A Carnivorous Plant Algorithm With Heuristic Decoding Method for Traveling Salesman Problem

The traveling salesman problem (TSP) is one of the most extensively studied problems in the combinatorial optimization area and still presents unsolved challenges due to its NP-hard attribute. Although many real-coded algorithms are available for TSP, they still have some performance challenges in the switch from continuous space to discrete space and perform at low convergence speed. This paper proposes a real-coded carnivorous plant algorithm with a heuristic decoding method (CPA-HDM) to solve the traveling salesman problem (TSP), which exhibits good convergence speed and solution accuracy. In this improved method, a new heuristic decoding method (HDM) is designed, which helps to map continuous variables to discrete ones without losing information, maintain population diversity, and enhance the solution quality after decoding. To balance the algorithm’s search capability and enhance the probability of preferable individuals generated, an adaptive attraction probability (AAP), an improved growth model of carnivorous plants (IGMOCP), and a position update method of prey (IPUMOP) are developed. Aiming to reduce the probability of premature and prevent search stagnation, an improved reproduction strategy (IRS) and an adaptive combination perturbation are reconstructed. Finally, a local search algorithm is employed to improve the exploitation capability. To verify its validation, CPA-HDM is compared with six algorithms, for solving 28 TSP instances. The simulation results and statistical analyses demonstrate the superior performance of the proposed algorithm.

the order-based arrangement [30], [31] and rounding method 98 [32], [33] are commonly employed for decoding, which is 99 easy to implement but plays a negative role in the solution 100 quality of its randomicity. The limitations of the decoding 101 method for TSPs, the lack of CPA in addressing combi-102 natorial optimization problems, and the promising results 103 achieved by CPA in continuous problems have severed as the 104 main motivation of this paper. 105 Considering the above problems, a new decoding method 106 HDM, which both considers the distance between cities and 107 the continuous variables of individuals, is designed at first. 108 For one thing, it can extract the outstanding features of parent 109 individuals, for another, it can diversify the population. Thus, 110 HDM can play a positive role in the convergence rate of the 111 algorithm. 112 The main effort of the proposed algorithm is to improve the 113 convergence rate and the search precision on TSP instances of 114 different sizes. To achieve it, CPA has been further optimized. 115 Firstly, in the existing literature, the growth of carnivorous 116 plants or the update of prey depends on attraction proba-117 bility, and attraction probability is a constant, which cannot 118 well balance the exploration and exploitation ability. Thus, 119 an AAP based on distance and city size is proposed, the 120 distance between the carnivorous plant and prey determines 121 whether the prey can be successfully attracted. Secondly, 122 the attraction may approach 0 in the growth phase, result-123 ing in a slow convergence speed. Then the CPA-HDM adds 124 the guidance of the best individual in the growth model of 125 carnivorous plants and the subgroup's best individual in the 126 position update rule of prey, which can improve the quality 127 of offspring. Thirdly, the reproduction phase only allows 128 the optimal individual to reproduce, which may enhance 129 the probability of the algorithm falling into the local opti-130 mum. Therefore, the IRS is proposed, and all carnivorous 131 plants are allowed to reproduce, which is helpful to increase 132 the information interaction among subgroups. Fourthly, the 133 2-opt exchange is generally used to detect a better solution, 134 however, the edges involved in the standard 2-opt exchange 135 are chosen at random, and the low probability of excellent 136 individuals is generated, which causes unnecessary search 137 times in the iteration. To address this problem, the neigh-138 borhood 2-opt and double-bridge exchange are presented 139 in this paper. Finally, to improve the search accuracy, the 140 2-Opt algorithm is adopted. Thus, the proposed algorithm 141 can both possess exploration and exploitation capabilities, 142 which obtain high-quality solutions and high convergence 143 speed. 144 The key contributions of this paper can be summarized as 145 follows: to enhance the exploitation capability and the self-escape 195 strategy helps to keep population diversity. Akhand et al. 196 [26] adopted a discrete spider monkey optimization (DSMO).

197
In DSMO, all the spider monkey was represented as TSP proposed by Eneko et al. [18]. Three strategies are employed 202 to improve the performance of the algorithm, which are:

203
(1) the Hamming distance is introduced to measure the 204 difference between two individuals; (2) the insert mutation 205 operator is adopted to emulate the evaporation and raining 206 process in the discrete solution space; (3) an adaptive mod-207 ification parameter is proposed to choose movement opera-208 tors. incorporated to enhance its performance. First, an improved 240 roulette selection is proposed to maintain the population 241 diversity; Second, the independent set is proposed to increase 242 the exploration ability; Third, the local optimum mutation 243 operator is presented to reduce the probability of stagnating; 244 Finally, the local search algorithm is employed to improve the 245 solution accuracy.

246
Some algorithms employ mathematical formulas suit-247 able for the continuous problem and need to adopt the 248 decoding method to generate legal TSP paths. Ezugwu and 249 Adewumi [33] adopted the rounding method and restruc-250 tured symbiotic organisms search by incorporating swap, 251 insert, and inverse operators to form a discrete symbi-252 otic organisms search (DSOS). Three mutation operators 253 are employed to improve its initial population. Zhang and 254 Han [30] applied the order-based arrangement to map contin-255 uous variables as discrete ones in the discrete sparrow search 256 algorithm (DSSA), the roulette wheel selection, Gaussian 257 mutation, and swap operator are introduced in DSSA to 258 increase the probability of jumping out of the local opti-259 mum, the 2-Opt algorithm is adopted to enhance the solution 260 quality.

317
The population size of CPA is nn, individuals in the pop-318 ulation are sorted from the smallest to largest according to 319 their fitness values for the minimization problem, the best 320 n individuals are regarded as carnivorous plants, and the 321 remaining n 1 individuals are regarded as prey (n 1 > n, n 1 322 is divisible by n). The group number is n, individuals in each 323 group are comprised of one carnivorous plant and n 1 /n prey. 324 The best prey is attracted by the best carnivorous plant, the 325 second-best prey is attracted by the second-best carnivorous 326 plant, the process is repeated, and the n th best prey is attracted 327 by the n th best carnivorous plant. It is noted that the (n + 1) th 328 best prey is attracted by the best carnivorous plant, and the 329 (n+2) th prey is attracted by the second-best carnivorous plant, 330 and the process is repeated until the n th 1 prey is attracted by 331 the n th carnivorous plant.

332
The grouping process is depicted by an example in Fig. 1, 333 where population size nn = 12, the number of carnivo-334 rous plants n = 3, the number of prey n 1 = 9, X = 335 (X 1 , X 2 , . . . , X 12 ) before sorting and it becomes X = 336 (X 1 , X 2 , . . . , X 12 ) after sorting, the objective function values 337 satisfied F (X 1 ) ≤ F (X 2 ) ≤ · · · ≤ F (X 12 ). The carnivorous plant lured prey by its scent, but prey may 340 successfully escape from the plants or not be attracted. Hence, 341 an attraction probability γ is introduced in CPA, if γ (γ = 342 0.8) is greater than a random number λ (λ is generated in the 343 range [0,1]), the carnivorous plant successfully lures the prey 344 to growth, and the model can be formulated as: where ⊗ represents multiplying the variables at the same 348 position in two vectors, xp i is the carnivorous plant in group 349 i, p iv is the v th prey in group i, rand is the random vector in 350 the range [0,1], gr is the growth rate, usually equals to 2.

351
If γ is less than λ, which stands for the prey escapes from 352 the trap or not being attracted by the plant and the growth 353 model of prey can be expressed as: where ⊗ represents multiplying the variables at the same The best carnivorous plant is allowed to perform the 363 reproduction operation, and the mathematical model is 364 summarized as: where ⊗ represents multiplying the variables at the same  This process is called recombination, which ensures that fitter 381 individuals can be selected for the later generation.

382
The pseudo-code of standard CPA is presented in 383 Algorithm 1.

384
Algorithm 1 CPA Input: the population size nn; the population size of carnivorous plants xp: n, the population size of prey p: n 1 , growth_rate, reproduction_rate, Maximum iteration: Maxgen; Output: Best solution and the optimal value; 1: Generate nn initial individuals in the population; 2: Calculate the fitness value and sort based on the fitness value; 3: While gen< Maxgen 4: Set n best individuals as carnivorous plants, the remaining n 1 individuals as prey, and sort a group as depicted in Fig.1;

5:
Newxp, Newp is updated with (1) and (3); 6: Newxp is updated with (5); 7: Newxp, Newp, xp, and p combined a new population; 8: Calculate the fitness of the population; 9: Sort according to the fitness value and select nn best individuals; 10: End While C. THE TRAVELING SALESMAN PROBLEM 385 TSP is usually described as a merchant who traverses m cities 386 to sell goods. In this process, one must pass through all the 387 cities, each city can only pass through once and finally return 388 to the original city. TSP can be represented as a weighted 389 graph G = (V , E), which goal is to find a Hamilton loop 390 with the smallest weights. V is the set of vertices, and E is 391 the set of edges. The vertices of the graph represent cities, the 392 edges denote the path between cities, and the weight of an 393 edge indicates the Euclidean distance between two cities. 394 Although the definition of TSP is simple, as the number 395 of cities increases, the number of possible tours increases 396 dramatically. The challenge is to solve this problem in an 397 acceptable time with the lowest travel costs. Until now, there 398 is still no effective way to solve this problem, and it can be 399 divided into symmetric and asymmetric TSP. If the distance 400 from city i to city j equals from j to i, it is considered 401 a symmetric problem, otherwise, an asymmetric problem. 402 Mathematically, the problem with m cities can be expressed 403 as: represents the distance between 407 the i th city and (i + 1) th city, which is calculated as where (x i , y i ) and (x i+1 , y i+1 ) are the coordinate of the i th city 410 and (i + 1) th city.

413
The CPA divides the population into several subgroups 414 according to the fitness of individuals. In each subgroup, 415 carnivorous plants and prey are applied to guide the pop-416 ulation to explore the solution space in various directions, 417 so that the algorithm has strong global searchability. The 418 optimal individual is allowed to reproduce in the reproduction 419 stage, which helps the convergence speed of the algorithm. 420 However, CPA is real-coded, and the solutions may with 421 decimal and repetitions, which are infeasible for solutions of 422 TSP. Therefore, it is necessary to find a suitable decoding 423 method to map continuous variables as legal TSP paths. 424 In addition, to enhance the performance of CPA, several 425 improvements are proposed to effectively balance the explo-426 ration and exploitation capabilities, and further improve the 427 convergence speed and solution quality. These improvements 428 will be discussed in the following subsections.   population size, m is the city size, a = 0, b = m. The variables 434 can be randomly and uniformly generated between a and b. 435 Therefore, the i th individual X 0 i in the initial population can 436 be initialized as Step 1: A uniformly distributed random integer µ is gener-463 ated between [1, m], µ is set as a home city;

472
Step 3: The minimum value in the vector Tp is determined, 473 and take the city i corresponding to the minimum value as the 474 next city to be visited;
Step 2 and Step 3 are repeated until the 476 order of visits for all cities is determined.

477
To facilitate an understanding of HDM, city size m = 5, 478 µ = 2, and X k = (1.3575, 4.5155, 3.4863, 0.7845, 0.0637) 479 are set as an example, the distance between each city is 480 defined in Table 1. The main steps of the HDM are shown 481 in Table 2.

485
It can be seen from (10) that Tp is both related to the dis-486 tance between cities and continuous variables of individuals. 487 For TSP, if the distance between the city i (i = 1, 2, . . . , m − 488 1) and city i+1 is small, the probability that the route could be 489 small will be increased, and the decoding integrates with the 490 greedy idea of the nearest neighbor, which helps to enhance 491 the solution quality. However, it may result in an increment in 492 the probability of premature convergence, thus the continuous 493 variables are incorporated in decoding. Therefore, the HDM 494 applies in TSP helps to improve the solution quality and 495 maintain population diversity.

497
The attraction probability γ of prey by carnivorous plants 498 is constant at 0.8 in CPA. However, the concentration of 499 scent released by carnivorous plants decreases with distance. 500 Therefore, the closer the distance between prey and carnivo-501 rous plant, the greater the attraction and vice versa.

502
Thus, an adaptive attraction probability γ based on dis-503 tance and number of cities is proposed, which is calculated 504 as: where m is the city number, r is the distance between xp i and 507 p iv , r is calculated as where n is the number of carnivorous plants, n 1 is the number 512 of prey, xp i is the carnivorous plant in group i, p iv is the v th 513 prey in group i, xp j i and p j iv are the j th components of xp i and 514 p iv , respectively. 515 VOLUME 10, 2022  Therefore, the prey is selected to update with high probability 529 and the exploration ability of the CPA is strong in the early 530 stage, the carnivorous plant is selected to grow with high 531 probability and the exploitation ability is strong in the late 532 stage.

533
The value of α is related to the search space of CPA, when 534 α is close to 0, the xp ip iv and p iup iv do not work at all, 535 and the global search ability of the algorithm becomes weak. 536 To address the above problems, the IGMOCP & IPUMOP are 537 proposed as follows: where ⊗ represents multiplying the variables at the same 543 position in two vectors, xp i is the carnivorous plant in group 544 i, xp 1 is the best individual in the population, p iv , p iu , p iw are 545 the v th , u th , w th prey in group i, respectively. rand (1, m) is an 546 m dimensional random vector in the range [0.2, 1], rand1(1, 547 m) is an m dimensional random vector in the range [−1, 1], gr 548 is the growth rate, m is the city size, r in (13) is the distance 549 between xp i and p iv , r in (14) is the distance between p iv and 550 p iu , t max is the maximum runtime, t is the current runtime.

556
It can be observed from (13) that the attractiveness of 557 the optimal carnivorous plant to the prey is added in the 558 carnivorous plant growth model, which helps to make the prey 559 move to the potential direction of the search space, improve 560 the probability of the excellent offspring, and strengthen the 561 exploitation ability. From (14), it can be known that the 562 attractiveness of the carnivorous plant to the prey in the same 563 group is added in the prey position update method, which 564 not only helps to enhance the exploration ability but also 565 improves the probability of the excellent offspring generated. 566 An optimization problem in 2 dimensions is taken as an 567 example to compare the difference between the basic and 568 improved update methods. Suppose the best carnivorous plant 569 xp 1 = (1.7, 2) T , the carnivorous plant in i th group xp i = 570 (1, 1) T , the prey in i th group p iu = (0.2, 0.2) T , p iv = (2.2, 571 0.5) T , and p iw = (0.1, 0.1) T , 1000 numbers of α in (1) 572 and (3) are randomly generated, 1000 numbers of α and σ 573 in (13) and (14) are generated according to (15) and (16), 574 respectively. The individuals' distribution in the search space 575 obtained according to (1) and (13) is shown in Fig. 3(a) and 576 Fig. 3(b), respectively. The individuals' distribution in the 577 search space obtained according to (3) and (14) is shown in 578 Fig. 4(a) and Fig. 4(b), respectively.

579
As is depicted in Fig. 3, the range of the abscissa of the 580 offspring produced by (1) Fig. 3(a), the number of individuals near the optimal 586 individual is increased in Fig. 3(b). In Fig. 4, the range of the 587 abscissa of the offspring produced by (3) Fig. 4(a), the number of indi-593 viduals near the carnivorous plant is increased in Fig. 4(b). 594 The analyses above show that the number of outstanding 595 offspring increases and the search space is expanded with the 596 IGMOCP & IPUMOP.

598
Every carnivorous plant can prey and absorb nutrients for 599 growth and reproduction in real-life. However, CPA only 600 allowed the best carnivorous plant to reproduce, which is 601 inconsistent with the law in nature. Besides, the range of β 602 in CPA is [0, 1.8], when β is close to 0, the xp i − xp j does 603 not work at all, then the reproduction is difficult to generate 604 excellent offspring, and the algorithm is prone to stick in 605 the local optimal. In response to the above problems, the 606 reproduction strategy is improved as each carnivorous plant 607 is allowed to reproduce, and the reproduction of the best 608 carnivorous plant is different from that of other carnivorous 609 plants. The IRS is as follows

621
The analysis of IRS shows that (17) has more exploitation 622 ability than (18), when the reproduction method with strong 623 exploitation capability is executed, the CPA can effectively 624 balance the exploration and exploitation abilities. When the 625 reproduction method with exploration capability is executed, 626 the CPA can decrease the probability of falling into the local 627 optimum.

629
The neighborhood 2-opt exchange and double-bridge 630 exchange are employed as perturbation methods in this paper 631 to find better individuals around nn * rr best individuals locally 632 (nn is the population size and rr is the selection ratio), and 633 the local search algorithm 2-Opt is adopted to improve the 634 quality of neighborhood solutions, if the new individual is 635 better, it will replace the original solution. To avoid expensive 636 computing, the maximum number of neighborhood solutions 637 is limited to 10 in this paper, and the nn * rr best individuals 638 VOLUME 10, 2022   The main steps are as follows:

665
Step 1: City a is randomly selected from m cities, a is the 666 center of the circle with radius r 1 as the neighborhood, which 667 is recorded as U (a, r 1 ). The calculation of r 1 is shown in (19). 668 where Z is the path length of the best individual in the 670 population.

749
The implementation method of the two operators is as Therefore, the algorithm plays a higher role in the early stage 762 than in the later phase, and the time complexity of 2-Opt is 763 O(n 2 ), thus, an adaptive probability P is designed, which is 764 calculated as where t max is the maximum runtime and t is the current 767 runtime.

768
The implementation method of the combination exchange 769 strategy is as follows: 1) ε is randomly and uniformly dis-770 tributed generated in the range [0, 1]; 2) if ε < P, the 771 2-Opt algorithm is executed; otherwise, do not execute the 772 algorithm.

773
As is shown in (21), the P is decreased with the iteration 774 time. Thus, the 2-Opt algorithm is executed in the early 775 stage with a higher probability, which helps to eliminate the 776 crossed path and significantly improve the solution quality, 777 and the algorithm is executed in the late phase with a smaller 778 probability, which helps to reduce the complexity of the 779 algorithm.

780
The adaptive combination perturbation and local search 781 algorithm are recorded as ACPLS, and the pseudo-code of 782 ACPLS is presented in Algorithm 2.

783
Algorithm 2 ACPLS Input: nn * rr best individuals X , the maximum number of neighborhood solutions are 10; Output: New nn * rr individuals X ; 1: For i = 1: nn * rr 2: If rand < P s 3: Execute 2-opt exchange on X i , record the new individuals as X i ; 4: elseif 5: Execute double-bridge exchange on X i , record the new individuals as X i ; 6: End 7: If rand < P 8: Execute 2-Opt algorithm on X i , record the new individuals as XX; 9: End 10: Calculate the fitness of XX, return the best XX to X i ; 11: End For The CPA has strong exploration ability, and ACPLS 784 exhibits strong exploitation ability. A hybrid algorithm inte-785 grating the ACPLS strategy into CPA can make a balance, 786 which can promote the convergence speed and enhance the 787 solution quality. The flowchart of CPA-HDM is depicted in Fig. 8. It can be 790 seen that CPA-HDM mainly consists of classification group-791 ing, growth phase, reproduction phase, recombination phase, 792 and ACPLS. Firstly, nn individuals are randomly generated 793 with (9); Secondly, the classification and grouping phase is 794 employed in Section II(B-1); Thirdly, the improved growth 795 in the search process of the algorithm, CPA-HDM can both 817 possess exploration and exploitation capabilities.

818
The pseudo-code of CPA-HDM is shown in Algorithm 3. 819 Algorithm 3 CPA-HDM Input: population size: nn; the population size of carnivorous plants xp: n, the population size of preys p: n 1 , Maxruntime; iteration times: t; I ; rr Output: Best solution and the optimal value; 1: Randomly generate nn initial individuals by (9); 2: Do HDM in nn individuals to map the continuous variables into discrete ones, the detail is depicted in Section III(B); 3: Calculate the fitness value and sort from small to large based on the fitness value; 4: While runtime < Maxruntime 5: t = t + 1; 6: Set the n best individuals as xp, the remaining n 1 individuals as p, and sort a group as depicted in Section II(B-1));

7:
Newxp and Newp are updated by (13) -(14); 8: Newxp is updated by (17) -(18); 9: Combined Newxp, Newp, and xp as a new population, which is recorded as A; 10: Calculate the fitness value of A, and select nn best individuals, which is recorded as B; 11: If t = 1 12: Select rr * nn best individuals from B and record as C; 13: elseif t mod I = 0 14: Select rr * nn best individuals from B and record as C; 15: End if 16: Do Algorithm 2on C; 17: Combined B and C, and select nn best individuals to continue iteration; 18: Record the running time; 19: End while 20: Output the shortest route and its length;

820
To measure the performance of CPA-HDM and its improve-821 ments, three sets of experiments were produced in this study. 822 The first set of experiments verifies the effectiveness of the 823 HDM, ACPLS, AAP, IGMOCP& IPUMOP, and IRS; the 824 second set of experiments applies to determine the optimal 825 parameters combination of n, n 1 , rr, and I ; the third set of 826 experiments discusses the superiority of CPA-HDM.  Table 3. where BKS is the theoretical optimal solution of the instance.

852
Friedman test [50], [51] is adopted in this paper to evalu-853 ate whether significant differences exist among participating 854 algorithms. The results are computed using the following 855 process.

859
Step 2: For the i th algorithm, the average rank of all 860 instance R i is calculated as:

862
Step 3: The R i of l algorithms are sorted from small to 863 large, and the final rank of l algorithms from 1 to l is obtained.

864
Step 4: Under the null hypothesis, the l participating algo-865 rithms perform similarly, The Friedman statistic χ 2 is com-866 puted as: 868 Step 5: The χ 2 α(l−1) is checked from the chi-square distri-869 bution table with the significance level α and k − 1 degrees 870 of freedom. If χ 2 > χ 2 α(l−1) , the null hypothesis (H 1 ) is 871 accepted, and the l participating algorithms are significantly 872 different; otherwise, hypothesis (H 0 ) is accepted, and the l 873 participating algorithms are similar.

874
To better verify the difference between the involved algo-875 rithm, Iman and Davenport [51] presented a better statistic 876 F ID , which is calculated as: where n represents the number of benchmark instances, and 879 l represents the number of participating algorithms.  The Friedman tests only can detect significant differences 886 over the whole multiple comparisons, being unable to find 887 the concrete pairwise comparisons which produce significant 888 differences. Thus, if the Friedman test shows that significant 889 differences exist in l algorithms, the post hoc test needs to 890 be employed to find out the concrete pairwise comparisons 891 which produce significant differences. Holm's procedure is 892 adopted in this paper and it can be divided into multiple com-893 parisons with a control algorithm and multiple comparisons 894 among all algorithms [52]. 895

1) MULTIPLE COMPARISONS WITH A CONTROL ALGORITHM 896
The significant difference of the control algorithm will be 897 contrasted against the rest of the l − 1 participating algo-898 rithms in this situation. Suppose the control algorithm is the 899 algorithm1 (Al 1 ), the adjusted p-value between Alg 1 and v th 900 algorithm (Al v ) is recorded as the APV v (v is the rank value 901 corresponding to the p-values sorted from small to large, 1 ≤ 902 v ≤ l − 1, the p-value of each hypothesis obtained through 903 the conversion of the results by the Friedman rank test by 904 adopting a normal approximation [53]), Holm's procedure 905 determines whether the two algorithms are significant by 906 comparing the APV v and the significance level α. If APV v < 907 α, the Al1 and Al v are significantly different. The APV v is 908 97154 VOLUME 10, 2022 calculated as follows:

913
In this situation, the significant difference of each algorithm 914 will be contrasted against the rest of the l − 1 algorithms 915 participating in the comparison, the possible pairwise com-916 parison between algorithms is M , and M = l * (l − 1) / 2. The 917 algorithm x is recorded as Al x , the algorithm y is recorded 918 as Al y (1 ≤ x ≤ l,1 ≤ y ≤ l). Suppose the rank of 919 the p-value sorted from small to large between Al x and Al y 920 among all pairwise comparisons is v (1 ≤ v ≤ M ), and the 921 adjusted p-value between Al x and Al y is recorded as APV v . 922 If APV v < α, the Al x and Al y are significantly different. The 923 APV v is calculated as follows:  that method is contained in the algorithm, and N repre-  The ICPA, Variant 1, and Variant 2 are adopted with 20 950 instances from TSPLIB to compare the performance of HDM. 951 To achieve a fair comparison, the same parameters are set in 952 the participating algorithms. The population size nn = 120, 953 the number of carnivorous plants n = 20, and the number of 954 carnivorous prey n 1 = 100. The performance of HDM and 955 the other two decoding methods considered for comparison 956 are displayed in Table 5.

957
It can be observed from Table 5 that   The performance of CPA-HDM is related to the number of 995 carnivorous plants n, the number of prey n 1 , the proportion rr 996 of the population to execute ACPLS, and the individuals per-997 forming the ACPLS are re-selected from the population every 998 I iterations. Therefore, to determine the optimal parameters 999 combination of n, n 1 , rr, and I , the orthogonal experiment in 1000 Table 8 is designed.

1051
The results of Friedman statistic χ 2 and F ID are shown in 1052 Table 13. The value of χ 2 is 107.83, and F ID is 48.39. When 1053 the degrees of freedom l − 1 = 6 and the significant level 1054 α = 0.05, the critical value of χ 2 α [6] is 12.59, and the critical 1055 value of F (6,162) at (l − 1)(n − 1) = 162 is 2.16. It can be 1056 observed that χ 2 > χ 2α [6], F ID > F (6,162) at α = 0.05, 1057 which verifies that the differences among the algorithms are 1058 significant, and the outperformance of CPA-HDM has been 1059 confirmed.

1060
For further statistical analysis, Holm's procedure is 1061 employed to evaluate the practical difference between 1062 CPA-HDM and the other six algorithms at a 95% confi-1063 dence level. The unadjusted and adjusted p-values returned 1064 is significantly different from all participating algorithms.   Table 15, and the graphical representation 1073 of the convergence analysis is depicted in Fig. 13. The T avg 1074 in Table 15 denotes the average time of t av , the x-axis in  Table 15 shows that CPA-HDM wins on 8 instances, 1078 whereas DSFLA performs better on att48 and pr107, DBAL 1079 performs better on pr124 and pr264, ABC performs better 1080 on ch130 and kroA150, and PACO-3Opt performs better 1081 on kroA100 and kroB150. The T avg of CPA-HDM achieved 1082 the shortest among seven algorithms, which means that the 1083 proposed algorithm exhibits superior performance.

1084
The figures depicted in Fig. 12 also confirm the efficiency 1085 of the CPA-HDM, as it achieved the fastest convergence 1086 speed among various comparison algorithms on six instances. 1087 Although CPA-HDM converges slower than D-GWO on six 1088 instances and DSFLA on pr299 in the early stage, it could 1089 keep the convergence speed at the highest level among the 1090 involved algorithms in the middle and later iteration for its 1091 balance the exploration and exploitation ability. He is currently a Researcher with the 1377 Engineering College, Northeast Agricultural Uni-1378 versity. He published over ten papers in domestic 1379 and international academic journals and confer-1380 ence proceedings. His current research interest 1381 includes intelligent agricultural equipment and technology.