1. 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
2Applied Biomathematics

2. Design Decision Modeling
Reality
Mathematical
Model
Computational
Method
Predictions
Decision

3. 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.
Computational Method ?
Black Box

4. 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

5. The Probability Box (p-box)
P-boxes can be used to represent:
Scalars
Intervals
Probability distributions
Imprecise probability distributions x -2 -1 1 2 3 4 5 6 7
0.5

6. 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)

7. Example of P-box Convolution
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

8. 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.

9. The Need for Alternative Methods
OPS
DLS
PCS
DBC
Parameterized
Non-parameterized
Rigorous
Black Box Compatible
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

10. Parameterized P-boxes-2 -1 1 2 3 4 5 6 7
0.5
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. -2 -1 1 2 3 4 5 6 7
0.5 x

11. Double Loop Sampling (DLS)
Sample 1 from P-box
Black
Box
Sample 2 from P-box
Black
Box
Input P-Box
Sample m from P-box
Black
Box
Lower and upper expected utilities are approximated by the minimum and maximum expected output values.
But sampling doesn’t work well for estimating extrema!

12. 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.

13. 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. 1 11
0.5 X

14. Classification of Methods
OPS
DLS
PCS
DBC
Parameterized
Non-parameterized
Rigorous
Black Box Compatible

15. 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

16. Transient Thermocouple Analysis
Estimating time until thermocouple junction reaches 99% of the measurand temperature
Parameterized methods:
Non-Parameterized methods:
DBC: Average Relative Error = 5.12% for 100 p-box slices
PBC: Average Relative Error = 1.06% for 1210 function evaluations

17. 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.

18. Modeling knowledge of dependence Black box interval propagation
Challenges
Global optimization
Modeling knowledge of dependence
Black box interval propagation
Would be BIG step forward in engineering design
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