Computational Methods for Decision Making Based on Imprecise Information

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Computational Methods for Decision Making Based
on Imprecise Information

• Morgan Bruns1, Chris Paredis1, and Scott Ferson2
• 1Systems Realization Laboratory
• Product and Systems Lifecycle Management Center
• G.W. Woodruff School of Mechanical Engineering
• Georgia Institute of Technology

www.srl.gatech.edu 2Applied Biomathematics
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Design Decision Modeling
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Design Computing with Uncertain Quantities

• For a given decision alternative,
• In practice, utility is dependent on uncertain
  quantities.
• In many engineering applications, utility is
  computed with a black box model.

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Representation of Uncertain Quantities

• Variability and imprecision
• Variability naturally random behavior,
  represented as probability distributions
• Imprecision (or incertitude) lack of knowledge,
  represented as intervals
• Has been argued that this distinction is useful
  in practice
• Trade-off between richness and tractability
• Decision analysis uses pdfs allows for
  straightforward Monte Carlo Analysis
• Richer representations result in computational
  difficulties
• Imprecise probabilities
• Assumes uncertainty is best represented by a set
  of probability distributions
• Bent quarter example
• Operational definition due to Walley
• corresponds to a minimum selling
  price
• corresponds to a maximum buying price

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The Probability Box (p-box)

• P-boxes can be used to represent
• Scalars
• Intervals
• Probability distributions
• Imprecise probability distributions

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State-of-the-Art for P-box Computations

• Discretizing the inputs
• Each input p-box is represented as a collection
  of interval-probability pairs (focal elements)
• Each interval-probability pair is propagated
  individually through a Cartesian product (Yager,
  1986 Williamson and Downs, 1990 Berleant, 1998)

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Example of P-box Convolution

• Example
• Where inputs are independent
• Focal Elements
• Cartesian Product

1,5, 1/3 3,6, 1/3 5,9, 1/3
6,7, 1/3 6,35, 1/9 18,42, 1/9 30,63, 1/9
7,9, 1/3 7,45, 1/9 21,54, 1/9 35,81, 1/9
8,9, 1/3 8,45, 1/9 24,54, 1/9 40,81, 1/9
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Dependency Bounds Convolution (DBC)

• DBC is a method of p-box convolution that
  determines best-possible and rigorous bounds on
  resultant p-box
• Best-possible in the sense of and
  being as close together as the given information
  allows.
• Rigorous in the sense of being guaranteed to
  contain the true result.
• DBC computes bounds on the resultant p-box under
  assumption of no knowledge about the dependence
  between the inputs.
• Williamson and Downs (1990) dependency bounds
  determined analytically for basic binary
  operations ,-,,/ using copulas
• Berleant (1993,1998) dependency bounds
  (distribution envelope) determined by linear
  programming
• DBC is implemented in the commercially available
  software Risk Calc 4.0.

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The Need for Alternative Methods

• DBC has two drawbacks
• (1) repeated variables
• (2) black-box propagation
• 3 non-deterministic methods
• (1) Double Loop Sampling (DLS)
• (2) Optimized Parameter Sampling (OPS)
• (3) P-box Convolution Sampling (PCS)
• Methods classified by
• Rigorous vs. stochastic
• Black box compatible vs. not easily black box
  compatible
• Representation of inputs parameterized vs.
  non-parameterized

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Parameterized P-boxes

• Assumes that the uncertain quantity follows some
  known distribution but with imprecise parameters.
• Definition
• A parameterized p-box with identical bounding
  curves is a subset of a general p-box.

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Double Loop Sampling (DLS)
Black Box
Black Box
Input P-Box
Black Box

• Lower and upper expected utilities are
  approximated by the minimum and maximum expected
  output values.
• But sampling doesnt work well for estimating
  extrema!

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Optimized Parameter Sampling (OPS)

• OPS is DLS with an optimization algorithm in the
  parameter loop.
• OPS solves the following two optimization
  problems
• where g represents the function of the
  probability loop.
• OPS results are less costly than DLS
• BUT g
• (1) is approximated non-deterministically, and
• (2) likely has many local extrema.
• Possible solutions
• (1) Use common random variates for each iteration
  of the probability loop.
• (2) Use multiple starting points for the
  optimizer in the parameter loop.

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P-box Convolution Sampling (PCS)

• PCS is compatible with non-parameterized p-box
  inputs.
• One iteration of PCS involves sampling an
  interval from each of the p-box inputs. This is
  repeated many times.
• These sampled intervals are then propagated
  through the black box model (in our research we
  have used optimization to accomplish this).
• Lower and upper expected values of the output
  quantity are then approximated by taking
  expectations of the resultant bounds for each set
  of sampled interval inputs.

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Classification of Methods
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Sum of Normal P-boxes

• Sum of two normal p-boxes Z A B

• Parameterized inputs
• DLS Average relative error 3.1 for 1000
  function evaluations
• OPS Average relative error 1.87 for 562
  function evaluations

• Non-parameterized inputs
• DBC Average relative error 6.2 for 100
  function evaluations
• PCS Average relative error 5.1 for 10
  function evaluations

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Transient Thermocouple Analysis

• Estimating time until thermocouple junction
  reaches 99 of the measurand temperature

• Non-Parameterized methods
• DBC Average Relative Error 5.12 for 100 p-box
  slices
• PBC Average Relative Error 1.06 for 1210
  function evaluations

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Summary

• Engineers must make decisions under uncertainty
• Value of decision is dependent on appropriateness
  of uncertainty formalism
• Tradeoff between richness of representation and
  computational cost
• P-boxes seem to be a good compromise
• Computational methods for propagating p-boxes are
  then needed that are
• Black box compatible
• Reasonably inexpensive
• Optimized Parameter Sampling (OPS) seems to be an
  improvement over Double Loop Sampling (DLS)
• Probability Bounds Convolution (PCS) propagates
  non-parameterized inputs through black box
  models.

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Challenges

• Global optimization
• Modeling knowledge of dependence
• Black box interval propagation
• Would be BIG step forward in engineering design
• Need your help