Multiobjective evolutionary algorithms: analyzing the state-of-the-art

Solving optimization problems with multiple (often conflicting) objectives is, generally, a very difficult goal. Evolutionary algorithms (EAs) were initially extended and applied during the mid-eighties in an attempt to stochastically solve problems of this generic class. During the past decade, a variety, of multiobjective EA (MOEA) techniques have been proposed and applied to many scientific and engineering applications. Our discussion's intent is to rigorously define multiobjective optimization problems and certain related concepts, present an MOEA classification scheme, and evaluate the variety of contemporary MOEAs. Current MOEA theoretical developments are evaluated; specific topics addressed include fitness functions, Pareto ranking, niching, fitness sharing, mating restriction, and secondary populations. Since the development and application of MOEAs is a dynamic and rapidly growing activity, we focus on key analytical insights based upon critical MOEA evaluation of current research and applications. Recommended MOEA designs are presented, along with conclusions and recommendations for future work.     

Comparison of multi-objective evolutionary algorithms in hybrid Kansei engineering system for product form design

Understanding the affective needs of customers is crucial to the success of product design. Hybrid Kansei engineering system (HKES) is an expert system capable of generating products in accordance with the affective responses. HKES consists of two subsystems: forward Kansei engineering system (FKES) and backward Kansei engineering system (BKES). In previous studies, HKES was based primarily on single-objective optimization, such that only one optimal design was obtained in a given simulation run. The use of multi-objective evolutionary algorithm (MOEA) in HKES was only attempted using the non-dominated sorting genetic algorithm-II (NSGA-II), such that very little work has been conducted to compare different MOEAs. In this paper, we propose an approach to HKES combining the methodologies of support vector regression (SVR) and MOEAs. In BKES, we constructed predictive models using SVR. In FKES, optimal design alternatives were generated using MOEAs. Representative designs were obtained using fuzzy c-means algorithm for clustering the Pareto front into groups. To enable comparison, we employed three typical MOEAs: NSGA-II, the Pareto envelope-based selection algorithm-II (PESA-II), and the strength Pareto evolutionary algorithm-2 (SPEA2). A case study of vase form design was provided to demonstrate the proposed approach. Our results suggest that NSGA-II has good convergence performance and hybrid performance; in contrast, SPEA2 provides the strong diversity required by designers. The proposed HKES is applicable to a wide variety of product design problems, while providing creative design ideas through the exploration of numerous Pareto optimal solutions.     

The balance between proximity and diversity in multiobjective evolutionary algorithms

Over the last decade, a variety of evolutionary algorithms (EAs) have been proposed for solving multiobjective optimization problems. Especially more recent multiobjective evolutionary algorithms (MOEAs) have been shown to be efficient and superior to earlier approaches. An important question however is whether we can expect such improvements to converge onto a specific efficient MOEA that behaves best on a large variety of problems. In this paper, we argue that the development of new MOEAs cannot converge onto a single new most efficient MOEA because the performance of MOEAs shows characteristics of multiobjective problems. While we point out the most important aspects for designing competent MOEAs in this paper, we also indicate the inherent multiobjective tradeoff in multiobjective optimization between proximity and diversity preservation. We discuss the impact of this tradeoff on the concepts and design of exploration and exploitation operators. We also present a general framework for competent MOEAs and show how current state-of-the-art MOEAs can be obtained by making choices within this framework. Furthermore, we show an example of how we can separate nondomination selection pressure from diversity preservation selection pressure and discuss the impact of changing the ratio between these components.     

A polar-based guided multi-objective evolutionary algorithm to search for optimal solutions interested by decision-makers in a logistics network design problem

In practical multi-objective optimization problems, respective decision-makers might be interested in some optimal solutions that have objective values closer to their specified values. Guided multi-objective evolutionary algorithms (guided MOEAs) have been significantly used to guide their evolutionary search direction toward these optimal solutions using by decision makers. However, most guided MOEAs need to be iteratively and interactively evaluated and then guided by decision-makers through re-formulating or re-weighting objectives, and it might negatively affect the algorithms performance. In this paper, a novel guided MOEA that uses a dynamic polar-based region around a particular point in objective space is proposed. Based on the region, new selection operations are designed such that the algorithm can guide the evolutionary search toward optimal solutions that are close to the particular point in objective space without the iterative and interactive efforts. The proposed guided MOEA is tested on the multi-criteria decision-making problem of flexible logistics network design with different desired points. Experimental results show that the proposed guided MOEA outperforms two most effective guided and non-guided MOEAs, R-NSGA-II and NSGA-II.     

Comparison of many-objective evolutionary algorithms using performance metrics ensemble

In this study, we have thoroughly researched on performance of six state-of-the-art Multiobjective Evolutionary Algorithms (MOEAs) under a number of carefully crafted many-objective optimization benchmark problems. Each MOEA apply different method to handle the difficulty of increasing objectives. Performance metrics ensemble exploits a number of performance metrics using double elimination tournament selection and provides a comprehensive measure revealing insights pertaining to specific problem characteristics that each MOEA could perform the best. Experimental results give detailed information for performance of each MOEA to solve many-objective optimization problems. More importantly, it shows that this performance depends on two distinct aspects: the ability of MOEA to address the specific characteristics of the problem and the ability of MOEA to handle high-dimensional objective space.     

A dominance tree and its application in evolutionary multi-objective optimization

Most contemporary multi-objective evolutionary algorithms (MOEAs) store and handle a population with a linear list, and this may impose high computational complexities on the comparisons of solutions and the fitness assignment processes. This paper presents a data structure for storing the whole population and their dominating information in MOEAs. This structure, called a Dominance Tree (DT), is a binary tree that can effectively and efficiently store three-valued relations (namely dominating, dominated or non-dominated) among vector values. This paper further demonstrates DT’s potential applications in evolutionary multi-objective optimization with two cases. The first case utilizes the DT to improve NSGA-II as a fitness assignment strategy. The second case demonstrates a DT-based MOEA (called a DTEA), which is designed by leveraging the favorable properties of the DT. The simulation results show that the DT-improved NSGA-II is significantly faster than NSGA-II. Meanwhile, DTEA is much faster than SPEA2, NSGA-II and an improved version of NSGA-II. On the other hand, in regard to converging to the Pareto optimal front and maintaining the diversity of solutions, DT-improved NSGA-II and DTEA are found to be competitive with NSGA-II and SPEA2.     

Simultaneous topology and sizing optimization of a water distribution network using a hybrid multiobjective evolutionary algorithm

This paper proposes a new direction for design optimization of a water distribution network (WDN). The new approach introduces an optimization process to the conceptual design stage of a WDN. The use of multiobjective evolutionary algorithms (MOEAs) for simultaneous topology and sizing design of piping networks is presented. The design problem includes both topological and sizing design variables while the objective functions are network cost and total head loss in pipes. The numerical technique, called a network repairing technique (NRT), is proposed to overcome difficulties in operating MOEAs for network topological design. The problem is then solved by using a number of established and newly developed MOEAs. Also, two new MOEAs namely multiobjective real code population-based incremental learning (RPBIL) and a hybrid algorithm of RPBIL with differential evolution (termed RPBIL–DE) are proposed to tackle the design problems. The optimum results obtained are illustrated and compared. It is shown that the proposed network repairing technique is an efficient and effective tool for topological design of WDNs. Based on the hypervolume indicator, the proposed RPBIL–DE is among the best MOEA performers.     

An effective model of multiple multi-objective evolutionary algorithms with the assistance of regional multi-objective evolutionary algorithms: VIPMOEAs

Division of the evolutionary search among multiple multi-objective evolutionary algorithms (MOEAs) is a recent advantage in MOEAs design, particularly in effective parallel and distributed MOEAs. However, most these algorithms rely on such a central (re) division that affects the algorithms’ efficiency. This paper first proposes a local MOEA that searches on a particular region of objective space with its novel evolutionary selections. It effectively searches for Pareto Fronts (PFs) inside the given polar-based region, while nearby the region is also explored, intelligently. The algorithm is deliberately designed to adjust its search direction to outside the region – but nearby – in the case of a region with no Pareto Front. With this contribution, a novel island model is proposed to run multiple forms of the local MOEA to improve a conventional MOEA (e.g. NSGA-II or MOEA/D) running along – in another island. To dividing the search, a new division technique is designed to give particular regions of objective space to the local MOEAs, frequently and effectively. Meanwhile, the islands benefit from a sophisticated immigration strategy without any central (re) collection, (re) division and (re) distribution acts. Results of three experiments have confirmed that the proposed island model mostly outperforms to the clustering MOEAs with similar division technique and similar island models on DTLZs. The model is also used and evaluated on a real-world combinational problem, flexible logistic network design problem. The model definitely outperforms to a similar island model with conventional MOEA (NSGA-II) used in each island.     

Using unconstrained elite archives for multiobjective optimization

Multiobjective evolutionary algorithms (MOEAs) have been the subject of numerous studies over the past 20 years. Recent work has highlighted the use of an active archive of elite, nondominated solutions to improve the optimization speed of these algorithms. However, preserving all elite individuals is costly in time (due to the linear comparison with all archived solutions needed before a new solution can be inserted into the archive). Maintaining an elite population of a fixed maximum size (by clustering or other means) alleviates this problem, but can cause retreating (or oscillitory) and shrinking estimated Pareto fronts - which can affect the efficiency of the search process. New data structures are introduced to facilitate the use of an unconstrained elite archive, without the need for a linear comparison to the elite set for every new individual inserted. The general applicability of these data structures is shown by their use in an evolution-strategy-based MOEA and a genetic-algorithm-based MOEA. It is demonstrated that MOEAs using the new data structures run significantly faster than standard, unconstrained archive MOEAs, and result in estimated Pareto fronts significantly ahead of MOEAs using a constrained archive. It is also shown that the use of an unconstrained elite archive permits robust criteria for algorithm termination to be used, and that the use of the data structure can also be used to increase the speed of algorithms using /spl epsi/-dominance methods.     

Decomposing the user-preference in multiobjective optimization

Preference information (such as the reference point) of the decision maker (DM) is often used in multiobjective optimization; however, the location of the specified reference point has a detrimental effect on the performance of multiobjective evolutionary algorithms (MOEAs). Inspired by multiobjective evolutionary algorithm-based decomposition (MOEA/D), this paper proposes an MOEA to decompose the preference information of the reference point specified by the DM into a number of scalar optimization subproblems and deals with them simultaneously (called MOEA/D-PRE). This paper presents an approach of iterative weight to map the desired region of the DM, which makes the algorithm easily obtain the desired region. Experimental results have demonstrated that the proposed algorithm outperforms two popular preference-based approaches, g-dominance and r-dominance, on continuous multiobjective optimization problems (MOPs), especially on many-objective optimization problems. Moreover, this study develops distinct models to satisfy different needs of the DM, thus providing a new way to deal with preference-based multiobjective optimization. Additionally, in terms of the shortcoming of MOEA/D-PRE, an improved MOEA/D-PRE that dynamically adjusts the size of the preferred region is proposed and has better performance on some problems.     

基于多目标进化算法的MOEA/D权重向量产生方法

在进化多目标优化研究领域,多目标优化是指对含有2个及以上目标的多目标问题的同时优化,其在近些年来受到越来越多的关注。随着MOEA/D的提出,基于聚合的多目标进化算法得到越来越多的研究,对MOEA/D算法的改进已有较多成果,但是很少有成果研究MOEA/D中权重的产生方法。提出一种使用多目标进化算法产生任意多个均匀分布的权重向量的方法,将其应用到MOEA/D,MSOPS和NSGA-III中,对这3个经典的基于聚合的多目标进化算法进行系统的比较研究。通过该类算法在DTLZ测试集、多目标旅行商问题MOTSP上的优化结果来分别研究该类算法在连续性问题、组合优化问题上的优化能力,以及使用矩形测试问题使得多目标进化算法的优化结果在决策空间可视化。实验结果表明,没有一个算法能适用于所有特性的问题。然而,MOEA/D采用不同聚合函数的两个算法MOEA/D_Tchebycheff和MOEA/D_PBI在多数情况下的性能比MSOPS和NSGA-III更好。     

基于改进K 支配排序的高维多目标进化算法

为提高4目标以上高维多目标优化问题的求解性能,提出一种基于改进K支配排序的高维多目标进化算法(KS-MODE).该算法针对K支配的支配关系和排序方法进行改进,避免循环支配并增强选择压力;设计新的全局密度估计方法提高局部密度估计精确性;设计新的精英选择策略和适应度值评价函数;采用CAO局部搜索算子加速收敛.在4～30个目标标准测试函数上的实验结果表明,KS-MODE能够在保证解集分布性的同时大幅提升收敛性和稳定性,能够有效求解高维多目标优化问题.     

多目标进化算法中选择策略的研究

在多目标进化算法(multiobjective evolutiorlsry algorithms,MOEAs)的文献中,对算法的选择策略进行系统研究的还很少,而MOEAs的选择策略不仅引导算法的搜索过程、决定搜索的方向而且对算法的收敛性有重要的影响,它是算法能否成功求解多目标优化问题的关键因素之一.在统一的框架下,首先讨论了多目标优化问题中适应度函数的构造问题,然后根据MOEAs的选择机制和原理将它们的选择策略重新分成了6种类型.一般文献中很少对多目标进化算法的操作算子采用符号化描述,这样不利于对算子的深层次理解,符号化描述了各类选择策略的操作机制和原理,并分析了各类策略的优劣性.最后,从理论上证明了具备一定特征的多目标进化算法的收敛性,证明的过程表明了将算法运行终止时得到的P known作为多目标优化问题的Pareto最优解集或近似最优解集的合理性.     

基于分解的多目标花朵授粉算法

在过去几十年里，许多多目标进化算法被广泛应用于解决多目标优化问题，其中一种比较流行的多目标进化算法是基于分解的多目标进化算法(MOEA/D)。花朵授粉算法是一种启发式优化算法，但迄今为止，花朵授粉算法在基于分解的多目标进化算法领域的研究还非常少。本文在基于分解的多目标进化算法的框架下，将花朵授粉算法拓展至多目标优化领域，提出一种基于分解的多目标花朵授粉算法(MOFPA/D)。此外，为了保证非支配解的多样性，本文提出一种基于网格的目标空间分割法，该方法从找到的Pareto最优解集中筛选出一定数量且分布均匀的Pareto最优解。实验结果表明，基于分解的多目标花朵授粉算法在收敛性与多样性方面均优于基于分解的多目标进化算法。     

改进多目标进化算法求解柔性作业车间调度问题

针对加工时间为模糊数的柔性作业车间调度问题,考虑最小化模糊最大完工时间、模糊机器总负荷、模糊关键机器负荷为优化目标,提出一种有效求解该类优化问题的多目标进化算法。算法采用一种混合不同机器分配和工序排序策略的方法产生初始种群,并采用插入空隙法对染色体进行解码。定义一种新的基于可能度的个体支配关系和一种基于决策空间的拥挤算子,并将所提支配关系和拥挤算子运用于快速非支配排序。接着,提出一种基于移动模糊关键工序的局部搜索策略对种群中的优势个体进行局部搜索。通过试验研究关键参数对算法性能的影响并将所提算法与3种不同的优化算法作对比。结果表明,所提算法能够比其它算法更有效解决多目标模糊柔性作业车间调度优化问题。     

Multiobjective evolutionary algorithm based on multimethod with dynamic resources allocation

In the last two decades, multiobjective optimization has become main stream and various multiobjective evolutionary algorithms (MOEAs) have been suggested in the field of evolutionary computing (EC) for solving hard combinatorial and continuous multiobjective optimization problems. Most MOEAs employ single evolutionary operators such as crossover, mutation and selection for population evolution. In this paper, we suggest a multiobjective evolutionary algorithm based on multimethods (MMTD) with dynamic resource allocation for coping with continuous multi-objective optimization problems (MOPs). The suggested algorithm employs two well known population based stochastic algorithms namely MOEA/D and NSGA-II as constituent algorithms for population evolution with a dynamic resource allocation scheme. We have examined the performance of the proposed MMTD on two different MOPs test suites: the widely used ZDT problems and the recently formulated test instances for the special session on MOEAs competition of the 2009 IEEE congress on evolutionary computation (CEC’09). Experimental results obtained by the suggested MMTD are more promising than those of some state-of-the-art MOEAs in terms of the inverted generational distance (IGD)-metric on most test problems.     

The impact of Quality Indicators on the rating of Multi-objective Evolutionary Algorithms

Evaluating and comparing multi-objective optimizers is an important issue. But, when doing a comparison, it has to be noted that the results can be influenced highly by the selected Quality Indicator. Therefore, the impact of individual Quality Indicators on the ranking of Multi-objective Optimizers in the proposed method must be analyzed beforehand. In this paper the comparison of several different Quality Indicators with a method called Chess Rating System for Evolutionary Algorithms (CRS4EAs) was conducted in order to get a better insight on their characteristics and how they affect the ranking of Multi-objective Evolutionary Algorithms (MOEAs). Although it is expected that Quality Indicators with the same optimization goals would yield a similar ranking of MOEAs, it has been shown that results can be contradictory and significantly different. Consequently, revealing that claims about the superiority of one MOEA over another can be misleading.     

Evaluating the epsilon-domination based multi-objective evolutionary algorithm for a quick computation of Pareto-optimal solutions

Since the suggestion of a computing procedure of multiple Pareto-optimal solutions in multi-objective optimization problems in the early Nineties, researchers have been on the look out for a procedure which is computationally fast and simultaneously capable of finding a well-converged and well-distributed set of solutions. Most multi-objective evolutionary algorithms (MOEAs) developed in the past decade are either good for achieving a well-distributed solutions at the expense of a large computational effort or computationally fast at the expense of achieving a not-so-good distribution of solutions. For example, although the Strength Pareto Evolutionary Algorithm or SPEA (Zitzler and Thiele, 1999) produces a much better distribution compared to the elitist non-dominated sorting GA or NSGA-II (Deb et al., 2002a), the computational time needed to run SPEA is much greater. In this paper, we evaluate a recently-proposed steady-state MOEA (Deb et al., 2003) which was developed based on the epsilon-dominance concept introduced earlier(Laumanns et al., 2002) and using efficient parent and archive update strategies for achieving a well-distributed and well-converged set of solutions quickly. Based on an extensive comparative study with four other state-of-the-art MOEAs on a number of two, three, and four objective test problems, it is observed that the steady-state MOEA is a good compromise in terms of convergence near to the Pareto-optimal front, diversity of solutions, and computational time. Moreover, the epsilon-MOEA is a step closer towards making MOEAs pragmatic, particularly allowing a decision-maker to control the achievable accuracy in the obtained Pareto-optimal solutions.     

A novel adaptive control strategy for decomposition-based multiobjective algorithm

Recently, evolutionary algorithm based on decomposition (MOEA/D) has been found to be very effective and efficient for solving complicated multiobjective optimization problems (MOPs). However, the selected differential evolution (DE) strategies and their parameter settings impact a lot on the performance of MOEA/D when tackling various kinds of MOPs. Therefore, in this paper, a novel adaptive control strategy is designed for a recently proposed MOEA/D with stable matching model, in which multiple DE strategies coupled with the parameter settings are adaptively conducted at different evolutionary stages and thus their advantages can be combined to further enhance the performance. By exploiting the historically successful experience, an execution probability is learned for each DE strategy to perform adaptive adjustment on the candidate solutions. The proposed adaptive strategies on operator selection and parameter settings are aimed at improving both of the convergence speed and population diversity, which are validated by our numerous experiments. When compared with several variants of MOEA/D such as MOEA/D, MOEA/D-DE, MOEA/D-DE+PSO, ENS-MOEA/D, MOEA/D-FRRMAB and MOEA/D-STM, our algorithm performs better on most of test problems.     

Incorporating domain knowledge into the optimization of energy systems

Energy plays a key factor in the advancement of humanity. As energy demands are mostly met by fossil fuels, the world-wide consciousness grows about their negative impact on the environment. Therefore, it becomes necessary to design sustainable energy systems by introducing renewable energies. Because of the intermittent availability of different renewable resources, the designing of a sustainable energy system should find an optimal mix of different resources. However, the optimization of this combination has to deal with a number of possibly contradictory objectives.Multi-objective evolutionary algorithms (MOEA) are widely used to solve this kind of problems. As optimizing an energy system by using a MOEA is computationally costly, it is necessary to solve the problem efficiently. For this purpose, we propose the incorporation of domain knowledge related to energy systems into different phases (i.e., initialization and mutation) of a MOEA run. The proposed approaches are implemented for two widely used MOEAs and evaluated on the Danish Aalborg test problem. The experimental results show that each approach individually achieves significant improvements of the energy systems, which is expressed in better trade-off sets. Moreover, a state-of-the-art stopping criterion is adapted to detect the convergence in order to save computational resources. Finally, all proposed techniques are merged within two MOEAs with the result that our combined approaches yield significantly better results in less time than generic approaches.