Weighted Sum (i.y. Kim) - 1

Struct Multidisc Optim (2006) 31: 105–116 DOI 10.1007/s00158-005-0557-6 RESEARCH PAPER I. Y. Kim · O. L. de Weck Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation Received: 21 January 2005 / Revised manuscript received: 6 July 2005 / Published online: 20 December 2005 © Springer-Verlag 2005 Abstract This paper presents an adaptive weighted sum (AWS) method for multiobjective optimization problems. The method extends the previously developed biobjective A WS met hod to pro ble ms wi th mor e tha n two obj ec ti ve fun c- tions. In the first phase, the usual weighted sum method is performed to approximate the Pareto surface quickly, and a mesh of Pareto front patches is identified. Each Pareto front patch is then refined by imposing additional equality constraints that connect the pseudonadir point and the expected Pareto optimal solutions on a piecewise planar hypersurface in the m-dimensional objective space. It is demonstrated that the meth od produ ces a well -dist rib uted P aret o fron t mesh for effecti ve visualization, and that it finds solutions in noncon- ve x re gio ns. T wo numeri ca l example s and a simple str uct ura l optimizatio n problem are solved as case studies. Keywords NBI · A WS · Multiobjective optimization · Adaptive weighted sum · Pareto front Nomenclature J = Objecti ve function vector x = Desi gn vec tor p = V ector of fixed parameters g = Inequality constraint vector h = Equality const raint vec tor m = Number of obje cti ves α i = W eighting factor J i = Normaliz ed objectiv e function Presented as paper AIAA-2004-4322 at the 10th AIAA-ISSMO Multi- discip linaryAnalysis and OptimizationConference , Alban y,New Y ork, August 30–September 1, 2004 O. L. de Weck ( B ) Department of Aeronautics and Astronautics, Engineering Systems Division, 33-410, Massachusetts Institute of Technology , Cambridge, MA 02139, USA e-mail: [email protected] e-mail: [email protected] I. Y. Kim Department of Mechanical Engineering, Queen’s University , Kingston, Ontario, K7L 3N6, Canada J Utopia = Ut opi a poi nt J Nadir = Nadir poi nt J i ∗ = i th anchor point P j = Pos iti on vec tor of the j th exp ecte d solut ion on the hyperplane 1 Introduct ion 1.1 Multiobjective optimization and literature review The goal of design optimization is to seek the best design that minimizes the objective function by changing design variables while satisfying design constraints. During design optimization, one often needs to consider severa l design criteria or objective functions simultaneously. For example, we may want to maximize range and payload mass whiletryin g to min imi ze lif e cy clecostforan air pla ne des ign . When there are mult iple obje cti ve func tion s to be consi dere d, the design problem becomes multiobjective. In this case, the usu al des ign opt imi zat ionme tho d for a sca lar obj ecti ve function cannot be used. Multiobjective optimization can be stated as follows: min J (x, p) = [ J 1 J 2 ... J m ] T s.t. g(x, p) ≤ 0 h(x, p) = 0 x i ,LB ≤ x i ≤ x i ,UB (i = 1, ..., n) (1) where the objective function vector J is a function of design vector x and a fixed parameter vector p; g and h are inequality and equality constraints, respectively; and x i ,LB and x i ,UB are the lower and upper bounds for the ith design var iabl e, respe ctively. Stad ler (197 9, 1984) appl ied the notion of Pareto optimality to the fields of engineering and science in the 1970s. The most widely used method for multiobjective optimizat ion is the weighted sum method. The method transforms multiple objectiv es into an aggregated scalar objective function by multiplying each objective function by a weighting factor and summing up all contributors: J weighted sum = w 1 J 1 + w 2 J 2 +···+ w m J m (2)

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Weighted Sum (i.y. Kim) - 1