu-timesec Error (%)
4 0.34 0.43 50 34715 16.40 â24.09
55 37499 18.20 â18.00
60 50034 19.23 9.40
65 31569 22.18 â30.97
70 33665 24.57 â26.39
75 46783 28.07 2.30
80 40573 27.23 â11.28
85 31703 31.86 â30.68
90 44526 32.87 â2.64
Table 4. Comparative result based on diïŹerent sizes cluster (pc = 0.34, pm=0.43,
popsize = 50)
Instance Cluster result cpu â timesec
kroA100 2 31186 17.37
3 38372 16.80
4 34715 16.40
5 51670 20.01
6 36372 19.73
7 53503 22.98
8 45696 22.52
9 49470 22.98
10 45106 27.63
5 Discussion
This article is a special attempt to ïŹnd out a way to solve a large scale TSP
in a convenient way. Here we have chosen the way as a cluster TSP (CTSP).
Our proposed HbGA algorithm is implemented by considering some parametric
values as probability of crossover (pc), probability of mutation (pm), maximum
number of chromosome as a population (pv) and maximum generation. This pro-
posed algorithm is written in C++. It is clear from Table 1 that our proposed
HbGA algorithm is much eïŹcient than LSA and HGA both. Results shown in
Table 1 based on 10 benchmark TSP references in TSPLIB. These ten instances
are between 16 and 51 cities. It is remarkable that our proposed HbGA is much
eïŹcient for bays29 for 29 cities problem and eil51 for 51 cities problem also. Com-
pare to both LSA and HGA using our proposed HbGA, we got better results
than existing, which are illustrated in Table 1. Table 2 is also prove the eïŹciency
of HbGA based on a comparative study of instances in TSPLIB between 52 and
Author
Proof
12. A Heuristic Approach for Cluster TSP 9
417 cities. So, all over performance of HbGA is better than HGA. Table 3 is
a parametric study based on standard TSPLIB instance of 100 cities. Table 3
represents better results considering four(4) clusters and all diïŹerent combina-
tion of parametric values by using our proposed HbGA. Also it is remarkably
mention that, we got these better results within less CPU time than existing.
From Table 4 we can observe that cluster size two(2) gives the better results
than cluster size four(4). From above discussion, we may come to an end that
our proposed HbGA is also applicable for solving real life optimization problems.
6 Conclusion
The present study, a heuristic based genetic algorithm modeled to solve cluster
TSP. Here we developed an alternative methodology, i.e., heuristic to the creation
and re-linking the inter-cluster and used GA for optimizing the path in intra-
cluster also. Finally, an optimized path is generated. Again diïŹerent numbers of
the cluster are investigated because of such realistic happening found in the small
scale tourism industry. In the tourism industry, it oftenly found that a diïŹerent
number of sight scenery are the demand by every group of tourist. Since tourism
is travel for pleasure and business, so management prepares diïŹerent kinds of
travel plan in that case such proposed cluster model eïŹectively works. Without
cluster attempt to solve such TSP using a heuristic process like using GA, is
a big headache regarding CPU time and complexity. The main motto of our
prescribed article is to demonstrate the eïŹciency of our proposed cluster TSP
algorithm than any other conventional Genetic Algorithms. We got a set of the
heuristic solution by applying our proposed GA on CTSP. The eïŹectiveness of
clustering method has been examined with both lexisearch algorithm (LSA) and
OCTSP [2] for few small TSPLIB instances. The experiment shows that CTSP
is better than LSA and HGA also. Few TSPLIB instances also compared with
HGA and the overall result is good enough. In the future, we can extend the
algorithm using fuzzy distance for cluster creation and dynamic relinking of the
inter-cluster also.
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13. 10 A. Manna et al.
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Author
Proof
14. MARKED PROOF
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