Artificial bee colony algorithm with generating neighbourhood solution for large scale coloured traveling salesman problem

IET Intelligent Transport SystemsVolume 13, Issue 10 p. 1483-1491 Research ArticleFree Access Artificial bee colony algorithm with generating neighbourhood solution for large scale coloured traveling salesman problem Xueshi Dong, Corresponding Author Xueshi Dong dxs_cs@163.com College of Computer Science and Technology, Qingdao University, Qingdao, People's Republic of China Computer School, Wuhan University, Wuhan, People's Republic of ChinaSearch for more papers by this authorQing Lin, Qing Lin College of Computer Science and Technology, Qingdao University, Qingdao, People's Republic of ChinaSearch for more papers by this authorMin Xu, Min Xu Chang Jiang Waterway Institute of Planning and Design, Wuhan, People's Republic of ChinaSearch for more papers by this authorYongle Cai, Yongle Cai Computer School, Wuhan University, Wuhan, People's Republic of ChinaSearch for more papers by this author Xueshi Dong, Corresponding Author Xueshi Dong dxs_cs@163.com College of Computer Science and Technology, Qingdao University, Qingdao, People's Republic of China Computer School, Wuhan University, Wuhan, People's Republic of ChinaSearch for more papers by this authorQing Lin, Qing Lin College of Computer Science and Technology, Qingdao University, Qingdao, People's Republic of ChinaSearch for more papers by this authorMin Xu, Min Xu Chang Jiang Waterway Institute of Planning and Design, Wuhan, People's Republic of ChinaSearch for more papers by this authorYongle Cai, Yongle Cai Computer School, Wuhan University, Wuhan, People's Republic of ChinaSearch for more papers by this author First published: 03 July 2019 https://doi.org/10.1049/iet-its.2018.5359Citations: 11AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Coloured travelling salesman problem (CTSP) is an extended model of multiple travelling salesman problems (MTSPs), as one kind of problem in combination optimisation problems, which has been applied to many real-world planning problems such as multi-machine engineering system (MES). Population-based algorithms such as genetic algorithm (GA) and simulated annealing (SA) algorithm are only used to solve small- or medium-scale cases in which the city number is <2000. Furthermore, in term of the convergence speed and solution quality, their performances have further improvement space. Since many real-world problems can be modelled by large-scale CTSP, it is necessary to study better algorithms to solve large-scale CTSP. Since artificial bee colony algorithm (ABC) can show good performance in solving combination optimisation problems, and therefore this study applied improved ABC algorithms to solve large-scale CTSP. The modified ABC algorithms use generating neighbourhood solution (GNS) to improve ABC for this problem, two limitation and optimisation conditions are applied into GNS during the process of reinserting cities. The extensive experiments verify that the improved algorithms can demonstrate better solution quality than the compared algorithms for large-scale CTSP. 1 Introduction Coloured travelling salesman problem (CTSP) [1-3] origins in multiple travelling salesman problem (MTSP) and travelling salesman problem (TSP) but is their generalisation, which can be applied in the planning problem of MES. It has a shared city set and several exclusive city sets, so that each individual not only implements the shared task, but also carries out its own exclusive task. CTSP is a more complex model than MTSP, and it can be transformed into MTSP and multiple single TSP problems under certain condition [2], therefore studying CTSP is significant for the research of other TSP variants. Up to now, some bio-inspired algorithms such as genetic algorithm (GA) and hill-climbing GA have been used to solve CTSP, but the traditional algorithms are easy to fall into local optimum. In the fields such as intelligent transport systems and multiple tasks cooperation, many real-world problems can be modelled by CTSP, and the scale of generated model is usually >2000 dimensions; thus it is necessary to study large-scale CTSP and its solving algorithms. For example, in the scheduling of MES, the basic elements of such problem, i.e. objective, machine individuals, and tasks can, respectively, match the objective, salesmen, and cities of CTSP, in many cases, the number of tasks easily tends to large-scale, where the corresponding city number is >2000. For handing the above-mentioned drawbacks, the paper studies large-scale CTSP, and applies the improved ABC algorithms to solve it. Since CTSP is a new problem, there are very few published papers in this field, Li et al. [1] firstly proposed the CTSP and used GA for solving the problem; after that they applied three algorithms including GA with greedy algorithm (GAG), GA by hill-climbing algorithm (HCGA) and GA using simulated annealing algorithm (SAGA) to solve CTSP, but the scale of the problem is limited, where the number of city is no more than 101 [2]; Dong et al. [3] applied the improved ITÖ algorithm to solve small-scale CTSP problem; Meng et al. [4] presented a more general CTSP which is a generalisation of both radial CTSP (R-CTSP) and serial CTSP (S-CTSP); Meng et al. [5] used population-based incremental learning algorithm for a serial CTSP; Li et al. [6] discussed the collision-free scheduling of multi-bridge machining systems; Li et al. [7] studied a two-stage approach to path planning and collision avoidance of multi-bridge machining systems; Dong et al. [8] proposed ant colony optimisation for CTSP by multi-task learning, where the city number is no more than 1002. In this paper, the CTSP that we investigated is a radial CTSP or R-CTSP for short. Although the above works have been done in the recent years, there are still some problems: (i) the scale of CTSP is still limited; (ii) the traditional algorithms for CTSP are easy to fall into local optimum. Aiming at the problems, the paper extends the scale of CTSP to large-scale, and uses improved ABC algorithms for large-scale CTSP. For these algorithms, generating neighbourhood solution is used to optimise the solution, which can delete and reinsert some cities, during the process of reinserting cities, the two limitation and optimisation conditions are utilised: (i) the exclusive city is inserted into the position of the tour for the appointed salesman, there is no such limitation for shared city; (ii) it is inserted into the position where the added tour cost is the least. The extensive experiments show that the modified ABC algorithms can demonstrate better performance than the former algorithms such as SAGA, HCGA and GAG in term of solution quality. The works of the paper mainly focus on the following aspects: on the one hand, this paper studies the large-scale CTSP problem where the city number is >2000; on the other hand, the paper uses the improved artificial bee colony (ABC) algorithms for solving large-scale CTSP. The other sections of the paper are as follows: the second section introduces the definition of CTSP; the third one gives the detail of improved ABC algorithms for CTSP; the fourth one is the experiment and analysis; the last one is the conclusion and future works. 2 Coloured travelling salesman problem Suppose that there are m salesmen and n cities for a CTSP [1, 2], where m ∈Z = {1, 2, 3, …}, m < n. The problem can be defined as a complete digraph G(V, E), V = {0, 1, 2, …, n − 1} stands for the cities set, each edge is (i, j)∈E, i ≠j, it is associated with weight wij representing the cost (e.g. distance) between the city i and city j. The cities set V is divided into m + l non-null sets, a set U represents the shared city set, the other sets mean Vi, ∀i ∈Zm = {1, 2, 3, … m }, it shows that only a salesman i can access to it. Vertex di, di ∈(U ∪Vi) is the depot where the salesmen i begins and ends. Colour i represents that the city is visited only by salesman i. The cities set Vi (i = 1, 2, 3, … m) is coloured by i, it means that only salesman i can visit it. Each city of the shared set U is coloured by colour set c (a), which means that only salesman i (i ∈c (a)) can visit it. The objective of CTSP is to find m Hamiltonian cycles in G with the minimal cost in all the tours, the restriction condition is that all cities can be visited only once and coloured city i is visited only by salesman i. For CTSP, there is a shared city set U. The integer coding model of the corresponding CTSP is as follows: The variable xijk (i ≠j, i, j ∈V, k ∈Zm) represents if the kth salesman passes city i to j, and the variable uik (i ∈V, k ∈Zm) is the city number that the kth salesman travels from depot to city i [1]. The objective function of CTSP is (1) The constraint condition of CTSP is as follows: (2) Formula (2) means that each salesman begins from and returns to the depot (city 0) (3) Formula (3) represents that salesman k cannot access other traveler's exclusive cities, furthermore, and other salesmen cannot visit the kth 's exclusive cities (4) Formula (4) shows that salesman l (≠k) can neither begin from the city of k exclusive nor go back to it (5) The formula (5) demonstrates each city except the depot 0 must be visited by a salesman exactly once (6) The formula (6) means that each salesman must access and withdraw from a shared city at the same time. 3 ABC algorithms for CTSP ABC algorithm [9-12] is a new swarm intelligence algorithm which can simulate the behaviour of honey bee swarms for foraging. For the algorithm, there are three kinds of bees for a colony: employed bees, onlooker bees and scout bees. Employed bees belong to the first half of the colony, and onlooker bees are in the second half. A solution of the optimisation problem is represented by the position of a food source, and the nectar amount for each food source stands for the fitness of the corresponding solution. The colony size is equal to the number of onlooker and employed bees. Firstly, an initialised population P is generated randomly that includes Np solutions with D -dimensional vector of decision variables Xi = {xi,1, xi,2, …, xi,D }. Xi defined by lower and upper bounds Xmin and Xmax is constructed by the following formula: (7) where i = 1, 2, …., Np, j = 1, 2, …, D. For the algorithm, an employed bee utilises the below search equation to produce a candidate solution Vi from the former one Xi : (8) where k ∈{1, 2, …, Np } and j ∈{1, 2, …, D } are randomly selected indexes, k is different from i ; Φi,j stands for a random number during [−1, 1]. What is different from the employed bees, onlooker bees choose a position to access by the probability value pi for the solution by the following equation: (9) where fiti represents the fitness value of the ith solution, the employed bees can communicate with onlooker bees using their information by the way. After the onlooker bee selecting a solution, each one produces a new solution by the (8) and a greedy selection is used for the new solution and former solution, which is in the case of the employed bees. While the solution can't be improved or optimised by the pre-determined cycles, it would be given up, and the corresponding employed bees can be transformed into scout bees. The scout bees produce a new solution by the formula (7). 3.1 Solution representation In the paper, we use m -tour encoding [9], take Fig. 1 for example. Fig. 1Open in figure viewerPowerPoint Example of CTSP m-tour encoding There is a CTSP with n = 18 and m = 4, city 1 to city 12 are exclusive cities: salesman 1 has exclusive city 1, city 2 and city 3; city 4, city 5 and city 6 are for salesman 2; city 7, city 8 and city 9 are for salesman 3; city 10, city 11 and city 12 are for salesman 4. Cities 13–18 are shared cities which can be assigned to any salesman. Cities 1, 2, 3, 13 and 14 are visited by salesman 1; cities 4, 5, 6 and 15 are accessed by salesman 2; cities 7, 8, 9, 17 and 18 can be visited by salesman 3; cities 10, 11, 12 and 16 are accessed by salesman 4. 3.2 Fitness and initialisation In this paper, the objective function is the fitness function, and the objective of CTSP is to find m tours with the minimum total travelling cost. It means that the smaller the fitness function is, the better the solution of CTSP becomes. In the initialisation, it initialises all the parameters of the algorithm. In this step, the initial employed bee solutions are produced. While the unassigned cities are shared cities, the cities are inserted into the position of the tour for all salesmen where the increased tour cost is the least; when the unassigned cities belong to exclusive city set, the cities should be inserted into the position of the tour for appointed salesman where the added distance is the shortest. 3.3 Generating neighbourhood solution The paper uses binary tournament selection method to choose a food source of onlooker bee, in order to generate a solution in the neighbourhood of solution S, the method copies each tour of S to the neighbouring solution one by one using the probability Pc, the unassigned cities are inserted into the unassigned city set. Pc is among the pre-defined initial Pmin c and the pre-defined final Pmax c [9]. In iteration, Pc is computed according to the following formula: (10) where itermax stands for the maximum number of iterations, while only fitness value is worse than neighbouring solution, the solution of employed bee is replaced by the neighbourhood solution, or it continues to be current solution. The algorithm utilises different number of employed bee and onlooker bee, if the solution of employed bee is not improved after the set number limitnoimp, employed bee and food source will be abandoned, the employed bee is transformed into scout bee [9]. In this paper, another method is used to determine the value of Pc by making experiments, because the value of Pc is during 0 to 1, while Pc = 0.1, 0.3, 0.5, 0.8, or formula (10), we do experiments to find that when Pc is 0.8 the improved ABC algorithms can obtain better solution quality. Algorithm 1 (Fig. 2) is the code of neighbourhood operator (generating neighbourhood solution). Fig. 2Open in figure viewerPowerPoint Algorithm 1: neighborhood operator In Algorithm 1 (Fig. 2), S is a solution, S' represents a neighbouring solution, r is a random number, it is during 0 to 1, Pc stands for the city keep probability, it is defined by the formula (10). While insert cc into S ′, there are two limitation and optimisation conditions: (i) the exclusive city is inserted into the position of the tour for the appointed salesman, there is no such limitation for shared city; (ii) it is inserted into the position where the added tour cost is the least. 3.4 Steps of the improved algorithms for CTSP The solution obtained by artificial bee colony algorithm can be optimised by using the generating neighbourhood solution, and it can repeatedly apply the method to all tour of the optimum solution. CTSP not only has shared city set, but also occupies exclusive city sets, and its objective is to minimise the total tours of m salesmen. In order to improve the solution obtained by the ABC algorithm, the paper applies generating neighbourhood solution to optimise the solution in employed bee phase or onlooker bee phase, which is different with the used improved ABC algorithm in literature [9]. Algorithm 2 is the steps of ABC algorithm for CTSP, ne stands for the number of employed bee, no represents the number of onlooker bee, NeighborhoodOperator (Sp) is generating neighbouring solution which is given in Algorithm 1 (Fig. 2), BinaryTournamentSelection (S1, S2, …, Sne) chooses a solution from the solutions S1, S2, …, Sne for an onlooker bee by binary tournament selection, the improved ABC algorithm utilises the generating neighbouring solution (algorithm 1) to improve the solution of CTSP. Algorithm 2 (Fig. 3) is the main steps of the used ABC algorithm for large-scale CTSP. Fig. 3Open in figure viewerPowerPoint Algorithm 2: The improved ABC algorithms for CTSP In Algorithm 2 (Fig. 3), the generating neighborhood solution is used in the employed bee phase, the algorithm is named EABC; the generating neighborhood solution is only used for optimisation in the onlooker bee phase, it is called OABC. If the algorithm does not use generating neighborhood solution neither in employed bee phase nor onlooker bee phase, the algorithm is the basic artificial bee colony algorithm. 4 Experiments and analysis We make experiments to analyse the performance of the different algorithms for large-scale CTSP. The computer environment is as follows: Intel® Core™ i7-6700 Processor and 8.00 GB RAM at 3.40 GHz. The experiments are developed based on Java. The initial parameters of GAs are from the CTSP paper [2]: the population size is 30, crossover probability as 0.7, and mutation probability is 0.1. Each algorithm runs ten times. The parameters of the SAGA are as follows: the initial temperature as 100, the total time of cooling is 60, the step length at each temperature is 30, and annealing coefficient as 0.9. The parameters of artificial bee colony algorithms and improved variants: the binary tournament selection probability parameter Pbts = 0.8, the maximum no-update number of each nectar source limitscout = 100, the disturbance rate of local search is from 0 to 0.8 with Pmin c = 0 and Pmax c = 0.8. All the algorithms have the same stopping condition. Table 1 is the small- and medium-scale experiment data. Table 2 is the large-scale data of CTSP. Table 1. Small and medium scale experiments data for CTSP Instance scale City count Salesman count Shared city Exclusive city Small n m s e 1 21 2 11 5 2 21 3 9 4 3 31 2 19 6 4 31 3 16 5 5 31 4 15 4 6 41 2 21 10 7 41 3 23 6 8 41 4 21 5 9 51 3 21 10 10 51 4 21 7, 8 11 51 5 21 6 12 76 3 31 15 13 76 4 36 10 14 76 5 26 10 15 76 6 40 6 medium 16 101 4 21 20 17 101 5 53 10 18 101 6 41 10 19 101 7 21 10 19 202 12 82 10 20 202 25 77 5 21 202 35 62 4 22 431 12 191 20 23 431 25 181 10 24 431 40 231 5 25 655 17 145 30 26 655 25 155 20 27 655 33 160 15 Table 2. Large-scale experiments data for CTSP Instance scale City count Salesman count Shared city Exclusive city Large n m s e 1 2463 3 661 600 2 2463 6 661 300 3 2463 12 661 150 4 2463 24 661 75 5 2463 30 661 60 6 3461 3 461 1000 7 3461 6 461 500 8 3461 12 461 250 9 3461 24 461 125 10 3461 30 461 100 11 3461 40 461 75 12 5397 20 397 250 13 5397 30 897 150 14 5397 40 1397 100 15 5397 50 397 100 16 5397 60 597 80 17 6397 20 397 300 18 6397 30 397 200 19 6397 40 397 150 20 6397 50 397 120 21 6397 60 397 100 22 7397 20 1397 300 23 7397 30 1397 200 24 7397 40 1397 150 25 7397 50 1397 120 26 7397 60 1397 100 In Table 1, n is the number of cities which ranges from 21 to 655, m is the number of salesmen, it is from 2 to 33, s is the city number of shared cities set, and e is the city number of an exclusive cities set. The data with n from 21 to 101 is provided in the literature [2]. Up to now, there is no any published literature for the CTSP problem with the city number >2000. In this paper, we study the large-scale CTSP problem. The stopping conditions such as fitness function evaluation and maximum number of iterations usually have problems, which are not fair stopping conditions [13, 14]. In order to make up the flaw of single stopping condition, there are two stopping conditions for large-scale CTSP experiments, one is running same time (60 s) for each algorithm, and the another is the running same max no-update iterations number (60 times). In Table 2, there are 26 instances, n is from 2463 to 7397, m is from 3 to 60, the large-scale data is made according to the original TSP (The TSPLIB Symmetric Travelling Salesman Problem Instances) data published on web. The steps of generating large-scale CTSP instances are as follows: the large-scale TSP instances are downloaded, for example, if the city number of TSP is 7397, it is divided into 60 sections one by one from the first city of TSP city list, each section has 100 cities, which is the exclusive city set, the remanent cities belong to shared city set, the last city is usually set as depot city, by the steps the CTSP instance with n = 7397 and m = 60 is generated. If it deletes the last 1000 cities of shared cities, it can be transformed into the instance with n = 6397 and m = 60, if it continues to delete the last 1000 cities again, by performing the same steps of the first instance, it is transformed into the instance with n = 5397 and m = 60. Other instances can be also generated by this rule. Fig. 4 is the best solution case of the four algorithms to solve CTSP by n = 101 and m = 5, the top left figure is the GAG for the problem, the upper right one is HCGA for CTSP, the bottom left one is SAGA for the problem, the lower right one is EABC for CTSP. The more information of the four algorithms for the problem is as follows: GAG: the average solution quality 949.5, the best quality solution 906.0; HCGA: mean solution quality 925.4, the best solution quality 887.0; SAGA: the mean solution quality 1057.7, the best solution 1017.0; EABC: the mean solution 798.2, the best solution 790.0. The experiments show that EABC demonstrates superiority over the other three algorithms in this case. Fig. 4Open in figure viewerPowerPoint Optimum routes of the four algorithms for CTSP with n = 101 and m = 5 In the following tables and figures, the algorithms GAG, HCGA and SAGA are from the literature [2], EABC is the improved ABC algorithm, which only uses the generating neighbourhood solution for optimisation in the employed bee phase, OABC is the modified ABC algorithm with generating neighbourhood solution in onlooker bee phase, EABC pc=0.8 is the optimised EABC with Pc = 0.8 replacing formula (10), OABC pc=0.8 is the improved OABC with Pc = 0.8 instead of the formula (10). Tables 3 and 4 are the experiments results of the algorithms for large-scale CTSP by running same time as the stopping condition. Table 3. the mean solution quality of the different algorithms for large-scale CTSP with running same time as the stopping condition (Unit: km) Instance n m GAG HCGA SAGA EABC OABC EABC pc =0.8 OABC pc =0.8 Large scale Mean Mean Mean Mean Mean Mean Mean 1 2463 3 154225.9 155442.1 151344.2 117824.7 118088.2 112847.1 113495.1 2 2463 6 196631.2 197768.4 193801.0 130194.3 129793.3 125310.5 126039.3 3 2463 12 263780.9 261928.2 256148.5 156449.4 156203.6 153208.9 153039.8 4 2463 24 234318.7 337881.8 333196.7 232669.5 233085.2 230916.7 231047.9 5 2463 30 385685.6 381341.3 373814.0 278543.2 279122.2 277004.8 276680.8 6 3461 3 201927.9 202061.5 200605.4 166456.3 165961.4 159679.3 159919.4 7 3461 6 238254.4 238865.0 235570.8 178026.6 178186.5 171913.7 172126.5 8 3461 12 297845.0 298478.1 292883.5 206774.1 206968.0 199570.8 200856.5 9 3461 24 396743.1 393986.4 386096.2 275844.5 276346.9 273491.3 273564.5 10 3461 30 410001.9 409216.1 404613.5 315394.0 315286.2 314038.2 313335.3 11 3461 40 491249.3 488342.0 484698.4 391472.0 391333.9 389086.4 389210.8 12 5397 20 5.15483369E7 5.15320901E7 5.13271238E7 3.95320639E7 3.93955057E7 3.92959528E7 3.92398825E7 13 5397 30 6.96071709E7 6.96245061E7 6.94671674E7 5.32797143E7 5.32799913E7 5.33341995E7 5.32598618E7 14 5397 40 5.63153041E7 5.59189163E7 5.48789369E7 3.20938429E7 3.21239031E7 3.21331832E7 3.20763471E7 15 5397 50 5.34596522E7 5.34176272E7 5.24364256E7 4.06759073E7 4.06769731E7 4.0593959E7 4.0580265E7 16 5397 60 6.69314805E7 6.65907524E7 6.55313765E7 4.75542739E7 4.75482612E7 4.74277166E7 4.74325622E7 17 6397 20 6.52828306E7 6.53026766E7 6.49373033E7 4.11121301E7 4.10752551E7 4.07931799E7 4.08666637E7 18 6397 30 7.98197278E7 7.94897053E7 7.9033804E7 5.26180251E7 5.2544455E7 5.23339669E7 5.24333979E7 19 6397 40 4.46905674E7 4.46424023E7 4.44016902E7 3.50089942E7 3.50124517E7 3.49176061E7 3.4926537E7 20 6397 50 5.2973838E7 5.28634701E7 5.22775626E7 4.25261875E7 4.25241944E7 4.24586369E7 4.24556713E7 21 6397 60 6.13275942E7 6.08879031E7 6.03594311E7 5.01921559E7 5.01769957E7 5.01047138E7 5.01006666E7 22 7397 20 9.94252888E7 9.94789879E7 9.94838509E7 4.52945783E7 4.5262409E7 4.49651418E7 4.4906541E7 23 7397 30 1.214188629E8 1.214937352E8 1.20685607E8 5.64477249E7 5.63766344E7 5.61988794E7 5.6220105E7 24 7397 40 5.38801996E7 5.42370045E7 5.39113479E7 3.55530592E7 3.55560607E7 3.56302989E7 3.55018818E7 25 7397 50 6.33236614E7 6.31817365E7 6.29262249E7 4.30298771E7 4.30297245E7 4.31921936E7 4.30909157E7 26 7397 60 7.2206097E7 7.29167058E7 7.22840858E7 5.07074553E7 5.07060842E7 5.08792422E7 5.07661747E7 Table 4. Best solution of the different algorithms for large-scale CTSP with running same time as the stopping condition (Unit: km) Instance N m GAG HCGA SAGA EABC OABC EABC pc =0.8 OABC pc =0.8 Large scale Best Best Best Best Best Best Best 1 2463 3 153317.0 153320.0 148772.0 116678.0 116949.0 111559.0 112235.0 2 2463 6 191482.0 194500.0 189651.0 128744.0 127623.0 122691.0 124461.0 3 2463 12 260992.0 258023.0 252499.0 154417.0 155021.0 151702.0 152036.0 4 2463 24 233185.0 331774.0 324662.0 231048.0 231546.0 228130.0 229826.0 5 2463 30 377601.0 374258.0 368734.0 277729.0 278436.0 275291.0 275496.0 6 3461 3 199557.0 199560.0 198009.0 165696.0 164615.0 159162.0 158916.0 7 3461 6 235535.0 236763.0 232817.0 177146.0 176833.0 170806.0 170771.0 8 3461 12 291663.0 291984.0 289531.0 205806.0 205024.0 198106.0 198035.0 9 3461 24 390231.0 388460.0 381267.0 274651.0 275458.0 272126.0 270548.0 10 3461 30 404133.0 400551.0 397018.0 314384.0 313952.0 312231.0 312370.0 11 3461 40 482666.0 483223.0 481590.0 390226.0 389776.0 387768.0 388034.0 12 5397 20 5.1014544E7 5.0772241E7 5.0861698E7 3.9314667E7 3.9207693E7 3.9127648E7 3.8978746E7 13 5397 30 6.8654566E7 6.9150399E7 6.8686543E7 5.3045213E7 5.3035696E7 5.3135044E7 5.2887801E7 14 5397 40 5.569987E7 5.4381535E7 5.3829356E7 3.1957906E7 3.2044776E7 3.2005794E7 3.1873049E7 15 5397 50 5.2836453E7 5.2461802E7 5.1876705E7 4.0652149E7 4.0655592E7 4.0547092E7 4.0562209E7 16 5397 60 6.630277E7 6.5880403E7 6.4946756E7 4.7527537E7 4.7515737E7 4.7398981E7 4.7368617E7 17 6397 20 6.4413115E7 6.4810128E7 6.4308089E7 4.0886077E7 4.0755792E7 4.0510717E7 4.0571988E7 18 6397 30 7.9383691E7 7.8171047E7 7.7883856E7 5.2490465E7 5.2194599E7 5.2071712E7 5.22376E7 19 6397 40 4.3792398E7 4.3756007E7 4.3707841E7 3.4968924E7 3.4987602E7 3.4885868E7 3.4890213E7 20 6397 50 5.2307037E7 5.2132899E7 5.160439E7 4.2506174E7 4.2509905E7 4.244102E7 4.2440067E7 21 6397 60 6.1061913E7 6.0028883E7 5.9853389E7 5.0172751E7 5.0165135E7 5.0085256E7 5.0071158E7 22 7397 20 9.8739735E7 9.8026518E7 9.8432422E7 4.4781409E7 4.4908977E7 4.4819194E7 4.4430944E7 23 7397 30 1.19856028E8 1.20384755E8 1.19875069E8 5.6066671E7 5.5982705E7 5.5830952E7 5.5830011E7 24 7397 40 5.3281309E7 5.3534898E7 5.2795667E7 3.5456544E7 3.5424078E7 3.5551525E7 3.5412646E7 25 7397 50 6.2273733E7 6.2103741E7 6.20778E7 4.289244E7 4.2973438E7 4.3079938E7 4.3027169E7 26 7397 60 7.0642834E7 7.1895538E7 7.0411756E7 5.0555011E7 5.0590796E7 5.065131E7 5.0647761E7 In Table 3, it is the experiment result of the algorithms for large-scale CTSP, the mean is the average solution quality of the algorithms for running ten times. GAG, HCGA and SAGA are from the CTSP paper [2]. From the data, it shows that EABC pc=0.8, OABC pc=0.8, EABC and OABC can show better solution quality than GAG, HCGA and SAGA, the improved algorithms EABC pc=0.8 and OABC pc=0.8 can show better solution quality than EABC and OABC. Table 4 is the best solution quality of the algorithms for large-scale CTSP with running same time as the stopping condition, and it shows that EABC pc=0.8, OABC pc=0.8, EABC and OABC display better solution quality than the compared algorithms, EABC pc=0.8 and OABC pc=0.8 can show better solution quality than EABC and OABC. Figs. 5 and 6 are the mean solution quality of the algorithms for solving CTSP by running same time as the stopping condition. They are made based on the mean quality solution of Table 3. Fig. 5Open in figure viewerPowerPoint Average solution quality of the algorithms for large-scale CTSP with instance numbers from 1 to 11 (Unit: km) Fig. 6Open in figure viewerPowerPoint Average solution quality of the algorithms for large-scale CTSP with instance numbers from 12 to 26 (Unit: km) In Figs. 5 and 6, the lateral axis represents the order number of the instance, for example, the number order 6 corresponds to n = 3461 and m = 3, the number order 22 is the corresponding data with n = 7397 and m = 20; vertical axis is the mean solution quality of the algorithms for large-scale CTSP problem. In order to effectively test the algorithms, we use the best solution deviation PDbest and the average solution deviation PDav to make significance test [15]. The deviation data is computed based on Tables 3 and 4, the computation process is given in the literature [15]. The computing result is shown in Table 5. Table 5. Deviation data of the different algorithms fo