Jason Aughenbaugh

1
Mathematical Formalisms for Handling Uncertainty
in Engineering Design

• Jason Aughenbaugh
• gtg224k_at_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
Vee model

• 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

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 Theory
Formal Representation Of Uncertainty
Formal Interpretation Of Output
Desired
In engineering design, probability is usually
used to model uncertainty
Probability Theory
Objective Probability Distribution
Objective Interpretation Of Output
Acceptable
Probability Theory
Subjective Probability Distribution
Subjective Interpretation Of Output
Acceptable
Unfortunately, the following is often done
Probability Theory
Subjective Probability Distribution
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 probability distributions
• 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