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The Broom Optimization Problem, or I Smell an Ig Noble!

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Previous Entry The Broom Optimization Problem, or I Smell an Ig Noble! Oct. 27th, 2004 @ 09:03 pm Next Entry
Everyday we clean the school. I am in charge of a stretch of hallway. I have to sweep the whole hallway in 5 days time. For quite some time I have become rather obsessed, while sweeping, of how to optimize my sweeping. There are several factors to contend with:
1) Different sections of the hall get dirtier at different rates, so merely dividing the hall into fifths might not be the best use of time
2) The broom looses efficiency over time of drag. Dust is absorbed into the broom and not the pile, dust in the broom will more than likely be redistributed on the floor. Short bursts of brushing seems more effective than long drags.
3) The pattern of brushing also makes a difference. The kids tend to sweep parallel to the walls, but sweeping perpendicular to the walls seems to yield more dust in the piles, because you can better get into the molding, but to clean the whole hall using perpendicular strokes would be tedious and probably have a net loss of efficiency.
4) Re-sweeping is also a problem. The brooms are very efficient, if used in short controlled sweeps, so re-sweeping an area is a waste of energy.
5) Sweeping in one direction seems to be more efficient when combine with short sweeps, but then energy is wasted walking back to the starting position from the ending position. So is only sweeping in one position truly more efficient?

Current solution:
Currently I sweep perpendicular to the walls to bring the dust more into the center of the hall. Then I sweep parallel to the halls but only in one direction. This creates a line of dust at the place where I start and stop my sweeps. I then sweep perpendicular to the walls again bringing all my lines of grime into one concise pile. That is how I currently do things, but I have been thinking of some new algorithms that might be more efficient. Who knows, maybe I may get an Ig Noble Award if I work this all the way out. A boy can dream, a boy can dream…
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Date:October 28th, 2004 01:25 am (UTC)

Response from Team Canada

It is interesting that you are required to sweep a stretch of hallway in short strokes perpendicular to the length vector of the hall. I recall a certain sport popularized in Canada called curling, in which half of the game is sweeping perpendicular to the "rock" velocity vector. This also happens to be along a long stretch of area. My sweeping theory (ha ha) follows below.

I feel that we must evoke the rules of bowling lane oiling: note, that as the sides of the hallway are solid, statistically speaking more people will tend to walk down the middle of the hall. This is analagous to people generally throwing the ball down the center of the lane. Assuming that this is the dominant source of dust (lint, dead skin cells, broken pencil lead), dust is deposited, in general, closer to the center of the hall. However, eddy currents and the "wake" cast by pedestrian traffic will tend to move the dust in the center out towards the sides, where it will tend to stay stationary relative to the dust in the center.

Average this over the course of 5 days and you have a statistically significant dust gradient in your hall space, with dust preferentially sitting towards the outside. Assuming the dust population drops off exponentially as you go from the wall to the center, you can figure out how far from the walls the dust population is equal to 1/e the population at the walls (the dust "lifetime"). This can be your raduis of sweeping (wrt the walls), because beyond that point, any residual dust is negligible compared to the total amount of dust in the hall.

Now what I propose is that you start at one end of the hall with two brooms with the bristles touching the walls and perpendicular to the walls. You could probably win style points by crossing the handles in front of you in an "X" formation. Run from one end of the hall to the other while keeping constant pressure on the brooms, and voila! All of the dust is now swept, to a first-order approximation.

I don't think you have to worry about the broom's effectiveness to sweep dust in this case. I think the dust dragging efficiency drops off exponentially with the amount of dust in the broom, because at some point the dust will saturate the bristles and form its own de facto bristle layer, pushing all the dust with which it comes in contact.

What do you think?
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