The How of Hurricane Response

After a Hurricane or other natural disaster, debris poses a huge problem; it knocks out vital infrastructure like electricity, and blocks roads — inhibiting the work of rescue crews and cutting off victims from access to hospitals, supplies, and evacuation routes. It is vitally important that disaster relief efforts have plans for using their limited resources to efficiently clear debris off roads — giving aid to as many people as possible, as quickly as possible.

This was the scenario the Harvard Institute for Applied Computational Sciences presented to two teams of graduate students last week, as part of their first computational challenge. They were given digitized road maps of Cambridge, MA, information about the population density, and a realistic projection of the road debris that would be left behind after a major Hurricane. Each team was given two weeks to design an algorithm to efficiently clear debris, minimizing the amount of time people are cut off from access to local hospitals. I was a member of one of these teams.

Within this scenario, we have enough resources to clear a limited amount of debris each day. All of the bulldozers start off at two local hospitals (admittedly somewhat contrived — in a real scenario, relief aid would likely work their way in from outside the disaster area). At any given time, we can only clear debris on the roads immediately adjacent to roads that we have already cleared — that is, we can’t magically airdrop bulldozers into the most heavily damaged areas. Instead, we have to clear our way to these areas.

“Solutions” to the problem consist of a schedule of which roads to clear, in which order. Whichever solution is most efficient in giving as many people fast access to hospitals, wins.

Coming up with a Solution

In principle, this problem can be solved very easily — just consider every possible schedule for clearing the roads, and choose the one which restores hospital access most quickly. Unfortunately, this approach is utterly infeasible — our map of Cambridge has 604 road segments, yielding about 604! or 10^{1420} options (thats a 1 with 1420 zeros after it). Even on the fastest computers in the world (or the future, really), this calculation would take far longer than the current age of the universe to complete. We need a more intelligent way of searching through possible solutions.

The approach we came up with turns out to be highly effective. It involves altering a given schedule to generate a new, similar, and possibly better solution. We called this “nudging” the schedule, and it works as follows:

  1. Take an initial schedule that solves the problem, but in a non-optimal way (these are easy to come up with).
  2. Truncate the schedule at some point, keeping only the first N decisions.
  3. Determine which points in the city do not have access to a hospital after these steps. Choose one at random.
  4. Clear out the most efficient path (the minimum-weight path, in Graph-algorithmic jargon) from one of the hospitals to this location. Add these decisions immediately after the partial-schedule from step 2.
  5. Add the rest of the decisions that were truncated in step 2 (paying careful attention to not duplicate any work that you did in step 4).

Nudging solutions has a lot of nice properties: its fast (we can easily do it ~100 times per second with our relatively slow python code), and provides a way to re-prioritize a schedule, since the location chosen step 3 is rescued earlier in the new plan than it was in the old. It isn’t too hard to convince yourself that, with enough nudging, it is possible to arrive at the globally optimum solution from any starting solution. And the choice of using the most efficient path in step 4 tends to create effective schedules which don’t waste resources.

Equipped with a strategy of nudging solutions, there are many algorithms which can search for the best strategy. We chose Simulated Annealing, which works more or less as follows:

  1. Start with a schedule S1
  2. Nudge S1 to generate a new schedule, S2
  3. If S2 is a better schedule, throw away S1
  4. If it is worse, then throw away S2 with some probability related to how much worse of a solution it is
  5. Repat the process with the solution that wasn’t discarded
The rejection probability in step 4 is gruadually lowered throughout the process — at the beginning, almost all solutions are accepted, allowing the algorithm to explore a wide range of possible scenarios. As the probability drops, the solution is gradually confined to better and better solutions.

The algorithmic showdown

After two weeks of development, each team applied their algorithm to a slightly modified map of cambridge. We were given 3 hours of computing time on one of Harvards super-computers to run our algorithm and come up with the best possible strategy.

Our solution turned out to be very effective. After only a few minutes of nudging, we had found a solution better than the competing team’s final answer. Furthermore, our strategy out-performed the solution generated by the competitions’ organizers from Georgia Tech, which was previously thought to be near-optimal.

How nudging solutions with simulated annealing decreases the penalty function, as a function of time. Each black line depicts a series of nudges as a function of computation time. The penalty function relates to how long each resident is stranded without hospital access (lower numbers are better). The penalty function corresponding to the organizers' solution is drawn in green. The red line depicts a strict lower limit to the penalty -- no solution can be better than this, given the amount of debris on the roads.

We can put our performance in more concrete terms: with our strategy, the average resident is stranded without hospital access for 2 days and 18 hours. The previous ‘optimal’ strategy kept the average resident waiting for 2 days and 21 hours, and naive strategies (i.e. clearing off roads at random) will keep residents waiting for over 4 days on average. These extra hours would cost many lives, since common post-disaster health problems like dehydration and cholera progress on the timescale of hours to days.

I’m pretty satisfied with our work these past few weeks. This approach isn’t too different from some of the data analysis tasks I tackle within astrophysics, and it was great to see these same techniques successfully handle a problem with real humanitarian benefit. I hope our solution gets taken under consideration in future disaster relief research — to encourage this, I’ve posted our code online.

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