Discrete Particle Swarm Optimization for TSP based on Neighborhood

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1 Journal of Computational Information Systems 6:0 (200) Available at Discrete Particle Swarm Optimization for TSP based on Neighborhood Huilian FAN School of Mathematics and Computer Science, Yangtze Normal University, Fuling Chongqing 40800, China Abstract Particle swarm optimization (PSO) is a kind of evolutionary algorithm to find optimal solutions for continuous optimization problems. Updating kinetic equations for particle swarm optimization algorithm are improved to solve traveling salesman problem (TSP) based on problem characteristics and discrete variable. Those strategies which are named heuristic factor, reversion mutant and adaptive noise factor, are designed and combined into a new hybrid discrete particle swam optimization algorithms. Experiments on low and high-dimensional data in TSPLIB show that, comparing with other hybrid discrete particle swarm (DPSO), the proposed algorithm can improve the search performance significantly no matter in convergent speed or precision. Keywords: Particle Swarm Optimization; Crossover; Reverse; Heuristic Methods. Introduction Particle swarm optimization (PSO) is a population based stochastic optimization technique developed by Dr. Eberhart and Dr. Kennedy[] in 995, inspired by social behavior of bird flocking or fish schooling. In PSO, the potential solutions, called particles which are initialized with a population of random solutions and searches for optima by updating generations, fly through the problem space by following the current optimum particles. Each particle keeps track of its coordinates in the problem space which are associated with the best solution (fitness) it has achieved so far. This value is called pbest. Another "best" value that is tracked by the particle swarm optimizer is the best value, obtained so far by any particle in the neighbors of the particle. This location is called lbest. When a particle takes all the population as its topological neighbors, the best value is a global best and is called gbest[2]. The particle swarm optimization concept consists of, at each time step, changing the velocity of (accelerating) each particle toward its pbest and lbest locations (local version of PSO). Acceleration is weighted by a random term, with separate random numbers being generated for acceleration toward pbest and lbest locations. Related work. The PSO algorithm has been successful in solving a number of continuous optimization problems, and attempts have been made recently to extend it to discrete optimization problems. Scholars have proposed a number of discrete particle swarm optimization (DPSO). Corresponding author. addresses: fhlmx@63.com (Huilian FAN) / Copyright 200 Binary Information Press October, 200

2 3408 H. Fan. /Journal of Computational Information Systems 6:0 (200) Kennedy and Eberhart[3] proposed the first discrete version (referred to as Quantum DiPSO in [4]) and Clerc [5,6,7] proposed a brief outline of the PSO method for solving TSP problems[8]. Hendtlass[9] applied the PSO algorithm to solve small-size TSP problems and improved its performance. Wang et al. [0] redefined the PSO operators by introducing the concepts of swap operator and swap sequence, therefore the TSP problems could be solved by the PSO in another way. By redefining the velocity and operation rules of PSO algorithm, above these methods can solve discrete optimization problems. However, because these algorithms lack of heuristic information for the solving problem, with the increase of dimension, the algorithms stability and convergence serious decline. Sizes of cities in [9] and [0] are both 4 (both of them selected Burma4, a benchmark problem in TSPLIB with 4 cities), and that of [8] is 7 (it selected br7, a benchmark problem in TSPLIB with 7 cities). That is to say, the sizes of cities are rather limited in their algorithms []. This paper proposes a novel hybrid discrete particle swarm optimization algorithm for traveling salesman problem, which incorporates problem characteristics in the search space. 2. TSP and Heuristic Factor Traveling Salesman Problem (TSP) is perhaps the most studied discrete optimization problem. Its popularity is due to the facts that TSP is easy to formulate, difficult to solve, and has a large number of applications. Given a list of cities and their pair wise distances, the task is to find a shortest possible tour that visits each city exactly once. The formal description is: given n cities and their coordinates, find an integer permutation π={c,c 2,,c n }, where c i denotes the city which number is i, that minimizes the sum f(π): n f ( π ) = d( ci, c ) d( cn, c ) i i+ + = where f(π) denotes the distance when the path is π, d(c i, c i+ ) denotes the distance between c i and c i+. In the final solution, best path is likely to contain the shortest distance path between adjacent cities. This feature is a property of TSP problem itself, also can be used in solving the problem. So, we can collect those nearest city for each city, and form the sequence vc=<c,c 2,,c m >, where c denotes the city number is c, and c is the nearest city to city number c, c 2 is the nearest city to city number c, and so on, the nearest city to c m is between c adn c m-. The sequence v c is heuristic factor, and all heuristic factors composed of heuristic factor set V={v c c=,2,,n}. Each v c has two characteristics: first, in the sequence, the next city is nearest city to its previous, and the second, the nearest neighbor of city c m is in from c to c m-. The heuristic factors structure of Eil5 in TSPLIB is shown in Fig.. ()

3 H. Fan. /Journal of Computational Information Systems 6:0 (200) Fig. The Heuristic Factors Set of Eil5 TSP Problem 3. Discrete Particle Swarm Optimization for TSP based on Neighborhood Standard PSO equations cannot be used to generate discrete values since positions are real-valued. For discrete optimization problem, conventional PSO algorithm must address the following two issues: How to change the position of particle? How to guarantee the position is reasonable? Inspired by the swap operator developed in [0], we redefine velocity and add heuristic factor, crossover operator and adaptive noise factor into our discrete PSO algorithm for the TSP. For an n-ordered TSP problem, n cities are numbered,2,3, and n respectively, denote X i = {x i, x i2,..., x in } as the ith particles position in the population of PSO, which represents the traveling circle of x i x i2 x in x i. Then, the fitness of the particle is the total tour length and given by Eq.(). Inspired by papers [0,2], we propose a self-adaptive hybrid discrete particle swarm optimization algorithm to tackle the discrete spaces, where particles are updated as follow three steps: 3.. Crossover Operator X = X ( t) (( c r) P ( t)) (( c2 r2) P ( t)) (2) i i i g where t is the number of algorithm iterations, x i (t) represents a traveling circle of ith particle at t iteration, P i (t) represents the best permutation which is the ith particle found after t iterates, P g (t) represents the best permutation which is the particle swarm found after t iterates, the values of c r and c2 r2 represents crossover length. The in update equation (2) is crossover operator. For n-ordered sequences X i (t)= {x t i, x t, i2..., x t in } and P i (t)= {p t i, p t t i2,..., p in }, the crossover operator acts on them is that: X i (t) P i (t)={p t i(k+}, p t i(k+2),...,p t i(k+m), x t il, x t i(l+),... },where k is a random integer in [,n-] and m=c r. For example, consider a particle instance with X i (t)= {5,3,7,2,8,0,6,4,9,},k=3,c r=4 and P i (t)={7,4,2,9,6,3,5,8,,0}. Fig.2 illustrates the crossover operator in detail:

4 340 H. Fan. /Journal of Computational Information Systems 6:0 (200) Fig.2 Crossover Operator 3.2. Inject or Reverse Operator X ( t ) inject( X, v ) reverse( X, s, e) i i c i + = (3) where v c in update Eq.(3) is heuristic factor which represents permutation start with city c, and all heuristic factors composed of heuristic factors set V={v c c=,2,,n}. Given particle s position X i = {x i, x i2,...,c,..., x in } and v c =<c,c 2,,c m >, define inject(x i,v c )={x i,x i2,..., c,c,c 2,,c m,...,x in }, and then eliminate all repetitive city in the permutation of original X i. The method for selecting inject point is: The farther distance between adjacent cities <a,b> in permutation, the more probability of selection. The probability is defined as: n pi = d( c, c ) d( c, c ) i i+ j= j j+ where d(c i, c i+ ) represents the distance between city c i and c i+. For instance, consider the X={9,6,3,5,7,2,8,0,4,},inject point is after city number 3 and heuristic factor v 3 ={6,8,2},then, after inject(x, v 3 ) is done, we get X={9,3,6,8,2,5,7,0,4,}. For an n-ordered sequence X i = {x i, x i2,..., x is, x i(s+),..., x i(e-), x ie,..., x in }, the reverse operator acts on X i is that: reverse(x i,s,e)= {x i, x i2,..., x ie, x i(e-),..., x i(s+), x is,..., x in }.It means to reverse the sub-position {x s, x s +,..., x ie } which start from element s and e elements are included, where s and e is random integer in[,n] and s<e. That is, given a position X = {,2,3,4,5,6,7}, after reverse(x,2,5) is done, we get X = {,5,4,3,2,6,7}. The in update Eq.(3) indicates either carry out inject or reverse operator and their probability is 50% Inject or Reverse Operator The noise factor is designed based on particle swarm diversity, which is defined as the similarity of particles position[0].use s ij represents similarity between particle X i and X j and defined as: s ij n = δ if(x ik ==X jk ) δ= elseδ=0 (5) n k= where (X ik ==X jk ) represents position of particle X i, X j have same value and the value of s ij is in [0,]. The diversity of ith particle is the similarity of the particle, Pi(t) and P g (t), can be computed as: diversityi = ( si, pi + si, pg + spi, pg ) (6) 3 The population diversity is the mean of diversity of all particles, is given by: diversity n diversityi n i = = (7) (4)

5 H. Fan. /Journal of Computational Information Systems 6:0 (200) So, position of particle at t+ iteration can by computed as: whereη defined as: X t+ = η disturb X t+ K (8) i( ) ( i( ), ) diversity < 0.4 η = 0 diversity 0.4 When diversity<0.4, carry disturb operator. Where, diversity is noise factor based on population diversity. For n-ordered sequences X i = {x i, x i2,..., x ik,..., x in },the disturb operator acts on X i is that: disturb(x i, K)={ x i, x i2,..., x ik,..., x in }, where x ik is the nearest to the x ik, K is a random integer in[0,n], and the operator may be carried more than one time where necessary. 4. Steps of the Algorithm The methodology, steps and strategies of my Self-Adaptive Hybrid Discrete Particle Swarm Optimization algorithm are given in detail: Step. Initialization Initialize the number of particles, the iteration times (Itermax) and each particle s position. Calculate the distance between cities and save to the distance matrix dist, and thus, heuristic factors can be built to array lv[..n], where lv[i] represents a chained list which start with i city. Step 2. Calculate fitness Calculate the fitness of each particle based on the above distance matrix dist. Every particle utilizes the current position as the present best position (pbest) of its own, and the global best position(gbest) is the best position in current particle swarms. Step 3. Start iteration Step 4. Crossover operator Execute the crossover operator for each particle according to Eq. (2). Step 5. Update pbest Calculate the fitness of each particle. Replace the pbest of the current particle with the current position of the particle if and only if current fitness is better than pbest. Step 6. Update gbest Update the gbest based on all particles s pbest. Step 7. Update position of particle Update position of particle for each particle according to Eq.(2),(3) and (8). Step 8. Judge the termination criterion If the current iteration iter<itermax, then go to STEP 3, else go to STEP 0. Step 9.: Output the global best position gbest. 5. Experimental Results and Discussion To show the efficiency of the proposed Improved Hybrid Discrete Particle Swarm Optimization (IHDPSO), we consider the following examples, low-dimensional TSP (Burma4), higher-dimensional TSP (Oliver30) and high-dimensional TSP (Eil5) from the TSPLIB library have been selected. (9)

6 342 H. Fan. /Journal of Computational Information Systems 6:0 (200) For Burma4 TSP, set the swarm size m=0, the number of iteration Iteramax=50. For Oliver30 TSP, set the swarm size m=30, the number of iteration Iteramax=300. For Eil5 TSP, set the swarm size m=60, the number of iteration Iteramax=000. Compare with the HDPSO proposed by reference [2], all simulations are run for 20 trials and the results are shown in Table. Table Comparison Results of IHDPSO and HDPSO for TSP Problem TSP Problem Algorithm Best value Average value Relative Error(%) Burma 4 Oliver30 Eil5 IHDPSO HDPSO IHDPSO HDPSO IHDPSO HDPSO In table, Average Value is the average fitness from 20 trials of each algorithm, and Relative Error is defined as Relative Error = (Average value Best value)/best value 00%. From the results of the two algorithms, it is clear that our algorithm (IHDPSO) is significantly better than the HDPSO algorithm. For Burma4 problem, although the best solution, average value and relative error is same, but our algorithm reached results requiring far fewer fitness evaluations than the HDPSO algorithms, using a smaller population size. Fig.3 compares the convergence performance of the two algorithms for Eil5 problem. Fig.3 Convergence Curves of IHDPSO and HDPSO for Eil5 For Oliver30 and Eil5 problem, in relative error and the average value, performance is also greatly improved by IHDPSO algorithm. Meanwhile, for Eil5 problem, the best fitness value achieved by IHDPSO is not only the smallest float value in these two algorithms but also the best value that has been obtained so far we know. It is shown that the IHDPSO algorithm is a better and more effective means to solve TSP problem. Fig. 4 illustrates the best solution of Eil5 by applying IHDPSO.

7 H. Fan. /Journal of Computational Information Systems 6:0 (200) Fig.4 The Best Solution of Eil5 by Applying IHDPSO 6. Conclusion and Future Work Focused on the TSP problems, a novel hybrid discrete PSO algorithm has been presented by add heuristic factor, crossover operator and adaptive disturbance factor into the approach. Numerical results show that the proposed algorithms are effective. The results presented in the previous section show that IHDPSO is a powerful algorithm that succeeds in solving the TSP problems using a small number of fitness evaluations to reach the best fitness. In the future work, we will study more exhaustive testing of my IHDPSO for other benchmark problems. Acknowledgement This work is supported by the Natural Science Project of Chongqing Municipal Commission of Education under Grant No.KJ We would like to thank to Prof. Xianli Li for providing the PSO algorithm code. Even though we developed our own code in Java, it was substantially helpful in grasping the components of the PSO algorithm in a great detail. We also appreciate his endless support and invaluable suggestions whenever needed. References [] J. Kennedy and R. C. Eberhart. Particle Swarm Optimization. In Proceedings of the IEEE International Conference on Neural Networks, pages ,995. [2] LÜ Yan-Ping, LI Shao-Zi, CHEN Shui-Li. Particle Swarm Optimization Based on Adaptive Diffusion and Hybrid Mutation. Journal of Software,8(): ,2007. [3] J. Kennedy and R. C. Eberhart. A discrete binary version of the particle swarm algorithm. In Proceedings of the IEEE International Conference on Computational Cybernetics and Simulation, pages ,997 [4] C. K. Mohan and B. Al-kazemi. Discrete particle swarm optimization. In Proceedings of the Workshop on Particle Swarm Optimization, Indianapolis, IN: Purdue School of Engineering and Technology, IUPUI, 200. [5] M. Clerc. The Swarm and the Queen: Towards a Deterministic and Adaptive Particle Swarm Optimization. In Proceedings of the Congress on Evolutionary Computation, pages ,999. [6] Clerc M, Kennedy J. The particle swarm-explosion, stability and convergence in a multi-dimensional complex space. IEEE Trans on Evolutionary Computation, 6(): 58 73,2002. [7] CLERCM. Discrete particle swarm optimization. New Optimization Techniques in Engineering. Berlin: Springer2Verlag, 2004: [8] M. Clerc. Discrete particle swarm optimization illustrated by the traveling salesman problem.

8 344 H. Fan. /Journal of Computational Information Systems 6:0 (200) [9] T. Hendtlass. Preserving Diversity in Particle Swarm Optimization.Lecture Notes in Computer Science, vol. 278: ,Springer,2003. [0] K.P. Wang, L. Huang, C.G. Zhou, W. Pang. Particle swarm optimization for traveling salesman problem. In Proceedings of the International Conference on Machine Learning and Cybernetics,pages ,2003 [] X.H. Shi, Y.C. Liang, H.P. Lee, C. Lu, Q.X. Wang. Particle swarm optimization-based algorithms for TSP and generalized TSP. Information Processing Letters,03(5):69-76,2007 [2] YU Ling-li, CAI Zi-xing, Multiple optimization strategies for improving hybrid discrete particle swarm. Journal of Central South University (Science and Technology),40(4): ,2009.