A novel genetic algorithm for large scale colored balanced traveling salesman problem

Future Generation Computer Systems 95 (2019) 727–742

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Future Generation Computer Systems journal homepage: www.elsevier.com/locate/fgcs

A novel genetic algorithm for large scale colored balanced traveling salesman problem ∗

Xueshi Dong a,b , , Yongle Cai b a b

College of Computer Science and Technology, Qingdao University, Qingdao 266071, China Computer School, Wuhan University, Wuhan 430072, China

highlights • • • •

This paper provides colored balanced traveling salesman problem (CBTSP). The large scale optimization for CBTSP is studied. A novel genetic algorithm is proposed for large scale CBTSP. Experiments show the superiority of the proposed algorithm.

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Article history: Received 22 July 2018 Received in revised form 20 October 2018 Accepted 29 December 2018 Available online 10 January 2019 Keywords: Novel genetic algorithm Large scale optimization Colored balanced traveling salesman problem Colored traveling salesman problem Balanced traveling salesman problem

a b s t r a c t The paper gives an applicable model called colored balanced traveling salesman problem (CBTSP), it is utilized to model optimization problems with partially overlapped workspace such as the scheduling and deploying of the resources and goods. CBTSP is NP-hard problem, the traditional nature-inspired algorithms, such as genetic algorithm (GA), hill-climbing GA and simulated annealing GA, are easy to fall into local optimum. In order to improve it, the paper proposes a novel genetic algorithm (NGA) based on ITÖ process to solve CBTSP. First of all, NGA utilizes the dual-chromosome coding to represent solution of this problem, and then updates the solution by the crossover and mutation operator. During the process of crossover operator, the length of crossover can be affected by activity intensity, which is directly proportional to environmental temperature and inversely proportional to particle radius. The experiments verify that NGA can demonstrate better solution quality than the compared algorithms for large scale CBTSP. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Colored traveling salesman problem (CTSP) [1,2] is a variant of multiple traveling salesman problems (MTSP) and traveling salesman problem (TSP), which can be applied in the planning problem of multi-machine engineering systems (MES), it has a shared city set and the exclusive city sets, so that each salesman not only implements the shared task, but also performs the exclusive task. This paper provides a new variant of CTSP named CBTSP, it can be used to model the optimization problems with partially overlapped workplace. In CBTSP problem, the multiple salesmen can not only carry out the independent task in own exclusive district, but also cooperatively perform the joint task with each other in the shared district, which refers to how to cooperate for performing the multiple tasks. In the fields such as intelligent transport systems and multiple tasks cooperation, some real-world problems can be modeled by CBTSP, and the scale of generated ∗ Corresponding author at: College of Computer Science and Technology, Qingdao University, Qingdao 266071, China. E-mail address: [email protected] (X. Dong). https://doi.org/10.1016/j.future.2018.12.065 0167-739X/© 2019 Elsevier B.V. All rights reserved.

model is usually up to large scale, thus it is necessary to study large scale CBTSP and related solving algorithms. Because CTSP is a new problem, there are very few published papers in this field, Li et al. [2] firstly proposed the CTSP and used genetic algorithm (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 [1]; Dong et al. [3] applied hybrid algorithms to solve the multiple balanced traveling salesmen problem, which displays that HCGA can show better performance than hybrid ITÖ algorithm, SAGA, GAG and GA. Genetic algorithm has been widely studied in recent years, the related literatures are as follows: Ardjmand et al. [4] applied GA to new bi-objective model; Metawa et al. [5] optimized bank lending decisions by using genetic algorithm based model; Ghosh et al. [6] used genetic algorithm by incorporating priors for medical image segmentation; Dong et al. [7] proposed a hybrid algorithm based on GA for colored bottleneck TSP; the

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literatures [8–13] applied genetic algorithm to solve some realworld problems; Zhang et al. [14] utilized parallel genetic algorithm for set cover problem and large scale wireless sensor networks; Friedrich el al. [15] gave a compact genetic algorithm; Rashid et al. [16] used enhanced genetic algorithm for protein structure prediction; Lakshmi et al. [17] made a genetic bankrupt ratio analysis tool by using genetic algorithm. According to the real-world applications, this paper provides a new model CBTSP, which can be applied in the optimization problems such as multiple tasks cooperation, the scheduling and deploying of the resources and goods. CBTSP is similar with CTSP, and the only difference is that they have different objective functions, therefore the nature-inspired algorithms, such as genetic algorithm, climbing hill genetic algorithm and simulated annealing genetic algorithm for CTSP, can be also used for solving CBTSP, however, while solve the problem, they are easy to fall into local optimum, which is not satisfying. In order to improve the problem, the paper proposes a new genetic algorithm called NGA to solve it, the algorithm uses dual-chromosome coding to generate the solution of problem, and crossover operator and mutation operator are used to update the solution, during the process, the crossover length can be affected by activity intensity, which is controlled by the particle radius and environment temperature, the mutation operator of NGA is same with the one of genetic algorithm. The extensive experiments show that NGA can demonstrate better performance than the compared algorithms such as SAGA, HCGA and GAG in term of solution quality. The contributions of the paper mainly focus on the following aspects: on the one hand, this paper provides a new model called CBTSP, and extends the scale of the model to large scale in which the city number is more than 1000; on the other hand, the paper proposes a novel genetic algorithm named NGA for solving CBTSP, the experiments show the superiority of the proposed algorithm. The other sections of the paper are as follows: the second section introduces the definition of CBTSP and relevant introduction; the third one gives the detail of NGA for solving CBTSP; the fourth one is the experiments and analysis; the last one is the conclusion and future works. 2. Colored balanced traveling salesman problem 2.1. The definition The definition of CBTSP is similar with CTSP [1], the only difference for them is that they have different objective functions. There are m traveling salesmen and n cities for CBTSP, 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, and each edge (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 set means 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. Color i represents that the city is visited only by salesman i. The cities set Vi (i = 1, 2, 3, . . . , m) is colored by i, it means that only salesman i can visit it [2]. For CBTSP, there is a shared city set U. The used common one is that U can be visited by all salesmen, i.e., ∀ a ∈ U, c (a) = Zm if di = 0, 0 ∈ U and ∀ a ∈ U, c (a ) = Zm , the integer coding model of the corresponding CBTSP is as follows: The variable xijk (i ̸ = j, i, j ∈ V , k ∈ Zm ) represents whether the kth traveling 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. The objective of CBTSP is to find m Hamiltonian cycles in G with the minimal difference of the maximum edge and minimum edge. The objective function of

Table 1 The small and medium scale experiments data for CBTSP. Instance scale

City count

Salesman count

Shared city

Small

n

m

s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exclusive city e

21 21 31 31 31 41 41 41 51 51 51 76 76 76 76

2 3 2 3 4 2 3 4 3 4 5 3 4 5 6

11 9 19 16 15 21 23 21 21 21 21 31 36 26 40

5 4 6 5 4 10 6 5 10 7, 8 6 15 10 10 6

101 101 101 101 202 202 202 431 431 431 655 655 655

4 5 6 7 12 25 35 12 25 40 17 25 33

21 53 41 21 82 77 62 191 181 231 145 155 160

20 10 10 10 10 5 4 20 10 5 30 20 15

Medium 16 17 18 19 20 21 22 23 24 25 26 27 28

CBTSP is as follow: Min f = max(wij ) − min(wzk ) (i, j = 1, 2, 3, . . ., n − 1; z , k = 1, 2, 3, . . ., n − 1)

(1)

The constraint conditions of CBTSP are as follows: n−1 ∑

x0ik =

i=1

n−1 ∑

xi0k = 1, k ∈ Zm

(2)

i=1

Formula (2) means that each salesman begins from and returns to the depot (city 0).

∑∑ i

xijk =

j

∑∑ i

xjik = 0, i ∈ Vk , j ∈ V \{U ∪ Vk }, k ∈ Zm

(3)

j

Formula (3) represents that salesman k cannot access other traveler’s exclusive cities, furthermore, and other salesmen cannot visit the kth’s exclusive cities.

∑∑ i

xijl =

j

∑∑ i

xjil = 0, i ̸ = j, k ̸ = l, i ∈ Vk , j ∈ V , l ∈ Zm (4)

j

Formula (4) shows that traveling salesman l (̸ = k ) can neither begin from the city of k exclusive nor go back to it. n−1 m ∑ ∑

xijk =

j=0 k=1

n−1 m ∑ ∑

xjik = 1, j ̸ = i, i ∈ V \{0}

(5)

j=0 k=1

The formula (5) demonstrates each city except the depot 0 must be visited by a salesman exactly once.

∑ l

xjlk =

∑

xijk , i ̸ = j ̸ = l, j ∈ U , i, l ∈ Vk ∪ U

(6)

i

The formula (6) means each salesman must access and withdraw from a shared city at the same time.

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Table 2 The experiment results of the five algorithms for CBTSP by running 20 s as the stopping condition (Unit: mean km). Instance

n

m

GA

GAG

HCGA

SAGA

NGA

Mean

Iteration

Mean

Iteration

Mean

Iteration

Mean

Iteration

Mean

Iteration

6.8 7.6 10.4 11.9 9.3 13.6 14.0 14.5 18.6 16.2 18.6 24.8 25.4 22.1 27.4

75 263 49 756 36 839 54 817 65 924 60 111 51 415 36 393 38 361 41 010 33 078 22 494 25 359 18 959 14 967

6.5 6.8 9.7 10.2 9.6 13.3 14.0 12.7 17.7 18.8 18.5 23.0 20.7 21.3 28.0

103 408 23 344 103 932 63 327 40 510 28 745 54 774 45 544 68 322 33 841 28 155 28 589 22 477 29 226 9 769

6.5 7.4 9.5 9.6 8.2 12.6 13.8 12.7 17.3 16.3 15.3 20.0 20.0 19.5 20.9

22 320 29 057 24 244 62 433 28 725 33 010 64 254 48 058 37 604 46 805 22 417 23 793 16 934 17 376 20 340

6.2 6.8 8.4 10.0 8.0 10.4 12.5 12.4 15.2 12.7 15.3 18.8 19.6 17.9 22.9

2982 6547 2159 1976 4228 1533 4628 2869 2168 5501 1953 3189 3702 2595 2290

8.4 9.4 11.7 13.9 13.5 11.6 13.5 15.9 15.8 17.3 19.2 15.7 16.9 18.5 21.0

26 907 21 610 15 727 15 036 14 982 11 504 11 337 11 188 9 334 8 932 8 679 5 922 5 556 5 723 5 538

29.1 37.4 29.9 26.4 2339.2 2393.6 2488.2 10 767.4 13 143.8 14 736.3 11 349.4 12 561.9 12 507.0

18 094 7 846 19 949 18 971 12 722 5 338 5 244 5 627 2 938 2 931 3 741 2 811 2 353

26.4 33.7 24.2 25.3 2257.4 2335.3 2498.7 9607.0 13 133.9 14 732.0 12 682.7 12 091.0 13 115.8

25.8 29.1 22.6 21.9 2267.2 2416.2 2506.8 8662.3 12 487.8 13 862.2 9914.7 10 457.4 11 435.4

17 724 17 134 18 430 17 154 10 962 10 050 6 425 3 924 2 885 2 695 2 528 2 360 1 537

21.5 29.5 25.6 24.0 2425.8 2490.0 2535.6 8017.4 12 786.7 14 749.2 10 694.8 9827.2 11 488.9

1474 1072 1845 2789 2761 2093 2143 1315 1265 1103 868 857 827

18.8 22.1 20.5 22.5 2626.6 2626.0 2647.0 7426.9 12 967.0 10 962.3 8873.1 10 470.9 11 250.6

Small 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

21 21 31 31 31 41 41 41 51 51 51 76 76 76 76

2 3 2 3 4 2 3 4 3 4 5 3 4 5 6

101 101 101 101 202 202 202 431 431 431 655 655 655

4 5 6 7 12 25 35 12 25 40 17 25 33

Medium 16 17 18 19 20 21 22 23 24 25 26 27 28

22 180 19 674 39 935 17 034 11 883 7 271 3 538 5 478 3 183 3 163 2 412 2 559 1 725

Table 3 The large scale experiments data for CBTSP. Instance scale

City count

Salesman count

Shared city

Exclusive city

Large

n

m

s

e

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2461 2461 2461 2461 2461 3461 3461 3461 3461 3461 3461 5397 5397 5397 5397 5397 7397 7397 7397 7397 7397

3 6 12 24 30 3 6 12 24 30 40 20 30 40 50 60 20 30 40 50 60

661 661 661 661 661 461 461 461 461 461 461 397 897 1397 397 597 1397 1397 1397 1397 1397

600 300 150 75 60 1000 500 250 125 100 75 250 150 100 100 80 300 200 150 120 100

2.2. CBTSP and CTSP CTSP and CBTSP are both a version of MTSP and TSP. CBTSP and CTSP not only have shared city set, but also occupy exclusive city sets, the shared cities can be visited by all salesmen, but exclusive cities are only accessed by the appointed salesman, other salesmen have no authority to visit them. CBTSP and CTSP have different objective functions: the objective of CBTSP is to find the tours

5 485 5 049 5 101 5 409 2 388 2 003 1 755 915 859 695 582 539 487

where the difference of the maximum edge and minimum edge is minimized, and the objective function of CTSP is to search the tours in which the total traveling distance is as small as possible [1]. 2.3. CBTSP related theory CBTSP is NP-hard problem. CTSP is a variant of MTSP and TSP, under some condition, CTSP can be transformed into MTSP or TSP, it has been proved that CTSP is a NP-hard problem [2]. With changing the objective function of CTSP, it can be transformed into CBTSP, the time complexity of this model will not change due to its operation, thus CBTSP is also NP-hard problem. 2.4. CBTSP applications The multiple balanced traveling salesmen problem [3] cannot be used to model the optimization problems with cooperative task. However, CBTSP can be applied in the optimization and planning problems of persons and vehicles with cooperative and exclusive tasks such as the multiple tasks cooperation, the planning and deploying of the resource and goods. For example, there are six people who will uniformly deploy goods, while the goods are too many in a district, the six persons need cooperate to allocate the goods, when the goods are few, each of them will independently carries out the task in six different districts, the objective is to find six tours in which the goods are uniformly deployed. The basic elements of such a problem, i.e., objective, persons, and tasks, can respectively match the objective, salesmen, and cities of CBTSP, thus it can be modeled by CBTSP. It is not limited to the given instance, and the model can be applied to the kinds of optimization problems with independent tasks and cooperative task.

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Fig. 1. An example of dual-chromosome coding for CBTSP.

3. NGA for CBTSP 3.1. Solution representation The literature [1] uses dual-chromosome coding to represent the solution of CTSP. This paper also utilizes the method to code the solution of CBTSP. As shown in Fig. 1, the city chromosome is the permutation of the n − 1 cities, while the salesman chromosome is the permutation of the salesmen corresponding to the cities. We give an example to show the coding, in Fig. 1, for example, there is a CBTSP problem with 9 cities and 2 salesmen, city 1 to city 3 are the exclusive cities of salesman 1, city 4 to city 6 belong to the exclusive cities of salesman 2, city 7 to city 9 are the shared cities of the two salesmen. If the starting and ending point (depot 0) is contained, the visiting path of salesman 1 is 0-9-3-1-7-2-0, and the access route of salesman 2 is 0-4-5-6-8-0. 3.2. The steps of NGA The proposed algorithm is a novel genetic algorithm based on ITÖ process [18,19], the solutions of problem are considered to be particles, and the activity intensity of particles is affected by particles radius and environment temperature. The crossover length of crossover operator is controlled by the activity intensity, the larger the intensity is, the longer the length becomes. During the process of solving, the ith particle is crossed with the best solution. The steps of NGA for CBTSP are shown in algorithm 1:

operator and mutation operator, where the best solution is memorized, the former is the key operator of NGA based on ITÖ process, which can randomly select some cities of city chromosome, then they are crossed with the corresponding ones of best solution, the crossover operator corresponds to the selection operator and crossover operator of genetic algorithm, the mutation operator is the same with the mutation operator of genetic algorithm, steps 16 to 17 update environment temperature and crossover length. 3.3. NGA algorithm NGA algorithm includes five parts: particle radius, environment temperature, activity intensity, crossover operator and mutation operator. Among them, particle radius and environment temperature are influence factors, which can affect the activity intensity, it can control the crossover length of the crossover operator [7]. For NGA, the crossover operator is redesigned based on ITÖ process, which corresponds to the selection operator and crossover operator of genetic algorithm; mutation operator is same with the mutation operator of genetic algorithm [1]. The details of particle radius, environment temperature, activity intensity, crossover operator for the proposed algorithm are as follows: (1) Particle radius The formula can be defined as follows: r(x) = g(f (x))

(7)

where x stands for the particle of current swarm, f (x) represents the fitness, g (x) is monotonic function. The paper computes the particle radius according to classification, N particles are classified by the fitness based on the best to worst order, which are represented by x1 , x2 , . . . ., xN , a version of particle radius is computed by: r(xi ) =

(N − i)(rmax − rmin ) N −1

+ rmin

(8)

where rmax and rmin respectively represent the maximum and minimum particle radius, all particle radii are uniformly distributed in rmax and rmin , by default, rmax is set as 1, rmin is 0. (2) Environment temperature For simulated annealing algorithm, during the process of iteration, the environment temperature is gradually reduced, it is defined by the below formula: Ti = ρ · Ti−1

(9)

where Ti represents the temperature at the ith scheduling time, ρ stands for the annealing coefficient, which can control the speed of the temperature dropping, by default, it is set as 0.9. (3) Activity intensity Activity intensity controls the movement intensity of particles, the activity intensity I of current particle xi is computed by the below formula: Ii =

(e−λri − e−λrmax ) (e−λrmin

−

e−λrmax )

1

· e− T

(10)

where I represents the activity intensity, ri is the radius of particle xi , T stands for temperature. In algorithm 1, step 1 is the parameters initialization, step 2 is to initialize the population M and initial temperature T , step 3 is to read the CBTSP data, step 4 is to compute the fitness of the particles, and save the best one, steps 4 to 6 calculate particle radius, environment temperature and activity intensity, step 8 is to classify the particles according to fitness, step 9 computes the crossover length by formula (11), steps 10 to 13 carry out crossover

(4) Crossover operator Because of crossover operator, particles can change in the solution space, under the environment influence, the particles (solutions) are crossed with the best solution, which can generate a new solution, it can display strong global search ability, and keep diversity of solution. The crossover operator contains two parts including crossover length and crossover process, the crossover length is controlled by the activity intensity, which is computed

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Fig. 2. An example of crossover operator.

Fig. 3. The optimal routes of the algorithms for CBTSP with n = 51 and m = 3 (Unit: km).

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Fig. 4. The optimal routes of the algorithms for CBTSP with n = 76 and m = 4 (Unit: km).

by environment temperature and particle radius [7]. The crossover length is calculated by the below formula: L i = γ · Ii · l

(11)

where Li represents the crossover length of the ith particle, γ is random obeying uniform distribution number, it is between 0 and 1, l stands for the length of dual-chromosome coding. As shown in Fig. 2, the crossover process of algorithm NGA is as follows: Step 1: A starting position in [1, l−Li ] in city chromosome is randomly selected, and the continuous Li positions in the gray segment are crossed with the best solution. Step 2: Replace the cities in gray segment by the corresponding cities of best solution. For example, the swap relationship is: 3-6, 5-8, 6-7, 1-4, 7-1. Step 3: According to the swap relationship, replace the redundant cities outside the gray segment in the ith particle until there are no redundant cities. For example, in the step 2, there is number 4 outside the gray segment, and number 4 is also in the gray segment, which is called redundancy. Therefore, the 4 outside the gray segment is needed to be replaced. According to the swap relationship, the 4 in best

solution corresponds to the 1 in ith particle, thus 4 is replaced by 1, but 1 is still redundant. Based on the rule, 1 is replaced by 7, it still does not meet the condition, then 7 is replaced by 6, which is also redundant, thus 6 is replaced by 3, which can meet the condition. Finally, the final value is 3. Step 4: Correct the wrong assignment value in salesman chromosome. For example, in the third step, the city chromosomes 3, 6 and 4 correspond the wrong salesmen, therefore it needs correction, and the revised result is shown in step 4. After the four steps, the particle (solution) finishes the crossover operator, and carries out the updating of the solution for solving CBTSP problem. 4. Experiments and analysis 4.1. The small and medium scale CBTSP We make experiments to analyze the performance of the different algorithms for CBTSP. The computer environment is as follow: R Intel⃝ CoreTM i7-6700 running Windows 7 with 3.40 GHz processor and 8.00 GB RAM. The experiments are developed based on Java.

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Fig. 5. The optimal routes of the algorithms for CBTSP with n = 101 and m = 5 (Unit: km).

The initial parameters of GAs are from the CTSP paper [1]: the population size is 30, crossover probability as 0.7, and mutation probability is 0.1. 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 to be 30, and annealing coefficient as 0.9. The parameters of NGA: the initial population is 150, and the initial environment temperature is 1000. All the algorithms have the same stopping condition. Each algorithm runs ten times. The below table is the small and medium scale experiment data. In Table 1, n is the number of city, m is the number of salesmen, s is the city number of shared set of CBTSP, and e is the city number of an exclusive data. The data with the city number during 21 to 101 is provided in the CTSP paper [1], and the data with city number from 202 to 655 is made by the original TSP data (The TSPLIB Symmetric Traveling Salesman Problem Instances) published on web. The following figures are the solving routes figure of the four algorithms for CBTSP. In Fig. 3, it is the best solution case of the four algorithms for CBTSP using n = 51 and m = 3, the upper-left small figure is GAG for CBTSP with n = 51 and m = 3, the upper-right one is HCGA for CBTSP, the bottom-left one means SAGA for the problem, the bottom-right one represents NGA for solving the CBTSP problem.

The detail of the four algorithms for CBTSP is as follows: GAG: average solution 17.7, iteration 68322; HCGA: average solution 17.3, iteration 37604; SAGA: average solution 15.2, iteration 2168; NGA: average solution 15.8, iteration 9334. Fig. 4 is the best solution case of the four algorithms for CBTSP by n = 76 and m = 4, the top-left small figure is GAG for the problem, the top-left one represents HCGA for CBTSP, the lower left one is SAGA for CBTSP, the lower right one is NGA for solving the problem. The details are as follows: GAG: average solution 20.7, iteration 22477; HCGA: average solution 20.0, iteration 16934; SAGA: average solution 19.6, iteration 3702; NGA: average solution 16.9, iteration 5556. Fig. 5 is the solving routes figures of the four algorithms for CBTSP by n = 101 and m = 5, the top-left small figure stands for GAG for the problem, the top-left one represents HCGA for CBTSP, the lower left one is SAGA for solving the problem, the lower right one is NGA for CBTSP. The details of the four algorithms are as follows: GAG: average solution 33.7, iteration 19674; HCGA: average solution 29.1, iteration 17134; SAGA: average solution 29.5, iteration 1072; NGA: average solution 22.1, iteration 5049. Fig. 6 is the solution case of the four algorithms for CBTSP by n = 202 and m = 12, the top-left small figure stands for GAG for

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Fig. 6. The optimal routes of the algorithms for CBTSP with n = 202 and m = 12 (Unit: km).

the problem, the top-left one represents HCGA for CBTSP, the lower left one is SAGA for the problem, the lower right one is NGA. Fig. 7 is the solving routes of the four algorithms to solve CBTSP by n = 431 and m = 25, the top left figure represents the GAG for solving the problem, the upper right small figure means HCGA for CBTSP, the bottom left one is SAGA for the problem, the lower right one is NGA for CBTSP. The details are as follows: GAG: average solution 13133.9, iteration 3183; HCGA: average solution 12487.8, iteration 2885; SAGA: average solution 12786.7, iteration 1265; NGA: average solution 12967.0, iteration 859 (see Fig. 8). Table 2 is the experiments of the five algorithms for small scale and medium scale CBTSP by the same stopping condition. Fig. 8 is the best routes of the four algorithms for CBTSP by n = 655 and m = 33, the top left figure is the GAG for solving the problem, the upper right one means HCGA for CBTSP, the bottom left one represents SAGA for the problem, the lower right one stands for NGA for CBTSP. The detail is as follows: GAG: average solution 13115.8, iteration 1725; HCGA: average solution 11435.4, iteration 1537; SAGA: average solution 11488.9, iteration 827; NGA: average solution 11250.6, iteration 487.

In Table 2, it is the experiment results of five algorithms to solve the small and medium scale CBTSP, the mean is the average solution quality of each algorithm running 10 times for the problem, the iteration represents the average iteration times of algorithm running 10 times for obtaining the best solution. Among them, GA, GAG, HCGA and SAGA are from the CTSP literature [1]. NGA is valid swarm intelligence algorithm for the optimization problems. By the experiments, it shows that NGA is effective to solve the small scale and medium scale problem. 4.2. The large scale CBTSP The computer environment and the parameters setting of all algorithms are the same with the one of the experiments for small scale and medium scale CBTSP problem. The stopping conditions such as fitness function evaluation and maximum number of iterations usually have problems, which are not fair stopping conditions [20,21]. In order to make up the flaw of single stopping condition, we use two stopping conditions to make experiments for large scale CBTSP, each algorithm runs same time for solving the problem, the first one is running 60 s for each algorithm, the second one runs 180 s. Table 3 is the large scale data of CBTSP.

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Fig. 7. The optimal routes of the algorithms for CBTSP with n = 431 and m = 25 (Unit: km).

In Table 3, there are 21 instances, n is from 2461 to 7397, m is from 3 to 60, the large scale data is made according to the original TSP (The TSPLIB Symmetric Traveling Salesman Problem Instances) data which is published on web. The following tables are the experiments of the algorithms for CBTSP by running 60 s as the stopping condition. The compared algorithms GA, GAG and HCGA are from the literatures [1,2], the modified genetic algorithm SAGA is from the literature [1], the hill-climbing algorithm are used to optimize genetic algorithm two times (the algorithm is named HHGA) [22], the simulated annealing algorithm is used for optimization after the operators of genetic algorithm (the algorithm is called GASA) [23], hill-climbing as the local search method (neighborhood search) is used after the crossover and mutation of genetic algorithm (the algorithm is GAHC) [24], the modified genetic algorithm GAQSA refers to the literature [25], NGA is the proposed algorithm in this paper. In Table 4, n is the city number, m stands for the number of salesmen, best and mean represent the best solution and average solution of the algorithms running 10 times for CBTSP. The table shows that NGA can show better solution quality than the compared algorithms.

From Table 5, it shows that NGA can display better solution quality than other algorithms. Figs. 9 and 10 are average solution quality of the algorithms for CBTSP. In Figs. 9 and 10, the lateral axis stands for the order number of the instance. For example, the number 2 corresponds to the instance with n = 2461 and m = 6; vertical axis represents the mean solution quality of the algorithms for the problem. Figs. 9 and 10 show that NGA demonstrates better mean solution quality than GAQSA, GAHC, GASA, HHGA, SAGA, HCGA, GAG and GA. In order to test the effectiveness of the algorithms for the problem, we use the best solution deviation PDbest and the average best solution deviation PDav to make significance test [26]. The percentage deviation is computed based on the data of Tables 4 and 5, and the computation formula and method are given in the literature [26]. The computation results are shown in Tables 6 and 7. From Table 6, we can see that the percentage deviation of NGA is smaller than the ones of other four algorithms, and it shows that the proposed algorithm NGA can show superiority over the SAGA, HCGA, GAG and GA in term of solution quality.

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X. Dong and Y. Cai / Future Generation Computer Systems 95 (2019) 727–742

Fig. 8. The optimal routes of the algorithms for CBTSP with n = 655 and m = 33 (Unit: km). Table 4 The solution quality of the five algorithms for large scale CBTSP with running 60 s as the stopping condition (Unit: km). Instance

n

m

Large 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2461 2461 2461 2461 2461 3461 3461 3461 3461 3461 3461 5397 5397 5397 5397 5397 7397 7397 7397 7397 7397

3 6 12 24 30 3 6 12 24 30 40 20 30 40 50 60 20 30 40 50 60

GA

GAG

HCGA

SAGA

NGA

Best

Mean

Best

Mean

Best

Mean

Best

Mean

Best

Mean

3939.0 3951.0 3923.0 3986.0 3938.0 4185.0 4205.0 4232.0 4189.0 4183.0 4196.0 735 474.0 750 620.0 431 785.0 435 987.0 435 639.0 698 756.0 727 724.0 462 935.0 462 308.0 462 830.0

3985.1 3991.4 4014.1 4040.5 4049.1 4276.9 4276.4 4273.4 4311.4 4283.6 4288.7 745 851.4 757 848.6 435 895.5 442 208.9 441 107.6 706 637.9 742 003.4 466 051.1 466 342.1 465 983.5

1322.0 1915.0 2831.0 3060.0 2930.0 1364.0 1968.0 2919.0 3603.0 3237.0 3456.0 702 354.0 731 087.0 379 393.0 379 333.0 391 221.0 586 709.0 620 816.0 380 267.0 383 740.0 384 435.0

1721.6 2201.9 3116.6 3382.1 3267.0 2040.9 2384.7 3201.0 3726.8 3592.7 3800.1 727 316.6 750 034.8 385 715.3 391 584.9 410 408.0 630 030.6 652 410.4 387 829.2 388 956.6 390 511.0

1103.0 2064.0 2892.0 3204.0 2992.0 1633.0 2040.0 2793.0 3603.0 3337.0 3667.0 707 444.0 726 710.0 378 446.0 380 971.0 380 520.0 594 284.0 618 871.0 378 808.0 385 567.0 386 792.0

1628.4 2481.1 3153.9 3432.3 3246.2 2012.5 2380.8 3131.1 3722.6 3612.4 3843.3 722 618.5 745 932.2 386 210.2 392 611.6 397 614.2 635 148.0 651 421.2 387 217.4 389 618.5 389 408.1

1324.0 1536.0 2078.0 1747.0 2372.0 1504.0 1770.0 2283.0 3040.0 3034.0 2620.0 674 393.0 714 826.0 378 006.0 378 198.0 379 429.0 594 832.0 632 820.0 379 482.0 378 016.0 383 588.0

1496.8 1944.4 2504.1 2672.1 2810.2 1917.2 2091.0 2762.1 3447.8 3489.2 3561.3 703 255.7 739 228.3 381 448.9 380 183.5 390 184.4 633 248.2 660 184.6 387 663.3 387 109.5 388 659.3

786.0 1322.0 1906.0 2479.0 2576.0 1235.0 1636.0 2061.0 2761.0 2911.0 3141.0 624 264.0 676 234.0 211 532.0 274 364.0 275 904.0 564 760.0 588 471.0 263 991.0 286 046.0 280 287.0

846.4 1439.0 2122.7 2667.1 2753.3 1278.3 1859.0 2366.2 2960.7 3170.3 3280.3 652 183.1 698 300.6 261 310.7 310 120.5 330 702.3 585 141.5 604 897.7 277 376.3 306 737.8 304 233.9

X. Dong and Y. Cai / Future Generation Computer Systems 95 (2019) 727–742

737

Table 5 The solution quality of the five algorithms for large scale CBTSP with running 60 s as the stopping condition (Unit: km). Instance

n

m

Large 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2461 2461 2461 2461 2461 3461 3461 3461 3461 3461 3461 5397 5397 5397 5397 5397 7397 7397 7397 7397 7397

3 6 12 24 30 3 6 12 24 30 40 20 30 40 50 60 20 30 40 50 60

HHGA

GASA

GAHC

GAQSA

NGA

Best

Mean

Best

Mean

Best

Mean

Best

Mean

Best

Mean

1412.0 1689.0 2447.0 3356.0 2774.0 1654.0 2173.0 2924.0 3407.0 3252.0 3458.0 711 914.0 732 445.0 377 224.0 378 413.0 382 978.0 567 116.0 634 302.0 383 743.0 383 740.0 383 743.0

1746.9 2341.1 2863.7 3492.6 3208.3 2051.0 2391.9 3187.9 3594.8 3558.0 3736.6 728 861.6 750 703.1 380 031.2 390 743.5 400 120.4 617 729.3 664 049.8 387 417.0 387 242.9 388 602.4

1230.0 1510.0 1754.0 2015.0 2899.0 1554.0 2122.0 2726.0 3011.0 2941.0 2962.0 680 495.0 710 104.0 379 412.0 377 818.0 379 414.0 599 513.0 636 528.0 384 432.0 379 693.0 383 585.0

1489.2 1894.7 2686.0 2878.7 3132.7 1985.5 2375.2 2989.3 3424.3 3437.8 3480.6 704 133.7 731 490.9 380 972.9 384 265.8 383 077.8 620 764.0 649 394.3 388 763.3 388 556.9 387 802.5

1434.0 1834.0 2696.0 3149.0 2741.0 1397.0 2015.0 2848.0 3524.0 3288.0 3468.0 695 960.0 719 362.0 378 961.0 379 408.0 379 474.0 592 693.0 627 827.0 383 588.0 381 267.0 384 151.0

1756.6 2391.0 3077.0 3363.4 3234.7 1769.0 2332.1 3161.6 3676.5 3602.4 3710.0 725 712.0 744 071.7 386 469.8 386 357.6 397 055.9 624 653.8 649 173.3 388 002.8 389 270.0 388 970.2

1333.0 1901.0 2189.0 2588.0 2607.0 1355.0 1973.0 2506.0 3030.0 3050.0 3125.0 693 410.0 685 882.0 377 605.0 378 394.0 379 325.0 589 409.0 624 777.0 379 482.0 381 267.0 385 559.0

1620.5 2174.7 2460.7 2691.7 2709.0 1877.4 2243.3 2796.9 3262.8 3416.0 3435.8 701 328.9 732 548.3 379 197.5 380 634.0 387 178.0 628 891.6 646 689.1 385 691.0 385 665.8 388 533.5

786.0 1322.0 1906.0 2479.0 2576.0 1235.0 1636.0 2061.0 2761.0 2911.0 3141.0 624 264.0 676 234.0 211 532.0 274 364.0 275 904.0 564 760.0 588 471.0 263 991.0 286 046.0 280 287.0

846.4 1439.0 2122.7 2667.1 2753.3 1278.3 1859.0 2366.2 2960.7 3170.3 3280.3 652 183.1 698 300.6 261 310.7 310 120.5 330 702.3 585 141.5 604 897.7 277 376.3 306 737.8 304 233.9

Table 6 The percentage deviation of the five algorithms for large scale CBTSP with running 60 s as the stopping condition. Instance

n

m

Large scale 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2461 2461 2461 2461 2461 3461 3461 3461 3461 3461 3461 5397 5397 5397 5397 5397 7397 7397 7397 7397 7397 Average

3 6 12 24 30 3 6 12 24 30 40 20 30 40 50 60 20 30 40 50 60

GA

GAG

HCGA

SAGA

NGA

PDbest

PDav

PDbest

PDav

PDbest

PDav

PDbest

PDav

PDbest

PDav

401.1 198.8 122.8 128.1 66.0 238.8 157.0 105.3 51.7 61.1 60.1 17.8 11.0 104.1 58.9 57.8 23.7 23.6 75.3 61.6 65.1

370.8 177.3 89.1 51.4 47.0 234.5 130.0 80.6 45.6 35.1 30.7 14.3 8.5 66.8 42.5 33.3 20.7 22.6 68.0 52.0 53.1

68.1 44.8 60.8 75.1 23.5 10.4 20.2 41.6 30.4 24.6 31.9 12.5 8.1 79.3 38.2 41.7 3.8 5.4 44.0 34.1 37.1

103.4 53.0 46.8 26.8 18.6 59.6 28.2 35.2 25.8 13.3 15.8 11.5 7.4 47.6 26.2 24.1 7.6 7.8 39.8 26.8 28.3

40.3 56.1 64.3 83.4 26.1 32.2 24.6 35.5 30.4 28.5 39.9 13.3 7.4 78.9 38.8 37.9 5.2 5.1 43.4 34.7 37.9

92.3 72.4 48.5 28.6 17.9 57.4 28.0 32.3 25.7 13.9 17.1 10.7 6.8 47.7 26.5 20.2 8.5 7.6 39.6 27.0 27.9

68.4 16.1 18.0 0.0 0.0 21.7 8.1 10.7 10.1 16.8 0.0 8.0 5.7 78.6 37.8 37.5 5.3 7.5 43.7 32.1 36.8

76.8 35.1 17.9 0.1 2.0 49.9 12.4 16.7 16.4 10.0 8.5 7.8 5.8 45.9 22.5 17.9 8.2 9.1 39.7 26.2 27.7

0.0 0.0 8.2 41.9 8.6 0.0 0.0 0.0 0.0 12.1 19.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

99.5

79.7

35.0

31.1

36.4

31.3

22.0

21.7

4.3

0.0

The Table 7 shows that NGA can show better solution quality than HHGA, GASA, GAHC and GAQSA for large scale CBTSP. The following tables are the experiments results of the algorithms for large scale CBTSP by running 180 s as the stopping condition. In Tables 8 and 9, GA, GAG and HCGA are all from the literatures [1,2]; the improved genetic algorithm SAGA is from the literature [1]; the modified genetic algorithms HHGA, GASA and GAHC respectively refer to the literature [22], literature [23] and literature [24]; the improved genetic algorithm GAQSA refers to the literature [25]; NGA is the proposed algorithm. In Tables 8 and 9, they are the experiment results of the algorithms for CBTSP by running 180 s as the stopping condition. The mean is also the average solution quality of the algorithms running 10 times for the problem. From the data, it shows that NGA can show better solution quality than the compared modified genetic algorithms. Figs. 11 and 12 are the mean solution quality of the five algorithms for solving CBTSP by running 180 s as the stopping condition.

In Fig. 11, 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 21 is the corresponding data with n = 7397 and m = 60; vertical axis is the mean solution quality of the algorithms for the problem. The lateral axis of Fig. 12 means the order number of instance data, and the vertical axis is the mean solution quality of the algorithms for CBTSP. The Figs. 11 and 12 show that NGA can demonstrate better mean solution quality than the modified genetic algorithms for solving large scale CBTSP. The below tables are the percentage deviation of the algorithms for CBTSP by running 180 s as stopping condition. In Table 10, it shows that the best solution percentage deviation PDbest and average best solution percentage deviation PDav of NGA is the smallest in the five algorithms, which means that NGA can demonstrate superiority over SAGA, HCGA, GAG and GA in term of solution quality.

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X. Dong and Y. Cai / Future Generation Computer Systems 95 (2019) 727–742 Table 7 The percentage deviation of the five algorithms for large scale CBTSP with running 60 s as the stopping condition. Instance

n

m

Large scale 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2461 2461 2461 2461 2461 3461 3461 3461 3461 3461 3461 5397 5397 5397 5397 5397 7397 7397 7397 7397 7397

3 6 12 24 30 3 6 12 24 30 40 20 30 40 50 60 20 30 40 50 60

Average

HHGA

GASA

GAHC

GAQSA

NGA

PDbest

PDav

PDbest

PDav

PDbest

PDav

PDbest

PDav

PDbest

PDav

79.6 27.7 39.5 66.5 7.6 33.9 32.8 41.8 23.3 11.7 16.7 14.0 8.3 78.3 37.9 38.8 0.4 7.7 45.3 34.1 36.9

106.3 62.6 34.9 30.9 18.4 60.4 28.6 34.7 21.4 12.2 13.9 11.7 7.5 45.4 25.9 20.9 5.5 9.7 39.6 26.2 27.7

56.4 14.2 0.0 0.0 12.5 25.8 29.7 32.2 9.0 1.0 0.0 9.0 5.0 79.3 37.7 37.5 6.1 8.1 45.6 32.7 36.8

75.9 31.6 26.5 7.9 15.6 55.3 27.7 26.3 15.6 8.4 6.1 7.9 4.7 45.7 23.9 15.8 6.0 7.3 40.1 26.6 27.4

82.4 38.7 53.7 56.2 6.4 13.1 23.1 38.1 27.6 12.9 17.0 11.4 6.3 79.1 38.2 37.5 4.9 6.6 45.3 33.2 37.0

107.5 66.1 44.9 26.1 19.4 38.3 25.4 33.6 24.1 13.6 13.0 11.2 6.5 47.8 24.5 20.0 6.7 7.3 39.8 26.9 27.8

69.5 43.7 24.8 28.4 1.2 9.7 20.5 21.5 9.7 4.7 5.5 11.0 1.4 78.5 37.9 37.4 4.36 6.1 43.7 33.2 37.5

91.4 51.1 15.9 0.9 0.0 46.8 20.6 18.2 10.2 7.7 4.7 7.5 4.9 45.1 22.7 17.0 7.4 6.9 39.0 25.7 27.7

0.0 0.0 8.6 23.0 0.0 0.0 0.0 0.0 0.0 0.0 6.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 1.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

32.5

30.7

22.8

23.9

31.8

30.0

25.3

22.4

1.7

0.07

Table 8 The solution quality of the five algorithms for large scale CBTSP with running 180 s as the stopping condition (Unit: km). Instance

n

m

Large 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2461 2461 2461 2461 2461 3461 3461 3461 3461 3461 3461 5397 5397 5397 5397 5397 7397 7397 7397 7397 7397

3 6 12 24 30 3 6 12 24 30 40 20 30 40 50 60 20 30 40 50 60

GA

GAG

HCGA

SAGA

NGA

Best

Mean

Best

Mean

Best

Mean

Best

Mean

Best

Mean

3782.0 3869.0 3831.0 3901.0 3906.0 4109.0 4155.0 4146.0 4124.0 4155.0 4186.0 728 821.0 740 720.0 426 569.0 428 986.0 430 699.0 689 050.0 731 807.0 459 451.0 462 300.0 461 342.0

3903.3 3914.9 3925.8 3961.5 3951.6 4200.4 4243.4 4196.4 4195.6 4193.9 4245.2 736 354.9 751 998.6 434 040.2 434 471.7 435 056.2 699 144.1 736 214.7 463 096.1 463 566.5 463 348.3

1152.0 1378.0 2428.0 2814.0 3064.0 1497.0 1870.0 2663.0 3603.0 3333.0 3325.0 703 417.0 719 075.0 377 954.0 379 344.0 379 385.0 603 540.0 614 787.0 386 784.0 384 435.0 386 784.0

1642.3 2199.1 2965.3 3357.5 3280.5 1897.3 2239.9 3135.0 3664.4 3663.2 3725.9 723 289.7 743 732.1 384 240.3 393 096.9 400 887.2 621 953.6 662 048.2 390 598.2 390 281.1 390 239.6

1345.0 1760.0 2316.0 2771.0 2639.0 1396.0 2032.0 2626.0 3335.0 3288.0 3609.0 677 681.0 732 496.0 377 937.0 378 736.0 378 383.0 587 012.0 631 410.0 383 743.0 384 424.0 384 151.0

1565.7 2242.6 2993.7 3201.9 3187.9 1932.3 2295.7 2988.3 3665.4 3606.5 3737.2 714 741.2 749 857.0 379 640.0 384 465.3 393 551.9 609 802.3 653 935.7 390 747.5 388 876.4 392 003.0

1132.0 1356.0 1721.0 1749.0 2372.0 1428.0 1796.0 2005.0 2493.0 2626.0 3139.0 651 856.0 675 320.0 377 201.0 377 168.0 377 149.0 528 691.0 602 838.0 379 482.0 381 267.0 378 147.0

1342.9 1570.3 2415.9 2645.4 2932.3 1828.0 2143.7 2536.5 3001.5 3160.8 3478.3 674 221.4 703 960.5 381 100.4 378 335.9 379 503.2 601 227.8 636 154.6 386 187.3 387 428.2 385 675.3

803.0 1395.0 1813.0 2442.0 2555.0 1209.0 1610.0 2171.0 2764.0 2859.0 3132.0 621 091.0 671 429.0 194 797.0 273 101.0 283 679.0 559 358.0 572 646.0 265 908.0 281 086.0 281 082.0

829.2 1490.1 2082.7 2577.0 2701.1 1268.1 1853.3 2360.2 2943.2 3123.1 3271.0 649 456.4 692 751.0 259 629.1 317 077.4 324 001.2 567 024.4 595 714.3 277 218.5 298 361.5 311 882.9

In Table 11, it shows that NGA can demonstrate better solution quality than the compared modified genetic algorithms HHGA, GASA, GAHC and GAQSA. 4.3. Discussion In intelligent transport systems and multiple tasks cooperation, many real-world problems can be modeled by CBTSP, the scale of constructed model is easy to tend to large scale, thus it is significant to study large scale CBTSP, however, the traditional modified genetic algorithms, such as GAG, HCGA and SAGA, are easy to fall into local optimum, in order to improve it, the NGA is proposed for this problem. For the small scale and medium scale CBTSP, the experiments show that NGA has no obvious superiority over the compared genetic algorithms, while the city number is more

than 2000, the traditional modified genetic algorithms are easy to fall into local optimum, NGA can display strong global search ability, it can demonstrate obvious superiority over the compared algorithms. The NGA and the modified genetic algorithms for large scale CBTSP are shown in Tables 4 and 5 by running 60 s as the stopping condition, Figs. 9 and 10 are the average solution quality of the nine algorithms for this problem, which is made based on the average solution of the former two tables, the percentage deviation of Tables 6 and 7 is made based on Tables 4 and 5. The Tables 4– 7 and Figs. 9–10 show that NGA has better solution quality than the compared modified genetic algorithms. Tables 8 and 9 are experiments results of the algorithms for large scale CBTSP by

X. Dong and Y. Cai / Future Generation Computer Systems 95 (2019) 727–742

739

Table 9 The solution quality of the five algorithms for large scale CBTSP with running 180 s as the stopping condition (Unit: km). Instance

n

m

HHGA

Large 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2461 2461 2461 2461 2461 3461 3461 3461 3461 3461 3461 5397 5397 5397 5397 5397 7397 7397 7397 7397 7397

3 6 12 24 30 3 6 12 24 30 40 20 30 40 50 60 20 30 40 50 60

GASA

GAHC

GAQSA

NGA

Best

Mean

Best

Mean

Best

Mean

Best

Mean

Best

Mean

1393.0 1991.0 2386.0 2672.0 2820.0 1712.0 1830.0 2588.0 3354.0 3011.0 3373.0 695 860.0 716 078.0 378 994.0 378 800.0 379 433.0 588 300.0 627 751.0 383 577.0 383 691.0 384 151.0

1717.8 2274.0 2896.6 3173.5 3199.1 2084.6 2328.0 3056.6 3566.2 3533.0 3689.4 715 289.6 741 008.1 383 869.1 387 059.5 396 287.0 621 945.7 656 876.8 386 670.4 387 736.2 388 113.9

970.0 1395.0 1424.0 1883.0 1915.0 1467.0 1799.0 2159.0 2466.0 3098.0 2799.0 665 442.0 679 130.0 377 138.0 375 260.0 377 844.0 588 677.0 602 168.0 379 482.0 381 264.0 379 468.0

1269.5 1568.9 2317.2 2802.2 2812.4 1764.8 2096.1 2614.7 3020.6 3305.6 3427.4 689 173.5 706 743.8 379 476.2 379 197.0 383 271.6 614 833.5 642 598.7 385 773.6 384 867.0 387 324.5

1196.0 1735.0 2414.0 2745.0 2874.0 1512.0 1938.0 2882.0 3324.0 3111.0 3346.0 700 339.0 709 360.0 379 344.0 377 840.0 379 410.0 599 363.0 603 003.0 383 732.0 385 547.0 383 588.0

1826.3 2244.2 2890.8 3230.6 3217.8 1956.7 2237.3 3086.0 3641.5 3514.7 3697.4 717 936.7 740 032.9 382 518.7 384 397.6 389 680.2 629 677.5 647 381.0 388 376.7 389 602.7 387 633.4

1384.0 1612.0 2235.0 2308.0 2324.0 1508.0 2027.0 2264.0 2517.0 2754.0 3078.0 675 930.0 674 711.0 377 142.0 375 496.0 376 825.0 607 557.0 608 134.0 372 940.0 375 989.0 384 107.0

1664.3 1989.2 2341.9 2589.3 2527.6 1830.1 2202.4 2515.9 3015.5 3172.1 3374.7 688 256.5 707 437.5 378 537.5 379 313.3 381 004.5 6327 47.0 641 923.1 383 990.7 383 986.7 387 665.3

803.0 1395.0 1813.0 2442.0 2555.0 1209.0 1610.0 2171.0 2764.0 2859.0 3132.0 621 091.0 671 429.0 194 797.0 273 101.0 283 679.0 559 358.0 572 646.0 265 908.0 281 086.0 281 082.0

829.2 1490.1 2082.7 2577.0 2701.1 1268.1 1853.3 2360.2 2943.2 3123.1 3271.0 649 456.4 692 751.0 259 629.1 317 077.4 324 001.2 567 024.4 595 714.3 277 218.5 298 361.5 311 882.9

Table 10 The percentage deviation of the five algorithms for large scale CBTSP with running 180 s as the stopping condition. Instance

n

m

Large 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2461 2461 2461 2461 2461 3461 3461 3461 3461 3461 3461 5397 5397 5397 5397 5397 7397 7397 7397 7397 7397 Average

3 6 12 24 30 3 6 12 24 30 40 20 30 40 50 60 20 30 40 50 60

GA

GAG

HCGA

SAGA

NGA

PDbest

PDav

PDbest

PDav

PDbest

PDav

PDbest

PDav

PDbest

PDav

370.9 185.3 122.6 123.0 64.6 239.8 243.6 106.7 65.4 58.2 47.9 17.3 10.3 118.9 57.0 51.8 30.3 27.7 72.7 64.4 64.1

370.7 162.7 88.4 53.7 46.2 231.2 128.9 77.7 42.5 34.2 29.7 13.3 8.5 67.1 37.0 34.2 23.3 23.5 67.0 55.3 48.5

43.4 1.6 41.0 60.8 29.1 23.8 54.6 32.8 44.5 26.9 17.5 13.25 7.0 94.0 38.9 33.7 14.1 7.3 45.4 36.7 37.6

98.0 47.5 42.3 30.2 21.4 49.6 20.8 32.8 24.5 17.2 13.9 11.3 7.3 47.9 23.9 23.7 9.6 11.1 40.8 30.8 25.1

67.4 29.7 34.5 58.4 11.2 15.4 68.0 30.9 33.7 25.2 27.5 9.1 9.0 94.0 38.6 33.3 11.0 10.2 44.3 36.7 36.6

88.8 50.4 43.7 24.2 18.0 52.3 23.8 26.6 24.5 15.4 14.2 10.0 8.2 46.2 21.2 21.4 7.5 9.7 40.9 30.3 25.6

40.9 0.0 0.0 0.0 0.0 18.1 48.5 0.0 0.0 0.0 10.9 4.9 0.5 93.6 38.1 32.9 0.0 5.2 42.7 35.6 34.5

61.9 5.3 15.9 2.6 8.5 44.1 15.6 7.46 1.9 1.2 6.3 3.8 1.6 46.7 19.3 17.1 6.0 6.7 39.3 29.8 23.6

0.0 2.8 5.3 39.6 7.7 0.0 33.1 8.2 10.8 8.8 10.7 0.0 0.0 0.0 0.0 0.0 5.8 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

102.0

78.3

33.5

30.0

34.5

28.7

19.3

17.4

6.3

0.0

running 180 s as the stopping condition, Figs. 11–12 and Tables 10– 11 are made based on Tables 8 and 9. Tables 8–11 and Figs. 11– 12 display that NGA can demonstrate better solution than the compared genetic algorithms. In the mentioned tables and figures, it shows that NGA can demonstrate better solution quality than the compared modified genetic algorithms GAG, HCGA, SAGA, HHGA, GASA, GAHC and GAQSA for large scale CBTSP. HCGA and SAGA have the similar solution quality, and GA displays the worst solution quality in the several algorithms. In term of the solution quality, the improved GAs such GAG, HCGA and SAGA have better performance than the basic GA. NGA has the best solution quality in the nine algorithms for solving the large-scale problem.

From the above experiments, it shows NGA can not only show better best solution quality than the former compared algorithms in most of instances for large scale CBTSP, but also display obvious superiority over the compared modified genetic algorithms in term of average solution quality. NGA is a new swarm intelligence algorithm which can be used in optimization problems such as planning and combination optimization problem. By the CBTSP experiments, it shows that the NGA is effective for large scale CBTSP, and can demonstrate superiority over the compared genetic algorithms.

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X. Dong and Y. Cai / Future Generation Computer Systems 95 (2019) 727–742 Table 11 The percentage deviation of the five algorithms for large scale CBTSP with running 180 s as the stopping condition. Instance

n

m

Large 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2461 2461 2461 2461 2461 3461 3461 3461 3461 3461 3461 5397 5397 5397 5397 5397 7397 7397 7397 7397 7397 Average

3 6 12 24 30 3 6 12 24 30 40 20 30 40 50 60 20 30 40 50 60

HHGA

GASA

GAHC

GAQSA

NGA

PDbest

PDav

PDbest

PDav

PDbest

PDav

PDbest

PDav

PDbest

PDav

73.4 42.7 67.5 41.9 47.2 41.6 13.6 19.8 36.0 9.3 20.5 12.0 6.6 94.5 38.7 33.7 5.1 9.6 44.2 36.5 36.6

107.1 52.6 39.0 23.1 26.5 64.3 25.6 29.5 21.1 13.1 12.7 10.1 6.9 47.8 22.0 22.3 9.6 10.2 39.4 29.9 24.4

20.7 0.0 0.0 0.0 0.0 21.3 11.7 0.0 0.0 12.4 0.0 7.1 1.1 93.6 37.4 33.1 5.2 5.1 42.7 35.6 35.0

53.0 5.2 11.2 8.7 11.2 39.1 13.1 10.7 2.6 5.8 4.7 6.1 2.0 46.1 19.5 18.2 8.43 7.8 39.1 28.9 24.1

48.9 24.3 69.5 45.7 50.0 25.0 20.3 33.4 34.7 12.9 19.5 12.7 5.6 94.7 38.3 33.7 7.1 5.3 44.3 37.1 36.4

120.2 50.6 38.8 25.3 27.3 54.3 20.7 30.7 23.7 12.5 13.0 10.5 6.8 47.3 21.2 20.2 11.0 8.6 40.0 30.5 24.2

72.3 15.5 56.9 22.5 21.3 24.7 25.9 4.8 2.0 0.0 9.9 8.8 0.4 93.6 37.4 32.8 8.6 6.1 40.2 33.7 36.6

100.7 33.4 12.4 0.4 0.0 44.3 18.8 6.5 2.4 1.5 3.1 5.9 2.1 45.7 19.6 17.5 11.5 7.7 38.5 28.6 24.2

0.0 0.0 27.3 29.6 33.4 0.0 0.0 0.5 12.0 3.8 11.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 6.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

34.8

30.3

17.2

17.4

33.3

30.3

26.4

20.2

5.6

0.3

Fig. 9. The average solution quality of the algorithms for large scale CBTSP (Unit: km).

Fig. 10. The average solution quality of the algorithms for large scale CBTSP (Unit: km).

5. Conclusion and future works

good performance in term of solution quality or solving speed; on the other hand, exploring and studying more applications of CBTSP model is another possible research work, and we can also use the proposed algorithm to solve other combinational optimization problems. In addition, the new learning strategies, such as multi-tasking learning [27–29], reinforcement learning [30– 33], social learning [34–36], or self-learning [37,38], should be introduced in the used algorithms for further improving their performance.

CBTSP is one of the combination optimization problems, which is from the applications where multiple salesmen work cooperatively in the workspaces that partially overlap with each other. The paper provides a new model called CBTSP, which can model real-world problems, and proposes a novel genetic algorithm for solving the problem. The extensive experiments show that NGA can demonstrate better performance than the compared modified genetic algorithms for large scale CBTSP in term of solution quality. The limitations of our works are as follows: although the city number of CBTSP is more than 7000, the scale is still limited; the given applications of CBTSP is not enough in this paper. The next possible works could be focused on the points: on the one hand, studying more advanced algorithms for larger scale CBTSP is a possible research area, the expected algorithms should show

Acknowledgments We acknowledge the financial support from National Natural Science Foundation of China under grant No. 61672024 and 61170305; thank the foundation committee for giving us an opportunity to apply for Shandong Provincial Natural Science Foundation, China, No. ZR201808190052; thank referees for their useful suggestions and comments that help us to improve this paper.

X. Dong and Y. Cai / Future Generation Computer Systems 95 (2019) 727–742

Fig. 11. The average solution quality of the algorithms for large scale CBTSP (Unit: km).

Fig. 12. The average solution quality of the algorithms for large scale CBTSP (Unit: km).

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X. Dong and Y. Cai / Future Generation Computer Systems 95 (2019) 727–742 Xueshi Dong is currently assistant professor in College of Computer Science and Technology, Qingdao University. He ever pursued Ph.D degree in Computer School, Wuhan University from 2013 to 2018. He worked in Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences from 2011 to 2012, and he also visited Institute of Computing Technology, Chinese Academy of Sciences and Department of Computer Science and Technology, Peking University from 2015 to 2016. His current research interests include artificial intelligence, intelligent computing and simulation optimization.

Yongle Cai received M.S. degree in Computer School, Wuhan University. His main research interests include intelligent computing and simulation optimization.