Evaluating Multiobjective Evolutionary Algorithms Using MCDM Methods

Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 9751783, 13 pages https://doi.org/10.1155/2018/9751783 Research Article Evaluating Multiobjective Evolutionary Algorithms Using MCDM Methods Xiaobing Yu ,1,2,3,4 YiQun Lu,4 and Xianrui Yu4 1 Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science & Technology, Nanjing 210044, China 2 Research Center for Prospering Jiangsu Province with Talents, Nanjing University of Information Science & Technology, Nanjing 210044, China 3 China Institute for Manufacture Developing, Nanjing University of Information Science & Technology, Nanjing 210044, China 4 School of Management Science and Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, China Correspondence should be addressed to Xiaobing Yu; Received 15 November 2017; Accepted 10 February 2018; Published 19 March 2018 Academic Editor: David Bigaud Copyright © 2018 Xiaobing Yu 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. The evaluation of multiobjective evolutionary algorithms (MOEAs) involves many metrics, it can be considered as a multiplecriteria decision making (MCDM) problem. A framework is proposed to estimate MOEAs, in which six MOEAs, five performance metrics, and two MCDM methods are used. An experimental study is designed and thirteen benchmark functions are selected to validate the proposed framework. The experimental results have indicated that the framework is effective in evaluating MOEAs. 1. Introduction Without a loss of generality, the mathematical formula of multiobjective problems (MOPs) can be expressed as follows: min 𝐹 (𝑥) = (𝑓1 (𝑥) , 𝑓2 (𝑥) , . . . , 𝑓𝑚 (𝑥)) 𝑥 ∈ Ω, (1) 󳨀 where → 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) is the decision vector in the decision space Ω. 𝐹(𝑥) is the objective function. Generally speaking, the objectives contradict each other. We cannot find a single solution to optimize all the objectives. Optimizing one objective often leads to deterioration in at least one objective. Over the past two decades, many multiobjective evolutionary algorithms (MOEAs) have been proposed, such as vector evaluated genetic algorithm (VEGA) [1], Pareto archived evolution strategy (PAES) [2], strength Pareto evolutionary algorithm (SPEA) [3], SPEA2 [4], Pareto envelope-based selection algorithm (PESA) [5, 6], nondominated sorting algorithm (NSGA) [7], NSGAII [8], multiobjective evolutionary algorithm based on decomposition (MOEAD) [9, 10], indicator-based evolutionary algorithm (IBEA) [11], epsilon-multiobjective evolutionary algorithm (epsilon-MOEA) [12], multiobjective particle swarm optimizer (MOPSO) [13], speed-constrained multiobjective particle swarm optimizer (SMPSO) [14], generalized differential evolution (GDE3) [15], ABYSS [16], multiobjective symbiotic organism search (MOSOS) [17], multiobjective differential evolution algorithm (MODEA) [18], grid-based adaptive MODE (GAMODE) [19]. These algorithms make great contributions to the development of evolutionary algorithms and optimization approaches. These methods try to make population move towards the optimal Pareto front region. In the single optimization, the algorithm performance can be evaluated by the difference between 𝑓(𝑥) and function optimal value. However, the method cannot be adopted in MOPs. In order to solve the problem, many criteria are proposed to evaluate the performance of MOEAs. In fact, the experiment results of almost every algorithm indicate that the proposed algorithm is competitive compared with the stateof-the-art algorithms. Nondominated objective space and box plot are adopted in SPEA2 [4]. NSGAII employs convergence and diversity metrics to compare with SPEA and PAES [8]. The set convergence and inverted generation distance (IGD) are used to evaluate the performance of MOEAD [9]. 2 Mathematical Problems in Engineering Six multiobjective optimization algorithms (1) NSGAII (2) PAES (3) SPEA2 (4) MOEAD (5) MOSPO (6) SMPSO Five performance measures (1) GD (2) IGD (3) HV (4) Space (5) Maximum Pareto front error Two MCDM methods (1) TOPSIS (2) VIKOR Empirical study Figure 1: Evaluation framework. Epsilon indicator is used in IBEA [11]. Convergence measurement, spread, hypervolume and computational time are selected as performance metrics in epsilon-MOEA [12]. To validate the proposed MOPSO, four quantitative performance indexes (success counting IGD, set coverage, two-set difference hypervolume) and qualitative performance index (plotting the Pareto fronts) are adopted [13]. Three quality indicators, additive unary epsilon indicator, spread, and hypervolume, are considered in SMPSO [14]. Spacing, binary metrics 𝐶 and 𝑉 are used in GD3 [15]. Three metrics, generation distance (GD), spread, and hypervolume, are used to estimate ABYSS [16]. GD, diversity, computational time, and box plot are considered as measurement in MOSOS [17]. GD and diversity metrics are adopted in MOEDA [18]. There are three metrics, GD, IGD, and hypervolume, in GAMODE [19]. Among these metrics, some focus on the convergence of MOEAs, while some pay attention to the diversity of MOEAs. Convergence is to measure the ability to attain global Pareto front and diversity is to measure the distribution along the Pareto front. It is observed that every proposed algorithm often introduces few metrics to estimate the performance based on the results from benchmarks. The conclusions of these MOEAs are that they are the best and competitive. However, it is unfaithfully to measure MOEAs performance by one or two metrics. Every metric can just demonstrate some specific qualification of performance while neglecting other information. For instance, the metric GD can provide information about the convergence of MOEAs, but it cannot evaluate the diversity of MOEAs. Therefore, these evaluations are not comprehensive. It cannot entirely estimate the whole performance of MOEAs. As evaluation of MOEAs involves many metrics, it can be regarded as a multiple-criteria decision making (MCDE) problem. MCDE techniques can be used to cope with the problem. In order to overcome the problem and make fair comparisons, a framework using MCDE methods is proposed. In the framework, comprehensive performance metrics are established, in which both convergence and diversity are considered. Two MCDE methods are employed to evaluate six MOEAs. The efforts can give more fair and faithful comparisons than single metric. The rest of this paper is organized as follows: Section 2 proposes the framework, in which six algorithms, five performance metrics, and two MCDM methods are briefly introduced. Experiments are presented in Section 3 and conclusions are illustrated in Section 4. 2. Evaluation Framework A framework is proposed to evaluate (...truncated)