We proposed a novel approach for solving TSP using PSO, namely edge-PSO by intelligent use of the edge recombination Operator. We observed that the edge recombination operator which was originally proposed for Genetic Algorithm can be used as a velocity operator for Particle Swarm Optimization so as to direct the search effectively to better corners of the hypercube corresponding to the solution space in each iteration thus significantly reducing the number of iterations required to find the optimum solution. The edge-PSO algorithm not only improved the convergence rate but also could produce near-optimal solutions, with accuracy better than those obtained from GA even without the use of a local search procedure for standard instances from TSPLIB.

1. Edge PSO : A Recombination Operator
Based PSO For Solving TSP
Vaisakh Shaj
Akhil P M
Asharaf S

2. PSO
Kennedy Ebehart (2001)

3. Particle Swarm Optimization
Current Direction
Personal Best
Global Best
Current Direction
Personal Best
Global Best
Arrows Representing Velocity Vector (Magnitude and Direction)

4. Particle Swarm Optimization
Current Direction
Personal Best
Global Best

5. History
 State of the art results in optimizing continuous non-linear functions.

 Motivated by success of PSO algorithms for continuous problems, there has been
attempts to adopt the above proposals to the discrete cases , often illustrated using
the popular TSP problem.

 TSP - Given a list of cities and the distances between each pair of cities, what is
the shortest possible route that visits each city exactly once and returns to the
origin city? It is an NP-hard problem in combinatorial optimization, important in
operations research and theoretical computer science.

6. History
 State of the art results in optimizing continuous non-linear functions.

 Motivated by success of PSO algorithms for continuous problems, there has been
attempts to adopt the above proposals to the discrete cases , often illustrated using
the popular TSP problem.

 TSP - Given a list of cities and the distances between each pair of cities, what is
the shortest possible route that visits each city exactly once and returns to the
origin city? It is an NP-hard problem in combinatorial optimization, important in
operations research and theoretical computer science.

7. How to fit PSO framework to a discrete
case ?
 How to perform meaningful position wise subtraction ?
 How to perform addition of a position and velocity required in the position update
stage ?
3 4 5 1 6 2 5 2 3 1 6 4 ?
3 4 5 1 6 2 ? ? ? ? ? ? ?

8. Past attempts to solve TSP using PSO
 Clerc (2004)
 Wang et all (2003)
 Approaches using swap sequences could only reasonably solve problem instances
upto 20 cities . Clerc in his 2004 paper even made the conclusion that PSO is not a
good technique to solve discrete optimization problems like PSO.
 Past few years further attempts using hybridized approaches – using ACO, GA and
problem specific high cost local search procedures. Problem instances <100 – 200 in
most cases.
Swap Sequences
3 4 5 1 6 2 5 2 3 1 6 4 (p1,p3) , (p2,p4)
5 2 3 1 6 4 (p1,p3) , (p2,p4) 3 4 5 1 6 2

9. Bringing In Recombination Operators to PSO Framework
 Why not use the recombination operator, traditionally proposed and used for
Genetic Algorithms, in discrete PSO to compute the dependent moves?
EDGE PSO

10. Edge PSO – Position Update Rule
 x(t+1) = x(t) + v(t+1)
=
x(t)
Pbest(t) x(t)
Gbest(t) x(t)
v(t+1) = c1*v(t) + c2 * ( Pbest(t) x(t) )
+ c3 * ( Gbest(t) x(t) )
Subjected to condition
c1,c2,c3 є { 0, 1 }
c1+c2+c3=1

11. Advantage
 Avoids explicitly calculation of the position – position subtraction, position - velocity
addition usually involved in the continuous PSO case.

 Good trade-off between exploitation and exploration, by careful activation of the c1,
c2 and c3 parameters.

 Complexity of the heuristic depends on the complexity of the recombination
operator and if we can use a low cost operator(eg: O(N)), it can scale to larger city
problems.

12. Experiment 1
 Problem Addressed : How Edge-PSO performs when compared with Edge-GA?
 Both Using O(N) complexity edge Recombination Operator.
 Why edge Recombination Operator? – Because it satisfies the Radcliffe and Surry
Conditions :

1. A recombination operator is said to respect a representation if and only if every
child it produces contains all the alleles common to its two parents.
2. A recombination operator is said to transmit alleles with respect to a given(formal)
allelic representation if and only if each allele in every child it produces is present
in at least one of its parents.

13. Result
 Convergence of edge PSO vs edge GA for 26 city.
Solution Error Iterations
Edge PSO 0.69 % 40
Edge GA 2 % 210

14. Result
 Convergence of edge PSO vs edge GA for 42 city .
Solution Error Iterations
Edge PSO 1.6 % 80

15. Result
 Edge PSO – Edge GA Comparison with larger city problems

16. Conjecture

17. Experiment 2
 How edge PSO performs one even larger city problems?

 The error rate was increasing for larger city problems, due to premature
convergence, a problem quite common with swarm intelligence based algorithms.

 In order to address this issue, we used two techniques :
1. Reverse Sequence Mutation Operator
2. Local Search Procedures

18. Result
 Was able to solve medium city problems ( 100 to 1000 Cities ) with reasonable
accuracy

19. Comparison with other evolutionary
algorithms from literature

20. Possibilities?
 A possible option Partition Crossover (2010 Whitley et all ), which also satisfies the
Radcliffe and Surry Conditions, at the same time quite efficient than edge
Recombination operator when the two parents are local optimum.
Velocity Update
Any Recombination Operator

21. Future Work
 Hope this would encourage researchers to develop PSO specific recombination
operators, keeping in mind the global vs local interaction that PSO encourages.