A Temporal Planning Framework for Disruption Aware Dynamic Route Optimization in Heterogeneous Railway Systems

# A Temporal Planning Framework for Disruption Aware Dynamic Route Optimization in Heterogeneous Railway Systems
Source: [https://arxiv.org/html/2606.14582](https://arxiv.org/html/2606.14582)
\[1\]\\fnmPollob Chandra\\surRay

1\]\\orgdivDepartment of Computer Science and Engineering,\\orgnameDhaka University of Engineering & Technology,\\orgaddress\\cityGazipur,\\postcode1707,\\countryBangladesh

###### Abstract

Efficient route optimization play a vital role in ensuring both safety and punctuality in railway operations\. It is very crucial particularly in heterogeneous multi\-gauge railway networks with varying train speed, stopping pattern, infrastructure compatibility constraints increase coordination complexity\. In single\-track systems these challenges are further intensify due to all trains to share the same track and requires frequent track switching\. Stochastic disruptions events including blocked tracks, blocked trains, engine failure and speed slowdowns introduces additional unpredictability in operations and deviate the timetable\. However, existing studies predominantly focuses on high\-level timetabling, omitting operational details such as track switching coordination\. As a result leaving decision to human operators, increasing safety risks into railway operations\. This study proposes a framework based on temporal planning for dynamic route optimization and disruption management in heterogeneous railway systems\. The framework formulates railway operations as a temporal planning problem using PDDL 2\.1 with explicitly modeling gauge compatibility constraints and diverse disruption scenarios\. It generates conflict\-free timestamped operational plans specifying both optimized schedules and executable action sequences\. To evaluate the proposed framework, we developed a benchmark problem set with 200 instances using up to 1,000 track points and 120 trains\. Two state\-of\-the\-art temporal planners and a plan validator were employed to assessed the framework\. The experimental results demonstrate that the framework effectively generates temporal operational plans for heterogeneous railway systems and handles multi\-gauge constraints, disruptions, and reduces dependence on manual decision making\.

###### keywords:

Temporal Planning, Automated Planning, Railway Route Optimization, Railway Scheduling, Disruption\-Aware Planning, Resource\-Constrained Scheduling\.

## 1INTRODUCTION

Railway is one of the more efficient ways to transport people and goods at scale, cheaper to run than roads, energy efficient, and capable of carrying large volume of freight\[islam2024optimizing\]\. With rising demand, railway infrastructure encounters significant challenges to ensure reliable and timely services due to network congestion\[bryan2007rail\]\. Maintaining reliable service in congestive network requires dynamic route optimization and optimal timetable to manage road congestion and reduce operational delays\.

Railway scheduling determines train timetables with departure and arrival time for each railway points\[kang2024critical\], while route optimization involves to find traveling path for trains to minimize travel duration and avoid conflicts\[wang2022train\]\. Both have decades of research behind them, genuinely hard to solve at scale and known to NP\-hard problem\[leutwiler2023review\]\. Real\-world systems introduce additional layers of complexity\. For instance, heterogeneous railway systems with varying speed profile trains and stopping pattern require higher level of coordination than uniform metro systems\[yan2019multi\]\. Multi\-gauge networks amplified this complexity by introducing multiple gauge types for track and train and require matching gauge types for train movement\[sanchis2021experimental\]\. While standard gauge \(1435 mm\) predominates in Europe and many other regions, several countries operate multi\-gauge systems\[sanchis2021experimental,su17188336\]\. For example, Spain uses three gauge types: Iberian gauge \(1668 mm\), standard gauge \(1435 mm\), and dual\-gauge tracks accommodating both\[sanchis2021experimental\]\. 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\. Similarly Bangladesh railway operates three track gauges: meter gauge \(1,000 mm\), broad gauge \(1,676 mm\), and dual gauge by combining both\[br2023\]\. The practical consequence is that gauge type determines which tracks a train can physically use\. To drive a train on a specific track must follow the gauge compatibility constraints\. A standard gauge train can only run on standard or dual gauge track\. An Iberian gauge train needs Iberian or dual gauge\. In Bangladesh, the same logic holds: meter gauge trains are confined to meter or dual gauge lines, broad gauge to broad or dual gauge\. The gauge compatibility constraint significantly limit route options, thereby increasing operational complexity\.

Moreover, railway operations get disrupted constantly by stochastic events: blocked tracks, blocked trains, engine failures, speed slowdowns\[xiu2024passenger\]\. These events deviates the railway schedule \(timetable\) and require dynamic rescheduling and route optimization to resume its operations\. Numerous studies have been conducted on various aspect of railway operations \(discussed in Section[2](https://arxiv.org/html/2606.14582#S2), such as scheduling, disruption handling, and rail station management\. However, most studies focused only on selected aspect\. Consequently, most of them generate railway schedule without executable actions, therefore leaving critical decisions like turnout \(track switching\) to human operators and contribute to safety incidents\[neves2024human\]\. For instance, human errors accounted for 50\.77% of operational incidents in Bangladesh railway during 2022\-2023\[br2023\]\. To bridge these gaps, this study proposes a framework based on temporal planning for railway route optimization in heterogeneous systems\.

Temporal planning, a branch of automated planning, generates a sequence of actions with specific timestamps while minimizing overall operation time\[benton2012temporal\]\. 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\. Existing temporal planners can handle the complexity inherent in railway networks efficiently since they employ optimal search algorithms \(e\.g\., A\*, IDA\*\) and domain\-independent heuristics\. In this work, we encode safety constraints, gauge compatibility, and turnout operations within a temporal planning domain that automates the generation of low\-level commands necessary to resolve disruptions\. We employ PDDL 2\.1 \(Planning Domain Definition Language\) to design our model\.

In this work, we also systematically design a 200\-instance benchmark \(100 nominal and 100 disrupted instances\), where we employ a monotonic scaling across four categories \(small, medium, large, and very large\) by incorporating up to 120 trains and 100 junctions\. This 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 framework and problem sets using two state\-of\-the\-art temporal planners: POPF\[coles2011popf2\], and OPTIC\[benton2012temporal\]\. 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 validated using a temporal validator VAL\[VAL\_1374201\], to confirm that the action sequence is consistent and correctly ordered, yielding executable action sequences ready for real\-world deployment\. By the verifiable planning domain, this framework offers a scalable solution to enhance the safety and efficiency of railway system\.

### 1\.1An Illustrative Example

We demonstrate how our framework models railway operations and generates executable plans using a simplified railway network, presented in Figure[1](https://arxiv.org/html/2606.14582#S1.F1)\. The network consists of six track points \(A, B, C, D, E, F\), whereBandEare junction \(turnout\) points\. JunctionBcan connect either to pointCorD\(currently connected toD\), while junctionEcan connect toCorD\(currently connected toD\)\.

The network also includes a heterogeneousmulti\-gaugeinfrastructure\. Track segmentsA\-BandE\-Fare dual gauge, allowing both meter\-gauge and broad\-gauge trains\. SegmentsB\-DandD\-Esupport only broad\-gauge trains, while segmentsB\-CandC\-Esupport only meter\-gauge trains\. StationS1, located between junctionsBandE, consists of two parallel platform pointsCandDwhere trains may stop for passenger boarding\. The network has two trains operating : trainT1\(broad gauge\) and trainT2\(meter gauge\)\.

The task is to move trainT1from pointAtoF, and to move trainT2from pointFtoA\. The trainT1also needs to stop at pointDfor a 2\-minute boarding operation, while trainT2requires a 3\-minute boarding at pointC\. Assume, trainsT1andT2travel 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\.

60 km10 km15 km10 km15 km40 kmABCDEFStation S1![Refer to caption](https://arxiv.org/html/2606.14582v1/fig/train.png)T1![Refer to caption](https://arxiv.org/html/2606.14582v1/fig/reverse_train.png)T2Broad gaugeMeter gaugeDual gauge \(3 rails\)Meter gauge \(inactive\)JSwitch / turnout pointFigure 1:Railway network with six track points \(A–F\)\. Circles denote stations and diamonds denote junction \(turnout\) points B and E\. The network contains broad\-gauge, meter\-gauge, and dual\-gauge \(three\-rail\) tracks\. Solid black lines represent broad\-gauge tracks, thin solid lines represent meter\-gauge tracks, and double lines represent dual\-gauge tracks; dashed lines indicate inactive turnout branches\. The curved blades at B and E show the current turnout setting toward D\. Station S1 contains points C and D\. Track distances are labeled along each segment\. Train T1 \(broad gauge, 60 km/h\) departs from A, and Train T2 \(meter gauge, 66 km/h\) departs from FTable 1:A temporal plan for the railway routing problem, presented in Figure[1](https://arxiv.org/html/2606.14582#S1.F1)Table[1](https://arxiv.org/html/2606.14582#S1.T1)shows a temporal operational plan, generated by our proposed model, for this simple railway network problem\. The plan specifies a sequence of timestamped actions, including train movements, turnout operations, and passenger boarding activities\. Each action is represented by its start time and execution duration\. For example, the action\(drive\-train T1 A B BROAD\)beginning at time 0\.000 with duration 60\.0 indicates that trainT1departs from pointAat time 0\.000 and arrives at pointBat time 60\.000, corresponding to the travel time of the track segment\. The labelBROADidentifies the trainT1as broad gauge, traveling through segmentA–B, a track compatible withT1\. This simple example demonstrates how the generated temporal plan provides a detailed operational schedule that specifies the exact sequence and timing of actions to co\-ordinate train movements, respecting gauge compatibility, station operations and turnout constraints\.

### 1\.2Contributions

This paper contributes the following:

1. 1\.A Temporal Planning Domain for Heterogeneous Railways: We model dynamic route optimization as a temporal planning problem, encoding gauge compatibility constraints and varying train profiles directly into the domain\.
2. 2\.Integrated Disruption Management:We incorporate disruption management within the temporal planning framework to handle disruptions: blocked trains, blocked tracks, engine failure, and slowdown\.
3. 3\.A Benchmark Problem Set:We systematically constructed benchmark dataset for heterogeneous railway system with 200 instances\. Instances scale from small networks up to 1,000 track points, following a monotonic progression designed to evaluate both scalability and robustness of the proposed model and existing planners\.

This work is an extended version of our conference paper accepted at ICSCA 2026\. This version substantially expands the work by introducing additional disruption scenarios, a large\-scale benchmark problem set, and extended experimental evaluation\.

### 1\.3Organization

The remainder of this paper is organized as follows\. In Section[2](https://arxiv.org/html/2606.14582#S2), we discuss the existing approaches to railway scheduling and optimization in the literature\. Section[3](https://arxiv.org/html/2606.14582#S3)provides the necessary background and semantics on temporal planning, and its modeling using PDDL\. In Section[4](https://arxiv.org/html/2606.14582#S4), 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](https://arxiv.org/html/2606.14582#S5)and conclude with our remarks in Section[6](https://arxiv.org/html/2606.14582#S6)\.

## 2RELATED WORKS

Researchers have proposed various optimization models and algorithms to address train scheduling, aiming to minimize delays, reduce waiting times, lower operational costs, and enhance system efficiency and service reliability\. These approaches can be broadly categorized based on the underlying methodology: MaxSAT, MILP, metaheuristic algorithms, and automated planning\.

Maximum Satisfiability \(MaxSAT\) formulations have recently gained attraction for optimizing scheduling under complex operational constraints\. Lemos et al\.\[lemos2024iterative\]proposed an iterative MaxSAT\-based framework for the Swiss Federal Railways \(SBB\) that minimizes delays and routing costs while handling disruptions such as track blockages and speed reductions through rerouting, waiting strategies, and velocity adjustments\. Gouveia et al\.\[gouveia2025maxsat\]developed a MaxSAT formulation for generating conflict\-free train timetables with route choice flexibility, evaluated using real\-world datasets from SBB and the Washington Metro system\. Gouveia et al\.\[gouveia2025maxsat\]built a MaxSAT formulation that generates conflict\-free timetables while preserving some flexibility in route choice, tested on real data from SBB and the Washington Metro\. Although these approaches work well within its scope for generating conflict\-free schedule, they generally consider only homogeneous train characteristics\. However, they fail to capture complexity of heterogeneous infrastructure involving different stop patterns, speeds and gauges across trains\. Furthermore, their primary focus remains to generate conflict free timetable rather than generation of operational plans with low level actions\.

Mixed Integer Linear Programming \(MILP\) has been the most common method for railway scheduling and rescheduling\. Chai et al\.\[chai2024branch\]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 et al\.\[zhuo2024demand\]took a similar demand\-driven angle, adapting both schedules and train compositions as passenger loads shift\. Ji et al\.\[ji2024optimization\]developed a two\-stage stochastic optimization model to address uncertainty in railway maintenance durations\. Ji et al\. stepped back from operations entirely, using two\-stage stochastic optimization to handle uncertainty in maintenance windows, when track work takes longer then schedule, the rest of the timetable has to flex around it\. Additional MILP and ILP\-based methods have been proposed for disruption management scenarios, including track blockages, rolling stock failures, and timetable adjustments\[acuna2011sapi,veelenturf2016railway,fischetti2017using,xiu2024passenger,li2025rolling\]\. While these models provide powerful optimization capabilities, they typically assume homogeneous train operations, often overlook heterogeneous infrastructure constraints such as multi\-gauge tracks, and primarily focus on timetable optimization rather than producing executable operational plans\.

Metaheuristic algorithms, including Genetic Algorithms\[nitisiri2019parallel\], Simulated Annealing\[zhang2022train\], and Ant Colony Optimization\[coviello2023integrated\], are frequently employed to navigate the non\-linear solution spaces of large\-scale networks\. These methods excel at minimizing passenger waiting times and operating cycles under fluctuating demand\. However, they frequently rely on simplified network abstractions\. By assuming uniform gauge compatibility and ignoring the physics of heterogeneous speeds across different track segments, these approaches struggle to maintain feasibility when applied to the complex, heterogeneous infrastructure addressed in this study\.

Automated planning offers a promising alternative by generating explicit action sequences for movement and resource utilization\. Cardellini et al\.\[cardellini2021station\]developed numeric planning based framework in PDDL\+ for train dispatching trains inside stations\. Louadah et al\.\[louadah2021translating\]used temporal planning to schedule maintenance in depots by translating ontological knowledge and SWRL rules into PDDL\. These methods optimize localized operations and can coordinate concurrent activities, but they do not extend to network\-wide route planning, heterogeneous train constraints, or integrated disruption handling\.

Despite these advancements, existing research often treats railway networks as uniform systems, overlooking the complexities of different train speeds, stopping patterns, and track gauge restrictions\. Most current methods reactively adjust timetables rather than proactively generating detailed, timestamped execution plans\. This paper addresses these gaps by introducing a temporal planning framework that accounts for diverse infrastructure and produces precise, actionable recovery plans for complex disruptions\.

## 3Preliminaries

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 Planning Domain Definition Language \(PDDL\) required for modeling the proposed framework\.

### 3\.1Temporal Planning

Automated planning \(AI planning\) is a branch of artificial intelligence concerned with the automatic generation of action sequences that enable an agent to achieve specified goals from a given initial state\[ghallab2004automated\]\. A planning system \(planner\) takes a formal description of the initial state and a set of goal conditions, and produces a plan that transforms the initial state into one satisfying the goals\.

In classical planning, actions are assumed to be instantaneous and sequentially executed\. The environment is typically modeled as fully observable, deterministic, static \(state changes occur only through agent actions\), and finite\. A solution to a classical planning problem is a sequence of actions \(i\.e\., a plan\) such that executing the actions successively from the initial state results in a state that satisfies the goal condition\. However, modeling 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\[cenamor2019predictability\]\. Intuitively, a temporal planning problem \(Definition[1](https://arxiv.org/html/2606.14582#Thmdefinition1)\) specifies the system state using logical and numeric variables, a set of time\-extended actions that modify the state, an initial configuration of the system, and a set of goal conditions that the planner must achieve\. A durative action \(Definition[2](https://arxiv.org/html/2606.14582#Thmdefinition2)\) represents an activity that unfolds over time, with conditions that must hold at different temporal stages and effects that update the system state at the beginning or end of the action\.

###### Definition 1\.

Atemporal planning problemis defined as a 5\-tuplePT=⟨F,V,Ad,I,G⟩P\_\{T\}=\\langle F,V,A\_\{d\},I,G\\rangle, where:

- •FFis a finite set of propositional fluents,
- •VVis a set of real\-valued numeric fluents,
- •AdA\_\{d\}is a finite set of*durative actions*,
- •I⊆FI\\subseteq Frepresents the initial state, and
- •G⊆FG\\subseteq Frepresents the goal condition\.

###### Definition 2\.

Adurative actiona∈Ada\\in A\_\{d\}is defined asa=⟨𝑝𝑟𝑒⊢​\(a\),𝑝𝑟𝑒↔​\(a\),𝑝𝑟𝑒⊣​\(a\),𝑒𝑓𝑓⊢​\(a\),𝑒𝑓𝑓⊣​\(a\),𝑑𝑢𝑟​\(a\)⟩a=\\langle\\mathit\{pre\}\_\{\\vdash\}\(a\),\\mathit\{pre\}\_\{\\leftrightarrow\}\(a\),\\mathit\{pre\}\_\{\\dashv\}\(a\),\\mathit\{eff\}\_\{\\vdash\}\(a\),\\mathit\{eff\}\_\{\\dashv\}\(a\),\\mathit\{dur\}\(a\)\\rangle, where

- •𝑝𝑟𝑒⊢​\(a\)\\mathit\{pre\}\_\{\\vdash\}\(a\)are preconditions required at action start,
- •𝑝𝑟𝑒↔​\(a\)\\mathit\{pre\}\_\{\\leftrightarrow\}\(a\)are invariant conditions that must hold throughout execution,
- •𝑝𝑟𝑒⊣​\(a\)\\mathit\{pre\}\_\{\\dashv\}\(a\)are preconditions required at action end,
- •𝑒𝑓𝑓⊢​\(a\)\\mathit\{eff\}\_\{\\vdash\}\(a\)are effects applied at action start,
- •𝑒𝑓𝑓⊣​\(a\)\\mathit\{eff\}\_\{\\dashv\}\(a\)are effects applied at action end, and
- •𝑑𝑢𝑟​\(a\)∈ℝ\+\\mathit\{dur\}\(a\)\\in\\mathbb\{R\}^\{\+\}denotes the action duration\.

Unlike classical planning, a temporal plan is not just a sequence of actions but a time\-stamped schedule of\(ti,ai\)\(t\_\{i\},a\_\{i\}\)pairs, wheretit\_\{i\}denotes the start time of actionaia\_\{i\}\[haslum2019introduction\]\. A temporal plan is valid if and only if \(i\) all preconditions are satisfied at their required times, \(ii\) over all \(invariant\) conditions hold throughout\[ti,ti\+dur​\(ai\)\]\[t\_\{i\},t\_\{i\}\+\\text\{dur\}\(a\_\{i\}\)\], no conflicting effects occur simultaneously, and the goalGGis satisfied at plan completion:tend=maxi⁡\(ti\+dur​\(ai\)\)t\_\{\\text\{end\}\}=\\max\_\{i\}\(t\_\{i\}\+\\text\{dur\}\(a\_\{i\}\)\)\. The makespan \(Definition[3](https://arxiv.org/html/2606.14582#Thmdefinition3)\) 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\.

###### Definition 3\.

Themakespanof a temporal planΠT\\Pi\_\{T\}is the total execution time of the plan, defined as

makespan​\(ΠT\)=max\(ai,ti\)∈ΠT⁡\(ti\+dur​\(ai\)\)\.\\mathrm\{makespan\}\(\\Pi\_\{T\}\)=\\max\_\{\(a\_\{i\},t\_\{i\}\)\\in\\Pi\_\{T\}\}\\left\(t\_\{i\}\+\\mathrm\{dur\}\(a\_\{i\}\)\\right\)\.

Minimizing the makespan is essential in time\-sensitive domains, such as railway operations, to ensure efficient use of resources and timely completion of all scheduled tasks\. Temporal planning allows actions to overlap, each with its own duration and constraints on when it can start or finish relative to others\. Which makes it particularly well\-suited for railway scheduling and route optimization, where multiple trains, track resources, and safety constraints must be coordinated over time\.

To formalize a planning problem and generate valid temporal plans, we need to represent the planning environment, such as states, actions, and goals\. For this purpose, the Planning Domain Definition Language \(PDDL\)\[haslum2019introduction\]is a widely adopted framework\.

### 3\.2Planning Domain Definition Language

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 the Planning Domain Definition Language \(PDDL\) 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\[fox2003pddl2\]\. 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](https://arxiv.org/html/2606.14582#LST1)and[2](https://arxiv.org/html/2606.14582#LST2), respectively\.

##### Domain Defintion

A temporal planning domain in PDDL is specified using the`define`construct and typically includes several components, such as`:requirements`,`:types`,`:predicates`,`:functions`, and`:durative actions`\.

The`:requirements`field specifies the language features used in the domain\. Our exemplary`simple\_temporal\_railway`domain includes`:durative\-actions`,`:fluents`, and`:typing`to support temporal operators, numeric functions, and typed parameters, respectively\. These declarations inform planners of the syntactic constructs that must be supported\. The`:types`section defines object categories \(e\.g\.,`train`,`location`\)\.

\(define\(domainsimple\_temporal\_railway\)

\(:requirements:strips:typing:durative\-actions:fluents\)

\(:typestrainlocation\)

\(:predicates

\(at?t\-train?l\-location\)\)

\(:functions

\(total\-cost\)\)

\(:durative\-actionmove

:parameters\(?t\-train?from\-location?to\-location\)

:duration\(=?duration5\)

:condition\(and

\(atstart\(at?t?from\)\)\)

:effect\(and

\(atstart\(not\(at?t?from\)\)\)

\(atend\(at?t?to\)\)

\(atend\(increase\(total\-cost\)10\)\)\)

\)

\)

Listing 1:Temporal Domain definition for a toyrailwaysystemPredicatesrepresent logical properties of the world state, defining relationships among objects and capturing dynamic conditions\. For instance,`\(at ?t ?l\)`specifies the location of a train in our example\.

Numeric functionsin PDDL act as state variables with numeric values, declared using`:functions`\. They enable quantitative reasoning for optimization, resource tracking, and cost evaluation\. The function`total\-cost`accumulates action costs during plan execution\. Numeric fluents may be increased, decreased, or assigned values within action effects\.

Durative actions, defined using the`:durative\-action`construct, include parameters, duration specifications, temporal conditions, and effects\. Temporal conditions specify when certain preconditions must hold\. In Listing[1](https://arxiv.org/html/2606.14582#LST1), the temporal conditions`\(at start <p\>\)`and`\(at end <p\>\)`must hold at action start and action completion, respectively, whereas`\(over all <p\>\)`must hold throughout execution of the corresponding action\. Effects may occur at the start or end of a durative action\.

In Listing[1](https://arxiv.org/html/2606.14582#LST1), the domain has one durative action`move`with two temporal conditions`\(at start \(at ?t ?from\)\)`and`\(at start \(track\_clear\)\)`, meaning the train must initially be at the departure location, and the track must be clear at the start of the action\. The action has three effects:`\(at start \(not \(at ?t ?from\)\)\)`,`\(at end \(at ?t ?to\)\)`and`\(at end \(increase \(total\-cost\) 10\)\)`\. Thus, the departure location is updated at the start, the arrival location is established at the end, and the numeric cost is incremented upon completion\. The`move`action also specifies a fixed execution time of 5 time units \(`:duration \(= ?duration 5\)`\)\. PDDL also allows duration inequalities and expressions involving numeric fluents\.

##### Problem Definition

A problem instance defines a specific planning scenario within the domain, declaring objects, initial state configuration, and goal conditions\. It provides the planner with the necessary data to instantiate domain actions and generate a valid plan\. Listing[2](https://arxiv.org/html/2606.14582#LST2)shows a corresponding toy problem for the domain in Listing[1](https://arxiv.org/html/2606.14582#LST1), where the task is to move a train from station A to station C in a simple railway network consisting of three nodes: A, B, and C\.

\(define\(problemtoy\_instance\)

\(:domainsimple\_temporal\_railway\)

\(:objects

train1\-train

ABC\-location\)

\(:init

\(attrain1A\)

\(=\(total\-cost\)0\)\)

\(:goal

\(attrain1C\)\)

\(:metricminimize\(total\-cost\)\)

\)

Listing 2:Problem definition for the railway domain in Listing[1](https://arxiv.org/html/2606.14582#LST1)The`:init`section specifies the initial state of the system, including logical facts \(e\.g\.,`\(at train1 A\)\)`and numeric initializations \(e\.g\.,`\(= \(total\-cost\) 0\)`\)\. The`:goal`specifies the logical conditions \(`\(at train1 C\)`\) that must hold at the end of the plan\.

PDDL also supports optimization objectives through the`:metric`field\. In this example, the metric`\(:metric minimize \(total\-cost\)\)`instructs the planner to minimize the total cost of executing actions\. Alternatively, one can minimize overall execution time by specifying`:metric minimize \(total\-time\)\)`, where the special variable`\(total\-time\)`denotes the makespan of the plan\. Temporal planners, therefore, search for a temporally valid plan that optimizes the specified objective function\.

A sample temporal plan for this problem, consisting of two sequential durative actions is shown below\. The first moves`train1`from station A to B, starting at time 0 and lasting 5 time units; the second action moves the`train2`from B to C, starting at time 5 and also lasting 5 time units\. This plan illustrates how temporal PDDL represents both the action sequencing and execution durations explicitly\.

0\.000:\(move\_A\_to\_Btrain1\)\[5\.000\]

5\.000:\(move\_B\_to\_Ctrain1\)\[5\.000\]

Such expressive capabilities of PDDL 2\.1, including durative actions with temporal annotations and numeric fluents, provide a solid basis for modeling railway systems\. They are especially useful in disruption management situations, where train movements, track allocations and recovery operations must be carefully coordinated under rigorous temporal and resource constraints\.

## 4METHODOLOGICAL FRAMEWORK AND PROBLEM FORMULATION

In this section, we introduce our framework, called Disruption Aware Railway Temporal Planning \(DART\) Framework, to represent railway systems as a*temporal planning problem*\.*DART*produces detailed, time\-stamped plans for coordinating train movements, infrastructure activities and disruption recovery, addressing gauge compatibility and coping with real\-time operational disturbances\.

DART systematically deals with operational challenges in heterogeneous railway networks, by proposing a disruption taxonomy and formalizing recovery strategies for each disruption type\. The framework encodes the network, rolling stock, and infrastructure elements in PDDL 2\.1, enabling temporal planners to generate executable plans that dynamically adapt to disruptions\. In the following sections, we detail the types of disruptions considered, the domain and problem modeling, and the durative actions that collectively allow DART to produce safe, efficient, and disruption\-resilient railway schedules\.

### 4\.1Disruption Taxonomy and Recovery Strategies

Railway operations in heterogeneous multi\-gauge networks are frequently affected by stochastic disruptions that deviate from nominal schedules and require dynamic rescheduling\. DART formalizes four primary disruption types, each represented in the PDDL 2\.1 domain using dedicated predicates and durative recovery actions\. These disruptions also form the basis for our benchmark instances in Section 4\.3:

1. 1\.Blocked Track Disruptionoccurs when a railway track segment becomes unavailable for train movements due to accidents, infrastructure failures, or maintenance activities\. Recovery strategies in our framework include \(a\) waiting at the current position until the track is cleared, or \(b\) rerouting via gauge\-compatible alternative paths using turnout operations\.
2. 2\.Blocked Train DisruptionOccurs 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. 3\.Slowdown Disruptionreduces train speeds over specific track segments due to external conditions such as adverse weather\. Travel time on a disrupted segment is computed as: traveling\-time=track\-distance​\(D,E\)\(1−α\)⋅train\-speed​\(t\)\\text\{traveling\-time\}=\\frac\{\\text\{track\-distance\}\(D,E\)\}\{\(1\-\\alpha\)\\cdot\\text\{train\-speed\}\(t\)\}\(1\)wherettdenotes the train traversing the disrupted segment, andα∈\(0,1\]\\alpha\\in\(0,1\]denotes the slowdown factor \(e\.g\.,α=0\.2\\alpha=0\.2for a 20% speed reduction\)\. Recovery strategies include traversing with increased travel time or rerouting to minimize makespan\.
4. 4\.Engine Failure Disruptionsoccurs 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\.

### 4\.2Temporal Railway Task Formulation

We model the railway environment as a tractable temporal planning problem, under four assumptions: the system is fully observable and deterministic, all trains have to reach their destinations and satisfy terminal conditions for the problem to be complete, the durations of disruptions \(e\.g\. blocked tracks, blocked trains, slowdowns\) are known a priori, and trains obey linear kinematic equations at constant speeds without acceleration and deceleration for computational tractability\. Under these assumptions a representation of railway tasks in PDDL as a domain description and corresponding problem instances is given\.

#### 4\.2\.1Domain Construction

We design the domain to capture the structural and operational heterogeneous railway systems\. It includes*infrastructure components*\(i\.e\., gauge types, track networks\),*railway operations*\(i\.e\., train movements, turnout operations, passenger boarding, auxiliary engine operations\),*operating rules*\(gauge compatibility constraints, safety constraints for collision avoidance\), and*disruption management mechanisms*\(track clearing, train blockage resolution, auxiliary engine attachment for engine failures\)\. This study formalizes these components in PDDL using type hierarchies, predicates, functions, and durative actions\.

The domain defines five object types for the railway task:`track\-point`,`station`,`engine`, and`gauge\-type`\. The type`track\-point`represents railway nodes in the network, while`station`,`train`, and`engine`represent railway stations, trains, and auxiliary engines, respectively\.`gauge\-type`represents gauge classifications \(i\.e\., meter, broad, standard\)\.

\(:typestrack\-pointtraingauge\-typeenginestation\)

##### Logical Variables

The predicates are designed to capture the logical conditions governing action applicability and state transitions in our Disruption\-Aware Railway System\.

###### Spatial Configuration

The first group of predicates captures the spatial configuration of operational entities within the railway network\. The predicates`train\-at`,`engine\-at`,`platform\-at`,`junction\-point`, and`boarding\-point`represent the locations of a train, an engine, a platform, a junction, and a boarding point in a specific railway point, respectively\.

\(train\-at?tr\-train?point\-track\-point\)

\(engine\-at?en\-engine?pt\-track\-point\)

\(platform\-at?pt\-track\-point?stn\-station\)

\(junction\-point?point\-track\-point\)

\(boarding\-point?tr\-train?point\-track\-point\)

###### Infrastructure Modeling

In addition to spatial configuration, the model captures infrastructure availability and connectivity conditions required for safe train movement using the following predicates\. The predicate`free`denotes the availability of a railway point\.`connected`indicates the physical connection \(i\.e\., track\) between two railway points in the network topology, while`track\-accessible`represents the operational accessibility of a track segment\. The predicate`passengers\-boarded`indicates the completion of passenger boarding at a railway point\.

\(free?point\-track\-point\)

\(connected?from\-track\-point?to\-track\-point\)

\(track\-accessible?from\-track\-point?to\-track\-point\)

\(turnout\-alternatives?t\-track\-point?tp1\-track\-point?tp2\-track\-point\)

\(passengers\-boarded?tr\-train?point\-track\-point\)

The predicate`turnout\-alternatives`models a railway turnout that provides two alternative continuation points from a track junction\. It specifies that the turnout`?t`\(illustrated in Figure[2](https://arxiv.org/html/2606.14582#S4.F2)\) can route a train toward either`?tp1`or`?tp2`, enabling the planner to reason about routing alternatives at track junctions\.

ttp1tp2![Refer to caption](https://arxiv.org/html/2606.14582v1/fig/train.png)Figure 2:A railway turnout at junctiontwith two alternative continuation track pointstp1andtp2\.
###### Gauge Compatibility

The following predicates associate gauge classifications with operational entities and infrastructure components, ensuring that vehicles operate only on compatible tracks\. The predicates`train\-gauge`,`track\-gauge`, and`engine\-gauge`define the gauge type associated with a train, a track segment, and an auxiliary engine, respectively\.

\(train\-gauge?tr\-train?gt\-gauge\-type\)

\(track\-gauge?from\-railway\-point?to\-railway\-point?gt\-gauge\-type\)

\(engine\-gauge?en\-engine?gt\-gauge\-type\)

###### Disruption Modeling

Finally, the following predicates model disruptions affecting tracks, trains, and engines\. The predicates`track\-clear`and`track\-blocked`denote the availability and blockage of track segments, respectively\. Similarly,`train\-clear`and`train\-blocked`represent whether a train is operational or immobilized due to disruption\.

\(track\-blocked?from\-track\-point?to\-track\-point\)

\(track\-clear?from\-track\-point?to\-track\-point\)

\(train\-blocked?tr\-train?point\-track\-point\)

\(train\-clear?tr\-train?point\-track\-point\)

Engine conditions are captured using the predicates`engine\-damaged`and`engine\-functional`, which represent failure and operational states of locomotive units\. The predicate`engine\-free`denotes the availability of an auxiliary engine, while`engine\-attached`represents its attachment to a train during recovery operations\.

\(engine\-damaged?tr\-train?point\-track\-point\)

\(engine\-functional?tr\-train?point\-track\-point\)

\(engine\-free?en\-engine\)

\(engine\-attached?en\-engine?tr\-train\)

Together, these predicates enable explicit representation of disruption occurrence and recovery processes, allowing the proposed railway system to generate recovery plans under dynamic railway conditions\.

##### Numeric Functions

Our domain represents quantitative characteristics of railway operations \( e\.g\., travel parameters, operational timing, and disruption recovery costs\) using*numeric functions*\(fluents\)\. The function`distance`specifies the travel distance between two railway points\. The functions`train\-speed`and`engine\-speed`represent the movement speeds of trains and auxiliary engines, respectively\. Passenger service operations are modeled using the function`boarding\-time`, which defines the duration required for passenger boarding at a station\. In addition,`turnout\-time`specifies the time required to reconfigure a railway turnout\.

\(distance?from\-track\-point?to\-track\-point\)

\(train\-speed?tr\-train\)

\(engine\-speed?en\-engine\)

\(boarding\-time?tr\-train?stn\-station\)

\(turnout\-time\)

Infrastructure disruptions and operational delays are captured through additional numeric functions\. The function`slowdown`represents delay factors on specific track segments, while`train\-blockage\-time`models delays caused by immobilized trains occupying railway points\.

\(slowdown?from\-track\-point?to\-track\-point\)

\(train\-blockage\-time?tr\-train?n\-railroad\-point\)

Recovery operations are represented through functions that quantify the time required to restore normal operations\. The function`track\-clear\-time`denotes the duration required to clear blocked tracks, whereas`engine\-attach\-time`specifies the time needed to attach an auxiliary engine to a disabled train\.

\(track\-clear\-time?r1\-track\-point?r2\-track\-point\)

\(engine\-attach\-time?en\)

These functions in our domain enable the planner to compute action durations dynamically based on infrastructure properties and operational conditions rather than assuming fixed execution times\.

##### Durative Actions

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\.

###### Movement Operations

The`drive\-train`and`drive\-engine`actions model the traversal of trains and auxiliary engines between railway points\. Both actions are similar in structure\. We describe the`drive\-train`action \(presented in Listing[3](https://arxiv.org/html/2606.14582#LST3)\) in detail as a representative example, while subsequent actions are highlighted only for their distinguishing features\. The action parameters include a train and a track segment defined by*from*and*to*track points\.

The temporal conditions of the`drive\-train`action ensure that, at initiation \(`at start`\), the train and track are gauge\-compatible, the track is clear and accessible, and the train is neither blocked nor operating with a damaged engine\. The destination point is also required to remain free throughout execution \(`over all`\) and upon completion \(`at end`\), preventing conflicting occupancy\. Its temporal effects mark the track segment inaccessible in both directions at the start of the action to avoid simultaneous use\. Upon completion \(`at end`\), the destination is marked occupied, and track accessibility is restored in both directions\.

\(:durative\-actiondrive\-train

:parameters\(?tr\-train?from\-track\-point

?to\-track\-point\)

:duration\(=?duration\(/\(distance?from?to\)

\(\*\(train\-speed?tr\)\(\-1\(slowdown?from?to\)\)\)\)\)

:condition\(and\(atstart\(train\-at?tr?from\)\)

\(atstart\(train\-gauge?tr?gt\)\)

\(atstart\(track\-gauge?from?to?gt\)\)

\(atstart\(train\-not\-blocked?tr?from\)\)

\(atstart\(track\-clear?from?to\)\)

\(atstart\(engine\-not\-damaged?tr?from\)\)

\(atstart\(track\-accessible?from?to\)\)

\(overall\(connected?from?to\)\)

\(overall\(free?to\)\)

\(atend\(free?to\)\)\)

:effect\(and\(atstart\(not\(train\-at?tr?from\)\)\)

\(atstart\(free?from\)\)

\(atstart\(not\(track\-accessible?from?to\)\)\)

\(atstart\(not\(track\-accessible?to?from\)\)\)

\(atend\(train\-at?tr?to\)\)

\(atend\(not\(free?to\)\)\)

\(atend\(track\-accessible?from?to\)\)

\(atend\(track\-accessible?to?from\)\)\)\)

Listing 3:Durative actiondrive\-trainmodeling train traversal between track points
###### Passenger Service Operations

Passenger service activities are represented through the`board\-passengers`action \(Listing[4](https://arxiv.org/html/2606.14582#LST4)\), ensuring that departure occurs only after boarding completion\.

\(:durative\-actionboard\-passengers

:parameters\(?tr\-train?stn\-station?pt\-track\-point\)

:duration\(=?duration\(boarding\-time?tr?stn\)\)

:condition\(and

\(atstart\(train\-at?tr?pt\)\)

\(atstart\(boarding\-point?tr?pt\)\)

\(atstart\(platform\-at?pt?stn\)\)

\(overall\(train\-at?tr?pt\)\)\)

:effect\(atend\(passengers\-boarded?tr?pt\)\)\)

Listing 4:Durative actionboard\-passengersrepresenting passenger boarding at a station
###### Disruption Recovery Operations

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\.

The`attach\-engine`action \(Listing[5](https://arxiv.org/html/2606.14582#LST5)\) 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\.

\(:durative\-actionattach\-engine

:parameters\(?en\-engine?tr\-train?pt\-track\-point\)

:duration\(=?duration\(engine\-attach\-time?en\)\)

:condition\(and

\(atstart\(engine\-at?en?pt\)\)

\(atstart\(train\-at?tr?pt\)\)

\(atstart\(engine\-free?en\)\)

\(atstart\(engine\-damaged?tr?pt\)\)

\(overall\(engine\-at?en?pt\)\)

\(overall\(train\-at?tr?pt\)\)\)

:effect\(and

\(atstart\(not\(engine\-free?en\)\)\)

\(atend\(engine\-attached?en?tr\)\)\)\)

Listing 5:Durative actionattach\-enginefor coupling an auxiliary engine to a damaged train\.Once attached, the`drive\-assisted\-train`action, shown in Listing[6](https://arxiv.org/html/2606.14582#LST6), enables the damaged train to move, with the auxiliary engine’s speed determining the duration\. The original train engine remains non\-functional, which is reflected at the destination\.

\(:durative\-actiondrive\-assisted\-train

:parameters\(?en\-engine?tr\-train?from\-track\-point?to\-track\-point?gt\-gauge\-type\)

:duration\(=?duration\(/\(distance?from?to\)\(\*\(engine\-speed?en\)\(\-1\(slowdown?from?to\)\)\)\)\)

:condition\(and

\(atstart\(engine\-attached?en?tr\)\)

…

\(atend\(free?to\)\)\)

:effect\(and

\(atstart\(not\(train\-at?tr?from\)\)\)

…

\(atend\(engine\-damaged?tr?to\)\)\)\)

Listing 6:Durative actiondrive\-assisted\-trainfor moving a damaged train with an attached auxiliary engine\.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 Operations

Infrastructure management primarily involves the`turnout`action \(Listing[7](https://arxiv.org/html/2606.14582#LST7)\), 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`\.

\(:durative\-actionturnout

:parameters\(?tp\-track\-point?alt1\-track\-point

?alt2\-track\-point\)

:duration\(=?duration\(turnout\-time\)\)

:condition\(and

\(atstart\(junction\-point?tp\)\)

\(atstart\(turnout\-alternatives?tp?alt1?alt2\)\)

\(atstart\(connected?tp?alt1\)\)

\(atstart\(not\-connected?tp?alt2\)\)\)

:effect\(and

\(atstart\(not\(connected?tp?alt1\)\)\)

…

\(atend\(not\(not\-connected?alt2?tp\)\)\)\)\)

Listing 7:Durative actionturnoutfor reconfiguring junction track connections\.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\.

#### 4\.2\.2Problem Instance Modeling

Each problem instance defines a specific railway scenario within the domain, including available resources \(e\.g\., trains and track points\), network configuration \(track layout\), and current and goal specifications\. It captures the current train positions, track and gauge types, network topology with track distances, train speeds, boarding times, any active disruptions, and destination points for each train\. This structure allows the dynamic generation of temporal plans tailored to the operational scenario\. The PDDL formulation of the problem corresponding to the network depicted in Figure[1](https://arxiv.org/html/2606.14582#S1.F1)is presented in Listing[2](https://arxiv.org/html/2606.14582#LST2)\.

\(:init\(train\-atT1A\)\(train\-atT2F\)

\(train\-gaugeT1broad\)\(train\-gaugeT2meter\)

\(freeB\)

…

\(freeE\)

\(connectedAB\)

…

\(connectedEF\)

\(accessible\-trackAB\)

…

\(accessible\-trackEF\)

\(junction\-pointB\)\(turnout\-alternativesBCD\)

\(junction\-pointE\)\(turnout\-alternativesECD\)

\(track\-gaugeABBROAD\)

…

\(track\-gaugeEFMETER\)

\(platform\-atCS1\)\(platform\-atDS1\)

\(boarding\-pointT1D\)\(boarding\-pointT2C\)

\(=\(road\-distanceAB\)60\)\(=\(road\-distanceBC\)10\)

\(=\(road\-distanceBD\)10\)\(=\(road\-distanceCE\)15\)

\(=\(road\-distanceDE\)15\)\(=\(road\-distanceEF\)40\)

\(=\(train\-speedT1\)1\.0\)\(=\(train\-speedT2\)1\.1\)

\(=\(boarding\-timeT1S1\)2\)\(=\(boarding\-timeT2S1\)3\)

\(=\(turnout\-time\)1\)\)

\(:goal\(train\-atT1F\)\(passengers\-boardedT1D\)

\(train\-atT2A\)\(passengers\-boardedT1C\)\)

\(:metricminimize\(total\-time\)\)

Together, the domain model and problem instances encode the railway scheduling and routing problem into a temporal planning framework, enabling temporal planners to automatically generate timestamped action sequences that ensure safety, minimize delays, handle disruptions, and optimize operational efficiency in heterogeneous \(multi\-gauge\) railway networks\.

### 4\.3Temporal Railway Planning Benchmark Generation

Based on this formulation, we developed a comprehensive benchmark problem set consisting of a single domain model and 200 problem instances\. Each problem instance models heterogeneous infrastructure with three track types \(meter, broad, and dual gauge\) and two train types \(meter and broad gauge\), each with distinct travel speeds, reflecting the complexity of real\-world railway systems\. The instances vary in complexity, characterized by different numbers of railway points, trains, junctions, and disruption scenarios\. We divided the benchmark into two operational regimes: nominal operation and disrupted operation\. The benchmark instances are organized into four primary scale categories: Small \(S\), Medium \(M\), Large \(L\), and Very Large \(VL\)\. Each category reflects a structured increase in infrastructure size and planning complexity\. Within every category, three progressively harder sub\-levels \(e\.g\., S1–S2, M1–M2, L1–L3, VL1–VL3\) further refine the growth in problem difficulty\.

*Nominal\-operation*instances represent standard railway planning scenarios without unexpected disturbances\. All infrastructure components are fully operational, no trains or track segments are blocked, and no speed reductions or engine failures occur\. These instances evaluate the planner’s ability to compute feasible and temporally consistent schedules under increasing infrastructure scale alone\. Table[2](https://arxiv.org/html/2606.14582#S4.T2)presents the characteristics of the nominal\-operation instances, including the number of trains, stations, track points, and junctions\.

Table 2:Nominal\-operation benchmark instances grouped by increasing problem size\. Instance complexity progressively increases with respect to the number of trains, stations, track points, and junctions\.In contrast, disrupted\-operation instances model adverse operational conditions\. In addition to the underlying infrastructure, these problems introduce exogenous disruptions, including blocked trains, blocked track segments, slowdown\-affected segments, engine failures, and limited auxiliary engines for recovery\. These instances therefore assess the planner’s robustness and recovery capabilities under escalating disruption severity combined with increasing network size\. Table[3](https://arxiv.org/html/2606.14582#S4.T3)summarizes the disrupted instances, detailing all problem size parameters alongside disruption\-specific characteristics\.

Table 3:Disrupted\-operation benchmark instances grouped by increasing problem and disruption complexity\. In addition to infrastructure size, instances vary in disruption severity including blocked trains, blocked tracks, engine failures, slowdown segments, and available auxiliary engines\.
Notably, the scaling is controlled and monotonic\. As the scale increases from S to VL, the number of trains, stations, track points, and junctions consistently grows\. In the disrupted benchmark, disruption severity increases in parallel with infrastructure size\. Larger instances not only contain more railway components but also feature a greater number of blocked trains, blocked track segments, slowdown\-affected segments, engine failures, and auxiliary engines\. This structured design ensures gradual infrastructure scaling for fine\-grained scalability evaluation, while disruption severity increases proportionally with network size to reflect realistic operational stress\. Consequently, the benchmark supports systematic analysis from small \(S\) to very large \(VL\) railway systems\.

## 5Experimental Results

This section presents a comprehensive experimental evaluation of the proposed temporal planning framework\. The railway domain and benchmark problem set were evaluated using two state\-of\-the\-art temporal planners: POPF3 and OPTIC\. The generated temporal plans were validated using the VAL plan validator\[VAL\_1374201\], which verifies the semantic correctness of plans with respect to the PDDL domain specification\.

These planners use different search mechanisms and heuristic techniques for solving temporal planning problems\. POPF\[coles2011popf2\]uses grounded forward search with linear programming, and a Temporal Relaxed Planning Graph \(TRPG\) heuristic to guide exploration\. OPTIC\[benton2012temporal\]extends POPF with mixed\-integer programming techniques for optimization, allowing preference\-aware temporal planning\. OPTIC also uses an improved TRPG heuristic that benefits

All planners were executed using their default parameter settings with a maximum time limit of 30 minutes per problem instance\. To ensure reproducibility, each problem instance was solved only once, as both planners operate deterministically\. All experiments were conducted on a workstation equipped with a 7th\-generation Intel Core i7 processor, 32 GB RAM, and an M\.2 SSD, running Ubuntu 24\.04\.3 LTS\.

To evaluate the validity and quality of the generated temporal plans, we consider several standard planning metrics, including*makespan*\(Definition[3](https://arxiv.org/html/2606.14582#Thmdefinition3)\),*plan length*, and*plan generation time*\. Plan length denotes the total number of actions contained in the generated operational plan, while plan generation time corresponds to the wall\-clock time required by a planner to produce a valid solution\.

While these metrics capture general planning efficiency, they do not directly reflect the operational impact of disruptions on railway performance\. Therefore, we additionally introduce domain\-specific delay metrics that quantify the deviation between the generated schedules and ideal uninterrupted train operations\.

##### Delay Metrics

To evaluate the effectiveness of the proposed temporal planning model under realistic operational conditions, we analyze the delays experienced by trains during the execution of generated schedules\. We define the delay metrics with respect to a temporal planΠT\\Pi\_\{T\}\. Let𝒯=\{1,…,n\}\\mathcal\{T\}=\\\{1,\\dots,n\\\}be the set of trains, and𝒥i=\{1,…,mi\}\\mathcal\{J\}\_\{i\}=\\\{1,\\dots,m\_\{i\}\\\}be the ordered set of track points traversed by trainiiinΠT\\Pi\_\{T\}\. Total Delay \(Definition[4](https://arxiv.org/html/2606.14582#Thmdefinition4)\) quantifies overall schedule degradation under operational conditions\.

###### Definition 4\(Total Delay\)\.

Thetotal delayof a temporal planΠT\\Pi\_\{T\}, defined with respect to our railway domain, is the cumulative deviation between actual and ideal arrival times over all trains and track points:

TotalDelay​\(ΠT\)=∑i=1n∑j=1mi\(Ti​jactual−Ti​jideal\),\\mathrm\{TotalDelay\}\(\\Pi\_\{T\}\)=\\sum\_\{i=1\}^\{n\}\\sum\_\{j=1\}^\{m\_\{i\}\}\\left\(T\_\{ij\}^\{\\mathrm\{actual\}\}\-T\_\{ij\}^\{\\mathrm\{ideal\}\}\\right\),whereTi​jactualT\_\{ij\}^\{\\mathrm\{actual\}\}andTi​jidealT\_\{ij\}^\{\\mathrm\{ideal\}\}are the actual and ideal arrival times of trainiiat track pointjjinΠT\\Pi\_\{T\}\. The ideal arrival time assumes uninterrupted travel at nominal speed:

Ti​jideal=Ti​0dep\+∑k=1jτi​k,T\_\{ij\}^\{\\mathrm\{ideal\}\}=T\_\{i0\}^\{\\mathrm\{dep\}\}\+\\sum\_\{k=1\}^\{j\}\\tau\_\{ik\},withTi​0depT\_\{i0\}^\{\\mathrm\{dep\}\}the departure time andτi​k\\tau\_\{ik\}the minimum traversal time on segmentkkas defined in the domain\.

Although total delay measures overall schedule degradation, it does not identify the underlying causes\. To capture these, we decompose total delay into three components, each defined with respect to actions in our railway domain: slowdown delay \(Definition[5](https://arxiv.org/html/2606.14582#Thmdefinition5)\), blockage delay \(Definition[6](https://arxiv.org/html/2606.14582#Thmdefinition6)\), and engine failure delay \(Definition[7](https://arxiv.org/html/2606.14582#Thmdefinition7)\)\.

###### Definition 5\.

Theslowdown delay, with respect to the domain DART, is the extra time incurred when trains traverse segments with reduced speed\. For eachdrive\-trainordrive\-assisted\-trainaction on segment\(p,q\)\(p,q\), letτnormal=dp​q/si\\tau^\{\\mathrm\{normal\}\}=d\_\{pq\}/s\_\{i\}be the nominal duration andτslow=dp​q/\(\(1−αp​q\)​si\)\\tau^\{\\mathrm\{slow\}\}=d\_\{pq\}/\(\(1\-\\alpha\_\{pq\}\)s\_\{i\}\)the actual duration under slowdown factorαp​q∈\[0,1\)\\alpha\_\{pq\}\\in\[0,1\)\. Then:

SlowdownDelay​\(ΠT\)=∑a∈ΠTa∈\{drive\-train,drive\-assisted\-train\}\(dur​\(a\)−τnormal\),\\mathrm\{SlowdownDelay\}\(\\Pi\_\{T\}\)=\\sum\_\{\\begin\{subarray\}\{c\}a\\in\\Pi\_\{T\}\\\\ a\\in\\\{\\texttt\{drive\-train\},\\,\\texttt\{drive\-assisted\-train\}\\\}\\end\{subarray\}\}\\bigl\(\\mathrm\{dur\}\(a\)\-\\tau^\{\\mathrm\{normal\}\}\\bigr\),where the difference is zero ifαp​q=0\\alpha\_\{pq\}=0\.

###### Definition 6\.

Theblockage delayin domain DART is the total waiting time due to train or track blockages, computed as the cumulative duration of allresolve\-train\-blockageandclear\-blocked\-trackactions:

BlockageDelay​\(ΠT\)=∑a∈ΠTa=resolve\-train\-blockagedur​\(a\)\+∑a∈ΠTa=clear\-blocked\-trackdur​\(a\),\\mathrm\{BlockageDelay\}\(\\Pi\_\{T\}\)=\\sum\_\{\\begin\{subarray\}\{c\}a\\in\\Pi\_\{T\}\\\\ a=\\texttt\{resolve\-train\-blockage\}\\end\{subarray\}\}\\mathrm\{dur\}\(a\)\+\\sum\_\{\\begin\{subarray\}\{c\}a\\in\\Pi\_\{T\}\\\\ a=\\texttt\{clear\-blocked\-track\}\\end\{subarray\}\}\\mathrm\{dur\}\(a\),where each action duration is defined in the domain\.

###### Definition 7\.

Theengine failure delay, with respect to the domain DART, is the total immobilization time of trains experiencing engine failure\. For each traini∈ℱi\\in\\mathcal\{F\}, recovery involves dispatching an auxiliary engine, attaching it, and resuming movement\. The delay is the sum of the dispatch and attach durations:

EngineFailureDelay​\(ΠT\)=∑i∈ℱ\(dur​\(aidispatch\)\+dur​\(aiattach\)\),\\mathrm\{EngineFailureDelay\}\(\\Pi\_\{T\}\)=\\sum\_\{i\\in\\mathcal\{F\}\}\\bigl\(\\mathrm\{dur\}\(a\_\{i\}^\{\\mathrm\{dispatch\}\}\)\+\\mathrm\{dur\}\(a\_\{i\}^\{\\mathrm\{attach\}\}\)\\bigr\),where durations are defined by the corresponding domain actions \(drive\-engine,drive\-engine\-to\-damaged\-trainandattach\-engine\)\.

By decomposing total delay into slowdown, blockage, and engine failure components with respect to our domain, this study provides a detailed assessment of how different types of disruptions affect the generated plans\.

### 5\.1Overall Experimental Results

Tables[4](https://arxiv.org/html/2606.14582#S5.T4)summarize the experimental results obtained by applying OPTIC and POPF temporal planners to the proposed railway domain and benchmark problem set\.

Table 4:Experimental results of OPTIC and POPF on nominal and disrupted\-operation instances\. Values reported as mean±\\pmstandard deviation over 10 instances per group\.Both planners produced identical makespan, plan length, and total delay values across all solved instances; therefore, a single unified set of results is reported\. We attribute this to OPTIC extending POPF with the same TRPG heuristic, and to the tightly constrained solution space imposed by gauge compatibility and single\-track mutual exclusion\. For nominal\-operation instances, makespan remains stable across all scales with low standard deviations, plan length grows proportionally with network size, and total delay remains exactly zero confirming conflict\-free scheduling under undisrupted conditions\. For disrupted\-operation instances, all metrics increase monotonically with problem scale\. plan length growth observed on recovery operations, while total delay exhibits the most pronounced increase driven by escalating disruption count and diversity\. The higher standard deviations, particularly in delay, reflect inter\-instance variability from differences in disruption severity, spatial distribution, and rerouting availability\. Both planners solved all 100 nominal instances and 99 of the 100 disrupted ones\. The single failure was the largest configuration in the benchmark: 120 trains, 1,000 track points, 108 concurrent disruptions\. All 199 generated plans were verified using VAL validator\. The validator confirmed that every plan satisfies all temporal constraints, precondition requirements, and goal conditions specified in the domain\. No mutex violations, precondition failures, or goal achievement errors were detected\. This confirms the semantic correctness of the domain encoding\.

Figure[3](https://arxiv.org/html/2606.14582#S5.F3)illustrates the average total delay across disrupted\-instance groups, showing that schedule degradation increases monotonically with problem scale in a clear and predictable pattern\. OPTIC and POPF produce identical delay values at every scale point, which suggests the domain is constrained tightly enough that both planners converge on the same solution rather than finding different valid alternatives\.

S1S2M1M2L1L2L3VL1VL2VL30200200400400Disrupted\-operation instance scaleAverage Delay per Instance \(minutes\)Average DelayFigure 3:Average delay for all trains in disrupted operation instances\. Both OPTIC and POPF generate identical coordination protocols with the same delay values\.Figure[4](https://arxiv.org/html/2606.14582#S5.F4)decomposes total delay by disruption types\. It reveals slowdowns are the biggest source of delay by far 60\.1% of total delay across all instances, compared to 30\.4% for engine failures and 9\.5% for blockages\. The slowdown share grows in larger networks, where higher disruption density leaves less room to reroute around affected segments\. Blockage delays stay low throughout, which indicates the planner resolved track conflicts without violating temporal constraints and find an alternative path\. This distribution confirms that coordination overhead arises from operational constraints encoded in the domain rather than from modeling artifacts\.

S1S2M1M2L1L2L3VL1VL2VL30200200400400Problem Instance Groups \(Disrupted\)Average Delay per Instance \(minutes\)Blockage DelayEngine Failure DelaySlowdown DelayFigure 4:Breakdown of average delay per instance by disruption type across problem scale\. Slowdown delays are the main contributor \(60\.1% overall\), followed by engine repair delays \(30\.4%\) and blockage delays \(9\.5%\)\. VL3 exhibits a surge in slowdown delay, reaching 69\.8%\.Figures[5](https://arxiv.org/html/2606.14582#S5.F5)and[6](https://arxiv.org/html/2606.14582#S5.F6)illustrate the growth of average computation time of the planners as problem size increases\. Under nominal conditions \(Figure[5](https://arxiv.org/html/2606.14582#S5.F5)\), OPTIC and POPF track each other closely, the curve is smooth and consistent as networks scale up\. For disrupted instances \(Figure[6](https://arxiv.org/html/2606.14582#S5.F6)\), computation times are substantially higher due to additional coordination requirements\. POPF is faster on smaller instances, but the gap closes as problems grow\. By the largest configurations, both planners are running at roughly the same speed\. The VL3 group plateaus because instance p200 hits the 30\-minute time limit, the curve flattens not because the problem gets easier, but because the clock runs out\.

S1S2M1M2L1L2L3VL1VL2VL3020204040Nominal\-operation instance scaleAverage computation time \(seconds\)OPTICPOPFFigure 5:Average computation time per instance for nominal\-operation instances\. Both OPTIC and POPF exhibit monotonically increasing computation times, ranging from approximately 0\.06 seconds for small instances \(S1\) to around 45 seconds for very large instances \(VL3\)\.S1S2M1M2L1L2L3VL1VL2VL310−110^\{\-1\}10010^\{0\}10110^\{1\}10210^\{2\}Disrupted\-operation instance scaleAverage computation time \(seconds\)OPTICPOPFFigure 6:Average computation time per instance for disrupted\-operation instances \(logarithmic scale\)\. Computation time increases monotonically from 0\.28 seconds \(S1\) to 619\.00 seconds \(VL3\), reflecting the growing combinatorial complexity of larger disrupted networks\.To further validate structural scalability, we conducted Spearman rank correlation analysis between problem size characteristics and performance indicators \(Tables[5](https://arxiv.org/html/2606.14582#S5.T5)and[6](https://arxiv.org/html/2606.14582#S5.T6)\)\. Table[5](https://arxiv.org/html/2606.14582#S5.T5)reveals perfect positive correlations\(ρ=1\.000,p<0\.001\)\(\\rho=1\.000,p<0\.001\)between all system size metrics and computation time, confirming controlled complexity scaling\. The absence of correlation between states evaluated and computation time\(ρ=−0\.176,p=0\.627\)\(\\rho=\-0\.176,p=0\.627\)suggests search efficiency is governed by heuristic pruning rather than state\-space size\.

Table 5:Spearman correlation heatmap: network characteristics vs\. computational performance\.Table[6](https://arxiv.org/html/2606.14582#S5.T6)presents the Spearman rank correlation matrix between disruption characteristics \(total disruptions, engine failures, blocked tracks, and blocked trains\) and coordination quality measured by total train delay\. All disruption quantities exhibit very strong to perfect positive correlations with total delay\(ρ≥0\.976,p<0\.001\)\(\\rho\\geq 0\.976,p<0\.001\), demonstrating that coordination degradation scales proportionally with disruption intensity across the benchmark\. This structured correlation pattern validates the monotonic benchmark design and confirms that the temporal planning framework responds predictably to increasing operational stress\.

Table 6:Spearman correlation heatmap: disruption characteristics vs\. performance impact

## 6CONCLUSIONS

In this study, a temporal planning\-based framework for dynamic route optimization and disruption management in heterogeneous multi\-gauge railway networks is developed\.

Our framework formulates railway operations in PDDL 2\.1 to produce timestamped action plans with explicit constraints for gauge compatibility, operational safety, and automated disruption resolution\. Extensive evaluation on 200 benchmark instance scenarios demonstrates robustness, optimal coordination across planners, and predictable scalability with increasing network size and disruption intensity\. The results confirm the potential of temporal planning to generate feasible and safety\-compliant operational strategies in real\-time even in large scale and disruption prone railway environments\.

Overall, this work establishes temporal planning as a practical and scalable foundation for intelligent railway management systems\. In future, we intend to extend this framework to support plan repair under uncertainty, enabling adaptive re\-planning in response to incomplete or evolving operational information\. Additional extensions, including derailment modeling and integration with real\-time data streams, will further enhance this framework’s resilience and deployment readiness\.

## 7Data Availability

## References