Analysis of aTaxi Supply and Demand in Hudson County
Karthik Dhore, Suraj Bhat
Fall 2014
Overview:
Placing constraints on the attributes of ride-sharing at Common Destination =
3p, Departure Delay = 300 seconds, and Maximum Circuity = 20%, we can
simulate the departures and arrivals of aTaxis on a ‘typical October weekday’ in
Hudson County. Following are the resulting trips, broken down by vehicle
capacity.
Vehicle Size: Departures Arrivals Net 50+ passenger 6,652 6,223 429 15 passenger 34,749 26,598 8,151 6 passenger 467,753 442,482 25,271 2 passenger 36,322 118,964 -­‐82,642 TOTAL: 545,476 594,267 -­‐48,791 Looking at the “Net” column, we can see the excess supply or demand for the
day, and draw general aTaxi repositioning observations. For example, we see
that we would need ~8,151 15-passenger vehicles to be repositioned into
Hudson County at the end of the day to accommodate the excess demand that
arises. In contrast, we see that we would need to reposition ~82,642 2passenger aTaxis out of Hudson County, as there are many more arrivals than
departures over the span of our simulated day.
Cumulative Net aTaxi Supply and Demand:
Given the time of each aTaxi departure and arrival, we can determine the net
supply and demand for aTaxis in Hudson County. The plot shown on the
following page displays the ‘net supply’ of aTaxis as the day progresses. At
any given point, we can determine the exact excess supply or excess demand in
the county. We see that by the end of the day, we arrive at a ~50,000 surplus.
This matches the 48,791 figure in the table above, confirming that we must
reposition this many aTaxis out of Hudson County at the end of the day.
Of course, this is a very ‘academic’ number, as we are grouping all aTaxis into one
category. A more practical approach would be to break down the departures and
arrivals by capacity. This would allow us to see, for example, how many 2-passenger
vehicles are available in or requested by the county at a given point in the day. The
plot below shows this breakdown by capacity.
As we had initially observed, we see that Hudson County quickly finds itself with
a great excess of 2-passenger vehicles, primarily achieved in the latter half of
the day. It also develops an overall deficit in 6-passenger aTaxis, though to a
lesser extent. When we consult the table shown on the first page, we see that
a deficit of ~25,000 is actually quite reasonable when we see the volume of
departures and arrivals of 6-passenger taxis. This is reassuring, as we observe
that the in-flow and out-flow of these taxis seem to be balancing themselves to
a reasonable extent. By taking a closer look at each capacity breakdown individually, shown on the
following page, we can pick up on some dynamics that were otherwise hidden
by the scale of the previous plot. In particular, we see there is a drastic rise in
demand for 50+ passenger aTaxis at around 6:00. We know that large spikes in
demand can be created by train arrivals at a station, so we suspect that this
may explain the noticeable jump here. Indeed, by inspecting the data set, we
see that pixels containing Hoboken and Secaucus railway stations account for
49.4% of all 50+ passenger aTaxi trips.
Case Study: Pixel {252, 170} – Newport, NJ
Pixel {252, 170} was responsible for generating the most aTaxi trips among all
pixels in Hudson County, with 13,270 trips originating there throughout the
simulated day. Here, we take a closer look at trips originating and terminating
at this pixel to see how they break down throughout the day and by vehicle
type.
The plot displayed to the left
shows a breakdown of aTaxi
supply and demand by vehicle
capacity. This unique pixel
displays trends quite different
from the overall
characteristics shown in the
county-wide plot above.
Immediately noticeable is the
fact that we do not see the
2-passenger vehicles develop
a drastic excess supply (from
incoming trips) as we found in
the county-wide dataset. By
consulting the first panel of
the second plot on the
following page, however, it becomes clear that an excess supply problem does
persist (since arrivals far exceed departures)—it is just on a much smaller scale.
This indicates, then, that there uniquely exist relatively few 2-passenger aTaxi
trips in the pixel. Indeed, consulting the histogram below, we can see the
number of 1 or 2 passenger trips is dwarfed by trips containing 3-5 passengers. We do see a trend of a deficit of 6-passenger vehicles for this pixel, similar to
the deficit we saw in the overall trend. Unlike the county overall, however, we
see here that 6-passenger taxis are being severely mismanaged. Of the 13,270
trips departing from the pixel, 12,164 of these were 6-passenger aTaxis. The
day finished with a deficit of ~10,000 6-passenger aTaxis, indicating that the
county would need to be well-prepared with vehicles of this capacity, and
significant rearrangement of empty vehicles would need to take place in order
for the system to run effectively.
For this specific pixel, we also see that 50-passenger vehicles rarely passed
through the county. Indeed, only 80 vehicles with greater than 15-passengers
departed or arrived in the county. Further, as shown in the Daily aTaxi Supply
by Capacity plot, all such aTaxis only appeared within a short window of time,
between 1:00 and 6:00 pm.
By examining the case of 6-passenger aTaxis in pixel {252, 170}, we see that
that a system of empty vehicle repositioning would be absolutely necessary.
How could such a system be implemented and achieved?
Empty Vehicle Repositioning
There are 196 pixels in Hudson County, with each having a demand for aTaxis at
any point in time. Furthermore, unused aTaxis at any particular pixel can be
reallocated to a different pixel to satisfy demand for an aTaxi at that particular
pixel. We move aTaxis from pixels with excess supply to pixels with excess
demand, looking to minimize the distance traveled over all aTaxis that are being
repositioned at any particular time block.
To do this, we first discretize the data of arrivals and departures into 5-minute
blocks because the departure delay of our aTaxi system is 300 seconds. We
can then set up an optimization problem for each time chunk to decide which
aTaxis to send to which pixels. An aTaxi that is arriving at a particular pixel
during this 5-minute interval can be used again for another trip departing from
that pixel in that same interval. However, that same aTaxi cannot be sent to
another pixel to be used for another departing trip in the same interval because
the time it would take to reallocate that aTaxi would exceed the departure delay
of 5 minutes.
The optimization problem involves allocating vehicles over a strongly connected
network of pixels while minimizing the total distance traveled by empty vehicles
for repositioning purposes. Although it is theoretically possible for a vehicle to
travel from a pixel to any other pixel, in reality we can only consider pixels that
are less than 3 miles away for other aTaxis that can be used in the same time
intervals for a departure trip because a departure delay of 300 seconds
precludes aTaxis traveling from farther than 3 miles away to depart from that
pixel in the same time block. Thus, each pixel can only have its demand
satisfied by a) empty aTaxis within the pixel itself, b) empty aTaxis in other
nearby pixels, or c) a “super source” pixel.
This super source pixel is used as a last resort, when there is excess demand
that cannot be reasonably satisfied by surrounding pixels. The source has a
“large” number of aTaxis that can be deployed to any pixel in the county, but
whose cost is higher than simply repositioning an already deployed aTaxi. Thus,
the decisions variables in this problem will be the number of vehicles that are
being repositioned from pixel i to pixel j. The solver is then run at each time
block to ascertain the repositioning of aTaxis from pixel i to pixel j. The solver
also needs to be run for each size aTaxi, as each is a distinct transportation
problem.
Below is the optimization problem to be solved:
For the purposes of our repositioning, we only consider the departures and
arrivals of pixels within Hudson County. By including other counties from which
excess supply of aTaxis could be used to satisfy demand within Hudson County,
we could use fewer aTaxis from the super source.