Heuristics for Optimising the Calculation of Hypervolume for Multi-objective Optimisation Problems

Transcription

1 Edith Cown University Reserch Online ECU Publictions Pre Heuristics for Optimising the Clcultion of Hypervolume for Multi-objective Optimistion Problems Lyndon While University of Western Austrli Lucs Brdstreet University of Western Austrli Luigi Brone University of Western Austrli Philip Hingston Edith Cown University /CEC This conference pper ws originlly published s: While, L., Brdstreet, L., Brone, L., & Hingston, P. F. (2005). Heuristics for Optimising the Clcultion of Hypervolume for Multi-objective Optimistion Problems. Proceedings of IEEE Congress on Evolutionry Computtion. (pp ). Edinburgh, Scotlnd. IEEE. Originl rticle vilble here 2005 IEEE. Personl use of this mteril is permitted. Permission from IEEE must be obtined for ll other uses, in ny current or future medi, including reprinting/republishing this mteril for dvertising or promotionl purposes, creting new collective works, for resle or redistribution to servers or lists, or reuse of ny copyrighted component of this work in other works. This Conference Proceeding is posted t Reserch Online.

2 2225 Heuristics for Optimising the Clcultion of Hypervolume for Multi-objective Optimistion Problems Lyndon While, Lucs Brdstreet, Luigi Brone The University of Western Austrli Nedlnds, Western Austrli 6009 { lyndon, lucs, Phil Hingston Edith Cown University Mount Lwley, Western Austrli 6050 p.hingston@ecu.edu.u Abstrct- The fstest known lgorithm for clculting the hypervolume of set of solutions to multiobjective optimistion problem is the HSO lgorithm (Hypervolume by Slicing Objectives). However, the performnce of HSO for given front vries lot depending on the order in which it processes the objectives in tht front. We present nd evlute two lterntive heuristics tht ech ttempt to identify good order for processing the objectives of given front. We show tht both heuristics mke substntil difference to the performnce of HSO for rndomly-generted nd benchmrk dt in 5-9 objectives, nd tht they both enble HSO to relibly void the worst-cse performnce for those fronts. The enhnced HSO will enble the use of hypervolume with lrger popultions in more objectives. 1 Introduction Multi-objective optimistion problems bound, nd mny evolutionry lgorithms hve been proposed to derive good solutions for such problems, e.g. [1, 2,,, 5]. However, the question of wht metrics to use in compring the performnce of these lgorithms remins difficult[6, 7, 1]. One metric tht hs been fvoured by mny people is hypervolume[8], lso known s the S-metric[9] or the Lebesgue mesure[10]. The hypervolume of set of solutions mesures the size of the portion of objective spce tht is dominted by those solutions collectively. Generlly, hypervolume is fvoured becuse it cptures in single sclr both the closeness of the solutions to the optiml set nd, to some extent, the spred of the solutions cross objective spce. Hypervolume lso hs nicer mthemticl properties thn mny other metrics[1 1, 12]. Hypervolume hs some non-idel properties too: it requires the (sometimes rbitrry) definition of reference point on which its clcultions re bsed, nd it is sensitive to the reltive scling of the objectives, nd to the presence or bsence of extreml points in front. While et l.[1] hve shown tht the fstest known lgorithm for clculting hypervolume exctly is HSO (Hypervolume by Slicing Objectives)[1, 15]. HSO works by processing the objectives in front, rther thn the points. It divides the nd-hypervolume to be mesured into seprte n - ID-slices through one of the objectives, then it clcultes the hypervolume of ech slice nd sums these vlues to derive the totl. In the worst cse HSO is exponentil in the number of objectives, but it still esily outperforms ll other known lgorithms for clculting hypervol- ume exctly[1, 16]. However, the performnce of HSO for given front depends on the order in which it processes the objectives in tht front. The number of points contributing hypervolume to ech n - 1D-slice depends on how mny points re dominted within tht slice: more dominted points implies smller set of points to process, which implies less work for tht slice. The principl contribution of this pper is the presenttion nd evlution of two lterntive heuristics tht ech enhnce HSO by trying to select good order in which to process the objectives for given front. We present performnce dt for bsic nd enhnced HSO showing tht both heuristics mke substntil difference to the typicl perfornnce of the lgorithm. The enhnced HSO will enble the use of hypervolume with lrger popultions in more objectives. The rest of this pper is structured s follows. Section 2 defines the concepts nd nottion used in multi-objective optimistion nd throughout this pper. Section describes the opertion of HSO, nd Section discusses its complexity nd performnce nd shows how heuristics might help. Section 5 defines our two (lterntive) heuristics, nd Section 6 gives empiricl dt for rnge of rndomlygenerted nd benchmrk fronts in 5-9 objectives showing how the heuristics improve the performnce of HSO. Section 7 concludes the pper nd outlines some future work. 2 Fundmentls In multi-objective optimistion problem, we im to find the set of optiml trde-off solutions known s the Preto optiml set. Preto optimlity is defined with respect to the concept of non-domintion between points in objective spce. Given two objective vectors x nd y, x domintes y iff x is t lest s good s y in ll objectives, nd better in t lest one. A vector x is non-dominted with respect to set of solutions X iff there is no vector in X tht domintes T. X is non-dominted set iff ll vectors in X re mutully non-dominting. Such set of objective vectors is sometimes clled non-domintedfront. A vector T is Preto optiml iff T is non-dominted with respect to the set of ll possible vectors. Preto optiml vectors re chrcterised by the fct tht improvement in ny one objective mens worsening t lest one other objective. The Preto optiml set is the set of ll possible Preto optiml vectors. The gol in multi-objective problem is to find the Preto optiml set, lthough for continuous problems representtive subset will usully suffice /05/$ IEEE.

3 2226 y +LZ x b c d x y 26 z Slice + E Slice -- Slice 2 Slice I Figure 1: One step in HSO for the four three-objective points shown. Objective x is processed, leving four two-objective shpes in y nd z. Points re mrked by circles nd lbelled with letters: unfilled circles represent points tht re dominted in y nd z. Slices re lbelled with numbers, nd re seprted on the min picture by dshed lines. Given set X of solutions returned by n lgorithm, the question rises how good the set X is, i.e. how well it pproximtes the Preto optiml set. One metric used for compring sets of solutions is to mesure the hypervolume of ech set. The hypervolume of X is the totl size of the spce tht is dominted by the solutions in X. The hypervolume of set is mesured reltive to reference point, usully the nti-optiml point or "worst possible" point in spce. (We do not ddress here the problem of choosing reference point, if the nti-optiml point is not known or does not exist: one suggestion is to tke, in ech objective, the worst vlue from ny of the fronts being compred.) If set X hs greter hypervolume thn set X', then X is tken to be better set of solutions thn X'. Precise definitions of these terms cn be found in [17]. The HSO Algorithm Given m mutully non-dominting points in n objectives, the HSO lgorithm is bsed on the ide of processing the set of points one objective t time. Initilly, the points re sorted by their vlues in the first objective to be processed. These vlues re then used to cut cross-sectionl "slices" through the hypervolume: ech slice will itself be n n - 1-objective hypervolume in the remining objectives. The n - 1-objective hypervolume in ech slice is clculted nd ech slice is multiplied by its depth in the first objective, then these n-objective vlues re summed to obtin the totl hypervolume. Ech slice through the hypervolume will contin different subset of the originl points. Becuse the points re sorted, they cn be llocted to the slices esily. The top slice cn contin only the point with the best vlue in the first objective; the second slice cn contin only the points with the two best vlues; the third slice cn contin only the points with the three best vlues; nd so on, until the bottom slice, which cn contin ll of the points. However, not ll points "contined" by slice will contribute volume to tht slice: some points my be dominted in whtever objectives remin nd will contribute nothing. After ech step (i.e. fter ech slicing ction), the number of objectives is reduced by one, the points re re-sorted in the next objective, nd newly-dominted points within ech slice re discrded. Figure 1 shows the opertion of one step in HSO, including the slicing of the hypervolume, the lloction of points to ech slice, nd the elimintion of newly-dominted points. The most nturl bse cse for HSO is when the points re reduced to one objective, when there cn be only one non-dominted point left in ech slice. The vlue of this point is then the one-objective hypervolume of its slice. However, in prctice, for efficiency resons, HSO termintes when the points re reduced to two objectives, which is n esy nd fst specil cse. Figure 2 gives pseudo-code for HSO. Note tht, for exposition purposes, the function hso builds explicitly set contining the slices to be processed fter ech itertion. We cn improve the performnce of the lgorithm by processing these slices on-the-fly, s they re generted. The Complexity nd Performnce of HSO While et l. [1] give recurrence reltion tht cptures the worst-cse complexity of HSO: f(m, 1) = 1 f(m,n) = m Ef(k,n-1) k=l (1) (2) 2226

4 2227 hso (ps): pl = sort ps worsening in Objective 1 s = {(1, pl)} for k = 1 to n-i s' = {} for ech (x, ql) in s for ech (x', ql') in slice (ql, k) dd (x * x', ql') into s' s = s5 vol = 0 for ech (x, ql) in s vol = vol + x * Ihed return vol (ql)[n] - refpoint[n] slice (pl, k): p = hed (pl) pl = til (pl) ql = [I s = {} while pl 1= [] ql = insert (p, k+l, ql) p' = hed (pl) dd (Ip[k] - p'[k]i, ql) into s P = P' pl = til (pl) ql = insert (p, k+l, ql) dd (Ip[k] - refpoint[k], ql) into s return s insert (p, k, pl): ql = [1 while pl /= [] && hed (pl)[k] bets p[k] ppend hed (pl) to ql pl = til (pl) ppend p to ql while pl /= [] if not (domintes (p, hed (pl), k)) ppend hed (pl) to ql pl = til (pl) return ql domintes (p, q, k): d = True while d && k <= n d = not (q[k] bets p[k]) k =k + 1 return d Figure 2: Pseudo-code for HSO. The summtion in (2) represents the fct tht ech slicing ction genertes m slices tht re processed independently to derive the hypervolume of the front. Furthermore, While et l. [1] solve this recurrence reltion to give the following identity: f m+n-2 ) f(ml,n) = - () Thus HSO is exponentil in the number of objectives n, in the worst cse (we ssume tht m > n). The "worst cse" in this context mens we ssume tht no (prtil) point is ever dominted during the execution of HSO, thus mximising the number of points in ech slice tht is processed. However, this is unlikely to be true for rel-world fronts. The mount of time required to process given front depends crucilly on how mny points re dominted t ech stge, nd, in ddition, on how erly in the process points dominte other points. From this fct, we cn infer tht the time to process given front vries with the order in which the objectives re processed. A simple exmple illustrtes how. Consider the set of points in Figure, in mximistion problem Figure : A pthologicl exmple for HSO. This pttern describes sets of five points in n objectives, n >. All columns except the lst re identicl. The pttern cn be generlised for other numbers of points. If we process the first objective (or in fct ny objective except the lst): no point domintes ny other point in the list in the remining n - 1 objectives. Thus we do indeed hve the worst cse for HSO, generting m slices contining respectively 1, 2,..., m points. If we process the lst objective: ech point domintes ll subsequent points in the list in the remining n - 1 objectives. Then we generte m slices ech contining only one point. Specificlly, the top slice (corresponding to the highest vlue in the lst objective) contins only the point , the second slice contins only the point , ll the wy down to the bottom slice, which contins only the point m... m. This is of course the best cse for HSO, nd the hypervolume is clculted much more quickly. Note tht, in generl, there is continuum of performnce improvement vilble: e.g. for the points in Figure, the erlier the lst objective is processed, the fster the hypervolume will be clculted. Thus it seems tht enhncing HSO with mechnism to help the lgorithm to identify good order in which to process the objectives in given front could mke substntil difference to the rel performnce of the lgorithm. 5 Heuristics We present nd evlute two lterntive heuristics tht ttempt to derive good order for HSO to process the objectives in given front. 5.1 Mximising the number of dominted points A good order for the objectives is one in which mny prtil points re dominted by other points erly in the process. One obvious tctic then is to clculte for ech objective how mny points will be dominted immeditely if tht objective is processed, nd to process first the objective tht will generte the most dominted points. We cll this heuristic MDP: Mximising the number of Dominted Points. We cn pply this ide in two wys. 2227

5 2228 * We cn simply clculte the heuristic once, then sort the objectives in decresing order of numbers of dominted points. * Alterntively, we cn clculte the heuristic once, eliminte the best objective, then re-clculte the heuristic to identify the next objective, nd so on, until ll the objectives hve been ordered. Our experience shows tht pplying the heuristic itertively works better, especilly for lrge numbers of objectives, but tht diminishing returns pply to some extent. We therefore iterte until four objectives remin, t which point we order those four objectives ccording to the lst clcultion. The complexity of MDP is esy to clculte: t ech itertion, for ech objective, we (nominlly) compre ech point with every other point for domintion. Thus for m points in n objectives, ech itertion of MDP hs complexity O(m2n2), nd pplying MDP itertively hs complexity O(m2rn), While this my sound expensive, remember tht HSO is exponentil in n in the worst cse, so good polynomil-time heuristic is likely to py lrge dividends. 5.2 Minimising the mount of worst-cse work For ech objective, MDP effectively counts the number of points tht will contribute to the bottom n - 1D-slice of the hypervolume. However, in some cses, this number might be misleding: it is theoreticlly possible to generte m slices where the first m - 1 slices contin respectively 1, 2,..., m - 1 points, but the bottom slice contins only 1 point. Exmple dt tht exhibits this behviour is given in Figure, for mximistion problem. Figure : A pthologicl exmple for MDP. MDP will choose to process the first objective, but processing the second objective would be fster. We cn void this possibility with slightly more involved heuristic tht clcultes explicitly for ech objective the number of non-dominted prtil points in ech slice, estimtes the mount of work required to process ech slice, nd sums these vlues to estimte the mount of work required if HSO processes tht objective first. This heuristic effectively models the recurrence reltion in (2), by summing the work required to process ech slice individully. For ech slice, we use the worst-cse complexity of HSO given in () to estimte the work required to process tht slice. Thus we cll this heuristic MWW: Minimising the Worst-cse Work. Agin, we cn pply this ide once only, or itertively, nd gin, our experience shows tht itertion works better. As with MDP, we pply MWW itertively until four objectives remin, t which point we order those four objectives ccording to the lst clcultion. The complexity of MWW is similr to tht of MDP. For ech objective, we sort the points in tht objective, then we build incrementlly the sets of points in ech slice, much s in the functions slice nd insert in Figure 2. This leds to the worst-cse complexity for ech itertion being O(n(m log m+m2n)), which gin simplifies to O(m2rn2). The need to mintin n explicit set of non-dominted points during the clcultion of MWW my mke it more expensive thn MDP in some cses, lthough ny difference is likely to be smll. 6 Empiricl Performnce Dt We evluted the performnce of the two heuristics vs. bsic HSO on two different types of dt: rndomlygenerted fronts, nd smples tken from the four distinct Preto optiml fronts of the problems in the well-known DTLZ test suite[18]. We evluted the heuristics (mostly) on dt in 5-9 objectives, so to estimte the best-, verge-, nd worst-cse timings for ech front using bsic HSO, we used the following procedure. we evluted ll n! permuttions of the objec- For n < 5 tives. For n > 5: we smpled the n! permuttions in two wys, nd we combined ll of the results in the clcultions. * We evluted ll n(n - 1) permuttions of the first two objectives (with the remining objectives rndomised). * Additionlly, we evluted 120 rndomlychosen permuttions. All timings were performed on dedicted 2.8Ghz Pentium IV mchine with 512Mb of RAM, running Red Ht Enterprise Linux.0. All lgorithms were implemented in C nd compiled with gcc -0. All times include the costs of clculting the heuristics, where pproprite. The dt used in the experiments re vilble t Benchmrk dt We evluted the heuristics on the four distinct fronts from the DTLZ test suite: the sphericl front, the liner front, the discontinuous front, nd the degenerte front. For ech front, we generted mthemticlly representtive set of 10,000 points from the (known) Preto optiml set: then to generte front of size m, we smpled this set rndomly. Ech hypervolume ws clculted s minimistion problem in every objective, reltive to the point Tbles 1()-1(c) nd Figures 5()-5(d) nd 6 show the resulting comprisons. Ech row of ech tble is bsed on runs with ten different fronts, nd it gives the following dt. * For HSO: wrst is the longest time for ny run on ny front. wst is the verge of the longest time for ech front. 2228

6 2229 n m wrst bsic HSO HSO+MDP HSO+MWW wst vrg bst best wrst vrg best wrst vrg best () The sphericl DTLZ front. n m wrst bsic HSO HSO+MDP HSO+MWW wst vrg bst best wrst vrg best wrst vrg best (b) The liner DTLZ front. bsic HSO HSO+MDP HSO+MWW n m wrst wst vrg bst best wrst vrg best wrst vrg best (c) The discontinuous DTLZ front. I 1 bsic HSO 1 HSO+MDP 1 HSO+MWW n m wrst wst vrg bst best J wrst vrg best J wrst vrg best (d) Rndomly-generted fronts. Tble 1: Comprison of the performnce of HSO, HSO+MDP, nd HSO+MWW on vrious fronts. Ech dtum is bsed on ten different dt sets: the figures for bsic HSO re clculted using the smpling procedure described in Section 6. For ech vlue of n, m is chosen so tht the HSO vrg los. 2229

7 D 5 E ' ux.0 () The sphericl DTLZ front in 5 nd 6 objectives. (b) The sphericl DTLZ front in 7 nd 9 objectives d smple 9d MWW 7d smple 7d MWW E F (c) The discontinuous DTLZ front in 5 nd 6 objectives. (d) The discontinuous DTLZ front in 7 nd 9 objectives. u 6 j_ d smple ~~~~~~~~~9d MWW 7d smple.... 7d MVWW (e) Rndomly-generted fronts in 5 nd 6 objectives. (f) Rndomly-generted fronts in 7 nd 9 objectives. Figure 5: Comprison of the performnce of HSO nd HSO+MWW on vrious fronts. Ech dtum is bsed on ten different dt sets: the figures for bsic HSO re clculted using the smpling procedure described in Section 6. The plot for the liner DTLZ front is similr to tht for the sphericl front nd is excluded for spce resons. 220

8 221 vrg is the verge time for ll of the runs. bst is the verge of the shortest time for ech front. best is the shortest time for ny run on ny front. * For ech heuristic: wrst is the longest time for ny run on ny front. vrg is the verge time for ll of the runs. best is the shortest time for ny run on ny front. As observed previously by While et l.[1], the best-cse objective order for the degenerte front gives performnce tht is polynomil in the number of objectives, so for tht front, we plot only the performnce of HSO+MWW. Ech other plot compres the performnce of HSO+MWW with the verge performnce of HSO over the smple of permuttions of the objectives. ( x) F 0.5 1d MWW 1 d MWW 9d MWW 7d MWW 8 5d MWW - -';W- -ml P Figure 6: The performnce of HSO+MWW on the degenerte front. Ech dtum is bsed on ten different dt sets. 6.2 Rndomly-generted dt We generted sets of m mutully non-dominting points in n objectives simply by generting points with rndom vlues x, 0.1 < x < 10, in ll objectives. In order to gurntee mutul non-domintion, we initilised S = X nd dded ech point x to S only if T U S would be mutully-nondominting. We kept dding points until ISI = m. Ech hypervolume ws clculted s mximistion problem in every objective, reltive to the origin. Tble 1(d) nd Figures 5(e)-5(f) show the resulting comprison. 6. Discussion.... X.. For ech heuristic in ech row of ech tble, we mke the following comprisons. * We compre vrg for the heuristic with the rnge wst... vrg... bst for bsic HSO, to determine how much improvement the heuristic delivers in typicl cses. * We compre wrst for the heuristic with wrst nd wst for bsic HSO, to determine how well the heuristic voids the worst-cse ordering. At' * We compre best for the heuristic with bst nd best for bsic HSO, to determine how close the heuristic gets to the best-cse ordering. (Note tht the best cses for the heuristics sometimes bet the best cse for bsic HSO: this is due to the incomplete nture of the smpling used for the bsic HSO figures.) We mke the following observtions. * For ll of the DTLZ fronts, both heuristics deliver mjor performnce gins, nd MWW in prticulr delivers performnce tht is not fr from optiml. The performnce gins for the sphericl nd liner fronts in prticulr re spectculr: speed-up fctors of in the verge cses. The performnce gin for the discontinuous front is somewht less (speed-up fctors of 2-): no doubt this is due to some property of the front itself. * Rndom fronts my be the worst-cse form of dt for the heuristics, but both heuristics still lwys outperform bsic HSO in the verge cse, with speed-up fctors up to 2.5. * Both heuristics void the worst-cse objective ordering in ll cses: in fct, the worst-cse for the heuristics is nerly lwys better thn the verge cse for bsic HSO, usully by substntil mount. * The performnce gin increses both with incresing number of objectives, nd with incresing number of points. * MWW generlly out-performs MDP. * The grphs however highlight the fct tht exponentil performnce mkes life tough: lthough the heuristics deliver useful speed-ups for processing fronts of given size, they do not lwys gretly improve the sizes of fronts tht cn be processed in given time. The question rises wht size of fronts the enhnced lgorithm cn process in vrious times. Tble 2 shows this dt for HSO+MWW on the sphericl front. We chose ten secn 10 seconds 1 second 5 1, Tble 2: Sizes of fronts in vrious numbers of objectives tht HSO+MWW cn process in the times indicted, for sphericl DTLZ dt. onds s indictive of the performnce required to use hypervolume in off-line metric clcultions fter the EA is complete, nd one second s indictive of the performnce required to use hypervolume in n on-line diversity or rchiving mechnism during the execution of the EA. 221

9 222 We lso performed some minor experimenttion to estimte the cost of clculting the heuristics themselves. Our experiments indicte tht these clcultions usully tke less thn 1% of the run-time of the enhnced lgorithm, nd tht they never exceed bout 6% of the run-time, even with popultions up to 2,000. This is of course to be expected, becuse of the exponentil complexity of HSO itself. 7 Conclusions nd Future Work We hve described two lterntive heuristics tht ech improve the performnce of the HSO lgorithm for clculting hypervolume, itself the fstest lgorithm described to dte. Ech heuristic works by re-ordering the objectives in front to reduce the sizes of the sets of points tht hve to be processed during the execution of the lgorithm. Both heuristics deliver significnt improvement to the performnce of HSO, with reductions in the run-time of the lgorithm of 25-98%. The enhnced HSO will enble the use of hypervolume with lrger popultions in more objectives. We intend to speed-up the clcultion of our heuristics, e.g. by minimising the cost of dominnce-checking, lthough we do not expect this to deliver serious further improvements. We lso intend to pursue other venues for mking HSO fster, such s reducing the mount of repeted work tht results from processing slices independently. We lso intend to design n incrementl version of HSO, for use s diversity or rchiving mechnism in n evolutionry lgorithm. Acknowledgments We thnk Simon Hubnd for discussions on hypervolume nd HSO, nd for providing the rw DTLZ dt. This work ws supported prtly by The University of Western Austrli Reserch Grnts Scheme, nd lso prtly by n ARC Linkge grnt. Bibliogrphy [1] S. Hubnd, P. Hingston, L. While, nd L. Brone, "An evolution strtegy with probbilistic muttion for multi-objective optimiztion," in CEC 200, H. Abbss nd B. Verm, Eds., vol.. IEEE, 200, pp [2] E. Zitzler, M. Lumnns, nd L. Thiele, "SPEA2: Improving the strength Preto evolutionry lgorithm for multiobjective optimiztion," in EUROGEN 2001, K. C. Ginnkoglou et l., Ed., 2001, pp [] R. C. Purshouse nd P. J. Fleming, "The MultiObjective Genetic Algorithm pplied to benchmrk problems - n nlysis," The University of Sheffield, UK, Reserch Report 796, [] K. Deb, A. Prtp, S. Agrwl, nd T. Meyrivn, "A fst nd elitist multiobjective genetic lgorithm: NSGA-II," IEEE Trnsctions on Evolutionry Computtion, vol. 6, no. 2, pp , [5] J. Knowles nd D. Come, "M-PAES: A memetic lgorithm for multiobjective optimiztion," in CEC 2000, vol. 1. IEEE, 2000, pp [6] T. Okbe, Y. Jin, nd B. Sendhoff, "A criticl survey of performnce indices for multi-objective optimistion," in CEC 200, H. Abbss nd B. Verm, Eds., vol. 2. IEEE, 200, pp [7] J. Wu nd S. Azrm, "Metrics for qulity ssessment of multiobjective design optimiztion solution set," Journl of Mechnicl Design, vol. 12, pp , [8] R. Purshouse, "On the evolutionry optimistion of mny objectives," Ph.D. disserttion, The University of Sheffield, Sheffield, UK, 200. [9] E. Zitzler, "Evolutionry lgorithms for multiobjective optimiztion: Methods nd pplictions," Ph.D. disserttion, Swiss Federl Inst of Technology (ETH) Zurich, [10] M. Lumnns, E. Zitzler, nd L. Thiele, "A unified model for multi-objective evolutionry lgorithms with elitism," in CEC 2000, vol. 1. IEEE, 2000, pp [11] E. Zitzler, L. Thiele, M. Lumnns, C. M. Fonsec, nd V. G. d Fonsec, "Performnce ssessment of multiobjective optimizers: An nlysis nd review," IEEE Trnsctions on Evolutionry Computtion, vol. 7, no. 2, pp , April 200. [12] M. Fleischer, "The mesure of Preto optim: Applictions to multi-objective metheuristics," Institute for Systems Reserch, University of Mrylnd, Tech. Rep. ISR TR , [1] L. While, P. Hingston, L. Brone, nd S. Hubnd, "A fster lgorithm for clculting hypervolume," IEEE Trnsctions on Evolutionry Computtion, [1] E. Zitzler, "Hypervolume metric clcultion," 2001, ftp://ftp.tik.ee.ethz.ch/pub/people/zitzler/hypervol.c. [15] J. Knowles, "Locl-serch nd hybrid evolutionry lgorithms for preto optimistion," Ph.D. disserttion, The University of Reding, [16] L. While, "A new nlysis of the Lebmesure lgorithm for clculting hypervolume," in EMO 2005, ser. LNCS, C. Coello Coello et l., Ed., vol. 10. Springer-Verlg, 2005, pp [17] T. Bck, D. Fogel, nd Z. Michlewicz, Eds., Hndbook of Evolutionry Computtion. lop Institute of Physics, [18] K. Deb, L. Thiele, M. Lumnns, nd E. Zitzler, "Sclble multi-objective optimiztion test problems," in CEC 2002, D. B. Fogel et l., Ed., vol. 1. IEEE, 2002, pp