A particle swarm optimizer for constrained multi-objective engineering design problems

This article presents a particle swarm optimizer (PSO) capable of handling constrained multi-objective optimization problems. The latter occur frequently in engineering design, especially when cost and performance are simultaneously optimized. The proposed algorithm combines the swarm intelligence fundamentals with elements from bio-inspired algorithms. A distinctive feature of the algorithm is the utilization of an arithmetic recombination operator, which allows interaction between non-dominated particles. Furthermore, there is no utilization of an external archive to store optimal solutions. The PSO algorithm is applied to multi-objective optimization benchmark problems and also to constrained multi-objective engineering design problems. The algorithmic effectiveness is demonstrated through comparisons of the PSO results with those obtained from other evolutionary optimization algorithms. The proposed particle swarm optimizer was able to perform in a very satisfactory manner in problems with multiple constraints and/or high dimensionality. Promising results were also obtained for a multi-objective engineering design problem with mixed variables.     

A new multi-objective particle swarm optimizer using empirical movement and diversified search strategies

Most real-world optimization problems involve the optimization task of more than a single objective function and, therefore, require a great amount of computational effort as the solution procedure is designed to anchor multiple compromised optimal solutions. Abundant multi-objective evolutionary algorithms (MOEAs) for multi-objective optimization have appeared in the literature over the past two decades. In this article, a new proposal by means of particle swarm optimization is addressed for solving multi-objective optimization problems. The proposed algorithm is constructed based on the concept of Pareto dominance, taking both the diversified search and empirical movement strategies into account. The proposed particle swarm MOEA with these two strategies is thus dubbed the empirical-movement diversified-search multi-objective particle swarm optimizer (EMDS-MOPSO). Its performance is assessed in terms of a suite of standard benchmark functions taken from the literature and compared to other four state-of-the-art MOEAs. The computational results demonstrate that the proposed algorithm shows great promise in solving multi-objective optimization problems.     

Decomposition-based multi-objective differential evolution particle swarm optimization for the design of a tubular permanent magnet linear synchronous motor

This article proposes a decomposition-based multi-objective differential evolution particle swarm optimization (DMDEPSO) algorithm for the design of a tubular permanent magnet linear synchronous motor (TPMLSM) which takes into account multiple conflicting objectives. In the optimization process, the objectives are evaluated by an artificial neural network response surface (ANNRS), which is trained by the samples of the TPMSLM whose performances are calculated by finite element analysis (FEA). DMDEPSO which hybridizes differential evolution (DE) and particle swarm optimization (PSO) together, first decomposes the multi-objective optimization problem into a number of single-objective optimization subproblems, each of which is associated with a Pareto optimal solution, and then optimizes these subproblems simultaneously. PSO updates the position of each particle (solution) according to the best information about itself and its neighbourhood. If any particle stagnates continuously, DE relocates its position by using two different particles randomly selected from the whole swarm. Finally, based on the DMDEPSO, optimization is gradually carried out to maximize the thrust of TPMLSM and minimize the ripple, permanent magnet volume, and winding volume simultaneously. The result shows that the optimized TPMLSM meets or exceeds the performance requirements. In addition, comparisons with chosen algorithms illustrate the effectiveness of DMDEPSO to find the Pareto optimal solutions for the TPMLSM optimization problem.     

基于粒子群与聚类的多目标优化算法

针对粒子群优化算法容易陷入局部最优的问题,提出了一种基于粒子群优化与分解聚类方法相结合的多目标优化算法。算法基于参考向量分解的方法,通过聚类优选粒子策略来更新全局最优解。首先,通过每条均匀分布的参考向量对粒子进行聚类操作,来促进粒子的多样性。从每个聚类中选择一个具有最小聚合函数适应度值的粒子,以平衡收敛性和多样性。动态更新全局最优解和个体最优解,引导种群均匀分布在帕累托前沿附近。通过仿真实验,与4种粒子群多目标优化算法进行对比。实验结果表明,提出的算法在27个选定的基准测试问题中获得了20个反世代距离(IGD)最优值。     

Swarm algorithms for single- and multi-objective optimization problems incorporating sensitivity analysis

Swarm algorithms such as particle swarm optimization (PSO) are non-gradient probabilistic optimization algorithms that have been successfully applied for global searches in complex problems such as multi-peak problems. However, application of these algorithms to structural and mechanical optimization problems still remains a complex matter since local optimization capability is still inferior to general numerical optimization methods. This article discusses new swarm metaphors that incorporate design sensitivities concerning objective and constraint functions and are applicable to structural and mechanical design optimization problems. Single- and multi-objective optimization techniques using swarm algorithms are combined with a gradient-based method. In the proposed techniques, swarm optimization algorithms and a sequential linear programming (SLP) method are conducted simultaneously. Finally, truss structure design optimization problems are solved by the proposed hybrid method to verify the optimization efficiency.     

A parallel particle swarm optimization algorithm for multi-objective optimization problems

In this article, a new proposal of using particle swarm optimization algorithms to solve multi-objective optimization problems is presented. The algorithm is constructed based on the concept of Pareto dominance, as well as a state-of-the-art ‘parallel’ computing technique that intends to improve algorithmic effectiveness and efficiency simultaneously. The proposed parallel particle swarm multi-objective evolutionary algorithm (PPS-MOEA) is tested through a variety of standard test functions taken from the literature; its performance is compared with six noted multi-objective algorithms. The computational experience gained from the first two experiments indicates that the algorithm proposed in this article is extremely competitive when compared with other MOEAs, being able to accurately, reliably and robustly approximate the true Pareto front in almost every tested case. To justify the motivation behind the research of the parallel swarm structure, the computational results of the third experiment confirm the PPS-MOEA's merit in solving really high-dimensional multi-objective optimization problems.     

Enhanced multiobjective particle swarm optimization in combination with adaptive weighted gradient-based searching

This article proposes a new multiobjective optimization method for structural problems based on multiobjective particle swarm optimization (MOPSO). A gradient-based optimization method is combined with MOPSO to alleviate constraint-handling difficulties. In this method, a group of particles is divided into two groups—a dominated solution group and a non-dominated solution group. The gradient-based method, utilizing a weighting coefficient method, is applied to the latter to conduct local searching that yields superior non-dominated solutions. In order to enhance the efficiency of exploration in a multiple objective function space, the weighting coefficients are adaptively assigned considering the distribution of non-dominated solutions. A linear optimization problem is solved to determine the optimal weighting coefficients for each non-dominated solution at each iteration. Finally, numerical and structural optimization problems are solved by the proposed method to verify the optimization efficiency.     

Biogeography-based particle swarm optimization with fuzzy elitism and its applications to constrained engineering problems

In evolutionary algorithms, elites are crucial to maintain good features in solutions. However, too many elites can make the evolutionary process stagnate and cannot enhance the performance. This article employs particle swarm optimization (PSO) and biogeography-based optimization (BBO) to propose a hybrid algorithm termed biogeography-based particle swarm optimization (BPSO) which could make a large number of elites effective in searching optima. In this algorithm, the whole population is split into several subgroups; BBO is employed to search within each subgroup and PSO for the global search. Since not all the population is used in PSO, this structure overcomes the premature convergence in the original PSO. Time complexity analysis shows that the novel algorithm does not increase the time consumption. Fourteen numerical benchmarks and four engineering problems with constraints are used to test the BPSO. To better deal with constraints, a fuzzy strategy for the number of elites is investigated. The simulation results validate the feasibility and effectiveness of the proposed algorithm.     

Multi-objective optimization of engineering systems using game theory and particle swarm optimization

This study proposes particle swarm optimization (PSO) based algorithms to solve multi-objective engineering optimization problems involving continuous, discrete and/or mixed design variables. The original PSO algorithm is modified to include dynamic maximum velocity function and bounce method to enhance the computational efficiency and solution accuracy. The algorithm uses a closest discrete approach (CDA) to solve optimization problems with discrete design variables. A modified game theory (MGT) approach, coupled with the modified PSO, is used to solve multi-objective optimization problems. A dynamic penalty function is used to handle constraints in the optimization problem. The methodologies proposed are illustrated by several engineering applications and the results obtained are compared with those reported in the literature.     

基于信息熵改进的粒子群算法在桁架结构

提出了信息熵改进的粒子群优化算法用于解决有应力约束、位移约束的桁架结构杆件截面尺寸优化设计问题.首先介绍了信息熵基本理论和基本粒子群优化算法理论,然后对粒子群优化算法作了合理的参数设置,并将信息熵引入粒子群优化算法的适应函数和停机判别准则中.最后对2个经典的优化问题进行求解并与其他算法进行了比较.数据结果表明信息熵改进后的粒子群优化算法在桁架结构优化设计中优于其他同类算法.     

Particle swarm optimization approach for multi-objective composite box-beam design

This paper presents a multi-agent search technique to design an optimal composite box-beam helicopter rotor blade. The search technique is called particle swarm optimization (‘inspired by the choreography of a bird flock’). The continuous geometry parameters (cross-sectional dimensions) and discrete ply angles of the box-beams are considered as design variables. The objective of the design problem is to achieve (a) specified stiffness value and (b) maximum elastic coupling. The presence of maximum elastic coupling in the composite box-beam increases the aero-elastic stability of the helicopter rotor blade. The multi-objective design problem is formulated as a combinatorial optimization problem and solved collectively using particle swarm optimization technique. The optimal geometry and ply angles are obtained for a composite box-beam design with ply angle discretizations of 10°, 15° and 45°. The performance and computational efficiency of the proposed particle swarm optimization approach is compared with various genetic algorithm based design approaches. The simulation results clearly show that the particle swarm optimization algorithm provides better solutions in terms of performance and computational time than the genetic algorithm based approaches.     

Hybridizing evolutionary strategies with continuation methods for solving multi-objective problems

Two techniques for the numerical treatment of multi-objective optimization problems—a continuation method and a particle swarm optimizer—are combined in order to unite their particular advantages. Continuation methods can be applied very efficiently to perform the search along the Pareto set, even for high-dimensional models, but are of local nature. In contrast, many multi-objective particle swarm optimizers tend to have slow convergence, but instead accomplish the ‘global task’ well. An algorithm which combines these two techniques is proposed, some convergence results for continuous models are provided, possible realizations are discussed, and finally some numerical results are presented indicating the strength of this novel approach.     

Feasibility-Guided Constraint-Handling Techniques for Engineering Optimization Problems

The particle swarm optimization (PSO) algorithm is an established nature-inspired population-based meta-heuristic that replicates the synchronizing movements of birds and fish. PSO is essentially an unconstrained algorithm and requires constraint handling techniques (CHTs) to solve constrained optimization problems (COPs). For this purpose, we integrate two CHTs, the superiority of feasibility (SF) and the violation constraint-handling (VCH), with a PSO. These CHTs distinguish feasible solutions from infeasible ones. Moreover, in SF, the selection of infeasible solutions is based on their degree of constraint violations, whereas in VCH, the number of constraint violations by an infeasible solution is of more importance. Therefore, a PSO is adapted for constrained optimization, yielding two constrained variants, denoted SF-PSO and VCH-PSO. Both SF-PSO and VCH-PSO are evaluated with respect to five engineering problems: the Himmelblau’s nonlinear optimization, the welded beam design, the spring design, the pressure vessel design, and the three-bar truss design. The simulation results show that both algorithms are consistent in terms of their solutions to these problems, including their different available versions. Comparison of the SF-PSO and the VCH-PSO with other existing algorithms on the tested problems shows that the proposed algorithms have lower computational cost in terms of the number of function evaluations used. We also report our disagreement with some unjust comparisons made by other researchers regarding the tested problems and their different variants.     

A generic framework for handling constraints with agent-based optimization algorithms and application to aerodynamic design

A generic constraint handling framework for use with any swarm-based optimization algorithm is presented. For swarm optimizers to solve constrained optimization problems effectively modifications have to be made to the optimizers to handle the constraints, however, these constraint handling frameworks are often not universally applicable to all swarm algorithms. A constraint handling framework is therefore presented in this paper that is compatible with any swarm optimizer, such that a user can wrap it around a chosen swarm algorithm and perform constrained optimization. The method, called separation-sub-swarm, works by dividing the population based on the feasibility of individual agents. This allows all feasible agents to move by existing swarm optimizer algorithms, hence promoting good performance and convergence characteristics of individual swarm algorithms. The framework is tested on a suite of analytical test function and a number of engineering benchmark problems, and compared to other generic constraint handling frameworks using four different swarm optimizers; particle swarm, gravitational search, a hybrid algorithm and differential evolution. It is shown that the new framework produces superior results compared to the established frameworks for all four swarm algorithms tested. Finally, the framework is applied to an aerodynamic shape optimization design problem where a shock-free solution is obtained.     

原油一次加工过程的多目标调度优化

在制定原油一次加工过程详细调度时,往往需要考虑多个优化目标。本文提出一种基于改进的骨干粒子群算法和II代非支配遗传算法协同进化的双种群算法,并通过Pareto差熵控制种群的交流,优化了供油罐使用成本、供油罐的切换成本、管道中原油混合成本以及供油罐罐底混合成本4个目标。通过一个工业实例,将本文算法与现有的几种具有代表性的进化多目标优化算法进行对比,验证本文算法的可行性和有效性。     

A surrogate assisted parallel multiobjective evolutionary algorithm for robust engineering design

A number of multi-objective evolutionary algorithms have been proposed in recent years and many of them have been used to solve engineering design optimization problems. However, designs need to be robust for real-life implementation, i.e. performance should not degrade substantially under expected variations in the variable values or operating conditions. Solutions of constrained robust design optimization problems should not be too close to the constraint boundaries so that they remain feasible under expected variations. A robust design optimization problem is far more computationally expensive than a design optimization problem as neighbourhood assessments of every solution are required to compute the performance variance and to ensure neighbourhood feasibility. A framework for robust design optimization using a surrogate model for neighbourhood assessments is introduced in this article. The robust design optimization problem is modelled as a multi-objective optimization problem with the aim of simultaneously maximizing performance and minimizing performance variance. A modified constraint-handling scheme is implemented to deal with neighbourhood feasibility. A radial basis function (RBF) network is used as a surrogate model and the accuracy of this model is maintained via periodic retraining. In addition to using surrogates to reduce computational time, the algorithm has been implemented on multiple processors using a master–slave topology. The preliminary results of two constrained robust design optimization problems indicate that substantial savings in the actual number of function evaluations are possible while maintaining an acceptable level of solution quality.     

Self-regulating and self-evolving particle swarm optimizer

In this article, a novel self-regulating and self-evolving particle swarm optimizer (SSPSO) is proposed. Learning from the idea of direction reversal, self-regulating behaviour is a modified position update rule for particles, according to which the algorithm improves the best position to accelerate convergence in situations where the traditional update rule does not work. Borrowing the idea of mutation from evolutionary computation, self-evolving behaviour acts on the current best particle in the swarm to prevent the algorithm from prematurely converging. The performance of SSPSO and four other improved particle swarm optimizers is numerically evaluated by unimodal, multimodal and rotated multimodal benchmark functions. The effectiveness of SSPSO in solving real-world problems is shown by the magnetic optimization of a Halbach-based permanent magnet machine. The results show that SSPSO has good convergence performance and high reliability, and is well matched to actual problems.     

Enhanced particle swarm optimization for size and shape optimization of truss structures

This article presents an enhanced particle swarm optimization (EPSO) algorithm for size and shape optimization of truss structures. The proposed EPSO introduces a particle categorization mechanism into the particle swarm optimization (PSO) to eliminate unnecessary structural analyses during the optimization process and improve the computational efficiency of the PSO-based structural optimization. The numerical investigation, including three benchmark truss optimization problems, examines the efficiency of the EPSO. The results demonstrate that the particle categorization mechanism greatly reduces the computational requirements of the PSO-based approaches while maintaining the original search capability of the algorithms in solving optimization problems with computationally cheap objective function and expensive constraints.     

基于多目标粒子群协同算法的状态参数优化

针对铅锌烧结过程综合透气性、烧结终点的优化具有强非线性、计算复杂等特点,提出了一种有效的多目标粒子群协同优化算法。首先,建立了有综合透气性、烧结终点两个目标的优化模型。接着,通过改进的约束比较方法、粒子极值选取方法,以及利用不同的粒子群来分别优化相应的变量,提出了一种改进的多目标粒子群协同优化算法。最后,利用提出的多目标优化算法进行综合透气性、烧结终点的优化。仿真结果表明,所提出的多目标优化算法能较好地解决综合透气性、烧结终点的优化问题。     

混流装配线调度问题的离散粒子群优化解

混流装配线调度问题是JIT生产中的一个重要问题。借鉴二进制遗传算法中的交叉操作过程，对传统的连续型粒子群算法进行改进，使其适用于离散问题的优化处理。然后以丰田公司的汽车组装调度函数作为目标函数，利用改进的离散粒子群算法进行求解。对比分析表明：新算法所得结果优于常用的目标追随法、遗传算法、模拟退火等方法。