Monday, February 4, 2013

Optimally Shoveling Snow

This post attempts to compare the efficiency of various approaches used to clear a rectangular area (L > B) of snow, using a good old manual shovel of width w, always moving in straight lines. We assume that the gradient of the area is zero, and that h inches of snow has fallen uniformly over the field. We also ignore the capacity of the shovel for now. The total snow that must be banked at the borders of the field is constant = LBh. Also total working distance traversed ~ LB/w.

Snow shoveling can be a grueling exercise for some, and public health departments always publish safe shoveling tips during the winter season. This is especially important for people with potential heart problems or back problems. Hopefully analyses like these will help (I'm sure people have looked at this before, but just in case ...).

Our objectives are to minimize:
1. Total work done = Total snow moved X distance moved.
2. Number of Stop-Starts with a fixed unit cost k incurred
3. Total work time in cold weather.

Given an area bx, and assuming we are moving along x, the snow nearest the starting point must be moved the maximum distance, and this distance linearly reduces to near-zero at the end of x. Thus (introducing an integral sign in this blog, yikes):
a) total work done = wxdx = wx2/2
b) Unit stop-start cost = k

Approach 1: Shoveling east-to-west (along L, banking at the west end) or bi-directional traversing:
total number of passes =  B/w
total work done = (B/w) * wL2/2 + kB/w = BL2/2 + kB/w

Note that the effort level is independent of the shovel width. A wider shovel requires fewer passes but more effort per pass.

Approach 2: Shoveling south-to-north (along B, banking at north end) or traversing in both directions:
total cost = LB2/2 + kL/w

Thus, the second approach reduces the total work done by a factor L/B. However, this reduction is achieved at the expense of more stops (which may not be a bad thing from a health perspective; k may be negative or zero for some). If L = 2B, the second approach requires 50% less effort, but requiring twice as many passes. 

Approach 3: I observed that my Canadian neighbor divides up the rectangle into two areas, always starts from the middle, and creates banks on both sides. If we bank the snow at the south and north ends:
total number of passes =  2L/w
total cost = LB2/4 + 2kL/w

The third approach is twice as effort-efficient as the second approach, at the expense of making twice as many banks, plus an additional penalty of traveling to the mid point after each pass. On the other hand, you are also less likely to run into capacity problems, since the material handled per pass is the lowest. We can also verify that among all starting points chosen for approach-3, the mid-point (B/2) yields the minimal effort. In general, any of these approaches become preferable in terms of total cost, depending on the chosen objective function. For example, we could modify approach-3 and bank snow close to the ends the field (< B/2) at their respective (west or east) ends at the expense of adding more stop-starts.

Time Considerations
If the shape of the field changes, the approaches may change, but it appears that a good principle to follow is to limit the material moved between successive bankings. In particular, we want to identify the shortest Euclidean path between every point in the field to the nearest feasible bank, and purely from an effort perspective, shovel snow along a path to a feasible bank that is optimal for all points on that path. This seems to reduce the working duration taken as well: 
Suppose we maintain constant momentum (i.e. mass X velocity = constant) during a pass, our velocity will inversely proportional to the amount of snow accumulated (e.g. 1/wx). Total time per pass ∝  wxdx = wx2/2, which is proportional to the effort.

For such a velocity model, approach-3 also appears to be optimal from the time perspective. If we assume constant velocity, then the three approaches take the same working time, ignoring the shovel-free travel time required for approach-3. Among these alternative time-optimal approaches, the third approach does the best on the effort-objective.

Lawn Mowing
Non-propelled, walking lawn mowers with bags? Regardless of the traversing method adopted, our effort (per our chosen model) increases as the square of distance until we empty our cut-grass. If we adopt approach-2 or 3, we have more banking opportunities.  However, approach-3 may result in a mowing pattern that won't look so nice, so approach 2 may be more preferable. Operating a self-propelled mower with built-in mulching may reduce our effort considerably.

Update 1: Post "Nemo" Shoveling, February 9, 2013
A truly optimal solution is to have a kind and wonderful neighbor. After three hours of efficient shoveling (h = 24 inches) and 40% completion, Kevin zipped in with his snow plough and finished the remainder in 5 minutes before going off to help others. I now have to find the best way to thank him.

Update 2: (WSJ, via @ThaddeusKTSim), February 10
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