Abstract. I. Introduction

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1 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conerence 8 - April 005, Austin, Texas AIAA M Intuitive Design Selection Using Visualized n-dimensional Pareto Frontier G. Agrawal *, C. L. Bloebaum, K. Lewis Department o Mechanical and Aerospace Engineering (MAE) and New York State Center or Engineering Design and Industrial Innovation (NYSCEDII) University at Bualo Bualo, New York Abstract A visualization methodology is presented in which a Pareto Frontier can be visualized in an intuitive and straightorward manner or an n-dimensional perormance space. An approach or preerence incorporation is presented that enables a designer to quickly identiy good points and regions o the perormance spaces or a multi-objective optimization application, regardless o space complexity, numbers o objectives, or numbers o Pareto points. Visualizing Pareto solutions or more than three objectives has long been a signiicant challenge to the multi-objective optimization community. The Hyper-space Diagonal Counting (HSDC) method described here enables the lossless visualization to be implemented to achieve a hyperspace Pareto rontier. In this paper, we demonstrate the incredible power o using the hyperspace Pareto rontier as a visualization tool or design concept selection in a multiobjective optimization environment. I. Introduction ultiobjective Optimization Problems (MOPs), in contrast to single objective problems, involve a set o objectives that might be cooperative, competitive or have no relationship. Typically, the case o competitive objectives is the most interesting, since the choice o an acceptable or best solution depends on the preerences on, compromises between, and trade-os o the objective unctions. Most engineering design problems can be categorized as being MOPs. When the objectives in an MOP are conlicting, the set o optimal solutions is known as the Pareto set, wherein no one solution is superior to the others. The region deined by the Pareto optimum is called the Pareto rontier (Figure). The concept o Pareto optima was irst introduced by Vilredo Pareto in the 9th century. Pareto optimality has been widely used in industry to aid designers in their decision-making processes. The decision-maker articulates his preerence pertaining to the dierent objectives once he has knowledge o the Pareto rontier. The approach o visualizing the Pareto rontier has been widely used in decision-making or two and three objective problems, since it can be readily visualized using traditional -D and 3-D graphical means. However, when the problem size is large (i.e. or more than three objective problems), the solution validation or MOPs becomes an extremely diicult issue, and there is no easy or intuitive method to visually represent the Pareto rontier. One standard approach is to use the parallel co-ordinate s representation, wherein a polyline is drawn or each set o Pareto points. Currently, General Motors, among other companies, use a program called Parallax, which uses parallel coordinates 3, 4 to represent multiple objectives. A sample representation rom a vehicle design problem is shown in Figure. Parallel coordinates has been one o the leading Design Space Perormance Space approaches used in industry to assist in ultimately choose X F leading design candidates to explore urther, but remains Pareto Frontier unwieldy or large numbers o unctions. As is seen in Figure, trying to plot only 4 objectives using even a small number o solutions becomes rather challenging. S * Research Assistant, Student Member AIAA Associate Proessor, Member AIAA Proessor, Associate Fellow AIAA X Figure. Mapping rom design space to perormance space F Copyright 005 by Agrawal, Bloebaum and Lewis. Published by the, Inc., with permission.

2 It is quite diicult to make any intuitive or eective conclusions regarding the perormance space and inherent tradeos. Being able to represent the multidimensional perormance space in an intuitive way would be o signiicant value to many companies and researchers. Other approaches include Cloud Visualization 5, which provides a means by which a designer can view all previously generated design inormation in both the design and the perormance spaces simultaneously. However, since all spaces are displayed in separate windows that are linked, the unctionality is compromised or large numbers o unctions, since it becomes tedious to work with too many windows. Mattson and Messac use a Pareto ilter to reduce the Pareto rontiers rom various disparate design concepts into a single Pareto rontier termed the s-pareto rontie 6. Figure. Parallel Axis or MOP Unortunately, the use o approximations in the s-pareto approach ultimately results in loss o dimension representation. In this paper, we present a new method or lossless dimension blending to enable development o an intuitive visualization capability or representing the Pareto Frontier or multidimensional perormance space in multiobjective optimization applications. The method proposed is termed the Hyper-Space Diagonal Counting (HSDC) Method 7 or Multidimensional Visualization. We will demonstrate the power o the HSDC approach, and will outline the means by which preerences can be incorporated into the visual representation to enable designers to intuitively choose the best Pareto points or concept development even in the presence o 0 or more objective unctions. We will also demonstrate that a preprocessing step to determine correlation coeicients or all objective unctions will provide a guide or grouping objective unctions in the correct way or visualization purposes. II. HSDC Methodology Development or Hyperspace Pareto Frontier As we established in a previous paper 7, the HSDC method can easily map n dimensions to a line, without loss o dimensional representation. In its application to multi-objective optimization, we can readily use the counting scheme described to map any number o objectives to a single line (i.e. axis) with any other number o unctions mapped to another line (i.e. a dierent axis) in order to generate the visual representation o the Pareto rontier or a multiobjective optimization problem. The counting scheme is implemented together with a simple but very signiicant binning technique in order to enable this meaningul visualization. The steps or visualizing the Pareto rontier using the HSDC approach are as ollows. Step. Obtain Pareto points using any appropriate optimization routine. Step. Identiy the minimum and maximum values or each o the objectives to establish a range. Divide these ranges into some inite number o compartments, resulting in small bins along each objective. Group objective unctions into two sets and count each set, producing indices or each. Step3. The indices o the bins created are plotted on two axes, where multiple unctions are represented on each single axis through the use o counting in the HSDC method. Note that this will result in a -D histogram that represents all dimensions in the perormance space. Step4. Determine what combination o indices correspond to which bins that contain each o the Pareto points. Table. Pareto Points or our objective MOP with HSDC Indices Pareto HSDC Indices 3 4 Points 34 A B C D E F G H I J Max Min Range Bin Size

3 The points are represented as a unit cylinder along the vertical axis. Multiple solutions might all at the same set o indices, resulting in a bin that might contain multiple Pareto points. We will use a our objective unction problem to demonstrate the HSDC-based binning approach. Assume that there are 0 Pareto Points (A-J) or this problem, organized by increasing order o. Table lists the ten Pareto points, with associated objective unction values and HSDC indices or each. Each objective has a dierent overall range (i.e. [Max Min]) resulting in dierent bin sizes or each (i.e. [Max Min]/Bins). Table shows the ranges associated with each bin or the our objectives. Figures 3 and 4 show how counting is perormed or the grouping and or the 34 grouping, where each is discretized into the 5 bins as shown in Table. For this problem, we obtain a set o Bin Index Table. Indexing or each range range range 3 range 4 range indices or and another or 34. Point F, or instance, has an that alls into the 4 th bin on that axis and alls into its nd bin. This corresponds to an index in the counted HSDC space o (Figure 3). From Figure 4, we see that 3 alls into the st bin and 4 alls into the nd bin, which results in an overall index o 3 in the 34 space. The same is done or all points, with indices as shown in Table. Using this set o indices that correspond to the unctions and the 34 unctions, a hyperspace Pareto rontier is constructed as shown in Figure 5. Figure 3. vs. indexing Figure 4. 3 vs. 4 indexing Figure 5. Four objective Pareto Frontier We see rom Figure5 that there is a location along the axis or which there are stacked points (i.e. index 4). This results rom the extremely coarse discretization used or the s or this example problem. The issue o discretization is one that will be addressed urther in the paper. However, one can easily imagine how additional bins in all s will result in more choices or a Pareto point to ultimately all elsewhere in the HSDC indexed space. A greater discretization will inevitably spread out the distribution o Pareto points into more bins. This HSDC-based binning approach results in an actual Pareto rontier or however many objective unctions one might have. There is theoretically no reason that this approach will not scale to as many objectives as one might want, except that there will be a greater spread o points associated with particular levels. Figure 6 shows a Pareto rontier or a six objective unction MOP with 3 indexed on one axis and 456 indexed on the other. We see that a relatively smooth hyperspace Pareto rontier can be generated, even or a very rough discretization o the s (0 bins). Another important point to this representation is that the points in the HSDC indexed space are actually bins that could conceivably contain numerous Pareto points. In Figure 6, we see a rotated view, which clearly shows a height associated with each bin, which tells us how many points reside in each. This is incredibly valuable, as we can quickly determine where there might be multiple solutions o interest in a single bin. Such a representation indicates that there might be multiple design conigurations yielding the same combinations o objectives. An issue here pertains to the act that dierent groupings o objectives will produce dierent representations in the HSDC-based Pareto rontier. This is absolutely true and dierent aspects o this issue will be discussed urther in this paper. We will state here, however, that extreme points remain extreme in any representation and good points remain clustered around the same region regardless o representation. The reason or this is that the way in which indexing is perormed in the HSDC ollows the trends o the objective unctions themselves (i.e. small to large 3

4 unction values generally correspond to small to large indexed values). Hence, Pareto points that correspond to upper ranges o all objectives will also have large indices in all objectives. I there were a Pareto point or which all objectives were small, this would translate to indices that are similarly small in the indexed space. As mentioned beore, there are two issues to be addressed here. The irst pertains to ensuring that a ull Pareto rontier has been generated. The second has to do with ensuring we have the best possible visual representation to enable eventual choices to be made or product concept development. In this paper, we will address the irst point briely, but will concentrate on the second as the major contribution. Figure 6. Six objective Pareto Frontier (0 bins or each ) III. Methodology or Preerence Incorporation in Hyperspace Pareto Frontier The hyperspace Pareto rontier serves essentially the same purpose as does that or two objectives. Just as a two objective (i.e. vs. ) Pareto rontier can provide insights into missing design concepts as well as implicit tradeos between unctions, so can the multidimensional representation generated rom HSDC (the hyperspace Pareto rontier). Messac and Mattson 8, 9, 0 reer to the approach o generating Pareto points, ollowed by an identiication o the most desirable members o this set, as the Generate First Choose Later (GFCL) approach. They point out that it s oten diicult to sort through the existing Pareto points to identiy what is really desirable, as well as the act that generation o an even distribution o the Pareto rontier is critical. They demonstrate that their Normal Constraint (NC) method can be used to ensure an even representation. They essentially deine a utopia line (or plane or hyper plane, depending on the number o objective unctions) which connects all endpoints o the Pareto rontier and then create a set o evenly distributed points on that line, plane or hyper plane which are essentially used to create the limits or the various objectives in the epsilon-constraint method. In this paper, we will be borrowing this concept to some extent. The hyperspace Pareto rontier representation has the tremendous beneit o blending all participating objectives in the indexed space in such a way that the nature o the original objectives is (basically) preserved once indexed. In other words, large objectives have large indices and small objectives have small indices. Further, i we think about what it means to have a cluster o Pareto points near the utopia point on a two dimensional Pareto rontier in perormance space (i.e. both objectives relatively small simultaneously), we see that the same holds or the hyperspace Pareto rontier. The designer might be interested in Pareto points in a certain area, but observes a gap in the rontier in that region. This might suggest that additional design concepts might be worth investigating. It would then be possible to generate additional Pareto points. Since we know the ranges o the objective unction bins being used or each index, we have a means o identiying limits or objectives using the epsilon-constraint method in order to generate additional points. This is a tremendous advantage o the HSDC-based approach. The ultimate goal o multiobjective optimization is to obtain one or a ew select design candidates that represent the very best combination o objectives to pursue urther, according to the preerences o the designer(s). In this work, we develop a mechanism or establishing preerences according to objective unction range, and then use these goodness indicators, through use o color and real-time interaction, to enable designers to make inormed choices about the best design candidates. Here, we will again borrow a concept rom Proessor Messac. In Messac s Physical Programming approach to multiobjective optimization, the designer speciies ranges or his objective unctions according to the ollowing classiications: Highly desirable; Desirable; Tolerable; Undesirable; Highly Undesirable; and Unacceptable. We propose a similar approach, prior to the optimization procedure, which will not aect the optimization itsel, but will rather be used in the visualization o the Pareto rontier or the multidimensional perormance space. To understand the proposed approach here, let us go back to the our objective examples. Since there were only ive bins originally speciied or this example, let us assign the preerences per bin, leaving out or now the Unacceptable range. This results in the assignment o goodness as seen in Table 4. Using this assignment results in the distribution o preerences or each Pareto Point as seen in Table 5. 4

5 We develop a procedure or identiying which o the Pareto points are best and then represent them according to their goodness values encoded as color. It is acknowledged that there might be many ways to achieve this. The approach presented here is a simple, straightorward way o representing and capturing these preerences. Additional investigations will explore alternative representations. At any rate, in this paper we assign a color-coding according to the most desirable (blue) to the least (orange), with ranges being set by the designer. Bin Index Table 4. Four objective MOP with Preerence Ranges speciied range range 3 range 4 range Preerence Highly Desirable (HD) Desirable (D) Tolerable (T) Undesirable (U) Highly Undesirable (HU) 0 Point Value Table 5. Pareto Points or our objective MOP with associated Preerences Pareto HSDC Indices Preerence Total 3 4 Points Points Color A HD HD HU HU 0 R B HD HD U HU 7 O C HD D T HU 6 O D D D D T 8 B E T T HD D G F U D HD HD 8 G G HU HU HD D R H HU T HD HD 3 O I HU HU D HD R J HU T HD HD 3 O We see rom Figure 8 that there is one point that appears to be best, corresponding to indices (5, 9) (Pareto point D). We see that point F also has a total point value o 8, as does D, but is colored green. This is because the presence o an undesirable, or this example, automatically reduced the desirability by one level. In other words, is it truly better to have all tolerable over many highly desirables coupled with a sole undesirable. This would be completely up to the designer to make such decisions. I it is observed that a green point has the same point value as a blue point, a designer can easily delve into the data to determine whether that would still be a good point. Color code used: Blue Green Yellow Orange Red Figure 8. Preerences in Pareto Frontier IV. Results: Validation o the Hyperspace Pareto Frontier In this paper, we look at two major issues. The irst involves a validation o the Hyperspace Pareto Frontier, beyond what was originally presented in reerence [7]. The second involves the incorporation o the preerence scheme into the hyperspace Pareto rontier, which is addressed in the next section. In this section, we will address several key issues, including how to choose the grouping o the objectives or counting using the HSDC and how unction discretization impacts the visual representation. Grouping Functions: Recall that the goal o this research is to develop a new way to intuitively visualize a hyperspace Pareto rontier that will then enable designers to make an inormed decision about which points are the best design candidates. It 5

6 is obviously critical that the visual representations be meaningul. One issue that exists with the proposed HSDCbased approach is that dierent groupings o objectives might produce very dierent visual representations. It is critical that there exists a reliable mechanism or making design choices, which means that there must be a way or designers to determine what unctions, should be grouped together or meaningul visualizations. We institute an approach whereby correlation coeicients are irst calculated to provide guidance or subsequent grouping o the unctions or counting. It is not necessary that unctions be evenly distributed on the axes (i.e. 4 unctions on one axis and 4 on the other). In act, we could easily have ive on one axis and on another. It is critical, however, that those unctions which are grouped be directly correlated with one another. Correlation is deined as a relation existing between two quantities that are somehow associated, or occur in a way not expected on the basis o chance alone. Subsequently, the correlation coeicient is a number that is indicative o the degree o correlation between two sets o data. Simple regression analysis illustrates how variables are linearly correlated. Depending on the nature o the data, quantities may be directly (positively) or inversely (negatively) correlated. A direct correlation between two quantities means that the quantities increase simultaneously and an inverse correlation means that when one grows the other degrades. The value o correlation coeicient ranges rom - to, where - denotes a perect inverse correlation and denotes a perect direct correlation between the quantities. To demonstrate the importance o correlation coeicients, consider a simple two objective problem that result in a traditional Pareto rontier plot (with Pareto points) as shown in Figure 9. We will replicate our two objectives to create a our objective problem by having 3 the same as and 4 the same as. Now, by plotting the hyperspace Pareto rontier or 3 vs. 4, we are essentially representing vs.. We would expect to obtain the same trend in the HSDC-based hyperspace Pareto rontier representation as that obtained in the traditional Pareto plot. This particularly makes sense, given that the correlation coeicient or a grouping o - and - is unity or each. Indeed, we see rom Figure 0 that this is the case, with two dierent discretizations o 0 bins and 0 bins shown. Pareto Frontier Plot F F Figure 9. Two objective Pareto rontier Figure 0. Four objectives Hyperspace Pareto rontier (0 bins, 0 bins) While the trend or the 0 bin discretization is close, it does not conorm exactly to the trend seen in the traditional Pareto rontier. This is a result o a low discretization, which is improved to capture the trend in the igure on the right, by having a higher discretization o the objective unction ranges (0 rather than 0). This clearly demonstrates that the hyperspace Pareto rontier resulting rom the HSDC counting not only looks the same but has the same meaning or more than three objective unctions. We will explore this concept urther by using the our objective multiobjective optimization problem previously discussed to demonstrate the importance o grouping objectives meaningully. Four-Objectives, Problem Description: The problem description or this constrained multiobjective problem is as on the right. For this problem, 800 non-dominated Pareto points were generated. Correlation coeicients were then calculated or each objective unction pair (, 3, etc.) and are shown in Table 6 below. This provides an interesting choice or the designer. While 3 have the largest positive correlation coeicient, 4 have a large negative correlation coeicient. This suggests an inappropriate grouping. The second highest positive correlation occurs between, while 34 has only a small negative correlation. 6

7 This grouping is thereore chosen or the hyperspace Pareto rontier representation. The resulting visualization, associated with a discretization o 5 bins (i.e. ranges) or each objective unction, is shown in Figure. Alternatively, a grouping o 3 versus 4 results in the visualization next to it, in Figure. One can see that there is a much greater distribution o points or the 3 versus 4 than or the versus 34. However, the small positive correlation in and small negative correlation in 34 do contribute to some spread in the inal hyperspace Pareto rontier. The discretization also impacts the visualization and is explored urther in the next subsection. I we were to group objectives that were highly inversely correlated (i.e. large negative correlation coeicients), then the impact o such a trend in the HSDC-based representation would be to invalidate the way in which counting is done, making the visual representation meaningless. Minimize : g 3 3 Where : ( X ) ( X ) ( X X 3) ( X X ) 75 ( 3X X 4) ( X X ) 8 ( 3X X 9) ( X ) X 4 34 Subjected to : 4X X g X g X X 4 3, X Table 6.Correlation Coeicients or Objective unction Pairs (4 Objectives) CORRELATIONS PAIRS COR. COEFF FF Yes FF FF No FF FF F3F OK Positive Negative Figure. FF vs. F3F4 Figure. FF3 vs. FF4 (4 objectives, 5 bins, 800 points) Recall that the indices increase rom let to right and rom bottom to top. Counting is perormed as in Figures 3 and 4, in a consistent spiraling manner, so as to ensure that any points on a particular level will not dominate one another. Further, as we move outward rom one level to another, there will be a sot dominance to the right or up (in -D) with a hard dominance diagonally. I we match two (or more) inversely related unctions (with large negative correlation coeicients), this will have the result o invalidating our presumptions o dominance, thereby invalidating our visual representation. We explore this with a six objective unction example problem as well. Six-Objectives, Problem Description: The problem description or this constrained multiobjective problem is as on the right. For this problem, 65 non-dominated Pareto points were generated. Correlation coeicients were then calculated or each objective unction pair (, 3, etc.) and are shown in Table 7 below. The grouping o 3 versus 456 provides the best positive correlation o unctions, while the second best is 56 versus 34 provides the second best combination. Figure 3a shows the hyperspace Pareto rontier or the 65 nondominated Pareto points associated with the six objective optimization problem or a coarse discretization o 5 bins per objective. Again, we see a band o points that result in something similar to the traditional Pareto rontier representation. It should be noted here that while we say points, this is not totally accurate. In reality, these are bins that might hold multiple points, as demonstrated in the rotated view in Figure 3b. The better correlation o unctions or this six objective problem results in a smaller band o resulting points in the hyperspace Pareto rontier below. Minimize : g Where : ( X ) ( X ) ( X X 3) ( X X ) 75 ( 3X X 4) ( X X ) 8 ( 3X X 9) ( X ) 34 ( 4X X 4) ( X ) ( X 4) ( X X ) 6 8 Subjected to : 3 X 4X X g X g X 4 X , X

8 As mentioned previously, however, the discretization also impacts this representation and is addressed in the next subsection. An alternative grouping o objectives or this six unction problem is explored in the preerence section. Impact o Discretization on Hyperspace Pareto Frontier: One can see rom Figures 4 a-c that increasing the discretization o each objective (rom 5 to 0 to 30 bins each) results in more bins being represented in the hyperspace Pareto rontier. This provides a greater opportunity or points to be more accurately represented, since the bins no longer capture as great a range. For the our objective problem represented below, this also correlates to a more compact band o points or higher discretizations. The same trend is seen in the six objective problem in Figures 5 a and b, where the ten bin representation is obviously more smooth than the ive. Even higher discretizations are explored or this problem in the next section, which deals with incorporation o preerences in the hyperspace Pareto rontier representation. Figures 3a-b. FFF3 vs. F4F5F6 (6 objectives, 5 bins, 65 points) From these results, one can conclude that it is possible to represent large numbers o Pareto solutions or optimization problems with many objective unctions using the hyperspace Pareto rontier. While we can say that points (i.e. bins) that are extreme in unction space remain extreme in indexed space, it is still not completely clear how this representation can be useul or identiying one or a select group o candidate designs rom the Pareto set. This is covered in the next section. Table 7. Correlation Coeicients or Objective unction Pairs (6 Objectives) CORRELATIONS PAIRS COR. COEFF FF FF FF FF FF FF FF FF FF F3F F3F F3F F4F F4F F5F Positive Negative Figure 4a-c. 5, 0, and 30 bins (4 objectives, FF vs. F3F4, 800 points) 8

9 Figure 5a-b. 5 and 0 bins (6 objectives, FFF3 vs. F4F5F6, 65 points) V. Results: Preerences in the Hyperspace Pareto Frontier As explained earlier, color is used as a means to incorporate the preerence strategies. Preerence Incorporation in Four and Six Objective Problems: Table 8 shows the point values assigned or unction desirability, as well as the ranges speciied or each unction in terms o desirability. Figures 6a-b show the our objective problem, where all the objectives have been discretized into bins o 0 and 5 each, and Pareto points are color coded in accordance with their total goodness values. Here, the points with a total goodness between 0 and 7 are blue, between 7 and 4 are green, between 4 and are yellow, and between and 40 are red. Recall that the lower the numerical value o the total goodness, the better it is or the designer, as a lower value means that more unctions have greater desirability. Table 8. Point Values associated with Preerences and Ranges or 4 Objective Problem Bin Point range range 3 range 4 range Preerence Index Value Highly Desirable (HD) Desirable (D) Tolerable (T) Undesirable (U) Highly Undesirable (HU) 0 Figure 6a-b. Preerences incorporated using colors (0 bins and 5 bins) The plot clearly shows many blue points (i.e. bins) that have low total goodness values. These points are more desirable and are close, as a general trend, to the utopia point whereas the red points that have very high total 9

10 goodness values (i.e. are least desirable) all at the extremes in the hyperspace Pareto rontier representation. As a general trend, thereore, we see a move rom blue to green to yellow to red. Figures 7a-b corresponds to the same preerence scheme applied to the six objective problem with a discretization o ive and eight bins per objective, respectively, and a total o 65 Pareto points. It would be absolutely impossible to achieve any kind o reasonable visual representation o this many Pareto solutions or six objectives using any o the other standard visualization approaches. The hyperspace Pareto rontier representation provides a highly intuitive mechanism or designers to better understand the Pareto solutions and to incorporate their preerences associated with dierent objectives. It also is suiciently lexible to enable the designer to easily change preerence ranges according to his desires, as well as to delve urther into the perormance space through a hierarchical scheme. This is explained urther in the next section. Figure 7a-b. Preerences incorporated using colors (6 objectives, 5 and 8 bins) Hierarchical Preerence Setting: At this point, a designer might want to tighten his range or the preerences, in order to eliminate some o the highly undesirable points at an early stage. Figure 8a shows the same representation (with a discretization o 0 bins per objective unction) with a tighter range on the preerences where total goodness ranges are colored as ollows: blue, 0-5; green, 5-0; yellow, 0-7; and red, -40. As one would expect, there are more red points that are undesirable to the designer, with ewer blue, as well. Figure 8b shows the same representation in which only 487 are let out o total 800 Pareto points, because the 33 undesired (red and yellow) points were eliminated. Figure 8 a-b. Tighter preerences and pruned representation. Once some clearly undesirable points are pruned, the designer might wish to again incorporate preerences with respect to the remaining Pareto points. Figures 9a-b show a second level application o designer s preerences, where ranges o the objectives rom the remaining 487 points were again discretized into bins o 0 each, keeping 0

11 the initial preerence structure. This type o hierarchical preerence investigation and implementation is easily accommodated with the hyperspace Pareto rontier representation and provides an intuitive and straightorward mechanism or designers to select one or more candidate Pareto solutions or concept development. Figure 9 a-b. Preerence incorporation or pruned points with rebinning. This same investigation was perormed or the six objective problem. In Figure 0a, all yellow and red points were removed rom Figure 7, leaving 447 Pareto points. The remaining points were then rediscretized with the same preerence scheme applied, resulting in the hyperspace Pareto rontier in Figure 0b. Here we see that there are no longer any blue points, since the ranges have essentially tightened up in the preerence scheme. Hence, the designer might wish to reset the preerence settings so that the ranges are as ollows: blue, 0-0; green, 0.0-0; yellow, ; and red, The resulting hyperspace Pareto rontier is seen in Figure rom a top view as well as the rotated view, which reminds us that these are actually bins that might hold multiple Pareto points. Figures 0 a-b. Pruned points and rediscretization o pruned points (5 bins)

12 Figure. New perormance ranges or pruned Pareto points (5 bins). Tailoring the Preerences or Individual Objectives: There are a host o investigations in which a designer might wish to engage in order to identiy a inal design or set o designs that would then move to the concept development stage. A designer might wish to emphasize one or more objectives over the others and see the resulting impact. Consider a case where a designer might decide to emphasize and 3 over and 4. One approach (o many) might be to change the goodness structure and point values associated with the various unctions so that the and 4 are essentially more highly weighted in the total goodness metric (as is seen in Table 9). The resulting hyperspace Pareto rontier is seen in Figure, where the total goodness breakdown corresponds to: blue, 0-6; green, 6.0-6; yellow, 6.0 6; and red, The resulting representation shows the blue band o points moving out away rom the utopia point and merging more with green and yellow points. This makes sense, since we have essentially weighted and 3 versus and 4, even though is not grouped with 3 (or with 4) on the indexed axes. This results in a loss o the banding type o behavior previously observed as a result o incorporating equally distributed preerences, but still enables the designer to pull o speciic points or urther investigation according to his particular interests and goals. Table 9. Tailored Preerence Setting or the 4 Objective Problem Point Index range 3 range Preerence Value Highly Desirable (HD) Desirable (D) Tolerable (T) Undesirable (U) Highly Undesirable (HU) 6 Point Index range 4 range Preerence Value Highly Desirable (HD) Desirable (D) Tolerable (T) Undesirable (U) Highly Undesirable (HU) 0

13 Figure. F and F3 emphasized (4 objective, 5 bins) Impact o Dierent Objective Groupings on Preerence Incorporation: Recall rom an earlier section that dierent groupings o objective unctions will produce dierent hyperspace Pareto rontier representations. However, it is critical to note that the colored preerence setting scheme ensures that the same points (i.e. bins) that are blue with one ordering will remain blue with another (as is also true with yellow, green, and red points). This eliminates the problem o potentially missing the good points due to choosing one grouping o objectives over another. Figures 3 and 4 show the six objective hyperspace Pareto rontier (5 bins) or two dierent groupings o objectives (3 vs. 456 and 34 vs. 56). The ranges on the color-coding are: blue, 0-7; green, 7.0-4; yellow, 4.0 ; and red, Even though we see that the representations are slightly dierent, still there are bands o goodness that are easily identiiable, with blue points still gathered close to the utopia point. Figure 3. Dierent F Groupings (6 objective, 5 bins) VI. Conclusions In this paper, the Hyperspace Diagonal Counting (HSDC) method is applied to multiobjective optimization with an incorporation o designer preerences. The goal is to create an easy and intuitive visual representation o the multidimensional perormance space. The resulting hyperspace Pareto rontier can easily represent any number o unctions and any number o Pareto solutions, while still enabling a designer to easily understand the visualization and its signiicance. In this paper, we have demonstrated that unctions can be grouped according to correlation coeicients, and that a iner discretization o objectives contributes to more accurate visual representations. It was demonstrated that a simple preerence setting scheme results in a powerul approach or enabling a designer to sort through hundreds or even thousands o Pareto points in order to identiy the most desirable points or possible product concept exploration. A hierarchical scheme can be used to delve down into iner and iner preerence 3

14 settings according to designer preerence. The paper demonstrates the substantial potentials o incorporating the HSDC method to achieve the hyperspace Pareto rontier. VI. Acknowledgments We grateully acknowledge support rom the New York State Center or Engineering Design and Industrial Innovation (NYSCEDII) and unding rom NYSTAR (New York State Oice o Science, Technology and Academic Research). VII. Reerences Pareto, V., Manual o Political Economy. A. M. Kelley, New York, NY, 97. A. Inselberg, Parallax: Sotware or Multidimensional Visualization and Automatic Classiication, Inselberg, A., 997, Parallel Coordinates or Visualizing Multidimensional Geometry, Proceedings o New Techniques and Technologies or Statistics, pp Inselberg, A., and Dimsdale, B., 990, Parallel Coordinates: A Tool or Visualizing Multidimensional Geometry, Proceedings o IEEE Visualization Conerence, San Francisco, CA, pp Eddy, J. P. and Lewis, K., 00, Visualization o Multi-Dimensional Design and Optimization Data Using Cloud Visualization, ASME Design Technical Conerences, Design Automation Conerence, DETC00/DAC Mattson, C. A. and Messac, A. 003, Concept selection using s-pareto rontiers. AIAA Journal, 4(6): Agrawal, G., Lewis, K.E., Chugh, K., Huang, C-H., Parashar, K., Bloebaum, C.L., Intuitive Visualization o Pareto Frontier or Multi-Objective Optimization in n-dimensional Perormance Space. 0 th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conerence, Albany, New York, August 30 September, 004, AIAA Messac, A. and Mattson, C. A., "Normal Constraint Method with Guarantee o Even Representation o Complete Pareto Frontier," AIAA Journal, In Press. 9 Messac, A., Ismail-Yahaya, A., and Mattson, C.A., "The Normalized Normal Constraint Method or Generating the Pareto Frontier," Structural and Multidisciplinary Optimization, Vol. 5, No., 003, pp Messac, A., and Mattson, C. A., Generating Well-Distributed Sets o Pareto Points or Engineering Design Using Physical Programming, Optimization and Engineering, Kluwer Publishers, Vol. 3, Issue 4, pp , December 00. Messac, A., and Ismail-Yahaya, A., "Multiobjective Robust Design Using Physical Programming," Structural and Multidisciplinary Optimization, Vol. 3, No. 5, 00, pp