Computational Intelligence and Neuroscience

Volume 2017 (2017), Article ID 6153951, 19 pages

https://doi.org/10.1155/2017/6153951

## AMOBH: Adaptive Multiobjective Black Hole Algorithm

^{1}School of Automation, China University of Geosciences, Wuhan 430074, China^{2}Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems, Wuhan 430074, China^{3}State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430070, China

Correspondence should be addressed to Tao Wu

Received 4 June 2017; Revised 1 October 2017; Accepted 22 October 2017; Published 23 November 2017

Academic Editor: José Alfredo Hernández-Pérez

Copyright © 2017 Chong Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes a new multiobjective evolutionary algorithm based on the black hole algorithm with a new individual density assessment (cell density), called “adaptive multiobjective black hole algorithm” (AMOBH). Cell density has the characteristics of low computational complexity and maintains a good balance of convergence and diversity of the Pareto front. The framework of AMOBH can be divided into three steps. Firstly, the Pareto front is mapped to a new objective space called parallel cell coordinate system. Then, to adjust the evolutionary strategies adaptively, Shannon entropy is employed to estimate the evolution status. At last, the cell density is combined with a dominance strength assessment called cell dominance to evaluate the fitness of solutions. Compared with the state-of-the-art methods SPEA-II, PESA-II, NSGA-II, and MOEA/D, experimental results show that AMOBH has a good performance in terms of convergence rate, population diversity, population convergence, subpopulation obtention of different Pareto regions, and time complexity to the latter in most cases.

#### 1. Introduction

Most problems can be considered as multiobjective optimization problems (MOPs) [1, 2] in the fields of social production [3], engineering design [4], path planning [5], product design [6], motor design [7], mechanics design [8], and so forth. For example, to design a new toll plaza of a highway, at least two objectives should be considered, which are traffic efficiency and land cost. Hence, the study of MOPs is very meaningful in many research fields. Unlike the single-objective optimization problems (SOPs), there are multiple global optimum solutions which are called Pareto optimal solutions in MOPs. And objectives are often conflicting with each other [1]. Some traditional approaches use the weight function to transform MOPs into SOPs. However, they require some prior knowledge and can only get one solution of the Pareto optimal solution set.

Over the past decades, bioinspired computation and swarm intelligence based algorithms have been introduced into solving MOPs called evolutionary multiobjective optimization. Bioinspired computation is motivated by the natural and social behavioral phenomena and can start with a set of initial variables and then evolve to find multiple optima simultaneously [9]. Therefore, bioinspired computation is suitable for solving MOPs. The most popular multiobjective evolutionary algorithms (MOEAs) are Pareto dominance based algorithms [2], such as nondominated sorting genetic algorithm II (NSGA-II) [10] and strength Pareto evolutionary algorithm II (SPEA-II) [11]. Besides the criterion of Pareto dominance, they also adopted a diversity related secondary criterion to promote a good distribution of the solutions [12].

The BH algorithm was first proposed to solve the complex, hard optimization and clustering problem which is NP-hard [13]. It simulates the phenomenon of the black hole absorption. It has evolved from the PSO algorithm with new mechanisms. The BH algorithm searches the entire space of solutions (stars) and finds the global optimum solution (black hole). In the PSO algorithm, particles cannot disappear, which results in a premature convergence problem. To overcome the weakness of the PSO algorithm, in the BH algorithm, a star will be reborn randomly in the search space if it comes too close to the black hole. So, compared with the PSO algorithm, the BH algorithm has a better performance in avoiding the premature convergence phenomenon and a less time complexity than PSOs and GAs. What is more, the BH algorithm has only one controlling parameter which is the radius of the black hole.

A lot of MOEAs have two main problems: (1) high selection pressure or low selection pressure and (2) how to balance the diversity and convergence. Therefore, in this paper, we proposed a new multiobjective evolutionary algorithm which is based on the black hole algorithm (BH algorithm) [13] to solve these problems. A new individual density criterion called cell density is proposed to improve the convergence and diversity of Pareto optimal solutions. In this paper, the global optimum selection strategy is based on the Shannon entropy and will be adjusted adaptively. The Shannon entropy is used to analyze the status of evolution. To calculate the entropy, the approximate Pareto front is mapped to PCCS [14]. The elite mutation strategy referred to in [14, 15] is used for improving the local search ability of the proposed algorithm. We rank solutions in an archive based on the cell density and the concepts of strength of cell dominance [14, 16], respectively. And we introduce adaptive evolution strategies to adjust the elite learning rate and the global optimum selection strategy according to evolution status adaptively. A simulation using seven test problems with different degrees of difficulty which are ZDT1, ZDT3, ZDT4 [17], DLTZ2, DLTZ4, DLTZ5, and DLTZ7 [18], respectively, is demonstrated to investigate the scalability of AMOBH. The simulation results showed that AMOBH has a good performance in both diversity and convergence during solving different multiobjective optimization problems. Furthermore, we perform a comparison with four well-known MOEAs—SPEA-II [11], PESA-II [19], NSGA-II [10], and MOEA/D [20]—for evaluation. Besides, we use the metric called inverted generation distance [20] which can evaluate convergence and diversity at the same time to analyze the results. In order to further verify the diversity or uniformity of the results, we use the metric called spacing [21–23] to corroborate it. As will be shown in this study, the proposed algorithm outperforms the other four algorithms in terms of diversity and convergence in most cases. And it has a good time complexity.

#### 2. Related Works and Motivation

After [24] first introduced vector evaluated genetic algorithms (VEGAs) to solve multiobjective optimization problems, a lot of MOEAs based on GAs have been proposed. Among them, the most representative algorithms are NSGA-II [10], SPEA-II [11], and MOEA/D [20]. And, recently, a lot of MOEAs based on GAs for solving many-objective problems were also proposed [1, 2, 12, 25, 26]. But all of them suffer from a slow convergence rate and a lot of time on generating new offspring, which are the main problems of GAs [27].

Compared with GAs, the particle swarm optimization (PSO) algorithm has the advantages of simplified formula, rather quick convergence rate, global optimization performance, fewer controlling parameters, and so forth. Based on the successful experience in the field of MOEAs using GAs, a multiobjective particle swarm optimization algorithm (MOPSO) was proposed soon [28]. Compared to some previous MOEAs, MOPSO has a faster convergence rate and it can get a rather satisfied and fully covered approximate Pareto front. Reference [29] introduced Shannon entropy [30] to analyze the MOPSO dynamics along the algorithm execution. Reference [14] introduced parallel coordinates in MOPSO to calculate entropy. However, the main problem of PSO algorithm is that it is easily trapped into a local optimum [31].

At present, a lot of MOEAs adopted the global optimum selection strategy based on the nondominated sorting [10, 11] or Pareto dominance [32] or hypervolume [33, 34] or niche [35, 36] and so on. But they all have some problems of high selection pressure or low selection pressure. In evolutionary multiobjective optimization, maintaining a good balance between convergence and diversity is particularly crucial to the performance of the evolutionary algorithms [37]. And most of the MOEAs face the problem of balancing the convergence and diversity. The evaluation of individual density is always based on one metric (such as Pareto dominance or crowding density), which results in the algorithm being unable to take into account both convergence and diversity [14].

Bearing these ideas and motivations in mind, an adaptive multiobjective black hole algorithm is proposed, investigated, and discussed in the following sections.

#### 3. Definitions and Some Concepts

##### 3.1. Multiobjective Optimization Problem

A minimum continuous unconstrained multiobjective optimization problem (in the optimization field, the maximization problems and minimization problems are dual problems) can be defined as follows:in which , is the number of decision variables, is the objective function, is the number of objectives, and is an -dimensional search space.

###### 3.1.1. Pareto Optimality

Normally, objectives are restricted or conflicting with each other [38], which results in the global optimum solution being not unique. There cannot be found a solution that is superior to all. But noninferior solutions exist; the so-called noninferior solutions are solutions that cannot be optimized for at least one objective; at the same time, other objectives will not deteriorate. And these solutions are called Pareto optimal.

Here are some definitions of Pareto optimality.

*Definition 1 (Pareto optimality). *In search space, and are the decision vectors; if the following conditions are satisfied: is said to be Pareto optimal.

*Definition 2 (Pareto dominance). *There are two objective vectors , , where is an -dimensional objective space; if the following conditions are met: is said to dominate .

*Definition 3 (Pareto front). *The set of all Pareto optimal solutions is called Pareto optimal set. The space composed of Pareto optimal objective vectors is called Pareto front.

##### 3.2. Parallel Cell Coordinate System

Parallel coordinates are a common way of visualizing high-dimensional geometry and analyzing multivariate data [39]. Reference [16] proposed the GrEA which mapped the approximate Pareto front to grid coordinates. Inspired by these, [14] proposed a concept called parallel cell coordinate system (PCCS) which mapped the approximate Pareto front to a two-dimensional plane from Cartesian coordinates to integer coordinates. The formula of the transformation is defined as follows:where is a two-dimensional plane grid composed of cells, is a top integral function, , is the number of members in archive in the current iteration , , is the number of objectives to be optimized, and are the minimum and maximum of objective in the current approximate Pareto optimal set, respectively, and is the integral coordinate or integral label of in PCCS.

To rank the solutions in archive, we need to introduce the following definitions [14, 16].

*Definition 4 (cell dominance). *If and are integral coordinates of any two solutions in an archive, at the same time, the following conditions are satisfied:Solution is said to cell-dominate solution .

*Definition 5 (strength of cell dominance). *The total number of solutions which are cell-dominated by solution is said to be the strength of cell dominance of solution .

##### 3.3. Shannon Entropy

In 1949, Shannon et al. in their paper [30] introduced a concept which is called Shannon entropy, which was proposed as a measure of the amount of information that is missing before reception.

Shannon entropy is defined as follows:where is the probability of occurrence of the th possible value of the source symbol, is a positive constant, and is the collection of events .

In this paper, we refer to [14, 29] to calculate as follows:in which is the number of objective vectors which are mapped to PCCS in the cell grid with indexes and , is the number of solutions in archive in the current iteration and it will be changed with the change of the solution’s number in archive, and is the number of objectives.

The update formula of entropy is as follows:

It is a measure to evaluate the status of evolution. It was mentioned in [29] that entropy is able to capture the convergence rate of the algorithm. Entropy represents uniformity and diversity of approximate Pareto optimal solutions. Larger entropy means better uniformity and diversity. The evolution status update algorithm is as shown in Algorithm 1 [14], where and are the number of solutions in archive in iterations and , respectively, and is the size of the archive. The initial entropy is and the initial delta entropy is .