Download Abstract - Lyle School of Engineering

A Systems’ View Methodology and Mathematical Model for
Identifying the Most Critical Links of a Highway Network with
Application to Critical Infrastructure Protection
Gerard Ibarra, Jerrell T. Stracener, and Steve Szygenda
Abstract
There is much research in the area of critical infrastructure protection (CIP).
Many researchers and practitioners are delving into the area by studying how to optimize
system recovery due to a disruption. Incidentally, the researchers fail to include a
system’s view for protecting critical infrastructures such that it helps improve system
recovery. The research does not include what element is the most critical to the system
for risk mitigation and resource allocation purposes. Hence, we introduce a methodology
and mathematical model for determining the most critical elements of a system. We
specifically look at a highway network and identify its most critical links.
1.0 Introduction
Since September 11 2001, the United States (US) shifted its way of thinking when
it comes to preventing terrorist attacks. Congress issued Public Law 107–56, better
known as the USA (United States of America) Patriot Act, to obviate such attacks
[United States Congress, 2001]. It is a law to “deter and punish terrorist acts in the
United States (US) and around the world, to enhance law enforcement investigatory tools,
and for other purposes” [US Department of Homeland Security, 2004]. This initiative
brought about security increases around the US with the most noticeable being at airports.
Furthermore, the Department of Homeland Security (DHS) created a task force to help
“identify state and local funding solutions that work effectively and can be extended to
situations where there are impediments to the efficient and effective distribution of state
and local homeland security funds” [DHS, 2004]. That is, the DHS has a Homeland
Security Grant Program (HSGP) to enhance the ability of states, cities, urban areas, and
territories to prepare for and respond to terrorist attacks and other disasters according to
applicant’s needs. For example, cities would ask for funding and then determine fund
4781525303
1 of 28
distribution based on city and regional planning [Dallas City Hall, 2005; Houston, 2004].
However, distribution of these funds by various city functions is at times questionable.
Because of this, the DHS changed its fund dispersion rules and procedures to abate
questionable city spending [National Public Radio, 2006]. This means city officials are
slightly more pressed for acquiring funding; however, they stand a better chance of
securing it by showing some type of cost effectiveness. A sure place for this lies in
protecting the nation’s critical infrastructures (CIs). Critical infrastructures are those
items necessary for individuals and companies to continue “life as usual” [Moteff and
Parfomak, 2004]. It consists of thirteen CIs as shown in Figure 1.0a in no particular
order [Office of Homeland Security, 2002]. Any major disruptions to these CIs have
severe consequences to citizens and businesses.
Figure 1.0a: Thirteen CI Defined by the US Government
Much of the CIP literature is devoted to responding to disruptions to CIs after
they occur. For example, Mendonca et al. [2004] consider the optimal deployment of
back-up generators after a blackout to supply power where it is needed most immediately.
Some CIP researchers, such as Sathe et al. and Pollack et al. [2005; 2004] suggest that CI
analysts should perform sensitivity analysis to improve emergency response to
disruptions of CIs. This type of analysis has long been a staple of CIP in the
telecommunications industry where the problem of finding the most cost-effective way to
install spare system components so that system functionality can be quickly and
automatically be restored after link failures has received considerable attention in the
4781525303
2 of 28
literature [Kennington et al. 2006]. A related issue in CIP, which is not explicitly
addressed by the research described above, is that of systematically determining the
critical component(s) of a CI.
Critical components are those components whose failure has the severest effect on
the system. Knowing the most critical components allows decision makers (DMs)
responsible for CIP to construct effective risk mitigation plans that help allocate
resources judiciously. That is, limited resources such as time and money can be
concentrated on the most critical components affecting the CIs. To this end, this paper
addresses the problem of determining the critical components of the transportation CI.
Specifically, we study the problem of determining which links in a highway network are
the most critical.
1.1 Paper Outline
We outline the rest of this paper as follows. In §2.0, we review the literature
relating systems engineering to CIP. We then present the problem overview in §3.0. We
describe in §4.0 our general methodology for finding the most critical links in a network.
In §5.0, we present our mathematical models. These models require significantly less
processing time than simulation tools. In §6.0 we validate the models using a highway
simulation tool, DYNASMART, and show our experiment’s results. Lastly, we provide
concluding remarks and direction for future research in §7.0.
2.0 Literature Review
We examine applications of systems engineering (SE) to the protection of CIs.
For a complete understanding of CIs, refer to Rinaldi et al., and Moteff and Parfomak
[2003, 2004]. We then examine simulations that focus on the interruption of traffic flow
in a highway network.
2.1 Critical Infrastructures
Currently, there is a great deal of work in identifying and analyzing CIs. For
example, Rinaldi et al. [2001] do an excellent job in expanding the CI knowledge area.
The article’s premise hinges around the six-dimensions for describing infrastructure
interdependencies. Infrastructure interdependencies are mutually dependent
4781525303
3 of 28
infrastructures where one infrastructure affects another. These dimensions are an ideal
place to start when attempting to identify the risk associated with infrastructures.
However, the paper presents no mathematical model or tool for decision makers (DMs) to
use to help streamline and analyze the CI interdependencies. Furthermore, it does not
provide any research or discussion on the elements most crucial to the CI. The article
focuses on the idea and concept of identifying CI interdependencies and leaves it to the
reader to investigate further the interaction between interdependencies. This would
involve developing a mathematical model that determines the CI’s most critical
components based on the article’s concepts of interdependency.
Quirk et al. [2005] discuss quantifying and analyzing infrastructures by trying to
estimate the robustness of the infrastructure during a disruption. Robustness is defined in
terms of the extent to which disruptions could cause the system in question to function
abnormally or fail completely. The work addresses robustness from a graph and network
theory where the study suggests evaluating the robustness of an infrastructure by
comparing possible outcomes. The outcomes that are the metrics for infrastructure
robustness provide the DMs with a preferred alternative. Incidentally, the model is from
a high-level view. The steps require analysts to evaluate outcomes from a specific
subsystem’s disruption. Once evaluated, the outcomes become part of a larger model that
measures the infrastructure’s robustness as a system. The drawback is the paper fails to
show a mathematical model or simulation that evaluates a disruption’s effect on the
system. Furthermore, the paper does not illustrate or discuss what the most important
edges are in the system. From a system’s perspective, this is an important consideration
when performing risk mitigation analysis.
Mendonca et al. [2004] consider a system where one infrastructure is
interdependent with another. In their paper, they model an electric power disruption.
The goal is to place generators strategically in the affected area such that the generators
provide electric power to the most needed and widest customer base. The DMs’ consider
the power’s priority based on constraints they determine in the model. For example,
restoring a hospitals’ or a police or fire station’s power, may be more important than
restoring power to businesses services such as retail, food, or fuel stations. Furthermore,
the model has input parameters that allow CI managers and emergency response officials
4781525303
4 of 28
to model different event scenarios. These individuals can then assess the scenarios’
impact on the services provided by the CI systems. However, the model does not provide
a way to rank scenarios based on criticality.
There is also Sathe et al. [2005] who uses optimization for location and relocation
of emergency response units (ERUs) guarding critical facilities. Because of computation
time, Sathe et al. uses heuristics to optimize the ERUs’ placement. This coincides with
our approach because to use simulation for determining the network’s most critical links
may be infeasible. This is especially true for large and complex systems. However,
Sathe et al. does not provide an avenue for determining the most critical locations for
coverage based on a systems’ perspective. Their criterion is to have at least one ERU at
each location of concern. This may not be prudent when DMs have limited ERUs. In
this case, DMs must decide the optimum use of resources that affects the most people in a
positive manner. Conversely, with sufficient ERUs it might be more feasible from a
system’s perspective not to provide an ERU to a particular facility because that ERU has
a greater system affect at another location.
Finally, Lee et al. [2004] uses a network flows’ approach to mathematically
model service restoration of interdependent infrastructure systems. For example, power
lines that cross a highway link are geographical interdependencies and are an
interdependent infrastructure system. Geographical interdependencies are infrastructures
close or adjacent to each other where a change in one infrastructure affects the other(s)
[Rinaldi et al., 2001; Ibarra et al., 2005]. Hence, the idea is to model the system’s
elements and provide DMs with the ability to perform sensitivity analysis. Overall, the
concept is to restore systems effectively by minimizing the objective function. The
function entails prioritizing emergency response organizations for maximum affect.
However, the paper does not mention identifying the most critical interdependencies for
resource allocation and risk mitigation. In addition, the paper suggests further research in
developing algorithms to speed up the computation. This is a tell tail sign that their
models may take a great deal of time for larger and more complex systems.
Other articles that analyze the CIs infrastructures include works by Brown et al.,
Latora et al., Pepyne et al., and Wolthusen, [2005; 2005; 2001; 2004]. The reason for
interest in modeling and analyzing CIs are the affects a disruption in CIs have on society.
4781525303
5 of 28
Consider the 911 attack or the water port strike in Los Angeles for example [US
Department of Labor, 2004; Parrott, 2002; Wikipedia, 2006, The San Francisco
Chronicle, 2002]. These disruptions severely affected our infrastructure and hence
caused unwanted rippling affects in our economy, nation, and world [The White House,
2003].
There is one final note on the models. Work by Latora et al. [2005] discusses
finding the most critical links of a network infrastructure. The work is similar to this
research. However, they limit the model to networks such as communications and transit
lines. Our model is more robust because analysts can use it for highway networks where
travelers change routes based on congestion and travel times. Furthermore, Latora et al.
gives a description of the model and shows its mathematical formulation. It uses the
shortest path concept. However, they do not verify their method. They present a case
based on network performance given a disconnected link.
In short, analysts are using mathematical models and simulations to mitigate risk
and improve efficiencies given a disruption in our CIs. They recognize the need for such
research because of the implications a major CI disruption would have to our society.
However, there is not sufficient research in identifying the most critical elements of a
system. Knowing this could help abate risk while using CIP funds efficiently. We thus
look next to any models and simulations from a SE prospective that consider the most
critical links of a highway network. This is the basis of our paper. That is, viewing the
disruption of a network link in terms of a system and identifying the link’s impact on the
network.
2.2 Application of SE to Traffic Flow Assignments
Research indicates there are few if any models and simulations that analyze from
a system view the criticality of disruptions in a highway network. In fact, based on
current literature review, there is no research outlining the systems’ impact of highway
traffic disruptions in a highway network.
Mirchandani et al. [2002] proposes an approach that incorporates the traveler’s
decision-making, traffic assignments, and real-world trip/traffic dynamics respectively. It
uses the method of successive averages (MSA) for obtaining traffic equilibrium. It keeps
4781525303
6 of 28
track of travel times experienced by the travelers impacted by disruptions such as a work
zone. Overall, their process can better simulate disruptions for traffic assignment
purposes. Incidentally, it does not demonstrate what impacted neighborhood network
(INN) would have the severest affect on the system. This is important for risk mitigation
purposes. In addition, Peytchev et al. [1996] defines a mathematical model that describes
the evolution of macroscopic traffic flows. The paper’s focus is viewing the network as a
system by including queues, drivers’ preferences, and turning movements on intervening
intersections. However, there is no indication describing the network’s most critical link.
Others who have done work in this area include Helbing [1995] and Bando et al. [1995].
The too omit determining the network’s most critical links.
In summary, there is much work needed in the area of evaluating transportation
networks as a system. Srinivasan [2002] best describes the need for advancement in this
area by discussing the evaluation of disruptions to the transportation network. He
illustrates a system’s perspective when it comes to CI disruptions in the transportation
sector. He discusses the impacts of 9/11 on the transportation system and describes the
need for technological and methodological tools to obviate and mitigate such events. He
suggests there is limited knowledge base understanding the likelihood and consequences
of such attacks on the transportation infrastructure. He thus calls for white papers for the
development of these tools from assessing the vulnerabilities to integrate security
considerations as an integral part of network planning. Overall, the paper is a detail
discussion of transportation network vulnerabilities to integrate security considerations as
an integral part of network planning. However, the discussion ends there. There is no
mathematical model describing how to assess the vulnerability of the network. The paper
focuses only on a framework for developing such models and methods.
3.0 Problem Overview
We divide the overall problem statement into two sections. First is the problem
definition. Second are the ground rules, assumptions, and definitions.
3.1 Problem Definition
We propose to investigate highway networks as a vital critical infrastructure (CI)
component. We want to determine the affect of a disruption in terms of a system based
4781525303
7 of 28
on the user’s average travel time. Specifically, we want to find the links that affect the
network the most in terms of the user’s average travel time. A link represents the
highway’s bi-directional lanes and we define it more clearly in §3.2. Ultimately, finding
the links affecting the network the most affords government officials, the decision makers
(DMs), to plan judiciously for mitigating risks. Instead of DMs allocating resources with
minimum consideration for the highway system, DMs have a methodology and
mathematical model that helps target the worst links where complete disruption of these
links causes the severest impact to the network. We rank these links with one being the
severest and refer to the link ranking as k-critical links.
There are many models and simulations in operations research (OR) used by
scientists, researchers, educators, and engineers alike to model various hypothetical and
real-life scenarios. One drawback of simulations is its optimality when applying them to
large and complex networks. Depending on the network’s size and objective, this could
take a great deal of time or even worse, the simulation may not be feasible. This is
unacceptable because in real-world problems, there are numerous large and complex
networks. If either the models or simulations cannot handle these types of networks, then
they are useless to the analysts. Another problem based on the current literature review is
none of the models and simulations systematically find the most critical links. It is not
part of the solution set. Analysts must perform the sensitivity analysis themselves to find
such links. However, our mathematical model finds these links without the user having
to perform rigorous sensitivity analysis.
Furthermore, our model is easy to use because
it requires the network’s link capacities, distances, and origin/destination (OD) matrix
while other models or simulations require various constraints and parameters to perform
the analysis. In addition, our model is capable of handling large and complex systems
because of its efficient computation time. This makes the model valuable to DMs for two
reasons. One, it is fast thus facilitating updates when changes occur in the network’s
capacities, link distances, or OD matrix. Two, DMs could model larger and more
complex systems in order to understand the systems’ criticality over a larger region
better.
Therefore, we define the problem to lie in two areas. First, highway networks are
increasing in size: links and interchanges. Second, from a system’s perspective DMs are
4781525303
8 of 28
not allocating homeland security funds efficiently for mitigating risks due to highway
networks disconnects. Hence, we must develop a methodology and mathematical model
to help process information of complex systems quickly, accurately, and feasibly, for
determining from a system view the network’s most critical links. Note our model is
applicable to any capacitated flow network. Concisely, Figure 3.1a depicts the problem
definition.
DHS
Resource
Allocation
k-Critical Links
Network
Analysis
Figure 3.1a: General Problem Definition
3.2 Ground Rules, Assumptions, and Definitions
In general, there are ground rules, assumptions, and definitions that exist. One set
applies to the DMs and the other to the mathematical model. For the DMs, they must
recognize the significance of viewing a problem as a system rather than an entity. In
other words, DMs must agree on what is important for their district or area may not be
important for the overall system, the city. The assumption is that they agree or at least
acquiesce to this premise. The model however has a total of eight ground rules,
assumptions, and definitions. They include highways, disconnects, steady state, shortest
path, system snapshot, major cities, OD matrix and link capacities, and k-critical links.
First, highway is the type of network we use in our mathematical model to
conduct the experiments and validate results. A highway, where tolls are inclusive,
consists of multiple bidirectional lanes with posted speeds of 45 MPH minimum, up to 70
MPH maximum. Furthermore, highways do not include their adjacent service roads.
Service roads are the access roads to the highways. Moreover, the highway network must
contain at least two nodes and more than on edge connecting the nodes. See Figure 3.3a
for three valid examples.
4781525303
9 of 28
Network 1
Network 2
Network 3
2
1
2
1
2
3
4
1
3
Figure 3.3a: Three Valid Network Examples
Second, we consider a disconnect as the complete obstruction of traffic flow on
the links connecting a pair of nodes in the highway network. A link is the stretch of
highway lanes in both directions connecting a pair of nodes. The node is the intersections
or transfer points connecting the links. The bidirectional link {i, j} represents the segment
of highway connecting intersections/transfer points i and j. Disconnects do not include
individually disrupted lanes, however. In other words, any disconnect must affect all
lanes in both directions.
Third, we use a steady state network when applying our mathematical model.
Steady state refers to the network operating under normal conditions. It means any
construction is well established and on going. There are also no outside forces, such as
weather, accidents, or terrorist threats or attacks, affecting the network. If one of these
occurs at t = 0, it changes the network so that it is no longer at steady state.
Fourth, we assume users of the network will take the shortest path (SP) to and
from their destinations based on travel times. This includes traveling under normal
conditions as well as during the event of disconnects. The SP optimizes going from one
or multiple to other single or multiple locations of a given network.
Fifth, we consider a snapshot of the network. Snapshot means the system is noncontinuous. That is, we omit time and time increments as a parameter in our model. Our
model performs its computation based on one period of the network.
Sixth, we limit our model to major cites. A major city is one with a population of
at least 100,000 or more. It has suburbs that are adjacent to the city making up a
metropolitan area. It is possible however to perform the analysis for a suburb or smaller
city given condition 1 exists.
Seventh, the analysts have or obtain the network’s OD matrix and link capacities
and distances. Refer to §4.3 for further discussion on these items. The OD matrix
4781525303
10 of 28
specifies the number of travelers from every origin to every destination. A link’s
capacity is the number of lanes on each arc between the two nodes that the link connects.
An arc is all highway lanes in a one-way direction. The distance is the length of the link
from node i to j.
Lastly, we define the k-critical links. These are links, edges in network flow
theory, that if disconnected would have the severest affect on the network. That is, the
network’s performance is at its lowest level base on the vehicle’s average travel time.
Hence, k-critical links are those links in the highway network if disconnected would have
the severest affect. k refers to the link’s rank from having the worst to negligible affect
on the network.
4.0 Methodology Framework
There are eight steps to our methodology as depicted in Figure 4.0a. The most
important is the first step, the input process. This consists of the customer needs and
problem definition. This is the basis for delivering a sound and valid solution to the
customer. The remaining steps include, constructing the network, collecting the data,
entering the data, auditing/verifying the data, applying the mathematical model, ranking
the links, and determining the most critical links, output.
Customer Needs &
Problem Definition
1
6
2
Construct Network
such that G = (N, A, x)
5
OUTPUT
Apply Model(s)
Audit/Verify Data 24
Hours Later Against G
3
Collect Data
No
4
Network
Verified?
Yes
Enter Data
8
k-critical links
INPUT
7
Rank Links
Figure 4.0a: Methodology
4.1 Step 1 – Input
The analysts begin with defining the customer needs, the decision maker’s (DMs).
This translates into the problem definition and is the input to our process as depicted in
Figure 4.0a. At this stage, analysts learn what the DMs are looking for and determine if
4781525303
11 of 28
the methodology and model are applicable to the DM’s problem. Our model is limited to
capacitated networks exhibiting maximum flow patterns or shortest path (SP). It is not a
production, blending, scheduling, or other type of optimization problem. For example,
maximum flow can represent items such as vehicles on a highway, data transmission on
communication networks, or passenger movement on subways to name a few. SP
represents taking the most efficient route based on costs, such as distances. The main
thing for the analysts to ensure is the problem definition represents a mathematical graph
with flow and distances on the edges. A mathematical graph models travel, flow and
adjacency patterns in a network [Radrin, 1998, p. 41].
4.2 Step 2 – Construct Network
Step two requires the analysts to represent the network as mathematical graph, G
= (N, A, x). N are the nodes, A are the links, and x is the link’s distances or capacities
depending on the model the analysts chooses. Formulating the graph helps the analysts
understand and collect the needed data for the model.
First analysts must determine the nodes of the network. In a highway network,
these are the interchanges where interchanges are sections of highways that allow
vehicles to change travel paths without exiting the network. Second analysts define the
links connecting the nodes. Links are the highway lanes connecting the interchanges.
Listed in Figure 4.2a is an example of nodes and links. The shaded dots represent the
highway’s interchanges, nodes. The dash lines represent the links. When complete, the
high-level view of Dallas’ Texas highway system in Figure 4.2b is equivalent to the
mathematical graph in Figure 4.2c. For the capacities and distances, the analysts assign
them according to the collected data as we discuss this in §4.3.
Figure 4.2a: Partial Representation of Dallas’ Highway Network
4781525303
12 of 28
Figure 4.2b: Dallas Texas’ Map
18
1
17
16
15
13
14
25
24
19
23
12
2
3
4
20
21
22
9
5
6
11
10
7
8
Figure 4.2c: Network Representation
4.3 Step 3 – Collect Data
Step three involves colleting the origin/destination (OD) matrix and link’s
capacities and distances. The analysts may acquire these through, but not limited to, the
Department of Transportation (DOT), the city planners, detail maps, and other agencies
or references. The analysts must make sure the data is complete and represents the
network’s operational and steady state. Operational state is the state DMs would like to
analyze the network. For example, the OD matrix and number of lane closures during
peak usage, or other usage times deemed crucial for planning purposes.
4781525303
13 of 28
In the OD matrix, analysts must use the most current data available. They must
assign the data, represented by zones, to nodes. Zones are geographical areas, such as zip
codes, businesses, communities, and shopping. Experience shows the OD matrix is often
depicted by zones. The analysts must convert the number of travelers from each zone
and represent them by the network’s nodes instead. This is the network’s OD matrix.
Depicted in Figure 4.3a is an example of converting zones into a node into the network.
Zones
Node
A
B
C
D
Network
Figure 4.3a: Zone to Node to Network Representation
Next, analysts assign the link capacities and distances to the network. Capacities
are the number of lanes from i to j and j to i where it is possible to have varying number
of lanes in either direction. For instance, suppose the capacity from x(i,j) = 4 and x(j,i) = 3.
The analysts would replace the single link into two arcs. Arc x(i,j) would get 4 and x(j,i) get
3. For distances, we use miles for our models. Figure 4.3b illustrates a completed
network with capacities and Figure 4.3c shows the distances. Table 4.3a is the network’s
OD matrix.
18
1
4
4
17
4
16
15
13
4
4
4
3
4
4 2
214
2
25
3
3
24
4
3
4
23
4
12
4
22
3
3
3
19
4
3
4
4
20
4
3
21
4
3
3
3
9
5
63
2
11
3
3
3
7
10
4781525303
3
3
8
14 of 28
Figure 4.3b: Network with Link Capacities
18
1
3.0
8.5
17
3.0
16
8.0
1.0
15
13.5
9.0
1.0
1.5 5.5
13
2
14
25
0.5
2.0
3
24
3.0
6.5
1.0
23
1.0
12
1.0
2.5
8.5
4
20
9.5
21
8.0
13.5
2.5
19
1.0 2.0
22
6.5
2.5
11.0
11.0
9
5
61.0
6.5
11
5.5
10.0
7
5.0
10
4.0
8.0
8
Figure 4.3c: Network with Link Distances
1
2
3
1 2 3 4 5 6 7 8 9 10
- 10 10 10 10 10 10 10 10 10
10 - 10 10 10 10 10 10 10 10
10 10 - 10 10 10 10 10 10 10
11
10
10
10
12
10
10
10
13
10
10
10
14
10
10
10
15
10
10
10
16
10
10
10
17
10
10
10
18
10
10
10
19
10
10
10
20
10
10
10
21
10
10
10
22
10
10
10
23
10
10
10
24
10
10
10
25
10
10
10
26
10
10
10
27
10
10
10
28
10
10
10
26 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 - 10 10
27 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 - 10
28 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 -
Table 4.3a: OD Matrix for Figure 4.3b and c
4.4 Step 4 – Enter Data
The fourth step involves the analysts to construct the AMPL™ data files as text
files. AMPL™ is a comprehensive algebraic modeling language for linear and nonlinear
optimization problems, in discrete or continuous variables. Depending on the model used
depends on the format of the data file. In our example, we use the Shortest Path Method.
We discuss this further in §5.1. The file consists of the network’s nodes, links, and
distances. Shown in Figure 4.4a is an example of the data file from the network in Figure
4.4b. Notice the terms used to set up the data file. To assign the nodes to the AMPL™
model, the analysts uses the term “set nodes :=”, for the network’s links “set links :=” and
“set arcs :=”, and use the link’s distances as “param distance :=”.
4781525303
15 of 28
Figure 4.4a: Data Text file for Figure 4.4b
1
2
4
1
1
1
1
1
1
6
1
1
3
1
5
Figure 4.4b: Network with Link Distances = 1
4.5 Step 5 – Review Data
The fifth step requires the analysts to review the data at least twenty-four hours
later. This helps the analysts reduce the likelihood of errors because of the numerous
data collected and transcribed from reports, individuals, and other indices, into a network
and data file configuration. Not completing this step could lead to errors. Moreover, it
could cause the analysts to spend unnecessary time in subsequent steps.
4781525303
16 of 28
4.6 Step 6 – Apply Model
The sixth step entails the analysts to apply our mathematical model(s) to the data
set. The model is a linear program (LP) written for AMPL™. The model solves for the
most critical links. We shall discuss this and our other mathematical models in §5.0.
4.7 Step 7 – Rank Links
From the results, the analysts must rank the links in descending order. The most
critical is at the top with the least critical on the bottom. These are the k-critical links of
the network. Note some links will have the same rank. We illustrate this in §6.2.
4.8 Step 8 – Present Results
Lastly, the analysts are done with the model. They present the results to the DMs.
The DMs review the results and address the links accordingly for mitigating potential
risks. This includes but is not limited to contingency plans, lane and link additions,
additional entrance and exits, and a myriad of other items.
5.0 Model Descriptions
We developed two models for determining the k-critical links. One is the shortest
path performance model. The other is the maximum concurrent flow problem (MCFP)
model.
5.1 Shortest Path Performance
We use the shortest path (SP) method to determine the network’s performance.
For a discussion on shortest path, refer to Ahuja et al. and Rardin [1993, p. 6; 1998, pp.
409 – 416]. We define performance, P, as the sum of the network’s SP costs for every
OD pair, Pi. That is, P = i 1 Pi . For example, depicted in Table 5.1a is the SP for every
n
OD pair of Figure 5.1a, where P = 33. The SP for {s – t} is {s, 1, 3, t} with a costs of
4.0. A discussion of the SP’s application for determining the network’s most critical
links follows.
4781525303
17 of 28
Path
{s - 1}
{s - 2}
{s - 3}
{s - 4}
{s - t}
Costs
1
2
2
3
4
Path
{1 - 2}
{1 - 3}
{1 - 4}
{1 - t}
{2 - 3}
Costs
2
1
2
3
2
Path
{2 - 4}
{2 - t}
{3 - 4}
{3 - t}
{4 - t}
Costs
1
4
1
2
3
Performance =
33
Table 5.1a: Network’s Performance for Figure 5.1a
i
c ij
j
1
1
3
1
2
2
s
2
1
2
t
3
2
1
4
Figure 7.2a: SP between s-t
We determine the network’s performance based on the SP concept as previously
discussed. Then we disconnect link {i, j}, ln, calculate the network’s performance based
on the disconnected link, P{i , j} , and then replace the link back in the network. A
disconnect is the removal of link ln by making its cost infinity such that no path uses ln.
We then disconnect another link, ln+1, determine the performance, and reconnect it to the
network. We continue this process {i, j}  A . When complete, we take the difference
between the disconnected link {i, j} and the base performance: P{i , j}  P . That is, we
subtract the larger performance from the smaller. The links with the greatest difference,
SP {i , j }  is the network’s most critical link. Table 5.1a illustrates the most critical links
for network Figure 5.1b. For instance, link {1, 3} is SP  {1,3} = P{1,3}  P = 44 – 33 = 11.
4781525303
18 of 28
Link
{1, 3}
{s , 1}
{3, t }
{3, 4}
{2, 4}
{s ,2}
{1, 2}
{2, 3}
{4, t }
SP{i , j }
11
7
6
3
2
1
1
0
0
Table 5.1b: Most Critical Links for Network 5.1a based on the SP Concept
5.2 Concurrent Flow Method
Our next method employs the maximum concurrent flow problem (MCFP)
concept. Briefly, the MCFP is the flow ratio between all OD pairs in a capacitated
network where flows move concurrently and distribution among the edges are even.
Furthermore, each OD pair acts as a supply and demand node and we assume each flow
unit uses one unit of each edge’s capacity. We also assume the cost on each edge is
linear in the edges’ flow. Succinctly, the MCFP’s objective is to satisfy all demands by
the same proportion. Equation 5.0 shows the mathematical model for determining the
CF, where  is the CF ratio, {i, j} are the links, x{i, j} is the link’s flow, u{i, j}is the link’s
capacities, and D(s, t) is the demand flow between node source s, and sink node t. To
illustrate its’ concept, we examine network Figure 5.2a. Assume all the OD pairs’ value
is 10.
Maximize 
s.t.
 (x
 x( j ,i,s,t ) )  u(i , j )
x

x

x
 u{i , j}
( i, j,s,t )
( s , t )D
{ s, j } A
{i , j } A
{i , j } A
4781525303
{ s,j,s,t }
{i,j,s,t }
{i,j }
x
{ j,s} A
{ j,s,s,t }
x
{ j , i} A
{ j,i,s,t }
(i, j )  A
  D{s ,t } , if source

    D{s ,t } , if sink
(i, j )  A
(i, j )  A
19 of 28
(5.0)
i
u ij
[S 1 T 1]
j
1
1
OD  10n  N
3
3
2
2
s
2
1
2
t
3
2
1
4
Figure 5.2a: Example Network showing Capacity
The ratio,  is the minimum s-t cut’s capacity divided by its traffic crossing the
cut. From the figure above, the minimum s-t cut capacity, [S1, T1], is 4. That is, |(S1, T1)|
= {1, 3} + {2, 3} + {2, 4} = 1 + 2 + 1 = 4. The traffic crossing this cut is 90. That is, S =
{s, 1, 2} = 3 and T = {3, 4, t} = 3, and |S ||T | = (3)(3)(10) = 90. Therefore,  = |(S1, T1)| /
|S ||T | = 4/90 = 0.044444. This is the maximum flow ratio between all OD pairs.
5.3 Link Capacity Zero
We determine the network’s maximum CF and use it as our base performance, .
We then choose a link from the network, reduce its capacity to zero, and determine the
network’s new CF. We record the ratio as {i , j} , where {i, j} identifies the link with zero
capacity. Afterwards, we reset link {i, j} to its original capacity, zero out the next link in
series, and determine the new CF. We continue this process {i, j}  A .
Lastly, we take the difference between  and {i , j} , called CF {i , j }  {i, j}  A .
We rank CF {i , j } in descending order. The links with the greatest difference are the
network’s most critical links. Table 5.3a illustrates the most critical links for network
Figure 5.2a. For example, CF  {2,3} =  {2,3}  = 0.066666 – 0.044444 = 0.022222.
4781525303
20 of 28
CF (Zero)
0.022222
0.019444
0.011111
0.011111
0.006944
0.006944
0.006944
0.004444
0.004444
Link
{2, 3}
{3, t }
{1, 3}
{2, 4}
{s , 2}
{1, 2}
{3, 4}
{s , 1}
{4, t }
Table 5.3a: Most Critical Links for Figure 5.2a based on the CF Concept
6.0 Validation and Results
We validate our concept against a simulation-assignment model called Dynamic
Network Assignment Simulation Model for Advance Road Telematics (DYNASMART)
[Jayakrishnan et al 1990, 1994]. Originally written in FORTRAN, Abdelghany and
Mahmassani updated the model to Java and incorporated the interaction between mode
choice and traffic assignment under different information provision strategies [2000].
We use four networks with the same number of node and edges, but vary their topology.
Shown in Figures 6.0a-d are the four networks. We illustrate how we set up the
simulation to capture the most critical links and then discuss the models’ results
compared to the simulation.
4
7
10
2
1
13
5
8
15
11
3
14
6
9
12
Figure6.0a: Network 1, 15 Nodes and 28 Edges
4781525303
21 of 28
7
10
4
11
2
5
8
1
13
14
12
3
9
6
15
Figure6.0b: Network 2, 15 Nodes and 28 Edges
1
2
3
4
5
6
8
9
12
7
10
13
14
11
15
Figure6.0c: Network 3, 15 Nodes and 28 Edges
3
9
6
11
14
1
4
7
12
15
2
5
8
10
13
Figure6.0d: Network 4, 15 Nodes and 28 Edges
6.1 Critical Links Simulation
To determine the k-critical links of the network, we must determine the effect of
each link on the network as a system. To do so, we disconnect link {i, j}, run the
simulation model, and then document the average travel time per vehicle. We then
connect the link, and disconnect a new link, run the simulation and document the time.
We do this for every link in the network. The greater the time, the more critical the link
is to the network because travelers spend on average more time in the network than they
would if the link were present.
4781525303
22 of 28
6.2 Model Results and Simulation Comparisons
Table 6.2a illustrates the results for Network 1 (N1). It shows the accuracy of
both models compared to the simulation. The remaining network results are in
Appendix A1.
The first column group, “Simulation”, shows the link, travel times, and rank for
N1. Link {13, 15} in sub column 1, row 1 is the most critical link based on the average
travel time. A disconnect in this link would cause each traveler to incur an average travel
time of 35.395. On the other hand, disconnecting link {5, 6} would have the least affect
on the network. It is 28 and has an average travel time of 28.177 minutes. In addition,
the dark shade in sub column 2 signifies the top 10% critical links based on 90% of
35.395 minutes. The range is from [35.395 - 31.856]. Any travel time within this range
is the network’s most critical links. Because the simulation is microscopic and detailed,
it is possible to have links that are relatively close to each other hence extending the raw
number of critical links. For instance, the difference between links ranked 3 – 8 is less
than or equal to 0.830 minutes. Furthermore, the light shade in sub column 3 shows the
top 20% most critical links. The average travel time range is from [35.395 – 29.216].
This accounts for 28 of the 27 links. Again, this is because of the simulation. Based on
this, the network’s topology is a semi uniform highway network where there is not much
difference between link disconnects.
4781525303
23 of 28
Simulation
Norml
MSE
Shortes Path
Norml
MSE
MCFP
Link
TrvTm
Rank
Link
Obj
Rank
Diff
Link
Obj
Rank
Diff
(13, 15)
(6, 8)
(4, 7)
(12, 14)
(4, 5)
(8, 11)
(10, 13)
(8, 9)
(7, 10)
(7, 8)
(1, 2)
(9, 12)
(11, 12)
(5, 8)
(7, 11)
(14, 15)
(2, 3)
(5, 7)
(11, 13)
(3, 6)
(3, 5)
(1, 3)
(2, 4)
(2, 5)
(10, 11)
(12, 13)
(6, 9)
(5, 6)
35.395
33.401
32.912
32.605
32.383
32.241
32.097
32.082
31.932
31.838
31.744
31.706
31.445
31.191
31.102
31.022
30.979
30.943
30.819
30.235
30.218
29.811
29.764
29.491
29.272
29.235
29.216
28.177
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
(9, 12)
(12, 14)
(13, 15)
(6, 9)
(11, 12)
(3, 6)
(4, 7)
(7, 10)
(7, 11)
(11, 13)
(5, 7)
(8, 11)
(1, 3)
(3, 5)
(1, 2)
(10, 13)
(12, 13)
(14, 15)
(2, 4)
(2, 5)
(4, 5)
(5, 6)
(5, 8)
(6, 8)
(7, 8)
(8, 9)
(10, 11)
(2, 3)
18
16
14
12
10
10
10
10
10
8
8
8
6
6
4
4
4
4
4
4
4
4
4
4
4
4
2
2
1
2
3
4
5
5
5
5
5
10
10
10
13
13
15
15
15
15
15
15
15
15
15
15
15
15
27
27
MSE
SSE
4
169
4
4
100
16
64
49
16
25
16
121
64
1
100
1
100
64
81
225
64
81
64
81
4
121
529
169
83.5
2,337
(12, 14)
(13, 15)
(7, 10)
(7, 11)
(8, 11)
(9, 12)
(10, 13)
(11, 12)
(11, 13)
(1, 2)
(1, 3)
(4, 7)
(5, 7)
(6, 9)
(7, 8)
(8, 9)
(14, 15)
(2, 3)
(2, 4)
(2, 5)
(3, 5)
(3, 6)
(4, 5)
(5, 6)
(5, 8)
(6, 8)
(10, 11)
(12, 13)
0.00237
0.00237
0.00124
0.00124
0.00124
0.00124
0.00039
0.00039
0.00039
0.00018
0.00018
0.00018
0.00018
0.00018
0.00018
0.00018
0.00018
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
1
3
3
3
3
7
7
7
10
10
10
10
10
10
10
10
18
18
18
18
18
18
18
18
18
18
18
MSE
SSE
0
256
49
9
169
9
0
4
36
0
1
81
36
16
144
36
1
64
144
4
9
144
25
36
49
64
289
100
63.4
1,775
Table 6.2a: Network 1 Results
The second column group is the “Shortest Path” model. It has four sub columns:
“Link”, “Obj”, “Rank”, and “Diff”. Column “Link” based on the shortest path (SP)
model depicts link {9, 12} as the most critical with its “Obj” function at 18. The least is
link {2, 3} with its objective at 2. Furthermore, the dark shade is the network’s top 10%
most critical links. The 10% stems from the number of links where we always round up
to the next integer. In this case, (0.10)( 28)  2.80  3.00 links. Hence, the three top
ranked links are {9, 12}, {12, 14}, and {13, 15}. The top 20% is twice the number of
10% links. We are looking for those links ranked from 1 to 6. Because there is no link
ranked 6, and 5 shows up five times, we take links ranked 1 – 5 as the most critical. This
4781525303
24 of 28
gives us nine links ranked as the top 20%. The last column in the SP model is “Normal
MSE”. This is the mean square error between the ranked simulation links’ versus the
models. For instance, link {13, 15} ranks 1 in the simulation and 3 with the SP model.
The square difference is (1 – 3)2 = 4.0. The sum of all squares is 2,337. The mean is
83.5. Appendix A1 shows the results for the remaining networks.
The third column group is the MCFP. The link, objective, ranking, and MSE
follow the same concept as the SP group. However, there is no top 20% links defined by
this model. This is because the first 6 links are in the top 10% where links {7, 10}, {7,
11}, {8, 11} and {9, 12} all tie for third. There are no links ranked 4, 5, or 6. We thus
consider every set of links to be in the top 10% criticality.
By comparing the simulation to the SP model, we see that two of the three links
from the SP model are in the top 10% of the simulation. This yields 67% accuracy.
Furthermore, all of the model’s top 20% links are in the simulation’s top 20%. This is
100% accuracy. In addition, all of the model’s top 10% links are in the simulation’s top
20%. This again yields 100% accuracy. For the MCFP, four of the six links are in the
simulation’s top 10%. This is 67% accuracy. There is no comparison of the model’s top
20% to the simulation because there is no top 20% for the model. On the other hand, all
of the model’s links are in the simulation’s top 20% for 100% accuracy. Table 6.2b
illustrates the results for the remaining models and experiments. Notice for each
network, the MCFP yields the best results. It is always over 90% and in the case of
comparing the model’s top 10% to the simulation’s top 20%, the model is always within
that range. This signifies the MCFP is suitable for finding a network’s top 10% and 20%
critical links versus using simulation.
4781525303
25 of 28
Top 10% Method Links Vs. Top 10% Simulation Links
Method
N1
N2
N3
N4
Shortest Path
66.7%
100.0%
50.0%
100.0%
MCFP
66.7%
100.0% 100.0% 100.0%
Top 20% Method Links Vs. Top 20% Simulation Links
Method
N1
N2
N3
N4
Shortest Path
100.0%
85.7%
80.0%
N/A
MCFP
N/A
N/A
100.0%
85.7%
Top 10% Method Links Vs. Top 20% Simulation Links
Method
N1
N2
N3
N4
Shortest Path
100.0% 100.0%
66.7%
50.0%
MCFP
100.0% 100.0% 100.0% 100.0%
Avg
79.2%
91.7%
91.7%
Avg
88.6%
92.9%
92.9%
Avg
72.2%
100.0%
100.0%
Table 6.2b: Results Comparing Networks N1 – N4 to the Simulation
7.0 Summary and Further Research
In summary, researchers are working to address CIP. They view it as a system,
and by modeling and simulation, they identify the best solutions for a given CI
disruption. In the transportation CI sector, researchers study the affects of traffic
congestion and the impact it has to the system. However, none of the literature review
discussed or showed any attempts to identify the most critical link of the system. This is
important for two reasons. One it helps with resource allocation. Two and more
importantly, it identifies the network’s most critical links. With this, DMs can effectively
expend resources in the most critical areas. They could focus there attention on risk
mitigation plans to obviate and abate the worst disruptions affecting the system.
Individuals could extend the breath of the research by expanding the test scenarios
for validation purpose. Use different network sizes, OD pairs, and link capacities.
Preliminary research shows for highway networks, the node to link ratio is linear. Refer
to Figure 7.0a. This makes sense because it is erroneous to build a highway network
where nodes i and j have multiple links connecting each other. It is infeasible as depicted
4781525303
26 of 28
by figure 7.0b. Furthermore, individuals could look at other mathematical models or a
combination of the SP and MCFP. This may lead to stronger results. One drawback of
both models is that they are snapshots of the network. Time is not a factor. Furthermore,
they may not give the precise link ranking. The models aggregate the system as a whole
and because of this, may not discriminate between 10 and 20%. This is evident in both
the SP and MCFP. The SP does not give the top 20% links for N4 and the MCFP omits
N1 and N2.
4781525303
27 of 28
Appendix A1
4781525303
28 of 28