Dynamic route optimization in disrupted heterogeneous railway systems via temporal planning

Railway is one of the most efficient ways to transport people and goods at scale, providing greater energy efficiency and the ability to carry large volumes of freight at lower cost (Ul Islam et al., 2024). With rising demand, railway infrastructure encounters significant challenges to ensure reliable and timely services due to network congestion (National Academies of Sciences, Engineering, & and Medicine, 2007). Thus, maintaining reliable service requires dynamic route optimization and efficient timetabling to manage network congestion and reduce operational delays.

Railway scheduling determines train timetables with departure and arrival time at each railway track point (Kang, Buhigiro, Sun, & Wu, 2024), while route optimization finds travel paths for trains to minimize travel duration and avoid conflicts (Wang, Song, He, & Song, 2022). Though both problems have decades of research behind them, yet they are hard to solve at scale and known to be NP-hard problem (Leutwiler & Corman, 2023). In addition, real-world systems introduce additional layers of complexity. For instance, heterogeneous railway systems, in which trains have varying speed profiles and stopping patterns, require a higher level of coordination than uniform metro systems (Yan, Bešinović, & Goverde, 2019). Multi-gauge networks amplify this complexity by introducing multiple gauge types for track and train, and require matching gauge types for train movement (Villalba Sanchis, Insa Franco, Martínez Fernández, & Salvador Zuriaga, 2021). While standard gauge (1,435 mm) predominates in Europe and many other regions, several countries operate multi-gauge systems (Grigonis, Kaušylas, & Palevičius, 2025; Villalba Sanchis et al., 2021). For example, Spain railway operates three track gauges: Iberian (1,668 mm) across most of its legacy network, standard (1,435 mm) on high-speed lines and dual-gauge sections that accommodate both Iberian and standard (Villalba Sanchis et al., 2021). Similarly, Bangladesh railway operates three track gauges: meter gauge (1,000 mm), broad gauge (1,676 mm) and dual gauge, which has a combination of both (Bangladesh Railway, 2023). The practical consequence is that gauge type determines which tracks a train can physically use. To drive a train on a specific track, it must follow the gauge compatibility constraints. For instance, a standard gauge train can only run on standard or dual gauge track. Similarly, an Iberian gauge train needs Iberian or dual gauge track. These constraints significantly limit route options, thereby increasing operational complexity.

Collectively, these characteristics define what we refer to as a heterogeneous railway system, characterized along four dimensions: (1) speed heterogeneity, in which trains with different speed profiles share the same track segments, (2) stopping-pattern heterogeneity, in which each express train may follow a distinct stopping pattern and different passenger boarding times at stations, (3) route-choice flexibility, in which multiple admissible paths exist between an origin and a destination and (4) gauge compatibility, in which track segments and rolling stock are constrained by differing gauges. Most prior studies address only a subset of these dimensions. In contrast, this work addresses all four jointly.

Moreover, railway operations get frequently disrupted by stochastic events such as blocked tracks, blocked trains, engine failures and speed slowdowns (Xiu, Pan, D’Ariano, Zhan, & Peng, 2024). These events cause deviations from the planned railway timetable and require dynamic rescheduling and route optimization to resume normal operations. Numerous studies have been conducted on various aspect of railway operations (discussed in Section 2), such as scheduling, disruption handling and rail station management. However, these studies typically focus on specific aspects of railway operations rather than addressing them in an integrated manner. Most existing approaches also generate optimized schedules without explicitly specifying the executable actions required to realize those schedules. Consequently, critical operational decisions, such as turnout (track-switching) operations, are often left to human operators, increasing the risk of operational errors and safety incidents (Neves, Ribeiro, Grilo, Infante, & Andrade, 2024). For instance, in Bangladesh, human errors accounted for 50.77% of operational incidents during 2022–2023 (Bangladesh Railway, 2023). These limitations raise the need for an integrated approach that can generate feasible and optimized schedules with explicit executable operational plans. To address this need, this study proposes a temporal planning framework for railway route optimization in heterogeneous railway systems.

Automated planning is a fundamental field of artificial intelligence that automatically synthesizes sequences of actions for achieving specified goals from a given initial state. Temporal planning extends this capability to produce timestamped plans by incorporating action durations and temporal constraints, and minimizes the overall operational time (Benton, Coles, & Coles, 2012). These timestamps specify precisely which actions to execute and when, making temporal planning particularly well-suited for railway scheduling and route optimization, where precise timing, coordination and sequencing of train movements are critical. Modern temporal planners can handle the complexity inherent in railway networks efficiently since they employ efficient search strategies (e.g. A*, IDA*) along with domain-independent heuristics. In this work, we model railway operations by encoding safety constraints, gauge compatibility and turnout operations using temporal planning, and automate the generation of low-level commands necessary to resolve disruptions. We employ PDDL 2.1 (Planning Domain Definition Language) to design our model.

We also systematically design a 200-instance benchmark (discussed in Section 4.3) comprising 100 nominal and 100 disrupted instances distributed across four problem-size categories: small, medium, large and very large. The instances follow a monotonic scaling scheme, with progressively increasing numbers of trains and junctions across the categories, reaching up to 120 trains and 100 junctions in the largest instances. This benchmark enables this study to analyze how the existing planners perform as the network density and disruption severity of the problem increase simultaneously.

We evaluate our proposed framework and benchmark using two state-of-the-art temporal planners: POPF (Coles, Coles, Fox, & Long, 2010) and OPTIC (Benton et al., 2012). Experimental results demonstrate that both planners efficiently generate conflict-free plans, with statistical analysis confirming predictable scaling even in highly disrupted scenarios. Every generated plan is also validated using a temporal validator VAL (Howey, Long, & Fox, 2004), to confirm that the action sequence is consistent and correctly ordered.

The proposed framework has important practical and research implications. Practically, it supports operational dispatching by generating executable plans rather than only revised timetables. These plans specify low-level actions, including train movements, turnout switching, passenger boarding and recovery operations, with explicit start times and durations, while minimizing overall operational time. Disruption recovery is also modeled at the operational level. For example, an engine failure is handled through explicit auxiliary-engine dispatch, attachment and assisted movement. Across the benchmark suite, large-scale disruptions require only 6–13% additional actions, maintaining computational efficiency on commodity hardware. These results establish the practical scope of the framework and motivate the research directions discussed in Section 6.

We demonstrate how our framework models railway operations and generates executable plans using a simplified railway network, presented in Figure 1. The network consists of six track points (A, B, C, D, E, F), where B and E are junction (turnout) points. Junction B can connect either to point C or D (currently connected to D), while junction E can connect to C or D (currently connected to D).

The network also includes a heterogeneous multi-gauge infrastructure. Track segments A-B and E-F are dual gauge, allowing both meter-gauge and broad-gauge trains. Segments B-D and D-E support only broad-gauge trains, while segments B-C and C-E support only meter-gauge trains. Station S1, located between junctions B and E, consists of two parallel platform points C and D where trains stop for passenger boarding. The network has two trains, train t1 (broad gauge) and train t2 (meter gauge).

The task is to move train t1 from point A to F, and to move train t2 from point F to A. Train t1 also needs to stop at point D for a 2-min boarding operation, while train t2 requires a 3-min boarding at point C. Assume trains t1 and t2 travel at 1 km/min (60 km/h) and 1.1 km/min (66 km/h), respectively. Turnout operations require 1 minute to switch track alignment. Table 1 presents a temporal plan, consisting of timestamped train movements, turnout operations and passenger boarding actions, each specified by its start time and duration (mm).

In the temporal plan in Table 1, $DriveTrain(t1,A,B,broad)$ starts at 00:00 with a duration of 60:00, indicating that train t1 departs point A at time 0 and arrives at point B after 60 minutes. The parameter broad specifies that t1 traverses the segment on a compatible broad-gauge track. The plan also illustrates the temporal coordination the model provides. Both trains and the turnout at $E$ act concurrently from time zero (00:00), and the junctions are reconfigured as the plan unfolds: the turnout at $E$ is first switched toward $C$ so that t2 can reach its boarding point, and switched back toward $D$ at 50:00 so that t1 can later pass through $E$ on its route to $F$⁠, each switch occupying the junction for its one-minute duration. This simple example demonstrates how the generated plan provides an executable operational schedule with precise action timing and satisfying gauge compatibility, station operations and turnout constraints.

This article makes the following contributions, organized into modeling, benchmarking and empirical evaluation:

1. A temporal planning model for heterogeneous railway routing: We formalize dynamic route optimization in multi-gauge networks as a temporal planning problem that jointly captures (1) gauge compatibility routing constraint, (2) turnouts as durative, resource-consuming actions, with switching time and junction unavailability factored into the temporal optimization and (3) track segments and points as exclusive temporal resources, yielding collision-free schedules by construction.

2. Integrated disruption management: We extend the temporal planning framework to support four classes of railway disruptions: blocked trains, blocked tracks, engine failures and slowdowns, via dedicated state predicates and durative recovery actions, including an auxiliary-engine rescue pipeline (dispatch, attachment, assisted movement).

3. A scalable benchmark for heterogeneous railway planning: We construct a 200-instance benchmark comprising 100 nominal and 100 disrupted instances across four progressively larger network categories, scaling up to 1,000 track points and 120 trains with increasing disruption severity.

4. A complete, open PDDL implementation and empirical evaluation: We provide the complete PDDL 2.1 domain, 200 benchmark instances and instance generator to reproduce the experimental results. We evaluate the benchmark using the OPTIC and POPF temporal planners and validate all generated plans using VAL. The evaluation compares planner coverage and runtime across problem sizes, analyzes delay according to disruption type and quantifies disruption resilience relative to nominal operations.

This work extends an earlier conference paper by introducing additional disruption scenarios, a large-scale benchmark problem set and an expanded experimental evaluation.

The remainder of this article is organized as follows. In Section 2, we discuss the existing approaches to railway scheduling and optimization in the literature. Section 3 provides the necessary background and semantics on temporal planning, and its modeling using PDDL. In Section 4, we present the proposed framework, including the design of the temporal planning domain and the formulation of railway planning problems. Following that, we discuss experimental evaluation and analysis in Section 5 and conclude with our remarks in Section 6.

Researchers have proposed a wide range of optimization models and algorithms for railway scheduling, aiming to minimize delays, reduce waiting times, lower operational costs, and enhance system efficiency and service reliability. These approaches can be broadly categorized into Maximum Satisfiability (MaxSAT), Mixed Integer Linear Programming (MILP), metaheuristic algorithms, and automated planning. MILP and MaxSAT formulate railway scheduling as mathematical optimization problems that produce optimal or near-optimal conflict-free timetables, whereas metaheuristic methods search efficiently for high-quality scheduling solutions under complex constraints. In contrast, temporal planning directly generates executable, timestamped action sequences that naturally capture action durations and concurrency. The remainder of this section reviews representative studies from each of these four methodological categories and discusses how they address the heterogeneity dimensions introduced in Section 1.

Maximum Satisfiability (MaxSAT) formulations have recently gained considerable attention for optimizing scheduling under complex operational constraints. Lemos, Gouveia, Monteiro, and Lynce (2024) proposed an iterative MaxSAT-based framework for the Swiss Federal Railways (SBB) that minimizes delays and routing costs while handling track blockages and speed reductions through rerouting, waiting strategies and velocity adjustments. Gouveia, Albino, and Saldanha (2025) developed a MaxSAT formulation to generate conflict-free timetables while preserving limited flexibility in route choice, and tested their model on real data from SBB and the Washington Metro. Although these approaches effectively generate conflict-free schedules, with Lemos et al. (2024) additionally supporting rerouting and speed adjustment, they generally assume relatively uniform train characteristics and do not jointly model stopping-pattern variation and gauge compatibility across heterogeneous rolling stock. Moreover, their solutions remain primarily timetable-based rather than action-level operational plans.

MILP has also been a widely adopted method for railway scheduling and rescheduling. Chai et al. (2024) built a branch-and-cut framework for urban rail that jointly optimizes train sequencing, station timing and coupling operations under time-varying passenger demand. Zhuo, Miao, Meng, Yang, and Shang (2024) took a similar demand-driven angle, adapting both schedules and train compositions as passenger loads shift. Ji et al. (2024) developed a two-stage stochastic optimization model to address uncertainty in railway maintenance durations. Additional MILP and ILP-based methods have been proposed for disruption management scenarios, including track blockages, rolling stock failures and timetable adjustments (Acuna-Agost, Michelon, Feillet, & Gueye, 2011; Fischetti & Monaci, 2017; Li, Zhao, Peng, Wang, & Zhong, 2025; Veelenturf, Kidd, Cacchiani, Kroon, & Toth, 2016; Xiu et al., 2024). These models provide powerful optimization capabilities and, in several cases, achieve global optimality for the problems they target. However, their formulations generally focus on timetable and sequencing decisions over a predefined infrastructure representation. Heterogeneity dimensions such as multi-gauge compatibility and route-level diversity typically fall outside these formulations. Consequently, the network-wide, action-level planning problem under combined speed, stopping-pattern, route and gauge heterogeneity remains largely unaddressed.

Metaheuristic approaches have also shown strong performance on scheduling and rescheduling instances (Fang, Yang, & Yao, 2015; Narayanaswami & Rangaraj, 2011). Genetic Algorithms (Nitisiri, Gen, & Ohwada, 2019), Simulated Annealing (Zhang & Ni, 2022), Large Neighborhood Search (Zhang & Ni, 2022) and Ant Colony Optimization (Coviello, Medeossi, Nygreen, Pellegrini, & Rodriguez, 2023) have been employed to navigate the nonlinear solution spaces of large-scale networks, with reported successes in reducing passenger waiting times and operating cycles under fluctuating demand. Representative examples include the Problem Space Search metaheuristic of Albrecht, Panton, and Lee (2013), which reschedules timetables under scheduled maintenance disruptions while preserving schedule quality, and the hybrid framework of Dündar and Şahin (2013), which couples a genetic algorithm for conflict resolution with an artificial neural network emulating dispatcher decision. In principle, such algorithms are flexible enough to encode heterogeneous infrastructure and rolling-stock constraints within their objective and feasibility functions. However, these studies frequently rely on simplified network abstractions (i.e. uniform gauge compatibility and homogeneous speed profiles) in order to keep the search space tractable. This reflects the particular modeling choices made in those studies, not any fundamental limitation of metaheuristics as a class. Consequently, while the paradigm itself remains a viable candidate, existing metaheuristic formulations in the literature do not directly produce feasible, action-level plans for the combined heterogeneity addressed here.

Automated planning offers an alternative paradigm that generates explicit action sequences for solving a particular task, and has been applied across several domains relevant to ours. Nyporko and Chrpa (2025) modeled industrial production and manufacturing with PDDL 2.1, selecting and scheduling activities on the resources that perform them, with each action representing an elementary production step. Yang and Liang (2022) applied automated planning to emergency-material logistics, regulating the transport and dispatch of supplies to demand points so as to reduce the losses caused by shortages. A temporal planning formulation of the temporally constrained journey problem was proposed by Caff, Di Mauro, and Scala (2014), producing solutions that satisfy cost, trajectory and maximum-travel-time constraints. Within the railway domain specifically, Cardellini, Maratea, Vallati, Boleto, and Oneto (2021) developed a numeric planning-based framework for train dispatching trains inside stations. Louadah et al. (2021) used temporal planning to schedule maintenance in depots by translating ontological knowledge and SWRL rules into PDDL. These works demonstrate that planning can coordinate concurrent activities and optimize localized operations. However, they are confined to station or depot level settings and do not extend to network-wide route planning, heterogeneous train constraints or integrated disruption handling, precisely the scope our framework targets.

Across these advancements, two critical gaps remain. First, most studies incorporate only a subset of the heterogeneity dimensions defined in Section 1, while multi-gauge compatibility and joint route choice are rarely modeled together. Second, the produced output is a conflict-free timetable, and disruption is handled by reactively adjusting that timetable, rather than by proactively generating detailed, timestamped execution plans of low-level actions. This study addresses both gaps by introducing a temporal planning framework that explicitly accounts for heterogeneity of railway and that produces precise, actionable recovery plans for complex disruptions.

This section presents the theoretical foundations of temporal planning that underpin our approach. It introduces the key concepts and definitions of temporal planning and the PDDL required for modeling the proposed framework.

Automated planning and scheduling is a core subfield of artificial intelligence concerned with the automatic generation of action sequences that transform a given initial state into one satisfying specified goal conditions (Ghallab, Nau, & Traverso, 2004). Learning-based approaches derive decision-making behavior from data through statistical training, whereas automated planning relies on explicit, symbolic models of actions, states and goals, and computes solutions through search and reasoning over these models. A planning system (planner), such as Optic or POPF, takes as input a formal description of the initial state, the available actions and the goal conditions, and produces a plan (a structured sequence of actions) that achieves the goals when executed.

In classical planning, actions are assumed to be instantaneous and sequentially executed. The environment is typically modeled as fully observable, deterministic, static (i.e. state changes occur only through agent actions) and finite. However, modeling many real-world domains that involve action durations, temporal dependencies or concurrent activities can be challenging within the classical planning framework. Railway operations, for example, involve activities such as train movements and turnout operations that require explicit representation of time and resource constraints.

Temporal planning extends classical planning by explicitly incorporating time into the model (Cenamor, Vallati, & Chrpa, 2019). Intuitively, a temporal planning problem (Definition 1) specifies the system state using logical and numeric variables, a set of durative actions that modify the state, an initial configuration of the system (given by an assignment over both fluent types) and a set of goal conditions that the planner must achieve. A durative action (Definition 2) represents an activity that unfolds over a time interval bounded by a start and an end instant. Three temporally distinguished sets of conditions govern its applicability: conditions required at the start instant, conditions required to hold throughout the entire interval and conditions required at the end instant. Correspondingly, its effects are partitioned into those applied at the start instant and those applied at the end instant, reflecting how the action's impact on the state may unfold at different points in its execution.

1. $F$ is a finite set of propositional fluents,

2. $X$ is a set of real-valued numeric fluents,

3. $Ad$ is a finite set of durative actions,

4. $I⊆F∪X$ represents the initial state, given by a truth assignment to the fluents in $F$ and a real-valued assignment to the numeric fluents in $X$ and

5. $G⊆F∪X$ represents the goal condition, expressed as a set of propositional fluents that must hold (optionally together with numeric constraints over $X$⁠)

1. $pre⊢(a)$ are start conditions, required to hold at the instant the action starts,

2. $pre↔(a)$ are over-all conditions, required to hold continuously throughout the action's execution duration,

3. $pre⊣(a)$ are end conditions, required to hold at the instant the action ends,

4. $eff⊢(a)$ are start effects (changes), applied at the instant the action starts,

5. $eff⊣(a)$ are end effects (changes), applied at the instant the action ends and

6. $dur(a)∈R+$ denotes the action duration.

Unlike classical planning, a temporal plan is not just a sequence of actions but a time-stamped schedule of $(ti,ai)$ pairs, where $ti$ denotes the start time of action $ai$ (Haslum, Lipovetzky, Magazzeni, & Muise, 2019). A temporal plan is valid if and only if (1) all preconditions are satisfied at their required times, (2) over all (invariant) conditions hold throughout $[ti,ti+dur(ai)]$⁠, no conflicting effects occur simultaneously and the goal $G$ is satisfied at plan completion, $tend=maxi(ti+dur(ai))$⁠. The makespan (Definition 3) of a temporal plan measures the total time required to complete all actions, from the start of the earliest action to the completion of the latest one.

To formalize a planning problem and generate valid temporal plans, we need to describe the planning environment, such as states, actions and goals. PDDL (Haslum et al., 2019) is the most widely adopted language for this purpose.

An Automated planning system requires a formal representation of the initial state, the actions to enable transitions between states and the goal conditions to generate a valid plan. For the past two decades, the planning community has widely adopted PDDL as the standard for modeling planning problems. PDDL divides a planning problem into two components: a domain and a problem instance. The domain captures the general structure of the planning environment, including state predicates and action definitions, while the problem instance specifies a particular scenario by defining the initial state and the desired goal conditions.

PDDL 2.1 extends classical PDDL by introducing temporal and numeric features (Fox & Long, 2003). In particular, it supports durative actions with explicit durations, temporal qualifiers (at start, over all, at end), numeric fluents and arithmetic expressions, and optimization metrics (e.g. makespan minimization).

We describe the syntax of PDDL 2.1 using the domain definition and the problem instance of a toy railway system, presented in Listings 1 and 2, respectively.

A temporal planning domain specifies the object types, state predicates, numeric fluents and durative-action schemas of the environment. Each durative action declares typed parameters, a duration, temporally annotated conditions that must hold at start, over all (as invariants) or at end of execution, and effects applied at the start or end of the action. Listing 1 presents a simplified railway domain illustrating these concepts. The domain defines two object types, train and location, two predicates, at and connected, representing the train's current location and the connectivity between locations, respectively, a numeric fluent total-cost and a single durative action, move.

The action has a fixed duration of five-time units and requires the train to be at the origin location when execution begins. In addition, the origin and destination locations need to be connected. At the start of execution, the action removes the train from the origin and places it at the destination upon completion. It then increases the accumulated travel cost by 10 units at the end of the action. Although this example uses a fixed action duration for simplicity, PDDL 2.1 also permits duration expressions over numeric fluents. Our proposed railway domain (Section 4) computes action durations dynamically from track distances, train speeds and slowdown factors.

A problem instance declares the objects, the initial state and the goal of a concrete scenario, providing the data from which the planner instantiates the domain actions. Listing 2 shows a toy instance for the domain in Listing 1. It defines one train, three locations, the initial location of the train, the connectivity between adjacent locations and initializes the accumulated cost to zero. The goal specifies that the train must reach location C, while the metric directs the planner to minimize the total action cost. Alternatively, specifying the special fluent total-time as the optimization metric minimizes the plan makespan. Temporal planners then search for a temporally valid plan optimizing the given objective.

A temporal plan for this instance consists of two sequential durative actions. Since locations A and C are not directly connected, the train must travel via the intermediate location B. Each plan entry specifies the action's start time, its ground instantiation and its duration. The temporal planners used in this work produce plans in the following format:

This plan has a makespan of $10$ time units and a total cost of $20$⁠, since each action has a duration of $5$ time units and incurs a cost of $10$ units.

Such expressive capabilities of PDDL 2.1, including durative actions with temporal annotations and numeric fluents, provide a solid basis for the railway domain model developed in this study.

This section presents the proposed Disruption Aware Railway Temporal Planning (DART) framework, which models railway systems as a temporal planning problem. The framework generates detailed, time-stamped plans for coordinating train movements, infrastructure operations and disruption recovery while satisfying gauge compatibility and operational constraints.

To address operational challenges in heterogeneous railway networks, DART incorporates a disruption taxonomy and corresponding recovery strategies for each disruption type. The railway network, rolling stock, infrastructure components and disruption scenarios are encoded in PDDL 2.1, enabling temporal planners to generate executable plans that dynamically adapt to operational disruptions.

The remainder of this section presents the disruption taxonomy, the formal temporal planning model, the railway action models and the benchmark generation methodology.

Railway operations in heterogeneous multi-gauge networks are frequently affected by stochastic disruptions that deviate from nominal schedules, thus requiring recovery planning. DART formalizes four primary disruption types, each represented by dedicated predicates and durative recovery actions (Section 4.2). These disruptions also form the basis for our benchmark instances in Section 4.3:

1. Blocked Track Disruption occurs when a railway track segment becomes unavailable for train movements due to accidents, infrastructure failures or maintenance activities. Recovery strategies in our framework include (1) waiting at the current position until the track is cleared or (2) rerouting via gauge-compatible alternative paths using turnout operations.

2. Blocked Train Disruption occurs when a train is temporarily immobilized (e.g. due to logistical issues, accidents or crew unavailability). When this disruption occurs, the affected train must remain at the disruption point until the blockage is resolved, while other trains continue operations

3. Slowdown Disruption reduces train speeds over specific track segments due to external conditions such as adverse weather. Travel time on a disrupted segment is computed as:

where $τ$ denotes the train (or engine) traversing the disrupted segment (⁠$ni$⁠, $nj$⁠), $v(t)$ its speed and $α(ni,nj)∈[0,1)$ the slowdown factor applied to that segment (e.g. α = 0.2 for a 20% speed reduction). Recovery strategies include traversing with increased travel time or rerouting to minimize makespan.

1. Engine Failure Disruptions occur when a locomotive becomes inoperative. Recovery involves dispatching an auxiliary engine to the affected train and coupling it to resume operations.

Together, these four disruption categories constitute a comprehensive operational stress model for the heterogeneous railway networks.

We formulate railway route optimization and disruption management as a temporal planning problem. The formulation assumes that the railway system is fully observable and deterministic, disruption durations (e.g. blocked tracks, blocked trains and slowdowns) are known a priori, and trains travel at constant speeds without acceleration or deceleration. For clarity and accessibility, we present the railway planning model using a first-order logical formalization, allowing readers to understand the underlying model independently of its PDDL implementation. The corresponding PDDL domain and problem specifications are then derived from this formalization.

1. $FR$ is the finite set of grounded propositional facts over the predicate set $PR$⁠, describing the railway states.

1. $XR$ is the finite set of grounded numeric fluents over the function set $XR$⁠, representing quantitative properties of the railway system.

1. $AR$ is the finite set of durative actions representing railway operations.

1. $IR$ denotes the initial railway state and

2. $GR$ specifies the goal conditions that every valid plan must satisfy.

The goal of the planning task is to ensure that every train reaches its designated destination while completing all required passenger-service operations. Depending on the disruption scenario, additional recovery conditions, such as clearing blocked tracks or recovering disabled trains, may also be required. Accordingly, the goal condition is defined as:

Among all valid temporal plans satisfying the goal conditions, the planner seeks the schedule with the minimum makespan,

where $π$ denotes the generated temporal plan.

The following subsections instantiate this formulation in detail, from the underlying type system and state predicates through the durative action schemas and temporal resource constraints, to the construction of a concrete problem instance.

This section summarizes the key modeling decisions that shape the predicates and action schemas introduced below, balancing expressiveness against planning tractability.

1. Segments and points as exclusive temporal resources: Rather than checking collisions post hoc, track segments and points are made temporarily unavailable for the duration a vehicle occupies them, so every plan is collision-free by construction rather than by validation.

2. Declarative gauge compatibility. Gauge type is attached to trains, engines and track segments as static facts, and every movement action's preconditions require a matching gauge. This keeps gauge reasoning local to each action rather than requiring a separate consistency check over the whole route.

3. Disruptions as explicit states, not stochastic events. Blocked tracks, blocked trains, engine damage and slowdowns are represented as ordinary state predicates and a fluent, not sampled during execution. This lets a temporal planner reason about recovery the same way it reasons about nominal operations, using the same search machinery.

4. Directional dispatch for engine rescue. Auxiliary-engine dispatch is split into an up-bound and a down-bound schema rather than one direction-agnostic action, so that a rescue engine is only ever routed to approach a disabled train against its direction of travel, matching how rescue operations work in practice.

Together, these decisions keep the domain compact while preserving the correctness guarantees the following action schemas rely on.

This subsection defines the type hierarchy and state predicates used to model the railway network, the operational entities within it and the numeric fluents governing action durations.

The domain is organized over five object types: track-point, train, gauge-type, engine and station. Track points form the nodes of the railway network and, together with the segments connecting them, define its topology. Trains and auxiliary engines represent the rolling stock; each train and engine has an associated gauge type and a cruising speed. Stations group the platform points where passenger boarding occurs. Gauge types capture the physical gauge classifications (meter, broad, dual) that constrain which trains may use which segments.

The predicates $PR$ collectively define the operational world state of the railway environment. They are organized into five groups by the aspect of the railway operations they describe.

1. Spatial configuration. Predicates $train_at$ and engine_at indicates the location of trains and auxiliary engines at track points; platform_at and boarding_point predicates identify where a train must stop for service, and passengers_boarded records that service is complete; junction_point marks a point as a turnout.

2. Infrastructure connectivity. The Predicate free marks a point as unoccupied; connected reflects the currently active turnout routing between two points; track_accessible marks a segment as available for traversal, and is withdrawn while a vehicle occupies it, making segments exclusive temporal resources; turnout_alternatives records the two routing choices available at a junction.

3. Gauge compatibility. Predicates train_gauge, engine_gauge and track_gauge attach a gauge type to a train, an auxiliary engine and a track segment, respectively, so that movement actions can enforce compatibility directly in their preconditions.

4. Disruption state. Predicates track_blocked and track_clear are complementary facts describing whether a segment is passable; train_blocked marks an immobilized train; engine_damaged, engine_free and engine_attached track a locomotive failure and its recovery via an auxiliary engine.

5. Directional coordination. The up and down predicates orient a segment; heading_up and heading_down record a train's or engine's current travel direction, so a rescue engine can be dispatched against the disabled train's heading.

6. Constants. A set of numeric fluents $XR$ governs action durations and remains fixed for a given instance. For instance, the distance and train_speed, engine_speed fluents determine travel time; boarding_time and turnout_time fix service and switching durations; slowdown scales travel time under a weather- or track-condition disruption; and train_blockage_time, track_clear_time and engine_attach_time fix the duration of the corresponding recovery actions. Each is computed from infrastructure and vehicle properties rather than hardcoded, so a single domain covers heterogeneous networks without per-instance modification.

This vocabulary (types, predicates and fluents) forms the basis for the durative action schemas introduced in the following subsections.

The proposed railway domain is modeled using ten durative actions to represent operational activities and disruption recovery procedures. We categorize the railway operations into four types: movement operations, passenger service operations, disruption recovery operations and infrastructure operations.

The $DriveTrain$ and $DriveEngine$ actions (Table 2) model the traversal of trains and auxiliary engines between railway points. We describe $DriveTrain$ in detail as the representative example and highlight only the distinguishing features of subsequent actions. The parameters of the action $DriveTrain$ are a train $t$⁠, a track segment defined by the points $ni$ and $nj$⁠, and a gauge type $g$⁠. The temporal conditions of $DriveTrain$ ensure that, at initiation, the train and track are gauge-compatible, the segment is clear and accessible, and the train is neither blocked nor operating with a damaged engine. Since OPTIC and POPF do not support negative preconditions, the $¬$ train_blocked(t, $ni$), $¬$ engine_damaged(t, $ni$) conditions are encoded using positive predicates train_clear(t, $ni$) and engine_functional(t, $ni$). The destination point must remain free throughout execution and upon completion, preventing conflicting occupancy. The temporal effects mark the segment inaccessible in both directions at the start of the action, avoiding simultaneous use. Upon completion, the destination is marked occupied and segment accessibility is restored in both directions.

Passenger service activities are represented through the board-passengers action (Table 3), which ensures that departure occurs only after boarding is complete. Boarding time is defined by $β(t,s)$⁠, the time (in minutes) required for train $t$ to board passengers at station $s$⁠.

To support disruption recovery, drive-engine-to-damaged-up-train and drive-engine-to-damaged-down-train actions dispatch auxiliary engines toward immobilized trains while respecting directional and connectivity constraints. These actions are similar in structure to movement operations, presented in Table 4. The attach-engine action (Table 4) models the coupling of an auxiliary engine to a damaged train on a track segment. Both entities (the corresponding engine and train) must be co-located throughout the operation, enforced via over all conditions. The engine is marked unavailable at the start (not (engine-free ?en)), and upon completion engine-attached is asserted, enabling the subsequent drive-assisted-train action. Once attached, the drive-assisted-train action, shown in Table 4, enables the damaged train to move, with the auxiliary engine's speed determining the duration. The original train engine remains nonfunctional, which is reflected at the destination.

Operational disruptions are resolved using two dedicated recovery actions. The resolve-train-blockage action restores mobility of blocked trains, while clear-blocked-track models infrastructure restoration following track obstructions.

Infrastructure management primarily involves the turnout action (Table 5), which switches routing at junction points. The active connection is severed at initiation, while the alternative is established at completion, modeling physical switching delays via turnout-time.

Together, these durative actions provide an integrated temporal representation of railway movement, service operations, infrastructure management and disruption recovery, enabling the planner to generate executable and disruption-resilient railway schedules.

Each problem instance instantiates $〈IR,GR〉$ for a concrete scenario over $PR$⁠. The initial state $IR⊆FR$⁠, together with the fluents $XR$⁠, fixes train positions and gauges, the network topology and segment gauges, junction alternatives, platform and boarding assignments, distances, speeds and service times; the goal follows the specification presented in Section 4.2, with metric min total-time (makespan minimization). Table 6 presents the instance corresponding to the network of Figure 1.

Together, the domain model and problem instances encode railway scheduling and routing as a temporal planning task, from which temporal planners automatically generate timestamped action sequences that respect safety and gauge constraints, handle disruptions and minimize makespan in heterogeneous multi-gauge networks.

Building on the proposed formulation, we constructed a benchmark comprising a single domain model and 200 problem instances, evenly divided into 100 nominal and 100 disrupted instances. Every instance models heterogeneous railway infrastructure with three track types (meter, broad and dual-gauge) and two train types (meter and broad-gauge) operating at different speeds, thereby capturing the gauge compatibility and speed heterogeneity of real railway systems. Table 7 summarizes the fixed parameters used during benchmark generation, including network characteristics, train movement properties, disruption settings and operational action durations. The benchmark generation procedures are presented in Algorithms 1 and 2.

GENERATENOMINAL function (Algorithm 1) takes the target numbers of trains, stations, track points and junctions together with a random seed as inputs. Then it constructs a connected railway network, assigns stations, junctions and trains, and generates the nominal planning task by producing the initial state $IR$ and goal state $GR$⁠. The generation process guarantees that every train has at least one gauge-compatible route between its origin and destination, ensuring that every nominal instance is solvable. Nominal-operation instances evaluate planning performance under normal operating conditions. Their complexity is determined solely by the railway infrastructure, with increasing numbers of trains, stations, track points and junctions.

On the other hand, GenerateDisrupted (Algorithm 2) starts from a nominal instance and introduces the specified numbers of blocked trains, blocked track segments, slowdown-affected segments and engine failures to generate a disrupted initial state $IR′$⁠, while preserving the original goal state $GR$⁠. During disruption generation, track blockages are placed without disconnecting the railway network, ensuring that every disrupted instance remains solvable. Disrupted-operation instances evaluate planning performance under both increasing infrastructure size and disruption severity. The benchmark instances are organized into four primary scale categories: Small (S), Medium (M), Large (L) and Very Large (VL). Each category represents a progressive increase in infrastructure size and planning complexity. Within each category, multiple sub-levels (e.g. S1–S2, M1–M2, L1–L3 and VL1–VL3) provide finer-grained variations in problem difficulty. Tables 8 and 9 summarize the characteristics of the nominal and disrupted benchmark instances, respectively. The benchmark is designed with controlled and monotonic scaling. As the scale increases from S to VL, the numbers of trains, stations, track points and junctions increase consistently.