ISPO : A New Way to Solve Traveling Salesman Problem

This paper first introduces the concepts of mobile operators and mobile sequence, with which it redefines the rate of particle swarm optimization algorithm and the formula of position updating. Combining this discrete PSO algorithm with neighbors, the paper puts forward Hybrd Particle Swarm Optimization Algorithm, whose effectiveness is verified at the end of this paper.


Introduction
The Particle Swarm Optimization algorithm [1] presented by psychologists Kennedy and Dr. Eberhart in 1995 is a new intelligent optimization algorithm which imitated the behaviors of birds.Compared with others, the PSO has shown its advantages.The PSO is intuitive and easy to realize and has high efficiency in complement.Nowadays, the PSO is widely used in many fields, such as functions optimization, training of neutral network, Fuzzy system control and so on.
Traveling salesman problem (TSP) is a typical combinatorial optimization problem, which may be described as follows.Given n cities and the distances between every two cities, which is the shortest path for travelling all the cities only once?
At first, PSO algorithm as an efficient method is mainly used to solve some continuous optimization problems.However, with the development of PSO algorithm, some experts and scholars began to think how to solve the discrete optimization problems, such as, Kennedy [2] proposed a discrete PSO algorithm used to solve binary problems in the year of 1997 and he proposed a dynamic probabilistic PSO in 2005 [3].In addition, some scholars presented the discrete PSO algorithm for solving TSP problem.Clerc [4] re-explained the velocity and position equation through replacement sequence, and Huang [5] proposed swapping sequence.Others combined crossover and mutation of genetic algorithm with PSO algorithm [6], Zeng and Cui presented a new unified model of PSO [7] which can be used in solving combinatorial optimization problem.This paper proposed mobile operators and mobile sequence for solving TSP problem.The arrangement of this paper is as follows.In Section 2, mobile operators and mobile sequence were defined.In Section 3, the velocity and position equation of PSO were re-explained and PSO algorithm was improved base on simulated annealing algorithm.In Section 4, TSP examples were used to evaluate the improved PSO algorithm, and the conclusions are given in Section 5.  (1,2,3,4,5), the mobile operator is sv (1,2), and s' can be presented as follows:

Mobile Operators and Mobile Sequence
, , , l v sv sv  , , , Definition 2. Mobile sequence is the ordered sequence of one or more mobile operators, and it can be expressed as follows: are mobile operators, and the order of each operators has the rigorous significance.
A mobile sequence is running on a TSP's solution which means all operators of the mobile sequence are running on the solution one by one.
Definition 3. Various mobile sequences running on one solution could bring same solution, all mobile sequences set with the same running result is called mobile sequences' equivalent set.Definition 4.More than one mobile sequence can be combined into a new sequence, is defined as the combining operator of two mobile sequences.
Definition 5. Assuming a mobile sequence , which is made up of n mobile operators, equals to , the length of this mobile sequence v is defined as n, i.e. the number of mobile operators.If v is one, v is called unit mobile sequence.Definition 6.Let be a real coefficient on the interval (0,1) and v be a mobile sequence and it equals to where rand is a random number between 0 and 1.It means, equals to on probability .

IPSO-Improved PSO Algorithm
Now, we can rewrite the iterative formula of PSO algorithm as follows: where := velocity of particle i at time step k + 1, As the PSO algorithm is easily trapped into local optimum in dealing with continuous problems, this paper applies the hybrid algorithm based on PSO to solve the TSP to avoid the same trouble.In each iteration, the PSO algorithm is running firstly, then produces l new positions in gk 's neighborhoods by using the simulated annealing (SA) algorithm, compares the fitness of l positions' and p gk 's, chooses the best position as p gk and assigns the new position of p gk to a particle as the first particle which is randomly selected.If the same solution is got in m times of iterations, the velocity of all particles must be re-initialized.

p p
The specific process of the improved algorithm is shown as follows: 1. Initialize the position and velocity of particle swarm, define velocity of each particle as the mobile sequence, and calculate , p and assign it to the first particle; 5. Determine whether the algorithm has found the same optimal value after m times' iteration, if the answer is yes, re-initialize all particle's velocity.
6. Determine whether the stop condition is met.If it's met, end the loop, otherwise, go to Step 2.

Experiments
The weight of inertia of PSO is linear, which can be expressed as follows: TSP with 14 nodes [4,8] is used to examine the effectiveness of the improved algorithm.Parameters setting: n-the number of particles is 40, the maximum times of iteration is 1000.The length of all particles' velocity is initialized to 1 randomly, and the value of m is 10 as well Copyright © 2013 SciRes.ICA as l.Weight initialization: max , min .All feasible solution sequence begins from the first node.We executed the algorithm randomly for 30 times, the results of the experiments are analyzed in the Table 1.
The maximum search space of the improved PSO algorithm is 50,000.The search space of [8] is 200000 and is 4 times of ours, which shows that our approach is more effective.And in all the 30 runs our approach has consistently found the optimal solution, which presents our algorithm convergence rate is 100 percent.
B so so so  equals to A. Normally, unit mobile sequence   is greater than zero and  is not greater than N. On the contrary, if is greater than N, it can be converted into a swapping sequence , sv i j i sv of length two, if j equals to I + 1, it can be converted into the unit mobile sequence sv i j i  .All of these imply that all problems the swapping sequence solved can be solved by mobile sequence, too.According to the length of exchange, mobile sequence has the advantage on swapping sequence.Moreover, if we take the definition  of swapping sequence into consideration, it has no identical operator, which means there is no unit swapping sequence A so A so can support the equation:   .However, the mobile sequence can overcome the shortcoming of swapping sequence with the equation is satisfied in all solution sequence.Therefore mobile sequence is more effective than swapping sequence.
For further verification of performance of improved algorithm, we apply the improved algorithm to solving Bays 29 [9] and Oliver 30 [10].The data come from TSPLIB, the weight decreases from 0.9 to 0.4 during the iteration and the maximum times of iteration is 1500, and other parameters are as same as TSP with 14 nodes.All experiments independently run 30 times and the results are shown in Table 2 and the best solutions of the two plans are shown in Figures 1 and 2.
From Table 2, it is obvious that the improved algorithm gets a high convergence rate either in Bays 29 or in Oliver 30, though the search space versus solution space is small (1.22996e−25, 4.24124e−27).It demonstrates that the improved algorithm has the advantages of small searching area and high convergence rate in solving the small TSP problems, and it is efficient.
problems.Through experiments in this paper, we can draw conclusions: 1) Using PSO algorithm with mobile operators and mobile sequence is an effective new way to solve TSP.
2) The improved algorithm has the advantages in small searching area and high convergence rate in solving the small TSP problems.
For comparison, three other artificial algorithms including basic SA, basic GA and Basic Ant Colony Algorithm are used to solve the same TSP problem.All four algorithms are run 20 times for Oliver30, the results are shown in Table 3.