This subsection presents three different variants of the problem of designing the backbone network of the public transportation system in Montevideo, which are solved in this article.
2.3.3. Variant 2: Bounded Travel Times Model
The maximum resilience problem variant is simple to solve. However, it pushes the physical independence of the designed LRT beyond problem premises, because the formulation does not permit that two lines share a single edge. However, in a real transportation system design, it is common for lines originating from different terminals to share tram sections. As a result, is it feasible to relax the requirement for physical independence, allowing for lower design costs while still maintaining the desired resilience design objectives. In addition, the bounded travel times problem variant takes into consideration passengers quality-of-experience, by integrating into the formulation a threshold for the duration of travels from terminals to the city center. The value represents the threshold for travel times between every terminal node in and the city center .
Equation (
2) presents the formulation of the bounded travel times problem variant. For each edge
a variable
(binary) is defined to denote if stations
i and
j are connected by an edge present in the considered solution (that is the case when
) or not (the case when
). The variable
(binary) is activated only when the
r-th line departing from the terminal station
utilizes the edge
to establish a connection to the city center. The variables
determine if the station
j is included in the route of the
r-th line departing from the terminal
p.
In Equation (
2), constraints
ensure that the number of edges used by the
r-th line to exit from terminal
p is one. The values of
r are 1 or 2, except for
(corresponding to
POC), for which it can also take the value 3. Constraints
ensure that intermediate nodes are either used twice, for both inward and outward edges, or not used. When applied simultaneously, constraints
and
establish the paths for lines of every terminal. Constraints
ensure that neither two nor three lines from the same remote terminal utilizes the same edge, thus guaranteering the physical independence of routes. The left term in constraints
represents the traveling times from origin to destination for lines of every terminal station
p. These travel times are limited by the threshold
. The right-hand side of constraints
calculates the number of times that an edge is utilized by any line. Due to the requirement that lines departing from the same terminal must not edges, and considering that only four terminals are defined in the proposed case study, constraints
can be satisfied by the assignment
. However, this setting implies that the defined objective function (infrastructure cost) increases. This is because the activation of variable
is sufficient if even one line uses edge
. When the values of the threshold
are large, travel times do not constraint the solutions and the problem in Equation (
2) turns into a relaxation of the one presented in Equation (
1), allowing for an additional level of physical dependence. This observation is useful to estimate appropriate lower bounds for the investments in infrastructure (tram sections).
The proposed bounded travel time model defines a problem that is NP-hard. It is related to the well-known Resource Constrained Shortest Path (RCSP) problem. The RCSP defines a mathematical model used in various applications within transportation and communications fields [
36]. In RCSP, considering a graph
, a cost function
, a pair of nodes
s (source) and
t (destination),
, a set of functions
that indicate the usage for diverse resource types
, and resource bounds
, the objective of the problem is to determine the path from
s to
t with the minimum cost, while ensuring that the usage of each resource type remains below the corresponding resource limit
. The RCSP is known to be NP-Hard [
37], even when considering only a single resource.
2.3.4. Variant 3: Multiobjective Optimization Model
This subsection presents the proposed multiobjective formulation for the studied problem.
Background. The problem formulations defined in
Section 2.3.2 and
Section 2.3.3 share their objective functions, which are the minimum basic infrastructure costs (i.e., tram rails and its complementary infrastructure) in both cases. Regarding control variables, Equation (
1) only concerns with the edges to be deployed to be able of having certain number of physically independent lines connecting the remote terminals with the city center. Those edges determine the basic infrastructure costs but not the assignment of lines, since in general there is more than one alternative to assign lines over a set of edges.
On the other hand, the formulation in Equation (
2) expands the set of control variables to identify each particular alignment, since they are needed to compute end-to-end time travels, which are to be upper bounded by
in this formulation. As it was already mentioned, for values of
sufficiently large, the formulation in Equation (
2) turns out to be a relaxed version of Equation (
1), allowing to compute lower bounds for the solution cost. In addition, if values of
are chosen appropriately, then the optimal solution for instances of the formulation in Equation (
1) are feasible for the formulation in Equation (
2), providing upper bounds to the infrastructure costs. The previous figures and the optimal solution of the problem formulated in Equation (
2) when available (recall that it is NP-Hard), provide good reference baseline values for the infrastructure costs.
From the economic point of view, basic infrastructure costs are the most important, as it is evinced in the experimental evaluation (see
Section 4). However, other costs must be included to build an accurate and realistic model. The infrastructure costs must include not only tram rails and its complementary investments, but must also consider the cost of the trams themselves. When analyzed over a repayment period and restated as amortizations, previous costs can be considered as annually fixed. Besides that, there are variable costs for having the trams rolling, which are proportional to the frequency and length assigned to each one of those lines.
Fixed and variable costs are of economic type, so they can simply be added when expressed over the same reference period. For instance, both could be expressed as annual amortizations and annual operation costs, just adding monetary quantities. However, the main goal of a public transportation system is being efficient, and that not only concerns with being economically competitive but also with providing good service level to end users, i.e., low travel times.
Quality of service was first considered in
Section 2.3.3 by setting upper bounds for the travel times of lines traveling from terminals to the city center. That approximation is simple and sets limits to the worst case travel times, but it does not capture the statistical distribution when substantial portions of the passengers do not pickup trams at the remote terminal stations, which is the case of the reference application case studied in this article. An alternative formulation, more suitable for such cases is the one proposed by Canca. Besides of considering fixed and variable costs, Canca introduced a metric for the service experience of the user, namely the users costs [
38].
Multiobjective formulation. The formulation proposed by Canca is intrinsically multiobjetive, since it has three components: infrastructure costs (fixed costs), operational costs (variable costs), and the users costs [
38]. In this article, for sake of simplicity, an approach applying linear combination is applied for solving the backbone network design for public transportation in Montevideo. Despite being surpassed by Pareto-based approaches in many cases of multiobjective optimization, aggregation methods remain widely used in the literature due to two key advantages. Firstly, it is well-suited for tackling multiobjective optimization problems characterized by a Pareto front that is convex. Secondly, the aggregation approach is efficient from a computational point of view, making it a recommended choice for complex problem models and instances [
39]. Furthermore, for the considered problem two of the three components can be expressed in terms of money annually expended, which can be added up directly.
Regarding
control variables, the multiobjetive approach incorporates those described in
Section 2.3.3, but new ones are needed also. A solution is determined by the configuration of lines between remote terminals and the center, complemented with the reference frequency for each one of them. Considering the studied real-world application as example, nine lines have to be crafted
. Each of these lines is determined by some sequence of nodes connecting a node in
with some node in
. These line configurations capture the information of variable
in Equation (
2), which in turn determines the values of variable
as in Equations (
1) and (
2). New variables
are included in the multiobjective formulation, corresponding to the frequency of each line, that is, to the expected number of times (per-hour) that each line arrives at each of its stations. Thus,
is complemented with
. Combined, sets
and
determine a solution. According to the current timetables for public transportation in Montevideo, the problem formulation assumes that
.
The consistency in the line configurations and the physical independence can be easily determined. For instance, if
and
correspond to Carrasco, suffices that the first node in both lists of nodes (
and
) to be 4, the last node to be in
, the existence of an edge in
E between each pair of consecutive nodes, and the independence between both sets of edges. For example purposes,
could be 4-10-27-33-2 and
= 4-8-28-30-42-49-3 (see
Figure 4).
Frequencies and also have to fulfil some conditions to attain a minimum level of service. For this formulation, the proposed model imposes that the number of trams arriving at each station per hour is sufficiently large for passengers to fit within wagons, in average. This depends of all the line configurations and the whole set of frequencies, since more than one line can use a station and be suitable as a trip for some users. Although a stochastic formulation of this subproblem would be more accurate, for simplicity, the proposed formulation considers the average and assumes the trams arrive at regular and uniform intervals to each station, which is consistent with the case studied in Montevideo.
In order to compute the previous constraints and the users costs, the proposed model needs a reference demand matrix. Let assume a demand matrix
, a square matrix with as many rows as the current number of bus stops, whose elements contain the number of passengers traveling between each combination of stops. The value of the cell at row
i and column
j corresponds to the number of passengers that are taking some bus at bus stop
i with destination to
j during the rush hour. During the peak hour most of the trips are towards the center (see the data reported in
Section 4). The return of passengers to their homes is distributed over a longer period, i.e., it is less intense. A configuration of lines (
and
) defines a set of demands matrices
over the LRTs, for
. The calculation of
requires the previous computation of
, namely the number of trips between any pair of stations. The previous matrix derives from
and
, since in the proposed two-level architecture, passengers board buses to get to/from some near station and use the backbone for the long-haul only. The value in the cell
corresponds to the average number of passengers taking a tram of the line
k at station
p with destination to station
q per hour, where both
p and
q are actual stations according the ones defined on
.
Considering the previously described variables and data, the terms of the multiobjective formulation of the problem are presented next.
The costs associated to infrastructure and investments (i.e., fixed costs,
are defined by Equation (
3).
The first term in Equation (
3) matches the objective functions in Equations (
1) and (
2), since it corresponds to the investments in tram rails. The second term complements the previous by adding up the acquisition costs of the tram units. Parameter
T in Equation (
3) is the cost of each tram. Strictly speaking, the second term in Equation (
3) is not the investments in trams but an upper bound for it. The number of rolling trams to attain
frequencies not only depends on these frequencies but also in the length of the alignment for each line, since when round trip times are below an hour, a single tram could serve more than one station within that period. It is assumed that round trip times are lower than 1 h, a hypothesis that is verified during the experimental analysis in
Section 4. The operation costs (
) depend on the number of kilometers trams have to travel to satisfy frequencies
over the line configurations determined by
(Equation (
4)).
simply adds up to the total number of per-hour kilometers trams must travel, which are proportional to the operational expenditures.
The final component corresponds to the cost of the users, which is the most ambiguous cost. The total travel times that the whole universe of users spend to fulfill their trips could be taken as reference. In such case, the design would be implicitly prioritising trips intrinsically long. Consider daily trips from the municipality
Ch to
B, which are around 100,000. The number of daily trips from
RInt to
B is five times lower (see
Table 1). However, Carrasco terminal is geographically much farther from center than Pocitos. Thus, passengers coming from
RInt (which would be boarding trams at Carrasco remote terminal) would be accounted with a substantially higher priority regarding those of Pocitos, even though the second group gathers a much larger number of passengers. In order to avoid such biases and account all passengers equally, the cost for each user is normalized by its own placement within the city.
The final term of the multiobjective formulation requires additional computations. Let
be a lower bound for the travel time between stations
p and
q in
E, which considers the shortest path over
E using
metric and computes the
p to
q trip time as if it had no intermediate stops. Consider the layout in
Figure 4. Sequences 21-34-35 and 21-37-36-35 connect nodes 21 and 35 with similar accumulated delays, respectively 445 and 438 s However, as described in the experimental section, delays are computed up from distances between stations combined with other parameters, such as: cruise speed, acceleration/deceleration times plus the time trams are stopped at each stations for passengers board/alight the train, all of them based on the regular hypotheses for the operation of lines.
The value
assumes that a personalized line is available for each combination of stations in the city, regardless of what stations are effectively used in the solution. Moreover, the time
presumes that a tram is awaiting at station
p for the next passenger to arrive, and when that happens, the tram departs immediately and does not make any intermediate stop until getting to
q. Thus, unlike end-to-end travel times,
times do not depend of the number of intermediate stations, but only of the total end-to-end distance. Following the previous example and using the parameters as in
Section 4, both sequences connecting stations 21 and 35 would have ideal travel times of respectively 317 and 242 s, so the optimal lower bound between 21 and 35 (i.e.,
) is 242 s. Based upon reference
values, a users cost metric that is unbiased by stations placements within the city can be computed. The expression is that of
defined in Equation (
5), which normalizes each trip time as relative to its optimal lower bound.
The outer sum in Equation (
5) adds up over all positive station-to-station demands in matrix
, whose values depend on the particular solution. Since
values only depend on the problem instance, not on the solution, they can be preprocessed. In each term of the sum (i.e., for each effective combination of
in
), the denominator corresponds to the sum of optimal lower bounds of travel times between every pair of stations
p and
q, which matches
multiplied by the number of per-hour trips between them. Conversely, the numerator of each term corresponds to actual average travel time between
p and
q for the current solution.
Being the set of lines connecting stations p and q, the product corresponds to the total number of passengers traveling per hour between stations p and q using line k. Since trams are supposed to arrive at regular and uniform intervals to each station, and assuming that each passenger picks up at p the following tram that takes him to its destination q, the average waiting time (in seconds) for passengers between p and q can be computed as (3600/. Finally, is the travel time between p and q for passengers using the line k, being the subsequence of the alignment for line k that connects p with q.
The proposed scalar multiobjective function results from a linear combination of Equations (
3)–(
5), considering positive coefficients. Coefficients of Equations (
3) and (
4) are adjusted to express hourly costs in both cases (see
Section 4). The infrastructure amortizations are scaled as within a time window of an hour for a repayment period of thirty years, while operation costs simply estimate the hourly consumption of energy for the whole system at the rush hour. As previously mentioned, the coefficient of Equation (
5) is adjusted to get to a solution with total infrastructure costs and terminal-to-center trip times similar to referential values resulting from formulations in
Section 2.3.2 and
Section 2.3.3.