## He Huang* , Min Zhu** and Jin Wang***## |

F | Mathematical formula | Range | Optimal value |

f1 | [TeX:] $$\sum_{i=1}^{n} x_{i}^{2}$$ | [-100,100] | 0 |

f2 | [TeX:] $$\sum_{i=1}^{n} i^{*} x_{i}^{2}$$ | [-100,100] | 0 |

f3 | [TeX:] $$\sum_{i=1}^{n} i x_{i}^{4}+r a n d o m[0,1)$$ | [-1.28,1.28] | 0 |

f4 | [TeX:] $$\sum_{i=1}^{n}\left|x_{i}\right|+\prod_{i=1}^{n}\left|x_{i}\right|$$ | [-10,10] | 0 |

f5 | [TeX:] $$\max \left(\left|x_{1}\right|,\left|x_{2}\right|, \ldots,\left|x_{n}\right|\right)$$ | [-100,100] | 0 |

f6 | [TeX:] $$\sum_{i=1}^{n}\left(\sum_{j=1}^{l} x_{j}\right)^{2}$$ | [-100,100] | 0 |

f7 | [TeX:] $$10^{6 *} x_{1}^{2}+\sum_{i=2}^{n} x_{i}^{2}$$ | [-100,100] | 0 |

In this subsection, this paper conducts experiments for the purpose of comparing the proposed ABCSI’s competitiveness with other evolutionary algorithms. In this comparison, the conventional PSO, DE, ABC algorithm and five ABC variants, including GABC, qABC [6], OCABC, ABCVSS [7] as well as ABCM [8] are adopted.

In this comparison, the population size is set to 40 and all the algorithms are repeated for 50 trails in order to be justice. Experiments are conducted on 30 dimensions, with the corresponding maximum iteration set as 3,000. For the maximum function evaluation number, it is set as the product of population size and maximum iteration number. When the maximum function evaluation number is achieved, the algorithm stops, and after 50 trials their final average fitness values (the first row) and standard deviations (the second row) result on these fourteen benchmarks are recorded, as presented in Table 2. Besides, the Wilcoxon rank-sum test [10] is also conducted to demonstrate the statistical effectiveness. In this measurement, the significant value is set to 5%. The symbol “=” means the proposed ABCIS algorithm attains results which are similar to the corresponding compared algorithm, while “+” means the corresponding algorithm exhibits better performance than ABCIS and “–” means worse. This outcome is also provided in Table 2. Further, Fig. 2 gives the convergence curves of these nine evolutionary algorithms on some typical benchmarks.

The results intuitively indicate that the proposed ABCIS algorithm exhibits best optimization performance almost on all the proposed benchmarks and dimensions. For the unimodal functions, the proposed ABCIS algorithm obtains the optimal value on several benchmarks and achieves best accuracy on other problems, which means that the two elite vectors guiding could better perfect the convergence speed. This can also be verified in the convergence curves presented in Fig. 2. For the multimodal functions, the proposed ABCIS algorithm also achieves great results, which may contribute to the update strategy of global best food source. Further demonstration will be discussed in the following subsection.

Table 2.

F | PSO | DE | ABC | GABC | ABCIS | |

f1 | Avg. fitness value | 2.05E-17 | 1.88E-47 | 5.29E-16 | 3.83E-16 | 0 |

Standard deviation | 5.06E-17 | 4.89E-47 | 7.13E-17 | 9.45E-17 | 0 | |

f2 | Avg. fitness value | 1.68E-16 | 1.28E-46 | 5.22E-16 | 3.98E-16 | 0 |

Standard deviation | 2.08E-16 | 1.67E-46 | 9.79E-17 | 8.55E-17 | 0 | |

f3 | Avg. fitness value | 2.20E-02 | 5.55E-03 | 3.97E-02 | 1.99E-02 | 2.76E-04 |

Standard deviation | 8.44E-03 | 1.68E-03 | 1.08E-02 | 5.98E-03 | 1.20E-04 | |

f4 | Avg. fitness value | 6.65E-13 | 1.96E-25 | 1.29E-15 | 1.21E-15 | 0 |

Standard deviation | 5.57E-13 | 1.93E-25 | 1.28E-16 | 1.47E-16 | 0 | |

f5 | Avg. fitness value | 2.20E-02 | 5.55E-03 | 3.97E-02 | 1.99E-02 | 2.76E-04 |

Standard deviation | 8.44E-03 | 1.68E-03 | 1.08E-02 | 5.98E-03 | 1.20E-04 | |

f6 | Avg. fitness value | 6.76E-17 | 1.02E-47 | 5.00E-16 | 3.69E-16 | 0 |

Standard deviation | 1.67E-16 | 1.24E-47 | 8.35E-17 | 7.14E-17 | 0 | |

f7 | Avg. fitness value | 3.63E+00 | 9.28E+00 | 2.19E+00 | 2.45E-01 | 5.79E-02 |

Standard deviation | 1.07E+00 | 6.66E+00 | 6.04E-01 | 5.08E-02 | 7.29E-02 |

In this subsection, we aim at verifying each component of ABCIS’s effectiveness. The total dimension update strategy is considered. For experimental settings, the population size and the test dimension is chose as 40 and 10, respectively. Each trial is repeated for 50 times and the average fitness value (the first row) and standard deviations (the second row) is recorded.

In the proposed ABCIS algorithm, food sources except the best one are updated on total dimension at each generation. To testify this component’s effectiveness, this subsection designs another ABC algorithm name as ABCIS_sd (ABCIS with single dimension) to perform their comparison. Comparison results are stated in Table 3, from which it could be observed that total dimension update strategy could achieve much better solution accuracy on unimodal functions.

In order to overcome the defects of the slow convergence of the conventional artificial bee colony algorithm, this paper first proposes an intellective search strategy for bees searching food source, and then to compensate the intellective search strategy’s drawback, we introduce a special division for the global best food source, which accelerate the solution accuracy. Experimental results show that the ABCIS algorithm proposed in this paper has made great improvements on both unimodal and multimodal functions.

He received B.S. degree from Hangzhou Dianzi University, China, in 2001 and M.S. degree from Hangzhou Dianzi University, China, in 2006. Now, he works at Department of Information Technology, Wenzhou Vocational & Technical College, China as associate professor. His research interests mainly include optimization algorithm design and artificial intelligence.

She received B.S. degree from Nanjing University of Posts and Telecommunications, China in 2002, M.S. degree from Beijing University of Posts and Telecommunications, China in 2005, and received Ph.D. degree in Nanjing University of Posts and Telecommunications, China in 2018. Now, he works at College of Information Science and Technology, Zhejiang Shuren University as lecturer. Her research interests mainly include routing protocol and optimization algorithm design.

He received the B.S. and M.S. degrees from Nanjing University of Posts and Telecommunications, China in 2002 and 2005, respectively. He received Ph.D. degree from Kyung Hee University Korea in 2010. Now, he is a professor in the College of Information Engineering, Yangzhou University. His research interests mainly include routing algorithm design, performance evaluation and optimization for wireless ad hoc and sensor networks. He is a Member of IEEE and ACM.

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