Assessing the value of information to design site investigation and construction quality assurance programs - Risk and Reliability in Geotechnical Engineering - page 500

C:Cost ($1000):

Implementation Failure

Design OK

20-0.4 x *

0

P ( X ≥ x *)

x *

E ( C | x *) = 20-0.4 x *
+ P ( X < x *)×100

Design fails

x *: Design
strength

20-0.4 x *

100

P ( X < x *)

0.06

μ X ~ N (μ′ μ ,σ′ μ )

0.05

Prior total
distribution of X

X |μ X ~ N (μ′ X ,σ′ X )

0.04

X ~ N (μ′ T X ,σ′ T X )

0.03

μ′ T X = μ′ μ

0.02

σ′ T X = ÷ ææææ

σ X 2 +σ′ μ 2

0.01

0.00

0 20 40 60 80 100

X, μ X (kPa)

Figure 13.22 Prior decision tree to select design shear strength for compacted fill.

Taking the derivative of EC ( ∗ with respect to x * and setting it equal to zero to find the
minimum produces the following result for the optimal value of x *:

100
204

2

−

x Opt

∗

=− + µ

′

( σ

2

σ

′2

) × ln ln( σσ

2

+

′2

) (13.21)

X X

π × .

2 µ

µ µ

The minimum cost that corresponds to the optimal x * is given by

100
204

2

EC () = ECx ( |

∗

) = −× + ( 20 04 . x

∗

) 100 × Φ − ln

−

ln(σσ µ

2

+ ′

2

Min Opt Opt

π × .

2

(13.22)

For example, the optimum value for x * is shown in Figure 13.23 .
The posterior decision tree to select the design strength for the compacted fill based on
QA/QC test results is shown in Figure 13.24 . Since this case with a normal prior distribution
for an uncertain mean value and a likelihood function corresponding to random sampling
from a normal distribution constitutes a conjugate pair ( Table 13.1 ), the posterior distribu-
tion is also normal with the following parameters:

µσ

′

(

2

/ nx

n

) +

+

() σ

′2

µ X µ

µ

″

=

(13.23)

µ

( σ

2

/ ) () σ

′2

X µ