Arenas, Smyles 1
Jean-Carlos Arenas and Adam Smyles
Professor Alain Kornhauser
ORF 467: Transportation Systems Analysis
Final Project: 2nd Assignment
So far, our analysis of autonomous taxi systems has not taken into account what to do when an
aTaxi is empty (other than just leaving it where it is). In a practical setting, we can only have a
finite number of aTaxis, all of which have more use than just providing a single ride. To
maximize the efficiency of aTaxis, they should be empty relatively infrequently. When they are
empty, they should be seeking other opportunities to transport people who are nearby. In order to
discern how to best utilize empty aTaxis, we need to know when and where they are needed the
most. In order to accomplish this, we analyze the supply/demand imbalance of aTaxis in Essex
County, paying specific attention to the 20 pixels with the most aTaxi departures per day. Supply
is measured by the number of aTaxis coming into a pixel at a given time, and demand is
measured by the number of aTaxis leaving each pixel. In other words, at time t, supply is what
has just been made available for use by the arrivals of aTaxis from other locations; demand is
equivalent to the number of departures, as that measures the number of people in that pixel who
wanted a ride to some other place. For each of the 20 pixels, we can create graphs for the
cumulative arrivals/departures and the net supply (arrivals minus departures) for all four aTaxi
sizes being examined: 2-passenger, 6-passenger, 20-passenger, and 50-passenger. For example,
for the pixel (150, 250), we have the following graphs1:
1
The rest of the graphs have been submitted along with this write-up.
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Figure 1: Graphs of cumulative demand/supply and net supply of aTaxis for pixel (150, 250).
We also have supply-demand imbalance graphs for some smaller pixels.
Figure 2: Net supply graphs for 2-passenger aTaxis in pixels (139, 249), (137, 256), and (160,
248).
A negative net supply implies that there are not enough aTaxis at that time – the number of cars
arriving at the station is not enough to give everyone there a ride. Basically, a negative net
supply is equivalent to unmet demand. A positive net supply means that there are excess aTaxis.
Without an empty aTaxi management plan, these vehicles simply stay where they are and remain
unused since the demand for that place at that time has already been met. The objective is to find
a precise way to take these excess aTaxis and route them to make them available to take care of
anticipated unmet demand at some future time.
We can approach this problem using networks. Pixels with a negative net supply can be treated
as demand nodes; pixels with a positive net supply can be treated as supply nodes. To effectively
route empty aTaxis, we move aTaxis from supply nodes (source nodes) to demand nodes (sink
nodes). By assigning the distances as costs for the possible trips between supply nodes and
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demand nodes, an operational plan can be developed by solving an adaptation of the general
transportation problem often studied in optimization. The adaptation we are looking to solve will
have a time component, unlike the general problem, since a decision about where to send an
empty vehicle must be made at time t so it can be useful in serving demand at future times. In a
qualitative sense, what we want is for all of the demand to be met while minimizing the distance
traveled by the autonomous taxis. This is comparable to using principles of optimization to
maximize utility, as we have been using distance as a measure of disutility throughout our study
of transportation.
At some point in solving this optimization problem, we could still run into supply/demand
imbalances as a result of the parameters. However, a super source and a super sink (for which the
cost to arrive from a source or sink, respectively, is zero) can be used to balance the supply or
demand.
Figure 3: Example of a general source (si)-sink (ti) graph with a super source (s) and a super
sink (t).
Examining the graphs for the county in general, we see that net supply is approximately zero
until about 6:00 a.m. for all of the aTaxi sizes. The 2-passenger aTaxis rarely experience unmet
demand overall. There is always excess supply after about 30,000-35,000 seconds (a start time of
between 8:20 a.m. and 9:45 a.m.). It is worth noting that 2-passenger aTaxis have a lot more use
in small pixels than in the ones with the most departures per day. This is indicated by these
aTaxis’ relatively low unmet demand in the smaller pixels in contrast to their large excess supply
in the larger pixels.
The 20-passenger and 50-passenger aTaxis, on the other hand, experience essentially only unmet
demand. This deficit needs to be addressed, either through routing that involves the aTaxis in
nearby counties (factoring in intercounty trips) or simply getting more aTaxis of these sizes.
Given the excess of 2-passenger aTaxis, it would likely be feasible to forgo spending on those
and redirect those resources to purchasing larger (20- and 50-passenger) aTaxis. If we focus our
efforts on rerouting larger aTaxis instead of having a bigger fleet, we need to be constantly
repositioning them to the larger pixels, since that is where they get their primary use. They could
be used to service some smaller trips on their way back to minimize their time being empty, but
the point is that as soon as one of these larger aTaxis finishes a many-passenger trip, it needs to
be ready to take on a new one immediately.
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Figure 4: Supply/demand and net supply graphs for Essex County (2-passenger, 6-passenger,
20-passenger, and 50-passenger aTaxis).
The 6-passenger aTaxis experience quite a large and relatively uniform deficit throughout the
county, which makes repositioning unlikely to be of major assistance. A large reserve supply
pool deployed during the day is a more viable option for resolving this issue. However, this
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requires coordinating factors external to Essex County, since there is a systemic leak of 6passenger aTaxis — more of them leave the county than come in. Another potential solution is to
use multiple surplus 2-passenger aTaxis to serve what were originally 6-passenger aTaxi trips.
While this technically would reduce the average vehicle occupancy for the whole system, it is
likely more efficient than having more vehicles from a real-world standpoint. This plan would
allow us to meet more of the demand without having to take on the additional cost of an
excessive number of 6-passenger aTaxis when so many 2-passenger aTaxis are going unused.
The approach just described of using smaller vehicles to help meet the demand for a larger type
of vehicle is only really practical for using 2-passenger aTaxis to help alleviate the stress on 6passenger aTaxis. It would be very inefficient to use smaller taxis to help with excess demand of
20- and 50-passenger aTaxis. Therefore, maintaining a large reserve pool of these aTaxis and
constantly routing them to immediately return to large pixels to provide more service is the only
feasible solution.