discrete optimization models for homeland security and disaster management

Discrete Optimization Models for Homeland Securityand Disaster Management

Laura Albert McLay

University of Wisconsin-MadisonIndustrial and Systems [email protected], @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

[email protected]

@lauramclay

Goal of this tutORial

The goal is to

a) introduce you to research in optimization research in disasters andhomeland security.

b) introduce you to important issues for doing research in optimizationresearch 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 Awards1444219, 1541165.


Discrete optimization can be used to address many typesof 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 impactand 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 notnational in scope. The event can be managed with local, regional, andsome national resources.

Examples: Hurricane Irene (2011), tornadoes, pandemic flu


Catastrophe

A severe event that affects an entire nation and where the local andregional responses are inadequate or impossible due to affected criticalinfrastructure. The event threatens the welfare of a substantial number ofpeople for an extended amount of time, and therefore, necessitates anational 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 responseis available.

Examples: Massive meteorite strike, Yellowstone Caldera super explosion(maybe), bi-national or multinational thermonuclear warfare, zombieoutbreak.


A note about terminology

While we use the term disasters generically in this paper to refer tointentional (man-made) and natural disasters with varying scopes ofdamage, we note that there are different scales associated with theseevents


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 theUnited States that the incapacity or destruction of such systemsand assets would have a debilitating impact on security, nationaleconomic security, national public health or safety, or anycombination 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 variouscharacteristics 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 areincluded 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 physicalinfrastructures and the natural environment. Social disruption andvulnerability 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 reducethe potential for harm, physical vulnerability, and social disruption.Mitigation can be structural, which involves designing, constructing,maintaining, and renovating physical infrastructure to withstand theimpact of a disaster, or nonstructural, which involves reducing exposure ofhuman 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 responseand recovery during and after a disaster. Preparedness actions ofteninclude the development of formal disaster plans and the maintenance ofresources 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 toprotect and maintain systems, rescue and respond to casualties andsurvivors, 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 servicesthrough restoration and repair of infrastructure and social and economicroutines, 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 anatural 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 ManagementAgency (FEMA)).


Who manages disasters?

Military organizations such as the National Guard and the U.S. CoastGuard provide support during disasters.

Numerous non-governmental organizations (NGOs) are also involved indisasters.

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 limitedresources, 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 andbetween 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 changingrequirements that are often difficult to recognize

Russell Ackoff: “Every problem interacts with other problems and istherefore part of a set of interrelated problems, a system of problems...Ichoose 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 adisaster

We focus on

1 Screening and pre-screening problems in homeland securityapplications that seek to reduce harm through detection andprevention, 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 explosivesI selectee and non-selectee screening

Goal was how to optimally deploy and use limited baggage screeningdevices.


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:I uncovered if 1+ bags on the flight has not been screenedI covered if all selectee bags on it have been screened

Baggage screening performance measures developed in conjunction withthe Federal Aviation Administration:

1 Uncovered flight segments (UFS), which captures the total number ofuncovered flights.

2 Uncovered passenger segments (UPS), which captures the totalnumber of passengers on uncovered flights.

3 Uncovered baggage segments (UBS), which captures the totalnumber 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 matterexperts)

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 offlights F ′ ∈ F such that

∑f ′i ∈F ′ C (f ′i ) ≤ S and the number of flights |F ′|

is maximized.

The Uncovered Passenger Segment Problem (UPSP): Find a subsetof flights F ′ ∈ F such that

∑f ′i ∈F ′ C (f ′i ) ≤ S and

∑f ′i ∈F ′ N(f ′i ) is

maximized.

The Uncovered Baggage Segment Problem (UBSP): Find a subset offlights F ′ ∈ F such that

∑f ′i ∈F ′ C (f ′i ) ≤ S and

∑f ′i ∈F ′ C (f ′i ) is maximized.



The Uncovered Flight Segment Problem (UFSP): Find a subset offlights F ′ ∈ F such that

∑f ′i ∈F ′ C (f ′i ) ≤ S and the number of flights |F ′|

is maximized.The Uncovered Passenger Segment Problem (UPSP): Find a subsetof flights F ′ ∈ F such that

∑f ′i ∈F ′ C (f ′i ) ≤ S and

∑f ′i ∈F ′ N(f ′i ) is

maximized.

The Uncovered Baggage Segment Problem (UBSP): Find a subset offlights F ′ ∈ F such that

∑f ′i ∈F ′ C (f ′i ) ≤ S and

∑f ′i ∈F ′ C (f ′i ) 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 stationi = 1, 2, ...,m, where F1,F2, ...,Fm are a partition of F .

Three models address general allocation sizes. Here is the modelassociated with the UPS measure.

The Multiple Station Passenger Segment Security Allocation andScreening Problem with General Allocation Sizes: Find the subsets offlights F ′i ∈ Fi , i = 1, 2, ...,m and a set of baggage screening securitydevice capacity allocations Si , i = 1, 2, ...,m that maximizes∑m

i=1

∑f ′i ∈F

′iN(f ′i ) such that

∑mi=1 Si ≤ S and

∑f ′i ∈F

′iC (f ′i ) ≤ 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 andScreening Problem with Predetermined Allocation Sizes: Find thesubsets of flights F ′i ∈ Fi , i = 1, 2, ...,m and a partition {A′i ,A′2, ...,A′m} ofsecurity device allocations A to stations that maximizes∑m

i=1

∑f ′i ∈F

′iN(f ′i ) such that

∑mi=1

∑f ′i ∈F

′iC (f ′i ) ≤

∑a′j∈A′ W (a′j).


Checked baggage security: uncovered targets

Extension to consider weapons of mass destruction (WMD), not justconventional weapons

scope of damage not just the airplane and its passengers

1 Uncovered targets (UT), which captures the total number ofuncovered targets (e.g., destination cities), where a target is coveredif all flights to the target are covered.

Leads to goal programming models that balance the UT measure with theUPS, 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-basedsecurity, which in led to TSA PreCheck.


Static checkpoint security: parameters and decisionvariables

N = number of passengers.

M = number of screening classes.

ATj = assessed threat value of passenger j = 1, ...,N, a riskassessment of passenger j returned by a passenger prescreeningsystem.

Li = the security level achieved by screening a passenger withscreening class i = 1, ...,M.

FCi = the fixed cost associated with screening class i = 1, ...,M.

MCi = the marginal cost of screening a passenger with screeningclass 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

LiATjxij (1)

subject to

M∑i=1

N∑j=1

MCixij +M∑i=1

FCiyi ≤ 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 expeditedscreening of the lower risk passengers.


Other risk based screening papers

How to optimally use security devices that are shared across securityclasses

Consider trade-offs between false alarm rate (proxy for passengerinconvenience) and number of screeners needed

How to sequentially “score” passengers and group passengersaccording to their scores based on cumulative device responses


Dynamic aviation passenger checkpoint security

Dynamic passenger screening provide insight into real-time screeningdecisions.

Modeled in the literature as a sequential process

Each passenger’s risk level becomes known upon arrival to a securitycheckpoint

Markov decision process models for assigning passengers (inreal-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 fornuclear material at ports of entry in a risk-based environment.

Ninety-five percent of international goods that enter the U.S. comethrough one of the ports of entry

Early screening was performed by physically unpacking cargo containersand inspecting their contents.


Cargo security

Effort to develop more effective screening methods based on imaging orpassive 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 inspectionused more sparingly and targeted at high-risk containers

Automated Targeting System (ATS) performs a risk assessment oncargo containers that can aid in designing risk-based screeningsystems.

Primary screening vs. secondary screening


Cargo security models

Linear programming model for performing primary screening on cargocontainers 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 primaryscreening devices for each risk group

Integrated model for primary and secondary screening

Secondary screening depends on the mix of primary screening devicesthat alarm

Large-scale linear programming model to sequence screening tests forscreening cargo containers.

How to supplement risk assessments with radiography-based images toidentify potential containerized threat scenarios



Network models address the need to design and fortify parts of a networkto 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 ashortest path from the source to the sink).


Several classes of network interdiction models that arewell-studied in the literature

Shortest-path network interdiction: the operator seeks to find theshortest path from a source to a sink after the leader lengthens arcsin the network,

Maximum reliability network interdiction: the operator seeks tomaximize the probability of evading detection after the leader reducesthe evasion probability on some of the arcs in the network,

Maximum flow interdiction: the operator seeks to maximize theflow achievable on the network after the leader reduces some of thearc capacities, and

p-median network interdiction: the operator seeks to minimize thetotal worst-case distance from the customers to the nearest facilityafter 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-medianand 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 6= j and dik > dij} = set of existing sites (notincluding 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 ∈ F0 otherwise

xij =

{1 if demand i is assigned to a facility at j , i ∈ I , j ∈ F0 otherwise


r -interdiction median problem integer programming model

z = max∑i∈I

∑j∈F

aidijxij (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 weighteddistance between demand locations and facilities that have not beeninterdicted.


r -interdiction covering problem

A demand location is no longer covered if all of the facilities that cover thelocation have been interdicted.

New decision variables in addition to sj , j ∈ J:

yi =

{1 if demand i is no longer covered0 otherwise


r -interdiction covering problem integer programming model

z = max∑i∈I

aiyi (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 thatis uncovered after interdiction.



Fortification: allow a decision-maker to first make some facilities immuneto failure

Fortification is one way to mitigate the worst-case vulnerabilities in thesetwo base models

This results in the r-interdiction median problem with fortificationModel has same parameters and decision variables as the r -interdictionmedian problem with a new set of decision variables:

zj =

{1 if facility j is fortified, j ∈ F0 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-fortifiedfacilities.


r -interdiction median problem with fortification

Lower level problem captures worst-case interdiction:

H(z) = max∑i∈I

∑j∈F

aidijxij (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 ther -interdiction median problem with fortification due to the structure of thelower level problem.

Scaparra and Church introduce an implicit enumeration algorithm to solvethe 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 aspossible 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 toan interdiction constraint

Solved by Benders decomposition and implement the algorithm usingoff-the-shelf project-management software.

Brown et al [2009]


Disaster preparedness

addresses planning that is done prior to a disaster that is implementedwhen 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) atpre-selected stations.

Jia et al. [2007] illustrate the models with examples that include a dirtybomb attack, a terrorist attack using anthrax, and a terrorist attack usingsmallpox.

The models provide insight into identifying tailored hospital locations forlarge-scale emergencies that take into account the unique attributes ofthese 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 emergencyscenario k , i ∈ I , k ∈ K .

pjk = degree of disruption of facility j under emergency scenario kwith 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 locationi 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 ∈ J0 otherwise

zij =

{1 if location i is assigned to a facility at j , i ∈ I , j ∈ F0 otherwise


p-median model for emergency scenario k ∈ K

Zk = min∑i∈I

∑j∈J

βikeikMidijzij (28)

subject to∑j∈J

xj = P (29)

∑j∈J

zijpjk = 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 demandlocations 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) demandlocation i ∈ I

Di is the longest distance from demand point i a facility may be to coveri ’s demand.

New decision variables (in addition to xj , j ∈ J):

uj =

{1 if location j is covered, i ∈ I0 otherwise


Covering model for emergency scenario k ∈ K

Zk = max∑i∈I

βikeikMiui (34)

subject to∑j∈J

xj ≤ P (35)

∑j∈Ni

xjpjk ≥ Qiui , 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 flowof traffic.

Evacuation and routing use maximum flow networks models, facilitylocation models, the traveling salesman problem, and the vehicle routingproblem.


Disaster response

Focuses on preventing a disaster or reducing the harmful effects of adisaster

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 forfire, police, or emergency medical service (EMS) service has been an activearea of research since the early 1970s.

The most relevant papers seek to assign vehicles to calls in real-time ascalls 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 ondelivering 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-sensitivecommodities to customers.

Ozdamar et al. [2004] integrates the multi-commodity network flowproblem and the vehicle routing problem (VRP) and seeks to deliver aid ona multi-modal transportation network

Mete and Zabinsky [2010] propose a stochastic optimization model forpre-locating (preparedness) and distributing (response) medical suppliesafter a disaster


Routing for relief

Campbell et al. [2008] provide two routing models for routing supplies forrelief 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 lastcustomer

Classic routing models such as the traveling salesman problem (TSP) andthe VRP do not reflect the operations and priorities in the aftermath of adisaster.


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 ≥ t0ix0i , i ∈ N (43)

xij ∈ {0, 1}, i , j ∈ N0 (44)

The objective function (39) is the minsum objective that captures the sumof the arrival times.

The model can be reformulated to capture the minmax objective byintroducing 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 toits pre-disaster capabilities.

Most of the optimization models in this area focus on networks and modelcommodities such as transportation, water, power, and communications.

How can we schedule the installation of arcs in a network over time in amulticommodity flow model? (Nurre et al [2012])

Flow can be directed over the new components once the installedcomponents become operational

The objective is to maximize the cumulative flow on the network overa 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 installnetwork 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 demandnode 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 timet = 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, ...,T .0 otherwise.


Network recovery: model (Part 1)

z = maxT∑t=1

∑i∈D

µtwivit (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 ≤ uijβijt , (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-sensitivecommodities

Discrete optimization models for disasters are not merely traditionaldiscrete optimization models with new objective functions


Mechanisms for new structural model components

1. several agencies often work together to respond to and delivercommodities after a disaster

multiple decision-makers with limited control over certain parts of theoperations

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 arereliable

may not be valid when facilities are damaged and may not be fullyoperational


Vulnerability

Models for disasters almost always focus on events with high consequences

There is often an interest in vulnerability and events that overwhelm theresources 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 systemfailures.

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 waysduring disaster management

e.g., ambulances that evacuate hospital patients after a hurricane

resource deployment strategies during and after disaster events maydiffer from those used for routine operations (e.g., reactivedeployment 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 fordisasters applications.

Data may be impractical to collect when events are rare (e.g., terroristattacks)

Data may not come from usual sources (e.g., the number of evacuees inshelters).

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 andpatients cannot call 911 for service, the calls may not be entered intocall logs.

Data incorrectly characterized

Computer aided dispatch software used by 911 centers do not havecodes for large-scale emergencies, such as carbon monoxide poisoningcaused by generators used during power outages. Instead, these 911calls 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 toparameterize the models that are solved ahead of time.

2. Models that address response and recovery may include real-timedecision 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 usingscarce resources and weigh multiple criteria.

I OR is a good tool!

Many of the problems involve discrete decisions such as location,assignment, and network flows.

I Discrete optimization is a good tool!

Important problems across the disasters lifecycle and across differenttypes of disasters.

Models for homeland security and disaster problems generally cannotuse canonical discrete optimization models


Resilience analytics

Data-driven interdependent physical, service, and community networksbefore, 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


http://resilienceanalytics.com

Thank you!

Additional reading from Punk Rock OR:

my national academies committee experience and risk-based floodinsurance

operations research, disasters, and science communication

staying safe from tornadoes

the forecasting models behind the power outages forecasts forHurricane Sandy

rivisiting September 11, 2001 fourteen years later

aviation security, there and back again


https://punkrockor.wordpress.com/2015/06/19/my-national-academies-committee-experience-risk-based-flood-insurance/

https://punkrockor.wordpress.com/2015/06/19/my-national-academies-committee-experience-risk-based-flood-insurance/

https://punkrockor.wordpress.com/2013/02/18/operations-research-disasters-and-science-communication/

https://punkrockor.wordpress.com/2014/04/28/staying-safe-from-tornadoes/

https://punkrockor.wordpress.com/2012/10/29/the-forecasting-models-behind-the-power-outages-forecasts-for-hurricane-sandy/

https://punkrockor.wordpress.com/2012/10/29/the-forecasting-models-behind-the-power-outages-forecasts-for-hurricane-sandy/

https://punkrockor.wordpress.com/2015/09/11/revisiting-aviation-security-on-september-11-fourteen-years-later/

https://punkrockor.wordpress.com/2015/07/24/aviation-security-there-and-back-again/

discrete optimization models for homeland security and disaster management

Data & Analytics