On the effect of reference point in MOEA/D for multi-objective optimization

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) has continuously proven effective for multi-objective optimization. So far, the effect of weight vectors and scalarizing methods in MOEA/D has been intensively studied. However, the reference point which serves as the starting point of reference lines (determined by weight vectors) is yet to be well studied. This study aims to fill in this research gap. Ideally, the ideal point of a multi-objective problem could serve as the reference point, however, since the ideal point is often unknown beforehand, the reference point has to be estimated (or specified). In this study, the effect of the reference point specified in three representative manners, i.e., pessimistic, optimistic and dynamic (from optimistic to pessimistic), is examined on three sets of benchmark problems. Each set of the problems has different degrees of difficulty in convergence and spread. Experimental results show that (i) the reference point implicitly impacts the convergence and spread performance of MOEA/D; (ii) the pessimistic specification emphasizes more of exploiting existing regions and the optimistic specification emphasizes more of exploring new regions; (iii) the dynamic specification can strike a good balance between exploitation and exploration, exhibiting good performance for most of the test problems, and thus, is commended to use for new problems.     

Attributed multi-objective comprehensive learning particle swarm optimization for optimal security of networks

In particle swarm optimization (PSO) each particle uses its personal and global or local best positions by linear summation. However, it is very time consuming to find the global or local best positions in case of complex problems. To overcome this problem, we propose a new multi-objective variant of PSO called attributed multi-objective comprehensive learning particle swarm optimizer (A-MOCLPSO). In this technique, we do not use global or local best positions to modify the velocity of a particle; instead, we use the best position of a randomly selected particle from the whole population to update the velocity of each dimension. This method not only increases the speed of the algorithm but also searches in more promising areas of the search space. We perform an extensive experimentation on well-known benchmark problems such as Schaffer (SCH), Kursawa (KUR), and Zitzler–Deb–Thiele (ZDT) functions. The experiments show very convincing results when the proposed technique is compared with existing versions of PSO known as multi-objective comprehensive learning particle swarm optimizer (MOCLPSO) and multi-objective particle swarm optimization (MOPSO), as well as non-dominated sorting genetic algorithm II (NSGA-II). As a case study, we apply our proposed A-MOCLPSO algorithm on an attack tree model for the security hardening problem of a networked system in order to optimize the total security cost and the residual damage, and provide diverse solutions for the problem. The results of our experiments show that the proposed algorithm outperforms the previous solutions obtained for the security hardening problem using NSGA-II, as well as MOCLPSO for the same problem. Hence, the proposed algorithm can be considered as a strong alternative to solve multi-objective optimization problems.     

Solving a new multi-objective multi-route flexible flow line problem by multi-objective particle swarm optimization and NSGA-II

Mass production, meeting the increasing demands of the customers is a necessity. Such a production is mainly dependent on a factory manufacturing called flow line production. This paper deals with special type of production by the name of flexible manufacturing system, assuming the presence of multi processors in each station of a multi-station arrangement. The model debated in the paper possesses three objective functions, the first of which attempts to minimize the weighted delays. The second objective function tries to minimize the capital for the purchase of the processors at stations and the third objective function minimizes the capital dedicated to select the optimum processing route of parts. For the validation of the mathematical model, use has been made of NSAGAII and MOPSO approaches.     

改进MOEA/D算法求解多目标模糊柔性车间调度问题

针对模糊柔性作业车间调度问题中关于求解多目标优化的研究中,利用模糊数表示相关参数,以最小化最大完工时间、总机器负载和最大机器负载为优化目标,提出一种改进MOEA/D算法的权重向量和初始化种群,以优化全局更新配对策略的多目标分解进化算法（I-MOEA/D）,以提高算法寻优能力。与MOEA/D、NSGA-Ⅱ和NSGA-Ⅲ算法相比,该方法优于其他算法,同时引入企业工程实例进行分析,证明I-MOEA/D算法具备良好的收敛性和分布性。     

MOEA/D线性插入方向向量策略研究

MOEA/D具有良好的收敛性、均匀的分布性、求解效率高等优点,普遍应用于求解多目标优化问题.然而对于Pareto前端复杂的多目标优化问题,预先设定均匀的权重向量并不能够维持Pareto最优解集的良好分布性.本文,首先分析均匀分布的权重向量、均匀分布的搜索方向二者与均匀分布的解集之间的关系,提出一种新的权重向量设置方式;其次基于进化过程中解集的分布,提出线性插入搜索方向策略,并将其转换为对应的权重向量,同时在MOEA/D中周期性应用该策略调整搜索方向,获取分布均匀的解集;最后将该算法在WFG系列测试问题上进行性能测试,并采用世代距离指标(GD)、Spacing指标(S)、超体积指标(HV)对算法收敛性和多样性进行对比分析,实验结果表明,与原始的MOEA/D、使用均匀分布的搜索方向MOEA/D、使用预处理的M OEA/D、M OEA/D-DU相比,改进的算法求出解集的多样性极大提高,收敛性明显增强,解集的整体质量显著提高.     

The effects of asymmetric neighborhood assignment in the MOEA/D algorithm

The Multiobjective Evolutionary Algorithm Based on Decomposition (MOEA/D) is a very efficient multiobjective evolutionary algorithm introduced in recent years. This algorithm works by decomposing a multiobjective optimization problem to many scalar optimization problems and by assigning each specimen in the population to a specific subproblem. The MOEA/D algorithm transfers information between specimens assigned to the subproblems using a neighborhood relation.In this paper it is shown that parameter settings commonly used in the literature cause an asymmetric neighbor assignment which in turn affects the selective pressure and consequently causes the population to converge asymmetrically. The paper contains theoretical explanation of how this bias is caused as well as an experimental verification. The described effect is undesirable, because a multiobjective optimizer should not introduce asymmetries not present in the optimization problem. The paper gives some guidelines on how to avoid such artificial asymmetries.     

Modeling a mixed-integer-binary small-population evolutionary particle swarm algorithm for solving the optimal power flow problem in electric power systems

This research discusses the application of a mixed-integer-binary small-population-based evolutionary particle swarm optimization to the problem of optimal power flow, where the optimization problem has been formulated taking into account four decision variables simultaneously: active power (continuous), voltage generator (continuous), tap position on transformers (integer) and shunt devices (binary). The constraint handling technique used in the algorithm is based on a strategy to generate and keep the decision variables in feasible space through the heuristic operators. The heuristic operators are applied in the active power stage and the reactive power stage sequentially. Firstly, the heuristic operator for the power balance is computed in order to maintain the power balance constraint through a re-dispatch of the thermal units. Secondly, the heuristic operators for the limit of active power flows and the bus voltage constraint at each generator bus are executed through the sensitivity factors. The advantage of our approach is that the algorithm focuses the search of the decision variables on the feasible solution space, obtaining a better cost in the objective function. Such operators not only improve the quality of the final solutions but also significantly improve the convergence of the search process. The methodology is verified in several electric power systems.     

基于多目标进化算法的MOEA/D权重向量产生方法

在进化多目标优化研究领域,多目标优化是指对含有2个及以上目标的多目标问题的同时优化,其在近些年来受到越来越多的关注。随着MOEA/D的提出,基于聚合的多目标进化算法得到越来越多的研究,对MOEA/D算法的改进已有较多成果,但是很少有成果研究MOEA/D中权重的产生方法。提出一种使用多目标进化算法产生任意多个均匀分布的权重向量的方法,将其应用到MOEA/D,MSOPS和NSGA-III中,对这3个经典的基于聚合的多目标进化算法进行系统的比较研究。通过该类算法在DTLZ测试集、多目标旅行商问题MOTSP上的优化结果来分别研究该类算法在连续性问题、组合优化问题上的优化能力,以及使用矩形测试问题使得多目标进化算法的优化结果在决策空间可视化。实验结果表明,没有一个算法能适用于所有特性的问题。然而,MOEA/D采用不同聚合函数的两个算法MOEA/D_Tchebycheff和MOEA/D_PBI在多数情况下的性能比MSOPS和NSGA-III更好。     

Multi-objective optimal power flow using quasi-oppositional teaching learning based optimization

This paper describes teaching learning based optimization (TLBO) algorithm to solve multi-objective optimal power flow (MOOPF) problems while satisfying various operational constraints. To improve the convergence speed and quality of solution, quasi-oppositional based learning (QOBL) is incorporated in original TLBO algorithm. The proposed quasi-oppositional teaching learning based optimization (QOTLBO) approach is implemented on IEEE 30-bus system, Indian utility 62-bus system and IEEE 118-bus system to solve four different single objectives, namely fuel cost minimization, system power loss minimization and voltage stability index minimization and emission minimization; three bi-objectives optimization namely minimization of fuel cost and transmission loss; minimization of fuel cost and L-index and minimization of fuel cost and emission and one tri-objective optimization namely fuel cost, minimization of transmission losses and improvement of voltage stability simultaneously. In this article, the results obtained using the QOTLBO algorithm, is comparable with those of TLBO and other algorithms reported in the literature. The numerical results demonstrate the capabilities of the proposed approach to generate true and well-distributed Pareto optimal non-dominated solutions of the multi-objective OPF problem. The simulation results also show that the proposed approach produces better quality of the individual as well as compromising solutions than other algorithms.     

Particle swarm optimization with an aging leader and challengers algorithm for the solution of optimal power flow problem

Solution of optimal power flow (OPF) problem aims to optimize a selected objective function such as fuel cost, active power loss, total voltage deviation (TVD) etc. via optimal adjustment of the power system control variables while at the same time satisfying various equality and inequality constraints. In the present work, a particle swarm optimization with an aging leader and challengers (ALC-PSO) is applied for the solution of the OPF problem of power systems. The proposed approach is examined and tested on modified IEEE 30-bus and IEEE 118-bus test power system with different objectives that reflect minimization of fuel cost or active power loss or TVD. The simulation results demonstrate the effectiveness of the proposed approach compared with other evolutionary optimization techniques surfaced in recent state-of-the-art literature. Statistical analysis, presented in this paper, indicates the robustness of the proposed ALC-PSO algorithm.     

基于动态多策略差分进化模型的MOEA/D算法*

在基于分解技术的多目标进化算法的框架中,引入一种动态多策略差分进化模型。该模型在分析不同差分进化策略的特点基础上,选择了三种差分进化策略,并对每种策略分配一子种群。在进化过程中,依据每种策略对邻域更新的贡献度,动态的调整其子种群的大小。对比分析采用不同差分进化算法的性能,结果表明运用多个策略之间相互协同进化,有利于提高算法性能。将新算法同NSG-II和MOEA/D算法在LZ09系列基准函数上进行性能对比,实验结果显示该算法的收敛性和多样性均优于对比算法。将新应用于I型梁多目标优化设计问题中,获得的Pareto前沿均匀,且解集域较宽广,对比分析表明算法的工程实用性。     

Multiagent based differential evolution approach to optimal power flow

This paper proposes a new differential evolution approach named as multiagent based differential evolution (MADE) based on multiagent systems, for solving optimal power flow problem with non-smooth and non-convex generator fuel cost curves. This method integrates multiagent systems (MAS) and differential evolution (DE) algorithm. An agent in MADE represents an individual to DE and a candidate solution to the optimization problem. All agents live in a lattice like environment, with each agent fixed on a lattice point. In order to obtain optimal solution quickly, each agent competes and cooperates with its neighbors and it can also use knowledge. Making use of these agent-agent interaction and DE mechanism, MADE realizes the purpose of minimizing the value of objective function. MADE applied to optimal power flow is evaluated on 6 bus system and IEEE 30 bus system with different generator characteristics. Simulation results show that the proposed method converges to better solutions much faster than earlier reported approaches.     

Simultaneously solving the transfer line balancing and buffer allocation problems with a multi-objective approach

Two of the most researched problems on transfer line, transfer line balancing problem (TLBP) and buffer allocation problem (BAP), are usually solved separately, although they are closely interrelated. When machine tools have different reliability, the traditional balancing approaches lead to a deviation of the production rate from the actual throughput, which is used as the objective of the following optimization on BAP. This may not only reduce the solution space of BAP, but also bring about a biased overall result.In this paper, the simultaneous solution of these two problems is presented, which includes transfer line balancing problem, BAP, and selection of line configuration, machine tools and fixtures. Production rate computed through simulation software and total cost considering machine tools and buffer capacities are used as two objective functions. The problem is solved applying a multi-objective optimization approach. Two well-known evolutionary algorithms are considered: Non-dominated Sorting Genetic Algorithm-II (NSGA-II) and Multi-Objective Particle Swarm Optimization (MOPSO). A real case study related to automotive sector is used to demonstrate the validity of the proposed approach.     

An optimal control approach to the optical flow problem

The optical flow problem is reduced to an optimal control problem governed by a linear parabolic equation having the unknown velocity field (the optical flow) as drift term. This model is derived from a new assumption, that is, the brightness intensity is conserved on a moving pattern driven by a Gaussian stochastic process. The optimality conditions are deduced by a passage to the limit technique in an approximating optimal control problem introduced for a regularization purpose. Finally, the controller uniqueness is addressed.     

Enhanced goal attainment method for solving multi-objective security-constrained optimal power flow considering dynamic thermal rating of lines

Security-constrained optimal power flow (SCOPF) is an important problem in power system operation. Dynamic thermal rating (DTR), as an effective method to increase transmission capacity of power systems, has been recently considered in some optimal power flow (OPF) and SCOPF models. Additionally, in today power systems, OPF problem involves various objectives leading to multi-objective OPF models. In this paper, a new multi-objective SCOPF model considering DTR of transmission lines is presented. In addition, a new multi-objective solution method is proposed to solve the multi-objective SCOPF problem. The proposed method is an enhanced version of goal attainment technique in which the search capability of this technique to cover borders of the Pareto frontier is enhanced. The proposed multi-objective DTR-included SCOPF model as well as the proposed multi-objective solution method are tested on the IEEE 118-bus test system and the obtained results are compared with the results of other alternatives.     

一种基于自适应选择策略的改进型MOEA/D算法

针对MOEA/D单纯使用邻域更新作为选择策略而造成的个体解的重复更新、缺乏全局适配性等问题,提出了一种兼及全局替换和局部更新策略的新算法,即基于自适应选择策略的改进型MOEA/D(MOEA/D-AS)。算法首先设计了一种新的基于最佳二分图匹配的选择策略(KMS),利用子问题和个体解的匹配关系,从全局角度实现精英个体集的最优选择;然后利用种群的进化信息构造一种匹配紊乱判断机制;最后利用紊乱判断机制,在综合分析邻域更新策略和KMS各自优势的基础上,使算法自适应地选择最合适的选择策略,以提高鲁棒性和优化效率。选取LZ09,DTLZ,CEC09等作为标准测试函数,将改进后的算法MOEA/D-AS与经典MOEA/D系列算法进行对比实验,并以Spread和IGD为性能评估指标。实验结果表明新算法具有更好的收敛性和分布性,验证了自适应选择策略能够有效地指导精英解的选择过程。     

Artificial bee colony algorithm solution for optimal reactive power flow

Artificial bee colony (ABC) algorithm is an optimization algorithm based on the intelligent foraging behavior of honeybee swarm. Optimal reactive power flow (ORPF) based on ABC algorithm to minimize active power loss in power systems is studied in this paper. The advantage of ABC algorithm is that it does not require these parameters, because it is very difficult to determine external parameters such as cross over rate and mutation rate as in case of genetic algorithm and differential evolution. The other advantage is that global search ability of the algorithm is implemented by introducing a neighborhood source production mechanism which is similar to mutation process. Because of these features, ABC algorithm attracts much attention in recent years and has been used successfully in many areas. ORPF problem is one of these areas. In this paper, proposed algorithm is tested on both standard IEEE 30-bus test system and IEEE 118-bus test system. To show the effectiveness of proposed algorithms, the obtained results are compared with different approaches as available in the literature.     

A multi-objective evolutionary approach for generator scheduling

This paper presents a novel, two-phase approach for optimal generation scheduling, taking into account the environmental issue of emission allowance trading in addition to the economic issue of operation cost. In the first phase, hourly-optimal scheduling is done to simultaneously minimize operation cost, emission, and transmission loss, while satisfying constraints such as power balance, spinning reserve and power generation limits. In the second phase, the minimum up/down time and ramp up/down rate constraints are considered, and a set of 24-h optimal schedules is obtained using the outputs of the first phase. Simulation results indicate effectiveness of the proposed approach.     

An improved artificial bee colony optimization algorithm based on orthogonal learning for optimal power flow problem

The increasing fuel price has led to high operational cost and therefore, advanced optimal dispatch schemes need to be developed to reduce the operational cost while maintaining the stability of grid. This study applies an improved heuristic approach, the improved Artificial Bee Colony (IABC) to optimal power flow (OPF) problem in electric power grids. Although original ABC has provided robust solutions for a range of problems, such as the university timetabling, training neural networks and optimal distributed generation allocation, its poor exploitation often causes solutions to be trapped in local minima. Therefore, in order to adjust the exploitation and exploration of ABC, the IABC based on the orthogonal learning is proposed. Orthogonal learning is a strategy to predict the best combination of two solution vectors based on limited trials instead of exhaustive trials, and to conduct deep search in the solution space. To assess the proposed method, two fuel cost objective functions with high non-linearity and non-convexity are selected for the OPF problem. The proposed IABC is verified by IEEE-30 and 118 bus test systems. In all case studies, the IABC has shown to consistently achieve a lower cost with smaller deviation over multiple runs than other modern heuristic optimization techniques. For example, the quadratic fuel cost with valve effect found by IABC for 30 bus system is 919.567 $/hour, saving 4.2% of original cost, with 0.666 standard deviation. Therefore, IABC can efficiently generate high quality solutions to nonlinear, nonconvex and mixed integer problems.     

A dynamic multi-objective approach for the reconfigurable multi-facility layout problem

The multi-facility layout problem involves the physical organization of departments inside several facilities, to allow flexible and efficient operations. This work studies the facility layout problem in a new perspective, considering a group of facilities, and two different concerns: the location of departments within a group of facilities, and the location of departments inside each facility itself. The problem is formulated as a Quadratic Programming Problem with multiple objectives and unequal areas, allowing layout reconfigurations in each planning period. The objectives of the model are: the minimization of costs (material handling inside facilities and between facilities, and re-layout); the maximization of adjacency between departments; and the minimization of the “unsuitability” of department positions and locations. This unsuitability measure is a new objective proposed in this work, to combine the characteristics of existing locations with the requirements of departments. The model was tested with data from the literature as well as with a problem inspired in a first tier supplier in the automotive industry. Preliminary results show that this work can be viewed as an innovative and promising integrated approach for tackling real, complex facility layout problems.