BAYESIAN NETWORKS FOR SET-BASED COLLABORATIVE DESIGN

Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2009 August 30 - September 2, 2009, San Diego, California, USA

DETC2009/87541

BAYESIAN NETWORKS FOR SET-BASED COLLABORATIVE DESIGN David Shahan [email protected]

Carolyn C. Seepersad [email protected]

Department of Mechanical Engineering The University of Texas at Austin Austin, TX 78712

ABSTRACT A set-based approach to collaborative design is presented, in which Bayesian networks are used to represent promising regions of the design space. In collaborative design exploration, complex multilevel design problems are often decomposed into distributed subproblems that are linked by shared or coupled parameters. Collaborating designers often prefer conflicting values for these coupled parameters, resulting in incompatibilities that require substantial iteration to resolve, extending the design process lead time without guarantee of achieving a good design. In the proposed approach to collaborative design, each designer builds a locally developed Bayesian network that represents regions of interest in his design space. Then, these local networks are shared and combined with those of collaborating designers to promote more efficient local design space search that takes into account the interests of one’s collaborators. The proposed method has the potential to capture a designer’s preferences for arbitrarily shaped and potentially disconnected regions of the design space in order to identify compatible or conflicting preferences between collaborators and to facilitate a compromise if necessary. It also sets the stage for a flexible and concurrent design process with varying degrees of designer involvement that can support different designer strategies such as hill-climbing or region identification. The potential benefits are the capture of expert knowledge for future use as well as conflict identification and resolution. This paper presents an overview of the proposed method as well as an example implementation for the design of an unmanned aerial vehicle.

1. INTRODUCTION Many engineering systems are complex enough to be decomposed into a set of interdependent and distributed design problems, each solved by designers with specialized knowledge and tools. Numerous approaches have been proposed for decomposing and solving these multilevel or multidisciplinary design problems, including analytical target cascading [1], simultaneous analysis and design [2], concurrent sub-space optimization [3,4], collaborative optimization [5], and BLISS [6]. When these approaches are implemented in a collaborative setting, they often lead to repeated iterations that are costly and time-consuming. The challenge is particularly acute in the early stages of design when teams of designers are trying to identify satisfactory regions of the design space, rather than refining a previously identified solution. In response to this challenge, set-based approaches have been proposed. These approaches dictate communicating and reasoning about sets of solutions, rather than single point solutions. Set-based strategies have been shown to reduce the likelihood of costly iterations without compromising the quality of resulting solutions [7,8]. This positive outcome is achieved by delaying commitment to a single solution, thereby preserving design freedom and increasing the diversity of options available for identifying satisfactory cross-disciplinary solutions. Set-based coordination strategies have taken several forms. Robust design techniques have been used for generating ranged sets or intervals of design specifications that

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can be shared with collaborating designers [9-11]. In recent work, Liu and coauthors [12] established a set theory-based quantization algorithm for dividing a design space into regions and a flexibility metric, based on aggregated achievability functions, for selecting the most achievable region or ranged set of design specifications. Like recent constraint-based approaches to set-based design [13], these interval-based approaches tend to provide relatively rigid representations of satisfactory solutions that do not capture arbitrarily shaped and potentially disconnected regions of the design space. In the Method of Imprecision [14,15], fuzzy set theory is used for modeling uncertain or imprecise parameters (such as preferences for performance variables) during negotiation. The method represents designer preferences but does not formally guide the search for new designs. Game theoretic approaches have been developed for coordinating the competitive reactions of designers to one another’s decisions [16-20]. These game theoretic techniques provide not only decision-theoretic strategies for guiding the search for solutions but also maps of individual design spaces, in the form of rational reaction sets or best reply correspondences. The difficulty is in generating these rational reaction sets, a process that may be expensive (e.g., creating metamodels of an entire design space). The interval-based constraint satisfaction (IBCS) method [21] combines the rational reaction sets of game theory with interval arithmetic, in a serial design process in which designers sequentially reduce the ranges of their design parameters. This method is not concurrent and does not accommodate arbitrarily shaped design regions. Genetic algorithms have been applied to decompose and/or solve multidisciplinary problems [22-24,21,25]. When they are used to solve multidisciplinary problems, they are setbased in terms of maintaining a diverse population of potential solutions, but they are resource intensive and do not provide a means of explicitly representing conditional relationships between variables. In this paper, an alternative set-based approach to

collaborative design is presented. The approach is based on Bayesian networks for representing interesting regions of a design space. In a distributed, collaborative setting, each designer (or design team) builds a Bayesian network of his/her local design space. These Bayesian networks can be shared between designers, as a means of identifying compatibilities and conflicts and improving the efficiency of local design space search. This approach differs from previous applications of Bayesian methods to multilevel design [26], in which Bayesian statistics are used simply for representing uncertainty. The proposed method uses Bayesian networks to capture a joint probability distribution defined over the design variables. The Bayesian network can be considered an estimate of the regions of the design space the designer believes are likely to contain high preference designs when a point from that region is sampled and analyzed. The use of Bayesian networks in this fashion is compatible with measures of preference defined over the results of analyses, such as utility, because, as will be seen, the joint probability distribution can be generated from a set of designs that have been analyzed for the designer’s preference according to an objective function. The joint probability distribution captured by the Bayesian network must evolve as more designs are evaluated and influence the designer’s interests in different regions of the design space where she believes prefered designs are located. Sections 2 and 3 demonstrate how this dynamic learning of interest in regions of the design space is accomplished. Other distributed design methods share this feature of providing approximations of collaborators’ interests over the design space. In particular, methods that share metamodels or rational reaction sets are similar in philosophy to the proposed method. However, these methods require a complete metamodel to exist in advance. In contrast, the proposed method can effectively communicate regions of interest of the design space at any stage of the design exploration process. In other words, the joint probability

System Level Design Group 1.1

Subsystem Design Group 2.1

Subsystem Design Group 2.2

Figure 1. Example Distributed Design Problem

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distribution of the Bayesian network should be considered an approximate model of the designer’s probability to choose designs from regions of the design space which can be sequentially constructed with every new set of samples and shared at any stage of its development. The beneficial features of the method include the capability of representing arbitrarily shaped (and potentially disconnected) regions of interest in the design space; the capability of sharing Bayesian networks in a fully concurrent manner, as they are being developed; and the option to interface with almost any automated or manual search process. The details of the method are presented in Section 2, followed by an example application in Section 3. 2. OVERVIEW OF THE METHOD There are two primary aspects of the proposed method for set-based collaborative design: 1) building a local Bayesian network to represent regions of interest in each designer’s design space and 2) sharing and combining local Bayesian networks with other designers to promote more efficient local design space search that takes into account the interests of other designers. Throughout this section, an analytical distributed design problem is solved as a demonstration of the techniques. As illustrated in Figure 1, the problem has the Himmelblau function as the system level design problem and two logistic functions as the subsystem level design problems. The objective of the system level designer is to achieve a value of 50 or lower for the Himmelblau function by choosing the values for the two decision variables, x1 and x2. The first subsystem level designer determines the system level decision variable, x1, through solution of a logistic function defined in terms of the sub-system level decision variable x3. Similarly, the second sub-system level designer determines the system level decision variable, x2, through solution of another logistic function defined in terms of the sub-system level decision variable, x3. The sub-system design activity is an equality constraint distributed between the two sub-system designers, reducing the degrees of freedom of the system level design problem from two dimensions to one. This example was constructed according to a recuring structure in distributed design. The assumption is that there exists a “system” level group that synthesizes the subsystem design results into a complete product for which some measure of performance can be maximized (or minimized). The system level group calculation of performance will often depend upon subsystem design calculation results (x1 and x2 in this example). The subsystem groups will often have an interface with common design variables (design variable x3 in this example) and/or analysis output variables. This structure, while typical, was chosen primarily as having the minimal amount and types of dependencies to illustrate the primary techniques of the proposed method without unnecessary complications. The proposed method is not limited to this type of distributed design architecture. The design process

begins with each designer building a Bayesian network representation of his/her local design space. 2.1 BUILDING LOCAL BAYESIAN NETWORKS Regions of interest in a local design space can be represented using a joint kernel probability distribution captured by a Bayesian network. A kernel probability distribution combines N probability density functions (PDF’s) as a uniformly weighted sum according to Eq. 1 [27]. (1) Each is a PDF defined over the decision variable vector, x, and a design point, x j. For this example we will use uniform distributions of width d centered on the D dimensional design point as shown in Eq. 2-3.

(2) (3) The kernel probability distribution works well for capturing arbitrarily shaped and potentially disconnected regions of the design space. The graph data structure of Bayesian networks provides an efficient means of capturing the conditional dependencies between the design parameters. Bayesian networks are directed acyclic graphs (DAG) that encode a joint PDF in factored form, according to Eq. 4 [27].

(4) This particular factorization requires the sequential calculation of the probabilities starting with parameter x1 and proceeding from there. Each subsequent probability is defined over one higher dimension. If one of the parameters, xa, is independent of another parameter, xb, then the dimension of one of the joint probabilities becomes smaller as shown in Eq. 5. (5) reducing the dimensionality of the PDF. The extreme case is independence of each variable as shown in Eq. 6. (6) Each node of the Bayesian network represents one of the variables, xi , and its conditional PDF. Each edge from a parent node to a child node represents the conditional dependence of

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the variable at the child node on the variable represented by the parent node. The Bayesian network on the left of Figure 2 represents the fully dependant joint PDF, while the Bayesian network on the right of Figure 2 represents the fully independent joint PDF. Any DAG in between these extremes is a viable possibility that captures different independencies of the variables. x1

x2 x3

x1

x2 x3

Figure 2. Fully Dependent (left) and Fully Independent (right) Bayesian Networks. Calculating the joint PDF represented by a Bayesian network involves calculating the PDF for the root nodes first and then proceeding to the child nodes such that ancestral PDF’s are all calculated prior to calculating the child node’s PDF. Given the full joint PDF, the conditional PDF’s can be calculated using Eq. 7.

objective function value so long as the resulting Bayesian network faithfully captures the designer’s interests. As a demonstration of the more automated approach, the method is applied to the Himmelblau function of Figure 3. The learning process begins with an initial uniform sampling over the entire problem domain. In general, this could be a sampling from a PDF that captures the current expert opinion for the design problem, or it can be a uniform sampling representing an equal interest in all values. For a hill-climbing strategy, the best performing designs are then selected and used to update the Bayesian network. With each generation, the variation of the PDF decreases as long as better designs are found. An alternative strategy for finding a design region that meets a constraint would be to base the PDF on every sample from every generation that meets the constraint. This latter variation was applied to the Himmelblau function for finding regions of 50 or less. The result is shown in Figure 3. The method does indeed learn the different shaped and disconnected regions of the design space that represent good places to search.

(7) Using the joint uniform kernel distribution, the PDF of the child node conditionally dependent on the n parent nodes can be expressed as Eq. 8. x1

x2

(8)

A procedure is needed for sampling the design space, to obtain design points for building the Bayesian network. We propose a design process that provides varying degrees of designer interaction. In the lightest implementation, the designer (or any other search method) samples the design space directly, instead of relying upon sampling the Bayesian network. In this case, the Bayesian network is built manually to represent the regions of interest to the designer. A more automated implementation would sample the resulting Bayesian network for the next generation of design points. The selection process could be conducted by the designer by either manually choosing or by creating rules for automatically choosing whether or not a sampled design point is used in the Bayesian network. In general, the designer should be free to use any evaluated design point independently of how long ago it was sampled and independently of its

selected point for kernel PDF Figure 3. The Himmelblau Function (top), the Bayesian Network (middle), and the Joint Uniform Kernel PDF (bottom). It should be noted that the procedure for building local Bayesian networks is similar to the iterated density estimation evolutionary algorithms (ID As) framework in its use of joint kernel probability density functions (PDFs) centered on selected design points [28-29]. While the ID As framework is intended to be a black-box optimizer, we propose a more general and flexible framework that supports a range of

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designer intervention throughout the process and a variety of different designer strategies. For example, design samples do not need to be uniformly weighted, and a more general joint kernel PDF can be defined as Eq. 9-10.

(9) (10) By choosing appropriate weights, different search strategies can be realized such as region identification, hill climbing, or a combination of both. For example, a region identification weighting scheme might uniformly weight all known design points that meet an inequality constraint while a hill climbing strategy might weight all known design points according to the objective function. Additionally, different sampling strategies can be used such that the next generation of points is guaranteed to contain some representatives of regions of special interest or to cover the PDF more efficiently. In general, local Bayesian networks can be shared with other designers to guide the collaborative design process to a mutually compatible and successful final result. With respect to the analytical example problem of this section, the system level designer has built a Bayesian network to represent his/her regions of interest of the Himmelblau function, in terms of parameters x1 and x2. The next section describes how this Bayesian network can be shared with sub-system level designers to influence their search for satisfactory systemwide solutions.

selected point for kernel PDF Figure 4. The System Level Kernel PDF for p(x1) (top), the First Sub-System Level Kernel PDF for p(x3) (bottom).

2.2 SHARING BAYESIAN NETWORKS The local Bayesian networks, which capture a designer’s regions of interest, are defined over a designer’s decision variables. Often, in multilevel design problems, those decision variables are coupled or shared with collaborating designers. In those cases, the collaborator’s regions of interest for the shared variable must be mapped to those of the designer. For example, in the analytical example problem, the sub-system level designers need to map the system level interests for x1 and x2 onto their own interests for their common decision parameter, x3. This can be accomplished by using the system level PDF as an objective function for the sub-system level design problem. The result is shown in Figures 4 and 5. The top plot depicts the regions of interest of the system level designer. The bottom plot shows the mapped regions of interest of the sub-system level designer using the shared PDF from the system level as an objective function. In this case, all samples of x3 that produced a value of x1 or x2 with a probability density greater than 0.05 were selected to create the kernel PDF for x3. This does not imply that the subsystem design activity can not begin until the system level design interests are learned. The subsystem level designers would simply uniformly sample their design space until

selected point for kernel PDF Figure 5. The System Level Kernel PDF for p(x1) (top), the Second Sub-System Level Kernel PDF for p(x3) (bottom). they are provided with more direction from the system level. While this might not represent the most efficient use of

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resources, it does present the possibility of a fully parallel and concurrent design process. While x1 and x2 are output parameters of the subsystems that serve as input parameters for the system, x3 is an input parameter for both subsystems. In this case, the PDF’s generated for x3 by the subsystem level designers can be combined through an intersection type of operation or a union type of operation. The intersection type of operation can be chosen to reduce the size of the search region, and the union operation can be used to increase the size of the search region. The intersection type of operation is accomplished according to Eq. 11-12.

(11) (12) The integral in the denominator of Eq. 11 ensures that the result is a viable PDF; it can be evaluated numerically at the same time that the numerator is evaluated. The result of this operation on the example problem is shown in the top plot of Figure 6. This operation is called the logarithmic opinion pool according to its use in the aggregation of beliefs [30]. The variation of the underlying kernel distributions may need to be increased in order to generate a meaningful result in the case that the regions of interest have been over learned. A

union type of operation is accomplished according to Eq. 1314. (13) (14) The result of this operation on the example problem is shown in the bottom plot of Figure 6. This operation is called the linear opinion pool according to its use in the aggregation of beliefs [30]. Once the two sub-system design teams have collaboratively combined their regions of interest over the design variable x3, they need to map this back into regions of interest for the system-level decision parameters, x1 and x2. This can be accomplished by sampling the combined PDF over x3 and evaluating each sample. The result would be another two PDF’s defined over x1 and x2 which would then be used at the system level to further reduce the design space.

Group 2.2 Combined

Figure 7. The Resulting Design Points after One Iteration of Collaboration.

Group 2.1

Group 2.2 Combined Group 2.1

The process would continue until the design space is small enough to suggest a single design point. Alternatively, a simpler procedure is to have the two subsystem level designers sample the combined PDF for the same n values of x3 at uniform probability intervals. The points for x3 can each be mapped to values of x1 and x2 by the two sub-system teams. The design pairs of x1 and x2 can then be communicated to the system level for evaluation. The resulting design points will lie on the one dimensional constraint line depicted in Figure 7, which represents the feasible combinations of x1 and x2 that satisfy subsystem level functions for common values of x3. For 10 sub-system samples, the results are plotted as blue points in Figure 7. The global optimum found through an exhaustive search of the all-in-one design problem is shown as a red point in Figure 7. Table 1 lists these results.

Figure 6. The “Intersection” of the Sub-System Learned Kernel PDF’s for p(x3) (top), the “Union” of the SubSystem Learned Kernel PDF’s for p(x3 ) (bottom).

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Table 1. Results of Example problem method x1 x2 x3 proposed method 1 2.82 -3.46 -.85 2 2.49 -3.12 -.73 3 2.23 -2.82 -.64 4 1.98 -2.54 -.56 5 1.79 -2.31 -.50 6 1.63 -2.11 -.45 7 1.42 -1.86 -.39 8 -1.25 1.64 .34 9 -2.26 2.86 .65 -2.79 3.43 .84 10 exhaustive search -2.64 3.26 .78

design method has also been applied to a more complicated example of unmanned aerial vehicle design. f 103 89 89 94 102 110 122 92 10 4 2

As can be seen, this application of the proposed method led to a set of solutions with one design point being particularly close to the global optimum, given the steepness of the problem. This simple test case demonstrates some of the techniques that can be used in collaborative design using Bayesian networks. Specifically, it demonstrates the method’s capability for capturing arbitrarily shaped and potentially disconnected regions of interest in the design space. In this case, the regions were disconnected due to the rejection of points with objective functions greater than 50. Examples of disconnected spaces are shown in Figure 6. This demonstrates that the joint PDF can have disconnected regions if desired. Because the sharp decision boundary for the objective function is arbitrary, this example does not motivate the usefulness of a disconnected probability distribution. Nevertheless, the capability exists and its usefulness can be established in future work. This example also demonstrates the use of aggregation techniques to combine Bayesian networks for the benefit of finding common interests, narrowing the design space, and suggesting a small set of samples to explore within the reduced region of interest. As described in the next section, the proposed collaborative

3.0 EXAMPLE APPLICATION In this section, we demonstrate the proposed method on the simplified design of an unmanned aerial vehicle (UAV) for the cruise condition. The objective of the problem is to maximize the UAV’s range subject to a maximum weight limit, a fixed cruise altitude, and a fixed cruise velocity. The problem has been distributed into three design groups as depicted in Figure 8. The systems performance group calculates the range, given the wing weight and drag, using the Breguet range equation [31]. The aerodynamics group calculates the wing drag as a function of its external geometry, subject to providing sufficient lift and keeping the pitching moment (the moment on the wing that points the UAV up or down) below a fixed level. They use a linear vortex panel method according to [32] and a boundary layer growth method according to [33]. The structures group calculates the wing weight subject to a constraint on structural failure, by determining the wing’s skin thickness. The architecture of this example distributed design process is similar to the analytical example of Section 2. The similarity lies primarily in the existence of a system level group that controls the overall objective function evaluation subject to subsystem level analyses, constraints, and compatibilities. However, this application is handled slightly differently, illustrating some alternative techniques of the proposed method. The design process begins with each design group exploring its local design space. The systems performance group uses a Bayesian network for its two decision parameters, with wing weight being dependent upon wing drag. A hill-climbing strategy is used to build the Bayesian network. First, ten design points are generated from a uniform Latin Hypercube sampling over the entire design space. From these ten samples, the five design points with the highest

Range

UAV Systems Wing Lift and Drag Aerodynamics

Wing Weight Structures

Wing External Geometry and Loads

Safety Factor

Wing Moment Figure 8. A UAV Distributed Design Decomposition 7

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wing drag

converged

wing weight

generation

converged

generation Figure 9. Statistics for the Drag and the Weight of the 5 Best Design Points per Generation for the Performance Group range are selected to construct the joint uniform kernel PDF of the Bayesian network for the systems performance group. The second generation of ten design points is obtained by sampling the resulting Bayesian network. The second generation

Bayesian network is constructed from the design points with the highest ranges in the first two generations. The process continuous until the diversity in the 5 best design points has been reduced, in this case, to zero. Figure 9 presents the five best wing drags and weights for each generation. After 3 generations the population has essentially converged on the lower limit of 0.1 Newtons for both the wing weight and drag, with a range of 1927 km. These results imply that the aerodynamics and structures teams should pursue minimum drag and minimum weight respectively. The aerodynamics group uses a drag minimization strategy that selects the fifteen best designs from thirty design samples per generation to construct their uniform kernel PDF’s for each decision variable in a fully independent Bayesian network. The Bayesian network representing the regions of interest for their decision variables is shown in black in Figure 10 after eight generations. Similarly, the Bayesian network representing regions of interest for the structures group is shown in blue in Figure 10 after eight generations. From this plot, one can see from the almost uniform distributions that the structures group’s weight calculation is relatively insensitive to the NACA parameters for the airfoil profile as well as to the angle of attack. The aerodynamics group has a stronger interest for specific values for these design parameters, and hence should set these values to their best performing airfoil. The structures group prefers a high chord and a low span: they’re interested in the high strength of a wide, short wing. The aerodynamics group on the other hand prefers a low chord and a high span: they’re interested in the low induced drag of a narrow, long wing. A compromise will need to be found. It is not justified to seek a reduced design space in this case because it is unclear what

NACA Parameter 1

NACA Parameter 2

NACA Parameter 3

Chord

Span

Angle of Attack Figure 10. PDF’s for the Aero (black) and Structures (blue) Groups’ Decision Parameters 8

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range the best performing chord will lie in, and so a linear opinion pooling [30] of equal weight and increased variation is applied to the two chord PDF’s to produce the union type of PDF of Figure 11. This PDF is sampled at ten evenly spaced probability intervals and evaluated for drag, weight, and range using the best airfoil aerodynamics has found to date. The resulting designs are recorded in Table 2 based upon the airfoil parameters that provide minimum drag: 4.8 for NACA parameter 1, 5.3 for NACA parameter 2, 5.0 for NACA parameter 3, and 1.9 degrees for the angle of attack. The span is a dependent variable that results from creating a wing with enough surface area to generate the required amount of lift. These results show how the compromise samples capture the tradeoff between drag and weight as affected by wing chord, creating a set of possible designs. chord 0.128 0.178 0.220 0.259 0.296 0.334 0.370 0.404 0.439 0.479

Table 2. Results of Example UAV problem span drag weight range 3.27 0.299 7.98 138 1.74 0.307 1.28 1221 1.18 0.406 0.481 1253 0.911 0.587 0.245 1103 0.752 0.869 0.147 907 0.645 1.30 0.097 711 0.578 1.87 0.071 554 0.531 2.59 0.056 435 0.496 3.56 0.045 342 0.466 4.97 0.037 263

Before sharing their interests, the systems group performed a total of 30 function calls, the aerodynamics group performed a total of 240 function calls over 8 generations, and the structures group performed a total of 240 function calls over 8 generations. After sharing their interests and generating the final set of designs to evaluate, each group executed 10 function calls, with the aerodynamics and the structures groups working in parallel and the systems group

frequency

chord Figure 11. The Linear Opinion Pool for a Chord Compromise (red) between Aerodynamics (black) and Structures (blue)

working last. As a point of comparison, a non-distributed multi-start sequential quadratic programming (SQP) solution with random starting points was run 2000 times. The SQP algorithm was stopped after a comparable number of function calls and the best feasible range found to date was recorded. If no feasible solution was found then a range of zero was assigned. The resulting distribution of ranges achieved is shown in Figure 12. The proposed approach performed comparably to the multi-start SQP solution, with two designs over 1200 km. Furthermore, the proposed set-based design method ends with a collection of design points after a single run that captures a tradeoff in the design space that is not evident from the multi-start SQP approach. Although a study of the multi-start SQP results might uncover the usefulness of performing subsequent parameter sweeps to capture the same tradeoff, the proposed method explicitly reveals the conflict of interests. In addition, more high range designs could be easily generated by sampling more airfoils from the aerodynamics group’s Bayesian network and performing the same type of tradeoff study over ten values for the chord parameter. The proposed method has captured the interests of the aerodynamics group on regions of the design space of reduced size, making it easier to generate more high range UAV’s.

range Figure 12. Ranges Achieved using an All-In-One Method 4. CLOSURE Bayesian networks have been applied to facilitate setbased collaborative design. The primary contribution is the mapping and communication of arbitrarily shaped and potentially disconnected design regions as represented by Bayesian networks. This research has demonstrated how a designer can locally build a Bayesian network by following either a region identification strategy, as was the case for the example of Section 2, or a hill climbing strategy, as was the case for the example of Section 3. As shown in the examples, the Bayesian networks can also be shared with collaborating designers to promote identification of mutually acceptable values of shared or coupled parameters. Logarithmic or linear opinion pooling can be used to combine shared Bayesian networks to narrow or broaden the search space, respectively, in terms of a coupled parameter.

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The potential benefits of the proposed design process include a flexible and highly concurrent framework for assisting distributed design that captures expert knowledge, identifies conflicts, and can aid decisions regarding which parameters to set and which parameters to explore further. Further work is needed to verify the efficiency of the proposed design process in terms of finding high quality solutions within a competitive period of time and with a competitive amount of resources relative to established MDO methods including both point-based and set-based methods. Of particular interest is how this method would compare to a setbased multi-objective genetic algorithm approach. The challenges of building, sharing, and combining Bayesian networks pose several opportunities for future work. For example, more work can be done to improve the efficiency of building local Bayesian networks and to make the process more robust to algorithm settings such as sample sizes, selection sizes, rate of reduction in variance, etc. Future work could also look at how different sampling strategies affect the method’s efficiency. It would be interesting to explore different aggregation techniques for combining Bayesian networks, along with strategies for identifying conflicts and determining how to best resolve them. The convergence properties of the method require further understanding regarding when to choose which technique such that the variety of conditions for which the method will converge as well as the probability of converging to the right solution are maximized. Finally, the extent to which a process is parallel or serial can be explored, allowing resources and lead time and design freedom to be traded off as required. ACKNOWLEDGMENTS We gratefully acknowledge support from the University of Texas at Austin and National Science Foundation grant DMI-0632766. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsors. REFERENCES [1] Kim, H. M., N. F. Michelena, P. Y. Papalambros and T. Jiang, 2003, “Target Cascading in Optimal System Design,” ASME Journal of Mechanical Design, Vol. 125, pp. 474-480. [2] Haftka, R. T., 1985, “Simultaneous Analysis and Design,” AIAA Journal, Vol. 23, No. 7, pp. 1099-1103. [3] Sobieszczanski-Sobieski, J., 1988, "Optimization by Decomposition: A Step from Hierarchic to NonHierarchic Systems," Second NASA/Air Force Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, Hampton, VA, NASA TM-101494. NASA CP-3031. [4] Wujek, B. A., J. E. Renaud, S. M. Batill and J. B. Brockman, 1996, “Concurrent Subspace Optimization Using Design Variable Sharing in a Distributed

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