Multi-objective optimization using evolutionary algorithms

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多目标遗传算法

A more appropriate approach to deal with multiple objectives is to use techniques that were originally designed for that purpose in the eld of Operations Research. Work in that area started a century ago, and many approaches have been re ned and commonly applied in economics and control theory.
Abstract: In this paper we propose the use of the genetic algorithm (GA) as a tool to solve multiobjective
optimization problems in structures. Using the concept of min-max optimum, a new GA-based multiobjective optimization technique is proposed and two truss design problems are solved using it. The results produced by this new approach are compared to those produced by other mathematical programminቤተ መጻሕፍቲ ባይዱ techniques and GA-based approaches, proving that this technique generates better trade-o s and that the genetic algorithm can be used as a reliable numerical optimization tool.

NSGAII

NSGAIIA F AST ELITIST MULTIOBJECTIVE GENETIC ALGORITHM:NSGA-IIARAVIND SESHADRI1.Multi-Objective Optimization Using NSGA-IINSGA([5])is a popular non-domination based genetic algorithm for multi-objective optimization.It is a very e?ective algorithm but has been generally criticized for its computational complexity,lack of elitism and for choosing the optimal parameter value for sharing parameterσshare.A modi?ed version,NSGA-II([3])was developed,which has a better sorting algorithm,incorporates elitism and no sharing parameter needs to be chosen a priori.NSGA-II is discussed in detail in this.2.General Description of NSGA-IIThe population is initialized as usual.Once the population in initialized the population is sorted based on non-domination into each front.The?rst front being completely non-dominant set in the current population and the second front being dominated by the individuals in the?rst front only and the front goes so on.Each individual in the each front are assigned rank(?tness)values or based on front in which they belong to.Individuals in?rst front are given a?tness value of1and individuals in second are assigned?tness value as2and so on.In addition to?tness value a new parameter called crowding distance is cal-culated for each individual.The crowding distance is a measure of how close an individual is to its /doc/6b11041199.html,rge average crowding distance will result in better diversity in the population.Parents are selected from the population by using binarytournament selection based on the rank and crowding distance.An individual is selected in the rank is lesser than the other or if crowding distance is greater than the other1.The selected population generates o?springs from crossover and mutation operators,which will be discussed in detail in a later section.The population with the current population and current o?springs is sorted again based on non-domination and only the best N individuals are selected,where N is the population size.The selection is based on rank and the on crowding distance on the last front.3.Detailed Description of NSGA-II3.1.Population Initialization.The population is initialized based on the prob-lem range and constraints if any.1Crowding distance is compared only if the rank for both individuals are same12ARA VIND SESHADRI3.2.Non-Dominated sort.The initialized population is sorted based on non-domination2.The fast sort algorithm[3]is described as below for each ?for each individual p in main population P do the following–Initialize S p=?.This set would contain all the individuals that is being dominated by p.–Initialize n p=0.This would be the number of individuals that domi-nate p.–for each individual q in Pif p dominated q then·add q to the set S p i.e.S p=S p∪{q}else if q dominates p then·increment the domination counter for p i.e.n p=n p+1–if n p=0i.e.no individuals dominate p then p belongs to the?rst front;Set rank of individual p to one i.e p rank=1.Update the?rst front set by adding p to front one i.e F1=F1∪{p}This is carried out for all the individuals in main population P.Initialize the front counter to one.i=1following is carried out while the i th front is nonempty i.e.F i=?–Q=?.The set for storing the individuals for(i+1)th front.–for each individual p in front F ifor each individual q in S p(S p is the set of individuals dominatedby p)·n q=n q?1,decrement the domination count for individualq.·if n q=0then none of the individuals in the subsequentfronts would dominate q.Hence set q rank=i+1.Updatethe set Q with individual q i.e.Q=Q∪q.–Increment the front counter by one.–Now the set Q is the next front and hence F i=Q.This algorithm is better than the original NSGA([5])since it utilize the infor-mation about the set that an individual dominate(S p)and number of individuals that dominate the individual(n p).3.3.Crowding Distance.Once the non-dominated sort is complete the crowding distance is assigned.Since the individuals are selected based on rank and crowding distance all the individuals in the population are assigned a crowding distance value. Crowding distance is assigned front wise and comparing the crowding distance between two individuals in di?erent front is meaning less.The crowing distance is calculated as belowFor each front F i,n is the number of individuals.–initialize the distance to be zero for all the individuals i.e.F i(d j)=0, where j corresponds to the j th individual in front F i.–for each objective function mSort the individuals in front F i based on objective m i.e.I= sort(F i,m).2An individual is said to dominate another if the objective functions of it is no worse than the other and at least in one of its objective functions it is better than the otherNSGA-II3?Assign in?nite distance to boundary values for each individual in F i i.e.I(d1)=∞and I(d n)=∞for k=2to(n?1)·I(d k)=I(d k)+I(k+1).m?I(k?1).mf maxmf minm·I(k).m is the value of the m th objective function of the k th individual in IThe basic idea behind the crowing distance is?nding the euclidian distance between each individual in a front based on their m objectives in the m dimensional hyper space.The individuals in the boundary are always selected since they have in?nite distance assignment.3.4.Selection.Once the individuals are sorted based on non-domination and with crowding distance assigned,the selection is carried out using a crowded-comparison-operator(?n).The comparison is carried out as below based on(1)non-domination rank p rank i.e.individuals in front F i will have their rankas p rank=i.(2)crowding distance F i(d j)p?n q if–p rank–or if p and q belong to the same front F i then F i(d p)>F i(d q)i.e.the crowing distance should be more.The individuals are selected by using a binary tournament selection with crowed-comparison-operator.3.5.Genetic Operators.Real-coded GA’s use Simulated Binary Crossover (SBX)[2],[1]operator for crossover and polynomial mutation[2],[4].3.5.1.Simulated Binary Crossover.Simulated binary crossover simulates the bi-nary crossover observed in nature and is give as below.c1,k=12[(1?βk)p1,k+(1+βk)p2,k]c2,k=12[(1+βk)p1,k+(1?βk)p2,k]where c i,k is the i th child with k th component,p i,k is the selected parent andβk (≥0)is a sample from a random number generated having the densityp(β)=12(ηc+1)βηc,if0≤β≤1p(β)=12(ηc+1)1,ifβ>1.This distribution can be obtained from a uniformly sampled random number u between(0,1).ηc is the distribution index for crossover3.That isβ(u)=(2u)1(η+1)β(u)=1[2(1?u)]1(η+1)3This determine how well spread the children will be from their parents.4ARA VIND SESHADRI3.5.2.Polynomial Mutation.c k=p k+(p u k?p l k)δkwhere c k is the child and p k is the parent with p uk being the upper bound4onthe parent component,p lk is the lower bound andδk is small variation which iscalculated from a polynomial distribution by usingδk=(2r k)1ηm+1?1,if rk<0.5δk=1?[2(1?r k)]1ηm+1if rkr k is an uniformly sampled random number between(0,1)andηm is mutation distribution index.3.6.Recombination and Selection.The o?spring population is combined with the current generation population and selection is performed to set the individuals of the next generation.Since all the previous and current best individuals are added in the population,elitism is ensured.Population is now sorted based on non-domination.The new generation is?lled by each front subsequently until the population size exceeds the current population size.If by adding all the individuals in front F j the population exceeds N then individuals in front F j are selected based on their crowding distance in the descending order until the population size is N. And hence the process repeats to generate the subsequent generations./doc/6b11041199.html,ing the functionPretty much everything is explained while you execute the code but the main arguments to get the function running are population and number of generations. Once these arguments are entered the user would be prompted for number of ob-jective functions and number of decision variables.Also the range for the decision variables will have to be entered.Once preliminary data is obtained,the user is prompted to modify the objective function.Have fun and feel free to modify the code to suit your need!References[1]Hans-Georg Beyer and Kalyanmoy Deb.On Self-Adaptive Features in Real-Parameter Evo-lutionary Algorithm.IEEE Trabsactions on EvolutionaryComputation,5(3):250–270,June 2001.[2]Kalyanmoy Deb and R.B.Agarwal.Simulated Binary Crossover for Continuous Search Space.Complex Systems,9:115–148,April1995.[3]Kalyanmoy Deb,Amrit Pratap,Sameer Agarwal,and T.Meyarivan.A Fast Elitist Multi-objective Genetic Algorithm:NSGA-II.IEEE Transactions on Evolutionary Computation, 6(2):182–197,April2002.[4]M.M.Raghuwanshi and O.G.Kakde.Survey on multiobjective evolutionary and real codedgenetic algorithms.In Proceedings of the8th Asia Paci?c Symposium on Intelligent and Evo-lutionasy Systems,pages150–161,2004.[5]N.Srinivas and Kalyanmoy Deb.Multiobjective Optimization Using Nondominated Sortingin Genetic Algorithms.Evolutionary Computation,2(3):221–248,1994.E-mailaddress:aravind.seshadri@/doc/6b110411 99.html,4The decision space upper bound and lower bound for that particular component.。

谢宁DOE

qsconsult www.qsconsult.be1 Willy VandenbrandeShainin: A concept for problemsolvingLecture at the Shainin conferenceAmelior11 December 20092Dorian Shainin (1914 –2000)•Aeronautical engineer (MIT –1936)•Design Engineer for United Aircraft Corporations •Mentored by his friend Joseph M. Juran•Reliability consultant for Grumman Aerospace (Lunar Excursion Module)•Reliability consultant for Pratt&Whitney (RL-10 rocket engine)•Developed over 20 statistical engineering techniques for problem solving and reliability •Started Shainin Consultants in 1984, his son Peter is currentCEO.Dorian Shainin and ASQ•15th ASQ Honorary Member (1996)•First person to win all four major ASQ medals•In 2004 ASQ created the Dorian Shainin Medal–For outstanding use of unique or creativeapplications of statistical techniques in thesolving of problems related to the quality of aproduct or service.3Dorian Shainin•Not very well known outside USA (compared to Deming, Juran)•1991: Publication of first edition of“World Class Quality”by Keki Bothe •2000: Second edition (Keki and Adi Bothe)•Books brought attention to Shainin methods, but are very biased.4Problem Solving •Focus is on variation reductionLSL USLLSL = Lower Specification Limit USL = Upper Specification LimitBeforeAfter5Problem Solving•But also …LSLAfterBefore6Basic Shainin assumption•The pareto principle of vital few and trivial many.•Only a few input variables are responsible for a large part of the output behavior.–Red X TM–Pink X TM–Pale Pink X TM•Problem solving becomes the hunt for the Red X TM7Shainin tools•Recipe like methods / statistics in the background•Comparing extremes allows easier detection of causes–BOB Best of Best–WOW Worst of Worse•Non parametrics with ranking tests in stead of calculations with hypothesis tests•Graphical Methods•Working with small sample sizes•The truth is in the parts, not in the drawing: let the parts talk!8Preliminary activities•Define the critical output variable(s) to be improved (called problem Green Y®)•Determine the quality of the Measurement System used to evaluate the Green Y®–A bad measurement system can in itself beresponsible for excessive variation–Improvements can only be seen if they can bemeasured910Overview of Shainin toolsComponents Search Multi-Vari chart Paired Comparisons Variables Search Full FactorialsB vs CScatter Plots PrecontrolProduct / ProcessSearchRSM methodsPositrolProcess CertificationClue generatingFormal Doe toolsValidationOptimization AssuranceOngoing controlControl20 –1000 variables5 –20 variables4 or less variablesNo interactionsInteractionsGeneral comments•Gradually narrowing down the search•Clear logic–Analyzing–Improving–Controlling•Not all tools are “Shainin”tools•“What’s in a name?”–Positrol versus Control Plan–Process Certification versus Process Audit11Tool details•Overview of methods•More info on B vs C TM and Scatter Plots in workshops•Some more detail on–Multi-Vari chart–Paired Comparison TM and Product/ProcessSearch–Pre Control1213Clue Generating / Multi-Vari Chart Very useful tool and best applied beforebrainstorming causes on excess variationComments Samples taken in production on current process Could be a big measurement investment Sample SizeDivide total variation in categories Search for causes of variation in the biggest category first PrinciplesProblem type: excess variation Wide applicability ApplicationUnderstand the pattern of variation Define areas where not to look for problems Allow a more specific brainstorm ObjectiveMulti-Vari Chart•Breakdown of variation in 3 families:–Positional(within piece, between cavities, …)–Cyclical(consecutive units, batch-to-batch, lot-to-lot)–Temporal (hour-to-hour, shift-to-shift, …)1415Multi-vari Chart•If one family of variationcontains a large part oftotal variation, we canconcentrate oninvestigating variablesrelated to this family ofvariation.16Clue Generating / Component SearchTM Disassembly / reassembly requirement limitsapplication.Comments 2 = 1 BOB and 1 WOW Sample SizeSelect BOB and WOW unit Exchange components and observe behavior. Components that change behavior are Red X comp PrinciplesProblem type: assembly does not perform to spec Limitation: Disassembly / Reassembly must be possible without product change ApplicationFind the component(s) of an assembly that is (are) responsible for bad behavior Objective17Clue Generating / Paired ComparisonTM Practical application of “let the parts talk”Comments5 to6 pairs of 1 BOB and 1 WOWSample SizeSelect pairs of BOB and WOW units Look for differences Consistent differences to be investigated further PrinciplesProblem type: occasional problems in production flow ApplicationFind directions for further investigationObjectivePaired Comparisons TM: method•Step1: take1 good and 1 bad unit–As close as possible in time–Aim for BOB and WOW units•Step2: note the differences between these units (visual, dimensional, mechanical, chemical, …). Let the parts talk!•Step3: take a second pair of good and bad units.Repeat step218Paired Comparisons TM: method •Step4: repeat this process with third, fourth, fith, …pair until a pattern of differences becomes apparent. •Step5: don’t take inconsistent differences into account. Generally after the fith or sixth pair theconsistent differences that cause the variationbecome clear.1920Clue Generating / Product/Process Search Tukey test is alternative for t-testWidely applicable methodProblem: available data (process parameters)Comments 8 BOB and 8 WOW units / batches Sample SizeSelect sets of BOB and WOW units –batches -..Add product data / process parameters and rank Apply Tukey test to determine important parameters PrinciplesProblem type: Various types of problems ApplicationPreselection of variables out of a large group of potential variables ObjectiveProduct/Process Search: example•Transmission assemblies rejected for noise.•Components search shows idler shaft as responsible component•One of the parameters of idler shaft is “out of round”•8 good / 8 bad units selected and measured for “out of round”2122Product/Process search: example0.0070.0110.0190.0170.0220.0140.0180.015Out of round good units(mm)0.0170.0210.0230.0240.0230.0160.0180.019Out of round bad units(mm)Tukey test procedure•Rank individual units by parameter and indicate Good / Bad.•Count number of “all good”or “all bad”from one side and vice versa from other side.•Make sum of both counts.•Determine confidence level to evaluate significance.2324Tukey test confidence levels99.9%1399%1095%790%6ConfidenceTotal end count25Tukey test: example0.0230.0230.0240.0160.0170.0180.0190.0210.0170.0180.0190.0220.0070.0110.0140.015BadGood Top end count (all good)4Bottom end count (all bad)3Overlap regionTukey test: example•Total end count = 4 + 3 = 7•95 % confidence that out-of-round idler shaft is important in explaining the difference in noise levels.2627Formal Doe tools / Variables SearchAlternative to fractional factorials on two levels Method comparable to components searchCommentsNumber of tests is determined by number of variables and quality of ordering.Sample Size List variables in order of criticality (process knowledge) and indicate good / bad level.Swap factor settings and observe behavior.Factors that change behavior (and interactions) are red X TM , Pink X TMPrinciplesProblem type: Various types of problemsAfter clue generating more then 4 potential variables leftApplicationDetermine Red X TM , Pink X TM including quantification of their effectObjective28Formal Doe tools / Full FactorialsWell established methodCommentsNumber of tests is determined by number of variables k (2k test combinations)Sample Size Classical DOE with Full Factorials at two levels Main Effects and interactions are calculated Principles Problem type: Various types of problems After clue generating 4 or less variables left Application Determine Red X TM , Pink X TM including quantification of their effectObjective29Formal Doe tools / B(etter) vs C(urrent)TMQuick validation that works well with big improvementsComments3 B and 3 C tests (each test can involve several units –test of variation reduction)All 3B’s must be better than all 3C’s Sample Size Create new process using optimum settings and compare optimum with current.Principles Problem type: Various types of problemsApplication Validation of Red X TM , Pink X TMObjective30Optimization / Scatter PlotsGraphical method that could easily be transformed to a statistical methodComments30 tests for each critical variableSample Size Do tests around optimum and use graphical regression to set tolerancePrinciples Problem type: Variation Reduction and optimizing signalApplication Fine tune best level and realistic tolerance for Red X TM , Pink X TM if no interactions are present Objective31Optimization / Response Surface Methods Method developed by George Box CommentsDepends on variables and surface.Sample SizeEvolutionary Operation (EVOP) to scan response surface in direction of steepest ascent PrinciplesProblem type: Variation Reduction and optimizing signal ApplicationFine tune best level and realistic tolerance for Red X TM , Pink X TM if interactions are present ObjectiveEVOP example3233Control / PositrolCan be compared with a Control Plan CommentsChecking frequency in the When columnSample SizeTable of What, How, Who, Where and When control has to be exercised. PrinciplesProblem type: all types ApplicationAssuring that optimum settings are kept Objective34Control / Process CertificationMix of 5S, Poka-Yoke, instructions, ISO 9000,audits,…Comments Checking frequency to be determined Sample SizeMake overview of things that could influence the process and install inspections, audits, …PrinciplesProblem type: all typesApplicationEliminating peripheral causes of poor quality Objective35Control / Pre ControlAlternative to classical SPCTraffic lights systemVery practical methodComments Checking frequency to be determinedSample SizeDivide total tolerance in colored zones and use prescribed sampling and rules to control the process.PrinciplesProblem type: control variation and setting of the process ApplicationContinuous checking of the quality of the process output Objective36Pre-Control: chart constructionUSL LSLTARGET ½TOL1/4 TOL1/4 TOLPre-control: use of chart1.Start process: five consecutive units ingreen needed as validation of set-up.2.If not possible: improve process.3.In production: 2 consecutive units4.Frequency: time interval between twostoppages (see action rules) / 6.3738Pre-control: action rulesStop and act 2 units in different yellow zoneStop and act1 unit in red zone Correct2 units in same yellow zoneContinue 1 unit in green and 1 unit in yellowzoneContinue 2 units in green zoneAction Result of samplesAfter an intervention: 5 consecutive units in green zone39Pre-control: exampleTime StartCorrect Startqsconsult www.qsconsult.be40 Willy VandenbrandeWilly Vandenbrande, Master TQM ASQ Fellow-Six Sigma Black Belt Montpellier 34B -8310 BruggeBelgië-BelgiumTel + 32 (0)479 36 03 75E-mail willy@qsconsult.be Website www.qsconsult.beQS Consult。

Samsung 动物诊所超声仪产品介绍说明书

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Samsung Medison hascommercialized the Live 3D technology in 2001 and since being part of Samsung Electronics in 2011, it is integrating IT, image processing, semiconductor and communication technologies into ultrasound devices for efficient and confident diagnosis.* The products, features, options, and transducers may not be commercially available in some countries.* Sales and Shipments are effective only after the approval by the regulatory affairs. Please contact your local sales representative for further details.* This product is a medical device, please read the user manual carefully before use.* Prestige is not a product name but is a marketing terminology.1. Optional feature which may require additional purchase.2. Strain value for ElastoScan+™ is not applicable in Canada and the United States.3. SonoSync™ is an image sharing solution.a. Image acquired by RS85 Prestige ultrasound system.b. Image acquired by V8 ultrasound system.c. Image acquired by HS60 ultrasound system.d. Image acquired by HS40 ultrasound system.© 2021 Samsung Medison All Rights Reserved.Samsung Medison reserves the right to modify the design, packaging,specifications, and features shown herein, without prior notice or obligation.SAMSUNG MEDISON CO., LTD.C T -T O T A L -V e t _I M C -211202-E NScan here or visit。

申请博士硕士专业学位授权点简况表-广东工业大学研究生院

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Decomposition of a multiobjective
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目标优化问题研究工业统计和质量控制问题。把生态位构建特征引入到生态与工业控制交叉学科邻域。本方向有教授 4 人，副教授 7 人，15 人获得博士学位，行业教师 5 人，其中高工 4 人。
注：专业学位领域（方向）按照各专业学位类别申请基本条件的要求填写。
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II-1 专任教师基本情况
专业技术职务
全国现有 103 个应用统计硕士专业学位点，其中北京 15 个，上海 6 个，江苏 10 个。广东省仅 2 个，难以支撑广东省制造业的转型升级和满足当前大数据行业对专业人才的需求。本学科聚集了一批学缘结构合理，老中青结合，学术水平较高的科研队伍，但缺乏专业硕士学位点的支撑，严重制约了学科发展。
3、与行业或职业发展的衔接学院已经与 10 家行业单位合作建立了实践教学基地，从合作办学单位外聘了 10 名行业教师，其中高工 7 名，在各自行业从业皆超过 8 年，完成了 10 余项专业实践活动与成果，合作完成实践教学专著 1 项。与单一的课堂教学形式相比，在实践教学基地接受具有行业指导的实践教学，有更强的吸引力，取得了较好的教学效果，更重要的是实现了与行业的对接，有利于学生未来职业的发展。 4、人才培养及思想政治教育在近 5 年中，共招收了 1299 名本科生，培养了 1396 名本科毕业生，平均就业率 95%以上，共招收了 116 名硕士研究生，获得硕士学位的毕业生有 112 名，就业率 100%。组织学生参加美国大学生数学建模竞赛、全国大学生数学建模竞赛和全国大学生数学竞赛等活动，共获得省级以上奖励 102 项，其中国际级奖励 31 项，国家级 15 项。大学生创新创业训练项目国家级 6 项，省级 16 项。学生毕业创业成功并成立公司 1 项。学生参与申请专利 4 项，发表核心期刊以上论文 50 余篇，其中 SCI 收录 7 篇。获广东省优秀硕士毕业论文 1 篇。

变负荷工况下NOx排放量预测控制

2018年第37卷第1期 CHEMICAL INDUSTRY AND ENGINEERING PROGRESS·343·化工进展变负荷工况下NO x 排放量预测控制唐振浩，张海洋，曹生现（东北电力大学自动化工程学院，吉林吉林 132012）摘要：NO x 是火电厂排放的主要污染物之一，降低NO x 的排放是火电厂面临的主要问题。

针对火电厂变负荷工况下的NO x 排放量最小化问题，本文提出了一种基于最小二乘支持向量机（LSSVM ）的非线性模型预测控制算法。

根据电站锅炉实际历史数据建立锅炉负荷预测模型和NO x 排放预测模型，并以交叉验证的方法优化模型参数，从而获得高精度模型。

在此基础上以NO x 的排放量最小为优化目标，考虑锅炉负荷约束，构建锅炉燃烧优化模型。

采用差分进化算法求解优化模型得到控制参数的最优设定值。

为了验证本文提出算法的有效性，采用实际生产数据进行实验。

实验结果表明本方法能够在变负荷工况下有效降低NO x 排放量，在不增加电厂改造成本上，为电厂提供了有效的控制手段，具有一定应用前景。

关键词：煤燃烧；优化；氮氧化物；差分算法；最小二乘支持向量机；模型预测控制中图分类号：TK224 文献标志码：A 文章编号：1000–6613（2018）01–0343–07 DOI ：10.16085/j.issn.1000-6613.2017-0716Model predictive control of NO x emission under variable load conditionTANG Zhenhao ，ZHANG Haiyang ，CAO Shengxian（School of Automation Engineering ，Northeast Electric Power University ，Jilin 132012，Jilin ，China ）Abstract: NO x is one of the main pollutants for coal-fired power plant emissions. The main problemfor the plants today is reducing NO x emission. A nonlinear model predictive control method based on least square support vector machine （LSSVM ）is proposed in this paper to solve the boiler NO x emission minimization problem considering varying load in coal-fired power plants. The boiler load model and NO x emissions model are constructed based on practical data. And then, the model parameters can be optimized by cross validation to obtain accuracy models. Based on these models, the boiler combustion optimization model is constructed. The optimization model aiming at minimizing the NO x emission considers the boiler load as a constraint. This optimization model is solved to obtain the optimal control variable settings by different evolution （DE ）algorithm. To testify the effectiveness of the proposed approach, the experiments based on real operational data are designed. The experiments results illustrate that the proposed method could reduce NO x emissions effectively under varying load. It provides an effective means at no additional cost and has a certain application prospect.Key words ：coal combustion ；optimization ；nitrogen oxide ；differential evolution algorithm ；least squares support vector ；model-predictive control为了解决我国面临的严峻的环境污染问题，由中华人民共和国环境保护部发布的《火电厂大气污染物排放标准》中要求自2012年1月1日起除个别地区外，火电厂NO x 的排放量不得高于100mg/m 3。

智能反射面增强的多无人机辅助语义通信资源优化

doi:10.3969/j.issn.1003-3114.2024.02.018引用格式:王浩博,吴伟,周福辉,等.智能反射面增强的多无人机辅助语义通信资源优化[J].无线电通信技术,2024,50(2): 366-372.[WANG Haobo,WU Wei,ZHOU Fuhui,et al.Optimization of Resource Allocation for Intelligent Reflecting Surface-enhanced Multi-UAV Assisted Semantic Communication[J].Radio Communications Technology,2024,50(2):366-372.]智能反射面增强的多无人机辅助语义通信资源优化王浩博1,吴㊀伟1,2∗,周福辉2,胡㊀冰3,田㊀峰1(1.南京邮电大学通信与信息工程学院,江苏南京210003;2.南京航空航天大学电子信息工程学院,江苏南京211106;3.南京邮电大学现代邮政学院,江苏南京210003)摘㊀要:无人机(Unmanned Aerial Vehicle,UAV)为无线通信系统提供了具有高成本效益的解决方案㊂进一步地,提出了一种新颖的智能反射面(Intelligent Reflecting Surface,IRS)增强多UAV语义通信系统㊂该系统包括配备IRS的UAV㊁移动边缘计算(Mobile Edge Computing,MEC)服务器和具有数据收集与局部语义特征提取功能的UAV㊂通过IRS 优化信号反射显著改善了UAV与MEC服务器的通信质量㊂所构建的问题涉及多UAV轨迹㊁IRS反射系数和语义符号数量联合优化,以最大限度地减少传输延迟㊂为解决该非凸优化问题,本文引入了深度强化学习(Deep Reinforce Learn-ing,DRL)算法,包括对偶双深度Q网络(Dueling Double Deep Q Network,D3QN)用于解决离散动作空间问题,如UAV轨迹优化和语义符号数量优化;深度确定性策略梯度(Deep Deterministic Policy Gradient,DDPG)用于解决连续动作空间问题,如IRS反射系数优化,以实现高效决策㊂仿真结果表明,与各个基准方案相比,提出的智能优化方案性能均有所提升,特别是在发射功率较小的情况下,且对于功率的变化,所提出的智能优化方案展示了良好的稳定性㊂关键词:无人机网络;智能反射面;语义通信;资源分配中图分类号:TN925㊀㊀㊀文献标志码:A㊀㊀㊀开放科学(资源服务)标识码(OSID):文章编号:1003-3114(2024)02-0366-07Optimization of Resource Allocation for Intelligent ReflectingSurface-enhanced Multi-UAV Assisted Semantic CommunicationWANG Haobo1,WU Wei1,2∗,ZHOU Fuhui2,HU Bing3,TIAN Feng1(1.School of Communications and Information Engineering,Nanjing University of Posts and Telecommunications,Nanjing210003,China;2.College of Electronic and Information Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing211106,China;3.School of Modern Posts,Nanjing University of Posts and Telecommunications,Nanjing210003,China)Abstract:Unmanned Aerial Vehicles(UAV)present a cost-effective solution for wireless communication systems.This article introduces a novel Intelligent Reflecting Surface(IRS)to augment the semantic communication system among multiple UAVs.The system encompasses UAV equipped with IRS,Mobile Edge Computing(MEC)servers,and UAV featuring data collection and local semantic feature extraction functions.Optimizing signal reflection through IRS significantly enhances communication quality between drones and MEC servers.The formulated problem entails joint optimization of multiple drone trajectories,IRS reflection coefficients,and the number of semantic symbols to minimize transmission delays.To address this non-convex optimization problem,this paper introduces a Deep收稿日期:2023-12-31基金项目:国家重点研发计划(2020YFB1807602);国家自然科学基金(62271267);广东省促进经济发展专项资金(粤自然资合[2023]24号);国家自然科学基金(青年项目)(62302237)Foundation Item:National K&D Program of China(2020YFB1807602);National Natural Science Foundation of China(62271267);Key Program of Marine Economy Development Special Foundation of Department of Natural Resources of Guangdong Province(GDNRC[2023]24);National Natural Sci-ence Foundation of China(Young Scientists Fund)(62302237)ＲｅｉｎｆｏｒｃｅｍｅｎｔＬｅａｒｎｉｎｇ（ＤＲＬ）ａｌｇｏｒｉｔｈｍ．Ｓｐｅｃｉｆｉｃａｌｌｙ，ｔｈｅＤｕｅｌｉｎｇＤｏｕｂｌｅＤｅｅｐＱＮｅｔｗｏｒｋ（Ｄ３ＱＮ）ｉｓｅｍｐｌｏｙｅｄｔｏａｄｄｒｅｓｓｄｉｓｃｒｅｔｅａｃｔｉｏｎｓｐａｃｅｐｒｏｂｌｅｍｓｓｕｃｈａｓｄｒｏｎｅｔｒａｊｅｃｔｏｒｙａｎｄｓｅｍａｎｔｉｃｓｙｍｂｏｌｑｕａｎｔｉｔｙｏｐｔｉｍｉｚａｔｉｏｎ．Ａｄｄｉｔｉｏｎａｌｌｙ，ＤｅｅｐＤｅｔｅｒｍｉｎｉｓｔｉｃＰｏｌｉｃｙＧｒａｄｉｅｎｔ（ＤＤＰＧ）ａｌｇｏｒｉｔｈｍｉｓｕｔｉｌｉｚｅｄｔｏｓｏｌｖｅｃｏｎｔｉｎｕｏｕｓａｃｔｉｏｎｓｐａｃｅｐｒｏｂｌｅｍｓ，ｓｕｃｈａｓＩＲＳｒｅｆｌｅｃｔｉｏｎｃｏｅｆｆｉｃｉｅｎｔｏｐｔｉｍｉｚａｔｉｏｎ，ｅｎａｂｌｉｎｇｅｆｆｉｃｉｅｎｔｄｅｃｉｓｉｏｎｍａｋｉｎｇ．Ｓｉｍｕｌａｔｉｏｎｒｅｓｕｌｔｓｄｅｍｏｎｓｔｒａｔｅｔｈａｔｔｈｅｐｒｏｐｏｓｅｄｉｎｔｅｌｌｉｇｅｎｔｏｐｔｉｍｉｚａｔｉｏｎｓｃｈｅｍｅｏｕｔｐｅｒｆｏｒｍｓｖａｒｉｏｕｓｂｅｎｃｈｍａｒｋｓｃｈｅｍｅｓ，ｐａｒｔｉｃｕｌａｒｌｙｉｎｓｃｅｎａｒｉｏｓｗｉｔｈｌｏｗｔｒａｎｓｍｉｓｓｉｏｎｐｏｗｅｒ．Ｆｕｒｔｈｅｒｍｏｒｅ，ｔｈｅｉｎｔｅｌｌｉｇｅｎｔｏｐｔｉｍｉｚａｔｉｏｎｓｃｈｅｍｅｐｒｏｐｏｓｅｄｉｎｔｈｉｓｐａｐｅｒｅｘｈｉｂｉｔｓｒｏｂｕｓｔｓｔａｂｉｌｉｔｙｉｎｒｅｓｐｏｎｓｅｔｏｐｏｗｅｒｃｈａｎｇｅｓ．Ｋｅｙｗｏｒｄｓ：ＵＡＶｎｅｔｗｏｒｋ；ＩＲＳ；ｓｅｍａｎｔｉｃｃｏｍｍｕｎｉｃａｔｉｏｎ；ｒｅｓｏｕｒｃｅａｌｌｏｃａｔｉｏｎ０　引言当前技术飞速发展的背景下，无人机（ＵｎｍａｎｎｅｄＡｅｒｉａｌＶｅｈｉｃｌｅ，ＵＡＶ）已经成为无线通信系统中一种重要的技术［１］。

Comparison of Multiobjective Evolutionary Algorithms Empirical Results1

Comparison of Multiobjective Evolutionary Algorithms:Empirical ResultsEckart ZitzlerDepartment of Electrical Engineering Swiss Federal Institute of T echnology 8092Zurich,Switzerlandzitzler@tik.ee.ethz.ch Kalyanmoy DebDepartment of Mechanical Engineering Indian Institute of T echnology Kanpur Kanpur,PIN208016,Indiadeb@iitk.ac.inLothar ThieleDepartment of Electrical EngineeringSwiss Federal Institute of T echnology8092Zurich,Switzerlandthiele@tik.ee.ethz.chAbstractIn this paper,we provide a systematic comparison of various evolutionary approaches tomultiobjective optimization using six carefully chosen test functions.Each test functioninvolves a particular feature that is known to cause difﬁculty in the evolutionary optimiza-tion process,mainly in converging to the Pareto-optimal front(e.g.,multimodality anddeception).By investigating these different problem features separately,it is possible topredict the kind of problems to which a certain technique is or is not well suited.However,in contrast to what was suspected beforehand,the experimental results indicate a hierarchyof the algorithms under consideration.Furthermore,the emerging effects are evidencethat the suggested test functions provide sufﬁcient complexity to compare multiobjectiveoptimizers.Finally,elitism is shown to be an important factor for improving evolutionarymultiobjective search.KeywordsEvolutionary algorithms,multiobjective optimization,Pareto optimality,test functions,elitism.1MotivationEvolutionary algorithms(EAs)have become established as the method at hand for exploring the Pareto-optimal front in multiobjective optimization problems that are too complex to be solved by exact methods,such as linear programming and gradient search.This is not only because there are few alternatives for searching intractably large spaces for multiple Pareto-optimal solutions.Due to their inherent parallelism and their capability to exploit similarities of solutions by recombination,they are able to approximate the Pareto-optimal front in a single optimization run.The numerous applications and the rapidly growing interest in the area of multiobjective EAs take this fact into account.After theﬁrst pioneering studies on evolutionary multiobjective optimization appeared in the mid-eighties(Schaffer,1984,1985;Fourman,1985)several different EA implementa-tions were proposed in the years1991–1994(Kursawe,1991;Hajela and Lin,1992;Fonseca c2000by the Massachusetts Institute of T echnology Evolutionary Computation8(2):173-195E.Zitzler,K.Deb,and L.Thieleand Fleming,1993;Horn et al.,1994;Srinivas and Deb,1994).Later,these approaches (and variations of them)were successfully applied to various multiobjective optimization problems(Ishibuchi and Murata,1996;Cunha et al.,1997;Valenzuela-Rend´on and Uresti-Charre,1997;Fonseca and Fleming,1998;Parks and Miller,1998).In recent years,some researchers have investigated particular topics of evolutionary multiobjective search,such as convergence to the Pareto-optimal front(Van Veldhuizen and Lamont,1998a;Rudolph, 1998),niching(Obayashi et al.,1998),and elitism(Parks and Miller,1998;Obayashi et al., 1998),while others have concentrated on developing new evolutionary techniques(Lau-manns et al.,1998;Zitzler and Thiele,1999).For a thorough discussion of evolutionary algorithms for multiobjective optimization,the interested reader is referred to Fonseca and Fleming(1995),Horn(1997),Van Veldhuizen and Lamont(1998b),and Coello(1999).In spite of this variety,there is a lack of studies that compare the performance and different aspects of these approaches.Consequently,the question arises:which imple-mentations are suited to which sort of problem,and what are the speciﬁc advantages and drawbacks of different techniques?First steps in this direction have been made in both theory and practice.On the theoretical side,Fonseca and Fleming(1995)discussed the inﬂuence of differentﬁtness assignment strategies on the selection process.On the practical side,Zitzler and Thiele (1998,1999)used a NP-hard0/1knapsack problem to compare several multiobjective EAs. In this paper,we provide a systematic comparison of six multiobjective EAs,including a random search strategy as well as a single-objective EA using objective aggregation.The basis of this empirical study is formed by a set of well-deﬁned,domain-independent test functions that allow the investigation of independent problem features.We thereby draw upon results presented in Deb(1999),where problem features that may make convergence of EAs to the Pareto-optimal front difﬁcult are identiﬁed and,furthermore,methods of constructing appropriate test functions are suggested.The functions considered here cover the range of convexity,nonconvexity,discrete Pareto fronts,multimodality,deception,and biased search spaces.Hence,we are able to systematically compare the approaches based on different kinds of difﬁculty and to determine more exactly where certain techniques are advantageous or have trouble.In this context,we also examine further factors such as population size and elitism.The paper is structured as follows:Section2introduces key concepts of multiobjective optimization and deﬁnes the terminology used in this paper mathematically.We then give a brief overview of the multiobjective EAs under consideration with special emphasis on the differences between them.The test functions,their construction,and their choice are the subject of Section4,which is followed by a discussion about performance metrics to assess the quality of trade-off fronts.Afterwards,we present the experimental results in Section6and investigate further aspects like elitism(Section7)and population size (Section8)separately.A discussion of the results as well as future perspectives are given in Section9.2DeﬁnitionsOptimization problems involving multiple,conﬂicting objectives are often approached by aggregating the objectives into a scalar function and solving the resulting single-objective optimization problem.In contrast,in this study,we are concerned withﬁnding a set of optimal trade-offs,the so-called Pareto-optimal set.In the following,we formalize this 174Evolutionary Computation Volume8,Number2Comparison of Multiobjective EAs well-known concept and also deﬁne the difference between local and global Pareto-optimalsets.A multiobjective search space is partially ordered in the sense that two arbitrary so-lutions are related to each other in two possible ways:either one dominates the other or neither dominates.D EFINITION1:Let us consider,without loss of generality,a multiobjective minimization problem with decision variables(parameters)and objectives:Minimizewhere(1) and where is called decision vector,parameter space,objective vector,and objective space.A decision vector is said to dominate a decision vector(also written as) if and only if(2)Additionally,in this study,we say covers()if and only if or.Based on the above relation,we can deﬁne nondominated and Pareto-optimal solutions: D EFINITION2:Let be an arbitrary decision vector.1.The decision vector is said to be nondominated regarding a set if and only if thereis no vector in which dominates;formally(3)If it is clear within the context which set is meant,we simply leave it out.2.The decision vector is Pareto-optimal if and only if is nondominated regarding.Pareto-optimal decision vectors cannot be improved in any objective without causing a degradation in at least one other objective;they represent,in our terminology,globally optimal solutions.However,analogous to single-objective optimization problems,there may also be local optima which constitute a nondominated set within a certain neighbor-hood.This corresponds to the concepts of global and local Pareto-optimal sets introduced by Deb(1999):D EFINITION3:Consider a set of decision vectors.1.The set is denoted as a local Pareto-optimal set if and only if(4)where is a corresponding distance metric and,.A slightly modiﬁed deﬁnition of local Pareto optimality is given here.Evolutionary Computation Volume8,Number2175E.Zitzler,K.Deb,and L.Thiele2.The set is called a global Pareto-optimal set if and only if(5) Note that a global Pareto-optimal set does not necessarily contain all Pareto-optimal solu-tions.If we refer to the entirety of the Pareto-optimal solutions,we simply write“Pareto-optimal set”;the corresponding set of objective vectors is denoted as“Pareto-optimal front”.3Evolutionary Multiobjective OptimizationT wo major problems must be addressed when an evolutionary algorithm is applied to multiobjective optimization:1.How to accomplishﬁtness assignment and selection,respectively,in order to guide thesearch towards the Pareto-optimal set.2.How to maintain a diverse population in order to prevent premature convergence andachieve a well distributed trade-off front.Often,different approaches are classiﬁed with regard to theﬁrst issue,where one can distinguish between criterion selection,aggregation selection,and Pareto selection(Horn, 1997).Methods performing criterion selection switch between the objectives during the selection phase.Each time an individual is chosen for reproduction,potentially a different objective will decide which member of the population will be copied into the mating pool. Aggregation selection is based on the traditional approaches to multiobjective optimization where the multiple objectives are combined into a parameterized single objective function. The parameters of the resulting function are systematically varied during the same run in order toﬁnd a set of Pareto-optimal solutions.Finally,Pareto selection makes direct use of the dominance relation from Deﬁnition1;Goldberg(1989)was theﬁrst to suggest a Pareto-basedﬁtness assignment strategy.In this study,six of the most salient multiobjective EAs are considered,where for each of the above categories,at least one representative was chosen.Nevertheless,there are many other methods that may be considered for the comparison(cf.Van Veldhuizen and Lamont(1998b)and Coello(1999)for an overview of different evolutionary techniques): Among the class of criterion selection approaches,the Vector Evaluated Genetic Al-gorithm(VEGA)(Schaffer,1984,1985)has been chosen.Although some serious drawbacks are known(Schaffer,1985;Fonseca and Fleming,1995;Horn,1997),this algorithm has been a strong point of reference up to now.Therefore,it has been included in this investigation.The EA proposed by Hajela and Lin(1992)is based on aggregation selection in combination withﬁtness sharing(Goldberg and Richardson,1987),where an individual is assessed by summing up the weighted objective values.As weighted-sum aggregation appears still to be widespread due to its simplicity,Hajela and Lin’s technique has been selected to represent this class of multiobjective EAs.Pareto-based techniques seem to be most popular in theﬁeld of evolutionary mul-tiobjective optimization(Van Veldhuizen and Lamont,1998b).In particular,the 176Evolutionary Computation Volume8,Number2Comparison of Multiobjective EAs algorithm presented by Fonseca and Fleming(1993),the Niched Pareto Genetic Algo-rithm(NPGA)(Horn and Nafpliotis,1993;Horn et al.,1994),and the Nondominated Sorting Genetic Algorithm(NSGA)(Srinivas and Deb,1994)appear to have achieved the most attention in the EA literature and have been used in various studies.Thus, they are also considered here.Furthermore,a recent elitist Pareto-based strategy,the Strength Pareto Evolutionary Algorithm(SPGA)(Zitzler and Thiele,1999),which outperformed four other multiobjective EAs on an extended0/1knapsack problem,is included in the comparison.4Test Functions for Multiobjective OptimizersDeb(1999)has identiﬁed several features that may cause difﬁculties for multiobjective EAs in1)converging to the Pareto-optimal front and2)maintaining diversity within the population.Concerning theﬁrst issue,multimodality,deception,and isolated optima are well-known problem areas in single-objective evolutionary optimization.The second issue is important in order to achieve a well distributed nondominated front.However,certain characteristics of the Pareto-optimal front may prevent an EA fromﬁnding diverse Pareto-optimal solutions:convexity or nonconvexity,discreteness,and nonuniformity.For each of the six problem features mentioned,a corresponding test function is constructed following the guidelines in Deb(1999).We thereby restrict ourselves to only two objectives in order to investigate the simplest caseﬁrst.In our opinion,two objectives are sufﬁcient to reﬂect essential aspects of multiobjective optimization.Moreover,we do not consider maximization or mixed minimization/maximization problems.Each of the test functions deﬁned below is structured in the same manner and consists itself of three functions(Deb,1999,216):Minimizesubject to(6)whereThe function is a function of theﬁrst decision variable only,is a function of the remaining variables,and the parameters of are the function values of and.The test functions differ in these three functions as well as in the number of variables and in the values the variables may take.D EFINITION4:We introduce six test functions that follow the scheme given in Equa-tion6:The test function has a convex Pareto-optimal front:(7)where,and.The Pareto-optimal front is formed with.The test function is the nonconvex counterpart to:(8) Evolutionary Computation Volume8,Number2177E.Zitzler,K.Deb,and L.Thielewhere,and.The Pareto-optimal front is formed with.The test function represents the discreteness feature;its Pareto-optimal front consists of several noncontiguous convex parts:(9)where,and.The Pareto-optimal front is formed with.The introduction of the sine function in causes discontinuity in the Pareto-optimal front.However, there is no discontinuity in the parameter space.The test function contains local Pareto-optimal fronts and,therefore,tests for the EA’s ability to deal with multimodality:(10)where,,and.The global Pareto-optimal front is formed with,the best local Pareto-optimal front with.Note that not all local Pareto-optimal sets are distinguishable in the objective space.The test function describes a deceptive problem and distinguishes itself from the other test functions in that represents a binary string:(11)where gives the number of ones in the bit vector(unitation),ififand,,and.The true Pareto-optimal front is formed with,while the best deceptive Pareto-optimal front is represented by the solutions for which.The global Pareto-optimal front as well as the local ones are convex.The test function includes two difﬁculties caused by the nonuniformity of the search space:ﬁrst,the Pareto-optimal solutions are nonuniformly distributed along the global Pareto front (the front is biased for solutions for which is near one);second,the density of the solutions is lowest near the Pareto-optimal front and highest away from the front:(12)where,.The Pareto-optimal front is formed with and is nonconvex.We will discuss each function in more detail in Section6,where the corresponding Pareto-optimal fronts are visualized as well(Figures1–6).178Evolutionary Computation Volume8,Number2Comparison of Multiobjective EAs5Metrics of PerformanceComparing different optimization techniques experimentally always involves the notion of performance.In the case of multiobjective optimization,the deﬁnition of quality is substantially more complex than for single-objective optimization problems,because the optimization goal itself consists of multiple objectives:The distance of the resulting nondominated set to the Pareto-optimal front should be minimized.A good(in most cases uniform)distribution of the solutions found is desirable.Theassessment of this criterion might be based on a certain distance metric.The extent of the obtained nondominated front should be maximized,i.e.,for each objective,a wide range of values should be covered by the nondominated solutions.In the literature,some attempts can be found to formalize the above deﬁnition(or parts of it)by means of quantitative metrics.Performance assessment by means of weighted-sum aggregation was introduced by Esbensen and Kuh(1996).Thereby,a set of decision vectors is evaluated regarding a given linear combination by determining the minimum weighted-sum of all corresponding objective vectors of.Based on this concept,a sample of linear combinations is chosen at random(with respect to a certain probability distribution),and the minimum weighted-sums for all linear combinations are summed up and averaged.The resulting value is taken as a measure of quality.A drawback of this metric is that only the“worst”solution determines the quality value per linear combination. Although several weight combinations are used,nonconvex regions of the trade-off surface contribute to the quality more than convex parts and may,as a consequence,dominate the performance assessment.Finally,the distribution,as well as the extent of the nondominated front,is not considered.Another interesting means of performance assessment was proposed by Fonseca and Fleming(1996).Given a set of nondominated solutions,a boundary function divides the objective space into two regions:the objective vectors for which the corre-sponding solutions are not covered by and the objective vectors for which the associated solutions are covered by.They call this particular function,which can also be seen as the locus of the family of tightest goal vectors known to be attainable,the attainment surface. T aking multiple optimization runs into account,a method is described to compute a median attainment surface by using auxiliary straight lines and sampling their intersections with the attainment surfaces obtained.As a result,the samples represented by the median attain-ment surface can be relatively assessed by means of statistical tests and,therefore,allow comparison of the performance of two or more multiobjective optimizers.A drawback of this approach is that it remains unclear how the quality difference can be expressed,i.e.,how much better one algorithm is than another.However,Fonseca and Fleming describe ways of meaningful statistical interpretation in contrast to the other studies considered here,and furthermore,their methodology seems to be well suited to visualization of the outcomes of several runs.In the context of investigations on convergence to the Pareto-optimal front,some authors(Rudolph,1998;Van Veldhuizen and Lamont,1998a)have considered the distance of a given set to the Pareto-optimal set in the same way as the function deﬁned below.The distribution was not taken into account,because the focus was not on this Evolutionary Computation Volume8,Number2179E.Zitzler,K.Deb,and L.Thielematter.However,in comparative studies,distance alone is not sufﬁcient for performance evaluation,since extremely differently distributed fronts may have the same distance to the Pareto-optimal front.T wo complementary metrics of performance were presented in Zitzler and Thiele (1998,1999).On one hand,the size of the dominated area in the objective space is taken under consideration;on the other hand,a pair of nondominated sets is compared by calculating the fraction of each set that is covered by the other set.The area combines all three criteria(distance,distribution,and extent)into one,and therefore,sets differing in more than one criterion may not be distinguished.The second metric is in some way similar to the comparison methodology proposed in Fonseca and Fleming(1996).It can be used to show that the outcomes of an algorithm dominate the outcomes of another algorithm, although,it does not tell how much better it is.We give its deﬁnition here,because it is used in the remainder of this paper.D EFINITION5:Let be two sets of decision vectors.The function maps the ordered pair to the interval:(13)The value means that all solutions in are dominated by or equal to solutions in.The opposite,,represents the situation when none of the solutions in are covered by the set.Note that both and have to be considered,since is not necessarily equal to.In summary,it may be said that performance metrics are hard to deﬁne and it probably will not be possible to deﬁne a single metric that allows for all criteria in a meaningful way.Along with that problem,the statistical interpretation associated with a performance comparison is rather difﬁcult and still needs to be answered,since multiple signiﬁcance tests are involved,and thus,tools from analysis of variance may be required.In this study,we have chosen a visual presentation of the results together with the application of the metric from Deﬁnition5.The reason for this is that we would like to in-vestigate1)whether test functions can adequately test speciﬁc aspects of each multiobjective algorithm and2)whether any visual hierarchy of the chosen algorithms exists.However, for a deeper investigation of some of the algorithms(which is the subject of future work), we suggest the following metrics that allow assessment of each of the criteria listed at the beginning of this section separately.D EFINITION6:Given a set of pairwise nondominating decision vectors,a neighborhood parameter(to be chosen appropriately),and a distance metric.We introduce three functions to assess the quality of regarding the parameter space:1.The function gives the average distance to the Pareto-optimal set:min(14)Recently,an alternative metric has been proposed in Zitzler(1999)in order to overcome this problem. 180Evolutionary Computation Volume8,Number2Comparison of Multiobjective EAs 2.The function takes the distribution in combination with the number of nondominatedsolutions found into account:(15) 3.The function considers the extent of the front described by:max(16) Analogously,we deﬁne three metrics,,and on the objective space.Letbe the sets of objective vectors that correspond to and,respectively,and and as before:min(17)(18)max(19)While and are intuitive,and(respectively and)need further explanation.The distribution metrics give a value within the interval()that reﬂects the number of-niches(-niches)in().Obviously,the higher the value,the better the distribution for an appropriate neighborhood parameter(e.g.,means that for each objective vector there is no other objective vector within-distance to it).The functions and use the maximum extent in each dimension to estimate the range to which the front spreads out.In the case of two objectives,this equals the distance of the two outer solutions.6Comparison of Different Evolutionary Approaches6.1MethodologyWe compare eight algorithms on the six proposed test functions:1.A random search algorithm.2.Fonseca and Fleming’s multiobjective EA.3.The Niched Pareto Genetic Algorithm.4.Hajela and Lin’s weighted-sum based approach.5.The Vector Evaluated Genetic Algorithm.6.The Nondominated Sorting Genetic Algorithm.Evolutionary Computation Volume8,Number2181E.Zitzler,K.Deb,and L.Thiele7.A single-objective evolutionary algorithm using weighted-sum aggregation.8.The Strength Pareto Evolutionary Algorithm.The multiobjective EAs,as well as,were executed times on each test problem, where the population was monitored for nondominated solutions,and the resulting non-dominated set was taken as the outcome of one optimization run.Here,serves as an additional point of reference and randomly generates a certain number of individuals per generation according to the rate of crossover and mutation(but neither crossover and mutation nor selection are performed).Hence,the number ofﬁtness evaluations was the same as for the EAs.In contrast,simulation runs were considered in the case of, each run optimizing towards another randomly chosen linear combination of the objec-tives.The nondominated solutions among all solutions generated in the runs form the trade-off front achieved by on a particular test function.Independent of the algorithm and the test function,each simulation run was carried out using the following parameters:Number of generations:250Population size:100Crossover rate:0.8Mutation rate:0.01Niching parameter share:0.48862Domination pressure dom:10The niching parameter was calculated using the guidelines given in Deb and Goldberg (1989)assuming the formation of ten independent niches.Since uses genotypic ﬁtness sharing on,a different value,share,was chosen for this particular case. Concerning,the recommended value for dom of the population size wastaken(Horn and Nafpliotis,1993).Furthermore,for reasons of fairness,ran with a population size of where the external nondominated set was restricted to.Regarding the implementations of the algorithms,one chromosome was used to en-code the parameters of the corresponding test problem.Each parameter is represented by bits;the parameters only comprise bits for the deceptive function. Moreover,all approaches except were realized using binary tournament selection with replacement in order to avoid effects caused by different selection schemes.Further-more,sinceﬁtness sharing may produce chaotic behavior in combination with tournament selection,a slightly modiﬁed method is incorporated here,named continuously updated shar-ing(Oei et al.,1991).As requires a generational selection mechanism,stochastic universal sampling was used in the implementation.6.2Simulation ResultsIn Figures1–6,the nondominated fronts achieved by the different algorithms are visualized. Per algorithm and test function,the outcomes of theﬁrstﬁve runs were uniﬁed,and then the dominated solutions were removed from the union set;the remaining points are plotted in theﬁgures.Also shown are the Pareto-optimal fronts(lower curves),as well as additional reference curves(upper curves).The latter curves allow a more precise evaluation of the obtained trade-off fronts and were calculated by adding max minto the values of the Pareto-optimal points.The space between Pareto-optimal and 182Evolutionary Computation Volume8,Number2f101234f2RANDFFGA NPGA HLGA VEGA NSGA SOEA SPEAFigure 1:T est function(convex).f101234f2RANDFFGA NPGA HLGA VEGA NSGA SOEA SPEAFigure 2:T est function(nonconvex).183f11234f2RANDFFGA NPGA HLGA VEGA NSGA SOEA SPEA Figure 3:T est function(discrete).f1010203040f2RANDFFGA NPGA HLGA VEGA NSGA SOEA SPEAFigure 4:T est function(multimodal).184f10246f2RANDFFGA NPGAHLGA VEGA NSGASOEA SPEA Figure 5:T est function (deceptive).f12468f2RANDFFGA NPGA HLGA VEGA NSGASOEA SPEA Figure 6:T est function(nonuniform).185reference fronts represents about of the corresponding objective space.However,the curve resulting from the deceptive function is not appropriate for our purposes,since it lies above the fronts produced by the random search algorithm.Instead,we consider all solutions with,i.e.,for which the parameters are set to the deceptive attractors (for).In addition to the graphical presentation,the different algorithms were assessed in pairs using the metric from Deﬁnition5.For an ordered algorithm pair,there is a sample of values according to the runs performed.Each value is computed on the basis of the nondominated sets achieved by and with the same initial population. Here,box plots are used to visualize the distribution of these samples(Figure7).A box plot consists of a box summarizing of the data.The upper and lower ends of the box are the upper and lower quartiles,while a thick line within the box encodes the median. Dashed appendages summarize the spread and shape of the distribution.Furthermore,the shortcut in Figure7stands for“reference set”and represents,for each test function,a set of equidistant points that are uniformly distributed on the corresponding reference curve.Generally,the simulation results prove that all multiobjective EAs do better than the random search algorithm.However,the box plots reveal that,,anddo not always cover the randomly created trade-off front completely.Furthermore,it can be observed that clearly outperforms the other nonelitist multiobjective EAs regarding both distance to the Pareto-optimal front and distribution of the nondominated solutions.This conﬁrms the results presented in Zitzler and Thiele(1998).Furthermore, it is remarkable that performs well compared to and,although some serious drawbacks of this approach are known(Fonseca and Fleming,1995).The reason for this might be that we consider the off-line performance here in contrast to other studies that examine the on-line performance(Horn and Nafpliotis,1993;Srinivas and Deb,1994). On-line performance means that only the nondominated solutions in theﬁnal population are considered as the outcome,while off-line performance takes the solutions nondominated among all solutions generated during the entire optimization run into account.Finally,the best performance is provided by,which makes explicit use of the concept of elitism. Apart from,it even outperforms in spite of substantially lower computational effort and although uses an elitist strategy as well.This observation leads to the question of whether elitism would increase the performance of the other multiobjective EAs.We will investigate this matter in the next section.Considering the different problem features separately,convexity seems to cause the least amount of difﬁculty for the multiobjective EAs.All algorithms evolved reasonably distributed fronts,although there was a difference in the distance to the Pareto-optimal set.On the nonconvex test function,however,,,and have difﬁculties ﬁnding intermediate solutions,as linear combinations of the objectives tend to prefer solutions strong in at least one objective(Fonseca and Fleming,1995,4).Pareto-based algorithms have advantages here,but only and evolved a sufﬁcient number of nondominated solutions.In the case of(discreteness),and are superior to both and.While the fronts achieved by the former cover about of the reference set on average,the latter come up with coverage.Among the considered test functions,and seem to be the hardest problems,since none of the algorithms was able to evolve a global Pareto-optimal set.The results on the multimodal problem indicateNote that outside values are not plotted in Figure7in order to prevent overloading of the presentation. 186。

精英竞争和综合控制的多目标粒子群算法

第41卷第2期Vol.41㊀No.2重庆工商大学学报(自然科学版)J Chongqing Technol &Business Univ(Nat Sci Ed)2024年4月Apr.2024精英竞争和综合控制的多目标粒子群算法陈㊀飞1,刘衍民2,刘㊀君3,张娴子31.贵州大学数学与统计学院,贵阳5500252.遵义师范学院数学学院,贵州遵义5630063.贵州民族大学数据科学与信息工程学院,贵阳550025摘㊀要:目的多目标粒子群算法虽然极易实现且收敛速度快,但在平衡其收敛性和多样性方面仍需进一步改善㊂方法针对上述问题,提出一种精英竞争和综合控制的多目标粒子群算法(ECMOPSO )㊂一方面,算法采用全局损害选择精英粒子集,然后将两两竞争引入多目标粒子群算法中,通过精英竞争选取优胜者粒子,将其与全局领导者融合形成更全面的社会综合信息,以增强种群中粒子之间信息的交互性,更好引导种群中的粒子飞行,提升算法全局探索能力;另一方面,结合全局损害和基于位移密度估计对外部存档进行维护,从而提高外部存档中非劣解的质量,平衡算法的收敛性和多样性㊂结果将ECMOPSO 算法与4个多目标粒子群算法和4个多目标进化算法在ZDT 和UF 系列基准测试问题上进行仿真实验,并采用Wilcoxon 秩和检验和Friedman 秩检验比较ECMOPSO 算法与所选对比算法的整体性能㊂实验结果表明:相比其他几个对比算法,ECMOPSO 算法的收敛能力㊁解的分布性以及稳定性都得到了一定的提升㊂结论ECMOPSO 算法可以很好地平衡收敛性和多样性,提升其整体性能,能有效求解大多数多目标优化问题㊂关键词:多目标粒子群算法;精英竞争;综合控制;全局损害;基于位移密度估计中图分类号:TP18㊀㊀文献标识码:A ㊀㊀doi:10.16055/j.issn.1672-058X.2024.0002.010㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀收稿日期:2022-12-27㊀修回日期:2023-03-02㊀文章编号:1672-058X(2024)02-0074-12基金项目:贵州省进化人工智能重点实验室([2022]059);贵州省数字经济重点人才计划(2022001).作者简介:陈飞(1999 ),女,贵州遵义人,硕士研究生,从事优化理论及智能算法研究.通讯作者:刘衍民(1978 ),男,山东临沂人,教授,博士研究生导师,从事优化理论及智能算法研究.Email:yanmin7813@.引用格式:陈飞,刘衍民,刘君,等.精英竞争和综合控制的多目标粒子群算法[J].重庆工商大学学报(自然科学版),2024,41(2):74 85.CHEN Fei LIU Yanmin LIU Jun et al.Multi-objective particle swarm optimization with elite competition and comprehensivecontrol J .Journal of Chongqing Technology and Business University Natural Science Edition 2024 41 2 74 85.Multi-objective Particle Swarm Optimization with Elite Competition and Comprehensive Control CHEN Fei 1 LIU Yanmin 2 LIU Jun 3 ZHANG Xianzi 31.School of Mathematics and Statistics Guizhou University Guiyang 550025 China2.School of Mathematics Zunyi Normal University Guizhou Zunyi 563006 China3.School of Data Science and Information Engineering Guizhou Minzu University Guiyang 550025 ChinaAbstract Objective Although multi-objective particle swarm optimization is easy to implement and has a fastconvergence speed it still needs to be further improved in the aspect of balancing convergence and diversity.Methods To solve the above problem a multi-objective particle swarm optimization with elite competition and comprehensive control ECMOPSO was proposed.On one hand the algorithm selected the elite particle set with global detriment and then the pairwise competitions were introduced into the multi-objective particle swarm optimization.Thewinner particles were selected through elite competition and the winner particles were fused with the global leaders toform more comprehensive social information so as to enhance the information interaction between particles in the第2期陈飞,等:精英竞争和综合控制的多目标粒子群算法population better guide the flight of particles in the population and improve the global exploration ability of the algorithm.On the other hand global detriment and shifted-based density estimation were combined to maintain the external archive so as to improve the quality of non-dominated solutions in the external archive and balance the convergence and diversity of the algorithm.Results The ECMOPSO algorithm four multi-objective particle swarm optimization algorithms and four multi-objective evolutionary algorithms were simulated for ZDT and UF series benchmark problems and the Wilcoxon rank sum test and the Friedman rank test were used to compare the overall performance of ECMOPSO algorithm with the selected comparison algorithms.The experimental results showed that compared with other comparison algorithms the convergence ability solution distribution and stability of the ECMOPSO algorithm have been improved to some extent.Conclusion ECMOPSO algorithm can balance convergence and diversity well improve its overall performance and can effectively solve most multi-objective optimization problems.Keywords multi-objective particle swarm optimization elite competition comprehensive control global detriment shifted-based density estimation1㊀引㊀言由于多目标优化问题(Multi-objective Optimization Problems,MOPs)[1]的目标之间相互冲突,一个目标的改进可能会导致其他目标退化,所以MOPs中通常得到的是一组折衷解,这个解被称为Pareto最优解[1]㊂粒子群算法(Particle Swarm Optimization,PSO)[2]具有收敛速度快㊁参数少和易实现等特点,在求解单目标优化问题时有着良好的优化性能,随之有学者将其扩展来求解MOPs,也取得了很好的效果㊂Coello等[3]在2002年提出多目标粒子群算法(Multi-Objective Particle Swarm Optimization,MOPSO),该算法使用外部存档来存储算法在搜索过程中发现的所有非劣解,为全局最优解(Global Best Solution,g best)的选取提供良好的候选者,从而与个体最优解(Personal Best Solution,p best)共同指导种群中的粒子飞行,这样可以加快算法的收敛速度㊂但是,在优化过程中,MOPSO算法全局搜索能力较弱,易导致早熟收敛而陷入局部最优及多样性差等问题㊂因此,许多研究人员针对这些问题提出了改进的MOPSO算法[4-8]㊂Han 等[4]使用多目标梯度法对外部存档进行维护,以提高进化过程中的收敛速度和局部开发能力,但在解决某些多模态问题时并没有表现出较大优势;Li等[5]提出一种主导差分方法来判别解之间的优势关系以及设计一个基于Lp范数的密度估计器,使算法不仅具有良好的收敛性和多样性,而且具有较低的复杂度,但在解决具有复杂Pareto前沿(Pareto Front,PF)问题时还有一定不足;Wu等[6]提出一种新的进化状态估计机制自适应地切换领导者的选择策略,平衡算法在演化过程中的开发与探索关系,但在某些MOPs上性能不太稳定; Han等[7]提出一种基于解分布熵和种群间距信息混合框架的自适应算法,在收敛速度和精度方面都有一定的提高,但增加了一定的计算复杂度;Zhu等[8]利用分解方法将MOPs转换为一组子问题,并且外部档案中的个体通过基于免疫的进化策略进行进化,有利于加快收敛速度,但其收敛性和多样性并未均衡㊂综上可知,以上各种算法在收敛性和多样性等方面都有一定的提升,但是在有效平衡其收敛性和多样性方面仍需改进㊂为进一步解决上述问题,本文提出一种精英竞争和综合控制的多目标粒子群算法(ECMOPSO)㊂首先,为了提高种群的搜索能力,采用全局损害选择精英粒子集;其次,将MOPSO算法与两两竞争结合起来,通过精英竞争选取优胜者粒子,将其与g best融合以便更好地引导种群中的粒子飞行;最后,结合全局损害和基于位移密度估计的综合平衡控制策略对外部存档进行维护,有效平衡算法的收敛性和多样性㊂2㊀相关工作2.1㊀多目标优化问题一般多目标优化问题[9]可以表述如下:min㊀F x()=f1x(),f2x(), ,f M x()()s.t.㊀g i x()ȡ0,i=1,2, ,qh j x()=0,j=1,2, ,r其中:x=x1,x2, ,x D()是决策空间X⊂R D中的一个D 维决策向量;F x()表示M维的目标向量;f i x()表示第i 维的目标函数;g i x()ȡ0i=1,2, ,q()和hjx()=0 j=1,2, ,r()分别是不等式约束和等式约束㊂57重庆工商大学学报(自然科学版)第41卷2.2㊀粒子群算法在PSO 算法[2]中,每个粒子通过跟踪两个领导者p best 和g best 更新其自身速度㊂p best 记录了每个粒子在历史搜索过程中的最优位置,而g best 则记录了整个种群当前搜索到的最优位置㊂假设第i 个粒子的速度矢量为V i =v i ,1,v i ,2, ,v i ,D (),位置矢量为X i =(x i ,1,x i ,2,,x i ,D ),其中i =1,2, ,N ,N 为种群规模,D 为决策空间的维数㊂则第i 个粒子在第t +1次迭代都遵循式(1)和式(2)来更新其速度和位置㊂V i t +1()=ωV i t ()+c 1r 1p best i t ()-X i t ()()+c 2r 2g best i t ()-X i t ()()(1)X i t +1()=X i t ()+V i t +1()(2)其中:t 是迭代次数,ω为惯性权重,c 1和c 2是分别用来控制p best 和g best 对粒子速度影响的加速度系数,r 1和r 2是[0,1]中均匀生成的两个随机数㊂2.3㊀竞争群优化器竞争群优化器(Competitive Swarm Optimizer,CSO)是由Cheng 等[10]提出的一个受PSO 算法启发且计算成本较低的算法,它与传统PSO 算法在本质上有很大的不同㊂CSO 算法不再使用p best 和g best 来引导粒子飞行,而是使用一种两两竞争学习来更新粒子的速度和位置㊂实验结果显示:CSO 算法比PSO 算法可以更好地平衡收敛性和多样性㊂第t +1次迭代过程中,第k 轮竞争之后失败粒子的速度和位置更新公式如下:V l ,k t +1()=r 1V l ,k t ()+r 2X w ,k t ()-X l ,k t ()()+φr 3X -k t ()-X l ,k t ()()X l ,k t +1()=X l ,k t ()+V l ,k t +1()其中:t 是迭代次数;k =1,2, ,N/2,N 为种群规模;r 1㊁r 2和r 3是[0,1]中均匀生成的3个随机数;V l ,k (t )和X l ,k (t )分别是第t 次迭代过程中第k 轮竞争之后失败粒子的速度和位置;X w ,k t ()和X -k t ()分别是第t 次迭代过程中第k 轮竞争之后优胜者粒子的位置和种群中所有粒子位置的均值;φ是用来控制X -k t ()影响的一个参数㊂3㊀MOPSO 算法设计为了更好平衡算法的收敛性和多样性,使粒子能够在较高维度的搜索空间中跳出局部最优而收敛到全局最优,本文提出一个改进的ECMOPSO 算法㊂与原始MOPSO 算法相比,ECMOPSO 算法提出了一种精英竞争的社会融合策略对粒子进行更新,并利用一种综合平衡控制策略对外部存档进行维护,有效地平衡了收敛性和多样性㊂3.1㊀精英竞争的社会融合在大多数元启发式算法中,如何有效平衡收敛性和多样性是一个至关重要的问题㊂在标准的MOPSO 算法中,种群中的每个粒子向p best 和g best 这两个领导者学习,在其社会学习部分只选择一个全局领导者㊂这种情况下,粒子不能从各种样本中学到更多有用的经验,从而降低了种群的多样性㊂基于上述这个问题,ECMOPSO 算法结合了CSO 算法中的两两竞争对种群中的粒子进行更新,以实现收敛性和多样性之间的平衡㊂首先,创建一个预定义大小为r 的精英粒子集E ,以获得综合性能较好的非劣解㊂由于精英粒子集是用来提供候选粒子的,用于两两竞争选择领导者来引导种群的搜索,所以选择的精英粒子应该在收敛性和多样性之间保持良好的平衡㊂因此,本文结合基于非支配排序和基于指标排序的方法来选择精英粒子㊂由于全局损害(Global Detriment,GD)[11]考虑解在不同目标上差异的绝对值累积,是一个考虑解之间差异有多显著的方法,所以选择它来作为E 中精英粒子的选择原则,最终选出排名前r 的粒子作为精英粒子集㊂GD 的描述具体如下:R GD NS i ()=1NM ðN j =1ðM m =1max f m NS i ()-f m NS j (),0()f m ,max -f m ,min其中:NS i 是第i 个非劣解;N 是非劣解的个数;M 是目标个数;f m ㊃()为第m 个目标的适应度函数;f m ,max 和f m ,min 分别是第m 个目标上的最大值和最小值㊂接下来,从精英粒子集E 中随机选出两个精英粒子竞争,优胜者粒子e 与g best 融合来引导种群中的粒子飞行,这样有利于更好地实现收敛性和多样性之间的平衡㊂假设V i =v i ,1,v i ,2, ,v i ,D ()为第i 个粒子的速度向量,X i =x i ,1,x i ,2, ,x i ,D ()为位置向量,优胜者粒子位置向量为e i =e i ,1,e i ,2, ,e i ,D (),其中:i =1,2, ,N ,N 为种群中的粒子数;D 为决策空间的维数㊂则第i 个粒子在第t +1次迭代时的速度和位置更新如式(3)和式(2)所示㊂V i t +1()=ωV i t ()+c 1r 1p best i t ()-X i t ()()+㊀c 2r 2g best i t ()+e i t ()()/2-X i t ()()(3)其中:t 为迭代次数,ω为惯性权重,p best 和g best 分别为个体最优解和全局最优解,c 1和c 2是分别用来控制p best 和g best 对粒子速度影响的加速度系数,r 1和r 2是[0,1]中均匀生成的两个随机数㊂3.2㊀综合平衡控制在大多数多目标粒子群算法中,随着算法迭代次67第2期陈飞,等:精英竞争和综合控制的多目标粒子群算法数的增加,为了能够保留搜索过程中发现的具有良好性能的非劣解,以及最后获得分布良好的Pareto 前沿,需要对外部存档进行有效维护㊂MOPSO [3]算法采用自适应网格策略对外部存档中的非劣解进行删除,该算法随机删除密度最大的粒子,这样很可能会删除一些性能相对较好的非劣解,降低了算法的收敛性㊂因此,本文提出GD 和基于位移密度估计(Shifted -based Density Estimation,SDE)[12]的综合平衡控制策略(其值由R SDE 表示)来估计存档中解的密度,从而对自适应网格中密度大的解进行删除,确保外部存档中非劣解保持良好的分布㊂SDE 同时涵盖个体的收敛性信息和分布性信息,当度量外部存档中的非劣解NS i 和其他非劣解NS j 之间的相似度时,首先按照式(4)来根据NS i 和NS j 之间的收敛性比较移动NS j 在目标空间中的位置,即f ᶄm NS j ()=f m NS j (),if㊀f m NS i ()<f m NS j ()f m NS i (),if㊀f m NS i ()ȡf m NS j (){(4)其中:i ,j =1,2, ,n ,i ʂj ,n 为当前外部存档中非劣解的个数;m =1,2, ,M ,M 表示目标个数;f m NS i ()和f m NS j ()分别为非劣解NS i 和NS j 的第m 个目标值;f ᶄm NS j ()表示解NS j 位移之后的目标值㊂移动位置后,根据式(5)中的欧氏距离来度量解NS i 和NS j 之间的相似度,d i ,j 表示NS i 与NS j 位移后的距离㊂再通过式(6)计算非劣解NS i 的密度D NS i ():d i ,j =ðMm =1f ᶄmNS j ()-f m NS i ()(5)D NS i ()=min d i ,1,d i ,2, ,d i ,n ()(6)下面描述外部存档维护的综合平衡控制(Comprehensive Balance Control,CBC)指标(其值用R CBC 表示)㊂如式(7)所示,R CBC 由R SDE 计算得到的密度和R GD 之比组成,可以更全面地评估外部存档中的非劣解,R CBC 的值越大说明解的收敛性和多样性越好㊂R CBC NS i ()=D NS i ()R GD NS i ()(7)为了验证R CBC 指标维护外部存档时的性能,图1显示了一个双目标的例子来说明分别采用指标R SDE ㊁R GD和R CBC 对密度最大的粒子进行删除时的效果㊂假设最密集的网格中有6个非劣解:N 1(0.0547,0.7131)㊁N 2(0.0550,0.7111)㊁N 3(0.0551,0.7110)㊁N 4(0.0553,0.7102)㊁N 5(0.0556,0.7081)和N 6(0.0559,0.7058),表1分别列出了由指标R SDE ㊁R GD 和R CBC 计算所获得的排序值,将要被删除解的值以粗体显示㊂从表中可以看出:N 2和N 3获得了相同的R SDE 值,这里随机删除其中一个,用R GD 评估时选择将极值点N 6删除,这将导致非劣解分布不均匀;而R CBC 则结合了这两个指标的优点,选择相对拥挤的解N 3删除㊂从图1可以直观地观察到:用R SED 和R CBC 来删除粒子时都获得了较均匀的分布,但R CBC 更能精准地删除综合性能不好的粒子,更有效地平衡算法的收敛性和多样性㊂图1㊀分别由R SDE ㊁R GD 和R CBC 策略删除外部存档中粒子的图解Fig.1㊀Illustrations of deleting particles in the externalarchive by R SDE ,R GD and R CBC ,respectively表1㊀由R SDE ㊁R GD 和R CBC 获得的排序值Table 1㊀The ranking values obtained by R SDE ,R GD and R CBC指㊀标N 1N 2N 3N 4N 5N 6R SDE 0.00030.00010.00010.00030.00030.0023R GD0.22030.12700.13630.15060.18600.2639R CBC 0.00140.00080.00070.00200.00160.00873.3㊀算法的主要流程算法的具体步骤如下:Step1㊀设置ECMOPSO 算法的相关参数,初始化种群中粒子的位置㊂Step2㊀对粒子进行变异操作,并计算其适应度值㊂Step3㊀结合非支配排序和指标排序选取精英粒子集E ㊂Step4㊀建立外部存档,存储非劣解㊂若外部存档没有达到最大存储数,则继续引入非劣解;若达到最大存储数,则利用综合平衡控制对外部存档进行维护㊂Step5㊀选取p best 和g best ,并利用两两竞争选择优胜者粒子e ㊂77重庆工商大学学报(自然科学版)第41卷Step6㊀通过式(3)和式(2)更新粒子的速度和位置㊂Step7㊀若当前迭代次数没有达到最大迭代次数,则返回Step2;否则终止迭代,输出最优解集㊂图2㊀ECMOPSO算法流程图Fig.2㊀Flow chart of ECMOPSO algorithm 4㊀实验仿真分析4.1㊀实验参数设置为了对ECMOPSO算法的性能更客观地评价,使用了ZDT[13]和UF[14]两个不同基准测试组的15个测试问题来评估算法的性能㊂其中包括ZDT测试组的5个双目标测试问题以及UF测试组的7个双目标测试问题和3个三目标测试问题㊂这些测试问题具有不同的特性和复杂的Pareto前沿特征,比如凹凸性㊁多模态和不规则Pareto前沿形状等,可以很好地验证算法的可靠性和效率㊂对于双目标测试问题,ZDT1 ZDT3和UF1 UF7的决策变量个数设置为30,ZDT4和ZDT6的决策变量个数设置为10㊂对于三目标测试问题, UF8 UF10的决策变量个数设置为30㊂另外,选取了8个算法进行对比,其中有4个先进的多目标粒子群算法:SMPSO[15]㊁dMOPSO[16]㊁MPSOD[17]㊁NMPSO[18],以及4个具有竞争力的多目标进化算法:NSGAIII[19]㊁MOEADD[20]㊁SPEAR[21]㊁DGEA[22]㊂为了保证算法性能比较的公平性,所有对比算法设置的相关参数都与原始参考文献一致,每个算法的主要参数设置如表2所示㊂表2㊀ECMOPSO与几个对比算法的参数设置Table2㊀Parameter settings of ECMOPSO and several comparison algorithms算法名称参数设置SMPSOωɪ[0.1,0.5],c1,c2ɪ[1.5,2.5],p m=1/n,ηm=20 dMOPSO T a=2,θ=5MPSODωɪ[0.1,0.5],c1,c2,c3ɪ[1.5,2.5],p c=0.9,F=0.5,CR=0.5,p m=1/n,ηm=20,ηc=20 NMPSOωɪ[0.1,0.5],c1,c2,c3ɪ[1.5,2.5],p m=1/n,ηm=20 NSGAIII p m=1/n,p c=1.0,ηm=20,ηc=20MOEADD p m=1/n,p c=1.0,ηm=20,ηc=30,T=20,δ=0.9 SPEAR p m=1/n,p c=1.0,ηm=20,ηc=20DGEA R=10ECMOPSOω=0.4,c1=c2=2,r=30㊀㊀所有算法种群规模设置为200,外部存档大小为200,最大函数评估次数为10000㊂对于参数r,即精英粒子集大小,将会在4.5节进行参数分析㊂各算法在每个测试问题上都是独立运行30次,并且所有算法的实验数据都是在Intel(R)Core(TM)i7-6700CPU@ 3.40GHz3.40GHz Windows7系统中运用PlatEMO平台[23]和MATLAB R2021b实现的㊂4.2㊀性能评价指标为了评估算法的性能,采用反世代距离(Inverted Generational Distance,IGD[24],其值用R IGD表示)和超体积(Hypervolume,HV[25],其值用R HV表示)作为算法的性能评价指标㊂(1)R IGD是一种综合性指标,用来衡量算法得到的Pareto最优解集与真实PF之间的距离,可以很好地检验算法的收敛性和多样性㊂一个算法的R IGD值越小,就说明算法的收敛性和多样性越好㊂R IGD的计算公式为R IGD P,S()=1PðxɪP dist x,S()其中:S表示算法得到的Pareto最优解集,P表示在Pareto前沿上均匀分布的解集,dist x,S()是P中的解x87第2期陈飞,等:精英竞争和综合控制的多目标粒子群算法与S 之间的最小欧氏距离㊂(2)R HV 也是一个综合的性能指标,表示算法获得的Pareto 最优解集与参照点围成的目标区域的体积㊂该指标可以估计算法所得到解集的收敛性和多样性,一个算法的R HV 值越大,就说明算法的收敛性和多样性越好㊂假设Z =Z 1,Z 2, ,Z m ()是目标空间中由所有Pareto 最优解支配的一个参考点,那么R HV 的计算公式为R HV S ()=δɣx ɪSf 1x (),z 1[]ˑf m x (),z m []()其中:S 表示算法得到的Pareto 最优解集,δ表示勒贝格测度㊂4.3㊀实验结果及数据分析表3和表4分别给出了ECMOPSO 算法与其他8个对比算法在15个测试问题上R IGD 值和R HV 值的均值(Mean)和标准差(Std),每个测试问题的最佳R IGD 值和R HV 值以粗体显示㊂此外,在α=0.05的显著性水平下,采用Wilcoxon 秩和检验,以显示检验结果之间的显著性差异㊂表中的符号 + ㊁ - 和 ʈ 分别表示其他算法的结果明显优于ECMOPSO 算法㊁明显差于ECMOPSO 算法以及与ECMOPSO 算法在统计上相似㊂表3㊀ECMOPSO 与8个对比算法在15个测试问题上的R IGD 值Table 3㊀R IGD values of ECMOPSO and eight comparison algorithms on fifteen test problems测试问题R IGD SMPSOdMOPSOMPSODNMPSONSGAIIIMOEADD SPEARDGEA ECMOPSOZDT1Mean StdWilcoxon 9.3128e -2(8.76e -2)- 5.7021e -2(1.72e -2)- 1.0539e -1(4.48e -2)- 2.9320e -2(1.09e -2)- 1.0075e -1(1.58e -2)- 1.2071e -1(1.84e -2)- 1.7041e -1(2.27e -2)- 1.1171e +0(2.91e -1)- 2.8807e -3(1.31e -4)ZDT2Mean Std Wilcoxon 5.3523e -2(1.07e -1)- 4.3899e -2(1.26e -2)- 1.7007e -1(1.02e -1)-1.9366e -2(6.61e -3)- 2.0184e -1(3.66e -2)- 1.5338e -1(2.65e -2)- 3.7517e -1(1.38e -1)-8.7982e -1(3.45e -1)- 3.4428e -3(1.24e -3)ZDT3Mean Std Wilcoxon 1.8286e -1(9.84e -2)- 3.5035e -2(6.31e -3)- 1.8725e -1(4.18e -2)-8.7429e -2(2.79e -2)-8.7967e -2(1.49e -2)- 1.8247e -1(1.50e -2)- 1.5654e -1(2.01e -2)-9.6677e -1(2.18e -1)- 3.4618e -3(3.00e -4)ZDT4Mean Std Wilcoxon 1.0525e +1(5.31e +0)- 5.4853e +0(6.11e +0)- 3.6093e +1(6.75e +0)- 1.7181e +1(1.02e +1)- 2.5028e +0(9.78e -1)-9.4142e -1(3.49e -1)ʈ2.0250e +0(6.56e -1)- 6.4764e +0(3.85e +0)- 1.4778e +0(1.53e +0)ZDT6Mean Std Wilcoxon 1.9172e -3(5.28e -5)+5.9543e -3(7.44e -3)- 2.7563e -2(2.87e -2)- 2.3135e -3(2.03e -4)+1.4465e +0(2.84e -1)- 5.8299e -1(1.72e -1)- 1.0509e +0(2.05e -1)- 2.2844e -2(1.09e -1)- 2.3842e -3(8.60e -5)UF1Mean Std Wilcoxon 3.8144e -1(9.05e -2)- 6.5685e -1(9.66e -2)- 2.8711e -1(5.02e -2)- 1.3308e -1(2.61e -2)ʈ1.3504e -1(3.81e -2)ʈ1.5518e -1(3.36e -2)- 1.3884e -1(2.47e -2)- 6.0941e -1(1.43e -1)- 1.2450e -1(8.69e -3)UF2Mean Std Wilcoxon 1.0184e -1(8.97e -3)-9.5765e -2(6.88e -3)- 1.1295e -1(1.01e -2)-8.3027e -2(7.26e -3)-8.1612e -2(5.33e -3)-7.5348e -2(6.19e -3)ʈ7.5129e -2(1.06e -2)ʈ1.8017e -1(2.18e -2)-7.4285e -2(4.41e -3)UF3Mean Std Wilcoxon 4.4586e -1(5.40e -2)- 3.3009e -1(6.15e -3)- 5.0047e -1(1.80e -2)- 3.5774e -1(6.03e -2)- 4.7958e -1(1.08e -2)- 4.6766e -1(1.32e -2)- 4.3291e -1(1.47e -2)- 5.6713e -1(3.36e -2)- 2.7446e -1(1.74e -2)UF4Mean Std Wilcoxon 1.1129e -1(8.05e -3)+1.3722e -1(4.87e -3)ʈ9.9054e -2(4.98e -3)+6.2896e -2(6.16e -3)+9.5857e -2(2.50e -3)+9.0998e -2(4.09e -3)+8.4957e -2(2.04e -3)+1.2029e -1(1.22e -2)+1.3613e -1(1.50e -2)97重庆工商大学学报(自然科学版)第41卷续表(表3)测试问题R IGD SMPSOdMOPSO MPSOD NMPSO NSGAIII MOEADD SPEAR DGEA ECMOPSOUF5Mean StdWilcoxon 2.8298e +0(5.59e -1)- 3.1949e +0(2.75e -1)- 2.7562e +0(2.66e -1)- 1.6868e +0(4.03e -1)- 1.6151e +0(3.91e -1)ʈ1.4347e +0(2.43e -1)ʈ1.1071e +0(1.88e -1)+3.0618e +0(7.08e -1)- 1.4249e +0(3.08e -1)UF6Mean Std Wilcoxon 1.2509e +0(4.17e -1)- 2.3449e +0(5.44e -1)-1.3904e +0(3.30e -1)-6.9338e -1(1.22e -1)-6.9635e -1(1.32e -1)-8.1561e -1(1.21e -1)-6.6148e -1(9.41e -2)-2.4887e +0(5.78e -1)-5.7726e -1(5.24e -2)UF7Mean StdWilcoxon 3.5088e -1(1.33e -1)-3.8396e -1(7.93e -2)-2.5133e -1(6.38e -2)-1.8663e -1(1.74e -1)-2.0383e -1(6.16e -2)-1.9100e -1(7.47e -2)-1.7285e -1(4.94e -2)-7.0149e -1(9.26e -2)-9.4107e -2(1.11e -2)UF8Mean StdWilcoxon 3.8732e -1(5.15e -2)- 3.4149e -1(3.37e -2)- 5.4728e -1(3.01e -2)- 4.7758e -1(1.01e -1)- 3.4595e -1(3.79e -2)- 3.3185e -1(2.17e -2)-3.2293e -1(1.95e -2)-7.2797e -1(1.26e -1)-2.7654e -1(3.88e -2)UF9Mean StdWilcoxon 5.5251e -1(3.74e -2)- 5.7965e -1(3.88e -2)- 6.6012e -1(3.81e -2)-4.6048e -1(5.27e -2)-5.1531e -1(4.58e -2)-5.2906e -1(6.04e -2)-4.6378e -1(4.87e -2)-7.5157e -1(1.39e -1)-1.4735e -1(1.99e -2)UF10Mean StdWilcoxon2.6494e +0(4.07e -1)-1.3204e +0(2.12e +0)ʈ4.2119e +0(4.16e -1)-1.4357e +0(3.60e -1)ʈ2.3930e +0(4.77e -1)-3.4430e +0(5.31e -1)-1.7848e +0(3.09e -1)-4.7389e +0(7.65e -1)-1.2703e +0(6.00e -1)+/-/ʈ2/13/00/13/21/14/02/11/21/12/21/11/32/12/11/14/0Best /All1/150/150/151/150/151/151/150/1511/15表4㊀ECMOPSO 与8个对比算法在15个测试问题上的R HV 值Table 4㊀R HV values of ECMOPSO and eight comparison algorithms on fifteen test problems测试问题R HV SMPSOdMOPSOMPSODNMPSO NSGAIII MOEADDSPEAR DGEA ECMOPSOZDT1Mean StdWilcoxon 6.0517e -1(1.05e -1)- 6.5272e -1(1.95e -2)- 5.7200e -1(5.85e -2)- 6.8969e -1(1.28e -2)- 5.8938e -1(1.96e -2)- 5.5955e -1(2.35e -2)- 4.9868e -1(2.42e -2)- 1.6149e -2(3.71e -2)-7.2136e -1(1.44e -4)ZDT2Mean Std Wilcoxon 3.9326e -1(9.98e -2)- 3.7831e -1(1.96e -2)- 2.5026e -1(1.02e -1)- 4.3397e -1(9.77e -3)- 2.0890e -1(2.99e -2)- 2.4709e -1(2.56e -2)-8.9477e -2(6.11e -2)- 2.9573e -2(9.92e -2)- 4.4532e -1(3.14e -3)ZDT3Mean Std Wilcoxon 5.2053e -1(6.83e -2)- 5.9777e -1(1.79e -2)ʈ4.6767e -1(4.14e -2)- 5.7228e -1(9.92e -3)- 5.4336e -1(1.03e -2)- 4.7287e -1(1.73e -2)- 5.0525e -1(3.00e -2)- 3.9698e -2(5.82e -2)- 6.0056e -1(7.13e -4)ZDT4Mean Std Wilcoxon 0.0000e +0(0.00e +0)- 3.9475e -2(9.21e -2)-0.0000e +0(0.00e +0)-0.0000e +0(0.00e +0)- 4.3538e -4(2.38e -3)- 3.3123e -2(4.47e -2)-0.0000e +0(0.00e +0)-0.0000e +0(0.00e +0)- 6.9196e -2(5.93e -2)ZDT6Mean Std Wilcoxon 3.9003e -1(5.86e -5)+3.8599e -1(7.41e -3)ʈ3.6732e -1(1.97e -2)-3.8975e -1(1.76e -4)+0.0000e +0(0.00e +0)-1.9708e -2(2.13e -2)-1.4687e -3(8.04e -3)-3.7990e -1(5.05e -2)-3.8942e -1(1.62e -4)8第2期陈飞,等:精英竞争和综合控制的多目标粒子群算法续表(表4)测试问题R HV SMPSOdMOPSOMPSODNMPSONSGAIIIMOEADDSPEARDGEAECMOPSOUF1Mean StdWilcoxon 2.6491e -1(7.17e -2)- 5.5084e -2(5.08e -2)- 3.3817e -1(5.07e -2)- 5.1537e -1(3.63e -2)ʈ5.1768e -1(4.46e -2)ʈ4.9279e -1(4.05e -2)- 5.1402e -1(3.05e -2)ʈ1.0999e -1(8.33e -2)- 5.2738e -1(1.50e -2)UF2Mean StdWilcoxon 6.0671e -1(8.62e -3)- 6.1545e -1(6.11e -3)- 5.7878e -1(1.04e -2)- 6.1923e -1(8.22e -3)- 6.1559e -1(6.98e -3)- 6.2093e -1(7.56e -3)- 6.2477e -1(9.70e -3)- 5.0246e -1(2.80e -2)- 6.3444e -1(4.21e -3)UF3Mean StdWilcoxon 2.1706e -1(5.15e -2)- 3.0963e -1(9.53e -3)- 1.7346e -1(1.59e -2)- 2.8302e -1(5.67e -2)- 1.8452e -1(1.12e -2)- 1.9992e -1(1.15e -2)- 2.2317e -1(8.52e -3)- 1.2176e -1(1.64e -2)- 3.7404e -1(1.88e -2)UF4Mean StdWilcoxon 2.9288e -1(9.25e -3)+2.5327e -1(4.58e -3)ʈ3.0800e -1(6.00e -3)+3.6040e -1(8.40e -3)+3.1268e -1(2.99e -3)+3.1886e -1(4.21e -3)+3.2527e -1(2.71e -3)+2.8110e -1(1.43e -2)+2.5715e -1(1.61e -2)UF5Mean StdWilcoxon 0.0000e +0(0.00e +0)ʈ0.0000e +0(0.00e +0)ʈ0.0000e +0(0.00e +0)ʈ0.0000e +0(0.00e +0)ʈ0.0000e +0(0.00e +0)ʈ0.0000e +0(0.00e +0)ʈ1.3453e -5(7.37e -5)ʈ0.0000e +0(0.00e +0)ʈ0.0000e +0(0.00e +0)UF6Mean StdWilcoxon 1.1432e -3(6.26e -3)-0.0000e +0(0.00e +0)-0.0000e +0(0.00e +0)- 1.8370e -2(2.67e -2)ʈ1.2244e -2(1.73e -2)- 2.8881e -3(6.71e -3)- 1.6659e -2(2.03e -2)ʈ0.0000e +0(0.00e +0)- 1.9962e -2(1.09e -2)UF7Mean Std Wilcoxon 2.0783e -1(1.03e -1)- 1.7261e -1(5.62e -2)- 2.6128e -1(6.09e -2)- 3.8775e -1(1.16e -1)ʈ3.1792e -1(6.35e -2)- 3.3427e -1(6.65e -2)- 3.5657e -1(4.50e -2)- 2.1735e -2(2.23e -2)- 4.4089e -1(1.70e -2)UF8Mean Std Wilcoxon 1.7390e -1(4.51e -2)- 2.6282e -1(2.32e -2)- 5.4523e -2(1.71e -2)- 2.8801e -1(5.17e -2)- 2.3078e -1(3.81e -2)- 1.5653e -1(3.31e -2)- 1.8219e -1(3.40e -2)- 2.4817e -2(2.82e -2)- 3.4706e -1(1.05e -2)UF9Mean Std Wilcoxon 2.1269e -1(3.11e -2)- 2.3294e -1(1.76e -2)- 1.1167e -1(2.52e -2)- 3.2189e -1(5.49e -2)- 2.3892e -1(4.10e -2)- 2.0844e -1(4.72e -2)- 2.7506e -1(4.50e -2)-8.2413e -2(5.59e -2)- 6.0995e -1(1.83e -2)UF10Mean Std Wilcoxon 0.0000e +0(0.00e +0)-9.1138e -2(2.49e -2)+0.0000e +0(0.00e +0)-0.0000e +0(0.00e +0)-0.0000e +0(0.00e +0)-0.0000e +0(0.00e +0)-0.0000e +0(0.00e +0)-0.0000e +0(0.00e +0)-3.0791e -2(5.35e -2)+/-/ʈ2/12/11/10/41/13/12/9/41/12/21/13/11/11/31/13/1Best /All1/151/150/151/150/150/151/150/1511/15㊀㊀从表3和表4可以看出:本文提出的ECMOPSO 算法综合性能明显优于对比的8个算法㊂从Wilcoxon 秩和检验结果来看:ECMOPSO 算法与对比算法SMPSO㊁dMOPSO㊁MPSOD㊁NMPSO㊁NSGAIII㊁MOEADD㊁SPEAR18重庆工商大学学报(自然科学版)第41卷和DGEA在15次比较中,分别有13㊁13㊁14㊁11㊁12㊁11㊁12㊁14次的表现显著优于这些算法,有0㊁2㊁0㊁2㊁2㊁3㊁1㊁0次得到了相似的结果㊂此外,从表3中的最佳值可以看出:ECMOPSO在15个测试问题上获得了11个最佳R IGD值,而算法SMPSO㊁dMOPSO㊁MPSOD㊁NMPSO㊁NSGAIII㊁MOEADD㊁SPEAR和DGEA获得的最佳R IGD值个数分别为1㊁0㊁0㊁1㊁0㊁1㊁1㊁0㊂表4中最佳R HV值所得结果与R IGD相似,ECMOPSO算法在15个测试问题上也获得了11个最佳值㊂所有统计结果都证明ECMOPSO算法和所选的算法比较仍然具有很强的竞争力㊂为进一步比较ECMOPSO算法与所选对比算法的整体性能,还采用Friedman秩检验计算所有算法的平均排名㊂从表5可以看出:无论是R IGD值还是R HV值,提出的ECMOPSO算法最终排名都是第一,这说明在和其他8个算法比较时,ECMOPSO算法在这些测试问题上的整体性能较为显著㊂表5㊀ECMOPSO算法和所有对比算法R IGD值和R HV值的Friedman秩检验Table5㊀Friedman rank test of R IGD values and R HV values ofECMOPSO and all comparison algorithms算㊀法R IGD R HV Friedman test Rank Friedman test RankSMPSO 5.737 5.436 dMOPSO 5.406 4.774 MPSOD7.008 6.708 NMPSO 3.402 3.102 NSGAIII 4.935 5.035 MOEADD 4.674 5.437 SPEAR 3.873 4.673 DGEA8.2797.909 ECMOPSO 1.731 1.971从R IGD㊁R HV㊁Wilcoxon秩和检验和Friedman秩检验结果可以得出结论:所提出的ECMOPSO算法与对比算法相比有更好的综合性能,在求解MOPs时可以获得更好的收敛性和多样性㊂4.4㊀图形比较分析为了比较各算法的收敛性和分布性,观察它们是否真正收敛到近似Pareto前沿,图3和图4分别展示ECMOPSO算法与其他8个对比算法在测试问题ZDT3和UF9上的Pareto最优解集分布㊂从图中可以看出: ECMOPSO算法在收敛性和分布性上的表现相对于其他算法来说都更有优势㊂这说明精英竞争和综合平衡控制策略能更好地平衡收敛性和多样性㊂㊀㊀(a)SMPSO(b)dMOPSO ㊀㊀(c)MPSOD(d)NMPSO ㊀㊀(e)NSGAIII(f)MOEADD ㊀㊀(g)SPEAR(h)DGEA(i)ECMOPSO图3㊀9个算法在ZDT3测试问题上的近似PFFig.3㊀Approximation PF of nine algorithms onZDT3test problems28第2期陈飞,等:精英竞争和综合控制的多目标粒子群算法㊀㊀(a)SMPSO(b)dMOPSO㊀㊀(c)MPSOD(d)NMPSO㊀㊀(e)NSGAIII(f)MOEADD(g)SPEAR(h)DGEA(i)ECMOPSO图4㊀9个算法在UF9测试问题上的近似PFFig.4㊀Approximation PF of nine algorithms on UF9test problems为了能够直观地比较各算法在ZDT和UF系列测试问题上的稳定性,图5展示了ECMOPSO算法与其他8个算法在一些测试问题上独立运行30次所得到的R IGD值分布统计箱型图㊂其中箱型图横坐标上的1㊁2㊁3㊁4㊁5㊁6㊁7㊁8㊁9依次表示算法SMPSO㊁dMOPSO㊁MPSOD㊁NMPSO㊁NSGAIII㊁MOEADD㊁SPEAR㊁DGEA和ECMOPSO,纵坐标表示算法的R IGD值㊂从这些图中可以观察到:图5展示了与表3一致的对比结果,在这些测试问题上,ECMOPSO算法无论在解的结果还是算法的稳定性上都比其他8个算法更为显著㊂(a)ZDT1(b)ZDT2(c)ZDT3(d)UF1(e)UF7(f)UF9图5㊀9个算法在不同测试问题上R IGD值的箱型统计图Fig.5㊀Box statistics of the R IGD values of nine algorithms ondifferent test problems除了解集的质量,算法的收敛速度也是一个重要的性能指标㊂图6为ECMOPSO算法和8个算法在ZDT3㊁UF7和UF9上评价10000次得到的R IGD值收敛轨迹㊂从图中可以看出,ECMOPSO算法具有很好的收敛速度,明显优于其他8个对比算法㊂(a)ZDT338重庆工商大学学报(自然科学版)第41卷(b )UF1(c )UF7图6㊀ECMOPSO 和8个对比算法在ZDT3㊁UF1和UF7测试问题上的R IGD 收敛轨迹Fig.6㊀R IGD convergence trajectories of ECMOPSO and eightcomparison algorithms on ZDT3,UF1and UF7test problems综上可知,从PF 图㊁箱型图和收敛轨迹图来看,所提出的ECMOPSO 算法与所选的几个有竞争力的算法相比,ECMOPSO 算法的综合性能要优于其他算法,说明所提出的精英竞争和综合平衡控制策略可以更好地平衡算法的收敛性和多样性㊂4.5㊀参数分析本节将对参数r 进行灵敏度分析㊂在提出的ECMOPSO 算法中,精英粒子集的大小r 对算法的性能有一定的影响:如果r 太小,可能会导致种群过早收敛,但如果r 太大,可能会降低种群收敛速度㊂因此,合适大小的r 可以很好地平衡收敛性和多样性㊂图7展示了ECMOPSO 算法在测试问题UF1㊁UF3㊁UF6㊁UF7和UF8上不同大小精英粒子集的R IGD 值,该算法在每个测试问题上都独立运行了30次㊂从图中可以看出,当精英粒子集的大小为30时,ECMOPSO 算法的性能最佳㊂图7㊀ECMOPSO 在一些测试问题上不同大小精英粒子集的R IGD 值Fig.7㊀The R IGD values of different sizes of elite particle setscalculated by ECMOPSO on some test problems5㊀结㊀论提出一种精英竞争和综合控制的多目标粒子群算法,有效平衡了算法的收敛性和多样性㊂ECMOPSO 算法将两两竞争引入多目标粒子群算法中,采用全局损害选择精英粒子集,然后通过精英竞争选择优胜者粒子,将其与全局领导者融合来引导种群中的粒子飞行,提升了算法的全局探索能力;并且结合全局损害和基于位移密度估计对外部存档进行维护,从而提高外部存档中非劣解的质量,平衡算法的收敛性和多样性;最后采用Wilcoxon 秩和检验和Friedman 秩检验来比较ECMOPSO算法与所选对比算法的整体性能,将ECMOPSO 算法与8个对比算法在ZDT 和UF 测试问题上进行仿真实验㊂实验结果表明:ECMOPSO 算法具有更好的综合性能,可以更好地平衡收敛性和多样性,能有效地求解大多数多目标优化问题㊂参考文献 References1 ㊀LUO Q WU G JI B et al.Hybrid multi-objective optimizationapproach with Pareto local search for collaborative truck-dronerouting problems considering flexible time windows J .IEEETransactions on Intelligent Transportation Systems 2022 23 813011 13025.2 ㊀KENNEDY J EBERHART R C.Particle swarm optimizationC //Proceedings of the International Conference on NeuralNetworks.Piscataway 1995 1942 1948.3 ㊀COELLO C A C LECHUGA M S.MOPSO A proposal formultiple objectiveparticleswarmoptimization C //Proceedingsofthe2002CongressonEvolutionaryComputation.CEC 02IEEE 2002 1051 1056.4 ㊀HAN H LU WZHANG L et al.Adaptive gradient48第2期陈飞,等:精英竞争和综合控制的多目标粒子群算法multiobjective particle swarm optimization J .IEEE Transactions on Cybernetics 2018 48 11 3067 3079.5 ㊀LI L CHANG L GU T et al.On the norm of dominant difference for many-objective particle swarm optimization J . 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Concepts

Rotordynamic Results
• System will run super-critical • Proper design choices needed to stay below first bending mode
Optimization Setup
• Variables:
– Blade thickness (8) – Impeller backface (5)
– Off the shelf packaged automated optimization of turbomachinery fluid dynamics using CFD and iSIGHT – Add FEA into automated optimization
• Next year
• Second optimization loop:
– Constant blade shape, solid FEA – Reduced stress to 105 ksi
• Highlights the stress / weight trade
Stress-Weight Trade
• Physical boundary imposed by geometry
Stress-Weight Trade
• Physical boundary imposed by geometry • Upper stress limit imposed by material properties
stress limit
Stress-Weight Trade
• Physical boundary imposed by geometry • Upper stress limit imposed by material properties • Upper weight limit imposed by bearings, rotordynamics, weight budget

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