A Conflict-Based Path-Generation Heuristic for Evacuation Planning 1 Victor Pillac a , Pascal Van Hentenryck a,b,* , Caroline Even a 2 a National ICT Australia (NICTA), 115 Batman Street, West Melbourne, Australia 3 b Australia National University (ANU), Canberra, Australia 4 Abstract 5 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-based 6 path generation 7 1. Introduction 8 Natural disasters, such as hurricanes, floods, and bushfires, affect numerous populated areas and may 9 endanger the lives and welfare of entire populations. Evacuation orders are some of the most important 10 decisions performed by emergency services: They ensure the safety of persons at risk by instructing them 11 to evacuate the threatened region, be it a building (e.g., fire), a neighborhood (e.g., industrial hazard), or a 12 whole region (e.g., flood). Evacuation planning also arises at strategic, tactical, and operational levels. At a 13 strategic level, the goal is to design evacuation plans for specific areas and possible threats (e.g., evacuation 14 plans for the surroundings of a nuclear power plant). At a tactical level, the goal is to design evacuation 15 * Corresponding author Preprint submitted to Elsevier February 4, 2015
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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
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