3. Numerical Examples
3.2 Numerical Results
3.2.1 Challenge Problem A Problem A-1
For this problem, both input variables a and b are described by single intervals [0.1, 1.0] and [0.0, 1.0], respectively. We follow the procedure outlined in Chapter IV to fit a family of bounded Johnson distributions to each single interval data set. As an example, samples of cumulative density functions for the family of Johnson distributions for input variable a are shown in Figure 5.
This problem belongs to Case 1 as described in Section 2 and is solved by both optimization and sampling-based strategies and the results are shown in Figure 6.
This particular problem can also be solved by a simple deterministic optimization approach as shown below:
( )
ub b lb
ub a lb t s
ba a y
≤
≤
≤
≤
+
=
) 6 ( .
. max min/
This optimization formulation yields the bounds on the system response as [0.6922, 2] which is exactly the same as the lowermost and uppermost bounds obtained by the proposed probabilistic approach, corresponding to CDF values of 0 and 1.
114 Figure 5: Family of Johnson distributions for input
variable a for Problem A-1
0.8 1 1.2 1.4 1.6 1.8 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
System response
CDF
PBO EBO Sampling
Figure 6: Bounds on CDF of system response for Problem A-1
Problem A-2
For this problem, the input variable a is described by a single interval [0.1, 1] and has the same uncertainty representation as shown in Figure 5. Input variable b is described by multiple interval data ([0.6, 0.8], [0.4, 0.85], [0.2, 0.9], [0.0, 1.0]) and we follow the procedure outlined in Chapter IV to fit a family of bounded Johnson distributions to the multiple interval data set of input variable b. Several sample cumulative density functions from the family of Johnson distributions for input variable b are shown in Figure 7.
This problem belongs to Case 1 as described in Section 2 and is solved by both optimization and sampling-based strategies and the results are shown in Figure 8.
115 Figure 7: Family of Johnson distributions for input
variable b for Problem A-2
0.8 1 1.2 1.4 1.6 1.8 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
System response
CDF
PBO EBO Sampling
Figure 8: Bounds on CDF of system response for Problem A-2
Problem A-3
For this problem, both input variables a and b are described by multiple interval data ([0.5, 0.7], [0.3, 0.8], [0.1, 1.0]) and ([0.6, 0.6], [0.4, 0.85], [0.2, 0.9], [0.0, 1.0]), respectively, and have similar representations of uncertainty as shown in Figure 7. This problem belongs to Case 1 as described in Section 2 and is solved by both optimization and sampling-based strategies and the results are shown in Figure 9.
116
0.8 1 1.2 1.4 1.6 1.8 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
System response
CDF
PBO EBO Sampling
Figure 9: Bounds on CDF of system response for Problem A-3
Problem A-4
For this problem, the input variable a is described by a single interval [0.1, 1.0]
and has the same uncertainty representation as shown in Figure 5. Input variable b is given by a log-normal probability distribution with imprecise parameters. These parameters are given by single intervals [0.0, 1.0] and [0.1, 0.5], respectively, and have similar uncertainty representations as shown in Figure 5. We follow the procedure outlined in Section 2.1 to obtain a family of log-normal distributions for input variable b given that the distribution parameters are represented as families of Johnson distributions.
As an example, samples of cumulative density functions of the family of log-normal distributions for input variable b are shown in Figure 10.
This problem belongs to Case 2 as described in Section 2 and is solved by both optimization and sampling-based strategies and the results are shown in Figure 11.
117 Figure 10: Family of log-normal distributions for input
variable b for Problem A-4
0 1 2 3 4 5 6 7 8 9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
System Response
CDF
PBO EBO Sampling
Figure 11: Bounds on CDF of system response for Problem A-4
Problem A-5
For this problem, the input variable a is described by multiple interval data ([0.5, 0.7], [0.3, 0.8], [0.1, 1.0]) and has a similar uncertainty representation as shown in Figure 7. Input variable b is given by a log-normal probability distribution with imprecise parameters. These parameters are described by multiple intervals ([0.6, 0.8], [0.2, 0.9], [0.0, 1.0]) and ([0.3, 0.4], [0.2, 0.45], [0.1, 0.5]), respectively, and have similar uncertainty representations as shown in Figure 7. We follow the procedure outlined in Section 2.1 to obtain a family of log-normal distributions for input variable b given that the distribution parameters are represented as families of Johnson distributions. Several sample cumulative density functions from the family of log-normal distributions for input variable b are shown Figure 12.
118 Figure 12: Family of log-normal distributions for input
variable b for Problem A-5
0 1 2 3 4 5 6 7 8 9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
System response
CDF
PBO EBO Sampling
Figure 13: Bounds on CDF of system response for Problem A-5
This problem belongs to Case 2 as described in Section 2 and is solved by both optimization and sampling-based strategies and the results are shown in Figure 13.
Problem A-6
For this problem, the input variable a is described by a single interval [0.1, 1.0]
and has the same uncertainty representation as shown in Figure 5. Input variable b is given by a log-normal probability distribution with precise parameters, 0.5 for each. This problem belongs to Case 1 as described in Section 2 and is solved by both optimization and sampling-based strategies and the results are shown in Figure 14.
119
0 1 2 3 4 5 6 7 8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
System response
CDF
PBO EBO Sampling
Figure 14: Bounds on CDF of system response for Problem A-6
It is seen in Figures 6, 11 and 14 that the bounds obtained by the expectation- based optimization (EBO) formulation and the percentile-based optimization (PBO) formulation almost coincide with each other. It is seen in Figures 8, 9 and 13 that the percentile-based optimization (PBO) formulation generates rigorous bounds compared to those obtained by the expectation-based optimization (EBO) formulation. The bounds obtained by EBO are still wider than those obtained by the sampling method.
The computational efforts for both PBO and EBO methods are listed in Table 1. It is seen from Table 1 that EBO is less expensive compared to PBO for each problem.
120
Table 1: Computation effort for Challenge Problem A Challenge
Problem A
PBO EBO
Function Evaluations Percentile
Points
Function Evaluations
A-1 21 7286 526
A-2 12 2962 148
A-3 11 3750 446
A-4 15 6123 349
A-5 15 10271 556
A-6 21 2494 127