Some issues of designing genetic algorithms for traveling salesman problems

Some issues of designing genetic algorithms for traveling

salesman problems

H.-K. Tsai, J.-M. Yang, Y.-F. Tsai, C.-Y. Kao

Abstract This paper demonstrates that a robust genetic algorithm for the traveling salesman problem (TSP) should preserve and add good edges efficiently, and at the same time, maintain the population diversity well. We analyzed the strengths and limitations of several well-known genetic operators for TSPs by the experiments. To evaluate these factors, we propose a new genetic algorithm integrating two genetic operators and a heterogeneous pairing selection. The former can preserve and add good edges efficiently and the later will be able to keep the population diversity. The proposed approach was evalu-ated on 15 well-known TSPs whose numbers of cities range from 101 to 13509. Experimental results indicated that our approach, somewhat slower, performs very robustly and is very competitive with other approaches in our best sur-veys. We believe that a genetic algorithm can be a stable approach for TSPs if its operators can preserve and add edges efficiently and it maintains population diversity. Keywords Edge assembly crossover, Heterogeneous pairing selection, Genetic algorithm, Neighbor-join mutation, Traveling salesman problem

1

Introduction

The traveling salesman problem (TSP) is a well-known NP-hard optimization problem which requires the deter-mination of the shortest route passing through a set of M cities under the condition that each city is visited exactly

once. TSPs raise important issues because many problems in science, engineering, and bioinformatics fields, such as routing, scheduling problems, flexible manufacturing systems, physical mapping problems [1], and phylogenetic tree construction [14] can be formulated as TSPs.

A large number of approaches have been developed for solving TSPs. A very promising direction is the genetic algorithm (GA) [8]. It is based on the ideas borrowed from genetics and natural selection. A genetic algorithm is a generally adaptable concept for problem solving. It is especially well suited for solving difficult optimization problems, where traditional optimization methods are less efficient. However, general problem-independent GAs are not very efficient in solving TSPs, especially for large problems.

To further improve the GAs for TSPs, many approaches have been proposed. Among these approaches, designing TSPs-specific operators, incorporating domain-specific local searches, and keeping population diversity are con-sidered as promising strategies. Designing TSPs-specified crossovers, such as cycle crossover [20], edge recombi-nation crossover [29], maximally preserving crossover [18], edge assembly crossover [19] and inver-over opera-tors [27], could raise the performances of GAs for solving TSPs. Incorporating domain-specific local search

techniques into GAs [12–13, 17] possess both the global optimality of the GAs as well as the convergence of the local search. Keeping population diversity is useful to avoid premature convergences [5, 23].

Several issues responsible for improving the solution quality of GA for solving TSPs were discussed. Here we addressed these issues by analyzing the behaviors of some crossover and mutation operators on some well-known TSPs. We found that genetic algorithms for TSPs should at least have three mechanisms: powerful genetic operators which can preserve and add ‘‘good edges’’ (i.e., the edges in the optimal tour) as well as a mechanism to keep the population diversity. Based on these factors we designed a new genetic algorithm for TSPs. It was evaluated on 15 well-known traveling salesman problems [22] whose numbers of cities range from 101 to 13509 cities. The experimental results indicated that the solution quality of the proposed GA stays within 0.00043 from the optima for testing problems. It is more robust than comparative approaches.

The rest of this paper is organized as follows. In Sect. 2, some issues of designing GA for TSP are discussed by analyzing crossover operators, mutation operators, and the mechanism of keeping population diversity. Section 3

Original paper Soft Computing 8 (2004) 689–697
Springer-Verlag 2003

DOI 10.1007/s00500-003-0317-8

Published online: 24 November 2003 H.-K. Tsai (&)

Department of Computer Science and Information Engineering, National Taiwan University,

Taipei 106, Taiwan

e-mail: d7526010@csie.ntu.edu.tw J.-M. Yang

Department of Biological Science and Technology & Institute of Bioinformatics,

National Chiao Tung University, Hsinchu 30050, Taiwan Y.-F. Tsai

Department of Information Management,

Chung Yu Junior College of Business Administration, Keelung 201, Taiwan

C.-Y. Kao

Bioinformatics Center,

National Taiwan University, Taipei 106, Taiwan

introduces a new genetic algorithm based on the analysis of these factors and compares it to other famous

approaches on some TSP benchmarks. Concluding comments are drawn in Sect. 4.

2

Some issues of designing GAs for TSPs

This section discusses some issues of designing GAs for TSPs by experiments. Some crossover and mutation operators are analyzed based on the mechanisms of pre-serving and adding good edges. The experimental results indicated that edge-based operators, such as edge assem-bly crossover and neighbor-join mutation, were able to meet these features. In order to keep the population diversity, a heterogeneous pairing selection was developed. 2.1

Analysis of crossover operators

Crossover operators for TSPs can be roughly divided into three categories: interval-preserving, position-based, and edge-based crossover. In an interval-preserving crossover, such as partially mapped crossover (PMX) [10], order crossover (OX) [6], order-based crossover (OBX) [26], and maximal preservative crossover (MPX) [18], a sub-path between two selected cities were copied from one parent to the offspring. Other cities were added into offspring according to the relative order of another parent. It exe-cutes just like typical two-point crossovers and repairs the solution based on parents’ information. A position-based crossover preserves the relative position of cities in par-ents and acts like traditional uniform crossover. It at-tempts to create an offspring from the parents where every position is occupied by a corresponding element from one of the parent. The most representative ones are cycle crossover (CX) [20] and position-based crossover (PBX) [26]. An edge-based crossover, such as heuristic crossover (HX) [11], edge recombination crossover (ERX) [29], distance preserving crossover (DPX) [7], and edge-assembly crossover (EAX) [19], generates offspring by preserving edges in parents and adding new edges heu-ristically. Different methods employ different preserving and adding mechanisms.

We used a simple GA to analyze the performances of these crossover operators. Fig. 1 shows the pseudo code of the simple GA, where the wheel selection [4] is used to select the pairing chromosomes and total generation replacement with elitism is adapted to select next popu-lation. The population size is set equally to N. Termination criteria are that the number of function evaluation exceeds 10000 N, where N is the number of cities.

Table 1 shows the experimental results for each tested crossover with the simple GA (without mutation) on five small TSP benchmarks, including att48, eil76, eil101, kroa150, and kroa200. In Table I, the values in parentheses of problems represent the optimal tour length. The column ‘‘Length’’ represents the brood size (L) defined in Fig. 1. brood size is the number of solutions a genetic operator generated. Each entry is the average error defined as
_average

optimum
1 

 100 where average is the average values of the best solutions obtained by testing approaches executed for 20 trials. As observed, when the brood size equals to 1, all the edge-based crossovers, except ERX, outperform the interval-based and position-based cross-overs. When the brood size is 20, the solution qualities of all operators are improved except CX, which is a position-based crossover. Another position-position-based crosssover, PBX, is not shown here beacuse the testing results is similar to CX. At the same time, the edge-based crossovers, including EAX, HX, DPX, and ERX are significantly better than those none edge-based crossovers. According to these experi-ments, we suggested that a good crossover operator should maintain the spirit of edge-based crossovers and execute for a larger brood size. The following describes that to preserve and add good edges is the main spirits of an edge-based operator.

In these surveyed edge-based crossover operators, EAX is the best one. We further discussed the results of these operators on several large TSPs. Figure 2 shows the results of these edge-based crossovers on eight TSP benchmarks, where the brood size is 20. Each problem is executed 20 trials. The relative error is defined as
_optimumaverage
1 100 where the ‘‘average’’ is the average values of the best solutions obtained by these methods. EAX still

outper-Fig. 1. The pseudo code of the tested simple GA

forms the others. HX and DPX get similar results, while ERX becomes worse when the number of cities increases.

In the following, we will analyze the characteristics of edge-based operators based on their abilities of preserving and adding edges to understand two issues: why edge-based operators are better than none edge-edge-based operators and why EAX is the best one among these tested operators. As mentioned in the previous paragraph, edge-based crossovers generate offspring by preserving and adding edges heuristically.

Table 2 briefly summarizes the mechanisms of pre-serving and adding edges of these four edge-based

cross-overs. Observing the mechanism of preserving edges of these crossovers, we found DPX only preserves the com-mon edges appeared in parents, HX preserves edges by concerning shorter edges from parents, and ERX inherits common edges from parents as much as possible. EAX inherits shorter edges from parents and considers the frequency of edges appeared in the current population. As to the aspect of adding edges, DPX modifies the

interme-diate solution by adding shorter edges, ERX and HX randomly add new edges to form feasible solutions, and EAX preserves edges heuristically and adds new edges with a greedy method which is analogous to a minimal span-ning tree.

To further examine the mechanisms of preserving and adding edges, we measured the abilities of adding and preserving ‘‘good edges’’ (i.e., the edges in the optimal tour) of EAX, DPX, HX, and ERX operators. Figure 3 shows the results of these four operators tested on prob-lem att532.tsp for adding and preserving ‘‘good edges’’ and average edge length. The average added ‘‘good edge’’ is defined as

where n is the popluation size. The average preserved ‘‘good edge’’ is defined in the similar way. The average edge length is defined as

Pn i f ðsiÞ

nm ;

where m is the number of cities, n is the popluation size, and f(si) is tour length of individual si.

Table 1. Comparison of nine crossovers with different search lengths using the simple GA (Fig. 1) on five TSPs benchmarks. The values in parentheses of problems represent the optimal tour length. ‘‘Length’’ denotes the brood size. Each entry is the average error defined as ð_optimumaverage
1Þ  100 where average is the average values of the best solutions obtained by testing approaches.for independent 20 trials

Type Operator Length att48 eil76 eil101 kroa150 kroa200 Edge based EAX 1 0.0263 0.0186 0.0000 0.0008 0.0136

crossover 20 0.0000 0.0000 0.0000 0.0000 0.0051 HX 1 1.2006 3.4944 4.8490 5.3314 8.1745 20 2.3758 2.5465 3.8474 6.0153 7.1700 DPX 1 6.2495 4.3123 14.3402 11.8323 10.8428 20 4.0581 2.4535 8.6963 5.8355 3.7602 ERX 1 43.6187 88.3086 178.6010 460.6364 635.6923 20 9.2416 11.2825 12.5278 23.2024 28.5654 Interval based MPX 1 82.9592 149.8885 208.1240 460.6858 602.9120 crossover 20 12.0342 11.8773 17.6471 50.0475 173.2563 PMX 1 87.2168 104.6097 124.1812 226.7577 275.0259 20 63.3628 82.9740 105.5803 190.0468 237.2678 OX 1 80.3218 152.8903 214.7059 474.0624 609.4436 20 12.8265 11.8680 40.5087 222.6821 359.7756 OBX 1 96.3361 111.3941 148.5692 168.8045 105.8833 20 49.0102 63.2528 72.0668 73.2160 44.7327 Position based CX 1 163.8700 188.3829 216.0890 392.3567 460.2087 crossover 20 160.1402 183.7732 224.2925 401.2091 479.9527

Fig. 2. Comparisons of four edge-based crossovers, including EAX, HX, ERX, and DPX using the simple GA (Fig. 1) on 8 TSP problems based on error defined as ð_optimumaverage
1Þ  100 where the ‘‘average’’ is the average values of the best solutions obtained by

these methods. EAX is the best one. The results of HX and DPX are similar, while the ERX is much worse when the number of cities increases

Pn

ið#of edges added after the genetic operator applied on solution iÞ

n ;

The population size and brood size are set to 500 and 20, respectively. Only the first 250 generations were ob-served. The similar results were obtained for the other problems. Figure 3(a) shows that EAX can continuously add more ‘‘good edges’’ than other operators in earlier stage (before the 70th generation). After the 70th genera-tion, the values of the average preserving ‘‘good edges’’ (Fig. 3(b)) and average edge length (Fig. 3(c)) of EAX are better than other operators. HX and DPX have similar behaviors for these three factors and ERX has the worst values. These results are consistent with our previous discussions. In summary, a good crossover operator should be edge-based and possess good mechanism of

adding and preserving ‘‘good edges’’. EAX is considered as a good crossover which meets these requirements. 2.2

Analysis of mutation operators

Some local search heuristics, such as 2-swap, 2-opt, 3-opt [15], neighbor-join [28] and Lin-Kernighan [16] have been used in GAs as mutation operations for TSPs. GAs and other evolutionary algorithms that use local search are often referred as Memetic algorithms. These mutations exchange some edges of parents to generate new children. Generally, the stronger the mutation (local search) used the better the performance of the GA.

Table 2. The mechanisms of preserving and adding edges of edge-based crossovers, including EAX, DPX, HX, and ERX

Operator Mechanisms

Preserving edge Adding edge EAX Offspring inherits edges from parents

according to the edge length and the frequencies appeared in the population

New edges are added into offspring to modify the intermediate solution by adding short edges through a spanning tree method

DPX Only the common edges in both parents are passed into offspring

Intuitively adding short edges to make the intermediate solution feasible

HX Offspring inherits edges from parents where the edge length is as short as possible

New edges are randomly added to become a feasible solution

ERX Offspring inherits edges from parents as much as possible, but the maximum number of inherited edges is not guaranteed

New edges are randomly added to become a feasible solution

Fig. 3. Comparison of edge-based crossovers, EAX, DPX, HX, and ERX, on att532.tsp based on the average added ‘‘good edges’’ a, the average preserved ‘‘good edges’’ b, and the average edge length c

Fig. 4. Comparisons of three mutations, including 2-swap, 2-opt, and NJ using the simple GA (Fig. 1) on 8 TSP problems based on ‘‘error’’ defined as

ð_optimumaverage
1Þ  100 where average is the average values of the best solutions obtained by these methods. NJ is the best one. The results of 2-opt and 2-swap performed poorly as the number of cities increased

Figure 4 shows the experimental results for three tested mutations, including NJ, 2-swap, and 2-opt, with the simple GA (Fig. 1) on eight small TSP benchmarks. These problems are including att48, eil76, eil101, kroa150, kroa200, lin318, pcb442, and att532. No crossover opera-tors were adapted in this test set. The error is defined as
_average

optimum
1 

 100 where average was the average values of the best solutions obtained by these methods in 20 trials. NJ performed well. 2-opt became worse when the number of cities increased. 2-swap was the worst one.

The mechanisms of adding and preserving edges of these mutations vary. The 2-swap mutation arbitrarily changes two cities at a time and thus four edges are randomly removed and four edges are randomly added. For 2-opt and NJ, they exchange some edges if the solution is better. In each iteration 2-opt and NJ exchange 2 and 4 edges, respectively.

We used the same strategies for crossover operators to measure the abilities of adding and preserving ‘‘good edges’’ of NJ, 2-opt, and 2-swap mutations. Figure 5 shows the results of these operators tested on problem att532.tsp for adding and preserving ‘‘good edges’’ and average edge length. The similar results were obtained for the other problems. Figure 5(a) shows that NJ can add more ‘‘good edges’’ than other operators in earlier stage as well as the average preserving ‘‘good edges’’ (Fig. 5(b)) in later. 2-opt and 2-swap have similar behaviors in adding edges, how-ever 2-swap is worse in preserving good edges. These results are consistent with our previous observations. In summary, a good mutation operator should possess good

mechanisms of adding and preserving ‘‘good edges’’. NJ is considered as a good mutation operator which meets these requirements.

Experimental results show that by combining NJ with EAX, TSPs can be well solved [28]. To examine the com-plementary characteristics of EAX and NJ, we measure the abilities of adding and preserving ‘‘good edges’’ (i.e., the edges in the optimal tour) by EAX and NJ operators. Figure 6 shows the results of EAX and NJ for adding and preserving ‘‘good edges’’. Figure 6(a) shows that NJ can add more ‘‘good edges’’ than EAX in early stage, while EAX outperforms NJ in late stage. The abilities of pre-serving ‘‘good edges’’ of these two operators are similar (Fig. 6(b)). Although we cannot theoretically prove the seamless of combining EAX and NJ, they indeed com-pensate each other in adding and preserving ‘‘good edges’’ in our experiments.

2.3

Mechanism of keeping diversity

Keeping population diversity is another issue in GA for TSPs. Here we designed a new pairing selection, named heterogeneous pairing selection (HpS), to select two parents for crossover operators to reduce the premature convergence effect based on the edge similarity of a population.

The formulation and implementation of the HpS is described as follows: Let {s1, s2,…sN} be the current pop-ulation, E(si) be the set of the edges of si, and ||E(si)|| be the number of the edges of E(si). The number of identical

Fig. 5. Comparison of three mutations, 2-swap, 2-opt, and NJ, on att532.tsp based on the average added ‘‘good edges’’ a, the average preserved ‘‘good edges’’ b, and the average edge length c

Fig. 6. The ability of adding and preserving good edges of EAX and NJ on the problem fnl4461. They seem be able to compensate each other on adding ‘‘good edge’’ and are similar on preserving ‘‘good edge’’

edges jjTi;jjj of two individuals (siand sj) is defined as jjTi;jjj ¼ jjEðsiÞ \ EðsjÞjj For each individual si, let tibe the average number of identical edges between siand the other individuals in the population ti¼_N
11 PNj¼1;j6¼ijjTi;jjj, where N is the population size.

For the given individual si, the HpS selects siand an-other individual sjwith jjTi;jjj  ti for the EAX operator. This similarity-based mechanism is useful for keeping the population diversity. Our experimental results were consistent with this claim. In practical implementation, we used another method to calculate tidue to the time complexity of getting all tithrough calculating ||Ti,j|| is O(N2M2). At the beginning of each generation, F(e), the count of edge e appearances in the current population, is calculated in advance where e 2 fEðs1Þ [ Eðs2Þ [ . . . [EðsNÞg. The sum of ||Ti,j|| of si in the population, can be reformulated as ti¼ 1 N
1 XN j¼1; j6¼i jjTi;jjj ¼ 1 N
1 XN j¼1 jjTi;jjj
jjTi;ijj ¼ 1 N
1 X e2EðsiÞ FðeÞ
M 0 @ 1 A ¼ 1 N
1 X e2EðsiÞ ðFðeÞ
1Þ :

Therefore, all tican be calculated in O(NM) by looking up the pre-calculated table. Since the EAX crossover also uses the information F(e), the extra effort of calculating tiwill be limited.

To show the performance of HpS, we compared HpS with five selections, including random pairing selection (RpS) [19], random selection [4], rank selection [3], rou-lette wheel selection [4], and tournament selection [9], on eight TSPs benchmarks. The simple GA (Fig. 1) and EAX were used and the search length was set to 20. Table 3 shows that HpS is better than the others in these eight tested problems with the same function evaluations. For att532, EAX with HpS can find the optimal solution for 19 times in 20 independent runs. For the other problems, EAX with HpS can always found the optima in each trial. In summary, we have demonstrated that the HpS is able to improve the solution quality for the EAX through

maintaining of population diversity and provision of a good pairing scheme.

3

System and results

Based on the discussion of section II, we investigated a new genetic algorithm combining the edge assembly crossover (EAX), neighbor-join mutation (NJ), and the heterogeneous pairing selection (HpS). Figure 7 shows the main steps of the proposed approach. N solutions are randomly generated as the initial population. After evaluating the fitness, each solution in the population sequentially uses the HpS to select itself (si) and another individual from the population based on the edge simi-larity. These two individuals become the parents of the EAX which generates only one intermediate offspring (Ii). The NJ mutation is then executed L times to generate a child (ci) by refining the intermediate solution Ii. In each

Fig. 7. Overview of our proposed genetic algorithm Table 3. Comparisons of various pairing selections, including

HpS, RpS, random, rank, wheel, and tournament selections, with the simple GA (Fig. 1) and EAX based on the ‘‘error’’ defined as ð_optimumaverage
1Þ  100.where average is the average values of the best

solutions obtained by testing approaches. These methods are tested on eight TSP problems based on the average solution qualities in 20 independent trials

HPS RPS Random Rank Wheel Tournament

att48 0.000000 0.000000 0.000000 0.000235 0.000000 0.000000 eil76 0.000000 0.000000 0.000000 0.000000 0.000000 0.000093 eil101 0.000000 0.000000 0.000130 0.000239 0.000000 0.000159 kroa150 0.000000 0.000002 0.000251 0.000004 0.000000 0.000008 kroa200 0.000000 0.000000 0.000342 0.000003 0.000510 0.000032 lin318 0.000000 0.000136 0.001006 0.000889 0.000834 0.001354 pcb442 0.000000 0.000000 0.000000 0.000000 0.000000 0.000007 Att532 0.000013 0.000224 0.000327 0.000354 0.000435 0.000502 694

pair of si and ci, the one with the better solution survives where 1  i  N. These N solutions become the new population of the next generation. Note that the proba-bilities of both the crossover and the mutation operators are 1.0. Our algorithm is terminated when one of the fol-lowing criteria is satisfied: 1) the maximum preset search time is exhausted, 2) all individuals of a population rep-resent the same solution, or 3) all of the children generated in five continuous generations are worse than their respective family parents.

We tested the proposed approach on 15 TSP benchmark problems whose numbers of cities range from 101 to 13509 cities and executed on a Pentium IV 1 GHz personal computer with single processor. Each problem was tested for 20 independent runs. According to the experiments,

the population size was set to the number of cities of a TSP whose number of cities is smaller than 1000 and set to the half of the number of cities of a TSP whose number of cities is larger than 1000 for the tradeoff between solution quality and convergence time. For usa13509, the popula-tion size was set to 2000 due to the memory constraint.

Table 4 shows the experimental results of our approach tested on 15 problems. The values in parentheses of problem represent the optimal tour length. The values in parentheses of the average tour length represent the percentage of error defined as ð_optimumaverage
1Þ  100, where average is the experimental value and optimum is the optimum of a TSP problem. The ‘‘optimal times’’ means the times our approach found the optima in 20 trials for a TSP problem.

Table 4. The experimental re-sults of our method on 15 TSP problems based on the average time (sec), generation, tour length, and the optimal times our approach found the optima. The error is defined as where average is the average values of the best solutions obtained by testing approaches

Problem Time Generation Average tour length (error) Optimal times eil101.tsp (629) 0.5 17.5 629 (0.000000) 20 Kroa200.tsp (29368) 4 35.9 29368 (0.000000) 20 Lin318.tsp (42029) 15 49.8 42029 (0.000000) 20 Pcb442.tsp (50778) 33 43.7 50778 (0.000000) 20 Att532.tsp (27686) 97 73.05 27686 (0.000000) 20 u574.tsp (36905) 106 70.75 36905 (0.000000) 20 rat783.tsp (8806) 376 83.25 8806 (0.000000) 20 vm1084.tsp (239297) 955 100.25 239304.8 (0.000033) 19 Pcb1173.tsp (56892) 1177 105.6 56892.5 (0.000009) 18 u1432.tsp (152970) 1875 82.8 152970 (0.000000) 20 pr2392.tsp (378032) 4968 147.5 378032 (0.000000) 20 Pcb3038.tsp (137694) 11617 178.2 137694 (0.00000) 20 Fnl4461.tsp (182566) 50033 255.4 182570.3 (0.000024) 16 rl5915.tsp (565530) 78527 195 565530.5 (0.000001) 19 Usa13509.tsp (19982859) 256186 452 19991528 (0.000434) 0

Table 5. Comparisons of our method (HSGA) with six methods, ant colony system (ACS) [25], the voronoi crossover genetic al-gorithm (VGA) [24], the compact genetic alal-gorithm (CGA) [5], iterative Lin-Kernighan (ILK) [12], chained Lin-Kernighan (CLK) [2], and Tabu search with Lin-Kernighan (LK) [30] on eight larger

TSP problems based on the average tour length. The length of the optimal tour of each problem is in brackets. Here the error is defined as where average is the average values of the best solutions obtained by testing approaches. N/A means the result is not available

Problem HSGA ACS VGA CGA ILK CLK Tabu with LK lin318 (42029) 42029 42029 42029 N/A N/A N/A

42029 (0.000000) (0.000000) (0.000000) (0.000000)

att532 27686 27718.2 27686.7 27686 N/A N/A N/A (27686) (0.000000) (0.001163) (0.000025) (0.000000)

rat783 8806 8837.9 N/A 8806 N/A N/A N/A

(8806) (0.000000) (0.003622) (0.000000)

vm1084 239304.8 N/A N/A N/A 239349 239301 240238 (239297) (0.000033) (0.000217) (0.000017) (0.003932) pcb1173 56892.5 N/A N/A N/A 56897 56984 57290

(56892) (0.000009) (0.000088) (0.001617)

u1432 152970 N/A N/A N/A 153122 153328 153727 (152970) (0.000000) (0.000994) (0.002340) (0.004949) pr2392 378032 N/A N/A N/A 378597 379629 380486

(378032) (0.000000) (0.001495) (0.004225) (0.006492) pcb3038 137694 N/A 137706.8 N/A 137861 138055 138893 (137694) (0.00000) (0.000093) (0.001213) (0.002622) (0.008708) fnl4461 182570.3 N/A 182605.9 182578.4 182814 182840 184373 (182566) (0.000024) (0.000219) (0.000068) (0.001358) (0.001501) (0.009898) frl5915 565530.5 N/A N/A 565554.0 565625 568570 570650 (565530) (0.000001) (0.000042) (0.000168) (0.005375) (0.009053) usa13509 19991528 N/A N/A 19991585 20015598 20022550 20160648

(19982859) (0.000434) (0.000437) (0.001638) (0.001986) (0.008897)

Table 4 shows the proposed approach which performs robustly for these tested problems. Except usa13509, our approach is able to find the optimum of the 14 tested benchmarks for at least 16 times in 20 trials. All the average tour length above the optima is within 0.00043. Our approach, somewhat slower, is able to find the optima and is stable for all tested problems.

To show the robustness of our proposed approach on large TSPs, we compared it with some methods, including the ant colony system (ACS) [25], the voronoi crossover genetic algorithm (VGA) [24], the compact genetic algo-rithm (CGA) [5], iterative Lin-Kernighan (ILK) [12], chained Lin-Kernighan (CLK) [2], and Tabu search with Lin-Kernighan [30], as shown in Table 5. These six appro-aches performed well on these test problems according to the original papers and the results on ‘‘8th DIMACS Imple-mentation Challenge: The Traveling Salesman Problem’’.

Table 5 shows that our method outperforms these surveyed approaches in these testing problems. It is able to find the optimum and the average solution quality is within 0.00043 above the optima value for each testing problem although the proposed GA is somewhat slower than these approaches. ILK is the fastest approach among these approaches and the testing result is slightly better than other surveyed approaches. For the larger problem such as usa13509, ILK is about 30 times faster than the proposed method with population size equal to 2000. Fortunately, when we set the population size to 100, the running time for our method is about the same as ILK and the average tour length is 20014159 (0.001566) which is slightly better than ILK.

4

Conclusion

This study demonstrates that GA can be a stable approach for TSPs via designing their components. From our analysis and experiences, we suggest that a genetic algorithm for TSPs should consist of local search strategies and maintain population diversity as well as implement the mechanisms of preserving good edges and inserting new edges into off-spring. In our proposed approach, the heterogeneous edge selection keeps the population diversity; the neighbor-join mutation is local search strategies. Our experiments indi-cated that the edge assembly crossover and the neighbor-join mutation are able to preserve good edges and add new edges. These strategies seem to be able to closely cooperate with each other to improve the overall search performance.

Experiments on 15 benchmark TSPs verify that the proposed approach is robust and is very competitive with algorithms of our best surveys. Our approach is able to find optimum and stable solutions for all testing TSPs; specifically, it finds out the optimum over 16 times in 20 independent runs for problems smaller than 6000 cities. For larger problem usa13509, the average solution is only 0.00043 above the optima. We believe that a GA with these claimed issues will become a robust tool for TSPs and potential applications.

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