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

Transport and Mobility Laboratory, EPFL - Ecole Polytechnique F

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

 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

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

(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

(k)

(b) simulate agent movements, obtain new τ

(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

(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

(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

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

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

ii. compute the beneﬁt of a move: δ =

s(n)d(n)

− c

s(n)d

iii. if δ > 0, assign d

as the new destination to n

and re-route n

• with P

switch

i. randomly select n

from {1, . . . , N}

ii. compute the minimum beneﬁt of a switch:

δ = min(c

s(n)d(n)

− c

s(n)d(n

)

, c

s(n

)d(n

)

−

s(n

)d(n)

)

iii. if δ > 0, then switch the destinations of n and

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

n’s 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)

s(n

)d(n

)

− c

s(n)d(n

)

− c

s(n

)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

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=1

∑

n=1

ˆy

• 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

indicates the allocation of agent

n to shelter d, i.e., x

= 1 ⇔ d(n) = d. y

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

= 1 and y

= 0 otherwise. ˆy

is the

estimated value of y

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”).

REFERENCES

Cetin, N., Burri, A., and Nagel, K. (2003). A large-scale

agent-based trafﬁc microsimulation based on queue

model. In Proceedings of the 3rd Swiss Transport

Research Conference, Monte Verita/Ascona, Switzer-

land.

Chalmet, L., Francis, R., and Saunders, P. (1982). Network

models for building evacuation. Management Science,

28:86–105.

Ford, L. and Fulkerson, D. (1962). Flows in Networks.

Princeton University Press.

Gawron, C. (1998). An iterative algorithm to determine

the dynamic user equilibrium in a trafﬁc simulation

model. International Journal of Modern Physics C,

9(3):393–407.

Jarvis, J. and Ratliff, H. (1982). Some equivalent objec-

tives for dynamic network ﬂow problems. Manage-

ment Science, 28:106–108.

ammel, G. and Fl

otter

od, G. (2009). Towards system op-

timum: Finding optimal routing strategies in time-

tependent networks for large-scale evacuation prob-

lems. volume 5803 of LNCS (LNAI), pages 532–539,

Berlin Heidelberg. Springer.

MATSim (accessed 2010). MATSim web site.

http://www.matsim.org.

Nash, J. (1951). Non-cooperative games. Annals of Mathe-

matics, 54(2):286–295.

Peeta, S. and Mahmassani, H. (1995). System optimal and

user equilibrium time-dependent trafﬁc assignment in

congested networks. Annals of Operations Research,

60:81–113.

Peng, L. (2006). A Dynamic Network Flow Optimization

for Large-scale Emergency Evacuation. PhD thesis,

City University of Hong Kong.

Sherali, H., Carter, T., and Hobeika, A. (1991). A location-

allocation model and algorithm for evacuation plan-

ning under hurricane/ﬂood conditions. Transportation

Research Part B: Methodological, 25(6):439 – 452.

ICEC 2010 - International Conference on Evolutionary Computation

254