A pplication to O ne-off P roducts
R. I. W H IT FIE L D , P. N . H . W RIG H T, G . C O AT E S & W . H ILL S
SU M M A RY Robust design is an activity of fund am ental im portance when de signing large, com plex, one-off engineering produ cts. W ork is de scribed which is concerned with the appli-cation of the theory of design of experim ents and stochastic optim ization m ethods to explore and op tim ize at the concept design stage. T he discussion begins with a description of state-of-the-art stochastic techniques and their application to robust design. The content then focuses on a generic m ethodolog y which is capable of m anipulating de sign algorithm s that can be used to describe a de sign concept. A n exam ple is presented, dem onstrating the use of the system for the robust de sign of a catam aran with respect to seakeeping.
1. Introduction
A p roduct’s robustness is a m easure of the variation in its utility experienced in a typical application. T hat is to say, the low er the sensitivity or variation in utility, the greater the robustness of the design. In this w ork, w e consider robust design to be the process by w hich a design is produced in w hich changes in the selected variables w hich de®ne the optim um design have relatively little effect on the perform an ce of the design, i.e. the beh aviour of the selected design is insensitive to m odest changes in the variab les. D uring the early stag es of the design p rocess, it is essential that m an y alternative proposals are exam ined in order to identify those designs w hich are robust. O ne of the dif®culties encountered during this stag e is that tim e is usu ally lim ited. In the case of large com plex products, particularly those classi®ed as m ade-to-order (M T O ) or one-off products, this shortage of tim e causes a further com plication since m odels w hich accurately represent th e design an d its behaviour or p erform an ce are, by necessity, large and com plex. Under these circum stances, designers often reso rt to using concep t design m odels which lack de®nition or decom po se the com p lex m odel into a set of su bm odels. T hese su bm odels are then optim ized an d the overall com pro-m ise `best design’ is assupro-m ed to be de®ned by a copro-m bination of those variables and criteria w hich op tim ize the individual constituent an d sub m odels.
T his approa ch can b e m isleading an d is ¯aw ed. A b etter approa ch is to seek m ethods w hich allow the fu ll, com plex m odel to be used bu t, by selecting a set of po ints in the design space, according to so m e prescribed strategy, a regression equ ation can be derived w hich accurately represents the response surface fo r the design space. T h is
D r I. W hit®eld, G. Coates and Professor W . Hills, Engineering D esign Centre, and P. W right, D epartm ent of M arine Technology, N ewcastle U niversity, Arm strong Building, N ewcastle upon T yne N E 1 7R U , UK . Correspondence should be addressed to W . Hills.
de®nition of the design space can then be searched to locate th e global optim um design (once the su rface has been de®ned). Search m ethods su ch as genetic algorithm s (G As) are particularly effective under su ch circum stances w hen the response surface m ay exhibit several local optim a. T raditionally, engineers have sought to de®ne their design m odels by representing th em in the form of m athem atical m odels an d then exploring the surface using som e system atic search technique. T his usu ally involves very large num bers of evaluations of the m odel and the perform ance criteria. Box an d H unter [1] and T aguchi an d W u [2] su ggested a m ethod fo r determ ining th e nature of the design sp ace based on the design of experim ents and statistical an alysis. Lu cas [3] extended this ap proach to achieve a robust process using response surface m ethodology. T hese techn iques have been ad opted as the basis for a new approa ch to `robust design’ wh ich is particularly applicable to M T O product design.
2. Ba ckgro u nd
In the 1940s, Japanese industry recognized that, if it w as to be com petitive there was a critical need to im p rove the quality of the products it produced an d the associated m anufacturing design process. G enichi T aguchi, a qu ality consultant, w as given the task of developing a m ethodology to m eet these requ irem ents. C onsequently, robust design was established as a system atic m ethodology involving the application of statis-tical techn iques to im prove product qu ality an d process design. Robust design im proves product design qu ality and enables m an ufacture at low cost by m aking product and process perform ance `rob ust’.
Su bsequently, m uch research has been carried out to im prove T aguchi’s robust design m ethodology; p articularly, the statistical techniques used. T he aim was to im prove product q uality w hile m aking signi®cant cost savings.
T aguchi’s m ethod for robust design is based on experim ental design and statistical analysis. T he approa ch to experim ental design involves a product array w hich com -prises a control array and a noise array. In an experim ent, each com bination of the control array is run with every com bination of the noise array. T aguchi’s robust design m ethodology is based on m axim izing signal-to-noise (SN ) ratios. T he SN ratio, or qu ality characteristic, is typically given by
S /N5 210log [M SD ] (1)
w here M SD refers to the m ean square deviation of the objective function. T h e qu ality characteristic is produced by differentiating design variables into control variab les and signal variab les. A robust optim um design is identi®ed by locating the optim um values for the control variables to reduce variation and then ad justing the signal variables to sh ift the m ean, achieved by m axim izing the SN ratio.
Several im p ortant im provem ents to T aguchi’s original wo rk have been su ggested by C hen et al. [4] w ho applied th e m ethods to top-level design speci®cations for the airfram e an d propulsion system of a high -speed civil transport system . W elchet al. [5] fu rther im prove robust design m ethodology using com bined arrays as op posed to T aguchi’s p roduct array. O ther usefu l im provem ents have also been su ggested by En gelund et al. [6] and U nal and Stanley [7].
3. T he Propo sed Robus t D esign M eth odology
FIG. 1(a). Robustness fram ew ork; (b) design coordination system .
fram ew ork, an d to allow fo r coordination during the design of robust M T O products. T he rob ustness fram ewo rk has been developed to a prototype stage to test the validity of th e statistical techn iques to the types of p roblem s envisaged, w hile the design coordination system is in the early stag es of developm ent. G raphical representations of the robustness fram ewo rk and design coordination system can be seen in Fig. 1. T he robust design process presented within Fig. 1 has been autom ated using research so ftw are produced b y the au thors. T his softw are is currently in the process of being utilized w ithin several industrial applications.
3.1 Robustness Fram ework
D epending up on the stag e of the an alysis, a suitable p oint ge nerator is selected using design of experim ents (D O E) theory. T he po int ge nerators available are fu ll and variable fractional factorial and central com posite design. T he fu ll an d fractional factorial designs are availab le for an y order of problem , w hereas the central com p osite design is designed to b e used for second-order problem s only.
T ypically, the ®rst run of the analysis uses a saturated fractional factorial design to provide an overview of th e design problem . W ith this inform ation, variables can be rem oved from the an alysis th at are not considered to be signi®cant w ith resp ect to the criteria. T his techn ique relies upon the assum ption that the m ain effects have greater signi®cance th an the interaction term s; hence, the rem oval of the variable from the analysis does not have any consequence on the response surface ge nerated. T h is m ethod allow s the design to be studied without a great deal of inform ation being required to de®ne th e design process. S ub sequently, m an y variables m ay be added to the problem de®nition and hence into th e an alysis w ithout dram atically increasing the size of the experim ent. A second-order response surface design can th en be used w ith the signi®cant variables to produce an im proved representation of the design concept. T he m ethodology involves th e use of com pu ter-based design algorithm s; h ow ever, it is an ticipated that experim ental or full-scale test runs could be undertaken an d data collected w hich could be utilized by the fram ew ork. Info rm ation is passed to th e design algorithm s which are then executed in the approp riate order to facilitate the correct ¯ow of info rm ation. Th e responses are then taken from the output ®les an d collated for use by the response su rface m odule.
T he control, noise an d response variables are used to produce a set of norm al equations. T he norm al equations can then be solved using a variety of different m ethods to produce a series of regression equations for the m ean an d variance of each response and constraint as fu nctions of th e inp ut variables. Analysis of variance is then perform ed on the regression equations to check for `goodness of ®t’.
C urrently, a single objective GA m ethod is used to obtain an optim um design using the regression equation. T he GA is preferred rather than m ore traditional hill-clim bing techn iques due to the ability of the D O E m odule to generate exp erim ents of orders having local optim a. T he GA m odule, ho we ver, is ob viously restrictive in its inability to deal w ith m ultiple objective functions an d constrained problem s. R esearch is nearing com pletion w ith in the N ew castle E ngineering D esign C entre, at the University of N ewc astle upon T yn e, on the use of a G A tool that enables m ultiple ob jectives and constraints to b e considered, an d it is intended that this tool w ill be incorporated for optim ization pu rp oses. R ather than producing an optim um design, th e m ulti-criteria G A produces a pareto-optim al set of designs.
3.2 D esign C oordination System
A design coordination system is currently being develop ed to facilitate the generation of rob ust designs w ithin a concurrent an d distributed com p uting environm ent. W ithin this system , a concurrent and distributed fram ew ork an d an agent com m unication architectu re are being developed [8]. T he aim of the design coordination system is to enable design algorithm s to be executed across a netw ork and a variety of com puter platform s. G iven the overall design requirem ents, a suite of design algorithm s will be available to enable som e com pu tation to be carried out. It is these design algorithm s w hich need to be coordinated.
generator, which selects th e design concepts to be evaluated, su pplies the concepts to the design coordination system . E valuated criteria produced b y th e design coordination system are collated and sub sequently used b y the robustness fram ew ork to construct response su rfaces.
4. Problem D escrip tion
For developm ent an d evaluation purposes, a case study was selected w ith w hich staff in the D epartm ent of M arine T echnology an d th e En gineering D esign C entre h ave undertaken considerab le wo rk. T he problem incorporates a single design algorithm w hich is capable of giving a num ber of m easurem ents for the seakeeping of a catam a-ran. T he objective of the w ork is to explore the design space for the catam aran and select a concept w hich is m ost rob ust w ith respect to selected seakeeping quantities at any particular w aveheading.
4.1 D esign For Seakeeping Strategies
For both m onohulls and m ultihulls, th ere is a need to develop strateg ies wh ich can clearly indicate how the designer m ay m odify the hullform geom etry so that im prove-m ents in th e vessels’ behaviour in waves can b e realized. T his prove-m ust be done in such a m anner th at other design considerations such as calm w ater resistan ce and intact stab ility are not com prom ised. H ow ever, no general rules exist to advise the designer how a hu llform m ay be m odi®ed to achieve speci®c im provem ents. T he existence of su ch rules w ould necessitate in-d epth `cause and effect’ understanding regarding changes in the hu llform ge om etry param eters of signi®cance and the various aspects of seakeeping w hich in¯uence the perform ance of a vessel in w aves, su ch as resultan t m otions an d accelerations at speci®c locations, deck we tn ess an d slam m ing. T here is then a need to provide su pport for th e designer at the conceptual design stag e to allow either m an ual or au tom atic searches fo r op tim al hullform s.
identi®cation of either prim ary, secondary or bo th groups of param eters. For catam aran hu llform s, this has b een based on the application of a G A [10].
E valuation of any selected m otion qu antity is dem anding at the early stages of design w hen ge om etry inform ation is not ®nalized and large num bers of alternatives are likely to be exam ined. T he norm al frequency dom ain approa ch is to provide solutions of m otion am plitudes, si, for m otion responses in a num ber of sine w aves of unit am plitude but differing frequency, w, for a particular w ave head ing relative to the vessel, Á, to provide a response am plitude operator (RA O ) for each m otion qu an tity desired, H(we,Á). T he rigid body m otions of particular interest are heave (i53), roll (i54) and pitch (i55). In conjunction w ith an encountered w ave spectrum represent-ing the irregu lar sea state of interest, Gw(we,Á), this allow s an approp riate response sp ectrum to be calculated,Gr(we,Á), from the area of w hich root m ean square (RM S ) values of response spectrum can be calculated.
Gr(we,Á)5([H(we,Á)]2´G
w(we,Á)) (2)
T o im p rove m otion characteristics, the need is then to m inim ize the area under the response spectrum . T o reduce the effort required, th e N ew castle approach to seeking im proved responses is to m inim ize the peak an d area of the R AO ,H(we,Á), rather than the area of the response spectrum , Gr(we,Á), directly, although the effect is the sam e and w ill provide im provem ent for any selected sea state w hile also reducing the level of calculation required fo r evaluating each alternative.
In order to provide the necessary RA O evaluations, the strip theory of Salvesen et al. [11] is utilized. T he explicit calculation of the necessary reactive hydrodynam ic coef®cients an d excitation forces fo r each alternative h ullform to p rovide the necessary so lution for each RA O w ould nevertheless be com plex an d dem an ding in term s of com putational effort and inapprop riate at the earliest stag es of design. In order to allow com plex evaluation of the alternative hullform s, but with considerably reduced effort, uniquely, the required hydrodynam ic data has b een pre-calculated for a series of generalized tw o-dim ensional ship sections. T he coef® cients approp riate to each alterna-tive hullform are then found by m app ing these stored solutions to the sections under investigation, w hich are de®ned from a three-dim ensional m odel of the particular underw ater su rface of the hull geom etry being investigated. T hese tw o-dim ensional values are then integrated over length to provide global coef® cients. T h is approa ch allow s the necessary com plex evaluation to be achieved for each alternative hu llform w ith a high degree of accuracy in the order of one-thirtieth of the tim e to undertake the explicit calculation of th e hydrodynam ic coef®cients [9]. Th is ability to provide a fast accurate design algorithm becom es even m ore im po rtant fo r catam aran design. For m onohulls, im proved m otions in head seas are accepted to give im provem ent at oth er w ave headings, bu t for catam arans, the sensitivities to hullform geom etry are m ore com plex an d the in¯uence on roll an d vertical m otions across oblique w ave headings needs to be explicitly considered. T he develop m ent of this app roach is describ ed by H earn et al. [10,12] along with its validation [13].
T his approach to evaluate the perform an ce of designs investigated is required because em p irically derived relationship s to relate changes in geom etry an d m otion characteristics are not satisfactory. T he cause an d effect relationships are particular to each ship type being investigated and show particular sensitivity in the case of catam a-ran design. H ow ever, previous experim ental studies as w ell as em pirical studies h ave validated this approach.
FIG. 2. D esign variables fo r a catam aran.
hu llform changes for heave, pitch and roll independently [10]. Th e relative m otion as a fu nction of all three m otions at a w aveheading of 135
°
was one qu an tity investigated by H earn et al. [12] in order to reduce th e problem atic catam aran phenom ena of `corkscrew ing’, w here pitch and roll m otions com bine. M ore recently, rather th an just including resistan ce perform an ce as a constraint, it w as included in a num ber of m ultiple criteria ob jective functions based on resistan ce an d m otion qu antities in H earn and W right [14]. T he application of a T aguchi-based approa ch allow ed the m otions of heave, pitch, roll an d relative m otion, as a fu nction of all th ree constraints to be assessed over a num ber of w aveheadings in order to identify a robust optim al form [15]. H ere, this approa ch is developed fu rther w ith th e application of the robustness fram e-w ork p resented in this paper.4.2 C atam aran D esign Space
In keeping with earlier w ork on seak eep ing fo r design, six controllable design variables w ere used to de®ne a basic catam aran design concept.
·
Prim ary design variables: H ull length , L;Breadth to draught ratio, B/T;
D istance betw een dem ihull centres, Hs;
·
Secondary design variables: Lo ngitudinal centre of b uoyancy, L C B;C oef®cient of waterplane, Cw p,
Lo ngitudinal centre of ¯oatation, L C F,
·
N oise variable: W aveh eading, Á.T he seakeeping qu an tities selected here to be m inim ized are th e p eak values of the RAO s associated with heave, roll, pitch an d the relative m otion at the bo w of each dem ihull, as a fu nction of all three m otion qu an tities com bined relative to the free su rface elevation,zx,y, at the bow located at (x,y).
sr5s31y´s42x´s52zx,y (3)
C riteria: m axim um heave am plitude,us3um ax; m axim um roll am plitude,us4um ax; m axim um pitch am plitude,us5um ax; m axim um relative bo w m otion, (RBM ),usrum ax. A diagram m atic representation of the design variab les and criteria can be seen in Fig. 2.
TAB LE I. D esign space for catam aran problem
V ariable T ype Parent Lower L im it M id Point Upper Lim it Sym bol
Á N oise 90 135 180 x1
L Control 104.0 m 29% 11% 111% x2
B/T Control 2.0 29% 11% 111% x3
Hs Control 31.0 m 29% 11% 11% x4
L CB Control 45.408 20.9% 10.1% 11.1% x5
Cwp Control 0.758 20.9% 10.1% 11.1% x6
L CF Control 43.306 20.9% 10.1% 11.1% x7
RBM Response ± ± ± Y
problem relatively sim ple, th e ob jective fu nction used w as the relative bo w m otion and w as constrained only by th e upper and low er bounds of the design space. T his is justi®ed as other investigations of the sam e hullform h ave dem onstrated equal or better resistan ce perform an ce fo r changes m ade to im p rove these m otion qu antities. T he robust design m ethodology, ho w ever, allow s fo r m ultiple objective fu nctions and m ultiple constraints.
T he design space w as explored relative to a parent design an d w as expressed as a percentage change for th e prim ary and secondary variables an d in absolute term s for the noise variable as shown in T able I.
4.3 E xperim ental R esults
A reso lution three fractional factorial design w as used for a screening run to establish the relative im po rtance of each of the design variables. T he follow ing regression equation w as ob tained using eight experim ental runs of the design algorithm .
Y50.9620.0003´x120.28´x220.3´x320.024´x420.22´x526.18´x615.93´x7 (4)
T he up per lim it fo r each variab le w as used to calculate the signi®cance with respect to the relative b ow m otion for each variable and can be seen in absolute term s in Fig. 3. It is ap parent from Fig. 3 th at the relative bow m otion obtained using a ®rst-order analysis is dom inated by L, B/T, Cw p an d L C F; h ow ever, Hs an d L C B w ere not considered to be su f®ciently sm all to rem ove th em from the an alysis.
FIG. 4. R epresentation of design po ints for central com posite design.
A num ber of different second-o rder experim ental plans we re created and tested using the D O E m odule including fu ll factorial, fractional factorial and central com -posite design (C C D ). T he C CD w as created using both full factorial an d reso lution ®ve fractional factorial designs for the ®rst-order design. It was discovered that the C C D produced the m ost accurate representation of the design concept w ith the fractional factorial C C D producing th e regression equation in 47 experim ental runs, as opposed to 2187 exp erim ental runs required fo r the fu ll factorial design. T he type of C C D used in this an alysis is know n as the face-centred cube and can b e seen in Fig. 4. T he experim ental design consists of a ®rst-order fractional or fu ll factorial design au gm ented w ith star an d centre points. T he star point distance, a, is given a value to allow the designer to achieve certain design properties. In this instance, ais given a value such that the factorial part de®nes the region contained w ithin th e design space. For com parison, Fig. 4 also sh ow s a C CD having a star po int distance greater than the factorial design space. T he experim ental plan w as then used with the design algorithm to ob tain a series of design concepts. T he design space w as subsequently explored in approx im ately 20 m in.
A set of norm al equ ations w ere produced using th e m ethod of least squares based up on the inform ation ge nerated from the second-order analysis. T hese norm al equ a-tions we re then so lved using Ch olesky L U factorization to p roduce the follow ing regression equation fo r the relative bow m otion as a function of the seven design variables obtained using the C C D .
Y53.646920.0786x111.1386x210.6558x320.0876x410.5590x512.5014x6 29.5733x710.0003x2120.0107x1x220.0073x1x310.0005x1x420.0051x1x5 20.0283x1x610.1020x1x720.0007x2210.0025x2x320.0002x2x420.0361x2x5 20.0094x2x610.0176x2x710.0070x2320.0006x3x420.0165x3x510.0018x3x6 20.0175x3x720.0013x
2
420.0041x4x520.0077x4x610.0013x4x720.0814x 2 5 20.0375x5x610.2128x5x710.0273x
2
610.7284x6x710.1844x 2
7 (5)
FIG. 5. RBM prim ary and secondary design charts.
in Fig. 5 for a w avehead ing of 135
°
. RBM w as plotted against hu ll length and breadth to draft ratio for the parentHs, as w ell as LC F an d LC B for the parentCw p. T he values of R BM are obtained w ith respect to the parent design, hence increasing negative values indicate increasing desirab ility. T he design algorithm w as used to ob tain results from the sam e design po ints for com parison. T he designer could m an ually select an optim al com bination of hu llform param eters based on the optim al individual com binations of prim ary an d secondary param eters indicated from the prim ary an d secondary design charts. T his su perp osition of the op tim al so lution for each group has been found previously to p rovide an indication of the global optim um , although not necessarily the optim al value, fo r each param eter.Based on this approa ch, the design charts w ould indicate that the optim um design for reducing the R BM for the parent values of Hsan d Cw p, at a w avehead ing of 135
°
, w ould be to m axim izeLandB/T, and m ove L C F fo rw ard and L C B aft, although L C B w as again show n to be of little signi®cance w ith resp ect to R BM .T he G A w as then used to search the design space fo r the optim um design. Since the G A w as only cap ab le of optim izing a single objective fu nction, the objective fu nction used com prised the su m m ation of the RBM at seven w aveheadings at 15
°
intervals. T he GA w as used initially to optim ize th e regression equations using the design space given in T able I. T he optim um design w as obtained in approx im ately 5 m in. For com parison, the optim ization w as rep eated using the design algorithm instead of the regression equation, again using the design space de®ned in T able I. T his optim ization process was com p leted in approx im ately 8 h. Finally, a m ore traditional T aguchi-typ e approa ch w as used to determ ine th e optim um design based upon the prim ary param e-ters, using the sam e m eth odology as that used by Senet al. [15]. T h e results from these optim ization processes can be seen in T able II.For these optim um designs, the design algorithm w as th en used to determ ine the RBM across the range of waveheadings (Fig. 6). It is apparent that the optim um design obtained using the regression equation produces a greater reduction in RBM across the w aveheadings than that using bo th the design algorithm an d the m ethod chosen b y S en
TA BLEII. D esign variable values selected by G A
M ethod d (L)% d(B/T)% d(Hs)% d(L CB)%L d(Cwp)% d(L CF)%L
Regression equation 7.69 10.70 10.50 0.85 8aft 1.10 0.894forward
Sim ulation tool 10.9 9.73 11.0 0.25forward 0.893 0.827forward
FIG. 6. V ariation in R BM w ith w aveheading.
et al. [15]. It is also clear that signi®cant im provem ents w ere obtained w ith an average reduction in R BM of approx im ately 18% across all waveheadings, as opp osed to a 12% reduction in RBM achieved b y S en et al. It was apparent that neither the optim um derived using the regression equation, the design algorithm or that found using the T aguchi approa ch, reduced th e RBM at w aveheadings around 90
°
.T he G A w as then used to identify th e optim um design at 90
°
using bo th the regression equ ation and design algorithm . Sim ilar results w ere obtained using th e two m ethods and indicated that the optim um design fo r a waveheading of 90°
lies in a com pletely different region of the design space than th at located fo r the range of w aveheadings.Finally, the an alysis w as repeated using h eave, pitch and roll separately as the objective fu nctions. T h e op tim um designs w ere again com pared w ith results obtained using the T aguchi m ethodology. T h e design algorithm w as again used to obtain values for heave, pitch and roll for the optim um designs obtained using bo th m ethodologies. T hese results can b e seen in Figs. 7±9, having average reductions in the peak am plitudes of the heave, pitch an d roll RAO s of 33% , 29% an d 25% respectively using the proposed m ethodology. T his com pares favourably w ith the reductions of 20% , 23% and 23% for heave pitch an d roll obtained using the T aguchi m ethodology of S enet al. [15].
5. D iscus sion of Results
FIG. 7. V ariation in heave with waveheading.
fu rther bene®t in reducing RBM fo r a w aveheading of 135
°
. For the secondary param eter design chart in Fig. 5, the conclusion of m oving L C F forw ard an d L C B aft, su bject to the parent Cw p, wo uld dem onstrate further im provem ent if Cw p w ere in-creased.T he approa ch ad opted has allow ed signi®cant im provem ent to be achieved in RBM across a w ide range of waveheadings, as dem onstrated by Fig. 6. T he lack of im prove-m ent for w avehead ings approa ching 90
°
is explained by the dom inance of th e roll com ponent over the vertical com p onent m otions of heave an d pitch. Param eter chan ges of bene®t to vertical m otions an d roll tend to con¯ict an d occupy different portions of the search sp ace, particularly w ith resp ect to B/T an d Cw p, w hich does not aid the identi®cation of a com prom ise so lution.T he prim ary param eter changes suggested to bene®t RBM are consistent fo r the
FIG. 9. V ariation in roll w ith w aveheading.
three m ethods investigated, as we ll as the best com p rom ising solution fo und previously [15]. T he secondary param eter changes suggested bo th by the regression equation and the design algorithm directly are also consistent w ith the exception of the LC B po sition indicated. T he secondary change s given by the regression equ ation based m ethod are recognizable from earlier w ork and are particularly bene®cial in reducing vertical m otions in head seas [9,10], w hereas the changes found by direct application of the design algorithm differ w ith respect to LC B. T he optim um found using th e design algorithm w ith L C B m oving forward of the parent p osition is com p arable to the com prom ise solution identi®ed by S enet al. [15] and earlier wo rk w here such a change w as found to bene®t pitch. T he optim um identi®ed from the regression equation w ould be preferred because of this consistency w ith earlier ®ndings in reducing vertical m otions, particularly in head seas w here lim iting responses su ch as slam m ing due to excessive relative m otion at the bo w result in loss of perform ance in w aves due to the necessity to involuntarily reduce speed.
Although there is a relative insensitivity to LC B over the design space investigated, the difference in L C B po sition indicated from the application of the regression- and sim ulation-based optim ization m ust be the m ajor factor in their relative perform ance as optim a in com parison to each other. T he fu rther im provem ent found by m oving it aft, as indicated by the regression-based optim um , also substa ntiates the previous p refer-ence fo r this optim a. H ow ever, if other considerations dependent on LC B, such as the trim of the vessel, becom e im po rtant, the results also dem onstrate that it could be m oved fo rward without incurring to o large a p enalty. T he difference in LC B selection m ight be explained by the regression equation; in this case, providing a m ore w ell de®ned op tim a in this region than w ould be dem onstrated by the design algorithm directly, w ith th e resu lt that w hen optim izing w ith the design algorith m explicitly, th is optim um is not identi®ed.
consequence of these m odi®cations to the prim ary an d secondary p aram eters can be rationalized by considering the individual geom etry of the hu ll sections, the in¯uence on the hydrodynam ic coef®cients and the overall consequence on the governing equations of m otion and hence, the resulting change to the R AO s.
T o p rovide further com parison w ith the Senet al. ap proach, the peak heave, roll and pitch we re also selected as individual objective fu nctions an d im proved solutions w ere again fo und over th e entire range of waveheadings considered.
T he results obtained indicate that signi®cant im provem ents in all asp ects of per-form an ce of the catam aran can be ob tained in considerably less tim e th an with oth er optim ization m ethods. T he entire process of concept exploration an d optim ization using this m ethodology took approx im ately 30 m in, as opposed to 8 h using the design algorithm only. Execution tim es w ere obtained using a Sun U ltraS parc platform . T he design algorithm is con®gured to allow oth er optim ization m ethods to be used, wh ich w ould reduce the tim e tak en to obtain th e optim um . W hile th e m ore traditional T aguchi approa ch achieved reductions in tim e to optim ize com pared w ith using the design algorithm , the approa ch was neither as fast as the proposed m ethodology nor did it produce as signi®cant an im provem ent in p erform an ce. T he reason for this w as due to th e differences in the m ethods of selection of the optim um designs. T he T aguchi approa ch of S enet al. explored the design space at discrete points using a fu ll factorial experim ental plan an d selected the optim um design as the po int w hich achieved the m ost desirable SN ratio. T he proposed m ethodology explored the design space in a sim ilar m anner using a m ore sop histicated and ef® cient experim ental plan to achieve a continuous representation of the design space w hich w as then used fo r optim ization pu rposes. G iven this continuous representation, po ints an ywh ere w ithin the design sp ace can be selected and checked for optim ality rather than the discrete po ints of the exploration stage.
6. Conclu sions
T he approach to robust design described in this paper has been show n to be ef®cient and effective w hen applied to a design prob lem in w hich the design m odel is com plex and so lutions com p utationally tim e consu m ing. Su ch m odels are com m on in th e M T O ®eld an d further wo rk is currently b eing carried out to determ ine the range of applicability of the proposed robust design m ethodology.
E xperience indicates that further im provem ent in ef®ciency can be achieved by incorporating a m ulti-criteria approa ch including an approp riate G A an d design selec-tion technique. T his approa ch is currently being investigated an d early results indicate that signi®cant im provem ents can be achieved.
Additional advantages of integrating the m ulticriteria G A and decision-m aking tools w ithin this fram ewo rk are that the pareto-optim al set w ill consist of designs w hich h ave each objective fu nction and constraint expressed in term s of both the m ean and variance. T he designer will then have th e ability to trade-off designs w hich h ave particular aspects of their perform ance that are on a ¯at region of the response su rface to designs w hose perform an ce is on a m ore peaked region.
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