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.     

Non-dominated archiving multi-colony ant algorithm for multi-objective optimization: Application to multi-purpose reservoir operation

Multi-objective optimization using heuristic methods has been established as a subdiscipline that combines the fields of heuristic computation and classical multiple criteria decision making. This article presents the Non-dominated Archiving Ant Colony Optimization (NA-ACO), which benefits from the concept of a multi-colony ant algorithm and incorporates a new information-exchange policy. In the proposed information-exchange policy, after a given number of iterations, different colonies exchange information on the assigned objective, resulting in a set of non-dominated solutions. The non-dominated solutions are moved into an offline archive for further pheromone updating. Performance of the NA-ACO is tested employing two well-known mathematical multi-objective benchmark problems. The results are promising and compare well with those of well-known NSGA-II algorithms used in real-world multi-objective-optimization problems. In addition, the optimization of reservoir operating policy with multiple objectives (i.e. flood control, hydropower generation and irrigation water supply) is considered and the associated Pareto front generated.     

Two efficient algorithms for constructing almost even approximations of the Pareto front in multi-objective optimization problems

In this article, two algorithms are proposed for constructing almost even approximations of the Pareto front of multi-objective optimization problems. The first algorithm is a hybrid of the ε-constraint and Pascoletti–Serafini scalarization methods for solving bi-objective problems. The second is a modification of the successive Pareto optimization (SPO) algorithm for solving three-objective problems. In these algorithms, the MATLAB fmincon solver is used to solve single-objective optimization problems, which returns a local optimal solution. Some metrics are considered to evaluate the quality of approximations obtained by the suggested algorithms on six test problems, and their results are compared with other algorithms (normal constraint, weighted constraint, SPO, differential evolution, multi-objective evolutionary algorithm/decomposition–differential evolution, non-dominated sorting genetic algorithm-II and S-metric selection evolutionary multi-objective algorithm). Experimental results show that the proposed algorithms provide almost even approximations of the whole Pareto front, and better quality of approximation and CPU time compared with established algorithms.     

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.     

A TABOO SEARCH BASED APPROACH TO FIND THE PARETO OPTIMAL SET IN MULTIPLE OBJECTIVE OPTIMIZATION

Taboo search is a heuristic optimization technique which works with a neighbourhood of solutions to optimize a given objective function. It is generally applied to single objective optimization problems. Taboo search has the potential for solving multiple objective optimization (MOO) problems, because it works with more than one solution at a time, and this gives it the opportunity to evaluate multiple objective functions simultaneously. In this paper, a taboo search based algorithm is developed to find Pareto optimal solutions in multiple objective optimization problems. The developed algorithm has been tested with a number of problems and compared with other techniques. Results obtained from this work have proved that a taboo search based algorithm can find Pareto optimal solutions in MOO effectively.     

带时间窗的时变多目标危险化学品道路运输路径优化

研究了危险化学品道路运输路径优化（VRP）问题,考虑了该问题的3个方面：1）路径选择涉及运输时间与运输风险两个目标;2）运输时间与风险具有时变特性;3）道路节点的服务时间窗限制。本文以运输时间和风险为多目标,综合考虑以上约束,建立了该问题的数学模型并设计了蚁群算法进行求解。求解结果表明该算法可以有效计算帕累托最优路径,决策者可结合实际问题和决策偏好作出最合适的决策,同时运输企业可依据不同时刻的运输结果制定车辆的出发时刻表,监管部门可通过合理规划各路径的服务时间窗及允许停留等待的节点来调节各路径运输时间及风险。     

Multi-objective hierarchical genetic algorithms for multilevel redundancy allocation optimization

Multilevel redundancy allocation optimization problems (MRAOPs) occur frequently when attempting to maximize the system reliability of a hierarchical system, and almost all complex engineering systems are hierarchical. Despite their practical significance, limited research has been done concerning the solving of simple MRAOPs. These problems are not only NP hard but also involve hierarchical design variables. Genetic algorithms (GAs) have been applied in solving MRAOPs, since they are computationally efficient in solving such problems, unlike exact methods, but their applications has been confined to single-objective formulation of MRAOPs. This paper proposes a multi-objective formulation of MRAOPs and a methodology for solving such problems. In this methodology, a hierarchical GA framework for multi-objective optimization is proposed by introducing hierarchical genotype encoding for design variables. In addition, we implement the proposed approach by integrating the hierarchical genotype encoding scheme with two popular multi-objective genetic algorithms (MOGAs)—the strength Pareto evolutionary genetic algorithm (SPEA2) and the non-dominated sorting genetic algorithm (NSGA-II). In the provided numerical examples, the proposed multi-objective hierarchical approach is applied to solve two hierarchical MRAOPs, a 4- and a 3-level problems. The proposed method is compared with a single-objective optimization method that uses a hierarchical genetic algorithm (HGA), also applied to solve the 3- and 4-level problems. The results show that a multi-objective hierarchical GA (MOHGA) that includes elitism and mechanism for diversity preserving performed better than a single-objective GA that only uses elitism, when solving large-scale MRAOPs. Additionally, the experimental results show that the proposed method with NSGA-II outperformed the proposed method with SPEA2 in finding useful Pareto optimal solution sets.     

A multi‐start threshold accepting algorithm for multiple objective continuous optimization problems

A multi‐start threshold accepting algorithm with an adaptive memory (MS‐TA) is proposed to solve multiple objective continuous optimization problems. The aim of this paper is to find efficiently multiple Pareto‐optimal solutions. Comparisons are carried out with multiple objective taboo search algorithm and genetic algorithm. Experiments on literature problems show that the proposed algorithm is more effective. The presented multi‐start adaptive algorithm improves the best‐known results by a significant margin. Copyright © 2010 John Wiley & Sons, Ltd.     

交叉变异的连续蚁群优化算法

研究了应用于连续空间优化问题的蚁群算法,给出了信息素的留存方式以及搜索策略.另外,针对蚁群算法易陷入局部最优的缺点,在最优蚂蚁周围进行了精细搜索,并加入了自适应的交叉变异算子,从而改进了蚁群算法的全局优化性能.数值仿真结果表明,该算法是一种有效的优化算法.     

A new algorithm for the robust optimization of rotor-bearing systems

This article presents a new algorithm for the robust optimization of rotor-bearing systems. The goal of the optimization problem is to find the values of a set of parameters for which the natural frequencies of the system are as far away as possible from the rotational speeds of the machine. To accomplish this, the penalization proposed by Ritto, Lopez, Sampaio, and Souza de Cursi in 2011 is employed. Since the rotor-bearing system is subject to uncertainties, such a penalization is modelled as a random variable. The robust optimization is performed by minimizing the expected value and variance of the penalization, resulting in a multi-objective optimization problem (MOP). The objective function of this MOP is known to be non-convex and it is shown that its resulting Pareto front (PF) is also non-convex. Thus, a new algorithm is proposed for solving the MOP: the normal boundary intersection (NBI) is employed to discretize the PF handling its non-convexity, and a global optimization algorithm based on a restart procedure and local searches are employed to minimize the NBI subproblems tackling the non-convexity of the objective function. A numerical analysis section shows the advantage of using the proposed algorithm over the weighted-sum (WS) and NSGA-II approaches. In comparison with the WS, the proposed approach obtains a much more even and useful set of Pareto points. Compared with the NSGA-II approach, the proposed algorithm provides a better approximation of the PF requiring much lower computational cost.     

Multi-objective optimization by a new hybridized method: applications to random mechanical systems

In this article two linear problems with random Gaussian loading are transformed into multi-objective optimization problems. The first problem is the design of a pillar geometry with respect to a compressive random load process. The second problem is the design of a truss structure with respect to a vertical random load process for several frequency bands. A new algorithm, motivated by the Pincus representation formula hybridized with the Nelder–Mead algorithm, is proposed to solve the two multi-objective optimization problems. To generate the Pareto curve, the normal boundary intersection method is used to produce a series of constrained single-objective optimizations. The second problem, depending on the frequency band of excitation, can have as Pareto curve a single point, a standard Pareto curve, or a discontinuous Pareto curve, a fact that has been reported here for the first time in the literature, to the best of the authors’ knowledge.     

System Reliability Optimization Using Gray Wolf Optimizer Algorithm

For the past two decades, nature‐inspired optimization algorithms have gained enormous popularity among the researchers. On the other hand, complex system reliability optimization problems, which are nonlinear programming problems in nature, are proved to be non‐deterministic polynomial‐time hard (NP‐hard) from a computational point of view. In this work, few complex reliability optimization problems are solved by using a very recent nature‐inspired metaheuristic called gray wolf optimizer (GWO) algorithm. GWO mimics the chasing, hunting, and the hierarchal behavior of gray wolves. The results obtained by GWO are compared with those of some recent and popular metaheuristic such as the cuckoo search algorithm, particle swarm optimization, ant colony optimization, and simulated annealing. This comparative study shows that the results obtained by GWO are either superior or competitive to the results that have been obtained by these well‐known metaheuristic mentioned earlier. Copyright © 2016 John Wiley & Sons, Ltd.     

Shape design of internal cooling passages within a turbine blade

The article concerns the optimization of the shape and location of non-circular passages cooling the blade of a gas turbine. To model the shape, four Bezier curves which form a closed profile of the passage were used. In order to match the shape of the passage to the blade profile, a technique was put forward to copy and scale the profile fragments into the component, and build the outline of the passage on the basis of them. For so-defined cooling passages, optimization calculations were carried out with a view to finding their optimal shape and location in terms of the assumed objectives. The task was solved as a multi-objective problem with the use of the Pareto method, for a cooling system composed of four and five passages. The tool employed for the optimization was the evolutionary algorithm. The article presents the impact of the population on the task convergence, and discusses the impact of different optimization objectives on the Pareto optimal solutions obtained. Due to the problem of different impacts of individual objectives on the position of the solution front which was noticed during the calculations, a two-step optimization procedure was introduced. Also, comparative optimization calculations for the scalar objective function were carried out and set up against the non-dominated solutions obtained in the Pareto approach. The optimization process resulted in a configuration of the cooling system that allows a significant reduction in the temperature of the blade and its thermal stress.     

Flower pollination algorithm: A novel approach for multiobjective optimization

Multiobjective design optimization problems require multiobjective optimization techniques to solve, and it is often very challenging to obtain high-quality Pareto fronts accurately. In this article, the recently developed flower pollination algorithm (FPA) is extended to solve multiobjective optimization problems. The proposed method is used to solve a set of multiobjective test functions and two bi-objective design benchmarks, and a comparison of the proposed algorithm with other algorithms has been made, which shows that the FPA is efficient with a good convergence rate. Finally, the importance for further parametric studies and theoretical analysis is highlighted and discussed.     

求解连续空间优化问题的扩展粒子蚁群算法

扩展蚁群算法是蚁群算法创始人Dorigo提出的一种用于求解连续空间优化问题的最新蚁群算法,但该算法的收敛速度参数和局部搜索参数取值缺乏理论指导,因此其性能受算法参数影响较大.本文提出一种求解连续空间优化的扩展粒子蚁群算法,将粒子群算法嵌入到扩展蚁群算法中用于在线优化扩展蚁群算法参数,减少了参数人为调整的盲目性.从而改善扩展蚁群算法的寻径行为.通过将本文提出的算法与遗传算法、克隆选择算法、蚁群算法、扩展蚁群算法对5种典型测试函数优化的结果对比表明,本文算法在搜索速度和全局搜索能力方面均优于其它算法.     

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。     

基于蚁群算法的多值属性系统测试序列优化

针对系统故障诊断中的多值属性系统测试序列优化问题,该文提出一种改进的蚁群算法,将成功运用在二值属性系统中的蚁群算法扩展到多值属性系统中,根据多值属性系统特点,设计相应的状态转移规则和信息素更新机制,并采用蚁群算法和遗传算法相融合的联合优化策略,解决了多值属性系统的序列优化问题,为多值属性系统的测试优化问题提供了一条新的解决途径。     

Multi-objective scheduling problems: Determination of pruned Pareto sets

There are often multiple competing objectives for industrial scheduling and production planning problems. Two practical methods are presented to efficiently identify promising solutions from among a Pareto optimal set for multi-objective scheduling problems. Generally, multi-objective optimization problems can be solved by combining the objectives into a single objective using equivalent cost conversions, utility theory, etc., or by determination of a Pareto optimal set. Pareto optimal sets or representative subsets can be found by using a multi-objective genetic algorithm or by other means. Then, in practice, the decision maker ultimately has to select one solution from this set for system implementation. However, the Pareto optimal set is often large and cumbersome, making the post-Pareto analysis phase potentially difficult, especially as the number of objectives increase. Our research involves the post Pareto analysis phase, and two methods are presented to filter the Pareto optimal set to determine a subset of promising or desirable solutions. The first method is pruning using non-numerical objective function ranking preferences. The second approach involves pruning by using data clustering. The k-means algorithm is used to find clusters of similar solutions in the Pareto optimal set. The clustered data allows the decision maker to have just k general solutions from which to choose. These methods are general, and they are demonstrated using two multi-objective problems involving the scheduling of the bottleneck operation of a printed wiring board manufacturing line and a more general scheduling problem.     

An extension of the directed search domain algorithm to bilevel optimization

A method is developed for generating a well-distributed Pareto set for the upper level in bilevel multiobjective optimization. The approach is based on the Directed Search Domain (DSD) algorithm, which is a classical approach for generation of a quasi-evenly distributed Pareto set in multiobjective optimization. The approach contains a double-layer optimizer designed in a specific way under the framework of the DSD method. The double-layer optimizer is based on bilevel single-objective optimization and aims to find a unique optimal Pareto solution rather than generate the whole Pareto frontier on the lower level in order to improve the optimization efficiency. The proposed bilevel DSD approach is verified on several test cases, and a relevant comparison against another classical approach is made. It is shown that the approach can generate a quasi-evenly distributed Pareto set for the upper level with relatively low time consumption.     

Practical solutions for multi-objective optimization: An application to system reliability design problems

For multiple-objective optimization problems, a common solution methodology is to determine a Pareto optimal set. Unfortunately, these sets are often large and can become difficult to comprehend and consider. Two methods are presented as practical approaches to reduce the size of the Pareto optimal set for multiple-objective system reliability design problems. The first method is a pseudo-ranking scheme that helps the decision maker select solutions that reflect his/her objective function priorities. In the second approach, we used data mining clustering techniques to group the data by using the k-means algorithm to find clusters of similar solutions. This provides the decision maker with just k general solutions to choose from. With this second method, from the clustered Pareto optimal set, we attempted to find solutions which are likely to be more relevant to the decision maker. These are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. To demonstrate how these methods work, the well-known redundancy allocation problem was solved as a multiple objective problem by using the NSGA genetic algorithm to initially find the Pareto optimal solutions, and then, the two proposed methods are applied to prune the Pareto set.