A Conflict-Based Path-Generation Heuristic for Evacuation Planning1

Victor Pillaca, Pascal Van Hentenrycka,b,∗, Caroline Evena2

aNational ICT Australia (NICTA), 115 Batman Street, West Melbourne, Australia3

bAustralia National University (ANU), Canberra, Australia4

Abstract5

Evacuation planning and scheduling is a critical aspect of disaster management and national security ap-

plications. This paper proposes a conflict-based path-generation approach for evacuation planning. Its key

idea is to decompose the evacuation planning problem into a master and a subproblem. The subproblem

generates new evacuation paths for each evacuated area, while the master problem optimizes the flow of

evacuees and produce an evacuation plan. Each new path is generated to remedy conflicts in the evac-

uation flows and adds new columns and a new row in the master problem. The algorithm is applied to

a set of large-scale evacuation scenarios ranging from the Hawkesbury-Nepean flood plain (West Sydney,

Australia) which require evacuating in the order of 70,000 persons, to the New Orleans metropolitan area

and its 1,000,000 residents. Experiments illustrate the scalability of the approach which is able to produce

evacuation for scenarios with more than 1,200 nodes, while a direct Mixed Integer Programming formula-

tion becomes intractable for instances with more than 5 nodes. With this approach, realistic evacuations

scenarios can be solved near-optimally in reasonable time, supporting both evacuation planning in strategic,

tactical, and operational environments.

Keywords: Evacuation planning and scheduling, regional evacuation, disaster management, conflict-based6

path generation7

1. Introduction8

Natural disasters, such as hurricanes, floods, and bushfires, affect numerous populated areas and may9

endanger the lives and welfare of entire populations. Evacuation orders are some of the most important10

decisions performed by emergency services: They ensure the safety of persons at risk by instructing them11

to evacuate the threatened region, be it a building (e.g., fire), a neighborhood (e.g., industrial hazard), or a12

whole region (e.g., flood). Evacuation planning also arises at strategic, tactical, and operational levels. At a13

strategic level, the goal is to design evacuation plans for specific areas and possible threats (e.g., evacuation14

plans for the surroundings of a nuclear power plant). At a tactical level, the goal is to design evacuation15

∗Corresponding author

Preprint submitted to Elsevier February 4, 2015

plans for an area facing an incoming threat (e.g., evacuation of a flood plain following high precipitations).16

Finally, at the operational level, the goal is to schedule an evacuation, possibly adjusting the evacuation17

plan in real-time as the threat unfolds.18

Existing work in evacuation planning typically consider free-flow models in which evacuees are dynami-19

cally routed in the network without constraining evacuees from a same area to follow a same path. However,20

free-flow models do not conform to existing evacuation methodologies followed by authorities in which evac-21

uated areas are assigned specific evacuation routes (see, for instance, (SES-NSW, 2005)).22

The critical challenge for evacuation planning is how to deal with congestion, which can rapidly propagate23

in the network, preventing people from evacuating and increasing the risk for evacuees of being trapped in24

the affected areas (Townsend, 2006; Fonseca et al., 2011). A first strategy to reduce congestion is to prepare25

evacuation plans that minimize the conflicts between evacuated areas, avoiding flows of evacuees that will26

saturate the road network. The key is to choose evacuation paths and to stage the evacuation to distribute27

the pressure on the road network evenly. A second strategy aims at increasing the capacity of the road28

network by reversing the direction of certain lanes on major roads. This procedure, known as contraflows29

(Matherly, 2012), is implemented on a regular basis for large-scale evacuations, such as during the evacuation30

of New Orleans in preparation for hurricane Katrina in 2005. Contraflows can virtually double the capacity31

of the selected roads without affecting traffic safety. However, they also increase the complexity of evacuation32

planning by adding a new degree of freedom: Kim et al. (2008) demonstrated that selecting contraflow roads33

is by itself NP-Hard. This work is the first to combine these two strategies and proposes the first scalable34

and comprehensive approach to simultaneously design evacuation plans and select which roads should be35

used in contraflow.36

From a technical standpoint, the algorithm can be broadly characterized as a Conflict-Based Path-37

Generation Heuristic (CPG for short), which decomposes the problem by considering separately the genera-38

tion of evacuation paths (subproblem) and the scheduling of the evacuation (master problem). The master39

problem selects exactly one evacuation path per evacuated area, and schedules the departure time of all40

evacuees. This leads to an evacuation plan in which different evacuation areas may be in conflict, in other41

words, their evacuation paths may converge to road segments which capacity is saturated. To remedy this,42

the path-generation subproblem finds new evacuation paths by solving a path of least cost under constraints,43

where the edge costs are derived from the conflicts and congestion in the incumbent solution. The procedure44

iterates between the subproblem and master problem until a certain convergence criterion is met.45

Experimental results show that the CPG algorithm is able to plan effective, large-scale evacuations with46

contraflows for threat scenarios involving up to 1,000,000 people and a network of 1,280 nodes, while a47

direct Mixed Integer Problem formulation becomes intractable for instances with more than 5 nodes. The48

approach can also be adapted to other contexts such as building or pedestrian evacuations.49

The remainder of this paper is organized as follows: Section 2 reviews related work, Section 3 formulates50

2

the evacuation planning problem, Section 4 presents the proposed approach, Section 5 presents computa-51

tional experiments on instances derived from real case studies. Finally, Section 6 concludes this paper.52

2. Related work53

According to Hamacher and Tjandra (2002), evacuation planning can be tackled using either microscopic54

or macroscopic approaches. Microscopic approaches focus on modeling and simulating the evacuees indi-55

vidual behaviors, movements, and interactions. Macroscopic approaches, such as the one presented in this56

study, aggregate evacuees and model their movements as a flow in the evacuation graph.57

The majority of macroscopic approaches solve the evacuation planning problem as a flow on a time-58

expanded graph. For instance, Lu et al. (2003, 2005) propose three heuristics to design an evacuation59

plan with multiple evacuation routes per evacuated node, minimizing the time of the last evacuation. The60

authors show that in the best case the proposed heuristic is able to solve randomly generated instances of61

up to 50,000 nodes and 150,000 edges in under 6 minutes. Liu et al. (2007) propose a Heuristic Algorithm62

for Staged Traffic Evacuation (HASTE), a similar algorithm that generates evacuation routes and schedule63

the evacuation of evacuated nodes in sequence. The main difference between HASTE and the previous64

algorithms is that it relies on a Cell Transmission Model (CTM)(Daganzo, 1994) to model more accurately65

the flow of evacuees.66

Acknowledging that all evacuated nodes may not be under the same level of threat, Lim et al. (2012)67

consider a short-notice regional evacuation maximizing the number of evacuees reaching safety weighted by68

the severity of the threat. The authors propose two solution approaches to solve the problem, and present69

computational experiments on instances derived from the Houston-Galveston region (USA) with up to 6670

nodes, 187 edges, and a horizon of 192 time steps.71

Other authors have focused on modeling more accurately the transportation network. For example,72

Bretschneider and Kimms (2011, 2012) present a free-flow mathematical model that describes in detail73

the street network and, in particular, the lane configuration at intersections of the network. They present74

computational experiments on generated instances with a grid topology of up to 240 nodes, 330 edges, and75

considering 150 times steps. Bish and Sherali (2013) present a model based on a CTM that assigns a single76

evacuation path to each evacuated node. Computational results include instances with up to 13 evacuated77

nodes, 2 safe nodes, and 72 edges.78

Dynamic aspects of evacuation have also been considered. For instance, Lin et al. (2008) present a time79

expanded graph in which they allow for time-dependent attributes such as varying capacity or demand.80

The authors apply their findings on a case study considering the evacuation of a 11-floor building with81

approximately 60 nodes, 100 edges, and 60 time steps.82

Contrarily to the present work, existing approaches do not provide evacuation plans that comply with83

the procedures in place in most emergency services and local authorities, in particular due to the fact that84

3

they do not guarantee that a single evacuation route will be assigned to a particular evacuated area. To85

the best of our knowledge, only a handful of studies design evacuation plans that produce both a set of86

evacuation routes and an evacuation schedule. Huibregtse et al. (2011) propose a two-stage algorithm that87

first generates a set of evacuation routes and feasible evacuation times, and then assigns a route and time88

to each evacuated area using an ant colony optimization algorithm. The main difference with the present89

work is that the approach does not explicitly schedule the evacuation but relies on a third party simulator90

(EVAQ) to simulate the departure time of evacuees depending on the evacuation time decided for each area91

and evaluate the quality of the solution. In later work, the authors studied the robustness of the produced92

solution (Huibregtse et al., 2010), and strategies to improve the compliance of evacuees (Huibregtse et al.,93

2012). Pillac et al. (2014) first introduced the CPG approach which was extended to contraflows by Even94

et al. (2014). Even et al. (2015) proposed a different approach to produce convergent evacuation plans that95

decomposes the evacuation problem in a tree-design problem and an evacuation scheduling problem.96

Contraflow strategies have been studied in the context of free-flow models. Kim et al. (2008) present a97

macroscopic optimization model that finds a contraflow network configuration such that the total evacuation98

time is minimized. The authors separate the contraflow network reconfiguration model, which decides which99

roads will be used in contraflow, from the route planner, which gives information on the flow of vehicles and100

is treated as a black box. They propose three approaches depending on the ratio of the number of evacuees101

over the network capacity (measure as its min-cut value), referred to as overload. The authors propose102

an Integer Program (IP) for low overload, and two greedy heuristics for medium and high overload. They103

compare the performance of three route planners from the literature. The main limitation of their approach104

is the fact the IP formulation becomes infeasible if not all evacuees can be evacuated. Xie and Turnquist105

(2011) present a detailed model of lane configuration for road segments and intersections, which includes106

contraflow strategies. They propose an integrated Lagrangian and tabu search approach that decides the107

lane configuration at each intersection to avoid crossings and maximize flow. However, the method does not108

produce actual evacuation plans but a configuration of the evacuation network.109

Microscopic approaches include the work by Richter et al. (2013) who challenge two assumptions generally110

made: The existence of a central planning entity with global knowledge, and the ability of this entity to111

communicate order to evacuees. They propose a decentralized decision making approach supported by112

smartphones and mobile applications. We note however that our target applications, such as evacuations113

for floods and hurricanes, use central decision making and have the time and ability to communicate their114

decisions.115

The approach described in the present work relates to the Column Generation (CG) optimization tech-116

nique. Its central concept is to decompose the problem at hand between a master and a subproblem. The117

master problem considers only a subset of columns and the method iteratively generates columns of negative118

reduced cost (assuming minimization) by solving the pricing subproblem. It has been widely used to solve119

4

large-scale MIP problems, and we refer the interested reader to the book by Desaulniers et al. (2005) and the120

study by Lubbecke and Desrosiers (2005) for a recent review of techniques and applications of column gen-121

eration. In particular, it has been used to solve multi-commodity network flow problems (MCNF) (Alvelos122

and Valerio De Carvalho, 2000), integer MCNF (Barnhart et al., 2000), origin-destination MCNF (Barnhart123

et al., 1997), and MCNF with side constraints on paths (Holmberg and Yuan, 2003). However, a distinctive124

feature of evacuation planning is the dependency between paths in the time-expanded network. More pre-125

cisely, a commodity (i.e., evacuees from a specific evacuated node) can only follow paths that correspond to126

the same physical path (sequence of edges in the evacuation graph). Therefore classical MCNF approaches127

cannot be applied directly, as one path in the evacuation model introduces multiple variables in the master128

problem. A straightforward CG decomposition of the evacuation problem described in this study consists in129

selecting in the master problem one evacuation plan for each evacuated area comprising of a an evacuation130

route and schedule. However, with that decomposition the complexity of the pricing subproblem is the same131

as the original problem, unless the scheduling of the evacuation is constrained to follow specific response132

curves (Pillac et al., 2015). In contrast, our approach does not consider the pricing problem explicitly, but133

heuristically generates new paths. Similar ideas were also used by Coffrin et al. (2011) and, to a lesser134

extent, in Massen et al. (2012).135

Contrary to existing approaches, the algorithm in this paper is the first to combine actionable evacuation136

plans, which provide an evacuation path and departure time to each evacuee, evacuation staging, which137

distributes the load on the network over time, and contraflow selection for roads. The result is an approach138

that provides high-quality actionable evacuation plans.139

3. Problem Formulation140

Figure 1 illustrates a general evacuation scenario. Figure 1(a) presents an evacuation scenario with one141

evacuated node (0) and two safe nodes (A and B). In this example, the evacuated node 0 has to be evacuated142

by 13:00, considering that certain links become unavailable at different times, (for instance, (2, 3) is cut at143

9:00), as indicated by the red labels. Figure 1(b) presents an abstraction of the evacuation scenario as an144

evacuation graph G = (N = E ∪T ∪S,A) where E , T , and S are the set of evacuated, transit, and safe nodes145

respectively, and A is the set of edges. Each evacuated node i is characterized by a number of evacuees di146

and an evacuation deadline fi (e.g., 20 and 13:00 for node 0 respectively), while each edge e is associated147

with a triple (se, ue, be), where se is the travel time, ue is the capacity, and be is the time at which the edge148

becomes unavailable, as indicated by the labels on edges and nodes. Finally, we note Ac ⊆ A the subset of149

edges that can be used in contraflow. In practice, Ac will contain major roads and one-way streets.150

A common way to deal with the space-time aspects of evacuation problems is to discretize the planning151

horizon into time steps of identical length, and to work on a time-expanded graph. Figure 2 illustrates the152

time-expanded graph generated from the evacuation graph of Figure 1(b). The labels on edges represent153

5

the edge capacity, while the labels on top of the nodes represent the corresponding time step. The node154

(2, 1) represents the node 2 at time step 1 (10:00). This graph Gx = (N x = Ex ∪T x ∪Sx,Ax) is constructed155

by duplicating each node from N for each time step. For each edge (i, j) ∈ A and for each time step t in156

which edge (i, j) is available, an edge (it, jt+s(i,j)) is created modeling the transfer of evacuees from node157

i at time t to node j at time t + s(i,j). In addition, edges with infinite capacity are added to model the158

evacuees waiting at evacuated and safe nodes. Finally, all evacuated nodes (resp. safe nodes) are connected159

to a virtual super-source vs (super-sink vt), modeling the inflow (outflow) of evacuees. The capacity of the160

edge between the super source and each evacuated node models the number of evacuees (demand) of that161

node. Note that some nodes may not be connected to either the super-source or super-sink (in light gray in162

this example), and can therefore be removed to reduce the graph size. The problem is then to find a flow163

from vs to vt that models the movements of evacuees in space and time.164

In this study, we consider a single threat scenario from which we derive the time when each evacuated165

node must be evacuated, and the time at which edges are closed. Following the practice in the field of166

emergency services operations, evacuees from a same node are evacuated to the same safe node following167

a single path, and evacuees can only wait or be held at their evacuated node or once they have reached a168

safe node. Both assumptions are motivated by practical considerations and to ensure the safety of evacuees.169

Finally, the edge capacity is fixed and does not depend on the flow. This is a common simplification that170

is compensated by the fact that edge capacities are set to ensure non-congested flow which reduces the171

complexity of the scheduling of the evacuation model.172

The objective is to first ensure that all evacuees reach a safe node, and then to schedule the evacuation.173

This second objective depends on the type of threat considered. For imminent threats, the goal is to get174

evacuees as fast as possible to one of the safe nodes. If we assume that evacuees safety is only threatened175

after the evacuation deadline, it is of practical interest to evacuate them as late as possible, as this leaves176

more time to potentially refine the threat forecast and hence avoids unnecessary evacuations. The decisions177

B2

0

3

1

A

11:00

11:009:00 8:00

11:00

13:00

13:00

(a) Evacuation Scenario

B 2

0

3

1

A

(1,10,9:00)

(1,10,9:00)(1,10,11:00)

(1,10,11:00) (1,10,11:00)

(2,5,13:00)

(1,10,∞)

(1,10,11:00)

(20,13:00)

(b) Evacuation Graph

Figure 1: Modeling of an Evacuation Planning Problem.

6

vs 0,020

0,1

2,1

B,2

0,2

1,2

B,3

0,3

1,3

A,3

∞ ∞ ∞

B,4

A,4

10

10 10

9:00 10:00 11:00 12:00 13:00

∞

∞ ∞

vt

∞

∞

2,0

B,0

1,0

A,0

3,0

B,1

1,1

A,1

3,1

A,2

1,410

5 5

Figure 2: Time-Expanded Graph for the Evacuation Scenario With 1-hour Time Steps.

that need to be made for each evacuated are the following: which safe node to evacuate to, which path to178

follow to reach the selected safe node, and how to schedule the departures over the horizon. In addition, the179

plan must specify which edge are used in contraflow. Finally, the global evacuation plan and schedule must180

respect the capacity of the network, and ensure that no evacuee travels on an edge that has been cut.181

We define e as the unique edge going in the opposite direction of edge e ∈ Ac. In addition, we define182

an arbitrary partition Ac = Ac|Ac such that ∀e ∈ Ac, e ∈ Ac. In order to control the contraflows, we183

create a binary variable ye for edge e ∈ Ac equal to 1 if edge e is used in its normal direction. With this184

definition, there are three possible configuration (ye, ye) for a road segment (e, e): (1, 1) if both edges are185

used in their normal direction, (1, 0) if edge e is in contraflow, (0, 1) if edge e is in contraflow. Let xke be186

a binary variable equal to 1 if and only if edge e ∈ A belongs to the evacuation path for evacuated node187

k, and ϕke be a continuous variable equal to the flow of evacuees from evacuated node k on edge e ∈ Ax.188

For convenience, we note e0 ∈ A the static edge associated with edge e ∈ Ax. Finally, we note δ−(i) (resp.189

δ+(i)) the set of incoming (resp. outgoing) edges of node i.190

Model (1-10) presents a Mixed Integer Program (MIP) formulation the Evacuation Planning Problem191

with Contraflows (EPP-CF). Constraints (2) ensure that exactly one path is used to route the flow coming192

from an evacuated node, while constraints (3) ensure the continuity of the path. Constraints (4) ensure the193

flow conservation through the time-expanded graph. Constraints (5) enforce the capacity of each edge in the194

time-expanded graph. Constraints (6) enforce the capacity on edges that allow contraflow. They allocates195

to e the capacity of edge e whenever e is used in contraflow, and forbid any flow on e when it is used in196

contraflow. Constraints (7) ensure that there is no flow of evacuees coming from an evacuated node k if197

edge e is not part of the evacuation path for k. Constraints (8) prohibit the simultaneous use of e and e in198

contraflow. The objective (1) maximizes the number of evacuees reaching safety.199

7

max∑k∈Ex

∑e∈δ−(vt)

ϕke (1)

s.t.∑

e∈δ+(k)

xke = 1 ∀k ∈ E (2)

∑e∈δ−(i)

xke −∑

e∈δ+(i)

xke = 0 ∀k ∈ E , i ∈ T (3)

∑e∈δ−(i)

ϕke −∑

e∈δ+(i)

ϕke = 0 ∀i ∈ N x \ vs, vt, k ∈ E (4)

∑k∈E

ϕke ≤ ue ∀e ∈ Ax \ Ac (5)

∑k∈E

ϕke ≤ ye0ue + (1− ye0)ue ∀e ∈ Ac (6)

ϕke ≤ ue ∗ xke ∀e ∈ Ax, k ∈ E (7)

ye + ye ≥ 1 ∀e ∈ Ac (8)

ye ∈ 0, 1 ∀e ∈ Ac (9)

ϕke ≥ 0, xke ∈ 0, 1 ∀e ∈ Ax, k ∈ E (10)

4. Conflict-Based Heuristic Path Generation200

Experimental results demonstrate that the EPP-CF model is computationally intractable for instances201

with over 5 evacuated node using a commercial MIP solver (Pillac et al., 2013). To address this issue,202

we propose a conflict-based heuristic path generation approach (CPG) that separates the generation of203

evacuation paths from the scheduling of the evacuation.204

Algorithm 1 gives an overview of the CPG approach. First, the algorithm generates an initial set of205

paths Ω′ (line 1) and solves the master problem to find an evacuation schedule that maximizes the number206

of evacuees reaching safety (line 2). The scheduleEvacuation function will be detailed in Section 4.1. The207

procedure then identifies critical evacuated nodes (line 4), which are not fully evacuated. This information is208

later used to generate new paths (line 5). Both findCriticalEvacuatedNodes and generatePaths methods209

will be detailed in Section 4.2. Finally, it solves the scheduling problem including the newly generated paths210

(line 6). The last four steps are repeated for a given number of iterations or until a predefined number of211

non-improving iterations has been reached (line 3).212

4.1. Evacuation scheduling master problem213

The master problem can be solved using a mixed integer program. Let Ω be the set of all feasible paths214

between evacuated nodes and safe nodes and Ωk be the subset of evacuation paths for evacuated node k.215

We define a binary variable xp which takes the value of 1 if and only if path p ∈ Ω is selected, a continuous216

8

Algorithm 1 The Conflict-Based Path Generation Approach (CPG).

HN-Input: G the evacuation graph, Gx the time-expanded graph.Output: S the best solution found

1: Ω′ ← generatePaths (G, ∅, E , ∅) . Subproblem2: S ← scheduleEvacuation (Ω′,G,Gx) . Master problem3: while stopping criterion not met do4: Ec ← findCriticalEvacuatedNodes (S)5: Ω′ ← Ω′ ∪ generatePaths (G,Ω′, Ec,S) . Subproblem6: S ← scheduleEvacuation (Ω′,G,Gx) . Master problem7: end while8: return S

variable ϕtp representing the number of evacuees to start evacuating on path p at time t, and a continuous217

variable ϕk accounting for the number of non-evacuated evacuees in node k. In addition, we denote by ω(e)218

the subset of paths that contain edge e and by τep the number of time steps required to reach edge e when219

following path p. Finally, we note Hp ⊆ H the subset of time steps in which path p is usable, and up the220

capacity of path p. Model (11)-(21) presents the evacuation scheduling master problem (CPG-MP):221

max∑p∈Ω

∑t∈Hp

ϕtp (11)

s.t.∑p∈Ωk

xp = 1 ∀k ∈ E (12)

∑p∈Ωk

∑t∈Hp

ϕtp + ϕk = dk ∀k ∈ E (13)

∑p∈ω(e)t−τe

p∈Hp

ϕt−τe

pp ≤ ue ∀e ∈ A \ Ac, t ∈ H (14)

∑p∈ω(e)t−τe

p∈Hp

ϕt−τe

pp ≤ yeue + (1− ye)ue e ∈ Ac, t ∈ H (15)

ye + ye ≥ 1 ∀e ∈ Ac (16)∑t∈Hp

ϕtp ≤ |Hp|xpup ∀p ∈ Ω (17)

ϕtp ≥ 0 ∀p ∈ Ω, t ∈ Hp (18)

ϕk ≥ 0 ∀k ∈ E (19)

ye ∈ 0, 1 ∀e ∈ Ac (20)

xp ∈ 0, 1 ∀p ∈ Ω (21)

The objective (11) maximizes the total flow of evacuees, which is equivalent to the number of evacuees222

reaching safety. Constraints (12) ensure that exactly one path is selected for each evacuated node, while223

9

(13-17)(18-19)

Figure 3: The Structure of the Evacuation Scheduling Master Problem Matrix.

constraints (13) account for the number of evacuated and non-evacuated evacuees. Constraints (14) and224

(15) enforce the capacity on the edges of the graph. Constraints (17) ensures that there is no flow on paths225

that are not selected. Constraints (16) prohibit the simultaneous use of e and e in contraflow.226

In practice, we only consider a subset of evacuation paths Ω′ ⊆ Ω each time we solve CPG-MP. Figure227

3 depicts the structure of the master problem matrix. Horizontal blocks represent groups of constraints228

numbered as in Model (11-21), while the shaded areas represent non-null coefficients in the matrix. Note229

that each constraint in group (17) only involves variables associated with the corresponding path and must230

be dynamically added to the model whenever a new path is considered. Nonetheless, a solution of CPG-MP231

considering the subset of paths Ω′′ ⊂ Ω′ is also a feasible when considering the set Ω′. Hence the solution232

from the previous iteration is used as starting solution for the current iteration. It is interesting to observe233

that the master model does not use a variable for each edge e and time step t. Instead, it reasons in terms234

of variables ϕtp which indicate how many evacuees leave along path p at time t. This leads to a reduction in235

the number of variable from |N ||A||H| to |Ω′||H|.236

4.2. Conflict-based path generation237

Traditionally, the generation of new columns searches for a column of positive reduced costs (assuming

maximization). With the decomposition proposed in this work, the subproblem aims at generating an

evacuation path, which in turns corresponds to multiple columns in the master problem. In addition, a new

path induces new constraints in the master problem. Consequently, the definition of the reduced cost of a

new path is non-trivial. To address this issue, we propose a conflict-based path generation which relies on

problem-specific knowledge to generate new columns that will potentially improve the objective function of

the master problem. First, we identify the subset E ′ ⊆ E of critical evacuated nodes, i.e., nodes that are not

fully evacuated in the current solution. Then, we include in E ′ all the evacuated nodes whose evacuation

paths share at least one edge with a node from E ′. Finally, we generate new paths for the critical evacuated

nodes E ′ by solving the following multiple-origins, multiple-destinations shortest path problem:

min∑k∈E′

∑e∈A

ceyke (22)

s.t.∑

e∈δ−(i)

yke −∑

e∈δ+(i)

yke = 0 ∀i ∈ T , k ∈ E ′ (23)

10

∑e∈δ+(k)

yke = 1 ∀k ∈ E ′ (24)

yke ∈ 0, 1 ∀k ∈ E ′, e ∈ A (25)

where yke is a binary variable taking the value of 1 if and only if edge e belongs to the path generated for

evacuated node k, and ce is the cost assigned to edge e. In order to generate diverse evacuation paths, the

cost ce of edge e is adjusted at each iteration using the following linear combination of the edge’s travel time

se, the number of occurrences of e in the current set of paths, and the utilization of e in the current solution:

ce = αtser

maxe∈E se+ αc

∑p∈Ω′

e∈p1

|Ω′|+ αu

∑p∈Ω′

e∈p

∑t∈Hp

ϕtp

ue(26)

where αt, αc, and αu are positive weights which sum is equal to 1, and r is a random noise factor initialized238

at 1, and increased in a range [1− ε, 1 + ε] when no improving solution is found.239

4.3. Scheduling the evacuation240

The model presented in the previous section maximizes the number of evacuees reaching safety. However,241

depending on the practical application considered, stakeholders may also be interested in minimizing the242

clearance time, or delaying the evacuation as much as possible (including a safety margin is needed).243

Minimizing the clearance time is of interest for and imminent threat, for instance in the case of a flash244

flood following a dam break. Delaying the evacuation is of relevance to minimize false alerts for slowly245

evolving threats with high uncertainty.246

In both cases, an explicit formulation can be derived by introducing a binary variable for each time

step controlling the flow of all evacuees. However this increases the complexity of the master problem by

adding |H| new binary variables. Consequently, in this section we propose a modification of the objective

function to improve the evacuation scheduling. The scheduling of the evacuation is achieved by associating

a time-dependent penalty with the flow of evacuees. Formally, let c be a high penalty for non-evacuated

evacuees, and ctp be the cost of 1 unit of flow on path p at time t, defined as:

ctp =

tH to minimize clearance time

H−(t+∑

e∈p se)

H to delay the evacuation

(27)

The modified CPG-MP model minimizes the total cost of the flow on all paths subject to the same

constraints:

min∑p∈Ω

ctpϕtp +

∑e∈Ex

cϕk (28)

s.t. (12)− (21)

11

(a) Map of the area of interest (b) Evacuation scenario

Figure 4: Geographical location of the case study

5. Computational experiments247

This section presents computational experiments on evacuation instances derived from real case studies.248

5.1. Hawkesbury Nepean case study249

We consider the evacuation of the Hawkesbury-Nepean (HN) floodplain, located North-West of Sydney250

(see Figure 4(a)), for which a 1-in-200 years flood will require the evacuation of 70,000 persons. The resulting251

evacuation graph, illustrated in Figure 4(b), contains 61 evacuated nodes, 5 safe nodes, 162 transit nodes,252

and 512 edges. We considered a horizon of 10 hours with a time step of 5 minutes (starting at 00h00). The253

evacuation deadlines and times at which edges are cut were derived from a 2D flood simulation assuming a254

inflow of 30,000 m3.s-1.255

In addition, we generated the instances HN60-Ix with a number of evacuees scaled by a factor of x ∈256

[1.1, 3.0]. The CPG and CPG-CF approaches are allowed 10 iterations, with a time limit of 300 seconds257

for the master problem at each iteration and 1800 seconds for the last iteration. Results are an average258

over 10 runs to capture the random nature of noise in the path generation heuristic. The path generation259

(subproblem) is performed in parallel for each evacuated node, and the MIP solver (master problem) uses260

parallel computing. In both cases the number of threads used is set to the number of available cores. All261

approaches were implemented using Java 7 and Gurobi 5.6.3, and experiments were conducted on a cluster262

of machines with AMD 6-Core Opteron 4334 processor (3.1GHz) and 32Gb of RAM.263

Table 1 presents computational results for the original HN61 instance and the HN61F-Ix instances for264

the Free-Flow model (see Appendix A), the CPG approach and the CPG approach allowing contraflows265

(CPG-CF). The first column reports the number of paths generated, the second and third give the number266

12

Num. Num. Num. CPU Perc. Evac.App. Instance Paths Cols (103) Rows (103) Time (s) Evac. Start

FF HN61F 73 29 4 100.0% 71HN61F-I1.1 73 29 4 100.0% 55HN61F-I1.2 73 29 3 100.0% 36HN61F-I1.4 73 29 4 100.0% 1HN61F-I1.7 73 29 4 98.4% 0HN61F-I2.0 73 29 4 95.5% 0HN61F-I2.5 73 29 4 92.2% 0HN61F-I3.0 73 29 4 90.1% 0

CPG HN61F 479 38 27 60 100.0% 27HN61F-I1.1 468 38 27 517 100.0% 10HN61F-I1.2 459 39 27 2262 100.0% 2HN61F-I1.4 427 34 26 348 98.4% 0HN61F-I1.7 394 33 24 369 95.3% 0HN61F-I2.0 368 31 24 52 92.2% 0HN61F-I2.5 478 42 28 118 88.4% 0HN61F-I3.0 466 40 28 448 81.5% 0

CPG-CF HN61F 494 31 38 55 100.0% 105HN61F-I1.1 495 32 38 74 100.0% 93HN61F-I1.2 508 37 39 133 100.0% 80HN61F-I1.4 510 39 42 224 100.0% 51HN61F-I1.7 488 42 43 250 100.0% 10HN61F-I2.0 450 39 42 2869 99.6% 0HN61F-I2.5 476 44 43 5472 97.9% 0HN61F-I3.0 479 41 43 4229 95.8% 0

Table 1: Experimental Results on Real-Size Instances.

of columns and rows in the MIP, the fourth reports CPU times, the sixth contains the percentage of evacuees267

reaching safety, and finally the seventh reports the time of the first evacuation.268

The results indicate that the evacuation plan produced by the CPG approach is very close to the bound269

provided by the FF model. On average, it evacuates 2.5% fewer vehicles, while the evacuation starts 15270

minutes earlier. These results are particularly compelling considering that the free-flow models are not271

realistic and are only useful to provide upper bounds on solution quality. In comparison, the plans produced272

by CPG-CF are able to evacuate 4.69% more vehicles than CPG and 37 minutes faster. The difference is273

explained by the additional capacity available when edges are used in contraflow. Note that both approaches274

generate a similar number of paths (465 on average). CPG and CPG-CF exhibit a different behavior in terms275

of CPU time. Instance HN61F-1.2 appears to be the most difficult for CPG, which can be explained by the276

tightness of the instance, in which the whole horizon is required to evacuate the 100% of the population.277

The remaining instances appear to be easier to solve, requiring 273s on average, regardless of whether the278

all vehicles can be evacuated in the horizon or not. This behavior can be explained by the high penalty279

costs for non-evacuated vehicles which may hinder the branch-and-bound solver. The CPG-CF approach280

on the other hand solves instance up to HN61F-I1.7 in 150s on average, but requires 4190s for instances in281

which not all evacuees can reach safety.282

13

0

10000

20000

0 200 400 600Time

Num

ber

of v

ehic

les

sim

Num. Veh

Plan

Quickest

Rnd. Quickest

Evacuation ProfileFree Flow

Figure 5: Comparison of evacuation profiles produced by the Free Flow (FF) model, Quickest and Random Quickest simulationscenarios on the HN61Finstance.

5.2. Validation through traffic simulation283

The optimization approaches presented in this work assume that the evacuees (or vehicles) flow over284

the evacuation network in a continuous and aggregated manner. In the real world however, evacuees are285

independent agents that move along the edges and show different behaviors in response to the evacuation286

plan. To assess the fitness and robustness of the results from the optimization, we introduce an agent-based287

traffic simulation based on the MATSIM simulation package (MATSim development Team, 2013).288

In this simulation, each evacuee is modeled as an agent with an individual plan composed by a start289

location (its evacuated area), a final destination (the chosen safe node), a path in the evacuation graph,290

and a departure time. Each individual plan can be either directly derived from the optimization results or291

generated by introducing random perturbations. The MATSIM simulation engine uses the set of plans to292

simulate the movement of evacuees in the evacuation graph. It models each edge of the graph as a queue,293

which realistically simulates a real-world transportation network, in particular by considering congestion.294

The first simulation experiment we conducted aims at studying the feasibility of the plan produced by295

the free flow model. Considering that FF does not produce a plan for each evacuee, we considered two296

scenarios, namely Quickest and Random Quickest. In the Quickest scenario, each evacuee goes to the closest297

accessible safe node at its departure time, and we ensure that the total volume of evacuees leaving each area298

is the same as the one produced by the optimization. In the Random Quickest scenario, we allow for more299

variation in the evacuees behaviors, and consider that 50% will go to the closest, 40% to one of the five300

closest, and 10% to a random safe node. In addition, we generate random departure times that depend on301

the earliest departure time of the neighboring areas and the latest departure time for the considered area.302

14

Instance Optim. Plan Rnd. Quickest Quickest

HN61F 100.0% 64.1% 38.6%HN61F-I1.1 100.0% 61.6% 37.4%HN61F-I1.2 100.0% 58.7% 37.3%HN61F-I1.4 100.0% 53.5% 36.9%HN61F-I1.7 98.4% 49.5% 33.0%HN61F-I2.0 95.5% 45.9% 29.1%HN61F-I2.5 92.2% 41.0% 28.1%HN61F-I3.0 90.1% 34.7% 25.0%

Table 2: Percentage of evacuees reaching a safe node when following the schedule produced by Free Flow (FF) under differentsimulation scenarios.

Instance Optim.Plan

Rush Rnd. Sched. Rnd. Plan Quickest

CPG HN61F 100.0% 99.9% 98.2% 91.9% 44.2%HN61F-I1.1 100.0% 99.9% 98.6% 91.2% 44.5%HN61F-I1.2 100.0% 100.0% 99.1% 89.4% 42.7%HN61F-I1.4 98.4% 97.5% 97.1% 88.0% 41.9%HN61F-I1.7 95.3% 94.7% 94.5% 83.1% 38.6%HN61F-I2.0 92.2% 90.9% 90.2% 77.2% 35.1%HN61F-I2.5 88.4% 84.6% 84.8% 69.3% 27.3%HN61F-I3.0 81.5% 76.0% 76.0% 59.3% 23.1%

CPG-CF HN61F 100.0% 100.0% 97.3% 68.3% 18.3%HN61F-I1.1 100.0% 100.0% 97.1% 65.8% 19.5%HN61F-I1.2 100.0% 99.9% 97.8% 62.3% 17.0%HN61F-I1.4 100.0% 99.9% 96.8% 59.5% 14.7%HN61F-I1.7 100.0% 99.7% 98.4% 54.9% 12.2%HN61F-I2.0 99.6% 97.6% 97.1% 52.0% 15.7%HN61F-I2.5 97.9% 97.7% 97.2% 49.7% 10.6%HN61F-I3.0 95.8% 92.9% 94.4% 46.1% 9.3%

Table 3: Percentage of evacuees reaching a safe node under different simulation scenarios.

Figure 5 presents the evacuation profile produced by the Free-Flow approach (Plan), and the Quickest303

and Random Quickest simulation scenarios. The solid line (“Num. Veh.”) represents the total number304

of vehicles to be evacuated in the considered area. This chart illustrates the fact that FF schedule is too305

optimistic when simulated, with only 39% of vehicles reaching a safe node by the end of the planning horizon306

in the Quickest scenario, and 64% in the Random Quickest. The curves indicate that the rate of evacuees307

reaching safety saturates in the simulation, which is a result of queuing occurring on the main roads where308

evacuees converge. This highlights the criticality of staging the evacuation such that queuing is avoided.309

Table 2 presents the percentage of evacuees reaching a safe node in the plan produced by FF, and in the310

Random Quickest and Quickest scenario. The results confirm the trend observed for the HN61F instance311

and reveal the low applicability of the evacuation plans produced by the FF model: on average, only 51%312

of evacuees would be evacuated by the end of the planning horizon in the Quickest scenario, and 33% in the313

Random Quickest scenario.314

15

0

10000

20000

0 200 400 600Time

Num

ber

of v

ehic

les sim

Num. Veh

Plan

Quickest

Rnd. Plan

Rnd. Schedule

Rush

Evacuation ProfileCPG

Figure 6: Comparison of evacuation profiles produced by CPG and different simulation scenarios on the HN61F instance.

The second experiment considers the evacuation plans produced by the CPG and CPG-CF approaches.315

Table 3 presents the number of evacuees reaching a safe node under different scenarios. The Optim. Plan316

column corresponds to the results of the optimization. The Rush column corresponds to a scenario in which317

all evacuees from a same area leave at the first departure time produced by the optimization. The Random318

Schedule column is a scenario in which the departure time of the evacuees is randomized depending on the319

earliest departure time of the neighboring areas and the latest departure time for the considered area. The320

Random Plan column adds an additional level of randomization by considering that 50% of the evacuees321

will follow the plan, 40% will go to one on the five closest safe nodes, and 10% will go to a random safe322

node. Finally, the Quickest column represents a scenario in which evacuees depart as instructed but go to323

the closest safe node.324

The results illustrate that the evacuation plans produced by the CPG and CPG-CF procedures are very325

close to the simulation and robust to evacuees behaviors. Of particular interest is the fact that Rush, and326

Random Schedule produce results within 2% of the plan produced by CPG and CPG-CF. Random Plan327

and Quickest introduces more perturbations by affecting the evacuation routes followed by evacuees, which328

leads to significantly less vehicles reaching safety in when simulating the CPG plans. The plans produced by329

CPG-CF appear to perform worse than CPG in this scenario, which is a consequence of evacuees selecting330

evacuation routes that use edges that are in contraflow. Note however that in practice, evacuees not following331

the plan will find an alternative route when facing a blocked road.332

Figure 6 illustrates the evacuation profiles produced by the CPG approach and the different simulation333

scenarios for the HN61F instance. These results illustrate the robustness of the plan produced by CPG:334

The curves representing the number of evacuees reaching safety are very close independently of the scenario335

16

Num. Num. Num. Num. Num. CPU Clear.Instance App. Nodes Veh. Paths Cols

(103)Rows(103)

Time(s)

Time

Hawkesbury-Nepean FF 277 38343 77 31 5 05h18CPG 277 38343 615 54 29 942 07h32CPG-CF 277 38343 651 33 34 468 04h37

Geelong FF 973 55518 669 271 136 08h07CPG 973 55518 2009 209 129 721 09h55CPG-CF 973 55518 2341 160 178 837 06h59

Gold Coast FF 320 188983 254 89 42 16h13CPG 320 188983 336 76 85 7181 19h40CPG-CF 320 188983 324 58 112 6791 15h10

New Orleans FF 1280 397558 955 356 275 50h40CPG 1280 397558 2832 568 386 8630 56h31CPG-CF 1280 397558 2828 288 382 10336 30h55

Table 4: Comparison of approaches on extended set of instances

considered. The only exception is Quickest which generates more congestion and for which only 44% of the336

evacuees reach a safe node. Note that the the shaded envelope captures the variability in results across the337

10 runs of the optimization approach.338

5.3. Additional experiments339

This section presents additional experiments conducted on a set of instances derived from 4 regional340

evacuation areas. The road network is a simplified version of the actual road network, while population341

estimates are based on official statistics. The resulting instances contain between 277 and 1280 nodes, 610342

and 3134 edges, and 38,000 to 400,000 vehicles to evacuate. In these experiments the objective is set to343

minimize the clearance time required to evacuate all vehicles.344

Table 4 presents the number of nodes, vehicles, paths, columns, rows, CPU time, and clearance time for345

the 4 instances and 3 approaches. The results illustrate the scalability of the CPG and CPG-CF approaches346

to instances with up to 1280 nodes and close to 400,000 vehicles. The relative gap between the solution of347

CPG and the lower bound of FF varies from 42% for the Hawkesbury-Nepean instance to 22% for Geelong348

and Gold Coast, and 11% for New Orleans. This can be explained by the differences in the network topology,349

which is closer to a tree structure in the instances with a lower gap, which implies that the FF model will350

have less degrees of freedom compared to CPG. Note also the significant reduction in clearance time achieved351

when allowing contraflows, which ranges from 23% for Gold Coast, to 45% for New Orleans.352

6. Conclusions353

This paper considered evacuation planning and scheduling, a critical aspect of disaster management and354

national security applications. It proposed a conflict-based path-generation approach (CPG[-CF]) whose key355

17

idea is to generate evacuation paths for evacuated areas iteratively and optimize the evacuation over these356

paths in a master problem, simultaneously considering the planning of the evacuation and the selection of357

contraflow roads. Each new path is generated to remedy conflicts in the evacuation and adds new columns358

and a new row in the master problem.359

The algorithm was applied to five case studies derived from real-world data. The first considers massive360

flood scenarios in the Hawkesbury-Nepean floodplain (Sydney area, Australia), which require evacuating361

about 70,000 persons. Computational results show that the proposed path-generation approach is able to362

design evacuation plans for such large-scale scenarios in under 60 seconds, contrary to a traditional MIP363

approach which does not scale to this problem size. Additional experiments demonstrated the ability of the364

method to cope with population growth. Bounds from a free-flow model demonstrate that the solutions365

produced by the CPG approach are of high quality with a marginal improvement of only 2.5% of evacuated366

vehicles and 15 minutes lead time. The results also demonstrate the benefits of allowing contraflow, which367

lead to an increase of 4.7% in the number of evacuees reaching safety, and a reduction of the lead time by 37368

minutes. Of particular interest is the fact that the proposed approach is able to produce evacuation plans369

for instances with 1,280 nodes, when a direct MIP formulation becomes intractable for instances with more370

than 5 nodes. Finally, validation through traffic simulation show that the solutions produced by CPG[-CF]371

are robust to a variety of evacuees behaviors. The remaining case studies consider four regional areas with372

up to 1280 nodes, 3134 edges, and 400,000 vehicles for the New-Orleans area. Results indicate that the373

CPG approach is able to produce evacuation plans in reasonable time even for the larger instances, while374

CPG-CF significantly reduces clearance times by using a subset of roads in contraflow. In the case of New375

Orleans, the evacuation plans produced by CPG require 56h30 to evacuate the whole area, compared with376

the optimistic estimate of 50h40 provided by the free-flow model.377

To the best of our knowledge, this is the first scalable evacuation-planning algorithm that conforms to378

evacuation methodologies and field requirements, and simultaneously considers the planning of an evacuation379

and the selection of contraflow roads. Our evacuation algorithm can be used in strategic, tactical, and380

operational environments, and is currently being deployed with Australian emergency services.381

Our current work aims at designing path generation algorithms that ensures the completeness of the382

search and provides an optimality guarantee on the solution. The approach proposed in this study is383

currently being extended for robust and stochastic optimization, in order to produce evacuation plans that384

take into account a range of threat scenarios. Future work will also focus on microscopic modeling of the385

transportation system, the inclusion of loading curves for notification, and models of human behavior in386

evacuation settings.387

Acknowledgments. NICTA is funded by the Australian Government as represented by the Department of388

Broadband, Communications and the Digital Economy and the Australian Research Council through the389

18

ICT Centre of Excellence program. The authors would like to thank the anonymous referees for their390

comments and suggestions.391

19

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Appendix A. Free-Flow Model461

The Free-Flow model (FF) assumes that evacuees can flow freely in the graph. From a practical per-462

spective, this corresponds to an ideal case in which the evacuees are given the order to evacuate and are463

then dynamically routed in the graph. In other words, evacuees do not know in advance the path to follow464

in the graph. Although of limited practical importance, this model provides a bound for more advanced465

evacuation planning.466

max Φ =∑

e∈δ−(vt)

ϕe (A.1)

s.t.∑

e∈δ−(i)

ϕe −∑

e∈δ+(i)

ϕe = 0 ∀i ∈ N x \ vs, vt (A.2)

ϕe ≤ ue ∀e ∈ Ax (A.3)

ϕe ≥ 0 ∀e ∈ Ax (A.4)

Model (A.1)-(A.4) presents the Free Flow model (FF). ϕe is the flow of evacuees on edge e ∈ Ax, and467

δ−(i) (δ+(i)) is the set of inbound (outbound) edges adjacent to node i. The objective (A.1) is to maximize468

the number of evacuees reaching a safe node. Constraints (A.2) ensure the conservation of the flow in469

the graph, while constraints (A.3) enforce the capacity on edges. The demand of the evacuations nodes is470

modeled implicitly as a capacity on the edges connecting them to the super-source (u(vs,i) = di). Note that471

the flow of evacuees is considered as a continuous quantity. This is motivated by the fact that the considered472

number of evacuees and edge capacity are already approximations of the reality, thus a unitary granularity473

is not necessary.474

Appendix B. Additional instances475

Figure B.7 presents the network of the 4 instances used in Section 5.3. The shaded orange area repre-476

sent the evacuated areas and their relative density of populations, the blue and green lines represent the477

evacuation network, with the green lines highlighting the roads that can be used in contraflow, finally, the478

green markers represent the location of the safe areas.479

21

(a) Hawkesbury-Nepean (b) Geelong

(c) Gold Coast (d) New Orleans

Figure B.7: Additional instances

22