TRANSPORTMETRICA A

Optimal Design of Rapid Evacuation Strategies in

Constrained Urban Transport Networks

Corresponding Author:

Jose Javier Escribano-Macias

PhD Student in Sustainable Civil Engineering

Department of Civil and Environmental Engineering

Imperial College London

SW7 2BU, UK

jose.escribano-macias11@imperial.ac.uk

Panagiotis Angeloudis

Senior Lecturer in Transport Systems and Logistics

Centre for Transport Studies

Department of Civil and Environmental Engineering

Imperial College London

SW7 2BU, UK

p.angeloudis@imperial.ac.uk

Ke Han

Lecturer in Transport Operations and Logistics

Centre for Transport Studies

Department of Civil and Environmental Engineering

Imperial College London

SW7 2BU, UK

k.han@imperial.ac.uk

Word Count: 11,541

Optimal Design of Rapid Evacuation Strategies in Constrained Urban Transportation Networks

Abstract

Large-scale evacuations constitute common life-saving exercises that are activated in many

disaster response campaigns. Their effectiveness is often inhibited by traffic congestion,

disrupted and imperfect coordination mechanisms, and the poor state of the underlying

transportation networks. To address this problem, this paper presents a hybrid simulation-

optimisation methodology to optimise evacuation response strategies through demand staging

and signal phasing. We introduce a pre-planning model that evaluates evacuation policies,

using a low-level dynamic traffic assignment model that captures the effects of congestion,

queuing and vehicle spillback. Optimal strategies are determined using derivative-free

optimisation algorithms, applied to an evacuation problem based on a benchmark dataset. The

effects of varying the number of activated paths and the frequency of departure under different

network conditions are observed. Our analysis indicates that combined departure time

scheduling and signal phasing is a promising method to improve evacuation efficiency when

compared to a worst-case benchmark scenario.

Keywords: simulation-optimisation, disaster evacuation, genetic algorithms, dynamic traffic

assignment

1. Introduction

Evacuations are disaster response mechanisms aiming to temporarily relocate populations

before, during and after large-scale disasters while seeking to minimise potential loss of life.

Given the inherent managerial, societal, financial, behavioural, and operational aspects of

evacuations, adequate planning is required to ensure their effectiveness and safety. Despite the

challenge of predicting the number of lives saved through a specific evacuation strategy given

the interrelated factors at play, evidence indicates that countries with extensive disaster

preparedness mechanisms have been able to reduce casualties substantially from natural

disasters (Hallegatte, 2012).

Key among the factors impeding evacuation processes is the unique traffic patterns that

arise due to the sudden surge in travel demand, which contravene nominal traveller behaviours.

Even when extensive warning is given, the mass mobilisation imposes excessive strains on the

transport network due to the small time window, the lack of resources and the number of

evacuees (Thomas and Kopczak, 2007). As a result, network congestion is among the most

significant challenges encountered during emergency evacuations and is aggravated by

spillbacks, shockwaves and gridlocks.

The effects of congestion were illustrated during the Tohoku earthquake and subsequent

tsunami in 2011, where 51% of casualties were evacuating, of which 30% had problems

reaching shelters due to network congestion (Yun and Hamada, 2015). An overview of recent

studies on high profile natural disasters, their impacts and problems encountered during

evacuations is provided in Table 1.

As is the case with ordinary traffic, many evacuation strategies rely on capacity

expansion measures to preclude the occurrence of congestion. Among them, contraflow and

crossing elimination have been found to result in capacity increases of up to 70% and 40%,

respectively (Wolshon, 2001; Cova and Johnson, 2003). However, such techniques rely on

prior familiarity with evacuation plans and are likely to increase safety risks in scenarios where

evacuees are unaware or non-compliant (Xie, Lin and Travis Waller, 2010).

Table 1 List of recent natural disasters and evacuation problems.

Disaster Area Date Deaths Evacuation Problems Reference

T PH 2013 6,300 800,000 Damage to evacuation sites. Mahar et al. (2015)

F IN 2013 4,000 24,000 Evacuation order not issued. Rautela (2013)

Rana et al. (2013)

EQ ID 2012 10 3 million Traffic congestion. Munawir (2012)

News24 (2012)

EQ-TS JP 2011 16,000 400,000 Damage to transport

infrastructure. Hasegawa (2013)

EQ HT 2010 230,000 482,000 Lack of warning, damage to

transport infrastructure.

Margesson & Taft-

Morales (2010)

H US 2005 1,800 400,000 Over 100,000 to 300,000 did

not or could not evacuate. Wolshon (2006)

EQ-TS SA 2004 220,000 N/A Lack of warning, damage to

transport infrastructure. Fehr et al. (2005)

Table Legend

Disaster Type: T – Typhoon, F – Floods, EQ – Earthquake, TS – Tsunami, H – Hurricane.

Area Affected: PH – Philippines, IN – India, ID – Indonesia, JP – Japan, HT – Haiti,

US – United States of America, SA – Southern Asia.

Demand management has been explored as an alternative method to limit congestion effects

by restructuring the distribution of evacuees across the network (Abdelgawad and Abdulhai,

2009). This is achieved by providing alternate safe sites to evacuees or by staging departures

at their source (Chen and Zhan, 2008).

The estimation of expected network congestion levels during an evacuation is not a trivial

task. The occurrence and scale of gridlocks, spillbacks and shockwaves are dependent upon

time and user behaviour, and as such cannot be integrated directly into linear optimisation

algorithms without affecting model accuracy or computational performance (Lu, George and

Shekhar, 2005). As a result, many previous studies on congestion prediction have adopted bi-

level optimisation models that involve Stackelberg games.

In such a model, the upper-layer usually focuses on decisions taken by a “super-user”,

who assumes the role of the decision-maker and possesses perfect knowledge of the evacuation

process. In practice, this role is assumed by emergency management and law enforcement

organisations, with support from transport agencies (Urbina and Wolshon, 2003). Conversely,

the lower-level focuses on modelling the effects of these decisions (Urbina and Wolshon, 2003;

Abdelgawad and Abdulhai, 2009). Typical implementations use traffic assignment models,

which may vary in scale and detail (Table 2).

Table 2 Traffic assignment model types

Model Type Description Examples

Microscopic

Considers individual behaviours and

congestion effects. Computationally

expensive.

OREMS (Li, Yang and Heng, 2006) and

MatSim (Nagel and Flötteröd, 2012)

Mesoscopic Models groups of agents. Compromise

between modelling detail and speed.

DYNASMART-P (Sbayti and

Mahmassani, 2006), DTALite (Zhou and

Taylor, 2014), and models by Szeto &

Lo (2004) and Ziliaskopoulos (2000)

Macroscopic

Vehicle movement modelled as fluid

flows. Less emphasis on individual

interactions.

DYNEV (Goldblatt, 2004) and models

by Friesz et al. (1993) and Han et al.

(2013)

Hobeika & Kim (1998) were among the first to use demand staging for evacuation

modelling of UE and SP assignments in nuclear power plant evacuations. However, their study

did not consider SO assignment, limiting the effectiveness of any evacuation plan.

Sbayti & Mahmassani (2006) developed a bi-level optimisation algorithm where the

upper-level generates all evacuee origin-destination choices and initial time-dependent historic

assignment paths to the lower level. The latter uses a mesoscopic cell transmission model

(CTM) featuring dynamic SO traffic assignment.

Another SO CTM-based solution is presented by Chiu et al. (2007) for no-notice

evacuations. Their formulation integrated departure schedule and traffic assignment decisions

into the single-destination bi-level evacuation optimisation model.

An optimal-control formulation is proposed by Liu et al. (2006), discretising the

evacuation area into staged departure zones, instead of the node-based departure staging

strategies described above. A similar problem is presented by Bish et al. (2014), who formulate

a CTM-based mixed integer linear problem for the reduction network of clearance times with

multiple evacuee types.

Mitchell & Radwan (2006) use a microscopic traffic assignment method with pre-trip

and en-route choice models to optimise network clearance time, highlighting the scalability

limitations derivative of microscopic assignment methods. Although considering single-route

evacuation, Chien & Korikanthimath (2007) presented a model that consider the total duration

and any congestion delays of multistage evacuations.

Chen & Zhan (2008) used agent-based simulation to evaluate the merits of evacuation

staging strategies under different network conditions, finding best results for geometric

transportation networks in high-density areas. Abdelgawad & Abdulhai (2010) proposed a

spatiotemporal multi-modal evacuation framework that minimises travel and wait times for

evacuees.

More recently, Bretschneider & Kimms (2012) presented a pattern-based dynamic flow

model solved using a bi-level optimisation heuristic to minimise evacuation clearance time. Li

et al. (2012) formulated a shortest-path evacuation staging algorithm that minimises evacuation

congestion. Extensions developed by Xie et al. (2014) and Li et al. (2018) considered multiple

destinations and incorporated a time-expanded network flow model respectively.

Kimms & Maiwald (2017) developed a CTM-based evacuation planning model

considering contraflow and staged evacuation. A further study by Kimms & Maiwald (2018)

investigated route resilience by minimising potential capacity using a CTM evacuation model.

An alternative formulation developed by Even et al. (2015) used a tree-design framework

to minimise converging and diverging delays and improve evacuation performance. A similar

framework is proposed by Pillac et al. (2016), where congestion and conflicts are minimised

through demand staging over pre-generated routes.

Another evacuation path generation algorithm is developed by Seekircher (2017), who

considers the selfish behaviour of the evacuees and manipulate evacuee behaviour through

strategic link blockages. The solution procedure consists of a bi-level approach where the

upper-level optimises the route choice and the lower-level manipulates the network structure.

While the optimisation of isolated components (demand staging, signalling or user

behaviour) during evacuation have been explored independently, no study in the reviewed

literature considers simultaneous mechanisms and their interactions, except Kimms & Maiwald

(2017) who explore contraflow assignment and demand staging. Additionally, the choice of

evacuation routes and origin-destination pairs have been commonly addressed as part of the

optimisation model, instead of being considered within the context of road user choice (Liu,

Lai and Chang, 2006; Sbayti and Mahmassani, 2006; Jabari, He and Liu, 2012).

This study focuses on the development of an evacuation planning algorithm in

constrained urban networks. We propose an assignment-optimisation framework composed of

two elements that contribute to the development of an evacuation strategy, namely an

evacuation decision model and a dynamic traffic model. The latter uses an efficient numerical

traffic model based on continuous time-dependent kinematic wave simulation, with

consideration of congestion effects and the impact of user information on road traffic. The

algorithm is structured as a Stackelberg game acting as a simulation-optimisation framework,

pursuing optimal evacuation strategies in the absence of derivative information (Nguyen,

Reiter and Rigo, 2014; Amaran et al., 2016).

Our main contribution to the state of the art comprises the development of an evacuation

decision model that simultaneously controls the evacuation and signal phasing strategy over a

specified time horizon. In addition, contrasting the literature, the model acknowledges the

evacuees’ ability to make independent route choices, only constrained by a pre-defined set of

available paths. In doing so, the study seeks to highlight the benefits of evacuation staging in

constrained road networks under varying conditions, including multiple sink nodes and

extensive link damage.

The mathematical formulation used to determine optimal evacuation strategies is

provided in Section 2, followed by the traffic and evacuation routing models presented in

Section 3. The implementation of the overall algorithm is described in Section 4, alongside any

modifications made to the underlying models to facilitate closer integration. A numerical

example is presented, accompanied by a discussion on performance and scalability.

Conclusions and recommendations for future work are provided in Section 5.

2. Evacuation Strategy Model

The upper-level model in our framework is used to capture the decisions involved in the design

of the evacuation strategy, with the objective to maximise evacuee numbers within a predefined

time span. In this section, we assume that the evacuation strategy adopts a departure staging

mechanism, with users prescribed a set of available routes and instructed when to vacate their

premises. Once users are allowed to depart, they are free to travel towards any of the evacuation

shelters (implemented as sink nodes) using any of the routes that have been made available. A

limitation on the total number of distinct routes selected at every time step is considered. The

following assumptions are adopted:

Assumption 1: Evacuees use their road vehicle as a mode of transportation.

We adopt this assumption to transform the model into a single-commodity assignment problem

where evacuees share the same attributes regarding driving behaviour (Kimms and Maiwald,

2018). We acknowledge that public transport can aid evacuation performance, especially

among specific groups such as limited mobility evacuees (Urbina and Wolshon, 2003) and that

limiting car usage may be beneficial to the evacuation performance (Shimamoto et al., 2018).

Assumption 2: Background traffic and shadow evacuation effects are ignored.

This assumption follows from literature (Tuydes-Yaman and Ziliaskopoulos, 2014; Xie et al.,

2014) and seeks to ensure deterministic results given its intended use as a strategic design tool.

However, we recognise that background traffic is likely to affect the performance of the

evacuation strategy alongside the availability of information systems and the nature of hazard

type (Zheng et al., 2010).

In practice, we expect that evacuation planners would apply the framework in a range

of problem instances, each pertaining to a different network state. Concerning shadow

evacuation or self-evacuation, it can account for 20% of evacuees for uncommon hazards, such

as chemical or nuclear threats (Zeigler et al., 1981; Mitchell, Cutter and Edmonds, 2007). We

assume that the citizens in our model are familiar with existing evacuation procedures.

Assumption 3: Intermediate trips are not considered.

Intermediate trips made to an alternate destination before evacuating increase network

clearance time, and are more common under short warning times (Liu and Murray-Tuite, 2011;

Murray-Tuite and Wolshon, 2013). An illustrative example is provided in the 2012 Indonesia

Earthquake, where the evacuation was undermined by families collecting their children from

the education centres (Munawir, 2012; News24, 2012). We assume that sufficient warning time

is given, minimising the impact of this phenomenon.

Assumption 4: The super-user has perfect information of the traffic network.

The assumption is based on the existence of Intelligent Transport Systems (ITS) that can update

information on the transport network during the evacuation. Therefore, authorities can collect

data on the network state in real-time and provide evacuees with instructions that reflect that

current state of the network. Assuming that sufficient warning is given, the ITS systems are

considered to be in operation during the evacuation. Additionally, no uncertainty related to

evacuee demand is considered, and thus a deterministic model is proposed (Sabouhi et al.,

2018).

Assumption 5: Users cannot change their evacuation route after departure.

During the evacuation, road user compliance with controllers is difficult to estimate and model,

as these depend on social and economic factors, among others (Sorensen, 2000). We partially

address the effects of this assumption by allowing users to select through a range of possible

routes during the traffic assignment stage of our algorithm.

The following notation is used in the remainder of this section:

Indices: Sets:

𝑖, 𝑗 links 𝔗𝑣 incoming links to node 𝑣

𝑝 paths 𝒪𝑣 outwards links from node 𝑣

𝑡 evacuation time step 𝒫 paths

𝑑 destination node 𝑇 time horizon

𝑠 source node 𝒟 destination nodes

𝑣 network node 𝒮 source nodes

𝒜 links in the network

𝒱 nodes in the network

Parameters:

𝐾 total number of paths [−] 𝑈 upper departure rate limit [𝑣𝑒ℎ/𝑠]

𝛾 fitness factor [−] 𝐸 lower departure rate limit [𝑣𝑒ℎ/𝑠]

𝑌𝑝,𝑠 path-source node relation [−] 𝑋𝑝,𝑑 path-destination node relation [−]

𝑀𝑑 destination node capacity [𝑣𝑒ℎ] 𝐵𝑠 source node total demand [𝑣𝑒ℎ/𝑠]

Δ𝑡𝑝𝑡 estimated travel time of path 𝑝 at

time 𝑡 [𝑠]

Decision variables:

𝑅𝑝𝑡 departure rate [𝑣𝑒ℎ/𝑠] 𝑎𝑝

𝑡 activated paths [−]

𝑁𝑢𝑝,𝑖𝑡 cumulative vehicles entering link 𝑖

[𝑣𝑒ℎ]

𝑞𝑖𝑛,𝑖𝑡 vehicle flow entering link 𝑖 [𝑣𝑒ℎ/𝑠]

𝑁𝑑𝑜𝑤𝑛,𝑖𝑡 cumulative vehicles exiting link 𝑖

[𝑣𝑒ℎ]

𝑞𝑜𝑢𝑡,𝑖𝑡 vehicle flow exiting link 𝑖 [𝑣𝑒ℎ/𝑠]

2.1.Integer Formulation

The overall objective of this model is to ensure that the maximum number of evacuees from a

given area complete their journeys within the course of a finite time horizon. The level of risk

which the evacuees are subjected to is assumed to increase with time. Therefore, we seek to

ensure that the journeys are completed as early as possible, using elapsed time as a surrogate

for risk level. We adopt a composite function (1) consisting of two factors that reflect these

objectives (1.1-1.2):

𝑍 = 𝐶 − 𝛾𝑊 (1)

𝐶 = ∑ (1

𝑡∑ 𝑁𝑢𝑝,𝑖

𝑡

𝑖∈𝔗𝑑

)

𝑡∈𝑇,𝑑∈𝒟

(1.1)

𝑊 =∑( ∑ 𝑁𝑑𝑜𝑤𝑛,𝑗𝑡

𝑠∈𝒮,𝑗∈𝒪𝑠

− ∑ 𝑁𝑢𝑝,𝑖𝑡

𝑑∈𝒟,𝑖∈𝔗𝑑

)

𝑡∈𝑇

(1.2)

Equation (1.1) penalises departures given the time step at which they are due to

complete their journeys. The 𝑁𝑢𝑝,𝑖𝑡 variable is used to capture the cumulative number of vehicle

departures at every time step 𝑡. The second component of the objective (1.2) seeks to minimise

the total wait time 𝑊, which captures the total duration of aggregate vehicle presence in the

network (which includes delays). A weight factor 𝛾 is used to control the priority allocated to

each of the two objectives. A maximum time horizon 𝑇 is defined along with discretised time

step Δ𝑡. The model formulation is as follows:

Objective:

𝑚𝑎𝑥𝑖𝑚𝑖𝑠𝑒 𝑍 (2)

Subject to:

𝑁𝑢𝑝,𝑗𝑡 = Δ𝑡∑𝑞𝑖𝑛,𝑗

𝑙

𝑡

𝑙=1

𝑗 ∈ 𝒜,

𝑡 ∈ 𝑇 (2.1)

𝑁𝑑𝑜𝑤𝑛,𝑗𝑡 = Δ𝑡∑𝑞𝑜𝑢𝑡,𝑗

𝑙

𝑡

𝑙=1

𝑗 ∈ 𝒜,

𝑡 ∈ 𝑇 (2.2)

∑ 𝑋𝑝,𝑑𝑁𝑢𝑝,𝑖𝑡

𝑝∈𝒫.𝑑∈𝒟,𝑖∈𝔗𝑑

= ∑ 𝑌𝑝,𝑠𝑞𝑜𝑢𝑡,𝑗𝑙 Δ𝑡

𝑝∈𝒫,𝑠∈𝒮,𝑗∈𝒪𝑠,𝑙∈𝑇:𝑙+Δ𝑡𝑝𝑙 ≤𝑡

𝑡 ∈ 𝑇 (2.3)

𝑞𝑜𝑢𝑡,𝑗𝑡 = ∑𝑎𝑝

𝑡 𝑅𝑝𝑡𝑌𝑝,𝑠

𝑝∈𝒫

𝑗 ∈ 𝒪𝑠 ,

𝑡 ∈ 𝑇 (2.4)

𝐸 ≤ 𝑅𝑝𝑡 ≤ 𝑈

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (2.5)

∑ 𝑌𝑝,𝑠 𝑅𝑝𝑡𝑎𝑝

𝑡

𝑡∈𝑇,𝑝∈𝒫

= 𝐵𝑠 𝑠 ∈ 𝒮 (2.6)

∑ 𝑁𝑑𝑜𝑤𝑛,𝑗𝑡

𝑗∈𝔗𝑑

≤ 𝑀𝑑 𝑡 ∈ 𝑇

𝑑 ∈ 𝒟 (2.7)

𝑎𝑝𝑡 = {0,1}

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (2.8)

∑𝑎𝑝𝑡

𝑝∈𝒫

= 𝐾 𝑡 ∈ 𝑇 (2.9)

Constraints (2.1) and (2.2) calculate the number of vehicles in every link 𝑗 among the

arcs 𝒜 of the network. Constraint (2.3) ensures that vehicles entering path 𝑝 will arrive at the

sink node within travel time Δ𝑡𝑝𝑡 . Constraint (2.4) calculates the vehicular flow entering the

network at time step 𝑡 given a vehicle inflow rate 𝑅𝑝𝑡 at the activated paths 𝑎𝑝

𝑡 .

Equation (2.5) ensures that the path rate is bounded by a lower limit 𝑉 and an upper limit

𝑈 at every time step 𝑡. Constraint (2.6) equates the total departures throughout the time horizon

𝑇 to the original demand. Equation (2.7) ensure capacity limitations at the destination nodes

are satisfied. Finally, (2.8) and (2.9) state that 𝑎𝑝𝑡 is a Boolean variable and that the paths

activated must be limited to 𝐾 for each origin, respectively.

The main limitations of the above formulation are its inability to consider signal control

and to evaluate Δ𝑡𝑝𝑡 , which is varies with vehicle flows due to congestion. Therefore, the value

of Δ𝑡𝑝𝑡 is treated as a dynamic variable with a non-linear relationship with the decision variables

𝑅𝑝𝑡 and 𝑎𝑝

𝑡 . Section 3 introduces a dynamic traffic assignment model that evaluates Δ𝑡𝑝𝑡 for each

time step and models nodes as signalled junctions.

2.2. Linear Model Formulation

A typical linearisation approach for the traffic assignment component of the model consists on

duplicating the network for every discrete time step of a time horizon 𝑇, increasing the problem

size to impractical levels for large transport networks (Lu, George and Shekhar, 2005; Shen,

Nie and Zhang, 2007; Li, Zhou and Ju, 2010). Mathematical simplifications including no-

overtaking (First-In-First-Out) and traffic holding require the inclusion of additional variables

and non-convex constraints that preclude the adoption of computationally efficient solution

algorithms (Doan and Ukkusuri, 2012).

Given these limitations, we develop a linear mathematical formulation that is solved in

tandem with the dynamic traffic simulator. The evacuation strategy developed using CPLEX

serves as an input for the traffic simulator, which updates path travel times Δ𝑡𝑝𝑡 considering the

effects of congestion. The linear problem is then solved using the updated network properties,

creating a deviation between the solutions before and after Δ𝑡𝑝𝑡 is updated. This iterative

process runs until the variation between solutions is negligible, thus estimating the effects of

congestion within the constraints of the linear formulation.

The main limitation of the proposed approach is the use of a static Δ𝑡𝑝𝑡 throughout the

time window for each iteration. We present the linear formulation, introducing the following

parameters in addition to the notation description in Section 2:

Decision Variables:

𝑏𝑝𝑡 slack variable to linearise relationship between 𝑎𝑝,𝑡 and 𝑁𝐷𝑝,𝑡 [𝑣𝑒ℎ/𝑠]

𝑉𝐷𝑡𝑝 vehicle departures from path 𝑝 at time-step 𝑡 [−]

𝑉𝐴𝑡𝑝 vehicle arrival from path 𝑝 at time-step 𝑡 [−]

Objective:

𝑚𝑎𝑥𝑖𝑚𝑖𝑠𝑒 𝑍 = 𝐶 − 𝛾𝑊 (3)

𝐶 =∑(1

𝑡∑ 𝑁𝐴

𝑡𝑝

𝑝 ∈ 𝑃

)

𝑡∈𝑇

(3.1)

𝑊 =∑(∑𝑁𝐷𝑡𝑝

𝑝∈𝑃

−∑𝑁𝐴𝑡𝑝

𝑝∈𝑃

)

𝑡∈𝑇

(3.2)

Subject to:

𝑉𝐴𝑡𝑝= ∑ 𝑉𝐷𝑝

𝑙

𝑙∈𝑇:𝑙+Δ𝑡𝑝𝑙 =𝑡

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (4.1)

𝑁𝐴𝑡𝑝=∑𝑉𝐴𝑝

𝑙

𝑡

𝑙=1

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (4.2)

𝑁𝐷𝑡𝑝=∑𝑉𝐷𝑝

𝑙

𝑡

𝑙=1

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (4.3)

𝑅𝑝𝑡Δ𝑡𝑠 = 𝑉𝐷

𝑡𝑝

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (4.4)

𝑎𝑝𝑡𝐸 ≤ 𝑅𝑝

𝑡 ≤ 𝑎𝑝𝑡 𝑈

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (4.5)

𝑏𝑝𝑡 − 𝐸(1 − 𝑎𝑝

𝑡 ) ≤ 𝑅𝑝𝑡 ≤ 𝑏𝑝

𝑡 − 𝑈(1 − 𝑎𝑝𝑡 )

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (4.6)

𝐵𝑠 = ∑ 𝑌𝑝,𝑠 𝑅𝑝𝑡

𝑡∈𝑇,𝑝∈𝑃

𝑠 ∈ 𝒮 (4.7)

𝑀𝑑 ≥ ∑𝑁𝐷𝑡𝑝 𝑋𝑝,𝑑

𝑝∈𝑃

𝑡 ∈ 𝑇

𝑑 ∈ 𝒟 (4.8)

∑𝑎𝑝𝑡

𝑝∈𝑃

𝑌𝑝,𝑠 = 𝐾 𝑠 ∈ 𝒮,

𝑡 ∈ 𝑇 (4.9)

𝑎𝑝𝑡 = {0,1}

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (4.10)

𝐸 ≤ 𝑏𝑝𝑡 ≤ 𝑈

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (4.11)

The objective function (3) is similar to equation (1), as it calculates the cumulative

weighted evacuations and the total wait time. Equation (4.1) ensures that vehicle departures

and arrivals are equal within a path 𝑝 given an expected travel time Δ𝑡𝑝𝑡 . Constraints (4.2-4.3)

calculates the cumulative vehicle departures 𝑁𝐷𝑡𝑝 and arrivals 𝑁𝐴

𝑡𝑝. Equation (4.4) considers

the departing flows 𝑅𝑝𝑡 and the total vehicles entering a path 𝑁𝐷𝑝

𝑡 . Constraints (4.5) and (4.6)

bounds the departing vehicles 𝑅𝑝𝑡 to 𝑉 and 𝑈 respectively while 𝑅𝑝

𝑡 = 0 for non-selected paths.

Equation (4.7) equates the total demand in each source node 𝑠 to the total departures of

each path, where 𝑌𝑝,𝑠 equates to 1 when source 𝑠 belongs to path 𝑝 and 0 otherwise. Equation

(4.8) considers sink node capacity similarly to (2.7). Constraint (4.9) ensures that the total

number of paths activated 𝑎𝑝𝑡 for each source node equals 𝐾. Finally, equations (4.10) and

(4.11) bound the decision variables 𝑎𝑝𝑡 and 𝑏𝑝

𝑡 to a Boolean and to 𝑉 and 𝑈 respectively.

3. Dynamic Network Loading Model

This section presents the model that addresses the inability of the above formulation to capture

the effects of congestion on vehicular flow by using a continuous time link-based kinematic

wave model that was initially developed by Han et al. (2016). The kinematic wave model

considers the effects of flow propagation and congestion, including physical queuing and

vehicle spillback, based on a reformulation of the scalar conservation law model (Lighthill and

Whitham, 1955; Richards, 1956), the Hamilton-Jacobi equation and variational theory. Such a

formulation has been independently derived by Jin (2015) and Han et al. (2016) and, when

discretised, leads to the well-known link transmission model (Yperman, Logghe and Immers,

2005). The model is path-based, with path selection and network properties provided

previously by the user.

The traffic flow model consists of three components: a junction model, a source model

and a link model. The junction model can be used to implement traffic signalling in the

network. The source model provides departure rates from specific nodes over the time-horizon

of the evacuation. Finally, the link model is based on the fundamental triangular diagram and

the concepts of demand and supply (Lebacque and Khoshyaran, 1999), allowing the simulation

to calculate in-flows and out-flows of links that are adjacent to the same junction and thus

simulate shockwave propagation and vehicle spillback.

The traffic simulator is described by a system of time-continuous differential algebraic

equations (DAE). It is also referred to as the dynamic network loading procedure in the

dynamic traffic assignment literature. The underlying mathematical model is presented below

following the notations introduced in Section 2 - the reader is referred to Han et al. (2016) for

its derivation and computational examples.

Parameters

𝐶𝑖 flow capacity of link 𝑖 [𝑣𝑒ℎ] 𝜌𝑖𝑗𝑎𝑚

jam density of link 𝑖 [𝑣𝑒ℎ/𝑚]

𝐿𝑖 length of link 𝑖 [𝑚] 𝑣𝑖 forward wave speed of link 𝑖 [𝑚/𝑠]

𝑤𝑖 backward wave speed of link 𝑖 [𝑚/𝑠]

Variables

𝐷𝑖𝑡 demand of link 𝑖 [𝑣𝑒ℎ] 𝑆𝑗

𝑡 supply of link 𝑗 [𝑣𝑒ℎ]

𝑞𝑖𝑛,𝑖𝑡 inflow of link 𝑖 [𝑣𝑒ℎ/𝑠] 𝑞𝑜𝑢𝑡,𝑖

𝑡 outflow of link 𝑖 [𝑣𝑒ℎ/𝑠]

𝑁𝑢𝑝,𝑖𝑡 cumulative link entry volume [𝑣𝑒ℎ] 𝑁𝑑𝑜𝑤𝑛,𝑖

𝑡 cumulative link exit volume [𝑣𝑒ℎ]

𝜇𝑖𝑝,𝑡(𝑥) path disaggregation variable [−] 𝜏𝑖

𝑡 link entry time function [𝑠]

𝛼𝑖𝑗𝐽,𝑡

flow split at junction 𝐽 [−] 𝜂𝑖𝑡 effective green time for link 𝑖 [−]

𝐴𝐽 flow distribution at 𝐽 [−]

The definitions of demand and supply that are used in this model conform with those

proposed by Lebacque & Khoshyaran (1999), and correspond to the inflow and outflow

capacity of a link. 𝑁𝑢𝑝,𝑖𝑡 and 𝑁𝑑𝑜𝑤𝑛,𝑖

𝑡 denote the cumulative number of vehicles that have entered

and exited the link 𝑖 by time 𝑡, respectively. In order to explicitly incorporate traveller route

choices, the path disaggregate variable 𝜇𝑖𝑝,𝑡(𝑥) represents the proportion of traffic at point 𝑥

and time step 𝑡 on link 𝑖 that is associated with path 𝑝, where 𝑖 ∈ 𝑝. Based on 𝑁𝑢𝑝,𝑖𝑡 and 𝑁𝑑𝑜𝑤𝑛,𝑖

𝑡 ,

the link entry time function can be calculated based on the following flow propagation identity:

𝑁𝑢𝑝,𝑖𝜏𝑖𝑡

= 𝑁𝑑𝑜𝑤𝑛,𝑖𝑡 ,

where 𝑡 denotes the link exit time. Finally, the flow split variable 𝛼𝑖𝑗𝐽,𝑡

indicates the probability

of traffic exiting upstream link 𝑖 that are heading towards downstream link 𝑗, where both 𝑖 and

𝑗 are adjacent to junction 𝐽. Given the above, and the formulation provided in the previous

section, the DAE system for the dynamic network loading model is as follows:

𝐷𝑖𝑡 =

{

𝑞𝑖𝑛,𝑖

𝑡−𝐿𝑖𝑣𝑖 if 𝑁

𝑢𝑝,𝑖

𝑡−𝐿𝑖𝑣𝑖 = 𝑁𝑑𝑜𝑤𝑛,𝑖

𝑡

𝐶𝑖 if 𝑁𝑢𝑝,𝑖𝑡−𝐿𝑖𝑣𝑖 > 𝑁𝑑𝑜𝑤𝑛,𝑖

𝑡

𝑖 ∈ 𝒜,

𝑡 ∈ 𝑇 (5.1)

𝑆𝑖𝑡 =

{

𝑞𝑜𝑢𝑡,𝑖

𝑡−𝐿𝑖𝑤𝑖 if 𝑁𝑢𝑝,𝑖

𝑡 = 𝑁𝑑𝑜𝑤𝑛,𝑖

𝑡−𝐿𝑖𝑤𝑖 + 𝜌𝑗

𝑗𝑎𝑚𝐿𝑗

𝐶𝑖 if 𝑁𝑢𝑝,𝑖𝑡 < 𝑁

𝑑𝑜𝑤𝑛,𝑖

𝑡−𝐿𝑖𝑤𝑖 + 𝜌𝑗

𝑗𝑎𝑚𝐿𝑗

𝑖 ∈ 𝒜,

𝑡 ∈ 𝑇 (5.2)

𝑁𝑑𝑜𝑤𝑛,𝑖𝑡 = 𝑁𝑢𝑝,𝑖

𝜏𝑖𝑡

𝑖 ∈ 𝒜,

𝑡 ∈ 𝑇 (5.3)

𝜇𝑗𝑝,𝑡(0) =

𝑞𝑜𝑢𝑡,𝑖𝑡 𝜇

𝑖

𝑝,𝜏𝑖𝑡

(0)

𝑞𝑖𝑛,𝑗𝑡 ∀𝑝 such that {𝑖, 𝑗} ∈ 𝑝

𝑡 ∈ 𝑇 (5.4)

𝐴𝐽(𝑡) = {𝛼𝑖𝑗𝐽,𝑡}, 𝛼𝑖𝑗

𝐽,𝑡 = ∑ 𝜇𝑖𝑝,𝜏𝑖

𝑡

(0)

𝑖,𝑗∈𝑝

𝑖 ∈ 𝔗𝐽, 𝑗 ∈ 𝒪𝐽,

𝑡 ∈ 𝑇 (5.5)

𝑞𝑖𝑛,𝑘𝑡 = 𝑅𝑆𝑘

𝐴𝐽 [(𝐷𝑖𝑡)𝑖∈𝔗J , (𝑆𝑗

𝑡)𝑗∈𝒪J

, (𝜂𝑖𝑡)𝑖∈𝔗J]

𝑘 ∈ 𝒪𝐽, 𝑡 ∈ 𝑇 (5.6)

𝑞𝑜𝑢𝑡,𝑙𝑡 = 𝑅𝑆𝑙

𝐴𝐽 [(𝐷𝑖𝑡)𝑖∈𝔗J , (𝑆𝑗

𝑡)𝑗∈𝒪J

, (𝜂𝑖𝑡)𝑖∈𝔗J]

𝑙 ∈ 𝔗𝐽, 𝑡 ∈ 𝑇 (5.7)

𝑁𝑢𝑝,𝑖𝑡 − 𝑁𝑢𝑝.𝑖

𝑡−1 =Δ𝑡 𝑞𝑖𝑛,𝑖𝑡 , 𝑁𝑑𝑜𝑤𝑛,𝑖

𝑡 − 𝑁𝑑𝑜𝑤𝑛,𝑖𝑡−1 = Δ𝑡 𝑞𝑜𝑢𝑡,𝑖

𝑡 𝑖 ∈ 𝒜, 𝑡 ∈ 𝑇 (5.8)

Equations (5.1)-(5.2) define the link demand and supply, given the cumulative entry and exit

volumes at earlier times. The relationship between entry and exit times at every link, based on

flow conservation and the First-In-First-Out principle is defined by equation (5.3). Equation

(5.4) propagates the path disaggregation variable between consecutive links, while equation

(5.5) states that the flow distribution at a junction with multiple downstream links can be

uniquely determined by the relevant path disaggregation variables.

Equations (5.6)-(5.7) determine the inflow and outflow of links that are incident to a

junction 𝐽 based on the Riemann Solver proposed by (Han et al., 2014; Han and Gayah, 2015),

which approximates the effect of on-off signal controls using a continuum approach based on

the kinematic wave theory, and depends on the junction topology and specific traffic

management scenarios such as signal control and priorities.

Finally, (5.8) links the two types of fundamental flow variables: flow and cumulative

volume. This DAE system can be solved in discrete time in a forward fashion; see Garavello

et al. (2016) for computational examples of this system in a number of traffic networks.

The traffic model uses a set of paths as primary inputs. While safety concerns in real-live

evacuation exercises may condition routing, we assume for this study that a reduction of vehicle

exposure in the network will increase user safety. This simplification allows us to use a k-

shortest path algorithm based on Yen’s algorithm to generate a set of multiple route-choices to

the exit node (Yen, 1971). Our implementation of the evacuation routing algorithm is detailed

in the Pseudocode 1 below and incorporates a Dijkstra’s-based solver to determine individual

paths (Dijkstra, 1959).

1. Algorithm Evacuation Routing is

2. Input: graph 𝐺, origins 𝑂𝑣, destinations 𝐷𝑣, number of paths 𝐾 3. Output: shortest paths 𝐿𝑝 4.

5. spurPath 𝑝𝑠 ← CreateEmptyList() 6. 𝑝∗ ← Dijkstra(𝐺, 𝑂𝑣 , 𝐷𝑣) 7. for each path 𝑝 in 𝐿𝑝:

8. for each vertex 𝑣 in 𝑝

9. spur vertex 𝑣𝑠 ← 𝑣

10. root path 𝑝𝑟 ← 𝑝[0, 𝑣] 11. for each subPath 𝑝𝑠 in 𝐿𝑝

12. if 𝑝𝑟 == 𝑝𝑠[0, 𝑣]

13. 𝐺. RemoveLink(𝑝𝑠[𝑣, 𝑣 + 1] 14. for each pathNode 𝑣𝑝 in 𝑝𝑟 except 𝑣𝑠

15. 𝐺. RemoveVertex(𝑝𝑟[𝑣𝑝])

16. 𝑝𝑠 ← Dijkstra(𝐺, 𝑂𝑣 , 𝐷𝑣) 17. newPath 𝑝𝑛 = 𝑝𝑟 + 𝑝𝑠 18. spur path list 𝐿𝑝𝑠 ← 𝑝𝑛

19. 𝐺. Restore() 20. if 𝐿𝑝𝑠 is empty 21. break

22. 𝐿𝑝𝑠 . SortByCost()

23. 𝐿𝑝 ← 𝐿𝑝𝑠 . First()

24. 𝐿𝑝𝑠 . RemoveFirst()

25. return 𝐿𝑝 Pseudocode 1 Shortest path algorithm pseudocode.

We integrate both models using a feedback loop as presented in Pseudocode 2. The traffic

model requests the initial paths and network conditions, a simulation time-step Δ𝑡𝑠 and decision

time-step Δ𝑡𝑑 as inputs. Between each Δ𝑡𝑠 the model updates the network properties based on

current vehicular flow, maintaining network properties constant during Δ𝑡𝑠.

The traffic model outputs the current network conditions and calls the evacuation routing

model after each Δ𝑡𝑑, returning the new shortest paths accounting for congestion. A list of

paths is updated in case the returned shortest paths have not been previously explored, allowing

the algorithm to respond to major changes in the state of network while minimising reductions

to the performance of the algorithm. The cost of all paths in the list is updated as the algorithm

runs for traceability purposes.

The proposed traffic assignment model has been applied to a range of network sizes and

configurations to determine its scalability. Networks A, B, C and D are defined in the

Appendix, while the Sioux-Falls and Enriched Sioux Falls networks are presented in further

detail in Section 4.2. Different junction configurations are tested to determine the effect in

computational time. As shown in Table 3, the traffic assignment model can operate in complex

networks under reasonable runtime. The computation time increases with network size and

network complexity.

1. Algorithm Traffic Model is

2. Input: network properties 𝑁, simulation time step Δ𝑡𝑠, decision time

step Δ𝑡𝑑 initial flows 𝑄, initial paths 𝐾, time horizon 𝑇, graph 𝐺,

origins 𝑂, destinations 𝐷, activated paths 𝑘 3.

4. Output: network flows 𝑄∗ 5.

6. 𝑡 = 0 7. while 𝑡 < 𝑇 8. 𝑄 = EvaluateFlows(𝑁, 𝑄, 𝐾) 9. if IsSimulationTimeStep(𝑡, Δ𝑡𝑠) 10. 𝑁 = UpdateNetworkProperties(𝑁, 𝑄) 11. if IsDecisionTimeStep(𝑡, Δ𝑡𝑑) 12. 𝐾 = YensAlgorithm(𝐺, 𝑂, 𝐷, 𝑘) 13. if AllVehiclesLeftNetwork(𝑄)

14. return 𝑄 as 𝑄∗ 15. 𝑡 = 𝑡 + 1 16. return 𝑄 as 𝑄∗

Pseudocode 2 Integrated traffic model.

Table 3 Traffic model scalability.

Network Nodes Links Origins Destinations Junctions Time

(seconds)

A 6 10 1 2 D 0.91

B 6 10 2 1 C 0.81

C 8 16 1 1 D-C 0.81

D 12 30 3 1 M 2.51

SF 24 76 4 1 M 3.88

ESF 282 334 21 2 M 33.5

Table Legend

Network Type: A, B,C,D – Provided in Appendix, SF – Sioux Falls, ESF – Enriched Sioux Falls.

Junction Type: D – Diverging, C – Converging, M – Multiple junction structures.

4. Formulation of Optimal Evacuation Strategy Model

In this section, we introduce and evaluate the evacuation optimisation model. We present the

final mathematical formulation to evaluate the effectiveness of the evacuation scheme as a

combination of equation sets (2) and (5):

Objective:

𝑚𝑎𝑥𝑖𝑚𝑖𝑠𝑒 𝑍 = 𝐶 − 𝛾𝑊 (6)

𝐶 = ∑ (1/𝑡 ∑ 𝑁𝑢𝑝,𝑖𝑡

𝑖∈𝔗𝑑

)

𝑡∈𝑇,𝑑∈𝒟

(6.1)

𝑊 =∑( ∑ 𝑁𝑑𝑜𝑤𝑛,𝑗𝑡

𝑠∈𝒮,𝑗∈𝒪𝑠

− ∑ 𝑁𝑢𝑝,𝑗𝑡

𝑑∈𝒟,𝑗∈𝔗𝑑

)

𝑡∈𝑇

(6.2)

Subject to:

𝑁𝑢𝑝,𝑗𝑡 = Δ𝑡𝑠∑𝑞𝑖𝑛,𝑗

𝑙

𝑡

𝑙=1

𝑗 ∈ 𝒜,

𝑡 ∈ 𝑇 (7.1)

𝑁𝑑𝑜𝑤𝑛,𝑗𝑡 = Δ𝑡𝑠∑𝑞𝑜𝑢𝑡,𝑗

𝑙

𝑡

𝑙=1

𝑖 ∈ 𝒜,

𝑡 ∈ 𝑇 (7.2)

𝐷𝑖𝑡 =

{

𝑞𝑖𝑛,𝑖

𝑡−𝐿𝑖𝑣𝑖 if 𝑁

𝑢𝑝,𝑖

𝑡−𝐿𝑖𝑣𝑖 = 𝑁𝑑𝑜𝑤𝑛,𝑖

𝑡

𝐶𝑖 if 𝑁𝑢𝑝,𝑖𝑡−𝐿𝑖𝑣𝑖 > 𝑁𝑑𝑜𝑤𝑛,𝑖

𝑡

𝑗 ∈ 𝒜,

𝑡 ∈ 𝑇 (7.3)

𝑆𝑗𝑡 =

{

𝑞𝑜𝑢𝑡,𝑗

𝑡−𝐿𝑖𝑤𝑖 if 𝑁𝑢𝑝,𝑖

𝑡 = 𝑁𝑑𝑜𝑤𝑛,𝑖

𝑡−𝐿𝑖𝑤𝑖 + 𝜌𝑗

𝑗𝑎𝑚𝐿𝑗

𝐶𝑖 if 𝑁𝑢𝑝,𝑖𝑡 < 𝑁

𝑑𝑜𝑤𝑛,𝑖

𝑡−𝐿𝑖𝑤𝑖 + 𝜌𝑗

𝑗𝑎𝑚𝐿𝑗

𝑗 ∈ 𝒜,

𝑡 ∈ 𝑇 (7.4)

𝜇𝑝,𝑗𝑡,0 =

𝑞𝑜𝑢𝑡,𝑖𝑡 𝜇

𝑝,𝑖

𝜏𝑖𝑡,0

𝑞𝑖𝑛,𝑗𝑡 ∀𝑝 such that {𝑖, 𝑗} ∈ 𝑝

𝑡 ∈ 𝑇 (7.5)

𝐴𝐽𝑡 = {𝛼𝐽,𝑖𝑗

𝑡 }, 𝛼𝐽,𝑖𝑗𝑡 = ∑ 𝜇𝑝,𝑖

𝜏𝑖𝑡,0

𝑝∋𝑖,𝑗

𝑖 ∈ 𝔗𝐽,

𝑗 ∈ 𝒪𝐽,

𝑡 ∈ 𝑇

(7.6)

𝑞𝑖𝑛,𝑘𝑡 = 𝑅𝑆𝑘

𝐴𝐽 [(𝐷𝑖𝑡)𝑖∈𝔗J , (𝑆𝑗

𝑡)𝑗∈𝒪J

, (𝜂𝑖𝑡)𝑖∈𝔗J]

𝑘 ∈ 𝒪𝐽,

𝑡 ∈ 𝑇 (7.7)

𝑞𝑜𝑢𝑡,𝑙𝑡 = 𝑅𝑆𝑙

𝐴𝐽 [(𝐷𝑖𝑡)𝑖∈𝔗J , (𝑆𝑗

𝑡)𝑗∈𝒪J,

, (𝜂𝑖𝑡)𝑖∈𝔗J]

𝑙 ∈ 𝔗𝐽,

𝑡 ∈ 𝑇 (7.8)

∑𝜂𝑖𝑡

𝑖∈𝔗J

= 1 𝐽 ∈ 𝒱 (7.9)

�̅�𝑚𝑖𝑛 ≤ 𝜂𝑖𝑡 ≤ �̅�𝑚𝑎𝑥

𝑖 ∈ 𝔗𝐽,

𝑡 ∈ 𝑇

𝐽 ∈ 𝒱

(7.10)

𝑞𝑜𝑢𝑡,𝑗𝑡 =∑𝑎𝑝

𝑡 𝑅𝑝𝑡

𝑝∈𝑃

𝑌𝑝,𝑠 𝑗 ∈ 𝒪𝑠 ,

𝑡 ∈ 𝑇 (7.11)

𝐸 ≤ 𝑅𝑝𝑡 ≤ 𝑈

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (7.12)

∑ 𝑌𝑝,𝑡 𝑅𝑝𝑡

𝑡∈𝑇,𝑝∈𝑃

= 𝐵𝑠 𝑠 ∈ 𝒮 (7.13)

∑ 𝑁𝑑𝑜𝑤𝑛,𝑗𝑡

𝑗∈𝔗𝑑

≤ 𝑀𝑑 𝑡 ∈ 𝑇

𝑑 ∈ 𝒟 (7.14)

𝑎𝑝𝑡 = {0,1}

𝑝 ∈ 𝒫,

𝑡 ∈ 𝑇 (7.15)

∑ 𝑎𝑝𝑡

𝑝 ∈𝒫

= 𝐾 𝑡 ∈ 𝑇 (7.16)

The objective function presented in Equation (6) remains analogous to Equation (2). It

aims to maximise the number of evacuees, promoting early evacuations, while reducing the

time vehicles remain in the network. Constraints (7.1-7.2) and (7.11-7.16) are carried over from

Section 2. Equations (7.1) and (7.2) describe the calculation of total vehicles in every link 𝐼𝑖

among the arcs of the network 𝒜. Equation (7.11) determines the vehicle flow entering the

network from each origin at a defined rate 𝑅𝑝𝑡 . Constraints (7.12-7.13) propose lower and upper

bounds on the departure rates based on available demand. Sink node capacity is captured by

constraint (7.14). Finally, the number of activated paths is constrained by Equations (7.15-

7.16).

The evacuation process is controlled by decision variables 𝜂𝑖𝑡, 𝑅𝑝

𝑡 and 𝑎𝑝𝑡 , where 𝜂𝑖

𝑡

determines the effective green time at critical junctions for all incoming links and is bounded

by (7.9-7.10). The evacuation staging strategy is defined by the number of users allowed to

evacuate at specified time-steps, 𝑅𝑝𝑡 , and paths allowed to be accessed by the evacuees, 𝑎𝑝

𝑡 .

Constraints (7.3-7.8) are carried over from the traffic model presented in Section 3.

Equations (7.3-7.4) define supply and demand for each link similar to (5.1-5.2), and (7.5-7.8)

reflect (5.4-5.7), determining flow distribution through junctions using the Riemann Solver.

The overall approach is structured as a Stackelberg game acting as a simulation-

optimisation framework (refer to Pseudocode 3), and proceeds as follows:

1. In the upper level, a metaheuristic algorithm generates a) an initial set of departure

vehicle rates 𝑅𝑝𝑡 , b) signal phasing strategies 𝜂𝑖

𝑡 based on (7.9-7.10) and (7.12) and c) a

set of K-shortest paths given network properties.

2. Following this, we execute the lower-level traffic model, evaluating a) the sets of

network flows 𝑞 as per (7.1-7.8, 7.11), b) activated paths 𝑎𝑝 described by (7.13-7.16) ,

and c) an overall fitness value 𝑍 as defined by equations (6-6.2).

3. Following every lower-level execution, we invoke the upper-level algorithm to obtain a

new set of departure rates and signal phases.

4. Steps 2-3 are repeated until the stopping criteria are met (either convergence or

maximum number of iterations).

1. Algorithm EvacuationOptimisationAlgorithm is

2. Input: network properties 𝑁, simulation time step Δ𝑡𝑠, decision

time step Δ𝑡𝑑 initial flows 𝑄, time horizon 𝑇, iterations χ, graph 𝐺,

origins 𝑂, destinations 𝐷, number of paths 𝑘 3. Output: optimal evacuation strategy 𝑠∗ 4.

5. initial evacuation strategy 𝑠 ← Generation()

6. initial network flows 𝑄 ← DetermineFlows(𝑠𝑖)

7. initial paths 𝐾 ← YensAlgorithm(𝐺, 𝑂, 𝐷, 𝑘) 8. best evacuation strategy 𝑠𝑓

∗ = −∞

9. while current iteration 𝑖 < 𝜒

10. vehicle flows 𝑄 ← TrafficModel(𝑁, 𝛥𝑡𝑠, 𝛥𝑡𝑑, 𝑄, 𝐾, 𝑇, 𝐺, 𝑂, 𝐷, 𝑘) 11. solution fitness 𝑠𝑓 ← EvaluateSolution(𝑄)

12. if 𝑠𝑓∗ ≤ 𝑠

13. 𝑠∗ ← 𝑠 14. 𝑠 ← GenerateNewSolution(𝑠) 15. 𝑖 = 𝑖 + 1 16. return 𝑠 as 𝑠∗

Pseudocode 3 Evacuation optimisation algorithm pseudocode.

4.1. Performance of Linear Formulation

The linearised problem presented in equations (3)-(4.9) is solved using the set of simple

networks provided in the Appendix. The key performance parameters explored include the

fitness value as described in equation (6); the weighted cumulative evacuation, defined as

∑ (𝑁𝑢𝑝,𝑖𝑡 /𝑡) 𝑡∈𝑇 in the same equation; the total number of departures ∑ 𝑁𝑑𝑜𝑤𝑛,𝑖

𝑡𝑡∈𝑇 , which

denotes the number of vehicles that are allowed to leave their residence; and the total number

of finished evacuations ∑ 𝑁𝑢𝑝,𝑖𝑡

𝑡∈𝑇 . Finally, we compute the mean travel time for each

evacuation strategy.

Figure 1 Percentage improvement of bi-level model approach in comparison with linear model. Networks A-D are presented

in the Appendix, while SF refers to the Sioux Falls network presented in Section 4.2.

We present the results in Figure 1 comparing the performance of the linear problem and

the bi-level solution algorithm presented in Section 4. In most networks, the linear formulation

provides lower fitness score than the bi-level model. Similar performance levels are only

obtained in the smallest network A – likely due to its simple layout, which only contains a

diverging junction. However, as the complexity of the network increases the deviation between

both solution increases. As an example, in network B which contains a single merging junction,

there is a 30% difference in fitness values between both methods. We attribute this behaviour

to the inability of the linear problem to consider dynamic travel time as a result of traffic

congestion.

Despite the model being solved with the traffic simulator iteratively until convergence,

optimisation techniques that can consider congestion effects during its evaluation provide a

better optimum. Therefore, the linear formulation proposes an insufficient framework to

evaluate and optimise evacuation strategies.

4.2. Test Network

The methodology is applied to the Sioux Falls Network (SFN), consisting of 24 nodes and 76

arcs (Bar-Gera, 2016). In this problem, a danger zone covers nodes 10, 11, 14 and 15, and a

single evacuation point is established in node 1 as shown in Figure 2a. Further iterations of the

model in Section 4.4 will contain additional sink nodes, damaged nodes and key junctions

presented in Figure 2a.

The minimum and maximum effective green time is set to �̅�𝑚𝑖𝑛 = 0.1 and �̅�𝑚𝑎𝑥 = 0.6

for this problem instance. These values are informed based on the network topology (see Figure

2a), in which all nodes contain at least 4 connecting links, with most having at least 3 incoming

links. The above constraints ensure that effective green time is given to all links.

1

8

4 5 63

2

15 19

17

18

7

12 11 10 16

9

20

23 22

14

13 24 21

3

1

2

6

8

9

11

5

15

122313

21

16 19

17

2018 54

55

50

48

29

51 49 52

58

24

27

32

33

36

7 35

4034

41

44

57

45

72

70

46 67

69 65

25

28 43

53

59 61

56 60

66 62

68

637673

30

7142

647539

74

37 38

26

4 14

22 47

10 31

a) b)

Figure 2 Experiment test networks. The evacuation area is enclosed in a rectangle, with the evacuation points highlighted

with a triangle, and additional sink nodes with a dashed triangle. The signal phasing of key junctions highlighted with a

dashed square are controlled by the algorithm. a) Sioux-Falls network (SFN) with closed link presented in a circle. b)

Enriched Sioux-Falls network (ESFN)

The model aims to evacuate users generated in the danger zone and take them to the safe

node. All nodes within the evacuation area are assumed to have an equal pre-defined demand

of 3,200 vehicles, flooding the network in case of simultaneous departure.

A large-scale version of the SFN is used to determine its scalability and applicability

(Figure 2b), consisting of 282 nodes and 334 arcs (Chakirov and Fourie, 2014). A small

alteration to the evacuation procedure is introduced to compensate for the larger network size

– we introduce a secondary sink node in node 2.

It is important to note that, due to the new network structure, the source nodes increase

from 4 to 20 nodes to create the same danger zone, each with a demand of 640 vehicles. For

the remainder of this paper, we will refer to this network as Enriched Sioux Falls Network

(ESFN).

We propose a selection of Derivative-Free Algorithms (DFO) to evaluate in Table 4.

Similar to simulation-optimisation strategies algorithms, DFOs are categorised into local

greedy and global elitist algorithms. Within local search, direct methods continuously evaluate

trial solutions, and model-based search creates a surrogate model to perform a guided search.

Global deterministic and model-based search are analogous to the local direct and model-based

search counterparts but partition the search space. Finally, stochastic methods allow

intermediate lower quality solutions to avoid local optima.

We consider both local and global methods in our test, including deterministic and

stochastic methods. A scenario with 2 activated paths 𝑘 and a decision time step size Δ𝑡𝑑 of 10

seconds was selected to test all algorithms, using the SFN.

As shown in Figure 3, local methods provide far from optimal solutions due to their

inability to escape local optima. On the performance of the global algorithms, the Simulated

Annealing and the Genetic Algorithm outperform other solvers. The latter provides the best

solution, with the highest number of departures, evacuation and shortest mean travel time.

Given the results of this analysis, we chose to adopt the Genetic Algorithm as the

optimisation solver used by the bi-level approach. This algorithm consists of three main

components: selection, crossover and mutation.

We use a binary tournament to pair solutions and select the best as indicated by their

fitness value. Although this process discards well-performing solutions as does not compare

candidates to the population, it can be parallelised to improve computational performance

(Noraini and Geraghty, 2011).

The crossover operator combines two parent solutions to create two new unique

solutions. We use a BLX-alpha crossover method as it allows exploration of solution further

than the parents (Eshelman and Schaffer, 1993). The size of the search space explored is

defined by parameter 𝛼, where a higher value increases the search space.

Finally, mutation alters the parent solutions to create unique solutions outside the range

of the parent solutions. We use the non-uniform mutation (Michalewicz, 1992), which reduces

the mutation range after every generation, encouraging convergence as the algorithm runs. The

decrease in range rate is controlled by the 𝛽 parameter, where a greater value of 𝛽 increases

the rate of decrease.

Figure 3 Key performance indicators for different Derivative Free Algorithms.

Table 4 Derivative-Free Optimisation algorithms used.

DFO Algorithm Type Solver Author

Nelder-Mead Simplex (NM) Local Direct fminsearch Mathworks

Implicit Filtering (IF) Local Model-Based imfil Kelley (2011)

Branch and Fit (BF) Global

Deterministic snobfit

Huyer & Neumaier

(2008)

Particle Swarm (PS) Global Stochastic Pswarm Vaz & Vicente (2007)

Simulated Annealing (SA) Global Stochastic Original -

Genetic Algorithm (GA) Global Stochastic Original -

4.3. Parameter Tuning

We proceed with manual tuning to determine suitable control parameters for the genetic

algorithm. The control parameters considered include 𝛼 and 𝛽, which control crossover and

mutation search spaces, and 𝑝𝑐 and 𝑝𝑚, which represent probability parameters for crossover

and mutation respectively. A total of 9 scenarios are considered, with their parameters

summarised in Table 5. The evacuation parameters decision time step Δ𝑡𝑑, and activated paths

𝐾 remain fixed in all cases. The scenario naming convention describes the evacuation

parameters used such that K1 denotes 𝐾 = 1 and D10 indicates that Δ𝑡𝑑 = 10.

As observed in Figure 4, higher values of 𝛼 and 𝛽 provide higher fitness scores than the

original K2D10. Both parameters provide initially a large search space that gradually decreases

in later iterations.

It can be observed that increasing crossover rate let to improvements in performance, as

well as reducing the mutation rate. The first provides more solution combinations, while the

second reduces the stochasticity of solutions. Despite mutation providing a means to deviate

from local optima, high mutation probabilities may result in a high-degree stochastic search

algorithm rather than elitist search mechanism. Conversely, providing a very low value does

not allow the algorithm to explore sufficient are of the solution surface to find a close to the

optimal solution, which is the case in m2. Table 5 Sioux Falls evacuation calibration scenarios.

Scenario 𝐾 Δ𝑡𝑑 𝛼 𝛽 𝑝𝑐 𝑝𝑚

K2D10-n 2 10 0.5 0.5 0.5 0.1

K2D10-a1 2 10 0.6 0.5 0.5 0.1

K2D10-a2 2 10 0.8 0.5 0.5 0.1

K2D10-b1 2 10 0.5 0.6 0.5 0.1

K2D10-b2 2 10 0.5 0.8 0.5 0.1

K2D10-c1 2 10 0.5 0.5 0.6 0.1

K2D10-c2 2 10 0.5 0.5 0.8 0.1

K2D10-m1 2 10 0.5 0.5 0.5 0.05

K2D10-m2 2 10 0.5 0.5 0.5 0.01

Figure 4 Key performance indicators for parameter calibration of the Genetic Algorithm. Refer to Table 5 for scenario

names. Scenario “n” refers to the standard K2D10-n.

4.4. Results and discussion

The algorithm is run under the scenarios presented in Table 9 under the network configurations

presented in Section 4.2. An initial run is executed without considering signal phasing to

evaluate the effectiveness of evacuation staging. A benchmark problem is used to simulate a

worst-case solution in which all demand is released simultaneously at the beginning of the

evacuation.

The model was run on a modest workstation with an Intel Core i7-4790 CPU and 8GB

RAM. The runtime increases given a higher value of 𝑘, but the variation is not substantial, as

all the scenarios required approximately 5 and 16 hours to complete in the SFN and the ESFN

respectively. This level of performance was deemed to be acceptable for pre-planning purposes

yet unsuitable for real-time implementation. Further efforts to accelerate the algorithm are to

increase parallelisation of the genetic algorithm or the use of high-performance cloud

computing facilities.

Table 6 Sioux Falls evacuation calibration scenarios.

Scenario 𝑘 Δ𝑡𝑑 𝛼 𝛽 𝑝𝑐 𝑝𝑚

Benchmark 5 10 - - - -

K1D10 1 10 0.6 0.8 0.6 0.05

K1D20 1 20 0.6 0.8 0.6 0.05

K2D10 2 10 0.6 0.8 0.6 0.05

K2D20 2 20 0.6 0.8 0.6 0.05

K3D10 3 10 0.6 0.8 0.6 0.05

K3D20 3 20 0.6 0.8 0.6 0.05

The results are summarised in Table 7. The benchmark scenario involves an

uncoordinated evacuation which saturates the network immediately after the evacuation

warning is emitted. As expected, this is the worst performing problem instance - the evacuation

process is not able to recover from the immediate surge in demand, with the rate of vehicle

arrivals at the evacuation point decreasing over time (Figure 5a). This scenario also yields the

lowest total departures and evacuations among all scenarios, meaning that congestion occurs

close to the source nodes, preventing vehicles from accessing the network.

The remaining instances performed better due to a combination of reduced decision time-

steps and increased activated route paths. The provision of additional paths increases the

network capacity, thus allowing a higher rate of evacuations to take place. However, the

evacuation improvement is reduced for 𝑘 > 2 due to greater convergence between the paths.

An alternate run is carried out with signal phasing control at node 3, which provided

improvements in fitness value, departure and arrival of evacuees in all the scenarios, at the cost

of a marginal increase in mean travel time.

The same experiment is carried out under an alternate Sioux Falls network containing

two more sink nodes (Figures 2a), as well as one where links 29 and 48 are unavailable. In

addition, every network is tested with signal phasing control at nodes 3 and 8 to evaluate its

benefit. The main benefit of adding more sink nodes is the increased proportion of evacuees

arriving to their destination, as well as a reduction in wait time. As a result, despite reduction

in the total departures and arrivals, an increase in fitness value is observed, particularly for

scenarios 𝑘 ≥ 2 activated paths. This is further reflected in a reduced mean evacuee travel time.

a) b)

c) d)

Figure 5 Performance parameters for the SFN case. a) Cumulative weighted sum of evacuees over time. b) Proportion of

evacuated users over time in case K2D10. c) Travel time and distance ratio. d) Travel distance deviation between users.

The introduction of signal phasing results in a general increase in departures and arrival

rates in all scenarios, yet mean travel time and distance deviation remain consistent (Figure 5c

and 5d). With the addition of a destroyed link at 29 and 48, longer evacuation times are seen

throughout the network. Figure 5c indicates that the increased travel time is directly related to

the increased path distance as a result of the link destruction, given that travel time and distance

ratio varies similarly in all network configurations.

Concerning the arrival and departure rates, a significant increment is observed for 𝑘 = 1

scenarios. However, the opposite behaviour is observed for larger values of 𝑘, resulting in a

reduction in completed evacuations.

Finally, as shown in Figure 5b, the damage network case results in reduced proportion of

completed evacuations throughout the evacuation process. Note the proportion of evacuations is

conceived using the number of users that have entered the network. Hence, a reduction in the

proportion of completed evacuations is possible if the numbers of entries to the network exceed

the number of exits.

With respect to the sink arrival rates, in all cases sink node 1 receives the greatest

proportion of users (Figure 7). As expected, the disparity increases when we introduce the

destroyed link, particularly in the absence of signal control optimisation. This is a result of

user’s route choice given perceived travel time throughout the network.

Another observation relates to the constant evacuation rate when 𝑡 > 100, as seen in

Figure 5a. Note that a similar behaviour is observed in all network configurations. Therefore,

the solutions not only significantly improve upon a do-nothing approach, but also create a

stable evacuation process that allows users to exit the network at a constant rate. Interestingly,

the do-nothing solution performs similarly to the rest at the beginning of the evacuation, under

free-flow network conditions. Evacuation rate promptly decreases as congestion starts

affecting the evacuation process.

The algorithm provides more stable results in the ESFN case study, where we observe an

improvement in the fitness value when increasing the number of activated paths and when

reducing the decision time step (Table 8). A more significant increase is observed when

introducing signal phasing, becoming more significant in the case where 𝑘 = 3.

The inclusion of additional sink nodes and signal phasing result in an increase in fitness,

total departures and evacuations in expense of travel time. Both also result in longer trips, and

greater deviation in the distance travelled between the evacuees (Figure 6c and 6d). As

observed in the SFN, signal phasing provides further improvements in evacuation completion

and fitness values.

a) b)

c) d) Figure 6 Performance parameters for the ESFN case. a) Cumulative weighted sum of evacuees over time. b) Proportion of

evacuated users over time in case K2D10. c) Travel time and distance ratio. d) Travel distance deviation between users.

The results suggest that demand management is an effective method to optimise

evacuations as all solutions improve evacuation upon the benchmark scenario and create stable

evacuation processes. The behaviour is represented by the constant gradient in Figure 5a and

Figure 6a.

It is interesting to note that the benchmark solution significantly reduces in effectiveness

as the evacuation is carried out, which may be particularly misleading for coordinating agencies

on the effectiveness of the evacuation. Of particular interest is the behaviour Figure 6a, where

the benchmark evacuation outperforms K1 strategies at the beginning of the evacuation.

Figure 7 Percentage evacuees accessing each node. K2D10 scenario.

a) b)

Figure 8 GA performance – fitness value variation over runtime under no signal phasing. a) SFN. b) ESFN.

On the evacuation performance under each network, it is evident that more vehicles

successfully evacuated the SFN, and the proportion of completed evacuation among departing

users is significantly smaller in the ESFN. This is a result of the smaller network density ESFN,

which contains several junctions (nodes) within what would be a single link in the SFN.

The provision of additional paths leads to reduced performance gains. The demand

monitoring scheme provides a more significant evacuation improvement than the subsequent

path provision. In all network configurations (except under the damaged network) greater

increase in fitness values is seen for 𝑘 ≤ 2. The variability suggests that higher path

convergence exists in the ESFN, and the performance increase is a result of the additional

capacity. Further performance gains are achieved using signal phasing despite the generalised

increase in travelling distance deviation, being more significant in the ESFN.

Figures 8a and 8b indicate that in both networks the algorithm converges into a stable

solution. As expected, problem instances with a more substantial number of activated paths

require a longer time to converge due to a higher number of decision variables. Although not

presented, a similar behaviour is observed when considering signal phasing and alternative

network conditions. Potential solutions include the use of a modified version of the k-shortest

path algorithm, or the consideration of alternative evacuation strategies that utilise crossing

elimination, which would reduce network congestion.

While the presented algorithm provides a platform to model urban evacuation strategies

using staging and signal phasing for pre-planning purposes, some improvements would provide

more robust results. Despite the high car ownership rates in Sioux Falls (United States Census

Bureau, 2017), the assumption that only a single mode of transport is used is unrealistic in

urban networks (Abdelgawad and Abdulhai, 2010). In addition, we consider a single type of

evacuee and do not identify priority groups.

While previous work limit destination choice by evacuee type (Bish et al. 2014),

distinctions between users regarding route choice and driving behaviours would provide more

realistic results. Furthermore, despite the increase in evacuation performance due to demand

staging, the study does not consider potential harm, physical or psychological, to evacuees are

ordered to wait at departure nodes (Hu et al., 2017). Finally, although some choice is given to

the users, the algorithm assumes that vehicles will not deviate from their initial route choice

and will comply with indications from evacuation controllers. Modelling compliance requires

research on educational and social factors, as well as providing greater uncertainty.

Table 7 Sioux Falls network results.

Network

Fitness Value

(vehicles/

time step)

Weighted

evacuations

(vehicles/

time step)

Total

departures

(vehicles)

Total

evacuations

(vehicles)

Mean travel

time

(time steps)

Value % Value % Value % Value % Value %

SF

N

BM 5,341 0 5,458 0 5,991 0 4,934 0 204.6 0

K1D10 8,664 62 8,831 62 11,302 89 9,771 98 192.1 6

K1D20 8,554 60 8,723 60 11,595 94 9,878 100 182.2 11

K2D10 9,091 70 9,259 70 11,393 90 10,254 108 187.6 8

K2D20 8,994 68 9,157 68 12,298 105 10,156 106 184.3 10

K3D10 9,199 72 9,347 71 11,679 95 10,150 106 180.6 12

K3D20 9,172 72 9,341 71 11,570 93 10,308 109 181.3 11

SF

N w

ith

sig

nal

ph

asin

g

BM 5,341 0 5,458 0 5,991 0 4,934 0 204.6 0

K1D10 8,604 61 8,775 60 11,075 85 9,460 92 187.2 9

K1D20 8,682 63 8,854 62 11,629 94 9,755 98 185.9 9

K2D10 9,164 72 9,318 71 11,890 98 10,319 109 188.3 8

K2D20 9,091 70 9,147 68 11,845 98 10,223 107 190.7 7

K3D10 9,218 73 9,401 72 11,501 92 10,223 107 186.5 9

Table 8 Enriched Sioux Falls network results.

Network

Fitness Value

(vehicles/

time step)

Weighted

evacuations

(vehicles/

time step)

Total

departures

(vehicles)

Total

evacuations

(vehicles)

Mean travel

time

(time steps)

Value % Value % Value % Value % Value %

ES

FN

BM 1,233 0 1,517 0 3,920 0 1,491 0 401.8 0

K1D10 1,949 58 2,321 53 8,025 105 3,584 140 460.7 -15

K1D20 1,933 57 2,273 50 7,495 91 3,557 139 470.5 -17

K2D10 2,185 77 2,534 67 6,856 75 3,705 148 447.5 -11

K2D20 2,187 77 2,521 66 6,819 74 3,647 145 449.2 -12

K3D10 2,269 84 2,572 70 6,185 58 3,562 139 429.1 -7

K3D20 2,234 81 2,546 68 6,346 62 3,487 134 437.7 -9

ES

FN

wit

h

sig

nal

lin

g

BM 1,233 0 1,517 0 3,920 0 1,491 0 401.8 0

K1D10 2,257 84 2,431 60 5,106 30 3,593 141 291.7 27

K1D20 2,224 80 2,388 57 5,313 36 3,586 141 318.9 21

K2D10 2,270 84 2,541 68 6,419 64 3,697 148 417.6 -4

K2D20 2,318 88 2,523 66 6,036 54 3,713 149 327.5 18

K3D10 3,004 144 3,266 115 6,824 74 4,457 199 386.1 4

K3D20 2,975 141 3,253 114 7,327 87 4,535 204 356.2 11

K3D20 9,268 74 9,463 73 11,963 100 10,328 109 190.2 7 BM 6,687 0 6,783 0 6,702 0 5,923 0 177.3 0

SF

N-m

K1D10 9,294 39 9,477 40 10,580 58 9,036 63 212.4 6

K1D20 9,483 42 9,634 42 10,071 50 9,625 65 183.7 17

K2D10 9,344 40 9,565 41 10,429 56 9,747 71 197.7 -12

K2D20 9,288 39 9,746 44 10,798 61 10,149 53 191.3 -8

K3D10 10,613 59 10,089 49 12,367 85 10,850 63 166.6 -20

K3D20 10,605 59 12,209 80 12,800 91 12,176 65 147.7 -4

SF

N-m

wit

h s

ign

al

ph

asin

g

BM 6,687 0 6,783 0 6,702 0 5,923 0 177.3 0

K1D10 10,108 51 10,277 52 10,344 54 9,291 57 202.1 -14

K1D20 10,143 52 10,265 51 10,918 63 9,364 58 144.4 19

K2D10 10,057 50 10,245 51 11,476 71 9,489 60 193.5 -9

K2D20 10,346 55 10,435 54 11,455 71 9,270 57 194.5 -10

K3D10 10,484 57 10,653 57 11,466 71 10,213 72 196.8 -11

K3D20 10,579 58 11,556 70 12,460 86 11,391 92 211.0 -19

SF

N-m

d

BM 6,286 0 6,376 0 6,771 0 5,872 0 181.0 0

K1D10 9,645 53 9,770 53 11,732 73 10,167 73 163.4 10

K1D20 10,310 64 10,435 64 10,569 56 9,578 63 171.5 5

K2D10 9,750 55 9,948 56 11,911 76 10,330 76 194.4 -7

K2D20 9,144 45 9,338 46 11,593 71 10,334 76 189.4 -5

K3D10 11,412 82 11,628 82 12,800 89 11,316 93 195.2 -8

K3D20 11,926 90 12,101 90 12,800 89 12,023 105 168.6 7

SF

N-m

d w

ith

sig

nal

ph

asin

g

BM 6,286 0 6,376 0 6,771 0 5,872 0 181.0 0

K1D10 10,281 64 10,442 64 12,428 84 11,190 91 184.9 -2

K1D20 10,729 71 10,886 71 12,800 89 11,050 88 167.1 8

K2D10 10,012 59 10,070 58 12,609 86 11,166 90 197.9 -9

K2D20 10,047 60 10,432 64 12,430 84 10,752 83 188.1 -4

K3D10 12,091 92 12,291 93 12,800 89 12,363 111 183.6 -1

K3D20 12,266 95 12,460 95 12,800 89 12,454 112 184.6 -2

ES

FN

-m

BM 1,751 0 2,015 0 4,039 0 1,346 0 272.4 0

K1D10 3,778 116 3,984 98 6,488 61 4,790 256 306.4 12

K1D20 3,701 111 4,004 99 7,993 98 4,937 267 365.1 34

K2D10 4,011 129 4,341 115 8,515 111 5,009 272 353.5 30

K2D20 3,923 124 4,219 109 8,118 101 4,950 268 346.9 27

ES

FN

-m w

ith

sig

nal

lin

g

BM 1,751 0 2,015 0 4,039 0 1,346 0 272.4 0

K1D10 4,257 143 4,481 122 7,060 75 5,475 307 310.1 14

K1D20 4,160 138 4,425 120 7,549 87 5,374 299 332.2 22

K2D10 4,573 161 4,886 142 9,233 129 5,835 333 333.9 23

K2D20 4,395 151 4,680 132 8,449 109 5,773 329 320.0 17

5. Conclusion

In this paper, we develop a bi-level modelling framework that is used to optimise evacuation

strategies using a network demand management approach. A driving principle that was adopted

in its design is that centralised control is difficult to apply in emergency evacuations, therefore

giving users an increased level of control over their choice of paths, when compared to previous

studies. An efficient traffic flow model is adopted, implemented as a black-box model that

captures traffic behaviour patterns during the evacuation.

The methodology was tested in two different networks under a variety of scenarios. The

analysis of the results indicates that demand management is an effective method to improve

the efficiency of evacuations – with improvements observed across all key performance

indicators when compared to the benchmark case. Further improvements in evacuation

parameters were observed when controlling signal phasing simultaneously. A range of

Derivative-Free Optimisation algorithms was evaluated, with Genetic Algorithms shown to

outperform other options and adopted as the solver that underpins the overall solution search

process.

Even though a degree of personal choice (for routing decisions) is considered by this

framework, further work is essential to establish the impact of varying rates of departure time

compliance and information quality. Our future research in this area will explore the

applicability of probabilistic decision-making techniques, including the use of myopic routing

models. Monte Carlo simulation can be used to model a range of user behaviours, which is

expected to come at the expense of longer runtimes. Furthermore, the use of a variety of

transport modes should be considered to aid population groups with accessibility problems or

lack of access to private vehicles.

Acknowledgements

The research was supported by the UK Engineering and Physical Sciences Research Council

(EPSRC) as part of the Sustainable Civil Engineering Centre for Doctoral Training (Grant

number EP/L016826/1).

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Appendix

We provide below the network configurations that were introduced in Section 4.1 for the

evaluation of the formulation in larger problem instances.

Networks

a) Network A.

b) Network B.

c) Network C.

d) Network D.

Figure A Sample networks for linear and bi-level algorithm testing. Source and sink nodes are marked with a square and

dashed circle, respectively.