1. The document describes a heuristic approach for solving the cluster traveling salesman problem (CTSP) using genetic algorithms. 2. The proposed algorithm divides nodes into pre-specified clusters, uses GA to find a Hamiltonian path for each cluster, then combines the optimized cluster paths to form a full tour. 3. The algorithm was tested on symmetric TSPLIB instances and shown to find high quality solutions faster than two other metaheuristic approaches for CTSP.

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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 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 clus-
ter 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 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 different variations. The clustered
traveling salesman problem (CTSP) is one of them. At first, CTSP was proposed
by Chisman [4]. Different 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 defined 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-
fied clusters. The prespecified 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 defined on the edge set E = {(vi, vj) : vi, vj ∈ V, i = j}.
Till now, different variants of CTSP is available based on different conditions.
c
Springer Nature Switzerland AG 2020
O. Castillo et al. (Eds.): ICITAM 2019, SCI 863, pp. 1–10, 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 effective number of cluster is deter-
mined dynamically, not determined by prespecified 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] first 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 specific 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, artificial 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 find the opti-
mal solution of given problem. The effectiveness 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:
• Cluster creation
• Genetic Algorithm (GA)
• Probabilistic selection
• Cyclic crossover
• Random crossing point
• Random mutation
• Routing between clusters
• 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 finished. 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-specified.
The pre-specified 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 defined on the edge set E = {(vi, vj) : vi, vj ∈ V, i = j}. There is a
Author
Proof

6. A Heuristic Approach for Cluster TSP 3
decision variable xij, xij = 1 iff 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
i∈vk
j∈vk
xij = |vk|, ∀|vk| ⊂ V, |vk| ≥ 1, k = 1, 2, 3, · · · , m
where xij ∈ {0, 1}, i, j = 1, 2, · · · , N
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(1)
Then the above CTSP reduces to
determine a complete tour (x1, x2, ..., xN , x1)
to minimize Z =
N−1
i=1
c(xi, xi+1) + c(xN , x1)
where xi = xj, i, j = 1, 2..., N.
⎫
⎪
⎪
⎬
⎪
⎪
⎭
(2)
along with sub tour elimination criteria
N
i∈S
N
j∈S
xij ≤ |S| − 1, ∀S ⊂ 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-specified 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-specified. At first, we ensure the size of each cluster.
Then, selected nodes are inserted into each specified cluster. It is clear that every
cluster must contain a unique set of nodes. That is after the optimization of each
cluster, generate a different 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 N−1).
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-specified 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 fittest 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 predefined population of the chromosome. All
selected chromosomes are used for the next operation. Our proposed algorithm
uses the Boltzman’s 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 significant probability, and as
a result, two new offspring 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 satisfied. Crossover creates
a new opportunity over GA generating new offspring 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 find 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 find 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 final 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. Define 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 specified 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

9. 6 A. Manna et al.
Table 1. A comparative study between LSA, HGA and HbGA for few symmetric
TSPLIB instances.
Instances Clusters Solution Error (%)
LSA HGA HbGA HGA vs HbGA LSA vs HbGA
ulysses16 2 7303 7303 7373 0.96 0.96
gr17 2 2517 2517 2256 −10.37 −10.37
gr21 2 3465 3465 3499 0.98 0.98
gr24 2 1558 1558 1526 −2.05 −2.05
fri26 2 957 957 1276 33.33 33.33
bay29 2 2144 2144 1931 −9.93 −9.93
3 2408 2408 2105 −12.58 −12.58
bayg29 2 2702 2702 2473 −8.48 −8.48
3 2991 2991 2608 −12.81 −12.81
swiss42 2 1605 1605 1801 12.21 12.21
3 1919 1919 1963 2.29 2.29
4 1944 1944 1982 1.95 1.95
gr48 2 6656 6433 6518 1.32 −2.07
3 7466 7466 7345 −1.62 −1.62
4 8554 8554 7820 −8.58 −8.58
eil51 2 570 564 555 −1.60 −2.63
3 689 681 647 −4.99 −6.10
4 714 714 659 −7.70 −7.70
Table 2. A comparative study between HGA and HbGA based on symmetric TSPLIB
instances
Instances Clusters Solution (HGA) Solution (HbGA) Error (%)
berlin52 2 10422 10257 −1.58
st70 2 916 895 −2.29
eil76 2 721 750 4.02
kroA100 4 45733 37650 −17.67
kroB100 4 45709 38855 −14.99
kroC100 4 46388 46558 0.37
kroD100 4 45681 42085 −7.87
kroE100 4 45431 43847 −3.49
rd100 4 15501 14628 −5.63
eil101 4 1080 1050 −2.78
pr107 4 51487 56136 9.03
bier127 4 174112 178416 2.47
(continued)
Author
Proof

10. A Heuristic Approach for Cluster TSP 7
Table 2. (continued)
Instances Clusters Solution (HGA) Solution (HbGA) Error (%)
ch130 4 12000 10566 −11.95
kroA150 4 52824 45355 −14.14
kroB150 4 54008 46955 −13.06
ch150 4 13042 12093 −7.28
d198 4 17320 20956 20.99
kroA200 4 62514 56442 −9.71
kroB200 4 62842 55145 −12.25
gil262 4 4874 4173 −14.38
pr264 4 60161 68531 13.91
rd400 4 30821 30223 −1.94
fl417 4 20457 22428 9.63
Table 3. Parameter study for kroA100 instance
Cluster pc pm popsize result cpu-timesec Error (%)
4 0.34 0.01 50 37747 12.97 −17.46
0.02 50 35119 13.04 −23.21
0.04 50 33109 12.75 −27.60
0.05 50 36775 13.74 −19.59
0.001 50 49489 14.49 8.21
0.003 50 44311 15.88 −3.11
0.004 50 43029 13.92 −5.91
0.005 50 42167 14.05 −7.80
0.007 50 37634 16.31 −17.71
0.009 50 36746 12.49 −19.65
4 0.10 0.43 50 40969 15.41 −10.42
0.30 50 38267 15.33 −16.33
0.35 50 33809 20 −26.07
0.40 50 42305 25.68 −7.40
0.45 50 36714 18 −19.72
0.50 50 32936 17.72 −27.98
0.55 50 38228 22.81 −16.41
0.65 50 42604 20.84 −6.84
0.70 50 37751 29.26 −17.45
0.75 50 35228 20.55 −22.97
0.80 50 34014 20.60 −25.62
(continued)
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 −24.09
55 37499 18.20 −18.00
60 50034 19.23 9.40
65 31569 22.18 −30.97
70 33665 24.57 −26.39
75 46783 28.07 2.30
80 40573 27.23 −11.28
85 31703 31.86 −30.68
90 44526 32.87 −2.64
Table 4. Comparative result based on different sizes cluster (pc = 0.34, pm=0.43,
popsize = 50)
Instance Cluster result cpu − 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 find 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 efficient 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
efficient 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 efficiency
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 different 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 different 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 different
number of sight scenery are the demand by every group of tourist. Since tourism
is travel for pleasure and business, so management prepares different kinds of
travel plan in that case such proposed cluster model effectively 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 efficiency 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 effectiveness 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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Proof

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