Intuitive Visualization of Pareto Frontier for Multi- Objective Optimization in n-dimensional Performance Space

Transcription

1 A Ituitive Visualizatio of Pareto Frotier for Multi- Objective Optimizatio i -Dimesioal Performace Space G. Agrawal *, K. Lewis, K. Chugh, C.-H. Huag **, S. Parashar *, C. L. Bloebaum Departmet of Mechaical ad Aerospace Egieerig (MAE) ad New York State Ceter for Egieerig Desig ad Idustrial Iovatio (NYSCEDII) Uiversity at Buffalo Buffalo, New York Abstract A visualizatio methodology is preseted i which a Pareto Frotier ca be visualized i a ituitive ad straightforward maer for a -dimesioal performace space. Based o this visualizatio, it is possible to quickly idetify good regios of the performace ad optimal desig spaces for a multi-objective optimizatio applicatio, regardless of space complexity. Visualizig Pareto solutios for more tha three objectives has log bee a sigificat challege to the multi-objective optimizatio commuity. The Hyper-space Diagoal Coutig (HSDC) method described here eables the lossless visualizatio to be implemeted. The proposed method requires o dimesio fixig. I this paper, we demostrate the usefuless of visualizig -f space (i.e. for more tha three objective fuctios i a multiobjective optimizatio problem). The visualizatio is show to aid i the fial decisio of what potetial optimal desig poit should be chose amogst all possible Pareto solutios. I. Itroductio lmost all real world egieerig desig problems are characterized by the presece of several coflictig ad/or cooperatig objectives, as opposed to havig a sigle objective. The term Multi-objective Optimizatio Problems (MOP) is used to broadly classify problems with more tha oe objective. A geeral multi-objective optimizatio problem ca be T mi F( x) ( f( x), f( x),... fk ( x) ) expressed by the equatios to the right where, k is the umber of where x S objective fuctios, is the umber of desig variables, l is the () T umber of costraits, ad S R is the solutio or desig x ( x, x, x,... x ) space. s. t. G( x) ( g ( x), g ( x),... g ( ) ) T l x Typically, as a solutio strategy, multi-objective problems are aggregated to form a sigle objective fuctio leadig to a sigle solutio for the aggregated fuctio. The solutio foud usig this approach is strogly depedet o the way the objectives have bee aggregated. A rather practical approach to deal with multi-objective problems is to fid a set of solutios, (called a Pareto set show as the Pareto optimal frotier i Figure ), istead of fidig a sigle aggregated objective-depedet global optimum. Vilfredo Pareto first itroduced the cocept of Pareto optima i the 9 th cetury. A solutio is said to be Pareto-optimal if it is ot domiated by ay other feasible solutio (i.e. there exists o other solutio that is better for at least oe objective fuctio value, ad equal or superior with respect to the other objective fuctios values). Thus, all of the Pareto solutios are equally importat ad all are the global optimal solutios. * Research Assistat, Studet Member AIAA ** Post Doctorate Fellow, Member AIAA Research Associate, Member AIAA Associate Professor, Member AIAA Professor, Associate Fellow AIAA Desig Space S Figure. Mappig from desig space to performace space F Performace Space Pareto Frotier F

2 The cocept of Pareto optimality has bee widely used i idustry to aid desigers i their decisio-makig processes. The decisio-maker articulates his preferece pertaiig to the differet objectives oce he has kowledge of the Pareto frotier. The approach of visualizig the Pareto frotier has bee widely used i decisiomakig for two ad three objective problems. The Pareto frotier for two or three objective fuctios ca be readily visualized usig traditioal -D ad -D graphical meas. However, whe the problem size is large (i.e. for more tha three objective problems), there is o easy or ituitive method to visually represet the Pareto frotier. I this paper, we preset a ew approach that eables ituitive visualizatio of the Pareto frotier for -objective problems. The method uses a ew techique of lossless dimesio bledig, termed the Hyper-Space Diagoal Coutig (HSDC) method for multidimesioal visualizatio. II. Backgroud First, a brief discussio of multidimesioal visualizatio methods is preseted, followed by a discussio of the types of these methods presetly used i visualizig solutios for multi-objective optimizatio. These discussios will provide a appropriate groudwork for uderstadig the Hyper-Space Diagoal Coutig (HSDC) method preseted here, together with its applicatio i visualizatio of the -f Pareto Frotier. A. Multidimesioal Visualizatio Techiques Visualizatio is a icredibly powerful mechaism for providig isight ito complex pheomea. By icorporatig the ability to iteract i real-time with resultig displays, a scietist ca achieve a rapid ad meaigful exploratio of a problem space or dataset. The goal of Multidimesioal Multivariate Visualizatio (MDMV) is to meaigfully traslate large amouts of multidimesioal data ito ituitive visual represetatios. Numerous methods ad applicatios have bee developed to achieve this goal, subject to hardware ad software limitatios of -D or -D visualizatio space. All multidimesioal visualizatio techiques attempt to trasform a multidimesioal problem or dataset so that it ca be mapped to a -D or -D visual space. A stadard approach that is used i multidimesioal visualizatio is to first process the multidimesioal data i such a way so as to reduce the dimesioality, while still preservig the itegrity of its meaig. This is geerally a particularly importat step for visualizatio, as the associated complexity of the problem or dataset will be sigificatly reduced. Oe of the most comprehesive resources for discussio of dimesio reductio techiques ca be foud i Carreira-Perpia. Most multidimesioal visualizatio techiques that deped o dimesio reductio i some way result i a loss of meaig, a loss of the cocept of a eighborhood, ad a associated loss of a ability to uderstad the represetatio i a ituitive way. Categorizig MDMV techiques is difficult for a variety of reasos. Possible criteria for such a categorizatio iclude the goal of visualizatio, the type ad/or dimesioality of the data, the use of color, the use of aimatio, ad the dimesioality of the visualizatio techique, amogst may other possibilities. We will use the three broad categories defied i referece [] to provide a overview of the best-kow methods. These categories are summarized as follows.. Techiques based o -variate displays as well as multiple views of -variate displays. This icludes fitted curves, referece grids, ad bakig.. Techiques based o multivariate displays. Here, color, symbols, ad matrices of views are ofte used as a key to the represetatio. Oe of the best-kow methods is the scatter plot matrix, where dimesios are projected oto *(-) scatter plots, i which each pair of dimesios has two scatterplots showig their relatio. HyperSlice 4 ad HyperBox 5 ca be cosidered extesios of scatterplot matrices, i which color ad iteractivity provide for greater isight ito the problem. Aother is Cheroff Faces 6 ad, more broadly, the use of glyphs 7,8, to represet characteristics of relatioships ad the space i questio. Hierarchical axis 9-, Dimesio Stackig, ad World withi Worlds, 4 all use the cocept of a hierarchical represetatio i some way for the dimesioal relatioships. Parallel Coordiates 5-7 has become a extesively used approach i a variety of fields ad applicatios, icludig for Multi-objective optimizatio. I this approach, a poit i N- dimesioal space is equivalet to a polylie through N parallel coordiates. Stump ad Simpso 8 use glyphs ad colors to represet multivariate data. These represet oly a small umber of the methods that have bee proposed for -dimesioal visualizatio, but are probably the most widely recogized.. Techiques usig time as a aimatio parameter. I this class of methods, time is geerally used as a fourth dimesio ad each set of data is applied to each step of a time series which is the represeted as a aimatio sequece.

3 Each of the techiques developed uder the three broad categories listed above have advatages ad limitatios. Some of them are somewhat difficult to uderstad, some of them are computatioally expesive, ad some are just awkward. A few of the more widely used techiques are show i Figure. We will attempt to demostrate the great potetials of visualizatio based o our HSDC method i this paper. This paper focuses o the applicatio of HSDC method to multi-objective optimizatio, i order to demostrate its usefuless i visualizig -f space (i.e. for more tha three objective fuctios i a multiobjective optimizatio problem). I the ext sectio, some of the leadig approaches for visualizig solutios for MOPs are preseted. Scatterplot Matrix Cheroff Faces 6 Parallel Aimated Mesh Coordiates 7 Figure. Represetative visualizatios associated with key MDMV visualizatios. B. Multidimesioal Visualizatio i Multiobjective Optimizatio Multi-Objective Optimizatio Problems (MOPs), i cotrast to sigle objective problems, ivolve a set of objectives that might be cooperative, competitive or have o relatioship. Typically, the case of competitive objectives is the most iterestig, sice the choice of a acceptable or best solutio depeds o the prefereces o, compromises betwee, ad trade-offs of the objective fuctios. Whe the objectives i a MOP are coflictig, the set of optimal solutios is kow as the Pareto set, wherei o oe solutio is superior to the others. The solutio validatio for multi-objective problems has cotiued to be a extremely difficult issue for all researchers. So far, the most practical way to validate MOP solutios are to use visualizatio to verify the obtaied results. Also, visualizatio allows desigers to iterpret results ad use desig iformatio i more meaigful ways. Sice visualizatio is recogized as a process that trasforms data ito visible objects, decisio-makers ca easily view ad aalyze a rich set of iformatio ad the use this to help make trade-offs so as to reduce the time ad cost of the desig process 8. Multidimesioal data visualizatio is a promisig research area for applicatio to MOPs. Beig able to visualize the Pareto frotier i a performace space with more tha three objective fuctios has bee a great challege to the optimizatio commuity. Parallel coordiates 5-7 has bee oe of the leadig approaches used i idustry to assist i ultimately choose leadig desig cadidates to explore further. Multivariate data is displayed usig a polylie that itersects equally spaced vertical axes. Each polylie represets oe desig possibility. The use of parallel coordiates to visualize the performace space still remais uwieldy for large umbers of fuctios. Cloud Visualizatio 9 provides a meas by which a desiger ca view all previously geerated desig iformatio i both the desig ad the performace spaces simultaeously. However, sice all spaces are displayed i separate widows that are liked, the fuctioality is compromised for large umbers of fuctios, sice it becomes tedious to work with too may widows. Mattso ad Messac 0 use a Pareto filter to reduce the Pareto frotiers from various disparate desig cocepts ito a sigle Pareto frotier termed the s-pareto frotier. Ufortuately, the use of approximatios i the s-pareto approach ultimately results i loss of dimesio represetatio. All these visualizatio techiques, which are essetially used as methods of solutio validatio, become cumbersome for very large problems. Hece, developig a appropriate multidimesioal visualizatio capability to facilitate the decisio-makig process for large multi-objective optimizatio problems is the mai focus of this work. III. Developmet of a Framework for Ituitive Multidimesioal Visualizatio (HSDC Method) I the previous sectios, a backgroud o multi-objective optimizatio, multidimesioal visualizatio techiques, dimesio reductio techiques, ad the applicatio of multidimesioal visualizatio to MOP was preseted to lay the groudwork for this research. Here, the Hyper-Space Diagoal Coutig (HSDC) method is developed, which eables a lossless mappig of dimesios usig a coutig strategy derived from complexity theory. Hece, complexity theory is briefly reviewed together with Cator s origial coutig method, which is critical for uderstadig the HSDC method developed i this work. The HSDC method ultimately eables a ituitive ad meaigful visualizatio capability for multidimesioal multi-objective optimizatio problems.

4 The goal of complexity theory is to uderstad what makes some problems algorithmically difficult to solve. Research pertaiig to coutig complexity classes has bee demostrated to be very promisig. Such research o coutig complexity classes has provided importat isights ito the iheret complexity of may atural computatioal problems. These classes essetially cosist of fuctios quatifyig the time or space required to eumerate them. Why is this importat to kow here? Because the cocept of coutig complexity classes origiated with the famous Germa mathematicia, Georg Ferdiad Ludwig Philipp Cator, who recogized that every poit o a surface has a correspodig poit o a lie, ad vice versa. The result of this observatio is that there is a oe to oe correspodece of poits o the iterval [0, ] ad poits i a -dimesioal space. It is this correspodece which provides the basis for the developmet of the Hyper-Space Diagoal Coutig method. Cator's discoveries i set theory rest upo a very simple idea that eve though you may ot be able to cout somethig, you ca equate it with somethig else that has the same cardiality (i.e. the same umber of elemets). Cator's isight was that eve though a ifiite set ca t be eumerated its cardiality ca be determied or at least equated to that of aother set. Two ifiite sets ca be show to have the same cardiality by fidig a oe-to-oe mappig betwee the elemets of each set. Cator s theorem for N N () mappig poits i a -dimesioal space o a lie ca be writte as i Eq.. The cardiality of N deotes ℵ 0 (the meta-letter alephzero or aleph-ull), which is equal to the cardiality of the itegers. N { ordered pairs of members of N} The very famous proof is represeted visually i Figure. Cosider a array that icludes all the ordered pairs where, N {,,, 4,...} of positive itegers ad the make a path through all the pairs i the spiral way show. I this way, every sigle elemet of set N ca be mapped to oe (ad oly oe) elemet of set N (i.e. the cardiality of set N is same as that of set N). Hece, for every poit o a -dimesioal surface there is a correspodig poit o a lie, which is uique. This cocept of coutig, which represets a very small portio of Cator s cotributios to the field of mathematics, is key to uderstadig the Hyper-Space Diagoal Coutig (HSDC) Method we propose here. The use of the HSDC eables us to achieve lossless dimesio bledig for multidimesioal visualizatio. We have exteded Cator s work o coutig to eable mappig poits i a -dimesioal space to a lie. Icredibly, this has ot bee doe, as others who use coutig i Complexity Theory have had o eed to move beyod two dimesios. This mappig ca be described by: Array of ordered pairs of positive itegers Path through all the pairs Figure. Graphic proof of Cator s theorem N N N N K where, N,,, 4,... N { } N 4 ℵ 0 () th { ordered combiatios of all the members of N} The proof for this is almost uecessary as Cator has already doe it for N ad what remais is a matter of makig a array of th ordered combiatios of all the members of N ad the creatig a path through all the poits to get a sequece. A extesio of Figure to show a mappig for three-dimesioal poits is show i Figure 4.I order to uderstad how this coutig will be geeralized, we have defied the followig terms that will be used throughout our method developmet sectio. The variable is used to deote the umber of dimesios, l is used to represet the level o which a particular poit falls (i.e. which loop out from the origi), E l is the umber of elemets (i.e. poits) o a particular level, C i is the coutig idex associated with elemet i (poit i), ad TE l are the total umber of elemets up to level l. Figure 4. Path for coutig i -d space There are five equatios which must be developed i order to be able to automate a coutig i ay multidimesioal space. We eed to kow the followig: ) the total umber of elemets at a particular level; ) the total elemets up to a particular level; ) the elemets at a particular level i terms of the elemets of the previous level; 4) the total umber of elemets up to a level i terms of the elemets i that level; ad 5) what coutig idices will correspod to a particular level. 4

5 The first four equatios ad the fifth formula have bee developed as is show i Equatios 4-8. k ( ) l + k k 0 ( )! (4) E l ( )! E TE l + l l + l E l (6) + l (7) El The fial formula provides more of a challege. It ca be expressed i terms of a iequality as: l TE l El (5) C i ( l + )! (8) ( l )! ( )! The fifth formula (Eq. 8) is a (-) th order polyomial iequality, which must be solved i costat time i order to automatically geerate ay combiatio of variables for ay size multidimesioal problem. Alteratively, a optimizatio routie or similar techique could be used i a preprocessig mode. This would ot be as desirable a approach, although it would certaily work. What this coutig ad idexig approach eables us to do is to map each ad every dimesio i a lossless way (i.e. o dimesio fixig) to a -D or -D graphical display. Because the coutig is doe i a outward spiralig maer, the cocept of a eighborhood is preserved with the HSDC, as is the cocept of goig from small to large as oe moves from left to right i the idexig. The details of how the HSDC method is applied to MOP is preseted i the ext sectio. IV. HSDC Applied to Visualizatio of Multidimesioal Performace Space i Multiobjective Optimizatio We first apply the HSDC approach to a two fuctio multi-objective optimizatio problem to demostrate that the resultig visualizatio of the Pareto frotier is what we are used to seeig for -D performace space. We will the exted the method to two differet three objective problems ad the a four objective problem. A. Two Objective Test Problem Miimize : Cosider a test problem with two objectives, four desig f + ( 7) ( 4 5) variables, ad two iequality costraits, i which the objectives are competitive ad to be miimized. The problem f 0 + ( 7) 5 4 statemet is show i Eq. 9. A typical -D visualizatio of (9) the Pareto frotier for this problem is a bi-objective plot as show i Fig. 5a or the parallel coordiate plot see i Fig. Subjected to : 5b. Both these visualizatios are icredibly useful, but they ( 5) + ( ) 75) have their ow limitatios. For the visualizatio of Figure 5, g 0 75 we ca plot the Pareto frotier for up to oly three objectives (if we move to a -D represetatio). Usig parallel ( 4 70) g 0 coordiates, we ca theoretically have as may objectives o 70 parallel axes as eeded. However, for large problems, we soo have so may of these plots that a ituitive Where : 0 i 0 iterpretatio becomes impossible. As we established i the previous sectios, the HSDC method ca easily map dimesios to a lie, without loss of dimesioal represetatio. I its applicatio to multi-objective optimizatio, we ca readily use the coutig scheme described to map ay umber of objectives to a sigle lie i order to geerate the visual represetatio of the Pareto frotier for ay umber of objectives. The procedure developed to eable visualizatio of the Pareto frotier for multidimesioal performace space is outlied ext. The coutig scheme is implemeted together with a simple but very sigificat biig techique i order to eable this meaigful visualizatio. For demostratio purposes, cosider the same bi-objective problem described previously. The steps for visualizig the Pareto frotier usig HSDC are outlied. 5

6 Step. Obtai Pareto poits usig ay appropriate optimizatio routie. Step. Idetify the miimum ad maximum values for each of the objectives to establish a rage. Divide these rages ito some fiite umber of compartmets, resultig i small bis alog each objective.(e.g. F rage divided ito five, resultig i five bis i which a poit may fall). Step. The idices of the bis created ca be plotted o a lie ad thus we ca have F bi idices o oe axis ad F bi idices o the other axis (e.g. for five rages associated with F, there will be five bis ad, hece, five idices through 5). For problems with more tha two objective fuctios, multiple fuctios are represeted o a sigle axis through the use of coutig i the HSDC method. Step 4. Each Pareto poit previously geerated i Step will fall uder some combiatio of these bis that are plotted usig their idices o the ad Y axes (e.g. if a Pareto poit has a F that falls ito the third rage of F ad a F that falls ito the fourth rage of F, this would correspod to idices (,4) ad would be placed i that particular bi). The poits are represeted as a uit cylider alog the vertical axis. Multiple solutios might fall at the same set of idices, resultig i a bi that cotais multiple Pareto poits. Figure 5a. Pareto frotier for the above problem usig the f, f performace space Figure 5b. Parallel coordiate plot of the performace space for the same problem Figure 6 shows the resultig -D represetatio of the Pareto frotier for the bi-objective problem usig the HSDC method. This provides a represetatio that is exactly what would be expected usig traditioal meas of plottig (i.e. Figure 5a). The oly differece betwee the two approaches is that the HSDC represetatio is actually plottig per idex, ot per fuctioal value. Hece, due to this discretizatio, some small chages i the visual represetatio might occur. What s importat to ote, however, is that the axes are actually eumeratig the idices. Hece, this is easily extesible to more tha two objective fuctios. For istace, i a four objective fuctio MOP, we ca idex two objectives per axis. For six objective fuctios, we would idex three objective fuctios per axis. The resultig plot, however, will still look very similar to the traditioal Pareto frotier show below. A slight variatio of this approach yields a extremely powerful capability. Rather tha ruig some umber of optimizatios i order to geerate Pareto poits, it is possible to sample poits i the origial desig space, evaluate the objectives at these poits, ad the idetify which of these are o-domiated. The, the same procedure outlied i Steps through 4 ca be performed. While ay type of approach could be used to sample the desig space, the HSDC method esures capture of all dimesios existig i the origial space. Figure 6. Pareto poits (bis) for two objective performace space usig HSDC method. For istace, cosider a case for the same bi-objective optimizatio problem we ve bee discussig, where a discretizatio grid of 0 is used alog all the dimesios (desig variables through 4 ). Objective fuctio values are evaluated at all the grid poits. All the feasible poits are compared to eumerate the o-domiated poits. The blue poits i Figures 7a ad 7b represet the o-domiated poits for the problem ad the gree poits show all the feasible poits. Figure 7a shows a more coarse discretizatio tha does Figure 7b, with the o-domiated blue poits fallig i lower idices bis. This is what we would ormally expect sice the o-domiated poits have low objective fuctio value for at least oe of the objectives. Figure 7b shows a more refied represetatio, with 50 bis alog 6

7 each objective. We ca readily see that o-domiated poits form a approximate Pareto curve by fallig towards the low objective valued bis. This represetatio is clearly aalogous to the ormal Pareto frotier represetatio from Figures 5a ad 6. The motivatio for usig this approximate approach would be for applicatios where obtaiig a full set of Pareto optimum poits is difficult or impractical. We ca see that the approximate Pareto frotier provides valuable iformatio to the desiger, suggestig where further refiemet ad ivestigatio might be warrated. Figure 7a. F bis o -axis, F bis o Y-axis ad occurrece of feasible ad o-domiated poits o Z-axis (Grid of 0 for F & F bis) Figure 7c shows the true Pareto poits (i black) for the problem alog side the o-domiated poits (i blue) obtaied usig the grid. Also, as we ca see from the previous figures that the larger the umber of divisios we have for our bis, the better the shape for the Pareto frotier. The divisio of the objectives for a optimum umber of bis remais a research issue. Aother issue pertais to the groupig of objectives for idexig. This is discussed further i subsequet sectios. Next, the HSDC approach for MOP Pareto frotier visualizatio is applied to a multi-objective problem with three objective fuctios. Figure 7b. F bis o -axis, F bis o Y-axis ad occurrece of feasible ad o-domiated poits o Z-axis (Grid of 50 for F & F bis) Figure 7c. F o -axis, F o Y-axis; Showig Pareto poits & No-domiated poits B. Three Objective Test Problem ( costraits) Miimize : ( ) ( + ) f This MOP has three objectives, three iequality + + costraits, side costraits, ad two desig variables. ( ) ( ) The problem statemet is give i Eq. 0. For this f problem, true Pareto poits were foud, together with results of a HSDC-based evaluatio to provide odomiated ad feasible poits (as explaied i the last 8 7 ( + 4) ( + ) f (0) sectio). Here, F ad F are idexed o oe axis while Subjected to : F is idexed o the secod. Figure shows feasible g (gree) ad o-domiated (blue) poits geerated g 0 through evaluatio at idex poits i the discretized g 0 desig space (usig HSDC), as well as true Pareto poits Where : 4, 4 (black) obtaied as a result of optimizatio rus. Figure shows that the HSDC-based idexed represetatio is quite ituitive, clearly showig the Pareto frotier. Further, usig the HSDC i the desig space allows for the evaluatio of objective fuctios with a subsequet idetificatio of o-domiated poits. These poits clearly approximate the behavior of the Pareto 7

8 frotier itself, without havig to actually perform ay formal optimizatio. Hece, for the purposes of idetifyig good regios for further detailed exploratio, this approximate Pareto frotier approach might be worthwhile. It should be oted that havig a odd umber of objectives ecessitates a slightly differet idexig strategy so that the resultig represetatio is still visually viable. This is a mior poit, which ca be see from the idices i the figures below. Aother poit to recall is that each poit i Figure 8a is actually a bi, which might cotai multiple poits (as show more clearly i Figure 8b). The tremedous advatage of usig the HSDC is that each poit withi a bi ca be liked back to its origial desig space values. Hece, a bi with multiple poits i the performace space allows the desiger to go back to the desig space values Figure 8a. F F o -axis, F o Y-axis, umber of poits o Z-axis (Grid of 0 for all three objectives) Figures 8c-8d show Pareto poits oly, with a discretizatio of 50 for each of the three objective fuctios. There are 49 Pareto poits for this particular problem, which are therefore beig distributed amogst 50x50x50 (5,000) bis. Oe ca see the slight differeces i the Pareto frotier betwee Figures 8a ad 8c, due to the differece i biig (7,000 versus 5,000). This is further demostrated i Figure 8e, i which each fuctio has 80 discretizatios (for a total of 5,000 possible bis ito which the Pareto poits might fall). However, the Pareto frotier is still easily ad ituitively recogized, eve for the 0 bi discretizatio. Figure 8b. F F o -axis, F o Y-axis, umber of poits o Z-axis (Grid of 0 for all three objectives), both o-domiated ad Pareto poits. Figure 8c. F F o -axis, F o Y-axis, umber of poits o Z-axis (Grid of 50 for all three objectives) Figure 8d. F F o -axis, F o Y-axis, umber of poits o Z-axis (Grid of 50 for all three objectives) Figure 8e. F F o -axis, F o Y-axis, umber of poits o Z-axis (Grid of 80 for all three objectives) 8

9 C. Three Objective Test Problem (oe costrait) Aother three objective MOP was ivestigated with oe costrait ad three desig variables. The problem formulatio is show i Eq. below. Agai, we see from Figs. 9a-c that a clear Pareto frotier is obtaied, eve while represetig this three objective problem i a - D graphical display. Recall that the third dimesio i the figures show correspods to the umber of Pareto poits fallig ito a particular bi. Also, we see from the three figures that a more refied discretizatio results i more bis with fewer poits fallig per bi. Miimize : f 5 Subjected to : g f 5 f 50 Where : ( ( + ) ) ( ( ) ( ( )) 0,, () Figure 9a. F F o -axis, F o Y- axis, umber of poits o Z-axis (Grid of 0 for all three objectives) Figure 9b. F F o -axis, F o Y- axis, umber of poits o Z-axis (Grid of 50 for all three objectives) Figure 9c. F F o -axis, F o Y- axis, umber of poits o Z-axis (Grid of 80 for all three objectives) D. Four Objective Test Problem Miimize : The four objective problem i this paper derives origially from ( a three objective test problem take from Veldhuize s dissertatio ) ( + ) f + +. Aother objective was added usig the same two desig variables. ( ) ( ) f + () The three iequality costraits for the test problem remaied the 75 7 same, as do the side costraits. The problem statemet is show i ( + 4) ( + ) Equatio to the right. f Figures 0a-b show two orietatios of a Pareto frotier for the ( + 9) ( + ) case where F ad F are idexed alog the x axis ad F ad F4 are f idexed alog the y axis, with a coarse 5 discretizatios for each of Subjected to : the objective fuctios. Figure 0c correspods to a fier g discretizatio of 40 alog each objective. These figures clearly show g 0 a Pareto frotier which has the exact same meaig as that for the g 0 previous bi-objective problem. As the grid is discretized to a fier Where : 4, 4 resolutio, we see that the greater umber of available bis for poits to fall i results i a smoother visual represetatio of the Pareto frotier. This type of visualizatio ca clearly provide a valuable meas for eablig desigers to determie which of the Pareto poits obtaied should be ivestigated further for a ultimate optimal solutio based o desiger prefereces. Figure 0a. F F o -axis, F F 4 o Y-axis, umber of poits o Z-axis (Grid of 5 for all four objectives) Figure 0b. F F o -axis, F F 4 o Y-axis, umber of poits o Z-axis (Grid of 5 for all four objectives) Figure 0c. F F o -axis, F F 4 o Y-axis, umber of poits o Z-axis (Grid of 40 for all four objectives) 9

10 Additioally, the approach of usig HSDC i the desig space to evaluate fuctios for approximate Pareto frotier visualizatio is also implemeted. A grid is created i the desig space, just as we did for the previous problem. Subsequetly the fuctio values for all the objectives are evaluated at each of the grid poits. The maximum ad miimum values for each objective are calculated to determie the rages. The rages obtaied are the divided to form the bis. It should be oted that while creatig bis for the various objectives we also associate idices with icreasig order of their values. The ext step is to use the HSDC method o the idexes of the objective bis to collapse the four objectives o to the horizotal plae, two each o the ad Y-axes. Each idex o the ad Y-axes would be associated with a combiatio the idices of the bis of two objectives usig the Hyper-Space Diagoal Coutig. I reality, we ca have as may objectives o oe axis as we eed by creatig a path through the idices of the bis created for each objective. The desig space pertaiig to each objective is show i Figure. The red poits are the ifeasible poits. Feasible poits are show i gree ad the blue poits are the o-domiated poits. The performace space plot for the four objective problems usig the HSDC method is show for a discretizatio of 0 i Fig. ad for a discretizatio of 5 i Figure. It ca be readily see from these figures that the odomiated poits form a approximate Pareto frotier alog the idices of the objective bis which have low objective fuctio values. The more the umber of bis alog the objectives, the better is the distributio of odomiated poits alog the idexed performace space. But it should be oted that there is a trade-off i havig a greater umber of bis, as we would eed to compare the values of the o-domiated poits with more ad more umbers of rages of these bis. This results i a associated computatioal cost which grows icredibly fast with more objective fuctios ad icreased discretizatios. Figure. ad o &Y-axis, ad Objective fuctios F, F, F, ad F4 o Z-axis Figure. F F o -axis, F F 4 o Y-axis, umber of poits o Z-axis (Grid of 0 for all four objectives) Figure. Two views of F F o -axis, F F 4 o Y- axis, umber of poits o Z-axis (Grid of 5 for all four objectives) V. Coclusios I this paper, we preseted a powerful ew approach for visualizig the Pareto frotier for problems with more tha three objectives. The results preseted here clearly demostrate that the HSDC method is a feasible ad usable approach for visualizig the Pareto frotier for MOPs with more tha three objectives. While the questio of scalability to larger MOPs is of issue, the method ca theoretically visualize -objectives meaigfully. The HSDCbased Pareto frotier visualizatio capability described here is particularly importat for large-scale multiobjective optimizatio problems. The method preseted overcomes the issues of awkwardess ad dimesio fixig so prevalet i the existig visualizatio strategies. The HSDC-based approach is a lossless represetatio, i which every sigle dimesio is represeted i a - or -D visual represetatio which is ituitive ad easy to uderstad. This method represets a sigificat breakthrough for visualizig the Pareto frotier for multi-objective optimizatio methods with more tha three objective fuctios. 0

11 VI. Ackowledgmets We gratefully ackowledge support from the New York State Ceter for Egieerig Desig ad Idustrial Iovatio (NYSCEDII). VII. Refereces. Pareto, V., Maual of Political Ecoomy. A. M. Kelley, New York, NY, 97.. Carreira-Perpia, M. A., 997, A Review of Dimesio Reductio Techiques, Techical Report CS 96-09, Departmet of Computer Sciece, Uiversity of Scheffield. ATKOSoft S.A., 997, Survey o Visualizatio Methods ad Software Tools, 4. Va Wijk, J. J., ad Va Liere, R., 99, HyperSlice - Visualizatio of Scalar Fuctios of May Variables, Proceedigs of IEEE Visualizatio Coferece, Sa Jose, CA, pp Alper, B., ad Carter, L., 99, The Hyperbox, Proceedigs of IEEE Visualizatio Coferece, Sa Diego, CA, pp Cheroff, H., 97, The Use of Faces to Represet Poits i k-dimesioal Space Graphically, Joural of America Statistical Associatio, Vol. 68, pp Beddow, J., 99, A Overview of Multidimesioal ioal Visualizatio: Elemets ad Methods, Desigig a Visualizatio Iterface for Multidimesioal Multivariate Data, IEEE Visualizatio 9 Tutorial 8, Bosto, MA. 8. Beddow, J., 990, Shape Codig of Multidimesioal Data o a Microcomputer Display, Proceedigs of IEEE Visualizatio Coferece, Sa Fracisco, CA, pp Mihalisi, T., Gawliski, E., Timli, J., ad Schwegler, J., 990, Visualizig a Scalar Field o a - Dimesioal Lattice, Proceedigs of IEEE Visualizatio Coferece, Sa Fracisco, CA, pp Mihalisi, T., Timli, J., ad Schwegler, J., 99, Visualizatio ad Aalysis of Multivariate Data: A Techique for All Fields, Proceedigs of IEEE Visualizatio Coferece, Sa Diego, CA, pp Mihalisi, T., Timli, J., ad Schwegler, J., 99, Visualizig Multivariate Fuctios, Data, ad Distributios, IEEE Joural of Computer Graphics ad Applicatios, Vol., No., pp LeBlac, J., Ward, M. O., ad Wittels, N., 990, Explorig -Dimesioal Databases, Proceedigs of IEEE Visualizatio Coferece, Sa Fracisco, CA, pp Feier, S., ad Beshers, C., 990, Worlds withi Worlds: Metaphors for Explorig -Dimesioal Virtual Worlds, Proceedigs of rd Aual ACM SIGGRAPH Symposium o User Iterface Software ad Techology, Sowbird, UT, pp Feier, S., ad Beshers, C., 990, Visualizig -Dimesioal Virtual Worlds with -Visio, Computer Graphics, ACM SIGGRAPH, Vol. 4, No., pp Iselberg, A., 997, Parallel Coordiates for Visualizig Multidimesioal Geometry, Proceedigs of New Techiques ad Techologies for Statistics, pp Iselberg, A., ad Dimsdale, B., 990, Parallel Coordiates: A Tool for Visualizig Multidimesioal Geometry, Proceedigs of IEEE Visualizatio Coferece, Sa Fracisco, CA, pp Wegma, E. J., 990, Hyper-dimesioal Data Aalysis Usig Parallel Coordiates, Joural of the America Statistical Associatio, Vol. 85, No. 4, pp Stump, G. M., Yukish, M., Simpso, T. W., ad Beett, L., 00, Multidimesioal Visualizatio ad Its Applicatio to a Desig by Shoppig Paradigm, 9th AIAA/USAF/NASA/ISSMO Symposium o Multidiscipliary Aalysis ad Optimizatio, Atlata, GA, AIAA Eddy, J. P. ad Lewis, K., 00, Visualizatio of Multi-Dimesioal Desig ad Optimizatio Data Usig Cloud Visualizatio, ASME Desig Techical Cofereces, Desig Automatio Coferece, DETC00/DAC Mattso, C. A. ad Messac, A. 00, Cocept selectio usig s-pareto frotiers. AIAA Joural, 4(6): Vadhuize, D. A. V., Multiobjective Evolutioary Algorithms: Classificatios, Aalyses ad New Iovatios, A Dissertatio Submitted to the Graduate School of Egieerig, Air Force Istitute of Techology.