Jason Aughenbaugh gtg224k@mail.gatech Advised by Dr. Chris Paredis

1. Mathematical Formalisms for Handling Uncertainty in Engineering Design Jason Aughenbaugh gtg224k@mail.gatech.edu Advised by Dr. Chris Paredis Systems Realization Laboratory

2. Manufacturing in 2020 • Product marketplace • Global markets for products • Global competition • Consumers with rapidly evolving demands • Technology • Computer speed, power, portability • Increased knowledge management • More powerful modeling • Product Systems • Complex and multidisciplinary • Need to identify and shape demands

3. Knowledge in a manufacturing enterprise • Manage employee knowledge • Get knowledge to engineers • Unlock the knowledge in the workforce • Provide means to communicate knowledge • Encourage others to listen and learn • Product lifecycle management (PLM) • Structural capital to support the intellectual capital • Often emphasizes individual tools • Interactions between pieces are important

4. Systems Design for 2020 Systems Engineering • Recognizes both the product system and human system • A holistic, hierarchical approach to decomposition, integration, verification, and validation • Limitations: decisions are • Coupled • Made under uncertainty Vee model Forsberg and Mooz (1992) "The Relationship of Systems Engineering to the Project Cycle," Engineering Management Journal, 4(3), pp. 36-43.

5. Uncertainty in Design Decisions • Aleatory Uncertainty: stochastic • Inherent randomness exists • Probability Distributions apply • Epistemic uncertainty: lack of knowledge • Not random • Intervals may apply • Arises in engineering design • System requirements • System environment • Future decisions • Emergent attributes

6. Possible Formalizations of Uncertainty • Probability theory: frequentist or subjective • Dempster-Shafer/Evidence theory • Possibility theory • Interval computations • Probability bounds analysis • A novel theory or combination of theories Bel(A) = 0.4 Pl(A) = 0.8 … ? [lower, upper]

7. Leveraging knowledge and models • Represent uncertainty • How good are the results or estimates? • Formalize model context • When does this model have stated uncertainty? • Integrate models • Use and reuse models together

8. Computations involving uncertainty Formal Representation Of Uncertainty Formal Theory Formal Interpretation Of Output Desired: In engineering design, probability is usually used to model uncertainty: Objective Probability Distribution Probability Theory Objective Interpretation Of Output Acceptable: Subjective Probability Distribution Probability Theory Subjective Interpretation Of Output Acceptable: Unfortunately, the following is often done: Subjective Probability Distribution Probability Theory Objective Interpretation Of Output Unacceptable:

9. Probability Bounds Analysis (P-boxes) • An enveloping of all possible CDFs • It represents aleatory uncertainty (variability) via the cumulative probabilitydistributions • It represents epistemic uncertainty (incertitude) via the interval on the parameters • Example p-boxes • Example 1 • N([0,1],1): Normal distribution • mean in the interval [0,1] • variance 1 • Example 2 • Minimum = 0 • Maximum = 100 • Mean = 50 Example 1 Example 2