Computational Methods for Decision Making Based on Imprecise Information

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 www.srl.gatech.edu 2Applied Biomathematics

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

3. Computational Method Black Box 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. ?

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. 1 0.5 0 -2 -1 0 1 2 3 4 5 6 7 x The Probability Box (p-box) • P-boxes can be used to represent: • Scalars • Intervals • Probability distributions • Imprecise probability distributions

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

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. Parameterized Black Box Compatible OPS DLS Rigorous PCS DBC Non-parameterized 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

10. 1 1 0.5 0.5 0 0 -2 -1 0 1 2 3 4 5 6 7 -2 -1 0 1 2 3 4 5 6 7 x x 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.

11. Sample 1 from P-box Sample 2 from P-box Sample m from P-box 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 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. 1 X 0.5 0 1 11 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.

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

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. Parameterized methods: 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

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