An artificial bee colony algorithm for multi-objective optimisation

Highlights

• Novel meta-heuristic to the multi-objective optimisation problem.

• The multi-objective optimization algorithm compared with other work in the literature.

• The algorithm possesses outstanding performance.

Abstract

In addition to dominance-based and decomposition-based algorithms, performance indicator-based algorithms have been widely used and investigated in the field of evolutionary multi-objective optimisation. This study proposes a multi-objective artificial bee colony optimisation method called $\epsilon \text{-MOABC}$ based on performance indicators to solve multi-objective and many-objective problems. The proposed algorithm develops an external archive on the basis of both Pareto dominance and preference indicators to save the non-dominated solutions produced in each generation. The population of the presented algorithm includes employed bees, onlooker bees, and scout bees. Employed bees adjust their trajectories according to the information provided by other employed bees. Motivated by employed bees, onlooker bees select food sources to update their positions according to a power law probability, with which the food sources with high quality have a high probability to be selected for exploration. The quality of food sources is calculated on the basis of the quality indicator $I_{\epsilon +}$. Scout bees dispose of food sources with poor quality. The proposed algorithm proves to be competitive in dealing with multi-objective and many-objective optimisation problems in comparison with other state-of-the-art algorithms for CEC09, LZ09, and DTLZ test instances.

Introduction

Problems composed of two or three conflicting objectives that must be satisfied simultaneously are called multi-objective optimisation problems (MOPs) [1], and problems with more than three conflicting objectives are called many-objective optimisation problems (MaOPs). MaOPs are common in the real world [2], [3]. As for MOPs, their solution is quite different from that of single-objective problems. When solving MOPs, the goal is to find the best possible trade-off among all the objectives. However, a single-objective problem always has a single solution. Pareto optimality is commonly used to deal with MOPs. The decision variables corresponding to all the solutions in the Pareto front are called the Pareto optimal set. The representative algorithms are Pareto-based algorithms, such as NSGA-II [4] and SPEA2 [5]. Pareto-based multi-objective evolutionary algorithms (MOEAs) are very effective when solving low dimension MOPs because they achieve an approximate set in a single run through their population-based property. However, MOEAs do not perform properly when handling MaOPs because the percentage of non-dominated solutions tends to reach 100% as the number of objectives increases. Thus, identifying the differences between solutions is difficult [6]. Furthermore, maintaining diversity for a high dimensional space is difficult because the similarity in such a space cannot be easily estimated. As a response to these challenges, new algorithms have been developed in recent years.

Given the disadvantages of Pareto-based algorithms in dealing with MaOPs, the intuitive idea is to modify the Pareto concept to augment the selection pressure in a high dimensional space. Some relaxed forms of Pareto dominance, such as $\epsilon \text{-dominance}$ [7] and $\alpha \text{-dominance}$ [8], are better approaches than the original Pareto dominance in terms of adding selection pressure towards the true Pareto front. Thus, $\epsilon \text{-dominance-based}$ MOEA ($\epsilon \text{-MOEA}$ [9]) is proposed; it shows good performance in solving MaOPs. Other strategies of modified dominance definitions include $L\text{-optimality}$ [10], fuzzy dominance [11] and preference order ranking [12]. In [13], a grid-based EA was proposed; a new grid dominance strategy ensures simultaneous convergence and distribution. Another idea is to enhance Pareto dominance by combining it with other convergence-metric approaches. In [14], the crowding distance assignment used in NSGA-II was replaced with substitute distance assignment; the new assignment could measure the degree to which a solution is nearly Pareto-dominated by another solution. In [15], the selection pressure for the solutions far from the Pareto front was increased by using an achievement function to compare the solutions that are close to the Pareto front.

The second type is decomposition-based algorithms; its representative algorithm is MOEA/D [16]. In MOEA/D, each solution in its population is associated with a different subproblem, which works in a collaborative way. Although a number of MOEA/D variants have recently been proposed [17], [18], [19], [20], [21], [22], the study of MOEA/D is still in its infancy. MOEA/D is not specially designed to solve MaOPs, but it is proved to be a good alternative to MaOPs.

The third class is performance indicator-based designs. The individual in the population is compared with other individuals in terms of fitness, which is determined by the indicator values between individuals. Indicator-based algorithms account for not only the convergence but also the diversity of the population. The representative algorithms are the indicator-based evolutionary algorithm (IBEA) [23], the S-metric selection-based multi-objective evolutionary algorithm (SMS-MOEA) [24], the many-objective metaheuristic algorithm based on R2 indicator (MOMBI) [25] and the hypervolume-based evolutionary algorithm (HyPE) [26]. These algorithms effectively handle convergence and diversity simultaneously when used to deal with MaOPs. However, their computational costs are huge, especially for HyPE.

A few approaches that do not belong to the above three categories exist. First proposed in 2014, NSGA-III [27] utilises a set of predefined, evenly distributed reference points to manage the diversity of candidate solutions. A reference vector-guided evolutionary algorithm [28] also adopts reference points that are used to generate a subset of preferred Pareto optimal solutions. In [29], the two-archive algorithm was presented with the following framework: two archives are developed with indicator and decomposition as bases, and a new diversity maintaining strategy is proposed. Other algorithms based on objective space reduction have been found to perform well in solving MaOPs [30], [31], [32], [33].

The above studies provide a variety of alternatives to address MaOPs, and many of them significantly contribute to the development of this domain. Inspired by the indicator-based algorithms for dealing with MaOPs, the present study proposes an indicator-based artificial bee colony (ABC) algorithm. As mentioned previously, balancing convergence and diversity to solve MaOPs is not an easy task. Most MOEAs share two common but conflicting goals: to minimise the distance to the Pareto-optimal set and to maximise the diversity within the approximation of the Pareto-optimal [34]. Two main performance assessment metrics measure convergence and diversity simultaneously for a high dimensional space: hypervolume (HV) [34] and quality indicator $I_{\epsilon +}$ [35]. The quality indicator $I_{\epsilon +}$ is adopted in the proposed approach because of the high computational cost of HV.

The ABC algorithm is an evolutionary algorithm introduced in 2005 [36]. It involves three types of bees that perform different tasks by cooperating with one another. Employed bees generate new food sources by exchanging information with its neighbours. If a new food source is better than the old one, then the new food source replace the old food source. Onlooker bees choose one of these food sources on the basis of the quality of the food sources shared by employed bees. The onlookers then try to improve the quality of such food sources. Scout bees find the poorest food sources which have not been optimised in a limited number of cycles and reinitialise them.

The standard ABC performs well when dealing with single-objective optimisation and achieves good solution quality and high convergence speed. In the present study, the ABC algorithm is combined with a quality indicator to solve MOPs and MaOPs. The proposed algorithm called $\epsilon \text{-MOABC}$ is based on the ABC algorithm; its basic colony is composed of three types of bees: employed bees, onlooker bees and scout bees. The main contributions of the present study are summarised as follows:

• The basic artificial bee mutation operator is easy to trap into local Pareto fronts. Thus, a combined mutation operator is suggested.

• The preference information based on a quality indicator $I_{\epsilon +}$ is used. Within $\epsilon \text{-MOABC}$, the updating methods of individuals in the population and archive are both based on the $I_{\epsilon +}$ indicator. A fixed-size archive is used to maintain the non-dominated solutions produced in each generation. The fitness assignment based on the quality indicator $I_{\epsilon +}$ attempts to rank population members according to their usefulness in relation to the optimisation goal.

• As to the uncertainty of the true Pareto front, the contributions of each individual to the population are different; hence, more computational resources should be allocated to outstanding individuals. In this study, the selection mechanism of individuals is based on their quality indicators combined with a power law probability when onlooker bees select food sources.

The rest of the paper is organised as follows. Section 2 introduces the basic concept of MOPs and the survey of related works about multi-objective ABC algorithms. Section 3 presents the details of the proposed multi-objective ABC algorithm called $\epsilon \text{-MOABC}$ for handling MOPs and MaOPs. Section 4 analyses and discusses the experimental results to solve DTLZ [37], LZ09 [50], and CEC09 [38] test instances. In this section, $\epsilon \text{-MOABC}$ is compared with several other state-of-the-art algorithms. Section 5 concludes the paper.