International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 3, Issue 1, January 2014
ISSN 2319 - 4847
AN OPTIMAL SOLUTION TO MULTIPLE
TRAVELLING SALESPERSON PROBLEM USING
MODIFIED GENETIC ALGORITHM
Varunika Arya1, Amit Goyal2 and Vibhuti Jaiswal3
1,2,3,
Department of Electronics and Communication Engineering, Maharishi Markandeshwar
University,Sadopur,Ambala,Haryana
Abstract
Travelling Salesman Problem (TSP) is an optimization problem. In its simplest form, a salesperson must make a circuit through
a certain number of cities, visiting each only once, while minimizing the total distance travelled. The multiple Traveling
Salesman Problem (mTSP) is the generalization of TSP and is a complex combinatorial optimization problem, where one or more
salesmen can be used in the solution. The Constraint in the optimization task is that, each salesman returns to starting point at
end of trip, travelling to a unique set of cities in between and except for the first, each city is visited by exactly one salesman. The
Cost Function is to search for the shortest route i.e. the least distance needed for each salesman to travel from the start location
to individual cities and back to the original starting place. mTSP is a complex, namely an NP-Hard problem and has a
multiplicity of applications mostly in the areas of routing and scheduling. The amount of computation time to solve this problem
grows exponentially as number of cities so, the heuristic optimization algorithms, like genetic algorithms (GAs) need to take into
account. GA generates a population of solutions at each iteration & the best point in the population approaches an optimal
solution. The aim of this paper is to review how genetic algorithm can be applied to solve these problems and propose an efficient
solution to mTSP .
Keywords: Multiple Travelling Salesperson Problem(MTSP), Optimization, Genetic Algorithm(GA), NP-Hard problems.
1. INTRODUCTION
The multiple Traveling Salesman Problem (mTSP) is the generalization of TSP and is a complex combinatorial
optimization problem, where one or more salesmen can be used in the solution. The Constraints in the optimization task
are:
1. Each salesman returns to starting point at end of trip
2. Each salesman travels to a unique set of cities in between
3. Except for the first, each city is visited by exactly one salesman
The Cost Function is to search for the shortest route i.e. the least distance needed for each salesman to travel from the
start location to individual cities and back to the original starting place. Hence the total cost of visiting all nodes is
minimized. In the MTSP, the n nodes must be partitioned into m tours, with each tour resulting in a TSP for one
salesman. The MTSP is more difficult than the TSP because it requires assigning nodes to each salesman, as well as the
optimal ordering of the nodes within each salesman’s tour.
The techniques used for solving the MTSP can be categorized into exact, heuristic and metaheuristic algorithms. Out of
these, the exact approaches are used only for relatively small problems, but they guarantee optimality. These techniques
apply algorithms that generate a lower and an upper bound on the true minimum value of the problem instance. Although
the MTSP is conceptually simple, it is difficult to obtain an optimal solution. In other words, when the problem size is
increased, the exact methods cannot solve it. So, heuristic or metaheuristic methods are necessary to be used for solving
them in a reasonable amount of time particularly with large sizes. Some of the well-known heuristic algorithms are
gravitational emulation search and local search. Metaheuristics methods tries to combine basic heuristic methods in
higher level frameworks which aim at efficient and effective exploration of a search space. The term metaheuristic derives
from the composition of two Greek words. Heuristic stems from the verb heuriskein which means “to find”, while the
prefix meta means “beyond in an upper level”. Before this term was widely adopted, metaheuristics were often called
modern. In general, it is vital to use metaheuristic algorithms to solve complex optimization problems when dealing with
them. Since the metaheuristic approaches are very efficient for escaping from local optimum, they are one of the best
group algorithms for solving combinatorial optimization problem. Some of the metaheuristic approaches which may be
used are genetic algorithm(GA), memetic algorithm (MA) , ant system (AS) and particle swarm optimization (PSO).
Volume 3, Issue 1, January 2014
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 3, Issue 1, January 2014
ISSN 2319 - 4847
In this paper, a modified GA is used for solving the MTSP. In the following sections, a mathematical model of MTSP is
presented, along with the basic GA and explanation of the proposed algorithm. Analysis of the result obtained is also
mentioned in the subsequent section.
2. MATHEMATICAL MODEL
Consider an undirected connected graph M(A,B) with a vertex set A={0,1,2,….,p} and an edge set B={ (i, j) : i, j ∈A, i ≠ j
}. The following variables are used for representing the integer linear programming model for MTSP:
p = number of nodes(cities) for each instance.
q = number of salesmen used for each instance.
C = cost matrix on graph M which is symmetric & true in triangle inequality. i.e. cij = cji and
cij + cji ≥ cik for each (
i,j,k = 1,2,….,p) .
So, the integer programming formulation for MTSP can be written as:
min∑∑ cij*xij
for i,j=1 to p
(1)
j=2,…,p
j=1
(2)
(3)
i=2,3,….,p
i=1
≥ 1 (ɸ ≠ S
∑ ∑
i
Sj
(5)
N = {2,…,p}), |S| ≥ 2
(6)
N = {2,…,p}), |S| ≥ 2
(7)
N-S
∑ ∑
≥ 1 (ɸ ≠ S
i
S
xij
(4)
N-S j
{0,1}
(8)
The objective function (1) minimizes the total distance traveled in a tour. Constraint sets (2) and (3) ensure that the
salesmen arrive once at each node and q times at the depot. Constraint sets (4) and (5) ensure that the salesmen leave
each node once and the depot q times. Constraint sets (6) and (7) are to avoid the presence of sub-tours for each salesman.
And the Constraint set (8) defines binary conditions on the variables.
3. METHODOLOGY
3.1 Genetic Algorithm
Genetic algorithm (GA) as a computational intelligence method is a search technique used in computer science to find
approximate solutions to combinatorial optimization problems. It includes the survival of the fittest idea algorithm. The
idea is to first ‘guess’ the solutions and then combining the fittest solution to create a new generation of solutions which
should be better than the previous generation
.
The genetic algorithm uses three main types of rules at each step to create the next generation from the current
population:
 Selection rules select the individuals, called parents that contribute to the population at the next generation.
 Crossover rules combine two parents to form children for the next generation.
 Mutation rules apply random changes to individual parents to form children.
In detail, the genetic algorithm process consists of the following:
1. Encoding: A suitable encoding is found for the solution to our problem so that each possible solution has unique
encoding and the encoding is some form of a string.
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Volume 3, Issue 1, January 2014
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2. Evaluation/Selection: The initial population is then selected, usually at random though alternative techniques using
heuristics have also been proposed. The fitness of each individual in the population is then computed that is, how well the
individual fits the problem and whether it is near the optimum compared to the other individuals in the population.
3. Crossover: The fitness is used to find the individual’s probability of crossover. Crossover is where the two individuals
are recombined to create new individuals which are copied into the new generation.
4. Mutation: Some individuals are chosen randomly to be mutated and then a mutation point is randomly chosen. The
character in the corresponding position of the string is changed.
5. Decoding: A new generation has been formed and the process is repeated until some stopping criteria has been
reached. At this point the individual who is closest to the optimum is decoded and the process is repeated until we get an
individual with high fitness value.
3.2 Proposed Algorithm
The proposed algorithm is a modified genetic algorithm in which only one kind of chromosome is commonly employed
for solving the MTSP. This technique involves using a single chromosome of length p + q and is referred to as the ‘‘one
chromosome’’ technique. In this technique, the nodes are represented by a permutation of the integers from 1 to p. This
permutation is partitioned into q sub-tours by the insertion of q negative integers (from 1 to q) that represent the change
from one salesman to the next.
A modified crossover is proposed based on the Order crossover. In this crossover, a randomly chosen crossover point
divides the parent strings in left and right substrings. The right substrings of the parents are selected. After selection of
nodes the process is the same as the order crossover.
The only difference is that instead of selecting random several positions in a parent tour, all the positions to the right of
the randomly chosen crossover point are selected. Two mutations are used in the proposed algorithm. These operators
select randomly two points in the string, and it replaces together or reverses the substring between these two cut points.
This algorithm is coupled with a local search technique to obtain a better solution as compared to previous iterations. The
search tries to improve route by replacing its two non-adjacent edges by two other edges. It should be noted that there are
several routes for connecting nodes and producing the tour again, but a state that satisfies the problem’s constraints is
acceptable.
So if, first, the above constraints are not violated and, second, the new tour produces a better value for the problem than
the previous solution then only the unique tour will be accepted. The process is repeated until there is no possibility of
reduction of route length.
4. COMPUTATIONAL RESULTS
The proposed algorithm is coded in MATLAB. To reveal the performance of the algorithm it was tested with a big
number of problems, only an illustrative result is presented here in the fig.1. As it was, tiny refinements in constraints are
in progress. The example represents a whole process of a real problem’s solution as an example same number of 5
salesperson with each salesperson were allowed to travel a minimum of 3 cities and fixed number of cities.
The first step is the determination of the distance matrix. The input data is given by a map as it can see on Fig. 1 and a
portion of the resulted distance table is shown on Table 1. It contains 50 locations (with the depot). The task is to
determine the optimal routes for these locations. After distance table determination, the optimizer algorithm can be
executed to determine the optimal routes using the proposed algorithm.
Fig.1 Plot of city locations
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 3, Issue 1, January 2014
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Fig.2 The result of the optimization in MATLAB
The first run optimized the solution to a distance of 67.9625 units after 1227 iterations. Table 1 shows the optimal
solution for the MTSP with 50 cities traversed by 3 salespersons. Results reveal the minimum distances travelled by the
salesperson along with the number of iterations.
Table 1: Performance of MTSP based on Iterations
Total
Iteration
Distanc
e
68.4053
1638
67.5306
1816
67.4341
1804
67.9625
1227
Fig.3 The Solution History.
5. Analysis
This paper presents the technique to solve the multiple travelling salesperson problem using a modified genetic
algorithm. As per the requirement & the application, the minimum number of cities travelled by each salesperson can be
varied and so could be the number of cities & the number of salesperson. As the number of cities is increased, the
computational complexity increases. So the proposed work will provide an effective solution to the problem in decreased
time. The time taken by the algorithm for the above mentioned iterations was also analyzed. In every case the running
time was nearly 30 seconds. The algorithm is sensitive for the number of iterations. The running time is proportional to
iteration number.
6. Conclusion & Future Directions
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
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In this paper, a modified Genetic Algorithm was proposed which, along with the local search technique, improvised the
solution at each iteration of the modified GA. For large-size problems, this proposed algorithm yields better results and
the run time is also optimized. It is also assumed that the combination of this algorithm with other algorithms including
prioritized GA, simulated annealing will yield better results. This proposed algorithm is also suggested for future research
in applications like vehicle routing, computer wiring, drilling of printed circuit boards, overhauling gas turbine engines,
X-Ray crystallography, order-picking problem in warehouses, mask plotting in PCB production, interview scheduling
problem, mission planning problem, design of global navigation satellite system surveying networks, manufacture
of microchips and
DNA sequencing.
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Volume 3, Issue 1, January 2014
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 3, Issue 1, January 2014
ISSN 2319 - 4847
AUTHORS
Varunika Arya received the B.Tech degree in Electronics and Communication Engineering from Kurukshetra
University in 2010. She is currently pursuing M.Tech in Electronics and Communication Engineering at
Maharishi Markandeshwar University, Sadopur, Ambala, Haryana, under the guidance of Mr. Amit Goyal.
Amit Goyal received the B.Tech and M.Tech degrees in Electronics and Communication Engineering from
Punjab Technical University in 2002 and 2010, respectively. During 2002-2010, he worked as Asstt. Prof. in
Electronics and Communication Engineering Department. His area of specialization is Neural Networks. His
other areas of interest includes Wireless & Mobile Communication, Microprocessors, Computer Networking
etc. He is currently working as Associate Professor in the Department of Electronics and Communication Engineering at
Maharishi Markandeshwar University, Sadopur, Ambala, Haryana
Vibhuti Jaiswal received the B.Tech degree in Electronics Engineering from Kurukshetra University and
M.Tech degree in Electronics and Communication Engineering from Maharishi Markandeshwar University.
She is working as Asstt. Prof. in the Department of Electronics and Communication Engineering at
Maharishi Markandeshwar University, Sadopur, Ambala, Haryana. Her areas of interest includes Neuro-fuzzy
System, Signals & Sytems, Network theory.
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