Survey paper
Multiobjective evolutionary algorithms: A survey of the state of the art

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Abstract

A multiobjective optimization problem involves several conflicting objectives and has a set of Pareto optimal solutions. By evolving a population of solutions, multiobjective evolutionary algorithms (MOEAs) are able to approximate the Pareto optimal set in a single run. MOEAs have attracted a lot of research effort during the last 20 years, and they are still one of the hottest research areas in the field of evolutionary computation. This paper surveys the development of MOEAs primarily during the last eight years. It covers algorithmic frameworks such as decomposition-based MOEAs (MOEA/Ds), memetic MOEAs, coevolutionary MOEAs, selection and offspring reproduction operators, MOEAs with specific search methods, MOEAs for multimodal problems, constraint handling and MOEAs, computationally expensive multiobjective optimization problems (MOPs), dynamic MOPs, noisy MOPs, combinatorial and discrete MOPs, benchmark problems, performance indicators, and applications. In addition, some future research issues are also presented.

Introduction

Many real-world optimization problems involve multiple objectives. A multiobjective optimization problem (MOP) can be mathematically formulated as minimizeF(x)=(f1(x),,fm(x))Ts.t.xΩ, where Ω is the decision space and xΩ is a decision vector. F(x) consists of m objective functions fi:ΩR,i=1,,m, where Rm is the objective space.

The objectives in (1) often conflict with each other. Improvement of one objective may lead to deterioration of another. Thus, a single solution, which can optimize all objectives simultaneously, does not exist. Instead, the best trade-off solutions, called the Pareto optimal solutions, are important to a decision maker (DM). The Pareto optimality concept, which was first proposed by Edgeworth and Pareto [1], is formally defined as follows [2], [3].

Definition 1

A vector u=(u1,,um)T is said to dominate another vector v=(v1,,vm)T, denoted as uv, iff i{1,,m}, uivi and uv.

Definition 2

A feasible solution xΩ of problem (1) is called a Pareto optimal solution, iff yΩ such that F(y)F(x). The set of all the Pareto optimal solutions is called the Pareto set (PS), denoted as PS={xΩ|yΩ,F(y)F(x)}. The image of the PS in the objective space is called the Pareto front (PF) PF={F(x)|xPS}.

Due to their population-based nature, evolutionary algorithms (EAs) are able to approximate the whole PS (PF) of an MOP in a single run. There has been a growing interest in applying EAs to deal with MOPs since Schaffer’s seminal work [4], and these EAs are called multiobjective evolutionary algorithms (MOEAs). By January 2011, more than 56001 publications have been published on evolutionary multiobjective optimization. Among these papers, 66.8% have been published in the last eight years, 38.4% are journal papers and 42.2% are conference papers. The research work on MOEAs has been surveyed from different aspects. Among these surveys, some are mainly on generic methodologies [5], [6], [7], [8], [9], [10], [11], [12]; some are on theoretical developments and applications [13], [14]; some work focus on special methods for MOPs, for example simulated annealing (SA) [15], particle swarm optimization (PSO) [16], and memetic algorithms [17]; some are on combinational problems [18], [19]; and others are on special applications, such as engineering problems [14], [20], [21], scheduling problems [22], economic and finance problems [23], automatic cell planning problems [24], traveling salesman problems [25], and preferences in MOPs [26]. However, no comprehensive survey has been conducted on MOEA development in recent years [6].

In this paper, we focus on recent developments on MOEAs. Our major concern is on continuous MOPs, while the works on combinational MOPs are covered in [19]. The remainder of this paper is organized as follows. Section 2 summarizes the advances in generic MOEA designs. Algorithm frameworks, selection strategies, and offspring reproduction operators are surveyed in this section. In Section 3, MOEAs for some complicated problems, such as constrained MOPs, multimodal problems, many-objective problems, expensive MOPs, and dynamic and noisy MOPs, are outlined. The benchmark problems and algorithm performance measures are surveyed in Section 4. Section 5 briefly discusses the applications of MOEAs. Finally, the paper is concluded in Section 6 with some potential directions for future research.

Section snippets

Advances in MOEA design

In this section, recent developments, including algorithm frameworks, selection and population updating strategies, offspring reproduction schemes, and other related issues, are surveyed.

Constraint handling in MOEAs

Although MOEAs have been more extensively investigated within the context of unconstrained and bound constrained MOPs, various general constraints are involved when solving real-world problems. Typically, the search space Ω of a constrained MOP can be formulated as follows: Ω={gj(x)=gj(x1,x2,,xn)0j=1,2,,Jhk(x)=hk(x1,x2,,xn)=0k=1,2,,KxiLxixiUi=1,2,,n, where gj(x) and hk(x) are inequality and equality constraint functions, respectively. Generally, equality constraints are transformed into

Benchmark problems

Benchmark problems are important for both assessing the qualities of MOEAs and designing MOEAs. Quite a few test problems have been designed in the early stages of MOEA research. Since they are relatively simple, they are not widely used nowadays. In recent years, several test suites have been designed, and some widely used ones are as follows.

In the report of the CEC 2007 Special Session and Competition [231], 19 multiobjective minimization problems are described, including bi-objective,

Applications

Due to the rapidly growing popularity of MOEAs as effective and robust multiobjective optimizers, researchers from several domains of science and engineering have been applying MOEAs to solve optimization problems arising in their own fields. The literature on MOEA applications is huge and multifaceted. Therefore, we summarize only the major applications of MOEAs in Table 2.

Conclusions and future directions

In this paper, research work on MOEAs has been surveyed. The advances in MOEA designs, MOEAs for complicated MOPs, benchmark problems, performance measures, and some applications, have been covered. Evolutionary multiobjective optimization is still in its early stage, although there have been a huge number of publications. The following issues, along with others, should define the future research trends of MOEAs.

  • New algorithmic frameworks: The current popular frameworks are Pareto-dominance

Acknowledgements

This work is partly supported by National Basic Research Program of China (No. 2011CB707104) and National Science Foundation of China (No. 61005050).

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