Evaluating Alternative Operations Strategies to Improve Travel Time ...

SHRP 2 L11: Final Appendices
Step 2 – Calculate Certainty-Equivalent Value of Unreliability associated with Rare
Event
A. For the given example,
V = the total present discounted value of the cost incurred
= Expected cost per event × Expected number of events = $100 million × x
*
V = the value of the event below which it is worth continuing to wait to invest K
K = the present value of cost invested to mitigate impacts
= $101 million
r = risk-free interest rate = 9%
π = the mean of the random process that generates V = 0.05
s = the standard deviation of the random process that generates V
= 0.10
The certainty-equivalent value of uncertainty is given by
−( x−µ
) −( x−µ
)
β
β
⎛
e σ
σ −
⎞
V ⎜e e ⎟
*
FV ( ; K, x, π , s) Gumbel
=
⎝
⎠
if V < V
β −1
⎛ Kβ
⎞
β ⎜ β 1
⎟
⎝ − ⎠
−( x−µ
)
⎛
−e
σ ⎞
−( x−µ
)
⎜ ⎟
σ
*
= V⎜e ⎟−K
if V ≥V
⎜ ⎟
⎝ ⎠
where
1 π ⎛ π 1⎞
2r
β = − + 1
2 ⎜ −
2 ⎟ + >
2
2 s ⎝s 2⎠
s
Kβ
V*
=
−( x−µ
)
⎛ −e
σ
−( x−µ
) ⎞
⎜ ⎟
σ
⎜e
⎟( β −1)
⎜ ⎟
⎝ ⎠
x = the number of hurricanes during the lifespan of the strategy/policy
2
VALUATION OF TRAVEL-TIME RELIABILITY FOR RARE EVENTS Page C-16