# Set-Based Design: A Decision-Theoretic Perspective

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1 Set-Based Design: A Decision-Theoretic Perspective Chris Paredis, Jason Aughenbaugh, Rich Malak, Steve Rekuc Product and Systems Lifecycle Management Center G.W. Woodruff School of Mechanical Engineering Georgia Institute of Technology

3 What is Set-Based Design? Point-based Design Set-based Design x 2 x 2 x 1 Marketing Eliminate Dominated Alternatives x 1 Propose Design Analyze & Critique Engineering Styling Marketing Styling Modify Manufacturing Manufacturing Engineering

4 Foundations by Ward, Sobek & Liker Ward (1989): Mechanical Design Compiler Compile high-level description into set of possible solutions Eliminate through labeled interval propagation Eliminate only alternatives that can be proven not to work Sobek, Ward & Liker (1999): Case study: Toyota Production system Engineers communicate in terms of sets Multiple design alternatives are developed in parallel Paradox: value despite apparent "inefficiencies"

5 Simple Example: Design of Pneumatic System (adapted from Ward, 1989) Catalog of Components: 50 motors 30 compressors, Interval-Based Characterization of Components Motor: RPM (nom load) = [1740,1800] Cylinder: Force = [0,100] N Design Requirements Load: Velocity = every [0,2] m/s Power-supply: 110V AC Propagation of set-based requirements Yields relatively small set of feasible solutions Motor Compressor Valve Cylinder Load

6 Some Short-Comings in Current State of the Art Only algebraic equations No differential or partial differential equations No black boxes equations need to be expressed symbolically Only for catalog design configuration of discrete options Significant extensions are needed to support continuous variables Only pure intervals no probabilities Ignoring probability information often leads to overly conservative designs Weak on optimization beyond constraint satisfaction What if satisfying all the constraints still leaves many alternatives?

7 Need for a Strong Foundation Our approach: Build on the foundation of Decision Theory Decision A 1 A 2 A n O 11 O 12 O 1k O 21 O 22 O 2k O n1 O n2 O nk UO ( 11) UO ( 12) UO ( 1 k ) UO ( 21) UO ( 22) UO ( 2 k ) UO ( n 1) UO ( n 2) ( ) UO nk Normative Decision Theory Select A i where E [ U ] is maximum + Reality of Design Context Bounded rationality Limited resources Incomplete knowledge Diverse information and knowledge needs Collaboration among geographically distributed stakeholders Formal, Systematic but Practical Methods for Engineering Design Formalisms Representations Methods Tools

8 Trade-off Between Information Cost and Value Information Economics Benefit Cost Information is only valuable to the extent that it leads to better decisions No change in the decision benefit of information is zero

9 Overview of Presentation What is Set-Based Design? Current Limitations of Set-Based Design Set-Based Design from a Decision-Theoretic Perspective Set-based design and sequential decision making Expressing the utility of decision alternatives as intervals Decision policy: eliminate non-dominated design alternatives Searching through a set of non-dominated alternatives: Branch & Bound Implication for Modeling and Simulation in Design Implications for PLM Conclusion

10 Set-Based Design and Sequential Decisions Sequential decisions: Decision 2 Decision 1 Vehicle type decision alternatives Engine/motor type decision alternatives gas car bike electric diesel Vehicle type decision alternatives In the first decisions, the designer chooses from a set car Gas car Electric car Diesel car bike Gas bike Electric bike Diesel bike Each decision alternative is a set of design alternatives Decision alternatives are imprecisely defined

11 Set-Based Decision Alternatives Cost =? \$ [400,1000] SET OF 150 hp Gas Engines Mass =? [100,300] kg Efficiency =? Reliability =? [25,30] % [95,99] % Imprecise Alternative Imprecise Performance

12 Imprecision and Variability P-Box Deterministic Probabilistic Precise cdf 0.5 cdf Precise Scalar X Precise Distribution X Imprecise cdf cdf P-box of interval X Probability-box X

13 Other Sources of Imprecision in Design Some other sources of uncertainty best represented by intervals Simulations and analysis models abstractions of reality Statistical data finite samples of environmental factors Bounded rationality imprecise subjective probabilities Expert opinion lack or conflict of information Preferences incomplete or non-stationary Numerical implementation limited machine precision Consequence: The performance (expected utility) of a decision alternative is best expressed in terms of intervals e.g. mass = [100,300] kg, cost = \$ [400,1000] utility = [, ]

14 Decision Making Under Interval Uncertainty In normative decision theory: Decision Policy = Maximize Expected Utility In set-based design: Uncertainty expressed as intervals or probability boxes Expected Utility Interval of Expected Utility How to make a decision when expected utilities are intervals?

15 Decision Making for Intervals of Expected Utility How can a decision be reached? Consider 3 design alternatives {A, B, C} with expected utility intervals as shown: Which alternative is the most preferred? Expected utility A B C Possible policies include: Interval Dominance Maximality Γ-maximin Γ-maximax Hurwicz criterion η E-admissibility

16 Interval Dominance Decision Policy Eliminate only alternatives that are provably dominated Consider 3 design alternatives {A, B, C} with expected utility intervals as shown: Expected utility X A B C upper bound of A < lower bound of C C dominates A eliminate A B and C continue to be considered (set-based design)

17 The Myth of "Optimal Design" Due to uncertainty, "optimal design" cannot be determined Set of non-dominated solutions When uncertainty is large, selecting only the "optimal design" often leads to back-tracking Expected Utility (\$) Interval Dominance: Expected Utility vs. Gear Ratio ELIMINATE non-dominated set ELIMINATE Gear Ratio, Ng

18 What If Non-Dominated Set is Too Large? cost bounds [ ] on cost Set of All Gas Engines [ bounds on mass ] mass Search non-dominated set using Branch and Bound approach

19 What If Non-Dominated Set is Too Large? cost [ ] [ ] Small Gas Engines Large Gas Engines [ small ] [ large ] mass Refine design alternatives Reduces imprecision in performance Allows for additional elimination

20 Γ-maximin Decision Policy Avoid a very bad outcome for sure. Consider 3 design alternatives {A, B, C} with expected utility intervals as shown: Chose the alternative with the highest lower-bound Expected utility * * * A B C Robust Solution Should only be used as a tie-breaker

21 Consequences for Set-Based Design Decision Alternatives and their Expected Utilities are Sets Unlikely that a single "point solution" will dominate "Point solutions" are often greedy result in expensive back-tracking "Point solutions" force us to make assumptions that are not supported by current information Constraint propagation versus non-domination Intersection of feasible sets for individual disciplines or sub-systems = elimination of dominated solutions Infeasible = overall utility is unacceptably low = dominated But: set of feasible solution is likely to be large need for efficient search Uncertain information should be represented accurately Without overstating what is known But also without omitting much information need for probabilistic or even hybrid (p-box) representations

22 Challenges for Modeling and Simulation Uncertainty quantification Every model is an abstraction of reality and thus wrong Model accuracy (systematic error) must be stated in terms of intervals Uncertainty quantification of model parameters / inputs / outputs Need for abstract models Allow designers to quickly eliminate large portions of the design space Currently not addressed opportunity: abstraction through data mining How to capture abstract models without losing much information? Capturing interval dependence is critical

23 Challenges for PLM Representations of design alternatives in terms of sets Most important at systems engineering level Set-based geometric representations leverage GD&T support Representations for communicating preferences Requirements are too limiting Better communication mechanism than requirements flow-down Methods for efficiently propagating constraints Interval arithmetic may yield hyper-conservative results Methods for efficiently searching set-based design spaces Branch and bound: How to branch efficiently?

24 Conclusions Set-Based Design Foundation developed by Ward et al. starting in late 80's Empirical evidence of superior results: Toyota Many remaining limitations and research issues Need for a strong foundation: Decision Theory All sequential design methods are set-based Expected utility of decision alternatives should be expressed as intervals Decision policy: eliminate non-dominated design alternatives Searching through a set of non-dominated alternatives: Branch & Bound

25 References Ward, "A Theory of Quantitative Inference Applied to a Mechanical Design Compiler", Ph.D. Thesis, MIT AI Lab, Sobek, Ward, and Liker, "Toyota s Principles of Set-based Concurrent Engineering" Sloan Management Review, 1999 Malak and Paredis, "An Investigation of Set-Based Design from a Decision Analysis Perspective," ASME Design Automation Conference, submitted. Rekuc, Aughenbaugh, Bruns, and Paredis, "Eliminating Design Alternatives Based on Imprecise Information," SAE World Congress, 06M-269, Panchal, Fernández, Allen, Paredis, and Mistree, "An Interval-Based Focalization Method for Decision-Making in Decentralized, Multi-Functional Design," Design Automation Conference, DETC , Long Beach, CA, September 24 28, Aughenbaugh and Paredis, "The Value of Using Imprecise Probabilities in Engineering Design," ASME Journal of Mechanical Design, to appear in July

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