Mathematical Problems in Engineering

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Advanced Techniques for Computational and Information Sciences

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Volume 2015 |Article ID 212794 | 6 pages | https://doi.org/10.1155/2015/212794

An Improved Genetic Algorithm with Initial Population Strategy for Symmetric TSP

Academic Editor: Chih-Cheng Hung
Received18 Jun 2015
Accepted09 Aug 2015
Published05 Oct 2015

Abstract

A new initial population strategy has been developed to improve the genetic algorithm for solving the well-known combinatorial optimization problem, traveling salesman problem. Based on the k-means algorithm, we propose a strategy to restructure the traveling route by reconnecting each cluster. The clusters, which randomly disconnect a link to connect its neighbors, have been ranked in advance according to the distance among cluster centers, so that the initial population can be composed of the random traveling routes. This process is -means initial population strategy. To test the performance of our strategy, a series of experiments on 14 different TSP examples selected from TSPLIB have been carried out. The results show that KIP can decrease best error value of random initial population strategy and greedy initial population strategy with the ratio of approximately between 29.15% and 37.87%, average error value between 25.16% and 34.39% in the same running time.

1. Introduction

Traveling salesman problem (TSP) is a well-known NP-hard problem in many real-world applications, such as job-shop scheduling and VLSI routing [1, 2]. The aim of TSP is to find a complete, minimal-cost tour when a salesman is required to visit each of given cities once and only once [3]. So far, TSP has often been a touchstone for new strategies and algorithms proposed to solve combinatorial optimization problem. In this paper, we consider the symmetric TSP, where the distance from city to city is the same as from city to city .

Many methods have been developed for solving TSP, including exact algorithms and approximate algorithms. The exact algorithms are carried out to find the optimal solution from all valid solutions in a number of steps. But, because of exponential complexity [3], they are always infeasible if the scale of TSP becomes large, for example, 100 cities with approximately different solutions. In contrast, the approximate algorithms, especially many bioinspired algorithms [49], can obtain accepted solutions for many NP-hard problems with (relatively) short running time. These approaches are usually very simple, like Lin-Kernigan [10], colony optimization (ACO) [11], and so on [12, 13]. All of them are efficient approaches in most of the problems, but ACO, for example, is not suitable for large-scale TSP because of its computational cost , in which is the ant numbers and is the number of cities.

Genetic algorithm (GA) [14] is a global search algorithm appropriate for problems with huge search, for example, TSP, in which the initial population decides iterations, the crossover realizes the construction of the offspring, and the mutation operator maintains the diversity of the individuals. So far, there is a lot of literature to improve the effectiveness of crossover and mutation [2, 1519]. Besides, as the first step of any GA, how to initialize an efficient population plays an important role in the process of solving a problem based on GA. Toǧan and Daloǧlu [20] adopted the member grouping strategy to reduce the size of the problem and the initial population strategy to reduce the number of generations. Chen et al. [21] developed a feature-based initial population method for the optimization of job-shop problem. Sharma et al. [22] proposed a domain-specific initial population strategy for compliant mechanisms. Ahmed [23] used a sequential sampling method for generating initial population.

In this paper, a new initial population method has been developed to increase the quality of initial population. Based on this method, the application performance of GA on TSP becomes more efficient. The rest of the paper is organized as follows. In Section 2, some related background theories are presented. In Section 3, the proposed method is described. In Section 4, experimental results are evaluated. In the final section, a brief conclusion is given.

2. Preliminaries

Some basic theories are shown in this section, including TSP, GA, and -means clustering.

2.1. Traveling Salesman Problem

TSP is one of the most widely studied combination optimization problems [3]. Mathematically, this problem can be stated as follows:where path set is a permutation of cities and represents the distance from city to city . The aim of TSP is to find a path from path set to minimize .

2.2. Genetic Algorithm

Genetic algorithm (GA) is an evolutionary algorithm based on natural election, developed by Holland [24]. It is to find approximate solutions for optimization and search problems by computer simulation [25]. The aim of GA is to achieve better results through selection, crossover, and mutation. Selection is to select the best solutions preferentially according to the fitness function from the population. The fitness function () is defined over the genetic representation and measures the quality of the represented solution. In this paper, is assigned to . Crossover and mutation are used to generate a second generation population of solutions from those selected. Crossover can vary the population from one generation to the next by recombining “parent” solutions. Mutation alters some gene of one solution to avoid local optimal solutions. In general, the basic steps of GA for a problem are shown in Algorithm 1.

// is a city-location matrix
// is the current iterations
initialize with chromosomes // is the pop size
while not (terminating condition) do
for to do
   // is the fitness function
for to do
    randomly choose with
 recombine with
// is the crossover probability
 mutate with
// is the mutation probability

2.3. -Means Clustering

-means clustering is as well-known as Lloyd’s algorithm [26], which aims to find centers from observations to minimize the mean distance from each observation to their nearest center. More details about the concepts and description of -means clustering can be found in a lot of literature [2628]. In this paper, we present a simple -means clustering to the initial population.

3. Proposed Method

The first step of GA is to generate an initial population in which a set of possible solutions is contained. The quality of this population plays an important role in solving a problem by GA [20, 29]. As can be seen from Figure 1, in which a TSP of 14 cities is considered, Figure 1(d) is the best solution obtained from 3000 solutions generated randomly (RIP) [30] and Figure 1(b) is the best solution from 3000 alternatives with the proposed method (KIP), respectively. Figure 1(c) is the best solution of this problem.

KIG is developed for TSP solution based on -means clustering and GA. -means clustering is used to divide a large-scale TSP into some small problems to obtain local optimal solutions. Then, GA is carried out to globally optimize the alternatives generated by randomly rewiring each local optimal solution. Without loss of generality, consider a TSP with cities, in which denotes the location of city , . An initial population can be obtained as follows.

Step 1. cities cluster into groups with based on -means clustering.

Step 2. GA is used to obtain the local optimal path of each group and a global optimal path of groups.

Step 3. According to the global optimal path, one edge of each local optimal path disconnects to rewire the front and back groups.

Step 4. Repeating Step 3, an initial population can be generated.

Considering the 14 cities example as shown in Figure 1, 14 cities cluster into 4 groups . A global optimal path (the black path) and 3 local optimal paths (the red and blue paths) can be seen from Figure 1(a). Then, select one edge from each group () and disconnect to rewire the front and back group, such as Figure 1(a) (disconnect) to Figure 1(b) (rewire). From Figure 1(b) (the best solution from the initial population) to Figure 1(c) (the best solution of this problem), one step () can be achieved. Besides, a performance about three methods on example a280 selected from TSPLIB [35] is shown in Figure 2.

4. Experimental Results

We conduct the experiments on a computer with Intel Core i3-2120 3.30 GHz processor using MATLAB R2013a. To test the efficiency of the proposed method, 14 different TSP are taken from the TSPLIB [35] and 10 trials are examined for each problem. Without loss of generality, an improved GA developed by Kirk [34] was used to test the efficiency of the proposed method. The basic steps of this improved method are shown in Algorithm 2.

// is a city-location matrix
// is the current iterations
initialize with chromosomes // is the pop size
while not (terminating condition) do
for to do
   // is the fitness function
for to do
    choose satisfy and
    mutate by flip mutation method [31]
    mutate by swap mutation method [32]
    mutate by slide mutation method [33]
  

Firstly, the same iteration (20000) is considered to analyze the time cost and the quality of the solutions. There is no crossover but three mutations occurred at each iteration in the software. As we can see from Table 1, KIP is superior to other methods: of best error results and of average error results. Then, the same running time is used to analyze the iterations and the quality of the solutions by three methods on each problem. of best error results and of average error results demonstrate KIP more efficiently as shown in Table 2. Besides, Figure 3 shows the best solution varies by time about example pcb442.


Method RIP GIP KIP
Examples Best err. (%) Ave. err. (%) Ave. time (s) Best err. (%) Ave. err. (%) Ave. time (s) Best err. (%) Ave. err. (%) Ave. time (s)

berlin520.0000 5.1418 16.9338 0.00001.9531 18.2832 0.0000 3.0120 16.3208
kroA100 2.3761 4.8328 20.3049 0.2787 4.0082 22.5599 0.1035 2.5505 20.3318
pr144 1.3889 4.7425 23.4091 3.1847 6.0775 26.6297 2.0176 3.5041 26.5335
ch150 5.6974 9.5032 23.84364.17805.9298 27.4067 5.1038 6.7850 25.4257
kroB150 2.9646 6.4505 23.8947 6.3054 8.2309 27.4617 3.3544 5.8072 25.5628
pr152 1.4383 4.4424 24.1544 3.1397 4.2628 27.4174 2.5998 3.5800 26.4204
rat195 6.77339.261827.2911 7.8855 11.2154 30.5484 8.7133 11.3242 30.0717
d198 3.9011 6.8199 27.5327 5.1889 6.3098 32.1868 2.81494.5911 32.5780
kroA200 10.4079 13.3538 27.8636 11.1678 12.2504 32.4812 5.37258.8791 31.8691
ts225 16.0522 17.9204 29.5267 14.6896 15.499 33.6234 10.300814.5409 34.1776
pr226 5.2754 9.1196 29.7816 8.6261 12.9046 35.4601 4.97416.1991 36.3008
pr299 20.1269 24.5098 35.0112 29.1241 30.3129 42.5489 14.658417.0753 44.6964
lin318 30.1053 34.0994 36.2227 24.9692 31.5281 45.8366 15.066917.2412 47.2462
pcb442 54.7724 58.4570 45.8070 44.7984 49.0039 57.0011 22.588724.0461 62.8187
Average 11.5200 14.9039 27.9698 11.6812 14.2490 32.8175 6.97639.2240 32.8824


Method RIP GIP KIP
Examples Best err. (%) Ave. err. (%) Ave. time (s) Best err. (%) Ave. err. (%) Ave. time (s) Best err. (%) Ave. err. (%) Ave. time (s)

berlin52 2.2890 5.6054 16.3333 4.5497 5.4338 16.3333 0.00003.0120 16.3208
kroA100 2.4233 4.8220 20.3425 0.7447 4.4327 20.3425 0.10352.5505 20.3318
pr144 1.57512.9365 26.5358 1.8611 4.8923 26.5358 2.0176 3.5041 26.5335
ch150 4.7941 9.8965 25.4282 5.0946 7.7747 25.4282 5.1038 6.7850 25.4257
kroB150 2.9470 5.9548 25.5686 3.4723 6.9908 25.5686 3.3544 5.8072 25.5628
pr152 3.0562 4.2483 26.4266 1.7454 3.9500 26.4266 2.59983.5800 26.4204
rat195 6.32518.6355 30.0770 11.4580 12.9902 30.0770 8.7133 11.3242 30.0717
d198 4.4745 5.3608 32.5886 4.4171 4.9603 32.5886 2.81494.5911 32.5780
kroA200 9.3804 11.7009 31.8710 8.4947 11.349 31.871 5.37258.8791 31.8691
ts225 9.762813.5387 34.1798 11.1856 16.8642 34.1798 10.3008 14.5409 34.1776
pr226 6.6253 7.0874 36.3014 9.6295 12.8062 36.3014 4.97416.1991 36.3008
pr299 18.0739 21.8767 44.7099 24.5210 28.4839 44.7099 14.658417.0753 44.6964
lin318 25.4203 27.0580 47.2527 28.4007 29.9471 47.2527 15.066917.2412 47.2462
pcb442 40.6996 43.8170 62.8216 41.6218 45.9415 62.8216 22.588724.0461 62.8187
Average 9.8462 12.3242 32.8884 11.2283 14.0583 32.8884 6.97639.2239 32.8824

5. Conclusion

In this paper, we present a new initial population strategy (KIP) to improve GA that is used to find the optimal solution for the well-known traveling salesman problem (TSP). To test the performance of this strategy, 14 different TSP examples selected from TSPLIB and two other methods (RIP and GIP) are used to experimentalize. The results show that the proposed method is efficient: of best error results and of average error results (Tables 1 and 2). KIP can decrease best error value of RIP and GIP with the ratio of approximately between and , average error value between and in the same iterations (Table 1). KIP can decrease best error value of RIP and GIP with the ratio of approximately between and , average error value between and in the same running time.

Conflict of Interests

No conflict of interests exists between the authors regarding the publication of this paper.

Acknowledgments

The work is partially supported by National High Technology Research and Development Program of China (863 Program) (Grant no. 2013AA013801) and the open funding project of State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant no. BUAA-VR-14KF-02).

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Copyright © 2015 Yong Deng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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