Performance evaluation of evolutionary multiobjective optimization algorithms for multiobjective fuzzy genetics-based machine learning

Recently, evolutionary multiobjective optimization (EMO) algorithms have been utilized for the design of accurate and interpretable fuzzy rule-based systems. This research area is often referred to as multiobjective genetic fuzzy systems (MoGFS), where EMO algorithms are used to search for non-dominated fuzzy rule-based systems with respect to their accuracy and interpretability. In this paper, we examine the ability of EMO algorithms to efficiently search for Pareto optimal or near Pareto optimal fuzzy rule-based systems for classification problems. We use NSGA-II (elitist non-dominated sorting genetic algorithm), its variants, and MOEA/D (multiobjective evolutionary algorithm based on decomposition) in our multiobjective fuzzy genetics-based machine learning (MoFGBML) algorithm. Classification performance of obtained fuzzy rule-based systems by each EMO algorithm is evaluated for training data and test data under various settings of the available computation load and the granularity of fuzzy partitions. Experimental results in this paper suggest that reported classification performance of MoGFS in the literature can be further improved using more computation load, more efficient EMO algorithms, and/or more antecedent fuzzy sets from finer fuzzy partitions.     

Balance between genetic search and local search in memetic algorithms for multiobjective permutation flowshop scheduling

This paper shows how the performance of evolutionary multiobjective optimization (EMO) algorithms can be improved by hybridization with local search. The main positive effect of the hybridization is the improvement in the convergence speed to the Pareto front. On the other hand, the main negative effect is the increase in the computation time per generation. Thus, the number of generations is decreased when the available computation time is limited. As a result, the global search ability of EMO algorithms is not fully utilized. These positive and negative effects are examined by computational experiments on multiobjective permutation flowshop scheduling problems. Results of our computational experiments clearly show the importance of striking a balance between genetic search and local search. In this paper, we first modify our former multiobjective genetic local search (MOGLS) algorithm by choosing only good individuals as initial solutions for local search and assigning an appropriate local search direction to each initial solution. Next, we demonstrate the importance of striking a balance between genetic search and local search through computational experiments. Then we compare the modified MOGLS with recently developed EMO algorithms: the strength Pareto evolutionary algorithm and revised nondominated sorting genetic algorithm. Finally, we demonstrate that a local search can be easily combined with those EMO algorithms for designing multiobjective memetic algorithms.     

Towards faster convergence of evolutionary multi-criterion optimization algorithms using Karush Kuhn Tucker optimality based local search

Evolutionary multi-criterion optimization (EMO) algorithms emphasize non-dominated and less crowded solutions in a population iteratively until the population converges close to the Pareto optimal set. During the search process, non-dominated solutions are differentiated only by their local crowding or contribution to hypervolume or using a similar other metric. Thus, during evolution and even at the final iteration, the true convergence behavior of each non-dominated solutions from the Pareto optimal set is unknown. Recent studies have used Karush Kuhn Tucker (KKT) optimality conditions to develop a KKT Proximity Measure (KKTPM) for estimating proximity of a solution from Pareto optimal set for a multi-objective optimization problem. In this paper, we integrate KKTPM with a recently proposed EMO algorithm to enhance its convergence properties towards the true Pareto optimal front. Specifically, we use KKTPM to identify poorly converged non-dominated solutions in every generation and apply an achievement scalarizing function based local search procedure to improve their convergence. Assisted by the KKTPM, the modified algorithm is designed in a way that maintains the total number of function evaluations as low as possible while making use of local search where it is most needed. Simulations on both constrained and unconstrained multi- and many objectives optimization problems demonstrate that the hybrid algorithm significantly improves the overall convergence properties. This study brings evolutionary optimization closer to mainstream optimization field and should motivate researchers to utilize KKTPM measure further within EMO and other numerical optimization algorithms.     

Dynamic multiobjective optimization problems: test cases, approximations, and applications

After demonstrating adequately the usefulness of evolutionary multiobjective optimization (EMO) algorithms in finding multiple Pareto-optimal solutions for static multiobjective optimization problems, there is now a growing need for solving dynamic multiobjective optimization problems in a similar manner. In this paper, we focus on addressing this issue by developing a number of test problems and by suggesting a baseline algorithm. Since in a dynamic multiobjective optimization problem, the resulting Pareto-optimal set is expected to change with time (or, iteration of the optimization process), a suite of five test problems offering different patterns of such changes and different difficulties in tracking the dynamic Pareto-optimal front by a multiobjective optimization algorithm is presented. Moreover, a simple example of a dynamic multiobjective optimization problem arising from a dynamic control loop is presented. An extension to a previously proposed direction-based search method is proposed for solving such problems and tested on the proposed test problems. The test problems introduced in this paper should encourage researchers interested in multiobjective optimization and dynamic optimization problems to develop more efficient algorithms in the near future.     

求解多目标问题的Memetic免疫优化算法

将基于Pareto支配关系的局部下山算子和差分算子引入免疫多目标优化算法之中,提出了一种求解多目标问题的Memetic免疫优化算法(Memetic immune algorithm for multiobjective optimization,简称MIAMO).该算法利用种群中抗体在决策空间上的位置关系设计了两种有效的启发式局部搜索策略,提高了免疫多目标优化算法的求解效率.仿真实验结果表明,MIAMO与其他4种有效的多目标优化算法相比,不仅在求得Pareto最优解集的逼近性、均匀性和宽广性上有明显优势,而且算法的收敛速度与免疫多目标优化算法相比明显加快.     

A preference multi-objective optimization based on adaptive rank clone and differential evolution

Evolutionary multi-objective optimization (EMO) algorithms have been used in various real-world applications. However, most of the Pareto domination based multi-objective optimization evolutionary algorithms are not suitable for many-objective optimization. Recently, EMO algorithm incorporated decision maker’s preferences became a new trend for solving many-objective problems and showed a good performance. In this paper, we first use a new selection scheme and an adaptive rank based clone scheme to exploit the dynamic information of the online antibody population. Moreover, a special differential evolution (DE) scheme is combined with directional information by selecting parents for the DE calculation according to the ranks of individuals within a population. So the dominated solutions can learn the information of the non-dominated ones by using directional information. The proposed method has been extensively compared with two-archive algorithm, light beam search non-dominated sorting genetic algorithm II and preference rank immune memory clone selection algorithm over several benchmark multi-objective optimization problems with from two to ten objectives. The experimental results indicate that the proposed algorithm achieves competitive results.     

Preference representation using Gaussian functions on a hyperplane in evolutionary multi-objective optimization

Many-objective optimization has attracted much attention in evolutionary multi-objective optimization (EMO). This is because EMO algorithms developed so far often degrade their search ability for optimization problems with four or more objectives, which are frequently referred to as many-objective problems. One of promising approaches to handle many objectives is to incorporate the preference of a decision maker (DM) into EMO algorithms. With the preference, EMO algorithms can focus the search on regions preferred by the DM, resulting in solutions close to the Pareto front around the preferred regions. Although a number of preference-based EMO algorithms have been proposed, it is not trivial for the DM to reflect his/her actual preference in the search. We previously proposed to represent the preference of the DM using Gaussian functions on a hyperplane. The DM specifies the center and spread vectors of the Gaussian functions so as to represent his/her preference. The preference handling is integrated into the framework of NSGA-II. This paper extends our previous work so that obtained solutions follow the distribution of Gaussian functions specified. The performance of our proposed method is demonstrated mainly for benchmark problems and real-world applications with a few objectives in this paper. We also show the applicability of our method to many-objective problems.     

多目标协调进化算法研究

进化算法适合解决多目标优化问题,但难以产生高维优化问题的最优解,文中针对此问题提出了一种求解高维目标优化问题的新进化方法,即多目标协调进化算法,主要特点是进化群体按协调模型使用偏好信息进行偏好排序,而不是基于Pareto优于关系进行了个体排序,实验结果表明,所提出的算法是可行而有效的,且能在有限进化代数内收敛。     

MOTGA: A multiobjective Tchebycheff based genetic algorithm for the multidimensional knapsack problem

This paper presents a new multiobjective genetic algorithm based on the Tchebycheff scalarizing function, which aims to generate a good approximation of the nondominated solution set of the multiobjective problem. The algorithm performs several stages, each one intended for searching potentially nondominated solutions in a different part of the Pareto front. Pre-defined weight vectors act as pivots to define the weighted-Tchebycheff scalarizing functions used in each stage. Therefore, each stage focuses the search on a specific region, leading to an iterative approximation of the entire nondominated set.     

Preferences and their application in evolutionary multiobjectiveoptimization

The paper describes a new preference method and its use in multiobjective optimization. These preferences are developed with a goal to reduce the cognitive overload associated with the relative importance of a certain criterion within a multiobjective design environment involving large numbers of objectives. Their successful integration with several genetic-algorithm-based design search and optimization techniques (weighted sums, weighted Pareto, weighted co-evolutionary methods, and weighted scenarios) are described and theoretical results relating to complexity and sensitivity of the algorithm are presented and discussed. Its usefulness was demonstrated in a real-world project of conceptual airframe design     

Solving multiobjective problems using cat swarm optimization

This paper proposes a new multiobjective evolutionary algorithm (MOEA) by extending the existing cat swarm optimization (CSO). It finds the nondominated solutions along the search process using the concept of Pareto dominance and uses an external archive for storing them. The performance of our proposed approach is demonstrated using standard test functions. A quantitative assessment of the proposed approach and the sensitivity test of different parameters is carried out using several performance metrics. The simulation results reveal that the proposed approach can be a better candidate for solving multiobjective problems (MOPs).     

Effects of nongeometric binary crossover on multiobjective 0/1 knapsack problems

Standard binary crossover operators (e.g., one-point, two-point, and uniform) tend to decrease the diversity of solutions while they improve the convergence to the Pareto front. This is because standard binary crossover operators, which are called geometric crossovers, always generate an offspring in the line segment between its parents under the Hamming distance in the genotype space. In our former study, we have already proposed a nongeometric binary crossover operator to generate an offspring outside the line segment between its parents. In this article, we examine the effect of our crossover operator on the performance of evolutionary multiobjective optimization (EMO) algorithms through computational experiments on various multiobjective knapsack problems. Experimental results show that our crossover operator improves the search ability of EMO algorithms for a wide range of test problems. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

Multiobjective evolutionary algorithms for electric power dispatch problem

The potential and effectiveness of the newly developed Pareto-based multiobjective evolutionary algorithms (MOEA) for solving a real-world power system multiobjective nonlinear optimization problem are comprehensively discussed and evaluated in this paper. Specifically, nondominated sorting genetic algorithm, niched Pareto genetic algorithm, and strength Pareto evolutionary algorithm (SPEA) have been developed and successfully applied to an environmental/economic electric power dispatch problem. A new procedure for quality measure is proposed in this paper in order to evaluate different techniques. A feasibility check procedure has been developed and superimposed on MOEA to restrict the search to the feasible region of the problem space. A hierarchical clustering algorithm is also imposed to provide the power system operator with a representative and manageable Pareto-optimal set. Moreover, an approach based on fuzzy set theory is developed to extract one of the Pareto-optimal solutions as the best compromise one. These multiobjective evolutionary algorithms have been individually examined and applied to the standard IEEE 30-bus six-generator test system. Several optimization runs have been carried out on different cases of problem complexity. The results of MOEA have been compared to those reported in the literature. The results confirm the potential and effectiveness of MOEA compared to the traditional multiobjective optimization techniques. In addition, the results demonstrate the superiority of the SPEA as a promising multiobjective evolutionary algorithm to solve different power system multiobjective optimization problems.     

A problem space genetic algorithm in multiobjective optimization

In this study, a problem space genetic algorithm (PSGA) is used to solve bicriteria tool management and scheduling problems simultaneously in flexible manufacturing systems. The PSGA is used to generate approximately efficient solutions minimizing both the manufacturing cost and total weighted tardiness. This is the first implementation of PSGA to solve a multiobjective optimization problem (MOP). In multiobjective search, the key issues are guiding the search towards the global Pareto-optimal set and maintaining diversity. A new fitness assignment method, which is used in PSGA, is proposed to find a well-diversified, uniformly distributed set of solutions that are close to the global Pareto set. The proposed fitness assignment method is a combination of a nondominated sorting based method which is most commonly used in multiobjective optimization literature and aggregation of objectives method which is popular in the operations research literature. The quality of the Pareto-optimal set is evaluated by using the performance measures developed for multiobjective optimization problems.     

Convergence of multi-objective evolutionary algorithms to a uniformly distributed representation of the Pareto front

In evolutionary multi-objective optimization (EMO), the convergence to the Pareto set of a multi-objective optimization problem (MOP) and the diversity of the final approximation of the Pareto front are two important issues. In the existing definitions and analyses of convergence in multi-objective evolutionary algorithms (MOEAs), convergence with probability is easily obtained because diversity is not considered. However, diversity cannot be guaranteed. By combining the convergence with diversity, this paper presents a new definition for the finite representation of a Pareto set, the B-Pareto set, and a convergence metric for MOEAs. Based on a new archive-updating strategy, the convergence of one such MOEA to the B-Pareto sets of MOPs is proved. Numerical results show that the obtained B-Pareto front is uniformly distributed along the Pareto front when, according to the new definition of convergence, the algorithm is convergent.     

Multiobjective optimization using an adaptive weighting scheme

A new Pareto front approximation method is proposed for multiobjective optimization problems (MOPs) with bound constraints. The method employs a hybrid optimization approach using two derivative-free direct search techniques, and intends to solve black box simulation-based MOPs where the analytical form of the objectives is not known and/or the evaluation of the objective function(s) is very expensive. A new adaptive weighting scheme is proposed to convert a multiobjective optimization problem to a single objective optimization problem. Another contribution of this paper is the generalization of the star discrepancy-based performance measure for problems with more than two objectives. The method is evaluated using five test problems from the literature, and a realistic engineering problem. Results show that the method achieves an arbitrarily close approximation to the Pareto front with a good collection of well-distributed nondominated points for all six test problems.     

基于精英进化导向的多目标PID参数优化

在多目标优化问题中,决策者必须对Pareto前沿的众多非劣解做出选择.本文将决策偏好融入Pareto优化过程,提出一种基于精英导向机制的多目标遗传算法,根据决策偏好选择Pareto最优解为精英,利用无损有限精度法和归一增量距离保持种群多样性,通过多种群进化机制将决策偏好的影响传播到整个种群.该方法成功应用于自动导引车(AGV)伺服系统的PID参数优化,可根据决策偏好快速有效地定向搜索Pareto最优解,保证伺服控制达到路径跟踪要求的速度响应性能.     

A hybrid evolutionary multiobjective optimization strategy for the dynamic power supply problem in magnesia grain manufacturing

The supply trajectory of electric power for submerged arc magnesia furnace determines the yields and grade of magnesia grain during the manufacture process. As the two production targets (i.e., the yields and the grade of magnesia grain) are conflicting and the process is subject to changing conditions, the supply of electric power needs to be dynamically optimized to track the moving Pareto optimal set with time. A hybrid evolutionary multiobjective optimization strategy is proposed to address the dynamic multiobjective optimization problem. The hybrid strategy is based on two techniques. The first one uses case-based reasoning to immediately generate good solutions to adjust the power supply once the environment changes, and then apply a multiobjective evolutionary algorithm to accurately solve the problem. The second one is to learn the case solutions to guide and promote the search of the evolutionary algorithm, and the best solutions found by the evolutionary algorithm can be used to update the case library to improve the accuracy of case-based reasoning in the following process. Due to the effectiveness of mutual promotion, the hybrid strategy can continuously adapt and search in dynamic environments. Two prominent multiobjective evolutionary algorithms are integrated into the hybrid strategy to solve the dynamic multiobjective power supply optimization problem. The results from a series of experiments show that the proposed hybrid algorithms perform better than their component multiobjective evolutionary algorithms for the tested problems.     

动态多目标优化的预测遗传算法

为了在动态环境中很好地跟踪最优解,考虑动态优化问题的特点,提出一种新的多目标预测遗传算法.首先对 Pareto 前沿面进行聚类以求得解集的质心;其次应用该质心与参考点描述 Pareto 前沿面;再次通过预测方法给出预测点集,使得算法在环境变化后能够有指导地增加种群多样性,以便快速跟踪最优解;最后应用标准动态测试问题进行算法测试,仿真分析结果表明所提出算法能适应动态环境,快速跟踪 Pareto 前沿面.     

基于智能体的多目标社会进化算法

提出了一种基于智能体的多目标社会进化算法用以求解多目标优化问题(multiobjective optimization problems,简称MOPs),通过多智能体进化的思想来完成Pareto 解集的寻优过程.该方法定义可信任度来表示智能体间的历史活动信息,并据此确定智能体的邻域、控制智能体间的行为.针对多目标问题的特点,设计了3 个进化算子分别体现适者生存、弱肉强食、多样性原则以及自学习的特性.同时采用擂台赛法则构造Pareto 解的存储种群.仿真实验结果表明,该算法能够较好地收敛到Pareto 最优解集上,并且具有良好的多样性.另外,通过对智能体局部邻域环境建立方式的分析结果表明引入“关系网模型”可有效提高算法的收敛速度,并能在一定程度上提高解的质量.