On the development of Bi-Level Integrated System Collaborative Optimization

Bi-Level Integrated System Collaborative Optimization (BLISCO) is a new multidisciplinary design optimization (MDO) method based on Bi-Level Integrated System Synthesis (BLISS) and Collaborative Optimization (CO). The key ideas of BLISCO are to replace compatibility constraint with the sum of coupled outputs as an integrated objective of subsystems and to decompose design variables into system design variables and subsystem design variables, while maintaining the collaborative mechanism of CO. One mathematical example and two engineering problems are used to test the effectiveness of BLISCO under the platform of iSIGHT^TM. Results from the test cases show that BLISCO has satisfactory convergence, accurate result and reliable robustness.     

基于NSGA-II 的具有多目标子学科的协同优化方法

针对子学科具有物理目标的多目标协同优化问题, 研究基于NSGA-II 的求解策略. 鉴于子学科个体满足约束可行性的进化过程与系统级分配期望值无关, 提出具有良好的可行性和多样性的初始种群生成方法, 以提高多目标子学科的计算效率和计算精度. 为了解决由一致性目标函数与物理目标函数的作用不同而造成的NSGA-II 非支配级排序困难, 提出将子学科一致性目标函数转化为子学科自身约束的策略. 最后, 利用工程算例对所提出方法的有效性进行了验证.

     

Multi-objective optimization problem under fuzzy rule constraints using particle swarm optimization

In this paper, a fuzzy multi-objective programming problem is considered where functional relationships between decision variables and objective functions are not completely known to us. Due to uncertainty in real decision situations sometimes it is difficult to find the exact functional relationship between objectives and decision variables. It is assumed that information source from where some knowledge may be obtained about the objective functions consists of a block of fuzzy if-then rules. In such situations, the decision making is difficult and the presence of multiple objectives gives rise to multi-objective optimization problem under fuzzy rule constraints. In order to tackle the problem, appropriate fuzzy reasoning schemes are used to determine crisp functional relationship between the objective functions and the decision variables. Thus a multi-objective optimization problem is formulated from the original fuzzy rule-based multi-objective optimization model. In order to solve the resultant problem, a deterministic single-objective non-linear optimization problem is reformulated with the help of fuzzy optimization technique. Finally, PSO (Particle Swarm Optimization) algorithm is employed to solve the resultant single-objective non-linear optimization model and the computation procedure is illustrated by means of numerical examples.     

Improving collaborative optimization for MDO problems with multi-objective subsystems

Structural and Multidisciplinary Optimization - This paper investigates collaborative optimization (CO) for multidisciplinary design optimization problems with multi-objective subsystems. A...     

一种具有全局快速寻优的多学科协同优化方法

针对协同优化算法迭代次数多、易收敛于局部极值点问题,提出一种全局快速寻优的协同优化算法。在系统级一致性等式约束中采用改进后松弛因子,改进动态松弛因子使优化设计点快速收敛于极值点,静态松弛因子使优化设计点跳出局部极值点,确保系统目标函数得到全局最优解;子学科目标函数由一致性目标函数和子学科最优目标函数两个部分以不同权重相加组成,考虑一致性的同时,又兼顾子学科独立性。采用减速器优化案例对改进协同优化算法进行验证。仿真结果表明,改进后算法在保证最大约束值较小的前提下,可快速得到全局最优解且鲁棒性好。     

Multi-objective genetic algorithms: problem difficulties and construction of test problems.

In this paper, we study the problem features that may cause a multi-objective genetic algorithm (GA) difficulty in converging to the true Pareto-optimal front. Identification of such features helps us develop difficult test problems for multi-objective optimization. Multi-objective test problems are constructed from single-objective optimization problems, thereby allowing known difficult features of single-objective problems (such as multi-modality, isolation, or deception) to be directly transferred to the corresponding multi-objective problem. In addition, test problems having features specific to multi-objective optimization are also constructed. More importantly, these difficult test problems will enable researchers to test their algorithms for specific aspects of multi-objective optimization.     

基于多点非均匀变异的多目标极值优化算法研究

多目标进化算法因其在解决含有多个矛盾目标函数的多目标优化问题中的强大处理能力,正受到越来越多的关注与研究。极值优化作为一种新型的进化算法,已在各种离散优化、连续优化测试函数以及工程优化问题中得到了较为成功的应用,但有关多目标EO算法的研究却十分有限。本文将采用Pareto优化的基本原理引入到极值优化算法中,提出一种求解连续多目标优化问题的基于多点非均匀变异的多目标极值优化算法。通过对六个国际公认的连续多目标优化测试函数的仿真实验结果表明：本文提出算法相比NSGA-II、 PAES、SPEA和SPEA2等经典多目标优化算法在收敛性和分布性方面均具有优势。     

A bi-objective model to optimize reliability and cost of system with a choice of redundancy strategies

Reliability problems are an important type of optimization problems that are motivated by different needs of real-world applications such as telecommunication systems, transformation systems, and electrical systems, so on. This paper studies a special type of these problems which is called redundancy allocation problem (RAP) and develops a bi-objective RAP (BORAP). The model includes non-repairable series–parallel systems in which the redundancy strategy is considered as a decision variable for individual subsystems. The objective functions of the model are (1) maximizing system reliability and (2) minimizing the system cost. Meanwhile, subject to system-level constraint, the best redundancy strategy among active or cold-standby, component type, and the redundancy level for each subsystem should be determined. To have a more practical model, we have also considered non-constant component hazard functions and imperfect switching of cold-standby redundant component. To solve the model, since RAP belong to the NP-hard class of the optimization problems, two effective multi-objective metaheuristic algorithms named non-dominated sorting genetic algorithms (NSGA-II) and multi-objective particle swarm optimization (MOPSO) are proposed. Finally, the performance of the algorithms is analyzed on a typical case and conclusions are demonstrated.     

Uniform adaptive scaling of equality and inequality constraints within hybrid evolutionary-cum-classical optimization

The holy grail of constrained optimization is the development of an efficient, scale invariant and generic constraint handling procedure. To address these, the present paper proposes a unified approach of constraint handling, which is capable of handling all inequality, equality and hybrid constraints in a coherent manner. The proposed method also automatically resolves the issue of constraint scaling which is critical in real world and engineering optimization problems. The proposed unified approach converts the single-objective constrained optimization problem into a multi-objective problem. Evolutionary multi-objective optimization is used to solve the modified bi-objective problem and to estimate the penalty parameter automatically. The constrained optimum is further improved using classical optimization. The efficiency of the proposed method is validated on a set of well-studied constrained test problems and compared against without using normalization technique to show the necessity of normalization. The results establish the importance of scaling , especially in constrained optimization and call for further investigation into its use in constrained optimization research.     

Estimation of the Pareto front in stochastic simulation through stochastic Kriging

Multi-objective optimization of simulated stochastic systems aims at estimating a representative set of Pareto optimal solutions and a common approach is to rely on metamodels to alleviate computational costs of the optimization process. In this article, both the objective and constraint functions are assumed to be smooth, highly non-linear and computationally expensive and are emulated by stochastic Kriging models. Then a novel global optimization algorithm, combing the expected hypervolume improvement of approximated Pareto front and the probability of feasibility of new points, is proposed to identify the Pareto front (set) with a minimal number of expensive simulations. The algorithm is suitable for the situations of having disconnected feasible regions and of having no feasible solution in initial design. Then, we also quantify the variability of estimated Pareto front caused by the intrinsic uncertainty of stochastic simulation using nonparametric bootstrapping method to better support decision making. One test function and an (s, S) inventory system experiment illustrate the potential and efficiency of the proposed sequential optimization algorithm for constrained multi-objective optimization problems in stochastic simulation, which is especially useful in Operations Research and Management Science.     

Evolutionary multi criteria design optimization of robot grippers

This paper explores the use of intelligent techniques to obtain optimum geometrical dimensions of a robot gripper. The optimization problem considered is a non-linear, complex, multi-constraint and multicriterion one. Three robot gripper configurations are optimized. The aim is to find Pareto optimal front for a problem that has five objective functions, nine constraints and seven variables. The problem is divided into three cases. Case 1 has first two objective functions, the case 2 considers last three objective functions and case 3 deals all the five objective functions. Intelligent optimization algorithms namely Multi-objective Genetic Algorithm (MOGA), Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) and Multi-objective Differential Evolution (MODE) are proposed to solve the problem. Normalized weighting objective functions method is used to select the best optimal solution from Pareto optimal front. Two multi-objective performance measures (solution spread measure (SSM) and ratio of non-dominated individuals (RNIs)) are used to evaluate the strength of the Pareto optimal fronts. Two more multi-objective performance measures namely optimizer overhead (OO) and algorithm effort are used to find the computational effort of MOGA, NSGA-II and MODE algorithms. The Pareto optimal fronts and results obtained from various techniques are compared and analyzed.     

基于改进遗传算法的生鲜多目标闭环物流网络模型

针对生鲜产品闭环物流网络中存在的经济成本高、碳排放量大、社会效益重视不足等问题,综合考虑退货量的不确定性,以经济成本最小、碳排放最小、社会效益最大为目标,建立了不确定条件下的生鲜多目标闭环物流网络模型。首先,利用改进的遗传算法(GA)求解该模型;然后,结合上海某生鲜企业运营管理数据,验证了模型的可行性;最后,将改进的GA的结果与粒子群优化(PSO)算法的结果对比,验证了算法的有效性,凸显了改进的GA在求解多目标的复杂约束问题时的优越性。算例结果表明,多目标优化满意度达到0.92,高于单目标优化满意度,展示了所提模型的有效性。     

Airfoil shape optimization using improved Multiobjective Territorial Particle Swarm algorithm with the objective of improving stall characteristics

In this paper, a new robust optimization technique with the ability of solving multi-objective constrained design optimization problems in aerodynamics is presented. This new technique is Multi-objective Territorial Particle Swarm Optimization (MOTPSO) algorithm in which diversity is actively preserved by avoiding overcrowded clusters of particles and encouraging broader exploration. Adaptively varying “territories” are formed around promising individuals to prevent many of the lesser individuals from premature clustering and encouraged them to explore new neighborhoods based on a hybrid self-social metric. Also, a new social interaction scheme is introduced which guided particles towards the weighted average of their “elite” neighbors’ best found positions instead of their own personal bests which in turn helps the particles to exploit the candidate local optima more effectively. The MOTPSO algorithm takes into account multiple objective functions using a Pareto-Based approach. The non-dominated solutions found by swarm are stored in an external archive and nearest neighbor density estimator method is used to select a leader for the individual particles in the swarm. Efficiency and robustness of the proposed algorithm is demonstrated using multiple traditional and newly-composed optimization benchmark functions and aerodynamic design problems. In final airfoil designs obtained from the Multi Objective Territorial Particle Swarm Optimization algorithm, separation point is delayed to make the airfoil less susceptible to stall in critical operating conditions and it also reveal an evident improvement over the test case airfoil across all objective functions presented.     

求解复杂耦合问题的多系统优化方法

由于复杂耦合问题具有多系统、多目标、多约束、多尺度和不确定等特点, 急需一种求解此类问题的高效 智能优化方法. 为此, 借鉴多种群进化算法的智能平行特征, 利用种群间进化信息的继承和交互作用, 提出一种多 系统优化方法. 首先以子种群来代表子系统的优化环境, 通过子系统内的进化操作求解各自的优化子问题; 然后通 过子系统间的迁移操作, 即利用变量共享、目标函数和约束条件的相似程度来实现子系统间的信息迁移与反馈, 加 速整个问题的全局优化; 最后将该方法应用到基准函数和具有多系统优化特征的三级供应链网络, 仿真实验表明 所提出的方法可行且有效.     

Implementation of polynomial multi-objective optimization in Mathematica

We describe implementation of main methods for solving polynomial multi-objective optimization problems by means of symbolic processing available in the programming language MATHEMATICA. Symbolic transformations of unevaluated expressions, representing objective functions and constraints, into the corresponding representation of the single-objective constrained problem are especially emphasized. We also describe a function for the verification of Pareto optimality conditions and a function for graphical illustration of Pareto optimal points and given constraint set.     

基于多点速度向量的多目标粒子群算法改进

针对多目标粒子群算法在高维条件下易早熟、迭代步骤数较多的问题,通过引入多点速度向量,提出一种基于多点速度向量的多目标粒子群改进算法,由于改进的多目标粒子群可以看成多个对于目标函数和当前种群的多目标最优点独立的速度和位置分量的叠加,减少了在目标函数最优值搜索之间相互的影响,从而有效地提高多目标粒子群在高维条件下的收敛速度以及准确性,理论证明这这种改进的有效性。实验结果证明了理论推导的正确性。     

An efficient evolutionary optimization algorithm for multiobjective distribution feeder reconfiguration

In this paper, a Multi-objective Modified Honey Bee Mating Optimization (MMHBMO) evolutionary algorithm is proposed to solve the multi-objective Distribution Feeder Reconfiguration (DFR). The real power loss, the number of the switching operations and the deviation of the voltage at each node are considered as the objective functions. Conventional algorithms for solving the multiobjective optimization problems convert the multiple objectives into a single objective using a vector of the user-predefined weights. This paper presents a new MHBMO algorithm for the DFR problem. In the proposed algorithm an external repository is utilized to save non-dominated solutions found during the search process. A fuzzy clustering technique is used to control the size of the repository within the limits because of the objective functions are not the same. The proposed algorithm is tested on a distribution test feeder.     

Formulation and parameter selection of multi-objective deterministic particle swarm for simulation-based optimization

Global derivative-free deterministic algorithms are particularly suitable for simulation-based optimization, where often the existence of multiple local optima cannot be excluded a priori, the derivatives of the objective functions are not available, and the evaluation of the objectives is computationally expensive, thus a statistical analysis of the optimization outcomes is not practicable. Among these algorithms, particle swarm optimization (PSO) is advantageous for the ease of implementation and the capability of providing good approximate solutions to the optimization problem at a reasonable computational cost. PSO has been introduced for single-objective problems and several extension to multi-objective optimization are available in the literature. The objective of the present work is the systematic assessment and selection of the most promising formulation and setup parameters of multi-objective deterministic particle swarm optimization (MODPSO) for simulation-based problems. A comparative study of six formulations (varying the definition of cognitive and social attractors) and three setting parameters (number of particles, initialization method, and coefficient set) is performed using 66 analytical test problems. The number of objective functions range from two to three and the number of variables from two to eight, as often encountered in simulation-based engineering problems. The desired Pareto fronts are convex, concave, continuous, and discontinuous. A full-factorial combination of formulations and parameters is investigated, leading to more than 60,000 optimization runs, and assessed by three performance metrics. The most promising MODPSO formulation/parameter is identified and applied to the hull-form optimization of a high-speed catamaran in realistic ocean conditions. Its performance is finally compared with four stochastic algorithms, namely three versions of multi-objective PSO and the genetic algorithm NSGA-II.     

一种改进的协同优化算法及其应用

针对协同优化方法收敛困难、优化效率低的问题,提出了一种改进的协同优化算法—ICO算法。通过引入自适应松弛因子将一致性等式约束转化为不等式约束,同时建立混合惩罚函数,将系统级约束优化问题转化为无约束优化问题,ICO算法较好地克服了传统协同优化算法难于收敛的缺点。标准算例实验结果表明,ICO算法能够有效提高优化的稳定性、可靠性和计算效率。优化结果显示了协同优化算法解决海洋供应船的设计优化问题的有效性,为解决更为复杂工程系统的设计优化问题奠定了基础。     

基于改进粒子群优化算法的约束多目标优化

针对约束多目标优化问题,提出一种改进的粒子群优化算法,采用距离量度和自适应惩罚函数相结合的约束处理技术,通过可行解比例有效均衡目标函数和约束条件,提高算法的边界搜索能力。定义新的k最近邻聚集密度,保持解集分布性,并将聚集密度和轮盘赌选择相结合选取全局最优粒子。仿真结果表明,该算法在Pareto解集均匀性及逼近性方面均具有优势。