A modified directed search domain algorithm for multiobjective engineering and design optimization

Transcription

1 Struct Multidisc Optim DOI.7/s RESEARCH PAPER A modified directed serch domin lgorithm for multiojective engineering nd design optimiztion Tohid Erfni Sergey V. Utyuzhnikov Brin Kolo Received: 25 July 22 / Revised: 5 My 23 / Accepted: 9 My 23 Springer-Verlg Berlin Heidelerg 23 Astrct Multiojective optimiztion is one of the key chllenges in engineering design process. Since the nswer to such prolem is not unique, set of evenly distriuted solutions is prticulrly importnt for designer. The Directed Serch Domin (DSD) method is numericl optimiztion pproch tht hs proven to e efficient enough to tckle such optimiztion prolems. In this pper, we propose two modifictions to the DSD pproch which mke the solution lgorithm simpler for progrm implementtion. These modifictions re relted to the control of the serch domin nd reformultion of the pproprite single ojective optimiztion prolem. As result, the computtionl efficiency of the method is incresed due to the lower numer of ojective function evlutions. The cpilities of the new pproch re demonstrted on set of test cses. Keywords Directed serch domin Multiojective optimiztion Engineering design Preto set T. Erfni ( ) Civil, Environmentl nd Geomtic Engineering Deprtment, University College London, Gower Street, London, WCE 6BT, UK e-mil: t.erfni@ucl.c.uk S. V. Utyuzhnikov School of Mechnicl, Aerospce nd Civil Engineering, The University of Mnchester, Mnchester, UK B. Kolo Zifin Inc., 26 Roger Bcon Drive, Reston, VA 29, USA Bckground nd motivtion Engineering design is n ctive nd wide-rnging reserch re in optimiztion. Design process usully includes simultneous optimiztions of different criteri (ojectives) which re often in conflict with one nother. As result, the idel (est) solution for ll ojectives usully lies outside of the fesile design spce. Therefore, the gol is to otin set of solutions which would represent the est trdeoff mong the ojectives. Such solutions re clled Preto solutions. Mthemticlly, solution is clled Preto solution if no improvement with respect to one ojective cn e mde without compromising the qulity of t lest one of the other ojectives. In multiojective optimiztion, Preto solutions in the ojective spce form surfce clled the Preto frontier. In rel-life design, decision mker is usully le to nlyze only very limited numer of solutions for trdeoff nlysis. Therefore, they should e provided with n evenly enough distriuted set of solutions which re representtive of the entire Preto frontier. In ddition, there my exist chllenges tht mke the genertion of well spred set of solutions difficult such s discontinuities in the Preto frontier, non-uniform density of fesile solutions, non-convexity nd non-linerity of ojective functions nd the constrints. In the literture, there re minly two ctegories of methods to pproximte the Preto frontier. A ctegory corresponds to the stochstic lgorithms popultion of points is evolved in ech prolem genertion using rndom opertors (e.g. muttion nd crossover) (Venter nd Hftk 2; De2). The second ctegory is represented y clssicl pproches which re the focus of this pper. In clssicl multiojective optimiztion, there exist severl methods tht cn efficiently otin Preto solutions.

2 T. Erfni et l. The weighted sum method nd constrint-sed lgorithms re mong the most used pproches. Often, these methods either cnnot provide n evenly distriuted set of solutions or fil to cpture prt of the Preto frontier s indicted in Mrler nd Aror (24). The well-known pproches, which re cple of generting the whole Preto frontier nd explined elow, include the Norml Boundry Intersection (NBI) (Ds nd Dennis 998), the Physicl Progrmming (PP) (Messc nd Mttson 22), the Normlized Norml Constrint (NNC) (Messc et l. 23), the Directed Serch Domin (DSD) (Erfni nd Utyuzhnikov 2; Utyuzhnikov et l. 29; Utyuzhnikov 2) nd the Successive Preto Optimiztion (SPO) (Mueller-Gritschneder et l. 29) methods. Ech of these lgorithms exploits the nchor points which re the minim of ech ojective in the ojective spce. All forementioned methods employ different sclriztion pproches. In prticulr, NBI nd NNC shre close strtegy for Preto genertion y introducing reduction constrint. In NBI, set of rys from the reference points nd orthogonl to the utopi hyperplne re introduced to determine the locl fesile spce. The distnce from the fesile oundry is tken s the ojective function, nd optimiztion long these rys provides n even genertion of solutions. However, the equlity constrint s the reduction criterion my not e esily stisfied in generl context. To ddress this prolem, the NNC improves computtionl stility y reducing the design spce. The reduction is mde y using n inequlity constrint defined y plne nd its orthogonl vector. Messc et l. (23) demonstrted tht in prctice the NNC method is less likely to generte non-preto nd loclly Preto solutions thn the NBI pproch. Siddiqui et l. (22) provide modifiction to the originl NBI lgorithm y scrificing some ccurcy in prolems with mny vriles nd/or constrints. Although test exmples show the strength of the proposed method, when compred to the originl NBI lgorithm, it ecomes computtionlly expensive for higher dimension prolems. From different perspective, the DSD introduces serch domin sed on the ffine trnsformtion of ojectives nd serches for the solution within ech domin. To gurntee well distriuted Preto set, DSD evenly spreds locl serch domins so tht they cn e esily controlled in the ojective spce. As reported y Utyuzhnikov et l. (29), on some test cses the DSD method outperforms the NNC nd NBI with regrds to the computtionl urden nd qulity of solutions. In ddition, it hs een shown tht on some chllenging test cses DSD is le to generte evenly distriuted solutions s NNC nd NBI my fil (Erfni nd Utyuzhnikov 2). However, due to introduction of new coordinte systems nd mtrix mnipultions, the method my e difficult to implement. In higher dimensionl cses, s recognized in Ds nd Dennis (998), the ove procedures my not cpture the entire Preto frontier if the orthogonl projection of the Preto surfce onto the utopi hyperplne exceeds the utopi polygon. The NNC, DSD nd SPO techniques ech provide different strtegies to cover the entire Preto frontier. SPO implements sequentil Preto genertiontechnique to pproximte the oundry of the Preto surfce nd evenly distriute reference points on the pproprite hyperplne not extending eyond the Preto oundry. Further optimiztion for ech reference point in SPO is similr to tht of NBI. This strtegy works well for proposed test cses, ut requires ssumptions tht my e violted in rel-world pplictions. Alterntively, for the NNC method, the utopi hyperplne is extended to cover the whole Preto surfce. This extension is implemented vi two nive optimiztion prolems. The ndir point is lso exploited to determine the extent of the Preto surfce. The DSD method pplies different technique for identifying points on the Preto frontier. The locl serch domin long the edges of the utopi polygon is rotted towrds the outer prt of the plne in order to cpture solution in the peripherl re. Both the SPO nd NNC methods my fil to generte the Preto frontier when nchor points coincide. This leds to degenertion of the utopi plne to lower dimension nd, hence, loss of the norml vector to the utopi plne. Menwhile, s is shown in Erfni nd Utyuzhnikov (2), DSD cn utilize the rottion technique nd continue the serch until the entire Preto surfce is otined. However, this my led to some difficulties due to the extent of rottion in higher dimension prolems. The gol of this pper is to introduce two modifictions to the DSD method to mke its implementtion esier. First, we chnge the formultion of the single ojective optimiztion prolem in serch domin nd replce it y simpler one. This llows us to void the trnsformtion etween coordinte systems. Next, we eliminte the rottion strtegy nd introduce geometricl technique to define new modified utopi hyperplne for the optimiztion process. The ltter tsk is simple to e implemented nd, contrry to the NNC, it does not rely on further optimiztion formultions. The rest of the pper is orgnized s follows. Section 2 provides generl overview of the multiojective optimiztion prolem. In ddition, the structure of the DSD method is reformulted nd its min steps re explined. The new modifictions to the DSD lgorithm re descried in Section 3 we introduce the modified DSD method clled DSD-II. Then, in Section 4, the method is tested on some chllenging nd well-studied numericl prolems s well s n engineering design prolem. Section 5 dels with some further remrks including the sensitivity of the method to the lgorithm prmeters. Finlly, the lgorithm is summrized in Section 6.

3 Directed serch domin lgorithm for multiojective optimiztion 2 Technicl preliminries In the following, old font is used to distinguish vectors from sclrs. 2. Multiojective optimiztion A generic form of multiojective optimiztion prolem is given y Min F(x)= [f (x), (x),...,f ns (x)], suject to x D, () f i (i =,...,ns) re the ojective functions nd D is the fesile spce defined y the prolem constrints. Usully, the corresponding minimum with respect to ll ojective function is locted outside D. Therefore, we look for set of solutions tht re clled Preto optiml solutions sed on the following definition (Mrler nd Aror 24; Erfni nd Utyuzhnikov 2): A point x is clled Preto optiml solution for () if nd only if there does not exist x D stisfying f i (x) f i (x ) for ll i =,...,nsnd f j (x)<f j (x ) for t lest one j. The vector F(x ) is then clled non-dominted or Preto point. The set of ll Preto points forms the Preto frontier. This oundry provides the est possile trde-off solutions to the multiojective optimiztion prolem (). The nchor point for ech ojective function is otined y optimizing the prolem () for ech ojective seprtely. An i-th nchor point is written s μ i = [f (x i), (x i) (,...,f ns x i)], x i = rg min f i (x). x The convex region defined y the nchor points is clled the utopi polygon. Further, the utopi point U is the est performnce vector given y U = [f (x ), (x 2) (,...,f ns x ns)]. The pseudo ndir point N, on the other hnd, is the worst ojective vlue of the nchor points formulted s N = [n,n 2,...,n ns ], n i = mx {f i (x ),f i (x 2),...,f i (x ns)} These definitions re shown grphiclly in Fig.. In this pper we modify the DSD method to mke its implementtion esier. We next provide rief description A generic description of multiojective serch spce New shrinking strtegy using the inner product ngles Fig. A generic two dimensionl multiojective prolem shrinkge strtegy of the DSD method in order to fcilitte comprison with DSD-II in the reminder of the pper. 2.2 Overview of the DSD DSD is method for the even genertion of the Preto frontier in generl multiojective optimiztion prolem with n ritrry numer of ojective functions nd constrints. It is sed on shrinking the serch domin nd finding the Preto solution in selected re on the Preto frontier. The lgorithm contins the following steps. First, the nchor points (μ i ) re generted for ech ojective function. Therefter, μ i re used to form the interior of the utopi hyperplne P y ns P = α i μ i, ns α i =, α i. (2)

4 T. Erfni et l. By vrying α i uniformly, it is possile to generte evenly distriuted reference points on P denoted y M. The points re evenly distriuted if they re eqully spced from ech other using pproprite metrics (Utyuzhnikov et l. 29). For ech M on P, the following single ojective optimiztion prolem is solved ns Min f i (x), suject to f i (x) M i, (i =,..., ns), (3) x D, M = [M,M 2,...,M ns ]. The ove set of constrints introduces locl serch domin. Solution of prolem (3) for ech reference point M leds to set of Preto solutions which re not necessrily evenly distriuted. This is due to the extent of the serch domin in which two neighour reference points my shre prt of the other s serch domin. This cn led to the sme solution for neighouring reference points (Utyuzhnikov et l. 29). To gurntee evenly distriuted solutions, distinct serch domin for ech reference point is introduced. To chieve this, the serch domin is reduced y introducing new coordinte system with the origin t point M. The xes of the coordinte system form given ngle with respect to unit vector l. This leds to the following modified constrint to the single ojective optimiztion for M in (3): ˆ f i ns M j B ji (i =,...,ns), (4) j= A = B is the trnsformtion mtrix from the Crtesin coordinte system with sis vectors e j (j=,...,ns) to the new locl coordinte system with sis vectors j (j=,...,ns), nd fˆ i is the i th ojective function of the new vriles. To determine the mtrix B in multidimensionl spce, the reder is referred to the originl ppers (Utyuzhnikov et l. 29; Erfni nd Utyuzhnikov 2). In the cse of non-convex oundry, it my hppen tht there will e no fesile solution in the serch domin. In this cse, the serch domin is flipped to the opposite side of the utopi hyperplne to cpture the points on the Preto frontier. This is done y reversing the inequlities in prolem (3). It should e noted tht the serch domin is flipped if no solution is ttined on one side of the utopi plne. Therefore, there is no need to know the shpe of the Preto frontier priori. To overcome the deficiency discussed in Section, the serch domin is rotted to the outer prt of the utopi polygon if the point M is locted on its edge. 2.3 DSD modifictions As illustrted in its originl formultion, the DSD method hs een used nd tested on different chllenging test cses with promising results (Erfni nd Utyuzhnikov 2). However, the DSD cn e improved for some pplictions with respect to: Shrinking the serch domin sed on trnsformtion from Crtesin to locl coordinte system my e computtionlly costly nd hrd to implement due to mtrix mnipultion. Flipping the serch domin my not e efficient when the serch spce is highly oscillting; tht is the Preto frontier vries repetedly from convex to non-convex shpe. Rottion strtegy my ecome hrd to implement specilly if the nchor points coincide (Utyuzhnikov et l. 29). In the current pper, we ddress ll of ove issues nd propose modified version of the DSD, which we cll DSD- II. 3 DSD-II: lgorithm sketch The mjor steps of DSD-II follow. Shrinking nd flipping In the proposed DSD-II, the shrinking constrint still exists. However, the shrinking procedure sed on coordinte system trnsformtion is replced s follows. A new vector ν ν = M c M, (5) is introduced F (x c ) = M c with x c eing the serch point in the current itertion. An dditionl vector n, orthogonl to the utopi hyperplne P, is lso exploited. Then, the shrinking process in prolem (3) is descried s ( ) ν n rccos θ, (6) ν n the left hnd side is the ngle etween ν nd n, nd (.) is the inner product etween two vectors see Fig.. By restricting the vlue for θ, it is esier to shrink the serch domin to generte solution in the desired re. It is

5 Directed serch domin lgorithm for multiojective optimiztion noted tht the ove eqution is vector-sed clcultion, which provides generic expression of the serch ngle. Thus, it is invrint with respect to the dimension of the prolem. To overcome the computtionl cost inherent to the flipping strtegy, t ech itertion, we lter the ove constrint s follows: γ = rccos ν n ν n θ, (7) which is considered s the replcement for the existing shrinking constrint in prolem (3). Inequlity (7) oviously elimintes the first two forementioned difficulties. On one hnd, the new development does not rely on the coordinte system, which elimintes the mtrix mnipultions, nd, on the other hnd, the flipping is not required. This reduces the computtionl cost. It should e noted tht, while the DSD introduces polyhedrl s the serch region, DSD-II implements cone directed towrd the Preto frontier. Rotting The rottion strtegy in DSD is implemented y introducing unit vector tht is the outer norml to the edge of the utopi hyperplne. The vector cn rotte from the norml direction of the utopi polygon to the direction lying outside the polygon. The rottion continues until no new solution is otined (Utyuzhnikov et l. 29; Erfni nd Utyuzhnikov 2). The DSD rottion is pplied in two different situtions. One cse is when the utopi hyperplne cnnot cover the whole Preto surfce. The other occsion is when degenertion of the utopi hyperplne results in the loss of the norml direction n. In the ltter cse, there is no strtegy to hndle such prolem in the literture. Although there exist some guidelines to optimize the rottion strtegy for the first sitution (Utyuzhnikov et l. 29), in the second cse the rottion might e cumersome nd inefficient. To improve the efficiency of the DSD method, we proceed s follows. We redefine the direction n from the pseudo ndir point N to the utopi point U. In the cse pproximte ounds of the ojective spce re known, one my tke the direction from the worst (possily pseudo ndir) point to the est (possily utopi) point. n = U N U N. (8) A hyper-ox ounding the serch spce is constructed s c d Bounding ox C nd vector n Plne PP nd its covering plne CP Intersection of CP nd PP C = [ f min,f mx ] [ f min 2,f2 mx ]... [ f min ns,fns mx ], (9) Modified utopi plne P Fig. 2 A generic description on defining the modified utopi plne P

6 T. Erfni et l. Here C is ounding hyper-ox with uniform distriution of nodes within it. Next, the nodes c C re projected onto hyperplne PP with its norml vector defined y n: PP = c t c t nn t. () Oviously, projected points on PP re not evenly distriuted. To ssure such set, covering hyperplne CP with the sme dimension s PP is constructed y the grid contining uniformly distriuted points. Then, P = CP PP () is hyperplne with evenly distriuted points M. The schemtic procedure for this step cn e found in Fig. 2. Given the ove mendments to the DSD method, we present the DSD-II in Algorithm, in which the flipping nd rottion strtegies re no longer required. Algorithm DSD-II Algorithm. Find the nchor points. 2. Find the modified utopi hyperplne P = CP PP 3. Generte evenly distriuted reference points M on P for ll M P do 4. Solve the following prolem: Min ns f i (x), s.t. γ θ, x D. end for 5. Apply filtering lgorithm to eliminte dominted points from the given set of Preto solutions. It is noted tht the filtering pproch presented in the step 5 of the lgorithm ims to remove the locl Preto solutions (Erfni nd Utyuzhnikov 2). The filtering is sed on the domintion concept illustrted in Preto definition in Section 2 tht returns suset of Preto points for which none will e dominted y ny other. Furthermore, in the implementtion of oth DSD nd DSD-II, we use the summtion over ll the ojective functions s the ggregte ojective function (AOF) for single ojective optimiztion suprolems. 4 Simultion results In this section, we demonstrte the performnce of the DSD- II on test cses nd compre the results with those of DSD. Test prolems re chosen from numer of pst studies in this re nd present vrious chllenges in multiojective optimiztion. For further illustrtion we investigte the performnce of the pproch s pplied to n engineering design prolem. 4. Performnce mesures To evlute the diversity of the solutions long the Preto frontier, we use the coefficient of evenness E s suggested in Utyuzhnikov et l. (29). First, introduce the Riemnn metric given y dr 2 = ns ns j= g ij dx i dx j, (2) {x i } (i =,...,ns ) is coordinte system on the Preto surfce nd g is its metric tensor. Then, the coefficient E defined y E = mx i min j r ij min i min j r ij, i = j, i =,...,numer of Preto points, (3) represents the rtio etween the mximl possile distnce from Preto point nd nother nerest point nd the miniml one. The symol r ij denotes the distnce etween Preto solutions i nd j (i = j) inmetric(2). In the cse Tle Comprison etween DSD nd DSD-II on E (evenness), O (the numer of ojective function evlutions) nd Ti(computing time in seconds) DSD DSD-II E O Ti(s) E O Ti(s) Spirl (.) (2.36) (.37) (.4) (9.3) (.44) ZDT (.3) (8.6) (.7) (.22) (7.) (.2) DTLZ (.) (25.4) (.) (.3) (9.38) (.5) DTLZ (.9) (2.44) (.8) (.2) (2.8) (.7) DTLZ (.3) (2.32) (.29) (.) (2.73) (.2) Vlues in the rckets re the vrince of the performnce over 3 runs. The est metric vlue etween the two methods is highlighted y the old font (the difference is sttisticlly significnt if ANOVA test s p-vlue is less thn.5)

7 Directed serch domin lgorithm for multiojective optimiztion of completely even set of solutions, we hve E =. In ddition to evenness, we report the numer of ojective functions evlutions O for the ske of comprison with the DSD method s well s the computtionl time (in seconds) required to solve the prolem (Ti). In ll test cses the numer of decision vriles, m, equls nd the suprolems re solved using grdient-sed lgorithm. Further, the prolems re solved 3 times to ccount for different rndom strting points for ech suprolem. The results in Tle re the men vlue of the totl runs nd the figures show the est performnce mong 3 runs. An ANOVA test is run to investigte whether the difference etween the DSD nd DSD-II lgorithms on the performnce metrics is significnt. We distinguish etween the performnce metrics vlues if the corresponding ANOVA test p-vlue is less thn.5. Therefore, the results tht re sttisticlly significnt show difference etween the performnce of the lgorithms on the test smples Preto frontier DSDII Spril f Preto solution for Spirl Preto frontier DSDII ZDT3.5 f Preto solution for ZDT3 Fig. 3 Results of DSD-II on two dimensionl test prolems 4.2 Numericl test prolems Spirl This test cse (Brnke et l. 24)exmines the efficiency of the method in otining the solution with highly oscillting Preto surfce s follows Min (f (x), (x)), s.t. x i (i =,..., m), f (x) = g(x)r(x )sin(πx()/2), (x) = g(x)r(x )cos(πx()/2), G(x) = + 9 m x i, m i=2 r(x ) = 5 + (x.5) cos(2πkx )), The results in Tle show tht due to fewer ojective function evlutions, DSD-II solves the prolem fster thn the originl version for the set o5 solutions. ZDT3 This test cse is proposed in De (2) ndis formulted s follows: Min (f (x), (x)), s.t. x i (i =,..., m), f (x) = x, ( x (x) = G(x) G(x) x ) G(x) sin(πx ), G(x) = + 9 m xi 2 m, i=2 As cn e seen in Fig. 3, the prolem hs discontinuous Preto frontier. Due to the numer of disconnected frontiers, the DSD flipping strtegy is hndled in some prt of the serch region if the solution is not ttined on one side of the utopi line. The high computtionl cost of this flipping is ovious when the performnce of DSD is compred with tht of DSD-II. In Tle, the two lgorithms find uniform spred of non-dominted solutions indicted y the Es. However, the set of solutions otined using DSD-II requires fewer ojective function evlutions (for 25 reference points). It should e noted tht some locl solutions re removed using the filtering strtegy in Section 3.

8 T. Erfni et l. DTLZ2 As reported in Utyuzhnikov et l. (29), NBI nd NNC methods generte numer of redundnt solutions for this three-dimensionl test cse Min (f (x), (x), f 3 (x)), s.t. x i (i =,..., m), f (x) = ( + G(x)) cos(x π/2) cos(x 2 π/2), (x) = ( + G(x)) cos(x π/2) sin(x 2 π/2), f 3 (x) = ( + G(x)) sin(x π/2), G(x) = m (x i.5) 2. i=3 As the Preto surfce is not convex, the numer of ojective function evlutions re smller in DSD-II s compred to DSD. Furthermore, due to the extension of the Preto surfce projected onto the utopi plne, the rottion strtegy is required y DSD to cpture ll 47 Preto solutions (See Fig. 4). DTLZ7 In the following test cse proposed y De et l. (25), hlf the Preto surfce cnnot e covered y the utopi hyperplne. Min (f (x), (x), f 3 (x)), s.t. x i (i =,..., m), f (x) = x(), (x) = x(2), f 3 (x) = ( + G(x))H (F,G), G(x) = H(F,G)= 3 m x i, i=3 2 ( ) f i + G(x) ( + sin(3πf i)). Moreover, the prolem hs four disconnected Pretooptiml regions in the serch spce. Although the results do not indicte ny significnt chnge from tht of the DSD method for the set of 5 reference points, the DSD-II voids the use of the rottion strtegy. Moreover, the discontinuity of the Preto frontier requires flipping the serch domin in DSD while DSD-II does not need this. The locl solutions in oth pproches re filtered out from the set of Preto solutions t the end of the optimiztion procedure (See Fig. 4). DTLZ5 This test prolem is reported in De et l. (25)s follows Min (f (x), (x), f 3 (x)), s.t. x i (i =,..., m), f (x) = ( + g(x 3 )) cos(θ ) cos(θ 2 ), (x) = ( + g(x 3 )) cos(θ ) sin(θ 2 ), f 3 (x) = ( + g(x 3 )) sin(θ ), m G(x) = (x i.5) 2, i=3 θ = π 2 (x ), π θ 2 = 4( + G(x)) ( + 2G(x)x 2). This test cse is chllenging for ll deterministic methods. It introduces sitution the numer of nchor points is smller thn the numer of ojective functions. Therefore, NNC nd NBI fil on this test cse. As reported in Erfni nd Utyuzhnikov (2), DSD successfully tckles this tsk due to its rottion nd flipping strtegy which comes with cost of the greter numer of ojective function evlutions in comprison to the DSD-II (see Tle ). In oth DSD nd DSD-II, 5 reference points re considered nd some locl solutions re generted nd removed y the filtering pproch (See Fig. 4). 4.3 Engineering design prolem I-Bem design This prolem is tken from Yng et l. (22). The gol is to find the dimensions of the concrete I-Bem which stisfy geometric nd strength constrints. The ojectives re simultneous minimiztion of f crosssectionl re of the em nd the sttic deflection of the I-Bem considering the orthogonl nd cross sectionl forces of P = 6 kn nd Q = 5 kn, respectively (Fig. 5). The prolem is formulted s follows: Min f = 2x 2 x 4 + x 3 (x 2x 4 ), Min = PL3 48EI, s.t. M y + M z σ, Z y Z z x 8, x 2 5,.9 x 3 5,.9 x 4 5,

9 Directed serch domin lgorithm for multiojective optimiztion Fig. 4 Results of DSD-II on three dimensionl test prolems c d e f I = ( )] [x 3 (x 2x 4 ) 3 +2x 2 x 4 4x x (x 2x 4 ), M y =.25PL, M z =.25QL, Z y = [ )] x 3 (x x 4 ) 3 +2x 2 x 4 (4x4 2 6x +3x (x 2x 4 ), Z z = [ ] (x x 4 )x3 3 6x +2x 4x2 3, 2 E = 2 4 kn/cm 2,σ = 6kN/cm 2. Figure 6 shows the performnce of oth DSD nd DSD- II on this prolem. DSD neglects the left hnd-side of the Preto frontier on this prolem due to the serch direction l. In DSD, l is usully tken s the norml to the utopi line, s explined in Section 2.2. Since the serch spce is highly skewed, the solution procedure ecomes highly sensitive. Even smll chnge in l leds to different solution. For DSD, the dversely scled ojective spce mkes the left prt of the Preto frontier lmost prllel to l nd, hence, the solutions re poorly cptured in this re. Menwhile, since DSD-II performs the shrinkge sed on the direction from U to N, (which inherent to the skewness), it does not fce these difficulties during Preto genertion.

10 T. Erfni et l. x 3 x 2 P Q x 4 x Fig. 5 Concrete I-Bem design prolem 5 Further considertion 5. Extr points on the modified utopi hyperplne One cn introduce test prolem in which some of the reference points generted on the modified utopi plne P led f 5 Preto solutions for I-Bem using DSD L P Q DSD I Bem (P) DSDII I Bem (P) f 5 Preto solutions for I-Bem using DSD-II Fig. 6 Preto solutions for I-Bem to exploring prt of the spce no fesile solution exists. This explortion occurs if the plne P exceeds the projection of the Preto surfce. Therefore, it is not necessry to solve the su-optimiztion prolems for those points M outside the Preto-surfce projection onto P. To identify those unnecessry points M, we need to investigte whether there exists ny fesile point inside the serch cone corresponding to point M. This cn e done y solving nive optimiztion prolem in which the ojective function is constnt vlue suject to the originl nd conic constrints. If there is no solution, we discrd the point M from further considertion. An lterntive pproch is to use lrger ngle θ ssocited with the shrinkge for points ner the edges of P. In doing so, lrger region is included in the serch domin. However, this cretes nother difficulty due to the extent of the serch spce which my led to the sme solution for different serch cones. 5.2 Sclility in the cse of lrge numer of ojective functions In multiojective optimiztion prolems the sclility to ny numer of decision vriles nd ojectives is n importnt issue. However, s ech suprolem in DSD-II is sclr optimiztion prolem, there is no difficulty with the method frmework. To demonstrte this, the DTLZ2 test cse is used to investigte DSD-II s ility to scle up its performnce with four nd eight ojective functions. From De et l. (25), it is known tht Preto optiml solutions must stisfy ns f i 2 =, in ojective spce. For oth the DTLZ2-4D nd DTLZ-8D test prolems, one cn see in Fig. 7 tht for ech solution the squre vlue of ech ojective function sums to one. Tle 2 shows the evenness nd the time required to solve the prolem in ech test cse. As efore, the solution of the prolems re computed for 3 times with rndomly chosen strting point for ech suprolem. It should e noted tht while sclility is chllenge in popultion-sed lgorithms, this is not the cse for multiojective sclriztion methods. This is ecuse single-ojective optimiztion prolem is solved in order to find ech Preto solution. Therefore, in the cse of good representtive set of reference points on the utopi hyperplne, the numer of function evlutions only depends on the sclriztion method in multiojective lgorithm. The AOF my require s mny ojective function evlutions s the dimension of the prolem.

11 i i Directed serch domin lgorithm for multiojective optimiztion Fig. 7,. For ech solution point the summtion over squre vlues of ojective functions equls one. Twenty Preto optiml solution vlues re mgnified showing the contriution of the squre vlue of ech ojective function for ech Preto point in the summtion 4 i = Preto points F F2 F3 F4 DTLZ2-4D.5 8 i = F F2 F3 F4 F5 F6 F7 F Preto points DTLZ2-8D 5.3 θ setting in the DSD-II The vrile θ is the only control prmeter in DSD-II nd defines the extent of shrinkge of ech individul serch domin. It plys criticl role in prolems contining highly skewed serch spce. The curvture of the Preto surfce my demnd more shrinkge to gurntee one distinct solution in ech serch cone. While the Preto frontier my ulk in the center of the utopi plne, it cn e flt frther wy nd ner the nchor points. For given θ, one finds tht the method my work well for M situted in the center of the utopi plne. However, the shrunk spce my e too lrge in reltively flt re ner the nchor points. The flt re ppers if the Preto frontier intersects the utopi hyperplne cutely. To llevite this prolem, either smller ngle is used t the initil serch stge for ll points M or n dptive shrinkge my e considered. Compred to the former, the ltter is recommended ecuse it is more roust due to smll serch spce for those points M which cn work resonly well with lrger ngles of shrinkge. The dptive shrinking cn e introduced y θ ˆθ = U M, 2 clcultes the distnce of M from the utopi point U. In this cse, the frther M is situted, the smller the shrinkge is. The dptive ngle ˆθ works in similr fshion s the scling strtegy the mximl nd miniml vlues of ech ojective re used to scle up the ojective prior to numericl clcultion. ˆθ ssures tht the proportionl extent of shrinkge with respect to the distnt of the reference point is used. To test the dptive shrinkge, we used the following test prolem ZDT- The originl form of this test cse is given in De (2). Here, we hve replced f y f to introduce n undesirle severe skewing of the serch spce. The prolem is given y Min (f (x), (x)), s.t. x i (i =,..., m), G(x) = + 9 m xi 2 m, f (x) = x, ( (x) = G(x) i=3 x G(x) ), Tle 2 DSD-II performnce on higher dimension prolem: E (evenness) nd Ti(rel time spent in seconds) E Ti(s) DTLZ2-4D.4 9. DTLZ2-8D

12 T. Erfni et l Preto frontier DSDII ZDT (P) 5 f Modified ZDT Preto frontier DSDII ZDT (P).2 5 f Modified ZDT with dptive shrinkge strtegy Fig. 8 Preto solutions for skewed ZDT using DSD-II Adversely dispersed ZDT s ojective function spce is shown in Fig. 8. Using the proposed shrinkge strtegy, it is visully evident in Fig. 8 tht the Preto frontier is evenly generted ((, ) is the utopi point). It should e noted tht the proposed dptive shrinkge is n lterntive remedy to the pre-scling of the ojective spce efore the optimiztion procedure. 6 Conclusion A new DSD-II lgorithm is proposed. By introducing n dditionl constrint in the shrinking strtegy, we hve modified the shrinking procedure used in the originl DSD method. The modifiction leds to the elimintion of the flipping strtegy in the lgorithm. In ddition, insted of the rottion strtegy, we hve considered modified utopi hyperplne which cn cover the entire Preto frontier. The pproch proves to e computtionlly efficient. The method hs een tested on different well known test cses nd results hve een compred with those of the DSD method. The comprison demonstrtes tht DSD- II my reduce the computtionl cost for finding set of Preto solutions. Menwhile, the evenness of the generted solutions remins on the sme level for oth methods. Considering the fct tht the DSD-II is esier to implement, one cn conclude tht oth pproches (DSD nd DSD- II) cn e used interchngely. For further considertion, nive optimiztion is introduced to eliminte those reference points on the utopi hyperplne tht led to unfesile solutions. As result, the efficiency of the method is further incresed. In ddition, the sclility of the method to prolems with mny ojectives hs een evluted. We hve lso studied the sensitivity of the DSD-II to shrinking the ngle θ. Empiriclly, it is demonstrted tht for dversely scled ojective functions, the choice of θ ffects the performnce of the method. Therefore, n dptive shrinkge strtegy sed on the distnce metric is introduced. On modified disprtely scled test cse, the proposed strtegy shows cler dvntge over the originl one. As future work, the existing clssicl methods my e compred to study the strengths nd weknesses of ech of the individul methods. Acknowledgments The uthors re grteful to the unknown referees for their very useful remrks which llowed us to improve the qulity of the pper. First uthor would like to thnk Luci Nigri who helped proofred prt of the pper. References Brnke J, De K, Dierolf H, Osswld M (24) Finding knees in multiojective optimiztion. In: Prllel prolem solving from Nture- PPSN VIII. Springer, pp Ds I, Dennis J (998) Norml-oundry intersection: new method for generting the Preto surfce in nonliner multicriteri optimiztion prolems. SIAM J Optim 8:63 De K (2) Multi-ojective optimiztion using evolutionry lgorithms. Wiley, New York De K, Thiele L, Lumnns M, Zitzler E (25) Sclle test prolems for evolutionry multiojective optimiztion. In: Evolutionry multiojective optimiztion. pp 5 45 Erfni T, Utyuzhnikov S (2) Directed serch domin: method for even genertion of the Preto frontier in multiojective optimiztion. Eng Optim 43(5): Mrler R, Aror J (24) Survey of multi-ojective optimiztion methods for engineering. Struct Multidiscip Optim 26(6): Messc A, Mttson C (22) Generting well-distriuted sets of Preto points for engineering design using physicl progrmming. Optim Eng 3(4):43 45 Messc A, Ismil-Yhy A, Mttson C (23) The normlized norml constrint method for generting the Preto frontier. Struct Multidiscip Optim 25(2):86 98 Mueller-Gritschneder D, Gre H, Schlichtmnn U (29) A successive pproch to compute the ounded Preto front of

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