EVACUATION SIMULATION WITH LIMITED CAPACITY SINKS
An Evolutionary Approach to Solve the Shelter Allocation and Capacity Assignment
Problem in a Multi-agent Evacuation Simulation
Gunnar Fl
¨
otter
¨
od
Transport and Mobility Laboratory, EPFL - Ecole Polytechnique F
´
ed
´
erale de Lausanne, Lausanne, Switzerland
Gregor L
¨
ammel
Transport Systems Planning and Transport Telematics, Berlin Institute of Technology, Berlin, Germany
Keywords:
Evacuation, Shelter allocation, Shelter capacity assignment, Iterative learning, Nash equilibrium, System op-
timum, Multi-agent simulation.
Abstract:
We heuristically solve an evacuation problem with limited capacity shelters. An evolutionary learning al-
gorithm is developed for the combined route- and shelter-assignment problem. It is complemented with a
heuristic method for the fair minimization of shelter capacities. Different behavioral assumptions “fair” vs.
“globally optimal”) are investigated. The proposed approaches are discussed in the context of a real-world
tsunami evacuation problem.
1 INTRODUCTION
The evacuation of whole cities or even regions is a
problem of substantial practical relevance, which is
demonstrated by recent events such as the evacuation
of Houston because of Hurricane Rita or the evacua-
tion of coastal cities in the case of tsunamis. Impor-
tant tools for the planning of organized reactions to
such events are model-based simulation systems.
The general evacuation problem is to minimize the
egress time of an endangered region or building by as-
signing a feasible escape route and destination to ev-
ery evacuee. This problem is complex because of con-
gestion effects that inevitably occur when many evac-
uees enter the transportation facilities (roads, hall-
ways, stairways) at once.
In some evacuation scenarios, there exist secure
areas of limited capacity within the evacuation zone,
such as shelter buildings in tsunami prone areas.
Concrete applications of mathematical programming
techniques to the evacuee–shelter–allocation problem
can be found in (Sherali et al., 1991; Peng, 2006).
This paper deploys a detailed microsimulation
for the representation, analysis, and optimization of
pedestrian evacuation dynamics for a tsunami situa-
tion in a large coastal metropolitan area. Building
on existing routing strategies (L
¨
ammel and Fl
¨
otter
¨
od,
2009), it provides new solutions to (i) the com-
bined route and shelter assignment problem and (ii)
the shelter capacity assignment problem, considering
both ”fair” and ”optimal” assignment rules.
The added value of the agent-based approach is its
natural representation of individual travelers as soft-
ware agents that interact in a simulated version of the
real world (a virtual environment). This is an ad-
vantage over analytical models in that it allows (at
least technically) for a much higher model resolution.
However, it comes at the price of greater difficulties in
the mathematical treatment of the problem. The opti-
mization results presented in this article are therefore
only of an approximate nature.
2 PROBLEM STATEMENT
In a first step, we investigate different strategies to
assign routes and destinations (shelters) to evacuees.
In a second step, we identify optimal dimensions of
the shelters. Overall, we consider two different ob-
jectives:
Fairness. No evacuee will agree to take an obvious
detour or to select an obviously faraway shelter in-
249
Flötteröd G. and Lämmel G..
EVACUATION SIMULATION WITH LIMITED CAPACITY SINKS - An Evolutionary Approach to Solve the Shelter Allocation and Capacity Assignment
Problem in a Multi-agent Evacuation Simulation.
DOI: 10.5220/0003086302490254
In Proceedings of the International Conference on Evolutionary Computation (ICEC-2010), pages 249-254
ISBN: 978-989-8425-31-7
Copyright
c
2010 SCITEPRESS (Science and Technology Publications, Lda.)
stead of a nearby one. This requires to identify route
and shelter assignments that are fair in that no evacuee
can obviously gain by switching to a different route
or shelter. It corresponds to a Nash equilibrium of all
evacuation strategies in the population.
Efficiency. It is desirable to evacuate the system as
quickly as possible. While a Nash strategy has the ob-
vious and important advantage of general acceptance,
it may be suboptimal in this regard because some
evacuees may do great damage to others by blocking
their ways/shelters. We therefore identify approxima-
tions of optimal evacuation strategies as benchmarks
to which fair solutions can be compared.
An important topic for future research is to com-
bine both approaches into evacuation strategies that
are more efficient than Nash equilibria without intro-
ducing obvious levels of unfairness.
2.1 Simulation Framework
We model the urban evacuation region and the pop-
ulation of evacuees with a multi-agent simulation,
where every single person is individually represented.
For this purpose, the MATSim simulation framework
is adopted (MATSim, 2010). MATSim is designed
for the computation of transport equilibria, and hence
it can be immediately deployed for the computation
of Nash evacuation strategies. Some adjustments are
necessary for approximately optimal strategies.
MATSim allows for adjustments in the different
choice dimensions of a simulated traveler through
modules, where, typically, one module is responsi-
ble for one choice dimension. In our application,
this requires to specify four modules: (1) Nash route
choice, (2) Nash destination (shelter) choice, (3) opti-
mal route choice, (4) optimal shelter choice.
MATSim computes approximate Nash equilibria
by iterating best-response behavior: in every itera-
tion, a fraction of the travelers recalculates a route or
a destination based on what would have been best in
the previous iteration, assuming that the behavior of
all other agents stays unchanged. After this replan-
ning, the resulting plans of all travelers are simulta-
neously executed in the mobility simulation and new
performance measures are computed. This process is
repeated many times. Once it stabilizes, no agent can
substantially improve through a route or destination
replanning, and an approximate Nash equilibrium is
obtained.
An alternative assignment logic is to not compute
best responses in every iteration but cooperative be-
havior that improves the situation of the population as
a whole. The more involved realization of such be-
havior follows essentially the same simulation logic
as the Nash assignment, but with a modified cost func-
tion being presented to the agents.
2.2 Network Modeling
The evacuation network consists of a set of nodes that
are connected by a set of directed links. Sources (ori-
gins) as well as sinks (destinations, shelters) are asso-
ciated with respective node subsets.
Every destination d has a capacity c
d
that repre-
sents the maximum number of evacuees it can shelter.
Destination nodes may also be located at the bound-
ary of the endangered area, in which case they do not
provide a limited shelter but access to a safe region,
which is modeled by assigning them an unlimited ca-
pacity.
We consider a pedestrian simulation scenario on a
road network, where the intersections correspond to
nodes and the street segments connecting the inter-
sections are modeled through links. Basic pedestrian
traffic flow dynamics are captured through a limited
number of link parameters: outflow capacity (maxi-
mum number of pedestrians the link can emit per time
unit), space capacity (maximum number of pedestri-
ans in the link), and maximum velocity (in uncon-
gested conditions). Note that this formalism can be
immediately transferred to vehicular evacuation prob-
lems (Cetin et al., 2003).
3 ROUTE ASSIGNMENT
Given that every evacuee n = 1. . . N is assigned to a
shelter d(n), the route assignment problem is to find a
feasible and in some sense best route from that evac-
uee’s origin s(n) to her shelter.
3.1 Nash Equilibrium Assignment
In the given context, a Nash equilibrium describes a
situation where no evacuee can gain by unilaterally
deviating from her current route (Nash, 1951). Since a
Nash equilibrium means that nobody has an incentive
to make a change, it can be considered as a socially
acceptable and hence implementable evacuation strat-
egy.
In a multi-agent (evacuation) simulation, the so-
lution can be moved towards a Nash equilibrium
through iterative learning (Gawron, 1998). As de-
scribed above, such an algorithm starts with a given
(routing) strategy for every agent, and then adjusts
this strategy through some trial and error mechanism.
In the given evacuation context, strategies are only
evaluated based on their travel times.
ICEC 2010 - International Conference on Evolutionary Computation
250
Algorithm 1: Nash equilibrium routing.
1. initialize τ
a
(k) with the free-flow travel time for
all links a and time steps k
2. repeat for many iterations:
(a) recalculate routes based on time-dependent link
costs τ
a
(k)
(b) simulate agent movements, obtain new τ
a
(k)
for all a and k
Formally, the real-valued time is discretized into
K segments (“bins”), which are indexed by k =
0...K 1. The time-dependent link travel time when
entering link a in time step k is denoted by τ
a
(k).
Alg. 1 drafts the Nash-equilibrium routing logic.
3.2 System Optimal Assignment
A system optimal routing solution minimizes the to-
tal travel time in the system. Classical solutions to
this problem apply mathematical programming tech-
niques, which are based on the theory of dynamic
network flows. The foundations of these techniques
have been laid in (Ford and Fulkerson, 1962), and dy-
namic flow models have been applied to evacuation
problems from the early 1980s on (see, e.g., (Chalmet
et al., 1982)).
In the multi-agent domain, an approximate sys-
tem optimum (SO) can be found through an iter-
ative learning approach that is closely related to
the simulation of a Nash equilibrium as described
above (L
¨
ammel and Fl
¨
otter
¨
od, 2009).
The only difference to the Nash routing logic
given in Algorithm 1 is that the travel time based on
which agents evaluate their routes is replaced by the
marginal travel time (Peeta and Mahmassani, 1995).
The marginal travel time of a link is the amount by
which the total system travel time changes if one ad-
ditional traveler enters that link. It is the sum of the
cost experienced by the added traveler (τ
a
(k)) and the
cost imposed on all other travelers. The latter is de-
noted as the time-dependent “social cost”.
Letting each evacuee individually minimize her
marginal travel time implicitly enforces a coopera-
tive behavior that also minimizes the total system
travel time. This maximizes the number of evacuees
who have reached their destinations in each time step,
which in turn minimizes the egress time (Jarvis and
Ratliff, 1982).
4 SHELTER ASSIGNMENT
The shelter assignment problem is to identify, for each
evacuee, if this evacuee should access a shelter or not
and, given that a shelter is accessed, to decide which
shelter. Again, both a Nash and an SO approach are
possible.
4.1 Nash Shelter Assignment
We extend the individual best-response logic to a pair-
wise best response, where for every agent n that re-
plans its shelter a ”switching partner” n
0
in another
shelter is randomly selected, and both agents switch
their shelters if and only if both benefit from this
switch. This decision is made based on the expected
travel time of a best-response re-routing to each new
destination; the resulting routes are also adopted in
the case of an accomplished switch.
The iterative simulation conducts a shelter switch
with a certain probability P
switch
, and it also maintains
the option of a plain route recomputation with P
reroute
.
Algorithm 2 defines the details of this logic.
Algorithm 2: Nash shelter allocation.
1. initialize routes and destinations for all agents
2. repeat many times
(a) load all agents on the network
(b) extract link travel times
(c) for every agent n = 1. . . N, do with P
replan
:
with P
reroute
, compute a new route from s(n)
to d(n).
with P
move
,
i. randomly select a non-full shelter d
0
ii. compute the benefit of a move: δ =
c
s(n)d(n)
c
s(n)d
0
iii. if δ > 0, assign d
0
as the new destination to n
and re-route n
with P
switch
,
i. randomly select n
0
from {1, . . . , N}
ii. compute the minimum benefit of a switch:
δ = min(c
s(n)d(n)
c
s(n)d(n
0
)
, c
s(n
0
)d(n
0
)
c
s(n
0
)d(n)
)
iii. if δ > 0, then switch the destinations of n and
n
0
and re-route both agents
The origin of agent n is denoted by s(n) and its
destination by d(n). The cost c
s(n)d(n)
corresponds to
the travel time from s(n) to d(n). Step 1., 2.(a), 2.(b)
and the first step of 2.(c) are symmetric to the routing
logic of Algorithm 1. An agent moves with probabil-
EVACUATION SIMULATION WITH LIMITED CAPACITY SINKS - An Evolutionary Approach to Solve the Shelter
Allocation and Capacity Assignment Problem in a Multi-agent Evacuation Simulation
251
ity P
replan
P
move
to a non-full shelter if it would ben-
efit from that move. With probability P
replan
P
switch
,
two agents switch their shelters if both of them would
benefit.
4.2 SO Shelter Assignment
Technically, the SO shelter assignment does not func-
tion differently from the Nash shelter assignment,
only that two agents now ”agree” to switch their shel-
ters if this reduces the total travel time in the system.
To decide this, the expected change in marginal travel
times is evaluated before and after the switch.
Algorithm 2 is still applicable with the follow-
ing modifications: c
s(n)d(n)
now represents agent
ns marginal travel time, cf. Section 3.2. A
switch is only performed if both agents would ben-
efit from it. Therefore, the switching benefit δ of
Step 2. (c) ii. needs to be computed as c
s(n)d(n)
+
c
s(n
0
)d(n
0
)
c
s(n)d(n
0
)
c
s(n
0
)d(n)
.
5 SHELTER CAPACITY
ASSIGNMENT
The shelter capacity assignment problem is to mini-
mize the total shelter capacity subject to the constraint
that no evacuee takes damage from being neither able
to reach the safe area nor to enter a shelter because of
lacking capacity. That is, a configuration is required
where only those evacuees are assigned to shelters
who would not make it to the safe region otherwise.
5.1 Shelter Capacity Assignment
Subject to Nash Constraints
We base our approach on the Nash simulation logic
of Algorithm 2. If there is more shelter capacity than
strictly needed, there are likely to be agents in the
shelters that could also make it to the safe region (be-
cause it can be assumed that for many such agents the
shelter still is closer than the safe area). It is not fea-
sible to ex post remove these agents from the shelters
and to constrain the shelter capacities accordingly be-
cause this would change the travel times and hence
the survival chances of the needy agents. The shel-
ter capacities therefore need to be gradually adjusted
during the iterations.
This effect is achieved by evaluating, in every it-
eration, the space occupied in every single shelter by
agents that would also have made it to the safe re-
gion. If this surplus is vanishing, the shelter is ur-
gently needed, and its capacity is increased by a rela-
tive amount (say, 5 percent). If, on the other hand, this
surplus is substantial, the shelter is too large, and it is
shrunk by a relative amount (say, again, 5 percent) of
its surplus capacity.
This mechanism, in combination with a strict pref-
erence for needy agents in the shelter allocation, even-
tually leads to a configuration where all available
shelter capacity is allocated to needy agents, given
otherwise fair Nash equilibrium conditions. Algo-
rithm 3 gives an overview.
Algorithm 3: Heuristic Nash shelter allocation and capacity
assignment.
1. initialize routes and destinations for all agents
2. repeat many times
(a) load all agents on the network
(b) extract link travel times
(c) for every agent n = 1 . . . N, replan with P
replan
its route/destination; give strict preference to
needy agents in shelter assignment
(d) for every shelter d = 1 . . . D, do:
o(d) = c(d)
N
n=1
x
nd
+
N
n=1
ˆy
n
if o(d) > 0, decrease c(d) by min(o(d), q
c(d)) and re-route non-needy agents to super-
shelter if necessary; else increase c(d) by
q c(d))
The variable x
nd
indicates the allocation of agent
n to shelter d, i.e., x
nd
= 1 d(n) = d. y
n
indicates
if agent n has enough time to evacuate to the super
shelter. If agent n has enough time to evacuate to the
super shelter y
n
= 1 and y
n
= 0 otherwise. ˆy
n
is the
estimated value of y
n
based on the experienced travel
costs from previous iteration. q denotes the relative
amount by which the capacity of a shelter can change
at most. The super-shelter represents the entire safe
area.
5.2 Shelter Capacity Assignment
Subject to SO Constraints
The only change when going from a shelter capacity
assignment subject to Nash constraints to one subject
to SO constraints is that the route choice and shelter
switching behavior of all re-planning agents is con-
ducted according to the SO logic described in Subsec-
tions 3.2 and 4.2. The conditions for shelter capacity
decreases and increases in the SO case are the same
as for the Nash case given in in Algorithm 3.
ICEC 2010 - International Conference on Evolutionary Computation
252
6 EXPERIMENTS
The Indonesian city of Padang, located at the West
Coast of Sumatra Island, is exposed to earth quake
triggered tsunamis. The evacuation street network
consist of approx. 12 500 unidirectional links and
4 500 nodes. There are in total 224 798 evacuees.
This corresponds to the number of persons living in
the evacuation area. 42 hypothetical shelter buildings
with a total capacity of roughly 31 500 evacuees are
placed in the network. A sketch of the network in-
cluding the shelters is given in Figure 1. The gray-
shaded area has to be evacuated.
0 1000 2000 meter
Figure 1: Map of the evacuation area.
We conduct four different simulations.
Run 1 implements the Nash equilibrium routing
and shelter allocation.
Run 2 implements the Nash equilibrium routing
and shelter allocation and the shelter capacity as-
signment.
Run 3 implements the SO routing and shelter al-
location.
Run 4 implements the SO routing and shelter al-
location and the shelter capacity assignment.
These runs are performed with a 10% sample of
the population. The shelter capacities and network
flow parameters are accordingly scaled down to 10%.
This procedure saves computing time while staying
reasonably realistic. With this setup, a simulation
with 2000 iterations takes between 05:30 h (Run 1)
and 10:30 h (Run 4) on a 2.66 GHz CPU running 64
bit Java on Linux. The memory consumption is below
3 GB in all experiments.
Evacuation time
avg. evacuation time in s
650
700
750
800
850
900
950
1,000
iteration
0 500 1,000 1,500 2,000
Run 1
Run 2
Run 3
Run 4
Figure 2: Average evacuation time per agent versus iteration
number.
At the beginning of all runs, the agents are as-
signed randomly to the shelters. Figure 2 shows the
average evacuation time per agent over the iteration
number. Run 1 shows that a Nash shelter assignment
leads to considerably better evacuation times than a
random shelter assignment. Adding the shelter capac-
ity assignment reduces the average evacuation times
further (Run 2). However, this comes at the cost of a
drastic increase in shelter capacities, which is shown
in Figure 3.
Shelter utilization
number of agents in shelters
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
iteration
0 500 1,000 1,500 2,000
Run 1
Run 2
Run 3
Run 4
Figure 3: Change of the shelters total utilization over the
learning iterations.
The SO in Run 3 and Run 4 is realized by adding
social costs to the travel time on each link.
EVACUATION SIMULATION WITH LIMITED CAPACITY SINKS - An Evolutionary Approach to Solve the Shelter
Allocation and Capacity Assignment Problem in a Multi-agent Evacuation Simulation
253
Figure 2 reveals that the SO experiments Run 3
and Run 4 yield higher travel times than the Nash
runs. However, the SO runs result in a substan-
tially lower shelter utilization, cf. Figure 3. The
higher travel times in the SO runs are likely to result
from those travelers that are kept out of the shelters
and hence have to take longer routes to the safe city
boundaries.
In theory, at least Run 3 should outperform Run 1.
The fact that this is not the case is due to the approxi-
mate nature of the deployed algorithms, which clearly
need further attention.
7 DISCUSSION AND SUMMARY
The safest strategy for a tsunami-threatened city like
Padang would arguably be to build a tsunami proof
shelter for every single person. However, this would
exceed any financial resources. The relevant question
thus is to identify which shelters are actually needed
by persons who cannot be evacuated out of the city in
time. The proposed algorithms help to identify these
shelters and their capacities under different behavioral
assumptions.
An important result of the simulation studies is
that the required shelter capacity in the Nash equilib-
rium case is much higher than in the SO case. If one
wanted to achieve the highest benefits with the least
effort, one could implement the shelter configuration
of Run 4 and distribute some kind of tickets to the peo-
ple that are allowed to enter a shelter. Those tickets
could be preferably handed out to the most vulnerable
people like the elderly or pregnant women.
Summarizing, a learning framework that approxi-
mately solves the shelter allocation and capacity op-
timization problem is presented and tested on a real-
world scenario. The learning framework can be con-
figured either to attain an approximate Nash equilib-
rium, where individual travel times are minimized
non-cooperatively, or an approximate system opti-
mum, where the global travel time is minimized in
a cooperative manner.
The experiments show that both approaches yield
reasonable results and that substantial savings in shel-
ter capacity are possible if the evacuation behavior
can be influenced to deviate from a perfectly fair Nash
equilibrium towards a system optimum.
In future work, the proposed algorithms could be
extended to also identify appropriate shelter locations.
This could be realized by starting with a high number
of shelter buildings, followed by a successive removal
of underutilized shelters. Another interesting topic for
further research would be to investigate the relative
effect of capacity improvements in the transportation
system when compared to investments in increased
shelter capacities.
ACKNOWLEDGEMENTS
This project was funded in part by the German Min-
istry for Education and Research (BMBF) under
grants 03G0666E (“last mile”) and 03NAPI4 (“Ad-
vest”).
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