Genetic algorithms optimization for normalized normal constraint method under Pareto construction

This paper presents the resolution of multiobjective optimization problems as a tool in engineering design. In the literature, the solutions of this problems are based on the Pareto frontier construction. Therefore, substantial efforts have been made in recent years to develop methods for the construction of Pareto frontiers that guarantee uniform distribution and exclude the non-Pareto and local Pareto points. The normalized normal constraint is a recent contribution that generates a well-distributed Pareto frontier. Nevertheless, these methods are susceptible of improvement or modifications to obtain the same level of results more efficiently. This paper proposes a modification of the original normalized normal constraint method using a genetic algorithms in the optimization task. The results presented in this paper show a suitable behavior for the genetic algorithms method compared to classical Gauss–Newton optimization methods which are used by the original normalized normal constraint method.     

The normalized normal constraint method for generating the Pareto frontier

The authors recently proposed the normal constraint (NC) method for generating a set of evenly spaced solutions on a Pareto frontier – for multiobjective optimization problems. Since few methods offer this desirable characteristic, the new method can be of significant practical use in the choice of an optimal solution in a multiobjective setting. This papers specific contribution is two-fold. First, it presents a new formulation of the NC method that incorporates a critical linear mapping of the design objectives. This mapping has the desirable property that the resulting performance of the method is entirely independent of the design objectives scales. We address here the fact that scaling issues can pose formidable difficulties. Secondly, the notion of a Pareto filter is presented and an algorithm thereof is developed. As its name suggests, a Pareto filter is an algorithm that retains only the global Pareto points, given a set of points in objective space. As is explained in the paper, the Pareto filter is useful in the application of the NC and other methods. Numerical examples are provided.     

Pareto frontier for simultaneous structural and manufacturing optimization of a composite part

Optimized design of composite structures requires simultaneous optimization of structural performance and manufacturing process. Such a challenge calls for a multi-objective optimization. Here, a generating multi-objective optimization method called normalized normal constraint method, which attains a set of optimal solutions and allows the designer to explore design alternatives before making the final decision, is coupled with a local-global search called constrained globalized bounded Nelder–Mead method. The proposed approach is applied to the design of a Z-shaped composite bracket for optimization of structural and manufacturing objectives. Comparison of the results with non-dominated sorting genetic algorithm (NSGA-II) shows that when only a small number of function evaluations are possible and a few Pareto optima are desired, the proposed method outperforms NSGA-II in terms of convergence to the true Pareto frontier. The results are validated by an enumeration search and by exploring the neighbourhood of the final solutions.     

Optimization of sonic crystal attenuation properties by <Emphasis Type="Italic">ev-MOGA</Emphasis> multiobjective evolutionary algorithm

This paper shows a promising method for acoustic barrier design using a new acoustic material called Sonic Crystals (SCs). The configuration of these SCs is set as a multiobjective optimization problem which is very difficult to solve with conventional optimization techniques. The paper presents a new parallel implementation of a Multiobjective Evolutionary Algorithm called ev-MOGA (also known as ) and its application in a complex design problem. ev-MOGA algorithm has been designed to converge towards a reduced, but well distributed, representation of the Pareto Front (solution of the multiobjective optimization problem). The algorithm is presented in detail and its most important properties are discussed. To reduce the ev-MOGA computational cost when objective functions are substantial, a basic parallelization has been implemented on a distributed platform. Partially supported by MEC (Spanish Government) and FEDER funds: projects DPI2005-07835, MAT2006-03097 and Generalitat Valenciana (Spain) projects GV06/026, GV/2007/191.     

Imparting desired attributes in structural design by means of multi-objective optimization

Commonly available optimization methods typically produce a single optimal design as a constrained minimum of a particular objective function. However, in engineering design practice it is quite often important to explore as much of the design space as possible, with respect to many attributes, to discover what behaviors are possible and not possible within the initially adopted design concept. This paper shows that the very simple method of the sum of weighted objectives is useful for such exploration. By geometrical argument it is demonstrated that if every weighting coefficient is allowed to change its magnitude and its sign then the method returns a set of designs that are all feasible, diverse in their attributes, and include the Pareto and non-Pareto solutions, at least for convex cases. Numerical examples in the paper include the case of an aircraft wing structural box with thousands of degrees of freedom and constraints, and over 100 design variables, whose attributes are structural mass, volume, displacement, and frequency. The weighted coefficients method is inherently suitable for parallel, coarse-grained implementation that enables exploration of the design space in the elapsed time of a single structural optimization.     

Pareto frontier exploration in multiobjective topology optimization using adaptive weighting and point selection schemes

Topology optimization has been used in many industries and applied to a variety of design problems. In real-world engineering design problems, topology optimization problems often include a number of conflicting objective functions, such to achieve maximum stiffness and minimum mass of a design target. The existence of conflicting objective functions causes the results of the topology optimization problem to appear as a set of non-dominated solutions, called a Pareto-optimal solution set. Within such a solution set, a design engineer can easily choose the particular solution that best meets the needs of the design problem at hand. Pareto-optimal solution sets can provide useful insights that enable the structural features corresponding to a certain objective function to be isolated and explored. This paper proposes a new Pareto frontier exploration methodology for multiobjective topology optimization problems. In our methodology, a level set-based topology optimization method for a single-objective function is extended for use in multiobjective problems, using a population-based approach in which multiple points in the objective space are updated and moved to the Pareto frontier. The following two schemes are introduced so that Pareto-optimal solution sets can be efficiently obtained. First, weighting coefficients are adaptively determined considering the relative position of each point. Second, points in sparsely populated areas are selected and their neighborhoods are explored. Several numerical examples are provided to illustrate the effectiveness of the proposed method.     

An efficient multi-objective optimization method for black-box functions using sequential approximate technique

Multi-objective optimization problems in practical engineering usually involve expensive black-box functions. How to reduce the number of function evaluations at a good approximation of Pareto frontier has been a crucial issue. To this aim, an efficient multi-objective optimization method based on a sequential approximate technique is suggested in this paper. In each iteration, according to the prediction of radial basis function with a micro multi-objective genetic algorithm, an extended trust region updating strategy is adopted to adjust the design region, a sample inheriting strategy is presented to reduce the number of new function evaluations, and then a local-densifying strategy is proposed to improve the accuracy of approximations in concerned regions. At the end of each iteration, the obtained actual Pareto optimal points are stored in an external archive and are updated as the iteration process. The effect of the present method is demonstrated by eight test functions. Finally, it is employed to perform the structure optimization of a vehicle door.     

多目标优化设计在杀螺剂配方研究中的应用

针对某生物杀螺剂制作中多目标约束问题,提出了一种应用Pareto遗传算法来解决问题的优化方法。建立了用于多目标优化的适应度函数,使用排列选择方法将带约束的多目标问题转换为无约束优化问题;并根据计算中的收敛情况引入了适当的移民算子,改善了遗传算法的进化性能,得到了Pareto最优解集,成功地解决了该生物杀螺剂的最优配方问题。     

吞吐量和延时约束下的片上通信结构的Pareto空间优化

SoC中各IP核之间的互连结构是决定片上系统性能的关键因素.近年来,片上互连通信结构的配置与优化成为SoC通信综合的研究重点和热点,而已有方法优化SoC互连通信结构的仿真速度较慢,支持设计自动化的能力较差,使用的单目标优化算法无法解决多个性能目标之间的冲突.针对以上不足提出了吞吐量和延时约束下的片上互连通信结构的自动配置与优化的方法,该方法提出了片上总线互连通信结构模板,使用事务级通信仿真和多目标演化算法,探索吞吐量和延时约束下的多目标Pareto空间.与已有的先进Srinivasan方法相比,该方法的吞吐量提高10%,传输延迟降低17%,有效提高了SoC互连通信结构的优化质量.     

Constrained multi-objective optimization algorithms: Review and comparison with application in reinforced concrete structures

Engineering design problems are often multi-objective in nature, which means trade-offs are required between conflicting objectives. In this study, we examine the multi-objective algorithms for the optimal design of reinforced concrete structures. We begin with a review of multi-objective optimization approaches in general and then present a more focused review on multi-objective optimization of reinforced concrete structures. We note that the existing literature uses metaheuristic algorithms as the most common approaches to solve the multi-objective optimization problems. Other efficient approaches, such as derivative-free optimization and gradient-based methods, are often ignored in structural engineering discipline. This paper presents a multi-objective model for the optimal design of reinforced concrete beams where the optimal solution is interested in trade-off between cost and deflection. We then examine the efficiency of six established multi-objective optimization algorithms, including one method based on purely random point selection, on the design problem. Ranking and consistency of the result reveals a derivative-free optimization algorithm as the most efficient one.     

A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization

New challenges in engineering design lead to multiobjective (multicriteria) problems. In this context, the Pareto front supplies a set of solutions where the designer (decision-maker) has to look for the best choice according to his preferences. Visualization techniques often play a key role in helping decision-makers, but they have important restrictions for more than two-dimensional Pareto fronts. In this work, a new graphical representation, called Level Diagrams, for n-dimensional Pareto front analysis is proposed. Level Diagrams consists of representing each objective and design parameter on separate diagrams. This new technique is based on two key points: classification of Pareto front points according to their proximity to ideal points measured with a specific norm of normalized objectives (several norms can be used); and synchronization of objective and parameter diagrams. Some of the new possibilities for analyzing Pareto fronts are shown. Additionally, in order to introduce designer preferences, Level Diagrams can be coloured, so establishing a visual representation of preferences that can help the decision-maker. Finally, an example of a robust control design is presented - a benchmark proposed at the American Control Conference. This design is set as a six-dimensional multiobjective problem.     

Approximating Pareto frontier using a hybrid line search approach

The aggregation of objectives in multiple criteria programming is one of the simplest and widely used approach. But it is well known that this technique sometimes fail in different aspects for determining the Pareto frontier. This paper proposes a new approach for multicriteria optimization, which aggregates the objective functions and uses a line search method in order to locate an approximate efficient point. Once the first Pareto solution is obtained, a simplified version of the former one is used in the context of Pareto dominance to obtain a set of efficient points, which will assure a thorough distribution of solutions on the Pareto frontier. In the current form, the proposed technique is well suitable for problems having multiple objectives (it is not limited to bi-objective problems) and require the functions to be continuous twice differentiable. In order to assess the effectiveness of this approach, some experiments were performed and compared with two recent well known population-based metaheuristics namely ParEGO and NSGA II. When compared to ParEGO and NSGA II, the proposed approach not only assures a better convergence to the Pareto frontier but also illustrates a good distribution of solutions. From a computational point of view, both stages of the line search converge within a short time (average about 150 ms for the first stage and about 20 ms for the second stage). Apart from this, the proposed technique is very simple, easy to implement and use to solve multiobjective problems.     

An interactive fuzzy multi-objective optimization method for engineering design

The coupling of performance functions due to common design variables and uncertainties in an engineering design process will result in difficulties in optimization design problems, such as poor collaboration among design objectives and poor resolution of design conflicts. To handle these problems, a fuzzy interactive multi-objective optimization model is developed based on Pareto solutions, where the metric function and some additional constraints are added to ensure the collaboration among design objectives. The trade-off matrix at the Pareto solutions was developed, and the method for selecting weighting coefficients of optimization objectives is also provided. The proposed method can generate a Pareto optimal set with the maximum satisfaction degree and the minimum distance from ideal solution. The favorable optimal solution can be then selected from the Pareto optimal set by analyzing the trade-off matrix and collaborative sensitivity. Two examples are presented to illustrate the proposed method.     

Normal-boundary intersection based parametric multi-objective optimization of green sand mould system

In manufacturing engineering optimization, it is often that one encounters scenarios that are multi-objective (where each of the objectives portray different aspects of the problem). Thus, it is crucial for the engineer to have access to multiple solution choices before selecting of the best solution. In this work, a novel approach that merges meta-heuristic algorithms with the Normal Boundary Intersection (NBI) method is introduced. This method then is used generate optimal solution options to the green sand mould system problem. This NBI based method provides a near-uniform spread of the Pareto frontier in which multiple solutions with gradual trade-offs in the objectives are obtained. Some comparative studies were then carried out with the algorithms developed and used in this work and that from some previous work. Analysis on the performance as well as the quality of the solutions produced by the algorithms is presented here.     

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。     

Ant colony approaches for multiobjective structural optimization problems with a cardinality constraint

Two Ant Colony Optimization algorithms are proposed to tackle multiobjective structural optimization problems with an additional constraint. A cardinality constraint is introduced in order to limit the number of distinct values of the design variables appearing in any candidate solution. Such constraint is directly enforced when an ant builds a candidate solution, while the other mechanical constraints are handled by means of an adaptive penalty method (APM). The test-problems are composed by structural optimization problems with discrete design variables, and the objectives are to minimize both the structure’s weight and its maximum nodal displacement. The Pareto sets generated in the computational experiments are evaluated by means of performance metrics, and the obtained designs are also compared with solutions available from single-objective studies in the literature.     

混合多目标粒子群优化算法在热精轧负荷分配优化中的应用

通过对热精轧负荷分配过程的分析,选取负荷均衡、板形良好和轧制功率最低为目标,建立了热精轧负荷分配多目标优化模型.为了提高多目标优化算法解集的分布性和收敛性,提出了一种混合多目标粒子群优化算法(HMOPSO),该算法根据Pareto支配关系得到Pareto前沿进而保证种群收敛;采用分解策略维护外部存档,该策略首先根据Pareto前沿求出上界点对目标空间进行归一化处理,然后对种群进行分区处理进而保证种群的分布性能.仿真结果表明,HMOPSO的收敛性和分布性都好于MOPSO和d MOPSO;采用模糊多属性决策的方法从Pareto最优解集中选择一个Pareto最优解,通过与经验负荷分配方法相比,表明该Pareto最优解可以使轧制方案更加合理.     

An evolutionary algorithm approach to generate distinct sets of non-dominated solutions for wicked problems

Many engineering design problems must optimize multiple objectives. While many objectives are explicit and can be mathematically modeled, some goals are subjective and cannot be included in a mathematical model of the optimization problem. A set of alternative non-dominated fronts that represent multiple optima for problem solution can be identified to provide insight about the decision space and to provide options and alternatives for decision-making. This paper presents a new algorithm, the Multi-objective Niching Co-evolutionary Algorithm (MNCA) that identifies distinct sets of non-dominated solutions which are maximally different in their decision vectors and are located in the same non-inferior regions of a Pareto front. MNCA is demonstrated to identify a set of non-dominated fronts with maximum difference in decision vectors for a set of real-valued problems.     

An extended study on multi-objective security games

The burgeoning area of security games has focused on real-world domains where security agencies protect critical infrastructure from a diverse set of adaptive adversaries. In such domains, decision makers have multiple competing objectives they must consider which may take different forms that are not readily comparable including safety, cost, and public perception. Thus, it can be difficult to know how to weigh the different objectives when deciding on a security strategy. To address the challenges of these domains, we propose a fundamentally different solution concept, multi-objective security games (MOSGs). Instead of a single optimal solution, MOSGs have a set of Pareto optimal (non-dominated) solutions referred to as the Pareto frontier, which can be generated by solving a sequence of constrained single-objective optimization problems (CSOPs). The Pareto frontier allows the decision maker to analyze the tradeoffs that exist between the multiple objectives. Our contributions include: (i) an algorithm, Iterative-ε-Constraints,, for generating the sequence of CSOPs; (ii) an exact approach for solving an mixed-integer linear program (MILP) formulation of a CSOP; (iii) heuristics that achieve speed up by exploiting the structure of security games to further constrain the MILP; (iv) an approximate approach for solving a CSOP built off those same heuristics, increasing the scalability of our approach with quality guarantees. Additional contributions of this paper include proofs on the level of approximation, detailed experimental evaluation of the proposed approaches and heuristics, as well as a discussion on techniques for visualizing the Pareto frontier.