a min–max method with adaptive weightings for uniformly spaced pareto optimum points -凯发官网入口

a min–max method with adaptive weightings for uniformlyspaced pareto optimum pointsw.h. zhang* , t. gaosino-french laboratory of concurrent engineering, northwestern polytechnical university, p.o. box 552, xi’an, shaanxi 710072, chinareceived 11 january 2005; accepted 27 april 2006available online 7 september 2006abstractthis work aims at obtaining uniformly spaced pareto optimum points in the objective space when multicriteria optimization problemsare solved. an original adaptive scheme is proposed to update automatically weighting coefficients involved in the min–max method. bymeans of a novel bilevel approach, it is shown that with the calculation of the tangent and normal directions of the pareto curve, paretooptimum points can be obtained sequentially with a uniformly spaced distribution. meanwhile, the distance between two adjacent paretooptimum points is controllable depending upon the prescribed step length along the tangent direction. to validate the method, numericalbicriteria examples are solved to show its effectiveness.? 2006 elsevier ltd. all rights reserved.keywords: multicriteria optimization; pareto optimum; min–max method1. introductionnowadays, multicriteria optimization is widely appliedat the design stage of mechanical structures and dynamicsystems. due to the basic nature of conflict among objectivefunctions, a set of compromising solutions called paretooptima exists. this solution set is essential for decision-making of designers including the selection of preferabledesign results and the reveal of the trade-off relationship.usually, multicriteria optimization problems are solved bytransforming the original vector-valued problem into a sca-lar one by means of scalarization methods such as theweighting method, the trade-off method and the min–maxmethod. however, an effective method is expected to gener-ate pareto points that are representative and able to capturethe shape and all parts of the pareto optimum curve in theobjective space. in this regard, pareto points obtained aredesired to be uniformly spaced with equal distance betweenadjacent points. in fact, this issue has received much atten-tion since a long time. as indicated by koski [1], the weight-ing method is based on a linear combination of objectivecriteria and unable to capture non-convex parts of the par-eto curve. lin [2], das and dennis [3] indicated that paretooptimum solutions obtained by the weighting method areoften found to be so few, or the distribution is so extremeand that it seems to exist no middle ground for any compro-mise although such a ground may actually exist. hence, theattainment of a uniform distribution of pareto optimumpoints is a property that the weighting method lacks eventhough weightings are uniformly discretized and used.actually, the normal-boundary intersection method (nbi)proposed by das and dennis [4] is devised to address theproblem of uniform distribution of pareto points.in this paper, the min–max method is preferable andstudiedindetail.firstly,itisshownthatallcriteriainthesca-larizing formulation are decoupled and transformed intoseparable inequality constraints so that problems can besolved as easily as single objective problems. secondly, anadaptiveweightingschemeisproposedtoensureauniformlyspaced distribution of generated pareto points. by compar-ison,thenbimethodupdatesonlythereferencepointonthe0045-7949/$ - see front matter ? 2006 elsevier ltd. all rights reserved.doi:10.1016/j.compstruc.2006.04.007*corresponding author. tel./fax: 86 29 88495774.e-mail address: zhangwh@nwpu.edu.cn (w.h. zhang).www.elsevier.com/locate/compstruccomputers and structures 84 (2006) 1760–1769