Computational Methods for Decision
Making Based on Imprecise
Information
Morgan Bruns1, Chris Paredis1, and Scott Ferson2
Systems Realization Laboratory
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
Design Decision Modeling
Design Decision N
Decision N+1
Probabilities
Selection
Simulations
Alternatives
Decision
Design
Alternative A
Environmental
Factors
Decision
Design
Alternative B
S1
S5
A
S2
A
S6
S9
Prefs.
S3
S7
A
S4
A
S8
S10
A
Decision
E[UA]
Decision
Decision
Frame
Frame
Selection or
Elimination
E[UB]
Probabilities
Systems Realization Laboratory
Selection
Decision N-1
Predictions
Simulations
Computational
Method
Alternatives
Mathematical
Model
Reality
Design Computing with Uncertain Quantities
 For a given decision alternative, Ai
Computational Method
X
?
E[U ( Ai )]
Black Box
Uij  f ( Ai , x j )
 In practice, utility is dependent on uncertain quantities.
 In many engineering applications, utility is computed with a black
box model.
Systems Realization Laboratory
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: P( H )   P( H ), P ( H )   0.3, 0.6 
• Operational definition due to Walley:


P( H ) corresponds to a minimum selling price
P ( H ) corresponds to a maximum buying price
Systems Realization Laboratory
The Probability Box (p-box)
 P-boxes can be used to
represent:
•
•
•
•
Scalars
Intervals
Probability distributions
Imprecise probability
distributions
1
P( X  x)
X
FX
0.5
FX
0
-2 -1 0
1
2
x
3
4
5
6
Systems Realization Laboratory
7
State-of-the-Art for P-box Computations
 Discretizing the inputs:
1
X
0.5
0
1
x2
x2
11
x
 Each input p-box is represented as a collection of intervalprobability pairs (focal elements):
 xi   xi , xi  , pi  , for i  1,..., m
 Each interval-probability pair is propagated individually through a
Cartesian product (Yager, 1986; Williamson and Downs, 1990;
Berleant, 1998)
Systems Realization Laboratory
Example of P-box Convolution
 Example:
1
Z  X Y
1
x
• Where inputs are independent
0.5
0.5
y
0
0
0
 Focal Elements:
Y
5
6
7
20
40
8
9
10
1
, x  [3, 6], p2  13  , and  x3  [5,9], p3  13 

3  2
:  y1  [6, 7], q1  13  ,  y2  [7,9], q2  13  , and  y3  [8,9], q3  13 
X : x1  [1,5], p1 
Y

10
Cartesian Product:
1
z
X
[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
0.5
0
0
60
80
Systems Realization Laboratory
Dependency Bounds Convolution (DBC)
 DBC is a method of p-box convolution that determines bestpossible and rigorous bounds on resultant p-box:
• Best-possible in the sense of FX and FX 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.
Systems Realization Laboratory
The Need for Alternative Methods
 DBC has two drawbacks:
(1) repeated variables
(2) black-box propagation
Parameterized
Black Box Compatible
OPS
DLS
 3 non-deterministic methods:
(1) Double Loop Sampling (DLS)
(2) Optimized Parameter Sampling (OPS)
(3) P-box Convolution Sampling (PCS)
 Methods classified by
Rigorous
PCS
DBC
Non-parameterized
• Rigorous vs. stochastic
• Black box compatible vs. not easily black box compatible
• Representation of inputs: parameterized vs. non-parameterized
Systems Realization Laboratory
Parameterized P-boxes
1
P( X  x)
 Assumes that the uncertain
quantity follows some known
distribution but with imprecise
parameters.
 Definition:
X
P
X
0.5
0
-2 -1 0 1 2 3 4 5 6 7
 FX ( x; θ) : θ  θ  θ
 A parameterized p-box with
identical bounding curves is a
subset of a general p-box.
x
P( X  x)
1
X
P
0.5
0
-2 -1 0 1 2 3 4 5 6 7
x
Systems Realization Laboratory
Double Loop Sampling (DLS)
Sample 1 from P-box
CDF
X
Input P-Box
Sample 2 from P-box
CDF
CDF
X
Black
Box
E1[U ]
Black
Box
E2 [U ]
Black
Box
Em[U ]
X
Sample m from P-box
CDF
X
 Lower and upper expected utilities are approximated by
the minimum and maximum expected output values.
 But sampling doesn’t work well for estimating extrema!
Systems Realization Laboratory
Optimized Parameter Sampling (OPS)
 OPS is DLS with an optimization algorithm in the parameter loop.
 OPS solves the following two optimization problems:
(1) minimize E  g  Θ   El
 ,  
(2) maximize E  g  Θ   Eu
 ,  
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.
Systems Realization Laboratory
P-box Convolution Sampling (PCS)

1
r3

X
r1

0.5

r2
0
1
x2 x1 x3 x2 x1 x3
11
PCS is compatible with nonparameterized 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.
Systems Realization Laboratory
Classification of Methods
Parameterized
Black Box Compatible
OPS
DLS
Rigorous
PCS
DBC
Non-parameterized
Systems Realization Laboratory
Sum of Normal P-boxes
 Sum of two normal p-boxes Z = A + B:
P(A<=a)
0.6
0.4
0.2
0
-5
0
5
10
15
20
1
1
0.8
0.8
0.6
0.6
P(Z <= z)

0.8
P(B<=b)
1
0.4
0.2
0
-20
0.4
0.2
-10
a
0
10
b
20
30
0
-20
-10
0
10
20
30
40
50
z
 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
Systems Realization Laboratory
Transient Thermocouple Analysis
 Estimating time until thermocouple junction reaches 99% of the
measurand temperature
t0.99 

T 
ln 100  100 i 
6h
T 

 Dc
 Parameterized methods:
10
 Non-Parameterized
methods:
0
• DBC: Average Relative Error
= 5.12% for 100 p-box slices
• PBC: Average Relative Error
= 1.06% for 1210 function
evaluations
Average Absolute Error
DLS
OPS
10
10
10
-1
-2
-3
10
2
10
3
4
10
10
Function Evaluations
5
10
6
Systems Realization Laboratory
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 nonparameterized inputs through black box models.
Systems Realization Laboratory
Challenges
 Global optimization
 Modeling knowledge of dependence
 Black box interval propagation
 Would be BIG step forward in
engineering design
 Need your help…
Systems Realization Laboratory