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Approach for uncertain multi-objective programming problems with correlated objective functions under CEV criterion 认领 引用 被引量：2
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作者 MENG Xiangfei WANG Ying +2 位作者 LI Chao WANG Xiaoyang LYU Maolong 《Journal of Systems Engineering and Electronics》 SCIE CSCD 2018年第6期1197-1208,共12页
An uncertain multi-objective programming problem is a special type of mathematical multi-objective programming involving uncertain variables. This type of problem is important because there are several uncertain varia... An uncertain multi-objective programming problem is a special type of mathematical multi-objective programming involving uncertain variables. This type of problem is important because there are several uncertain variables in real-world problems.Therefore, research on the uncertain multi-objective programming problem is highly relevant, particularly those problems whose objective functions are correlated. In this paper, an approach that solves an uncertain multi-objective programming problem under the expected-variance value criterion is proposed. First, we define the basic framework of the approach and review concepts such as a Pareto efficient solution and expected-variance value criterion using an order relation between various uncertain variables.Second, the uncertain multi-objective problem is converted into an uncertain single-objective programming problem via a linear weighted method or ideal point method. Then the problem is transformed into a deterministic single objective programming problem under the expected-variance value criterion. Third, four lemmas and two theorems are proved to illustrate that the optimal solution of the deterministic single-objective programming problem is an efficient solution to the original uncertainty problem. Finally, two numerical examples are presented to validate the effectiveness of the proposed approach. 展开更多
关键词 uncertainty theory uncertain multi-objective programming expected-variance value criterion
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New approach for uncertain random multi-objective programming problems based on CESD criterion 认领 引用 被引量：1
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作者 SUN Yun WANG Ying +2 位作者 MENG Xiangfei FU Chaoqi LUO Chengkun 《Journal of Systems Engineering and Electronics》 SCIE CSCD 2021年第3期619-630,共12页
To overcome the defects that the traditional ap-proach for multi-objective programming under uncertain ran-dom environment(URMOP)neglects the randomness and uncer-tainty of the problem and the volatility of the result... To overcome the defects that the traditional ap-proach for multi-objective programming under uncertain ran-dom environment(URMOP)neglects the randomness and uncer-tainty of the problem and the volatility of the results,a new ap-proach is proposed based on expected value-standard devi-ation value criterion(CESD criterion).Firstly,the effective solution to the URMOP problem is defined;then,by applying sequence relationship between the uncertain random variables,the UR-MOP problem is transformed into a single-objective program-ming(SOP)under uncertain random environment(URSOP),which are transformed into a deterministic counterpart based on the CESD criterion.Then the validity of the new approach is proved that the optimal solution to the SOP problem is also effi-cient for the URMOP problem;finally,a numerical example and a case application are presented to show the effectiveness of the new approach. 展开更多
关键词 chance theory independent-uncertain random multi-objective programming expected value-standard derivation value criterion(CESDcriterion)
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Generating Efficient Solutions in Bilevel Multi-Objective Programming Problems 认领 引用 被引量：1
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作者 Calice Olivier Pieume Patrice Marcotte +1 位作者 Laure Pauline Fotso Patrick Siarry 《American Journal of Operations Research》 2013年第2期289-298,共10页
In this paper, we address bilevel multi-objective programming problems (BMPP) in which the decision maker at each level has multiple objective functions conflicting with each other. Given a BMPP, we show how to constr... In this paper, we address bilevel multi-objective programming problems (BMPP) in which the decision maker at each level has multiple objective functions conflicting with each other. Given a BMPP, we show how to construct two artificial multiobjective programming problems such that any point that is efficient for both the two problems is an efficient solution of the BMPP. Some necessary and sufficient conditions for which the obtained result is applicable are provided. A complete procedure of the implementation of an algorithm for generating efficient solutions for the linear case of BMPP is presented. A numerical example is provided to illustrate how the algorithm operates. 展开更多
关键词 Multi-Objective Programming Bilevel Programming Efficient Solution Efficient Edge Hierarchical Systems
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Roughly B-invex Multi-Objective Programming Problems 认领 引用 被引量：1
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作者 Tarek Emam 《Open Journal of Optimization》 2012年第1期1-7,共7页
In this paper, we shall be interested in characterization of efficient solutions for special classes of problems. These classes consider roughly B-invexity of involved functions. Sufficient and necessary conditions fo... In this paper, we shall be interested in characterization of efficient solutions for special classes of problems. These classes consider roughly B-invexity of involved functions. Sufficient and necessary conditions for a feasible solution to be an efficient or properly efficient solution are obtained. 展开更多
关键词 Multi-Objective Programming Problems Roughly B-invex Efficient Solutions Properly Efficient Solutions
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Quality-guaranteed Dubins Path Planning for USV Based on Mixed-integer Piecewise linear Programming for Addressing the Extended Minimum-time Intercept Problem 认领 引用 被引量：1
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作者 Xing Zhou Kelin Zhu +3 位作者 Shuang Liu Zhaoqing Li Wenxin Zhang Kang Du 《哈尔滨工程大学学报（英文版）》 CSCD 2026年第1期216-227,共12页
During the use of robotics in applications such as antiterrorism or combat,a motion-constrained pursuer vehicle,such as a Dubins unmanned surface vehicle(USV),must get close enough(within a prescribed zero or positive... During the use of robotics in applications such as antiterrorism or combat,a motion-constrained pursuer vehicle,such as a Dubins unmanned surface vehicle(USV),must get close enough(within a prescribed zero or positive distance)to a moving target as quickly as possible,resulting in the extended minimum-time intercept problem(EMTIP).Existing research has primarily focused on the zero-distance intercept problem,MTIP,establishing the necessary or sufficient conditions for MTIP optimality,and utilizing analytic algorithms,such as root-finding algorithms,to calculate the optimal solutions.However,these approaches depend heavily on the properties of the analytic algorithm,making them inapplicable when problem settings change,such as in the case of a positive effective range or complicated target motions outside uniform rectilinear motion.In this study,an approach employing a high-accuracy and quality-guaranteed mixed-integer piecewise-linear program(QG-PWL)is proposed for the EMTIP.This program can accommodate different effective interception ranges and complicated target motions(variable velocity or complicated trajectories).The high accuracy and quality guarantees of QG-PWL originate from elegant strategies such as piecewise linearization and other developed operation strategies.The approximate error in the intercept path length is proved to be bounded to h2/(4√2),where h is the piecewise length. 展开更多
关键词 Minimum-time intercept problem Dubins vehicle Mixed-integer piecewise-linear program Linearization Approximate error trigonometric function USV
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Solving Fuzzy Multi-Objective Linear Programming Problem by Applying Statistical Method 认领 引用 被引量：1
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作者 Samsun Nahar Marin Akter Md. Abdul Alim 《American Journal of Operations Research》 2022年第6期293-309,共17页
In this paper, the statistical averaging method and the new statistical averaging methods have been used to solve the fuzzy multi-objective linear programming problems. These methods have been applied to form a single... In this paper, the statistical averaging method and the new statistical averaging methods have been used to solve the fuzzy multi-objective linear programming problems. These methods have been applied to form a single objective function from the fuzzy multi-objective linear programming problems. At first, a numerical example of solving fuzzy multi-objective linear programming problem has been provided to validate the maximum risk reduction by the proposed method. The proposed method has been applied to assess the risk of damage due to natural calamities like flood, cyclone, sidor, and storms at the coastal areas in Bangladesh. The proposed method of solving the fuzzy multi-objective linear programming problems by the statistical method has been compared with the Chandra Sen’s method. The numerical results show that the proposed method maximizes the risk reduction capacity better than Chandra Sen’s method. 展开更多
关键词 Fuzzy Multi-Objective Linear Programming Problem Fuzzy Linear Programming Problem Chandra Sen’s Method Statistical Averaging Method New Statistical Averaging Method
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Solving Multi-Objective Linear Programming Problem by Statistical Averaging Method with the Help of Fuzzy Programming Method 认领 引用
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作者 Samsun Nahar Marin Akter Md. Abdul Alim 《American Journal of Operations Research》 2023年第2期19-32,共14页
A multi-objective linear programming problem is made from fuzzy linear programming problem. It is due the fact that it is used fuzzy programming method during the solution. The Multi objective linear programming probl... A multi-objective linear programming problem is made from fuzzy linear programming problem. It is due the fact that it is used fuzzy programming method during the solution. The Multi objective linear programming problem can be converted into the single objective function by various methods as Chandra Sen’s method, weighted sum method, ranking function method, statistical averaging method. In this paper, Chandra Sen’s method and statistical averaging method both are used here for making single objective function from multi-objective function. Two multi-objective programming problems are solved to verify the result. One is numerical example and the other is real life example. Then the problems are solved by ordinary simplex method and fuzzy programming method. It can be seen that fuzzy programming method gives better optimal values than the ordinary simplex method. 展开更多
关键词 Fuzzy Programming Method Fuzzy Linear Programming Problem Multi-Objective Linear Programming Problem Statistical Averaging Method New Statistical Averaging Method
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Determining Efficient Solutions of Multi-Objective Linear Fractional Programming Problems and Application 认领 引用
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作者 Farhana Akond Pramy Md. Ainul Islam 《Open Journal of Optimization》 2017年第4期164-175,共12页
In this paper, a modified method to find the efficient solutions of multi-objective linear fractional programming (MOLFP) problems is presented. While some of the previously proposed methods provide only one efficient... In this paper, a modified method to find the efficient solutions of multi-objective linear fractional programming (MOLFP) problems is presented. While some of the previously proposed methods provide only one efficient solution to the MOLFP problem, this modified method provides multiple efficient solutions to the problem. As a result, it provides the decision makers flexibility to choose a better option from alternatives according to their financial position and their level of satisfaction of objectives. A numerical example is provided to illustrate the modified method and also a real life oriented production problem is modeled and solved. 展开更多
关键词 Linear Programming (LP) Linear Fractional Programming (LFP) Multi-Objective Linear Programming (MOLP) Multi-Objective Linear Fractional Programming (MOLFP)
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MULTI-OBJECTIVE PROGRAMMING FOR AIRPORT GATE REASSIGNMENT 认领 引用 被引量：1
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作者 李军会 陈欣 朱金福 《Transactions of Nanjing University of Aeronautics and Astronautics》 EI 2013年第2期209-215,共7页
To improve the efficiency of gate reassignment and optimize the plan of gate reassignment,the concept of disruption management is introduced,and a multi-objective programming model for airport gate reassignment is pro... To improve the efficiency of gate reassignment and optimize the plan of gate reassignment,the concept of disruption management is introduced,and a multi-objective programming model for airport gate reassignment is proposed.Considering the interests of passengers and the airport,the model minimizes the total flight delay,the total passengers′walking distance and the number of flights reassigned to other gates different from the planned ones.According to the characteristics of the gate reassignment,the model is simplified.As the multi-objective programming model is hard to reach the optimal solutions simultaneously,a threshold of satisfactory solutions of the model is set.Then a simulated annealing algorithm is designed for the model.Case studies show that the model decreases the total flight delay to the satisfactory solutions,and minimizes the total passengers′walking distance.The least change of planned assignment is also reached.The results achieve the goals of disruption management.Therefore,the model is verified to be effective. 展开更多
关键词 gate assignment multi-objective programming simulated annealing algorithm disruption management
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Incorporating Fully Fuzzy Logic in Multi-Objective Transshipment Problems:A Study of Alternative Path Selection Using LR Flat Fuzzy Numbers 认领 引用
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作者 Vishwas Deep Joshi Priya Agarwal +4 位作者 Lenka Cepová Huda Alsaud Ajay Kumar B.Swarna Ashish Kumar 《Computer Modeling in Engineering & Sciences》 SCIE EI 2025年第7期969-1011,共43页
In a world where supply chains are increasingly complex and unpredictable,finding the optimal way to move goods through transshipment networks is more important and challenging than ever.In addition to addressing the ... In a world where supply chains are increasingly complex and unpredictable,finding the optimal way to move goods through transshipment networks is more important and challenging than ever.In addition to addressing the complexity of transportation costs and demand,this study presents a novel method that offers flexible routing alternatives to manage these complexities.When real-world variables such as fluctuating costs,variable capacity,and unpredictable demand are considered,traditional transshipment models often prove inadequate.To overcome these challenges,we propose an innovative fully fuzzy-based framework using LR flat fuzzy numbers.This framework allows for more adaptable and flexible decision-making in multi-objective transshipment situations by effectively capturing uncertain parameters.To overcome these challenges,we develop an innovative,fully fuzzy-based framework using LR flat fuzzy numbers to effectively capture uncertainty in key parameters,offering more flexible and adaptive decision-making in multi-objective transshipment problems.The proposed model also presents alternative route options,giving decisionmakers a range of choices to satisfy multiple requirements,including reducing costs,improving service quality,and expediting delivery.Through extensive numerical experiments,we demonstrate that the model can achieve greater adaptability,efficiency,and flexibility than standard approaches.This multi-path structure provides additional flexibility to adapt to dynamic network conditions.Using ranking strategies,we compared our multi-objective transshipment model with existing methods.The results indicate that,while traditional methods such as goal and fuzzy programming generate results close to the anti-ideal value,thus reducing their efficiency,our model produces solutions close to the ideal value,thereby facilitating better decision making.By combining dynamic routing alternatives with a fully fuzzybased approach,this study offers an effective tool to improve decision-making and optimize complex networks under real-world conditions in practical settings.In this paper,we utilize LINGO 18 software to solve the provided numerical example,demonstrating the effectiveness of the proposed method. 展开更多
关键词 Multi-objective transshipment problem LR flat fuzzy numbers fully fuzzy optimization flexible routing solutions ranking approach
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Estimating the Macroeconomic Costs of CO2Emission Reduction in China Based on Multi-objective Programming 认领 引用
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作者 Ying Fan Xiaobing Zhang Lei Zhu 《Advances in Climate Change Research》 2010年第1期27-33,共7页
This paper estimates the macroeconornic costs of CO2emission reduction in China employing the input-output analysis with the multi-objective programming approach.The results show that the effect of reducing CO2e... This paper estimates the macroeconornic costs of CO2emission reduction in China employing the input-output analysis with the multi-objective programming approach.The results show that the effect of reducing CO2emissions on China's economy is significant.Under the present conditions,the estimated macroeconomic costs of CO2emission reduction in 2010 for China are approximately 3,100-4,024 RMB t-1.The stronger the abatement actions,the higher the macroeconomic costs of per unit emission reduction would be.Excavation industry,oil industry,chemical industry,and metal smelting industry have high potential to abate their CO2emissions. 展开更多
关键词 CO2emission reduction macroeconomic cost multi-objective programming input-output analysis
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Principal-subordinate hierarchical multi-objective programming model of initial water rights allocation 认领 引用 被引量：6
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作者 Dan WU Feng-ping WU Yan-ping CHEN 《Water Science and Engineering》 CAS 2009年第2期105-116,共12页
The principal-subordinate hierarchical multi-objective programming model of initial water rights allocation was developed based on the principle of coordinated and sustainable development of different regions and wate... The principal-subordinate hierarchical multi-objective programming model of initial water rights allocation was developed based on the principle of coordinated and sustainable development of different regions and water sectors within a basin.With the precondition of strictly controlling maximum emissions rights,initial water rights were allocated between the first and the second levels of the hierarchy in order to promote fair and coordinated development across different regions of the basin and coordinated and efficient water use across different water sectors,realize the maximum comprehensive benefits to the basin,promote the unity of quantity and quality of initial water rights allocation,and eliminate water conflict across different regions and water sectors.According to interactive decision-making theory,a principal-subordinate hierarchical interactive iterative algorithm based on the satisfaction degree was developed and used to solve the initial water rights allocation model.A case study verified the validity of the model. 展开更多
关键词 initial water rights allocation principal-subordinate hierarchy multi-objective programming model satisfaction degree
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Path Selection of Multimodal Transport Based on Multi-Objective Mixed Integer Programming 认领 引用
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作者 Zhongxin Luo 《Journal of Transportation Technologies》 2019年第4期462-473,共12页
Based on “One Belt and One Road”, this paper studies the path selection of multimodal transport by using the method of multi-objective mixed integer programming. Therefore, this paper studies the factors of transpor... Based on “One Belt and One Road”, this paper studies the path selection of multimodal transport by using the method of multi-objective mixed integer programming. Therefore, this paper studies the factors of transportation time, transportation cost and transportation safety performance, and establishes a mathematical model. In addition, the method of multi-objective mixed integer programming is used to comprehensively consider the different emphasis and differences of customers on cargo transportation. Then we use planning tools of Microsoft Excel to solve path selection and to determine whether the chosen path is economical and reliable. Finally, a relatively complex road network is built as an example to verify the accuracy of this planning method. 展开更多
关键词 One Belt and One Road Multi-Objective Multimodal Transport Mixed Integer Programming Path Selection
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Optimality for Multi-Objective Programming Involving Arcwise Connected d-Type-I Functions 认领 引用
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作者 Guolin Yu Min Wang 《American Journal of Operations Research》 2011年第4期243-248,共6页
This paper deals with the optimality conditions and dual theory of multi-objective programming problems involving generalized convexity. New classes of generalized type-I functions are introduced for arcwise connected... This paper deals with the optimality conditions and dual theory of multi-objective programming problems involving generalized convexity. New classes of generalized type-I functions are introduced for arcwise connected functions, and examples are given to show the existence of these functions. By utilizing the new concepts, several sufficient optimality conditions and Mond-Weir type duality results are proposed for non-differentiable multi-objective programming problem. 展开更多
关键词 Multi-Objective Programming Pareto Efficient Solution Arcwise Connected d-Type-I Functions Optimality Conditions Duality
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A Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming 认领 引用
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作者 Zhiqing Meng Rui Shen Min Jiang 《American Journal of Operations Research》 2014年第6期331-339,共9页
In this paper, we present an algorithm to solve the inequality constrained multi-objective programming (MP) by using a penalty function with objective parameters and constraint penalty parameter. First, the penalty fu... In this paper, we present an algorithm to solve the inequality constrained multi-objective programming (MP) by using a penalty function with objective parameters and constraint penalty parameter. First, the penalty function with objective parameters and constraint penalty parameter for MP and the corresponding unconstraint penalty optimization problem (UPOP) is defined. Under some conditions, a Pareto efficient solution (or a weakly-efficient solution) to UPOP is proved to be a Pareto efficient solution (or a weakly-efficient solution) to MP. The penalty function is proved to be exact under a stable condition. Then, we design an algorithm to solve MP and prove its convergence. Finally, numerical examples show that the algorithm may help decision makers to find a satisfactory solution to MP. 展开更多
关键词 Multi-Objective Programming Penalty Function Objective Parameters Constraint Penalty Parameter Pareto Weakly-Efficient Solution
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A Survey of Evolutionary Algorithms for Multi-Objective Optimization Problems With Irregular Pareto Fronts 认领 引用 被引量：53
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作者 Yicun Hua Qiqi Liu +1 位作者 Kuangrong Hao Yaochu Jin 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2021年第2期303-318,I0001-I0004,共16页
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 remed... 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. 展开更多
关键词 Evolutionary algorithm machine learning multi-objective optimization problems(MOPs) irregular Pareto fronts
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Improved genetic algorithm for nonlinear programming problems 认领 引用 被引量：10
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作者 Kezong Tang Jingyu Yang +1 位作者 Haiyan Chen Shang Gao 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2011年第3期540-546,共7页
An improved genetic algorithm（IGA） based on a novel selection strategy to handle nonlinear programming problems is proposed.Each individual in selection process is represented as a three-dimensional feature vector w... An improved genetic algorithm（IGA） based on a novel selection strategy to handle nonlinear programming problems is proposed.Each individual in selection process is represented as a three-dimensional feature vector which is composed of objective function value,the degree of constraints violations and the number of constraints violations.It is easy to distinguish excellent individuals from general individuals by using an individuals＇ feature vector.Additionally,a local search（LS） process is incorporated into selection operation so as to find feasible solutions located in the neighboring areas of some infeasible solutions.The combination of IGA and LS should offer the advantage of both the quality of solutions and diversity of solutions.Experimental results over a set of benchmark problems demonstrate that IGA has better performance than other algorithms. 展开更多
关键词 genetic algorithm（GA） nonlinear programming problem constraint handling non-dominated solution optimization problem.
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Hybrid particle swarm optimization with chaotic search for solving integer and mixed integer programming problems 认领 引用 被引量：21
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作者 谭跃 谭冠政 邓曙光 《Journal of Central South University》 SCIE EI CAS 2014年第7期2731-2742,共12页
A novel chaotic search method is proposed,and a hybrid algorithm combining particle swarm optimization(PSO) with this new method,called CLSPSO,is put forward to solve 14 integer and mixed integer programming problems.... A novel chaotic search method is proposed,and a hybrid algorithm combining particle swarm optimization(PSO) with this new method,called CLSPSO,is put forward to solve 14 integer and mixed integer programming problems.The performances of CLSPSO are compared with those of other five hybrid algorithms combining PSO with chaotic search methods.Experimental results indicate that in terms of robustness and final convergence speed,CLSPSO is better than other five algorithms in solving many of these problems.Furthermore,CLSPSO exhibits good performance in solving two high-dimensional problems,and it finds better solutions than the known ones.A performance index(PI) is introduced to fairly compare the above six algorithms,and the obtained values of(PI) in three cases demonstrate that CLSPSO is superior to all the other five algorithms under the same conditions. 展开更多
关键词 particle swarm optimization chaotic search integer programming problem mixed integer programming problem
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Solving material distribution routing problem in mixed manufacturing systems with a hybrid multi-objective evolutionary algorithm 认领 引用 被引量：7
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作者 高贵兵 张国军 +2 位作者 黄刚 朱海平 顾佩华 《Journal of Central South University》 SCIE EI CAS 2012年第2期433-442,共10页
The material distribution routing problem in the manufacturing system is a complex combinatorial optimization problem and its main task is to deliver materials to the working stations with low cost and high efficiency... The material distribution routing problem in the manufacturing system is a complex combinatorial optimization problem and its main task is to deliver materials to the working stations with low cost and high efficiency. A multi-objective model was presented for the material distribution routing problem in mixed manufacturing systems, and it was solved by a hybrid multi-objective evolutionary algorithm (HMOEA). The characteristics of the HMOEA are as follows: 1) A route pool is employed to preserve the best routes for the population initiation; 2) A specialized best?worst route crossover (BWRC) mode is designed to perform the crossover operators for selecting the best route from Chromosomes 1 to exchange with the worst one in Chromosomes 2, so that the better genes are inherited to the offspring; 3) A route swap mode is used to perform the mutation for improving the convergence speed and preserving the better gene; 4) Local heuristics search methods are applied in this algorithm. Computational study of a practical case shows that the proposed algorithm can decrease the total travel distance by 51.66%, enhance the average vehicle load rate by 37.85%, cut down 15 routes and reduce a deliver vehicle. The convergence speed of HMOEA is faster than that of famous NSGA-II. 展开更多
关键词 material distribution routing problem multi-objective optimization evolutionary algorithm local search
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Compactness, Contractibility and Fixed Point Properties of the Pareto Sets in Multi-Objective Programming 认领 引用
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作者 Zdravko Dimitrov Slavov Christina Slavova Evans 《Applied Mathematics》 2011年第5期556-561,共6页
This paper presents the Pareto solutions in continuous multi-objective mathematical programming. We discuss the role of some assumptions on the objective functions and feasible domain, the relationship between them, a... This paper presents the Pareto solutions in continuous multi-objective mathematical programming. We discuss the role of some assumptions on the objective functions and feasible domain, the relationship between them, and compactness, contractibility and fixed point properties of the Pareto sets. The authors have tried to remove the concavity assumptions on the objective functions which are usually used in multi-objective maximization problems. The results are based on constructing a retraction from the feasible domain onto the Pareto-optimal set. 展开更多
关键词 Multi-Objective Programming Pareto-Optimal Pareto-Front Compact Contractible Fixed Point Retraction
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