Planning for evacuation: insights from an efficient ...

Planning for evacuation: insights from an efficient network design

model

Georgios Kalafatas, [email protected]

Srinivas Peeta1, [email protected]

School of Civil Engineering

Purdue University, USA

550 Stadium Mall Drive

West Lafayette, IN 47907-2051

Office: CIVL 4148

Phone: 765-494-2209

Fax: 765-496-7996

1. Corresponding author

Kalafatas and Peeta 2

Abstract: In the context of evacuation planning at a strategic level, an efficient model is

proposed to augment network capacity under a budget constraint so as to enhance operational

performance under a security threat. Contra-flow strategies and lane additions are considered for

capacity augmentation. The corresponding mathematical programming model is obtained

through a case-specific graph theoretic transformation of Daganzo’s cell transmission model

(CTM), exploiting problem characteristics to enhance computational efficiency. This paper

focuses on experiments and sensitivity analyses to provide insights: (i) on the computational

efficiency of the proposed model, and (ii) for strategies for evacuation planning and operations.

Sensitivity analyses are performed for a test network in terms of the budget for contra-flow

operations, the evacuee population size, and the amount and spatial distribution of origin-

destination (O-D) demand. The results suggest that there is a threshold budget beyond which

benefits, in terms of the network clearance time, are negligible. They also indicate that the

network clearance time varies linearly with uniformly distributed population size. Finally,

greater spatial uniformity of O-D demand reduces clearance times, suggesting that specifying

multiple destinations in the safety zone may lead to more efficient evacuation in many instances.

From a response standpoint, the study suggests that operational effectiveness requires pre-

determined contra-flow strategy plans with designated and well-trained personnel, rather than

just reacting to a security event in real-time.

CE Database Subject Headings: Evacuation, network design, traffic models, budget allocation,

spatial analysis


Introduction

Contemporary research in evacuation planning has modeled and proposed solutions for several

aspects of the evacuation problem (Sheffi et al. 1982, Southworth 1991, Wolshon 2001, Cova

and Johnson 2003, Tuydes and Ziliaskopoulos 2004, Kwon and Pit 2005, Sbayti and

Mahmassani 2006, Kalafatas and Peeta 2006). The studies have addressed the optimal routing

and allocation of contra-flow options for specific population sizes and spatial distributions of

origin-destination (O-D) demand. However, this problem is computationally burdensome (Li et

al. 2003, Tuydes and Ziliaskopoulos 2006) and has been proven to be NP-Hard (Johnson et al.

1978). Hence, a comprehensive sensitivity analysis is a challenging proposition because of the

computational requirements involved as well as the need for exact knowledge on all parameters

of each evacuation scenario. However, not all parameters of the evacuation scenario are readily

available a priori, at least currently. This suggests the need for extensive sensitivity analysis so

as to provide emergency response planners useful insights to develop response deployment

plans, which motivates the need for a computationally efficient formulation.

The study formulation derives from the authors’ previous work (Kalafatas and Peeta,

2006) based on the cell transmission model (Daganzo 1994, 1995). It differs in that a

mathematical transformation is proposed for extensive sensitivity analysis, as described later.

Kalafatas and Peeta (2006) proposed an exact formulation for the evacuation network design

problem (ENDP) to augment network capacity under a budget constraint so as to enhance

operational performance under a security threat. They considered contra-flow strategies and lane

additions as the network design options for capacity augmentation. However, the formulation is

computationally burdensome and cannot enable extensive sensitivity analysis at a strategic

planning level; strategic planning requires a significant number of experiments in order to


capture the uncertainty on the exact values of problem parameters (for instance, the size and the

spatial distribution of the population).

The experiments performed in the current study focus only on the contra-flow option for

capacity augmentation. Each contra-flow option is associated with budgetary costs and number

of trained personnel requirements. However, data on budget and personnel needs for contra-flow

options require a dedicated study on the part of the responsible transportation agency. Hence,

these data are difficult to obtain currently, though this may not be an issue in the future as

security/disaster preparedness plans become more commonplace. To circumvent this issue, the

budget and trained personnel constraints are substituted by an equivalent constraint on the

number of contra-flow options allowed: the number of reversed links (RL). Reversed links are

links in which at least one lane has its traffic flow direction reversed so as to enhance the

capacity of traffic flow in the opposite direction. Hence, the number of reversed links allowed is

used as a proxy for the budget and number of trained personnel required for the corresponding

contra-flow option.

Another characteristic of the model is the use of a static traffic signal control strategy,

whereby a single phase operates at an intersection for the entire evacuation period. Since

evacuation can lead to near-gridlock conditions, our strategy allows flow at any intersection on

only one of multiple conflicting right-of-ways. Hence, crossing flows are precluded when

determining the capacity enhancements to optimize operational performance. The actual flow

direction (for example, north-south or east-west) at each intersection is determined as part of the

optimization process. The resulting re-designed network is time-invariant, which on one hand it

indicates a straight forward planning/application proposal to planners, emergency officers and


practitioners, and on the other hand it is not capable of directly modeling time-depending link

capacities.

The paper is organized as follows. The mathematical transformation methodology of the

Kalafatas and Peeta (2006) formulation is summarized here in terms of the properties that allow

the efficient formulation for the ENDP as well as the associated mathematical program of the

formulation. Then, the experiment setup for the study experiments is described, including the

description of the test network, the selected scenario sets, and the computational resources used.

The first scenario set addresses the impact of the magnitude of the available budget resources on

evacuation response performance. The second set analyzes the effect of the evacuee population

size, which is also used as a proxy for problem size, to obtain insights on how the formulation

scales. The third set studies the impact of the quantity and the spatial distribution of the

evacuation O-D demand. Other scenario sets, not examined in the present study, like traffic

conditions at the time of evacuation or other combinations of the above scenarios, can attract the

interest of specialized evacuation planners. It is exactly this need that motivates the

computationally efficient formulation introduced in this paper; the need to evaluate and draw

insights from multiple potential situations in a timely manner. The paper concludes with a

summary of the key insights on evacuation planning obtained through the sensitivity analyses.

Methodology

The primary barriers to efficient sensitivity analysis in the evacuation planning problem context

are its time-dependent nature, complexity, and large size in terms of network topology and

population. Initially, the authors proposed an exact mixed-integer time-dependent formulation

(Kalafatas and Peeta 2006) based on the cell transmission model for the evacuation network

design problem. However, given the NP-Hard nature of the problem, the solutions for various


instances (scenarios) required computational times of the order of days before optimality was

achieved. This motivated the need to develop a more efficient formulation.

Traffic flow characteristics and the planning focus of the evacuation problem of interest

were used to develop a more efficient formulation. The key observation which led to the new

efficient formulation is that the maximum capacity is achieved at free-flow speed conditions in

the flow-density relationship of the cell transmission model (Fig. 1). While this is not necessarily

true as per the Highway Capacity Manual (Highway Capacity Manual 2000), it is a reasonable

and convenient assumption for planning applications due to their long-term focus. That is, the

planning stage of the evacuation problem is focused primarily on generating action plans that

allow the strengthening of the evacuation preparedness in the intermediate stage. For example, a

key planning objective could be the identification of network links where capacity should be

added strategically to enhance the expected evacuation-related performance under an actual

disaster. Hence, traffic need not be assigned at traffic densities greater than the traffic density

where maximum flow is achieved. This leads to better bounds for the variables representing

traffic density. Also, the modeling of backward propagating traffic waves becomes redundant

under low traffic densities. Under these conditions, a time-expanded network representation is

revealed (Fig. 2) that enables the more efficient formulation.

Extending the aforementioned notion, we assign traffic to intermediate cells up to the

minimum traffic density at which maximum flow is achieved. This is because we can still

achieve the optimal solution in a stricter solution space, since the traffic density variables have

stricter bounds and optimality depends on the maximum flow out of the network rather than the

maximum density of individual cells.


Hence, the following propositions aided the development of the more efficient

formulation:

Proposition 1. Backward propagating traffic waves are not meaningful at traffic densities of light

traffic conditions. Therefore, the constraints explicitly handling the backward propagating traffic

waves are redundant.

Proposition 2. The maximum occupancy (density) variable in our network design problem

becomes equivalent to the maximum flow variable. Hence, in the formulation, the maximum

occupancy variable and the associated equation can be eliminated.

Proposition 3. Flow in intermediate cells is constrained at free-flow speed only since congestion

does not contribute towards optimality. Physically speaking, evacuees will be allowed to exit the

source cells only if free-flow conditions are guaranteed along the entire route from the origin to

the destination cell. Interestingly, this can be interpreted as an extreme application of staged

evacuation at the level of the individual evacuee.

In summary, since the traffic network in the evacuation zone is expected to be highly

congested in an actual scenario, the planning focus is on enhancing capacity to maximize flow

out of the evacuation zone, and not on operational routing patterns. Hence, while these

propositions may, at times, seem to lack in routing realism vis-à-vis actual evacuation conditions,

they are part of the non-inferior frontier of the initial network design formulation. Thus, the

propositions are proposed not for the exact representation of the traffic realities of the evacuation

problem, but as an efficient mathematical programming transformation which enables the

development of a minimum cost flow routing sub-structure for the evacuation planning problem

(as illustrated by the modifications to the CTM cell representations in Fig. 2), thereby

significantly reducing the computational cost. It should be noted here that the insights from the


inner graph sub-structure of the evacuation problem represent the stimulus for an exact Graph

Theoretic version of the CTM (or GTCTM) proposed by Kalafatas and Peeta (2007).

The resulting formulation, labeled iENDP, is mathematically described hereafter.

Parameters

The parameters of the formulation are:

B The total available budget.

,C i C∈ The set of all cells.

RC The subset of source cells (origin cells).

SC The subset of destination cells.

,E j E∈ The set of cell connectors.

,M m M∈ The set of network design options.

,T t T∈ The set of discrete and constant time intervals.

0iQ The initial maximum number of vehicles that can flow into or out of cell i C∈ .

miQ The maximum number of vehicles that can flow into or out of cell

( )\ R Si C C C∈ ∪ , if network design option m M∈ is implemented.

jQ The maximum number of vehicles that can flow through cell connector j E∈ .

( )iΓ The set of the successor cells of cell i C∈ .

1( )i−Γ The set of the predecessor cells to cell i C∈ .

mc The cost of implementing design option m M∈ .

tid The demand (inflow) at source cell Ri C∈ in time interval t T∈ .

mia The binary indicator showing if the network design option m is associated with


the cell i C∈ .

jlβ The binary indicator showing if the flow in cell connector j E∈ can be restricted

by intersection l L∈ .

τ The constant discrete time interval’s length.

Variables

The variables of the formulation are:

iQ The maximum number of vehicles that can flow into or out of cell i C∈ .

jp The binary variable indicating whether the flow in cell connector j E∈ is

restricted by an intersection constraint.

tis The number of vehicles in an origin or destination cell ( )R Si C C∈ ∪ not

propagated to the next cell in time interval t T∈ .

tix The number of vehicles in cell i C∈ in time interval t T∈ .

tjy The number of vehicles moved by cell connector j E∈ in time interval t T∈ .

mz The binary decision variable indicating if the network design option m M∈ is

selected.

Formulation

The objective of the formulation (1) is to minimize the total time spent in the network.

minimize

\ S

ti

t T i C Cxτ

∈ ∈

⋅∑ ∑ (1)


subject to:

( )1

1t ti j

j i

x y−

−

∈Γ

= ∑ ( )\ ,R Si C C C t T∀ ∈ ∪ ∀ ∈ (2)

( )

t tj i

j iy x

∈Γ

=∑ ( )\ ,R Si C C C t T∀ ∈ ∪ ∀ ∈ (3)

( )1

1 1t t t ti i j i

j i

s d y x−

− −

∈Γ

+ + =∑ ( ),R Si C C t T∀ ∈ ∪ ∀ ∈ (4)

( )

t t ti j i

j ix y s

∈Γ

= +∑ ( ),R Si C C t T∀ ∈ ∪ ∀ ∈ (5)

R

T ti i

i C t T i Cx d

∈ ∈ ∈

=∑ ∑∑ (6)

ti ix Q≤ ( )\ ,R Si C C C t T∀ ∈ ∪ ∀ ∈ (7)

( ) 1mm i

m Mz a

∈

⋅ ≤∑

i C∀ ∈ (8)

( ) ( )01 m m mi i m i i m i

m M m MQ a z Q a z Q

∈ ∈

⎛ ⎞= − ⋅ ⋅ + ⋅ ⋅⎜ ⎟⎝ ⎠∑ ∑

( )\ ,R Si C C C t T∀ ∈ ∪ ∀ ∈ (9)

( )m mm M

z c B∈

⋅ ≤∑

(10)

tj j jy p Q≤ ⋅ ,j E t T∀ ∈ ∀ ∈ (11)

1jl jj E

pβ∈

⋅ ≤∑

l L∀ ∈ (12)

0tix ≥ ,i C t T∀ ∈ ∀ ∈ (13)

0tjy ≥ ,j E t T∀ ∈ ∀ ∈ (14)

0tis ≥ ( ),R Si C C t T∀ ∈ ∪ ∀ ∈ (15)


0iQ ≥ i C∀ ∈ (16)

{ }0,1jp ∈ j E∀ ∈ (17)

{ }0,1mz ∈ m M∀ ∈ (18)

The constraints are discussed here. Constraints (2) to (7) address the traffic flow

modeling to route evacuees. Constraints (8) to (12) model the network design options, and

constraints (13) to (18) represent the non-negativity and integrality constraints.

Constraints (2) and (3) are the mass conservation constraints between cell and cell

connectors for intermediate cells (all cells other than the source and destination cells). Each

equation corresponds to a conservation of flow equation at each of the two nodes of an

intermediate cell’s graph representation, as illustrated in Fig. 2. The number of vehicles tix in

cell i C∈ in time interval t T∈ equals the total cell inflow (constraint (2)) and the total cell

outflow (constraint (3)). They establish traffic movement at free-flow conditions for intermediate

cells; all vehicles are propagated to the next cells. Constraints (4), (5) and (6) are the mass

balance equations for origin and destination cells. The characteristic difference between the mass

balance constraints for intermediate ((2) and (3)) and origin and destination cells ((4) and (5)) is

the existence of congestion, as represented by the number of vehicles tis in cell R Si C C∈ ∪ in

time interval t T∈ ; holding of traffic is performed at origin cells till free-flow speed conditions

are guaranteed for the rest of the trip. Accordingly, equations (4) and (5) correspond to a

conservation of flow equation at each of the two nodes of an origin or destination cell’s graph

representation, as illustrated in Fig. 2. Constraint (6) represents conservation of demand; it

ensures consistency between demand generated at the origin cells and the evacuees exiting the


network at destination cells. It also holds even if all evacuees are not able to reach destination

cells by the last time interval T . Summarizing, constraints (2) to (6) are the mass balance

constraints of the minimum cost flow sub-structure. Constraint (7) is the maximum cell

occupancy constraint for intermediate cells, and at the same time denotes the maximum flow

propagated to downstream cells. These two statements hold simultaneously due to the

mathematical transformation previously described; traffic density is restricted to be less than or

equal to the minimum traffic density at which the maximum flow is achieved (Fig. 1, Point P2).

It is noted that the occupancy tix and congestion t

is variables for origin and destination cells do

not have a finite upper bound, as is typical in the CTM-related literature.

Constraint (8) restricts the selection of network design options to be at most one for each

cell, since a unique value for the maximum flow iQ must be assigned to every cell. If no

network design option is selected, a cell retains its initial maximum flow 0iQ . This can be seen in

constraint (9), where a cell’s maximum inflow/outflow iQ take values (right hand side) that

correspond either to its default value 0iQ (left component of the right hand side) or the maximum

flow miQ of the selected network design option mz . Constraint (10) is the budget constraint; the

total budgetary cost cannot exceed the total available budget. Constraint (11) reflects both the

cell connector’s individual flow capacity and the intersection flow restriction. Constraint (12)

allows at most one crossing flow to be realized at an intersection; it ensures a static traffic signal

control strategy where a single phase – green or red – is assigned for the whole planning period.


Finally, constraints (13) to (16) ensure that the routing variables tix , t

jy , tis and the

maximum cell inflow/outflow capacity variable iQ are real non-negative numbers. Constraints

(17) and (18) state that the network design variables jp and mz are binary.

Experimental Setup

The test network

The test network for the study is illustrated in Figure 3 (Kalafatas and Peeta 2006). It consists of

a 3x4 grid network that replicates a dense urban environment with highways (long cells, 1-22),

arterials (medium cells, 24-78) and side streets (short cells, 79-126). From an evacuation

standpoint, the bottom of the network represents the boundary of the evacuation zone (and the

safety zone) to which evacuees are routed (short cells, 147-150) from the origin (source) cells

(short cells, 127-146). The 20 potential origin cells are attached to arterial and side street cell.

The number of evacuees assigned to each source cell depends on the assumed scenario. The cell

parameters for each scenario are shown in Tables 1 to 4 using terminology from the cell

transmission model.

The selected 3x4 grid network appears to be topologically small (neighborhood or small

district size) in comparison to potential evacuation zones (large districts, metropolitan area).

Nevertheless, when thousands of evacuees are assigned in the selected network, it adequately

models evacuation schemes of populated areas under threat, like stadiums, exhibition centers,

governmental buildings, and shopping centers, and proposes certain link capacity additions

which improve the operational performance of the evacuation operation. Also, the proposed

formulation models any topologically large evacuation area and again offers the advantage of

significantly reduced computational times, compared to existing formulations (Tuydes and

Ziliaskopoulos 2004, Kalafatas and Peeta 2006). It can even solve topologically large evacuation


areas with the spatio-temporal decomposition of the mass balance constraints, which is outside

the scope of this paper.

Design of experiments

There are three major scenario sets according to the three major parameters studied. They are: (i)

the number of reversed links, (ii) population size, and (iii) the spatial distribution of evacuation

O-D demand. The characteristic parameters of each scenario type are summarized in Table 3.

The first scenario set examines the effect of different levels of resource allocation for the

network design options. Only contra-flow operations are assumed as a network design option,

and the resource allocation is quantified by using the number of reversed links as a proxy. Hence,

reversed links ranging from 0 to 20 are examined for a uniform distribution of 5000 evacuees to

20 sources. It is expected that these experiments will provide insights on the “ideal” levels of

resource allocation for the decision-makers (or planners). The experiments also analyze the

computational time efficiencies.

The second scenario set examines the effect of the population size on the evacuation

performance for a constant number of reversed links, acquired after the analysis of the first

scenario set. The number of reversed links used is the one which leads to the most improvement

in network performance in the first scenario set. For this number of reversed links (eight),

population sizes of 500 to 5000 evacuees are assigned to 20 sources.

The third scenario set examines the effect of the spatial distribution of the transportation

demand for evacuation. 5000 evacuees are assigned to 1 source, 2 sources uniformly, and 20

sources uniformly and randomly, and routed to 1 destination, 2 destinations, and 4 destinations.

It seeks insights on the topological properties of the selected reversed links. Table 4 illustrates

the distribution of demand under the random demand distribution scenario.


The three scenarios are evaluated using the network clearance time (as in Fig. 4, 8 and

11), the evacuation rate (as in Fig. 5, 9 and 10), and the graphical view of the test network with

the selected reversed links (as in Fig. 6 and 12). Also, the computational time is analyzed in

Figure 7.

Computational resources

The computing environment consists of a Sun Ultra Enterprise server E6500 with 26 400-MHz

UltraSparc II processors under the multi-user Solaris 7 operating environment with 23 GB of

RAM, 131 GB of swap space, and 8 MB of cache. The GAMS modeling language and CPLEX’s

mixed integer solver were used.

Computational Results

Effect of resource allocation on evacuation performance for uniformly distributed population

The network clearance time is used to analyze the network performance under various resource

(number of reversed links) constraints, as illustrated in Figure 4. The initial network of 0

reversed links entails a clearance time of 22 minutes. The corresponding value for 20 reversed

links is 14 minutes, representing a 36% reduction in network clearance time. This implies that

using the contra-flow option in dense urban environments can lead to significant performance

enhancements under security-related mass evacuation scenarios. An important practical insight is

that most of the potential benefits through contra-flow operations are realized when 8 reversed

links are allocated, which results in a network clearance time of 16 minutes (Fig. 4). It illustrates

that there may be an optimal level of resource allocation beyond which additional benefits are

insignificant. This implies that decision-makers (traffic and/or emergency response operators)


can determine effective contra-flow strategies by identifying the best level of resource allocation

from a cost-benefit perspective.

The evacuation rate, illustrated in Figure 5, indirectly illustrates the sigmoid nature of the

cumulative network clearance time curve. The evacuation rate (rate of arrivals to the destination

cells), representing the tangent of the cumulative curve, initially increases. It reaches a maximum

rate, and then keeps decreasing. The various characteristics of the results can be explained by

tracking the time-dependent nature of traffic congestion. Initially, the clearance rate increases as

demand is being serviced below capacity, that is, the network is not congested to capacity. As

further demand is serviced, the network links reach their capacities and that is represented by the

region of the stabilized maximum evacuation rate. As time progresses, demand decreases leading

to reduced evacuation rates until all traffic is cleared from the evacuation zone.

The network design options for 8 RL and 20 RL are illustrated graphically in Figures

6(A) and 6(B), respectively. The selected reversed links for contra-flow operations are indicated

with thick lines. There is a clear topological trend in terms of the formation of contra-flow

corridors with increased resources. The reversed links start forming close to the destination cells

of the test network and extend inwards to form corridors as the number of reversed links

increase. This is because of the directionality of the evacuation flows which makes the capacity

closer to the destinations critical in terms of enabling efficient evacuation rates. After the 8

reversed links case, a general tree structure is exhibited. It is important to note that most of the

improvement of the network clearance time is achieved at the level of 8 reversed links.

Figure 7 indicates the exponential increase in computational times when the number of

reversed links increases. The exponential increase of the computational time is attributed to the

exponentially greater solution space (larger branch and bound tree) generated by the increased


number of the feasible combinations of the network design binary variables. Hence, the insights

from the computational times also suggest that beyond some resource allocation levels, the

benefits are marginal.

Effect of uniformly distributed population size on evacuation performance

Figure 8 suggests a linear relationship between the evacuee population size and the network

clearance time (under the uniform distribution on the time-invariant network). The linearity is

explained by the constant evacuation rate for a long time period. The clearance time for 5000

evacuees is 15 minutes, while for 500 evacuees is 2.5 minutes. The correlated linear relation

between population size x and clearance time y in minutes is y = ax+b, where a = 1/360

evacuations/min and b = 1.12 minutes. In this expression, the capacity of the bottleneck is

directly recognized as 1/a = 360evacuations/min = 21,600 evacuations /hour. We can view the b

(= 1.12 minutes) coefficient as the least clearance time for a single “equivalent” evacuee. Since a

single evacuee would meet no congestion, this time is his/her travel time. Concluding, the 1/a is

the maximum evacuation rate, or the capacity of the bottleneck, and the b coefficient pinpoints

the effect of travel time from the origins to the network exits on the clearance time. It is further

noted here that this linear relationship corresponds to evacuation cases where steady-state flow

conditions prevail at the bottleneck locations, or heavily populated areas are evacuated in non-

time-dependent network topologies. It is relevant for metropolitan area, town and city

evacuations.

The evacuation rate of evacuees exiting the evacuation zone for various population sizes

(under uniform spatial distribution) are illustrated in Figure 9. It is useful to note that for small

evacuee population sizes, the evacuation rate does not reach the network capacity. However, for

large population sizes, the maximum capacity of the network is reached and retained constant for


a long time period, substantially constraining the evacuation performance. In terms of the

solution methodology, this observation also indicates that it is adequate to solve for a population

size that activates the bottleneck capacity and not for the potentially millions of the actual

evacuees in order to design for the optimal contra-flow options. The observation highlights the

direct usage of the minimum-cut maximum-flow theorem for large population sizes that

maintains steady-state flows at the bottlenecks for long periods.

Effect of spatial distribution of evacuation demand on network performance

The evacuation rates of evacuees exiting the evacuation zone for various scenarios of spatial

distribution under the 6 reversed links case are illustrated in Figure 10. The examined scenarios

are combinations of single or multiple sources and/or destinations, as illustrated in Table 3.

These scenarios are discrete and cannot be physically examined in a continuous manner. The

results suggest that the patterns with multiple sources, multiple destinations, and uniform spatial

distributions lead to better clearance times (Fig. 11). This trend is reasonable as multiple sources

and destinations avoid local congestion hotspots that can occur due to concentration of demand

at few locations. Further, as expected, the uniform distribution scenario performs better than the

random distribution scenario.

The topology of the selected reversed links is illustrated in Figure 12. The selected

reversed links for contra-flow operations are indicated with thick lines. They indicate that

capacity is added where bottlenecks exist. This is easily observed especially in the scenarios with

1 or 2 sources or sinks. It is important to note that for the scenarios with 1-2 sources to many

destinations, there exist links that do not reduce bottlenecks, like links 52 and 54 for both

scenarios (Fig. 12, part A and B). That is, the bottleneck has been optimally improved, and the


reversal of links 52 and 54 does not provide additional benefits (it simply satisfies the 6 reversed

links requirement).

Concluding Comments

This paper presents an extensive sensitivity analysis for the evacuation planning problem. The

parameters studied were the magnitude of the available budget, the population size, and the

amount and spatial distribution of the evacuation demand. The results suggest that there is a

threshold budget beyond which benefits (in terms of the network clearance time) are negligible.

They also indicate that the network clearance time varies linearly with population size. Finally,

greater spatial uniformity of O-D demand reduces clearance times, suggesting that specifying

multiple destinations in the safety zone may lead to more efficient evacuation in many instances.

The study results consistently suggest the presence of efficient sizes of resource

allocation (budget); there is a critical level of resource allocation (in terms of the number of

reversed links) beyond which benefits are trivial (in terms of network clearance time). The level

of resource allocation is directly related to the resources required to efficiently handling the

capacity bottleneck in the studied network. More specifically, it was observed that capacity is

allocated to the exact location of the bottlenecks. Under uniformly distributed population,

capacity was allocated near the evacuation zone exits leading to the formation of corridors and,

eventually, trees. In the case of 1-2 sources or destinations, capacity was allocated to the links in

their vicinity. Furthermore, the network clearance time was observed to be linearly related to the

population size for the same level of resource allocation. Again, the effect of the activated

bottleneck was dominant, especially for large population sizes. Finally, the analysis of the effects

of the spatial distribution of the population indicates that uniformity in the spatial distribution

and multiple origins/destinations lead to lesser clearance times. This was attributed to the fact


that a pattern of distributed demand could avoid local congestion hotspots, therefore utilizing

more capacity spatially in the network.

From a response standpoint, the study suggests that operational effectiveness requires

pre-determined contra-flow strategy plans with designated and well-trained personnel, rather

than just reacting to a security event in real-time in an ad-hoc manner.

In summary, it appears that the capacity bottleneck plays a dominant role for large-scale

evacuation problems: (i) it is responsible for the efficient size of resource allocation, (ii) it

strongly influences the network clearance time, and (ii) even for different spatially distributed

patterns, the location of the capacity bottleneck determines the performance. In future research,

we plan to analyze the effects of different network topologies, various driving behavior patterns

under evacuation conditions, the effects of various sigmoid network loading curves, and

alternative traffic conditions, and develop heuristic solution methodologies to further reduce

computational times.


References

Cova, T. J., and Johnson, J. P. (2003). “A network flow model for lane-based evacuation

routing.” Trans. Res., Part A, 37, 579-604.

Daganzo, C.F. (1994). “The cell transmission model: A simple dynamic representation of

highway traffic consistent with the hydrodynamic theory.” Trans. Res., Part A, 28(4), 269-

287.

Daganzo, C.F. (1995). “The cell transmission model, Part II: Network traffic.” Trans. Res.,

Part B, 29(2), 79-93.

Highway Capacity Manual. (2000).

Johnson, D.S., Lenstra, J. K., and Rinnoy, A. H. (1978). “The complexity of the network

design problem.” Networks, 8, 279-285.

Kalafatas, G., and Peeta, S. (2006). “Network design for evacuation planning.” Proc., 11th

IFAC Symposium on Control in Transportation System, Delft, Netherlands.

Kalafatas, G., and Peeta, S. (2007). “An exact graph structure for dynamic traffic assignment:

Formulation, properties, computational experience.” Proc., 86th Annu. Meeting, Transp. Res.

Board.

Kwon, E., and Pitt, S. (2005). “Evaluation of emergency evacuation strategies for downtown

event traffic using a dynamic network model.” Proc., 84th Annu. Meeting, Transp. Res.

Board.

Li, Y., T. S. Waller, and Ziliaskopoulos, A. (2003). “A decomposition scheme for system

optimal dynamic traffic assignment models.” Networks and Spatial Economics, 3(4), 441-

455.


Sbayti, H., and Mahmassani, H. S. (2006). “Optimal scheduling of evacuation operations.”

Proc., 85th Annu. Meeting, Transp. Res. Board.

Sheffi, Y., Mahmassani, H. S. and Powell, W. B. (1982). “A transportation network

evacuation model.” Trans. Res., Part A, 16(3), 209-218.

Southworth, F. (1991). “Regional evacuation modeling: a state-of-the-art review.” Oak Ridge

National Labs, ORNL/TM-11740.

Tuydes, H., and Ziliaskopoulos, A. (2004). “Network re-design to optimize evacuation

contra-flow.” Proc., 83rd Annu. Meeting, Transp. Res. Board.

Tuydes, H., and Ziliaskopoulos, A. (2006). “Tabu-based heuristic for optimization of

network evacuation contraflow.” Proc., 85th Annu. Meeting, Transp. Res. Board.

Wolshon, B. (2001). “One-way-out: Contra flow freeway operation for hurricane

evacuation.” Natural Hazards Review, 2(3), 105-112.


Table 1. Cell characteristic parameters of the test network.

Cell Type Highway Arterial Side Street Source Destination

Cell IDs 1-22 23-78 79-126 127-146 147-150

Free flow speed (miles/h) 70 35 20 - -

Time interval (sec) 10 10 10 10 10

Cell length (feet) 1000 500 250 - -

Number of lanes 3 2 1 3 3

Maximum flow per lane

(veh/hour/lane) 2160 1800 1800 2160 2160

Maximum cell flow 18 10 5 infinite infinite

Reduced maximum cell flow

(veh/time step) (due to the

evacuation operations)

12 7 3 12 12

Number of vehicles per cell

(veh/cell) 108 36 9 infinite infinite

Table 2. Cell characteristic parameters in each direction according to contra-flow options.

Final number of

lanes per direction

Maximum cell flow

Qi (veh/h)

Maximum cell flow Qi

(veh / time step of 10 sec)

Direction Improved Reduced Improved Reduced Improved Reduced

Highway cell

(3 lanes per

direction)

5 1 5760 900 16 2.5

6 0 7200 0 20 0

Arterial cell

(2 lanes per

direction)

3 1 3240 720 9 2.0

4 0 4320 0 12 0

Side street cell

(1 lane per direction) 2 0 1800 0 5 0


Table 3. Characteristic parameters of the experiment scenarios.

Scenario

ID

(SID)

Description Population

size

Spatial

distribution of

the population

Number

of

sources

Number of

destinations

Number of

reversed

links

1.1

Number of

reversed

links

5000 Uniform 20 4 0

1.2 5000 Uniform 20 4 2

1.3 5000 Uniform 20 4 4

1.4 5000 Uniform 20 4 6

1.5 5000 Uniform 20 4 8

1.6 5000 Uniform 20 4 10

1.7 5000 Uniform 20 4 12

1.8 5000 Uniform 20 4 14

1.9 5000 Uniform 20 4 18

1.10 5000 Uniform 20 4 20

2.1

Population

size

500 Uniform 20 4 8

2.2 1000 Uniform 20 4 8

2.3 1500 Uniform 20 4 8

2.4 2000 Uniform 20 4 8

2.5 2500 Uniform 20 4 8

2.6 3000 Uniform 20 4 8

2.7 3500 Uniform 20 4 8

2.8 4000 Uniform 20 4 8

2.9 4500 Uniform 20 4 8

2.10 5000 Uniform 20 4 8

3.1 Spatial

distribution

of

evacuation

demand

5000 Uniform 20 4 6

3.2 5000 Random 20 4 6

3.3 5000 Uniform 1 4 6

3.4 5000 Uniform 2 4 6

3.5 5000 Uniform 20 1 6

3.6 5000 Uniform 20 2 6


Table 4. 5000 evacuees randomly distributed to source cells in the random distribution scenario.

Source

ID Population

Source

ID Population

127 235 137 429

128 48 138 81

129 27 139 234

130 73 140 398

131 167 141 111

132 390 142 67

133 214 143 343

134 268 144 97

135 1042 145 436

136 203 146 137


Fig. 1. Characteristic points of the CTM traffic flow-density relationship.


Fig. 2. Refined cell representations for planning applications.


Fig. 3. Test network.


Fig. 4. Clearance time as a function of the number of reversed links.


0

100

200

300

400

500

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Time elapsed from the beginning of the evacuation (min)

Evacuees reaching the safety zone per minute

(evacuees/minute)

0 RL 2 RL4 RL 6 RL8 RL 10 RL12 RL 14 RL16 RL 18 RL20 RL

Fig. 5. Evacuation rate per minute for different numbers of reversed links.


Part

B. I

mpr

oved

net

wor

k w

ith 2

0 re

vers

ed li

nks f

or 5

000

evac

uees

unifo

rmly

dis

tribu

ted

to 2

0 so

urce

s and

rout

ed to

4 d

estin

atio

ns

(SID

1.1

0).

Fig.

6. T

opol

ogic

al c

hara

cter

istic

s of t

he o

ptim

al so

lutio

n fo

r var

ious

leve

ls o

f con

tra-f

low

reso

urce

s.

Part

A. I

mpr

oved

net

wor

k w

ith 8

reve

rsed

link

s for

500

0 ev

acue

es

unifo

rmly

dis

tribu

ted

to 2

0 so

urce

s and

rout

ed to

4 d

estin

atio

ns

(SID

1.4

)


Fig. 7. Computational time for 1% optimality gap for different number of reversed links.


Fig. 8. Clearance time as a function of evacuee population with 8 reversed links.


0

50

100

150

200

250

300

350

400

450

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Time elapsed from the beginning of the evacuation order (min)

Evac

uatio

n ra

te p

er m

inut

e (e

vacu

ees/m

in.)

500 Evacuees1000 Evacuees1500 Evacuees2000 Evacuees2500 Evacuees3000 Evacuees

3500 Evacuees4000 Evacuees4500 Evacuees5000 Evacuees

Fig. 9. Evacuation rate per minute for different evacuee population sizes with 8 reversed links.


0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44

Time elapsed from the beginning of the evacuation order (min)

Evac

uatio

n ra

te p

er m

inut

e (e

vacu

ees/m

in.) Multiple sources - multiple destinations / uniform

Multiple sources - multiple destinations / random1 source - multiple destinations2 sources - multiple destinationsMultiple sources - 1 destinationMultiple sources - 2 destinations

Fig. 10. Evacuation rate per minute for different scenarios of spatial evacuation distribution with

6 reversed links.

Fig. 11. Clearance time as a function of the spatial distribution of evacuation demand for 6

reversed links.


Pa

rt C

. Im

prov

ed n

etw

ork

with

6 re

vers

ed li

nks f

or 5

000

evac

uees

uni

form

ly d

istri

bute

d to

2 so

urce

s (ce

lls 1

42 a

nd 1

34,

high

light

ed) a

nd ro

uted

to 4

des

tinat

ions

(SID

3.4

).

Pa

rt E.

Impr

oved

net

wor

k w

ith 6

reve

rsed

link

s for

500

0

evac

uees

uni

form

ly d

istri

bute

d to

20

sour

ces a

nd ro

uted

to 2

dest

inat

ions

(cel

ls 1

47 a

nd 1

50, h

ighl

ight

ed),

(SID

3.6

).

Pa

rt B

. Im

prov

ed n

etw

ork

with

6 re

vers

ed li

nks f

or 5

000

evac

uees

in a

1 so

urce

(cel

l 142

, hig

hlig

hted

) cas

e an

d

rout

ed to

4 d

estin

atio

ns (S

ID 3

.3).

Pa

rt D

. Im

prov

ed n

etw

ork

with

6 re

vers

ed li

nks f

or 5

000

evac

uees

uni

form

ly d

istri

bute

d to

20

sour

ces a

nd ro

uted

to 1

dest

inat

ion

(cel

l 148

, hig

hlig

hted

), (S

ID 3

.5).

Pa

rt A

. Im

prov

ed n

etw

ork

with

6 re

vers

ed li

nks f

or 5

000

evac

uees

rand

omly

dis

tribu

ted

to 2

0 so

urce

s and

rout

ed to

4

dest

inat

ions

(SID

3.2

). H

ighl

ight

ed o

rigin

cel

ls in

dica

te

larg

er p

opul

atio

n ce

nter

s.

Fig.

12.

Opt

imal

net

wor

k co

nfig

urat

ion

for v

ario

us sp

atia

l

dist

ribut

ions

of t

he e

vacu

atio

n de

man

d.

Planning for evacuation: insights from an efficient ...

Documents

Transcript of Planning for evacuation: insights from an efficient ...