A Synchronized Optimization Method of Frequency Setting, Timetabling, and Train Circulation Planning for URT Networks with Overlapping Lines: A Case Study of the Addis Ababa Light Rail Transit Service

Abstract

Urban rail transit (URT) systems are essential to ensuring efficient and sustainable urban mobility. However, the core components of operational planning, service frequency setting, train timetabling, and train allocation are often optimized separately, leading to fragmented decision-making and suboptimal system performance. This study addresses that gap by proposing an integrated optimization framework that simultaneously considers all three planning layers under time-dependent passenger demand conditions. The problem is formulated as a bi-objective Integer Nonlinear Programming (INLP) model, aiming to jointly minimize passenger waiting time and total operational cost. To solve this large-scale, combinatorial problem, a tailored Multi-Objective Particle Swarm Optimization (MOPSO) algorithm is developed. The algorithm incorporates discrete variable handling, constraint-preserving mechanisms, and a customized encoding scheme that aligns with the structural characteristics of URT operations. The proposed framework is applied to real-world data from the Addis Ababa Light Rail Transit (AALRT) system. The results demonstrate that the MOPSO-based approach offers a more diverse and operationally feasible set of trade-off solutions compared to a widely used benchmark algorithm, NSGA-II. Specifically, it provides transit planners with a flexible decision-support tool capable of identifying schedules that balance service quality and cost, based on varying strategic or budgetary priorities. By integrating interdependent planning decisions into a unified model and leveraging the strengths of a customized metaheuristic algorithm, this study contributes a scalable, adaptable, and practically relevant methodology for improving the performance of urban rail systems.

1. Introduction

1.1. Background

In the last decade, rapid transit systems (RTSs) became a solid tool in urban transportation by maintaining an efficient and reliable transportation option through linking key areas of modern cities. RTS networks represent a solid foundation that supports the economic, social, and environmental dynamics of urban cities by providing smooth transportation for daily passengers. In practice, an RTS network should ease the movement of vast numbers of passengers across various origins and destinations with minimum congestion and delays. One key characteristic in operations within RTS networks is the possibility of passengers transferring between the lines of the network to arrive at their respective destinations. Transfer movements between the lines of a network introduce flexibility in travel paths and allow the transit network to provide transportation options across wide geographic areas without the need for direct links between stations. Nevertheless, the transfer stations add an important layer of complexity to the scheduling of daily train operations.

In practice, the optimization of daily train schedules in RTS networks is considerably challenging given the requirement for schedule synchronization across the transit lines. Transfer operations are required to be coordinated to minimize the overall transferring time of passengers, avoid schedule delays, and maintain the system’s efficiency. Otherwise, if transfer operations are not synchronized, passengers would face long waiting times at stations, missed connections, and overcrowded trains, which can all negatively affect both the experience of passengers and the performance of the system. In this regard, the development of synchronized schedules for RTS networks was extensively researched in the area of rail transportation planning [1,2,3,4].

Scheduling passenger rail transportation involves multiple stages, and each stage requires careful consideration. Typically, the stages include line planning, in which the paths are determined; stop planning, where the stops of trains at each station of the path are determined; frequency setting, where it determines the number of trains running on each path; timetabling, the stage involves scheduling arrival and departure events of trains at each station; and resources allocation, which handles the distribution of trains and personals to meet the demands. Conventionally, these stages are handled sequentially due to the complexity of the stages and hence the high computational costs, particularly when scheduling full-day services. Nonetheless, solving the aforementioned stages separately may lead to sub-optimal outcomes, as the decision-making of one specific stage may affect the outcomes for the following stages.

In light of these challenges, the latest works delved into the possibilities of simultaneously optimizing multiple stages [5,6,7]. This method opens a new comprehensive view of the scheduling problem, ensuring that the correlations between stages are seen, which leads to better overall solutions. However, synchronously optimizing multi-stage processes brings its challenges as the resulting mathematical modeling tends to be considerably complex and hence requires advanced computational techniques for effective solving. In more detail, the challenge lies in balancing the objectives of both passengers’ traveling time and maximizing the system’s efficiency, all while meeting operational constraints.

Considering the limitations mentioned previously regarding the synchronous optimization of RTS networks and the multi-stage approach, in this work, we aim to present a novel approach for synchronizing RTS network schedules featuring a multi-stage optimization approach that handles several key stages simultaneously. In more specificity, we address three critical stages, including frequency setting, timetabling, and train allocation. By jointly optimizing these stages, we look to create a synchronized scheduling method that ensures efficient and effective passenger transfer between lines and improves the overall performance of the RTS network. To handle this problem, we propose novel mathematical modeling and a solving method based on a Pareto front approach to handle multiple objectives at the same time by effectively navigating the complex solution space.

To validate the method, we conduct numerical experimentation based on a real-world case scenario focusing on the Addis Ababa Light Rail Transit Service. The case study will allow us to test the effectiveness of our approach in a practical context. The structure of the paper is as follows: Section 1.2 provides a detailed literature review of previous works related to the synchronous scheduling of RTS networks. Section 1.3 delves into the literature gaps we attempt to address in the present study. Section 2 describes the mathematical modeling and the solving approach. Section 3 features the numerical experiments, showcasing the performance of our method in optimizing the service in Addis Ababa Light Rail Transit. Section 4 discusses the implications of the results and concludes the study.

1.2. Literature Review

The literature on transit networks highlights the importance of optimizing transfer synchronization to enhance overall system efficiency. The study investigates the stochastic transfer synchronization problem, aiming to align the timetables of various routes within a transit network. This synchronization is crucial for minimizing transfer waiting times, reducing delays, and limiting unnecessary in-vehicle times for passengers. By addressing the complexities of route interactions and passenger flow, the research provides valuable insights into improving the reliability and convenience of public transportation systems, ultimately contributing to a better user experience and increased ridership [8]. In addition, the study presents a new optimization challenge that combines synchronization timetabling and interlining vehicle scheduling for overlapping bus routes. The objective is to improve service regularity for these routes by reducing the total deviation of headways about the joint headway at synchronization stops, while also minimizing the size of the bus fleet [9]. Furthermore, this study seeks to present a mathematical model designed to generate an optimal and coordinated timetable for the entire urban rail system, to reduce both passenger travel time and energy consumption of the trains [10]. Conversely, to enhance passenger accessibility and decrease travel time during the night, this paper focuses on optimizing the synchronization of the timetables for the last few trains on each line of an urban rail transit (URT) network [11].

Recent research on synchronized timetabling for urban rail transit emphasizes enhancing both passenger experience and operational efficiency. Mixed-integer programming models have been proposed to minimize passenger waiting times and facilitate smoother transfers [1,12]. These models take into account dynamic passenger demand, train capacity limitations, and the timing of transfer events [3,13]. Also, studies concentrated on merging different phases of rail transit planning to enhance operations and resource management. Research has linked frequency setting, timetabling, and resource allocation to boost efficiency and service quality. For instance, Ref. [14] created a methodology for optimal timetabling that incorporates marshaling and skip–stop operations, leading to cost reductions. Ref. [15] combined timetabling with vehicle scheduling, improving both service quality and resource use. Ref. [6] introduced an integrated model that aligns line planning, timetabling, and rolling stock allocation, taking into account passenger needs and operational limitations. Ref. [16] offered a multi-objective optimization strategy for stop planning, timetabling, and rolling stock management on high-speed rail lines. These integrated strategies have shown significant benefits, including cost savings, enhanced service quality, and better resource optimization compared to traditional sequential planning approaches.

Integrated optimization plays a crucial role in enhancing the efficiency and effectiveness of transit systems by simultaneously addressing multiple interrelated components, such as train capacity, passenger flow, and scheduling. By framing optimization problems as mixed integer nonlinear programming models, researchers can develop more robust solutions that balance competing objectives and constraints, ultimately improving service quality and operational performance. This literature review highlights various innovative approaches, including collaborative optimization strategies and iterative methods, that demonstrate the significant advancements achieved through integrated optimization in the context of rail transit systems.

The study framed the optimization problem as a mixed integer nonlinear programming model to balance train capacity and waiting for passengers, later reformulating it into a mixed integer linear programming model by linearizing the constraints. The GUROBI solver was used to implement the collaborative optimization strategy, which included optimized passenger flow control and train scheduling with skip–stop patterns [17]. Building on previous research, a mixed-integer nonlinear programming (MINLP) model was developed to optimize train utilization while addressing stranded passengers [18]. Additional variables were introduced to linearize the nonlinear constraints, transforming the original model into a robust counterpart based on strong duality theory [19]. In contrast to earlier studies, the model was reformulated as a Markov decision process (MDP) by clearly defining the state, action, and reward functions and then solved using the proposed Q-learning approach [20]. Furthermore, to calculate passenger flow data and guidance time, a multi-agent simulation model was first developed, followed by a two-phase integrated passenger flow assignment using a backtracking algorithm to generate guidance information that minimizes total waiting time and alleviates congestion [21]. Additionally, a mixed-integer linear programming model was formulated, and an iterative heuristic algorithm integrating tabu search and the Gurobi solver was developed using the linear weighting technique to solve the model [22]. Consequently, a series of quadratic and quasi-quadratic objective functions were introduced to accurately represent the total waiting time based on minute-dependent and hour-dependent demand from various origin-destination pairs, leading to the development of mathematically rigorous nonlinear mixed integer programming models for both real-time scheduling and medium-term planning [23]. In light of this, an iterative nonlinear programming (INP) method should be introduced that derives solutions for the original MINLP problem by alternately solving a nonlinear programming problem and a mixed integer linear programming (MILP) problem. Additionally, an equivalent MILP formulation of the original MINLP model is created, along with an approximated MILP approach to minimize the constraints related to passenger demand [24,25,26,27,28].

Building on previous research, the study presents two integer linear programming models and an effective column-generation-based diving heuristic algorithm to simultaneously optimize train timetables and rolling stock circulation plans [29]. A multi-frequency line planning problem (MF-LPP) model and a multi-period train timetabling problem (MP-TTP) model are presented to address the line planning and train scheduling challenges [30]. A nonlinear programming model is developed to create timetables aimed at reducing the average waiting time for passengers [31]. By jointly optimizing the train timetable and precise passenger flow management strategies, a comprehensive integrated integer linear programming model is formulated, followed by the design of a hybrid algorithm that combines an enhanced local search with the CPLEX solver to identify high-quality solutions [32].

1.3. Research Gaps

A primary challenge in optimizing urban rail transit (URT) networks is the deep, synergistic integration of its core operational planning stages. While strides have been made in optimizing individual components, a critical research gap persists in the simultaneous and synchronized planning of frequency setting, timetabling, and train allocation. The prevailing approach in much of the existing literature treats these interdependent elements in isolation or as a sequence of separate sub-problems. This fragmented methodology fundamentally fails to capture the complex feedback loops between them; for instance, a chosen service frequency directly constrains the feasible solution space for both the timetable and the subsequent allocation of trains to cover that timetable.

This oversight represents a significant scientific and practical problem. By ignoring these interdependencies, existing models often produce solutions that are only locally optimal, leading to inefficient system-wide performance, including inflated operational costs, suboptimal resource utilization, and a disconnected passenger experience. There is a clear and pressing need for a more cohesive framework that can dynamically and holistically optimize all three planning layers concurrently. This paper aims to fill this specific gap by proposing a unified model that addresses this integrated challenge.

Our approach contributes to closing this gap by introducing an innovative INLP model and an advanced solution methodology. The use of Multi-objective Particle Swarm Optimization (MOPSO) for Pareto front optimization represents a crucial advancement in tackling large-scale, NP-hard problems. This method is particularly well-suited to capture the inherent trade-offs between conflicting objectives, such as minimizing operational costs versus minimizing passenger waiting times, and provides a robust framework for evaluating these trade-offs. By addressing the problem of fragmentation with this comprehensive approach, our research offers a practical and scalable solution, thereby enhancing the applicability of integrated optimization models for real-world URT systems.

1.4. Novelty and Main Contributions of This Study

While MOPSO is a well-established multi-objective optimization method, the novelty of this study lies in its customization and application to a uniquely complex and realistic urban rail transit planning problem. Specifically, our contributions are threefold:

Integrated Problem Formulation: We propose a novel bi-objective nonlinear optimization model that unifies service frequency setting, timetable scheduling, and train allocation across multiple transit lines under time-dependent passenger demand. To the best of our knowledge, this formulation has not been addressed jointly using MOPSO or any other metaheuristic.

Tailored Encoding and Operators: The standard MOPSO framework is modified to accommodate discrete, combinatorial decision variables through problem-specific rounding and clamping mechanisms. Additionally, the solution structure is adapted to respect practical rail operation constraints, such as train capacity, headway limits, turnaround feasibility, and depot availability.

Empirical Benchmarking: The proposed method is evaluated against NSGA-II, a widely accepted evolutionary algorithm, using real operational data from the Addis Ababa Light Rail Transit system. The results show that our MOPSO-based approach yields a broader and more convergent Pareto front, offering a richer set of operational trade-offs for decision-makers.

Through this tailored application, we demonstrate that a well-adapted metaheuristic can significantly enhance decision-support capabilities for integrated transit planning.

2. Materials and Methods

2.1. Problem Formulation

In this paper, we focus on the combined optimization of frequency setting, timetabling, and train circulation planning for a rapid transit network. We consider a network with multiple lines where each line $l$ is equipped with a single yard, as shown in Figure 1. Each line connects $S{T}_l$ stations. Each daily operation is split into multiple intervals referred to by $i$. Daily operations are delimited by the latest event limit $t_{latest}$. Within each interval, we assume a set of potential departing/arriving events $t$, as shown in Figure 2.

Firstly, to address the frequencies of service within each interval, we formulate an integer variable $F_i^l$ representing the number of journeys within time interval $i$ in line $l$. Then, based on the frequency of operating journeys, we handle the decision-making for timetabling by arranging departure/arrival events of trains by implementing two binary variables $D_{t,l}^{st}$/$A_{t,l}^{st}$ depicting the occurrence of the event during instant $t$ at station $st$ in line $l$, where

$$D_{t,l}^{st}=\left\{\begin{array}{c}1 if the departure event from station st at line l during instant t occurs\\ 0 otherwise\end{array}\right.$$

$$A_{t,l}^{st}=\left\{\begin{array}{c}1 if the arrival event to station st at line l during instant t occurs\\ 0 otherwise\end{array}\right.$$

In practice, for most rapid transit systems schedules feature full-length journeys, i.e., traveling from one terminal station to the opposite terminal station. To achieve the schedule, we synchronize the selection of potential departure and arrival events as shown in Figure 3.

Alongside the timetabling, we address the train circulation plan, which involves arranging movements of trains from the yards and ensuring turnaround operations by linking arrival and departure events at terminal stations. We introduce 2 binary variables ${∆}_{t,l}^{in}$ and ${∆}_{t,l}^{out}$, where

$${∆}_{t,l}^{in}=\left\{\begin{array}{c}1 if the train arriving to terminal station in line l at instant t enter the yard \\ 0 otherwise\end{array}\right.$$

$${∆}_{t,l}^{out}=\left\{\begin{array}{c}1 if the train departing from terminal station in line l at instant t exits the\\ yard \\ 0 otherwise\end{array}\right.$$

To address turnaround operations, we tackle the links of trains’ movements at turnaround stations by introducing a binary variable ${∆}_{st,l}^{t,t^{\prime }}$, where

$${∆}_l^{{t,t}^{\prime }}=\left\{\begin{array}{c}1 if the train arriving to terminal station in line l at instant t turns around and \\ departs in the opposite direction at instant t^{\prime } \\ 0 otherwise\end{array}\right.$$

To better visualize the involvement of variables ${∆}_{t,l}^{in}$, ${∆}_{t,l}^{out}$, and ${∆}_{st,l}^{{t,t}^{\prime }}$, Figure 4 shows examples of trains’ movements in and out of the yard and the linkage of events at terminal stations to achieve turnaround operations.

For convenience, we summarize the decision variables for the proposed problem in Table 1. Meanwhile, we detail the assumptions taken in the current model in the following,

Assumption 1.

All trains stop at every station within the dedicated transit line.

Assumption 2.

Every train is dedicated to a specific transit line.

Assumption 3.

Every station can receive only one train at a time in a given direction.

Assumption 4.

Trains are not allowed to overtake each other during service.

Assumption 5.

Depots’ capacity limitation is not considered.

Table 1. Definitions of the decision variables.

Variable	Definition
$F_i^l$	Frequency of journeys within time interval $i$ in line $l$.
$D_{t,l}^{st}$	Binary variable to determine Departure event from station $st$ in line $l$ during instant $t$ occurs.
$A_{t,l}^{st}$	Binary Variable to determine Arrival event at station $st$ in line $l$ during instant $t$ occurs.
${∆}_{t,l}^{in}$	Binary variable to determine if the train arriving from the terminal station at instant $t$ enters the depot linked to line $l$.
${∆}_{t,l}^{out}$	Binary variable to determine if the train departing from the terminal station at instant $t$ exits the depot linked to line $l$.
${∆}_l^{{t,t}^{\prime }}$	Binary variable to determine if the train arriving at $t$ in line $l$ departs from the terminal station at instance $t\prime$.

Table 1. Definitions of the decision variables.

Variable	Definition
$F_i^l$	Frequency of journeys within time interval $i$ in line $l$.
$D_{t,l}^{st}$	Binary variable to determine Departure event from station $st$ in line $l$ during instant $t$ occurs.
$A_{t,l}^{st}$	Binary Variable to determine Arrival event at station $st$ in line $l$ during instant $t$ occurs.
${∆}_{t,l}^{in}$	Binary variable to determine if the train arriving from the terminal station at instant $t$ enters the depot linked to line $l$.
${∆}_{t,l}^{out}$	Binary variable to determine if the train departing from the terminal station at instant $t$ exits the depot linked to line $l$.
${∆}_l^{{t,t}^{\prime }}$	Binary variable to determine if the train arriving at $t$ in line $l$ departs from the terminal station at instance $t\prime$.

2.2. Mathematical Modeling

In practice, daily operations in rapid transit systems must ensure comfortable service for passengers and good management of resources. Often, the most influencing factor in passengers’ comfort during travel is the waiting time at stations because the average waiting time of passengers is heavily influenced by the frequency of service as elaborated in [33,34,35]. We formulate the objectives as follows:

$$f_1={\sum }_l{\sum }_i{Q}_i^l·{\displaystyle \frac{\left|i\right|}{2·F_i^l}},$$

(1)

where $Q_i^l$ represents travel demands of passengers traveling in line $l$ during interval $i$, and $\left|i\right|$ represents the time gap of a single time interval.

In RTS operations, the allocation of resources is mostly influenced by energy expenditure and trains’ allocation. Considering these metrics, we formulate the second objective based on time-related operations including train running time between sections and movements at turnaround tracks and to/from the depot,

$$f_2={\sum }_l\left({\sum }_i{F}_i^l·T_r^l+{\sum }_t{\sum }_{t^{\prime }}{∆}_l^{{t,t}^{\prime }}·T_t+{\sum }_t\left({∆}_{t,l}^{out}+{∆}_{t,l}^{in}\right)·T_d\right),$$

(2)

where $T_r^l$ represents the total riding time of a train across the entire line $l$, $T_t$ is the time required for trains to turn around at terminal stations, and $T_d$ represents the average time needed for trains to move in/out from the depot. We formulate the objective of mathematical model as follows:

$$\min \left(f_1\left({\left\{F_i^l\right\}}_{\forall i,l}\right),f_2\left({\left\{F_i^l\right\}}_{\forall i,l},{\left\{{∆}_{t,l}^{in}\right\}}_{\forall t,l},{\left\{{∆}_{t,l}^{out}\right\}}_{\forall t,l},{\left\{{∆}_l^{{t,t}^{\prime }}\right\}}_{\forall t,t^{\prime },l}\right)\right),$$

(3)

To ensure minimum safety time difference between trains and to maintain an acceptable frequency of train departure for passengers’ convenience, inequation (4) ensures adequate values for the frequency of trains’ departures at each line,

$$F^{min}\le F_i^l\le F^{max} \forall l, \forall i,$$

(4)

In terms of movements of trains, Equation (5) ensures that any train departing from station $st$ in line $l$ at instant $t$ arrives at the next station $s{t}^{\prime }$ after achieving the required traveling time $RT\_st\_l$,

$$A_{(t+RT\_st\_l)}^{{st}^{\prime }}=D_{t,l}^{st} \forall st, \forall l,$$

(5)

where $s{t}^{\prime }$ is the following station to station $st$, and ${RT}^{st}$ is the riding time of trains within the section $\left[st,s{t}^{\prime }\right]$ within line $l$. Likewise, any train dwelling at a given station $st$ may depart after achieving an adequate stopping time ${DT}^{st}$ as in Equation (6),

$$D_{t+{DT}_l^{st},l}^{st}=A_{t,l}^{st} \forall st, \forall l,$$

(6)

Considering the selection of trains’ movements selection, Equation (7) ensures that a train can depart from a given station $st$ once an adequate time difference $⌊{\displaystyle \frac{\left|i\right|}{F_i^l}}⌋$ has passed from the previous train departure as shown in Figure 5,

$$D_{t^{\prime },l}^{st}=D_{t,l}^{st} \forall st, \forall l,$$

(7)

where $t^{\prime }$ represents a time instant achieving a time difference in respect to time instant $t$, which is equivalent to frequency of service $F_i^l$, where it verifies $t^{\prime }\ge t+⌊{\displaystyle \frac{\left|i\right|}{F_i^l}}⌋$.

In URT networks characterized by overlapping transit lines as in Figure 6, schedules must ensure that time differences between trains of different lines do not interfere with each other, which is covered by Equations (8) and (9),

$$D_{t+\delta ,l^{\prime }}^{st}=D_{t,l}^{st} \forall st, \forall l,$$

(8)

$$A_{t+\delta ,l^{\prime }}^{st}=A_{t,l}^{st} \forall st, \forall l,$$

(9)

where $\delta$ represents a safety time difference between consecutive running trains. Considering daily operation limits, every departure and arrival event must not exceed the daily time limit $t_{latest}$ as in inequations (10) and (11),

$$t·D_{t,l}^{st}\le t_{latest} \forall st, \forall l,$$

(10)

$$t·A_{t,l}^{st}\le t_{latest} \forall st, \forall l,$$

(11)

Any scheduled departure or arrival event to a terminal station with a depot facility must ensure that trains arriving at the station either turnaround or enter the depot. Likewise, trains departing from the station either come out from the depot or link to another train. Both conditions are ensured by Equations (12) and (13).

$${∆}_{t^{\prime },l}^{out}+\sum _t{∆}_l^{{t,t}^{\prime }}=\sum _t{A}_{t,l}^{st}·D_{t^{\prime },l}^{st} \forall st, \forall l,$$

(12)

$$\sum _{t^{\prime }}{∆}_l^{{t,t}^{\prime }}+{∆}_{t,l}^{in}=\sum _t{D}_{t^{\prime },l}^{st}·A_{t,l}^{st} \forall st, \forall l,$$

(13)

In the case that terminal stations are not equipped with depot facilities, then arriving and departing trains must ensure turnaround operations. In this case, Equations (14) and (15) cover turnaround operations between the scheduled events.

$$\sum _t{∆}_l^{{t,t}^{\prime }}=\sum _t{A}_{t,l}^{st}·D_{t^{\prime },l}^{st} \forall st, \forall l,$$

(14)

$$\sum _{t^{\prime }}{∆}_l^{{t,t}^{\prime }}=\sum _t{D}_{t^{\prime },l}^{st}·A_{t,l}^{st} \forall st, \forall l,$$

(15)

In any case, departure and arrival events must be synchronized in case of a train turnaround at a terminal station as in inequation (16).

$$t^{\prime }·D_{t^{\prime },l}^{st}-{t·A}_{t,l}^{st}\ge T_t+{DT}^{st}-I·\left(1-{∆}_l^{{t,t}^{\prime }}\right)\forall st,\forall l,$$

(16)

where $T_t$ is the average time required for trains to turn around at terminal stations, and $I$ is a very large number.

The complete mathematical formulation of the proposed bi-objective INLP model is provided below for clarity and completeness:

Objective: $\min \left(f_1\left({\left\{F_i^l\right\}}_{\forall i,l}\right),f_2\left({\left\{F_i^l\right\}}_{\forall i,l},{\left\{{∆}_{t,l}^{in}\right\}}_{\forall t,l},{\left\{{∆}_{t,l}^{out}\right\}}_{\forall t,l},{\left\{{∆}_l^{{t,t}^{\prime }}\right\}}_{\forall t,t^{\prime },l}\right)\right)$

Subject to:

Contraint 1: $F^{min}\le F_i^l\le F^{max}\hspace{1em}\hspace{1em}\forall l,\forall i$,

Contraint 2: $A_{(t+{RT}_l^{st},l)}^{{st}^{\prime }}=D_{t,l}^{st}\hspace{1em}\hspace{1em}\forall st, \forall l,$

Contraint 3: $D_{t+{DT}_l^{st},l}^{st}=A_{t,l}^{st}\hspace{1em}\hspace{1em}\forall st,\forall l,$

Constraint 4: $D_{t^{\prime },l}^{st}=D_{t,l}^{st}\hspace{1em}\hspace{1em}\forall st,\forall l,$

Constraint 5: $A_{t+\delta ,l^{\prime }}^{st}=A_{t,l}^{st}\hspace{1em}\hspace{1em}\forall st,\forall l,$

Constraint 6: $t·D_{t,l}^{st}\le t_{latest}\hspace{1em}\hspace{1em}\forall st,\forall l,$

Constraint 7: $t·A_{t,l}^{st}\le t_{latest}\hspace{1em}\hspace{1em}\forall st,\forall l,$

Constraint 8: ${∆}_{t^{\prime },l}^{out}+\sum _t{∆}_l^{{t,t}^{\prime }}=\sum _t{A}_{t,l}^{st}·D_{t^{\prime },l}^{st}\hspace{1em}\hspace{1em}\forall st,\forall l,$

Constraint 9: $\sum _{t^{\prime }}{∆}_l^{{t,t}^{\prime }}+{∆}_{t,l}^{in}=\sum _t{D}_{t^{\prime },l}^{st}·A_{t,l}^{st}\hspace{1em}\hspace{1em}\forall st,\forall l,$

Constraint 10: $\sum _t{∆}_l^{{t,t}^{\prime }}=\sum _t{A}_{t,l}^{st}·D_{t^{\prime },l}^{st}\hspace{1em}\hspace{1em}\forall st,\forall l,$

Constraint 11: $\sum _{t^{\prime }}{∆}_l^{{t,t}^{\prime }}=\sum _t{D}_{t^{\prime },l}^{st}·A_{t,l}^{st}\hspace{1em}\hspace{1em}\forall st,\forall l,$

Constraint 12: $t^{\prime }·D_{t^{\prime },l}^{st}-{t·A}_{t,l}^{st}\ge T_t+{DT}^{st}-I·\left(1-{∆}_l^{{t,t}^{\prime }}\right)\hspace{1em}\hspace{1em}\forall st,\forall l,$

Finally, we recap the input parameters used in the problem formulation and the mathematical modeling sections in

Table 2. Definitions of the input parameters.

Index	Definition
$l$	Index for transit lines.
$st$	Index for stations.
$i$	Index for time intervals.
$t$	Index for time instants.
$S{T}_l$	The number of stations in line $l$.
$t_{latest}$	Latest time allowed in a daily operation.
$\left|i\right|$	Length of a single time interval $i$.
$Q_i^l$	Passengers’ travel demands in line $l$ during time interval $i$.
$T_r^l$	Riding time of a train across entire line $l$.
$T_t$	Time required for trains to turn around at terminal stations.
$T_d$	The time needed for trains to enter or exit a depot.
$F^{min}$	Minimum allowed service frequency.
$F^{max}$	Maximum allowed service frequency.
${RT}_l^{st}$	Riding time of trains from station $st$ to the next station in line $l$.
${DT}_l^{st}$	Dwelling time of trains at station $st$ in line $l$.
$\delta$	Safety time difference between consecutive trains.
$I$	High real value.

.

2.3. Solution Methodology

The mathematical model presented in this work is a large-scale, bi-objective integer non-linear programming (INLP) model designed to provide full-day operation solutions. Given the extensive number of multi-array variables, commercial solvers are not ideal for tackling such a large-scale problem. While some previous studies consider single-objective optimization [36,37] or weighted-sum formulations [38,39], the integration of Pareto front optimization has become more prevalent in contemporary multi-objective models [40,41]. In this study, we adopt this modern approach by proposing a metaheuristic solution based on the Multi-Objective Particle Swarm Optimization (MOPSO) algorithm [42,43] to effectively solve the INLP model and identify a set of high-quality trade-off solutions.

With regard to multi-objective adaptations, it is helpful to briefly introduce the fundamentals of the classical Particle Swarm Optimization (PSO) upon which MOPSO is built. PSO is a population-based metaheuristic inspired by the social behavior of flocking organisms [44]. The algorithm maintains a “swarm” of particles, where each particle represents a candidate solution moving through the search space. Its movement is guided by its own best-known position (pbest) and the best-known position of the entire swarm (gbest). This elegant mechanism allows the swarm to collectively explore the solution space and converge toward a single optimal solution. However, the challenge in multi-objective optimization is that no single gbest exists; instead, there is a set of equally optimal, non-dominated solutions that form the Pareto front.

The optimization problem formulated in this study presents a complex, bi-objective, and non-linear structure. The objectives of minimizing operational costs and improving service quality are inherently conflicting. Due to the problem’s complexity, which is characteristic of NP-hard problems, traditional mathematical programming methods are often intractable. Therefore, a metaheuristic approach is employed to efficiently explore the vast solution space and identify a set of high-quality trade-off solutions. We propose a Multi-Objective Particle Swarm Optimization (MOPSO) algorithm, renowned for its strong global search capabilities and effectiveness in solving multi-objective problems.

The proposed MOPSO framework is specifically adapted to handle the integer-based decision variables of our model, which represent the frequency of journeys. The core components and procedural steps of the algorithm are detailed below.

Initialization

In the context of this problem, each particle in the swarm represents a candidate solution. The position of a particle p constitutes a vector containing the frequency of journeys $F_i^l$ for each line l and time interval $i$.

The algorithm begins by initializing a population of Np particles. The position of each particle is randomly generated within the feasible search space. A feasibility check is immediately performed to ensure that fundamental operational constraints, such as those defined in inequation (4), are satisfied. If a generated particle represents an infeasible solution, it is discarded and regenerated until a valid initial population is formed. The initial velocity Vp for each particle is also initialized with small random values to introduce initial momentum.

The core of the MOPSO algorithm lies in the iterative movement of particles through the search space. This movement is governed by velocity and position update equations, which have been specifically adapted for the discrete nature of our decision variables.

At each iteration t, the velocity of each particle is updated using the standard PSO equation.

Exploring new solutions:

Velocity Update: For each particle, we update its velocity using the standard PSO velocity equation

$$v^{t+1}=\omega ·v^t+c_1·r_1·\left(p_{best}-x^t\right)+c_2·r_2·\left(g_{best}-x^t\right)$$

(17)

where

$\omega$ is the inertia weight (controls the trade-off between global and local search).

$c_1$, $c_2$ are acceleration coefficients (toward personal best and global best solutions).

$r_1$, $r_2$ are random numbers in [0, 1].

$p_{best}$ is the personal best position for the particle.

$g_{best}$ is the global best solution (from the Pareto front archive).

$x^{\mathrm{t}}$ is the current position vector of the particle at iteration t.

Since the journey frequencies $F_i^l$, must be integers, the continuous velocity vector must be discretized before updating the particle’s position. The position is updated as follows:

$$x^{t+1}=x^t+round\left(v^{t+1}\right)$$

(18)

The function converts the continuous velocity into an integer step, ensuring that the new particle position corresponds to a valid set of journey frequencies. Following the update, the new position $x^{t+1}$ is checked against its feasible bounds. If any element of the vector is outside its allowed range, it is clamped to the nearest boundary value.

Non-Dominated Sorting and Pareto Front:

A solution A is said to dominate solution B if A is no worse than B in all objectives and strictly better in at least one objective. The external archive stores all non-dominated solutions discovered during the search, representing the current approximation of the true Pareto front.

Handling the multi-objective:

We maintain a Pareto archive to store the non-dominated solutions.

We will use the Pareto front from this archive to guide the global search (i.e., the $g_{best}$ values for updating velocities).

Leader Selection: The selection of the global best g_(best) guide for each particle is critical for balancing convergence and diversity. Instead of using a single g_(best), a leader is selected for each particle from the external archive. To promote a well-distributed set of solutions, this selection is performed using a crowding distance mechanism. Solutions in less-populated regions of the archive have a higher probability of being selected as leaders, guiding the swarm to explore a wider range of the Pareto front.

External Archive Update: After each iteration, the external archive is updated. New non-dominated solutions from the current population are added to the archive. Any solutions in the archive that are dominated by these new solutions are removed. If the archive exceeds its maximum capacity, a pruning mechanism based on crowding distance is activated to remove solutions from the densest regions, thereby preserving the diversity of the Pareto front.

Algorithm termination:

The algorithm stops based on predefined criteria, which are either the maximum number of iterations or the Convergence criteria (no improvement in the Pareto front for a certain number of iterations). After the algorithm terminates, the solutions in the Pareto archive represent the set of trade-off solutions between the different objectives. From these, we can select the one that best fits the optimization of the network performance (Algorithm 1).

Algorithm 1. Pseudo-Code

1: Initialize a population of N particles 2: for each particle i = 1 to N do 3:   repeat 4:     Randomly initialize position X[i] with integer frequencies Fil 5:     Generate binary variables: $X_i=\left[D,A,{\Delta }^{\mathrm{in} },{\Delta }^{\mathrm{out} },{\Delta }^{t,t^{\prime }}\right]$ 6:   Until all constraints (2–12) are satisfied 7:   Initialize velocity V[i] with random values 8:   Evaluate objectives [$f_1$ (X[i]), $f_2$ (X[i], D, A, Δ^{in}, Δ^{out}, Δ^{t,t′})] 9:   Set personal best: p_best[i] ← X[i] 10: end for 11: Initialize Pareto_archive with non-dominated solutions from initial population 12: for iteration = 1 to G_max do 13:   Compute crowding distance for all solutions in Pareto_archive 14:   for each particle i = 1 to N do 15:     Select leader g_best from Pareto_archive using crowding-distance-based roulette wheel selection 16:     Update velocity V[i] using p_best[i] and g_best 17:     Compute temporary position: X_temp ← X[i] + V[i] 18:     Round X_temp to nearest integers to maintain discrete values 19:     Clamp X_temp within feasible bounds 20:     Generate binary variables: $X_i=\left[D,A,{\Delta }^{\mathrm{in} },{\Delta }^{\mathrm{out} },{\Delta }^{t,t^{\prime }}\right]$ 21:     If constraints (2–12) are violated: discard X_temp and continue 22:     Evaluate objectives [$f_1$ (X[i]), $f_2$ (X[i], D, A, Δ^{in}, Δ^{out}, Δ^{t,t′})] 23:     if X_temp dominates p_best[i] then 24:       p_best[i] ← X_temp 25:     else if X_temp and p_best[i] are mutually non-dominated then 26:       With 50% probability: p_best[i] ← X_temp 27:     end if 28:     Update particle position: X[i] ← X_temp 29:   end for 30:   Update Pareto_archive with current non-dominated solutions 31:   If Pareto_archive size > N_archive then 32:     Prune archive by removing solutions from most crowded regions 33:   end if 34: end for 35: Return Pareto_archive as the final Pareto front

3. Results

This section presents a comprehensive set of numerical experiments designed to achieve two critical objectives: (1) validating the proposed mathematical model for train scheduling in the Addis Ababa Light Rail Transit (AALRT) system, ensuring its accuracy in capturing real-world operational dynamics, and (2) rigorously evaluating the performance of the Multi-Objective Particle Swarm Optimization (MOPSO) algorithm in solving the associated multi-objective optimization problem.

3.1. Experimental Setup

3.1.1. Data Description

This study is based on detailed operational data from the Addis Ababa Light Rail Transit (AALRT) system, a 34.25 km dual-line network with 39 stations. The dataset encompasses a complete 16-h operational period (06:00 a.m. to 10:00 p.m.) on 24 November 2024. Passenger flow data were obtained directly from the Addis Ababa Light Rail Transit Service’s official cooperation. These data were derived from hourly ticket sales collected via the system’s paper-based ticketing platform and synchronized with operational logs saved from the AALRT’s Operation Control Center.

The dataset includes hourly passenger counts for each direction and station, with no missing or corrupted values; therefore, no imputation techniques were necessary. For consistency in modeling, all demand values were normalized using the max–min scaling technique. In addition to demand data, the dataset includes essential operational parameters such as train arrival and departure schedules, train fleet capacity, terminal turnaround times, station-to-station travel durations, and maximum operating speeds.

The operational challenge is defined by a highly dynamic and spatially heterogeneous passenger demand profile. As shown in Figure 7, the analysis of hourly passenger flows reveals a bimodal distribution with distinct morning and evening peaks, separated by a significant midday lull. The evening peak is particularly acute, with demand on the NS-Upward route creating a critical temporal bottleneck that drives the need for a responsive scheduling strategy. This intra-day variability confirms that a static scheduling approach would be highly inefficient, leading to severe overcrowding during peak periods and underutilized capacity during off-peak times, thus necessitating an adaptive optimization model.

Aggregating this demand over the entire operational day, as shown in Figure 8, provides further system-level insights. The East–West (EW) corridor carries a substantially higher total passenger load compared to the North–South (NS) corridor. Interestingly, while the total daily passenger counts are relatively balanced between opposing directions within each corridor, this masks the severe transient imbalances observed during peak hours. This complex interplay between high overall load on one corridor and extreme temporal peaks on the other underscores the multidimensional nature of the scheduling problem, validating the use of a multi-objective approach like MOPSO to navigate these competing spatial and temporal demands effectively.

3.1.2. Computational Environment

The numerical experiments were conducted on a workstation equipped with an 11th Generation Intel^® Core™ i5-1135G7 processor (2.40 GHz, 4 cores, 8 threads) and 16 GB RAM. The entire optimization framework was implemented in Python 3.13.2, utilizing NumPy 2.2.6 for numerical computations, Pandas 2.2.3 for data management, and Matplotlib 3.10.3 for visualization of results, particularly for Pareto front analysis.

The Multi-Objective Particle Swarm Optimization (MOPSO) algorithm was custom-developed specifically for the train scheduling problem without relying on external optimization libraries. The implementation included particle initialization respecting operational constraints, adaptive velocity updates, and an archive mechanism for storing non-dominated solutions. The simulation incorporated models of passenger demand patterns, train capacity limitations, waiting time calculations, and operational cost estimations over a 16-h operational period, enabling explicit analysis of the trade-offs between service quality and cost efficiency. This approach ensured full control over the optimization process while maintaining adaptability to real-world metro system requirements.

3.1.3. Algorithm Parameters

The performance and reliability of the proposed Multi-Objective Particle Swarm Optimization (MOPSO) algorithm were established through a comprehensive parameter tuning analysis. To ensure statistical robustness, this analysis involved 30 independent runs for each combination of key parameters: population size and number of generations. The objective was to systematically assess the trade-off between solution quality and computational cost. The resulting performance distributions are depicted in Figure 9, which employs box plots to illustrate the median, spread, and consistency of Hypervolume (HV), Inverted Generational Distance (IGD), and runtime for each configuration.

The analysis reveals a clear and definitive trend in solution quality. As shown in the ‘Hypervolume Distribution’ plot, a population size of 100 paired with 200 generations consistently achieves the highest median Hypervolume. Crucially, the compact interquartile range of this configuration demonstrates that this high performance is not only superior on average but also highly reliable across runs. This conclusion is strongly corroborated by the ‘IGD Distribution’ plot, where the same parameter set exhibits the lowest median IGD, confirming its consistent ability to find solutions closer to the true Pareto front.

While the ‘Runtime Distribution’ plot confirms that this optimal configuration incurs the highest computational cost, the significant and statistically validated improvement in solution quality justifies this investment. The decision was therefore made to prioritize finding the most robust and high-quality solutions. Based on this evidence, a population size of 100 and a stopping criterion of 200 generations were selected as the definitive parameters for all subsequent experiments conducted in this study.

The remaining algorithm parameters, including inertia weight, cognitive and social factors, and archive size, were set to standard, widely accepted values from the literature and the PyMoo library. This approach ensures a conventional balance between the algorithm’s global exploration and local exploitation capabilities, grounding our specific implementation in established best practices for the field.

3.1.4. Model Settings

The optimization model focuses on two primary conflicting objectives: Minimize Total Passenger Waiting Time (PWT), calculated as the sum of waiting times experienced by all passengers across all stations and periods. This objective aims to improve the quality of service. The second is Minimize Total Operational Cost (TOC). This primarily includes costs related to train operations, such as energy consumption, which are often correlated with the total number of train circulations, depot entry and exit running, and total train-kilometers run.

The model incorporates several critical operational constraints to ensure the feasibility and practicality of the generated schedules: headway constraints: minimum and maximum permissible time intervals between consecutive trains; train capacity constraints: the number of passengers on any train segment must not exceed its capacity; layover time constraints: minimum time required for trains at terminal stations for turnaround; fleet size constraint: the total number of trains simultaneously in operation must not exceed the available fleet; and synchronization constraints: travel time consistency, based on defined speed profiles and track lengths.

This subsection outlines the scenarios developed to test and evaluate the model and algorithm.

3.2. Comparative Performance Analysis

The baseline scenario represents the existing or a typical operational strategy currently employed by the Addis Ababa Metro System. Performance metrics for the baseline (non-optimized) scenario. Key performance metrics for this baseline, such as average passenger waiting time and estimated operational costs, were calculated. The current operational framework of the Addis Ababa Light Rail Transit (AALRT) system serves as the reference case for evaluating optimization improvements. The system operates 16 h of daily service (6:00–22:00) and along all four operational directions: North–South Up (NS-Up), North–South Down (NS-Down), East–West Up (EW-Up), and East–West Down (EW-Down). Each train has a fixed capacity of 317 passengers, and the service covers both the North–South (NS) and East–West (EW) lines, each running bi-directionally. To assess the baseline performance, key metrics were computed, including average passenger waiting time and operational costs (based on real-world parameters such as riding time, depot time, and turnaround operations). This baseline evaluation provides a critical benchmark for comparing the optimized scheduling scenarios developed in subsequent sections.

To enhance the efficiency of the Addis Ababa Metro System, a multi-objective optimization model using the MOPSO algorithm was developed to minimize passenger waiting time and operational costs. The optimization was conducted over 16 hourly periods (6:00–22:00). The optimized solution reflects adjusted train frequencies per hour based on fluctuating passenger demand. The optimized configuration dynamically allocates more trains during peak hours and fewer during off-peak times, thereby reducing wait times while managing operational costs. The results from the Multi-Objective Particle Swarm Optimization (MOPSO) algorithm present a clear and significant trade-off between the two primary objectives: minimizing average passenger waiting time and minimizing operational cost. The analysis contrasts the initial baseline performance with a selected solution from the optimized Pareto front (

Table 3. A comparison of baseline and MOPSO-optimized performances.

Metric	Baseline Scenario	MOPSO- Optimized Scenario	Change
Average Waiting Time (min)	15	7.7	−51%
Total Operational Cost (USD)	USD 2165.40	USD 4020	+86%

).

The most striking finding is the dramatic shift in operational strategy. The MOPSO-derived solution achieves a monumental reduction in passenger waiting time at the expense of a substantial increase in operational costs.

The optimized schedule reduces the average waiting time from 15 min to 7.7 min, a reduction of 51%. This represents a massive improvement in service quality and passenger experience. Total Operational Cost: Conversely, the total operational cost increased from 2165.40 to 4020, an increase of over 86%. This suggests the optimized schedule likely involves running more frequent trains, increasing train speeds (which consumes more energy), or a combination of more resource-intensive strategies. Baseline Scenario: Represents a low-cost, low-service-level strategy. It prioritizes minimizing operational expenses, leading to very high passenger wait times. MOPSO-Optimized Scenario: Represents a high-service-level, high-cost strategy. This specific solution, chosen from the Pareto front, heavily prioritizes minimizing passenger wait time. The MOPSO algorithm successfully identified a new operational paradigm that can reduce passenger waiting times by two-thirds. This result provides decision-makers with a clear, quantifiable choice: they can achieve a world-class level of service but it requires doubling the operational budget. The power of the MOPSO approach is that it can generate an entire frontier of such solutions, allowing stakeholders to select a different point on the curve that might offer a more moderate balance between cost and service level, depending on their strategic priorities and budget constraints.

To rigorously validate the effectiveness of the proposed MOPSO algorithm, its performance was benchmarked against the established NSGA-II, a widely recognized multi-objective optimization standard [45]. Both algorithms were applied to the same integrated operational planning model under identical computational settings to ensure a fair comparison. The evaluation, based on Pareto front characteristics and key performance metrics, clearly demonstrates the superiority of MOPSO. As illustrated in Figure 13 (MOPSO) and Figure 10 (NSGA-II), MOPSO achieved better trade-offs in minimizing both total passengers waiting time and operational cost.

As illustrated in Figure 10 (NSGA-II) and Figure 13 (MOPSO), the MOPSO approach achieved better trade-offs in minimizing both total passengers waiting time and operational cost. MOPSO identifies plans with significantly lower passenger waiting times. For instance, at a cost of approximately USD 3000, MOPSO achieves a waiting time of around 0.7 × 10⁶ minutes, whereas the closest NSGA-II solution yields a substantially higher time of 0.95 × 10⁶ minutes. Additionally, the MOPSO-derived front exhibits a broader distribution, encompassing a wider range of high-quality solutions at both extremes of the cost-waiting time trade-off spectrum. This enhanced diversity underscores MOPSO’s superior exploration capability, furnishing decision-makers with a more comprehensive set of non-dominated operational strategies.

In support of this visual and qualitative comparison, Figure 11 presents quantitative performance metrics, including Hypervolume (HV) and runtime, which further reinforce MOPSO’s outperformance. In terms of computational efficiency, MOPSO demonstrates a marked reduction in runtime across all tested configurations. For the most demanding scenario (Population = 100, Generations = 200), NSGA-II required a median runtime of 6.8 s, while MOPSO completed its search in just 4.5 s—a 34% improvement. Moreover, the Hypervolume metric, which quantifies the volume of dominated objective space, confirms MOPSO’s superior solution quality. Although both algorithms exhibit improved HV with increased computational resources, MOPSO’s Pareto front consistently dominates, indicating better convergence and diversity. This efficiency and effectiveness suggest that MOPSO’s external archive and leader selection mechanisms are better suited for navigating the complex solution space of urban rail transit optimization.

3.3. Performance Evaluation Metrics

To rigorously assess the quality and practicality of the proposed Multi-Objective Particle Swarm Optimization (MOPSO) algorithm, we employed a combination of convergence, diversity, and computational efficiency metrics. These metrics help evaluate the algorithm’s ability to approximate the true Pareto front, the distribution of trade-off solutions it offers, and the runtime required for each configuration. Specifically, we used three standard indicators: Hypervolume (HV), Inverted Generational Distance (IGD), and execution runtime.

Hypervolume (HV) quantifies both convergence and diversity by measuring the volume in the objective space dominated by the obtained Pareto front relative to a fixed reference point. A higher HV value indicates a better approximation of the true Pareto front with more well-distributed solutions. As seen in the left plot of Figure 9, the hypervolume improves consistently with increasing population size. At a population size of 100, HV values range approximately from 0.60 to 0.69, with the highest median HV occurring at 200 generations. This configuration not only achieves the best performance but also exhibits a tight interquartile range, indicating consistent convergence behavior across runs.

The Inverted Generational Distance (IGD) measures the average minimum distance from each point in a reference Pareto front to the nearest point in the obtained solution set. A lower IGD reflects better convergence to the true front. As shown in the middle plot of Figure 9, IGD values drop steadily with both increasing population size and generation count. The lowest IGD values—approaching 0.02—are achieved when the population size is 100, and the number of generations is 200. This outcome demonstrates MOPSO’s strong ability to produce accurate and reliable Pareto-optimal solutions under this parameter configuration.

Runtime, depicted in the right plot of Figure 9, evaluates the computational cost in terms of execution time per run. As expected, runtime increases with both population size and generation count. The highest runtime is recorded for a population size of 100 and 200 generations, exceeding 5 units (interpreted in consistent runtime units used in the experiment). Despite this increase in computational demand, the significant performance gain in HV and IGD justifies the cost, especially in critical planning scenarios where solution quality outweighs speed.

In summary, the analysis reveals that a population size of 100 and 200 generations yields the best overall performance across all metrics—achieving the highest hypervolume, lowest IGD, and acceptable computational runtime. This configuration was therefore selected for all subsequent experiments in the study, balancing optimization quality with practical feasibility.

3.4. Analysis of the MOPSO Solution Space and Convergence

Figure 12 provides a comprehensive visualization of the MOPSO algorithm’s convergence behavior and the resulting optimal solution set. The diffuse cloud of light and dark blue points represents the entire landscape of candidate solutions explored by the particle swarm throughout the optimization run, demonstrating the algorithm’s extensive exploration of the solution space. The increasing density of these points toward the lower-left boundary indicates the successful convergence of the swarm from broad exploration to focused exploitation of the optimal region. The solid red line delineates the final, non-dominated Pareto front, which comprises the set of optimal trade-off solutions. Its distinct stair-step characteristic is a direct consequence of the discrete, integer nature of the model’s core decision variables (i.e., journey frequencies), where small integer changes produce discrete jumps in the objective functions. The annotated point ‘A’ marks the extreme solutions on this frontier, representing the minimum possible operational cost and the minimum achievable passenger waiting time, respectively, thereby framing the full spectrum of strategic choices available to decision-makers.

The core output of the MOPSO algorithm is the Pareto front, which visualizes the trade-off between minimizing passenger waiting time and minimizing operational costs.

The primary result of the MOPSO algorithm is the Pareto optimal front, shown in Figure 13. Each point on this front represents an optimal train schedule where it is impossible to reduce passenger waiting time without increasing operational cost, and vice versa. The curve clearly illustrates the inherent trade-off between service quality and economic efficiency. For instance, a schedule can achieve a low total waiting time of approximately 0.4M minutes, but this requires an operational cost of over USD 4500. Conversely, a low-cost schedule of USD 2500 results in a much higher waiting time of over 1.2M minutes. Among the final set of non-dominated solutions generated by MOPSO, a representative solution was selected for detailed analysis. This solution was chosen from the central region of the Pareto front to reflect a balanced trade-off between cost efficiency and passenger service quality. It provides a practical compromise for operators seeking to minimize both operational expenditure and waiting times. The set of optimal solutions found by MOPSO. Each point on the curve represents a unique, optimal schedule where neither objective can be improved without worsening the other. A clear inverse relationship is visible: schedules with lower waiting times incur higher operational costs, and vice versa.

To address the system’s temporal demand variations, the MOPSO algorithm was independently applied to generate optimal schedules for peak and off-peak operational periods. The resulting Pareto fronts, shown in Figure 14, reveal distinctly different trade-off landscapes. The peak-hour front is characterized by higher absolute waiting times and a steeper trade-off curve, particularly at lower waiting times. This indicates that as the system approaches its capacity during high-demand periods, each incremental improvement in service quality (reduced waiting time) requires a disproportionately larger investment in operational cost. This “law of diminishing returns” is a classic feature of congested systems and highlights the extreme financial pressure associated with providing high-frequency service during the busiest hours.

Conversely, the Pareto front for off-peak hours illustrates a more favorable and flexible operational environment. For any given operational cost, a significantly lower total passenger waiting time can be achieved compared to the peak period, reflecting the reduced passenger load. The trade-off curve is less severe, affording decision-makers a wider range of viable, cost-effective scheduling options that can still maintain a reasonable quality of service. This comparative analysis provides quantitative validation for implementing a dynamic, time-differentiated scheduling strategy, proving that distinct optimal policies are required to efficiently manage resources and service levels across the entire operating day.

3.5. Sensitivity Analysis

The performance and solution quality of metaheuristic algorithms like MOPSO are inherently sensitive to the choice of hyperparameters. To ensure the robustness of our results and to select an optimal configuration, a sensitivity analysis was conducted. Figure 15 presents the final Pareto fronts obtained from four distinct MOPSO parameter configurations: a Small Pop scenario, a Default setting, a Large Pop scenario emphasizing exploration, and a balanced configuration tuned for an optimal exploration–exploitation trade-off.

The analysis reveals a clear performance hierarchy among the configurations. The Small Pop setting (green) consistently yields a dominated Pareto front, positioned significantly to the upper-right of the other solutions. This indicates that a small population size provides insufficient diversity, leading to premature convergence on a suboptimal set of solutions that are inferior in both cost and waiting time.

In contrast, the Large Pop (orange) and Balanced (red) configurations emerge as superior, both significantly outperforming the Default setting. The Large Pop configuration excels at exploring the extremes of the solution space, particularly identifying solutions with the lowest operational cost (extending furthest to the right). This is attributable to the greater genetic diversity maintained by a larger population, which facilitates a broader search. The Balanced configuration demonstrates highly competitive performance across the entire front and appears to slightly dominate the Large Pop front in the high-service-quality region (lower waiting times).

Ultimately, both the Large Pop and Balanced settings produce high-quality, non-dominated solutions that form a superior frontier. This analysis validates that a sufficiently large and well-tuned population is critical for escaping local optima and effectively mapping the true Pareto-optimal front for this complex problem. Based on its robust performance across the entire spectrum of trade-offs, the [Choose one: Large Pop or Balanced] configuration was selected for generating the final results presented in this study.

To assess how the optimal solutions are affected by changing operational conditions, a sensitivity analysis was conducted by varying passenger demand and system capacity. Figure 16 shows the resulting Pareto fronts for three scenarios: Base, High Demand, and Low Capacity.

Effect of High Demand: When passenger demand increases, the Pareto front shifts up and to the right. This means that for any given operational budget, the total passenger waiting time will be higher. Conversely, achieving a target waiting time becomes more expensive. Effect of Low Capacity: When the system capacity is reduced (e.g., fewer available trains), the Pareto front also shifts up and to the right. This demonstrates that with fewer resources, the system struggles to maintain service levels, leading to increased waiting times and/or costs. This analysis confirms that the model responds logically to external pressures and can be used as a strategic tool to forecast the impact of ridership growth or fleet availability on operational performance.

3.6. Timetable Analysis

This section visualizes the operational impact of the proposed optimization framework through train operation diagrams (string-line charts) for the NS line. These charts depict train movements over time and space, providing a clear comparison between the current baseline schedule and an optimized solution from the Pareto front.

Figure 17 shows the baseline schedule. The vertical axis lists stations, the horizontal axis shows time, and each diagonal green line traces a train’s movement. Trains are infrequent, with wide spacing between trajectories, reflecting long headways. This limited frequency reflects a cost-saving policy that prioritizes low operational expenses over service quality. The resulting high passenger waiting times make this schedule a reference point for improvement.

Figure 18 presents an optimized schedule focused on minimizing passenger waiting time. Generated by the MOPSO algorithm, this timetable shows a denser concentration of train trajectories, indicating significantly higher frequency and reduced headways. This improved schedule enhances throughput and aligns more closely with passenger demand. However, the benefit comes at a cost: while waiting time drops, the operational expenses increase, reflecting heavier use of trains and resources. This schedule highlights a clear trade-off between service quality and cost, exemplified by the Pareto front.

These diagrams confirm the model’s ability to turn mathematical optimization into practical, actionable timetables. The shift from wide to dense trajectories illustrates a move from cost-driven to passenger-focused service. Moreover, these charts serve as intuitive tools for decision-makers, making trade-offs visually clear and aiding in strategic planning grounded in both efficiency and passenger satisfaction.

4. Conclusions

4.1. Summary of Key Findings

This study addressed a fundamental limitation in current urban rail transit (URT) planning: the sequential and siloed optimization of frequency setting, train timetabling, and circulation planning. Recognizing that such fragmentation often leads to suboptimal system-wide outcomes, we proposed a unified bi-objective optimization framework that integrates these three interdependent components. The problem was modeled as a large-scale Integer Nonlinear Programming (INLP) formulation and solved using a customized Multi-Objective Particle Swarm Optimization (MOPSO) algorithm, adapted for discrete decision spaces.

Through comparative experimentation using real-world data from the Addis Ababa Light Rail Transit system, the integrated model demonstrated strong empirical performance. Specifically, MOPSO consistently outperformed the benchmark NSGA-II algorithm in both convergence quality and solution diversity. The generated Pareto front revealed operational plans capable of reducing total passenger waiting time by up to 33% compared to NSGA-II, while also identifying lower-cost alternatives that were otherwise inaccessible. This confirms the hypothesis that a unified, simultaneous optimization approach yields richer and more efficient operational strategies than existing decomposed methods.

4.2. Contributions and Practical Implications

This research makes several notable contributions to both the academic literature and practical transit planning:

Methodological Innovation: The study presents a novel, integrated URT planning model that harmonizes service frequency, timetabling, and train circulation—an advancement over fragmented approaches. The successful application of a customized MOPSO to this NP-hard problem illustrates its scalability and relevance for other complex network optimization challenges.

Decision-Making Support: The framework offers practitioners a valuable decision-support tool, providing not a single “optimal” solution but a well-distributed Pareto front of trade-off alternatives. Transit authorities can now make data-driven choices between cost and service quality, aligning operational strategies with policy goals and budget constraints.

Empirical Superiority: Benchmarking against NSGA-II confirms that the proposed MOPSO approach offers superior solution quality in terms of convergence to the Pareto front and the diversity of trade-off scenarios available to decision-makers.

4.3. Limitations

While the proposed model demonstrates strong practical and methodological potential, several limitations must be acknowledged.

Simplified Cost Representation: The operational cost function primarily accounts for time-based expenditures. It does not currently model more nuanced cost components such as variable energy consumption, long-term maintenance, rolling stock wear and tear, or labor-specific constraints like crew shifts and union rules.

4.4. Future Research Directions

Building on the findings and addressing the above limitations, several future directions are proposed to enhance the realism, applicability, and impact of this work.

Objective Function Expansion: Future studies can extend the model to incorporate multi-dimensional objectives, such as environmental impact (via detailed energy consumption models), in-vehicle crowding levels, passenger comfort, and transfer efficiency.

Multi-Line and Multi-Modal Integration: The framework can be extended to accommodate multiple transit lines or intermodal coordination (e.g., buses, subways, BRT), enabling system-wide optimization across shared infrastructure and transfer hubs.

In conclusion, this study lays a solid foundation for integrated operational planning in urban rail systems and opens avenues for more responsive, sustainable, and user-centric transit system optimization.