# No Transfers Required: Integrating Last Mile with Public Transit Using Opti-
Mile

Raashid Altaf1 and Pravesh Biyani2 1Raashid Altaf is a PhD Candidate in
Department of Computer Science and Engineering, Indraprashta Institute of
Information Technology, Delhi, India<EMAIL_ADDRESS>Biyani is an
Associate Professor with the Department of Electronics and Communications
Engineering, Indraprashta Institute of Information Technology, Delhi, India
<EMAIL_ADDRESS>
###### Abstract

Public transit is a popular mode of transit due to its affordability, despite
the inconveniences due to the necessity of transfers required to reach most
areas. For example, in the bus and metro network of New Delhi, only 30% of
stops can be directly accessed from any starting point, thus requiring
transfers for most commutes. Additionally, last-mile services like rickshaws,
tuk-tuks or shuttles are commonly used as feeders to the nearest public
transit access points, which further adds to the complexity and inefficiency
of a journey. Ultimately, users often face a tradeoff between coverage and
transfers to reach their destination, regardless of the mode of transit or the
use of last-mile services.

To address the problem of limited accessibility and inefficiency due to
transfers in public transit systems, we propose “opti-mile,” a novel trip
planning approach that combines last-mile services with public transit such
that no transfers are required. Opti-mile allows users to customise trip
parameters such as maximum walking distance, and acceptable fare range. We
analyse the transit network of New Delhi, evaluating the efficiency,
feasibility and advantages of opti-mile for optimal multi-modal trips between
randomly selected source-destination pairs. We demonstrate that opti-mile
trips lead to a 10% reduction in distance travelled for 18% increase in price
compared to traditional shortest paths. We also show that opti-mile trips
provide better coverage of the city than public transit, without a significant
fare increase.

##  I Introduction

Public transportation is a cost-efficient mode of travel. However, in
developing countries like India, it may not always be the most reliable option
due to delays and other operational issues. An ever-increasing number of
private vehicles in a rapidly growing urban environment has led to typical
transportation problems such as high traffic, crowded public transit, noise
and air pollution, among other things. Rapid urbanisation also leads to
unplanned expansion of residential areas in the city, which are not always
easily accessible through public transit.

Although public transit offers broad coverage across the serviced region,
achieving this coverage often necessitates transfers between different routes
or modes of transportation. In Delhi, for example, only approximately 30% of
the stops can be directly accessed from any starting stop. Such transfers are
inconvenient and introduce uncertainty into travel plans due to potential
delays. Moreover, the extent of coverage provided by public transit heavily
relies on the willingness of users to walk significant distances.

The coverage of a transit mode in a geographical region is the fraction of the
area of the region that is accessible by the transit mode. A location is
accessible by a transit mode in a distance of $k$ kilometres if the nearest
transit access point is no more than $k$ kilometres away from it. For
instance, public transit covers 97.40% of the area for the 10% most densely
populated municipal wards in New Delhi if the commuters walk up to 1 kilometre
to/from transit. However, this coverage drops by up to 35% if users only walk
up to 500 metres. This drop in coverage is because public transit(PT) does not
always connect directly to homes, offices and other places of interest for
commuters. A last-mile (LM) service is usually needed to bridge the distance
from a location of interest to the nearest public transit point.

The last mile refers to the initial/final leg of a person’s journey from a
transit stop to their destination and is often the most critical part of the
commute. Last-mile services are of two types: dedicated last-mile, which
includes bike/car taxis and shared last-mile, which include rickshaws, tuk tuk
and other modes of transportation common in South-East Asia and other
developing countries worldwide as a cheap and efficient way to get around a
city.

While the last mile is an obvious way to increase the coverage provided by
public transit, it is often just used as a feeder to the nearest public
transit access point. Most people walk or take a last-mile service to the
nearest stop, as it makes the most intuitive sense. However, adding a last-
mile leg to the public transit adds a layer of complexity to the trip. When
combined with potential transfers in public transit, this can make a trip
inefficient. A user often has to make a trade-off between the convenience of a
trip and the number of transfers they have to make to reach their destination.
There is seemingly only the option of either taking an end-to-end cab or
enduring the transfers and delays associated with PT, which decreases the
efficiency of the trip. We posit that a trip that includes a last-mile and no
transfers in the public transit leg would be a more convenient way to travel.

###  I-A Opti-Mile

This paper introduces the term “opti-mile” to describe a trip where a commuter
chooses to walk or use a last-mile service to reach a stop that ensures a
direct public transit (PT) connection to the destination (Fig. 1). Opti-Mile
optimises the overall travel experience by strategically selecting a stop that
offers a more efficient public transit journey rather than simply boarding the
nearest stop to the commuter’s location.

Figure 1: An Opti-Mile trip ensures a direct public transit connection with
the help of last-mile services

We measure the efficiency of numerous multi-modal trips (LM + PT) in Delhi and
illustrate that the most efficient trips are more likely to be opti-mile
trips. We also show that an opti-mile trip does not require a user to
compromise on the coverage (convenience) as an opti-mile trip provides similar
or better coverage than public transit without needing any transfers in the
public transit leg. We perform two experiments to draw our conclusions:

  1. 1.

We randomly sample 1000 source-destination pairs from bounding boxes selected
based on the population density of the different municipal wards in the city.
For each pair, we record the paths having the highest and the lowest
efficiency and observe their attributes.

  2. 2.

We also measure the coverage provided by opti-mile trips by randomly sampling
1000 locations and measuring the area coverage achievable from the location
through just opti-mile trips.

###  I-B Contributions and Insights

Main objectives in this paper are:

  1. 1.

To showcase opti-mile as an effective strategy for enhancing the accessibility
and usability of public transit networks. We take Delhi as our case study.

  2. 2.

To highlight the effect of user commuting preferences such as fare, number of
transfers, or preference of public transit over the last-mile on the optimal
path in a multi-modal transit network.

Based on our experiments, we argue that the opti-mile trip is a more efficient
way of integrating public transit vis-a-vis traditional LM. Our results
indicate that:

  1. 1.

The most efficient paths tend to be opti-mile, with 90% of them preferring
longer last-mile ranges and penalising transfers in public transit more than
the least efficient paths. This implies that the paths favour longer last-mile
services over transfers in public transit, as expected for an opti-mile trip.

  2. 2.

An opti-mile trip of two kilometre range provides similar area coverage as the
transit network coverage with a 500-metre walk.

  3. 3.

Opti-Mile trips offer a 10% reduction in distance traveled but come with an
18% increase in fare compared to unrestricted trips with no transfer
limitations. The lack of transfers and lower travel distance may make opti-
mile a compelling choice for non-regular users of public transit, contributing
to increased adoption of public transportation.

##  II Related Works

Integrating first- and last-mile services with a main transit mode has long
been a crucial research topic in Operations Research. Previous studies have
mainly focused on improving the efficiency and sustainability of last-mile
delivery for goods [1, 2] or identifying the factors that influence the demand
and satisfaction of last-mile services for passengers [3, 4, 5, 6, 7].
However, Silva [2] points out that currently there is no ideal solution for
the last-mile problem in practice, especially when considering the conflicting
objectives of different stakeholders.

Existing approaches for solving the last-mile problem in public transit can be
classified into three categories: scheduled services [8, 9, 10], flexible and
curb-to-curb services [11], or a hybrid of both [12]. These approaches are
variations of the Transit Route Network Design Problem (TRNDP), where the
routes and frequencies of last-mile services are optimised according to the
demand at the nodes and the availability of vehicles [13]. Grahn [8] proposed
a system that integrates public transit shuttles with private transportation
network companies like Uber to connect riders to mainline transit. Xiong [9]
and Xie [10] design a community shuttle system that provides flexible mobility
to public transit passengers to access metro/rapid transit networks. Pei [11]
developed a flexible routing model that offers alternative options to
commuters based on their willingness to pay, aiming to balance the goals of
passengers and bus companies. Authors in [14, 5, 15] aim to reduce the
vehicle-hours in the system and increase transit ridership by providing a fast
and reliable mobility service at a low cost.

Many research works on last-mile integration concentrates on providing access
to public transit rather than optimising the entire journey. This approach may
improve the connectivity of shuttle services to public transit stops, but it
does not guarantee that public transit is used efficiently. Our experiments
show that connecting users to the nearest public transit stops often result in
more transfers, leading to inefficient trips and low satisfaction with public
transit [3].

We consider the entire trip as a unit of analysis and suggest policy changes
that would improve the end-to-end journey for a commuter instead of focusing
on the last-mile and public transit legs separately.

##  III Trip Planning Overview

Trip Planning is a method of providing an optimal path between two points in a
transit network where optimality is defined under some pre-defined contraints,
e.g. travel time. Opti-mile is a trip planning method with constraints on
number of transfers and fare. To explain opti-mile, we first provide a general
overview of the trip planning paradigm and describe the trip planning model
used.

A trip planning system consists of two parts: a network model, and an
optimisation problem.

###  III-A Trip Planning in Public Transit

####  III-A1 Network Model

We model the transit network as a graph G (V, E) such that V denotes a set
$\\{v_{1},v_{2},\ldots v_{n}\\}$ of public transit stops and
$E=\\{e_{1},e_{2},\ldots,e_{m}\\}$ denotes the edges between any two stops
with weight $w_{i}\forall e_{i}\in E$. We also define a set of $|R|$ routes
indexed by unique IDs $R\subset N$.

Each route $r\in R$ is defined as a sequence of edges
$\\{e_{i_{1}},e_{i_{2}},\dots,e_{i_{k}}\\}$ for $e_{i_{j}}\in E$,
$i_{j}\in\\{1,2,\ldots,m\\}$, and $j\in\\{1,2,...,k\\}$.

####  III-A2 Cost of a Path

The cost of a path in a graph is the sum of edge weights. In case of trip
planning, the travel time between two nodes is considered to be the weight of
the edge. Let $p(v_{i},v_{j})=\\{e_{b_{1}},e_{b_{2}},\dots,e_{b_{a}}\\}$ be a
sequence of $a$ edges from $v_{i}$ to $v_{j}$. The cost $c(p)$ of the path
$p(v_{i},v_{j})$ is:

$c(p)=\sum_{e_{b_{k}}\in p}w_{b_{k}}$

####  III-A3 Optimisation Problem

The shortest path problem is the most commonly used method to obtain the
optimal path in a transit network graph. If
$P\left(u,v\right)=\left\\{p_{1}\left(u,v\right),p_{2}\left(u,v\right),\ldots,p_{t}\left(u,v\right)\right\\}$
is the set of $t$ possible paths between a source-destination pair
$\left(u,v\right)$, the optimal path is a path $\tilde{p}\left(u,v\right)$
such that;

$c\left(\tilde{p}\right)=\min_{p\in P\left(u,v\right)}c(p)$

###  III-B Last-Mile Integrated Trip Planning

Delhi’s public transit network includes buses and metro. The bus network has
6700+ stops and 7000+ buses on 2000+ routes. The metro network consists of
250+ stops and 14 lines. Modeling the PT network using the approach described
in section III-A1 results in a dense network with 150,000+ edges. Traditional
shortest path algorithms on such a graph are computationally expensive due the
numerous exploration paths. To address this, we remodel the transit network as
a bipartite graph, reducing the maximum path depth to 1 for any source-
destination pair. This significantly reduces computational complexity, despite
possibly having an equal or greater number of edges than the original graph
$G$.

####  III-B1 Remodelled PTN Graph:

We take two sets $V^{s}$ and $V^{d}$ such that $V^{s}=V^{d}=V$. The bipartite
graph is then defined as $G_{b}(V^{s}\cup V^{d},E_{b})$ (Fig. 2(a)). We define
a directed edge $e_{ij}\in E_{b}$ for $i,j\in\\{1,\ldots,m\\}$ if: there
exists a direct route between $v^{s}_{i}\in V^{s}$ and $v^{d}_{j}\in V^{d}$. A
tuple $(a_{ij},b_{ij})$ is also associated with every edge $e_{ij}\in E_{b}$
where $a_{ij}$ and $b_{ij}$ are the travel time and the number of transfer
between nodes $v^{s}_{i}\in V^{s}$ and $v^{d}_{j}\in V^{d}$ respectively. As
the node $v^{s}_{i}=v^{d}_{i}$, edges $e_{ii}$ are not included in the graph.

####  III-B2 Last Mile Integration

To enable integrated trip planning that combines last mile and public transit
services, we make two key observations. First, the route for last mile
services depends on the origin and destination and cannot be pre-computed.
Second, public transit connections between stops remain static and do not
undergo regular changes.

Based on these observations, we propose a dynamic graph construction method
that preserves the stationary component of the Last Mile Public Transit
Network (LMPTN): the PTN graph $G_{b}$, for each source-destination pair. This
eliminates the need to reconstruct the entire LMPTN for every new pair. The
graph construction method follows these steps: Given a query, we introduce two
dummy nodes, $s$ and $d$, into the graph $G_{b}$, representing the origin and
destination locations, respectively. We define a new set of edges, $E_{LM}$,
where directed edges $e_{si}\in E_{LM}$ connect $s$ to some $v_{i}^{s}\in
V^{s}$, and $e_{id}\in E_{LM}$ connect some $v_{i}^{d}\in V^{d}$ to $d$. These
edges represent the last mile trips to or from the transit mode stops.
Consequently, we define the integrated last mile and public transit network
graph (LMPTN) as $G_{b}(V^{s}\cup V^{d}\cup\\{s,d\\},E_{b}\cup E_{LM})$ (Fig
2(b)).

(a) Remodelled PTN Graph

(b) LMPTN Graph

Figure 2: Public Transit Network Model

To ensure practical last mile connections in a multi-modal transit scenario,
we restrict the edges between the dummy source and destination nodes and
nearby transit nodes by avoiding connections to distant locations. The
distance range can be dynamically adjusted according to user preferences
without compromising optimality.

####  III-B3 Cost of a Path

In our path-cost definition (as in section III-A2), we account for multiple
transit modes and acknowledge that users may have preferences among them. To
reflect this, we introduce penalties for each transit mode. Let $w_{LM}$ and
$w_{PT}$ represent the penalties for last mile travel time and public transit
travel time, respectively. In a multi-modal network, a journey consists of two
last mile legs and one public transit leg. For a path $p(s,d)$ between
$(s,d)$, with $a\in V^{s}$ and $b\in V^{d}$ as the entry and exit points of
the public transit leg, we define the following parameters:

  1. 1.

$tt_{PT}(a,b)$: the travel time through public transit between $a\in V^{s}$
and $b\in V^{s}$.

  2. 2.

$tt_{LM}(s,a)$ and $tt_{LM}(b,d)$: the travel time through a last mile service
from the source to a transit stop $a\in V^{s}$ and from a transit stop $b\in
V^{s}$ to the destination respectively.

  3. 3.

$f(p)$: the total fare of a multi-modal path $p$

For a source destination pair $(s,d)$, we define a path $p\left(s,d\right)$ in
$G_{b}$ as a vector $\left(s,a^{*},b^{*},d\right)$, where $a^{*}\in V^{s}$ and
$b^{*}\in V^{d}$. The cost of a path $c\left(p\right)$ is then a convex
combination of the travel times incurred through last mile and public transit,
i.e:

$c\left(p\right)=c\left(s,a^{*},b^{*},d\right)=w_{LM}tt_{LM}+w_{PT}tt_{PT}(a^{*},b^{*})$
(1)

where $w_{LM}+w_{PT}=1$ and $w_{LM},w_{PT}>0$.

Here, $tt_{LM}$ is the total time spent in a last-mile service while
travelling on path $p$ such that:

$tt_{LM}=tt_{LM}(s,a^{*})+tt_{LM}(b^{*},d)$

####  III-B4 Optimisation Problem

To obtain the optimal path between a source and destination, our objective is
to find the transit stops $a^{*}$ and $b^{*}$ that minimise the time-cost
$c\left(p\right)$ while satisfying the defined constraints. Let $MAX\\_FARE$
be the upper limit of fare of the journey set by the user.

The optimisation problem is defined as:

$\displaystyle\min_{\begin{subarray}{c}a^{*}\in V^{s}\\\ b^{*}\in
V^{d}\end{subarray}}$ $\displaystyle\quad c\left(s,a^{*},b^{*},d\right)$ s.t.
$\displaystyle\quad f(s,a^{*},b^{*},d)<MAX\\_FARE$ $\displaystyle\quad
w_{LM}+w_{PT}=1$ $\displaystyle\quad
tt_{LM}=tt_{LM}(s,a^{*})+tt_{LM}(b^{*},d)$ $\displaystyle\quad
w_{LM},w_{PT},f(s,a^{*},d)>0$

The solution to the above problem is a path starting from $s$ connecting to
$a^{*}$ through a last-mile, followed by a public transit connection from
$a^{*}$ to $b^{*}$ and finally a last mile connection from $b^{*}$ to the
destination $d$. The algorithm may suggest walking to or from the transit stop
towards the destination or source respectively. We include the provision of
walking whenever mentioning last-mile for simplicity.

####  III-B5 Input Parameters

The optimal path in a multi-modal network includes penalties and constraints
that may vary between users. So, we formally define the following parameters
as user-specific dynamic input parameters:

  1. 1.

Max-Fare: $\in S_{1}=\left\\{50+10n\mid n\in\mathbb{N},0\leq n\leq
45\right\\}$

  2. 2.

Last Mile Penalty:

$w_{LM}\in S_{LM}=\left\\{0.1+0.1n\mid n\in\mathbb{R^{+}},0\leq n\leq
4\right\\}$

  3. 3.

Public Transit Penalty, $w_{PT}\in\left\\{1-2*x\mid x\in S_{LM}\right\\}$

  4. 4.

LM range, $r\in\left\\{2,5,10\right\\}$: The maximum distance that the user is
willing to travel by last mile services.

####  III-B6 Path attributes

We assign different range of values to the input parameters and observe the
effect on the chosen path between a source and a destination. Specifically, we
measure the following attributes of the optimal path:

  1. 1.

Fare/km: The average fare per kilometer of the trip.

  2. 2.

Travel Time

  3. 3.

Total Distance

  4. 4.

Public Transit Fare: as per transit agency fare rules.

  5. 5.

Last-Mile Fare: Rs. 25 for first km and Rs. 10 for subsequent km.

##  IV Experiment Details

We opine that opti-mile trips lead to improved city coverage, improved
accessibility of public transit, and more efficient journeys. To demonstrate
this, we conduct two experiments: analyse path efficiency based on output
attributes and study the impact of opti-mile trips on city coverage.

###  IV-A Experiment 1: Measuring the Efficiency of a Path

For a path $p\left(u,v\right)$ between a source-destination pair
$\left(u,v\right)$ in the LMPTN, efficiency $\Lambda\left(p\right)$ is a
function of normalised convenience and cost-effectiveness of a path.
Convenience measures the ease and comfort of using a transportation option,
while cost-effectiveness evaluates the economic efficiency of the path in
terms of value provided to the passenger. Formally, Convenience,
$C\left(p\right)$ is defined as the ratio of the total distance covered
$d\left(p\right)$ to the cost of the path, while Cost-Effectiveness,
$\left(E\left(p\right)\right)$ is the ratio of the total fare
$\left(f\left(p\right)\right)$ to the total distance traveled:

$\displaystyle C(p)$ $\displaystyle=\frac{d(p)}{c(p)},\quad$
$\displaystyle\quad E(p)$ $\displaystyle=\frac{d(p)}{f(p)}$

Since $E\left(p\right)$ and $C\left(p\right)$ have different units
(kilometer/Rupee and km/s, respectively), we normalise them as
$E_{norm}\left(p\right)$ and $C_{norm}\left(p\right)$ to facilitate a linear
combination of the two factors.

$E_{norm}\left(p\right)=\frac{E\left(p\right)-\min\left(E\left(p^{*}\right)\right)}{\max\left(E\left(p^{*}\right)\right)-\min\left(E\left(p^{*}\right)\right)}$
$C_{norm}\left(p\right)=\frac{C\left(p\right)-\min\left(C\left(p^{*}\right)\right)}{\max\left(C\left(p^{*}\right)\right)-\min\left(C\left(p^{*}\right)\right)}$

The maximum and minimum values of $E\left(p\right)$ and $C\left(p\right)$ are
obtained from the results of our experiments for all the possible paths
between a given source-destination pair.

####  IV-A1 Efficiency of a Path

The efficiency of a path $\Lambda\left(p\right)$ is the linear combination of
$E_{norm}\left(p\right)$ and $C_{norm}\left(p\right)$:

$\Lambda\left(p\right)=w_{C}*C_{norm}\left(p\right)+w_{E}*E_{norm}\left(p\right)$

where $w_{C},w_{E}>0$ denote the weights assigned to convenience and cost-
effectiveness, respectively, and $w_{C}+w_{E}=1$.

A path $p_{a}$ is considered to be better than path $p_{b}$ if and only if
$\Lambda\left(p_{a}\right)>\Lambda\left(p_{b}\right)$

####  IV-A2 Methodology

The following steps were taken to perform the experiment:

  1. 1.

We first create the PTN graph mentioned in section III.

  2. 2.

1000 pairs of source-destination locations were randomly sampled from Delhi-
NCR region. For every source-destination pair:

    1. (a)

Let the MAX_FARE = 60, $w_{LM}=0.2$, $w_{PT}=0.8$, and LM Range = 5km.

    2. (b)

Take graph $G_{b}$ and construct edges $e_{si}$ from $s$ to $v_{i}^{s}\in
V^{s}$ where the distance between the stop $v_{i}^{s}$ and
$s$,$d^{s}\left(v_{i}^{s},s\right)\leq\text{LM Range}$

    3. (c)

Also construct edges $e_{jd}$ from $v_{j}^{d}\in V^{d}$ to $d$, where the
distance between the stop $v_{j}^{d}$ and $d$,
$d^{d}\left(v_{j}^{d},d\right)\leq\text{LM Range}$

    4. (d)

In the resultant graph, solve the optimisation problem given in section III-B4

    5. (e)

Record the output attributes (section III-B6) of the optimal solution.

    6. (f)

Repeat from 2b to 2e for all combinations of the input parameter values given
in section III-B6.

The results from this experiment were used to make the observations detailed
in section V.

###  IV-B Experiment 2: Effect of Opti-Mile on Coverage

To evaluate the coverage of the Public Transit Network (PTN) in Delhi, we
employ a random sampling approach using a collection of bounding boxes on the
map of the Delhi-NCR region. We establish four bounding boxes (Fig. 3) based
on the boundaries that enclose the most densely populated municipal wards,
representing population densities of 10%, 50%, 80%, and 90% of the total city
population [16].

To analyse the impact of opti-mile trips on coverage, we calculate the
combined area covered by the bus and metro network in Delhi. This coverage is
then compared to the area covered through last mile trips. The coverage is
measured for various walking/last mile distances, as indicated in Table III.

####  IV-B1 Transit network Coverage

To measure the area coverage of the transit network, we perform the following
steps for every bounding box:

  1. 1.

Mark the locations of all the bus and metro stops.

  2. 2.

For each stop, draw a circle with a radius of 500 meters. Exclude any area
outside the bounding box to obtain the stop coverage area.

  3. 3.

Combine all the individual stop coverage areas to create a coverage map.

  4. 4.

Calculate the total area coverage by dividing the area of the coverage map by
the area of the bounding box.

####  IV-B2 Opti-Mile coverage

For every bounding box, we perform the following steps :

  1. 1.

Randomly sample 1000 random locations within the bounding box. For every
location:

    1. (a)

Identify the nearest bus and metro stops within a 2-kilometer range. These are
the source stops.

    2. (b)

For each source stop, determine all possible stops that can be reached through
a direct connection and record their locations as destination stops.

    3. (c)

Create a circle with a radius of 2 kilometers around each source and
destination stop. Exclude any area outside the bounding box to obtain the stop
coverage area.

    4. (d)

Combine all the individual stop coverage areas to create a coverage map.

    5. (e)

Calculate the area coverage by dividing the area of the coverage map by the
area of the bounding box. This represents the coverage for one location.

    6. (f)

Repeat steps 1b to 1e for different radius ranges and note the corresponding
coverage values.

  2. 2.

The approximate opti-mile coverage for the bounding box is determined as the
average of all the area coverage values recorded in 1

##  V Observations

Figure 3: Coverage map showing the 90% population density bounding box. Blue
represents the transit network, while red represents the coverage area

###  V-A Experiment 1: Measuring the Efficiency of a Path

####  V-A1 Fare and Distance Trade-Off

An opti-mile trip involves trade-off between and distance. Table I hows that
opti-mile trips have an 18% higher median fare while covering 10% less
distance compared to optimal trips without transfer constraints, for the same
source and destination. This trade-off may be attractive to non-regular users
of public transit, who would otherwise rely on end-to-end cab services, as it
offers a more efficient use of public transit without the inconvenience of
transfers.

####  V-A2 Factors Affecting Path Efficiency

For some source-destination pairs, the paths with the highest and lowest
$\Lambda$ may have the same fare even though the paths themselves may be
different. In these cases, the $\Lambda$ values depends largely on the
following factors:

  * •

Transfer Penalty: Introducing a transfer penalty encourages the selection of
opti-mile paths, i.e only direct public transit connections. We observed that
78% of the paths with the highest $\Lambda$ preferred a high transfer penalty,
while 71% of the paths with low $\Lambda$ had no transfer penalty (Fig 4 (a)).

  * •

Last Mile Range: From Fig. 4 (b), we observe that 83% of the paths with the
lowest $\Lambda$ have the lowest last mile range (2-3km), whereas the ones
with higher $\Lambda$ prefer a larger last-mile range ($>$ 3km) 90% of the
times. A larger range for last mile services implies longer distances traveled
to reach a stop with a direct connection to the destination.

The findings described above support the notion that opti-mile journeys, with
fewer transfers, sometimes even at the cost of longer last-mile distances, are
the most efficient paths.

TABLE I: Fare v/s Distance Comparison

Median Deviation Opti-Mile Non-Opti-Mile Opti-Mile Non-Opti-Mile Total
Fare(Rs) 130.00 110.00 20.09 17.71 Total Distance (km) 21.71 24.29 8.05 7.72

Figure 4: Factors Affecting Path Efficiency

###  V-B Experiment 2: Effect of Opti-Mile on Coverage

From our analysis on the coverage provided by the transit network and opti-
mile, we draw the following conclusions

  1. 1.

Opti-Mile trips, with a maximum last-mile range of 2km, provide 4% less
coverage for areas with a 10% of the total population density. However, for
areas with a 90% of the total population density, there is an observed
increase in coverage of over 8%. Hence, it can be confidently stated that
utilising Opti-Mile trips does not result in a significant loss of coverage.
Further, increasing the range of last-mile to 5km provides over 99.5% coverage
in all the cases.

TABLE II: PTN and Opti-Mile coverage comparison

Population (%Density) PTN Coverage Opti-Mile Coverage 10% 78.56 75.31 50%
68.11 68.49 80% 59.73 64.93 90% 55.57 60.71

TABLE III: PTN coverage by population and walk distance

Population (%Density) Walk Distance (km) PTN Coverage (%) 10% 0.5 78.56 10% 1
97.40 50% 0.5 68.11 50% 1 91.67 80% 0.5 59.73 80% 1 83.24 90% 0.5 55.67 90% 1
80.78

  2. 2.

Table III shows a significant decrease in coverage ranging from 20% to 35%
when the maximum walking distance is reduced to 500m. This reduction in
coverage greatly affects the accessibility of the public transit network.
Depending solely on public transit in such cases may leave certain areas with
limited coverage.

##  VI Conclusion

Through our experiments, we demonstrate how opti-mile trips contribute to a
cost-effective and efficient transportation network. Last-mile services for
short-distance trips can reduce traffic congestion on longer routes, and
further improvements in ride-sharing services can help alleviate this issue.
Integrating last-mile options also provides commuters with flexibility to
choose transportation modes based on personal preferences.

Implementing opti-mile trips to enhance public transit accessibility in Delhi
poses significant challenges. Extensive research on last-mile availability
[17, 18, 19], can provide valuable insights and recommendations for
policymakers. By leveraging these findings and combining them with our transit
model, policymakers can make informed decisions to enhance the effectiveness
and efficiency of public transit systems.

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