Optimized Design of a Backbone Network for Public Transportation in Montevideo, Uruguay ^†

Abstract

This article presents the application of exact and metaheuristic approaches to the problem of designing the backbone network of a hierarchically public transportation system for Montevideo, Uruguay. This is a very relevant problem in nowadays smart cities, as it accounts for many social and environmental impacts and also affects the dynamics of the cities. The design of the proposed backbone network is conceived in combination with the bus network, with the main objective of improving the overall quality of service and reducing travel times. Three different variants of the problem are solved, considering different design premises. Exact solvers are proposed for simpler variants of the problem, which account for maximum resilience and bounded travel times. An evolutionary algorithm is proposed for a multiobjectie version of the problem that optimizes cost and quality of service. The main results indicate that the computed optimized designs provide reduced end-to-end travel times, which improve up to five times over the current system, and are economically viable to be implemented.

1. Introduction

Public transportation is an important component of nowadays societies, which have direct implication on many aspects (e.g., commercial, work, recreation, culture) of the activities performed by citizens [1]. Furthermore, public transportation has a directly impact sustainability by reducing air pollution, optimizing energy usage, and enhancing the livability of cities [2,3]. The study of public transportation systems falls in the realm of smart mobility [4] in the emerging smart cities paradigm [5]. Under this scope, smart transportation systems leverage technologies to improve transportation systems [6,7,8,9]. with a particular emphasis on public modes of transportation such as buses and trains.

In general, public transportation serves a broader purpose and provides greater advantages compared to private transportation. Investing in public transportation is critical for reducing the dependence of automobiles and lessening their impact on society and the environment, while also enabling transportation to continue to play a critical role in sustainable development [10].

In order to provide a successful service, travel times of public transportation should be better than those of private vehicles, but at the same time, public transportation has to be competitive in economic terms, and must provide both reliability and robust connectivity. Achieving a well-designed public transportation system involves striking a delicate balance between numerous trade-offs, a task that has been acknowledged as challenging [11].

In major cities, a a well-functioning public transportation system consists of different types of modes of transportation, resulting in a substantial infrastructure. For example, in Paris and London, the rapid transit systems Métro and Underground transport approximately one and a half billion passengers each year; whereas Moscow Metro is even more extensive, serving around two and a half billion passengers each year. In order to account for effective designs for large cities, intelligent resolution methods must be applied to solve the underlying optimization problem.

This article presents the development of exact and metaheuristic approaches applied to the problem of designing the backbone network of a hierarchical public transportation system for a specific case study on Montevideo, Uruguay. The specific goals and objectives of the research are developing and providing accurate and reliable computational models to be used by city administration and planners to enhance nowadays public transportation systems. A specific objective is also validating the proposed approaches and algorithms in the considered real case study. Three different problem variants are solved, taking into account different design considerations. Exact solvers are used for simpler problem variants, such as maximizing resilience and bounding travel times at minimum infrastructure costs (i.e., tram rails and its complementary infrastructure). An Evolutionary Algorithm (EA) is proposed for a multiobjective version of the problem that considers the simultaneous optimization of infrastructure, operation and user experience costs.

The resolution approach integrates topological constraints to improve reliability of the public transportation network [12,13]. This is a specific contribution over works reported in related literature. The research considers a real-world case in Montevideo, Uruguay, which has not been previously explored, thereby making it another valuable contribution to the field. The primary findings demonstrate that the newly designed public transportation system in Montevideo has the capacity to deliver significantly improved quality of service, evaluated in terms of end-to-end travel times, and is economically viable. Travel times improve up to five times for particular cases respect to the current design in Montevideo. Furthermore, the designed solution can be implemented using sustainable clean technologies (e.g., electric trams), significantly improving the sustainability of the public transportation system.

This article extends the previous conference article “Designing a backbone trunk for the public transportation network in Montevideo, Uruguay”, presented in II Ibero-American Conference on Smart Cities, Soria, Spain, 2019. The new content in this article includes: (i) an extended description of the methodology and the case study, (ii) a multiobjective formulation of the problem considering the joint optimization of the design cost and the quality of service, and (iii) a metaheuristic method (EA) developed for solving the proposed multiobjective optimization problem and new results for this problem variant.

The article is organized as follows. Section 2 introduces the problem, describes the case study in Montevideo, and the three studied problem formulations. Section 3 outlines the proposed methodology for designing the backbone transportation network. Experimental results are reported in Section 4. Finally, Section 5 presents the conclusions and the main lines of future work.

2. Problem Definition and Case Study

This section describes the problem of designing the backbone network of a public transportation system, presents the considered case study, and introduces the formulation of three different variants of the problem.

2.1. General Considerations

The task of designing a public transportation network belongs to the realm of network design [14]. Typically, large networks are organized into hierarchical levels that employ diverse technologies to cater to varying scales of needs. Some examples of this hierarchical structure are: (i) access networks and backbone networks in telecommunications, (ii) electrical networks, from high to low voltage, and (iii) (in the context of public transportation systems) subways, trams, and buses.

Several articles in the literature of public transportation networks design have proposed to integrate those technologies, aiming at improving some quality-of-service related metrics [15,16,17,18,19]. One of the most relevant metrics considered is passenger per hour when connecting (strategical) geographically distant locations in a city, which is to be maximized. Many published manuscripts approached the problem as a combination of transportation technologies, where certain segments of the existing infrastructure are considered fixed or definitive.

Cities with large transportation systems often develop their services by incorporating additional elements to face specific challenges or issues. This process allows city administrators to progressively improve the quality-of-service without applying a coordinated approach for selecting and configuring all the design variables [20]. However, the outcome of such an incremental process is typically suboptimal and tends to be more costly compared to designs that are computed using an integrated approach. Furthermore, the use of sustainable technologies is not always considered.

This article studies an efficient methodology for designing the backbone network within a hierarchical public transportation system [21,22]. The design principles draw inspiration from the fundamental attributes of scalable, resilient networks [23]. Efficient optimization methods are proposed to address various problem variations within this context. The underlying model assumes that designers initiate the planning process from a clean slate, wherein a few stations are strategically positioned based on real demand data and quality-of-service objectives. The focus is on designing the trunk network topology in a manner that minimizes the required infrastructure costs while simultaneously reducing travel times for citizens.

A methodology applying two phases is adopted for the development of the public transportation network. Two separate subproblems are defined, related to two different transportation media. In the first stage, the backbone trunk is designed to build a Rapid Transit System using a sustainable electric light railway tram (LRT). This is the subproblem that demands the highest infrastructure investment. It is addressed and solved in this article, considering different problem variants. In the second stage, bus lines are purposefully designed to be used as feeders for the pre-established backbone network. This second problem is out of the scope of this article and it is presented as the main line for future work.

The followed hierarchical approach makes the newly designed backbone critical, as any disruption in its operation can lead to the disconnection of different zones within the city [24]. To address this concern, the design topological constraints are incorporated, aimed at minimizing such risks. This is an innovative contribution of the proposed research, over those reported in existing literature. A real-world application case is considered in the research: the public transportation system of Montevideo (Uruguay), presented in the next subsection.

2.2. The Public Transportation System in Montevideo

This article studies the specific case of building an LRT network for the public transportation system of Montevideo, Uruguay [25].

The public transportation system in Montevideo is operated by four companies. Their fleets account for a total number of 1527 buses (data for June 2018, according to the Mobility Observatory of the city administration [26]). In Montevideo, the bus network comprises 145 bus lines including variations for outward and return trips, and also shorter or longer sub-lines of given lines. When considering each individual variant and sub-lines, the total number is 1383, which is a significantly high number for a city with a population of approximately one and a half million residents.

Figure 1 presents the bus lines (blue lines) of the public transportation system in Montevideo, on top of the road map (grey lines). The graphic was built using open data from the Geographic Information System of the City Hall of Montevideo [27].

Bus lines in Montevideo are notably long, with an average length of 16.7 km (standard deviation 7.1 km) and median 16.5 km. The longest bus line stretches for 39.6 km. These numbers appear notably extensive, especially when considering that the surface of Montevideo is 530 km² and the city can be encompassed within a rectangle measuring 26 × 37 km [28].

The pie chart in Figure 2 summarizes the means of transportation used by citizens of Montevideo, according to a mobility survey performed in 2017 in Montevideo [29].

Results show that just 25% of the travels make use of the public transportation system for mobility in the city. Automobiles account for the largest share among private vehicles (32% of the trips in the city). Furthermore, the number of cars has been significantly increasing in the last fifteen years in Montevideo, with the subsequent environmental and health issues. These figures and facts suggest that public transportation in Montevideo still has room to improve, by providing better quality of service to citizens, in order to increase the share of trips performed using this public service.

Figure 3 presents the eight municipalities of Montevideo (labeled ‘A’ to ‘F’) and the five most important sources of trips performed from outside the city, mainly from locations in the metropolitan area. These locations are connected to Montevideo by five important national routes: R1, R5, R6/7, R8, and RInt.

Table 1. Origin and destination of trips for zones in Montevideo and nearby locations, according to data from the metropolitan mobility survey.

		Destination of Trips
		A	B	C	Ch	D	E	F	G	R1	R5	R6/7	R8	RInt	Total
Origin of trips	A	249,690	31,578	20,632	15,239	6493	4074	4224	26,308	1121	915	140	0	1285	361,699
	B	30,040	395,971	58,687	108,525	22,695	50,501	21,039	21,939	2686	10,342	3240	1321	19,775	746,761
	C	22,523	60,039	179,910	23,949	22,213	12,420	10,507	27,319	1082	2541	3940	234	3693	370,370
	Ch	15,770	99,818	25,244	239,919	24,776	53,164	17,432	12,108	1690	3067	1986	318	12,639	507,931
	D	4492	25,010	19,156	23,617	214,225	13,484	41,526	10,877	801	398	2362	349	2935	359,232
	E	4216	53,784	15,085	51,642	14,511	97,917	19,354	5346	158	995	1184	162	27,726	292,080
	F	4505	20,462	10,637	18,560	39,957	19,330	209,980	5553	137	1333	777	166	6958	338,355
	G	23,200	19,480	29,574	13,017	7510	8850	4897	136,145	684	6979	312	0	0	250,648
	R1	966	3803	1172	639	727	78	137	684	67,299	405	90	0	285	76,285
	R5	915	9394	2615	3384	398	1365	1402	8057	405	235,823	1999	462	789	267,008
	R6/7	152	3326	3723	2142	2275	1114	561	381	90	2369	70,024	3129	3352	92,638
	R8	0	861	234	318	349	401	327	0	0	599	3178	35,342	9421	51,030
	RInt	1669	19,161	3827	11,876	3027	28,998	7083	0	205	858	3082	10,073	350,559	440,418
	Total	358,138	742,687	370,496	512,827	359,156	291,696	338,469	254,717	76,358	266,624	92,314	51,556	439,417	4,154,455

reports the matrix of daily origin-destination trips between zones, considering municipalities and the demand from satellite cities. Values account for total trips performed by any mean of transportation.

Following literature recommendations, this research study focuses on designing a network that aims to integrate Montevideo and the surrounding metropolitan area [30]. Consequently, key-nodes are selected taking into account their relevance as sources of trips [31].

Results in Table 1 reveal that the city center (municipality B) accounts for the largest number of trips. The second position is for municipality Ch, located nearby the city center. These results are consistent with the design of bus lines reported in Figure 1, which clearly shows that the city center serves as a central network hub, with numerous bus lines that converge towards this area. Considering municipalities over the city border as connected with near satellite cities, the analysis allows concluding that the areas with the largest transportation demands are E + RInt (East), G + R5 (North), D + R6/7 (Northeast), and A + R1 (West). Therefore, the case study examined in this manuscript focuses on the design of the LRT, which aims to connect five specific zones. The design includes seven key nodes (terminal nodes) and forty-eight intermediate (optional) nodes. The design premises for the backbone of the considered bus network comprise the following considerations:

• the architecture of the backbone follows a hub-type design [32]; the city center (zone B) serves as the central hub. This choice is justified by the fact that the city center is the primary destination for trips within the city. Furthermore, due to its strategic position as the geographic urban center, the city center is well-suited for facilitating travel demands between the other zones in Montevideo.
• the city center includes three nodes: Independence Square Plaza Independencia, PZI, node1), Three Crossings (Tres Cruces, 3XX, node2), and City Hall (Palacio Municipal, PLM, node3); it is assumed that a fast interconnection media exists between them, allowing citizens to rapidly reach any of them from any other. This assumption is realistic since they are separated by less than 2 km.
• Also included as key-node are terminals Carrasco (CAR, node4), Cerro (CRO, node5), Pocitos (POC, node6), and Colón (TCO, node7). POC (zone Ch) and CAR (zone E + RInt) are introduced as new stations, due to the large number of trips originating in them. Pocitos (municipality Ch) is the zone with the second largest demand within the city, and Carrasco is near Ciudad de la Costa, the most populated conglomerate of the country after Montevideo with a population of about 150.000 citizens, most of them working in Montevideo and coming by RInt. Together, they are assimilable to RInt+E, the third source of traffic. CRO (zone A + R1) and TCO (zone G + R5) are actual bus stations, which account for a significant number of trips.
• All lines connect the city center (nodes 1 to 3) with some remote terminal (nodes 4 to 7). Terminals TCO, CAR, and CER are the gateways for trips towards North, East, and West of the country.

The possible connections (edges) are defined considering the significance of streets and avenues in the road network of Montevideo. Except for the fixed key-nodes detailed during Design Premises, remaining stations are optional and they are mostly defined by intersections of edges. The chosen architecture requires the connection of terminals to the city center through the deployment of LRT lines. The inclusion of other locations as part of the solution, either as stations of the proposed LRT or not, depends on the feasibility and convenience of their usage. When a passenger addresses a location that is not the city center or that is not accesible from any other intermediate station, the hub architecture implies that that passenger transfers in the center to another line that takes him to his final destination.

The proposed approach assumes that buses will be reorganized and assigned as feeders for the backbone network. Therefore, the backbone becomes critical to keep the system operating. Additionally, the proposed design for the LRT network takes into consideration the fact that surface-level vehicles are more susceptible to failures compared to vehicles used in underground/subway systems. Thus, an organically resilient design approach was chosen for the network. Instead of relying on predetermined geometric patterns to construct the network topology [33], the applied methodology follows a common criteria used in network design: resilience is assumed to be a result of the connectivity degree, specifically the number of physically independent (disjoint) paths that connect different points in the network. As a condition of this design, it is required that there be more than one disjoint line between each remote concentrator (CAR, CRO, POC or TCO) and the hub (i.e., the city center).

In the case study, the required number of independent paths was defined as 2 for terminals CAR, CRO, and TCO, and is set 3 for POC, due to the larger number of passengers traveling from/to that zone. Nevertheless, the formulation remains general, as the specific number of independent paths required is considered a parameter of each solved problem instance. The proposed approach assumes the existence of a rapid transport subsystem connecting stations within the city center. Therefore, the approach assumes that reaching any of the stations in that zone (PZI, PLM, or 3XX) does not impact the quality of service provided to users. Figure 4 presents the considered network, showing terminal stations (in red) and optional stations (in yellow) over a map of Montevideo, and a possible solution to connect them. Blue stations represent the three proposed stations for the rapid transport subsystem within the city center.

2.3. Problem Formulation and Its Variants

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.

2.3.1. General Model

The principal objective is, given a network of possible tram connections, to determine an optimal sub-network, along with the assignment of lines, in order to meet travel demands and satisfy quality-of-service constraints. The problem formulation represents the bus network using an undirected graph, denoted as $G=(V,E)$. The set of nodes V represents the important stations and the set of edges E represents the potential connections between stations in V. Some nodes/stations belong to the city center (set $\mathcal{C}$), whereas other nodes serve as terminals (set $\mathcal{T}$). Other nodes in the network, denoted as $V\setminus \mathcal{C}\setminus \mathcal{T}$, are known as Steiner nodes. In the example depicted in Figure 4, $\mathcal{C}=\{1,2,3\}$ and $\mathcal{T}=\{4,5,6,7\}$. The problem formulation incorporates additional functions for cost ($C:E\to {\mathbb{R}}^{+}$), delay ($D:E\to {\mathbb{R}}^{+}$), and length ($L:E\to {\mathbb{R}}^{+}$). The costs and the lengths may not have a linear relationship and can be adjusted to account for specific conditions, such as urbanization rate and/or population density. As reported in the description of the problem instance in Section 4, geographical lengths are considered for computing the delays and costs of trips. As a consequence, the costs ($c_{ij}$), delays ($d_{ij}$), and lengths ($l_{ij}$) of rail stretches can be pre-computed as given attributes for each stretch (edge) $ij\in E$.

The goal is to build a maximum coverage subgraph $G^{\prime }=(V,E^{\prime },C)$ with $E^{\prime }\subseteq E$ subject to specific constraints such as: minimum redundancy (i.e., existence of several transportation lines between specific pairs of locations), maximum values for the travel times between locations, maximum design and/or operation cost, etc. In this article, the problem is approached in three different versions: the maximum resilience model, the bounded travel time model, and the multiobjective optimization model. Each variant of the problem is described in detail, including its formulation, in the subsequent subsections.

2.3.2. Variant 1: Maximum Resilience Model

In a simplified problem variant, a predetermined number of lines is assigned from each remote terminal to the center, with the additional constraint that no line can share any edge with another line. This constraint is more stringent than the one stated in the general formulation, where the independence constraint only applies to lines originating from the same terminal. A relevant example is the realistic scenario that specifies that two lines are needed fromCAR (node 4), CRO (node 5) and TCO (node 7) to the center, to the center. Additionally, three lines are required from POC (node 6) to the center. This is an important realistic scenario taking into account the population in the considered zones and the mobility demands, according to the mobility survey and the origin-destination matrix of the city.

Let $G=(V,E)$ be the non-directed representing a real problem instance. It is transformed into a directed graph $\tilde{G}=(V,\tilde{E})$ as follows. Each edge in E is duplicated to generate edges in both directions. The only exception to this duplication rule is for edges with endpoints at nodes PZI (node 1), 3XX (node 2) or PLM (node 3), where only the incoming edge is utilized. Let variable $x_{ij}$ represents the amount of flow traversing edge $ij$ in the directed graph $\tilde{G}$ (with edges denoted as $\tilde{E}$). Equation (1) presents the problem formulation for the variant 1: maximum resilience model.

$$\left\{\begin{array}{cc}{\displaystyle min}\hfill & {\displaystyle \sum _{ij\in \tilde{E}}{c}_{ij}{x}_{ij}}\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & {\displaystyle \sum _{jk\in \tilde{E}}{x}_{jk}-\sum _{ij\in \tilde{E}}{x}_{ij}=2,}\hfill & j=4,5,7\hfill & \left(i\right)\\ \hfill & {\displaystyle \sum _{6k\in \tilde{E}}{x}_{6k}-\sum _{i6\in \tilde{E}}{x}_{i6}=3,}\hfill & \hfill & \left(ii\right)\\ \hfill & {\displaystyle \sum _{ij\in \tilde{E}}{x}_{ij}-\sum _{jk\in \tilde{E}}{x}_{jk}=0,}\hfill & \forall j\ge 8\hfill & \left(iii\right)\\ \hfill & 0\le x_{ij}\le 1,\hfill & \forall ij\in \tilde{E}\hfill & \left(iv\right)\end{array}\right.$$

(1)

The formulation presented in Equation (1) revolves around the concept of flow paths, which define the sequence of edges utilized by each line. The objective function aims to minimize the total investment cost. Constraints $\left(i\right)$ enforce that the outgoing flow from nodes 4, 5, and 7 is of 2 units, whereas constraint $\left(ii\right)$ requires a flow of 3 units from node 6. Constraints $\left(iii\right)$ ensure the balance of flow for Steiner nodes, meaning that traffic in the network can only exit through nodes 1, 2, and 3. Given that for a a network flow problem the extremes of the feasible region are integer [34], no contraints are needed to enforce the integrality of variables $x_{ij}$. Therefore, constraints $\left(iv\right)$ ensure that each arc is either utilized ($x_{ij}=1$) or not ($x_{ij}=0$), with a maximum flow of one unit (representing a bus line).

The problem variant 1 (maximum resilience model) belongs to the polynomial time complexity class, because the formulation presented in Equation (1) is a linear programming problem. It is known that there exists one polynomial-time algorithm, as described by Karmarkar [35], to solve this type of problems.

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 $T{D}_p$ represents the threshold for travel times between every terminal node in $\mathcal{T}$ and the city center $\mathcal{C}$.

Equation (2) presents the formulation of the bounded travel times problem variant. For each edge $ij\in E$ a variable $x_{ij}$ (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 $x_{ij}=1$) or not (the case when $x_{ij}=0$). The variable $y_{ij}^{pr}$ (binary) is activated only when the r-th line departing from the terminal station $p\in \mathcal{T}$ utilizes the edge $ij\in E$ to establish a connection to the city center. The variables ${\theta }_j^{pr}$ determine if the station j is included in the route of the r-th line departing from the terminal p.

$$\left\{\begin{array}{cc}{\displaystyle min}\hfill & {\displaystyle \sum _{ij\in E}{c}_{ij}{x}_{ij}}\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & {\displaystyle \sum _{pj\in E}{y}_{pj}^{pr}=1,}\hfill & 4\le p\le 7,1\le r\le 2\left(3\right)\hfill & \left(i\right)\\ \hfill & {\displaystyle \sum _{ij\in E}{y}_{ij}^{pr}+\sum _{jk\in E}{y}_{jk}^{pr}=2{\theta }_j^{pr},}\hfill & \begin{array}{c}j\ge 4,j\ne p,4\le p\le 7,\hfill \\ 1\le r\le 2\left(3\right)\hfill \end{array},\hfill & \left(ii\right)\\ \hfill & {\displaystyle \sum _{r=1}^{2\left(3\right)}{y}_{ij}^{pr}\le 1,}\hfill & ij\in E,4\le p\le 7,\hfill & \left(iii\right)\\ \hfill & {\displaystyle \sum _{ij\in E}{d}_{ij}{y}_{ij}^{pr}\le T{D}_p,}\hfill & 4\le p\le 7,1\le r\le 2\left(3\right),\hfill & \left(iv\right)\\ \hfill & {\displaystyle 4x_{ij}\ge \sum _{p=4}^7\sum _{r=1}^{2\left(3\right)}{y}_{ij}^{pr},}\hfill & ij\in E,\hfill & \left(v\right)\\ \hfill & x_{ij},y_{ij}^{pr},{\theta }_j^{pr}\in \{0,1\}\hfill & \begin{array}{c}\forall ij\in E,4\le p\le 7,\hfill \\ 1\le r\le 2\left(3\right)\hfill \end{array}\hfill & \left(vi\right)\end{array}\right.$$

(2)

In Equation (2), constraints $\left(i\right)$ 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 $p=6$ (corresponding to POC), for which it can also take the value 3. Constraints $\left(ii\right)$ ensure that intermediate nodes are either used twice, for both inward and outward edges, or not used. When applied simultaneously, constraints $\left(i\right)$ and $\left(ii\right)$ establish the paths for lines of every terminal. Constraints $\left(iii\right)$ 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 $\left(iv\right)$ represents the traveling times from origin to destination for lines of every terminal station p. These travel times are limited by the threshold $T{D}_p$. The right-hand side of constraints $\left(v\right)$ 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 $\left(v\right)$ can be satisfied by the assignment $x_{ij}=1$. However, this setting implies that the defined objective function (infrastructure cost) increases. This is because the activation of variable $x_{ij}$ is sufficient if even one line uses edge $ij$. When the values of the threshold $T{D}_p$ 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 $G=(V,E)$, a cost function $C:E\to {\mathbb{R}}^{+}$, a pair of nodes s (source) and t (destination), $s,tinV$, a set of functions $R_k:E\to {\mathbb{R}}^{+}$ that indicate the usage for diverse resource types $k\in 1,\dots ,K$, and resource bounds $R{L}_k$, 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 $R{L}_k$. 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 $T{D}_p$ in this formulation. As it was already mentioned, for values of $T{D}_p$ 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 $T{D}_p$ 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 $\mathcal{L}=\{{\mathsf{\ell }}_1,{\mathsf{\ell }}_2,{\mathsf{\ell }}_3,{\mathsf{\ell }}_4,{\mathsf{\ell }}_5,{\mathsf{\ell }}_6,{\mathsf{\ell }}_7,{\mathsf{\ell }}_8,{\mathsf{\ell }}_9\}$. Each of these lines is determined by some sequence of nodes connecting a node in $\mathcal{T}$ with some node in $\mathcal{C}$. These line configurations capture the information of variable $y_{ij}^{pr}$ in Equation (2), which in turn determines the values of variable $x_{ij}$ as in Equations (1) and (2). New variables $f_{\mathsf{\ell }}$ 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, $\mathcal{L}$ is complemented with $\mathcal{F}=\{f_1,f_2,\dots ,f_9\}$. Combined, sets $\mathcal{L}$ and $\mathcal{F}$ determine a solution. According to the current timetables for public transportation in Montevideo, the problem formulation assumes that $f_{\mathsf{\ell }}\in \mathbb{N}$.

The consistency in the line configurations and the physical independence can be easily determined. For instance, if ${\mathsf{\ell }}_1$ and ${\mathsf{\ell }}_2$ correspond to Carrasco, suffices that the first node in both lists of nodes (${\mathsf{\ell }}_1$ and ${\mathsf{\ell }}_2$) to be 4, the last node to be in $\{1,2,3\}$, the existence of an edge in E between each pair of consecutive nodes, and the independence between both sets of edges. For example purposes, ${\mathsf{\ell }}_1$ could be 4-10-27-33-2 and ${\mathsf{\ell }}_2$ = 4-8-28-30-42-49-3 (see Figure 4).

Frequencies $f_1$ and $f_2$ 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 $\overline{D}$, 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 ($\mathcal{L}$ and $\mathcal{F}$) defines a set of demands matrices $D_{\mathcal{L},\mathcal{F}}=\left\{D_{\mathcal{L},\mathcal{F}}^k\right\}$ over the LRTs, for $k=1,\dots ,9$. The calculation of $D_{\mathcal{L},\mathcal{F}}$ requires the previous computation of $D_{\mathcal{L}}$, namely the number of trips between any pair of stations. The previous matrix derives from $\overline{D}$ and $\mathcal{L}$, 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 ${\left(D_{\mathcal{L},\mathcal{F}}^k\right)}_{pq}$ 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 $\mathcal{L}$.

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, $CC(\mathcal{L},\mathcal{F}))$ are defined by Equation (3).

$${\displaystyle CC(\mathcal{L},\mathcal{F})=\sum _{ij\in E}{x}_{ij}+T\times \sum _{f_k\in \mathcal{F}}{f}_k}$$

(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 $\mathcal{F}$ 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 ($OC(\mathcal{L},\mathcal{F})$) depend on the number of kilometers trams have to travel to satisfy frequencies $\mathcal{F}$ over the line configurations determined by $\mathcal{L}$ (Equation (4)). $OC(\mathcal{L},\mathcal{F})$ simply adds up to the total number of per-hour kilometers trams must travel, which are proportional to the operational expenditures.

$${\displaystyle OC(\mathcal{L},\mathcal{F})=\sum _{\begin{array}{c}(l_k,f_k)\\ \in (\mathcal{L},\mathcal{F})\end{array}}\sum _{ij\in {\mathsf{\ell }}_k}{l}_{ij}{f}_k=\sum _{\begin{array}{c}(l_k,f_k)\in (\mathcal{L},\mathcal{F})\\ ij\in {\mathsf{\ell }}_k\end{array}}{l}_{ij}{f}_k}$$

(4)

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.

$${\displaystyle UC(\mathcal{L},\mathcal{F})=\sum _{{\displaystyle {\left(D_{\mathcal{L}}\right)}_{pq}>0}}\left(\frac{{\displaystyle \sum _{{\mathsf{\ell }}_k\in J_{pq}}{f}_k·{\left(D_{\mathcal{L},\mathcal{F}}^k\right)}_{pq}(\frac{1800}{{\sum }_{{\mathsf{\ell }}_m\in J_{pq}}{f}_m}+\sum _{ij\in {\mathsf{\ell }}_{k\left(pq\right)}}{d}_{ij})}}{t_{pq}·{\left(D_{\mathcal{L}}\right)}_{pq}}\right)}$$

(5)

The final term of the multiobjective formulation requires additional computations. Let $t_{pq}$ be a lower bound for the travel time between stations p and q in E, which considers the shortest path over E using $l_{ij}$ 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 $t_{pq}$ 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 $t_{pq}$ 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, $t_{pq}$ 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., $t_{21,35}$) is 242 s. Based upon reference $t_{pq}$ values, a users cost metric that is unbiased by stations placements within the city can be computed. The expression is that of $UC(\mathcal{L},\mathcal{F})$ 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 $D_{\mathcal{L}}$, whose values depend on the particular solution. Since $t_{pq}$ 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 $pq$ in $\mathcal{L}$), the denominator corresponds to the sum of optimal lower bounds of travel times between every pair of stations p and q, which matches $t_{pq}$ 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 $J_{pq}$ the set of lines connecting stations p and q, the product $f_k·{\left(D_{\mathcal{L},\mathcal{F}}^k\right)}_{pq}$ 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/${\sum }_{{\mathsf{\ell }}_m\in J_{pq}}{f}_m)/2$. Finally, ${\sum }_{ij\in {\mathsf{\ell }}_{k\left(pq\right)}}{d}_{ij}$ is the travel time between p and q for passengers using the line k, being ${\mathsf{\ell }}_{k\left(pq\right)}$ 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.

3. Exact and Metaheuristic Approaches for LRT Design

This section presents the exact and metaheuristic approaches applied to solve the three considered variants versions of the proposed problem.

3.1. Maximum Resilience Problem

The maximum resilience problem version has a reduced intrinsic complexity. Thus, solvers applying linear programming methods can effectively handle such kind of problem, even for instances including many millions variables and many constraints, within very short timeframes.

By instantiating the mathematical formulation in Equation (1) for the specific problem instance considered in Montevideo (as shown in Figure 4), the resulting linear programming problem has a reduced number of variables (189) and constraints (52). The considered instance was solved using the linear programming solver available in Matlab 8.5.0, which employs the interior point method [40]. Remarkably, the resolution was extremely fast: the optimal solution was found in just 0.04 s. The computation was performed on a personal computer equipped with an Intel Core i5 processor at 1.8 GHz and using 8 GB RAM.

3.2. Bounded Travel Time Problem

In turn, it was stated that the bounded travel time problem version has a significantly harder complexity than the maximum resilience version (in fact, the bounded travel time version is NP-Hard). Thus, it is not expected that exact methods are able to compute the optimal solution for large instances of the problem in reasonable computing times, regardless of specific features of the formulation. Indeed, the linear programming solver in Matlab 8.5.0 was unable to determine the optimal solution in a one-hour execution time limit for the instantiated problem formulation in Equation (2) that describes the case study in Montevideo. This problem instance corresponds to an integer programming problem with a significantly larger number of variables (1515) and constraints (987) than the previous problem variant.

The proposed approach to solve that more challenging problem variant applied the commercial optimization software IBM ILOG CPLEX Interactive Optimizer, version 12.6.3. Accomplishing this task required a significantly more powerful hardware setup, consisting of an HP ProLiant DL385 G7 high performance computing server with 24 AMD Opteron processors operating at 2.1 GHz and using 64 GB of RAM, which was accessed through the National Supercomputing Center, Uruguay [41]. With the described hardware and software configuration, the problem instance for Montevideo was resolved in an overall execution time of 30 s.

The previously commented empirical result has shown that for the realistic case study in Montevideo, the bounded travel times version of the problem is solvable using existing software. As a result, there was no need to develop heuristic methods to compute high-quality solutions for this specific problem variant. The exact solution could be easily obtained in less than a minute. However, it is important to acknowledge that for more complex problem variants or larger scenarios, such as the multiobjective problem introduced in this article, heuristic or metaheuristic approaches are likely necessary. These approaches, as discussed in previous research [42], provide effective means to solve more complex problem variations and handle larger problem instances.

3.3. Evolutionary Algorithm for the Multiobjective Problem Version

The main features of the proposed EA for solving the multiobjective problem formulation are explained in this subsection.

3.3.1. Solution Representation

A solution for the multiobjetive formulation is determined by the configuration of lines from remote terminal to center ($\mathcal{L}$) and the frequencies of operation ($\mathcal{F}$). Variables $x_{ij}$ and $y_{ij}^{pr}$ in MIP formulations for the maximum resilience and the bounded travel times problems derive from $\mathcal{L}$. Additional variables $\mathcal{F}$ must be added to determine frequencies, only present in the multiobjetive problem variant. Taking into account the previous considerations, solutions of the problem are characterized by paths from each terminal node to the city center, combined with the frequency for each line. Thus, the developed EA applies a specific ad-hoc encoding for representing solutions for the multiobjective problem variant. Figure 5 presents an example of solution representation.

The map in Figure 5, sketches a possible set of line configurations. Lines from the same remote terminal do not share any edge, thus guaranteeing reliability. Paths from each remote terminal towards the center are expressed as lists of nodes and grouped into genes (one gene per each remote station). To avoid redundancy, the node_id of terminals is not included in the list of nodes. The frequency for each line is also part of the encoding, associated to each terminal node. For example, for lines whose remote terminal is Pocitos (POC), the encoding in Figure 5 is interpreted as follows: the first line of Pocitos has a frequency of 3 trams by hour and spans stations 6-29-28-27-26-25-2; the second line spans 6-30-42-41-40-2 and has a frequency of 4 trams by hour; the third line follows the sequence 6-43-45-46-47-49-3 three times per hour.

3.3.2. Fitness Function

The considered fitness function in the EA to solve the multiobjective problem variant is presented in Equation (6).

$${\displaystyle f(\mathcal{L},\mathcal{F})=\frac{1}{{\lambda }_{CC}·CC+{\lambda }_{OC}·OC+{\lambda }_{UC}·UC}}$$

(6)

The fitness function considers a linear combination of the problem objectives defined in Section 2.3.4: construction costs $CC(\mathcal{L},\mathcal{F})$, operation costs $OC(\mathcal{L},\mathcal{F})$, and users costs $UC(\mathcal{L},\mathcal{F})$. All relevant values for defining these functions are fully determined up from $\mathcal{L}$ and $\mathcal{F}$. Thus, the fitness function is well defined for any solution represented in the proposed EA.

In Equation (6), ${\lambda }_{CC}$, ${\lambda }_{OC}$ and ${\lambda }_{UC}$ are fixed parameters that set the relative weight of each cost component. The cost minimization problem is transformed to the maximization of the inverse of the cost function. Weights ${\lambda }_{CC}$ and ${\lambda }_{OC}$ are adjusted to account for the per-hour cost for a given solution. The total infrastructure costs of $CC$ (Equation (3)) are prorated as for a repayment period of thirty years, without considering interest rates. Thus, ${\lambda }_{CC}$ equals $1/(30\times 365\times 24)$, a figure that intends to represent hourly amortizations. In turn, $OC$ (Equation (4)) expresses the operation costs as the number of kilometers traveled per hour by the whole set of trams in the system, when they are getting to stations at frequencies as in $\mathcal{F}$. ${\lambda }_{OC}$ scales the previous values to an estimate of the economic costs associated to those tours. They are adjusted using as a reference, the average cost incurred by a tram to keep on rolling for another kilometer. Details are elaborated in Section 4.

So far, parameters ${\lambda }_{CC}$ and ${\lambda }_{OC}$ are plenty determined for the reference period over which both costs are estimated. As a result, ${\lambda }_{CC}·CC$ and ${\lambda }_{OC}·OC$ can be directly added up because both of them represent the same magnitude, namely the USD-per-hour for that solution. On the other hand and because of its definition, the users cost function $UC$ is a dimensionless magnitude, that is, a quantity without an associated physical dimension. This is because travel times are accounted as from the relative perspective for each user regarding its optimal condition. Unlike Equation (2), where travel times where bounded by constraints, this multiobjective formulation introduces the quality of service of the system by means of a component in the objective function. So, ${\lambda }_{UC}\ge 0$ is the only constraint regarding this parameter. Setting ${\lambda }_{UC}$ to 0 allows the design to focus in the cost of the system. On the other hand, high values of ${\lambda }_{UC}$ would mean that the design focuses in computing solutions with lower average travel times.

Although in theory there is freedom to set ${\lambda }_{UC}$ values, adjusted to meet whatever goal, a realistic design should provide a balance between economic costs and quality of service. In the studied real-world application case, the ${\lambda }_{UC}$ value is manually adjusted to reach a solution with infrastructure costs and travel times from terminal to destination similar to those obtained for the same instance but over other problem variants, namely, the maximum resilience problem and the bounded travel times problem. Concrete figures for ${\lambda }_{OC}$ and ${\lambda }_{UC}$ depend of the particulars of the instance, so they are elaborated in Section 4.

3.3.3. Evolution Model

The proposed method is a micro evolutionary algorithm, i.e., an EA with rather few elements in the population [39]. This decision is based on the significantly large execution time needed to evaluate the population in each generation step, due to the intrinsic complexity of the fitness function calculation, which needs to compute the full demand matrices for each candidate solution. Thus, a population with few elements (between 10 and 20) is used to keep computation cost tractable. Micro EAs have provided good exploration patterns for hard combinatorial optimization problems [43].

Furthermore, the proposed EA follows a ($\mu +\lambda$) evolutionary approach [44], which allowed computing better results than a standard generational model in preliminary experiments. An elitist strategy is applied to preserve the main features of the best solutions, as presented in the description of the replacement operator in next section.

3.3.4. Evolutionary Operators

Population initialization. The EA initializes its population executing a randomized procedure to build solutions based on the exact method for the already described variant with maximum resilience. The exact method for optimizing construction costs (presented in Equation (1)) is modified in two key aspects, which are described next.

Terminal nodes are iteratively processed, one at the time, with independence of other terminal nodes in the scenario. Thus, Equation (1) is modified to exclusively craft the two (or three) lines for one terminal node (node_ids 4, 5, 6 or 7). This decision sustains the physical independence of lines departing from the same remote terminal, allowing those from different terminals to share edges. The number of computations required to build lines for an individual ($\mathcal{L}$) is of polynomial time complexity, as demonstrated in Section 2.3.2. The complete solution is simply built by gathering the lines of each terminal node.

Another difference with respect to the maximum resilience problem is that in the objective function, instead of using $c_{ij}$, the optimization considers random values $d_{ij}^{\prime }$. Values of $d_{ij}^{\prime }$ are independent and identically distributed random variables with uniform distribution in $[0,2\times d_{ij}^{\prime }]$. This randomization process is conceived to provide diversity to the original population, and at the same time, it creates (in average) individuals with low terminal-to-center delays. Another option for creating individuals would be setting upper limits to travel times, but that would have led to a population initialization method of non-polynomial time complexity, as it is proven in Section 2.3.3.

Finally, frequencies in $\mathcal{F}$ are generated by applying the corrective function defined in the description of the feasibility check, in order to generate feasible initial solutions.

Selection. Preliminary experiments shown that the selection operator did not affect significantly the search capabilities of the proposed EA. This result is mainly explained because the selection of appropriate individuals and features to guide the search is guaranteed by the ($\mu +\lambda$) evolution model and the elitist replacement strategy applied. For this reason, the applied selection operator simply chooses solutions from the population applying a uniform distribution. These solutions have useful features for both exploitaition and exploration, performed by the recombination and mutation operators.

Recombination. The applied recombination is a variation of Uniform Crossover applied in the list of remote terminals. Given two parent individuals, the proposed recombination operator selects genes from each parent with a uniform distribution (probability 0.5) and generates two offspring by exchanging the whole paths corresponding to each terminal node for each position. The level of application (terminal nodes) was defined because recombining paths or information inside the paths associated to each terminal node causes a high number of non-feasible individuals, due to the physical independence between the two paths connecting each terminal node and the city center. Frequencies for each line are generated by applying the corrective function, in order to generate feasible solutions. The proposed recombination operator provides a simple procedure for exploitation by sharing information between two solutions. Figure 6 presents a specific example of the recombination operator.

Mutation. The proposed mutation operator is a variant of delete mutation, applied in the paths defined between terminal nodes and the city center. The mutation selects a station in a given path using a uniform distribution between 1 and the maximum number of stations in the path. After that, the selected station and all stations after it in the corresponding path are removed from the solution. Finally, new information is generated by searching a new path between the last disconnected station and the city center. The search is performed applying the well-known Dijkstra’s Shortest Path First (SPF) algorithm, where actual costs $c_{ij}$ are used for edges, except for those already built for some other line, for which their marginal cost is set to 0. In order to keep feasibility, the new path cannot use links of other lines for that terminal. This is accomplished by pre-filtering those links prior to run the SPF algorithm. Frequencies for each line are generated by applying the corrective function, in order to generate feasible solutions.

Figure 7 presents an example of the mutation operator. Station 46 in the third path connecting Pocitos (POC) and the city center is selected for applying the mutation. Station 46 and the following stations in that path (46-47-50-51) are removed from the solution, and a new path connecting station 45 and the city center is constructed (45-20-21-34-2). Frequencies for each line are generated by applying the corrective function.

Replacement. A specific method is applied for replacement, with the main purpose of guiding the search to solutions including useful features. Replacement has been recognized as one of the most important strategies for computing high-quality results in micro evolutionary algorithms. In the proposed ($\mu +\lambda$) evolution model, the base population of ten individuals is expanded up to a maximum number of twenty individuals. When the upper limit is reached, the replacement operator selects individuals considering their fitness values (elitist selection), but 10% (i.e., two individuals) are selected among the worst in the population, in order to provide diversity to the new generation. The described operator outperformed other studied replacement methods in preliminary evaluation experiments, allowing the proposed EA to compute the optimal solution in a specific instance of the bounded travel times problem variant.

Feasibility check and corrective function. Regarding paths in lines, the proposed evolutionary operators are conceived to always generate topologically feasible solutions. Thus, no explicit feasibility check is needed for information in $\mathcal{L}$. On the other hand, once the line configurations are defined, frequencies $\mathcal{F}$ must be computed to guarantee feasibility regarding quality of service. A basic premise for quality of service is that trams must not be crowded, i.e., passengers must be able to board the next tram of their convenience, in average.

To compute a consistent pair configuration ($\mathcal{L}$, $\mathcal{F}$), a greedy approach is applied. The process operates as follows. First, all frequencies are initialized with a minimum value of one (i.e., corresponding to one trip of that line per hour). After that, the number of users in each path component for each line is computed and if the capacity of a tram is exceeded, the frequency is incremented in one unit for the most congested tram; (iii) the process is repeated until all path component for each line operates at a feasible capacity.

The described greedy approach is also used like a corrective function for possible non-feasible individuals generated in the evolutionary search (regarding frequencies).

4. Experimental Analysis

This section presents the experimental analysis of the exact and metaheuristic approaches developed for designing the backbone network of the public transport system of Montevideo.

4.1. Problem Instance

The considered instance of the problem represents the public transportation network in Montevideo, for whom many of its generalities have beed described in Section 2.2.

Table 2. Reference costs ($c_{ij}$) [in million USD], lengths ($l_{ij}$) [in meters] and travel times/delays ($d_{ij}$) [in seconds] for edges that model the considered case study.

i	j	${\mathit{c}}_{\mathit{ij}}$	${\mathit{l}}_{\mathit{ij}}$	${\mathit{d}}_{\mathit{ij}}$	i	j	${\mathit{c}}_{\mathit{ij}}$	${\mathit{l}}_{\mathit{ij}}$	${\mathit{d}}_{\mathit{ij}}$	i	j	${\mathit{c}}_{\mathit{ij}}$	${\mathit{l}}_{\mathit{ij}}$	${\mathit{d}}_{\mathit{ij}}$
1	52	24	263	84	12	13	34	3448	275	30	42	81	1478	157
1	53	65	1133	136	12	23	64	3218	262	31	32	36	509	99
1	54	45	690	110	12	25	82	4679	349	32	33	32	460	96
1	55	33	476	97	13	14	29	3283	265	32	41	38	772	115
2	25	116	2660	228	13	22	48	5500	398	34	35	12	410	93
2	33	49	821	118	14	22	30	3103	255	35	36	28	1231	142
2	34	135	2660	228	15	16	26	3037	251	35	39	126	2495	218
2	36	166	4203	321	15	20	27	2676	229	36	37	27	1182	139
2	39	97	2167	199	15	22	20	1888	182	36	39	107	2069	193
2	40	10	82	73	16	17	19	1822	178	37	38	118	2758	234
3	39	78	1642	167	17	19	29	2807	237	37	39	126	2463	216
3	40	100	1921	184	18	19	24	2282	205	38	39	37	542	101
3	49	103	1986	188	18	37	71	3727	292	38	54	31	410	93
3	51	54	1248	143	19	20	14	460	96	39	55	81	1543	161
3	55	31	410	93	19	21	23	1198	140	40	41	25	279	85
4	8	17	2380	211	20	21	26	1100	134	40	49	73	1297	146
4	9	14	1231	142	21	34	84	4728	352	41	42	72	1281	145
4	10	22	3267	265	21	37	31	1478	157	42	45	43	673	109
5	17	36	3776	295	22	23	78	3924	304	42	49	42	640	107
5	18	18	1707	171	22	36	74	4022	310	43	44	33	460	96
6	29	5	230	82	23	24	16	591	104	43	45	42	624	106
6	30	5	279	85	23	35	19	739	113	45	46	28	345	89
6	43	7	460	96	24	25	26	1116	135	46	47	31	410	93
6	44	4	230	82	24	34	10	279	85	46	48	71	1264	144
7	14	154	3185	260	25	26	10	279	85	47	49	55	1018	130
7	15	160	3201	261	26	27	59	1018	130	47	50	61	1379	151
7	16	186	3956	306	26	33	74	1445	155	48	49	99	1872	181
8	9	7	493	98	27	28	38	1806	177	48	50	74	1789	176
8	28	88	4712	351	27	31	52	1395	152	49	51	78	1740	173
9	10	12	1166	138	27	33	81	1527	160	50	51	49	1001	129
10	11	104	6600	464	28	29	21	1018	130	51	52	93	1806	177
10	27	98	5040	371	28	30	68	1395	152	52	53	62	1248	143
11	12	24	4285	326	29	44	29	361	90	53	54	99	2233	202
11	26	61	3300	266	30	31	37	542	101	54	55	38	542	101

reports values for attributes of lines and tram stretches ($c_{ij}$, $l_{ij}$ and $d_{ij}$), for the topology presented in Figure 4.

Values reported in Table 2 were determined applying the procedures described next:

• The lengths of paths were determined using the Google Maps service, available at maps.google.com, accessed on 15 May 2023.
• The travel times between stations (referred to as delays, $d_{ij}$ in Table 2) were calculated as follows: (i) The acceleration of the LRT was set to $1.96\mathrm{m}/{\mathrm{s}}^2$ (0.2 g), ensuring that standing commuters are able to maintain equilibrium; (ii) the deceleration was also set to $1.9\mathrm{m}/{\mathrm{s}}^2$ for passenger comfort; (iii) the time needed to board and alight the train at each station was set to 60 s; this value was added to the delay for every edge in the model; (iv) the top speed (cruise speed) was set to 60 km/h; the LRT takes 8.5 s to to reach this speed. The reference values for constants in (i) to (iii) were determined based on information from experts and personal experience. Acceleration and deceleration values were set according to reference values from the related literature [45,46,47] and the personal experience in the public transportation system of Montevideo. Larger than standard values were considered, taking into account that in Montevideo, buses have no exclusive infrastructure and shares lines with private transportation, which often lead to abrupt acceleration and deceleration.
• In terms of the costs of rail stretches, the proposed approach is based on the study conducted by Flyvbjerg et al. [48] regarding capital costs (per kilometer) in urban rails. The model assumes that elevated crossing points will be required at certain key intersections. Consequently, the highest per-kilometer cost reported in the base study was adopted.
• The construction costs are determined based on the lengths of the rail stretches, but they are additionally penalized by the density of urbanization in each zone. A specific penalty model is applied, which is described next. The penalty model divides the city into zones (grouping similar neighborhoods), and assigns each zone a coefficient (between 0 and 1). The higher the density of urbanization, the higher the coefficient. The urbanization level is assigned according to aerial images, available as open data, and considering the width of the avenues where the tram stretches were to be deployed. The resulting average penalty is 0.26, and 70% of the data around that average value does not differ by more than a factor of 5. As a reference, the model considers that the ratio for the civil cost (per meter) of channeling in telecommunications between urban and rural areas is 10. In a detailed engineering phase, urban planners can change any of these values, but those models and the methodology applied in this article are still valid, disregarding the parameter values. Furthermore, as the model assumes that both cost and delay are magnitudes derived from the distance (i.e., there is a high correlation between them, whose value is 0.53), solutions computed from either optimizing the total cost or delay they are similar.
• Origin-destination matrix, demand matrix ($\overline{D}$), and the actual travel times are computed using real information from the public transportation system in Montevideo. Information was obtained applying a data analysis approach [49].

Once the optimal (maximum resilience) solution was computed for the case study (as described in Section 4.3.1), the obtained cost values were adjusted by a factor to ensure an average cost of 30 million USD per km. This adjustment followed the approach outlined in the baseline work by Flyvbjerg et al. [48]. Considering that the objective function in the bounded travel times problem (Equation (2)) scales linearly, making proportional changes to the function affects the objective value without altering the solution. Consequently, applying a posteriori tuning is both feasible and valid approach in this context.

Regarding the multiobjective problem, the steps to sequentially compute $D_{\mathcal{L}}$ up from $\overline{D}$ are as follows:

• Each demand in $\overline{D}$ is associated to the geographically nearest station, among all stations in $\mathcal{L}$. For instance, when $G=(V,E)$ is as presented in Figure 4 and all the stations are used, a possible bus-stop correspondence is shown in Figure 8. From this association results a first approximation, namely $\overline{{\mathcal{E}}_{\mathcal{L}}}$;
• Diagonal elements in $\overline{{\mathcal{E}}_{\mathcal{L}}}$ are removed. This is because those intra-zone trips are not to be attended by the backbone, but for local buses. These buses span many stops while gathering some passengers towards backbone stations, concretely, those that demand making a long trip. Short-trips are to be attended by zonal buses themselves, eventually by combinations of them;
• Demands between nodes in $\mathcal{C}$ are also removed from $\overline{{\mathcal{E}}_{\mathcal{L}}}$, since one of the considered design premises is that they are solved with another media, e.g., a metro line connecting the stations in the city center;
• It is intended that the system operates with an appropriate quality of service, allowing meeting all the demand of passengers during peak hours. During that time window, the dominant direction is towards the city center. Trams must follow the line routes back and forth, so if they are dimensioned to fulfill the dominant course, they will have plenty of capacity to serve passengers in the opposite direction. Hence, demands from the center towards remote stations are removed as well, because they are residual when compared with those in the dominant direction;
• Remains to be seen how to consider those trips that need to transfer in the center. Surviving demands in $\overline{{\mathcal{E}}_{\mathcal{L}}}$ whose destination station is not spanned by in any of the lines reaching the origin station are precisely those that need to transfer. In the hub architecture is assumed that such passengers are connecting in the center with another line compliant with its final destination. Since backwards traffic is disregarded, all those trips are transformed as if they had the center as destination, and $\overline{{\mathcal{E}}_{\mathcal{L}}}$ should be updated according to these changes.

The resulting value for $\overline{{\mathcal{E}}_{\mathcal{L}}}$ corresponds to matrix $D_{\mathcal{L}}$. A set of matrices $D_{\mathcal{L},\mathcal{F}}$ accounts for the number of travels per hour for each particular tram of each line during the rush hour. Let $D_{\mathcal{L},\mathcal{F}}^k$ be the matrix for the k-th line. The period between tram arrivals for each line is determined by frequencies in $\mathcal{F}$. Just as any periodic function, tram arrivals have frequency and phase. While the frequency of a line accounts how many arrivals per-hour are expected and therefore defines inter-arrival times (i.e., the period), the phase shifts those instants forwards or backwards. As an ideal reference hypothesis, assume that phases are coordinated in such a way that arrivals are uniformly distributed at each station. For instance, if the sum of the frequencies per hour of lines passing through a station is 10, the model assumes that the time between consecutive arrivals is fixed in 6 min. This coordination could be provided as a service of a management/control facility for the public transportation system.

The model also assumes that during the rush hour, passengers arrive to stations uniformly. Those uniformity hypotheses allow equally prorating demands among those trams capable of serving some trips. Thus, given any two stations p and q in $\mathcal{L}$ with ${\left(D_{\mathcal{L}}\right)}_{p,q}>0$, an additional demand is assigned to each ${\left(D_{\mathcal{L},\mathcal{F}}^k\right)}_{pq}$ with $k\in J_{pq}$, whose value results from calculating ${\left(D_{\mathcal{L}}\right)}_{p,q}/{\sum }_{j\in J_{pq}}{f}_j$, being $J_{pq}$ the set of lines connecting stations p and q. Repeating the process for every pair of stations p and q such that ${\left(D_{\mathcal{L}}\right)}_{p,q}>0$, all $D_{\mathcal{L},\mathcal{F}}^k$ matrices are computed.

4.2. Software and Hardware Platform

The software applied in the proposed resolution approaches include:

• Matlab version 8.5.0 for the maximum resilience problem variant;
• IBMILOGCPLEX for the bounded travel times problem variant;
• Java SE Runtime for the proposed EA.
• Python and Pandas for data analysis and computation of origin-estimation demand matrices.

The experimental analysis was made on a standalone server for the bounded travel times problem variant (HP ProLiant DL385 G7, AMD Opteron processor 6172, 24 cores at 2.1 GHz and 64 GB of RAM), and a laptop (ASUS UX305CA, with an Intel m3-6Y30, 0.9 GHz, with 8 GB of RAM) for the EA applied to solve the multiobjective problem variant. Data analysis was performed applying parallel computing on servers of the National Supercomputing Center (Cluster-UY) (Intel Xeon-Gold 6138 nodes, up to 1120 CPU cores at 3.7 GHz, 3.5 TB RAM, and 28 GPU Nvidia Tesla P100, www.cluster.uy, accessed on 15 September 2023) [41].

4.3. Results

This subsection presents and discusses the empirical results of the exact and metaheuristic approaches for designing the backbone of the public transportation network in Montevideo, Uruguay.

4.3.1. Maximum Resilience Problem

A standard solver computed the optimal solution using an exact method. Figure 9 sketches the computed solution considering the reference cost, time, and distance values presented in Table 2. The maximum resilience solution yielded a total cost of 2744 million USD. Although the solution optimizes the rail stretch costs, it allows for multiple configurations of the lines. For example, in the optimal solution, it is possible to swap sub-tours between node 4 for line CA1 and node 10 for line CA2, while maintaining feasibility of the solution and the same computed cost.

The formulation presented in Equation (1) does not specify a specific configuration of lines. However, the solution guarantees the existence of at least one feasible configuration of lines and ensures that the rail costs are optimized regardless of the given characteristics and attributes of such configuration. The map in Figure 9 presents a proposed configuration for the lines, depicted in different colors. The configurations were selected to achieve balanced travel times between lines originating from the same terminal station.

Table 3. Attributes of lines for the maximum resilience problem.

Line	CA1	CA2	CE1	CE2	PC1	PC2	PC3	TC1	TC2
Cost (million USD)	254	313	283	323	225	231	272	423	420
Length (m)	9785	16,106	9292	13,364	6091	4383	5467	13,742	13,314
Delay (s)	929	1308	900	1212	913	537	807	1236	1074
Avg. speed (km/h)	37.9	44.3	37.2	39.7	24.0	29.4	24.4	40.0	44.6
Stations dist. (m)	1957	3221	1858	2227	761	1096	781	2290	3329
Stations time. (s)	186	262	180	202	114	134	115	206	269

shows the specific attributes for each line, based on the presented configuration.

4.3.2. Bounded Travel Time Problem

Given the formulation of the problem presented in Equation (2), and assuming that no specified limits on traveling times between stations, i.e., effectively assuming an infinite time limit ($T{D}_p=\infty$). By eliminating constraints $\left(iv\right)$ from Equation (2), a relaxation of the original formulation is obtained. The relaxed problem admits as solution a cycle that spans all terminal stations in $\mathcal{T}$ and includes node 3XX (node2). Additionally, such solution includes an additional path in Pocitos, for the required third line for that station. The cycle essentially follows the border of the map shown in Figure 4. However, this configuration is not suitable for a real setting of lines as the secondary line has an excessively long traveling time, making it impractical for actual use from passengers. Nonetheless, the cost of the cycle solution holds significance as it serves as a lower bound for improved, more useful in practice, and passenger-friendly solutions. The cost of the solution obtained from the relaxation of Equation (2) amounts to 1383 million USD. The average speeds are almost the same for the solution of the maximum resilience problem variant and the solution of the bounded travel time problem variant, but the speed for the different lines are different, as reported in Table 3 and

Table 4. Attributes of lines for the bounded travel times problem variant.

Line	CA1	CA2	CE1	CE2	PC1	PC2	PC3	TC1	TC2
Length (m)	10,951	12,264	12,412	15,169	7011	2611	3399	14,513	14,365
Delay (s)	1273	1079	1223	1252	832	499	614	1146	1204
Avg. speed (km/h)	31.0	40.9	36.5	43.6	30.3	18.8	19.9	45.6	43.0
Stations dist. (m)	1217	2453	1773	3034	1169	522	567	3628	2873
Stations time. (s)	141	216	175	250	139	100	102	287	241

. However, the main difference is that the overall cost is significantly lower for the solution of the bounded travel time problem variant, as a consequence of the fact that lines are allowed to share edges/tram sections.

The values taken as reference for $T{D}_p$ are determined based on the graph presented in Figure 9. This choice is made because the travel times provided in Table 3 are significantly shorter than those observed in the current transportation system in Montevideo, as reported in our previous study [28]. To allow for a better exploration pattern, while keeping traveling times near to those in Table 3, the value $T{D}_p$ for a terminal station p is defined as the overall time of the worst configuration of lines, over all lines associated to p. For example, for CAR (node4) the worst configuration is 4–9–10–11–26–25–2, with an overall travel time of 1323 s. By following the described procedure for the other terminal stations, the resulting travel time bounds are as follows: 1266 s for terminal CER, 913 s for terminal POC, and 1236 s for terminal TCO. With these values for $T{D}_p$, the configuration of lines shown in Figure 9 are feasible within the framework of Equation (2). Therefore, the cost of the computed maximum resilience solution serves as an upper bound for the bounded travel time variant. Consequently, the optimal cost for this problem lies between a minimum vale of 1383 million USD and a maximum value of 2744 million USD. The cost of the optimal solution of the bounded travel time problem is 1890 million USD, which is actually below the average of the previously mentioned cost range. By dividing the total cost among the number of annual ticket sold in Montevideo (i.e., 300 million tickets) and taking into account a repayment period of 30 years, the extra cost per-ticket is calculated to be 0.21 USD. This extra cost is just 18% of the current average ticket cost (1.12 USD, as of October 2023 [50]).

The optimal configuration of lines, based on the computed solution and following the defined premises for the model, is depicted in Figure 10. The corresponding key attributes for the lines are presented in Table 4. It is important to note that in this problem version, there is not a concept of per-line cost, as the lines are admitted to share segments (tram rails).

In this problem variant, the travel times between stations in the solution are lower to those computed for the maximum resilience solution for three lines, namely CA2, PC1, and TC1. The average traveling costs per-terminal in the computed solution are higher in three stations (Colón, $1.7\%$; Carrasco, $5.1\%$; and Cerro, 17.2%), but lower ($-13.8\%$) for the important station of Pocitos that has a higher passenger volume compared to the other stations. A significant disparity is observed in the rail stretch costs, as the cost of bounded time solution is 45.2% lower than the cost of maximum resilience solution.

The average speeds are nearly identical for both solutions: 36.9 km/h for the previous problem variant and 36.6 km/h in the bounded travel time problem. As a result, the quality of service in both cases is similar.

4.3.3. Multiobjective Problem

Just as in the maximum resilience problem, the multiobjetive formulation requires the graph of potential connections and their complementary attributes. The corresponding information is reported in Table 2.

Unlike the bounded travel time problem, where $T{D}_p$ values are needed to set upper bounds for the travel times between remote terminals and the center, the multiobjetive formulation integrates travel times as a term in the objective function.

Equation (6) states the fitness function as a linear combination of the construction costs $CC(\mathcal{L},\mathcal{F})$, operation costs $OC(\mathcal{L},\mathcal{F})$, and users costs $UC(\mathcal{L},\mathcal{F})$ (with coefficients ${\lambda }_{CC}$, ${\lambda }_{OC}$ and ${\lambda }_{UC}$, respectively). As stated in Section 3, using ${\lambda }_{CC}=1/(30\times 365\times 24)$ the product ${\lambda }_{CC}\times CC(\mathcal{L},\mathcal{F})$ corresponds to hourly amortizations. The computation of $CC(\mathcal{L},\mathcal{F})$ requires an additional parameter T, which is the acquisition costs of a tram (one unit). It is also necessary to know the maximum capacity $MC$ of a tram unit, because lines frequencies must be set to values such that the average number of passengers per train does not exceed its capacity. Both parameters T and $MC$ are related, since they depend on the number of wagons. The reference values used are $T=3$ million USD and $MC=350$ passengers.

Section 3 stated that ${\lambda }_{OC}$ is the per-kilometer energy consumption of a tram, and the term ${\lambda }_{OC}\times OC(\mathcal{L},\mathcal{F})$ matches the per-hour expenditures. Using as reference that the consumption of a diesel bus is 2.5 km/L, and that the price of fuel is 1.39 USD/L (2023), it is concluded that the consumption of a bus is about 0.58 USD/km. Trams move four times more passengers than a bus, so the formulation assumes that they have an expense of ${\lambda }_{OC}=2.24$ USD/km. The previous formulation needs a demand matrix $\overline{D}$. The matrix was computed using actual information of the Metropolitan Transportation System of Montevideo [41]. Matrix $\overline{D}$ has 6150 rows and columns, that correspond to bus stops at Montevideo and its metropolitan area.

Finally, the only constraint of the third parameter in the fitness function is ${\lambda }_{UC}\ge 0$. The proposed EA was configured by performing multiple executions using the instance defined by the previous dataset for different values of ${\lambda }_{UC}$, until computing a solution with infrastructure costs and terminal-to-center travel times comparables with those of previous formulations. Line configurations are represented in Figure 11. Lines in $\mathcal{L}$ must be complemented with the set of corresponding frequencies $\mathcal{F}$. The computed values are: 3 (CA1), 9 (CA2), 11 (CE1), 9 (CE2), 6 (PC1), 7 (PC2), 3 (PC3), 3 (TC1), and 6 (TC2).

The total investment costs for the computed solution is 1995 million USD. Costs are split into 1824 million USD corresponding to tram rails (with its complementary infrastructure) and 171 million USD corresponding to trams themselves. The last figure results from adding up frequencies and multiplying the result by T. The cost of the tram rails of the solution found by the EA reduced in 66 million USD (i.e., 3%) the value computed for the bounded travel times problem variant (i.e., 1824 vs. 1890 million USD).

Furthermore, operation costs of the solution computed by the EA are significantly lower than construction costs. Considering the computed frequencies and the lengths of each line, the whole set of trams travel 1170 km at each hour. Assuming an hourly frequency distribution of 8 h at 100% (peak), 6 h at 67% (3 before and 3 after the peak), and 10 h at 50% (demand valley), on average, the set of trams travel about 19,890 km every day. Taking into account the reference efficiency of 2.24 USD per kilometer, the annual cost of operation is 16.2 million USD, which represents 25% of the annual depreciation costs in infrastructure.

A compendium of the economic values in the result is as follows. The annual cost of the optimized backbone is 82.7 million USD, the distribution of these costs is 80% for infrastructure (73% in tram railways and 7% in tram units), and 20% for operation (i.e., energy consumption). This distribution confirms hypotheses for the maximum resilience and bounded travel times versions of the problem, which mainly focus upon optimizing tram railways costs.

Table 5. Attributes of lines for the multiobjective problem variant.

Line	CA1	CA2	CE1	CE2	PC1	PC2	PC3	TC1	TC2
Length (m)	10,803	10,655	11,132	15,169	5402	2463	4383	14,365	15,366
Delay (s)	1332	914	1146	1252	667	558	606	1204	1401
avg. speed (km/h)	29.2	42.0	35.0	43.6	29.2	15.9	26.0	43.0	39.5
Stations dist. (m)	1080	2664	1590	3034	1080	411	877	2873	2195
Stations time. (s)	133	229	164	250	133	93	121	241	200

reports the main attributes for the configuration of lines. Overall, travel times are similar to those for the bounded travel times (see Table 4), in fact the accumulated delays are −0.46% apart. Moreover, the new solution improves or matches previous travel times between stations in five out of the nine lines, and the tram rail investments in this case (1824 million USD) are slightly below that cost for previous solution (1890 million USD). The multiobjective version improves that cost by not considering the travel times constraints in Equation (2), e.g., terminal-to-center travel times for CA1 (1332 s) and TC2 (1401 s) are above $T{P}_4$ and $T{P}_7$ values, (1308 and 1236 s, respectively).

From the point of view of commuters, a relevant metric is the waiting time at stations for the computed solution. As a consequence of the frequencies $\mathcal{F}$ and line configurations $\mathcal{L}$, at the peak hour, in average, passengers making a trip from remote terminal stations will have to wait 2.5 min in Carrasco, 1.5 min in Cerro, 1.6 min in Pocitos, and 3.3 min in Colón. Line configurations determine that passengers from Pocitos can be served by their own lines and also by CA1. That effect applies to several other stations, for instance, the station whose node_id is 21 is used by four different lines (CE1, CE2, TC1, TC2), whose frequencies combined add up to 29. The previous number implies that a tram is arriving at that station every 2 min, so the average waiting time is of 1 min. Even being geographically close, the station with node_id is 15 is only served by TC1, so the average waiting time in this case is 10 min.

The approach to asses user benefits (Equation (5)) aims at maximizing the average travel time. As a consequence, some users might be privileged over others. A more detailed analysis of particular cases is proposed as future work. Comparing travel times between the current public transportation system in Montevideo and that presented in this work is subtle. The proposed system is hierarchical, with a trunk backbone network (designed in this article) hierarchically integrated with buses, that are now reassigned as feeders (i.e., access function). The problem of designing a consistent and optimized structure for new bus lines is currently been addressed by our research group. Once this ongoing work is completed, detailed conclusions regarding the statistical distribution of end-to-end travel times will be obtained. As a reference, some relevant cases are reported in

Table 6. Travel time comparison for selected relevant trips in Montevideo.

Origin	Destination	Current Time	LRT	Speed up
Cerro	Carrasco	1 h 35 min	33 m	2.9
Cerro	Centro	1 h 10 min	17 min	4.1
Colón	Centro	1 h 40 min	19 min	5.3
Colón	Pocitos	1 h 20 min	29 min	2.8
Pocitos	Centro	25 min	9 min	2.8

. Real GPS data from travels by passengers between points near to terminals are used to compute the travel times using the proposed LRT design and the current public transportation system in Montevideo. The speedup (i.e., relative reduction on travel times) is reported. Results demonstrate that travel time reduces up to five time with respect to the current times, for the specific case of a travel between Colón and Pocitos (1 h 40 m to 19 m). In average, travel time reductions are over a factor of three.

As a final comment, over a basis of 300 million tickets sold per year and taking into account a repayment period of 30 years, the total cost for the solution of the multiobjective problem variant (which accounts infrastructure and operating expenditures) is 0.28 USD. Considering the average ticket cost (1.12 USD in October 2023), the backbone design implies an increment of 25%. On the other hand, savings are expected from buses reassignments. Given that bus lines in Montevideo are notably long, with an average length is 16.7 km and standard deviation of 7.1 km, when reconverted to fulfill the access service, buses will have shorter tours and there will be savings from this transformation, which will compensate the increment in budget due to the backbone, at least partially. Once finished the current stage of this work, i.e., the design of new routes for buses, the effectiveness of the new hierarchical architecture as a plenty competitive and economically viable design for the public transportation system in Montevideo will be demonstrated. Regarding travel times, values reported in Table 6 evidence that the quality of service of the new system will be plenty superior to the one provided by the current flat network structure.

5. Conclusions and Future Work

This article described exact and metaheuristic approaches applied to different variants of the network design problem applied to a trunk backbone network for a hierarchically integrated public transportation system. The case study analyzed in the article was the public transportation system of Montevideo, Uruguay, for which no previous proposals have been presented. Real data were considered to build origin-destination and demand matrices, and also the real travel times.

Exact solvers were proposed for solving the simpler variants of the problem, to minimize infrastructure costs considering maximum resilience and bounded travel times. In turn, an evolutionary approach was proposed for solving a significantly harder multiobjective version of the problem that considers the optimization of infrastructure costs, operation costs, and quality of service. As a specific contribution over previous articles in the related literature, the developed resolution approaches integrated topological constraints to improve reliability of the public transportation network.

The results of the developed models and algorithms indicate that new designs for the public transportation system in Montevideo provide significantly better quality of service (i.e., lower end-to-end travel time) at a reasonable cost. Computed travel times improved up to five times for relevant cases respect to the current design of the public transportation system in the city; while estimates of additional infrastructure and operational costs for the new backbone round 25% of the actual ticket price. In turn, the resulting costs, between 26 and 29 million USD/km, are significantly lower than the reference costs for metro systems (between 50 and 100 million USD/km [48], a prohibitive cost for a third world country such as Uruguay).

The reported values are to be complemented with a proper optimized design for the access network, that is, with new alignments for bus lines, which constitutes the main line of future work. Shorter bus lines will compensate additional backbone costs and will spur the whole electrification of the public transportation infrastructure [51]. Conversely, the lower average speed of buses will decrease speedup ratios when accounted in the total bus-stop to bus-stop time. Another relevant issue concerns to designing a fully integrated multi-modal public transportation system, integrating bicycles ini the most important bus hubs, to be used as feeders for first and last mile trips.

Regarding the algorithmic contribution, parallel models of the proposed evolutionary algorithm must be explored in order to consider larger populations to extend the search capabilities by using the computing power of modern high performance computing facilities. An explicit multiobjective approach should also be considered to find solutions with different trade-offs to be considered by authorities and decision makers. The application of the developed methodology and algorithms to other relevant case studies is a promising line of work too.