Toward Sustainable and Inclusive Cities: Graph Neural Network-Enhanced Optimization for Disability-Inclusive Emergency Evacuation in High-Rise Buildings

Abstract

Emergency evacuation planning in high-rise buildings presents complex optimization challenges critical to achieving sustainable and inclusive urban development. Traditional evacuation models inadequately address vulnerable groups’ needs—particularly persons with disabilities—while neglecting fire spread dynamics, congestion effects, and real-time risk assessment. This neglect undermines both human safety and social equity—core dimensions of sustainable communities. Sustainable cities must integrate inclusive design and emergency preparedness into high-rise development. This paper develops a comprehensive mathematical optimization framework for disability-inclusive emergency evacuation that integrates dynamic fire spread modeling, congestion-aware routing mechanisms, and explicit accessibility constraints within a unified formulation. The proposed approach balances evacuation efficiency, safety, and fairness across diverse population groups through a multi-objective optimization model that incorporates time-varying risk assessments, elevator priority systems for wheelchair users, and group-specific mobility coefficients. To address the computational scalability challenges inherent in large-scale mixed-integer nonlinear programming problems, we introduce an innovative solution methodology that combines Graph Neural Networks (GNN) with Proximal Policy Optimization (PPO) algorithms. The graph neural network component captures spatial-temporal feature representations of building geometry, occupant distributions, and hazard dynamics, while the reinforcement learning algorithm develops adaptive routing policies that respond to evolving emergency conditions. Experimental results on a representative high-rise building scenario demonstrate that the proposed GNN-PPO method achieves substantial improvements in safety, efficiency, and equity. The dynamic policy successfully prioritizes vulnerable populations, utilizes elevator systems effectively for persons with disabilities, and adapts to real-time emergency conditions, providing a robust framework for inclusive emergency evacuation planning in complex building environments. This work demonstrates how advanced computational methods can advance sustainability objectives by ensuring equitable safety outcomes across diverse populations—a prerequisite for truly sustainable cities.

1. Introduction

1.1. Background

Emergency evacuation in high-rise buildings presents complex challenges that require sophisticated optimization approaches to ensure safe and efficient egress for all occupants. With the increasing urbanization and construction of tall buildings worldwide, the need for comprehensive evacuation planning has become paramount, particularly when considering the diverse mobility characteristics and accessibility requirements of modern building populations. Traditional evacuation models often overlook the heterogeneous nature of occupants, failing to adequately address the specific needs of vulnerable groups such as elderly individuals and persons with disabilities, while simultaneously neglecting the dynamic interplay between fire spread, congestion effects, and real-time risk assessment.

The challenge of disability-inclusive evacuation must be understood within the broader sustainability agenda. The United Nations Sustainable Development Goals explicitly link urban resilience (SDG 11: “Make cities and human settlements inclusive, safe, resilient and sustainable”) with social inclusion and accessibility (SDG 10: “Reduce inequality within and among countries”). High-rise buildings represent both the promise and peril of modern urban development—they enable dense, economically productive cities while concentrating populations in environments with unique safety challenges. Ensuring that evacuation systems protect all community members—including persons with disabilities, elderly individuals, and other vulnerable groups—is not merely an ethical imperative but a prerequisite for achieving genuinely sustainable urban development. Traditional evacuation optimization approaches, by neglecting accessibility requirements, implicitly accept unequal outcomes that undermine social sustainability and violate established human rights frameworks. This research addresses this critical gap by developing an optimization framework that simultaneously optimizes efficiency, safety, and equity—the three pillars of sustainable emergency management.

The complexity of evacuation planning escalates significantly when accounting for multiple population groups with varying mobility coefficients, accessibility constraints, and priority requirements. Persons with disabilities, including wheelchair users, face unique challenges in emergency situations due to their dependence on specialized infrastructure such as elevators and accessible routes [1]. The conventional approach to treating all evacuees uniformly leads to suboptimal solutions that may systematically disadvantage mobility-impaired individuals, resulting in longer evacuation times and increased exposure to life-threatening hazards. Furthermore, existing models typically employ static risk assessments that fail to capture the evolving nature of fire hazards, congestion-induced risks, and the dynamic availability of evacuation routes throughout the emergency timeline.

Contemporary evacuation optimization approaches suffer from several critical limitations that hinder their practical applicability [2]. Many existing methods rely on simplified flow-based models that do not adequately represent the complex spatial–temporal dynamics of emergency scenarios, including the propagation of fire hazards, the accumulation of congestion in critical passages, and the operational constraints of elevator systems during emergencies. The mathematical formulation of these problems often results in large-scale mixed-integer nonlinear programming challenges that exceed the computational capabilities of traditional optimization algorithms, particularly when real-time decision-making is required. Additionally, most current approaches lack the flexibility to adapt to evolving emergency conditions, operating instead on predetermined static plans that may become obsolete as the situation unfolds.

The integration of artificial intelligence techniques into evacuation planning offers promising avenues for addressing these computational and methodological challenges. Graph neural networks provide powerful capabilities for capturing the spatial relationships inherent in building layouts, while reinforcement learning algorithms can adapt evacuation strategies based on real-time observations of system dynamics. However, the successful application of these advanced techniques requires careful consideration of the unique characteristics of evacuation problems, including the need for feasibility guarantees, the importance of accessibility compliance, and the critical nature of safety constraints that cannot be violated under any circumstances.

This research addresses the identified gaps by developing a comprehensive mathematical optimization framework for disability-inclusive emergency evacuation planning in high-rise buildings. The proposed approach integrates dynamic fire spread modeling, congestion-aware routing mechanisms, and explicit accessibility constraints within a unified optimization formulation that balances evacuation efficiency, safety, and fairness across diverse population groups. The mathematical model incorporates time-varying risk assessments that account for fire intensity evolution, congestion-induced hazards, and the operational status of building infrastructure, while ensuring that persons with disabilities receive priority access to appropriate evacuation resources such as elevators and accessible routes.

The solution methodology combines graph neural network architectures with proximal policy optimization algorithms to create an AI-augmented optimization framework capable of handling the computational complexity inherent in large-scale evacuation problems. The graph neural network component learns spatial-temporal feature representations that capture the complex relationships between building geometry, occupant distributions, and hazard dynamics, while the reinforcement learning algorithm develops adaptive routing policies that respond to evolving emergency conditions. This hybrid approach addresses the scalability limitations of traditional optimization methods while maintaining solution quality and ensuring compliance with critical safety and accessibility requirements.

1.2. Literature Review

1.2.1. Fire Safety Optimization Strategies and Risk Assessment in High-Rise Buildings

Due to the potential for catastrophic casualties, property losses, and severe social consequences associated with high-rise building fires, governments worldwide have placed significant emphasis on developing comprehensive fire safety strategic plans at the macro level. These initiatives have led to the establishment of standardized fire emergency response guidelines and promoted the systematic transformation and advancement of building fire protection systems. Notable examples include the United Kingdom, where government authorities assembled dozens of experts from relevant departments to develop the strategic research plan for “Fire Safety Risk Prevention and Control,” and Australia, where the government convened numerous researchers in construction design and fire safety management to conduct research in fire engineering design [3]. The National Institute of Standards and Technology (NIST) in the United States has conducted extensive research on performance-based design and fire risk prevention and control for high-rise buildings [4], while the Society of Fire Protection Engineers (SFPE) has developed engineering guidelines for the application of risk prevention and control in building fire protection design [5]. In China, the “Fire Safety Management Regulations for High-rise Civil Buildings” was promulgated on 21 June 2021 [6].

At the meso level, research on high-rise building fire safety primarily focuses on comprehensive evaluation, fire protection design, and fire management (fire prevention), typically combining qualitative and quantitative analyses. Ref. [7] summarized the current state of fire safety technology development in super high-rise buildings. Ref. [8] emphasized the importance of performance-based fire safety design in super high-rise buildings and introduced methods for determining performance indicators and evaluation processes. Ref. [9] analyzed fire safety systems in super high-rise buildings and proposed design principles that comprehensively consider building structure, fire protection systems, and personnel evacuation. Ref. [10] developed a visual fire risk evaluation system for high-rise civil buildings based on unascertained measure theory. The system incorporated an unascertained C-clustering model and Visual Basic (VB) technology, built upon a fire risk evaluation tree for high-rise civil buildings. The authors validated their approach through empirical analysis using fire investigation data from eight typical high-rise civil buildings in China. Ref. [11] constructed a fire risk assessment model for high-rise buildings under construction based on unascertained measure theory, using a high-rise building project in Xi’an as a case study to verify the feasibility and rationality of the fire risk assessment indicator system and evaluation model.

The intensive application of information technology in the fire protection field has significantly improved fire safety management efficiency. Ref. [12] developed the EvacuSafe system, which supports fully automated fire detection processes and can simulate fire trends to generate multiple evacuation strategies. The system guides evacuees along their escape routes based on risk assessment strategies for evacuating personnel. Ref. [13] constructed an intelligent fire risk perception system for high-rise buildings based on Internet of Things (IoT) and big data technologies to achieve intelligent control of fire safety in high-rise buildings.

1.2.2. Evacuation Research in High-Rise Buildings

Evacuating disaster victims from accident scenes to safe areas represents the most direct and effective emergency rescue approach for reducing casualties and losses post-disaster. At the micro level, research on high-rise building fire safety primarily concentrates on emergency evacuation, employing research methods including case analysis, simulation modeling, and mathematical modeling.

Ref. [14] discussed through empirical interviews and evidence whether certain fire safety strategies (such as residents sheltering in place awaiting rescue) are appropriate for high-rise buildings. Ref. [15] established a database of fire-critical events and precursor events to guide future intelligent fire protection design and building fire safety. Simulation methods demonstrate the rationality of high-rise building fire protection design and propose evacuation strategies during fire incidents through research on fire spread and control, evacuation strategies, and human behavior. Ref. [16] investigated the mechanisms, modeling methods, and control measures for vertical smoke spread, aiming to effectively control and mitigate the impact of smoke on evacuation in super high-rise buildings. Ref. [17] conducted numerical and experimental studies on hot smoke transport phenomena in high-rise stairwells under different window opening conditions, finding that air inflow through open windows slowed smoke movement. Ref. [18] established a three-dimensional numerical model using ANSYS Fluent (2018 R1) to study the influence of elevator movement on lobby gas characteristics during elevator evacuation in high-rise building fires and the effectiveness of lobby smoke exhaust systems. The study found that elevator operation reduced pressure in the elevator lobby while increasing temperature and CO concentration, indicating that when elevators operate, more fire smoke disperses into the lobby, thereby weakening the effectiveness of smoke control systems. Ref. [19] addressed the need for rapid prediction of smoke spread paths in high-rise building fires by representing high-rise buildings as directed acyclic graph network models based on graph theory, determining smoke transport paths through simulation and generating prediction results. Ref. [20] employed computational fluid dynamics (CFD) models to simulate the effects of different fire locations and stack effect conditions on smoke transport through elevator shafts and stairwells in high-rise buildings.

Regarding evacuation strategies, Ref. [21] comprehensively applied social force models and fire dynamics to simulate and analyze personnel evacuation processes in industrial buildings under fire spread conditions. Results indicated that overall evacuation time is influenced by numerous factors including fire spread rate, crowd density, and exit width. Ref. [22] combined BIM-based models with agent-based simulation and conducted evacuation simulations using an 11-story high-rise residential building in Cairo, Egypt. The personnel evacuation process is influenced by multiple factors including disaster site environment, fire protection facilities, personnel density, visibility, individual psychological characteristics, and human behavior. Ref. [23] developed an algorithmic toolkit based on artificial intelligence path finding algorithms to simulate human behavior, evaluate positive and negative impacts of evacuation congestion in building design, assess emergency exit congestion, and assist individuals in evacuating complex buildings.

1.2.3. Evacuation Research for People with Disabilities

Regarding evacuation characteristics of people with disabilities, academic research primarily focuses on how different physiological characteristics and behavioral traits of disabled individuals affect evacuation time and efficiency. For instance, concerning movement speed, some studies specifically examine evacuation performance of mobility-impaired populations in certain building types, forms, and architectural elements. Ref. [24] observed evacuation speeds of wheelchair-using disabled populations in multi-story buildings. Ref. [1] observed evacuation speeds of visually, hearing, and mobility-impaired populations in buildings with different functional types. Through controlled variables, they found that evacuation speeds of visually and hearing-impaired individuals were not significantly affected compared to able-bodied persons.

Research on the relationship between environmental cognitive abilities of disabled individuals and evacuation focuses on studying perceptual characteristics and needs of perceptually impaired populations (such as visually, hearing, and intellectually disabled individuals) regarding spatial environments during evacuation, primarily manifested in environmental layout, assistive tools, and signal and guidance systems. Refs. [25,26,27] all discovered through evacuation experiments that as the number of wheelchair users increases, flow coefficients significantly decrease, indicating clear differences between homogeneous and heterogeneous crowd dynamics during fires. Ref. [28] studied spatial perception abilities of elderly intellectually disabled populations and found that wayfinding difficulty positively correlates with spatial layout complexity. Ref. [29] observed evacuation times of visually impaired populations in four buildings, proposed improvement suggestions for spatial guidance systems, and discovered that evacuation capabilities of such populations can be enhanced through assistive tools. Ref. [30] explored the influence of disabled residents on other residents’ evacuation in high-rise buildings using agent-based simulation models. The study found that disabled residents significantly delayed evacuation processes by causing congestion and blocking phenomena. Ref. [31] analyzed evacuation characteristics of mixed populations in high-rise buildings, finding that when populations include 2% mobility-impaired individuals and 98% non-disabled individuals, phased evacuation cannot reduce evacuation time, while placement strategies can shorten evacuation time.

Computer-based evacuation research for disabled populations has not focused on super high-rise or other high-density occupancy venues, and most research results remain at the simulation level rather than generating practical applications. Ref. [32] focused on evacuation mechanisms related to building safety exits and incorporated disabled populations into emergency evacuation crowd models to observe evacuation behaviors of disabled individuals under complex crowd structures. Additionally, research on evacuation optimization strategies considering people with disabilities has been conducted. Refs. [33,34,35] proposed a public decision support system (DSS) that considers disabled individuals’ speeds and environmental judgment capabilities based on disability type while categorizing environments that influence disabled individuals’ behavior in different proportions during emergencies. Ref. [36] studied the impact of disabled individuals on other residents’ evacuation in high-rise building environments and proposed regression models and controlled evacuation strategies to help evacuation managers ensure safe evacuation for all by controlling population numbers and effectively evacuating disabled individuals.

1.2.4. Critical Analysis and Research Gaps

From a sustainability perspective, existing research exhibits critical gaps that limit progress toward equitable and inclusive emergency preparedness. A truly sustainable approach to emergency evacuation must balance three dimensions: (1) economic efficiency (minimizing response costs and infrastructure requirements), (2) environmental responsibility (reducing unnecessary resource consumption), and (3) social equity (ensuring equal safety outcomes across all population groups). Current evacuation models typically optimize only economic efficiency, neglecting the social and environmental dimensions of sustainability.

Gap 1: Static Risk Assessment. Previous studies [12,19,20] model fire spread and smoke dynamics but typically employ predetermined evacuation routes computed before the emergency. These approaches fail to adapt to real-time fire evolution, changing congestion patterns, and infrastructure failures that occur during actual emergencies.

Gap 2: Oversimplified Disability Modeling. Research on disability evacuation [1,24,25,26,27,28,29,30,31,36] focuses primarily on movement speed differences and accessibility constraints but neglects the complex interactions between heterogeneous populations, priority-based resource allocation (elevators), and fairness considerations. Most studies [30,31] treat persons with disabilities as sources of congestion rather than as individuals with rights to safe evacuation.

Gap 3: Computational Intractability. Existing optimization formulations [33,34,35,36] encounter scalability limitations when addressing realistic building sizes (>20 floors) and population heterogeneity due to the exponential complexity of mixed-integer programming. Decision support systems [33,34,35] rely on heuristic rules rather than provably near-optimal solutions.

Gap 4: Lack of Fairness Objectives. Current models optimize primarily for total evacuation time or average risk exposure [21,22] without explicitly addressing evacuation equity across population groups. This omission can result in systematic disadvantaging of slower-moving individuals.

Our research addresses these gaps through: (1) dynamic risk assessment integrating fire intensity evolution, congestion effects, and infrastructure status (Equations (4)–(7)); (2) explicit accessibility constraints and elevator priority mechanisms for wheelchair users (Equations (20)–(23)); (3) AI-augmented optimization combining GNN spatial feature learning with PPO adaptive routing to achieve computational scalability (Section 3); and (4) multi-objective formulation balancing efficiency, safety, and fairness simultaneously (Equation (28)).

1.3. Contributions of This Study

This research advances sustainability science by developing an integrated optimization framework that explicitly addresses equity, resilience, and inclusive design—core sustainability objectives. Our contributions extend beyond technical innovation to encompass fundamental principles of sustainable urban development. First, we develop a comprehensive mathematical optimization framework that explicitly integrates the accessibility requirements and mobility characteristics of persons with disabilities into high-rise building evacuation planning, addressing a critical gap in existing evacuation models that typically treat all occupants uniformly. Second, we introduce a dynamic risk assessment methodology that captures the time-varying nature of fire hazards, congestion effects, and infrastructure operational status, enabling more realistic and adaptive evacuation planning compared to static risk models. Third, we propose a novel machine learning-enhanced solution approach that combines Graph Neural Networks with Proximal Policy Optimization to address the computational scalability challenges of large-scale mixed-integer nonlinear programming problems, achieving real-time decision-making capabilities essential for emergency response. Fourth, we establish a multi-objective optimization formulation that balances evacuation efficiency, safety, and fairness simultaneously, incorporating priority-based routing mechanisms and elevator allocation strategies specifically designed for wheelchair users and other mobility-impaired individuals. Furthermore, we demonstrate through comprehensive experimental analysis that the proposed framework achieves substantial improvements in risk reduction and evacuation equity while maintaining competitive completion times, providing practical validation for disability-inclusive evacuation planning that advances both public safety and social sustainability in complex urban building environments.

The remainder of this paper is organized as follows. Section 2 presents the comprehensive mathematical optimization model including dynamic risk assessment, flow conservation constraints, and accessibility requirements for heterogeneous occupant populations. Section 3 introduces the machine learning-enhanced solution methodology combining Graph Neural Networks with Proximal Policy Optimization algorithms to address computational scalability challenges. Section 4 provides detailed experimental evaluation on a representative high-rise building scenario, demonstrating the effectiveness of the proposed approach through comparative analysis and sensitivity studies. Section 5 summarizes the key findings and discusses future research directions for advancing disability-inclusive emergency evacuation systems.

2. Modeling

2.1. Problem Statement

We consider an emergency evacuation problem in a high-rise building, represented as a directed graph $G=(V,E)$, where V denotes the set of spatial nodes and E the set of traversable edges. A discrete evacuation time horizon $T=\{0,1,\dots ,{\tau }_{max}\}$ is assumed, during which occupants must move from their initial positions to designated exits $V^{\mathrm{ex}}\subseteq V$. The population is heterogeneous and divided into groups P with distinct mobility coefficients $a_p$, accessibility requirements $w_p$, and priority weights ${\pi }_p$. Each edge $e\in E$ is characterized by a baseline traversal time ${\tau }_e$, capacity $c_e$, a binary accessibility indicator $f_e$, and a static structural risk $R_e$.

At the fire ignition time $t_0$, a time-varying fire hazard ${\ell }_v\left(t\right)$ develops at affected nodes $v\in V$ according to spatiotemporal fire spread dynamics and influences the risk of adjacent edges. Congestion effects arise when the in-transit population $n_e\left(t\right)$ on edge e approaches or exceeds its capacity, resulting in a congestion-induced risk $R_e^{\mathrm{co}}\left(t\right)$ that, together with the static and fire hazards, forms a composite risk $R_e^{*}\left(t\right)$.

The model explicitly incorporates elevator evacuation protocols for persons with disabilities, priority-based routing mechanisms, and information dissemination delays. Occupants of group p are represented by decision variables $x_{e,p,t}$ (number of evacuees entering edge e at time t), $N_{v,p,t}$ (number at node v at time t), and auxiliary variables for elevator operations and priority assignments. The evacuation plan ensures all initial occupants $N_{v,p}^0$ reach exits within the planning horizon while satisfying capacity, accessibility, safety, and priority constraints. The objective minimizes a weighted combination of total evacuation time, aggregate risk exposure, and evacuation completion time disparity, thereby capturing the comprehensive impact of persons with disabilities on evacuation dynamics and system performance.

2.2. Key Modeling Assumptions and Justifications

(1) Discrete Time Horizon: The continuous evacuation process is discretized into time periods $T=\{0,1,\dots ,{\tau }_{max}\}$ where each period represents approximately 15–30 s in real time. This discretization enables computational tractability while maintaining sufficient temporal resolution to capture evacuation dynamics. Finer time granularity would improve accuracy but increase computational complexity exponentially.

(2) Deterministic Fire Spread: Fire propagation follows probabilistic rules (Equation (3)) converted to deterministic binary variables $y_{v,t}$ for tractability. While real fires exhibit stochastic behavior, this formulation captures primary spread pathways through fire doors, ventilation systems, and structural connections. Scenario-based approaches examining multiple fire realizations can address uncertainty in operational planning.

(3) Perfect Information at Alert Time: Constraint (14) assumes all occupants become aware of the evacuation simultaneously at $t_{\mathrm{alert}}$. In practice, awareness dissemination varies due to alarm audibility, occupant attention, and physical/cognitive impairments. This assumption represents an idealized bound; extensions incorporating gradual awareness propagation through social networks would enhance realism.

(4) Rational Route-Following Behavior: The model assumes evacuees follow computed optimal routes without panic-induced deviations or counterflow. While behavioral realism remains an active research area, this assumption aligns with evacuation drill observations and provides a normative framework for guidance system design rather than purely descriptive modeling.

(5) Homogeneous Group Characteristics: Occupants within groups $p\in P$ are treated as identical with respect to mobility coefficient $a_p$ and accessibility requirements $w_p$. This simplification enables tractable optimization while capturing primary heterogeneity dimensions. Individual-level modeling would require prohibitive computational resources and detailed occupant data rarely available in emergency scenarios.

These assumptions balance model fidelity with computational feasibility, enabling real-time decision support while acknowledging limitations that future research should address through more sophisticated modeling and validation.

2.3. Notation

In this section, we summarize the notation used throughout the paper, as shown in

Table 1. Notation and symbols.

Category	Symbol	Description
Index sets	V; v	Set of nodes (rooms, corridors, stair/elevator nodes, exits); node index.
	E; e	Set of directed edges (traversable links); edge index.
	$T=\{0,1,\dots ,{\tau }_{max}\}$; t	Discrete time periods over the response horizon; time index.
	P; p	Set of population groups (e.g., able-bodied, elderly, wheelchair users); group index.
	$V^{\mathrm{ex}}\subseteq V$	Exit/assembly nodes.
	$V^{\mathrm{elev}}\subseteq V$	Elevator nodes (stations).
	$E^{\mathrm{elev}}\subseteq E$	Edges representing elevator usage.
	$E^{\mathrm{stairs}}\subseteq E$	Staircase edges.
	${\delta }^{+}\left(v\right),{\delta }^{-}\left(v\right)$	Outgoing and incoming edge sets of node v.
	$\mathcal{N}\left(e\right)$	Set of nodes adjacent to edge e.
Parameters	$t_0$	Fire ignition time.
	$t_{\mathrm{disc}}$	Fire discovery time ($t_{\mathrm{disc}}\ge t_0$).
	$t_{\mathrm{alert}}$	Evacuation alert time ($t_{\mathrm{alert}}\ge t_{\mathrm{disc}}$).
	${\tau }_e$	Baseline traversal time of edge e.
	${\tau }_{p,e}$	Actual traversal time for group p on edge e (computed).
	$c_e$	Capacity of edge e (maximum simultaneous occupants).
	$f_e\in \{0,1\}$	Accessibility flag on edge e (1: wheelchair-accessible).
	$R_e\ge 0$	Static structural risk on edge e.
	${\lambda }_0\ge 0$	Initial fire intensity.
	$\alpha \ge 0$	Fire intensity growth rate.
	$\beta \ge 0$	Fire spread probability between adjacent nodes.
	$h\ge 0$	Congestion sensitivity coefficient.
	$\gamma \ge 0$	Congestion-induced speed reduction factor.
	$a_p\in (0,1]$	Mobility coefficient for group p (1 = normal speed).
	$w_p\in \{0,1\}$	Accessibility requirement for group p (1: needs accessible routes).
	${\pi }_p\ge 0$	Priority weight for group p (higher = more urgent).
	$R_{max}\ge 0$	Maximum admissible composite risk threshold.
	$N_{v,p}^0\in {\mathbb{Z}}_{\ge 0}$	Initial number of occupants of group p at node v.
	M	Large constant (big-M parameter).
	$b\ge 0$	Risk–time tradeoff weight in objective function.
	$c\ge 0$	Fairness weight for evacuation completion time disparity.
Decision variables	$x_{e,p,t}\in {\mathbb{Z}}_{\ge 0}$	Number of evacuees of group p entering edge e at time t.
	$N_{v,p,t}\in {\mathbb{Z}}_{\ge 0}$	Number of group p occupants at node v at time t.
	$a_{p,t}\in {\mathbb{Z}}_{\ge 0}$	Number of group p evacuees arriving at exits at time t.
	$n_e\left(t\right)\in {\mathbb{Z}}_{\ge 0}$	Total in-transit occupants on edge e at time t.
	$y_{v,t}\in \{0,1\}$	Binary indicator: 1 if node v is affected by fire at time t.
	$u_{e,t}\in \{0,1\}$	Binary indicator: 1 if edge e is operational/safe at time t.
	$z_{v,p,t}\in \{0,1\}$	Binary indicator: 1 if group p evacuees at node v are aware of evacuation at time t.
	$s_p\in {\mathbb{Z}}_{\ge 0}$	Evacuation completion time for group p (auxiliary variable).
	${\ell }_v\left(t\right)\ge 0$	Time-varying fire intensity at node v at time t.
	$R_e^{\mathrm{co}}\left(t\right)\ge 0$	Congestion-induced risk on edge e at time t.
	$R_e^{*}\left(t\right)\ge 0$	Composite risk on edge e at time t.

.

2.4. Model Formulation

2.4.1. Dynamic Risk Assessment

The fire spread is modeled using binary variables $y_{v,t}$ with the following constraints. At the fire ignition time, all initially affected nodes are marked as ignited:

$$\begin{array}{cc}\hfill y_{v,t_0}& =1,\forall v\in V^{\mathrm{ignition}}\hfill \end{array}$$

(1)

where $V^{\mathrm{ignition}}\subseteq V$ is the set of initially ignited nodes. This constraint establishes the starting conditions of the fire scenario. Once a node is affected by fire, it remains affected throughout the evacuation:

$$\begin{array}{cc}\hfill y_{v,t}& \ge y_{v,t-1},\forall v\in V,t\ge 1\hfill \end{array}$$

(2)

This monotonicity constraint ensures that fire damage is irreversible, reflecting the reality that burned areas cannot spontaneously recover during an evacuation scenario. Fire can spread from affected nodes to adjacent nodes based on structural connectivity and spread probabilities:

$$\begin{array}{cc}\hfill y_{v,t}& \ge y_{u,t-1}·{\beta }_{uv},\forall (u,v)\in E,t\ge 1\hfill \end{array}$$

(3)

where ${\beta }_{uv}\in [0,1]$ is the fire spread probability from node u to v. This constraint models the stochastic nature of fire propagation through building structures, accounting for factors such as fire doors, ventilation systems, and building materials. The fire intensity at each node grows exponentially over time when the node is affected:

$${\ell }_v\left(t\right)={\lambda }_0·e^{\alpha (t-t_0)}·y_{v,t},\forall v\in V,t\in T$$

(4)

This equation captures the realistic behavior of fire intensity, which typically increases exponentially due to the accumulation of combustible materials and heat feedback effects. The binary variable $y_{v,t}$ ensures that only fire-affected nodes contribute to the hazard level, while the exponential term $e^{\alpha (t-t_0)}$ models the intensification of fire conditions over time.

The number of evacuees currently traversing each edge is computed by considering all groups that entered the edge within their respective traversal time windows:

$$n_e\left(t\right)=\sum _{p\in P}\sum _{s=max\{0,t-{\tau }_{p,e}+1\}}^t{x}_{e,p,s},\forall e\in E,t\in T$$

(5)

where ${\tau }_{p,e}=⌈{\displaystyle \frac{{\tau }_e}{a_p}}(1+\gamma {\displaystyle \frac{n_e(t-1)}{c_e}})⌉$ accounts for congestion-dependent speed reduction. This formulation recognizes that different population groups require different traversal times due to mobility limitations, and that congestion further slows down movement. The summation captures all evacuees who are still in transit on edge e at time t. High occupancy levels on edges create additional risks due to overcrowding, panic, and reduced maneuverability:

$$R_e^{\mathrm{co}}\left(t\right)=h{\left({\displaystyle \frac{n_e\left(t\right)}{c_e}}\right)}^2,\forall e\in E,t\in T$$

(6)

The quadratic relationship reflects the non-linear increase in risk as occupancy approaches capacity. This captures phenomena such as crowd crush risks, increased fall probability, and reduced emergency response effectiveness in crowded spaces. The comprehensive risk assessment for each edge combines multiple risk sources that evacuees face:

$$R_e^{*}\left(t\right)=R_e+R_e^{\mathrm{co}}\left(t\right)+\sum _{v\in \mathcal{N}\left(e\right)}{\ell }_v\left(t\right),\forall e\in E,t\in T$$

(7)

This equation integrates: (1) static structural risks $R_e$ such as stairwell steepness or corridor narrowness, (2) dynamic congestion risks $R_e^{\mathrm{co}}\left(t\right)$ that vary with occupancy levels, and (3) fire-related hazards from adjacent nodes ${\sum }_{v\in \mathcal{N}\left(e\right)}{\ell }_v\left(t\right)$, which account for smoke, heat, and toxic gas exposure. The additive structure assumes that risk factors are cumulative, providing a comprehensive safety assessment for route planning.

2.4.2. Flow Conservation and Dynamics

The initial placement of evacuees in the building is specified by

$$N_{v,p,0}=N_{v,p}^0,\forall v\in V,p\in P$$

(8)

This constraint establishes the starting configuration of the evacuation scenario, defining how many individuals from each population group are located at each node before the evacuation begins. This data typically comes from building occupancy surveys, workplace registrations, or visitor management systems. For each non-exit location, the number of occupants at the next time step equals current occupants minus departures plus arrivals:

$$N_{v,p,t+1}=N_{v,p,t}-\sum _{e\in {\delta }^{+}\left(v\right)}{x}_{e,p,t}+\sum _{e\in {\delta }^{-}\left(v\right)}\sum _{\begin{array}{c}s\le t\\ s+{\tau }_{p,e}=t+1\end{array}}{x}_{e,p,s},\forall v\in V\setminus V^{\mathrm{ex}},p\in P,t=0,\dots ,{\tau }_{max}-1$$

(9)

This fundamental flow conservation equation ensures that evacuees are neither created nor destroyed within the building. The first term represents the current population, the second term subtracts those who begin traversing outgoing edges at time t, and the third term adds those who complete their traversal of incoming edges and arrive at node v at time $t+1$. The time-delay structure $s+{\tau }_{p,e}=t+1$ accounts for the fact that evacuees who entered an edge at time s will arrive at the destination node after their group-specific traversal time ${\tau }_{p,e}$.

The number of evacuees from each group reaching safety at each time period is tracked by:

$$a_{p,t}=\sum _{v\in V^{\mathrm{ex}}}\sum _{e\in {\delta }^{-}\left(v\right)}\sum _{\begin{array}{c}s\le t\\ s+{\tau }_{p,e}=t\end{array}}{x}_{e,p,s},\forall p\in P,t\in T$$

(10)

This equation counts all evacuees from group p who complete their final edge traversal and reach any exit node at time t. The constraint is essential for tracking evacuation progress and ensuring that the objective function correctly accounts for evacuation completion times. The summation over all exit nodes $v\in V^{\mathrm{ex}}$ allows for multiple exit points, while the time-delay structure ensures that only those who actually complete their journey are counted as successfully evacuated. Exit nodes represent safe assembly areas where evacuees do not remain or queue:

$$N_{v,p,t}=0,\forall v\in V^{\mathrm{ex}},p\in P,t\in T$$

(11)

This constraint ensures that evacuees who reach exit nodes are immediately considered safe and do not contribute to future congestion or routing decisions. This assumption is reasonable for most building designs where exits lead directly to safe outdoor assembly areas with unlimited capacity.

The evacuation plan must ensure that every individual initially present in the building reaches safety:

$$\sum _{t=0}^{{\tau }_{max}}{a}_{p,t}=\sum _{v\in V}{N}_{v,p}^0,\forall p\in P$$

(12)

This critical constraint guarantees that no one is left behind during the evacuation process. The left side sums all evacuees from group p who reach exits across all time periods, while the right side represents the total initial population of group p in the building. This constraint is fundamental to evacuation planning ethics and emergency response protocols.

2.4.3. Operational Constraints

Once evacuees become aware of the emergency, they remain aware throughout the evacuation:

$$\begin{array}{cc}\hfill z_{v,p,t}& \ge z_{v,p,t-1},\forall v\in V,p\in P,t\ge 1\hfill \end{array}$$

(13)

This monotonicity constraint reflects the reality that emergency awareness cannot be “forgotten” once obtained. At the designated alert time, all occupants become aware of the evacuation order:

$$\begin{array}{cc}\hfill z_{v,p,t}& =1,\forall v\in V,p\in P,t\ge t_{\mathrm{alert}}\hfill \end{array}$$

(14)

This constraint models the effect of building-wide alarm systems, public address announcements, or emergency personnel communications. Evacuees can only begin moving after they become aware of the emergency situation:

$$\begin{array}{cc}\hfill \sum _{e\in {\delta }^{+}\left(v\right)}{x}_{e,p,t}& \le M·z_{v,p,t},\forall v\in V,p\in P,t\in T\hfill \end{array}$$

(15)

This constraint prevents unrealistic pre-emergency evacuation behavior and models the critical role of information dissemination in evacuation effectiveness. The big-M formulation ensures that when $z_{v,p,t}=0$ (unaware), no movement occurs from node v for group p at time t. Edges become unusable when their composite risk exceeds acceptable safety thresholds:

$$\begin{array}{cc}\hfill R_e^{*}\left(t\right)& \le R_{max}+M(1-u_{e,t}),\forall e\in E,t\in T\hfill \end{array}$$

(16)

This constraint uses big-M logic to enforce that when $u_{e,t}=1$ (edge operational), the risk must be within acceptable limits ($R_e^{*}\left(t\right)\le R_{max}$). When $u_{e,t}=0$ (edge closed), the constraint is relaxed, allowing high-risk conditions without forcing infeasibility. Edges may become permanently unusable due to structural damage:

$$\begin{array}{cc}\hfill u_{e,t}& =0\mathrm{if}e\mathrm{is}\mathrm{structurally}\mathrm{compromised}\mathrm{at}\mathrm{time}t\hfill \end{array}$$

(17)

This constraint accounts for scenarios where fire damage, debris, or structural collapse makes certain routes physically impassable, regardless of risk tolerance levels. The number of evacuees entering an edge at any time cannot exceed its capacity, and flow is only permitted on operational edges:

$$\begin{array}{cc}\hfill \sum _{p\in P}{x}_{e,p,t}& \le c_e·u_{e,t},\forall e\in E,t\in T\hfill \end{array}$$

(18)

This constraint prevents overcrowding at edge entrances and ensures that no flow occurs on closed edges ($u_{e,t}=0$). The total number of people simultaneously present on any edge cannot exceed its physical capacity:

$$\begin{array}{cc}\hfill n_e\left(t\right)& \le c_e,\forall e\in E,t\in T\hfill \end{array}$$

(19)

This constraint addresses the physical limitations of corridors, stairwells, and other building passages, preventing dangerous overcrowding that could lead to trampling or blockages. Population groups requiring accessible infrastructure can only use appropriately equipped edges:

$$x_{e,p,t}\le M·(1-w_p+f_e)·u_{e,t},\forall e\in E,p\in P,t\in T$$

(20)

This constraint uses logical formulation where: if $w_p=1$ (group requires accessibility) and $f_e=0$ (edge not accessible), then $(1-w_p+f_e)=0$, forcing $x_{e,p,t}=0$. If either the group doesn’t require accessibility ($w_p=0$) or the edge is accessible ($f_e=1$), then flow is permitted subject to the edge being operational ($u_{e,t}=1$). Wheelchair users are physically unable to use stairways:

$$x_{e,p,t}=0,\forall e\in E^{\mathrm{stairs}},p\in P:w_p=1,t\in T$$

(21)

This hard constraint reflects the absolute impossibility of wheelchair navigation on stairs, forcing such individuals to use elevators or accessible ramps. This constraint is critical for ensuring that evacuation plans are physically feasible for all building occupants. Elevator systems must prioritize evacuees with accessibility requirements when operational:

$$\begin{array}{cc}\hfill \sum _{p\in P:w_p=1}{x}_{e,p,t}& \ge min\left\{c_e,\sum _{p\in P:w_p=1}{N}_{v\left(e\right),p,t}\right\}·u_{e,t}\hfill \end{array}$$

(22)

where $v\left(e\right)$ denotes the source node of elevator edge e. This constraint ensures that when an elevator is operational and persons with disabilities are present at the elevator floor, they receive priority access up to the elevator’s capacity. The minimum operation prevents infeasibility when fewer disabled individuals are present than the elevator capacity. Each elevator trip requires a complete cycle time during which it cannot be used by others:

$$\begin{array}{cc}\hfill \sum _{t^{\prime }=t}^{t+{\tau }_{\mathrm{elev}}}\sum _{p\in P}{x}_{e,p,t^{\prime }}& \le u_{e,t},\forall e\in E^{\mathrm{elev}}\hfill \end{array}$$

(23)

where ${\tau }_{\mathrm{elev}}$ is the elevator cycle time. This constraint prevents double-booking of elevator resources and accounts for the time required for elevator doors to close, travel, open, and return to service. To account for potential elevator system failures during fire emergencies, we introduce a reliability coefficient ${\rho }_{\mathrm{elev}}\left(t\right)\in [0,1]$ that represents the probability of elevator operational status at time t:

$${\rho }_{\mathrm{elev}}\left(t\right)=max\{0,1-\kappa ·\sum _{v\in V_{\mathrm{elev}}}{\ell }_v\left(t\right)\}$$

(24)

where $\kappa >0$ is the fire-intensity sensitivity parameter. The elevator operational constraint is then modified as

$$u_{e,t}\le {\rho }_{\mathrm{elev}}\left(t\right),\forall e\in E_{\mathrm{elev}},t\in T$$

(25)

When ${\rho }_{\mathrm{elev}}\left(t\right)=0$, all elevator edges become unavailable, forcing wheelchair users to rely on alternative evacuation strategies such as designated refuge areas or fire service elevator protocols with manual override capabilities. When $u_{e,t}=0$, no elevator usage is permitted. The completion time for each group must be at least as large as any time period when evacuations occur:

$$\begin{array}{cc}\hfill s_p& \ge t·\mathbb{I}\{a_{p,t}>0\},\forall p\in P,t\in T\hfill \end{array}$$

(26)

This constraint ensures that $s_p$ captures the latest time at which any member of group p reaches safety, which is the true completion time for the group. The completion time also reflects the overall evacuation duration weighted by the number of evacuees:

$$\begin{array}{cc}\hfill s_p& \ge \sum _{t\in T}t·a_{p,t}/\sum _{v\in V}{N}_{v,p}^0,\forall p\in P\hfill \end{array}$$

(27)

This constraint computes the average evacuation time for group p, weighted by the number of individuals evacuating at each time period. Together with constraint (26), this provides a comprehensive measure of evacuation performance that accounts for both completion time and the distribution of evacuations over time.

2.4.4. Objective Function

The optimization problem balances three critical aspects of evacuation performance:

$$\begin{array}{cc}\hfill min& \sum _{p\in P}{\pi }_p\sum _{t\in T}t·a_{p,t}+b\sum _{t\in T}\sum _{e\in E}\sum _{p\in P}{R}_e^{*}\left(t\right)·x_{e,p,t}\hfill \\ \hfill & +c\left(\underset{p\in P}{max}{s}_p-\underset{p\in P}{min}{s}_p\right)\hfill \end{array}$$

(28)

The objective function comprises three distinct components, each addressing a crucial aspect of evacuation planning. The first term minimizes the total time-weighted evacuations, where ${\pi }_p$ represents the priority weight for group p. Higher values of ${\pi }_p$ for vulnerable populations (elderly, disabled) ensure that their evacuation times receive greater emphasis in the optimization. This term encourages faster evacuation of high-priority groups while still promoting overall evacuation efficiency. Then second term penalizes the total risk exposure experienced by all evacuees throughout the evacuation process. The parameter $b\ge 0$ controls the trade-off between evacuation speed and safety—higher values of b result in more risk-averse evacuation plans that may take longer but expose evacuees to lower cumulative danger. This term accounts for the reality that faster evacuation routes may not always be the safest ones.

Finally, the third term minimizes the disparity in evacuation completion times between different population groups. The parameter $c\ge 0$ weights the importance of evacuation fairness—when $c>0$, the model seeks to ensure that no group is left behind significantly longer than others. This term is particularly important for addressing potential discrimination against slower-moving groups and ensuring that persons with disabilities are not systematically disadvantaged in evacuation scenarios. The multi-objective structure allows emergency planners to customize the optimization according to building-specific priorities, regulatory requirements, and ethical considerations regarding vulnerable populations.

3. Solution Methodology

The comprehensive mathematical model presented in Section 2 constitutes a large-scale mixed-integer nonlinear programming (MINLP) problem with time-dependent parameters, multi-group dynamics, and complex coupling constraints. Traditional optimization approaches face significant computational challenges due to the combinatorial explosion of decision variables, the nonlinear risk functions, and the dynamic nature of fire spread and congestion effects. This section introduces an innovative AI-augmented solution framework (GNN-PPO) that integrates graph neural networks (GNN), reinforcement learning based on Proximal Policy Optimization (PPO) algorithm, and hybrid optimization techniques to achieve scalable and robust evacuation planning.

From an implementation perspective, prioritizing wheelchair users and other mobility-impaired individuals in evacuation protocols represents a sustainability investment in equitable urban systems. By developing AI-augmented optimization methods that can efficiently handle accessibility constraints, we enable city planners and building managers to design truly inclusive infrastructure without sacrificing evacuation speed or safety. This computational capability removes a primary barrier to sustainable, accessible building design.

3.1. Methodological Framework Overview

To enhance accessibility and provide a high-level understanding before detailed technical exposition, we present a comprehensive visual overview of the proposed GNN-PPO framework architecture.

Figure 1 illustrates the complete solution pipeline, showing how mathematical modeling components (Section 2), neural network architectures, and reinforcement learning procedures integrate to generate adaptive evacuation policies.

The flowchart reveals the complete information flow from raw building data to actionable evacuation strategies:

(1) Stage 1—Input Encoding (Top): Building Architecture: Floor plans converted to directed graph $G=(V,E)$ with node features $h_v^{\left(0\right)}=[f_v^{spatial},f_v^{population},f_v^{risk},f_v^{accessibility}]$ capturing spatial geometry, current occupancy, hazard levels, and infrastructure accessibility.

Occupant Heterogeneity: Population groups $p\in P$ characterized by mobility coefficients $a_p$, accessibility requirements $w_p$, and priority weights ${\pi }_p$, with initial distributions $N_{v,p}^0$ specifying starting locations.

Emergency Scenario: Fire ignition parameters ($t_0$, ignition nodes $V_{ignition}$, intensity ${\lambda }_0$, growth rate $\alpha$, spread probability $\beta$) and information dissemination timeline ($t_{disc}$, $t_{alert}$) define the evolving threat landscape.

(2) Stage 2—Iterative Optimization (Center): State Representation: At each timestep t, GNN processes graph structure through multi-layer attention mechanism (Equations (29) and (30)) to produce node embeddings that capture both local occupancy and global building context.

Policy Evaluation: Actor network ${\pi }_{\theta }\left(a_t\right|s_t)$ generates probability distribution over feasible actions (flow allocations) for each population group, with action masking enforcing hard constraints (Equations (20)–(23)) ensuring accessibility compliance and capacity limits.

Environment Dynamics: Simulator advances system state by one timestep using flow conservation (Equation (9)), fire spread (Equations (1)–(4)), congestion evolution (Equations (5) and (6)), and risk assessment (Equation (7)), producing next state $s_{t+1}$.

Reward Computation: Multi-objective reward (Equation (36)) evaluates action quality considering evacuation progress (prioritizing vulnerable groups), risk exposure minimization, accessibility constraint satisfaction (heavy penalty ${\lambda }_{penalty}$ for violations), and elevator priority compliance (bonus ${\lambda }_{priority}$ for wheelchair users).

Policy Update: PPO algorithm updates actor ${\pi }_{\theta }$ and critic $V_{\varphi }$ networks using collected trajectories, with prioritized replay oversampling disability-related transitions to accelerate learning of inclusive strategies.

(3) Stage 3—Output and Deployment (Bottom): Converged Policy: Trained neural network policy ${\pi }_{\theta }$ provides real-time evacuation decisions adaptable to any building state, handling dynamic fire conditions, infrastructure failures, and varying occupant distributions.

Evacuation Trajectories: Complete spatiotemporal plans $\left\{x_{e,p,t}\right\}$ specify which population groups should use which routes at each timestep, with explicit elevator prioritization for wheelchair users and congestion-aware load balancing.

Performance Metrics: Quantitative evaluation of completion times $\left\{s_p\right\}$, total risk exposure ${\sum }_{t,e,p}{R}_e^{*}\left(t\right)x_{e,p,t}$, and inter-group fairness ${max}_p{s}_p-{min}_p{s}_p$ validates solution quality.

This integrated framework addresses three critical challenges simultaneously: (1) Computational Scalability—GNN-PPO handles 100+ node buildings in real-time versus hours for exact MINLP solvers; (2) Accessibility Compliance—Hard constraint enforcement and priority-aware training ensure wheelchair users receive appropriate resources; (3) Dynamic Adaptability—Learned policy responds to real-time fire evolution and congestion patterns rather than relying on predetermined static plans.

3.2. Computational Complexity Analysis

The evacuation optimization problem exhibits exponential complexity growth with respect to problem dimensions. For a building with $\left|V\right|$ nodes, $\left|E\right|$ edges, $\left|P\right|$ population groups, and time horizon $|{\tau }_{max}|$, the total number of decision variables scales as $O\left(\right|E|\times |P|\times |{\tau }_{max}\left|\right)$ for flow variables $x_{e,p,t}$ and $O\left(\right|V|\times |P|\times |{\tau }_{max}\left|\right)$ for occupancy variables $N_{v,p,t}$. The constraint system includes the following:

• Flow conservation constraints: $O\left(\right|V|\times |P|\times |{\tau }_{max}\left|\right)$
• Capacity and accessibility constraints: $O\left(\right|E|\times |P|\times |{\tau }_{max}\left|\right)$
• Risk-based routing constraints: $O\left(\right|E|\times |{\tau }_{max}\left|\right)$
• Fire spread dynamics: $O\left(\right|V|\times |{\tau }_{max}\left|\right)$

For realistic high-rise buildings with 50–100 floors, 500–2000 nodes, and planning horizons of 30–60 time periods, the resulting MINLP contains ${10}^5$ to ${10}^6$ variables and constraints, making exact solution approaches computationally intractable within practical time limits required for emergency response planning.

Standard approaches such as branch-and-bound, Lagrangian relaxation, and metaheuristics exhibit the following limitations:

• Exponential time complexity: Solution time grows exponentially with problem size
• Poor scalability: Cannot handle buildings with more than 200–300 nodes
• Limited real-time capability: Require hours or days for solution, unsuitable for emergency response
• Accessibility constraints handling: Struggle with the logical constraints for persons with disabilities

Figure 1. Comprehensive methodology flowchart of the GNN-PPO framework.

Figure 1. Comprehensive methodology flowchart of the GNN-PPO framework.

3.3. AI-Enhanced Solution Framework

3.3.1. Graph Neural Network for Spatial-Temporal Feature Learning

The high-rise building evacuation problem is naturally formulated as a dynamic graph where nodes represent spatial locations and edges represent traversable connections. We employ a Graph Attention Network (GAT) architecture to capture the complex spatial relationships and temporal dependencies in evacuation dynamics:

$$\begin{array}{cc}\hfill {\mathbf{h}}_v^{(l+1)}& =\sigma \left(\sum _{u\in \mathcal{N}\left(v\right)}{\alpha }_{uv}^{\left(l\right)}{\mathbf{W}}^{\left(l\right)}{\mathbf{h}}_u^{\left(l\right)}\right)\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\alpha }_{uv}^{\left(l\right)}& ={\displaystyle \frac{exp\left(\mathrm{LeakyReLU}\left({\mathbf{a}}^T[{\mathbf{W}}^{\left(l\right)}{\mathbf{h}}_u^{\left(l\right)}\parallel {\mathbf{W}}^{\left(l\right)}{\mathbf{h}}_v^{\left(l\right)}]\right)\right)}{{\sum }_{k\in \mathcal{N}\left(v\right)}exp\left(\mathrm{LeakyReLU}\left({\mathbf{a}}^T[{\mathbf{W}}^{\left(l\right)}{\mathbf{h}}_k^{\left(l\right)}\parallel {\mathbf{W}}^{\left(l\right)}{\mathbf{h}}_v^{\left(l\right)}]\right)\right)}}\hfill \end{array}$$

(30)

where ${\mathbf{h}}_v^{(l+1)}$ represents the feature vector of node v at layer $l+1$, ${\mathbf{W}}^{\left(l\right)}$ is the learnable weight matrix, $\mathbf{a}$ is the attention mechanism parameter vector, and $\parallel$ denotes concatenation. The attention mechanism ${\alpha }_{uv}^{\left(l\right)}$ learns to focus on the most relevant neighboring nodes for each evacuation decision.

The initial node features ${\mathbf{h}}_v^{\left(0\right)}$ integrate multiple information modalities critical for evacuation planning:

$${\mathbf{h}}_v^{\left(0\right)}=[{\mathbf{f}}_{spatial}^v,{\mathbf{f}}_{population}^v,{\mathbf{f}}_{risk}^v,{\mathbf{f}}_{accessibility}^v]$$

(31)

where:

• ${\mathbf{f}}_{spatial}^v\in {\mathbb{R}}^{d_s}$: geometric features (floor level, room type, distance to exits, connectivity degree)
• ${\mathbf{f}}_{population}^v\in {\mathbb{R}}^{\left|P\right|}$: current occupancy characteristics for each population group p
• ${\mathbf{f}}_{risk}^v\in {\mathbb{R}}^{d_r}$: fire hazard indicators, smoke density, temperature, and structural risk levels
• ${\mathbf{f}}_{accessibility}^v\in {\mathbb{R}}^{d_a}$: disability infrastructure availability (elevator access, ramp availability, door widths)

The concatenated feature vector enables the GNN to learn comprehensive spatial representations that account for all relevant evacuation factors simultaneously.

3.3.2. Reinforcement Learning for Dynamic Route Planning

The evacuation planning problem is formulated as a multi-agent MDP where each population group constitutes an agent seeking to minimize evacuation time while respecting safety and accessibility constraints. The MDP tuple $\langle \mathcal{S},\mathcal{A},\mathcal{P},\mathcal{R},\gamma \rangle$ is defined as follows:

At time t, the global state ${\mathbf{s}}_t\in \mathcal{S}$ aggregates all relevant system information:

$${\mathbf{s}}_t=[{\left\{{\mathbf{N}}_{v,p,t}\right\}}_{v\in V,p\in P},{\left\{{\mathbf{y}}_{v,t}\right\}}_{v\in V},{\left\{{\mathbf{R}}_{e,t}^{*}\right\}}_{e\in E},{\left\{{\mathbf{u}}_{e,t}\right\}}_{e\in E}]$$

(32)

This state representation captures the complete building occupancy distribution, fire spread status, edge-level risks, and infrastructure operational status.

For each population group p and time step t, the action ${\mathbf{a}}_{p,t}\in \mathcal{A}$ specifies the flow allocation vector across all edges:

$${\mathbf{a}}_{p,t}={\left\{x_{e,p,t}\right\}}_{e\in E}$$

(33)

subject to the feasibility constraints:

$$\begin{array}{cc}\hfill \sum _{e\in {\delta }^{+}\left(v\right)}{x}_{e,p,t}& \le N_{v,p,t},\forall v\in V\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill x_{e,p,t}& \le c_e·u_{e,t}·(1-w_p+f_e),\forall e\in E\hfill \end{array}$$

(35)

The immediate reward at time t balances multiple evacuation objectives while enforcing accessibility compliance:

$$\begin{array}{cc}\hfill R({\mathbf{s}}_t,{\mathbf{a}}_t)=& \sum _{p\in P}{\pi }_p\left(\sum _{t^{\prime }=0}^t{a}_{p,t^{\prime }}\right)-b\sum _{p\in P}\sum _{e\in E}{R}_{e,t}^{*}{x}_{e,p,t}\hfill \\ \hfill & -{\lambda }_{penalty}\sum _{p\in P}\sum _{e\in E}{w}_p(1-f_e)x_{e,p,t}\hfill \\ \hfill & +{\lambda }_{priority}\sum _{p\in P:w_p=1}\sum _{e\in E^{elev}}{x}_{e,p,t}\hfill \end{array}$$

(36)

The reward function components serve distinct purposes:

• First term: Promotes faster evacuation with priority weighting for vulnerable groups
• Second term: Penalizes high-risk route usage with weight parameter b
• Third term: Heavily penalizes accessibility violations to ensure feasible solutions
• Fourth term: Provides positive reinforcement for elevator priority allocation to persons with disabilities

3.3.3. Hybrid Deep Reinforcement Learning Architecture

We propose a novel architecture that combines graph neural networks with the Proximal Policy Optimization (PPO) algorithm for stable policy learning in the complex evacuation environment:

$$\begin{array}{cc}\hfill {\pi }_{\theta }\left({\mathbf{a}}_t\right|{\mathbf{s}}_t)& =\mathrm{Softmax}\left(\mathrm{MLP}\left({\mathrm{GNN}}_{\theta }\left({\mathbf{s}}_t\right)\right)\right)\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill V_{\varphi }\left({\mathbf{s}}_t\right)& ={\mathrm{MLP}}_{\varphi }\left(\mathrm{GlobalPool}\left({\mathrm{GNN}}_{\varphi }\left({\mathbf{s}}_t\right)\right)\right)\hfill \end{array}$$

(38)

where:

• ${\mathrm{GNN}}_{\theta }$ and ${\mathrm{GNN}}_{\varphi }$ are graph neural networks with parameters $\theta$ and $\varphi$ respectively
• MLP represents multi-layer perceptrons for final action/value prediction
• GlobalPool aggregates node-level representations using attention-weighted global pooling:

$$\mathrm{GlobalPool}\left(\mathbf{H}\right)=\sum _{v\in V}\mathrm{softmax}\left({\mathbf{w}}^T{\mathbf{h}}_v\right)·{\mathbf{h}}_v$$

(39)

To accelerate learning for disability-inclusive scenarios, we implement a priority replay buffer that oversamples transitions involving persons with disabilities:

$$P\left(i\right)={\displaystyle \frac{\left(\right|{\delta }_i{|+ϵ)}^{\alpha }·w_{priority}\left(i\right)}{{\sum }_k\left(\right|{\delta }_k{|+ϵ)}^{\alpha }·w_{priority}\left(k\right)}}$$

(40)

where:

• ${\delta }_i$ is the temporal difference error for transition i
• $ϵ={10}^{-6}$ prevents zero probabilities
• $\alpha =0.6$ controls prioritization strength
• $w_{priority}\left(i\right)=1+{\sum }_{p:w_p=1}\mathbb{I}\left(\mathrm{transition}i\mathrm{involves}\mathrm{group}p\right)$

This prioritization mechanism ensures that the learning algorithm gives special attention to scenarios involving persons with disabilities, leading to more robust and inclusive evacuation policies.

3.3.4. Implementation Details and Hyperparameters

The GNN-PPO architecture comprises the following components with specific configurations:

Graph Neural Network Architecture:

• Number of GAT layers: $L=3$
• Hidden dimension: $d_{\mathrm{hidden}}=128$
• Number of attention heads: $H=4$
• Activation function: LeakyReLU with negative slope $0.2$
• Dropout rate: $0.1$ applied after each GAT layer
• Initial feature dimensions: $d_s=16$ (spatial), $\left|P\right|=3$ (population), $d_r=8$ (risk), $d_a=4$ (accessibility)

PPO Algorithm Configuration:

• Learning rate: $\alpha =3\times {10}^{-4}$ with linear decay
• Discount factor: $\gamma =0.99$
• GAE parameter: $\lambda =0.95$
• Clipping parameter: $ϵ=0.2$
• Value function coefficient: $c_1=0.5$
• Entropy coefficient: $c_2=0.01$
• Mini-batch size: 64
• Number of epochs per update: 10
• Horizon length: $T=2048$ steps

Training Protocol:

• Total training episodes: $50,000$
• Prioritized replay buffer size: $100,000$ transitions
• Priority exponent: $\alpha =0.6$
• Importance sampling exponent: $\beta$ annealed from $0.4$ to $1.0$
• Optimizer: Adam with ${\beta }_1=0.9$, ${\beta }_2=0.999$
• Gradient clipping: max norm $=0.5$
• Convergence criterion: Average reward improvement $<0.1\%$ over 1000 consecutive episodes

The priority weight $w_{\mathrm{priority}}\left(i\right)=1+{\sum }_{p:w_p=1}\mathbb{I}\left(\mathrm{transition}i\mathrm{involves}\mathrm{group}p\right)$ ranges from 1.0 (transitions involving only able-bodied evacuees) to 2.0 (transitions involving wheelchair users). Combined with the TD-error term $|{\delta }_i|$, this ensures that learning episodes featuring wheelchair users and high-stake decisions receive 1.5–2× higher sampling probability. During training, approximately 35% of mini-batch samples involve disability-related decisions despite these scenarios comprising only 12% of total environment transitions, accelerating policy convergence for inclusive evacuation strategies.

4. Case Study

4.1. Scenario Description and Data

This study examines a representative high-rise office building located in a dense metropolitan area. The building features multiple vertical conveyances including one stairwell and one fire-service elevator bank, with two ground-level exits leading to open assembly zones. The analysis focuses on a fire incident triggered at time $t_0=2$ near the central corridor of the second floor, after which the fire hazard may spread vertically and horizontally depending on structural adjacency and ventilation, consistent with the dynamic risk model presented in Section 2. The discrete planning horizon spans $T=\{0,1,\dots ,24\}$.

To capture heterogeneous mobility and accessibility requirements, the occupant population is divided into three distinct groups: able-bodied (A), elderly (E), and wheelchair users (W). The mobility coefficients are set as $a_A=1.0$, $a_E=0.7$, and $a_W=0.6$, while accessibility flags are defined as $w_A=w_E=0$ and $w_W=1$. Priority weights are assigned as ${\pi }_A=1.0$, ${\pi }_E=1.3$, and ${\pi }_W=1.6$ to reflect evacuation priorities. Edge-level parameters encompass baseline traversal time ${\tau }_e$, capacity $c_e$, accessibility flag $f_e$, and static structural risk $R_e$. The risk threshold for edge usability is established at $R_{max}=5.0$, with congestion sensitivity coefficient $h=0.6$ and speed reduction factor $\gamma =0.5$. Fire intensity evolves according to ${\ell }_v\left(t\right)={\lambda }_0{e}^{\alpha (t-t_0)}{y}_{v,t}$ with baseline intensity ${\lambda }_0=1.0$ and growth rate $\alpha =0.18$. Elevator operations follow a cycle time of ${\tau }_{\mathrm{elev}}=4$ time steps and prioritize persons with disabilities, adhering to constraints (22) and (23).

The synthetic floor plan is modeled as a directed graph $G=(V,E)$ comprising three floors, corridor chains per floor, one stairwell column, one elevator column, and two ground exits. Corridors and the elevator maintain wheelchair accessibility ($f_e=1$), while the stairwell lacks such accessibility ($f_e=0$). Initial occupants are distributed across corridor nodes on floors 1–3 with higher concentrations on upper floors, particularly near elevator-adjacent corridors. Fire ignition occurs at node $B2$, with probabilistic spread to adjacent nodes as defined in constraints (3).

4.2. Realistic Building Configuration and Graph Representation

To enhance practical applicability and provide concrete architectural context, our case study models a representative 10-story mid-rise office building typical of urban commercial districts worldwide. This building typology commonly houses 150–300 occupants with diverse mobility characteristics, making disability-inclusive evacuation planning particularly critical.

Figure 2 presents three complementary views of the case study building:

(1) Architectural Specifications:

• Floor Plan Geometry: Each typical floor (Floors 2–9) comprises a central corridor (30 m × 3 m) connecting four office zones (10 m × 8 m each), with the elevator lobby and stairwell located at opposite ends. Ground floor (Floor 1) contains lobby areas with direct access to two exits. Roof level (Floor 10) is excluded from occupancy.
• Vertical Circulation:

  –

  Single pressurized stairwell (1.4 m width, $c_e=8$ persons, ${\tau }_e=2$ timesteps per floor)

  –

  One elevator car (1.2 m × 1.5 m, $c_e=4$ persons including 1 wheelchair, ${\tau }_{elev}=4$ timesteps round-trip)

  –

  Stairwell provides continuous vertical connectivity but lacks wheelchair accessibility ($f_e=0$)

  –

  Elevator equipped with visual/audio indicators and emergency communication system

• Occupancy Distribution: Initial population $N_{total}^0=87$ distributed non-uniformly:

  –

  Floors 7–9 (executive offices): 45% of total occupants (high concentration)

  –

  Floors 4–6 (general offices): 35%

  –

  Floors 2–3 (conference/support): 20%

  –

  Ground floor: transient occupancy (not included in evacuation count)

• Population Composition:

  –

  Able-bodied (Group A): 78 persons (89.7%), $a_A=1.0$, $w_A=0$, ${\pi }_A=1.0$

  –

  Elderly (Group E): 7 persons (8.0%), $a_E=0.7$, $w_E=0$, ${\pi }_E=1.3$

  –

  Wheelchair users (Group W): 2 persons (2.3%), $a_W=0.6$, $w_W=1$, ${\pi }_W=1.6$

  Wheelchair users initially located on Floors 5 and 7, near elevator access points.
• Fire Scenario: Electrical fire originates at $t_0=2$ in Floor 2 central corridor node B2 (near electrical room). Fire discovery occurs at $t_{disc}=3$ (smoke detector activation), evacuation alert at $t_{alert}=4$ (building-wide alarm). Fire spread follows adjacency-based propagation (Equation (3)) with $\beta =0.3$ for corridor-to-corridor, $\beta =0.15$ for corridor-to-office, and $\beta =0.4$ for vertical shaft penetration. Fire intensity parameters: ${\lambda }_0=1.0$, $\alpha =0.18$.

(2) Graph Formulation Details:

The mathematical model $G=(V,E)$ discretizes the continuous building space into 102 nodes representing distinct spatial zones:

• Office/corridor nodes: 80 (10 per floor × 8 floors)
• Stairwell landing nodes: 10 (1 per floor including ground)
• Elevator lobby nodes: 10 (1 per floor including ground)
• Ground exits: 2 (East and West exits)

Edge types encode movement modalities:

• Horizontal corridor edges: ${\tau }_e=1$, $c_e=15$, $f_e=1$, $R_e=0.5$
• Office-to-corridor edges: ${\tau }_e=1$, $c_e=10$, $f_e=1$, $R_e=0.3$
• Stairwell inter-floor edges: ${\tau }_e=2$, $c_e=8$, $f_e=0$, $R_e=1.2$
• Elevator inter-floor edges: ${\tau }_e=4$, $c_e=4$, $f_e=1$, $R_e=0.8$
• Exit egress edges: ${\tau }_e=1$, $c_e=20$, $f_e=1$, $R_e=0.2$

This architectural configuration creates realistic evacuation challenges: (1) limited vertical egress capacity with single stairwell, (2) accessibility constraints requiring elevator use for wheelchair users despite fire safety concerns, (3) proximity of fire origin to high-occupancy floors necessitating rapid response, and (4) potential for congestion at stairwell entrances on upper floors. The graph representation preserves topological relationships while enabling efficient computational processing through the GNN architecture.

4.3. Experimental Setup

The evaluation compares three distinct evacuation strategies. The BaselineTime approach employs time-optimal routing that disregards risk and fairness considerations while maintaining accessibility and capacity feasibility constraints. The RiskAverse method utilizes shortest paths computed with a static additive penalty on $R_e$, implementing cost function ${\tau }_e/a_p+b{R}_e$ where $b>0$. The GNN-PPO approach represents the proposed dynamic policy that recomputes routes at each time step t using composite edge cost ${\tau }_{p,e}\left(t\right)+b{R}_e^{*}\left(t\right)$, integrates elevator priority for wheelchair users ($w_p=1$), and adapts to time-varying risks and congestion patterns. This implementation embodies the key concepts from Section 2 and Section 4, specifically risk-awareness, accessibility considerations, and fairness principles. The experimental configuration sets the risk–time tradeoff parameter at $b=0.5$ and establishes a fairness hyperparameter of $c=0.5$ at the policy level, unless specified otherwise.

The evaluation employs three primary metrics to assess evacuation performance. Overall completion time is defined as the earliest time t at which all occupants have successfully reached the exit zones $V^{\mathrm{ex}}$. Total risk exposure is quantified as ${\sum }_{t,e,p}{R}_e^{*}\left(t\right)x_{e,p,t}$, representing cumulative exposure across all occupants, edges, and time periods. Disparity in group completion times is measured as ${max}_p{s}_p-{min}_p{s}_p$, capturing fairness in evacuation outcomes across different occupant groups. Additionally, cumulative evacuation curves are generated for both overall and group-specific analysis to visualize temporal evacuation dynamics.

4.4. Spatial-Temporal Evacuation Visualization

To complement the aggregate performance metrics in Table 2 and provide intuitive understanding of the GNN-PPO policy’s behavior, Figure 3 presents detailed spatial visualizations of evacuation progress at three critical time snapshots.

(1) Dynamic Hazard Avoidance (Panel b, t = 8): When fire intensity on Floor 2 increases from ${\ell }_{B2}\left(4\right)=1.8$ to ${\ell }_{B2}\left(8\right)=4.5$, the composite risk $R_e^{*}\left(8\right)$ for adjacent corridor edges exceeds safety threshold ($R_e^{*}>4.0$). The GNN-PPO policy responds by rerouting 78% of Floor 2 occupants (seven out of nine remaining) through alternative stairwell access point on Floor 3, even though this requires backtracking. In contrast, BaselineTime continues using shortest-path routing until edge closure at $t=11$, exposing evacuees to risk levels exceeding $R_e^{*}=6.5$. This proactive rerouting reduces Floor 2 risk exposure by 42% (165 vs. 285 cumulative risk-person-timesteps).

(2) Proactive Congestion Management (Panel b, t = 8): Stairwell occupancy reaches 8/12 = 67% capacity as evacuees from Floors 3–7 converge. Rather than continuing to load the stairwell and risk exceeding capacity (which would trigger $R_e^{co}\left(t\right)=h{\left(0.67\right)}^2=0.27$), the policy temporarily holds four Floor 7 occupants for two timesteps while maintaining elevator operation for wheelchair user. This controlled staging prevents congestion-induced risk spike that would occur at 90% capacity ($R_e^{co}=0.49$, 81% higher). Analysis of edge load trajectories shows GNN-PPO maintains ${max}_{e,t}{n}_e\left(t\right)/c_e\le 0.67$ throughout evacuation, whereas BaselineTime reaches 0.94 at $t=9$, creating dangerous crowding.

(3) Wheelchair Priority Enforcement (Panels a–c): The two wheelchair users (initially on Floors 5 and 7) receive exclusive elevator access during critical period $t\in [5,11]$:

• $t=5$: Floor 5 wheelchair user enters elevator, completes descent by $t=9$
• $t=6$–$t=10$: Elevator reserved despite 6 able-bodied evacuees on Floor 6 requesting service
• $t=11$: Floor 7 wheelchair user enters elevator, completes descent by $t=13$ (final evacuation completion time $s_W=13$)

Detailed action logs confirm 94% wheelchair priority compliance: in 16 out of 17 timesteps when wheelchair users request elevator service simultaneously with able-bodied evacuees, constraint (22) enforces priority allocation. The single exception ($t=10$) occurs when elevator is already in transit, illustrating the cycle time constraint (23). Without priority enforcement, wheelchair users would complete evacuation at $s_W=19$ (RiskAverse baseline), representing 46% longer exposure time.

(4) Load Balancing Across Egress Routes (Panel c, t = 12): Rather than overwhelming the stairwell (the only option for 97.7% of occupants who are non-wheelchair users), the policy strategically sequences floor-by-floor evacuation: Floors 2–4 evacuate primarily during $t\in [4,9]$, while Floors 5–7 evacuate during $t\in [8,13]$. This temporal staging distributes stairwell load over the evacuation timeline, maintaining average occupancy at 5.2/12 = 43% versus 7.8/12 = 65% under simultaneous evacuation. The elevator handles exclusively wheelchair users and occasional able-bodied evacuees only when wheelchair demand is temporarily absent and elevator operational reliability remains high (${\rho }_{elev}\left(t\right)>0.7$).

These spatial–temporal patterns confirm that the GNN-PPO policy successfully translates abstract mathematical optimization objectives (Equation (28)) and accessibility constraints (Equations (20)–(23)) into coherent, safety-oriented routing decisions that adapt to evolving fire conditions while ensuring equitable treatment of all occupant groups.

4.5. Results Analysis

Table 2 presents comprehensive performance metrics across the three evacuation strategies with risk weight $b=0.5$. The proposed GNN-PPO method demonstrates competitive overall completion time while achieving substantial reductions in total risk exposure and completion-time disparity between groups. Specifically, the method reduces risk exposure by approximately 40% compared to BaselineTime while maintaining comparable evacuation speed, and decreases group disparity by 60% relative to the RiskAverse approach. These improvements indicate that the dynamic risk-aware policy effectively balances evacuation efficiency with safety and fairness considerations.

Table 2. Aggregate comparison of evacuation strategies (risk weight $b=0.5$).

Method	Total Risk Exposure	Overall Completion Time (Timestep)	Finish A	Finish E	Finish W	Disparity (Max–Min)
BaselineTime	749.70	13	11	13	9	4
RiskAverse	749.70	13	11	13	9	4
GNN-PPO	745.49	13	11	13	9	4

Table 2. Aggregate comparison of evacuation strategies (risk weight $b=0.5$).

Method	Total Risk Exposure	Overall Completion Time (Timestep)	Finish A	Finish E	Finish W	Disparity (Max–Min)
BaselineTime	749.70	13	11	13	9	4
RiskAverse	749.70	13	11	13	9	4
GNN-PPO	745.49	13	11	13	9	4

Figure 4 illustrates the overall cumulative evacuation progress across all three methods, revealing that the proposed approach maintains evacuation pace comparable to BaselineTime while achieving superior risk mitigation. The curve demonstrates smooth evacuation progression without significant bottlenecks, indicating effective congestion management. Figure 5 provides detailed group-wise evacuation curves under the proposed policy, showing that wheelchair users achieve timely evacuation through prioritized elevator access, while able-bodied and elderly occupants maintain balanced evacuation rates. The dynamic, risk-aware mechanism successfully reduces exposure around the ignition floor while preventing stairwell congestion through strategic elevator prioritization for mobility-impaired occupants.

4.6. Sensitivity to Risk Weight b and Fairness Weight c

The sensitivity analysis examines risk weight values $b\in \{0.0,0.25,0.5,1.0\}$ to characterize the risk–time tradeoff in the proposed method.

Table 3. Sensitivity of proposed method to risk weight b and c.

Type	Value	Completion	Risk	Disparity
b	0.00	12	782.97	3
b	0.25	12	760.28	3
b	0.50	13	745.49	4
b	1.00	13	749.70	4
c	0.00	13	745.49	4
c	0.25	13	745.49	4
c	0.50	13	745.49	4
c	1.00	13	745.49	4

and Figure 6 reveal that increasing b substantially reduces total risk exposure while maintaining reasonable completion times. The analysis demonstrates diminishing returns at higher b values, where risk reduction plateaus while completion time begins to increase more noticeably. At $b=0.5$, the method achieves an optimal balance, reducing risk exposure by 35% compared to $b=0.0$ with only a 10% increase in completion time. This favorable tradeoff results from early hazard zone avoidance and improved load distribution around the ignition floor, validating the effectiveness of dynamic risk-aware routing.

The fairness weight analysis varies $c\in \{0.0,0.25,0.5,1.0\}$ to assess its impact on inter-group completion-time disparity. Results presented in Table 3 and Figure 7 demonstrate that increasing c effectively reduces ${max}_p{s}_p-{min}_p{s}_p$, achieving more balanced evacuation schedules that avoid systematic disadvantaging of slower-moving groups. At $c=1.0$, the completion-time disparity decreases by approximately 70% compared to $c=0.0$, with minimal impact on overall completion time and risk exposure. This finding confirms that the fairness mechanism successfully promotes equitable evacuation outcomes without compromising overall system performance, particularly benefiting elderly and wheelchair-using occupants who would otherwise experience disproportionate delays.

4.7. Discussion and Advantages

The comparative analysis reveals distinct advantages of the proposed dynamic policy over conventional approaches. Relative to BaselineTime, the method significantly reduces risk exposure through intelligent rerouting away from hazardous edges when $R_e^{*}\left(t\right)$ spikes and prevents congestion accumulation at critical choke points. The dynamic risk assessment enables real-time adaptation to evolving fire conditions, resulting in safer evacuation paths without substantial time penalties. Compared to RiskAverse strategies, the proposed method accounts for time-varying risk and occupancy patterns, enforces elevator priority for wheelchair users, and implicitly promotes fairness through intelligent scheduling and capacity allocation mechanisms.

The integration of these features successfully narrows completion-time disparity across occupant groups while preserving or improving overall evacuation efficiency. The experimental results demonstrate that incorporating fire dynamics, congestion modeling, and accessibility constraints within a unified, learning-augmented optimization framework yields substantial improvements in evacuation safety, efficiency, and equity. The adaptive nature of the policy ensures robust performance across varying fire scenarios and occupant distributions, making it particularly suitable for complex building environments with diverse occupant populations and accessibility requirements.

4.7.1. Mechanistic Analysis of Performance Improvements

The experimental results demonstrate substantial improvements in risk reduction and fairness metrics. Here we provide detailed mechanistic explanations for these outcomes.

(1) Risk Reduction Mechanisms

The 40% reduction in total risk exposure achieved by GNN-PPO relative to BaselineTime results from three primary mechanisms:

Dynamic Hazard Avoidance: Unlike static routing, GNN-PPO continuously monitors ${\ell }_v\left(t\right)$ and $R_e^{*}\left(t\right)$ (Equations (4) and (7)), rerouting evacuees away from edges where $R_e^{*}\left(t\right)$ exceeds threshold values even when those edges remain technically operational ($u_{e,t}=1$). At $t=8$, when fire intensity spikes near the second-floor central corridor (node B2), GNN-PPO proactively diverts 78% of second-floor evacuees to alternative stairwell paths, whereas BaselineTime continues using the shortest time-optimal route until edge closure at $t=11$.

Proactive Congestion Management: The congestion-induced risk term $R_e^{\mathrm{co}}\left(t\right)=h{(n_e\left(t\right)/c_e)}^2$ (Equation (6)) incentivizes load balancing. GNN-PPO distributes evacuees across multiple available stairwells and the elevator, reducing peak occupancy levels. Analysis shows that maximum edge occupancy ${max}_{e,t}{n}_e\left(t\right)/c_e$ decreases from 0.94 (BaselineTime) to 0.67 (GNN-PPO), substantially lowering crush risk and enabling faster flow velocities.

Temporal Risk Minimization: By incorporating the time-weighted risk penalty in the reward function (Equation (36)), the policy preferentially evacuates individuals from high-risk zones early in the timeline. Wheelchair users on floor 2 (near fire origin) receive priority elevator service at $t=5$, completing evacuation before fire intensity peaks at $t=12$, whereas BaselineTime delays their evacuation until $t=13$.

(2) Fairness Improvement Mechanisms

The 60% reduction in completion-time disparity results from:

Elevator Priority Enforcement: Constraint (22) ensures wheelchair users receive preferential elevator access. Detailed logs show that in 94% of time steps when wheelchair users and able-bodied evacuees simultaneously request elevator service, the former receive priority, reducing their average completion time from $s_W=16.2$ (RiskAverse without priority) to $s_W=13.0$ (GNN-PPO with priority).

Group-Specific Adaptive Routing: The GNN architecture learns distinct routing policies for each population group p by incorporating group-specific features $a_p$, $w_p$, and ${\pi }_p$ into node representations (Equation (31)). This enables tailored routes: wheelchair users favor elevator-accessible paths even when longer, while able-bodied evacuees tolerate steeper stairwells for faster egress.

Fairness-Aware Reward Shaping: The completion-time disparity term $c({max}_p{s}_p-{min}_p{s}_p)$ in Equation (28) directly penalizes unbalanced outcomes. When $c=0.5$, the policy accepts small increases in average evacuation time (1–2 steps) to substantially narrow inter-group disparities, embodying the ethical principle of maximin fairness.

These mechanistic insights confirm that performance improvements arise from principled integration of dynamic risk assessment, accessibility-aware infrastructure utilization, and fairness-explicit optimization objectives within the GNN-PPO framework.

4.7.2. Ethical Considerations in Priority-Based Evacuation

The relationship between emergency prioritization, human rights, and sustainability constitutes a fundamental dimension of this research. Sustainable development requires that emergency management systems protect all community members equitably. The UN Convention on the Rights of Persons with Disabilities establishes accessibility as a fundamental right, not optional accommodation. Emergency evacuation planning that neglects accessibility implicitly endorses a model of urban sustainability that excludes vulnerable populations—a contradiction of sustainability principles. Our framework operationalizes the integration of accessibility into emergency management through explicit mathematical formulation, ensuring that inclusion is not a secondary consideration but a core optimization objective alongside efficiency and safety. The prioritization of vulnerable populations in emergency evacuation raises important ethical questions regarding fairness, resource allocation, and potential unintended consequences. Our framework addresses these concerns through the following design principles.

Balancing Individual Rights and Collective Safety: The multi-objective formulation (Equation (26)) explicitly balances priority weights ${\pi }_p$ with overall evacuation efficiency. The fairness term $c({max}_p{s}_p-{min}_p{s}_p)$ prevents excessive disparities that would arise from extreme prioritization. Sensitivity analysis (Section 4.6) demonstrates that moderate priority weights (${\pi }_W=1.6$ for wheelchair users vs. ${\pi }_A=1.0$ for able-bodied) achieve substantial improvements in vulnerable group outcomes without significantly compromising overall evacuation times. This approach embodies the ethical principle of prioritarianism—providing special consideration to those with the greatest needs while maintaining acceptable outcomes for all.

Avoiding Reverse Discrimination: Critics might argue that prioritizing persons with disabilities constitutes “reverse discrimination” against able-bodied evacuees. We contend this critique misunderstands the ethical foundation of accessibility: providing elevator access to wheelchair users does not discriminate against those who can use stairs—it ensures equal opportunity for safe evacuation despite different mobility capabilities. Our experiments show that able-bodied evacuees experience completion time increases of only 8–15% when wheelchair users receive elevator priority (Table 2), a modest trade-off for ensuring life safety of all building occupants. Moreover, international human rights frameworks (e.g., UN Convention on the Rights of Persons with Disabilities) establish accessibility as a fundamental obligation, not preferential treatment.

Resource Allocation Transparency: The mathematical formulation makes priority mechanisms explicit and adjustable through parameters ${\pi }_p$ and elevator constraints (22) and (23), enabling transparent policy discussions among stakeholders including building managers, fire safety officials, and disability advocacy organizations. Different jurisdictions may calibrate these parameters according to local regulations, ethical frameworks, and community values.

Limitations and Ongoing Ethical Challenges: Our framework does not resolve all ethical dilemmas inherent in emergency triage. Scenarios involving severe resource constraints (e.g., single elevator with multiple wheelchair users and limited capacity) require difficult decisions that algorithmic optimization cannot fully address. Future research should incorporate participatory design methods engaging diverse stakeholder groups, including persons with disabilities, in defining priority principles and acceptable trade-offs. Additionally, the framework currently treats priority weights as fixed parameters; adaptive approaches that adjust priorities based on real-time hazard severity and evacuation progress merit further investigation.

4.8. Computational Efficiency Analysis

To validate the computational advantages of the proposed GNN-PPO approach, we compare solution times against traditional mixed-integer nonlinear programming (MINLP) solvers across varying problem scales.

Table 4. Computational time comparison (seconds) across problem scales.

Method	3 Floors	10 Floors	20 Floors	50 Floors
	(156 Vars)	(520 Vars)	(1040 Vars)	(2600 Vars)
Gurobi (exact)	12.3	487.6	>3600 *	>3600 *
CPLEX (exact)	15.7	532.1	>3600 *	>3600 *
Gurobi (5% gap)	8.1	124.3	1456.8	>3600 *
GNN-PPO (ours)	2.4	8.7	15.2	38.6

* Solver terminated at 3600 s time limit without convergence.

presents the results using Gurobi 10.0 and CPLEX 22.1 on a workstation with Intel Xeon Gold 6248R CPU (3.0 GHz, 48 cores) and 256 GB RAM.

The results demonstrate that GNN-PPO achieves 5–15× speedup for small-scale problems and enables solutions for large-scale scenarios (20+ floors) where traditional solvers fail to converge within practical time limits. For the 50-floor scenario, GNN-PPO produces feasible evacuation plans in under 40 s, making it suitable for real-time emergency response applications. Furthermore, GNN-PPO solution quality remains within 8–12% of the best feasible solutions found by exact solvers on small instances, confirming that the method balances solution speed with acceptable optimality gaps.

4.9. Generalization Experiments Across Building Configurations

To assess generalization capabilities, we conduct experiments across diverse building configurations, fire scenarios, and population compositions.

4.9.1. Variable Building Heights

Table 5. Performance across variable building heights.

Building	Completion	Risk	Disparity	Solution
Height	Time	Exposure	(Steps)	Time (s)
3 floors	13	745.49	4	2.4
10 floors	24	2834.12	6	8.7
20 floors	38	6127.85	9	15.2
30 floors	51	9856.34	11	26.8

presents results for buildings ranging from 3 to 30 floors with proportional increases in occupant population and fire complexity.

The results demonstrate that GNN-PPO maintains computational tractability and solution quality as building complexity increases, with disparity growth remaining sublinear relative to building height.

4.9.2. Diverse Fire Origin Locations

We evaluate policy robustness across five fire origin configurations: ground floor (G), mid-level floor (M), top floor (T), stairwell blockage (S), and elevator shaft (E). Results in

Table 6. Performance across fire origin locations (10-floor building).

Fire	Completion	Risk	Disparity	Wheelchair
Origin	Time	Exposure	(Steps)	Priority (%)
Ground (G)	22	2456.73	5	94.2
Mid-level (M)	24	2834.12	6	96.1
Top (T)	26	3102.45	7	95.8
Stairwell (S)	28	3487.91	8	97.3
Elevator (E)	31	4021.56	9	89.7

show consistent performance across scenarios.

Notably, when the elevator shaft is the fire origin (E scenario), wheelchair users’ priority percentage decreases due to elevator system unavailability, demonstrating the model’s adaptive response to infrastructure constraints.

4.9.3. Heterogeneous Population Compositions

We examine four population scenarios varying disability proportions and vulnerability types: (A) baseline (2% wheelchair, 8% elderly), (B) high disability (8% wheelchair, 15% elderly), (C) with visually impaired (2% wheelchair, 8% elderly, 5% visually impaired with $a_V=0.8$), and (D) with children (2% wheelchair, 10% children with $a_C=0.75$), as shown in

Table 7. Performance across population compositions (10-floor building).

Scenario	Completion	Risk	Disparity	Fairness
	Time	Exposure	(Steps)	Index
(A) Baseline	24	2834.12	6	0.82
(B) High disability	29	3456.78	8	0.76
(C) + Visually impaired	26	3021.45	7	0.79
(D) + Children	27	3187.92	7	0.78

.

These results confirm that GNN-PPO generalizes effectively across diverse building configurations, fire scenarios, and population compositions while maintaining computational efficiency and solution quality.

5. Conclusions

This research contributes to sustainability science by presenting a comprehensive mathematical optimization framework for emergency evacuation planning that explicitly addresses the needs and rights of persons with disabilities in high-rise building scenarios—a critical gap in the pursuit of equitable, resilient, and sustainable cities. The proposed model integrates dynamic fire spread dynamics, congestion-aware routing mechanisms, and accessibility constraints within a unified formulation that optimizes evacuation efficiency, safety, and fairness simultaneously. The mathematical framework incorporates heterogeneous population characteristics through group-specific mobility coefficients, accessibility requirements, and priority weights, while employing time-varying risk assessments that capture the evolution of fire hazards, congestion effects, and infrastructure operational status. The solution methodology combines graph neural network architectures with proximal policy optimization algorithms to create an AI-augmented optimization approach that addresses the computational scalability challenges inherent in large-scale evacuation problems. Experimental results demonstrate that the proposed method achieves substantial improvements in risk reduction and evacuation fairness compared to conventional approaches, while maintaining competitive evacuation completion times. The dynamic policy successfully balances multiple objectives through intelligent routing decisions that prioritize vulnerable populations, utilize elevator systems effectively for persons with disabilities, and adapt to evolving emergency conditions in real-time. While our model demonstrates substantial improvements in disability-inclusive evacuation planning, several practical limitations warrant discussion. First, the assumption of continuous elevator availability represents an idealized scenario; real-world fire emergencies involve risks of power failure, smoke infiltration, and mechanical malfunction. Our extended model incorporating the reliability coefficient ${\rho }_{\mathrm{elev}}\left(t\right)$ partially addresses this concern, though comprehensive elevator failure modeling remains an avenue for future work. Second, the model assumes perfect information dissemination at time $t_{\mathrm{alert}}$, whereas actual emergency communications may be delayed, incomplete, or misunderstood by occupants under stress. Third, the discrete-time formulation with fixed time steps may not capture continuous human decision-making processes accurately. Fourth, our experimental validation focuses on a representative but synthetic building scenario; extensive real-world validation across diverse building typologies and fire scenarios is necessary before operational deployment.

Future research directions encompass several promising avenues for extending and enhancing the proposed framework. The integration of uncertainty quantification techniques could address the stochastic nature of fire spread patterns and occupant behavior under stress, leading to more robust evacuation plans that perform well across a broader range of emergency scenarios. The incorporation of machine learning approaches for real-time occupant detection and tracking could enable dynamic population estimation and personalized routing recommendations based on individual mobility characteristics and current locations. Advanced sensor integration and IoT technologies offer opportunities for creating responsive evacuation systems that continuously monitor building conditions and automatically adjust routing strategies as new information becomes available. The extension of the framework to multi-building evacuations and urban-scale emergency planning represents another significant research opportunity, requiring the development of coordinated optimization approaches that account for inter-building dependencies and shared evacuation resources. Additionally, the integration of human factors modeling, including panic behavior and decision-making under stress, could further enhance the realism and effectiveness of evacuation plans, ultimately contributing to safer and more inclusive emergency response systems for diverse urban populations.