An improved multi-objective evolutionary algorithm based on point of reference

Transcription

1 IOP Conference Seres: Materals Scence and Engneerng PAPER OPEN ACCESS An mproved mult-objectve evolutonary algorthm based on pont of reference To cte ths artcle: Boy Zhang et al 08 IOP Conf. Ser.: Mater. Sc. Eng Vew the artcle onlne for updates and enhancements. Ths content was downloaded from IP address on 3/0/08 at 00:0

2 IOP Conf. Seres: Materals Scence and Engneerng (08) 0605 do:0.088/ x/3/6/0605 An mproved mult-objectve evolutonary algorthm based on pont of reference Boy Zhang, Xue Zhou,Yuqng Lu, Xangl Xu, Lbao Zhang * School of Informaton Scence and technology, Northeast Normal Unversty, Changchun, Chna Department of Informaton Operaton and Command Tranng, NDU of PLA, Bejng, Chna *Correspondng author e-mal: lbzhang@nenu.edu.cn Abstract. In the artcle presents a new evolutonary algorthms, ths algorthm s based on reference ponts. Ths algorthm accordng to a archve fnd the non-domnated soluton, and make ths soluton from large to small order accordng to crowdng dstance and let the space s dvded to several small space equdstant, n each the subspace to obtan ts centrod as the reference pont. The method used n ths paper are not predefned reference pont but n the course of evoluton accordng to the current state of dynamc reference pont partcles to gude the evoluton, fnally usng fve effectveness test functons prove algorthm.. Introducton The mult-objectve optmzaton problems(mop) has run through our lves. For eample the scentfc practce, the engneerng system desgn and the socal producton actvtes. The process of solvng MOP and sngle objectve optmzaton(sop) s dfferent. About ths, we can fnd the frst-rate soluton, whch s the mnmum or mamum value correspondng to the objectve functon. About the MOP, non-estent sngle optmum soluton [] because ths objectves are mutually conflctng. The resultng soluton s a set of compromse solutons, whch can not make the optmzaton on any object functon free from one other objectve functon at least. Ths soluton can be called a Pareto optmal soluton []. The tradtonal method to solve MOP needs a pror understandng about the problem such as transform nto a SOP. However such evolutonary based on operatng populaton may parallel seek multple soluton n the soluton space, and can use dfferent smlarty between the solutons to mprove the effcency of solvng, therefore ts very sutable for evolutonary computaton to solvng MOP. The typcal ntellgent optmzaton algorthm s partcle swarm optmzaton(pso) whch utlze the swarm teratve search. Advantage of ths algorthm are smple operaton and fast convergence. So the method whch applyng to solve mult-objectve problems has receved etensve attenton n academa [3]. At present, many scholars have proposed some mproved algorthms, such as Wckramasnghe [4] proposed a algorthm utlze the dstance measurement, no longer use Pareto domnance relaton to avod the problems caused by Pareto control. Brtto [5] proposed an mproved Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, journal ctaton and DOI. Publshed under lcence by Ltd

3 IOP Conf. Seres: Materals Scence and Engneerng (08) 0605 do:0.088/ x/3/6/0605 eternal fle mantenance strategy to ncrease the selecton pressure of swarm to solvng MOP.. The problem of Mult-objectve optmzaton The MOP mathematcally epress as follows: mn/ ma y { f( ), f( ),..., f g ( ) 0,,3,..., n n ( )} () The decson vector, The target vecton y The objectve functon, The constrant functon The followng defntons are gven: Defne : For any ponts n the space, for a certan pont y, f fy f always s true, call the Pareto optmal soluton. Defne : The sets consstng of all non-domnated optmal soluton can be called as non nferor optmum soluton set of MOP, the curve constructed by the soluton set s the Pareto optmal front. Defne 3: Let O be a collecton, the sze s m, each ndvdual has d attrbutes n O, the epresson s the evaluaton functon of the j attrbute of the frst I ndvdual, the relatonshps between ndvduals n the O can dvded nto the followng two: Domnance relaton: Any, y O arbtrary attrbute, there s and there s at least one property k, make, called the y s domnated by, s the domnant ndvdual, y s the domnated ndvdual. Non domnated relaton: f, y y, called and non domnant or rrelevant, the epress domnance relaton. Defne 4: For a gven ndvdual O, f there s no arbtrary y O, make y, and the s the non domnatng body. There s a collecton of non domnated ndvduals, can be called the non domnatng set [6]. 3. Related concepts of PSO About PSO, there are two concepts. One s the best soluton called pbest, the other one s the global soluton called gbest. In the PSO algorthm, f the partcle populaton sze as S, poston as (=,,...,S) partcle can epress as, the ndvdual etreme value s denoted as, The speed epress as v, global etremum s represented by. So the speed and poston [7] of the partcle wll update accordng to the followng formula: η, :learnng factor; rand, rand : random numbers on [0,]; w: nerta weght. () 4. A MOP based on reference ponts 4. related concepts of reference ponts In the target space, f the functon value s not bad for all the solutons, t becomes the global deal pont. As shown n (4): mn,,,, (3) When s the current populaton,,,, s the deal pont.

4 IOP Conf. Seres: Materals Scence and Engneerng (08) 0605 do:0.088/ x/3/6/ The reference pont The reference pont n ths artcle s shown n fgure : Fgure. the generaton of reference ponts The partcle s sorted accordng to the crowdng dstance, optmzaton rules may be deleted accordng to the ndvdual domnated, then the target space s dvded nto equal spacng subspace, accordng to the dstrbuton of partcles and partcle crowdng dstance n the target space, respectvely, to fnd the mamum and the mnmum crowdng dstance values to generate the reference pont. Thus, n each subspace there are a number of unequal partcles, where there may be multple partcles n each unt nterval, and no partcles est. In the nterval of reference ponts, after clusterng, the center s taken as the fnal reference pont, and the reference pont s generated by usng remanng reference pont mappng at ntervals wthout reference ponts. 4.3 The ndvdual selecton based on reference ponts As mentoned earler, the evaluaton of partcles s based on the relatve poston between the partcle and the reference pont. In ths secton, ntroduce a method for calculatng the dstance between reference ponts and partcle. The Chebyshev dstance formula s used to calculate [8] : ma,,, (4) s Chebyshev dstance between reference ponts partcle and. And f domnates, s less than. About the PSO, we need select global optmum from fle to gude the populaton convergence, so the reference pont acts on the fle set. Algorthm flow: Step: ntalzaton, set the populaton, populaton number N and mamum teraton number ; set the fle. Step: adds the non domnated soluton of to form, let k k, updates the locaton of the partcles, and calculate the target functon values. Step3: compares n a populaton wth non domnated solutons n. If partcles domnate the old solutons n, or both do not domnate, the partcles are added to the, and the old solutons n the are deleted. Step4: sorts the soluton n accordng to crowdng dstance, and selects αn partcles wth large dstance. Step5: generates reference ponts accordng to solutons n, then connects all solutons to the correspondng reference ponts. Step6: calculates Chebyshev dstance between partcle and reference ponts, and chooses ndvdual wth small dstance as the fnal soluton. Step7: termnates the algorthm, and fnally outputs the non-domnated solutons n. 3

5 IOP Conf. Seres: Materals Scence and Engneerng (08) 0605 do:0.088/ x/3/6/ Smulaton eperment and result analyss 5. The test functon and parameter settngs We use fve test functons to detect the performance. These functons are used n document [9], and are the most commonly used test functons for most mult-objectve evolutonary algorthms. Table. Test functons. Name Decson space ZDT 0, 30 ZDT 0, 30 ZDT3 0, 30 objectve functon mn f mn f g g g n mn f mn f g g g n mn f mn f g g sn 0 g g n Non nferorty optmal surface feature Conve Non conve Conve and dscontnuous ZDT4 0, and 5,5 9 n 0 0 cos 4 mn f mn f g g g n Non conve, locally optmal ZDT6 0, mn f ep sn 6 mn f g f g g n Non conve nhomogeneous 5. The algorthm performance evaluaton crtera Used followng two crtera to compare wth other algorthms and llustrate the performance[0]. For tested problem, t s assumed that the non domnated optmal target regon obtaned by the algorthm s z, and the theoretcally non domnated optmal target doman s z. ().The convergence, the convergence measure by the z and z wth the mnmum dstance between the average value, r mnz Z, z Z z zz The soluton wll close to the Pareto optmal soluton set when r s smaller. (). Measure by the type of dversty, 4

6 IOP Conf. Seres: Materals Scence and Engneerng (08) 0605 do:0.088/ x/3/6/0605 d d f f d l d l Z d d ( Z ) d The better the optmal dversty when s smaller. 5.3 The epermental results and analyss The target number s 0, 50. The algorthm s run 0 tmes, each teraton of the 00 generaton and the dversty of the standard and the mnmum value n Table SPEA algorthm usng bnary encodng, run 0 tmes, each teraton of the 50 generaton results; NSGA s run 0 tmes, the 50 generaton teraton; tables and of ths algorthm s the mean, varance s below the last tme. Table. Convergence standard algorthms zdt zdt zdt3 zdt4 zdt6 SPEA NSGA Ths artcle E Table 3. Dversty standard algorthms zdt zdt zdt3 zdt4 zdt6 SPEA NSGA Ths artcle E E As can be seen from table, the convergence of the algorthm to the fve test functons s obvously better than that of SPEA, especally n ZDT, ZDT4 and ZDT6. For the NSGA algorthm, there are obvous advantages over ZDT3, ZDT4, and ZDT6, whle the other two functons are closer. Can be seen from table two, based on fve test functons than SPEA and NSGA n the soluton of dversty performance, look for SPEA n the functon of ZDT4 and ZDT6 have obvous advantages, for the NSGA algorthm n ZDT, ZDT, ZDT4 and ZDT6 have an obvous advantage n ZDT3 s close to. 6. Concluson Propose a mult-objectve optmzaton algorthm n vew of reference ponts. Accordng to a seres of ponts of reference eternal archve n the partcle space, accordng to relatve poston of the non domnated solutons, and select partcles wth better dversty and convergence as a reference pont to as global wzard, to gude the partcle to the true Pareto front unform convergence. In the eperment, we can see that although the algorthm needs to be mproved, the algorthm also ensures the convergence and dversty wth two benchmarks. Through eperment fve test functons and comparson wth other epermental results, the algorthm s llustrated. Acknowledgments Ths work was fnancally supported by the Natonal Nature Scence Fund of Chna(No ). 5

7 IOP Conf. Seres: Materals Scence and Engneerng (08) 0605 do:0.088/ x/3/6/0605 References [] Guoqang Deng,Zhangcan Hang, Objectve Optmzaton Algorthms and Performance metrcs[d], wuhan:wuhan Unversty of Technology, 007. [] C A Coello, A Comprehensve survey of evolutonary-based mult-objectve optmzaton, technques, Know ledge and Informaton Systems. (999) [3] Guang Lu, Desgn and Implementaton of the Port Schedulng System Based on the Mult-Objects PSO[D], Harbn:Harbn Engneerng Unversty. (008). [4] Wckramasnghe UK, Carrese R, L X, Desgnng arfols usng a reference pont based evolutonary many-objectve partcle swarm optmzaton algorthm[c], In Proceedngs of IEEE Congress on Evolutonary Computaton. IEEE. (00)-8. [5] Brtto A, Pozo A, Usng archvng methods to control convergence and dversty for many-objectve problems n partcle swarm optmzaton[c], In Proceedngs of IEEE Congress on Evolutonary Computaton, IEEE, (0) -8. [6] Guang Lu, Desgn and Implementaton of the Port Schedulng System Based on the Mult-Objects PSO[D], Harbn:Harbn Engneerng Unversty. (008). [7] Y S hand, R Eberhart, A modfed partcle swarm optmzer, IEEE Int' l Conf on Evolutonary Computaton, Anchorage, Alaska. (998). [8] K. Deb, J. Sundar, N, Udaya Bhaskara Rao, S, Chaudhur, Reference pont based mult-objectve optmzaton usng evolutonary algorthms, Internatonal Journal of Computatonal Intellgence Research (3) (006) [9] Ztzler E, Deb K, Thele L, Comparson of mult-objectve evolutonary algorthms: Emprcal results, Evolutonary Computaton. (000) [0] Deb K, Pratap A, Agarwal S, Meyarvan T, A fast and eltst mult-objectve genetc algorthm: NSGA, IEEE Transactons on Evolutonary Computaton. (00)