## Zhanxiang Ye* , Min Zhu** and Jin Wang***## |

F | Formula | Range | Optimal value |

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

f2 | [TeX:] $$10 ^ { 6 } * x _ { 1 } ^ { 2 } + \sum _ { i = 2 } ^ { n } x _ { i } ^ { 2 }$$ | [-100, 100] | 0 |

f3 | [TeX:] $$\sum _ { i = 1 } ^ { n } \left[ x _ { i } ^ { 2 } - 10 \cos \left( 2 \pi x _ { i } \right) + 10 \right]$$ | [-5.12, 5.12] | 0 |

f4 | [TeX:] $$\frac { 1 } { 4000 } \sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } - \prod _ { i = 1 } ^ { n } \cos \left( \frac { x _ { i } } { \sqrt { i } } \right) + 1$$ | [-600, 600] | 0 |

f5 | [TeX:] $$\left( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } \right) ^ { 0.25 } \left\langle \left\{ \sin \left[ 50 \left( 3 x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } \right) ^ { 0.1 } \right] \right\} ^ { 2 } + 1 \right\rangle$$ | [-100,1000] | 0 |

To highlight the effectiveness and competitiveness of our proposed MABC algorithm, comparison experiments will be carried out in this subsection. The traditional PSO, DE, ABC, as well as four ABC variants, including the STOC-ABC [5], dABC [6], distABC [7] and NSABC [8], are selected as the comparison algorithms.

In this experiment, the population size is set as 40. For the first ten benchmarks presented in Table 1, all the algorithms are tested on 10 dimensions. For the last benchmark, comparison algorithms are tested on 2 dimensions. The corresponding maximum iteration number and function evaluation number are presented in Table 2. To be fair, all the trials are repeated 50 times, and the corresponding mean fitness value (the first row) and the standard deviation (the second row) are recorded, as presented in Tables 3–4.

Table 2.

Dimension | Maximum iteration number | |

2 | 100 | 4,000 |

10 | 1,000 | 40,000 |

Table 3.

F | PSO | DE | ABC | STOC-ABC | dABC | distABC | NSABC | MABC |

f1 | 3.44E-25 | 1.67E-49 | 1.04E-16 | 1.45E-41 | 3.59E-52 | 4.12E-76 | 3.80E-80 | 1.38E-149 |

(8.09E-25) | (2.46E-49) | (3.47E-17) | (3.51E-41) | (6.72E-52) | (1.19E-75) | (1.13E-79) | (6.14E-149) | |

f2 | 5.82E-25 | 6.75E-49 | 1.38E-16 | 4.58E-38 | 7.34E-48 | 2.46E-74 | 9.29E-78 | 2.08E-147 |

(1.25E-24) | (1.45E-48) | (5.62E-17) | (1.40E-37) | (3.08E-47) | (7.98E-74) | (3.23E-77) | (9.27E-147) | |

f6 | 2.74E+00 | 1.14E+01 | 0 | 5.33E-16 | 0 | 0 | 0 | 0 |

(1.53E+00) | (2.87E+00) | (0) | (2.38E-15) | (0) | (0) | (0) | (0) | |

f7 | 9.51E-02 | 8.43E-02 | 1.03E-02 | 9.04E-03 | 1.01E-02 | 9.79E-03 | 1.07E-02 | 9.70E-03 |

(4.65E-02) | (6.87E-02) | (3.19E-03) | (1.64E-03) | (2.64E-03) | (2.03E-03) | (4.08E-03) | (2.74E-03) |

Table 4.

F | PSO | DE | ABC | STOC-ABC | dABC | distABC | NSABC | MABC |

f5 | 8.05E-10 | 6.92E-24 | 1.12E-17 | 2.09E-20 | 1.37E-25 | 1.40E-49 | 2.08E-26 | 6.00E-76 |

(1.51E-09) | (2.10E-23) | (1.17E-17) | (5.91E-20) | (5.38E-25) | (2.72E-49) | (8.35E-26) | (1.80E-75) |

From the above comparison, it could be easily observed that the proposed MABC algorithm achieves better performance on most cases. For the multimodal functions, as demonstrate in the ABC variants’ experiments, ABC exhibits stronger global search ability than PSO and DE. For the proposed MABC algorithm, it also achieves similar of better outcome than other ABC variants on multimodal functions. For the unimodal functions, MABC obtains obviously improvement compared to the traditional artificial bee colony, and also shows better performance than other comparison algorithms. For the last 2-dimensional test functions, the proposed MABC algorithm attains the best solution accuracy among all the algorithms. All these experimental results demonstrate that the decrease of employed bees’ number does not cripple the exploration ability and the increase of onlooker bees’ number improves the exploitation capability. And the successful performance on low dimensional functions indicates that the proposed MABC algorithm is not limited by problem’s dimension. Besides, for the unimodal functions, except for the solution accuracy presented in the results table, another important evaluation criterion is the convergence speed. Thus we plot the convergence curves of *f1*, as shown in Fig. 2. *f2* is a sphere function, the most representative unimodal function, the convergence speed is the main evaluation criterion for it.

Regarding the insufficiency of traditional ABC algorithm, in this paper, we proposed a modified artificial bee colony to accelerate the convergence speed and enhance the local search capability. Two modifications were executed. One of them is to reallocate the employed bees and onlooker bees’ number, and the other one is to perfect the search equation. Experimental results demonstrated the superior performance of our proposed MABC algorithm from the perspective of both theory and practicability.

She received B.S. degree from Nanjing University of Posts and Telecommunications, China in 2002 and M.S. degree from Beijing University of Posts and Telecommunications, China in 2005. Now, she is working toward the Ph.D. degree in Nanjing University of Posts and Telecommunications, China. Her research interests mainly include routing protocol and optimization algorithm design.

He received the B.S. and M.S. degree 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 School of Computer Communication Engineering, Changsha University of Science Technology. 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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