ROBUST AND DYNAMIC MODELS FOR SUPPLY CHAIN AND ...
143
The Pennsylvania State University The Graduate School College of Engineering ROBUST AND DYNAMIC MODELS FOR SUPPLY CHAIN AND TRANSPORTATION NETWORKS A Dissertation in Industrial Engineering by Byung Do Chung 2010 Byung Do Chung Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2010
Transcript of ROBUST AND DYNAMIC MODELS FOR SUPPLY CHAIN AND ...
The Pennsylvania State University
The Graduate School
College of Engineering
ROBUST AND DYNAMIC MODELS FOR SUPPLY CHAIN
AND TRANSPORTATION NETWORKS
A Dissertation in
Industrial Engineering
by
Byung Do Chung
2010 Byung Do Chung
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
December 2010
The dissertation of Byung Do Chung was reviewed and approved* by the following:
Tao Yao
Assistant Professor of Industrial Engineering
Dissertation Co-Advisor
Co-Chair of Committee
Terry L. Friesz
Harold & Inge Marcus Chaired Professor of Industrial Engineering
Dissertation Co-Advisor
Co-Chair of Committee
Soundar R.T. Kumara
Allen E. Pearce/Allen M. Pearce Chaired Professor of Industrial Engineering
Venky Shankar
Associate Professor of Civil Engineering
Paul Griffin
Peter and Angela Dal Pezzo Department Head Chair
Head of the Industrial and Manufacturing Engineering
*Signatures are on file in the Graduate School
iii
ABSTRACT
This dissertation considers operation and planning issues of dynamic supply chain and
transportation networks in an uncertain environment. In particular, robust optimization
approaches are applied to 1) emergency logistics planning, 2) network design and 3) congestion
pricing problems under demand uncertainty residing in an appropriate uncertainty set such as
box or polyhedral uncertainty set.
First of all, we develop a robust linear programming model of the cell transmission model
(CTM) based on a robust optimization approach. Then, an affinely adjustable robust linear
programming model is derived to study the multi-period problem. As an application area, we
propose a methodology to generate a robust logistics plan that can mitigate demand uncertainty in
humanitarian relief supply chains using a CTM based system optimum dynamic traffic
assignment (SO DTA) model. Next, the proposed framework for SO DTA is extended to a
dynamic network design problem. Finally, we consider robust congestion pricing problems under
user equilibrium in static networks and extend it to consider robust dynamic user equilibrium
optimal toll, which is formulated as a differential mathematical program with equilibrium
constraints (DMPEC). Also, a cutting plane algorithm and a simulated annealing algorithm are
proposed to solve the DMPEC problems.
Theoretically, the tractability and conservativeness of robust counterparts are discussed.
Also, numerical experiments show that the robust optimization approach leads to high quality
solutions compared to the deterministic problem or the sampling based stochastic problem. The
results of the numerical experiments justify the modeling advantage of the robust optimization
approach and provide useful managerial insights, which may have wider applicability in supply
chain and transportation networks.
iv
TABLE OF CONTENTS
LIST OF FIGURES ................................................................................................................. vi
LIST OF TABLES ................................................................................................................... viii
ACKNOWLEDGEMENTS ..................................................................................................... x
Chapter 1 Introduction ............................................................................................................ 1
1.1 Robust Optimization .................................................................................................. 4
1.2 Dynamic Traffic Assignment ..................................................................................... 6
1.3 Emergency Logistics Planning ................................................................................... 8
1.4 Dymaic Network Design Problem ............................................................................. 10
1.5 Dynamic Congestion Pricing Problem ....................................................................... 12
Chapter 2 Robust Optimization Model for CTM .................................................................... 16
2.1 CTM for DTA problem .............................................................................................. 17
2.2 RC of the CTM .......................................................................................................... 22
2.3 RC with Inequality Relaxation ................................................................................... 31
Chapter 3 Affinely Adjustable Robust Optimization Model for CTM ................................... 34
3.1 The RO Approach for Multi-period Problems ........................................................... 34
3.2 AARC with Box Uncertainty Set ............................................................................... 37
3.3 AARC with Polyhedral Uncertainty Set .................................................................... 39
Chapter 4 Emergency Logistics Planning ............................................................................... 44
4.1 Demand Modeling ...................................................................................................... 45
4.2 Small Network Example ............................................................................................ 47
4.2.1 RC vs. DLP ..................................................................................................... 50
4.2.2 AARC vs. DLP ................................................................................................ 54
4.2.3 AARC vs. Sampling based Stochastic Programming...................................... 56
4.3 Cape May County Network Example ........................................................................ 59
Chapter 5 Robust Dynamic Network Design Problem ........................................................... 62
5.1 Deterministic Model .................................................................................................. 65
5.2 Robust Fomulation ..................................................................................................... 67
5.3 Numerical Analysis .................................................................................................... 70
5.3.1 A Toy Network................................................................................................ 71
5.3.1.1 Optimal solution under Different Uncertainty Levels .......................... 72
5.3.1.2 Worst Case Analysis............................................................................. 74
5.3.1.3 Simulation Results ................................................................................ 77
5.3.2 The Nauyen-Dupis Network ........................................................................... 79
v
Chapter 6 Robust Congestion Pricing Problem ...................................................................... 84
6.1 Motivation .................................................................................................................. 85
6.2 Robust Congestion Pricing for Static Traffic Networks ............................................ 88
6.2.1 Deterministic Problem ..................................................................................... 88
6.2.2 RC of MPEC ................................................................................................... 90
6.3 Robust Congestion Pricing for Dynamic Traffic Networks ....................................... 94
6.3.1 Dynamic Network Loading ............................................................................. 94
6.3.1.1 The Arc Delay Model ........................................................................... 95
6.3.1.2 The DAE System .................................................................................. 95
6.3.1.3 A Simplified Network Loading Procedure ........................................... 96
6.3.1.4 Constructing the Path Delay for a Given kh ........................................ 97
6.3.2 Robust Dynamic Congestion Pricing Formulation ......................................... 98
6.3.2.1 Dynamic Optimal Toll Problem with Equilbrium Constraints ............. 100
6.3.2.2 Robust DOTPEC Problem .................................................................... 103
6.3.2.3 Cutting Plane Algorithm ....................................................................... 105
6.3.2.4 Simulated Annealing Algorithm ........................................................... 107
6.4 Numerical Experiments .............................................................................................. 110
6.4.1 Static Two-route Network ............................................................................... 110
6.4.2 Static Braess Network ..................................................................................... 112
6.4.3 Dynamic Three-arc Four-node Network ......................................................... 115
Chapter 7 Conclusion .............................................................................................................. 118
References .............................................................................................................................. 122
vi
LIST OF FIGURES
Figure 4-1: Three response curves (Fu et al. 2007). ................................................................ 46
Figure 4-2: S-curve - box uncertainty. ..................................................................................... 47
Figure 4-3: S-curve - polyhedral uncertainty. .......................................................................... 47
Figure 4-4: Example network (Chiu et al. 2007). .................................................................... 48
Figure 4-5: Consequence of data uncertainty for nominal solution. ........................................ 51
Figure 4-6: Relative performance of robust solution. .............................................................. 53
Figure 4-7: Cape May county evacuation network (Yazici and Ozbay, 2007). ...................... 60
Figure 5-1: Cell representation of the toy network (Ukkusuri and Waller 2008). ................... 72
Figure 5-2: The objective-budget relationship under different demand uncertainty levels ..... 73
Figure 5-3: Optimal investment distributions over the network. ............................................. 74
Figure 5-4: Relative improvement of travel cost in worst-case scenarios under different
demand uncertainty levels. ............................................................................................... 76
Figure 5-5: Relative improvement of travel cost in worst-case scenarios under different
investment budget levels. ................................................................................................. 77
Figure 5-6: The node-link topology of the Nguyen-Dupis network. ....................................... 80
Figure 5-7: The objective-budget relationship under different demand uncertainty levels. .... 80
Figure 5-8: Relative improvement of travel cost in worst-case scenarios under different
demand uncertainty levels. ............................................................................................... 81
Figure 5-9: Relative improvement of travel cost in worst-case scenarios under different
investment budget levels. ................................................................................................. 81
Figure 6-1: Two-route network. ............................................................................................... 86
Figure 6-2: Braess network ...................................................................................................... 113
Figure 6-3: 3-arc 4-node network. ........................................................................................... 115
Figure 6-4: Toll price on arc 2. ................................................................................................ 116
vii
Figure 6-5: Departure rate and tolled unit travel cost for path 1. ............................................. 116
Figure 6-6: Departure rate and tolled unit travel cost for path 2. ............................................. 117
viii
LIST OF TABLES
Table 1-1: Notations. ............................................................................................................... 18
Table 4-1: Time invariant cell properties. ................................................................................ 49
Table 4-2: Time dependent data. ............................................................................................. 49
Table 4-3: Degradation of nominal solution under uncertain demand. ................................... 51
Table 4-4: Improvement of robust solution relative to the nominal solution........................... 53
Table 4-5: Objective value – polyhedral uncertainty. .............................................................. 55
Table 4-6: AARC vs. SP when changes (Beta(5,2), L =50, M =100). ............................... 58
Table 4-7: AARC vs. SP when changes (Beta(1,1), L =50, M =100). ............................... 58
Table 4-8: AARC vs. SP when M changes (Beta(5,2), L =50, =0.1). ................................ 59
Table 4-9: Cell properties. ....................................................................................................... 60
Table 4-10: Objective value – polyhedral uncertainty. ............................................................ 61
Table 4-11: AARC vs. SP when changes (Beta(5,2), L =50, M =100).............................. 61
Table 5-1: Notations. ............................................................................................................... 65
Table 5-2: Cell characteristics of the toy network (Ukkusuri and Waller 2008). .................... 72
Table 5-3: Travel cost of robust and nominal solutions in worst-case scenarios. .................... 76
Table 5-4: Comparison of simulation results. .......................................................................... 78
Table 5-5: Comparison of the robust optimization results and simulation results. .................. 82
Table 6-1: Unit arc cost............................................................................................................ 86
Table 6-2: Solutions form deterministic MPECs. .................................................................... 87
Table 6-3: Realized revenue. ................................................................................................... 87
Table 6-4: Notations. ............................................................................................................... 88
Table 6-5: Objective value and optimal toll. ............................................................................ 111
ix
Table 6-6: Simulation results. .................................................................................................. 112
Table 6-7: Unit arc cost............................................................................................................ 113
Table 6-8: Objective value and optimal toll. ............................................................................ 114
Table 6-9: Simulation results. .................................................................................................. 114
Table 6-10: Simulation results. ................................................................................................ 117
x
ACKNOWLEDGEMENTS
It is a pleasure to thank those who made this dissertation possible. First of all, I would
like to express my gratitude to my advisors, Prof. Tao Yao and Prof. Terry L. Friesz for their
guidance, support and supervision. I have benefited not only from their knowledge and
experience but also from the dedication and kindness they have shown. I wish to specially thank
Prof. Yao for deep understanding and valuable comments. He encouraged me to find a solution
whenever I faced difficulties. I would also like to thank Prof. Soundar R.T. Kumara and Prof.
Venky Shankar for their support in my dissertation.
Next, I would to thank Prof. Aharon Ben-Tal and Dr. Chi Xie for their insightful
suggestions on my work. I am grateful to Supreet R. Mandala, Andreas Thorsen and Taeil Kim
for the active collaboration and great discussion. I also want to thank Bo Zhang, Sai Zhang,
Xiaohuang Wu and PSUIEKSA members who always support me.
Last but not least, I am forever indebted to my parents so much for love and
encouragement they have given me. I thank my wife, Hae Young Kim, for always believing in
me and respecting my decision. Many thanks go to parents-in-law, my sons and my friends.
Chapter 1
Introduction
An increasing number of researchers and practitioners in several fields are
concerned with multi-period problems as well as information uncertainty. In the field of
supply chain and transportation, it includes decision making problems on network
capacity design and commodity (or traffic) flow planning and control. Also, these
problems usually face uncertain information in parameters such as demand, capacity, etc.
So far, stochastic programming models have been widely applied to supply chain and
transportation networks to find expected minimum cost.
In this thesis, we apply robust optimization (RO) to consider operation and
planning issues of dynamic supply chain and transportation networks in an uncertain
environment. In particular, RO approaches are applied to 1) emergency logistics planning
2) network design and 3) congestion pricing problems under demand uncertainty residing
in an appropriate uncertainty set such as box or polyhedral uncertainty set.
RO approach has been developed to deal with linear programming (LP) or conic-
quadratic problems (CQP) using crude uncertainty with hard constraints. It means that
uncertainty is assumed to reside in an appropriate set and RO guarantees the feasibility of
the solution within the prescribed uncertainty set by adopting a min-max approach. In
contrast, traditional approaches such as stochastic programming and dynamic
programming require the probability distribution for the underlying uncertain data to
obtain expected minimum cost of the objective function. However, in many cases, it is
2
very difficult to accurately identify the distribution required to solve a problem. In
addition, constraints of the problems may be violated. The RO technique can overcome
such limitations and has been successfully applied in engineering design and optimization
problems similar as robust control in control theory. (Ben-Tal and Nemirovski 1999,
2002)
The main contributions of this thesis are summarized as follows:
In Chapter 2 and 3, we develop a robust optimization framework for a system
optimum dynamic traffic assignment (SO DTA) problem. The framework
incorporates a linear programming (LP) formulation based on the Cell
Transmission Model (CTM) (Daganzo 1994, 1995; Ziliaskopoulos 2000). Robust
counterpart (RC) and Affinely Adjustable Robust Counterpart (AARC) are
formulated as LP problems and computationally tractable. Also, the
characteristics of a robust solution are analyzed by comparing RC and AARC in a
box uncertainty set as well as a polyhedral uncertainty set.
In Chapter 4, the proposed RO framework is applied to an emergency response
and logistics planning problem. Numerical examples are provided to illustrate the
value of the RO in the context of emergency logistics and demonstrate the
computational viability of the developed framework. Simulation experiments
show that the AARC solution provides excellent results when compared to the
solutions from deterministic LP and Monte Carlo sampling based stochastic
programming.
In Chapter 5, RC of the SO DTA problem is extended to a robust dynamic
network design problem (RDNDP). For simplicity, we present our RO model only
3
for single-destination, system-optimal networks. However, the basic RO
counterpart formulation method can be readily transferred to the multi-destination
problem case. This work adds to the body of knowledge in the dynamic network
design by presenting an emerging method related to the solution robustness. The
numerical analysis for the impact of the investment budget bound and the demand
uncertainty level on network design solutions justifies the solution robustness.
In Chapter 6, we consider a robust optimization approach to user equilibrium
optimal toll problems in static and dynamic transportation networks. First, the
properties of the static robust congestion pricing problem are shown. Next,
dynamic congestion pricing problem is formulated as a differential mathematical
program with equilibrium constraints (DMPEC) incorporating dynamic user
equilibrium and approximated network loading by using the second order Taylor
expansion. Finally, a cutting plane algorithm and a simulated annealing algorithm
are proposed to solve the DMPEC problems.
This thesis obtains some general insights that may have wider applicability for
supply chain and transportation managers: 1) A robust solution may improve both
feasibility and performance when infeasibility costs are significant. Intuitively,
the usual nominal optimal solution may be not far from the robust solution, but
the usual optimal solution can perform much worse in the worst case. 2) An
integration of RO and transportation modeling will improve the generation,
communication, and potential use of uncertainty data in logistics and
transportation management. The intuition for this insight is twofold. First, in
many application areas, the set-based uncertainty (used by RO) is an appropriate
4
notion of data uncertainty. Second, computational tractability (resulting from this
set-based uncertainty and dynamic traffic flow modeling in LP formulations)
allows for identification of efficient solutions for logistics and transportation
management under uncertainty.
1.1 Robust Optimization
The idea of RO is not new, as Soyster (1973) first studied it. His paper considered
a linear program where the column vectors from the constraint coefficient matrix are
within prescribed convex sets. Unfortunately, the column-wise uncertainty case is
extremely conservative which means that too much optimality has been traded off to
guarantee robustness. The issue of robustness was relatively silent in the optimization
community until the recent works of Ben-Tal and Nemirovski (1998, 1999, 2000),
Ghaoui et al. (1997, 2003) and Bertsimas and Sim (2003, 2004). These papers make a
significant step forward and propose less conservative models by considering tractable
robust counterparts for nominal problems (Ben-Tal and Nemirovski, 2002) . These
works, with the development of efficient interior point algorithms for convex
optimization and improvements in computation technology, have provided computational
tractability for RO in both theory and practice, and hence have reinvigorated a sudden
burst of interest in the RO field. For applications of RO to the problems of
transportation systems, refer to Ordonez and Zhao (2007), Atamturk and Zhang (2007),
Mudchanatongsuk et al. (2008) and Erera et al. (2009) . Also, refer to Ben-Tal et al.
(2004, 2005) and Bertsimas and Thiele (2004, 2006) for supply chain applications.
5
The underlying assumption of RO is “here and now” decisions and all decision
variables are determined before any uncertain data are realized. However, in multi-period
problems, “wait and see” decisions can be made, which means some decision variables
are affected by part of the realized data. Ben-Tal et al. (2004, 2005) have extended the
RO approach and developed an affinely adjustable robust optimization (AARO) approach
to consider “wait and see” decisions. The new approach provided excellent results in a
multi-period inventory problem and a retailer-supplier flexible commitment problem.
Recently, two-stage robust optimization approaches have been proposed for network
problems. Atamturk and Zhang (2007) considered a two-stage robust optimization
problem for network flow and design. Erera et al. (2009) dealt with integer programming
problem for repositioning empty transportation resources. In both of the above studies,
the time-space network is decoupled into two steps. Decision variables of the first step
are decided before the realization of uncertainty, and then second step variables are
determined as recourse or recovery actions while maintaining feasibility after the
uncertain data is known.
Even though the affinely adjustable robust counterpart (AARC) of an original
deterministic problem is more flexible and gives better results than robust counterpart
(RC), there are some limitations. Most of all, AARC approximates the solution with
linear decision rules in order to maintain computational tractability,so we cannot identify
whether the rule is close to optimal or not. Also, AARC can only be used for finite-
horizon linear programming with exogenous uncertainty. It is not applicable if decision
variables recursively effect the uncertainty or if the problem cannot be formulated using
linear expressions. For example, AARC is not suitable for the transportation problem
6
where the uncertain demands for origin-destination (OD) pairs are influenced by the
decision variables on traffic flow. However, it is a proper approach when a decision
maker seeks an efficient robust solution guaranteeing feasibility without knowing the
distribution of underlying uncertainty.
1.2 Dynamic Traffic Assignment
The dynamic traffic assignment problem describes a traffic system with time-
varying flow. The problem has evolved and been modeled by numerous analytical and
heuristic approaches. The research can be classified into four main categories:
mathematical programming, optimal control, variational inequality and simulation-based
approaches (see Peeta and Ziliaskopoulos (2001) for a review).
In the DTA literature many studies use link performance (for example, the Bureau
of Public Roads approach) to propagate traffic. Such functions often tend to overestimate
the time required to travel as they are convex functions of flow on the links. An attractive
alternative to using link performance functions is the Cell Transmission Model (CTM).
This model was originally proposed by Daganzo (1994, 1995) to simulate traffic flow
based on hydrodynamic flow. The transportation network is decomposed into cells whose
length corresponds to the maximum distance that can be traveled in a unit time and given
speed. The direction of traffic flow is represented by the connectors. Based on the CTM
model, Ziliaskopoulos (2000) formulated the single destination SO DTA problem as a LP
by reducing difference equations to linear relationships. Li et al. (2003) proposed an
effective decomposition scheme to reduce computational complexity.
7
Recently, the CTM based SO DTA model has been applied to Supply Chain
Management (SCM) and disaster management. Kalafatas and Peeta (2009) showed the
conceptual equivalence between SCM and the CTM based SO DTA problem. In the
disaster management, Chiu et al. (2007) proposed a network transformation and demand
modeling technique for solving the evacuation traffic assignment planning problem using
the CTM based SO DTA model. The main difference from the original CTM model is
that time dependent OD demand is not input data but a solution of the model. Chiu and
Zheng (2007) extended the evacuation problem by considering multi-priority groups.
Yazici and Ozbay (2007) introduced probabilistic capacity constraints and solved the
CTM based SO DTA problem for a hurricane evacuation setting. Tuydes (2005)
proposed CTM with capacity reversibility to include the temporary capacity design
problem by reversing direction and exchange capacity under disaster conditions.
One thing to notice is that most research in DTA has assumed deterministic input
parameters. This is surprising because demand uncertainty, capacity reductions and
implementation errors of the optimal solution may have a drastic impact on the optimality
and even the feasibility of the solution. Waller et al. (2001) and Waller and
Ziliaskopoulos (2006) addressed the impact of demand uncertainty and the importance of
robust solutions. However, there is limited research on transportation management under
uncertainty. To deal with the uncertainty, Peeta and Zhou (1999) used Monte Carlo
simulation to compute a robust initial solution for the real-time online traffic management
problem. Chance constraint programming for the SO DTA problem was developed by
Waller and Ziliaskopoulos (2006), in which the chance constraint with known cumulative
distribution function is reformulated as an equivalent deterministic constraint for
8
describing uncertain demand. Karoonsoontawong and Waller (2007) proposed a DTA
based network design problem formulated as a two stage stochastic programming
problem and a scenario-based robust optimization problem. Ukkusuri and Waller (2008)
proposed a two stage stochastic programming model with recourse to take into account
demand uncertainty.
However, as mentioned before, the limitations of prior research is the assumption
that demand arise from a known distribution. Also, a sampling-based Monte Carlo
simulation or scenario-based robust optimization may be prohibitively expensive due to
the requirement for large samples for statistical significance. This thesis, in contrast,
considers crude uncertainty set without requiring exact information such as mean and
standard deviation on the distribution of uncertain demand. We study RC and AARC of a
SO DTA problem with various uncertainty sets in Chapter 2 and 3, respectively.
1.3 Emergency Logistics Planning
Over the past three decades, the number of reported disasters has risen threefold.
Roughly, 5 billion people have been affected by disasters with estimated damages of
about 1.28 trillion dollars (Guha-Sapir et al. 2004). Although most of these disasters
could not have been avoided, significant improvements in death counts and reported
property losses could have been made by efficient distribution of supplies. The supplies
here could mean personnel, medicine and food which are critical in emergency situations.
The supply chains involved in providing emergency services in the wake of a disaster are
referred to as Humanitarian Relief Supply Chains. Humanitarian Relief supply chains are
9
formed within a short time period after a disaster with governments and NGO’s being the
major drivers of the supply chain. Clearly, emergency logistics is an important
component of humanitarian relief supply chains.
Most literature in emergency logistics focuses on generating transportation plans
for rapid dissemination of medical supplies inbound to the disaster hit region (Sheu 2007,
Ozdamar et al. 2004, Lodree Jr and Taskin 2008). There is, however, another aspect of
emergency logistics which is often ignored - outbound logistics. The outbound logistics
considers a situation where people and emergency supplies (e.g. medical facilities and
services for special need evacuees) need to be sent from a particular location affected by
disaster within a given time horizon.
In the outbound emergency logistics, the demand of traffic flows is usually highly
uncertain and depends on a number of factors including the nature of the disaster (natural/
man-made) and time of impact. This uncertainty in the demand causes disruptions in
emergency logistics and hence disruptions in humanitarian relief supply chains leading to
severe sub-optimality or even infeasibility which may ultimately lead to loss of life and
property. In order to mitigate the risk of uncertain demand, we study the problem of
generating evacuation transportation plans which are robust to uncertainty in outgoing
demand. More specifically, in Chapter 4, we solve a dynamic (multi-period) emergency
response and evacuation traffic assignment problem based on SO DTA with uncertain
demand at source nodes.
1.4 Dynamic Network Design Problem
Network design consists of a broad spectrum of problems, each corresponding to
different sets of objectives, decision variables and resource constraints, implying different
behavioral and system assumptions, and possessing varying data requirements and
capabilities in terms of representing network supplies and demands. Network design
models have been extensively used as various types of strategic, tactical and operational
decision-making tools and spanned over a variety of applications in, for example,
transportation, production, distribution, and communication fields. In a transportation
network, traffic congestion has long been a major concern of the network operator, which
occurs when traffic volumes exceed the road capacity. Network design problems (NDP)
for transportation networks in general aim at minimizing network traffic congestions (or
minimizing some general network-wide traveler costs) by implementing an optimal
capacity expansion policy in the network.
An optimal capacity expansion policy, however, may not be reached without
properly considering the behavioral nature of travel demands, which are inherently time-
variant and uncertain. Travel demands are an aggregate result of individual travel
activities, which are determined by various observed and unobserved socioeconomic
factors and subject to geographical, technological and temporal constraints. The vast
body of the literature has focused on static deterministic NDPs (see, for example,
Magnanti and Wong 1984; Minoux 1989; Yang and Bell 1998). A major limitation of
static network design models is the inability to capture traffic dynamics, such as traffic
11
shockwave propagation and the build-up and avoidance of queues. Dynamic models, on
the other hand, allow us to model the time-dependent variation of traffic flows and travel
behaviors and hence better describe traffic evolution and interaction phenomena over the
network (Peeta and Ziliaskopoulos 2001). Travel demand uncertainty is not only the
underlying characteristic of travel activities but also a likely result of our inaccurate or
inconsistent travel demand estimation procedures. Without explicit and rigorous
recognition of uncertainty in travel demands, any transportation network development
plans and policies may take on unnecessary risk and even result in misleading outcomes
(Zhao and Kockelman 2002).
In terms of their mathematical functional forms, dynamic traffic assignment
(DTA) based NDPs can be classified into two major groups: single-level models and bi-
level models (see the discussion in Lin et al. 2008). The focus of Chapter 5 is on an
application of robust optimization (RO) for dynamic NDPs under demand uncertainty, or
more succinctly, a robust dynamic NDP (RDNDP), which has a single-level structure.
The single-level structure provides an easier way to manipulate robust counterpart and
make RDNDP computationally tractable.
The research community has observed a number of recent network design studies
that explicitly incorporate demand uncertainty into NDPs with time-varying flows (see
Waller and Ziliaskopoulos 2001; Karpoonsoontawong and Waller 2007; Ukkusuri and
Waller 2008; Karoonsoontawong and Waller 2008). The common feature of these
problems is that time-varying flows are described by the cell transmission model (CTM)
(Daganzo 1994, 1995) and the network flow pattern is then characterized by CTM-based
DTA methods, under either the system-optimal (Ziliaskopoulos 2000) or user-optimal
12
assignment mechanism (Ukkusuri and Waller 2008). The demand uncertainty of these
problems is accommodated by a chance constraint setting, a two-stage recourse model, or
a scenario-based simulation method. These techniques, however, suffer from
deficiencies related to lack of data availability and problem tractability, which limit their
applicability to a broad range of applications. Resulting models from these stochastic
modeling methods are often computationally intractable and require known probability
distributions.
In Chapter 5, we follow a similar fashion to form our RDNDP using the CTM-
based system-optimal DTA model, but employ the RO approach to account for demand
uncertainty. Given the fact that the CTM-based DTA model has a LP formulation, we
use the set-based RO method (Ben-Tal and Nemirovski 1998, 1999, 2000, 2002) to form
a tractable LP model for the RDNDP, which overcomes the limitations of previous
stochastic optimization methods.
1.5 Dynamic Congestion Pricing Problem
Congestion pricing has been regarded as an efficient method to manage travel
demand by affecting travel behavior to minimize social cost or maximize a private firm’s
revenue. Typically, congestion pricing models assume that demand is known in advance
and deterministic values of demand are used in solving for optimal tolls. However,
system performance can be negatively impacted when deterministic demands are
employed, especially when demands depart significantly from their expected nominal
values (Waller et al. (2001) and Gardner et al. (2008)). Also, precise travel demands are
13
virtually impossible to obtain, due to specification errors and imperfect data that plague
real-world forecasting. Accordingly, in Chapter 6, we consider robust congestion pricing
problems in the presence of transportation demand uncertainty.
After the initial idea of road pricing by Pigou (1920), the literature on congestion
pricing is growing rapidly in theory and practice. The congestion pricing problem can be
classified to four criteria: 1) first-best or second-best pricing, 2) static or dynamic traffic
assignment 3) homogeneous or heterogeneous users and 4) deterministic or stochastic
parameters. The literature review presented below focuses on second-best congestion
pricing problems, particularly in dynamic traffic assignment with homogeneous users,
instead of providing a comprehensive survey on congestion pricing problems. Refer to
Yang and Huang (2004) for a survey and the references therein.
Since the first-best pricing problem calculates tolls based on the difference
between social and private marginal cost over all links in a network, it may not be
applicable in the real world. Therefore, second-best pricing, which means a subset of arcs
can be tolled, is gathering more attention from researchers and practitioners for practical
issues(e.g. Lindsey and Verhoef (2001); Lawphongpanich and Hearn (2004)). In the case
of a dynamic transportation network, Henderson (1974) explained the importance of
departure time decisions and showed the influence of time varying congestion tolls with
the single bottleneck model. Later, congestion pricing for the bottleneck model was
investigated by various researchers. (Arnott et al. (1990), Arnott and Kraus (1998), Yang
and Huang (1998), Braid (1996), De Palma and Lindsey (2000)) However, the limitation
of these works is that only simple networks are considered to analyze the impact of
congestion pricing.
14
In the context of general networks, Carey and Srinivasen (1993) provided
analytical approximate expressions for congestion tolls using Kuhn-Tucker optimality
conditions. Wie and Tobin (1998) formulated the convex optimal control problem for
first-best dynamic marginal tolls. A simulation-based analysis to determine the impact of
six types of link tolling schemes is conducted by De Palma et al. (2005). Lin et al. (2010)
proposed a heuristic combining dual variable approximation techniques using a cell
transmission model based linear programming model. There are several papers
considering Bi-level or MPEC formulation for second-best pricing. Viti et al. (2003)
proposed a framework for the joint choice of route and departure time. In their paper, the
departure time and route choice were modeled sequentially and simple grid search
approach was used to find optimal uniform tolls. In Joksimovic et al. (2005), the
departure time and route choice were modeled simultaneously. Simple grid search
approach was used for finding optimal uniform and time varying tolls. Wie (2007)
assumed triangular shaped multi-step congestion tolls to maximize consumer surplus and
proposed the Hooke-Jeeves algorithm.
In the area of congestion pricing under uncertainty, Gardner et al. (2008)
proposed a stochastic mathematical programming model with equilibrium constraints to
determine robust first-best tolls under uncertain demand. The objective of that effort was
to minimize a weighted sum of expected total travel time and standard deviation for a
finite number of pre-determined demand scenarios. Nagae and Akamatsu (2006)
formulated a stochastic singular control problem for second-best toll pricing. In their
paper, toll price was selected from a set of tolls to maximize expected net profit value.
15
Recently, RO approach was applied by Ban et al. (2009) to find a robust road
pricing in the case of having multiple traffic assignment solutions with fixed demand.
Lou et al. (2010) studied robust congestion pricing to minimize total system travel time
among all possible boundedly rational user equilibrium distribution. The main focus of
this research is the formulation and solution of robust congestion pricing problems in
which only a subset of the links in a transportation network can be tolled. In Chapter 6,
we propose to apply a robust optimization (RO) approach to user equilibrium optimal toll
problems under demand uncertainty.
Chapter 2
Robust Optimization Model for CTM
The CTM models freeway traffic flow using a finite difference approximation of
the kinematic wave model. Such a model naturally incorporates congestion effect in
traffic flows via shock waves in fluid flow. The finite difference approximation ensures
piecewise linear dependence between traffic flow and density on the link which forms the
foundation for linear programming based approaches.
More formally, let q and k denote the traffic flow and density on a link in a
traffic network. The following equation describes the relationship between q and k in
terms of v (free flow velocity), maxk (maximum possible density), w (velocity of shock
wave) and maxq (maximum allowable flow on the link).
max maxmin , ,q vk q w k k
Based on the free flow velocity and length of discrete time step, a segment of a
freeway is decomposed into cells so that traffic can move only to adjacent cells in unit
time. The connectors between cells are dummy arcs indicating the direction of flow
between cells. Ziliaskopoulos (2000) extends the original CTM model of Daganzo (1994,
1995) by formulating the DTA problem as a linear program.
In the remaining of this paper, we will formulate a deterministic linear
programming (DLP) in the line of CTM by incorporating the infeasible cost due to
uncertain demand in Section 2.1. Then, Section 2.2 presents and analyzes a robust
17
solution by developing the robust counterpart formulation of the DLP to consider data
uncertainty. To overcome the conservativeness of the robust solution, an inequality flow
control constraint is proposed in section 2.3
2.1 CTM for DTA Problem
Our reformulation of the LP based deterministic CTM model includes the
characteristics of time-space dependent costs and an adjacency matrix. In the traditional
CTM research, it is assumed that the coefficient of cost is a constant value within the
time-space network. However, in this thesis, the coefficient is assumed to be dependent
on time horizon and demand nodes. This situation is more common and is necessary to
study dynamic transportation planning under uncertainty. This generalization is
particularly appropriate for evacuation and emergency logistic planning problems.
A typical CTM objective is a measure of the total time taken for all vehicles (or
evacuees) to reach a destination (or shelter). But, in an evacuation scenario not all places
are equally prone to the disaster. For example, hurricane direction determines the areas
which have to be evacuated before any other. In addition, as the hurricane changes
direction, the threat level faced by evacuees changes across time. To represent those
characteristics, a measure called coefficient of threat level, is introduced, which is an
estimate of the susceptibility of an area to disaster at a particular time. Such a
generalization allows us to capture spatial-temporal priorities during evacuation. While
this coefficient of threat level is assumed to be constant across space and time in a
traditional CTM objective, such a modification provides a natural way to incorporate
18
infeasibility cost into the objective function, hence opening the door to study the
significance of robustness. More importantly, such modification presents a unique way to
compare the robust solution with the nominal solution. In this thesis we focus on demand
uncertainty, but this modeling framework can be extended to study the effect of
uncertainty on other factors including capacity, cost, or threat levels. Also, an adjacency
matrix A = [ ija ] is defined for representing the connectivity of the cells. More formally,
the value of ija is equal to 1 if cell i is connected to cell j, otherwise 0.
Sets Description
Set of discrete time intervals, {1,..., }T
C Set of cells, },...,1{ I , including the set of sink cells ( S
C ) and the set of source
cells ( RC )
A
Adjacency matrix, }{ ijaA , where each ),( ji component, ija , equals 1 if cell i
is connected to cell j , and equals 0 otherwise
Parameters Description
t
id Demand generated in cell i at time t
t
ic Travel cost in cell i at time t
t
iN Capacity in cell i at time t
t
iQ Inflow/outflow capacity of cell i at time t
t
i Ratio of the free-flow speed over the backward propagation speed of cell i at
time t
ix Initial number of vehicles of cell i
Variables Description
t
ix Number of vehicles staying in cell i at time t
t
ijy Number of vehicles moving from cell i to cell j at time t
Table 1-1: Notations
19
Based on the notations in Table (1-1), we present the deterministic linear
programming (DLP) model:
,\
(M-DLP1)s
t t
i ix y
t i C C
min c x
(2.1)
subject to
1 1 1 1 ,t t t t t
i i ki ki ij ij i
k C j C
x x a y a y d i C t
(2.2)
,t t
ki ki i
k C
a y Q i C t
(2.3)
,t t t t t
ki ki i i i i
k C
a y x N i C t
(2.4)
,t t
ij ij i
j C
a y Q i C t
(2.5)
0 ,t t
ij ij i
j C
a y x i C t
(2.6)
Cixx ii ˆ0 (2.7)
CCjiyij ),( 00 (2.8)
0 ,t
ix i C t (2.9)
0 ( , ) ,t
ijy i j C C t (2.10)
The dynamics of the system is that the change of traffic level is determined by
traffic flow and demand at each node and in each time period. By letting demand be 0
everywhere except source cells, the formulation can be generalized by Eq. (2.2). The total
inflow into a cell is bound by not only the inflow capacity (Eq. (2.3)) but the remaining
capacity of cell (Eq. (2.4)). Similarly, total output flow from a cell is limited by the
outflow capacity (Eq. (2.5)) and the current occupancy of the cell (Eq. (2.6)). It is
assumed that the capacities of source and sink cells are assumed to be infinite, i.e.,
20
, t
i S RN i C C . The initial conditions and non-negativity conditions are
considered as the remaining constraints.
The cost parameter t
ic depends on time in order to give a penalty when any
people cannot arrive at the destination at the end of time horizon T. i.e.
1 ,
,
t
i
i C t Tc
M i C t T
where M is assumed to be a positive large number to represent the unsatisfied demand
cost. By using the time dependent cost parameter, the objective function measures the
total cost incurred, which consist of travel cost and penalty cost. This introduction of time
dependent cost coefficients is distinct from the penalty function proposed by Mulvey et al.
(1995). In their paper, the slack variables for each constraint appear in the objective
function. A scenario-based robust optimization approach is used, hence the violation of
constraints may still be observed. In contrast, we develop a set-based robust optimization
approach where feasibility in a prescribed uncertainty set is guaranteed. While Chiu et al.
(2007) consider “no-notice evacuation” by focusing on deterministic demand realized at
time 0, the present model can also be used for short-notice evacuation (hurricane,
wildfire, and flooding) by considering time dependent demand.
The next step for applying RO approach is to reformulate M-DLP1 so that it
consists only of inequality constraints except initial value assignments. In our model, the
state variable t
ix is represented as
1' ' '
' 0
ˆ ( )t
t t t t
i i ki ki ij ij i
t k C j C
x x a y a y d
21
using Eq. (2.2) as well as Eq. (2.7), and then substituted in both Eq. (2.4) and Eq. (2.6).
Note that Eq. (2.9) is a redundant constraint, since 0t t
ij ij i
j C
a y x
, 0 t
ijy and 0 ija . It
is evident that 0 t t
ij ij i
j C
a y x
and the Eq.(2.9) can be eliminated. Thus, we can get the
following equivalent DLP formulation:
,
1' ' '
\ ' 0
(M-DLP2)
subject to
ˆ( ( ))
ˆ(
s
y z
tt t t t
i i ki ki ij ij i
t i C C t k C j C
t t
ki ki i
k C
t t
ki ki i i
k C
min z
c x a y a y d z
a y Q
a y x
1
' ' '
' 0
1' ' '
' 0
( ))
ˆ( ( )) 0
tt t t t t
ki ki ij ij i i i
t k C j C
t t
ij ij i
j C
tt t t t
ij ij i ki ki ij ij i
j C t k C j C
a y a y d N
ia y Q
a y x a y a y d
0
0 ,
0 ,
ij
t
ij
C t
y i j C C
y i j C C t
In the model shown above, we study the effect of uncertain demand information.
Traditionally, the demand at source nodes is assumed to be known at the beginning and
used as input data of CTM model. However, our model assumes demand arise in
predefined uncertainty sets. Our basic aim is to study the effect of uncertain demand on
22
the value of the objective function. We assume the demand d belong to a prescribed
uncertainty set dU .
To simplify the notation, in order to address demand uncertainty, we can denote
the objective function ( , )V y d , as a function of flow on the links, y ,and the demand
variable, d . Given a deterministic demand dd U , the nominal solution becomes
( ) argmin ( , ) 2.11N Ny
y y d V y d
Proposition 1 The nominal solution for a deterministic demand is not necessarily optimal
when the demand changes.
Proof: From Eq. (2.11), we have 1 1 2 1 1 2( ( ), ) ( ( ), ), ,N N dV y d d V y d d d d U . ■
2.2 RC of the CTM
Now, it is clear that uncertainty needs to be taken into account to create a robust
transportation plan. In this section, we apply a RO methodology to deal with uncertainty
and illustrate this approach with demand uncertainty.
Given the defined demand uncertainty set dU , the robust counterpart of the M-
DLP2 is formulated as shown below:
23
,
1' ' '
\ ' 0
'
(M-RC1)
subject to
ˆ( ( ))
ˆ( (
s
y z
tt t t t t
i i ki ki ij ij i i dt i C C t k C j C
t t
ki ki i
k C
t t t
ki ki i i ki ki
k C k C
min z
c x a y a y d z d U
a y Q
a y x a y
1
' '
' 0
1' ' '
' 0
))
ˆ( ( )) 0
tt t t t t
ij ij i i i i d
t j C
t t
ij ij i
j C
tt t t t t
ij ij i ki ki ij ij i i d
j C t k C j C
a y d N d U
a y Q
a y x a y a y d d U
0
0 ,
0 ,
ij
t
ij
i C t
y i j C C
y i j C C t
The formulation M-RC1 is a semi-infinite linear problem and it has a finite
number of decision variables and infinite number of constraints. When the uncertainty set
dU is a compact and convex set, it can be reformulated as a tractable mathematical
optimization problem. The following three theorems show that it can be converted into a
tractable equivalent deterministic problem.
Theorem 2 Given that dU is a polyhedral set { : , 0}d d Ad b d where ( )I Td R ,
( )I Td R , ( )I TR , ( )m I TA R and mb R , the robust counterpart with uncertain
demand data is equivalent to the following deterministic problem.
24
, ,
1' ' ' 1
' '
\ ' 0 ' 1
1 '
' ',( ', ') ' '
' 1
1
'
(M-RC2)
subject to
ˆ( ( ))
' , '
0
s
y z
t mt t t t
i i ki ki ij ij i m m
t i CC C t k C j C m
mt t
m m i t i i
m t
m
min z
c x a y a y d z
c i C t
1' ' ' 2
' '
' 0 ' 1
2 '
' ',( ', ')
' 1
' {1,.., }
ˆ( ( ))
t t
ki ki i
k C
t mt t t t t t t
ki ki i i ki ki ij ij i m it m i i
k C t k C j C m
mt t t
m i m i t i i
m t
m m
a y Q
a y x a y a y d N
2
' ',( ', ')
' 1
2
'
' , ' , '
0 ' , ' , '
0 ' {1,.., }
ˆ( (
mt
m i m i t
m
t
m i
t t
ij ij i
j C
t
ij ij i ki
j C
i C t i i
i C t i i
m m
a y Q
a y x a
1
' ' ' 3
' '
' 0 ' 1
3 '
' ',( ', ')
' 1
3
' ',( ', ')
' 1
'
)) 0
' , ' , '
0 ' , ' , '
t mt t t
ki ij ij i m it m
t k C j C m
mt t t
m i m i t i i
m t
mt
m i m i t
m
m i
y a y d
i C t i i
i C t i i
3
0
0 ' {1,.., }
0 ,
0 ,
t
ij
t
ij
i C t
m m
y i j C C
y i j C C t
where ',( ', ')m i t and 'm are entries of A and b respectively.
25
Proof: For notational simplicity, the each constraint affected by the demand uncertainty
of M-RC1 can be generalized by 1
I
i i
i
c d
for { : , 0}i dd d Ad b dd U . The
equation is equivalent to 1
( )i
I
i id
i
Max c d
where { : , 0}i dd d Ad b dd U .
Then, we consider the following primal linear programming (P) and dual linear
programming (D).
( ) (P)
subject to
0
j j j jd
i
ij j i
j
j
Max c d d
d
d
(D)
subject to
0
j j i id
j i
i ij j j
i
i
Min c d
c
where i is a dual variable.
Note that based on the fundamental theorem of duality (Bazaraa et al., 2005), one of the
following is true
(1) If one problem has an optimal solution, then the other problem also has an optimal
solution and two values are equal.
(2) If one problem has a bounded optimal solution, then the other problem is infeasible.
(3) Both problems are infeasible.
26
Therefore, if M-RC1 has an optimal solution, the dual linear programming M-RC2 has an
equal optimal solution. M-RC2 is a linear programming problem, hence is tractable. ■
Below we present similar results for ellipsoid and box uncertainty sets.
Theorem 3 Given that dU is ellipsoid set 1 2{ : ( ) ( ) }d d d S d d where ( )I Td R ,
( )I Td R , 1R and ( ) ( )I T I TS R , the robust counterpart with uncertain demand data is
equivalent to the following deterministic problem
,
1' ' '
1 1
\ ' 0
'
(M-RC3)
subject to
ˆ( ( ))
ˆ( (
s
y z
tt t t t T
i i ki ki ij ij i
t i C C t k C j C
t t
ki ki i
k C
t t t
ki ki i i ki ki
k C k
min z
c x a y a y d C SC z
a y Q
a y x a y
1
' '
2 2
' 0
1' ' '
3 3
' 0
))
ˆ( ( )) 0
tt t T t t
ij ij i it it i i
t C j C
t t
ij ij i
j C
tt t t t T
ij ij i ki ki ij ij i it it
j C t k C j C
a y d C SC N
a y Q
a y x a y a y d C SC
0
0 ,
0 ,
ij
t
ij
i C t
y i j C C
y i j C C t
where ( )
1
I TC R is a matrix, of which ( ', ')thi t entries are '
'
t
i
t
c
,
( )
2
I T
itC R is a matrix, of which ( ', ')thi t entries are '
'
t
i if 'i i , otherwise 0,
and ( )
3
I T
itC R is a matrix, of which ( ', ')thi t entries are '
'
t
i if 'i i , otherwise 0.
27
Proof: Similar to the proof of Theorem 2, an equivalent formulation of the each
constraint affected by the demand uncertainty becomes 1
( )I
i id
i
Max c d
where
1 2{( ) ( ) }d
d d S d dd U and our interest is the optimal solution of the following
mathematical programming
1
1 2
subject to
( ) ( )
I
i id
i
Max c d
d d S d d
By the Karush-Kuhn-Tucker (KKT) conditions, the solution of the problem is
Td d SC
C SC
and
1
I
T
i i i id
i i
Max c d c d C SC
where IC R is a matrix, of
which ( )thi entries are ic . Now, we have following relationship and M-RC3 can be
formulated.
1 2
1
{( ) ( ) } I
T
i i i i idi i
c d d d S d d c d C SCd U
■
To illustrate the RO approach, let us consider a box uncertainty set
1 , 1dU d d
where t
id is the nominal demand in cell i at time t . As shown in Theorem 2 and 3, this
simple interval uncertainty set can be extended into a more general form of uncertainty
set. (see Bertsimas et al. (2007) for a survey).
28
Theorem 4 Given that dU is box set { : 1 1 }d d d d where ( )I Td R ,
( )I Td R , ( )I TR , the robust counterpart with uncertain demand data is equivalent to
the following deterministic problem
,
1' ' ' '
\ ' 0
'
(M-RC4)
subject to
ˆ( ( (1 )))
ˆ( (
s
y z
tt t t t t
i i ki ki ij ij i i
t i C C t k C j C
t t
ki ki i
k C
t t t
ki ki i i ki ki
k C k C
min z
c x a y a y d z
a y Q
a y x a y
1
' ' '
' 0
1' ' ' '
' 0
(1 )))
ˆ( ( (1 ))) 0
tt t t t t
ij ij i i i i
t j C
t t
ij ij i
j C
tt t t t t
ij ij i ki ki ij ij i i
j C t k C j C
a y d N
a y Q
a y x a y a y d
0
0 ,
0 ,
ij
t
ij
i C t
y i j C C
y i j C C t
Proof: Note the following relation for any real numbers i
and v (see Ben-Tal et al.
(2004) for more details).
{ : 1 1 }
( )
( )
t
it
t
i i i i i i i i i
I
i id
I
i i i i
I
d v d d d d d
Max d v
d v
Using the equivalence of equations, we can obtain M-RC4. ■
29
The solution satisfying all constraints of RC is called a robust feasible solution
and the optimal solution minimizing the objective value is called a robust optimal
solution. The robust optimal solution can be interpreted as the solution being feasible for
any realization of the uncertain data and achieving best worst case objective value. In
other words, the objective value ( RCz , i.e., z of (M-RC1)) of RC is guaranteed for any
demand realization within an appropriate uncertainty set. Hence, RCz is an upper bound of
a realized (or simulated) objective value ( R RCz ). The realized robust objective value
( R RCz ) refers to the objective value we can obtain when the robust optimal solution ( RCy )
is used and a data scenario ( d ) is realized, i.e. ( , )R RC
RCz V y d (see proposition 1).
However, in some cases, RC is only feasible at unrealistic small uncertainty levels or
generates solutions that are too conservative (Ben-Tal et al., 2004, 2005).
Theorem 5 The robust optimal solution ( RCy ) of M-RC4 corresponds to a deterministic
LP (M-DLP1) considering possible minimum demand in the box uncertainty set.
Proof: Let us consider the constraints for source nodes.
1' ' ' '
\ ' 0
1' '
' 0
ˆ( ( (1 ))) (2.12)
ˆ( ( (1
s
tt t t t t
i i ki ki ij ij i i
t i C C t k C j C
t t
ij ij i
j C
tt t t
ij ij i ij ij i
j C t j C
c x a y a y d z
a y Q
a y x a y d
'
0
))) 0 (2.13)
0 ,
0 ,
R
t
i
ij
t
ij
i C t
y i j C C
y i j C C t
30
Eq. (2.12) and (2.13) are related to the uncertain demand. The constraint (2.13) is
equivalent to 1
' ' '
' 0 ' 0
ˆ (1 )t t
t t t
ij ij i i i
t j C t
a y x d
, which means total number of evacuees from
a source node i at time t cannot exceed the sum of initial occupancy of the source node
and total minimum demands until time 1t . In other words, any additional demand
exceeding possible minimum demand ' '(1 )t t
i id cannot be controlled by the RC. ■
Note that Theorem 5 does not mean the objective value of M-RC4 is equal to
deterministic LP with minimum possible demand. The number of vehicles in the source
nodes will also be different.
Proposition 6 If the least demand is realized and the ideal solution exists, realized robust
objective value( R RCz ) is equal to the ideal objective value( Leastz ).
Proof: Let ' '(0, 2 )t t
i i be the additional demand exceeding minimum demand ' '(1 )t t
i id
within the box uncertainty set. The constraint (2.12) can be reformulated as
1 1
' ' ' ' '
\ ' 0 ' 0
ˆ( ( (1 )))s
t tt t t t t t t
i i ki ki ij ij i i i i
t i C C t k C j C t i C t
c x a y a y d c z
. It means that
1'
' 0
2t
RC Least t t
i i
t i C t
z z c
and 1
'
' 0
2t
R RC Least t t
i i
t i C t
z z c
when uncertain demand data
' '(0, 2 )t t
i i are realized. Also, R RCz becomes Leastz when the least demands are
realized. ■
31
Proposition 6 also shows that RC is always feasible as long as Leastz exists.
However, as the uncertainty level increases, it becomes too conservative to adopt the
solution in the real world. Also, according to Theorem 5, the solution of (M-RC4) means
that only minimum possible vehicles are allowed to move to destination. The solution
may be worthless since we have to give up the transportation planning (or evacuation in a
disaster management problem) to find the uncertainty immunized solution. In the next
section, we will show that an inequality constraint will improve the performance of the
robust solution.
2.3 RC with Inequality Relaxation
Returning to our CTM based evacuation problem (M-DLP1), let us recall the flow
control constraint, Eq. (2.2). The equality constraint can be written as an inequality
constraint 1 1 1 1t t t t t
i i ki ki ij ij i
k C j C
x x a y a y d
(e.g., Ukkusuri and Waller, 2008). Clearly,
for a given deterministic demand t t
i id d , the inequality flow control constraint is always
binding and therefore becomes an equality based flow constraint. However, the actual
realized demand can be lower or higher than the expected anticipated demand, t
id .
Intuitively, if the realized demand is lower than expected, the nominal solution should
remain feasible by allocating the realized demand proportionally to the planned routes.
Therefore, we formulate the flow constraint as an inequality to accommodate for
uncertainty in demand. Under the assumption of box uncertainty set, a tractable robust
counterpart is written as M-RC5 similar to Theorem 4.
32
,\
1 1 1 1
(M-RC5)
subject to
1
s
t t
i ix y
t i C C
t t t t t
i i ki ki ij ij i i
k C j C
t t
ki ki i
k C
t t t t t
ki ki i i i i
k C
min c x
x x a y a y d
a y Q
a y x N
0
0
ˆ
t t
ij ij i
j C
t t
ij ij i
j C
i i
i C t
a y Q
a y x
x x i C
0 0 ,
0
0 ,
ij
t
i
t
ij
y i j C C
x i C t
y i j C C t
Clearly, the robust counterpart corresponds to the case when there is maximum
demand at each of the cells. Intuitively, the worst case should correspond to maximum
demand at each node. It is worthwhile to note that our original DLP considers expected
demand in all cells. Based on this finding, we can propose the following theorem (for
similar discussions see Bertsimas and Perakis (2005)).
Theorem 7 For the CTM with inequality flow control constraint, RC with uncertain
demand in box uncertainty set corresponds to a deterministic LP (M-RC5) considering
maximum possible demand in the uncertainty set.
33
From Theorem 7, we have the following:
4 argmin max ( , ) ( ) argmin ( , )RC N max maxy yd
y V y d y d V y d
where 1t
max id d , the maximum possible demand within demand uncertainty set
defined in box uncertainty set. Then, we can have the following proposition.
Proposition 8 In the Cell Transmission Model with uncertain demand in box uncertainty
set, the robust solution of the robust counterpart has performance better than or same as
any nominal solution.
Proof: From Proposition 1 and 6, we have
( ( ), ) ( ( ), ) ( , )N max N max max R maxV y d d V y d d V y d . ■
The implication of Proposition 8 is that a robust solution performs better than any
nominal solution under worst scenario demand. A natural question is that, on average,
which solution will have better performance? We will conduct numerical experiments in
section 4 to investigate this question.
34
Chapter 3
Affinely Adjustable Robust Optimization Model for CTM
CTM based DTA problem is a generic multi-period linear programming problem.
In this section, we apply AARO methodology to deal with uncertainty in demand and
find a less conservative robust solution than what we found at the Chapter 2 for the multi-
period decision problem. In particular, in Section 3.1, we introduce a linear decision rule
and derive AARC formulation with the consideration of uncertain demand data. Based on
the model developed, tractable AARC problems are formulated by considering box
uncertainty set and polyhedral uncertainty set in Section 3.2 and 3.3, respectively.
3.1 The RO Approach for Multi-period Problems
As shown in the previous chapter, the RC solution is uncertainty-immunized and
feasible for all realization of uncertain value in predefined uncertainty set dU . The
optimal objective value is guaranteed and the objective value with any realized uncertain
data in dU never exceeds the optimal objective value. However, in some cases, the RC
solution is too conservative since the uncertainty-immunized solution for entire planning
horizon is developed at the beginning of planning horizon before any uncertainty is
realized. In our model, RC for CTM with equality constraint finds optimal dynamic flow
with minimum demand and it is too conservative and practically meaningless (see
Theorem 5). In order to avoid the conservativeness of robust counterpart, we apply
affinely adjustable robust counterpart. When transportation planning and operation face a
35
sequential decision environment, information of demand, traffic condition, weather, etc.
can be updated as time evolves.
The adjustable control variables, t
ijy , can be represented as an affine function of
previously observed demand values, i.e., 1
'
R t
t s
ij ijt ijt s
s C I
y d
, where 1ijt and s
ijt are a
set of non-adjustable variables and 0,.., 1tI t . Then the state variable t
ix can be given
as follows:
' '
1
1' ' '
' 0
11 1 '
' ' ' '
' 0
1 1
ˆ ( )
ˆ ( ( ) ( ) )
ˆ
R t R t
R t
tt t t t
i i ki ki ij ij i
t k C j C
ts s t
i ki kit kit s ij ijt ijt s i
t k C s C I j C s C I
s t
i it it s i
s C I
x x a y a y d
x a d a d d
x d d
where 1
1 1 1
' '
' 0
( )t
it ki kit ij ijt
t k C j C
a a
,
1
' ' { }
' 1
( )t
s s s
it ki kit ij ijt s i
t k C j C
a a I
and { }
1 if i=s
0 otherwisei sI
Similar as before, by substituting the state and control variables, we have AARC
formulation (M-AARC1). The formulation (M-AARC1) is intractable like M-RC1 since
it is a semi-infinite program but it can be reformulated as a tractable optimization
problem. The minimum objective value AARCz is also guaranteed value for all realization
36
of uncertain data under the assumption of linear dependency. The only difference is that
decision variables of AARC is not adjustable control variables, t
ijy , but a set of
coefficient of affine function of the control variables including 1
ijt , sijt , 1
it and sit . It
means that the solution of AARC is a linear decision rule.
1
, ,
1 1
1
(M-AARC1)
ˆ( )
( )
R t
R t
z
t s t t
i i it it s i i dt i C s C I
s
ki kit kit s
k s C I
min z
subject to
c x d d z d U
a d
1
1
1 1
1
( )
ˆ ( )
(
R t
R t
t t
i i dC
s
ki kit kit s
k C s C I
t s t t t t
i i it it s i i i i ds C I
s
ij ijt ijt s
Q d U
a d
x d d N d U
a d
1
1
1 1
)
( )
ˆ ( ) 0
R t
R t
R t
t t
i i dj C s C I
s
ij ijt ijt s
j C s C I
s t t
i it it s i i ds C I
Q d U
a d
x d d d U
1
0
1
0 ,
0 , R t
ij
s t
ijt ijt s is C I
i C t
i j C C
d i j C C t d
11 1 1
' '
' 0
1
' ' { } 1
' 1
( )
( )
d
t
it ki kit ij ijt
t k C j C
ts s s
it ki kit ij ijt s i R t
t k C j C
U
a a i C t
a a I i C t s C I
37
3.2 AARC with Box Uncertainty Set
In robust optimization approach, it is assumed that demand t
id is unknown and it
belongs to a prescribed uncertainty set. In this section, the box uncertainty set is
considered since it is easy to adopt and generally used. When the box uncertainty set is
considered, according to theorem 4, we know the following relationship.
{ : 1 1 }
( )
( )
t
it
t
i i i i i i i i i
I
i id
I
i i i i
I
d v d d d d d
Max d v
d v
wherei
is coefficient of uncertain vector, v is right-hand side not containing the
uncertain vector. The absolute value function can be manipulated by introducing a new
decision variable i
( )
t
i i i i
I
i i i
d v
Now, we can write the equivalent LP of the AARC formulation (M-AARC2).
, , , ,
1
1 1,
{0,.. 1} { 1... }
0 0 0 0
2
(M-AARC2)
subject to
( ) ( * )
,
z
t s
i it t t s i s
t i t s i t T
Ts t s s s s
i it
t i C
min z
c x c I I d z
c
2 R Ts C I
38
1
1 1s 1s
1s 1s 1s 1s
1 1 2s 2s
( )
,
ˆ( ) (
R t
R t
t
ki kit it it s s i
k C s C I
s
it ki kit it it it R t
k C
t
ki kit i i it it it s
k C s C I
a d Q
a s C I
a x
3s 3s 1 1
2s 2s 2s 2s
1
3s 3s 3s 3s
{ }
)
( )
,
,
R
s
t t t t
it it s s i i
s C
s t s
it ki kit i it it it it R t
k C
t
it ki i s i it it it
k C
d
d N
a s C I
a I
1
1 4s 4s
4s 4s 4s 4s
1 1 5s 5s
( ) )
,
ˆ (
R t
t
R
t
ij ijt it it s s i
j C s C I
s
it ij ijt it it it R t
j C
ij ijt i it it it s
s I
s C
a d Q
a s C I
a x
6s 6s 1 1
5s 5s 5s 5s
1
6s 6s 6s 6s
{ }
)
( ) 0
,
,
R
R
s
j C C
t t
it it s s
s C
s s
it ij ijt it it it it R t
j C
s
it ij ijt s i it it it
j C
d
d
a s C I
a I
1
0
1 7s
7s 7s
0 ,
( ) 0 ,
R t
R
ij
s
ijt ijt ijt s s
s C I
s
ijt ijt ijt
s C
i j C C
d i j C C t
1
11 1 1
' '
' 0
' '
'
,
( )
( )
R t
t
it ki kit ij ijt
t k C j C
s s s
it ki kit ij ijt
t k C j C
i j C C t s C I
a a i C t
a a
1
{ } 1
1
t
s i R t
i C t
I i C t s C I
39
Note that when s
ijt is set to 0, the AARC problem becomes RC. Since AARC
problem has larger feasible region, the objective value of AARC is less than or equal to
RC, which means we may have a less conservative optimal solution.
AARC formulation for the CTM with inequality flow control constraint is
considered by Theorem 9.
Theorem 9 For the CTM with inequality flow control constraint, RC, which is a
maximum demand case, is equivalent to AARC.
Proof: Robust counterpart of the model is formulated as
,\
1 1 1 1
(M-RC4)
subject to
1
.(3 10)
s
t t
i ix y
t i C C
t t t t t
i i ki ki ij ij i i
k C j C
min c x
x x a y a y d i C t
and Eq
The flow control constraint is the only constraint affected by uncertainty and each
demand uncertainty is independent. As a result, by Ben-tal et al. (2004), RC is equivalent
to AARC and they have same optimal objective value, which is maximum demand case.
■
3.3 AARC with Polyhedral Uncertainty Set
In previous section, Theorem 9 shows a case where AARC provides the exactly
same solution as RC as the uncertainty is “constraint-wise”. In this section, to find out a
40
less conservative solution, we consider a joint constraint where the demands are upper
limited. Let us consider t
i i R
t T
d D i C
, which refers to a joint budget for demand
uncertainty. This represents the situation that the total demand ( t
i
t T
d
) from a source
node is limited by an upper bound ( iD ). A box uncertainty set in conjunction with a
budget uncertainty set becomes a polyhedral uncertainty set. Now, we have the following
uncertain data set.
, ,
min max{ : , }t t i t t i t t
i d i i i i
t T
d U d d d d d D
Theorem 10 For the CTM with inequality flow control constraint, RC with box
uncertainty set is equivalent to RC with polyhedral uncertainty set, when the maximum
value of the projection of dU onto the data space of each uncertain constraint is equal to
,
max
i td .
Proof: Let us consider the flow control constraint, since it is the only constraint
containing uncertain data.
1 1 1 1t t t t t
i i ki ki ij ij i
k C j C
x x a y a y d
where , ,
min max{ : , }t t i t t i t t
i d i i i i
t T
d U d d d d d D
Since we want to find uncertainty immunized solution, we have following sub problem.
1 1 1 1 ,
maxmaxti d
t t t t t i t
i i ki ki ij ij id U
k C j C
x x a y a y d d
.
Clearly, this is equivalent to RC with box uncertainty set.■
41
Similar to the previous section, for AARC formulation, it is assumed that decision
variables are depend on the previously realized data, that is, 1
R t
t s
ij ijt ijt s
s C I
y d
and
1
R t
t s
i it it s
s C I
x d
. Then the AARC formulation is written as
, ,
1
{0,.., 1} { 1... } \ \
{ 1, } 1 1 1 { 1}
z (M-AARC3)
subject to
( ( ) *
R s s
z
t s t
i it s i it d
T s C t T i C C t i C C
s s s s
t s i it it ki kit ij ijt t
k C
min
c d z c d U
I a a I
1 1 1 1
1 1 1
1
)
R t
R t
s
s C I j C
it it ki kit ij ijt d
k C j C
s t
ki kit s i ki kit
s C I k C k C
d
a a d U
a d Q a
1 1
1
( ) ( )
R t
R t
d
s t s t t
ki kit i it s i i it ki kit d
s C I k C k C
s t
ij ijt s i ij ijt
s C I j C j C
d U
a d N a d U
a d Q a
1 1
1
( )
,
R t
R t
d
s s
ij ijt it s it ij ijt d
s C I j C j C
s
ijt s ijt d
s C I
i C t
d U
a d a d U
d d U i j C C
1 1
0 0
ˆ , 0 ,i i ki
t
x i C i j C C
By using the LP duality as shown in Theorem 1, we can reformulate each
constraint affected by uncertain data as an equivalent LP problem. Therefore, the
equivalent tractable AARC of the SO based CTM becomes
42
, ,
11 12 13 1
max min
\
11 12 13
{ 1... } \
11 12 13
z (M-AARC4)
subject to
( )
{0,.. 1},
0
R R s
s
z
s s t
s s s s i it
s C s C t i C C
t s
s s s i it R
t T i C C
s s s
min
d d D z c
c T s C
11 12 13
21 22 23 1 1 1 1
max min 1 1 1
21 22 23
{ 1, }
{ },
, , 0
( )R R
R
s s s
s s
its its s its it it ki kit ij ijt
s C s C k C j C
s
its its its t s i it
T s C
d d D a a
I
1 1 1 { 1}
21 22 23
( ) *
{0... 1},
0
s s s
it ki kit ij ijt t
k C j C
R
its its its
a a I
t s C
21 22 23
31 32 33 1
max min
31 32 33
31
{ ... },
, , 0
( )
{0... 1},
R R
R
its its its
s s t
its its s its i ki kit
s C s C k C
s
its its its ki kit R
k C
its
t T s C
d d D Q a
a t s C
32 33
31 32 33
41 42 43 1 1
max min
41 42 43
0 { ... },
, , 0
( ) ( )
R R
its its R
its its its
s s t t
its its s its i i it ki kit
s C s C k C
its its its
t T s C
d d D N a
41 42 43
41 42 43
51 52 53
max min
{0... 1},
0 { ... },
, , 0
( )R
s t s
ki kit i it R
k C
its its its R
its its its
s s
its its s its
s C s
a t s C
t T s C
d d D
1
51 52 53
51 52 53
51 52
{0... 1},
0 { ... },
, ,
R
t
i ij ijt
C j C
s
its its its ij ijt R
j C
its its its R
its its
Q a
a t s C
t T s C
53
0its
i C t
43
61 62 63 1 1
max min
61 62 63
61 62 63
( )
{0... 1},
0
R R
s s
its its s its it ij ijt
s C s C j C
s s
its its its ij ijt it R
j C
its its its
d d D a
a t s C
61 62 63
71 72 73 1
max min
71 72 73
{ ... },
, , 0
( )
R R
R
its its its
s s
ijts ijts s ijts ijt
s C s C
ijts ijts ijts ij
t T s C
d d D
71 72 73
71 72 73
0
,
{0... 1},
0 { ... },
, , 0
s
t R
ijts ijts ijts R
ijts ijts ijts
i
i j C C t
t s C
t T s C
1
1
0
ˆ
0 ,
i
ki
x i C
i j C C
where is dual variable. Note that the numerical index of is used for notational
simplicity.
44
Chapter 4
Emergency Logistics Planning
Emergency logistics during extreme events based on large-scale transportation
systems is of critical importance. It is challenging to develop a model due to the inherent
complexity and uncertainty. Moreover, distinct from typical transportation networks,
transportation networks for emergency logistics bear significant infeasibility cost,
resulting from the potential loss of life and property in extreme events. The infeasibility
cost refers to the cost incurred when the routing policy is rendered infeasible due to
uncertain demand. In a general traffic assignment scenario, the infeasibility cost under
uncertain demand is much smaller compared to an emergency scenario. Therefore, robust
solutions play a major role in evacuation transportation planning.
In the emergency operations literature, the region to be evacuated is typically
represented as a transportation network, where nodes correspond to the regions and arcs
represent the roads. In the outbound emergency logistics consists of a flow over time on
the transportation network which satisfies the evacuee demand from source nodes to sink
nodes. Therefore, the proper approaches may come from variety of fields such as
dynamic network flows (see Ahuja et al. (2003) for a complete survey on network flow
theory), dynamic traffic assignment (see Peeta and Ziliaskopoulos (2001) for a review)
and simulation (see Mahmassani (2001) for a survey).
Emergency management is one of the best application areas for applying robust
optimization due to the uncertainty of human beings and disaster. Robust solution,
especially AARC solution, can play an important role for emergency logistics planning
45
for several reasons. First of all, the role of hard constraints is emphasized since the
penalty cost for an infeasible solution is loss of life or property. Next, it is very difficult
to estimate or forecast the demand model in the to-be-affected areas due to unexpected
human behavior and nature of disaster. Finally, we can take advantage of updated or
realized data on demand by employing AARC solutions. When we solve AARC
problems, the optimal coefficients of the Linear Decision Rule (LDR) are computed
offline. Going online, the actual decision variables (flows) are determined for period t by
inserting the revealed uncertainties from previous periods in the LDR. A fully online
version of the method can be also implemented. In such a version, at period t only the t-
period design variables are activated. The horizon is then rolled forward and the problem
is resolved after adjusting the state variables revealed in previous periods.
The structure of the chapter is as follows. In Section 4.1, an emergency logistic
planning problem is considered and the meaning of demand uncertainty sets is explained.
Then, we present a summary of experiments to test the performance of the AARC
approach. The AARC solution is benchmarked against the RC solution, sampling based
stochastic programming and an ideal solution with complete future information. Two test
networks are chosen from Chiu et al. (2007) and Yazici and Ozbay (2007) for the
numerical analysis in Section 4.2 and 4.3, respectively.
4.1 Demand Modeling
In an emergency logistics problem, a general approach to model time-varying
evacuee demand is captured by the following steps: The first step of demand modeling is
46
calculation of total demand. Next, demand arrival or vehicle departure rate is determined
for describing a dynamic environment. For an example, S-shape curve can be used for
representing cumulative percentage of demand arrival. In most studies, it is assumed that
the parameters (e.g. slope) of S-curve are unknown but deterministic in value. Since the
parameters can be estimated with empirical data or simulation results, different research
showed different values (Radwan et al. 1985, Lindell 2008). The S-shape loading curves
can be classified as quick, medium or slow (see Figure (4-1)).
However, in the real world, both total number of demand and departure rate are
uncertain. By considering box uncertainty or polyhedral uncertainty set, we can
overcome the limitation of deterministic S-curve and cover infinite number of S-curve
including fast, medium and slow response. Figure (4-2) shows the S-curve with upper
and lower bound defined by box uncertainty. In Figure (4-3), polyhedral uncertainty set
(box uncertainty & budget uncertainty) is shown and the upper bound of S-curve is
limited by total demand.
Figure 4-1: Three response curves (Fu et al. 2007)
47
4.2 Small Network Example
In the first numerical experiment, a small network configuration is drawn in
Figure (4-4) to verify the performance of RC and AARC from the illustrative example of
Figure 4-2: S-curve - box uncertainty
Figure 4-3: S-curve - polyhedral uncertainty
48
Chiu et al. (2007). The network consists of 14 nodes including 3 source cells (1,5 and 9)
and 1 super sink cell (14)
The input data consists of topology or connectivity, demand estimates at source
nodes and geometric characteristics of the transportation network. The geometric
characteristics include length, no. of lanes, speed limits and capacity limits on the road.
Using this information, an equivalent cell model is constructed in which length of a cell
corresponds to the maximum distance traveled by a vehicle in a unit time interval. The
capacity and demand information is appropriately reflected in the cell transmission
model. The model parameter, , is assumed to be unity, i.e., 1, ,t
i i C t .
The data of the transportation network is adopted from Chiu et al. (2007) except
demand data since deterministic demand was used in the original model. Data for the
small network are summarized in Table (4-1) and Table (4-2). In the example, the flow
capacity of node 3 is time dependent and changes from time 1 to 6.
Figure 4-4: Example network (Chiu et al. 2007)
49
As mentioned before, we consider uncertain multi-period demand. In particular,
the following mathematical formulation of S-curve (Radwan et al., 1985) is adopted for
demand loading.
( ) 1/ (1 exp( ( )))P t t
where ( )P t is the cumulative distribution with =1, the slope of curve, and =3, the
median departure time. In both box and polyhedral uncertainty set, nominal demand at
time t is calculated by multiplying ( ( ) ( 1))P t P t with expected total demand. The
nominal solution is obtained by assuming a deterministic demand, where the realized
demand is equal to the expected demand d . Also, the joint budget of demand uncertainty
is assumed to be one and half times of the sum of expected total demand,
t
i i R
t T
d D i C
.
Table 4-1: Time invariant cell properties
Cell 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Ni ∞ 20 20 20 ∞ 20 20 20 ∞ 20 20 20 20 ∞
Qi 12 12 * 12 12 12 12 12 12 12 12 12 12 12
ˆix 0 0 0 0 0 0 0 0 0 0 0 0 0 0
* See Table 4-2 for time-dependent data
Table 4-2: Time dependent data
Time 0 1 2 3 4 5 6 7 8 9 10
3
tQ 12 6 6 0 0 0 12 12 12 12 12
50
4.2.1 RC vs. DLP
In numerical experiments, at first, we focus on the importance of robustness by
analyzing the impact of infeasibility. Since we do not know exact future demand, the
realized demand can be less than or greater than the expected demand and one has to
adjust the nominal solution in order to implement it. Waller and Ziliaskopouls (2006)
choose to duplicate extra demand randomly when more realized than expected appear.
Mudchatongsuk et al. (2008) propose artificial arcs with high cost to avoid higher
demand. By considering vehicle holding (vehicle may wait in some places as long as they
are not violating the constraints), we choose the following rules to adjust nominal
solution:
1) If the realized demand is greater than the expected, then the excess demand remains at
the source cell. Although this is a restrictive assumption, we expect that this will provide
a good starting point for further work.
2) If there is less demand than expected then proportionate demand is allocated to each
path.
We assume i i C and M =10. For a given , 100 random demand
samples were generated and the average objective function value ( nomz ) obtained by
implementing the nominal solution. It was compared to objective function value ( 0
nomz
=414) in a deterministic scenario. The average degradation relative to the nominal
solution is calculated and plotted as is varied from 0 to 30% in intervals of 2%. The
degradation is calculated as follows
51
0
0degradation( )
nom nom
nom
z z
z
As shown in Table (4-3) and Figure (4-5), an average degradation of 10-15% was
observed when the nominal solution was subject to uncertain demand. This may be
significant as the increase in objective cost function value corresponds to loss of human
life and property. Also, uncertainty in demand seems to be proportional to the
degradation in the nominal solution.
Table 4-3: Degradation of nominal solution under uncertain demand
Degradation Degradation
0 0 16 12.3
2 1.47 18 13.84
4 3.01 20 15.39
6 4.55 22 16.95
8 6.1 24 18.5
10 7.65 26 20.05
12 9.2 28 21.61
14 10.75 30 23.16
Figure 4-5: Consequence of data uncertainty for nominal solution
0
5
10
15
20
25
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Maximum Uncertainty (%)
Relative Degradation (%)
52
One can argue that the analysis seems to be dependent on the policy we used to
deal with excess/less demand. But, we note that policies used to adjust solutions must be
computationally inexpensive and relatively simple for the traffic controllers to implement
it in real time. Although different policies can exhibit different results, a similar trend can
be expected as seen in Table (4-3). This experiment provides a clear motivation to
consider uncertainty in evacuation problems.
Similar setting is used to compare the robust solution to the nominal solution. For
a given , 100 random demand samples were generated and the objective function
values obtained by implementing the robust ( robz ) and the nominal solution ( nomz ) were
compared. The average relative improvement over the nominal solution is calculated and
plotted as is varied between 0 to 30% in intervals of 2%. The improvement is
calculated as follows
( )
nom rob
nom
z zimprovement
z
Table (4-4) and Figure (4-6) show that an average improvement of 6-8% is over the
nominal solution by the robust solution under varying level of uncertainty. Although the
improvement observed is not monotone, the robust solution seemingly performs better at
higher uncertainty levels in demand. Similar non-monotone results were reported by
Bertsimas et al. (2007) when RO was applied to an inventory control problem. Although,
the results obtained are based on several assumptions such as excess demand being left at
source nodes, we feel that the robust solution is conceptually superior to a deterministic
solution.
53
In addition, one can see that the robust solution itself can be conservative as it
deals with the worst case scenario which corresponds to maximum demand at each of the
source nodes. In reality, there is a small chance of this scenario to occur. We argue that a
conservative solution such as a robust solution will provide a guaranteed bound and be
preferable to a nominal solution which does not guarantee feasibility or solution quality
Table 4-4: Improvement of robust solution relative to the nominal solution
Improvement Improvement
0 0 16 5.07
2 0.87 18 8.64
4 1.74 20 9.05
6 2.4 22 6.99
8 2.86 24 10.68
10 4 26 10.08
12 3.56 28 11.51
14 3.82 30 11.34
Figure 4-6: Relative performance of robust solution
0
2
4
6
8
10
12
14
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Maximum Uncertainty (%)
Relative Performance (%)
54
under all demand realizations. This may be particularly relevant in an evacuation scenario
where solution infeasibility may result in loss of life and property. Also, one can restrict
the uncertainty set to obtain robust solutions which will provide more realistic guarantees
during evacuation. In this section, we tested whether the evacuation problem is an
appropriate application area for optimization under uncertainty. (see Ben-Tal et al. (2007)
for a similar analysis of a drug development example). Clearly, RO is a promising
approach to develop evacuation plans which are immune to uncertainty.
4.2.2 AARC vs. DLP
Based on the nominal data, uncertain demand in a polyhedral set is generated and
tested. The uncertainty level θ is increased from 2.5% to 30%. First, objective values are
calculated and emergency logistics plans are generated using M-DLP1 and M-AARC4.
Next, given the uncertainty level and evacuation plan, simulated (or realized) objective
value from Eq. (2.1) is computed by generating random demand in the specified
uncertainty set. Average values, standard deviation and worst case solution of 1000
simulated objective values are used to compare the traffic assignment solutions.
Our first objective of the experiment is comparison of AARC and DLP under a
polyhedral uncertainty set. Objective values of robust optimization approaches, which
measure the worst case solution of the vehicle control plan, are computed and compared
at Table (4-5) by perturbing the uncertainty level. The DLP solution shows the cost when
only deterministic nominal demand is dealt with. It is natural that the objective value of
55
AARC increases with larger uncertainty level. Also, the objective value of DLP is
smaller since it is equivalent to AARC with zero uncertainty level.
In simulation, the emergency logistics plan has to be adjusted in some way since
we relaxed the constraint in Eq. (2.2). We assumed that if there are fewer vehicles in a
node than the vehicle flow plan, proportionate flow is allocated to each path. Also, any
vehicles exceeding the plan will remain at the node and pay a penalty for not moving
them to the destination cells. Table (4-5) shows the simulated objective value of ideal
DLP, DLP, and AARC. Ideal DLP is the case where perfect future demand information is
known at the beginning of the planning horizon. It is the lower bound of simulated
objective value. The average improvement of AARC over DLP is significant at higher
uncertainty level. The AARC problem with 14 nodes and 15 planning horizon has 36,600
Table 4-5: Objective value – polyhedral uncertainty
0.025 0.05 0.075 0.1 0.15 0.2 0.25 0.3
Obj. DLP 350.1 350.1 350.1 350.1 350.1 350.1 350.1 350.1
AARC 358.18 366.27 375.24 384.35 402.57 420.8 439.03 457.25
Avg. Ideal 354.35 358.59 362.85 367.33 376.78 386.35 395.95 405.55
DLP 417.13 484.25 551.49 618.81 753.49 888.23 1023 1157.77
AARC 355.34 359.7 364.88 370.72 381.79 397.5 410.17 420.57
Std. Ideal 1.08 2.16 3.26 4.57 7.2 9.7 12.17 14.63
DLP 15.14 30.19 45.1 59.95 89.63 119.29 148.94 178.59
AARC 0.91 1.76 2.97 3.78 5.35 6.14 7.9 10.12
Worst Ideal 356.54 362.97 369.83 377.14 391.76 406.38 421 435.61
DLP 452.33 554.56 656.8 759.03 963.49 1167.95 1372.42 1576.88
AARC 357.17 363.27 370.73 378.44 393.29 410.64 427.02 441.57
56
constraints and 190,428 variables. It is solved in about 44 seconds on a PC with Intel
processor 1.87 Ghz and 2GB of memory.
4.2.3 AARC vs. Sampling based Stochastic Programming
Sampling based stochastic programming (SSP), or Monte Carlo sampling method,
is an important approximation approach. Stochastic problems are solved by generating
random samples and solving a deterministic problem to optimize sample average
objective value. For comparison with stochastic programming, a beta distribution is
assumed, and if the sum of the sampling demand is bigger than the upper bound of total
demand, it is ignored and re-sampled. The following equation represents SSP.
,\
1 1 1 1
1
subject to
s
t t
i ilx y
l t i C C
t t t t t
il il ki ki ij ij il
k C j C
t t
ki ki i
k C
t t t
ki ki i il i
k C
Min c xL
x x a y a y d l
a y Q
a y x
0
t t
i
t t
ij ij i
j C
t t
ij ij il
j C
N l i C
a y Q
a y x l
0
0
ˆ
0 ,
0
0 ,
il i
ij
t
il
t
ij
t
x x i C l
y i j C C
x i C t l
y i j C C t
57
where independent sampling scenario {1,2,..., }l L , t
ilx is the number of people
contained in cell i at time t for sampling scenario l , t
ild demand generated in cell i at
time t for sampling scenario l .
For the comparison, first 50 samples are generated using beta distribution
function, Beta(1,2). Next, Beta(5,2) and Uniform distribution (i.e. Beta(1,1)) are used for
generating uncertain demand for simulation. This may be reasonable when we do not
have exact information on the distribution. The objective value of SSP is lower than
AARC since it finds the average of minimum cost with given sample data. It has a
different meaning from the RO approach generating best worst case solution. However,
the simulated objective values can be compared since they show the performance of the
emergency logistics plan. When Beta(5,2) is used for simulation, we can see that AARC
is better than SSP in terms of the average of the simulated objective value in Table (4-6).
The gap between AARC and the ideal solution is very small even with higher
uncertainty, e.g. it is less than 4% when the uncertain level is 30 %! In contrast for SSP,
the gap is increased drastically. As shown in Table (4-7), the average values of the
simulated objective value from AARC and SSP are comparable with the random demand
from Beta(1,1).
Under both demand scenarios, AARC provides more stable and robust solution
than SSP in the aspect of standard deviation and worst case solution. In all cases, the
worst case costs of SSP exceed the worst case value of AARC. Moreover, the AARC
solution guarantees the feasibility and provides a guaranteed upper bound on the optimal
cost. The SSP solution does not guarantee either of the above.
58
Next, we test and summarize the effect of penalty value on the performance of
each approach. Table (4-8) shows that as the value of M changes, AARC always provides
more stable and robust solution than SSP in the aspect of standard deviation and worst
Table 4-6: AARC vs. SP when changes (Beta(5,2), L =50, M =100)
0.025 0.05 0.075 0.1 0.15 0.2 0.25 0.3
Obj. AARC 358.18 366.27 375.24 384.35 402.8 420.8 439.03 457.25
SSP 350.5 350.91 351.31 351.71 352.58 353.82 355.59 357.74
Avg. AARC 355.34 359.7 364.88 370.72 381.79 397.5 410.17 420.57
SSP 368.84 387.57 406.31 425.04 463.3 501.69 540.09 578.5
Gap AARC 0.28% 0.31% 0.56% 0.92% 1.33% 2.89% 3.95% 3.70%
SSP 4% 8% 12% 16% 23% 30% 36% 43%
Std. AARC 0.91 1.76 2.97 3.78 5.35 6.14 7.9 10.12
SSP 9.02 18.04 27.06 36.08 54.3 72.44 90.59 108.72
Worst AARC 357.17 363.27 370.73 378.44 393.29 410.64 427.02 441.57
SSP 392.49 434.88 477.27 519.66 605.5 691.37 777.27 863.15
Table 4-7: AARC vs. SP when changes (Beta(1,1), L =50, M =100)
0.025 0.05 0.075 0.1 0.15 0.2 0.25 0.3
Obj. AARC 358.18 366.27 375.24 384.35 402.8 420.8 439.03 457.25
SSP 350.2 350.3 350.4 350.5 350.74 351.34 352.8 354.63
Avg. AARC 350.86 350.68 350.29 360.15 362.09 364.82 376.34 380.24
SSP 352.51 354.93 357.34 359.75 364.78 370.16 375.96 381.92
Gap AARC 0.41% 0.55% 0.63% 3.66% 4.61% 5.75% 9.37% 10.71%
SSP 1% 2% 3% 4% 5% 7% 9% 11%
Std. AARC 2.16 4.15 6.39 7.12 10.39 14.43 15.8 20.33
SSP 6.13 12.26 18.4 24.53 36.99 48.78 60.93 73.06
Worst AARC 356.04 361.62 367.7 377.49 389.53 403.76 419.08 433.76
SSP 357.97 401.84 427.71 453.59 505.9 560.14 618.25 677.22
59
case solution, and provides an evacuation solution that leads to small gap from the ideal
solution and can meet all the demand.
4.3 Cape May County Network Example
We select another network from Yazici and Ozbay (2007) to increase the size of
the problem and see the robust performance of our approach. An official evacuation route
of Cape May county, New Jersey is considered in Figure (4-7), which is composed of 27
nodes including 3 origin nodes (1,2, and 3) and 1 super destination node (27). All data
except uncertain demand set are adopted from Yazici and Ozbay (2007) and listed in
Table (4-9). For the departure time distribution function, the demand loading equation in
the previous section is used with different parameters, =1 and =6. Also, the penalty
cost ( M ) for unmet demand is 100.
Table 4-8: AARC vs. SP when M changes (Beta(5,2), L =50, =0.1)
M 25 50 75 100
Obj. AARC 384.35 384.35 384.35 384.35
SSP 345.04 347.63 349.46 350.5
Avg. AARC 370.48 370.27 371.06 373.69
SSP 435.43 474.35 470.81 430.37
Gap AARC 0.86% 0.80% 1.02% 1.73%
SSP 19% 29% 28% 17%
Std. AARC 3.91 4.22 3.87 3.34
SSP 19.88 31.82 36.19 37.03
Worst AARC 379.35 379.13 379.61 380.42
SSP 474.46 539.28 552.61 524.02
60
Tables (4-10) and (4-11) show similar results as the previous small example.
AARC approach improves the transportation solution compared to the deterministic
model. Also, we can observe that AARC solution provides better results than SSP in
Table 4-9 Cell properties
Nodes 11-16 The others
t
iN 600 450
t
iQ 1440 1080
ˆix 0 0
Figure 4-7: Cape May county evacuation network (Yazici and Ozbay, 2007)
61
terms of the worse case solution as well as solution stability. The AARC problem with 27
nodes and 45 planning horizon has 4,096,941 constraints and 9,079,890 variables. It is
solved in about 4 hours on a PC with Intel processor 3.0 Ghz and 32 GB of memory.
Table 4-10: Objective value – polyhedral uncertainty
0.025 0.05 0.075 0.1 0.15 0.2 0.25 0.3
Obj. DLP 306901 306901 306901 306901 306901 306901 306901 306901
AARC 370641 379965 389686 399438 419351 439908 460738 511177
Avg. Ideal 366938 373144 379451 385697 398391 411315 424475 437901
DLP 387104 413517 439826 466134 518751 571368 623985 676602
AARC 368196 375368 382618 389101 403235 417660 432095 456290
Std. Ideal 1041 2131 3222 4306 6533 8844 11218 13700
DLP 4367 8769 13154 17538 26307 35077 43846 52615
AARC 598 1437 2143 3307 5449 7731 10066 17039
Worst Ideal 369062 377521 386006 394492 411802 429603 447677 467532
DLP 395998 431356 466584 501812 572267 642723 713178 783633
AARC 369472 378343 387086 396030 414668 433757 452758 497991
Table 4-11: AARC vs. SP when changes (Beta(5,2), L =50, M =100)
0.025 0.05 0.075 0.1 0.15 0.2 0.25 0.3
Obj. AARC 370641 379965 389686 399438 419351 439908 460738 511177
SSP 356996 353143 349344 345584 338247 331121 324304 317763
Avg. AARC 368196 375368 382618 389101 403235 417660 432095 456290
SSP 387661 414563 441409 468254 521959 575665 629294 682927
Gap AARC 0.34% 0.60% 0.84% 0.88% 1.22% 1.54% 1.79% 4.20%
SSP 6% 11% 16% 21% 31% 40% 48% 56%
Std. AARC 598 1437 2143 3307 5449 7731 10066 17039
SSP 4363 8761 13141 17522 26282 35043 43805 52565
Worst AARC 369472 378343 387086 396030 414668 433757 452758 497991
SSP 396562 432436 468218 503999 575578 647151 718651 709139
Chapter 5
Robust Dynamic Network Design Problem
Numerous NDPs for transportation applications have been presented in the past
three decades (see Magnanti and Wong 1984; Minoux 1989; Yang and Bell 1998). These
NDPs are distinguished by a variety of problem settings and supply and demand
assumptions. The literature review presented below by no means provides a
comprehensive survey to general network design problems or to network design
applications in the transportation field; instead, our discussion is focused on those
network design models and solution methods with data uncertainty, particularly network
design problems with time-varying flows.
A great amount of attention has been paid to NDPs with data uncertainty in past
years and various modeling techniques are used for dealing with uncertain input data and
parameters. The main approaches can be classified into two groups: stochastic
programming (SP) and robust optimization (RO). The SP approach requires known
probability distributions of the uncertain data and includes techniques such as the Monte
Carlo sampling approach and chance-constrained programming. For example, Waller
and Ziliaskopoulos (2001) solved a NDP under uncertain demands where the probability
distributions of demand rates are known a priori. They used a CTM-based system-
optimal NDP formulation with chance constraints. Ukkusuri and Waller (2008) extended
the CTM to model both the system-optimal and user-optimal NDPs and presented the
63
formulations of a chance-constrained NDP model and a two-stage resource NDP model
to account for demand uncertainty.
Mulvey et al. (1995) proposed a scenario-based RO approach for general LP
problems. Karoonsoontawong and Waller (2007) applied this approach to a CTM-based
dynamic NDP with stochastic demands under both the system-optimal and user-optimal
conditions. A similar RO model formulation approach was employed by Ukkusuri et al.
(2007), in which a scenario-based robust NDP with discrete decision variables was
tackled by a genetic algorithm. The limitations of scenario-based RO approach are
similar to stochastic programming in that we must know the probability of each scenario
in advance and it is computationally expensive when there are a large number of
scenarios.
Recently, a variety of papers have used the set-based RO technique to characterize
optimization models with data uncertainty. Interested readers are referred to Ben-Tal and
Nemirovski (2002) and Bertsimas et al. (2007) for reviews of the set-based RO methods.
For NDPs with uncertain demands, Yin and Lawpongpanich (2007) considered a static
continuous equilibrium NDP under demand uncertainty. Ordonez and Zhao (2007)
formulated and solved a static multi-commodity NDP with demand and travel time
uncertainties bounded by polyhedral sets. Mudchanatongsuk et al. (2008) extended the
work by considering some generalized assumptions on demand uncertainty, in which they
discussed a path-constrained NDP and introduced a column generation method to solve
the robust NDP with polyhedral uncertainty sets. Ban et al. (2009) considered a robust
road pricing problem (which is an NDP in the broader definition) that contains multiple
traffic assignment solutions. Atamturk and Zhang (2007) formulated and solved a NDP
64
by using the two-stage RO method and taking advantage of the network structure for its
solutions. To characterize their uncertainty sets, they used a budget of uncertainty which
limits the number of observed demand values that can differ from nominal values. They
also discussed the numerical results for a simple location-transportation problem and
compared the two-stage robust approach with the single-stage robust approach as well as
two-stage scenario-based stochastic programming.
There have also been approaches where the set-based RO approach is used to
construct discrete network design models. For example, Lou et al. (2009) described a
discrete NDP with user-equilibrium flows based on the concept of uncertainty budget and
proposed a cutting-plane method for problem solutions; Lu (2007) addressed a discrete
user-equilibrium NDP with polyhedral uncertainty sets using the RO approach and used
an iterative solution algorithm to solve the problem.
To the best of our knowledge, no work has been done in applying the set-based
RO technique to investigate a NDP with dynamic flows and uncertain demands. In this
chapter, our effort is given to analytically developing and numerically analyzing the
robust counterpart model of such an NDP in the context of transportation network design.
The remaining part of this chapter is structured as follows. In Section 5.1, we
generalize the formulation given by Ukkusuri and Waller (2008) as a CTM-based
deterministic dynamic NDP (DDNDP). We then in Section 5.2 propose a robust
counterpart formulation of the DDNDP to account for demand uncertainty, which we
name the RDNDP. Computational experiments and result analyses from applying the
RDNDP model to a few numerical examples are elaborated in Section 5.3.
65
5.1 Deterministic Model
This section presents the deterministic version of the dynamic NDP model we
have discussed, or the DDNDP model in abbreviation, which provides the basic modeling
platform and functional form for the RDNDP model we will introduce in the next section.
For discussion convenience, let us first present the additional notation used throughout
these models (see Table 5-1).
Sets Description
EC Set of cells that can be expanded, SE CCC \
Parameters Description
B Total investment budget available for capacity expansion
if Conversion coefficient of investment cost of cell i for a unit increase of ib
i Increase in capacity of cell i for a unit increase of ib
i Increase in inflow/outflow capacity of cell i for a unit increase of ib
Variables Description
ib Investment cost spent on cell i
The network design problem aims at minimizing the sum of the total system travel
cost and the capacity expansion cost. By assuming the system-optimal principle and the
linear relationship between investment and capacity increase, the DDNDP model can be
written as a LP program with the notation listed in Table (5-1):
ES Ci
ii
t CCi
t
i
t
ibyx
bfxc \
,,min
subject to
Table 5-1: Notations
66
tCidyayaxx t
i
Cj
t
ijij
Ck
t
kiki
t
i
t
i , 1111 (5.1)
tCCiQya E
t
i
Ck
t
kiki ,\ (5.2)
tCibQya Eii
t
i
Ck
t
kiki ,
(5.3)
tCCiNxya E
t
i
t
i
t
i
t
i
Ck
t
kiki ,\ (5.4)
tCibNxya Eii
t
i
t
i
t
i
t
i
Ck
t
kiki , )(
(5.5)
tCCiQya E
t
i
Cj
t
ijij ,\ (5.6)
tCibQya Eii
t
i
Cj
t
ijij ,
(5.7)
tCixya t
i
Cj
t
ijij , 0 (5.8)
Bb
ECi
i
(5.9)
Cixx ii ˆ0 (5.10)
CCjiyij ),( 00 (5.11)
tCixt
i , 0 (5.12)
tCCjiy t
ij ,),( 0 (5.13)
Ei Cib 0 (5.14)
The objective function includes both the travel cost and expansion cost1. The
coefficient if converts the investment cost (money measure) to the travel cost (time
measure). In fact, such coefficient is the reciprocal of value of time, which can be
1 Costs in general do not vary linearly with respect to the transportation facility capacity or size. Typically,
scale economies or diseconomies exist. Abdulaal and LeBlanc (1979) discussed the cases of linear
relationship, scale economies and scale diseconomies in the context of transportation network design
problems. If the average investment cost per unit of capacity is declining, then scale economies exist.
Empirical data are needed to establish the economies of scale for road construction. This paper assumes a
linear relationship between the investment cost and the capacity, for the reasons of simplicity and the
requirement of the linear model. The linear case can be regarded as an approximation to the case of scale
economies in an expected capacity-increasing range. The problem has more to do with cost uncertainty.
67
measured by empirical methods (Wardman (1998)). Note here that the expansion cost
appears in the objective function and is subject to the investment budget constraint, which
makes this formulation different from the traditional charge design problem (where the
expansion cost term is only included in the objective function) and budget design
problem (where the expansion cost terms only appears in the investment budget
constraint).
The constraint set of the DDNDP model specifies the capacity expansion limit, flow
conservation and propagation relationships, initial network conditions and flow non-
negativity conditions. The flow conservation constraint (i.e., Eq. (5.1)) for cell i at time
t can be generalized by setting t
id to be zero in ordinary and sink cells. Constraints (5.2)
and (5.3) are the bounds for the total inflow rate of non-expandable and expandable cell i
at time t , respectively. Similarly, the total outflow rate of cell i at time t is restricted by
constraints (5.6) and (5.7). Constraints (5.4) and (5.4) bound the total inflow rate into a
cell by its remaining space. Constraint (5.8) bounds the total outflow rate of a cell by its
current occupancy, and constraint (5.9) sets the upper bound on the sum of capacity
investments over all cells. The remaining constraints from Eq. (5.10) to (5.14) set initial
network conditions and flow non-negativity conditions.
5.2 Robust Formulation
Now we develop the robust counterpart of the DDNDP model, which incorporates
the demand uncertainty into a LP program via the RO approach. In the deterministic
68
version, Eq. (5.1) is the only set of constraints related to the demand generation. This
equality constraint can be rewritten as an inequality constraint as shown in Section 2.3
1 1 1 1 5.15t t t t t
i i ki ki ij ij i
k C j C
x x a y a y d
It is assumed that all possible demand instances of t
id belong to a box uncertainty
set )],1(),1([ t
i
t
i
t
i
t
idddU t
i
where t
id is the nominal demand level and t
i is the demand uncertainty level. Then, the
robust counterpart of the Eq. (5.15) with demand uncertainty becomes
1
11111 ,
tid
t
i
t
i
Cj
t
ijij
Ck
t
kiki
t
i
t
i Uddyayaxx
This is equivalent to the following inequality,
1
111
11
max
t
iUd
Cj
t
ijij
Ck
t
kiki
t
i
t
i dyayaxxtid
ti
which becomes the flow conservation constraint for the RDNDP model. The above
conversion of the flow conservation constraint leads the RDNDP to be in a deterministic
functional form with the maximum possible demand in the box uncertainty set. Given
that other constraints can be directly transferred from the DDNDP model to the RDNDP
model, the RDNDP formulation can be written into the following LP form:
SS CCi
ii
t CCi
t
i
t
ibyx
bfxc\\
,,min
subject to
tCidyayaxx t
i
t
i
Cj
t
ijij
Ck
t
kiki
t
i
t
i , )1( 11111 (5.16)
tCCiQya E
t
i
Ck
t
kiki ,\
tCibQya Eii
t
i
Ck
t
kiki ,
69
tCCiNxya E
t
i
t
i
t
i
t
i
Ck
t
kiki ,\
tCibNxya Eii
t
i
t
i
t
i
t
i
Ck
t
kiki , )(
tCCiQya E
t
i
Cj
t
ijij ,\
tCibQya Eii
t
i
Cj
t
ijij ,
tCixya t
i
Cj
t
ijij , 0
Bb
ECi
i
CCjiyij ),( 00
tCixt
i , 0
tCCjiy t
ij ,),( 0
Ei Cib 0
In Eq. (5.16), the value of )1( 11 t
i
t
id is the maximum possible demand in cell i at
time 1t , according to the uncertainty set 1tid
U , which represents the worst-case scenario.
Therefore, the optimal solution will remain feasible for all instances of demand. In other
words, we will obtain an optimal solution with the cell capacity values that are adequate
for any realized demand scenarios within the uncertainty set 1tid
U .
We make the following observation between the optimal objective value and the total
budget level B from the RDNDP model. The implication of this property is that the
network designers should consider a budget level as large as possible even if the
objective function minimizes the money used for network expansion together with the
travel cost.
70
Property 12 The optimal objective function value of the RDNDP monotonically
decreases with respect to the investment budget level.
Proof: Let the objective function of RDNDP be )(* Bzr , given the total budget level B
Without loss of generality, we assume that two budget levels 1B and 2B are given as
21 BB . Since the RDNDP with 2B has a larger feasible region than the RDNDP with 1B ,
)( 1
* Bzr is smaller than or equal to )( 2
* Bzr , i.e. )()( 2
*
1
* BzBz rr . ■
When other types of uncertainty sets such as an ellipsoidal uncertainty set or a
polyhedral uncertainty set are assumed, different deterministic formulations are derived.
For example, the equivalent tractable robust counterpart with an ellipsoidal uncertainty
set is a conic quadratic problem; if a polyhedral uncertainty set is assumed, it becomes a
linear problem as explained in Section 2.
5.3 Numerical Analysis
The purpose of presenting computational experiments in this section is twofold: 1)
to demonstrate the difference between robust network design solutions and corresponding
nominal solutions from DDNPP; and 2) to illustrate the advantage of the RO approach for
network design under demand uncertainty. Two numerical examples are selected from
the literature for the experiments: 1) a smaller network with 16 cells and 15 time intervals;
and 2) a larger network with 167 cells and 300 time intervals. For each example, under
the assumption that all cells except destination cells can be invested, we derived the
71
optimal capacity investment solutions and the objective function values from the DDNPP
and RDNDP models with various demand uncertainty levels. To evaluate the solution
robustness, we also conducted a parallel simulation experiment to randomly generate 100
demand instances within the given box uncertainty set. The objective function values
from the simulation experiment are also evaluated by solving the embedded SO DTA
problem based on the same capacity expansion scheme as the one derived by the RDNDP
model.
5.3.1 A Toy Network
The first experiment uses the test network shown in Figure (5-1) and the data set
in Table (5-2), which are from Ukkusuri and Waller (2008). Since they considered a set
of deterministic demands, it is assumed that the demand data in their paper are nominal
values of the network design problem under uncertainty. Let us assume that uncertain
demands from source cell 1 and 14 are [2(1- ), 2(1+ )] at time 0 and 1, and [1(1- ),
1(1+ )] at time 3. Note that when is equal to 0, the uncertainty sets become the
nominal values. The investment cost coefficient ( if ) and penalty cost ( M ) for this
example are set to 0.1 and 10, respectively. The resulting RDNDP model has 748
constraints and 344 variables, which has been solved within 3 seconds on a PC with an
Intel 1.87GHz CPU and 2GB RAM using GAMS/CPLEX.
72
5.3.1.1 Optimal Solutions under Different Uncertainty Levels
The objective function value is calculated and plotted as the total budget level is
varied from 0 to 80 in the interval of 1 unit. Figure (5-2) shows the change of the
objective function values of the DDNDP and RDNDP models with three different
uncertainty levels (including, = 0.1, 0.2 and 0.3). As the budget level increases, the
objective function value of the RDNDP model decreases and it converges to a certain
value (see Property 12). Robust solutions are the best worst-case solutions and thus their
objective function values are greater than those of the corresponding deterministic cases.
Note that any nominal solution is equivalent to its robust solution with the zero
uncertainty level ( = 0).
Figure 5-1: Cell representation of the toy network (Ukkusuri and Waller 2008)
Table 5-2: Cell characteristics of the toy network (Ukkusuri and Waller 2008)
Cell 2 3 4 5 6 7 8 9 10 11 12 15 16
Ni 4 4 4 4 4 2 4 4 4 4 4 4 4
Qi 1 2 2 2 1 2 1 2 1 2 1 1 1
ˆix 0 0 0 0 0 0 0 0 0 0 0 0 0
73
In all the above cases, the same cells (including cells 7, 9, 11, 15 and 16) are chosen
for capacity expansion, which indicates that they are bottleneck cells in the network.
However, the proportions of the investment on the cells are dependent on the investment
budget level and the demand uncertainty level. Figure (5-3) shows the investment
distribution over the cells. The implication behind these distribution curves is that the
investment strategy should be changed depending on the budget bound we set and the
demand uncertainty degree we expect to face.
It is readily observed that there is a critical/maximum investment point associated
with the investment budget level, beyond which a higher investment does not reduce the
travel cost, or a higher investment even increases the objective function value if it is used
for capacity expansion in the network. For example, this maximum investment point is
between 30 and 40 monetary units in the DDNDP case, and the point is about 70
monetary units in the RDNDP case with = 0.3. The critical investment point can be
interpreted as the threshold for investment: when the budget is less than this threshold,
Figure 5-2: The objective-budget relationship under different demand uncertainty levels
74
the marginal travel cost (reduction) is greater than the marginal construction cost
(increase); when the budget is greater than the threshold, the marginal travel cost
(reduction) is less than the marginal construction cost (increase).
5.3.1.2 Worst Case Analysis
After obtaining the investment solutions from the DDNDP and RDNDP models,
we then evaluated the relative improvement of robust solutions from their corresponding
(a) The DDNDP Model (b) The RDNDP Model ( = 0.1)
(c) The RDNDP Model ( = 0.2) (d) The RDNDP Model ( = 0.3)
Figure 5-3: Optimal investment distributions over the network
75
nominal solutions under the worst-case scenario. The relative improvement ( RI ) in this
study is defined as:
5.17d r
r
TC TCRI
TC
where dTC is the total travel cost from the nominal solution and rTC is the total travel
cost from the robust solution.
The following worst-case analysis consists of two parts. First, we fixed the demand
uncertain level and increased the investment budget level B . The computation results
are shown in Table (5-3) and Figure (5-4). When the budget level is low, it is natural that
there is little difference between the nominal and robust solutions. Moreover, when the
investment budget is less than 10 monetary units, the model always selects cell 7 as the
site for capacity expansion, in that it is a merging cell and the bottleneck of the network.
The total travel cost associated from the robust design solutions is slightly lower than that
of the corresponding nominal solutions when the total budget is between 10 and 35 units.
We can also see that the robust solutions significantly outperform the nominal solutions
when the budget is large enough and the demand uncertainty is on a sufficiently high
level. However, the relative improvement of the robust solution against the nominal
solution shown in Figure (5-4) is not necessarily a monotonically increasing function
with respect to the investment budget level. Though rTC and dTC both decrease as the
investment budget level increases, the travel cost reduction rates of the two terms change
over the budget level, which is a result of the tradeoff between marginal investment costs
and marginal travel costs in the two different problem cases.
76
Next, we fixed the total budget level B at four different levels (including 30, 40, 50
or 60 monetary units) with the demand uncertainty level ranging from 0 to 0.5. The
computation result is depicted in Figure (5-5). We can see that with a lower budget level,
the demand uncertainty has a weaker affect on the performance of the RDNDP model.
Table 5-3: Travel cost of robust and nominal solutions in worst-case scenarios
Budget = 0.1 = 0.2 = 0.3
DDNDP RDNDP DDNDP RDNDP DDNDP RDNDP
0 86.7 86.7 97.4 97.4 109.1 109.1 86.7 86.7 97.4 0 86.7
10 79.7 79.7 89.4 89.4 99.6 99.6
20 77.2 76.7 86.4 85.77 95.7 95.35
30 75.03 74.87 83.73 83.65 92.85 92.8
40 74.7 73.53 83.4 82.15 92.35 90.98
50 74.7 72.9 83.4 80.73 92.35 89.48
60 74.7 72.9 83.4 79.8 92.35 87.98
70 74.7 72.9 83.4 79.8 92.35 86.7
80 74.7 72.9 83.4 79.8 92.35 86.7
Figure 5-4: Relative improvement of travel cost in worst-case scenarios under different
demand uncertainty levels
77
However, the solution of the RDNDP model may be largely different from the solution of
the corresponding DDNDP model when the budget level is relatively high. Similar to
Figure (5-4), we can also observe that the relative improvement of the total travel cost of
the robust solution against the nominal solution is not always a monotonically increasing
function with respect to the demand uncertainty level.
5.3.1.3 Simulation Results
Finally, we evaluated the objective function by implementing the robust network
design solutions and nominal solution with random demands generated by the given box
uncertainty sets. Specifically, 100 sets of random data generated from a beta distribution
(i.e., Beta(5, 2)) are used for this evaluation. Note that we only know the support of the
primitive uncertain data and accordingly use box uncertainty sets to characterize the
Figure 5-5: Relative improvement of travel cost in worst-case scenarios under different
investment budget levels
78
bounded uncertain demand. The beta distribution is a reasonable choice for simulating
bounded uncertain data.
Table 5-4: Comparison of simulation results
(a) = 0.1
Budget Mean Standard Deviation Maximum
DDNDP RDNDP DDNDP RDNDP DDNDP RDNDP
0 80.66 80.66 1.52 1.52 83.55 83.55 86.7 86.7 97.4 0 86.7
10 74.36 74.36 1.27 1.27 77.10 77.10
20 72.01 71.78 1.35 1.29 75.16 74.81
30 70.06 70.18 1.38 1.30 72.78 72.72
40 69.82 68.95 1.07 1.01 72.55 71.55
50 69.62 68.34 1.20 1.07 72.14 70.56
60 69.64 68.35 1.16 1.04 71.88 70.39
70 69.62 68.34 1.09 0.98 71.61 70.02
80 69.64 68.36 1.32 1.18 72.79 71.18
(b) = 0.2
Budget Mean Standard Deviation Maximum
DDNDP RDNDP DDNDP RDNDP DDNDP RDNDP
0 85.27 85.27 2.96 2.96 93.02 93.02 86.7 86.7 97.4 0 86.7
10 78.22 78.22 2.51 2.51 83.35 83.35
20 76.16 75.69 2.50 2.41 82.14 81.51
30 73.90 73.83 2.47 2.44 78.84 78.76
40 73.56 72.62 2.49 2.34 78.58 77.33
50 73.51 71.49 2.35 2.18 78.82 76.48
60 72.92 70.39 2.68 2.44 77.92 74.89
70 73.41 70.83 2.20 1.96 78.75 75.57
80 73.65 71.05 2.63 2.35 80.03 76.78
79
The mean, standard deviation, and maximum values of the objective function
values generated from the simulation experiment are shown and compared in Table (5-4).
It can be seen that, in almost every case, the mean objective function value of the robust
solutions is better than that of the nominal solutions; in all cases, the standard deviation
and maximum values of the robust solutions are less than or equal to those of the nominal
solutions.
5.3.2 The Nguyen-Dupis Network
Now we present a second numerical example to show the computational
tractability and the performance consistency of with the RDNDP model in larger
networks. The Nguyen-Dupis network with 13 nodes in total (including 2 source nodes
and 1 super sink node) is considered here (see Figure (5-6)). An equivalent cell network
(c) = 0.3
Budget Mean Standard Deviation Maximum
DDNDP RDNDP DDNDP RDNDP DDNDP RDNDP
0 90.21 90.21 4.24 4.24 99.54 99.54 86.7 86.7 97.4 0 86.7
10 82.88 82.88 4.20 4.20 90.37 90.37
20 79.86 79.44 3.64 3.61 87.50 86.97
30 77.30 77.24 3.99 3.98 86.49 86.58
40 77.42 76.34 3.57 3.52 84.98 83.85
50 76.83 74.57 3.45 3.23 84.96 82.33
60 76.88 73.72 3.92 3.57 85.75 81.93
70 76.88 73.72 3.92 3.57 85.75 81.93
80 77.57 73.64 3.52 3.19 84.35 79.85
80
with 167 cells is created from the original node-link version. The resulting RDNDP
model from the Nguyen-Dupis network has 96,794 constraints and 190,694 variables,
which has been solved in about 60 seconds on a PC with an Intel 1.87GHz CPU and 2GB
RAM using GAMS/CPLEX.
Figure (5-7) shows the optimal objective function values of the DDNDP and
RDNDP models with three different demand uncertainty levels. As similar to the
previous example, there is a set of cells that are chosen for capacity expansion (where, in
Figure 5-6: The node-link topology of the Nguyen-Dupis network
Figure 5-7: The objective-budget relationship under different demand uncertainty levels
5 6 74 8
9 10 11 2
13 3
1 12r
r
s
s
14
81
this case, there are 36 cells in total) in the Nguyen-Dupis network, which delivers a
similar objective-budget relationship to the previous toy example. Investment decisions
vary with different demand uncertainty levels.
The relative improvement of the robust solutions from the corresponding nominal
solutions in the worst-case scenarios is aggregated in Figures (5-8) and (5-9). It is shown
Figure 5-8: Relative improvement of travel cost in worst-case scenarios under different
demand uncertainty levels
Figure 5-9: Relative improvement of travel cost in worst-case scenarios under different
investment budget levels
82
from Figure (5-8) that the robust solution significantly improves the nominal solution
when the investment budget level is greater than 2,200 monetary units, in particular when
the demand uncertainty level is high. A similar phenomenon can be observed from
Figure (5-9).
Table 5-5: Comparison of the robust optimization results and simulation results
(a) = 0.1
Budget Mean Standard Deviation Maximum
DDNDP RDNDP DDNDP RDNDP DDNDP RDNDP
0 7631.63 7631.63 111.42 111.42 7846.08 7846.08 86.7 86.7 97.4 0 86.7
1500 6488.78 6488.61 91.84 91.42 6685.74 6684.25
2500 6391.82 6366.79 78.54 75.08 6549.64 6517.81
3500 6380.82 6353.72 78.88 75.25 6530.00 6497.45
(b) = 0.2
Budget Mean Standard Deviation Maximum
DDNDP RDNDP DDNDP RDNDP DDNDP RDNDP
0 7976.02 7976.02 204.57 204.57 8395.07 8395.07 86.7 86.7 97.4 0 86.7
1500 6788.14 6784.49 182.44 179.68 7192.98 7185.69
2500 6671.00 6645.71 149.01 143.96 7030.38 6990.74
3500 6666.09 6614.52 154.87 146.25 7006.21 6937.27
(c) = 0.3
Budget Mean Standard Deviation Maximum
DDNDP RDNDP DDNDP RDNDP DDNDP RDNDP
0 8389.44 8389.44 369.82 369.82 9046.43 9046.43 86.7 86.7 97.4 0 86.7
1500 7096.40 7089.95 272.83 269.47 7847.98 7838.24
2500 6963.85 6931.39 251.54 241.73 7494.99 7441.00
3500 6957.23 6878.45 297.76 278.49 7520.06 7414.78
83
Finally, the simulation results are compared in Table (5-5). It is found that the
simulated objective function values from DDNDP and RDNPD are comparable when the
investment budget level is less than 1,500 monetary units. However, the robust solutions
provide a lower travel cost when the investment budget goes higher. Our computational
results show that the robust solution is more attractive than the nominal solution from the
simulation experiment. We note that the total travel cost is affected by 1) the network
capacity expansion policy and 2) the underlying traffic flow pattern. Since our focus is
the network design problem, we only tested the impact of robust network capacity
expansion solution in simulation with the deterministic traffic assignment solutions. We
expect the improvement be more significant when both a robust capacity expansion
policy and a robust traffic assignment procedure are used.
Chapter 6
Robust Congestion Pricing Problem
In this chapter, we begin with a robust static user equilibrium optimal toll problem in
general networks. One challenging aspect of robust congestion pricing is the user
equilibrium condition. Under user equilibrium, the nominal problem can be modeled with
a bi-level formulation as a mathematical program with equilibrium constraints (MPEC).
To address this issue, user equilibrium condition is stated as variational inequality (VI) in
the constraint. Using the duality based reformulation technique proposed by Aghassi et al.
(2006); VI is reformulated as a set of equations and a single level optimization problem
equivalent to an MPEC is presented. Under the assumption of linear cost function, we
show the mathematical structure of the reaction function. Finally, the reaction function is
used for the reformulation of robust counterpart of deterministic MPEC and it can be
solved more efficiently than a cutting plane algorithm, which is introduced in Section 3
and can be used for all cases including general cost function cases and dynamic
transportation network cases.
Next, we extend the static problem to consider robust dynamic user equilibrium
optimal tolls. The deterministic mathematical formulation of dynamic optimal toll
problem with equilibrium constraints (DOTPEC) was introduced and studied by Friesz et
al. (2007). In dynamic traffic network, the main purpose of a toll system is to create a
roadway that provide better road situation for commuters within the planning horizon.
This means the limited or extra expense charged to access roads can provide commuters
85
with a short travel time with a faster way of getting from one place to another, especially
during the peak time of the day. Choosing a toll schedule (i.e. toll shape and toll price)
has the impact on the flow pattern. For simplicity, we assume that the toll shapes are
determined in advance through an external selection process, specifically the triangular-
shaped tolls with pre-determined starting and ending period. We note that there are
several types of toll collecting policy in the references (e.g. the uniform toll in Viti et al.
(2003), the time-varying toll selection from two pre-determined price in Nagae and
Akamatsu (2006), the triangular shape in Wie (2007), etc.). Due to the characteristic of
triangular-shaped toll and the pre-determined time interval, it is necessary to decide only
the value of the maximum toll for each tolled arc. According to the triangular shaped toll,
the maximum toll leads the rate of toll within the pre-determined time interval. With the
deterministic DOTPEC problem, we formulate robust counterpart and employ a cutting
plane algorithm and a simulated annealing algorithm for dynamic user equilibrium tolls.
6.1 Motivation
The main focus of this study is to find robust congestion price under demand
uncertainty, which performs well even in the worst case. In this section, we only consider
two demand scenarios for a small network consisting of two competing congested routes
as shown in Figure (6-1) in order to see the impact of uncertain demand. A private firm
wants to maximize the revenue from arc 1 by finding optimal toll price (y) under the
assumption that future demand for origin-destination (OD) pair (1,3) is either 8 or 10.
Unit arc cost ( aic ) is given in Table (6-1).
86
The second-best congestion pricing problem for maximizing a private firm’s revenue
becomes a mathematical program with equilibrium constraint (MPEC) as following:
1,
max
subject to
0
y fyf
f UE flow
y
For each possible future demand, optimal congestion price is calculated from the
deterministic MPEC. When total demand is 8, the optimal toll is 4.5 and the objective
value is 6.75. When future demand is forecasted as 10, the optimal toll is 5.5 and the
objective value is 10.08. The objective values can be interpreted as forecasted revenue
given optimal toll price. The results are summarized in Table (6-2).
Figure 6-1: Two-route network
Table 6-1: Unit arc cost
Index (i) Arc From To Unit cost ( aic )
1 a1 1 2 11 2 f y
2 a2 1 2 22 f
3 a3 1 3 31 f
87
Next, we calculate realized revenue from true demand (Q*), which is either 8 or 10.
As shown in Table (6-3), when we set toll price as 4.5, realized revenue becomes 6.57 or
9.75 depending on the realization of demand. Realized revenue becomes 6.41 or 10.08
when toll price is 5.5. In this example, we can say that toll price 4.5 is more robust than
5.5 since the objective value is less sensitive to uncertain demand and makes more
revenue in worst case.
The remaining part of this chapter is structured as follows. Under the assumption that
demands belong to uncertainty set, we adopt the robust optimization approach and
propose solution algorithms to find robust optimal congestion price for both static and
dynamic transportation networks in Section 6.2 and 6.3, respectively. Computational
experiments and results from three examples are in Section 6.4.
Table 6-2: Solutions from deterministic MPECs
Total Demand Optimal Toll Flow on Path 1 Flow on Path 2 Total Revenue
8 4.5 1.5 6.5 6.75
10 5.5 1.83 8.17 10.08
Table 6-3: Realized revenue
Toll Total Revenue (Q*=8) Total Revenue (Q*=10)
4.5 6.75 9.75
5.5 6.41 10.08
88
6.2 Robust Congestion Pricing for Static Traffic Networks
6.2.1 Deterministic Problem
This section presents the deterministic version and robust counterpart of the static
congestion pricing problem. For discussion convenience, the notations used throughout
these models are presented in Table (6-4).
As mentioned in the previous section, a congestion pricing problem can be
formulated as MPEC, where the upper level aims at minimizing total travel cost or
Table 6-4: Notations
Symbol Description
,i j N Nodes in the network
a A An arc in the network
w W An origin-destination pair
wp P A path between OD pair
[ ]ap The arc-path incidence matrix
[ ]wp The OD pair-path incidence matrix
[ ]wQ Q The vector of traffic demand
[ ]ph h The vector of path flows
[ ]af f The vector of arc flows
( ) [ ( )]ac f c f The vector of arc cost function
( ) [ ( )]pc h c h The vector of path cost function
[ ]ay y The vector of congestion toll on arc a
[ ]a The vector of defining tolled arc; 1a if arc a is tolled, otherwise 0
89
maximizing total revenue. When the goal of a government is the improvement of social
welfare, the upper level objective function becomes
min p p
p P
z c h h
Also, in the view of a private firm operating tolled roads, optimal tolls can be
calculated using the following objective function for revenue maximization. In this case,
the upper boundary of toll price can be restricted by the agreement with a government.
min a a
a A
z y f
The constraints of the congestion pricing problems consist of feasible user
equilibrium (UE) flow and the boundary of toll price given demand vector for OD pairs.
It is well known that the UE problem is an instance of the variational inequality (VI)
problem and thus following constraints are considered.
* *, 0p p p
p P
LB UB
c h y h h h
y y y
where , 0p w p
p P
h Q h
.
VI is equivalent to Linear Programming (LP) problem given *h
* * *, min ,
. .
0
p p p ph
p P p P
p w
p P
p
c h y h c h y h
s t
h Q w
h h
By LP strong duality theorem (Aghassi et al. (2006)), a system of equation equivalent to
the VI is derived.
90
,
,
0
T T
p
T
c h y h Q
c h y
h Q
h
where is dual variable.
6.2.2 RC of MPEC
Based on the deterministic MPEC problem, the robust optimization method is
applied to a congestion pricing problem to deal with uncertainty in demand. We assume
that uncertain demand belongs to box uncertainty sets as defined in Eq. (6.1). The box
uncertainty set is used in RO when the support of uncertain data is known, which is a
relatively mild assumption on distributions.
(1 ), (1 ) 6.1ww Q w wQ U Q Q
where wQ is nominal demand associated with OD pair w and is uncertainty level.
Given the uncertainty set, robust congestion pricing problem (M-RCPP) can be
formulated as the following min-max problem:
, ,min max ( )
subject to
, 6.2
,
0
w
T
h y Q
T T
T
LB UB
w Q
z c h h M RCPP
c h y h Q
c h y
h Q
h
y y y
Q U
91
In this section, we assume linear cost functions to investigate the property of the
robust counterpart and solve M.RCPP more efficiently. Under the assumption of linear
cost, the mathematical structure of reaction function is identified as shown in Lemma 13.
Lemma 13 Under the assumption of linear cost function, reaction function (path flow)
becomes an affine function of demand and toll price.
0 1 2
p p pw w pa a a
w W a A
h Q y
Proof: Let a and a be given cost parameters. Then, arc cost function can be expressed
as follows:
a a a a a ac y f
Let iju be minimum path cost. According to the UE definition, if path flow, *
ph , is
positive, path cost is equal to iju .
* *
1 2 1 2, , 0. 6.3p p ij p pc c u whereh h
An arbitrary path of the network is defined as 1 2, ,...,
m pp a a a , where m p is the
number of arc of path p . Since path cost is sum of associating arc cost, path cost
becomes linear function of toll price and path flow.
1 2
1 2
1 2
, ,...,
, ,...,
'
', ,...,
6.4
m p
m p
m p
p a
a a a a
a a a a a
a a a a
a a a a ap p
p Pa a a a
c c
y f
y h
92
Also, we have flow conservation equation, which is linear w.r.t. total demand Q
6.5w p
p P
Q h w
By solving Eq. (6.3-6.5), we can conclude that
0 1 2
p p pw w pa a a
w W a A
h Q y
. ■
The following results provide a characterization of robust congestion pricing
problems.
Theorem 14 Under the assumption of linear cost function, robust congestion pricing
problem for minimizing total travel cost is equivalent to a deterministic problem with
either minimum or maximum demand.
Proof: Let py be congestion price charged in path p . Path toll is easily calculated with
arc-path incidence matrix and arc toll. By the Eq. (6.2), we know that
T T Tc h h y f Q .
Therefore, the objective function of the robust counterpart can be reformulated as
followings
, ,
, ,
0 1 2
, ,
1 0 2
, ,
min max
min max
min max
min max
T
h y Q
w w p ph y Q
w W p P
w w p p pw w pa a ah y Q
w W p P w W a A
w p pw w p p pa a ah y Q
w W p P p P a A
c h h
Q y h
Q y Q y
y Q y y
93
Depending on the coefficient of wQ , optimal solution (i.e. worst case demand) becomes
1
*
1
(1 ) 0
(1 ) 0.
w p pww
p P
w
w p pww
p P
Q if y
QQ if y
■
Theorem 15 Under the assumption of linear cost function, robust congestion pricing
problem for maximizing total revenue is equivalent to a deterministic problem with either
minimum demand or maximum demand.
Proof: Objective function can be reformulated as followings
, ,
0 1 2
, ,
1 0 2
, ,
min max
min max
min max
p ph y Q
p P
p p pw w pa a ah y Q
p P w W a A
p pw w p p pa a ah y Q
w W p P p P a A
y h
y Q y
y Q y y
Depending on the coefficient of wQ , optimal solution (i.e. worst case demand) becomes
1
*
1
(1 ) 0
(1 ) 0.
p pww
p P
w
p pww
p P
Q if y
QQ if y
■
By Theorem 14 and15, we can conclude that the static robust congestion pricing
problem with linear arc cost is equivalent to a deterministic problem with either
minimum demand or maximum demand. Also, robust congestion price can be found by
solving finite number of deterministic problems. We note that a cutting plane algorithm
94
described in the next section can be used when it is hard to derive the reaction function
(e.g. general networks with nonlinear arc cost function.)
6.3 Robust Congestion Pricing for Dynamic Traffic Networks
In this section, as a foundation of robust dynamic congestion pricing problem, we
first introduce a deterministic DOTPEC problem which has been studied by Friesz et al.
(2007): The key portions of the DOTPEC problem is the time-shifted DUE formulation
in network loading part given in Friesz et al. (2001). As flow propagation constraints hold
the time-shift in equation, it is difficult to handle them especially in computation
perspective. However, Friesz et al. (2010) derive the DAE system which describes the
network loading when the point queue model is invoked, and it may be efficiently and
accurately approximated using a related system of ordinary differential equations by
using the second order Talyor expansion for flow propagation constraints. In following
sections, we describe the Friesz et al. (2010) network loading approach and introduce the
deterministic DOTPEC problem as well as robust counterpart and a solution method.
6.3.1 Dynamic Network Loading
The purpose of the dynamic network loading is to find arc activity when travel
demand and departure rates (path flows) are given. Effective path delays are constructed
from arc delays that, directly or indirectly, depend on arc activity; moreover, activity on a
95
given arc is influenced by the delays on paths traversing that arc. Thus, dynamic network
loading is quite intertwined with the determination of path delays.
6.3.1.1 The Arc Delay Model
Arc volume is the sum of volumes associated with individual paths using an arc; that
is
p
a ap a
p P
x t x t a A
where p
ax t denotes the volume on arc a associated with path p and
1 if arc belongs to path
0 otherwise. ap
a p
We make use of the simple deterministic arc delay model suggested by Friesz et al.
(1993). In that model, we denote the time to traverse arc ia for drivers who arrive at its
tail node at time t by i ia aD x t .
6.3.1.2 The DAE System
It is well known that application of the chain rule to
1 1 1 6.6p
a a at D x t p P
where i
p
a t is time of exit from arc 1,i m p for path p P , and
1 1
, 2, 6.7i i i i i
p p p
a a a a at D x t p P i m p
allows one to derive the flow propagation constraints
96
1 1 1 1 1 1
1
'
'
1 6.8
1 , 2, 6.9i i i i i i i
a a a a a a p
p p
a a a a a a a
g t D x t D x t x h t
g t D x t D x t x t g t p P i m p
from Eq. (6.6) and (6.7); see for example Friesz et al. (2001). Therefore, we acquire the
following differential algebraic equation (DAE) system describing dynamic network
loading:
1
1 1 1 1 1
1
,0 1
, ;
'
, 1, 6.10
0 , 1, 6.11
1 6.12
1 , 2, 6.13
i
i i
i i
i
i i i i i i i
p
a p p
a a
p p
a a
k
p a a a a a a
p p
a a a a a a a
dx tg t g t p P i m p
dt
x x p P i m p
h t g t D x t D x t x
g t g t D x t D x t x t p P i m p
where i
p
ag t is the flow along path p that exits arc ia at time t ; by convention
0
,p k k
a pg h p P
is departure rate from the origin of path p P .
6.3.1.3 A simplified Network Loading Procedure
The main concern in the DAE system (6.10), (6.11), (6.12) and (6.13) in computation
perspective is to solve the time shifts appearing in the flow propagation constraints. A
second order Taylor series approximation of the time shifted term of the flow propagation
constraints yields
97
1 11 1
1 1 1 1 1 1
22
2
22
2
2
, 2,2
i
i i i i i i
i ii
p pa aa ap p
a a a a a a
p
ap p
a a a a a a
pa aa
D x tdg t d g tg t D x t g t D x t p P
dt dt
dg tg t D x t g t D x t
dt
D x td g tp P i m p
dt
When the above approximations are made and appropriate dummy variables are
introduced, the DAE system may be approximated by a system of first order ordinary
equations having specific initial conditions. That system of equations is articulated in full
in Friesz et al. (2010).
6.3.1.4 Constructing the Path Delay for a Given kh
Using the recursive relationships of Eq. (6.6) and (6.7), the total traversal time for
path p may be expressed in terms of the final exit time function and the departure time:
1
1i i m p
m p
p p p
p a a a
i
D t t t t p P
where i
p
a t is the time of exit from arc 1,i m p for path p P given departure from
the origin at time t . We assume that the effective delay includes a potentially asymmetric
arrival penalty operator F ; thus, the effective delay operator is
p p p Ac D F t D T p P
where AT is the desired arrival time. If the path flows kh are known, it is possible to find
the arc exit flows, volumes and delays. Let us denote the traffic volumes from the DAE
solution for given kh by
98
, : , 1,i
k p k
ax x p P i m p
and define
,k p k
a ap a
p P
x t x t a A
.
The arc exit time functions may be computed for a path by first noting that
1 1 1
, , ,p k k
a a at t D x t
where t is the departure time. Once the arc exit time function of the first arc has been
computed, the arc exit time function for the next arc in the path may be computed as
2 1 2 2 1
, , , ,p k p k k p k
a a a a at D x t
and so forth until the arc exit times of all arcs have been computed. This procedure is
carried out for each path p P . When the arc exit time functions ,
i
p k
a are kwon for all
p P and 1,i m p , the effective path delay may be computed as pure functions of
time following
, ,
m p m p
k p k p k
p a a Ac t t t F t T
.
6.3.2 Robust Dynamic Congestion Pricing Formulation
We have studied the approximated network loading in the previous section, which
allows us to solve the dynamic user equilibrium efficiently. Furthermore, it is certain that
we have to consider the efficient toll which should exist in the form of effective path
delay operator. Hearn and Yildirim (2002) studied the efficient toll in the static
congestion pricing with the linear traveling cost for traffic flow. The objective of the
99
efficient toll is to make the user equilibrium traffic flow equivalent to the system
optimum by appropriate congestion pricing. To study the dynamic efficient toll problem,
we introduce the notion of a tolled effective delay operator:
, ,p p p p At h t y t y t D F t D T p P
where py t denotes the toll for path p . It is easy to observe that
, , ,p p pt h t y t y t c t h t .
In order to apply the toll meaningful, we set the boundary of toll
LB UBy y y
where LBy and UBy are lower bound and upper bound, respectively. Therefore, a dynamic
tolled user equilibrium must obey
0
* *, , 0,
ft
p
p P t
t h t y t h t h t h
where
0
, 0,
ft
p w p
p P t
h t Q h t w W
.
Furthermore, the dynamic system optimum can be achieved by solving the
0
0
min ,
subject to
0
f
f
w
t
p p
p P t
t
p w
p P t
p
z c t h t h t
h t Q w W
h t p P
100
6.3.2.1 Dynamic Optimal Toll Problem with Equilibrium Constraints
In order to consider the dynamic optimal toll problem, it is obvious that the dynamic
tolled user equilibrium and the dynamic system optimum problem should be considered
at the same time. Consequently, DOTPEC problem has the form of a dynamic system
optimum objective function with the dynamic tolled user equilibrium as constraints. As
mentioned before, this is called mathematical programming with equilibrium constraints
(MPECs). Furthermore, we can notice that the state dynamics as well as all other state
and control constraints in the dynamic tolled user equilibrium are identical to those
introduced above for the dynamic system optimum.
Now we introduce the dynamic optimal toll problem with equilibrium constraints as
following
0
0
* *
min ,
. .
, , 0, 6.14
6.15
f
f
t
p p
p P t
t
p
p P t
LB UB
z c t h t h t
s t
t h t y t h t h t h
y y y
where
0
, 0,
ft
p w p
p P t
h t Q h t w W
.
The DOTPEC is a type of dynamic network design problem for which a central
authority (upper level objective function) tries to minimize congestion in order to
maximize the social welfare in transport network whose flow obey a dynamic network
user equilibrium by dynamically adjusting tolls. Dynamic user equilibrium is the solution
101
from variational inequality equations, Eq. (6.14). However, we have not mentioned that
Eq. (6.14) is equivalent to a differential variational inequality. This can be shown easily
by noting that the flow conservation constraints may be re-stated as
0
,
( ) 0,
( ) ,
w
wp
p P
w
w f w
dsh t w W
dt
s t w W
s t Q w W
which is recognized as two-point boundary problem value problem. Therefore, the
constraints (6.14) and (6.15) may be expressed as following including differential
variational inequality
0
* *, , 0, 6.16
ft
p
p P t
LB UB
t h t y t h t h t h
y y y
where
00; , ( ) 0, ( ) ,w
wp w w f w
p P
dsh h t s t s t Q w W
dt
.
Theorem 16 Dynamic user equilibrium equivalent to a differential variational inequality.
Assume 1
0, , : ,p fh y t t is measurable and strictly positive for all p P and all
h . A vector of departure rates (path flows) *h is a dynamic user equilibrium if
and only if *h solves differential variational inequality, as defined by (6.16).
Proof: See Friesz et al. (2010). ■
102
It is quite complicated to solve the differential variational inequality due to the
fact that the effective path delay operator *, ,p t h t y t is typically neither monotonic
nor differentiable. Friesz et al. (2010) study this and suggest an iterative method in
Hilbert space for a fixed point problem equivalent to the differential variational inequality.
Therefore the solution of differential variational inequality may be obtained by solving an
appropriate fixed point problem.
Theorem 17 Fixed point statement. Assume that 1
0, , : ,p fh y t t is measurable
for all p P , h Q . Then the fixed point problem
, ,Q
h P h t h y
is equivalent to (6.16) where QP
is the minimum norm projection onto Q and
1 .
Proof: See Friesz et al. (2010). ■
Naturally Theorem 5 suggests the algorithm
1 , ,k k k
Qh P h t h y
which is clearly a particular instance of the abstract algorithm
1k kx M x
for solving the fixed point problem
x M x
103
where x V , a Hilbert space. Convergence of the fixed point algorithm has been shown
in Friesz et al. (2010). Finally we may show that the algorithm itself has the form given
below:
Fixed Point Algorithm for DUE , ,r
Step 0. Initialization
Select 0h and the rule for generating the sequence k . Also select a stopping tolerance
1 . Set 0k
Step 1. Major Iteration
Compute 1 0 1 ,k k k
k k Qh h P h t h
Step 2. Stopping test
If 1|| ||k kh h , stop and declare ,* , 1kh h . Otherwise, set 1k k and go to Step 1.
In addition, 1
1
q
kk
, where q is a parameter to be designed and which must satisfy
0 1q .
6.3.2.2 Robust DOTPEC Problem
As we have considered uncertainty for total demand in static case problem, here
we define an uncertainty set for total demand during planning horizon
(1 ), (1 ) 6.17ww Q w wQ U Q Q
104
where wQ is nominal demand associated with OD pair w and is uncertainty level.
With considering the uncertainty of total demand, we may formulate our robust DOTPEC
problem as following
1
0
0
1, ,
* *
1
min
subject to
, , 0, 6.18
, 6.19
f
f
h y z
t
i i i i i
p
p P t
t
i i
p p
p P t
LB UB
z
t h t y t h t h t h Q
c t h t h t z
y y y
where
00; , ( ) 0, ( ) , ,w
w
ii i i i i iw
p w w f w w Q
p P
dsh h t s t s t Q Q U w W
dt
.
In addition, ih and i
ws are departure rate and dummy variable for flow conservation
constrains with respect to total demand i
wQ , respectively. The variable 1z in objective
function represents the maximum total travel cost. Therefore, the objective function seeks
to minimize the maximum possible total travel cost. A vector *h is the dynamic user
equilibrium departure rate with respect to a given uncertain total demand vector. We may
recognize that Eq. (6.18) is DVI and the number of constraints for DVI is infinite due to
the uncertainty set QU . Furthermore, Eq. (6.19) in conjunction with the objective function
try to solve the maximum total travel cost over iQ .
The given formulation for Robust DOTPEC problem is also an MPEC problem
which is a class of non-convex optimization problem. Furthermore, as we mentioned
above, the number of constraints for DVI is infinite as the number of decision variables is
105
infinite. This fact compounds the difficulty of MPEC problem. In order to overcome the
difficulties, Yin and Lawphongpanich (2007) propose a cutting plane algorithm based on
scenarios especially for robust network design problem and also provide the convergence
of the optimal solution under some assumptions. This algorithm allows us to adapt Yin
and Lawphongpanich (2007) to robust DOTPEC problem.
6.3.2.3 Cutting Plane Algorithm
The algorithm used in this section is called a cutting constraint or plane algorithm
which have been used by Kelley Jr (1960), Marcotte (1983), and Lawphongpanich and
Hearn (2004). In order to apply a cutting plane algorithm approach to solve robust
DOTPEC problem, we need to assume a finite number of candidate total demand from
Eq. (6.17)
1 2ˆ , ,..., nQ Q Q Q
then relaxed robust DOTPEC (RR-DOTPEC) problem can be written as
1
0
0
1, ,
* *
1
min
subject to
, , 0, , 1,...,
, 1,...,
f
f
h y z
t
i i i i i
p
p P t
t
i i
p p
p P t
LB UB
z
t h t y t h t h t h Q i n
c t h t h t z i n
y y y
where
0ˆ0; , ( ) 0, ( ) , ,
w
ii i i i i iw
p w w f w w w
p P
dsh h t s t s t Q Q Q w W
dt
106
Let *
1z be a global optimal solution to relaxed robust DOTPEC, then *
1z solves the
original robust DOTPEC problem only if the dynamic user equilibrium *ih associated
with every ˆiQ Q has a total travel cost no larger than *
1z such as
0
* * *
1, 1,..., 6.20
ft
i i
p p
p P t
c t h t h t z i n
.
In order to verify the inequality Eq. (6.20), let us consider the following the worst case
demand problem (WCD) with
0
0
2
* *
max ,
. .
ˆ, , 0, , 1,...,
f
i
f
t
i i
p pQ
p P t
t
i i i i i
p
p P t
z c t h t h t
s t
t h t y t h t h t h Q i n
where
00; , ( ) 0, ( ) , ,w
w
ii i i i i iw
p w w f w w Q
p P
dsh h t s t s t Q Q U w W
dt
.
For given y t , the objective function of the worst case demand problem is to find a total
demand in Q whose dynamic user equilibrium flow yields the maximum total travel cost.
We may observe the two cases based on the worst case demand problem with the relaxed
robust DOTPEC problem. If ˆ iQ is the global solution to the worst case demand problem
and 2z with ˆ iQ is less than equal to *
1z , then y is an optimal toll. On the contrary, if 2z
with ˆ iQ is larger than *
1z , then an improved solution may be obtained by solving the
relaxed robust DOTPEC problem with adding the ˆ iQ to the demand scenario set such as
ˆ ˆ ˆ iQ Q Q
107
Finally, we may show that the algorithm itself has the form given below:
Cutting Plane Algorithm
Step 0 Set 1k and determine initial demand scenario 1Q .
Step 1 Solve R-DOTPEC with finite number of demand scenarios Q . Let 1 ,k kz y be the
objective value and optimal congestion price.
Step 2 Solve WCD with given toll price ky . Let 2 ,k kz d be the objective value and worst
case demand.
Step 3 If 2 1
k kz z , stop and ky is a robust congestion price vector. Otherwise set
1ˆ ˆk k kQ Q d and 1k k . Go to Step 1.
6.3.2.4 Simulated Annealing Algorithm
Relaxed robust DOTPEC problem is mathematical programming with equilibrium
constraints (MPECs). The MPECs problem is known to be non-convex; hence it is
difficult to solve for a global optimal solution. There are a few papers for the MPECs
problem to solve the dynamic transportation network problem with our knowledge. In
this section, we adapt a simulated annealing approach by Friesz et al. (1992) as a sub-
problem of a cutting plane algorithm, which would have significant application to large
scale problems. Kirkpatrick et al. (1983) propose a simulated annealing approach to find
out a relation to the mechanics of annealing solids. The concept of a simulated annealing
starts with atoms state in the system. For example, if the system is in high temperature
108
state, then the atoms in the system stay in a highly disordered state, which leads the
overall energy of the system is high too. The way to get the atoms into a highly orderly
state is to reduce the energy of the system. This is accomplished by lowering the
temperature of the system. Consequently, atoms may reach an equilibrium state at every
temperature stage. Interestingly, the scheme employed to reduce temperature may be
applied to MPECs problem. In this section, a simulated annealing algorithm is used to
solve R-DOTPEC and WCD. Now, we may show that the algorithm itself has the form
given below:
Simulated Annealing Algorithm
Step 0 Determine an initial value kT (temperature at stage k ), kQ (step size
distribution) , ( , )m ky , and M (the number of iteration at each temperature stage); set 1k
for temperature stage and 1m where m is the iteration at each temperature stage.
Step 1 Solve the variational inequality problem for given ( , )m ky where ( , )m ky is the value
of y on the thm step at thk temperature stage; otherwise go to Step 6.
Step 2 If m M ; in order to determine a candidate optimal solution, the enhancement
variables are randomly perturbed from their current values according to
1, , ,m k m k m ky y y
where ,m k ky Q u . u is a random vector and each iu is randomly and independently
chosen from the normalized interval 3, 3
.
Step 3 Solve the variational inequality problem for given y(m+1;k):
109
Step 4 If ( 1, ) ( 1, ) ( , ) ( , ) ( , ) ( 1, ),m k m k m k m k m k m ky y y y and 1m m . Then go to Step 1.
Otherwise go to Step 5.
Step 5 Calculate the
1, ,
exp
m k m k
k
B
Pk T
where Bk is the Boltzman constant and compare with a random number 0,1R . If R is
less than or equal to kP , then the ( , ) ( 1, )m k m ky y and 1m m . Then go to Step 2.
Otherwise the ( , ) ( , )m k m ky y and 1m m and go to Step 2.
Step 6 Calculate the
( , )
1
( , ) ( , )
1
16.21
16.22
i
i j
Mk m k
i a
m
Mk m k k m k k
ij a i a j
m
A yM
S y A y AM
The covariance matrix s for the next temperature stage 1k is chosen as follows
1k kss SM
where s is the growth factor, typically > 1.
Step 7 Obtain 1kQ corresponding to any desired covariance matrix s
1 1 1T
k k ks Q Q
Then (1, ) ( , ) 1, , 0.8k M k k k k ky y Q Q T T and 1k k .Then go to Step 1
In the Vanderbilt and Louie (1984) procedure, a measure of the local topography
is developed using information from excursions of the random walk at a given
110
temperature stage. At the end of thl temperature stage, the first and second moments of
the walk segment are calculated for each enhancement variables k
iA and k
ijS in Eq. (6.21)
and Eq. (6.22), respectively. S describes the actual segment of a random walk. For
determining the step size, Vanderbilt and Louie (1984) suggest a self-regulating
mechanism for the step size determination which insures an efficient choice of step size
as shown in step 7 in the simulated annealing algorithm.
6.4 Numerical Experiments
The purpose of the numerical experiments in this section is to illustrate the
advantage of the RO approach for congestion pricing under demand uncertainty. We
implemented algorithm with MATLAB and GAMS and solved three problems: two static
problems and one dynamic problem. Since the purpose of this section is to find robust toll
price, we assume that congestion price is positive. That is, LBy is 0 and UBy is . For
each example, a parallel simulation experiment is conducted to evaluate the robust
solution with 100 demand instances randomly generated by the given box uncertainty sets.
6.4.1 Static Two-route Network
First, we select a static two-route problem, which is a classical example of the
second best problem. One tolled route and one untolled alternative route are available in
the network as shown in Figure (6-1). Arc cost summarized in Table (6-1) is used for this
example. It is assumed that uncertain demand from node 1 to node 3 is 10(1 ),10(1 ) .
111
Uncertainty level is increased from 0.1 to 0.3. We note that uncertain data set becomes
nominal data when is equal to 0.
The objective value of the revenue maximization problem and the optimal
congestion price are calculated with nominal, minimum and maximum demand scenarios
in Table (6-5). A deterministic problem with nominal demand is solved as a bench mark
problem and the other demand scenarios are solved to find the robust optimal solution.
According to the Theorem 14, optimal toll price from a deterministic problem with
minimum demand becomes robust optimal solution since it has lower (i.e. worse)
objective value than maximum demand case. We also implemented the cutting plane
algorithm (CPA) to find the robust toll price. As shown in Table (6-5), the optimal
solution from the cutting plane algorithm is very close to the deterministic solution of
minimum demand case.
In simulation, realized revenue is calculated by implementing the robust
congestion pricing solution and the nominal solution with random demand from uniform
distribution (i.e. 10(1 ),10(1 )U ). Worst case revenue, mean and standard deviation
are summarized in Table (6-6). Relative improvement with robust solution is also
calculated and compared. It can be seen that robust solution provides more stable
Table 6-5: Objective value and optimal toll
Objective Value Optimal Toll
Nom Min Max CPA Nom Min Max CPA
0.1 10.08 8.33 12.00 8.33 5.5 5.0 6.0 4.99
0.2 10.08 6.75 14.08 6.75 5.5 4.5 6.5 4.51
03. 10.08 5.33 16.33 5.33 5.5 4.0 7.0 3.99
112
objective value and performs well in the worst case. However, mean value with robust
toll price is slightly worse than that with nominal toll price due to the conservativeness of
the robust solution.
6.4.1 Static Braess Network
In general, we think that arc flow is reduced as total demand decreases and thus
minimum demand becomes worst case scenario in a revenue maximization problem.
However, this is not always true in a congestion pricing problem. A second static
problem is presented to show the case where a robust congestion pricing problem is
equivalent to a deterministic problem with maximum demand. The network configuration
and unit arc cost for each arc are given in Figure (6-2) and Table (6-7), respectively. The
total demand associating with OD pair (1,2) is given as 12 20(1 ),20(1 )Q
Table 6-6: Simulation results
Worst Case Mean Standard Deviation
Nom. Rob. Nom. Rob. Imp. Nom. Rob. Imp.
0.1 8.27 8.35 9.82 9.76 -1% 1.01 0.92 10%
0.2 6.45 6.78 9.55 9.32 -3% 2.02 1.65 22%
03. 4.64 5.37 9.29 8.76 -6% 3.03 2.20 33%
113
Similar to the previous section, the objective value and optimal solution of three
deterministic problems are calculated. It is shown from Table (6-8) that the objective
value with maximum demand is less than that of minimum demand. For example, when
demand is 18 (i.e. minimum demand case when 0.1 ), optimal toll price is 8 and total
revenue is 25.6. However, in the case of maximum demand, optimal toll price is 7 and
total revenue becomes 19.6, which is worse than minimum demand case. It can be
interpreted as Braess paradox. Originally, the paradox was observed in the equilibrium
network design problem (Braess (1968)) and later in congestion pricing problem (Tan et
Figure 6-2: Braess network
Table 6-7: Unit arc cost
Index (i) Arc From To Unit cost ( aic )
1 a1 1 3 115 2 f
2 a2 3 4 210 2y f
3 a3 4 2 315 2 f
4 a4 1 4 450 f
5 a5 3 2 550 f
114
al. (1979)). In this problem, flow on 2a decreases as the demand increases. Therefore, in
the view of a private firm collecting toll on 2a , maximum demand case becomes worst
scenario. In this example, the cutting plane algorithm also provides an optimal solution
close to the deterministic solution with maximum demand.
The simulation results are compared in Table (6-9). In this example, it is also
found that robust solution provides better results in the worst case. The realized objective
value with robust solution is more stable than using the deterministic counterpart solution.
Table 6-8: Objective value and optimal toll
Objective Value Optimal Toll
Nom Min Max CPA Nom Min Max CPA
0.1 22.5 25.6 19.6 19.56 7.5 8 7 7.02
0.2 22.5 28.9 16.9 16.9 7.5 8.5 6.5 6.51
03. 22.5 32.4 14.4 14.39 7.5 9 6 6.02
Table 6-9: Simulation results
Worst Case Mean Standard Deviation
Nom. Rob. Nom. Rob. Imp. Nom. Rob. Imp.
0.1 19.51 19.61 22.91 22.78 -1% 1.64 1.54 7%
0.2 16.52 16.92 23.32 22.81 -2% 3.29 2.85 15%
03. 13.53 14.43 23.72 22.58 -5% 4.93 3.94 25%
115
6.4.3 Dynamic Three-arc Four-node Network
In this section, the 3-arc 4-node network illustrated in Figure (6-3) is considered
to illustrate a robust DOTPEC problem. The set of OD pairs is 1,4 , 2,4W . Note
that there is only one path for each OD pair of this test problem, namely
1 14
2 24
1,3
2,3
p P
p P
There is uncertain travel demand for each OD pair; that is 14 24, 90,100Q Q .
The commuting period is between 0t =08:00AM and ft =09:00AM. The desired
arrival time is 08:30 AM for OD pairs (1,3) and (1, 4). The parameters used for the linear
arc delay function a a a a aD A y B x are given by 14 0.003 f , 13 0.0025y f and
12 0.002y f from arc 1 to 3, respectively.
We solved this problem using the cutting plane algorithm proposed in Section 3,
which is solved within 9 iteration and each DUE problem is solved around 10 seconds on
a PC with Intel processor 2.00 Ghz and 3GB of memory. The congestion toll is charged
from 08:15 AM to 08:45 AM and the maximum toll of the R-DOTPEC problem is 9.78
with total cost of 5,740.4. It means that the total travel cost is at most 5,740.4 with the
Figure 6-3: 3-arc 4-node network
116
robust toll price. In contrast, the maximum toll of the deterministic solution is 8.71 and
objective value is 5154.3. Figure (6-4) shows the optimal dynamic congestion price on
arc 2 for both the deterministic problem and the robust counterpart. The path flow and
tolled travel cost for the deterministic problem are shown in Figure (6-5) and (6-6).
Figure 6-4: Toll price on arc 2
Figure 6-5: Departure rate and tolled unit travel cost for path 1
117
Simulation results are summarized in Table (6-10). Using the robust solution, the
worst case objective value among 100 random demands is 5740.4. The worst objective
value is 5799.1 with deterministic DOTPEC solution. We can see that robust solution
provides guaranteed upper bound of objective value and performs well in terms of worst
case solution, mean and standard deviation even though the improvement of the robust
solution is not significant in this toy problem.
Figure 6-6: Departure rate and tolled unit travel cost for path 2
Table 6-10: Simulation results
Nom. Rob. Imp.
Worst Case 5780.59 5722.41 1%
Mean 5215.89 5182.85 1%
Standard Deviation 355.26 330.42 8%
118
Chapter 7
Conclusion
This thesis applied robust optimization approach to operation and planning
problems of dynamic supply chain and transportation networks under uncertainty in
demand. First, we have formulated a deterministic SO DTA problem based on CTM and
develop the robust counterparts (RC) by considering various uncertainty sets including
box, polyhedral and ellipsoidal uncertainty set. Theoretically, it is shown that the robust
counterparts are computationally tractable but robust solution may be conservative. We
also proposed the affinely adjustable robust counterparts (AARC) to provide less
conservative solution than the RC. It was shown that the affinely adjustable formulation
is computationally tractable.
Based on the RC and AARC developed, we have studied an application in
emergency management and provide numerical experiment results for two emergency
logistics planning examples. Two S-shaped curves with upper and lower bound was
introduced by considering uncertainty sets, which are appropriate for modeling uncertain
demand. The AARC solution was benchmarked against an ideal solution with complete
future information, deterministic LP, and sampling based stochastic programming. It was
shown that AARC approach leads to high quality solutions compared to the deterministic
problem and the sampling based stochastic problem.
Next, we have described a robust optimization approach for a network design
problem explicitly incorporating CTM based SO DTA model and demand uncertainty.
Numerical experiments showed that the robust optimization approach can provide better
119
network design solutions that produce lower objective function values than the
corresponding deterministic approach, especially at a high demand uncertainty level and
a high investment budget level.
Finally, we considered congestion pricing under demand uncertainty. We
proposed to apply a robust optimization (RO) approach to optimal congestion pricing
problems under user equilibrium. In particular, we first discussed robust static user
equilibrium optimal toll. Next, we extended to consider robust dynamic user equilibrium
optimal toll based on a formulation of the dynamic optimal tool problem with equilibrium
constraints, or DOTPEC. Finally, we conducted numerical experiments and qualitative
analysis to investigate the performance and robustness of the solutions obtained. It was
shown that the robust solutions provides guaranteed upper bounds or objective values and
performs better than nominal solutions.
However, through the various examples, we do not argue that the AARC
approach always outperforms the stochastic programming. The proposed method is
favorable when either reliable information on probability distribution of uncertain
parameter is not available or decision makers want to find a strongly guaranteed
performance without having to face infeasible solutions even in extreme cases. In those
cases, RO can outperform the traditional stochastic programming approach.
Numerous future research directions remain. In the emergency logistics problem,
our work has focused on the CTM based SO-DTA problem by using affine control rule
for uncertain demand. The reason for using the linear decision rule is to derive
computational tractable problem. However, theoretically, we do not know how the
approximation makes the robust solution be deviated from the optimal solution. The
120
approximation approach is used based on the belief that it is important to provide a
solvable problem in emergency logistics field (Shapiro and Nemirovski (2005), Remark
2). The scope of future work could be extended to consider control beyond linear
decision rule. Robust optimization approach can be applied to different uncertainty
sources (e.g. capacity uncertainty or cost uncertainty) and other supply chain and
transportation problems.
Second, there are other issues raised from the CTM based SO-DTA problem. One
of these issues is that LP based CTM model allows vehicle holding, which may be
unrealistic. RO approach can be applied to alternative deterministic mathematical
formulations (e.g. Zheng, 2009; Nie, 2010) to overcome this issue.
In some cases, the RC do not have fixed recourse and it becomes an intractable
problem and rolling horizon RC can be considered to find a robust solution. Extension to
considering unbounded uncertainty set with globalized robust optimization (Ben-tal. et al.,
2006), chance constraint programming or other approximation methods can be another
straightforward research direction.
Next, the ambiguous chance-constrained programming can be applied to the
models when we have more information about the uncertain data. For example, this
approach may be particularly interesting when we only know the support and mean of
uncertain parameters or when we know that demand can arise from a set of distributions.
Also, in this thesis, we demonstrated the performance of robust solution with
rather small examples. As a future search direction, RO method will be applied to
realistic large-scale networks. Theoretically, the proposed mathematical formulations are
solvable in polynomial time. However, they still require powerful computational resource.
121
We may reduce computational burden by incorporating graph based CTM formulation
and efficient large scale methods such as network simplex method, interior point
algorithm, first order methods or alternative heuristic approaches.
Finally, agent-based simulation (ABS) can be used to see what uncertain factors
affect the robust solution and the benefit of the robust solution. Even though we
developed a robust solution for system optimization, individual entities in the system may
not follow the solution for selfishness. In such a complex system, it is very hard to derive
analytical solutions and find managerial implications. An ABS approach may be
plausible in this case since it is able to model interactions of robust solution, human
behavior and dynamics of uncertain factors.
References
Abdulaal, M., and L. J. LeBlanc. 1979. Continuous equilibrium network design models.
Transportation Research Part B 13: 19-32.
Aghassi, M., D. Bertsimas, and G. Perakis. 2006. Solving asymmetric variational
inequalities via convex optimization. Operations Research Letters 34 (5): 481-
490.
Ahuja, R.K., T.L. Magnanti, and J.B. Orlin, 2003. Network flows: Theory, algorithms and
applications. New Jersey: Pretince Hall.
Arnott, R., A. De Palma, and R. Lindsey. 1990. Departure time and route choice for the
morning commute. Transportation Research Part B 24 (3): 209-228.
Arnott, R. and M. Kraus. 1998. When are anonymous congestion charges consistent with
marginal cost pricing? Journal of Public Economics 67 (1): 45-64.
Atamturk, A. and M. Zhang. 2007. Two-Stage robust network flow and design under
demand uncertainty. Operations Research 55(4): 662-673.
Ban, X., S. Lu, M. Ferris, and H. Liu. 2009. Risk averse second best toll pricing.
Transportation and Traffic Theory 2009, Golden Jubilee, 197-218.
Bazaraa, M.S., J.J.Jarvis, and H.D. Sherali. 2005. Linear programming and network
flows. New Jersey: Wiley.
Ben-Tal, A. and A. Nemirovski. 1998. Robust convex optimization. Mathematics of
Operations Research 23(4): 769-805.
123
Ben-Tal, A. and A. Nemirovski. 1999. Robust solution of uncertain linear programs. OR
Letters 25: 1-13.
Ben-Tal, A. and A. Nemirovski. 2000. Robust solutions of linear programming problems
bontaminated with uncertain data. Mathematical Programming 88(3): 411-424.
Ben-Tal, A. and A. Nemirovski. 2002. A robust optimization- methodology and
applications. Mathematical Programming 92(3): 453-480.
Ben-Tal, A., A. Goryashko, E. Guslitzer and A. Nemirovski. 2004. Adjustable robust
solutions of uncertain linear programs. Mathematical Programming 99(2): 351-
376.
Ben-Tal, A., B. Golany and A. Nemirovski. 2005. Retailer-supplier flexible
commitments contracts: a robust optimization approach. Manufacturing & Service
Operations Management 7(3): 248-271.
Ben-Tal, A., L.E, Ghaoui and A. Nemirovski. 2007. Robust optimization.
http://www2.isye.gatech.edu /~nemirovs /RBIntroTOC.pdf.
Ben-Tal, A., S. Boyd and A. Nemirovski. 2006. Extending scope of robust optimization:
comprehensive robust counterparts of uncertain problems. Mathematical
Programming 107 (1-2): 63-89.
Bertsimas, D., D.B. Brown and C. Caramanis. 2007. Theory and applications of robust
optimization.
http://users.ece.utexas.edu/~cmcaram/pubs/RobustOptimizationSV.pdf.
Bertsimas, D. and G. Perakis. 2005. Robust and adaptive optimization: a tractable
approach to optimization under uncertainty. NSF/CMMI/OR 0556106.
124
Bertsimas, D. and M. Sim. 2003. Robust discrete optimization and network flows.
Mathematical Programming 98(1-3): 49-71.
Bertsimas, D. and M. Sim. 2004. The price of robustness. Operations Research 52 (1):
35-53.
Bertsimas, D. and A. Thiele. 2004. A robust optimization approach to supply chain
management. Integer programming and combinatorial optimization, 3604, 145-
156.
Bertsimas, D. and A. Thiele. 2006. A robust optimization approach to inventory theory.
Operations Research 54 (1): 150-168.
Braid, R. 1996. Peak-load pricing of a transportation route with an unpriced substitute.
Journal of Urban Economics 40 (2): 179-197.
Braess, D. 1968. Uber ein paradoxon der verkehrsplanung. Unternehmensforschung. 12:
258-268.
Carey, M. and A. Srinivasen. 1993. Externalities, average and marginal cost, and toll on
congested network with time-varying flows. Operations Research 41: 217-231.
Chiu, Y. C. and H. Zheng. 2007. Real-time mobilization decisions for multi-priority
emergency response resources and evacuation groups: model formulation and
solution. Transportation Research Part E 43(6): 710-736.
Chiu, Y.C., H. Zheng, J. Villalobos and B. Gautam. 2007. Modeling no-notice mass
evacuation using a dynamic traffic flow optimization model. IIE Transactions
39(1): 83-94.
Daganzo, C.F. 1994. The cell transmission model part I: a simple dynamic representation
of highway traffic. Transportation Research Part B 28(4): 269-287.
125
Daganzo, C.F. 1995. The cell transmission model part II: network traffic. Transportation
Research Part B 29(2): 79-93.
De Palma, A. and R. Lindsey. 2000. Private toll roads: Competition under various
ownership regimes. The Annals of Regional Science 34 (1): 13-35.
De Palma, A., M. Kilani and R. Lindsey. 2005. Congestion pricing on a road network: A
study using the dynamic equilibrium simulator metropolis. Transportation
Research Part A: Policy and Practice 39 (7-9): 588-611.
Erera, A. L., J.C. Morales and M. Savelsbergh. 2009. Robust optimization for empty
repositioning problems. Operations Research 57(2): 468-483.
Friesz, T., H. Cho, N. Mehta, R. Tobin, and G. Anandalingam. 1992. A simulated
annealing approach to the network design problem with variational inequality
constraints. Transportation Science 26 (1): 18-26.
Friesz, T. L., D. Bernstein, T. Smith, R. Tobin, and B. Wie. 1993. A variational
inequality formulation of the dynamic network user equilibrium problem.
Operations Research 41 (1): 80-91.
Friesz, T., D. Bernstein, Z. Suo, and R. Tobin. 2001. Dynamic network user equilibrium
with state-dependent time lags. Networks and Spatial Economics 1(3/4): 319-347.
Friesz, T. L., C. Kwon, and R. Mookherjee. 2007. A computable theory of dynamic
congestion pricing. In 17th International Symposium on Transportation and
Traffic Theory.
Friesz, T., M. Rigdon, T. Kim, and C. Kwon. 2010. Approximate network loading and
dual-time-scale dynamic user equilibrium. Transportation Research Part B
(forthcoming).
126
Fu, H., C.G. Wilmot, H. Zhang and E.J. Baker. 2007. Modeling the hurricane evacuation
response curve. Transportation Research Record: Journal of the Transportation
Research Board 2022: 94-102.
Gardner, L., A. Unnikrishnan, and S. Waller. 2008. Robust pricing of transportation
networks under uncertain demand. Transportation Research Record: Journal of
the Transportation Research Board 2085: 21-30.
Ghaoui, L.E., M. Oks and F. Oustry. 2003. Worst-case value-at-risk and robust portfolio
optimization: a conic programming approach. Operations Research 51(4); 543-
556.
Ghaoui, L.E., F. Oustry and M. AitRami. 1997. A cone complementarity linearization
algorithm for static output-feedback and related problems. IEEE Transactions on
Automatic Control 42(8): 1171-1176.
Guha-Sapir, D., G.S. Debarati, D. Hargitt, P. Hoyois, R. Below and D. Brechet. 2004.
Thirty years of natural disasters 1974-2003. Presses univ. de Louvain.
Hearn, D. W. and M. B. Yildirim. 2002. A toll pricing framework for traffic assignment
problems with elastic demand. Kluwer Academic Publishers.
Henderson, J. 1974. Road congestion: A reconsideration of pricing theory. Journal of
Urban Economics 1 (3): 346-365.
Joksimovic, D., M. Bliemer and P. Bovy. 2005. Optimal toll design problem in dynamic
traffic networks with joint route and departure time choice. Transportation
Research Record: Journal of the Transportation Research Board 1923: 61-72.
127
Kalafatas, G. and S. Peeta. 2009. A common modeling framework for dynamic traffic
assignment and supply chain management systems with congestion phenomena.
Transportation and Traffic Theory, 541-557.
Karoonsoontawong, A. and S.T. Waller. 2007. Robust dynamic continuous network
design problem. Transportation Research Record: Journal of the Transportation
Research Board 2029: 58-71.
Karoonsoontawong. A. and S.T. Waller. 2008. Integrated network capacity expansion
and traffic signal optimization: robust bi-level dynamic formulation. Network and
Spatial Economics doi: 10.1007/s11067-008-9071-x.
Kelley Jr, J. 1960. The cutting-plane method for solving convex programs. Journal of the
Society for Industrial and Applied Mathematics 8 (4): 703-712.
Kirkpatrick, S., C. Gelatt Jr, and M. Vecchi. 1983. Optimization by simulated annealing.
Science 220 (4598): 671-680.
Lawphongpanich, S. and D. Hearn. 2004. An MPEC approach to second-best toll pricing.
Mathematical Programming 101 (1): 33-55.
Li, Y., S.T. Waller and T. Ziliaskopoulos. 2003. A decomposition scheme for system
optimal dynamic traffic assignment models. Networks and Spatial Economics 3(4):
441-455.
Lin D, A. Karoonsoontawong and S.T. Waller. 2008. A Dantzig-Wolfe decomposition
based heuristic scheme for bi-level dynamic network design problem. Networks
and Spatial Economics doi: 10.1007/s11067-008-9093-4.
128
Lin, D.Y., A. Unnikrishnan, and S.T. Waller. 2010. A dual variable approximation based
heuristic for dynamic congestion pricing. Networks and Spatial Economics
(forthcoming).
Lindell, M. K. 2008. EMBLEM2: An empirically based large scale evacuation time
estimate model. Transportation Research Part A 42: 140-154.
Lindsey, R. and E. Verhoef. 2001. Traffic congestion and congestion pricing. Handbook
of transport systems and traffic control: 77-105.
Lodree, Jr, EJ and S. Taskin. 2008. An insurance risk management framework for
disaster relief and supply chain disruption inventory planning. Journal of the
Operational Research Society 59(5): 674–684.
Lou, Y., Y. Yin and S. Lawpongpanich. 2009. A robust approach to discrete network
designs with demand uncertainty. Transportation Research Record: Journal of
the Transportation Research Board 2090: 86-94.
Lou, Y., Y. Yin and S. Lawphongpanich. 2010. Robust congestion pricing under
boundedly rational user equilibrium. Transportation Research Part B:
Methodological 44 (1): 15.28.
Lu, Y. 2007. Robust transportation network design under user equilibrium, Master’s
Thesis, MIT, Cambridge, MA.
Magnanti, T.L. and R.T. Wong. 1984. Network design and transportation planning:
models and algorithms. Transportation Science 18:1–55.
Mahmassani, H.S. 2001. Dynamic network traffic assignment and simulation
methodology for advanced System management applications. Networks and
Spatial Economics 1(3-4): 267-292.
129
Marcotte, P. 1983. Network optimization with continuous control parameters.
Transportation Science 17 (2): 181-197.
Minoux, M. 1989. Network synthesis and optimum network design problems: Models,
solution methods and applications. Networks 19:313-360
Mudchanatongsuk, S., F. Ordonez and J. Liu. 2008. Robust solutions for network design
under transportation cost and demand uncertainty. The Journal of Operational
Research Society 59(5): 652-662.
Mulvey, J.M., R.J. Vanderbei and S.A. Zenios. 1995. Robust optimization of large-scale
systems. Operations Research 43(22): 264-281.
Nagae, T. and T. Akamatsu. 2006. Dynamic revenue management of a toll road project
under transportation demand uncertainty. Networks and Spatial Economics 6 (3):
345-357.
Nie, Y. 2010. A Cell-based Merchant-Nemhauser Model for the System Optimum
Dynamic Traffic Assignment Problem. Transportation Research Board 89th
Annual Meeting .
Ordonez, F. and J, Zhao. 2007. Robust capacity expansion of network flows. Networks
50:136-145.
Ozdamar, L., E. Ekinci and B. Kucukyazici. 2004. Emergency logistics planning in
natural disasters. Annals of Operations Research 129(1): 217–245.
Peeta, S. and C. Zhou. 1999. Robustness of the off-line a priori stochastic dynamic traffic
assignment solution for on-line operations. Transportation Research Part C 7(5):
281-303.
130
Peeta, S. and A. K. Ziliaskopoulos. 2001. Foundations of dynamic traffic assignment: the
past, the present and the future. Networks and Spatial Economics 1(3-4): 233-265.
Pigou, A. 1920. The economics of welfare. London: MacMillan.
Radwan, A., A. Hobeika and D. Sivasailam. 1985. A computer simulation model for rural
network evacuation under natural disasters. ITE journal 55(9): 25-30.
Shapiro, A. and A. Nemirovski. 2005. On complexity of stochastic programming
problems . Springer.
Sheu, J.B. 2007. An emergency logistics distribution approach for quick response to
urgent relief demand in disasters. Transportation Research Part E 43(6): 687–
709.
Soyster, A.L. 1973. Convex programming with set-inclusive constraints and applications
to inexact linear programming. Operations Research 21(5): 1154–1157.
Tan, H. N., S. Gershwin and M. Athans. 1979. Hybrid optimization in urban traffic
networks. LIDS Technical Report, MIT, Cambridge.
Tuydes, H. 2005. Network traffic management under disaster conditions. Ph.D.
Dissertation, Northwestern University, Evanston, IL.
Ukkusuri, S.V., T. Mathew and S.T. Waller. 2007. Robust transportation network design
under demand uncertainty. Computer-aided civil and infrastructure engineering
22: 6-18.
Ukkusuri, S.V. and S.T. Waller. 2008. Linear programming models for the user and
system optimal dynamic network design problem: formulations, comparisons and
extensions. Networks and Spatial Economics 8(4): 383-406.
131
Vanderbilt, D. and S. Louie. 1984. A monte carlo simulated annealing approach to
optimization over continuous variables. Journal of Computational Physics 56 (2):
259.271.
Viti, F., S. Catalano, M. Li, C. Lindveld, and H. Van Zuylen. 2003. An optimization
problem with dynamic route-departure time choice and pricing. In proceedings of
82nd Annual Meeting of the Transportation Research Board, Washington D.C.
Waller, S.T., J.L. Schofer A.K. Ziliaskopoulos. 2001. Evaluation with traffic assignment
under demand uncertainty. Journal of the Transportation Research Board 1771:
69–75.
Waller, S.T. and A.K. Ziliaskopoulos. 2001. Stochastic dynamic network design problem.
Journal of the Transportation Research Board 1771:106-113.
Waller, S.T. and A.K. Ziliaskopoulos. 2006. A Chance-constrained based stochastic
dynamic traffic assignment model: analysis, formulation and solution algorithms.
Transportation Research Part E 14(6): 418-427.
Wardman, M. 1998. The value of travel time: a review of British evidence. Journal of
Transport Economics and Policy 32: 285-316.
Wie, B. 2007. Dynamic stackelberg equilibrium congestion pricing. Transportation
Research Part C 15 (3): 154-174.
Wie, B. and R. Tobin (1998). Dynamic congestion pricing models for general traffic
networks. Transportation Research Part B 32 (5): 313-327.
Yang, H. and M.G.H. Bell. 1998. Models and algorithms for road network design: a
review and some new developments. Transport Reviews 18:257–278.
132
Yang, H. and H. Huang. 1998. Principle of marginal-cost pricing: how does it work in a
general road network? Transportation Research Part A 32 (1): 45-54.
Yang, H. and H. Huang. 2004. The multi-class, multi-criteria traffic network equilibrium
and systems optimum problem. Transportation Research Part B 38 (1): 1-15.
Yazici, M.A. and K. Ozbay, K. 2007. Impact of probabilistic road capacity constraints on
the spatial distribution of hurricane evacuation shelter capacities. Journal of the
Transportation Research Board 2022: 55-62.
Yin, Y. and S. Lawpongpanich. 2007. A robust approach to continuous network design
with demand uncertainty. In 17th International Symposium on Transportation
and Traffic Theory: 111-126.
Zhao, Y. and K. Kockelman. 2002. The propagation of uncertainty through travel
demand models: An exploratory analysis. The Annals of regional science 36:145-
163.
Zheng, H. 2009. Efficient algorithms for the cell based single destination system optimal
dynamic traffic assignment problem. Ph.D. Dissertation, The University of
Arizona, Tucson, AZ.
Ziliaskopoulos, A.K. 2000. A linear programming model for the single destination
system optimum dynamic traffic assignment problem. Transportation Science
34(1): 37-49.
VITA
Byung Do Chung
Byung Do Chung is currently completing his dissertation from the Department of
Industrial and Manufacturing Engineering at The Pennsylvania State University. He received his
B.S. and M.S. degrees in Industrial and Systems Engineering at Yonsei University. Before joining
The Pennsylvania State University, he worked for over 5 years in Information Technology
consulting and Supply Chain Management areas. During the Ph.D. degree, he worked as a
research assistant at the Center for Service Enterprise Engineering and a teaching assistant in
Smeal College of Business. His research interests are 1) optimization theory and modeling
including robust optimization, stochastic programming and game theory and 2) application of
optimization to logistics and transportation system, service engineering, revenue management and
dynamic pricing, and supply chain management.
