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1

Multi-objective task offloading optimization using deep reinforcement learning with resource distribution clustering

Yang Qin, Yoo Sang-Jo

[NRF 연계] 한국통신학회 ICT Express Vol.11 No.4 2025.08 pp.734-742

※ 협약을 통해 무료로 제공되는 자료로, 원문이용 방식은 연계기관의 정책을 따르고 있습니다.

원문보기

Task offloading in multi-access edge computing (MEC) systems is critical for managing computational tasks in dynamic urban environments. Existing strategies face challenges such as high communication overheads and regional performance deviations including centralized and distributed methods. Clustering approaches have been explored to address these issues, yet they often rely on physical proximity to form clusters, overlooking the variability in task rate distributions across edges. To overcome these limitations, this paper proposes a graph-driven inter-cluster resource distribution (GIRD) clustering scheme that clusters edge nodes based on task request distribution and computing resource status, ensuring similar resource utilization across clusters. Building on this, a proximal policy optimization (PPO)-enabled intra-cluster task offloading algorithm (PITO) is introduced to determine one execution server for task offloading?either an edge server within a cluster or a cloud server?using various network state information. This dynamic decision-making process optimizes a multi-objective function that includes task processing delay, consumed energy, success rate, and cloud cost. Simulation results demonstrate the proposed GIRD-PITO framework achieves superior task success rates, reduced delays, and improved regional performance fairness, making it a promising solution for large-scale MEC systems. 2018 The Korean Institute of Communications and Information Sciences. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

2

다목적 최적화 알고리듬을 이용한 SAGD 공법 운영조건 최적화

이재윤, 민배현, 조수렴, 김재준

[NRF 연계] 한국자원공학회 한국자원공학회지 Vol.55 No.5 2018.10 pp.421-430

※ 협약을 통해 무료로 제공되는 자료로, 원문이용 방식은 연계기관의 정책을 따르고 있습니다.

원문보기

이 논문은 다목적 최적화 알고리듬을 이용하여 오일샌드 저류층에서 생산성이 높으면서도 최적의 에너지효 율을 갖는 SAGD(Steam Assisted Gravity Drainage) 공법의 운영조건을 선정하는 기법을 제안한다. 기존의 SAGD 공 법 최적화 연구들은 경제성인자가 고정된 단목적 최적화에 집중하였다. 이 연구에서는 회수율 최대화와 누적증기오 일비 최소화 관점에서 비지배관계에 놓이는 최적 생산운영 시나리오를 선정하였다. 이를 통하여 특정 목적함수에 편 향된 하나의 최종해만을 도출하는 기존 방법의 한계를 개선하였다. 제안한 방법으로 선정된 최적운영 시나리오들은 기존 방법과 달리 유가상황 및 비용 변화에 따라 추가적인 최적화 작업 없이 광구운영 의사결정에 유용하게 활용할 수 있다.

This paper proposes multi-objective optimization of the Steam Assisted Gravity Drainage (SAGD) process for improving the energy efficiency and recovery factor of oil sand reservoirs. Previous studies conducted on optimizing the SAGD process have focused on single-objective optimization with fixed economic factors. In this study, multiple trade-off operating scenarios were selected by applying a multi-objective optimization algorithm that aims at maximizing the recovery factor and minimizing the cumulative steam-oil ratio for efficiently addressing volatile market conditions. Thus, the proposed method can overcome the limitations of conventional optimization methods that not only yield a single solution based on a particular objective function but also are hard to adapt to fluctuating oil prices. Furthermore, the proposed method can provide optimum trade-off operating scenarios, and hence can aid in planning operating (i.e., marketing) strategies according to the variation in oil prices and operating costs, without the need for an additional optimization process.

3

다목적 최적화 기반 CO2 수송-주입-저장 통합 시스템의 기본 설계

최수인, 김태우, 최병인, 민배현

[NRF 연계] 한국자원공학회 한국자원공학회지 Vol.61 No.5 2024.10 pp.347-362

※ 협약을 통해 무료로 제공되는 자료로, 원문이용 방식은 연계기관의 정책을 따르고 있습니다.

원문보기

이 연구는 CO2 수송-주입-저장 통합 시스템의 기본 설계 단계에서 다목적 최적화에 기반한 설비및 운영 인자 선정 방법을 제안한다. CO2 지중저장 통합 시스템의 설계 인자에 대한 다목적 최적화를 수행하여 최적의 비지배 해집합인 파레토 최적면을 탐색함으로써 상호 비지배 관계인 등가의설계안을 다수 도출한다. 방법론의 성능은 대염수층 대상 CO2 지중저장 통합 시스템의 기본 설계에 적용하여 검증하였다. 결정변수는 CO2 송출 압력, 수송관 내경, 주입정 튜빙 내경, 주입정 상단에서의 CO2 온도 4종류를 설정하였고, 목적함수는 CO2 저장량, 주입정 공저압, 편익비용비 3종류를 설정하였다. 연구 결과, 전체 탐색공간 대비 약 5% 수준의 계산 비용으로 파레토 최적면을 도출하여 설계 인자 범위를 확인하였다. 또한, 최적화 목적함수에 대한 사전 선호도와 사후 선호도 적용영향을 비교 분석하였다. 제안하는 방법론은 향후 CO2 지중저장 통합 시스템 설계 과정에서 검토할 수 있는 우수해를 다수 제공할 수 있음을 확인하였다.

This study proposes a method based on multi-objective optimization to select facility and operational parameters for the basic design of an integrated CO2 transportation-injection-storage system. Multi- objective optimization was conducted to explore a Pareto-optimal front composed of non-dominated design solutions. The performance of the proposed approach was validated by applying it to CO2 storage at a saline aquifer. We set four decision variables (CO2 source pressure, pipeline inner diameter, injection well tubing inner diameter, and CO2 temperature at wellhead) and three objective functions (CO2 storage capacity, injection pressure, and benefit-cost ratio). The Pareto-optimal front was derived at approximately 5% of the computational cost compared to the exhaustive search. A comparative analysis was conducted to analyze the effects of prior and posterior preferences toward objectives on optimization performance. This method demonstrates the potential to provide effective solutions for designing an integrated CCS (carbon capture and storage) system.

4

다중최적화기법을 이용한 분포형 수문모형의 최적화 KCI 등재

김정호, 김태균

한국습지학회 한국습지학회지 제21권 제1호 2019.02 pp.1-8

※ 기관로그인 시 무료 이용이 가능합니다.

4,000원

본 연구에서는 다중최적화기법을 이용하여 2가지 수문학적 과정을 통하여 유출량을 산정하는 수문모형의 모형 최적화를 시도하였으며, 수문모형으로는 융설량과 유출량을 동시에 산정할 수 있는 분포형 수문모형인 HL-RDHM을 이용하였다. 대상유역으로는 융설량 자료를 수집할 수 있는 미국 콜로라도의 Durango River 유역을 선정하였다. 다중최적화기법으로는 MOSCEM을 활용하였으며, 융설과 관련된 매개변수 5개와 유출에 관련된 매개변수 13개를 선정하여 매개변수 보정과 수문모형 최적화를 시도하였다. 모형 최적화를 위해 2004 – 2005년의 자료가 활용되었고, 2001 – 2004년 자료를 이용하여 검증하였다. 융설량과 유출량을 동시에 최적화함으로써 RMSE 기준으로, 3개의 SNOTEL 지점에서 초기해에 의한 모의치 보다 7% - 40%까지 RMSE 오차를 줄일 수 있었고, 유출구의 USGS 관측점에서 초기해에 비해 약 40% 값이 개선됨을 확인하였다.

In this study, the multi-objective optimization method is attemped to optimize the hydrological model to estimate the runoff through two hydrological processes. HL-RDHM, a distributed hydrological model that can simultaneously estimate the amount of snowfall and runoff, was used as the distributed hydrological model. The Durango River basin in Colorado, USA, was selected as the watershed. MOSCEM was used as a multi-objective optimization method and parameter calibration and hydrologic model optimization were tried by selecting 5 parameters related to snow melting and 13 parameters related to runoff. Data from 2004 to 2005 were used to optimize the model and verified using data from 2001 to 2004. By optimizing both the amount of snow and the amount of runoff, the RMSE error can be reduced from 7% to 40% of the simulation value based on the initial solution at three SNOTEL points based on the RMSE. The USGS observation point of the outflow is improved about 40%.

5

基于混合进化算法的多目标路径优化问题

程娜, 崔荣

한국어정보학회 한국어정보학 제10권 1호 2008.06 pp.1-6

※ 기관로그인 시 무료 이용이 가능합니다.

4,000원

According to Genetic algorithms principle, the new hybrid evolutionary algorithm (HEA) is proposed in this paper by combining the Immune algorithm, Genetic algorithm and Pareto optimal solutions. The HEA has high convergence precision and improved the diversity of population. Multiple near optimization paths can be developed by the algorithm with multiobjective restriction, and satisfy to minimize the routing of transportation and the numbers of the vehicles. The HEA has been used to solve the vehicle routing problem, the results of simulation experiment show that the HEA can gain higher global convergence rate and higher speed.

6

4,000원

8

4,000원

In various machines used in industrial sites and transportation equipment, fastening structures of bolts and nuts are widely employed. However, conventional Steel sockets, classified as non-explosion-proof materials, have a high likelihood of generating sparks due to friction with components, which can lead to explosions or large-scale fires. To address this issue, this study developed a lightweight explosion-protection socket using AL-7075-T6 aluminum alloy, which is known for its excellent explosion-proof properties. However, due to the inherent characteristics of aluminum, it has lower rigidity compared to Steel, requiring the use of more expensive alloy materials. Therefore, our research team utilized Finite Element Analysis (FEA) and Multi-Objective Genetic Algorithm (MOGA) to optimize the mass and safety factor of the socket, proposing a design that simultaneously achieves both weight reduction and structural stability. The socket developed in this study is approximately 30% lighter than traditional Steel-based sockets while maintaining a safety factor of 1.2 or higher, significantly enhancing operational safety in explosive environments. This research sets a new standard in the design and manufacturing process of explosion-proof sockets and is expected to contribute to the optimization of various explosion-proof equipment in the future.

9

Multi-objective Optimization of the Steering Linkages Considering Transmitting Ratios SCOPUS

Gu Yufeng, Dong Fulong

보안공학연구지원센터(IJDTA) International Journal of Database Theory and Application Vol.9 No.6 2016.06 pp.53-66

※ 원문제공기관과의 협약기간이 종료되어 열람이 제한될 수 있습니다.

Trapezoidal mechanisms and swing arm mechanisms in the steering linkages were usually designed separately, and only the steered wheel deflection errors were fixed as the optimization objective. To deal with these design faults, multi-objective optimization model of the steering linkages were developed in this paper. The model considered not only the steered wheel deflection errors but also the unevenness of the steering linkages transmitting ratios. In the model, the optimization variables were the link dimensions of the steering linkages and the optimization was processed in Adams software using SPQ algorithm. The optimization results show that the unevenness of the first axle steering mechanism transmitting ratio reduced from9.7% to 4.9%, the unevenness of the second axle steering mechanism transmitting ratio reduced from3.8% to 2.1%, the maximum deflection error of the first axle right steered wheel reduced 39.3%, the maximum deflection error of the second axle left steered wheel reduced 21.4%, the maximum deflection error of the second axle right steered wheel reduced 48.4%. Benefitting from the unevenness improvement of the steering linkages transmitting ratios, the maximum difference between the bilateral deflection errors of the first axle right steered wheel reduced from1 to 0.5 , the maximum difference between the bilateral deflection errors of the second axle left steered wheel reduced from 4 to 0.8 , the homologous maximum difference of the second axle right steered wheel reduced from 6 to 0.3 . For the handiness, the maximum difference of the bilateral steering forces reduced from1.44Nㆍm to1.1Nㆍm, the maximum difference of the bilateral number of the steering wheel turns reduced from 0.15 to 0.11. For the lemniscate simulations, the difference between the positive and negative amplitude of the yaw angular velocity reduced from1.67 / s to 0.58 / s , the difference between the positive and negative amplitude of the lateral accelerometer reduced from 2 83.3mm/ s2 to 49.6mm/ s2 . Because the unevenness of the steering linkages transmitting ratios was optimized, the steered wheel deflection errors tended to be bilateral symmetry and the vehicle handiness and control stability were both improved.

10

Multi-objective Optimization of Machined Surface Integrity for Hard Turning Process

Caixu Yue, Liquan Wang, Jun Liu, Shengyu Hao

보안공학연구지원센터(IJSH) International Journal of Smart Home Vol.10 No.6 2016.06 pp.71-76

※ 원문제공기관과의 협약기간이 종료되어 열람이 제한될 수 있습니다.

Hard turning has characteristics of good processing flexibility and environmental performance. Under certain processing conditions it has more advantages than grinding process. Also, it has become the new process which has broad prospects for development. After process of hard cutting, the integrity of machined surface plays a vital role for performance of the workpiece. In this paper, the integrity of machined surface (surface roughness and the thickness of the plastic deformation) is focused on for hard turning process of die steel Cr12MoV. The multi-objective optimization was adopted. On the basis of revealing the relationship between cutting conditions and surface integrity indicators, combined with response surface methodology (RSM) the correspondence between the surface integrity evaluation and cutting parameters is established. By improved particle swarm algorithm a multi-objective optimization of surface integrity prediction was achieved, and the relative optimal cutting parameters were obtained. The research results of this paper provide the theoretical basis and reference for optimization of the experimental conditions.

11

Asymptotically Optimal Scenario-based Multi-objective Optimization for Distributed Generation Allocation and Sizing in Distribution Systems SCOPUS

Lizhen Wu, Xusheng Yang, Hu Zhou, Xiaohong Hao

보안공학연구지원센터(IJGDC) International Journal of Grid and Distributed Computing Vol.9 No.4 2016.04 pp.75-86

※ 원문제공기관과의 협약기간이 종료되어 열람이 제한될 수 있습니다.

Suitable location and optimal sizing are impact on voltage stability margin of the distributed system. It is important to accurately simulate the random output active power of Distributed Generation (DG). In order to model uncertainties of intermittent distributed generation and load, this paper proposes a multi-scenario tree model of wind-photovoltaic-load using multiple scenarios technique based on the Wasserstein distance metrics, which generates asymptotically optimal scenario. And in this paper, a multi-objective optimizes control model with scenario tree is presented, which including objectives that are the total active power losses and the voltage deviations of the bus. Moreover, a new hybrid Honey Bee Mating Optimization and Particle Swarm Optimization (HBMO-PSO) algorithm is proposed to solved the problems. In the HBMO-PSO algorithm, the mating process is corrected, which the PSO algorithm is combined with the HBMO algorithm to improve the performance of HBMO. Finally, a typical IEEE 33-bus distribution test system is used to investigate the feasibility and effectiveness of the proposed method. Simulation results illustrate the correctness and adaptability of the proposed model and the improved algorithm.

12

Multi-Objective Particle Swarm Optimization of Regenerative Intercooled Gas Turbine Cycle

Karim Salahshoor, Kamal Jafarian

보안공학연구지원센터(IJHIT) International Journal of Hybrid Information Technology Vol.6 No.5 2013.09 pp.269-276

※ 원문제공기관과의 협약기간이 종료되어 열람이 제한될 수 있습니다.

Gas turbines can be found in many industrial application areas. Gas turbine generation is limited by some undesirable effects which can be incorporated as operational constraints. Because of importance of energy, optimization of power generation systems is necessary. In order to achieving higher efficiencies, some propositions are preferred such as recovery of heat from exhaust gases in a regenerator, utilization of intercooler in a multistage compressor, steam injection to combustion chamber and etc. In this article multi-objective particle swarm optimization are employed for Pareto approach optimization of Gas Turbine cycle. In the multi-objective optimization a number of conflicting objective functions are to be optimized simultaneously. Multi-objective optimization offers a candidate scheme whose solution can satisfy the foregoing major requirements. At the first stage single objective optimization has been investigated and then MOPSO has been used for multi-objective optimization. The sets of selected decision variables based on this Pareto front, will cause the best possible combination of corresponding objective functions. The obtained results show that the output of multi-objective optimization scheme confirms that of single objective results.

13

An Archived Multi-Objective Simulated Annealing Algorithm for Vehicle Routing Problem with Time Windows

Yang Gao, Chao Wang, Chao Liu

보안공학연구지원센터(IJUNESST) International Journal of u- and e- Service, Science and Technology Vol.9 No.12 2016.12 pp.187-198

※ 원문제공기관과의 협약기간이 종료되어 열람이 제한될 수 있습니다.

The vehicle routing problem with time window (VRPTW) is a well-known combinatorial optimization problem, which is to find the lowest-cost routes from a central depot to a set of geographically scattered points with various demands. This paper deals with a multi-objective variant of the VRPTW that simultaneously minimizes the number of the vehicles and the traveled distance. A metaheuristic based on simulated annealing was proposed, and the concept of archive was introduced, in order to provide a set of tradeoff solutions for the problem. The accuracy of solutions is defined as their proximity to the best known solution of Solomon’s benchmarking tests. Computational results demonstrate that the proposed approach is quite effective, as it provides solutions competitive with the best known in the literature.

14

Optimal Placement of SVC using NSGA-II SCOPUS

Shishir Dixit, Laxmi Srivastava, Ganga Agnihotri

보안공학연구지원센터(IJGDC) International Journal of Grid and Distributed Computing Vol.9 No.9 2016.09 pp.347-368

※ 원문제공기관과의 협약기간이 종료되어 열람이 제한될 수 있습니다.

Improving voltage stability, reducing real power loss (PL) and voltage deviation (VD) are the most important tasks in the operation of electrical power systems. Voltage instability and voltage collapse are the severe problems which may take place because of deficit reactive power at load buses due to increased loading or contingencies. In this paper, the problem of obtaining optimal location and size of SVC is formulated as true multi-objective optimization problem for simultaneous minimization of the two objectives namely real power losses and load bus voltage deviation. The two algorithms real coded genetic algorithm (RCGA) and non-dominated sorting genetic algorithm-II (NSGA-II) with a feature of adoptive crowding distance have been used for solving nonlinear constrained multi-objective optimization problem. Both the algorithms have been used for obtaining optimal location and sizing of SVC. Voltage security of the power system has also been analyzed separately for all placement of SVC to ensure secure operation of the system. The proposed approaches have been implemented on IEEE 30-bus test system. The simulation results of the two algorithms have been compared for solution quality, computational complexity and computational time. It has been found that NSGA-II presents better performance in solving multi-objective optimization problem and also in obtaining a diverse set of solutions which converge near the true Pareto-optimal front. The simulation results of NSGA-II have also been presented to exhibit the capabilities of the algorithm to generate well-distributed Pareto-optimal front.

15

Based on the Final Decision of Particle Group Algorithm Applied Research

Fenghua Liu

보안공학연구지원센터(IJHIT) International Journal of Hybrid Information Technology Vol.8 No.8 2015.08 pp.343-354

※ 원문제공기관과의 협약기간이 종료되어 열람이 제한될 수 있습니다.

Particle swarm optimization algorithm is a species of intelligent algorithm, it can solve the problem of multiple end of decision making. But the algorithm is based on each group of particles would have been the effective information hypothesis. For most of the optimization problem, by the convergence speed, set the parameters of the limit, so this paper proposes a new more volume particle group algorithm. Crowding mechanism algorithm was applied to select group of particles in the process of the optimal value, thus maintaining the dispersion, the selection of the global optimal value is more reasonable. To introduce the concept of half a feasible region, and then to avoid the traditional processing method only considers particles in area the disadvantages of the boundary value processing precision is not high. In respect of time complexity, the grouping method is adopted to choose random switching strategy, improve the ef

16

Multi-objective Integration of Flexible Collaborative Planning and Fuzzy Flexible Lot-Splitting Scheduling Based on the Pareto Optimal

Zhenqiang Bao, Xiaoqing Ren, Yulu Yang, Junwu Zhu, Cheng Wang, Richao Yin

보안공학연구지원센터(IJHIT) International Journal of Hybrid Information Technology Vol.8 No.7 2015.07 pp.305-318

※ 원문제공기관과의 협약기간이 종료되어 열람이 제한될 수 있습니다.

Since that a job usually contains several working procedures in actual production, and it’s hard to optimize the flexible collaborative planning, the flexible lot splitting and scheduling of the job simultaneously in batch production mode, a model for multi-objective integration of flexible collaborative planning and fuzzy lot-splitting scheduling is established. We take four performance indicators below which are the most common as standards to optimize the model: average delivery satisfaction, fuzzy total cost, fuzzy completion time and average credibility of job tardiness, and then a multi-objective algorithm based on the Pareto optimal is established. In this algorithm, we design the integrated coding scheme, which include collaboration chromosome, lot-splitting chromosome and scheduling chromosome, meanwhile the Pareto optimal scheme is designed. Finally, the efficiency of the model and algorithm is proved by the simulation.

17

An Improved Multi-objective Evolutionary Algorithm for Multi-Objective 0/1 Knapsack Problem SCOPUS

Zhanguo Li, Qiming Wang

보안공학연구지원센터(IJMUE) International Journal of Multimedia and Ubiquitous Engineering Vol.10 No.5 2015.05 pp.383-394

※ 원문제공기관과의 협약기간이 종료되어 열람이 제한될 수 있습니다.

To further enhance the distribution uniformity and extensiveness of the solution sets and to ensure effective convergence of the solution sets to the Pareto front, we proposed a MOEA approach based on a clustering mechanism. We named this approach improved multi-objective evolutionary algorithm (LMOEA). This algorithm uses a clustering technology to compute and maintain the distribution and diversity of the solution sets. A fuzzy C-means clustering algorithm is used for clustering individuals. Finally, the LMOEA is applied to solve the classical multi-objective knapsack problems. The algorithm performance was evaluated using convergence and diversity indicators. The proposed algorithm achieved significant improvements in terms of algorithm convergence and population diversity compared with the classical NSGA-II and the MOEA/D.

18

Optimal Allocation of SVC for Minimization of Power Loss and Voltage Deviation using NSGA-II

Shishir Dixit, Laxmi Srivastava, Ganga Agnihotri

보안공학연구지원센터(IJAST) International Journal of Advanced Science and Technology Vol.71 2014.10 pp.67-80

※ 원문제공기관과의 협약기간이 종료되어 열람이 제한될 수 있습니다.

Modern this paper proposes non dominated sorting genetic algorithm (NSGA-II) which has feature of adaptive crowding distance for finding optimal location and sizing of Static Var Compensators (SVC) in order to minimize real power losses and voltage deviation and also to improve voltage profile of a power system at the same time. While finding the optimal location and size of SVC, single line outages are considered as contingencies and voltage limits for the buses are taken as security constraints. To demonstrate the effectiveness of the proposed approach, NSGA-II has been applied for finding optimal location and sizing of SVC on IEEE 30-bus test system. The obtained results are highly encouraging and reveal the capability of the NSGA-II to generate well-distributed non-dominated Pareto front.

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In present research work the effect of insert nose radius and machining parameters including cutting speed, feed rate and depth of cut on surface roughness(Ra) and material removal rate(MRR) in a turning of HSS(M2) are investigated using the Taguchi method and ANOVA. A three level, four parameter design of experiment,L9 orthogonal array using Minitab 14 software, the signal-to-noise (S/N) ratio is employed to study the performance characteristics in the turning of HSS(M2) by taking nose radius of Tin coated carbide inserts tool of 0.4,0.8 and 1.2 mm on CNC turning centre. The analysis of variance (ANOVA) is applied to study the percentage contribution of each machining parameters while CNCturning of HSS (M2) material. The all experimental trials are conducted in dry machining environment and at a constant spindle speed 2800 rpm.The results are verified by taking confirmation experiments. The present investigation indicates that feed rate and nose radius are the most significant factors in case of material removal rate and surface roughness for turning of HSS (M2) material.

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Multi-Objective Optimization Algorithm based on Biogeography with Chaos

Xu Wang, Zhidan Xu

보안공학연구지원센터(IJHIT) International Journal of Hybrid Information Technology Vol.7 No.3 2014.05 pp.225-234

※ 원문제공기관과의 협약기간이 종료되어 열람이 제한될 수 있습니다.

Biogeography-based optimization (BBO) has shown excellent exploitation ability of the population information for simple-objective optimization problem. But if BBO is directly applied in multi-objective optimization problems (MOPs), optimal solution set gained by BBO has worse diversity and distribution. To overcome these shortcomings, a chaos migration operator is put forwards to improve the diversity of the population. And then based on the new chaos migration operator, Chaos biogeography multi-objective optimization algorithm (CBBMO) is proposed for MOPs. In CBBMO, the chaos migration operator and original mutation operator of BBO are applied to produce the next generation population. The archive is used to conserve the Pareto optimal solutions. The experiment results show that the proposed algorithm CBBMO is feasible and effective for MOPs.

 
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