Evaluating Alternative Operations Strategies to Improve Travel Time ...

SHRP 2 L11: Final Appendices
The Koh and Paxson approach is very flexible, because it allows for the case in which events are
only meaningful if multiple instances occur simultaneously. For example, a highway network
might be such that there are three alternative passes at risk of avalanche, but significant delays only
occur if all three experience an avalanche. The Koh and Paxson formulation can be used to
examine what the project value is for different quantities of simultaneous avalanches.
As shown in Equation 4, the greater the number of events, x, that can occur at one point in time,
the lower the project value. This is because the probability of many simultaneous rare events is
low. In the example of avalanche closures, we find the highest values of F(V) occurs for values of
x near the location parameter from the Gumbel distribution—around two or three closures per
month. As the number of closures per month, x, increases, F(V), the project value declines.
In the case of a perpetual horizon evaluation using the Koh and Paxson approach (using an
extreme value distribution and a perpetual, American option formulation), the present value of
benefits and costs are incorporated (as V and K, respectively) directly in the project option value
function. The reason is that the timing of the unreliability process is allowed to occur any time
within the perpetual life of the option. Thus, the certainty equivalent value of the costs of the
unreliability (or the benefits of remediating those costs) is associated with the (stochastic) event
timing. Although it would be desirable to model such circumstances more simply (i.e., separating
the project valuation from the rare event process), a closed-form representation of an American put
option using rare-event distributions and a perpetual life is unknown at this time.
In this appendix we have extended an approach first proposed by Koh and Paxson to the context of
transportation investment decision-making under uncertainty due to rare events. The method relies
on extreme value distributions for rare events to help guide investment, incorporating not only the
uncertainty surrounding the frequency of the event, but also the uncertain timing of such an event
in a way that addresses the discounted expected benefits (savings) due to the investment. Though
the options theoretic approach is more standard under the assumption of log-normal distribution,
these rare events can cause considerable disruption to the transportation network and methods
should be improved to help make the necessary investments to mitigate the effects of rare events.
PROBLEM 2
RARE-EVENT RELIABILITY VALUATION EXAMPLE
The Issue
A transportation agency in the northeastern region of the U.S. oversees a section of a network on
which typical rain conditions have minimal impact on highway network performance. However, an
extremely rare hurricane event would overwhelm and damage this section of roadway, imposing
significant trip and schedule delays on users of the network. The planners have proposed that the
agency install and maintain a pumping system for the affected network section that will pump out
the water in the rare event that the hurricane rainfalls flood the roads. The agency would like to
compare the investment cost and expected losses. The exact size and burden of these hypothetical
delays is not known because damaging hurricanes in the region are rare. Analysts have examined
the possible range of user travel and economic impacts using standard capital planning models.
They have calculated the future stream of costs from a damaging hurricane and converted the
estimated costs to present value by using a discount rate that reflects their uncertainty of the
timing and damage associated with the event. The present value of the damage calculated using
this method equals $100 million (“impact cost”) and would be borne by the agency and/or the
VALUATION OF TRAVEL-TIME RELIABILITY FOR RARE EVENTS Page C-13