A FLEXIBLE OBJECTIVE-CONSTRAINT APPROACH AND A NEW ALGORITHM FOR CONSTRUCTING THE PARETO FRONT OF MULTIOBJECTIVE OPTIMIZATION PROBLEMS

In this article, a novel scalarization technique, called the improved objective-constraint approach, is introduced to find efficient solutions of a given multiobjective programming problem. The presented scalarized problem extends the objective-constraint problem. It is demonstrated that how adding variables to the scalarized problem, can lead to find conditions for (weakly, properly) Pareto optimal solutions. Applying the obtained necessary and sufficient conditions, two algorithms for generating the Pareto front approximation of bi-objective and three-objective programming problems are designed. These algorithms are easy to implement and can achieve an even approximation of (weakly, properly) Pareto optimal solutions. These algorithms can be generalized for optimization problems with more than three criterion functions, too. The effectiveness and capability of the algorithms are demonstrated in test problems.

Multi-Objective Optimization Algorithm for Grouping Decision Variables Based on Extreme Point Pareto Frontier

The existing algorithms for solving multi-objective optimization problems fall into three main categories:Decomposition-based,dominance-based,and indicator-based.Traditional multi-objective optimization problemsmainly focus on objectives,treating decision variables as a total variable to solve the problem without consideringthe critical role of decision variables in objective optimization.As seen,a variety of decision variable groupingalgorithms have been proposed.However,these algorithms are relatively broad for the changes of most decisionvariables in the evolution process and are time-consuming in the process of finding the Pareto frontier.To solvethese problems,a multi-objective optimization algorithm for grouping decision variables based on extreme pointPareto frontier(MOEA-DV/EPF)is proposed.This algorithm adopts a preprocessing rule to solve the Paretooptimal solution set of extreme points generated by simultaneous evolution in various target directions,obtainsthe basic Pareto front surface to determine the convergence effect,and analyzes the convergence and distributioneffects of decision variables.In the later stages of algorithm optimization,different mutation strategies are adoptedaccording to the nature of the decision variables to speed up the rate of evolution to obtain excellent individuals,thusenhancing the performance of the algorithm.Evaluation validation of the test functions shows that this algorithmcan solve the multi-objective optimization problem more efficiently.

基于Pareto-GA多目标的企业管理系统优化研究——以某造纸厂为例

传统造纸厂管理优化通常只针对单一目标,忽略了质量和安全等重要方面。因此,提出了一种面向造纸厂的多个目标优化方法,并结合Pareto排序以及遗传算法搜索机制改进的Pareto-遗传算法作为求解方法,以实现对造纸厂监管系统的优化设计。研究结果显示,使用Schaffer's F6 Function进行测试时,改进的Pareto-遗传算法在72次迭代后达到最大适应度值0.93,优于其他两种算法。进一步将工期、成本、质量和安全多目标问题分解为两个子问题,成功获得3组Pareto最优解,为管理者提供不同需求下的优化方案。同时,提出的造纸厂管理系统优化设计方案能够提升造纸厂管理的效率和安全性,具有重要的理论价值和实际应用前景。

A Reference Vector-Assisted Many-Objective Optimization Algorithm with Adaptive Niche Dominance Relation

It is still a huge challenge for traditional Pareto-dominatedmany-objective optimization algorithms to solve manyobjective optimization problems because these algorithms hardly maintain the balance between convergence and diversity and can only find a group of solutions focused on a small area on the Pareto front,resulting in poor performance of those algorithms.For this reason,we propose a reference vector-assisted algorithmwith an adaptive niche dominance relation,for short MaOEA-AR.The new dominance relation forms a niche based on the angle between candidate solutions.By comparing these solutions,the solutionwith the best convergence is found to be the non-dominated solution to improve the selection pressure.In reproduction,a mutation strategy of k-bit crossover and hybrid mutation is used to generate high-quality offspring.On 23 test problems with up to 15-objective,we compared the proposed algorithm with five state-of-the-art algorithms.The experimental results verified that the proposed algorithm is competitive.

Bi-Objective Optimization: A Pareto Method with Analytical Solutions

Multiple objectives to be optimized simultaneously are prevalent in real-life problems. This paper develops a new Pareto Method for bi-objective optimization which yields analytical solutions. The Pareto optimal front is obtained in closed-form, enabling the derivation of various solutions in a convenient and efficient way. The advantage of analytical solution is the possibility of deriving accurate, exact and well-understood solutions, which is especially useful for policy analysis. An extension of the method to include multiple objectives is provided with the objectives being classified into two types. Such an extension expands the applicability of the developed techniques.

Pareto-Optimal Reinsurance Based on TVaR Premium Principle and Vajda Condition

Reinsurance is an effective risk management tool for insurers to stabilize their profitability. In a typical reinsurance treaty, an insurer cedes part of the loss to a reinsurer. As the insurer faces an increasing number of total losses in the insurance market, the insurer might expect the reinsurer to bear an increasing proportion of the total loss, that is the insurer might expect the reinsurer to pay an increasing proportion of the total claim amount when he faces an increasing number of total claims in the insurance market. Motivated by this, we study the optimal reinsurance problem under the Vajda condition. To prevent moral hazard and reflect the spirit of reinsurance, we assume that the retained loss function is increasing and the ceded loss function satisfies the Vajda condition. We derive the explicit expression of the optimal reinsurance under the TVaR risk measure and TVaR premium principle from the perspective of both an insurer and a reinsurer. Our results show that the explicit expression of the optimal reinsurance is in the form of two or three interconnected line segments. Under an additional mild constraint, we get the optimal parameters and find the optimal reinsurance strategy is full reinsurance, no reinsurance, stop loss reinsurance, or quota-share reinsurance. Finally, we gave an example to analyze the impact of the weighting factor on optimal reinsurance.

考虑Pareto最优的列车运行图与维修天窗协调优化

列车运行图与维修天窗之间的冲突始终无法避免,且维修天窗开设时间的长短显著影响列车总运行时间。针对此问题,综合考虑维修天窗对列车运行造成的限速约束、车站到发线数量约束等,建立列车总运行时间最小,以及维修天窗实际开设时长与理想时长总偏差最小的双目标混合整数规划模型;对困难约束设置中间辅助变量将模型线性化以提高求解效率,并设计约束转换算法求解双目标模型的Pareto最优;微观化处理铁路线,将站内资源和站间资源细化为一系列行车资源单元,得到更加符合实际旅客运输需求的运行图。以某地区铁路线夜间开行列车及维修天窗开设计划为研究背景,调用商业软件求解双目标函数模型的Pareto最优,并对双目标模型的最小支配解和最优支配解进行对比分析;针对最优支配解下的列车进入、离开行车资源单元的时间、停站作业时间及维修天窗的开始时间及开设时长,绘制列车运行图。求解结果表明:模型在满足维修天窗最小开设时长的同时,能够兼顾列车运行总时间最小和维修天窗开设时长更充裕。基于最优支配解绘制的列车运行图表明:微观路网下的列车运行时刻表优化结果更符合实际旅客运输生产作业需要。研究结果可为铁路运营管理部门进一步优化列车运行图编制与维修天窗开设提供参考。

基于Pareto Optimality的PPP项目三大主参与方效用探讨

在PPP项目的众多参与方中,选取项目发起人、SPC和贷款银行三个最主要的参与方,通过分析项目发起人与SPC、SPC与贷款银行和贷款银行与项目发起人之间的内在联系,在Pareto Optimality理论分析的基础上,得到三者之间两两优势互补的埃奇沃思方框图,并根据瓦尔拉斯均衡理论得出PPP项目能够实现帕累托最优,实现项目效用的最大化,最后拟画出PPP项目三大主参与方的效用可能性曲线。

PARETO FRONT CAPTURE USING DETERMINISTIC OPTIMIZATION METHODS IN MULTI-CRITERION AERODYNAMIC DESIGN

Deterministic optimization methods are combined with the Pareto front concept to solve multi-criterion design problems. The algorithm and the numerical implementation are applied to aerodynamic designs. Evolutionary algorithms （EAs） and the Pareto front concept are used to solve practical design problems in industry for its robustness in capturing convex, concave, discrete or discontinuous Pareto fronts of multi-objective optimization problems. However, the process is time-consuming. Therefore, deterministic optimization methods are introduced to capture the Pareto front, and the types of the captured Pareto front are explained. Numerical experiments show that the deterministic optimization method is a good alternative to EAs for capturing any convex and some concave Pareto fronts in multi-criterion aerodynamic optimization problems due to its efficiency.

Pareto解集旋转的分类多策略预测动态多目标优化

为更有效地解决Pareto解集(Pareto set,PS)旋转的动态多目标优化问题,提出PS旋转的分类多策略预测方法(rotation-based forecasting method,RFM)。将PS的旋转类型分为PS中心点旋转、PS原点旋转和非标准旋转;针对以上不同的PS旋转类型,自适应地选择合适的预测模型,建立不同点集的时间序列,为后续进化提供初始种群。引入拉丁超立方策略(Latin hypercube strategy,LHS)生成的随机种群与上述预测种群共同构建新种群,保证种群的多样性。为验证算法的有效性,将RFM算法与DNSGA-II、PPS、SPPS和MMP算法在8个标准的动态测试函数上进行实验对比。实验结果表明,RFM算法取得了6个最优IGD值、7个最优SP值、3个最优MS值,证明了RFM算法可以更有效地解决基于PS旋转的动态多目标优化问题。同时验证了RFM算法的普适性,在FDA系列函数上进行实验对比,实验结果表明,该算法在处理非旋转的动态多目标优化问题中仍具有较优性能。

Tourism Route Recommendation Based on A Multi-Objective Evolutionary Algorithm Using Two-Stage Decomposition and Pareto Layering

Tourism route planning is widely applied in the smart tourism field.The Pareto-optimal front obtained by the traditional multi-objective evolutionary algorithm exhibits long tails,sharp peaks and disconnected regions problems,which leads to uneven distribution and weak diversity of optimization solutions of tourism routes.Inspired by these limitations,we propose a multi-objective evolutionary algorithm for tourism route recommendation(MOTRR)with two-stage and Pareto layering based on decomposition.The method decomposes the multiobjective problem into several subproblems,and improves the distribution of solutions through a two-stage method.The crowding degree mechanism between extreme and intermediate populations is used in the two-stage method.The neighborhood is determined according to the weight of the subproblem for crossover mutation.Finally,Pareto layering is used to improve the updating efficiency and population diversity of the solution.The two-stage method is combined with the Pareto layering structure,which not only maintains the distribution and diversity of the algorithm,but also avoids the same solutions.Compared with several classical benchmark algorithms,the experimental results demonstrate competitive advantages on five test functions,hypervolume(HV)and inverted generational distance(IGD)metrics.Using the experimental results of real scenic spot datasets from two famous tourism social networking sites with vast amounts of users and large-scale online comments in Beijing,our proposed algorithm shows better distribution.It proves that the tourism routes recommended by our proposed algorithm have better distribution and diversity,so that the recommended routes can better meet the personalized needs of tourists.

Surrogate-Assisted Particle Swarm Optimization Algorithm With Pareto Active Learning for Expensive Multi-Objective Optimization

For multi-objective optimization problems, particle swarm optimization(PSO) algorithm generally needs a large number of fitness evaluations to obtain the Pareto optimal solutions. However, it will become substantially time-consuming when handling computationally expensive fitness functions. In order to save the computational cost, a surrogate-assisted PSO with Pareto active learning is proposed. In real physical space(the objective functions are computationally expensive), PSO is used as an optimizer, and its optimization results are used to construct the surrogate models. In virtual space, objective functions are replaced by the cheaper surrogate models, PSO is viewed as a sampler to produce the candidate solutions. To enhance the quality of candidate solutions, a hybrid mutation sampling method based on the simulated evolution is proposed, which combines the advantage of fast convergence of PSO and implements mutation to increase diversity. Furthermore, ε-Pareto active learning(ε-PAL)method is employed to pre-select candidate solutions to guide PSO in the real physical space. However, little work has considered the method of determining parameter ε. Therefore, a greedy search method is presented to determine the value ofεwhere the number of active sampling is employed as the evaluation criteria of classification cost. Experimental studies involving application on a number of benchmark test problems and parameter determination for multi-input multi-output least squares support vector machines(MLSSVM) are given, in which the results demonstrate promising performance of the proposed algorithm compared with other representative multi-objective particle swarm optimization(MOPSO) algorithms.

Multi-objective optimization of membrane structures based on Pareto Genetic Algorithm

A multi-objective optimization method based on Pareto Genetic Algorithm is presented for shape design of membrane structures from a structural view point.Several non-dimensional variables are defined as optimization variables,which are decision factors of shapes of membrane structures.Three objectives are proposed including maximization of stiffness,maximum uniformity of stress and minimum reaction under external loads.Pareto Multi-objective Genetic Algorithm is introduced to solve the Pareto solutions.Consequently,the dependence of the optimality upon the optimization variables is derived to provide guidelines on how to determine design parameters.Moreover,several examples illustrate the proposed methods and applications.The study shows that the multi-objective optimization method in this paper is feasible and efficient for membrane structures;the research on Pareto solutions can provide explicit and useful guidelines for shape design of membrane structures.

A Survey of Evolutionary Algorithms for Multi-Objective Optimization Problems With Irregular Pareto Fronts

Evolutionary algorithms have been shown to be very successful in solving multi-objective optimization problems(MOPs).However,their performance often deteriorates when solving MOPs with irregular Pareto fronts.To remedy this issue,a large body of research has been performed in recent years and many new algorithms have been proposed.This paper provides a comprehensive survey of the research on MOPs with irregular Pareto fronts.We start with a brief introduction to the basic concepts,followed by a summary of the benchmark test problems with irregular problems,an analysis of the causes of the irregularity,and real-world optimization problems with irregular Pareto fronts.Then,a taxonomy of the existing methodologies for handling irregular problems is given and representative algorithms are reviewed with a discussion of their strengths and weaknesses.Finally,open challenges are pointed out and a few promising future directions are suggested.

Optimal setting and placement of FACTS devices using strength Pareto multi-objective evolutionary algorithm

This work proposes a novel approach for multi-type optimal placement of flexible AC transmission system(FACTS) devices so as to optimize multi-objective voltage stability problem. The current study discusses a way for locating and setting of thyristor controlled series capacitor(TCSC) and static var compensator(SVC) using the multi-objective optimization approach named strength pareto multi-objective evolutionary algorithm(SPMOEA). Maximization of the static voltage stability margin(SVSM) and minimizations of real power losses(RPL) and load voltage deviation(LVD) are taken as the goals or three objective functions, when optimally locating multi-type FACTS devices. The performance and effectiveness of the proposed approach has been validated by the simulation results of the IEEE 30-bus and IEEE 118-bus test systems. The proposed approach is compared with non-dominated sorting particle swarm optimization(NSPSO) algorithm. This comparison confirms the usefulness of the multi-objective proposed technique that makes it promising for determination of combinatorial problems of FACTS devices location and setting in large scale power systems.

Sufficient Optimality Conditions for Multiobjective Programming Involving (V, ρ)h,ψ-type Ⅰ Functions

New classes of functions namely (V, ρ)_(h,φ)-type I, quasi (V, ρ)_(h,φ)-type I and pseudo (V, ρ)_(h,φ)-type I functions are defined for multiobjective programming problem by using BenTal's generalized algebraic operation. The examples of (V, ρ)_(h,φ)-type I functions are given. The sufficient optimality conditions are obtained for multi-objective programming problem involving above new generalized convexity.

Genetic algorithm for pareto optimum-based route selection

A quality of service （QoS） or constraint-based routing selection needs to find a path subject to multiple constraints through a network. The problem of finding such a path is known as the multi-constrained path （MCP） problem, and has been proven to be NP-complete that cannot be exactly solved in a polynomial time. The NPC problem is converted into a multiobjective optimization problem with constraints to be solved with a genetic algorithm. Based on the Pareto optimum, a constrained routing computation method is proposed to generate a set of nondominated optimal routes with the genetic algorithm mechanism. The convergence and time complexity of the novel algorithm is analyzed. Experimental results show that multiobjective evolution is highly responsive and competent for the Pareto optimum-based route selection. When this method is applied to a MPLS and metropolitan-area network, it will be capable of optimizing the transmission performance.

基于AG-MOPSO的含风电配电网无功优化

针对风电机组并网出力的不确定性,采用基于概率发生的场景分析法将不确定性模型转换为不同发生概率的多场景问题,建立以有功网损和电压偏差最小为目标的无功优化模型。针对传统方法得到的Pareto前沿多样性较差的问题,提出基于自适应网格的多目标粒子群优化AG-MOPSO(adaptive grid multi-objective particle swarm optimization)算法。该算法采用自适应网格得到外部档案库中粒子的密度,并根据密度信息以轮盘赌机制选取全局最优粒子和维护外部存储库的规模,有效地保证了Pareto前沿分布的均匀性和多样性。运用该算法对含风电的IEEE 33节点系统进行无功优化计算,并与已有NSGA-Ⅱ算法进行比较,结果表明所提算法得到的Pareto前沿较好,验证了该模型和算法的可行性。

A genetic algorithm for the pareto optimal solution set of multi-objective shortest path problem

Unlike the shortest path problem that has only one optimal solution and can be solved in polynomial time, the muhi-objective shortest path problem （ MSPP ） has a set of pareto optimal solutions and cannot be solved in polynomial time. The present algorithms focused mainly on how to obtain a precisely pareto optimal solution for MSPP resulting in a long time to obtain multiple pareto optimal solutions with them. In order to obtain a set of satisfied solutions for MSPP in reasonable time to meet the demand of a decision maker, a genetic algo- rithm MSPP-GA is presented to solve the MSPP with typically competing objectives, cost and time, in this pa- per. The encoding of the solution and the operators such as crossover, mutation and selection are developed. The algorithm introduced pareto domination tournament and sharing based selection operator, which can not only directly search the pareto optimal frontier but also maintain the diversity of populations in the process of evolutionary computation. Experimental results show that MSPP-GA can obtain most efficient solutions distributed all along the pareto frontier in less time than an exact algorithm. The algorithm proposed in this paper provides a new and effective method of how to obtain the set of pareto optimal solutions for other multiple objective optimization problems in a short time.

基于Pareto控制的多目标PSO算法在铣削参数优化中的应用

工艺参数是影响零件加工质量和效率的关键因素,工艺参数的优化和调节是改善加工工艺的最有效方法。针对铣削加工参数优化问题,提出了一种基于应用实例的多目标切削参数优化方法。首先,以材料去除率、切削力和刀具寿命为目标函数,建立了统一的切削工艺参数多目标优化数学模型。随后,使用切削数据对目标函数进行组合,建立了该问题的数学模型,研究了适用的求解方法以获得最优解,并通过实验验证了参数优化的有效性。该方法可为加工参数的选择提供指导和依据。