Numerical comparisons of migration models for Multi-objective Biogeography-Based Optimization
Introduction
Nowadays, Artificial Intelligence (AI) plays a very active role in many areas including science, engineering, finance, and even our daily lives [8], [9], [19], [39], [49], [51]. A hot issue in AI is to solve optimization problems by intelligent approaches. In practice, Multi-Objective Problems (MOPs) are very common, and therefore many researchers do a lot of work on this topic. In MOPs, more than one objectives are to be optimized simultaneously and in general the objectives are coupled or conflicting [2], [21], [23], [52]. To solve MOPs, it is crucial to find true Pareto front (PF) composed by trade-off solutions. Since Evolutionary Algorithms (EAs) are very skilled in approximating the whole true PF in a single run, it gains popularity in dealing with MOPs. The first real application of Evolutionary Algorithms (EAs) in finding multiple trade-off solutions was suggested and worked in 1984 [31], where Multi-objective Genetic Algorithm (MOGA) was proposed. After that, increasing kinds of Genetic Algorithms for MOPs were proposed such as Non-dominated Sorting Genetic Algorithm (NSGA), Niched-Pareto Genetic Algorithm (NPGA), NSGA-II and so forth [11], [18], [37]. Besides, up to now, many classic EAs are designed to solve MOPs, including Pareto-frontier Differential Evolution (PDE), Non-dominated Neighbor Immune Algorithm (NNIA), Multi-Objective Particle Swarm Optimization (MOPSO), Multi-Objective Simulated Annealing (MOSA) and so on [1], [14], [28], [30], where all achievements demonstrate that MOEAs are feasible and effective in dealing with MOPs.
With a rapid development of EAs, novel algorithms inspired from nature phenomena are proposed in current years. By mimicking species’ migration and mutation in the science of biogeography, Biogeography-Based Optimization (BBO) was proposed in 2008 [33]. According to experiment studies and applications in [16], [24], [33], BBO is very competitive in optimization. Besides, in [36], to investigate the performances of BBO, Simon compared this algorithm with GA using Markov analysis and emphasized the specialties of BBO [35]. He also proved, in a mathematical way, BBO outperforms Genetic Algorithm (GA) for some problems. In our previous work [16], Guo revealed the effects of migration models to BBO’s performance using probability analysis and the conclusions are useful to design migration models. In addition to theoretical analysis, BBO has also been well applied to many areas including network, power system, mechanical design and so forth [3], [5], [6], [7], [15]. However to date, most researches of migration models focus on SOPs but few on MOPs. Considering that MOP is more common in practice, it is necessary to extend our previous work [16] to carry out the research on BBO’s migration models for MOPs.
In this paper our contributions are given as follows. First, we propose a multi-objective BBO (MOBBO) and demonstrate its superiority in dealing with MOPs by numerical simulations on standard benchmarks. Second, though the comparisons of migration models for SOPs have been conducted in our previous work [16], there are a few investigations about migration models for MOPs. In this paper we find that the conclusion for SOPs does not hold for MOPs. The migration models that perform well for SOPs may lose for MOPs. Based on the previous migration models [16], we extend them to solve MOPs and reevaluate their impacts to MOBBO’s performance.
The remainder of this paper is organized as follows. Section 2 generally illustrates the multi-objective optimization. In Section 3, after a short overview of BBO, we propose Multi-objective BBO by constructing non-dominated solution set. Meanwhile six migration models are proposed in this section. In Section 4, two sets involving total 12 classic MOPs with low and high dimensions are used to investigate the performances of the proposed migration models. The analysis, comparisons and discussions are presented in this section. Besides, we also compare MOBBO with several other kinds of popular MOEAs and the results validate the superiority of MOBBO. We conclude this paper in Section 5 and present future work.
Section snippets
Brief of multi-objective optimization
A multi-objective optimization problem has two or more objectives which are to be minimized or maximized simultaneously. Without loss of generality, we only consider minimization problem in this paper since a maximization problem is equivalent to minimize its negative. In mathematical terms, a multi-objective optimization problem can be formulated as shown in (1), where Ω is the decision space and x ∈ Ω is a decision vector. F(x) consists of m objective
Overview of Biogeography-Based Optimization
Biogeography-Based Optimization mimics the species distribution in nature biogeography. In the science of biogeography, a criterion to judge whether geographical areas are well suited as residences for biological species is habitat suitability index (HSI), which refers to the factors including climate, temperature, humidity and topographic features. All the factors that characterize habitability are called suitability index variables (SIVs). Fertile areas with a high HSI have a large number of
Benchmark functions
In this section, we have selected two sets of benchmarks for simulations. The first set of benchmarks are ZDT problems shown in Table 1 where the benchmarks are defined in [10]. By combining the initials of authors family names (E. Zitzler, K. Deb and L. Thiele), the benchmark set is named ZDT. The second set concludes MOPs benchmarks shown in Table 2 which are defined in [41].
The ZDT set is a family of bi-objective MOPs which have been widely used to assess the performance of meta-heuristics
Conclusions
In this paper, Multi-objective Biogeography-Based Optimization is proposed with six migration models. To investigate the migration models’ performances, 12 classic benchmarks are employed. By analyzing the simulation results, we draw the conclusions that: (1) emigration rate and immigration rate play different roles in optimization. According to our previous work, for SOPs, constant emigration rate or constant immigration rate will lead to an inferior performances. However, constant emigration
Acknowledgment
This work was sponsored by the National Natural Science Foundation of China under Grant no. 61503287, No. 71371142 and No. 61203250, Program for Young Excellent Talents in Tongji University (2014KJ046), Â Program for New Century Excellent Talents in University of Ministry of Education of China, Ph.D. Programs Foundation of Ministry of Education of China (20100072110038). The first author Weian Guo would express his thanks to Jiali Wang, his beloved wife, for her continuous strong support.
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