ii
Copyright © Alireza Naimi
All rights reserved

iii
Acknowledgements
I would like to take this opportunity to express my gratitude to all those who
helped me during the various stages of my life to understand the potential of scientific
thinking and conducting advanced research.
First and foremost, I would like to thank my advisors, Dr. Mihalis M. Golias (the
University of Memphis) and Dr. Sabya Mishra (the University of Memphis), for their
wonderful knowledge, guidance, patience and support throughout this research. I have
benefited greatly from their advice on many research and projects over the years.
I am grateful to the members of my dissertation advisory committee: Dr. Mihalis
M. Golias (committee chair), Dr. Sabya Mishra (committee member), Dr. Charles Camp
(committee member), Dr. Kyriacos Mouskos (committee member), Dr. Dincer Konur
(committee member), and Dr. Bryan Higgs. Their comments and advice were very
helpful for improving the quality of this dissertation.
Last but not the least, I reserve my deepest gratitude for my family, who always
believed in me and stood by me through all times.

iv
Abstract
Naimi, Alireza. The University of Memphis. May 2016. The Robust Urban
Transportation Network Design Problem (UTNDP). Major Professor: Dr. Mihalis M.
Golias.
In today’s congested transportation networks, disruptions like crashes may cause
unexpected and significant delays. All transportation networks are vulnerable to
disruptions, to some extent, with temporary or permanent effects. Vulnerability is more
important in urban transportation networks, due to heavy use and road segments that are
close to each other. Small disturbances on an urban transportation network segment can
have a huge impact on its accessibility. Intelligent adversaries may take advantage of
these vulnerable parts of the network and disrupt transportation operations, increasing the
overall transportation cost for the users.
Often, the decision about improving the networks in transportation planning and
management is made without adequately considering the possible vulnerabilities. By
considering the factor of vulnerability in their decision, planners could prevent or limit
the impact of severe unforeseen disruptions.
This dissertation proposes two models for designing robust networks against
intelligent attackers. In both models, three stakeholders are considered: i) the network
manager/designer, ii) the adversary (intelligent attacker), and iii) the network users. The
frameworks of both models and some other possible models are presented in this
dissertation.
The first framework is a bi-objective designer model. The designer in this model
has two objectives at the top level: to reduce the total system cost and to reduce the

v
vulnerability of the network. The Sioux Falls network consists of 24 nodes and 76 links
was chosen for to evaluate this framework. The decision of the designer and attacker was
improving or destroying the links. Metaheuristic algorithm was used to solve the designer
and attacker problems. For the user equilibrium problem, the Frank-Wolfe algorithm was
implemented. The objective of the designer of the network in the first model, consist of
two goals. The two goals may conflict on the amount of amount of limited available
budget to be invested on the desired project/links. Therefore, a trade off solutions
between these two objectives may forms. The results proved that the proposed multi-level
model is able to find the Pareto front solutions for the two objectives of the designer. The
second framework is a three-level zero-sum game model. In this framework, the payoffs
from the designer are assumed to have the same value to the adversary entity. Therefore,
the goal of this framework is to minimize the maximum gain that the adversary can
achieve. An example network with 6 nodes and 16 links was used to examine this
framework. The results showed that the model could be a valuable tool to reduce the
potential vulnerability of networks. Other indicators of system performance can be
implemented in the upper-level of this framework, in order to examine different goals.
Both frameworks were tested using a medium size network with applications to larger
scale networks as a future research direction.

vi
Table of Contents
Chapter Page
1 INTRODUCTION ........................................................................................................ 1
Contributions .................................................................................................... 2
Structure of the Manuscript.............................................................................. 3
2 LITERATURE REVIEW ............................................................................................. 5
Introduction ...................................................................................................... 5
Modeling Traffic Flow ..................................................................................... 6
Network Design Problem ................................................................................. 8
Bi-level and Multilevel Optimization ..................................................... 10
Performance measures.................................................................................... 15
Reliability................................................................................................ 16
Resiliency................................................................................................ 18
Vulnerability............................................................................................ 19
Robust network design ................................................................................... 30
Summary and Future Research Needs............................................................ 35
3 METHODOLOGY ..................................................................................................... 37
Introduction .................................................................................................... 37
Game theory ................................................................................................... 37
Players ............................................................................................................ 41
Users........................................................................................................ 42
Adversary ................................................................................................ 44
Designer .................................................................................................. 45
Frameworks for vulnerability/robustness ....................................................... 50

vii
Hierarchy of Decision Flows................................................................... 52
Multi-period plan for NDP...................................................................... 57
Model 1: Bi-objective Designer Model................................................... 61
Model 2: A Zero-Sum Model.................................................................. 65
4 SOLUTION METHODOLOGIES ............................................................................. 68
Introduction .................................................................................................... 68
Algorithm for Users........................................................................................ 76
Algorithm for Designer and Adversary.......................................................... 78
Decoding and Chromosomal Representation ................................................. 80
Genetic Algorithm Operations........................................................................ 83
Example Network 1 - Braess Network.................................................... 83
Example Network 2 – 16 links network.................................................. 85
Decoding/Encoding Genotype-Phenotype Space.................................... 85
Elitism ..................................................................................................... 87
Crossover Operators................................................................................ 88
Mutation Operator................................................................................... 92
Sensitivity Analysis of Demand on Test Network 1 and 2 ..................... 94
5 NUMERICAL EXPERIMENTS ................................................................................ 97
Model 1: Bi-objective Designer Model.......................................................... 97
Model 2: Zero-Sum Game Model ................................................................ 133
6 CONCLUSIONS AND FUTURE RESEARCH ...................................................... 139

viii
List of Tables
Table Page
1. Notations - Sets and Indices:.................................................................................xii
2. Notations – Parameters .........................................................................................xii
3. Notations – Variables...........................................................................................xiii
4. Summary of methods for robust transportation network design........................... 34
5. Examples of adversarial games............................................................................. 40
6. Metaheuristics algorithms for NDP ...................................................................... 73
7. Data for Test Network 1 (5-Link)......................................................................... 84
8. The trip rates for the Sioux Falls network (1000 veh/time unit)......................... 100
9. The local optimum solution for the first scenario on the 16 link network, Bz = 1
............................................................................................................................. 135
10. The local optimum solution for the second scenario on the 16 link network, Bz = 2
............................................................................................................................. 135
11. The local optimum solution for the first scenario on the Sioux Falls network, Bz=1
............................................................................................................................. 137
12. The local optimum solution for the first scenario on the Sioux Falls network, Bz=2
............................................................................................................................. 138

ix
List of Figures
Figure Page
1. Complexity versus level of details in traffic flow modeling. Adopted from
Washington (2008)........................................................................................................ 7
2. Operations of Network Design Problems ..................................................................... 9
3. Vulnerability versus reliability. The thick line is the “risk curve” of Kaplan et al.
(1981).......................................................................................................................... 20
4. Example of investing and getting attack on the same link.......................................... 51
5. Examples of hierarchy of sequences of player’s moves and structures of games ...... 56
6. Examples of a hierarchy of three players’ decision flow............................................ 57
7. An example of the multi-period NDP results for Braess network.............................. 60
8. Four possible combinations of undirected graph with three nodes............................. 68
9. Number of possible undirected networks by number of nodes................................... 69
10. Domain size of Lane Addition (Discrete Variable) .................................................... 70
11. Flowchart of the Solution Approach........................................................................... 75
12. Decoding Procedure.................................................................................................... 81
13. Genotype chromosome representation of adversary entity......................................... 82
14. Test Network 1 – Braess Paradox Network (5-Link) ................................................. 84
15. Test Network 2 (16-Link)........................................................................................... 85

x
16. Variation of reaching the solution by different size of bit-string in binary chromosome
representation.............................................................................................................. 86
17. Elitist selection with different size, for population size 100....................................... 87
18. Different Crossover operators (Crossover rate 10%).................................................. 88
19. Convergence of capacity expansion vector to optimum values by crossover rate
(crossover type: Uniform Crossover (UPX), mutation rate: 2%, population size 30) 90
20. Crossover values by Mating rate (one-point Crossover) ............................................ 91
21. Sensitivity test of the convergence to mutation rate ................................................... 93
22. Test Network 2 (16-Link) Results .............................................................................. 95
23. Total System Travel Time for Test Network 2 - (16-Link)........................................ 96
24. Sioux Falls network configuration.............................................................................. 99
25. Links Included in Expansion (links with orange color) for the Sioux Falls network101
26. Improvement of the two objectives at the designer level by generations................. 103
27. Individuals of the two objectives at the designer level by generations..................... 105
28. Decision of the designer (Number of lanes to be added to the network).................. 106
29. The optimal decisions of the attacker ....................................................................... 107
30. Improvement of the capacity-expanded network compare to the initial conditions. 109
31. Flows in the Initial and Improved network, before and after the disruptions (veh/day)
................................................................................................................................... 110
32. Travel times in the Initial and Improved network, before and after the disruptions
(min).......................................................................................................................... 112

xi
33. Individuals of the two objectives at the designer level by generations..................... 114
34. Decision of the designer (Number of lanes to be added to the network).................. 115
35. The optimal decisions of the attacker for the initial and improved networks........... 116
36. Flows in the Initial and Improved network, before and after the disruptions (veh/day)
................................................................................................................................... 117
37. Travel times in the Initial and Improved network, before and after the disruptions
(min).......................................................................................................................... 119
38. The optimal decisions of the designer and the attacker ............................................ 123
39. Flow on the links after the disruptions (veh/day) ..................................................... 127
40. Travel Time of the links after disruption .................................................................. 131
41. Travel system travel times by 𝐵𝑧 in the Initial and Improved network after the
disruptions................................................................................................................. 132
42. Test Network 1 (16-Link)......................................................................................... 134

xii
Abbreviations
Table 1
Notations - Sets and Indices
𝒜 Set of links
𝒩 Set of Nodes
ℛ Set of origin nodes; ∀ℛ ∈ 𝒩
𝒮 Set of destination nodes; ∀𝒮 ∈ 𝒩
𝒦𝑟𝑠
The complete set of available paths connecting (O/D) pairs 𝑟 − 𝑠, ∀𝑟 ∈ ℛ, ∀𝑠 ∈ 𝒮 in
the network
𝑞 𝑟𝑠 Demand between each Origin-Destination (O/D) pair 𝑟 − 𝑠, ∀𝑟 ∈ ℛ, ∀𝑠 ∈ 𝒮
𝜆 𝑟𝑠 Shortest path for O/D pair 𝑟𝑠
𝑛 𝑟𝑠 Number of O/D pairs in the network
Table 2
Notations – Parameters
𝛼 𝑎 Constant, varying by facility type (BPR function)
𝛽 𝑎 Constant, varying by facility type (BPR function)
ta
o Free flow path travel time for link 𝑎 (hr)
ha Capacity of each lane (veh/hr/ln)
MP A multiplier constant number to give high cost for vehicles for using the target link
𝐵 𝑑 Total budget/resources available to the designer
𝐵𝑧 Total budget/resources available to the adversary
𝐶 𝑎 Capacity for link 𝑎
𝑙 𝑎 Length for link 𝑎
𝑣 𝑎 Space mean speed for link 𝑎

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Table 3
Notations – Variables
𝑔 𝑎(𝑦𝑎) improvement cost function for link 𝑎
ta Flow-dependent link travel time on link 𝑎 (hr)
𝑓𝑘
𝑟𝑠 Flow on path k, connecting each origin-destination (O-D) pair 𝑟 − 𝑠, ∀𝑟 ∈ ℛ, ∀𝑠 ∈
𝒮
𝛿 𝑎𝑘
𝑟𝑠
𝛿 𝑎,𝑘
𝑟𝑠
= 1 if route 𝑘 between OD pair 𝑟 − 𝑠 contains link 𝑎, and equal to 0
otherwise
𝑥 𝑎 Total link flow (vph) on link 𝑎
𝑦𝑎 Integer decision variable; total number of lanes to be added/expanded to link 𝑎
𝑧 𝑎 Binary decision variable; 1 if link 𝑎 is disabled, and 0 if it is not

1
1 INTRODUCTION
Transportation networks are indispensable components of daily life in today’s
society. Traffic engineers try to utilize available resources to provide an efficient
transportation system for all users (both passenger and freight). However, a transportation
network is usually not designed from scratch. The network design problem (NDP) aims
to modify an existing or implement a new network to improve system performance
(based on various and often conflicting objectives). It proves to be one of the most
challenging problems for researchers in the field of transportation. There are various
uncertainties that may be unknowns to the designer of a transportation network including
uncertain input parameters (e.g., demand and supply) and disruptions (natural or man-
made). The latter (i.e., disruptions) may reduce the supply of the network, change
demand patterns, and may even completely interrupt the operations of a set of network
elements. The research presented herein aims to develop mathematical models and
solution algorithms that design a robust network considering intelligent disruptions.
Assessing vulnerability and optimizing network robustness have been studied in
the literature using a variety of approaches. To date, no generally accepted indicator of
robustness exists. Furthermore, there is a gap in designing robust transportation networks
considering an intelligent adversary/enemy entity. This research aims to fill the latter gap
in the literature and propose game theory-based frameworks to study the strategic robust
network design against intelligent attackers. Two models are proposed in this dissertation
for designing robust network. The proposed models consider the following three
stakeholders:
1) Traffic management agency or government;

2
2) Users of transportation network; and
3) An adversary (or attacker).
In order to design a robust network, various distinct ways of representing games
based on the order of play, the information available to each player and structure of
formulation can be defined. In the frameworks provided in this dissertation, the traffic
management agency is interested in designing a network that is less vulnerable to enemy
entity moves. On the other hand, the adversary (or evil entity) is assumed to maximize
the disruption to the network. The users respond to the adjusted network by the
transportation agency and the evil entity. The proposed model can be customized and
applied to other similar network designs, like telecommunications and biology networks.
As it will be described in more details in chapter 2 and 4, the corresponding
discrete optimization problems for the designer and attacker are combinatorial and NP-
hard to solve optimally (Feremans & Laporte, 2003). Hence, no efficient exact or
heuristic methods are available to solve these problems in reasonable computational time.
Therefore, metaheuristic approaches were used to solve these problems. On the other
hand, traffic flow at the user level was modeled using Nash equilibrium concepts. The
user equilibrium problem is convex and can be efficiently solved using methods like
Frank-Wolfe or origin-based algorithm (Bar-gera, 1999).
Contributions
The main contribution of this research is to provide various frameworks for
designing robust networks strategically, by considering an intelligent adversary entity,
who attempts to exploit the vulnerabilities of the network to the maximum of his or her
capabilities. As it also discussed earlier, no generally accepted vulnerability index exists

3
(to the author’s knowledge). Therefore, the goal and achievements of the intelligent entity
also need to be defined. One of the appropriate approaches to analyze and model the
intelligent adversary’s rule in vulnerability could be modeling it as a player in a game
who is interested in achieving his objective(s).
While designing a robust transportation network against stochastic vulnerabilities
(due to the stochastic events, for instance, natural disasters) have been studied extensively
in the literature, this research aims to provide additional insights by considering the
network elements that are vulnerable to the intelligent adversary entity. Two frameworks
are presented to model the interactions between the players.
To summarize, the main objectives of this research are:
1) To provide new frameworks to model robust UTNDP against potential
intelligent attacks;
2) Develop metaheuristic methodology using genetic algorithm to solve the
discrete problems, and a convex combination method to solve the user equilibrium
problem; and
3) Demonstrate the proposed methodologies on small and medium sized
networks.
Structure of the Manuscript
The structure of the rest of this dissertation is as follows: Chapter 2 presents a
literature review of the related studies. The general design of transportation networks and
the evaluation of vulnerabilities as a performance measure are provided in this chapter.
Chapter 3 briefly explains the game theory and its application in the proposed
frameworks. The mathematical formulation for each model and the role of each player in

4
modeling the robust network design problems is presented. Chapter 4 discusses the
algorithms to solve each player optimization model. The Genetic Algorithms parameter
settings are also discussed. Numerical experiments were conducted in chapter 5 to test the
performance of the proposed models. Lastly, chapter 6 concludes the dissertation and
provides the possible topics for future research.

5
2 LITERATURE REVIEW
Introduction
In this chapter, a review of the literature is presented in order to provide the
foundation to understand the models and algorithms proposed in this dissertation. Since
three decision makers are considered in this research, the possible models to account for
the interactions, objectives, and decision-making hierarchy are reviewed.
First, a brief introduction to game theory is provided followed by a review of the
network design problem (NDP). The focus of this research is on NDP considering
vulnerabilities against intelligent disruptions. Therefore, the robustness and the
vulnerability as a network performance measure were reviewed in this chapter. The
design of robust networks has captured the attention of many researchers. As it is often
the case with popular terms, there is not a generally adopted notion for the vulnerability
of networks (Jenelius, 2010). Robustness is the opposite of vulnerability. Therefore, a
network that is vulnerable is not robust and vice versa (Snelder, 2010).
An appropriate approach to model the vulnerabilities of intelligent disruptors
could be to model them as a player in a game, which is interested in achieving his
objective(s). Therefore, the following objectives for each player can be defined: From the
designer side, it might have various objectives to improve the performance of a network.
The performance of a network can be represented as the total system cost, robustness
against reliability and vulnerability, reduction of pollution emission and multiyear
investments. On the other hand, from a user’s perspective, they look for their optimal
route choice, mode, and destination. From an adversary viewpoint, the objective is to
degrade the performance of the network to the maximum of his capabilities. Hence, a

6
robust network design model must consider the alleviation of the potential disruptions. In
the next section, first a general summary of three common methodologies to model traffic
flow is described, followed by a brief introduction to network design problem and its
methodology.
Figure 1 demonstrates the relationship between the complexity of
modeling/simulation of traffic flow and the considered level of details in the models. In
Modeling Traffic Flow
The application of computer technology provided engineers with the capability to
model complex transportation systems. Various types of models have been published
during the last decades. They can be categorized based on the level of detail and their
complexity into three categories: Microscopic, Mesoscopic, and Macroscopic models.

7
Figure 1. Complexity versus level of details in traffic flow modeling. Adopted from
Washington (2008).
In microscopic models, the smallest unit in the simulation is the driver-vehicle
unit (or other types of flowing items, for example, vessels, airplanes, packages).
Microscopic models provide an adequate amount of information to analyze most of the
operational (e.g., operational lane changing models) and tactical (e.g., tactical lane
changing models and tactical overtaking models) systems (Michon, 1985; Moridpour,
Sarvi, & Rose, 2006). In the case of modeling driver-vehicle, some behaviors (like lane
changing and overtaking) requires large amounts of information, and modeling the
decisions drivers are making based on these data, is difficult (Chamieh & El-kouatly,
n.d.; Kano, Shiraishi, & Kuwahara, 2007; Suzuki & Mori, n.d.; Wheeler & Llc, n.d.). In
ms seconds minutes hours days months years
Complexity
Microscopic
Mesoscopic
Macroscopic
ScaleFeet Miles
Time
Unit
Level of details

8
addition, defining, simulating and validating rich cognitive driver behavior models,
requires significant effort (Numrich & Tolk, 2010; Yilmaz, 2009).
Mesoscopic models, further simplify assumptions of microscopic models by
combining the driver-vehicle units into groups of driver-vehicle (or other transport flow
units). The Cellular Automata (CA) models usually model the transport units in groups
that are moving from one cell to the others by advancing in simulation steps. Thus, these
types of models fall into mesoscopic classification.
Macroscopic traffic flow models try to formulate the relationships between traffic
deterministic relationships of the speed, flow, and density of a traffic stream
(Washington, 2008). These types of models originated under a theory that traffic flows,
as a whole, are similar to fluid streams systems. The characteristics of traffic flow in the
network are typically considered homogeneous in a specific time unit (which usually
ranges from a few minutes to days).
Network Design Problem
The natural population growth, and other factors such as the increase in income
and employment, will result in the increase in travel demand on transportation networks.
This may lead to problems such as congestion and safety in the system. The
transportation agencies need to plan transport networks properly to alleviate these
problems. This will require new infrastructures for serving the new transportation
networks or improve the existing system. The planning, design, and managing these
issues are traditionally addressed in network design problem (NDP). NDP is usually used
for determining the optimal sub-network, which will result in improvement of the whole
network. Various definitions of NDP are provided in the literature. For example, Friesz

9
(1985) defined it as: “network design problem is to determine the optimal locations of
facilities to be added to a transportation network, or to determine the optimal capacity
enhancements of existing facilities in a network” (p. 413).
Modeling the transportation planning problems is typically complex (Beimborn,
1995). Hence, in practice, these problems are typically decomposed into a sequence of
subproblems. Some of the examples of decomposing the transportation problems into
independent sub-problems are the classical four-step planning process, network design
problem, and traffic signal setting design.
The network design problem can be described using graph theory. Likewise, a
complex network can be represented by a graph. A graph 𝑮 = (𝑵, 𝑳) is characterized by
a set of links 𝑳 and a set of nodes 𝑵. Each link connecting two nodes, and can be directed
or undirected. Attributes like weight/cost (𝑪) can be assigned to each nodes and link
(Figure 2). NDP transforms an existing network (graph 𝐺 = {𝑁, 𝐿, 𝐶}) into a new
improved network (graph 𝐹 = {𝑁’, 𝐿’, 𝐶’}). In road transportation network, distances
between end points of links or travel time are well known attributes of links.
Figure 2. Operations of Network Design Problems
G=(N,L,C) NDP F=(N’,L’,C’)

10
Finding the optimal road design has been the subject of transportation studies for
a long while, and is known to be one of the most complicated problems in transportation.
A large number of methodologies and solution algorithms have been presented over the
last 50 years to provide solutions to these complex mathematical problems (S.-W. Chiou,
2005a; Leblanc, 1973; Murray, Davis, Stimson, & Ferreira, 1998; Suwansirikul, Friesz,
& Tobin, 1987).
Bi-level and Multilevel Optimization
Advances in computer technology gave researchers ability to study the design of
the networks in new aspects, and in more analytical details. Among the possible
modeling approaches, bi-level programming received more attention. It provides a
comprehensible representation of the designer and the users of the network as
independent sub-problems. The bi-level programming problem is a subcategory of multi-
level programming problem, with two level. In problems with conflicting objectives
within a hierarchical structure based on the sequential order of two decision makers, bi-
level optimization is an effective solution approach. It originated from the fields of game
theory and it can describe a number of problems in transportation planning and modeling.
Its hierarchal framework involves two separate optimization problems at different levels.
In case of Stackelberg competition, the first problem - called the upper-level or leader
problem - has a feasible solution set. The solution set is determined by the optimization
problem at the second level. The second problem is the lower-level problem or the
follower problem. This concept can be expanded to define multi-level programs with any
number of levels (Vicente & Calamai, 1994a). The bi-level program is an NP-hard
problem; hence, it is difficult to solve using exact algorithms. Ben-Ayed (1993) and

11
Ayala (2013) investigated on bi-level problems and concluded that even a simple bi-level
problem with both linear upper-level and lower-level problems is also NP-hard. One
reason is that bi-level model for NDP is non-convex (Gangi, Pianificazione, & Luongo,
2005). Luo, Pang, and Ralph (1996) also mentioned that even if both problems at upper-
level and lower-level is convex, the convexity of the bi-level problem is not guaranteed.
The no convexity of the problem makes it difficult to solve optimally.
Multi-level programming, which has received significant attention during the last
few decades, is a branch of mathematical programming that can be viewed as either a
generalization of minimization-maximization problems or as a particular class of
Stackelberg games. The network design problem can be cast into such a framework.
Marcotte (1986) presented a formal description of the problem and developed various
suboptimal procedures to solve it. Multilevel optimization problems have shown to be
(usually) non-convex and are thus difficult to solve using exact optimization algorithms
(Konur, Golias, & Darks, 2013).
The very first studies in bi-level NDP were investigated by Leblanc (1973),
Bruynooghe (1972), and Ochoa-Rosso (1969). They used the branch and bound
techniques for solving the NDP. Moreover, Poorzahedy and Turnquist (1982) studied a
typical heuristic algorithm to find the solution using integer programming model.
Further research has been done to find more efficient heuristic algorithms, which
may give near optimal solutions or local optimum solutions (Allsop, 1974; Steenbrink,
1974). Methods like equilibrium decomposed optimization EDO (Suwansirikul, 1987),
which are computationally efficient but result in suboptimal solutions and not suitable for
large real networks problems. Gershwin and Tan (1979) formulated the continuous

12
network design problem (CNDP) as a constrained optimization problem in which the
constrained set was expressed in terms of the path flows. Patrice Marcotte & Marquis
(1992) presented heuristics for CNDP on the basis of system optimal approach and
obtained good numerical results. However, these heuristics have not been extensively
tested on large-scale networks generally.
Advances in metaheuristic models, (e.g. evolutionary algorithms, and simulated
annealing) drew the attention of researchers in mid-90s and 2000s. The benefit of using
metaheuristics is their globality, parallelism, robustness and ease in implementation
(Mathew & Sharma, 2006). For an example, Friesz (1985) and Meng (2009) utilized
simulated annealing (SA) method to solve the upper-level problem. Despite the faster
runs of SA, especially for the larger problems, the solution quality of Genetic Algorithm
(GA) was found to be better than SA and other metaheuristic algorithms (Adewole,
2012). Mathew and Sharma (2006) performed a study on using GA in CNDP. They
applied their model to the small to large size problems. Mouskos (1991) utilized the Tabu
search to solve the single class bi-level UTNDP with a Budget constraint where the
decision variable was to improve (or not) each roadway link by one lane using the static
traffic assignment as the lower level. Furthermore, Zeng (1998) utilized a hybrid SA-
Tabu search method to solve the two-class (automobiles plus trucks) to solve the bi-level
UTNDP for large networks.
In many governments and public transportation projects, the cost-benefit analysis
is utilized to determine if the estimated benefits provide an acceptable return on the
expected costs and investments. From this point of view, Meng and Yang (2002) solved
the bi-level benefit distribution for network design problem using the ratio of the benefits

13
gained from the capacity expansion for each link. Their model was non-convex, non-
differentiable, and continuous, so they chose simulated annealing method to solve their
optimization problem. Their multi-objectives trying to maximize the total benefit among
the links, while also trying to minimize the differential between beneficial gained by each
link. In the next section, a brief introduction to resiliency, reliability, and vulnerability is
presented.
The majority of planner’s decisions dealing with project selection involve single
initial costs while benefits could spread over many years in the future. Brown (1980)
applied dynamic programming to obtain a set of projects which provide an optimum,
taking into consideration not only present costs but also the benefits that accrue over
several years into the future. Moreover, Baskan performed a study utilizing bi-level
optimization that took into consideration the increasing future congestion and limited
budget constraints (Baskan, 2013). Optimal link capacity expansion values were found by
minimizing the total system travel time as well as the associated link investment costs
within roadway networks.
Regarding the sensitivity based approach applied to bi-level optimization
problem, Falk and Liu (1995) investigated theoretic analysis for general non-linear bi-
level optimization problem and proposed a descent approach in terms of the bundle
method to solve the non-linear bi-level problem where the gradient of the objective
function can be obtained when the subgradient information of the lower level is available.
Chiou (2005) explored a mixed search procedure to solve an area traffic control
optimization problem confined to equilibrium network flows, where good local optima
can be effectively found via the gradient projection method.

14
Several attempts were made in the last few years to find the global optimum
solution for network design problems. Wang, Meng, and Yang (2010) partitioned the
feasible space of nonlinear travel time function into several regions and provided a path
based MILP. Each region represents a piecewise linear function that can approximate the
original nonlinear travel time function. The model required a heavy computational time
and required using a large memory storage. Paramet Lauthep (2011) mentioned that
Wang’s approach is inapplicable to the case of DNDP and MNDP, because the paths in
their network structure are generated in advance, where it should change during the
design process. He modified the model to a linked-based and provided an efficient mixed
integer linear program. Li, Yang, Zhu, and Meng (2012) presented a model to convert bi-
level CNDP into a sequence of single level concave programs, based on the concept of
gap and penalty function. Furthermore, Wang, Meng, and Yang (2013) presented a global
optimization method for DNDP. The presented model was not computationally efficient
and may not be practical to large problems.
A comprehensive and scientific review of the literature of the various type of
urban transportation network design problems was written by Farahani, Miandoabchi,
Szeto, and Rashidi (2013). They classified the available models from different aspects: by
type of performance measures, decision variables, transport mode, and solution
algorithms. A list of the possible future roadmaps was also provided. The decision
variables used in the previous studies were categorized into (1) strategic, (2) tactical, and
(3) operational. Strategic decisions are about adding new links and expanding capacities.
The two later are about to maintain the current network. In the next section, a review of
the performance measures of transportation networks is presented.

15
Performance measures
The goal of transportation planning and management generally involves finding a
set of optimal solutions, for certain decision variables by optimizing different system
performance measures. Performance measures, which are defined as indicators of system
efficiency, are progressively becoming an important factor in transportation planning
(Pei, Fischer, & Amekudzi, 2010). They are the main factor in determining whether a
roadway network is viable for the future. Some of the important performance measures in
planning problems are congestion, emissions, accessibility, mobility, reliability, pollution
emission, noise, and safety.
The typical performance measure in the basic structure of transportation NDP is
the Total System Travel Time (TSTT). The level of congestion directly affects travel
times. When a part of a network becomes overly congested, travel times will increase and
level of services (LOS) will decrease. The effects of congestion can then spill into other
portions of the network and increase the system-wide travel times. Another important
performance measure is accessibility. Accessibility refers to how suitable is a public
transport network for letting travelers go from the point that they enter the network to the
point that they exit the network in a reasonable amount of time (Murray, 1998).
Similarly, mobility has attributes like having access to the point of interest, maintaining
networks, benefiting from travel to social contacts and potential travel (Alsnih &
Hensher, 2003). Safety is of critical importance in transportation. The key objective of
the safety of a network is to reduce the annual number of crashes to a fraction of the
current levels (Dijkstra, 2013). Resiliency, reliability, and vulnerability are three other
comparable performance measures that are discussed in the following sections.

16
Reliability
The reliability of the transportation network refers to the probability that a system
can perform its expected function to an acceptable level of performance for a given
period of time (Bell, 2000). Berdica (2002) defined the reliability of a network as the
possibility of moving freight or passengers from one place to another successfully. Yim,
Wong, Chen, Wong, and Lam (2011) further defined reliability as the ability of the
network and its elements to operate under capacity. Reliability gained more attention
during the 90’s when the natural disasters like earthquakes damaged or completely lost
the connectivity of some of the major roadways around the world (Yim, Wong, Chen,
Wong, & Lam, 2011). Following the development of transportation networks, the
reliability studies focus on alleviating the damage effects on the network and investigate
the unpredictable variations caused by the uncertainties (Nicholson, Schmöcker, Bell, &
Iida, 2003). Some of the main measure of the reliability of transportation networks are
connectivity reliability, travel time reliability, and capacity reliability (Chen, Yang, Lo, &
Tang, 2002). The types of measuring reliability in transportation can be categorized as
following types:
1. Statistical range method, e.g. Standard deviation (STD) and Coefﬁcient of
variation (COV) of travel time.
2. Buffer time methods: The extra time a user has to add to the average travel time
to arrive on time 95% of the time.
3. Tardy trip measures: The amount of trips that are late.
4. Probabilistic measures: The probability that travel times occur within a
speciﬁed time interval.

17
The importance of the measuring reliability can lie on the fact that most of the
users get a resiliency of the cost (Travel Time) over the time, and a sudden change on it
can have a big effect on the network. This can be seen usually during the need for fast
evacuation.
Transportation network planning efforts have traditionally relied on the localized
level-of-service (LOS) measures such as the v/c (volume/capacity) ratio, to identify
highly congested links that are considered as critical links. The problem with this
approach is looking at the individual segments’ performance; the individual elements
may not enable planners to identify the most critical highway segments or corridors in
terms of maximizing system-wide, especially for system travel-time benefits by
implementing highway improvement project.
Various approaches have been studied to analyze the reliability of networks. A
reliability of transportation network by changing the cost or disutility function as the
standard deviation of travel time was examined by Fosgerau and Karlström (2010). They
found out that the maximum expected utility has linear mean and standard relationship
correspondingly to the travel time. Markov Chain has been used by several researchers to
replace the conventional transportation planning and predicting the future pattern of flow
(Antoniou, Koutsopoulos, Yannis, & Model-based, 2007). Indrei (2006) tried to model
the traffic flow system using Markov Chain theorem. However, his work only limits to a
unit car. Iyer, Nakayama, and Gerbessiotis (2009) also used a continuous-time Markov
chain (CTMC) model for predicting the reliability of a system by evaluating cascading
failure procedure. They distinguished all the possible cascading failures of different sets

18
of elements that lead to breakdown the whole system, and based on the Markov
probabilities, they rank the elements by their contribution to system breakdown.
Resiliency
Resiliency is defined as the ability to resist, absorb, recover from, or successfully
adapt to adversity or a change in conditions (Bhushan, Narasimhan, & Rengaswamy,
2008). In a transportation network or in a sequence of events, it can be seen how the
different elements work together to recover after a disruption (such as flood, hurricane,
tornado, etc.) happens. Resilience can be viewed as the opposite of brittleness, which
describes a system that cannot tolerate disruptions, and loses the capacity (or
functionality or other words that describe the productivity of a system). In this context,
therefore, resiliency is an attribute that contributes to achieving the required reliability.
However, resiliency is not an independent measure of reliability.
Resiliency gained more attention during the last decade, and various studies have
been performed on this topic. Some of the studies focused on the resiliency of the
maritime systems. In order to examine the resilience of ports, Kamal Achuthan (2012)
developed a simulation model and performed a variety of analysis. He considered the
interactions between different elements in a port and saw how statically they can
incorporate in the resiliency of the port due to a disruption. He also considered the
stakeholders contribution in his interdependencies model. The output of his model
includes resilience matrices for before, during, and after disruption, the number of ships
served by each resource and also queues and delays. Some of the important vulnerability
indicators in literature are described in the rest of this section. To manage the resilience
strategies in maritime systems, Mansouri and Mostashari (2009) developed a decision

19
analysis methodology. They mainly focused on the costs (probable disruptions,
investments in resilience strategies, losses, and gaining from using resilience strategies).
Therefore, the study can be considered as a business work with a monetary focus.
A three-stage framework to analyze infrastructure resilience was defined by
Ouyang (2012). The first stage was defined as a consistent mode, which can be used as a
representation of disasters. The second stage defined as damage propagation, and the last
two stages defined as a situation which the authorities trying to stop the damage
propagation and recover it. He and his co-researchers chosen power grid model as a case
study, and then they consider several disasters (in their case, random hazards and
hurricane hazards) and different approaches to recovering the damages. They figured out
that the annual resilience mainly happens due to its higher frequency of occurrence
compare to hurricane hazards. In addition, they found out that the type of recovery
sequences is important.
Vulnerability
As it was discussed in chapter 1, there is no universally accepted definition of
vulnerability. Therefore, vulnerabilities can be evaluated and viewed from different
aspects. From a transport side, the vulnerability can be defined as how vulnerable the
transport system is in the case of failure of one or several of components of transportation
systems (Erath, Birdsall, Axhausen, & Hajdin, 2009). In another word, it defined as
sensitivity to attack or injury. According to Jenelius (2010) the technological and social
aspect of transportation networks can be distinguished from their perspective: From the
technological point of view, he defined the infrastructure component’s importance by the
impact of the failure of that element. Furthermore, criticality is defined as the

20
combination of probability and the importance of failure. From the social side, exposure
is defined as the equivalent of importance which shows the failure impact to an individual
user. Likewise, vulnerability is defined as the combination of exposure and the
probability of failure.
Vulnerability should be differentiated from reliability. One of the main
differences between these two concepts is their focus on the magnitude and the
probability of the adverse consequences. Figure 3 shows the “risk curve” of (Kaplan &
Garrick, 1981) in probability format. The probability of occurrence of a scenario and its
level of damage can be found by looking at this curve. The frequencies of occurrence of
regular events are lower when its impact is higher.
Figure 3. Vulnerability versus reliability. The thick line is the “risk curve” of Kaplan et
al. (1981).
Level of Damage
Probability

21
The interrelationships between infrastructures, impact of risks within the system,
and consequence of events has not been studied well in the literature. A failure of a
network component could also cause the breakdown of other critical infrastructures in a
disruption event. For example, a disruption in a fuel transport network for a period time
of several days to several weeks could have a sequence of further disruptions in other
networks, such as transportation, energy; or a breakdown of telecommunication or energy
network could affect the transportation system for foods (Murray & Grubesic, 2007). It
should be noted that different transport materials also do not have the same importance in
term of overcoming the critical situation. For instance, the transports of medicine and
foods usually have a more crucial impact than the farm products in severe events. In
terms of time, the interruptions in the service of an infrastructure may last for a short
period (e.g., few hours), or longer periods (e.g., several days or weeks), or in extreme
conditions, they can be permanent and last for an indefinite time.
The concept of vulnerability can be classified in the following ways: static which
evaluates the vulnerability based on a physical property of a network, and does not
depend on traffic flow; and dynamic that directly refers to the robustness of a network.
Most of the works were focused on graph theory and their property correspond the
possible vulnerability of the network. However, in road transportation network, more
realistic model considers traffic flows, as they are the main concern of the designer of
network if the impact for a single user under a specific scenario is to be evaluated, this
may call for exposure of the user to that scenario Jenelius, Petersen, & Mattsson, 2006).
Kröger and Zio (2011) also categorized different approaches for assessing the
vulnerabilities of Critical Infrastructures (CI). According to their research, vulnerability

22
evaluation focuses on three main elements: degree of loss, degree of exposure, and
degree of resilience.
The availability and quality of alternative routes are a very important indicator of
vulnerability. The availability of spare capacity (capacity minus the flow) also could be
an important indicator of vulnerability. Other examples could be v/c ratio, the number of
OD-pairs that use a link, number of vehicles affected by spill back (the spare capacity
can be used to bypass an incident), extra vehicle kilometers traveled as a result of link
closure, travel time losses as a result of crashes.
Crucitti, Latora, and Marchiori (2004) and Latora and Marchiori (2001) provided
a measure for the performance of a network, called ‘network efficiency’. The network
efficiency 𝐸(𝐺) of a network 𝐺 is only depends on topological characteristics of a
network. Their performance indicator is based on shortest path between nodes and
number of nodes. Efficiency of the network defined based on the number of possible
edges (higher number of edges increase the efficiency and reduce the disruptions in
network), and shortest path between all the nodes (smaller shortest paths means higher
efficiency).
Jenelius et al. (2006) presented a mathematical model to evaluate the vulnerability
of link, nodes, and whole network. In his model traffic flow is considered as the source of
vulnerability indicator, and was based on changes in the cost of travel and unsatisfied
demands of links or nodes. He further transformed his model, by incorporating changes
on the cost of travel and unsatisfied demands of elements covered in grids (Jenelius &
Mattsson, 2012). In denser network areas, grids can be defined smaller to provide better
accuracy to analysis performance ratio.

23
Equity of impacts if network degradation among all the users is considered as a
key in analyzing and design network. Jenelius (2010) presented a methodology for link
performance measure considering equity measures. In his model, equity importance
measure in disruption events is defined as the coefficient of variation of increase in travel
times. The degradation is measured using the total changes in travel time.
The link usage proportion-based algorithms are applied to solve bi-level
transportation problems in which demands act as upper-level decision variables (H. Yang
& H. Bell, 1998). In this algorithm, an influencing factor for each link is a ratio between
its usage and its capacity. In this case, the link that is used to its capacity or over is likely
to receive an improvement. This algorithm is applied to ramp (H. Yang & H. Bell, 1998),
zone reserve capacity (H. A. Yang, 1997) and O-D matrix estimation (Jin & Yang, 2014).
Snelder (2011) presented a topological vulnerability indicator, based on the
availability of alternative (backup) links, which can be translated to alternative routes. In
her model, the links that cross a line perpendicular to the target link 𝑎 are considered
alternatives for the link 𝑎, if they meet the following requirements: The absolute angle
between the link 𝑎 and the alternative link must be smaller than 60 degrees. The
vulnerability index in her model is based on: (1) the ratio of capacity of link 𝑎, over the
summation of capacities of alternative links for link 𝑎, (2) a function of shortest path
between link 𝑎 and its alternative links, (3) and a parameter for the importance of the
distance.
Reniers and Dullaert (2013) used a scoring system in GIS to evaluate the
vulnerability in transport hazardous materials in four different modes. They subcategorize
each type of materials by the mode type and give each route segments a score that

24
represents the vulnerability on its transportation. They also used a score factor for the
number of people that are influenced by the consequence grade.
Accessibility also considered in literature to evaluate vulnerability. In Chen
(2007) model, vulnerability is assessed based on changes in accessibility measure (a
utility function) due to the degradation of network structure. He combined trip
distribution, mode, and assignment in the model. Taylor and D’Este (2007) defined a
node to be vulnerable, if loss (or substantial degradation) of a small number of links
significantly reduces the accessibility of the node, as measured by a standard index of
accessibility. In their model, they did not consider traffic flow.
Gregoriades and Mouskos (2013) utilized a combination of the mesoscopic traffic
simulator called VISTA and Bayesian Network to model the accident potential in links of
the network. They proposed an index called ARI (accident risk index) which was the
result of the Bayesian Networks (BN) output. The topography of BN comprised from
different variables such as the pavement quality and link attributes; and includes two new
parameters from VISTA: flow and speed. Having these two parameters, they claimed it
would improve the prediction power of BN. The validation process shows about 81
percent prediction validity, which the authors mentioned it can improve by improving
different input variable to BN. Berdica and Mattsson (2007) also developed a simulation-
based method to evaluated vulnerability at the link or network level. They predefined
twelve scenarios defined (i.e., Lane/link closure, change the BPR function element) to
study the vulnerability.
Murray, Matisziw, and Grubesic (2007) developed four bi-objective optimization
models to study the possible disruptions in a graph network. Furthermore, the bi-

25
objective converted into a single objective using a weighted combination of the two
objectives. The objective is to min/max the bandwidth of the network and the impacted
population. She further studied vulnerability indicators and provided a structured
approach to optimization to evaluate vulnerability for a set of nodes, or total interacted
O/D pairs. The model was based on selected number of nodes to be interdicted, the
optimization model seeks to find worst (best) nodes to be interdicted, such a way that the
total O/D disconnected would be maximum (or minimum) (Murray et al., 2007). Bell
(2000) and Bell, Kanturska, Schmocker, and Fonzone (2008) also developed a game
theory approach to identify the vulnerable elements. They provided a min-max
optimization model of the worst-case scenario.
A vulnerability index for each node/link or a set of node/link in a region or
multiple regions could be defined by evaluating the total loss. The vulnerable
infrastructure then could be ranked to improve its robustness against disruptions. The
Network Robustness Index (NRI) was presented in Scott, Novak, and Guo (2005) and
Scott, Novak, Aultman-Hall, and Guo (2006). This index provides a performance
measure to assess the vulnerability of the link or the whole network. The NRI value is
obtained by comparing the total changes before a link removal, to the state before
disruptions. Therefore, the alternative route and the additional cost would be considered
in the model. The Scott et al. (2005) model was further developed by Sullivan, Aultman-
Hall, and Novak (2009). The new robustness index (NRI-m) is similar to the original NRI
index, and the only difference is in the partial capacity reduction of the elements.
Qiang and Nagurney (2007) further improved Latora’s performance indicator by
involving the flow of traffic. They proposed a new unified model to present the network

26
performance measure. Their performance indicator provides importance identification
and the ranking of network components. The model is based on equilibrium demand and
disutility for O/D pairs. The defined efficiency/performance measure incorporates
vulnerability and reliability in their model. A summary of vulnerability indicators is
presented in Table 4.
Table 4
Summary of Vulnerability Indicators
________________________________________________________________________
Reference
Static /
dynamic
Evaluate
level Model formulation and Notations Notes
Latora and
Marchiori (2001,
2004)
Static
(topological)
Network
𝑬(𝑮) =
𝟏
𝒏(𝒏 − 𝟏)
∑
𝟏
𝒅𝒊𝒋
𝒊≠𝒋∈𝑮
E(g): network efficiency of a given network g
𝒏: number of nodes in g
𝒅𝒊𝒋: expresses the shortest path length (the geodesic
distance) between nodes i and j
Based on Graph theoretic
Properties:
 Shortest paths
 Density of edges
Advantages:
 Fast
Disadvantages:
 Does not consider flow
Qiang and
Nagurney (2007)
Dynamic Node, link,
network
𝛆 = 𝛆(𝐆, 𝐝) =
∑
𝒅 𝒘
𝝀 𝒘
𝒘𝝐𝑾
𝒏 𝒘
𝛆(𝐆, 𝐝): network performance/efficiency measure for
a given graph g and the equilibrium demand vector d
𝒏 𝒘: number of o/d pairs in the network
𝒅 𝒘: equilibrium demand for o/d
Pair w
𝝀 𝒘: equilibrium disutility for o/d
Pair w
Optimization Based
Properties:
 Flow
 Shortest paths
 Density of edges
Advantages:
 Unified
Disadvantages:
 Computationally expensive (worst
among this list)
Snelder (2010) Static
(topological)
Link
expandable
to network
𝑽𝒖𝒍𝒏𝒆𝒓𝒂𝒃𝒊𝒍𝒊𝒕𝒚 𝒊𝒏𝒅𝒊𝒄𝒂𝒕𝒐𝒓
𝒇𝒐𝒓 𝒂𝒍𝒕𝒆𝒓𝒏𝒂𝒕𝒊𝒗𝒆 𝒓𝒐𝒖𝒕𝒆𝒔
=
𝒄𝒂𝒑 𝒂
∑ (𝒄𝒂𝒑 𝒂𝒂 ∗ 𝝇 𝒅𝒊𝒔𝒕 𝒂,𝒂𝒂)𝒂∈𝑨 𝒂
𝒄𝒂𝒑 𝒂: link capacity
𝒅𝒊𝒔𝒕 𝒂,𝒂𝒂: shortest distance over the network between
link a and link aa
𝑨 𝒂: 𝒔𝒆𝒕 𝑨 𝒂 is determined by taking a line
perpendicular to link a
𝒄𝒂𝒑 𝒂: link capacity
𝝇: parameters that represent the importance of the
distance from alternative routes
Based on Graph theoretic
Properties:
 Angular degree between available
alternative link(s) (considered as
representation of routes) for each link
 Distance between each link and its
alternative routes
Advantages:
 Fast
 Easy to implement
Disadvantages:
 Does not consider flow

27
Reference
Static /
dynamic
Evaluate
Chen, Yang,
Kongsomsaksakul,
and Lee (2007)
Dynamic Link,
expandable
to network
Properties:
 Considering hierarchy structure for
making decisions
Advantages:
 Combined travel-destination-mode-
route model
 Considering modes of transfer
Murray-Tuite and
Mahmassani
(2004)
Dynamic Link,
expandable
to network 𝑽 𝒂
𝒓,𝒔
=
{
𝟏 , 𝒊𝒇 𝒌 𝒓,𝒔
> 𝑲 𝒓,𝒔
𝟏 − ∑ 𝒈𝒋
𝒓,𝒔 𝑿 𝒂,𝒋
𝒙 𝒂
𝒓,𝒔
𝒌 𝒓,𝒔
𝒋=𝟏
, 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔
𝑽 𝒂
𝒓,𝒔
: vulnerability index for link 𝒂
𝒈𝒋
𝒓,𝒔
: utility of alternative path j 𝒈𝒋
𝒓,𝒔
= (
𝑪𝒋
𝒓,𝒔
𝝆𝒉 𝒋
)(
𝑻𝒋
𝟎
𝝉𝒋
)
𝒌 𝒓,𝒔
: number of alternate paths needed to
accommodate flor from r to s on link l (xlr,s)
𝑪𝒋
𝒓,𝒔
: excess capacity on path j available to r,s
𝝉𝒋: marginal path travel time
𝑻𝒋
𝟎
: free flow path travel time for path j
𝝆: maximum service rate of path
𝒉𝒋: bottleneck link of path j
Optimization Based
Properties:
 Flow
 Alternative routes
 Available capacity on alternative
routes
Advantages:
 Consider alternative path(s), flows and
quality of service on current and
alternative path(s)
Disadvantages:
 Analysis is computationally expensive
Murray et al.
(2007)
Dynamic Set of
nodes, total
interacted
o–d pairs
Structured approaches (optimization-based)
Properties:
Find worst(best) case scenarios of possible
degradations
Advantages:
 Fast
Darren M. Scott et
al. (2005), Darren
M. Scott et al.
(2006)
Dynamic
(indirect)
Link,
expandable
to network
𝒒 𝒂 = 𝒄 𝒂 − 𝒄
𝒄 𝒂 = ∑ 𝒕 𝒂 𝒙 𝒂 𝜹 𝒂
𝒂
𝒄 = ∑ 𝒕 𝒂 𝒙 𝒂
𝒂
𝜹 𝒂 = {
𝟏, 𝒊𝒇 𝒍𝒊𝒏𝒌 𝒂 𝒊𝒔 𝒏𝒐𝒕 𝒕𝒉𝒆 𝒍𝒊𝒏𝒌 𝒓𝒆𝒎𝒐𝒗𝒆𝒅
𝟎, 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆
𝒒 𝒂: the value of the nri for link a
𝒙 𝒂: flow on each link a
𝒕 𝒂: travel time on each link a
𝒕 𝒂: travel time on each link a
𝒄 𝒂: the system-wide, travel-time cost of removing the
link a
𝒄 𝒂: total system, travel-time
Optimization Based
Properties:
 Flow
Advantages:
 Measures the effects to overall users
Disadvantages:
 Computationally expensive
Grubesic, Murray,
and Mefford
(2007)
Dynamic Node Optimization Based
Properties:
 Flow
 Node attribute (in their case,
population impacted)
Advantages:
 Fast
 Direct dynamic vulnerability measure

28
Reference
Static /
dynamic
Evaluate
Disadvantages:
 Quality of weighting approach
Erik Jenelius et al.
(2006)
Dynamic Node, link,
network
∆𝒄𝒊𝒋
(𝒆)
= 𝒄𝒊𝒋
(𝒆)
− 𝒄𝒊𝒋
(𝟎)
𝒖𝒊𝒋
(𝒆)
= {
𝒙𝒊𝒋 𝒊𝒇 𝒄𝒊𝒋
(𝒆)
= ∞
𝟎 𝒊𝒇 𝒄𝒊𝒋
(𝒆)
< ∞
𝑰𝒎𝒑𝒐𝒓𝒕𝒂𝒏𝒄𝒆 𝒏𝒆𝒕(𝒌) =
∑ ∑ (𝒄𝒊𝒋
(𝒌)
− 𝒄𝒊𝒋
(𝟎)
)𝒋≠𝒊𝒊
∑ ∑ (𝒘𝒊𝒋)𝒋≠𝒊𝒊
𝑰𝒎𝒑𝒐𝒓𝒕𝒂𝒏𝒄𝒆 𝒏𝒆𝒕
𝒖𝒏𝒔
(𝒌) =
∑ ∑ (𝒖𝒊𝒋
(𝒌)
)𝒋≠𝒊𝒊
∑ ∑ (𝒙𝒊𝒋)𝒋≠𝒊𝒊
𝒄𝒊𝒋
(𝒆)
: the cost of travel from demand
Node i to demand node j when element 𝒆 has failed
𝒄𝒊𝒋
(𝟎)
: the cost of the initial, undam- aged network
𝒖𝒊𝒋
(𝒆)
: unsatisfied demand
𝒘𝒊𝒋: weight assigned to each od pair that reflects its
significance in relation to the other pairs
Properties:
 Flow
Advantages:
 High-quality analysis
Disadvantages:
 Computationally expensive
Erik Jenelius and
Mattsson (2012)
Dynamic Grids,
network
Optimization Based
Properties:
 Flow
 Size of grids
Advantages:
 Quality
 Adjustable grid size for faster/slower
and lower/higher details of analysis
Disadvantages:
 Lower quality on denser grid, while
having redundant grids on the same
link on rural are
M. Li (2006) Dynamic
µ =
∑ 𝐍𝐋(𝐭)𝒕
∑ 𝐍𝐋∗(𝐭)𝒕
𝑵𝑨𝑺(𝒕) =
∑ 𝐯 𝒂(𝐭)𝐟 𝒂(𝐭)𝒂
∑ 𝐟 𝒂(𝐭)𝒂
µ: robustness indicator (loading multiplier)
𝐍𝐋(𝐭): network load within period t (veh/h),
(𝐍𝐋(𝐭) = ∑ 𝐟 𝐚(𝐭)𝐚
𝐟 𝐚(𝐭): flow of link a
𝐯 𝒂(𝐭): average link speed
Simulation Based
Advantages:
 DTA and SUE
Berdica and
Mattsson (2007)
Dynamic link,
network,
comparing
scenarios
Simulation Based
Properties:
 Flow
 BPR function elements
Advantages:
 Ease and flexibility in defining
various scenarios

29
Reference
Static /
dynamic
Evaluate
Disadvantages:
 Needs to perform simulations
Tampère et al.
(2007)
Dynamic Link,
network
𝑰 𝟏
=
𝒒
𝟏 − 𝒒/𝑪
𝒒: flow
𝑪: capacity
𝑰 𝟏
: shows he influence of the flow.
Tampère et al.
(2007)
Dynamic Link,
network
𝑰 𝟐
=
𝟏
𝑻 𝒃
𝑻 𝒃 = 𝑳𝒊/𝒒𝒊(𝒍𝒊. 𝒌𝒋𝒊 −
𝒒𝒊
𝒗 𝒇 𝒊
)
𝒒: flow
𝑻 𝒃: the time it take before the tail of a queue reaches
the upstream junction. The higher 𝑻 𝒃 is the lower
will be the impact of an blockage
𝑰 𝟐
: indicator showing the impact of an blockage
Simulation Based
Tampère et al.
(2007)
Dynamic Link,
network
𝑰 𝟑
= 𝑰𝒊
𝟏
. 𝝑(𝒒 − 𝟐𝟓𝟎𝟎)
𝝑(𝒙) = {
𝟎 𝒙 < 𝟎
𝟏 𝒙 > 𝟎
𝒒: flow
𝑰 𝟑
: similar to 𝑰 𝟏
however, limited to links with a
capacity of 2500 pcu/hour
Tampère et al.
(2007)
Dynamic Link,
network
𝑰 𝟒
= 𝑰 𝟏
× 𝒒
𝒒: flow
𝑰 𝟒
: similar to 𝑰 𝟏
, aims at expressing the effects of an
incident. Related to the probability that an incident
occurs. In the formulation for 𝑰 𝟒
this probability is
taken proportional with flow q
Tampère et al.
(2007)
Dynamic Link,
network
𝑰 𝟓
= 𝑰𝒊
𝟐
× 𝒒𝒊 × ∑ 𝑰𝒊
𝟏
𝒋∈𝑼 𝒊
𝒒: flow
𝑰 𝟓
: similar to 𝑰 𝟒
, capturing both effects and incident
probability. However, i5 also takes the possible
effect of blocking back into account
Tampère et al.
(2007)
Dynamic Link,
network
𝑰 𝟔
= 𝑰𝒊
𝟑
× 𝒒𝒊 × ∑ 𝑰𝒊
𝟏
𝒋∈𝑼 𝒊
𝒒: flow
𝑰 𝟔
: same as 𝑰 𝟓
, however restricted to lower-capacity
links. (good for e.g. Risk-prone off ramps just
downstream) of a junction.

30
Reference
Static /
dynamic
Evaluate
Tampère et al.
(2007)
Dynamic Link,
network
𝑰 𝟕
= ∑ 𝑰𝒊
𝟏
𝒋∈𝑼 𝒊
𝑰 𝟕
: sum of the effects (estimated by i1) on all
upstream links j of link i, which might be blocked
due to spillback of congestion of a blocking on links
i. This shows the links that cause large problems in
blocking back effects
M. Li (2006) Dynamic Link,
network
𝑽 =
𝒒
𝑪
𝒒: flow
𝑪: capacity
𝑽: captures the links that have a large volume
compared to their capacity. This usually is an
indication that the link is heavily used, and that if an
blockade happens, the queue will grow quickly.
Advantages:
 Tradition indicator for critical links
Tamminga (2005) Dynamic Link,
network
𝑽 = 𝒒𝒊 − 𝑪𝒊
𝒃
𝒒: flow
𝑪: capacity
𝑽: shows rate at which cars arrive in the queue when
an incident occurs on a link and therefore shows the
direct consequences
Robust network design
In the previous sections, the concept of robust network design has been discussed.
The goal is to design reliable and robust networks that are less vulnerable to disruptions.
Robust network design focus is on the reduction of the impact of disruptions in terms of
reliability, vulnerability, and resiliency (Snelder, n.d., 2011). Disruptions could occur in
the travel times, trip rates, capacity, traffic signals, and even the change in direction of a
link. A network is more robust if it can withstand unexpected disruptions. Robust
network design can be categorized based on design approaches:
1. Scenario specific (some manually selected scenarios)
2. Strategy-specific (same as 1, but guided, which means some arc/nodes are more
likely to be disrupted.

31
3. Structured approaches (optimization-based)
Approach 1 and 2 are usually utilized in micro- or mesoscopic simulations based
models. Several studies in literature utilized dynamic traffic assignment (DTA) in robust
network design problem. A framework was presented by (Snelder, n.d., 2011) for robust
network design problem, considering combined route choice, mode choice and trip
distribution in the lower level DTA problem. In the model, the designer has the decision
of adding capacity to link, route or buffer lanes. The vulnerability is considering using a
term in the top-level objective function. This term, comprised from the multiplication of
expected number of incidents, by total system travel time loss, and multiplying by the
value of robustness (weight of robustness in top-level objective function). She assumed
that the probability of an incident is a function that depends on the number of vehicle
kilometers driven. The short-term variation in supply caused by incidents is also included
in the model. They solved the model formulation using the genetic algorithm on several
test networks.
Chiou (2015) presented a model for designing a robust network strategically. He
transformed the bi-level hierarchical problem into a single level. The vector of link
capacity expansion can be optimally determined in a worst-case scenario of travel
demand growth for its equilibrium flow. The lower level UE problem is solved by
parametric variational inequality, and a single level minimax model was provided.
Dziubiński and Goyal (2013) studied various games between a designer and an
adversary. The designer tries to form a network consisting of n links - which are costly to
construct - and protect a set of them against disruptions. On the other hand, the adversary

32
entity is interested in damaging the network to the maximum of its capabilities. Perfect
and imperfect information in different scenarios is assumed available to the designer. The
difference is considered as the knowledge of the designer of the possible moves of the
adversary, which depends on their payoffs. Their main finding was with limited available
resources, the best defense would be in sparse networks, rather than centralized.
Murray-Tuite and Mahmassani (2004) studied four types of games between a
transportation operation manager of a network, and an adversary entity who tries to
damage the network, using bi-level formulation. In their method, the vulnerability index
value is based on the utility of alternative routes, considering the current flow, and ratio
of flow over demand. The utility is based on the ratio of free flow travel time over
marginal travel time and the relative capacity. The utility values range from 0 to 1, where
1 indicates that the link is extremely important to the connectivity of specific O/D route.
Martin (2007) studied various types of network design against attacks and
developed a tri-level defender-attacker defender model to design a robust network, which
the defender in the inner model tries to minimize the users’ costs. The proposed
framework assumes that the defender at the outer level uses limited defensive resources
to protect a system from attacks. At the middle level, the attacker uses their limited
resources to attack the unprotected components while at the inner level the defender
operates the system to minimize operating costs from damage (resulting from the
attacker).
Zhang, Xu, Hong, Wang, & Fei, (2012) tried to utilize the unified performance
indicator defined by Qiang and Nagurney (2007) in order to provide a robust network

33
design. The model is formulated using bi-level optimization. In the upper level, the
designer of the network interested in maximizing the performance of the network,
constraint by his available budget/resources. The lower level problem is user equilibrium.
Maximizing the network performance will result in a more robust network.
Chen, Zhou, Chootinan, Ryu, Yang, and Wong (2011) presented a bi-objective
model that optimized capacity reliability and travel time reliability. These performance
measures give the supply and demand of a roadway network’s reliability. The
minimization of total system travel time is a key objective when using bi-level
optimization in transportation planning. Multiple works have been conducted on this
topic (Ben-Ayed et al., 1988; Gao, Wu, & Sun, 2005; Yang & Bell, 1998).
Melachrinoudis and Kozanidis (2002) presented a mixed integer knapsack solution to
find the optimal set of projects which maximize the total reduction in the expected
number of accidents constrained by a limited budget.
Furthermore, Dziubiński and Goyal (2013a) studied on the various shape of
networks. They considered two players in the game, designer, and adversary. The
designer forms link between a set of defined nodes. The adversary attacks on the nodes
based on his resources. They found out that the best shape of the network in terms of
affordability and reliability, is sparse and heterogeneous, and either fully or centrally
protected (Dziubiński & Goyal, 2013a).
Table 5 presents a list of models for designing robust transportation networks.

34
Table 5
summary of methods for robust transportation network design
Author(s)
Designer/Defender
Users Intelligent
Adversary
note
Decisions Goal Decisions Goal Decisions Goal
Strategic/Tactical/Operational
Variable(s)
C/D/MI1
MaxRobustness
Min.ConstructionCost
MaxConsumerSurplus
Single/Multipleoptimization
SolutionmethodUL
Route
ModeChoice
DestinationChoice
TimedepartureChoice
RouteChoicemodel
SolutionmethodLL
Intelligent/Stochastic/Nodisruptions
Variable(s)
MaxDamage
SolutionmethodAdversary
Snelder (2010) S
link / route /
buffer cap
D ● ● S GA ● ● ● ● DTA
Micro
Simulation
S - -
Murray-Tuite and Mahmassani
(2004)
T/
O
Rerouting C ● S ● SO I
Dis.
link
● BF2
S. Chiou (2015) S
link / route /
buffer cap
C ● S GB ● SUE S - -
Converted to
single level
Dziubiński and Goyal (2013) S adding links D ● ● S BF ● - I
Dis.
link
● BF Topological
Wu, Guo, Sun, and Wang (2014) S link-cap C ● S
PS
O ● DUE N - -
G. Brown, Carlyle, Salmerón, and
Wood (2006)
T Defense a link D ● S BF ●
flow-balance
equilibrium
LP I
Dis.
link
BF
1
Continues, Discrete and Mixed Integer
2
Brute force - Enumeration

35
Summary and Future Research Needs
The works studied in this chapter provide a basis for the research in this
dissertation, which focuses on the development of models for identifying vulnerabilities
within a network, and design the robust networks against intelligent disruptions.
Growing demand has forced the transportation authorities to improve the
performance of transportation networks. They try to find a solution to improve the
existing network under the budget constraint such a way that the social welfare and the
network robustness is maximized while accounting for the equilibrium of the route choice
of the network users. Improving a transportation network, with limited resources, could
be done by considering various network performance measures. The historical
approaches to model these types of problem and the solution methods have been
reviewed and studied. Furthermore, an introduction to game theory and network design
problem based on its concepts was presented. Likewise, the vulnerability indicators in
transportation networks were reviewed. Vulnerabilities in the networks have been
considered in many studies focusing on the initiate of the problem base on stochastic
events. However, considering vulnerabilities due to the intelligent adversary entities’
behavior in network design process should be more studied.
In this study, three decision makers are considered to form the robust network
design models. This type of optimization problem is hard to solve since sets of decision
makers with different objectives are inherently involved. There is no clear urban
transportation model based on the three earlier aforementioned decision makers.
Therefore, attaining models that consider these players into the game forms the basis for

36
future areas of research. In the next chapter, the frameworks to model this problem is
provided.

37
3 METHODOLOGY
Introduction
This chapter first discusses the concept of game theory, followed by introducing
the players that are used in the proposed frameworks. Furthermore, the proposed
framework and the mathematical formulation of the models are presented. To formulate
the models, the sets, parameters, and variables are defined in Table 1 through Table 3.
The notations are similar to model and graph representations in (Sheffi, 1985), and are
adopted for the proposed models. To have a better understanding of the methodology,
some essential information about game theory concepts are presented in the next section.
Game theory
Game theory provides mathematical tools for analyzing situations in which parties
- called players - make independent decisions. A game is defined as a finite game when
each player has a finite number of options, the number of players is finite, and the game
cannot go on indefinitely. It can be defined as the study of mathematical models of
conflict and cooperation between intelligent rational decision-makers. A solution to a
game is the optimal decisions of the players, who may have similar, conflicting, or mixed
interest and the outcomes that may result from these decisions.
Hence, a game is a set of strategies for each player that does depend on other
players’ strategy. If the solution of any player does not depend on other players’ decision,
the problem is not a game. In a game of regular network design problem, the decision of
designer depends on the users of the network, and vice versa.
Games can be classified based on the information available to players:

38
 Perfect information available. A player that has perfect information knows
everything about the moves in the game at all the time. They player with
perfect information may not some information on other players payoff or the
structure of their optimization. An example of this type of games is a game of
chess. If one player is aware of another one, (i.e. human be aware of computer
moves), the human can reduce the final computer score (or improve her
ultimate score), but may not avoid lose/draw.
Imperfect information. Oppose of the previous one. An example is the game
of poker. Each player does not know all of their opponents’ cards. The payoffs
in this game could be represented by money.
Based on the bind between decision of the players in variable-sum games, games
can be categorized into cooperative games and non-cooperative:
 Cooperative games: players in this type of game can communicate and
have bound in their decision.
 Non-cooperative games: players in this game may communicate; however,
they cannot make a binding agreement with their decision.
Furthermore, games can be classified into categories of simultaneous and
sequential, based on order of moves:
 Simultaneous games are games where both players move simultaneously,
or if they do not move simultaneously, the later players are unaware of the
earlier players' actions (making them effectively simultaneous)

39
 Sequential games (or dynamic games) are games where later players have
some knowledge about earlier actions. This need not be perfect
information about every action of earlier players; it might be very little
knowledge.
The mathematical representation of the problem can be provided in two formats:
zero sum game and non-zero sum game. A zero-sum game is a situation in which a gain
or loss in utility of each player is exactly balanced by the losses or gains of other players’
utility. In other words, if the total gains of the players are added up and the total losses
are subtracted, the summation will be zero. On the other hand, in non-zero-sum games,
the summation of losses and gains is not equal to zero. Zero sum games are strictly
competitive, which means there exist some losses associated with each gain. Non-zero
sum games can be competitive or non-competitive. One of the common approaches for
solving zero-sum games is Minimax theorem. In game theory, Minimax is a decision rule
for minimizing the worst-case scenario loss (maximum loss). For the two players finite
zero-sum games, the solution from Minimax, Maximin, and Nash equilibrium are
equivalent. Therefore, in a zero-sum game, the participant’s loss of utility is exactly equal
to the gain of the utility of the other participant, while it is not the case for the non-zero
sum games. Many conventional games are considered in this category. A list of some of
the adversarial zero-sum games is provided in Table 6.

40
Table 6
Examples of adversarial games
Deterministic Stochastic
Perfect information Chess, Checkers, Go, Othello Backgammon, Monopoly
Imperfect information Poker, Bridge, Scrabble
A player, who is playing against a perfect opponent, knows that his opponent is
unpredictable. An adversarial search algorithm should for a search through possible
sequences of moves to find the best next move. However, due to the time limits of most
of running most of the known algorithms, it is usually impractical to consider all the
possibilities. Considering chess game, the initial state is the current board configuration.
Therefore, the operations to move in the search space are the moves of the players on the
board. The final state of search could be won or lost. Exact minimax analysis for the
chess game is infeasible with current solution algorithms and computational power in a
reasonable time. In order to reduce the complexity of minimax search, technics like α-β
pruning are being utilized, which works by pruning portions of the search space. To
evaluate the moves in the search space, a utility function (or payoff function) needs to be
defined. The aim of the utility function is to measure how badly the opponent is beaten.
The quantitative typical values are usually +1 (win) and -1 (lose), -infinity and +infinity
and 0 and 1. However, based on problem specific requirements, other quantitative
payoffs could be defined.
Stratego

41
Dealing with conflicting objectives in optimization problems has always been
challenging for scientists. Multilevel optimization problems are referred to mathematical
programs, which a subset of their variables constrained to an optimal solution of other
programs parameterized by their remaining variables. When the other programs are pure
mathematical programs, the problem is bi-level programming problem (BPP). When the
other programs are bi-level themselves, the problem becomes three level programming.
This notion can be extended to multilevel programs with any number of levels (Vicente
& Calamai, 1994b). The simplest form of BPP in the linear form shown to be NP-Hard
(Bard, 1998; Ben-Ayed et al., 1988; Vicente & Calamai, 1994b). Programs like linear
integer problem, bilinear, quadratic, and minimax programs can be stated as a special
instance of bi-level programs. There have been studies conducted to provide a link
between bi-objective problems and BPP (Bard, 1998; Ünlü, 1987). However, they were
not succeeded in a sense that the optimal solution of a given bi-level program is Pareto
optimal or efficient for both lower and upper-level problems. Game theory approaches
are widely accepted solutions to deal with the conflicting problems. The main problem
can be decomposed into sub-problems, which are the players in the game. In the next
section, a review of the players involved in the proposed frameworks is provided.
Players
The problem and the solutions could be viewed from the aspect of three main
players involved: the designer of the network, the users of it, and the adversary entity.
Frameworks can be defined based on one or more of these players. The usage of any of
the frameworks differs by the involved players and the order of moves.

42
Other elements that can be considered in the frameworks are the non-intelligent
disruptors. Examples of these disruptions are natural disasters and unintended human
mistakes. Natural disaster refers to natural events like earthquakes, tornados, and
flooding. Other stochastic disruptions could be traffic accidents, telecommunication, and
electrical interruption. These type of disruptions could be considered as components in
the frameworks. However, it should be noted that these types of non-intelligent
component, are not optimization models. The goal and decisions of each player are
described in the following sections.
Users
Users are the travelers in the transportation network. They can be defined as
persons, driver-vehicle units or any other modes of transportation. The goal of the user
level problem is to decide upon their route, in order to minimize the individuals’ travel
cost. The decision variables are flows on paths or bushes between the origin and the
destination(s). Users may have a choice of transport modes, and the departure time
choice.
Therefore, the problem at the user level is to assign the trip matrix into the
network using the route choice algorithm. A first mathematical investigation of the
problem has been done by Wardrop (as cited in Sheffi, 1985). He developed the so-called
Wardrop’s first and second principle of equilibrium model that are based on the concept
of Nash equilibrium in game theory. His model denotes that no user can experience a
lower travel time by unilaterally changing routes. In simple terms, the equilibrium is
achieved when the travel cost on all used paths is equal. This principle is behaviorally

43
robust, computationally efficient, and possesses the unique solution. A convenient way to
model the travel time of a roadway link is the Bureau of Public Roads (BPR) travel time
function that is has been used widely to model the static traffic assignment problem. It is
noted that this function is simplistic as it cannot capture the time dimension and the
traffic control characteristics of a roadway link as well as the impact of other adjoining
roadway links on the travel time of the subject roadway link and the interactions among
various classes of vehicles. The BPR travel time function 𝑡 𝑎 specific to a given link 𝑎 is
given by
𝑡 𝑎(𝑥 𝑎) = 𝑡 𝑜 (1 + 𝛼 𝑎 (
𝑥 𝑎
𝐶 𝑎
))
𝛽 𝑎
(1)
Where 𝛼 𝑎 and 𝛽 𝑎 are link specific constants, and 𝑡 𝑜 is the free flow time on
link 𝑎. Generally, the constants 𝛼 𝑎 and 𝛽 𝑎 are calibrated using the observed field data.
One important feature of BPR function is its monotonically increasing convex format.
The nonlinear programming model for the user level problem is provided in
equations (2) through (6):
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑇𝑇 = ∑ ∫ 𝑡 𝑎(𝑤 𝑎, 𝑦𝑎, 𝑧 𝑎)𝑑𝑤
𝑥 𝑎
0𝑎∈𝐴
(2)
Subject to:
∑ 𝑓𝑘
𝑟𝑠
𝑘∈𝒦
= 𝑞 𝑟𝑠 ∀𝑟, 𝑠 (3)

44
𝑥 𝑎 = ∑ ∑ 𝛿 𝑎,𝑘
𝑟𝑠
𝑓𝑘
𝑟𝑠
∀𝑎 ∈ 𝐴
𝑘∈𝒦𝑟−𝑠
(4)
𝑓𝑘
𝑟𝑠
≥ 0 ∀𝑘, 𝑟, 𝑠 (5)
𝑞 𝑟𝑠 ≥ 0 ∀𝑟, 𝑠 (6)
Equation (2) denotes the objective function of the UE problem. Constraint (3)
describes the demand conservation condition. That is to say, all trips should be assigned
to the network. The flow on all routes between each OD pair has to be equal to the OD
trip rate. Constraint (4) outlines the relation between flows on links and route(s), for each
OD. The binary value of 𝛿 𝑎,𝑘
𝑟𝑠
is 1, when link 𝑎 is on the path 𝑘, and it is zero otherwise.
Constraint (5) and (6) satisfies the non-negativity of path flow and travel demand
correspondingly, so the solutions are physically meaningful.
Adversary
Adversary or attacker is the entity who is trying to degrade the performance of the
network. Its objective can be maximizing total system travel cost, the number of travelers
who cannot reach their destinations, and other vulnerability related factors. Examples for
the decision of this player are disabling/reducing the capacity of nodes/links/grids,
manipulate VMSs/DMSs, and signal timing. The adversary entity possibly can increase
his total gain by having some information about the vulnerability of the network.
The degradation of links due to the damages from adversary can be modeled in
various approaches: 1) removing the link(s) from the graph, 2) Adding a big number as a
constant term to the cost of using the disabled link(s), 3) decreasing the capacity of
link(s), 4) increasing the free flow travel time of the link(s). The first approach is not

45
suitable for this problem since it make the graph disconnected. One tactic to deal with
this issue is eliminating the OD pairs that cannot make their trips and associating a cost to
this removal. The second approach does not consider any property of links. The third
approach would conflict with the capacity expansion procedure of the model. The last
approach could provide a better modeling of degraded links.
Designer
This category belongs to conventional transportation NDP. The objective of the
designer (or the defender) can be defined as a single level process to improve the
performance of the network. Performance indicator can be any or combinations of
vulnerability, reliability, system cost or etc. The objective of the designer is related to
economic growth and is about to improve the desired performance measures. For
instance, he may try to lower travel times, provides more reliable travel times, improve
social equity, improve environmental conditions, and improve livability. Other objectives
usually include, but are not limited to the following examples:
• Minimizing cost of construction
• Minimizing total system travel cost
• Minimizing environmental effects (e.g. emission pollution, noise)
• Maximizing indicator(s) of robustness (or similarly minimization of vulnerability)
• Maximizing of the consumer surplus
• Maximizing of spare capacity
• Maximizing the reliability of the network (or minimizing the probability of
overloading the network links)

46
The decision variables available to the designer to achieve the mentioned
objectives could be:
• Capacity
• New links
• Transit schedules
• Number of lanes
• Toll pricing
• Signal timing
• Safety related components
• Protection related components
• Maintenance
3.3.3.1 Conflicting problems of Designer and Adversary
In the games with asynchronous moves like Stackelberg games, each player has
different benefits from the hierarchy structure of the optimizations. For example, in the
Stackelberg game with two players, the follower player that moves after the leader, has
the advantage of having perfect information regards the leader’s moves. On the other
hand, the first player has the benefit of implicitly control the next player’s move, such a
way that optimize his/her own goal. In this case, the evil entity has the advantage of
moving after the designer move. The decisions by the designer already are in place, and
at the time of adversary’s move, the designer cannot take any more actions. Hence,
adversary entity has a perfect information about the designer’s move, while the designer
does not. The objective can be defined in different approaches. For instance, it can be a

47
maximization of the average travel time, minimizing the connectivity, maiming the total
system travel time of a specific region or etc. However, a more realistic objective is to
maximize the total travel time of the whole network.
As it mentioned previously, three players are involved in the proposed game
method. The designer of the network does not have perfect information about the possible
moves of the adversary. Therefore, an appropriate model should search over the possible
moves of the adversary, and try to alleviate them.
The first move is completed by the designer of the network who has the advantage
of putting his decision in place, and observing the reaction of the other players. The
designer decision is defined by vector y. The value of 𝑦𝑎 shows the amount of expanding
the capacity of link a. In this research, 𝑦𝑎 is the number of lanes to be added to link a. In
the proposed model, it is assumed that the adversary entity finds the maximum possible
damage to the network. His decision in the model is defined as vector z. The value of
𝑧 𝑎 = 1 shows the state to which link 𝑎 is damged and not available to the users;
otherwise, the link is not affected. The damage is evaluated as the increase in the total
system travel time. After the decisions of the designer and the adversary were made, the
users of the network complete the next move. The reaction of the users is modeled using
user equilibrium principles. The bi-level formulation models the relationship between the
network manipulated by designer and adversary at the upper level, and the users at the
lower level problem. These problems are described in details in the rest of this chapter.
As discussed in chapter 2, considering the vulnerabilities of intelligent disruptions
in the process of network design is crucial to alleviate the consequences of such events.

48
Furthermore, a selection of vulnerability indicators was presented. As it was studied in
the literature, the vulnerability can be evaluated by measuring the increase in the Total
System Travel Time (TSTT). This way, the damage due to the disturbance is evaluated
over the whole system. One approach to deal with the robust design of the network would
be modeling the problem as a bi-level formulation. Therefore, the solutions of NDP can
be compared to the increase in their TSTT, after the disruption occurs; a network that its
TSTT has less variation would be considered more robust. Hence, the goal of the NDP is
defined as to reduce the maximum possible damages that the adversary entity can imply.
From the designer side, the model formulated as a min-max optimization, where the
designer tries to minimize the maximum damages which enemy entity would put on the
network.
At designer level, the objective is to maximize the robustness (similarly minimize
At designer level, the objective is to minimize the vulnerability of the network, by
investing the available budget/resources in the expansion of the current capacity of links,
by adding new lanes. Therefore, the designer makes his decision by adding new lanes to
the network, considering his budget as a constraint. Then, the model examines the
maximum damage that an adversary can inflict on the network, by incapacitating the
links. Again, the model considers the limitation on adversary’s available
resources/budget. Hence, the goal at upper level can be modeled as equation (7).
𝑼𝑳: min
𝑦
(max
𝑧
𝐷 𝑦,𝑧) (7)

49
where 𝐷 𝑦,𝑧 represents the total payoff to the adversary. The value of the payoff
can be considered as the increase in total system travel time. Therefore, the adversary can
look for the damage which results in the maximum possible travel time of users of the
system. In this case, the objective at the upper level may be written as:
𝐷(𝑥 𝑎(𝑦, 𝑧), 𝑦𝑎, 𝑧 𝑎) = ∑ 𝑥 𝑎(𝑦, 𝑧). 𝑡 𝑎(𝑥 𝑎(𝑦, 𝑧), 𝑦𝑎, 𝑧 𝑎)
𝑎 ∈𝐴
(8)
The decision of the designer and adversary are constrained by the following
limits:
∑ 𝑔 𝑎(𝑦𝑎) ≤ 𝐵 𝑑
𝑎∈𝐴
(9)
∑ 𝑧 𝑎 ≤ 𝐵𝑧
𝑎∈𝐴
(10)
𝑔 𝑎(𝑦𝑎) = 𝑦𝑎. 𝑑 𝑎 , ∀𝑎 ∈ 𝐴 (11)
Where 𝐵 𝑑 and 𝐵𝑧 respectively represent the budget available to designer and
adversary, 𝑔 𝑎 is the total cost of adding 𝑦𝑎 lanes to link 𝑎, and 𝑑 𝑎 is the cost of
constructing one lane for link 𝑎 (eq. (11)). Finally, constraint (12) and (13) respectively
requires non-negativity of designer’s decision, and the binary decision of the adversary
entity:

50
𝑦𝑎 ≥ 0, ∀𝑎 ∈ 𝐴 (12)
𝑧 𝑎 = {0,1}: ∀𝑎 ∈ 𝐴 (13)
where 𝑧 𝑎 = 1 shows that link 𝑎 is disabled, and 𝑧 𝑎 = 0 indicates that link 𝑎 is not
affected. The result of the upper-level of bi-level model is available to the users. The
users’ move, is done after the first two players find their decisions, and passed it to the
user level. In the next section, the behavior of users in reaction to the decisions made at
upper-level is discussed.
Frameworks for vulnerability/robustness
In the proposed model, the infrastructure that is selected for investment by the
designer might also be selected by the adversary. The model does not prevent the
designer from investing in these type of infrastructures that are attractive to the adversary
entity. One question may arise concerning why investing in a component, which the
investment also makes it more attractive to the adversary.
A short answer can be provided by a simple example. Figure 4a shows a basic
network of two links, connecting the same origin to a destination. There is a total flow of
100 units, which at the equilibrium, 70 units take link 1 and 30 units take link 2.If the
decision of the designer is to increase the capacity of the links, he has two options:
scenario 1 to invest on link 1, and scenario 2 to invest on link 2. If the investment goes to
link 1, it will be more attractive for the current users, which will result in diverting more
traffic to link 1. If the adversary wants to damage this network by disabling one of the
links, the worst-case scenario would be attacking link 1, since it carries the mainstream

51
flow. Hence, the potential damage to the network will be even higher than the original
network since link 1 carries more traffic (Figure 4b).
a) Base network
b) Scenario 1: Improvement on link 1 c) Scenario 2: Improvement on link 2
Figure 4. Example of investing and getting attack on the same link
On the other hand, if the designer decided to invest on link 2, the flow will be
distributed more uniformly over the network, which decreases the total possible damage
(Figure 4c). In this case, if the adversary decides to disable link 2, the maximum flow that
he can affect is 60. On the contrary, if he decides to disable link 1, he will be able to gain
more than the payoff from disabling link 2. A similar pattern could exist in networks that
are more complex. By comparing the two scenarios, it can be concluded that even if the
1
2
1
2
1
2

THE ROBUST URBAN TRANSPORTATION NETWORK DESIGN PROBLEM

THE ROBUST URBAN TRANSPORTATION NETWORK DESIGN PROBLEM

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