Discrete Optimization Models for Homeland Security and Disaster Management

Discrete Optimization Models for Homeland Security
and Disaster Management
Laura Albert McLay
University of Wisconsin-Madison
Industrial and Systems Engineering
laura@engr.wisc.edu, @lauramclay
This work was funded by the National Science Foundation [Awards 1444219, 1541165]. The views and conclusions contained in
this document are those of the author and should not be interpreted as necessarily representing the oﬃcial policies, either
expressed or implied, of the National Science Foundation.
Laura Albert McLay () Disaster Management tutORial 2015 1 / 97

Goal of this tutORial
The goal is to
a) introduce you to research in optimization research in disasters and
homeland security.
b) introduce you to important issues for doing research in optimization
research in disasters and homeland security (modeling and data).
Not a comprehensive review of all research models in disasters!

Thank you National Science Foundation!
Enabling the Next Generation of Hazards and Disasters Researchers,
2009-2010.
This work was funded by the National Science Foundation Awards
1444219, 1541165.

Discrete optimization can be used to address many types
of disasters, including“acts of God” and acts of terrorism

Types of hazards we can address
Courtesy of FEMA

Scales of disasters
Disaster metrics must capture the magnitude and scope of physical impact
and social disruption.
Community, regional, or societal levels
Not at the individual level

Emergency
An event with local eﬀects that can be managed at the local level.
Examples: bus crash, local ﬂoods, building collapses, etc.

Disaster
A severe event that affects a region but whose damaging effects are not
national in scope. The event can be managed with local, regional, and
some national resources.
Examples: Hurricane Irene (2011), tornadoes, pandemic flu

Catastrophe
A severe event that affects an entire nation and where the local and
regional responses are inadequate or impossible due to affected critical
infrastructure. The event threatens the welfare of a substantial number of
people for an extended amount of time, and therefore, necessitates a
national or international response.
Examples: Massive New Madrid zone earthquake, Spanish flu, 9/11,
Hurricane Katrina, 1976 Tang-Shen earthquake in China.

Extinction level event
Results (or could result) in the loss of all human life. No eﬀective response
is available.
Examples: Massive meteorite strike, Yellowstone Caldera super explosion
(maybe), bi-national or multinational thermonuclear warfare, zombie
outbreak.

A note about terminology
While we use the term disasters generically in this paper to refer to
intentional (man-made) and natural disasters with varying scopes of
damage, we note that there are diﬀerent scales associated with these
events

Disaster: Joplin, Missouri
Courtesy of Jose Holguin-Veras at Rensselaer Polytechnic Institute

Disaster: Joplin, Missouri

Disaster: F-4 Tornado in Illinois, 11/17/2013

Catastrophe: Minami Sanriku, Japan

Catastrophe: Japan

Critical infrastructure: The U.S. PATRIOT Act
“Systems and assets, whether physical or virtual, so vital to the
United States that the incapacity or destruction of such systems
and assets would have a debilitating impact on security, national
economic security, national public health or safety, or any
combination of those matters.”

Disasters cycles
Courtesy of FEMA

Disasters Lifecycle
Vulnerability
Mitigation
Preparedness
Emergency response
Recovery
The NAE notes that classiﬁcation schemes are based on various
characteristics of disasters, such as their length of forewarning,
detectability, speed of onset, magnitude, scope, and duration of impact.
Additionally, the NAE recommends that willful acts of terrorism are
included in this lifecycle.

Disasters Lifecycle: Vulnerability
Vulnerability is the potential for physical harm and social disruption.
Physical harm represents the damage and threat of damage to physical
infrastructures and the natural environment. Social disruption and
vulnerability represent threats to human populations, such as death, injury,
morbidity, and disruption to behavior and social functioning.
Note: vulnerability does not lend itself to optimization.

Disasters Lifecycle: Mitigation
Mitigation includes actions taken prior to a disaster to prevent or reduce
the potential for harm, physical vulnerability, and social disruption.
Mitigation can be structural, which involves designing, constructing,
maintaining, and renovating physical infrastructure to withstand the
impact of a disaster, or nonstructural, which involves reducing exposure of
human populations and physical infrastructure to hazard conditions.
Examples
Checkpoint screening for security
Network design and fortiﬁcation
Pre-locating medical facilities and response stations

Disasters Lifecycle: Preparedness
Preparedness includes actions taken prior to a disaster to aid in response
and recovery during and after a disaster. Preparedness actions often
include the development of formal disaster plans and the maintenance of
resources that are used in response and recovery.
Examples
Pre-positioning crews and supplies in advance of a disaster
Evacuation planning
Emergency crew scheduling

Disasters Lifecycle: Emergency response
Emergency response includes actions during and after a disaster to
protect and maintain systems, rescue and respond to casualties and
survivors, and restore essential public services.
Examples
Urban search and rescue
Routing and distribution of supplies and commodities
Hospital evacuation

Disasters Lifecycle: Recovery
Recovery includes eﬀorts to reestablish pre-disaster systems and services
through restoration and repair of infrastructure and social and economic
routines, such as education, consumption, and healthcare.
Examples
Debris clean up and removal
Roads, bridge, and facility repair and restoration
Replanting and restoration of forests and wetlands aﬀected by a
natural disaster.

Who manages disasters?
Department of Homeland Security (DHS) manages disasters in the U.S.
DHS directorates manage diﬀerent hazards and disasters, including
transportation security (Transportation Security Administration
(TSA)),
nuclear security (Domestic Nuclear Detection Oﬃce),
border security (Customs and Border Protection (CBP)),
maritime security (U.S. Coast Guard),
all natural hazards and disasters (Federal Emergency Management
Agency (FEMA)).

Who manages disasters?
Military organizations such as the National Guard and the U.S. Coast
Guard provide support during disasters.
Numerous non-governmental organizations (NGOs) are also involved in
disasters.
The Red Cross, etc.
Usually, many organizations work together.

Discrete optimization is a good tool for disasters
Eﬀorts to improve disaster management often involve managing limited
resources, such as
pre-locating scarce resources and supplies
routing response vehicles
scheduling recovery crews
Disaster management often involves discrete choices
But we do not end up with textbook problems

Timeframe over which we model and inform decisions
Different types of disasters and times
Hurricane and wildfires give advanced warning
Climate change vs. tornado timeframes (slow / fast)
Mitigation vs. response (data collection ahead of time / on-the-fly)

Discrete optimization has its challenges
Challenges:
informational uncertainties in terms of supply and demand
damage to critical infrastructure and transportation networks
limited resources
challenges in communication between organizations providing aid and
between those who need help
data!

I recognize that disaster problems are “wicked” problems
There are tame problems
Some problems need just operations research solutions
And then there are wicked problems
Problems that have a strong social, legal, and political components
Many stakeholders, incomplete, contradictory, and changing
requirements that are often diﬃcult to recognize

I recognize that disaster problems are “wicked” problems
There are tame problems
Some problems need just operations research solutions
And then there are wicked problems
Problems that have a strong social, legal, and political components
Many stakeholders, incomplete, contradictory, and changing
requirements that are often diﬃcult to recognize
Russell Ackoﬀ: “Every problem interacts with other problems and is
therefore part of a set of interrelated problems, a system of problems...I
choose to call such a system a mess.”

Now let’s introduce some models

Disaster mitigation
Focuses on preventing a disaster or reducing the harmful eﬀects of a
disaster
We focus on
1 Screening and pre-screening problems in homeland security
applications that seek to reduce harm through detection and
prevention, and
2 Network interdiction and fortiﬁcation models

Checked baggage security
Early discrete optimization research dates back prior to September 11,
2001
Checked baggage for high-risk passengers screened for explosives
selectee and non-selectee screening
Goal was how to optimally deploy and use limited baggage screening
devices.

Checked baggage security: covering models
Definitions:
Station = set of airport facilities that share security resources
Flight segment = flight between takeoff and landing of an aircraft
A flight segment is:
uncovered if 1+ bags on the flight has not been screened
covered if all selectee bags on it have been screened
Baggage screening performance measures developed in conjunction with
the Federal Aviation Administration:
1 Uncovered flight segments (UFS), which captures the total number of
uncovered flights.
2 Uncovered passenger segments (UPS), which captures the total
number of passengers on uncovered flights.
3 Uncovered baggage segments (UBS), which captures the total
number of unscreened selectee bags (regardless of flight).

Checked baggage security: covering models
Side note: coverage is a useful objective
in low-data situations,
when better data collection is cumbersome (e.g., subject matter
experts)
when objectives are not clear
for mitigation (want to “touch” as many vulnerable parts of a system)
Second side note: we have limited resources:
Maximal coverage subject to budget or cardinality constraints

Checked baggage security: parameters
F = {f1, f2, ..., fn} = a set of n flights.
C(fi ) = the number of selectee bags on flight fi , i = 1, 2, ..., n.
N(fi ) = the number of passengers on flight fi , i = 1, 2, ..., n.
S = the upper bound on the number of bags that can be screened.

The Uncovered Flight Segment Problem (UFSP): Find a subset of
ﬂights F ∈ F such that fi ∈F C(fi ) ≤ S and the number of ﬂights |F |
is maximized.

The Uncovered Flight Segment Problem (UFSP): Find a subset of
flights F ∈ F such that fi ∈F C(fi ) ≤ S and the number of flights |F |
is maximized.
The Uncovered Passenger Segment Problem (UPSP): Find a subset
of flights F ∈ F such that fi ∈F C(fi ) ≤ S and fi ∈F N(fi ) is
maximized.
The Uncovered Baggage Segment Problem (UBSP): Find a subset of
flights F ∈ F such that fi ∈F C(fi ) ≤ S and fi ∈F C(fi ) is maximized.

Checked baggage security: multiple stations
New parameters include:
m = the number of screening stations.
Fi ∈ F = the set of ﬂights that are associated with station
i = 1, 2, ..., m, where F1, F2, ..., Fm are a partition of F.
Three models address general allocation sizes. Here is the model
associated with the UPS measure.
The Multiple Station Passenger Segment Security Allocation and
Screening Problem with General Allocation Sizes: Find the subsets of
ﬂights Fi ∈ Fi , i = 1, 2, ..., m and a set of baggage screening security
device capacity allocations Si , i = 1, 2, ..., m that maximizes
m
i=1 fi ∈Fi
N(fi ) such that m
i=1 Si ≤ S and fi ∈Fi
C(fi ) ≤ Si ,
i = 1, 2, ..., m.

Checked baggage security: predetermined allocation sizes
Requires capacity to be assigned to stations in ﬁxed, discrete amounts.
New parameters include:
A = {a1, a2, ..., a } = set of available security device allocations.
W (aj ) = the capacity associated with security device allocation aj ,
j = 1, 2, ..., .
The Multiple Station Passenger Segment Security Allocation and
Screening Problem with Predetermined Allocation Sizes: Find the
subsets of ﬂights Fi ∈ Fi , i = 1, 2, ..., m and a partition {Ai , A2, ..., Am} of
security device allocations A to stations that maximizes
m
i=1 fi ∈Fi
N(fi ) such that m
i=1 fi ∈Fi
C(fi ) ≤ aj ∈A W (aj ).

Checked baggage security: uncovered targets
Extension to consider weapons of mass destruction (WMD), not just
conventional weapons
scope of damage not just the airplane and its passengers
1 Uncovered targets (UT), which captures the total number of
uncovered targets (e.g., destination cities), where a target is covered
if all ﬂights to the target are covered.
Leads to goal programming models that balance the UT measure with the
UPS, UFS, and UBS measures.

Risk-based methods for screening passengers

Aviation passenger checkpoint security
Risk-based multilevel screening considers
2+ levels of security to screen passengers
Passenger risk assessments
Static vs. dynamic security
This research provided the fundamental technical analysis for risk-based
security, which in led to TSA PreCheck.

Static checkpoint security: parameters and decision
variables
N = number of passengers.
M = number of screening classes.
ATj = assessed threat value of passenger j = 1, ..., N, a risk
assessment of passenger j returned by a passenger prescreening
system.
Li = the security level achieved by screening a passenger with
screening class i = 1, ..., M.
FCi = the ﬁxed cost associated with screening class i = 1, ..., M.
MCi = the marginal cost of screening a passenger with screening
class i = 1, ..., M.
B = the screening budget to be used for the time horizon.
yi = 1 if screening class i is used and 0 otherwise, i = 1, 2, ..., M.
xij = 1 if passenger j is screened by class i and 0 otherwise,
i = 1, 2, ..., M, j = 1, 2, ..., N.

Static checkpoint security: model
max
M
i=1
N
j=1
Li ATj xij (1)
subject to
M
i=1
N
j=1
MCi xij +
M
i=1
FCi yi ≤ B (2)
M
i=1
xij = 1, j = 1, 2, . . . , N (3)
xij − yi ≤ 0, j = 1, 2, ..., N, i = 1, 2, . . . , M (4)
yi ∈ {0, 1}, i = 1, 2, . . . , M (5)
xij ∈ {0, 1}, i = 1, 2, . . . , M, j = 1, 2, . . . , N. (6)

Static checkpoint security: model
max
M
i=1
N
j=1
Li ATj xij (1)
subject to
M
i=1
N
j=1
MCi xij +
M
i=1
FCi yi ≤ B (2)
M
i=1
xij = 1, j = 1, 2, . . . , N (3)
xij − yi ≤ 0, j = 1, 2, ..., N, i = 1, 2, . . . , M (4)
yi ∈ {0, 1}, i = 1, 2, . . . , M (5)
xij ∈ {0, 1}, i = 1, 2, . . . , M, j = 1, 2, . . . , N. (6)
Insight: passengers tend to be assigned to few classes, with expedited
screening of the lower risk passengers.

Other risk based screening papers
How to optimally use security devices that are shared across security
classes
Consider trade-oﬀs between false alarm rate (proxy for passenger
inconvenience) and number of screeners needed
How to sequentially “score” passengers and group passengers
according to their scores based on cumulative device responses

Dynamic aviation passenger checkpoint security
Dynamic passenger screening provide insight into real-time screening
decisions.
Modeled in the literature as a sequential process
Each passenger’s risk level becomes known upon arrival to a security
checkpoint
Markov decision process models for assigning passengers (in
real-time) to aviation security resources.
Selectee/nonselectee vs. multi-level screening
Two-stage models for device purchase/usage decisions

Cargo security
A stream of literature focuses on how to screen cargo containers for
nuclear material at ports of entry in a risk-based environment.
Ninety-ﬁve percent of international goods that enter the U.S. come
through one of the ports of entry
Early screening was performed by physically unpacking cargo containers
and inspecting their contents.

Cargo security
Effort to develop more effective screening methods based on imaging or
passive radiation detection to efficiently screen more containers.
Nearly all cargo containers are screened by radiation portal monitors
(RPMs)
Advanced screening methods such as imaging and physical inspection
used more sparingly and targeted at high-risk containers
Automated Targeting System (ATS) performs a risk assessment on
cargo containers that can aid in designing risk-based screening
systems.
Primary screening vs. secondary screening

Cargo security models
Linear programming model for performing primary screening on cargo
containers exiting a security checkpoint based on sensor responses
Which containers should be selected for secondary screening?
Linear programming models for selecting the right mix of primary
screening devices for each risk group
Integrated model for primary and secondary screening
Secondary screening depends on the mix of primary screening devices
that alarm
Large-scale linear programming model to sequence screening tests for
screening cargo containers.
How to supplement risk assessments with radiography-based images to
identify potential containerized threat scenarios

Network models address the need to design and fortify parts of a network
to mitigate vulnerabilities
Network interdiction models focus on worst-case failures
Failures could be due to natural or intentional causes
Does not model random component failure or reliability
Nearly all network interdiction problems modeled as Stackelberg games
Leader acts ﬁrst by interdicting components of the network (e.g.,
lengthening arcs)
Follower acts second by performing recourse actions (e.g., selecting a
shortest path from the source to the sink).

Several classes of network interdiction models that are
well-studied in the literature
Shortest-path network interdiction: the operator seeks to find the
shortest path from a source to a sink after the leader lengthens arcs
in the network,
Maximum reliability network interdiction: the operator seeks to
maximize the probability of evading detection after the leader reduces
the evasion probability on some of the arcs in the network,
Maximum flow interdiction: the operator seeks to maximize the
flow achievable on the network after the leader reduces some of the
arc capacities, and
p-median network interdiction: the operator seeks to minimize the
total worst-case distance from the customers to the nearest facility
after the leader removes some of the facilities.
The interdiction model literature considers applications in transportation,
power, and other networks.

Network design and interdiction
Facility location interdiction support disaster mitigation eﬀorts: p-median
and covering.
Parameters:
F = set of existing facilities.
I = set of demand locations.
ai = demand at location i ∈ I.
dij = shortest distance between i and j, i ∈ I, j ∈ F.
r = number of facilities to be interdicted or eliminated.
Tij = {k ∈ F|k = j and dik > dij } = set of existing sites (not
including j) that are as far or farther than j is from demand i.
Ni ⊂ F = set of facility sites that cover demand i, i ∈ I.

Network design and interdiction
The r-interdiction median problem admits the following decision variables:
sj =
1 if facility j is eliminated, j ∈ F
0 otherwise
xij =
1 if demand i is assigned to a facility at j, i ∈ I, j ∈ F
0 otherwise

r-interdiction median problem integer programming model
z = max
i∈I j∈F
ai dij xij (7)
subject to
j∈F
sj = r (8)
j∈F
xij = 1, i ∈ I (9)
k∈Tij
xik ≤ sj , j ∈ F (10)
xij ∈ {0, 1}, i ∈ I, j ∈ F (11)
sj ∈ {0, 1}, j ∈ F (12)
The objective function (7) seeks to maximize the resulting total weighted
distance between demand locations and facilities that have not been
interdicted.

r-interdiction covering problem
A demand location is no longer covered if all of the facilities that cover the
location have been interdicted.
New decision variables in addition to sj , j ∈ J:
yi =
1 if demand i is no longer covered
0 otherwise

r-interdiction covering problem integer programming model
z = max
i∈I
ai yi (13)
subject to
j∈F
sj = r (14)
yi ≤ sj , i ∈ I, j ∈ Ni ∩ F (15)
yi ∈ {0, 1}, i ∈ I (16)
sj ∈ {0, 1}, j ∈ F (17)
The objective function seeks to maximize the resulting total demand that
is uncovered after interdiction.

Fortification: allow a decision-maker to first make some facilities immune
to failure
Fortification is one way to mitigate the worst-case vulnerabilities in these
two base models
This results in the r-interdiction median problem with fortification
Model has same parameters and decision variables as the r-interdiction
median problem with a new set of decision variables:
zj =
1 if facility j is fortified, j ∈ F
0 otherwise

r-interdiction median problem with fortification
Formulated as a bi-level programming model.
Upper level problem captures fortification:
min H(z) (18)
subject to
j∈F
zj = k (19)
zj ∈ {0, 1}, j ∈ F (20)
Lower level problem (next slide) captures interdiction of non-fortified
facilities.

r-interdiction median problem with fortiﬁcation
Lower level problem captures worst-case interdiction:
H(z) = max
i∈I j∈F
ai dij xij (21)
subject to
j∈F
sj = r (22)
j∈F
xij = 1, i ∈ I (23)
h∈Tij
xih ≤ sj , j ∈ F (24)
sj ≤ 1 − zj , j ∈ F (25)
xij ∈ {0, 1}, i ∈ I, j ∈ F (26)
sj ∈ {0, 1}, j ∈ F (27)
Note: a fortiﬁed facility cannot be interdicted

Solving the bi-level programming model is hard.
Decomposition, duality, or reformulation are not suitable for solving the
r-interdiction median problem with fortiﬁcation due to the structure of the
lower level problem.
Scaparra and Church introduce an implicit enumeration algorithm to solve
the r-interdiction median problem with fortiﬁcation.
A stream of papers examine algorithms and new models

Interdicting a nuclear weapons project
Attacker wishes to create a ﬁssion weapon as quickly as possible
Interdictor seeks to delay the completion of the project for as long as
possible by delaying certain activities needed to create a nuclear weapon.
The attacker’s model is a project management (critical shortest path)
problem
Precedence constraints and sequencing tasks
Attacker can expedite (shorten) tasks by using additional resources
Interdictor can delay the project by delaying certain tasks subject to
an interdiction constraint
Solved by Benders decomposition and implement the algorithm using
oﬀ-the-shelf project-management software.
Brown et al [2009]

Disaster preparedness
addresses planning that is done prior to a disaster that is implemented
when the disaster is imminent.
We focus on
1 pre-positioning supplies, and
2 Evacuation planning
Note: Pre-positioning way in advance is more on the mitigation side.

Pre-positioning resources for a large-scale emergency
The resulting models can be used to pre-locate ambulances (facilities) at
pre-selected stations.
Jia et al. [2007] illustrate the models with examples that include a dirty
bomb attack, a terrorist attack using anthrax, and a terrorist attack using
smallpox.
The models provide insight into identifying tailored hospital locations for
large-scale emergencies that take into account the unique attributes of
these emergencies.

Pre-positioning resources for a large-scale emergency:
parameters
J = set facility locations.
I = set of demand locations.
K = set of possible emergency situations.
Mi = population of location i ∈ I.
eik = the weight of location i under emergency scenario k,
i ∈ I, k ∈ K.
βik = the likelihood that location i suﬀers large-scale emergency
scenario k, i ∈ I, k ∈ K.
pjk = degree of disruption of facility j under emergency scenario k
with 0 ≤ pjk ≤ 1, j ∈ J, k ∈ K.
dij = distance from demand location i to j, i ∈ I, j ∈ J.
Qi = minimum number of facilities that must be assigned to location
i for it to be considered covered, i ∈ I.
P = the maximal number of facilities to locate.

Pre-positioning resources for a large-scale emergency
The decision variables are as follows:
xj =
1 if a facility is located at site j ∈ J
0 otherwise
zij =
1 if location i is assigned to a facility at j, i ∈ I, j ∈ F
0 otherwise

p-median model for emergency scenario k ∈ K
Zk = min
i∈I j∈J
βikeikMi dij zij (28)
subject to
j∈J
xj = P (29)
j∈J
zij pjk = Qi , i ∈ I (30)
zij ≤ xj , i ∈ I, j ∈ J (31)
zij ∈ {0, 1}, i ∈ I, j ∈ J (32)
xj ∈ {0, 1}, j ∈ J (33)
The model minimizes the demand-weighted distance between the demand
locations and the opened facilities.

Pre-positioning resources for a large-scale emergency:
coverage
Let Ni = {j|dij ≤ Di } be the set of facilities that cover (service) demand
location i ∈ I
Di is the longest distance from demand point i a facility may be to cover
i’s demand.
New decision variables (in addition to xj , j ∈ J):
uj =
1 if location j is covered, i ∈ I
0 otherwise

Covering model for emergency scenario k ∈ K
Zk = max
i∈I
βikeikMi ui (34)
subject to
j∈J
xj ≤ P (35)
j∈Ni
xj pjk ≥ Qi ui , i ∈ I (36)
ui ∈ {0, 1}, i ∈ I (37)
xj ∈ {0, 1}, j ∈ J (38)
The model maximizes weighted coverage.

Evacuation
Evacuation is needed for many disasters:
hurricanes
tropical storms
wildfires
floods
nuclear emergencies
Evacuation starts with the decision to evacuate
Evacuation models must yield detailed plans for evacuation guide the flow
of traffic.
Evacuation and routing use maximum flow networks models, facility
location models, the traveling salesman problem, and the vehicle routing
problem.

Disaster response
Focuses on preventing a disaster or reducing the harmful eﬀects of a
disaster
We focus on
1 response,
2 relief, and
3 routing

Emergency response to routine emergencies
Emergency response to routine emergencies that arise from 911 calls for
ﬁre, police, or emergency medical service (EMS) service has been an active
area of research since the early 1970s.
The most relevant papers seek to assign vehicles to calls in real-time as
calls for service arrive to the system.
Few papers in this area focus on large-scale emergencies.
The ideas and methods translate to disasters since both focus on
delivering time-sensitive service to customers.
Both focus on public welfare and equity

Emergency response: limitations
All of the papers for routine emergencies implicitly assume that
data are available and
data are accurate
system experiences low traﬃc (i.e., long queues do not form)
These assumptions are not valid for disaster response!

Routing for relief
Routing for relief and delivering aid requires delivering time-sensitive
commodities to customers.
Ozdamar et al. [2004] integrates the multi-commodity network ﬂow
problem and the vehicle routing problem (VRP) and seeks to deliver aid on
a multi-modal transportation network
Mete and Zabinsky [2010] propose a stochastic optimization model for
pre-locating (preparedness) and distributing (response) medical supplies
after a disaster

Routing for relief
Campbell et al. [2008] provide two routing models for routing supplies for
relief eﬀorts that are variants of the TSP and VRP.
(1) a minsum objective that minimizes the sum of arrival times
(2) a minmax objective that captures the beginning time of the last
customer
Classic routing models such as the traveling salesman problem (TSP) and
the VRP do not reﬂect the operations and priorities in the aftermath of a
disaster.

Routing for relief: parameters
N = {1, ..., n} = set of customers (nodes).
N0 = N ∪ {0} = total set of nodes including the depot (0).
tij = travel time between the nodes i and j in N0.
T = a suﬃciently large number.
The decision variables include:
xij =
1 if the vehicle travels from i to j.
0 otherwise.
ai = the arrival time at customer i.

Routing for relief: model with minsum objective
z = min
i∈N
ai (39)
subject to
j∈N0
xij = 1, i ∈ N (40)
j∈N0
xij −
j∈N0
xji = 0, i ∈ N0 (41)
tij + ai ≤ aj + T(1 − xij ), i, j ∈ N (42)
ai ≥ t0i x0i , i ∈ N (43)
xij ∈ {0, 1}, i, j ∈ N0 (44)
The objective function (39) is the minsum objective that captures the sum
of the arrival times.
The model can be reformulated to capture the minmax objective by
introducing auxiliary variable ¯a to capture the last delivery time.

Disaster recovery
helping the system eﬃciently return to its pre-disaster capabilities
We focus on
1 Network recovery, and
2 Interdependent infrastructure

Recovery
Disaster recovery models focus on helping the system efficiently return to
its pre-disaster capabilities.
Most of the optimization models in this area focus on networks and model
commodities such as transportation, water, power, and communications.
How can we schedule the installation of arcs in a network over time in a
multicommodity flow model? (Nurre et al [2012])
Flow can be directed over the new components once the installed
components become operational
The objective is to maximize the cumulative flow on the network over
a fixed time horizon.

Network recovery: parameters
G = (N, A) = a network with nodes N and arcs A.
A = set of arcs that can be installed on the network.
K = the number of parallel identical work groups that can install
network components
S ⊂ N = set of supply nodes with supply si , i ∈ S.
D ⊂ N = set of demand nodes with demand di , i ∈ D.
wi = weight associated with a unit of ﬂow that arrives at demand
node i ∈ D.
uij = arc capacity for arc (i, j) ∈ A ∪ A .
pij = processing time to install arc (i, j) ∈ A .
T = length of the time horizon.
µt = weight associated with the performance of the network at time
t = 1, 2, ..., T.

Network recovery: decision variables
The decision variables include:
1 xijt = the amount of ﬂow on arc (i, j) ∈ A ∪ A at time t = 1, ..., T.
2 vit = the amount of demand met at node i ∈ D at time t = 1, ..., T.
3 βijt =
1 if arc (i, j) ∈ A is operational at time, t = 1, ..., T.
0 otherwise.
4 αijkt =
1 if workgroup k completes arc (i, j) ∈ A in time period , t = 1, ...
0 otherwise.

Network recovery: model (Part 1)
z = max
T
t=1 i∈D
µtwi vit (45)
subject to
(i,j)∈A∪A
xijt −
(j,i)∈A∪A
xjit ≤ si , i ∈ S, t = 1, ..., T (46)
(i,j)∈A∪A
xijt −
(j,i)∈A∪A
xjit = 0, i ∈ N{S ∪ D}t = 1, ..., T(47)
(i,j)∈A∪A
xijt −
(j,i)∈A∪A
xjit = −vit, i ∈ D, t = 1, ..., T (48)
0 ≤ vit ≤ di , i ∈ D, t = 1, ..., T (49)
0 ≤ xijt ≤ uij , (i, j) ∈ A, t = 1, ..., T (50)
0 ≤ xijt ≤ uij βijt, (i, j) ∈ A , t = 1, ..., T (51)

Network recovery: model (Part 2)
(i,j)∈A
min{T,t+pij −1}
s=t
αkijs ≤ 1, k = 1, ..., K, t =, ..., T (52)
βijt −
t
s=1
K
k=1
αkijs ≤ 0, (i, j) ∈ A , t = 1, ..., T (53)
pij −1
t=1
βijt = 0, (i, j) ∈ A (54)
K
k=1
pij −1
t=1
αkijt = 0, (i, j) ∈ A (55)
αkijt, βijt ∈ {0, 1}, (i, j) ∈ A , k = 1, ..., K, t = 1, ..., T. (56)

Data, information, and modeling challenges

New criteria
Traditional criteria:
Quality
Proﬁt
Cost
Distance
Disasters criteria:
Loss of life
Morbidity
Coverage
Delivery of critical time-sensitive
commodities
Discrete optimization models for disasters are not merely traditional
discrete optimization models with new objective functions

Mechanisms for new structural model components
1. several agencies often work together to respond to and deliver
commodities after a disaster
multiple decision-makers with limited control over certain parts of the
operations
2. issues such as fairness often emerge in models for disasters.
Lots of ways to model equity
3. traditional models often implicitly assume the network components are
reliable
may not be valid when facilities are damaged and may not be fully
operational

Vulnerability
Models for disasters almost always focus on events with high consequences
There is often an interest in vulnerability and events that overwhelm the
resources and capacities in the system.
Vulnerability can be modeled with:
max-min models to capture worst-case performance
the inclusion of uncertain and unpredictable demands and system
failures.
models with cascading failures

Resource allocation
In a disaster setting, resources may include:
1. new resources specific to the disaster at hand
e.g., anthrax vaccines dispensed after an anthrax attack
2. resources used for non-disaster scenarios that may be used in new ways
during disaster management
e.g., ambulances that evacuate hospital patients after a hurricane
resource deployment strategies during and after disaster events may
differ from those used for routine operations (e.g., reactive
deployment of first responders after a large-scale emergency)

Data collection is hard
Getting data is hard.
This is true for many application areas, but it is particularly true for
disasters applications.
Data may be impractical to collect when events are rare (e.g., terrorist
attacks)
Data may not come from usual sources (e.g., the number of evacuees in
shelters).
Data may be perishable and therefore must be collected in a timely manner

Data collection sometimes cannot be done in advance

Data have bad quality
Data that are collected may be inaccurate and incomplete.
Censored data
E.g., If there are communications failures after a hurricane strikes and
patients cannot call 911 for service, the calls may not be entered into
call logs.
Data incorrectly characterized
Computer aided dispatch software used by 911 centers do not have
codes for large-scale emergencies, such as carbon monoxide poisoning
caused by generators used during power outages. Instead, these 911
calls are mapped to codes used for typical emergencies (e.g.,
overdose).

Data challenges
1. Data can be used in a static manner, where it is collected and used to
parameterize the models that are solved ahead of time.
2. Models that address response and recovery may include real-time
decision making and may integrate data from multiple sources (variety)
that arrive in real-time (velocity).
“Big Data” issues are a national research priority.

Conclusions
Managing disasters and improving homeland security involve using
scarce resources and weigh multiple criteria.
OR is a good tool!
Many of the problems involve discrete decisions such as location,
assignment, and network ﬂows.
Discrete optimization is a good tool!
Important problems across the disasters lifecycle and across diﬀerent
types of disasters.
Models for homeland security and disaster problems generally cannot
use canonical discrete optimization models

Resilience analytics
Data-driven interdependent physical, service, and community networks
before, during, and after a disruption.
Descriptive analytics: what is happening?
Predictive analytics: what will happen?
Prescriptive analytics: what do we do about it?
For more: http://resilienceanalytics.com

Thank you!
Additional reading from Punk Rock OR:
my national academies committee experience and risk-based ﬂood
insurance
operations research, disasters, and science communication
staying safe from tornadoes
the forecasting models behind the power outages forecasts for
Hurricane Sandy
rivisiting September 11, 2001 fourteen years later
aviation security, there and back again

Discrete Optimization Models for Homeland Security and Disaster Management

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