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Optimum multimodal routing under normal condition and disruptions
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Optimum multimodal routing under normal condition and disruptions
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Content
OPTIMUM MULTIMODAL ROUTING UNDER
NORMAL CONDITION AND DISRUPTIONS
by
Afshin Abadi
December 2014
i
Acknowledgments
It has been an honor to work as a research assistant with Professor Petros A. Ioannou
during the course of my graduate studies. It has been a wonderful experience for me to
benefit from his knowledge and expertise. I am also grateful to be a member of the
Center for Advanced Transportation Technologies at the University of Southern
California. I would like to thank my dissertation committee members Professor Maged
M. Dessouky and Professor Viktor K. Prasanna for their valuable suggestions.
ii
TABLE OF CONTENTS
Acknowledgments ....................................................................................................... ii
List of Tables .............................................................................................................. vi
List of Figures ............................................................................................................ vii
Abstract ....................................................................................................................... ix
Chapter 1: Introduction ................................................................................................ 1
1.1 The Problem .................................................................................................. 1
1.2 Literature Review .......................................................................................... 4
1.3 Contribution .................................................................................................. 9
1.4 Outline ......................................................................................................... 10
Chapter 2: Lack of Data: Traffic Flow Data Completion .......................................... 12
2.1 Introduction ................................................................................................. 12
2.2 Traffic Flow Data Completion .................................................................... 13
2.2.1 Initial Traffic Flow Estimation ............................................................ 14
2.2.2 Offline Traffic Flow Estimation .......................................................... 18
2.2.3 Online Traffic Flow Estimation ........................................................... 23
2.3 Computational Results ................................................................................ 23
2.4 Conclusions ................................................................................................. 25
Chapter 3: Optimum Routing: Road Network ........................................................... 27
3.1 Introduction ................................................................................................. 27
iii
3.2 Finding the Shortest Path ............................................................................ 29
3.3 Estimating the Best Route Given Major Events ......................................... 32
3.3.1 On-Road Events ................................................................................... 32
3.3.2 Off-Road Events .................................................................................. 34
3.4 Computational Results ................................................................................ 35
3.5 Conclusions ................................................................................................. 36
Chapter 4: Optimum Multimodal Routing ................................................................ 38
4.1 Introduction ................................................................................................. 38
4.2 Optimization Formulation ........................................................................... 46
4.3 Simulation Model Framework .................................................................... 51
4.4 Case Simulation Study ................................................................................ 55
4.5 Conclusions ................................................................................................. 66
Chapter 5: Multimodal Routing Under Disruptions .................................................. 67
5.1 Introduction .................................................................................................. 67
5.2 Multimodal Routing under Road Network Disruptions due to Earthquake 70
5.3 Multimodal Routing under Port Disruptions .............................................. 73
5.4 Conclusions ................................................................................................. 85
Chapter 6: Concluding Remarks and Proposed Future Topics .................................. 86
6.1 Final Conclusions........................................................................................ 86
6.2 Proposed Future Topics .............................................................................. 89
iv
Bibliography .............................................................................................................. 90
v
List of Tables
Table 3.1: Comparison of the shortest path algorithms ............................................. 31
Table 3.2: Computational results ............................................................................... 36
Table 4.1: Demands (number of containers) ............................................................. 56
Table 4.2: Comparison between uncoordinated and coordinated MDFLB ............... 59
Table 4.3: Rush hour .................................................................................................. 62
Table 4.4: Noon ......................................................................................................... 62
Table 4.5: Night ......................................................................................................... 62
Table 4.6: Accidents .................................................................................................. 64
Table 5.1: Vehicles’ delay in the Newport-Inglewood earthquake scenario ............. 72
Table 5.2: Ranking of North American ports by TEUs processed annually ............. 78
Table 5.3: Cargo growth in value (Port of Los Angeles) .......................................... 79
Table 5.4: Corresponding regions of FAF map ......................................................... 81
Table 5.5: Volume (ton/day) of imported goods to the domestic regions ................. 82
Table 5.6: Computational results ............................................................................... 83
vi
List of Figures
Figure 1.1: Proposed modeling approach .................................................................... 3
Figure 2.1: Dynamic traffic flow generation ............................................................. 14
Figure 2.2: Four stage model ..................................................................................... 15
Figure 2.3: Measured and estimated volumes (no initial solution) ........................... 24
Figure 2.4: Measured and estimated volumes (initial solution) ................................ 24
Figure 3.1: Layout of region of interest ..................................................................... 35
Figure 4.1: (a) Uncoordinated distribution system (b) Coordinated distribution
system ............................................................................................................................... 39
Figure 4.2: Framework of the proposed methodology ............................................. 42
Figure 4.3: Uncoordinated MDLFB procedure ......................................................... 49
Figure 4.4: Coordinated MDFLB procedure ............................................................. 50
Figure 4.5: Balancing procedure ................................................................................ 51
Figure 4.6: Layout of VISUM network .................................................................... 53
Figure 4.7: Region of study ....................................................................................... 55
Figure 4.8: Locations of terminals, shipping companies, and coordinator ................ 57
Figure 4.9: Possible links between a pair of origin-destination................................. 58
Figure 4.10: Iterating approach convergence (coordinated rush hour) ...................... 60
Figure 4.11: (a) Uncoordinated MDFLB (b) Coordinated MDFLB.......................... 61
Figure 4.12: Location of the accidents ...................................................................... 64
Figure 4.13: Number of containers assigned ............................................................. 65
Figure 5.1: Multimodal transportation planning levels ............................................. 68
Figure 5.2: Multimodal physical network ................................................................. 69
vii
Figure 5.3: Bridge damage estimation in the Newport-Inglewood earthquake ......... 70
Figure 5.4: Link closure 3 days after the Newport-Inglewood earthquake ............... 71
Figure 5.5: Link closure 12 days after the Newport-Inglewood earthquake ............. 71
Figure 5.6: Link closure 49 days after the Newport-Inglewood earthquake ............. 71
Figure 5.7: Time-space transportation network ......................................................... 74
Figure 5.8: Original service network ......................................................................... 75
Figure 5.9: Reconfiguration of service network ........................................................ 76
Figure 5.10: The selected domestic regions in the west coast ................................... 80
Figure 5.11: Aggregated graph network for west coast ............................................. 81
viii
Abstract
The road and rail networks, ports and terminals, and sea carriers are part of the
multimodal freight transportation chain that handles most imports and exports of goods in
the United States. Most of the ports and terminals are located adjacent to urban regions.
Urban traffic conditions are extremely time-dependent, and change drastically throughout
a day. A major problem in getting traffic flow information in real time is that the vast
majority of links are not equipped with traffic sensors. Another problem is that factors
affecting traffic flows such as accidents, public events, and road closures are often
unforeseen, suggesting that traffic flow forecast is a challenging task. On the other hand,
any disruption to the freight transportation chain may have devastating effects on the
economy and society. Reconfiguration and routing strategies are needed to mitigate the
impact of potential disruptions along the freight transportation chain.
In this research, we use an optimization modeling approach to address the
multimodal routing under normal condition and disruptions. The optimization modeling
approach consists of traffic flow data completion and optimum routing in road network,
optimum multimodal routing, and multimodal routing under disruptions. First, we use a
dynamic traffic flow simulator to generate flows in all links in real time using available
traffic information, estimated demand, and historical traffic data from the links equipped
with sensors. Using the traffic flow data completion methodology and an optimization
model, the optimum routing in road network is developed for an individual driver to
reach destination in a complex urban network. Then, we use the developed optimum
routing in road network and a railway simulation system to address the optimum
ix
multimodal routing. The proposed model evaluates volume and travel time of each route
based on current traffic conditions and number of trucks or trains that have been assigned
dynamically on each route by the optimization models using the optimum routing in road
network and a railway simulation system. Finally, the proposed optimization modeling
approach will make freight distribution process become more resilient in the presence of
disruptions. Our approach is based on reconfiguration to include available resources and
alternative routes which in the absence of disruption were not cost effective, and
therefore under normal conditions were not feasible choices. Several case studies are
presented to demonstrate the effectiveness of the proposed optimization modeling
approach.
x
Chapter 1
Introduction
1.1 The Problem
Urban traffic conditions are extremely time-dependent and change drastically during a
day. Urban transportation network is a complex and living organism that is breeding with
inefficiencies. Drivers face inevitable uncertainties during their car trips in the urban
environments. Factors that affect traffic conditions such as accidents, major public
events, and road closures are not only outside the driver’s control, but are also often
unforeseen. Only a few numbers of links in the transportation network are equipped with
traffic sensors which collect real time traffic data. These links include freeways and
major streets, and there is a lack of data for the rest of links in the transportation network.
The lack of data would not only affect drivers in the urban area; it will also have a huge
impact on freight movements especially in the presence of major events which change
traffic flows in the region drastically.
The research starts with addressing traffic flow data completion in which the general
information from the region, historical and real time traffic data are used to estimate
traffic flows for all the links in the transportation network in real time. In the proposed
methodology, traffic flow estimation methodology is classified into three categories
including initial traffic flow estimation, off-line traffic flow estimation, and real-time
traffic flow estimation. A four-stage model based on the general information within the
region such as location of schools and number of students commuting is used to estimate
1
the initial traffic flows in the transportation network. The column generation method is
used to solve the offline traffic flow estimation problem, and the real-time traffic flow
estimation is expressed as the least squares problem. A macroscopic traffic flow
simulator is used to generate real-time traffic flows as well as prediction of flows on the
road network, since the macroscopic traffic flow simulator is much faster than the
microscopic traffic flow simulator. The microscopic traffic flow simulator cannot be used
in real-time due to long simulation running time. Traffic events (accident, road closure,
disabled car, etc.) and social events (sport games, concerts, etc.) are also implemented in
the macroscopic traffic flow simulator to evaluate their impacts on the steady flow in the
transportation network.
Once the traffic flow data completion procedure is done, a shortest route algorithm is
used to find the optimum route from the driver’s location (origin) to his/her final
destination in the network. The problem requires us to analyze the effects of potential
traffic events on the links’ traffic flows and update the real-time traffic flows accordingly.
The goal of the proposed model is to provide a driver with an optimal route to navigate
through the congested urban environment as well as a far more accurate prediction of
travel time by taking into account the historical and current traffic conditions, and the
traffic and social events.
Furthermore, an optimization problem with appropriate constraints is defined to
address the optimum multimodal routing using the optimum routing in road network and
a railway simulation system. In the road transportation network, travel time at some break
point increases as a nonlinear function of the volume. Moreover, travel times in the
railway transportation follow a non-linear model. The proposed model evaluates volume
2
and travel time of each route based on current traffic condition and number of trucks or
trains that have been assigned dynamically on each route by the optimization model.
. Finally, we use optimization models to address multimodal routing under
disruptions. During disruption, demands may not be met and the total cost will increase.
The proposed optimization modeling approach will make freight distribution process
become more resilient in the presence of disruptions. Our approach is based on
reconfiguration of the service network to include available resources and alternative
routes which in the absence of disruption were not cost effective, and therefore under
normal conditions were not feasible choices. Our approach identifies which of these
resources and routes are best choices for the kind of disruption that took place. The
proposed modeling approach is summarized in Figure 1.1.
Figure 1.1: Proposed modeling approach
Macroscopic Traffic Flow Simulator
(Calibrated)
Historical/Real
Time Traffic
Flow Data and
Available
Information
from the Region
Traffic Flow Data Completion
(Road Network)
Optimum Routing (Road Network)
Traffic and Social
Events Data
Optimum Multimodal Routing
Railway Simulation
System
Optimum Multimodal Routing Under Disruptions
Disruption Data
3
1.2 Literature Review
The Intelligent Transportation System (ITS) strategies not only depend on the availability
of data, but also on the quality of the estimation models. Most of traffic models use an
Origin-Destination matrix (OD) to describe travel demands. The OD matrix determines
the number of trips from a specific region (zone) to another one within a transportation
network. There exist various OD matrix estimation methods for a transportation network.
Turnquist et al. [96] and Van Zuylen et al. [104] developed the static OD matrix
estimation model based on data from conductive loops embedded on the surface of streets
and highways. Unfortunately, the loop data is only available for small portion of streets,
and there is no traffic flow data for major part of a transportation network. Therefore,
some other source of information such as surveys and probe vehicles can be taken into
account to improve the OD matrix estimation algorithm. Cascetta [23] demonstrated the
OD matrix estimation algorithm using traffic count data and survey data.
Service Network Design Problem (SNDP) has various applications in
telecommunication, transportation, and even in computer networks. We use the
framework of the SNDP to formulate the optimization problem for the optimum
multimodal routing. Magnanti et al. [67] and Minoux [71] developed the service network
design models and algorithms. Moreover, Crainic [30] improved the SNDP formulation
for the capacitated freight transportation. Powell et al. [80] and Farvolden et al. [39]
presented the SNDP formulation by taking into account variable transportation cost. They
obtained optimal solutions with the sub-gradient optimization method and a gradient-
based search algorithm.
4
Some efforts have been made on the express shipment service employing the SNDP.
Barnhart et al. [14, 15] introduced the service network design with time window for the
express shipment service. They presented the column generation method and the location
elimination model using the Linear Programming (LP) relaxation. Numerous works
concentrated on the less-than-truckload (LTL) transportation. Cunha et al. [32] developed
a genetic algorithm for configuring the transportation network for a LTL trucking
company in Brazil. The computational experiments for 15 cities demonstrated the quality
of the developed algorithms running in the reasonable time. Crainic et al. [29] and Hane
et al. [49] used a general multi-commodity flow problem for the freight transportation
solved by dynamic programming and the steepest edge simplex algorithm.
Many publications addressed the railroad planning and scheduling. Van Dyke [37]
demonstrated a practical approach to the freight railroad scheduling by developing the
automated blocking model. Barnhart et al. [14] proposed an application of the SNDP in
the railroad blocking problem. The proposed methodology reduced the total cost as well
as number of classifications in the real size railroad blocking problems. Marin et al. [69]
presented both exact and heuristic methods in the design of rail freight networks. A
general decomposition algorithm is used for the heuristics methods including the gradient
descending method, the simulated annealing, and the tabu search method. The
reformulation of general decomposition algorithm is required to solve the design and
schedule of rail freight networks using the exact method.
Much attention has been devoted to the ship scheduling and network design in the
last two decades, since most of commodities in the international container transportation
are transferred by ocean-going ships. Compared to the air transportation, ships are much
5
cheaper and safer. Agarwal et al. [2] presented a network design and ship scheduling
model for the cargo transportation. The proposed network design formulation is time
dependent assuming that ports operating seven days per week. Different approaches are
used to solve the Mixed Integer Programming (MIP) including the greedy heuristic, the
column generation, and the benders decomposition methods. Song et al. [93] showed the
strategic planning level can be designed and simulated engaging the game theory
approaches. Christiansen et al. [28], Rana et al. [83], and Ronen [86,87] introduced novel
algorithms addressing the maritime cargo transportation.
The multimodal transportation system is very complex due to different
characteristics of transportation services and the routes on which these services are
offered. Transportation restrictions need to be taken into account in the SNDP
formulation as additional constraints. Jansen et al. [53] presented a general multimodal
freight transportation formulation by taking into account time, capacity, and container
repositioning. The proposed algorithm consists of modality choice, repositioning, order
combination, order planning, and planning improvement. Many tests are presented to
evaluate the results of the proposed algorithm. Yamada et al. [100] introduced a heuristic
approach based on the genetic local search algorithm to find the optimal solution in the
multimodal freight transportation. The problem is divided into two sub-problems: the
lower level and the upper level. The modal split assignment is performed in the lower
level, whereas the upper level uses the results in the lower level to compute the best
solution for the multimodal design problem. Arnold et al. [8], Arroyo et al. [9], Guelat et
al. [46], Priemus [81], and Russ et al. [88] have made some contributions in the
multimodal freight transportation system design and scheduling.
6
The SNDP formulation is the NP-hard problem for a large network and computing
the best feasible solution is very complex. Various heuristics algorithms have been
developed and used to solve the NP-hard problems in a manageable amount of time.
Aarts et al. [1] used a simulated annealing approach to combinatorial optimization and
neural computing. The Tabu search method and the genetic algorithm are the most
common approaches in the network optimization problems. Yuval [101], Diaz et al. [35] ,
Goldberg [45], Jaszkiewicz et al. [54], [55], Merz et al. [70], Shepherd et al. [91],
Syswerda [95], and Jula et al. [58] presented some applications using the heuristics
algorithms.
Since 1990, extensive efforts have been made on the Branch-and-Price-and-cut
algorithm to compute solutions for the Vehicle Routing Problem (VRP) and Split
Delivery Vehicle Navigating Problem (SDVRP). Fontes et al. [42] and Holmberg et al.
[51] demonstrated a Lagrangian relaxation approach-based on the branch-and-bound
algorithm. The branch-and-price method is the branch-and-bound method with the linear
relaxation which is solved by the column generation method; whereas in the branch-and-
cut method valid inequalities are added to separate the solutions (row generation).
Barnhart et al. [15] developed a branch-and-price method for solving a large integer
programming problem. Andersen et al. [4] applied the branch-and-price method for the
SNDP with asset management. Kohl et al. [60] introduced the k-path cut for the Vehicle
Routing Problem with Time Window (VRPTW) in the form of valid inequalities added to
the column generation method. Combining the branch-and-price and the branch-and-cut
methods is very challenging to implement. Barnhart et al. [13] presented the branch-and-
7
price-and-cut method for the VRP, and Desaulniers [33] added time window constraints
to the generalized SNDP formulation.
Finally, Archetti et al. [7] developed an enhanced branch-and-price-and-cut method
for the split deliveries vehicle routing problem with time windows. They used the
heuristics search algorithm for the column generation method, and the pool of solutions is
separated by adding valid inequalities which violate the linear programming constraints.
The dynamic OD matrix estimation has been investigated in the last two decades
significantly, and used in the traffic management. Various methodologies have been
developed to address the time-dependent OD matrix estimation. Cremer et al. [31] and
Josefsson et al. [56] solved the optimization problem by using the bi-level programming
methodology. The cost function is defined in two levels: the upper level and the lower
level.
The dynamic OD matrix estimation can be classified into two categories:
Assignment-based OD matrix estimation and non-assignment-based estimation. Traffic
flow simulators are used in the assignment-based OD matrix estimation to generate the
link flows based on the traffic data. Cascetta et al. [24] presented a generalized least
square model to minimize the difference between the traffic data and corresponding
simulated flows. Sherali et al. [92] presented a new methodology that takes into account
the cost of the entire transportation network and generated link flow on each individual
link. Gajewski et al. [44] used the least square error problem to derive the OD matrix
according to the real time traffic data. As noted, bi-level programming has been
extensively used to estimate the OD matrix. Zhou et al. [103] introduced an improved
model of the OD matrix estimation using time window constraints. Balakrishna et al. [11]
8
and Kim [59] developed offline generalized least square OD matrix estimation
methodologies which are the improved versions of Zhou et al. which was developed in
2003 [103]. Also, the state-space estimation model has been used for the dynamic OD
matrix estimation. Okutani [77], Ashok et al. [10] ,and Antoniou et al. [6] developed the
dynamic OD matrix estimation methodologies based on the state-space models.
In the non-assignment OD matrix estimation, the traffic flow is generated according
to the transportation flow equations. The non-assignment OD matrix estimation can be
used for a small transportation network such as intersections. Bell [18] presented the OD
matrix estimation model with regard to the transportation flow equations. Chang et al.
[26] also developed a non-assignment OD matrix estimation model based on traffic flows
on a small section of freeway. Fernandez et al. [41] described several methodologies
which are used to impute of missing traffic flow data from conductor loops on freeways,
and Zhong et al. [102] compared some of these methodologies. Muralidharan et al. [75]
provided the need to estimate traffic flow for the links without traffic sensor. They
presented a model-based approach to estimate traffic flows on ramps of a freeway.
Despite all the mentioned works above, there exists a need to develop a model-based
approach to estimate traffic flow for all links in a transportation network in real time.
Traffic sensors are only available on a small portion of links in a transportation network.
Also, the mentioned works do not take into account the dynamic nature of link travel time
which is non-linear with respect to the volume assigned to the link.
1.3 Contribution
In this research, we propose an optimization modeling approach for the multimodal
routing under normal condition and disruptions. The main contribution of this research is
9
to identify all the cost parameters in the optimization formulation for an optimum
multimodal routing using an optimum routing in road network and a railway simulation
system to find the best feasible freight transferring decision by taking into account
constraints and the non-linear nature of links’ travel time with respect to the volume. As
mentioned earlier, for a certain range of flow the cost of a link may change linearly, but
at some break point the congestion increases as a nonlinear function of the volume. We
capture these nonlinearities in the objective function of an optimum multimodal routing
optimization formulation. To address the optimum multimodal routing, the optimum
routing in road network is required. The main issue in the optimum routing in road
network is lack of traffic data. The major problem in recording traffic flow information in
real time is that the vast majority of links are not equipped with traffic sensors. The
traffic flow completion methodology estimates the link flows and corresponding OD
matrices in real time for the entire transportation network based on the historical and the
real time traffic flow data and other available information from the region of study.
Furthermore, the proposed optimization modeling approach will make freight distribution
process become more resilient in the presence of disruptions. Our approach is based on
reconfiguration of the service network to include other resources and routes which in the
absence of disruption were not cost effective, and therefore under normal conditions were
not feasible choices.
1.4 Outline
We will begin with a traffic flow data completion model. We will describe the method of
real time estimating the link flows and corresponding OD matrices for the entire links in
a transportation network. In Chapter 3, we will discuss a methodology to find the
10
shortest path for an individual driver. Then, we will discuss the impacts of major social
events, and on-road and off-road traffic events on the estimated link flows. In Chapter 4,
the optimum multimodal routing is explained in details. The multimodal routing under
disruptions is presented in Chapter 5. Disruptions are assumed to occur both on the road
network and inside the sea terminals. Several case studies are presented to demonstrate
the effectiveness of the proposed optimization modeling approach.
11
Chapter 2
Lack of Data: Traffic Flow Data Completion
2.1 Introduction
The loop conductors embedded on the surface of streets are used to collect traffic flow
data. Unfortunately, the loop conductors are only implemented on the surface of
highways and major streets, and there is a lack of data for majority of roads in the
transportation network. Guiding drivers in the urban transportation network requires
using the methodologies to estimate flows where traffic flow data is unavailable. In
traffic flow simulators, the link traffic flows are evaluated by the dynamic traffic
assignment and the OD matrices. We divide the network into zones; where zones
represent certain trips from the origins to the destinations. The OD matrix contains
information regarding the volume of traffic travelling between any two zones in the
network. The aggregate sum of all trips originating from a zone is referred to as the
zone’s production, and the aggregate sum of all the trips that terminate in a zone is
referred to as the zone’s attraction. The OD matrix indicates travel demand, but it does
not specify traveler’s route choices.
One of the most significant challenges in the transportation planning and
management is the task of accurately modeling the true OD matrix for a transportation
network. Roadside traffic studies are not only expensive and laborious, but are also more
importantly lengthy. In a world of changing traffic patterns, results could be meaningless
within a few months of the conclusion of the study; that is, the relevance of the OD
matrix estimated may be short-lived and no longer reflect the reality. The loop conductors
12
can provide real-time and round-the-clock information that can be used to estimate and
update the transportation network’s OD matrix dynamically.
To estimate the real-time OD matrix and corresponding link flows, we estimate the
initial traffic flows in the transportation network based on the general information from
the region of interest such as location of businesses and number of commuters. In our
proposed methodology, the initial traffic flow estimation is important since it is used as
the initial solution for the offline traffic flow estimation model. The offline traffic flow
estimation model estimates traffic flows based on data from monitoring the past events
and the historical traffic flow data. Finally, we update the traffic flow estimation results
based on the real-time traffic flow data. This continual adjustment helps to ensure that
the modeled system is as close as possible to the actual traffic conditions. The level of
accuracy of traffic flow estimation and the ability to adjust it in real time have a direct
impact on the quality of our goal output: optimum route in road network and accurate
travel time prediction for an individual driver.
2.2 Traffic Flow Data Completion
The transportation network consists of several elements such as links, nodes, zones, etc.
The nodes are connected by the links, and the links represent streets or freeways. The
zones are places that considerable numbers of people visit such as schools, stadiums,
commercial buildings, and so on. Also, one zone is defined for each residential district.
The OD matrix determines the number of trips within zones in each time interval. Each
OD matrix is assigned for one transportation choice. The links represent the range of
paths (e.g. specific street blocks) between the nodes. The links are characterized by their
maximum allowable speed limit, the number and type of lanes, and their length.
13
Ultimately, these factors lead to a value of 𝑡 0
which is the travel time along the link given
free-flow condition. Finally, the nodes represent intersections between the links. The
nodes give drivers choices between certain links; that is, there is often more than one set
of links and nodes enable a driver to travel from one zone to another. When traffic flow
on a certain link exceeds the link’s capacity, congestion ensues.
The dynamic traffic flow estimation is an important issue in the transportation
planning. The physical network model consists of many objects such as links, nodes,
zones, etc. The demand matrix represents the aggregated trips from one zone to another.
The traffic flow on each link can be obtained by applying the traffic assignment on a
dynamic traffic flow simulator with regard to the physical network characteristics and the
OD matrix. Clearly, the OD matrix can be estimated based on the loop conductor traffic
flow data. Figure 2.1 depicts the diagram for the dynamic traffic flow generation.
Figure 2.1: Dynamic traffic flow generation
In the Proposed methodology, the traffic flow data completion is modeled in three steps:
An initial traffic flow estimation, an off-line traffic flow estimation, and a real-time
traffic flow estimation as discussed below.
2.2.1 Initial Traffic Flow Estimation
The purpose of the initial OD matrices estimation is to estimate the OD matrices based on
the estimated demand data from the region. For instance, students go to schools at 8 AM
Physical Network
Model
OD Matrix
Dynamic Traffic
Flow Simulator
Link Flows
14
and leave schools at 2 PM. We can roughly estimate the demand of each school in the
morning based on the number of students. We use the four stage model (gravity model) to
estimate the initial OD matrices as explained below and illustrated in Figure 2.2 [38]:
Figure 2.2: Four stage model
The first stage in the model, referred to as trip generation, is the aggregated travel
demand for each zone. Each zone has a certain production, the number of trips that begin
at that zone, and a certain attraction, the number of trips that culminate at that zone. The
information (i.e. production and attraction) can be obtained through surveys and
information from the region. For instance, in a big sport game, the attraction includes
thousands of cars and possibly hundreds of buses heading to the parking lots adjacent to
the stadium before the starting of the game. The structure is the stadium and one zone is
assigned for each parking lot. The production/attraction ratio of each zone (parking lot)
for this specific structure (stadium) can be estimated based on the capacity of the parking
lots. Also, each residential district is considered as one structure within one zone in the
center of the district. Let us denote the trip production of zone 𝑖𝑖 and the trip attraction of
zone 𝑗 in the time interval 𝑛𝑛 ∈ {1, … . , 𝑇 }, as 𝑃 𝑖𝑖 𝑛 and 𝐴 𝑗 𝑛 respectively. 𝑃 𝑖𝑖 𝑛 determines the
total number of trips originating from zone 𝑖𝑖 , and 𝐴 𝑗 𝑛 is the total number of trips ending
to zone 𝑗 . The production and attraction of each zone can be expressed as follows:
𝑃 𝑖𝑖 𝑛 = � 𝛼 𝑞 𝑛 ( 𝑖𝑖 )
𝑞 ∈ 𝑄 𝑝 𝑞 𝑛
∀ 𝑖𝑖 ∈ 𝑍 , ∀ 𝑛𝑛 ∈ {1, … . , 𝑇 } (2.1)
Trip Generation
Trip
Distribution
Transportation
Choice
Traffic
Assignment
15
𝐴 𝑗 𝑛 = � 𝛽 𝑞 𝑛 ( 𝑗 )
𝑞 ∈ 𝑄 𝑎 𝑞 𝑛
∀ 𝑗 ∈ 𝑍 , 𝑛𝑛 ∈ {1, … . , 𝑇 } (2.2)
where 𝑎 𝑞 𝑛 and 𝑝 𝑞 𝑛 are the total attraction and production for structure 𝑞 in time interval 𝑛𝑛 .
The structure 𝑞 is a subset of 𝑄 which contains all the places in the region of study that
people commute. The attraction/production of each structure is distributed to the adjacent
zones. Each structure has a production/attraction ratio for each zone. 𝛼 𝑞 𝑛 ( 𝑖𝑖 ) and 𝛽 𝑞 𝑛 ( 𝑗 )
represent the production/attraction ratios for zones 𝑖𝑖 and 𝑗 respectively. The parameter 𝑍
indicates the set of all zones in the network.
The second stage of the model referred to as, trip distribution, determines the number
of trips traverse from one zone to another. The number of trips can be estimated as
follows:
𝑉 𝑖𝑖 𝑗 𝑛 = 𝜌 𝑖𝑖 𝑗 𝑛 . 𝑃 𝑖𝑖 𝑛 . 𝐴 𝑗 𝑛 . 𝑒 − 𝑑 𝑖𝑖𝑗
∀ 𝑖𝑖 , 𝑗 ∈ 𝑍 , ∀ 𝑛𝑛 ∈ {1, … . , 𝑇 } (2.3)
where 𝑉 𝑖𝑖 𝑗 𝑛 is the number of trips from zone 𝑖𝑖 to zone 𝑗 in time interval 𝑛𝑛 . The correlation
between two zones can be illustrated in different formats such as exponential, linear, and
so on. In this research, we choose the exponential format since it provides the best fit for
the selected region which is an urban region. 𝑑 𝑖𝑖 𝑗 is defined as the shortest distance
between zones 𝑖𝑖 and 𝑗 . The parameter 𝜌 𝑖𝑖 𝑗 𝑛 is a scaling factor to adjust the total number of
trips from zone 𝑖𝑖 to zone 𝑗 in the time interval 𝑛𝑛 to satisfy the following constraint
equations.
16
� 𝑉 𝑖𝑖 𝑗 𝑛 = 𝑃 𝑖𝑖 𝑛 1 ≤ 𝑗 ≤ 𝑍
∀ 𝑖𝑖 ∈ 𝑍 , ∀ 𝑛𝑛 ∈ {1, … . , 𝑇 } (2.4)
� 𝑉 𝑖𝑖 𝑗 𝑛 = 𝐴 𝑗 𝑛 1 ≤ 𝑖𝑖 ≤ 𝑍
∀ 𝑗 ∈ 𝑍 , ∀ 𝑛𝑛 ∈ {1, … . , 𝑇 } (2.5)
The next step in Figure 2.2, referred to as, transportation choice stage, determines the
proportion of 𝑉 𝑖𝑖 𝑗 𝑛 that uses a specific transportation mode such as bicycles, single or
multi-passenger cars, or public transit vehicle. The proportion of 𝑉 𝑖𝑖 𝑗 𝑛 that uses car and
truck are used to generate the OD matrices of trips at different intervals of time. For
instance, trucks represent X% of the total number of vehicles and their proportion is
denoted as 𝑉 𝑖𝑖 𝑗 , 𝑏 𝑛 where 𝑉 𝑖𝑖 𝑗 , 𝑏 𝑛 = 0.01 ∗ 𝑋 ∗ 𝑉 𝑖𝑖 𝑗 𝑛 , and the rest of vehicles (100-X)% are cars
( 𝑉 𝑖𝑖 𝑗 , 𝑐 𝑛 = 𝑉 𝑖𝑖 𝑗 𝑛 − 𝑉 𝑖𝑖 𝑗 , 𝑏 𝑛 ) for a pair of zones ( 𝑖𝑖 , 𝑗 ). The ratios of trucks and cars are variable
depending on the location of zones.
Finally, we can assign the estimated OD matrices onto the links. The assignment of
OD matrices onto links allows us to calculate the link flows and estimate travel time
between zones. In this stage, we assign 10% of the total demand on the links with the
minimum impedance, and evaluate new link impedance values in each step. The process
continues until the total demand is assigned in the network. It generates the initial routes
from origins to destinations as well as the initial link flows. In the proposed methodology,
travel time in the loaded network is considered as the impedance which can be presented
as follows [12]:
𝑡 𝑙 − 𝑙𝑃𝑃 𝑂𝑂𝑑
= 𝑡 𝑙 − 𝑓𝑓𝑟𝑟𝑆𝑆 𝑆𝑆 . [1 −
𝑞 𝑙 𝑞 𝑙 − 𝑚𝑂𝑂𝑥
]
𝛼
(2.6)
17
where 𝑡 𝑙 − 𝑓𝑓 𝑟𝑟𝑆𝑆 𝑆𝑆 represents travel time of link 𝑙 ∈ 𝐿 in the unloaded network and
𝑞 𝑙 𝑞 𝑙 − 𝑚𝑎𝑥 is
the ratio of the density to the density jam of a link. The density jam for a link occurs
when the link’s volume and the link’s speed reach zero. The parameter 𝛼 is a negative
number less than 1 which can be estimated using historical traffic data. The capacities of
links are difficult to estimate, since the capacity depends on many parameters as
presented below [50].
𝐶 = ( 𝐶 0
)( 𝑛𝑛 ). ( 𝑎 )( 𝑏 )( 𝑐 )( 𝑑 )( 𝑒 ) (2.7)
𝐶 0
: 1900 vehicles per hour per lane
𝑛𝑛 : Number of lanes
𝑎 : Lane width factor
𝑏 : Truck factor
𝑐 : Parking factor
𝑑 : Distance to downtown
𝑒 : Bus stop factor
The initial OD matrix and the initial link flows for all the links in a transportation
network are generated via the methodology that was explained in this subsection. The
following subsection provides an optimization algorithm and solution methodology to
estimate the traffic flows based on the historical traffic flow data of loop conductors.
2.2.2 Offline Traffic Flow Estimation
The traffic flow estimation for a large-scale network as it is in our case cannot be done in
the real time due to the computational time. We use an offline traffic flow estimation to
calibrate the parameters in the network, and then apply it for the real-time traffic flow
estimation. In the offline estimation, we aim to evaluate the proportion of demand
18
assigned on each link in the network. In other words, we derive the Link-to-Link
Dividing Ratios (LLDRs) of the link flows based on the initial traffic flow estimation and
the historical link flows from the loop conductor traffic flow data. The LLDRs determine
the ratio of traffic flows propagating from a specific link to the adjacent ones. For the
majority of the links in a transportation network, there exist no loop conductor; therefore,
we solely rely on the four stage model and recalibrations of it. We define the offline
traffic flow estimation problem with an optimization formulation by taking into account
appropriate constraints.
The physical transportation network is defined on a set of links and nodes, 𝑔 =
( 𝑁𝑁 , 𝐿 ) where N and L represent nodes and links in the network respectively. Let S denote
the OD pairs and 𝜃 be a set of all routes connecting OD pair 𝑠 ∈ 𝑆 . The cost of using
route r for OD pair s at time t, 𝑐 𝑠 , 𝑟𝑟 , 𝑡 is variable and depends on the traffic flow of the
route at time t. We define cost as travel time of loaded route which is formulated in (2.6).
We also denote 𝑣 ̅ 𝑙 , 𝑛 the mean of observed flow on link 𝑙 in time slot n, and 𝑣 𝑙 , 𝑛 the
estimated link flows with regard to the historical data and information from the region
including the initial link flow estimation. Note that 𝑛𝑛 = {1, … . , 𝑇 } and t ∈ [0, 𝑇 ].
Unfortunately, loop conductors are only available for a small portion of the links in the
network; therefore, we use the initial traffic flow estimation solutions to generate flows
on all the links in the network. Let 𝑣 𝑠 , 𝑟𝑟 , 𝑡 be the generated volume of route r for OD pair s
at time t. Therefore, we have:
𝑣 𝑙 , 𝑛 = � � � 𝜌 𝑠 , 𝑟𝑟 , 𝑛 𝑙 , 𝑡 𝑣 𝑠 , 𝑟𝑟 , 𝑡 𝑡 ∈ 𝑇 𝑟𝑟 ∈ 𝜃 𝑠 ∈ 𝑆𝑆
∀ 𝑙 ∈ 𝐿 ,
𝑛𝑛 ∈ {1, … . , 𝑇 } (2.8)
19
where 𝜌 𝑠 , 𝑟𝑟 . 𝑛 𝑙 , 𝑡 is the decision variable, 1 if link 𝑙 is on route r connecting OD pair s during
time t in time slot n and 0 otherwise. Note that 𝑣 𝑙 , 𝑛 must follow flow conservation, so we
have:
𝑣 𝑙 , 𝑛 = � 𝑝 𝑙 , 𝑂𝑂 , 𝑛 𝑣 𝑂𝑂 , 𝑛 𝑂𝑂 ∈ 𝐿𝐿
∀ 𝑙 ∈ 𝐿 ,
𝑛𝑛 ∈ {1, … . , 𝑇 } (2.9)
where 𝑣 𝑂𝑂 , 𝑛 represents the flow of links feeding 𝑙 and 𝑝 𝑙 , 𝑂𝑂 , 𝑛 is defined as the set of LLDRs
of link 𝑘 to 𝑙 . We define 𝜑 𝑛 ( 𝑅𝑅 ) as a set of 𝑣 𝑙 , 𝑛 and corresponding 𝑣 𝑠 , 𝑟𝑟 , 𝑡 as well as 𝑝 𝑙 , 𝑂𝑂 , 𝑛
satisfying (2.8) and (2.9). The historical link flow 𝑣 �
𝑙 , 𝑛 can be expressed as follows.
𝑊𝑊𝐴 𝐻𝑇𝑉
𝑙 ( ℎ, 𝑑 ) = �
(| 𝑀 | − 𝑚 + 1)
2
∑ (| 𝑀 | − 𝑘 + 1)
2 𝑀 𝑂𝑂 = 1
∑ 𝐻 𝑇𝑉
𝑙 ( ℎ, 𝑑 , 𝑤𝑤 , 𝑚 )
𝑤𝑤 ∑ 1
𝑤 ∈ 𝑊𝑊 𝑀 𝑚 = 1
∀ ℎ ∈ 𝐻 , 𝑑 ∈ 𝐷𝐷
(2.10)
where 𝑊𝑊𝐴𝐻 𝑇 𝑉 is the weighted average hourly traffic volume and 𝐻𝑇 𝑉 is the hourly
traffic volume. In the rest of this research, the historical link flow 𝑣 �
𝑙 , 𝑛 is calculated by
(2.10). Parameters h, d, w, and m represent hour, day, week, and month respectively. It is
assumed that the link traffic flows are recorded for M consecutive months. The offline
traffic flow estimation problem can be formulated as an optimization problem as
presented below, and we call it the master problem.
minimize
𝑣 𝑠 , 𝑓𝑓 , 𝑡 𝑔 ( 𝑉 ) = 𝛼 1
� � �
𝑣 �
𝑙 , 𝑛 − 𝑣 𝑙 , 𝑛 𝑣 �
𝑙 , 𝑛 �
2
+
𝑙 ∈ 𝐿𝐿 𝑛 ∈{ 1,.., 𝑇 }
𝛼 2
� � � 𝑐 𝑠 , 𝑟𝑟 , 𝑡 𝑣 𝑠 , 𝑟𝑟 , 𝑡 𝑡 ∈ 𝑇 𝑟𝑟 ∈ 𝜃 𝑠 ∈ 𝑆𝑆 (2.11)
subject to 𝑣 𝑠 , 𝑟𝑟 , 𝑡 ≥ 0 (2.12)
Objective function (2.11) minimizes the normalized variation between the historical link
flows and the simulated ones as well as the total cost of the network. Constraint (2.12)
imposes non-negative value for link flows. The coefficients 0 ≤ 𝛼 1
, 𝛼 2
≤ 1 weight the
20
relative importance of average historical count data and the total cost of the network. We
choose 𝛼 1
to be much larger than 𝛼 2
, since the historical link flows obtained from the
loop detectors are much more reliable than the total cost of the network to derive the
LLDRs. Also, it can be presented in the following format using (2.8).
minimize
𝑣 𝑠 , 𝑓𝑓 , 𝑡
𝑔 ( 𝑉 ) = 𝛼 1
� � �
𝑣 �
𝑙 , 𝑛 − ∑ ∑ ∑ 𝜌 𝑠 , 𝑟𝑟 , 𝑛 𝑙 , 𝑡 𝑣 𝑠 , 𝑟𝑟 , 𝑡 𝑡 ∈ 𝑇 𝑟𝑟 ∈ 𝜃 𝑠 ∈ 𝑆𝑆 𝑣 �
𝑙 , 𝑛 �
2
+
𝑙 ∈ 𝐿𝐿 𝑛 ∈{ 1,.., 𝑇 }
𝛼 2
� � � 𝑐 𝑠 , 𝑟𝑟 , 𝑡 𝑣 𝑠 , 𝑟𝑟 , 𝑡 𝑡 ∈ 𝑇 𝑟𝑟 ∈ 𝜃 𝑠 ∈ 𝑆𝑆 (2.13)
subject to 𝑣 𝑠 , 𝑟𝑟 , 𝑡 ≥ 0 (2.14)
Therefore, the link flows are estimated using the offline link flow estimation model for
each time interval. For example, we would like to know what the estimated flow on link
l is on Saturdays, 8-9 AM. During an observation interval such as 1 year, we can
continually adjust the model parameters to develop an accurate model for the traffic flow
estimation.
Solution Methodology
We use the column generation method [34] to solve the problem; therefore, we need to
start with a subset of solution 𝜃 1
⊂ 𝜃 . The restricted master problem is a way to expedite
the process of finding the lowest cost route. The restricted master problem is defined as a
subset of master problem ( 𝜃 1
). The initial solution would be the solution of the four-
stage model as explained in detail in the previous section. The column generation method
adds new routes to the initial subset of solution in the restricted master problem until the
stopping condition is reached. The column generation method reaches a reliable solution
in a large-scale network consisting of a large amount of variables. Initially, the column
generation method begins with the restricted master problem with respect to only a small
subset of routes (initial solution). Then, it adds new eligible routes corresponding to the
21
solutions of the sub-problem. The master problem is solved based on the solutions
derived from the restricted master problem for each step in the column generation
process. The restricted master problem is defined as follows:
minimize
𝑣 𝑠 , 𝑓𝑓 , 𝑡 𝑔 ( 𝑉 ) = 𝛼 1
� � �
𝑣 �
𝑙 , 𝑛 − 𝑣 𝑙 , 𝑛 𝑣 �
𝑙 , 𝑛 �
2
+
𝑙 ∈ 𝐿𝐿 𝑛 ∈{ 1,.., 𝑇 }
𝛼 2
� � � 𝑐 𝑠 , 𝑟𝑟 , 𝑡 𝑣 𝑠 , 𝑟𝑟 , 𝑡 𝑡 ∈ 𝑇 𝑟𝑟 ∈ 𝜃 1
𝑠 ∈ 𝑆𝑆 (2.15)
subject to 𝑣 𝑠 , 𝑟𝑟 , 𝑡 ≥ 0 (2.16)
where 𝜃 1
⊂ 𝜃 and 𝑣 𝑙 , 𝑛 = ∑ ∑ ∑ 𝜌 𝑠 , 𝑟𝑟 , 𝑛 𝑙 , 𝑡 𝑣 𝑠 , 𝑟𝑟 , 𝑡 𝑡 ∈ 𝑇 𝑟𝑟 ∈ 𝜃 1
𝑠 ∈ 𝑆𝑆 for the restricted master problem.
The initial solution 𝜃 1
is derived from the four-stage model. The column generation
method can be expressed as follows.
Step 1: Find the initial solution
Step 2: Calculate the total cost
Step 3: Add new routes
Step 4: Total cost improved?
Step 5: Continue until the stopping condition is satisfied
Now, we need to find a model to find new eligible routes to add to the initial solution. In
this problem, we define “cost” as a measure of time required to travel from origin to
destination in the loaded network which is dynamic. Therefore, the minimum cost trip
would also be the shortest and most desirable trip.
𝑔 𝑚 , 𝑝 , 𝑤 =
𝜕𝑔 ( 𝑉 )
𝜕 𝑣 𝑚 , 𝑝 , 𝑤 = 2 𝛼 1
� � �
𝑣 �
𝑙 , 𝑛 − ∑ ∑ ∑ 𝜌 𝑠 , 𝑟𝑟 , 𝑛 𝑙 , 𝑡 𝑣 𝑠 , 𝑟𝑟 , 𝑡 𝑡 ∈ 𝑇 𝑟𝑟 ∈ 𝜃 𝑠 ∈ 𝑆𝑆 𝑣 �
𝑙 , 𝑛 �
𝑙 ∈ 𝐿𝐿 𝑛 ∈[ 0, 𝑇 ]
𝜌 𝑚 , 𝑝 , 𝑛 𝑙 , 𝑤 + 𝛼 2
𝑐 𝑚 , 𝑝 , 𝑤
(2.17)
The routes with the negative value of 𝑔 𝑚 , 𝑝 , 𝑤 are eligible routes and are added to the
previous routes. As a result, the problem of finding a new eligible route to add to the
22
previous routes has changed to the dynamic shortest route problem with new link costs.
The process of adding eligible routes continues until there is no eligible route (stopping
condition).
2.2.3 Online Traffic Flow Estimation
The estimation model must be constantly recalibrated to have a better traffic flow
estimation. However, the results from the previous steps can fill gaps where the real time
data is not available for certain portions of the network. Therefore, we take into account
both the historical and real time data to minimize travel time.
minimize
𝑣 �
𝑙 , 𝑛
� Β
1
� �
𝑣 𝑙 , 𝑟𝑟 − 𝑣 �
𝑙 , 𝑛 𝑣 𝑙 , 𝑟𝑟 �
2
+ Β
2
� �
𝑣 𝑙 , 𝑛 , ℎ
− 𝑣 �
𝑙 , 𝑛 𝑣 𝑙 , 𝑛 , ℎ
�
2
𝑙 ∈ 𝐿𝐿 𝑙 ∈ 𝐿𝐿 1
� (2.18)
subject to 𝑣 �
𝑙 , 𝑛 ∈ 𝜑 𝑛 ( 𝑅𝑅 ) (2.19)
where 0 < B
2
< B
1
< 1, 𝑣 𝑙 , 𝑟𝑟 is real time flow on link 𝑙 for 𝐿 1
⊂ 𝐿 , and 𝑣 𝑙 , 𝑛 , ℎ
is the
estimated link flows based on the offline estimation for link 𝑙 in time interval 𝑛𝑛 . It is
worth noting that we take B1>B2 because of the relative important role of the real-time
data in computing the link flows. The objective is to find 𝑣 �
𝑙 , 𝑛 and the corresponding OD
matrix. The problem is the linearly least-squares problem and strictly convex; therefore,
it can be solved in real time.
2.3 Computational Results
Downtown Los Angeles is selected as testing area, since it is one of the congested regions
in the United States. It consists of 22,659 links, 8,621 nodes, and 573 zones. Figures 2.3
and 2.4 demonstrate the relationship between the estimated link flows and the measured
23
ones during one hour of a random day by taking into account no initial solutions and
initial solutions respectively.
Figure 2.3: Measured and estimated volumes (no initial solution)
Figure 2.4: Measured and estimated volumes (initial solution)
The Root Mean Square Percentage Error (RMSPE) is defined
as 𝑅𝑅 𝑀𝑆 𝑃𝐸 =
�
1
𝐿𝐿 ∑ �
𝑣 �
𝑙 , 𝑛 − 𝑣 𝑙 , 𝑛 𝑣 𝑙 , 𝑛 �
2
𝐿𝐿 𝑙 = 1
. The RMSPE between the estimated and the measured
link flows is 0.17 when no initial traffic flow estimation is used; whereas, the RMSPE is
24
0.09 by using the initial traffic flow estimation in the proposed methodology. The results
demonstrate the important role that the initial traffic flow estimation play in the traffic
flow data completion model. Also, the following Table provides the RMSPE at each
stage of the traffic flow data completion. The random links from the transportation
network are selected to evaluate the proposed methodology.
Table 2.1: The RMSPE for each step of the traffic flow data completion
Link ID ITFE (RMSPE) OTFE (RMSPE) RTFE (RMSPE)
37 0.12 0.03 0.009
137 0.17 0.07 0.012
1,295 0.19 0.04 0.003
1,509 0.23 0.07 0.009
2,784 0.18 0.09 0
3,765 0.19 0.02 0
5,110 0.15 0.03 0.005
where the ITFE indicates the initial traffic flow estimation, the OTFE represents the off-
line traffic flow estimation, and the RTFE is the real-time traffic flow estimation. We
assumed that there is no traffic flow data for these links to evaluate the model, but in fact
there exist loop conductors on these links. The results yield the effectiveness of the
proposed methodology by taking into account the initial solutions derived from the four
stage model.
2.4 Conclusions
The complexity of modern transportation networks is so high. Failures in a fraction of
the system will cascade throughout the system and cause delays that impact the entire
transportation system. Management can take place on the macroscopic level in the form
25
of congestion management technology such as traffic signal coordination, congestion
pricing systems, and changeable message signs. Management can also occur on the
microscopic level in the form of directions given to individual drivers.
We developed a methodology for the traffic flow data completion. At its core, the
challenge behind the problem is an effective real time estimation of the link flows and
corresponding OD matrices for the entire transportation network. If we can effectively
estimate the travel time from one zone to another one within a network, we can assign
traffic to the links (roads) in the network, and generate traffic flows given roadway
capacities. To accurately estimate traffic flows, we first make the initial traffic flow
estimation based on the general information within the region of study. Then, we take
into account the historical loop conductor traffic flow data to feed the offline estimation
model. Finally, we add the real-time traffic data and try to match the final results with
the existing traffic conditions. The traffic flow data completion model needs continuous
and dynamic adjustment, so that the generated flow in the macroscopic traffic flow
simulator (VISUM) is as much as close to the real world. The computational results
demonstrate the important role of using the initial solutions in the proposed methodology.
26
Chapter 3
Optimum Routing: Road Network
3.1 Introduction
On-board navigation technology has become commonplace in recent years. Current
navigation technology is not relatively fast to adapt to real-time traffic information nor
does it take into account potential traffic disruptions when planning travel routes for the
user [90]. The optimum routing methodology seeks to synthesize the variables that affect
traffic conditions to provide an optimal route as well as accurate destination arrival time
for drivers.
There are several different types of traffic disruptions that affect traffic flows in
varying degrees. Some of these disruptions are predictable such as traffic signals, stop
signs, or public transit service. These disruptions typically affect a few roads (links)
within the immediate area. The impacts of these predictable and regularly scheduled
disruptions can be implemented into the optimum routing methodology. Other
disruptions are scheduled events that affect entire neighborhoods. These could include
sporting events, music concerts, or farmer’s markets. The size of the event and the
proportion of attendees choosing the automobile as their mode choice would determine
the amount of links that experience traffic flows above historically normal levels. For
example, a large scale sporting event with tens of thousands of attendees could affect
drivers adjacent to the stadium parking lot. The optimum routing methodology takes into
account these events, and evaluates their subsequent impacts to the traffic network.
27
Finally, there are unpredictable events such as car accidents, breakdowns, and road
closures. These incidents would have typical disruption patterns that correspond to the
time elapsed after the incident; that is, the flows on the road network immediately
following the incident will differ from flows ten minutes, twenty minutes, and thirty
minutes after the incident and so forth. The optimum routing methodology would be able
to estimate the time delay and impact of these events, and adjust the OD matrix in the
real-time accordingly.
We will focus on the road network in the USC-Downtown Los Angeles area, what
we will call the Downtown Los Angeles sub-network. The sub-network is about 15
square miles in size and contains small neighborhood streets, primary and secondary
arterials, and several freeways. Major entertainment venues within the region including
the Staples Center, which hosts nearly four million guests in a year, L.A. Live, a 5.6
million square-foot entertainment complex and home of the Nokia Theater, and the Los
Angeles Music Center, which is one of the three largest performing arts centers in the
United States. It is one of the most congested regions in the United States containing
various entertainment venues within the region.
The system will use the historical and the real-time traffic flows to analyze the
dynamic traffic flow estimation in the designated area. We will take these data and
generate the real time flow using VISUM, a macroscopic traffic flow simulator. We will
also apply optimization algorithms to find the fastest route between a given origin and
destination pair as well as help predict traffic congestion due to flow-disrupting events.
28
3.2 Finding the Shortest Path
Finding the shortest path is necessary in the offline traffic flow estimation step. Also,
once we have estimated the OD matrix, it becomes necessary to find the fastest route
from the driver’s current position (A) to their final destination (B). The Dijkstra’s
algorithm [36] is a popular search algorithm for routing problems. The algorithm works
by finding the path with the lowest cost between two nodes in a network. It stops once
the shortest path is determined between A and B. With the real-time traffic flow
estimation, it will be necessary to continuously perform the modified Dijkstra’s
algorithm, so the optimum routing methodology can give the driver the best estimation of
travel time (cost) and the optimum route to his or her destination. The reason behind
modifying Dijkstra’s algorithm is that the optimum routing methodology must perform in
real-time, so that we need to modify Dijkstra’s algorithm to find the best route much
faster than the original algorithm. Of course, not always the algorithm finds the best
solution, since we limit the searching region. The Dijkstra’s algorithm and the proposed
approach are described in below.
In terms of this research, all links between nodes will have an associated cost.
Initially, we set all the cost values for travel between nodes as infinite in order to denote
them as unvisited nodes. Together all these nodes represent the unvisited set. Even if a
traveler is not at a certain node, we can call his/her initial position as a new node in the
middle of a link. From this initial node (A), we calculate the cost of traversing all the
links to the neighboring nodes. Then, we examine all the nodes neighboring the initial
set of neighboring nodes in a road network, it is likely that there are multiple paths
connecting the initial node to the secondary set of neighboring nodes.
29
Then, the algorithm finds a first cost of travelling from node (A) to the secondary set
of neighboring nodes. The algorithm finds a second cost using a different pair of links.
If this cost is less than the first cost, the cost is overwritten with the new low cost. Once
all combinations of links have been queried to go from initial node (A) to the node in the
secondary set of nodes, the node is marked as “visited” and the lowest cost is recorded.
The algorithm will not check the travel from the initial node to the visited node again. If
the destination node (B) becomes part of the visited set, the algorithm stops and provides
the driver with the shortest path.
The Dijkstra algorithm ensures that all possible combinations of links connecting
initial node (A) and destination node (B) have been examined and that the lowest cost set
of links is selected as the shortest path. Since it examines all possible path possibilities
before marking a node as visited, the algorithm is inherently cumbersome and relatively
slow in finding the shortest path solution. Furthermore, urban road networks are highly
complex with thousands of links connecting thousands of nodes making the runtime for
the Dijkstra’s algorithm too lengthy to meet the demands of the urban navigation system
(Dijkstra’s algorithm checks every possibility regardless of its practicality). Therefore,
we need to modify the Dijkstra’s algorithm to speed program run-time; so that, we
examine the shortest path algorithms that reduce the size of the network examined. We
restrict the search space with the geometrical shapes such as square, rectangle, and
hexagon. The experimental results illustrate that hexagon is the best geometrical figure
that fits our network (LA downtown). The computational time is reduced more than 10
times by restricting the graph network with hexagon. The pseudo-code is presented in the
following [36].
30
Dijkstra(G,S,D):
for each node n in G:
cost[n] := infinity ;
parent[n] := undefined ;
cost[s]:=0;
P:= { ∪ 𝑛𝑛 };
H:= {};
end for
while P has node:
m := the smallest cost in P;
P := P \{m};
H := H ∪ {m};
for each adjacent node n of m:
if n is not in the designated bounds:
find another n
else
n_cost := cost[m] + cost(m, n) ;
if n_cost < cost[n]:
cost[n] := n_cost ;
parent[n] := m ;
end if
end if
end for
end while
return n_cost;
The results are demonstrated in the following table.
Table 3.1: Comparison of the shortest path algorithms
Algorithm Visited Nodes
Computational time
(Sec)
Total Cost (Sec)
Dijkstra 5,359 12.3 1,711
Dijkstra Restricted by
Square
1,367 2.5 1,711
Dijkstra Restricted by
Rectangle
924 1.8 1,711
Dijkstra Restricted by
Hexagon
411 0.9 1,711
31
The first column indicates the restriction of the region of interest by the specific
geometrical shapes. The second column represents the visited nodes (checked nodes) by
the algorithm, and the third one is the computational time in terms of second. The fourth
column indicates the total cost (travel time) in terms of second. All four methods find the
same route for the user, but the Dijkstra’s algorithm restricted by the hexagon provides
much faster result.
3.3 Estimating the Best Route Given Major Events
One of the shortcomings of contemporary navigation systems is their response to events,
both of on-road and off-road nature. Some navigation systems respond to inputs of
traffic jams or slow speeds by re-routing drivers to nearby detours routes. However,
depending on the severity of the event, these detour routes may not be the most efficient
routes since drivers in the midst of a traffic jam will choose adjacent links as their
primary alternatives. We seek to understand how major events such as accidents or
social events change traffic conditions on not only the primary road in question, but
rather on the entire sub-network of links surrounding the incident of event. In turn, this
will allow the route guidance system to estimate the best route for the driver.
3.3.1 On-Road Events
First, it is necessary to estimate the changes to the network capacity and the OD matrix
when an incident occurs on the links. This could include car accidents involving one or
more cars, road and lane closures, stalled or disabled vehicles impeding the traffic flow.
The easiest of these events to adjust are road closures. In the event of road closures, we
can simply set the capacity of the affected link to zero. Since, road closures are fairly
32
commonplace and temporary by nature, it is relatively easy to study the effects of traffic
on the links surrounding the closed roads and then compare it to the baseline flow on
those links when the road is open to traffic.
In the same way, lane closures are also relatively easy to implement; however, the
nature of the lane closure could have a significant effect on capacity. For example, a lane
closure due to an unusual social event could slow drivers down significantly more than a
lane closure due to a routine overnight paving operation. These events are preplanned, so
that the OD matrix does not need to be dynamically updated to evaluate the effects of
road closures.
It is much harder to estimate the effects of accidents on the road network. Unlike
preplanned closures, accidents are not preplanned and their effects on the road network
are highly unpredictable. There are several factors which accidents effect the traffic
flows in the network such as the number of cars involved, the extent of the damage, or
even the visibility of the accidents (in fact, the California The Department of
Transportation has constructed “Accident Investigation Sites” off the Interstate-10
mainline in Los Angeles in an attempt to mitigate the effects of “rubbernecking” drivers
slowing down to observe automobile collisions). Usually, it takes several minutes for
accidents to appear in a database, so that we hope expedite the response time and
effectiveness of the optimum routing methodology by identifying the traffic pattern
“signature” associated with a specific type of accident or disabled vehicle. After the
initial adjustments to the physical network are made based on the significant variations
on the traffic flow, confirmation from the accident databases are used to update and fine-
tune the adjustments made for the initial accident hypothesis
33
3.3.2 Off-Road Events
Accidents and road closures affect the network, because they occur directly on links or on
nodes. However, major events that occur adjacent to the road network and draw
hundreds, if not thousands, of attendees can have as large or larger impact on the network
flows. Major events could include athletic matches, concerts, or other cultural activities.
Attendees could affect flows on the links that are miles away from the actual event. In
terms of the OD matrix, major events create huge attractions in the zones that may
normally have relatively insignificant levels of the trip attractions. Likewise at the end of
these events, the production from that zone would be even larger than its attraction since
attendees tend to arrive at events over a several hour period, but return to their respective
origins almost simultaneously.
To estimate traffic flows for these types of events, we use the initial traffic flow
estimation model based on the surveys, historical attendance figures, and typical demand
in region at times in the past when events occur. Next, we use the offline traffic flow
estimation by monitoring traffic at several events around Los Angeles such as University
of Southern California (USC) football games. Since traffic at these events change
rapidly, we will attempt to create unique OD matrices for ten-minute time intervals as
opposed to the typical hour-by-hour intervals for the usual offline traffic flow estimation
model. Finally, once the system goes online, the model updates the estimation model
based on the real time traffic flow data dynamically.
34
3.4 Computational Results
As we mentioned earlier, we focus on the USC and downtown LA to perform the real
experiments and verify the developed algorithm. The following Figure depicts the layout
of the region of experiment in VISUM format.
Figure 3.1: Layout of region of interest
Numbers 1-5 in Figure 3.1 indicate large venues that have big impact on the traffic flows
of streets adjacent to the specific venue when an event is being held. We estimated traffic
flows in the transportation network for the interested region one day before an event, and
then compare with the real time flows where traffic data is available. Table 3.2 illustrates
the computational results.
1
2
3
5
4
35
Table 3.2: Computational results
EVENT
RMSPE (BEFORE
EVENT)
RMSPE (DURING
EVENT)
RMSPE (AFTER EVENT)
EVENT
DATA
NO EVENT
DATA
EVENT
DATA
NO EVENT
DATA
EVENT
DATA
NO EVENT
DATA
1
2
3
4
5
2 &3
4 & 5
0.19
0.18
0.15
0.09
0.09
0.28
0.25
0.67
0.51
0.39
0.27
0.31
0.84
0.50
0.11
0.13
0.11
0.05
0.04
0.17
0.19
0.31
0.44
0.22
0.18
0.26
0.45
0.37
0.28
0.26
0.19
0.12
0.11
0.33
0.29
0.83
0.60
0.46
0.48
0.49
0.98
0.61
The first column represents the location of events corresponding to Figure 3.1. Columns
2-4 demonstrate the Root Mean Square Percentage Error (RMSPE) of the estimated and
the measured link flows. The RMSPE is defined as 𝑅𝑅 𝑀𝑆 𝑃𝐸 =
�
1
𝐿𝐿 ∑ �
𝑣 �
𝑙 , 𝑛 − 𝑣 𝑙 , 𝑛 𝑣 𝑙 , 𝑛 �
2
𝐿𝐿 𝑙 = 1
. The
results in Table 3.2 present the effectiveness of the proposed methodology to estimate the
traffic flows in the urban region in case of social events. In each case, the comparison of
using the historical data for the events and ignoring those data in the traffic flow
estimation is provided.
3.5 Conclusions
In the optimum routing methodology, we attempt not only to refine the ability of finding
the optimal route, but also to provide the driver with an accurate prediction of travel time
to his/her destination. To find the optimal route, we focused on refining and enhancing
the estimation of link flows in the network by looking at the historical data and the real
time traffic data as well as by trying to estimate the impacts of on road/off road events
36
such as road closures, accidents, and large-scale cultural events like sporting events and
concerts on traffic flows.
The optimum routing methodology needs to be able to handily adapt to a rapidly
changing link flows; otherwise, the dynamic traffic flow estimation is worthless for the
routing purposes. To optimize the performance of the optimum routing methodology, we
analyzed several different path search algorithms such as the Dijkstra’s algorithm. In the
proposed approach, we developed a new algorithm by modifying the Dijkstra’s algorithm
restricting the search region to reduce the computational time. The overall goal is to find
the best route for a driver by taking into account past and current traffic conditions as
well as on-road and off-road events.
37
Chapter 4
Optimum Multimodal Routing
4.1 Introduction
In an increasingly complex physical distribution system, a world in which time is tightly
correlated with money; it becomes more critical to effectively manage the way that we
interact with the transportation infrastructure. Furthermore, as the population grows and
the infrastructure ages, the demands on the built environment are increased. The routing
of goods involves complex operations in a complex dynamical network with uncertainties
and limited resources. This complex system leads to congestion and delays that have a
significant impact on the quality of life and the environment as well as the economy. On
the other hand, sea-ports play a crucial role in the global economy, since the majority of
containers are transported through sea-ports. Most sea-ports are located adjacent to an
urban area with time dependent traffic. Therefore, the local freight transportation process
has to be done in the best possible way to avoid creating additional traffic congestion.
Distribution systems offer the potential of aggregation and sharing of multiple
resources. These resources could be owned and operated by different organizations. Each
individual organization may have different objectives, goals, and strategies from other
organizations. Load balancing in a distribution system is a process of balancing
workloads by taking into account the organization objectives. The load balancing process
can be either static or dynamic. In the static load balancing process, statistical behavior of
the system is only considered and transfer decisions are made without taking into account
38
the current system state. Historical and current states of the system are used to make
transfer decisions in the dynamic load balancing process. The main advantage of the
static load balancing is its simplicity; whereas, in the dynamic load balancing the current
system state must be fed to the solution procedure. However, dynamic load balancing has
the potential to provide better solutions for the distribution system. The dynamic load
balancing process can be performed coordinated or uncoordinated. In the coordinated
dynamic load balancing, a coordinator collects and updates the information regarding the
historical and current states of the distribution system to make transfer decisions. The
solution obtained from the coordinated dynamic load balancing works for the overall
system but it may not satisfy an individual distributor system’s goals. However, the
uncoordinated dynamic load balancing satisfies the individual distributor system’s
objective at possibly the expense of the overall system’s performance. The following
figure illustrates the concept of the coordinated and uncoordinated distribution systems.
(a) (b)
Figure 4.1: (a) Uncoordinated distribution system (b) Coordinated distribution system
The transportation network can be represented as a graph network consists of a set of
nodes where arcs connecting the nodes. Nodes (N) represent origins ( 𝑛𝑛 0
), intermediary
Distributor system B
Distributor system A
Distributor system B
Distributor system A
Coordinator
39
( 𝑛𝑛 𝐼 ), and destinations ( 𝑛𝑛 𝐷𝐷 ). A set of arcs in the network is characterized by the set of
transportation links L (roads/railways) offered between the origins and destinations.
Let 𝑋 ( 𝑘 ) = [𝑥 1
( 𝑘 ) … 𝑥 𝑠 ( 𝑘 )]
𝑇 be an array which determines the volume traveling on
the links at time 𝑘 ∈ 𝐾 where 𝐾 = {0,1, … , 𝑇 }, and 𝑑 𝑖𝑖 ( 𝑘 ) denotes the number of units to
be deployed from origin 𝑖𝑖 at time 𝑘 .The historical links’ volumes { 𝑋 (0), … , 𝑋 ( 𝑘 )} and
future numbers of units to be deployed { 𝑑 𝑖𝑖 ( 𝑘 + 1), … , 𝑑 𝑖𝑖 ( 𝑘 + 𝑚 )} are assumed to be
available. The model aims to estimate the links’ volume based on the historical/real time
volumes and future deploying units which can be represented as follows for the
uncoordinated dynamic load balancing.
𝑋 ( 𝑘 + 1) = 𝑋 ( 𝑘 ) + 𝑓𝑓 � 𝑋 ( 𝑘 ), 𝑘 , 𝑑 ̂ ( 𝑘 ), 𝐴 𝑖𝑖 𝑑 𝑖𝑖 ( 𝑘 ) �
(4.1)
where 𝐴 𝑖𝑖 = [𝑎 𝑖𝑖 1
… . 𝑎 𝑖𝑖 𝑠 ]
𝑇 is an array which determines proportions of 𝑑 𝑖𝑖 ( 𝑘 ) which are
assigned to the links, and 𝑓𝑓 ( 𝑋 ( 𝑘 ), 𝑘 , 𝑝 ̂( 𝑘 ), 𝐴 𝑖𝑖 𝑑 𝑖𝑖 ( 𝑘 )) is a non-linear function of the links’
volume, time, estimated impact of the other vehicles (e.g. cars, buses) from the adjacent
links ( 𝑝 ̂( 𝑘 )), and the deployed number of units from origin 𝑖𝑖 . The links’ volumes in the
network are time-dependent and change due to various reasons such as load departed and
incidents. Let 𝐶 ( 𝑘 + 1) = [𝑐 1
( 𝑘 + 1) … 𝑐 𝑠 ( 𝑘 + 1)]
𝑇 be an array which specifies the
total costs per unit for the links. Also, let 𝑢 𝑙 denote the maximum capacity of the link.
The total cost per unit for each link is a function of volume, travel time or other
parameters depending on the objective of the individual distributor system. The travel
time is a function of the volume assigned to a link. The uncoordinated dynamic load
balancing model aims to find the best solution based on the information of an individual
40
distributor system. Hence, the uncoordinated dynamic load balancing problem for origin 𝑖𝑖
can be formulated as below.
𝑚 𝑖𝑖𝑛𝑛 𝑖𝑖𝑚 𝑖𝑖𝑧 𝑒 𝐿𝐿 𝑖𝑖 𝐶 𝑇 ( 𝑘 + 1) 𝐴 𝑖𝑖 𝑑 𝑖𝑖 ( 𝑘 ) (4.2)
subject to
𝑋 ( 𝑘 + 1) = 𝑋 ( 𝑘 ) + 𝑓𝑓 � 𝑋 ( 𝑘 ), 𝑘 , 𝑑 ̂ ( 𝑘 ), 𝐴 𝑖𝑖 𝑑 𝑖𝑖 ( 𝑘 ) �
(4.3)
0 ≤ 𝑥 𝑙 ( 𝑘 + 1) ≤ 𝑢 𝑙 ∀ 𝑙 ∈ 𝐿 (4.4)
� 𝑎 𝑖𝑖 𝑙 = 1
𝑙 ∈ 𝐿𝐿
(4.5)
given { 𝑋 (0), … , 𝑋 ( 𝑘 )} (4.6)
and { 𝑑 𝑖𝑖 ( 𝑘 + 1), … , 𝑑 𝑖𝑖 ( 𝑘 + 𝑚 )} (4.7)
Objective function (4.2) aims to minimize the total cost of the network by taking into
account only the number of units to be deployed from origin 𝑖𝑖 . Constraint (4.3) estimates
the links’ volume, constraint (4.4) corresponds to the maximum volume of the links, and
constraint (4.5) ensures distribution of 𝑑 𝑖𝑖 . In the coordinated dynamic load balancing, a
coordinator is responsible to collect and update the information from all distribution
systems in order to make a final transfer decision. Therefore, the coordinated dynamic
load balancing for 𝑁𝑁 0
origins can be presented as follows:
𝑚 𝑖𝑖𝑛𝑛 𝑖𝑖𝑚 𝑖𝑖𝑧 𝑒 𝐿𝐿 1
,…, 𝐿𝐿 𝑁 0
𝐶 𝑇 ( 𝑘 + 1) � 𝐴 𝑖𝑖 𝑑 𝑖𝑖 ( 𝑘 )
𝑁 0
𝑖𝑖 = 1
(4.8)
subject to 𝑋 ( 𝑘 + 1) = 𝑋 ( 𝑘 ) + 𝑓𝑓 � 𝑋 ( 𝑘 ), 𝑘 , 𝑑 ̂ ( 𝑘 ), � 𝐴 𝑖𝑖 𝑑 𝑖𝑖 ( 𝑘 )
𝑁 0
𝑖𝑖 = 1
� (4.9)
41
0 ≤ 𝑥 𝑙 ( 𝑘 + 1) ≤ 𝑢 𝑙 ∀ 𝑙 ∈ 𝐿 (4.10)
� 𝑎 𝑖𝑖 𝑙 = 1
𝑙 ∈ 𝐿𝐿
∀ 𝑖𝑖 ∈ 𝑛𝑛 0
(4.11)
given { 𝑋 (0), … , 𝑋 ( 𝑘 )} (4.12)
and { 𝑑 𝑖𝑖 ( 𝑘 + 1), … , 𝑑 𝑖𝑖 ( 𝑘 + 𝑚 )} ∀ 𝑖𝑖 ∈ 𝑛𝑛 0
(4.13)
The parameters of the non-linear function 𝑓𝑓 are unknown. One approach is to estimate
the parameters of function 𝑓𝑓 to estimate the links’ volume which is complicated for a
large network due to its complexity. The other approach is to use cost evaluators to
estimate the links’ volume based on the historical/real time volumes as well as future
units to be deployed form the origins. The cost evaluators are chosen to estimate links’
travel time in the proposed methodology. Figure 4.2 illustrates the framework of the
proposed methodology.
Figure 4.2: Framework of the proposed methodology
The initial condition involves a set of initial links’ volume in the network. First, the
links’ travel time corresponding to the initial condition are estimated by the cost
Initial
Condition
Cost
Evaluators
Optimization Real System
Final Solution
Cost Estimations
Satisfy stopping
condition?
NO
Yes
Update solution
42
evaluators and fed into the optimization algorithm. Then, the optimization algorithm
generates the transfer decision which improves the total cost based on links’ travel time
obtained from the cost evaluators iteratively. This process continues until the stopping
conditions are satisfied which yields the best feasible solution. The cost evaluators use
nonlinear models to estimate travel times. The final result can be fed to a real system in
order to calibrate the cost evaluators’ parameters and verify the solution.
Due to the important role of freight transportation on the economy, we focus on the
Multimodal Dynamic Freight Load Balancing (MDFLB). Numerous papers have been
published to address the multimodal freight transportation. Formulation and solution of a
multimodal network flow model were presented by Haghani et al. [47]. The multimodal
network was based on a time-space network, and two heuristics were proposed to solve
the problem. Jourquin et al. [57] analyzed a multimodal freight transportation policy in
Europe. They introduced a model for freight moving, loading, and unloading. The
proposed optimization algorithm aimed to minimize the total transportation cost
including route choices, modes, and means. Modesti et al. [73] provided a utility measure
for finding multi-objective shortest paths in urban multimodal transportation networks.
They presented an approach based on the classical shortest path problem for a
multimodal transportation network. The experiments were performed on the urban
transportation network of an Italian city. Lozano et al. [62] extended the previous work
by using label correcting techniques to find the shortest viable path in a multimodal
transportation network. Intermodal and international freight network modeling was
studied by Southworth et al. [94]. They constructed a network that covered five million
origin-to-destination freight shipments reported as part of the 1997 United States
43
commodity flow survey. Multimodal transportation, logistics and the environment were
studied by Rondinelli et al. [85]. They looked into the interactions among transportation
activities and the types of environmental impacts emanating from multimodal
transportation operations and facilities. Supply-demand equilibrium in a multimodal
transportation network was proposed by Fernandez et al. [40]. They developed a new
approach for an intercity freight transportation system that takes into account supply-
demand equilibrium. Arnold et al. [8] presented an approach for modeling a rail/road
intermodal transportation system. The experiments were focused on the rail/road
transportation system in the Iberian Peninsula.
Macharis et al. [65] provided a comprehensive review of an intermodal freight
transportation system. They argued that the intermodal freight transportation research is
emerging as a new research application field which needs different types of models than
those applied to uni-modal transport system. Also, Nijkamp et al. [76] presented a
comparative modeling of interregional transport flows which was applied to the
multimodal European freight transport. They compared the descriptive and predictive
power of two classes of statistical estimation models for multimodal network flows.
Scheduling of the multimodal freight transportation systems is an important issue
which has been addressed by many researchers. Castelli et al. [25] used a Lagrangian
based heuristic procedure for scheduling multimodal transportation networks. At each
step, their proposed algorithm schedules a single line by correcting the previous
decisions. Ham et al. [48] provided an estimation of a combined model of interregional,
multimodal commodity shipment and transportation network flows. They described the
formulations and solutions of the model using U.S. interregional commodity shipment
44
data, and evaluated the model with the observed data. Optimizing the design of
multimodal freight transport network in Indonesia was proposed by Russ et al. [89]. A
mathematical model is developed within the framework of a bi-level programming
problem, where a multimodal multi-user assignment forms the lower level problem and
upper level includes the combination of actions for capacity expansion.
Environmental issues are taken into account in multimodal freight transportation
system modeling. Janic [52] developed an approach for modeling the full costs of a
multimodal and road freight transport network. A heuristic approach was presented by
Yamada et al. [100] for designing a multimodal freight transportation network. The
proposed model determines a suitable set of actions for improving the existing
infrastructure or establishing new railways, roads, and freight terminals. A mathematical
model of selecting transport facilities for multimodal freight transportation is provided by
Lingaitiene [61]. The cost function takes into account overall technological costs of
transportation using road, rail, and sea links. Caris et al. [22] studied planning problems
in multimodal freight transportation systems. They extended the previous work by
proposing decision support models for network operators, drayage operators, and
terminal operators as well as for public actors such as policy makers and port authorities.
Many other publications addressed the multimodal freight transportation system
[5,20,27,66,68,72,78,84].
However, none of the mentioned works addresses the coordinated/uncoordinated
MDFLB. Moreover, the mentioned works do not take into account the dynamic nature of
traffic condition which is non-linear with respect to the volume. For a certain range of
flow, the cost of using a link and consumption of capacity may change linearly, but at
45
some break point the congestion increases as a nonlinear function of the volume. To
overcome these limitations, we will formulate the optimization using the framework of
the service network design problem [30] with an appropriate cost function to find the
optimal solution for the coordinated /uncoordinated MDFLB.
The main contribution of this chapter is to identify the cost parameters for the
uncoordinated /coordinated MDFLB using the optimum routing model in road network
and a railway simulation system to find the best feasible solution by taking into account
constraints as well as the non-linear nature of travel times with respect to the volumes.
For a certain range of flow, the cost and consumption of capacity of a link may change
linearly, but at some break point the congestion increases as a nonlinear function of the
volume. We capture these nonlinearities in the objective function using the simulators.
4.2 Optimization Formulation
In this section, we present a mathematical formulation of the problem. Consider a graph
G = (N, L) which represents a physical network, where N is a set of nodes and L is a set
of links. Nodes (N) are classified into three categories which are origins ( 𝑛𝑛 0
),
intermediary ( 𝑛𝑛 𝐼 ), and destinations ( 𝑛𝑛 𝐷𝐷 ). A set of arcs in the network is characterized by
the set of transportation links L (roads/railways) offered between the origins and
destinations. The objective is to select the flow on the links in a network in order to
satisfy the demands for customers at minimum system cost with regard to capacity
constraints. Let 𝑐 𝑖𝑖 𝑗 ( 𝑘 ) indicate the transportation cost per unit flow for pair ( 𝑖𝑖 , 𝑗 ) at
time 𝑘 ∈ 𝐾 . The amount of flow between pair ( 𝑖𝑖 , 𝑗 ) at time 𝑘 ∈ 𝐾 is represented
46
by 𝑥 𝑖𝑖 𝑗 ( 𝑘 ). Let 𝑢 𝑖𝑖 𝑗 ( 𝑘 ) be the residual capacity of the link connecting pair ( 𝑖𝑖 , 𝑗 ), and 𝐷𝐷 𝑖𝑖
represents the amounts of supply or demand of node 𝑖𝑖 which can be defined as follows:
�
𝐷𝐷 𝑖𝑖 = 𝑠 𝑢 𝑝𝑝𝑙 𝑦 𝑖𝑖𝑓𝑓 𝑖𝑖 ∈ 𝑛𝑛 0
𝐷𝐷 𝑖𝑖 = − 𝑑𝑒 𝑚𝑎 𝑛𝑛𝑑 𝑖𝑖𝑓𝑓 𝑖𝑖 ∈ 𝑛𝑛 𝐷𝐷 𝐷𝐷 𝑖𝑖 = 0 𝑖𝑖𝑓𝑓 𝑖𝑖 ∈ 𝑛𝑛 𝐼 (4.14)
Note that ∑ 𝐷𝐷 𝑖𝑖 = 0
𝑖𝑖 ∈ 𝑁 and 𝑥 𝑖𝑖 𝑖𝑖 ( 𝑘 ) = 0 for all nodes in the network. In the coordinated
MDFLB, the coordinator collects and updates information from all the origins in the
network. Therefore, the optimization formulation for the coordinated MDFLB can
expressed as follows:
minimize
� � 𝑐 𝑖𝑖 𝑗 ( 𝑘 + 1)
( 𝑖𝑖 , 𝑗 ) ∈ 𝐿𝐿 𝑂𝑂 ∈ 𝐾 𝑥 𝑖𝑖 𝑗 ( 𝑘 )
(4.15)
subject to
� � 𝑥 𝑖𝑖 𝑗 ( 𝑘 )
𝑗 :( 𝑖𝑖 , 𝑗 ) ∈ 𝐿𝐿 − � � 𝑥 𝑗𝑖𝑖
( 𝑘 )
𝑗 :( 𝑗 , 𝑖𝑖 ) ∈ 𝐿𝐿 =
𝑂𝑂 ∈ 𝐾 𝑂𝑂 ∈ 𝐾 𝐷𝐷 𝑖𝑖
∀ 𝑖𝑖 ∈ 𝑁𝑁 (4.16)
0 ≤ 𝑥 𝑖𝑖 𝑗 ( 𝑘 ) ≤ 𝑢 𝑖𝑖 𝑗 ( 𝑘 ) ∀( 𝑖𝑖 , 𝑗 ) ∈ 𝐿 , ∀ 𝑘 ∈ 𝐾 (4.17)
𝑥 𝑖𝑖 𝑗 ( 𝑘 ) ∈ {0,1,2, … } ∀( 𝑖𝑖 , 𝑗 ) ∈ 𝐿 , ∀ 𝑘 ∈ 𝐾 (4.18)
Objective function (4.15) aims to minimize the total cost of the flow. Constraint (4.16)
provides demand satisfaction requirements, and constraint (4.17) is the capacity
restriction of the flow on each link. In the uncoordinated MDFLB, transfer decisions are
made based on the individual origin information. Therefore, only the supply and demands
of the assigned origin are taken into account in (4.16) for the uncoordinated MDFLB, and
the other origins’ supplies and demands are set to zero.
47
In the above formulations, the cost of using a link is evaluated by a nonlinear
function with respect to the volume assigned to the link. We associate dual variable 𝜆
corresponding to (4.16) where 𝜆 is unrestricted. The dual linear function of (4.15)-(4.18)
is defined as follows [74]:
𝑞 ( 𝜆 ) = 𝑖𝑖 𝑛𝑛𝑓𝑓
𝑥 𝑖𝑖𝑗
( 𝑂𝑂 ) ∈[ 0, 𝑢 𝑖𝑖𝑗
( 𝑂𝑂 )]
� � � 𝑐 𝑖𝑖 𝑗 ( 𝑘 + 1)
( 𝑖𝑖 , 𝑗 ) ∈ 𝐿𝐿 𝑂𝑂 ∈ 𝐾 𝑥 𝑖𝑖 𝑗 ( 𝑘 )
+ � � 𝜆 𝑖𝑖 ( 𝑘 )
𝑖𝑖 ∈ 𝑁 𝑂𝑂 ∈ 𝐾 � � 𝑥 𝑗𝑖𝑖
( 𝑘 )
𝑗 :( 𝑖𝑖 , 𝑗 ) ∈ 𝐿𝐿 − � 𝑥 𝑖𝑖 𝑗 ( 𝑘 ) + 𝐷𝐷 𝑖𝑖 𝑗 :( 𝑗 , 𝑖𝑖 ) ∈ 𝐿𝐿 � �
(4.19)
By the duality theorem [99], we have:
𝑥 𝑖𝑖 𝑗 ∗
( 𝑘 ) ∈ 𝑎𝑓𝑓 𝑔 𝑚 𝑖𝑖𝑛𝑛
𝑥 𝑖𝑖𝑗
( 𝑂𝑂 ) ∈[ 0, 𝑢 𝑖𝑖𝑗
( 𝑂𝑂 )]
� � 𝑐 𝑖𝑖 𝑗 ( 𝑘 + 1) − 𝜆 𝑖𝑖 ( 𝑘 ) + 𝜆 𝑗 ( 𝑘 ) � 𝑥 𝑖𝑖 𝑗 ( 𝑘 )
+ � 𝐷𝐷 𝑖𝑖 𝑖𝑖 ∈ 𝑁 𝜆 𝑖𝑖 ( 𝑘 ) �
∀ 𝑘 ∈ 𝐾
(4.20)
Therefore, the non-basic link with the most negative reduced cost ( 𝑐 𝑖𝑖 𝑗 ( 𝑘 + 1) − 𝜆 𝑖𝑖 ( 𝑘 ) +
𝜆 𝑗 ( 𝑘 )) by taking into account the capacity constraint is added to the basic links. The
process continues until all the non-basic links with negative reduced cost are covered
within the network. The simulation model framework updates 𝑐 𝑖𝑖 𝑗 iteratively which is
non-linear with respect to the volume assigned to the link. The objective function is
nonlinear; therefore, the optimization problem is not solved directly but iteratively. The
uncoordinated/coordinated MDFLB procedures are presented as follows [19]:
48
Figure 4.3: Uncoordinated MDLFB procedure
Split supply of origin 𝑖𝑖 ∈ 𝑛𝑛 0
: 𝐷𝐷 𝑖𝑖 𝑤𝑤 =
𝐷𝐷 𝑖𝑖 𝑊𝑊 𝑓𝑓 𝑓𝑓𝑓𝑓 𝑤𝑤 = 1, . . , 𝑊𝑊
𝑤𝑤 = 0
Apply the traffic flow data completion methodology
Update links’ cost
Find the most negative reduced cost route for all OD pairs initiating from
origin 𝑖𝑖
Assign 𝐷𝐷 𝑖𝑖 𝑤𝑤 to the route
𝑤𝑤 = 𝑤𝑤 + 1
𝑤𝑤 = 𝑊𝑊 ?
Update links’ cost
Balance all OD pairs initiating form origin 𝑖𝑖
Update links’ cost
New routes found?
Find the most negative reduced cost route for all OD pairs initiating from
origin 𝑖𝑖
End
YES
YES
49
Figure 4.4: Coordinated MDFLB procedure
Split supplies of all OD pairs: 𝐷𝐷 𝑖𝑖 𝑤𝑤 =
𝐷𝐷 𝑖𝑖 𝑊𝑊 𝑓𝑓 𝑓𝑓𝑓𝑓 𝑤𝑤 = 1, . . , 𝑊𝑊 ; 𝑖𝑖 = 1, . . , 𝑁𝑁 0
𝑤𝑤 = 0
Apply the traffic flow data completion methodology
Update links’ cost
Find the most negative reduced cost route for all OD pairs
Assign 𝐷𝐷 𝑖𝑖 𝑤𝑤 to the routes
𝑤𝑤 = 𝑤𝑤 + 1
𝑤𝑤 = 𝑊𝑊 ?
Update links’ cost
Balance all OD pairs
Update links’ cost
New routes found?
Find the most negative reduced cost route for all OD pairs
End
YES
YES
50
Note that the capacity constraint must be satisfied in each step. The balancing is part of
both the coordinated/uncoordinated MDFLB procedures which can be expressed as
follows:
Figure 4.5: Balancing procedure
The iterative procedures aim to find the final solution from a finite set of possible
solutions. The links’ costs are updated based on the solution of the optimization model
iteratively. The simulation model framework is used to estimate costs using the results of
the optimization model.
4.3 Simulation Model Framework
The simulation model framework consists of a macroscopic traffic flow and a railway
simulation system. Various traffic flow simulators have been developed to simulate and
For an OD pair, there exist 1,…,P routes
Select the minimum and maximum route costs P1 and P2
Balance the volume of P1 and P2 such that |cost(P1)-cost(P2)|< 𝜖𝜖
where 𝜖𝜖 is a positive small constant
Select the new minimum and maximum route costs P1 and P2
Check |cost(P1)-cost(P2)| ≥ 𝜖𝜖 ?
Update links’ cost
End
YES
51
analyze traffic flow on highways and surface streets. These simulators are developed
using software tools such as Corsim, Paramics, VISUM, and others. In this research, we
use the macroscopic traffic simulator (VISUM) [82] to simulate traffic flows of a road
network in the designated region. VISUM is a commercial software package that allows
the development of macroscopic traffic flow simulation models of the selected roadway
networks. VISUM allows the evaluation of different road configurations, traffic flow
control techniques, infrastructure technologies, etc. without having to build them and/or
perform actual experiments which are costly and may significantly disrupt traffic in an
adverse way. The inputs of VISUM are the OD matrices, and the outputs are the
estimated links’ volume and corresponding travel time. The transportation network
consists of several elements such as links, nodes, zones, etc. Nodes are connected by
links, and links represent streets or freeways. Zones are places that considerable numbers
of people visit such as schools, stadiums, commercial buildings, and so on. Also, one
zone is defined for each residential district. The OD matrix determines the number of
trips within zones in each time interval. Figure 4.6 illustrates the layout of a VISUM
network.
52
Figure 4.6: Layout of VISUM network
Note that vehicles’ travel time has a nonlinear relationship with the volume. We capture
these non-linearity using VISUM. Travel time of link 𝑙 ∈ 𝐿 in the loaded network
( 𝑡 𝑙 − 𝑙𝑃𝑃 𝑂𝑂𝑑
) can be calculated as follows [50]:
𝑡 𝑙 − 𝑙𝑃𝑃 𝑂𝑂𝑑
( 𝑘 + 1) = 𝑡 𝑙 − 𝑓𝑓𝑟𝑟𝑆𝑆 𝑆𝑆 . [1 −
𝑞 𝑙 ( 𝑘 + 1)
𝑞 𝑙 − 𝑚𝑂𝑂𝑥
]
𝛼 (4.21)
where 𝑡 𝑙 − 𝑓𝑓 𝑟𝑟𝑆𝑆 𝑆𝑆 represents travel time of link 𝑙 ∈ 𝐿 in the unloaded network and
𝑞 𝑙 ( 𝑂𝑂 + 1)
𝑞 𝑙 − 𝑚𝑎𝑥 is
the ratio of the density to the density jam of a link at time 𝑘 + 1. The density jam for a
link occurs when the link’s volume and the link’s speed reach zero. The parameter 𝛼 is a
negative number less than 1 which can be estimated using historical traffic data. We use
the optimum routing in road network methodology to estimate flow for all links in the
network. Railway simulation system was developed to evaluate train movements in a
complex network. In the developed railway simulation system, railway track is
decomposed into different segments. Then, the abstract graph is constructed where each
node includes segments, junctions or stations, and each arc represents the connection
53
between different segments or stations. Trains’ movement in the physical railway system
is simulated as movements in the constructed graph. Train travel time is calculated
according to [64] by taking into account trains’ dynamics. Given the acceleration rate 𝑏 1
,
deceleration rate 𝑏 2
, link length 𝐻 , the entering speed 𝛽 1
, exiting speed 𝛽 2
and the speed
limit 𝛽 for one link, train travel time 𝑡 within this link is calculated as follows [64]:
𝑡 =
𝛽 2
− 𝛽 1
𝑏 1
If 𝛽 1
≤ 𝛽 2
≤ 𝛽 and 𝛽 2
2
− 𝛽 1
2
= 2 𝑏 1
𝐻 (4.22)
𝑡 =
𝛽 1
− 𝛽 2
𝑏 1
If 𝛽 2
≤ 𝛽 1
≤ 𝛽 and 𝛽 1
2
− 𝛽 2
2
= 2 𝑏 2
𝐻 (4.23)
𝑡 = −
𝛽 1
𝑏 1
−
𝛽 2
𝑏 2
+ (
1
𝑏 1
+
1
𝑏 2
)
�
𝑏 1
𝛽 2
2
+ 𝑏 2
𝛽 1
2
+ 2 𝑏 1
𝑏 2
𝐻 𝑏 1
+ 𝑏 2
If 𝛽 1
≤ 𝛽 , 𝛽 2
≤ 𝛽 ,
𝛽 2
2
− 𝛽 1
2
≤ 2 𝑏 1
𝐻 ,
𝛽 1
2
− 𝛽 2
2
≤ 2 𝑏 2
𝐻 ,
�
𝑏 1
𝛽 2
2
+ 𝑏 2
𝛽 1
2
+ 2 𝑏 1
𝑏 2
𝐻 𝑏 1
+ 𝑏 2
≤ 𝛽
(4.24)
𝑡 =
𝛽 − 𝛽 1
𝑏 1
+
𝛽 − 𝛽 2
𝑏 2
+
𝐻 𝛽 −
1
𝛽 (
𝛽 2
− 𝛽 1
2
2 𝑏 1
+
𝛽 2
− 𝛽 2
2
2 𝑏 2
)
If 𝛽 1
≤ 𝛽 , 𝛽 2
≤ 𝛽 ,
𝛽 2
2
− 𝛽 1
2
≤ 2 𝑏 1
𝐻 ,
𝛽 1
2
− 𝛽 2
2
≤ 2 𝑏 2
𝐻 ,
�
𝑏 1
𝛽 2
2
+ 𝑏 2
𝛽 1
2
+ 2 𝑏 1
𝑏 2
𝐻 𝑏 1
+ 𝑏 2
> 𝛽
(4.25)
54
Travel times of trucks and trains are time dependent and non-linear with respect to the
volumes. The outputs of the simulation framework are fed in to the optimization
formulation to find the best feasible solution for the coordinated/uncoordinated MDFLB
with regard to constraints.
4.4 Case Simulation Study
The simulator framework covers an area that includes the Port complex LA/LB, the
freeway and the surface street network of about 20 square miles. We have formulated the
simulation as a work-flow composition and configure the system for evaluating various
scenarios for LA/LB. Figure 4.7 illustrates the region of study.
Figure 4.7: Region of study
We assigned 6 main destinations in the region with different demand requirements which
are supplied by three terminals in the Port complex. The first three destinations are
located along I-405 highway between the I-110 and I-710 freeways; whereas, the other
three destinations are placed along CA-91 as shown in Figure 4.7 by 𝐷𝐷 . The simulation
D
D
D
D
D
D
Port of LA
Port of LB
D1 D2 D3
D4
D5
D6
55
framework consists of VISUM and the railway simulation system is used to the estimate
links’ travel time in the region of study. Many assumptions are taken into consideration
for the experiments. We considered 5 trains with homogenous capacities of 50 containers
each are available at the port complex. Average weight of the containers is assumed to be
25 tonnages. In the free flow condition, the transportation cost per unit (price/(ton.mile))
for truck transportation is estimated to be 0.37 cents; whereas, train transportation is
much cheaper to be about 0.03 cents [21]. The requirements of the destinations are
provided in Table 4.1 in terms of containers.
Table 4.1: Demands (number of containers)
Dest. 1 Dest. 2 Dest. 3 Dest. 4 Dest. 5 Dest. 6
Port of LA/LB 350 450 400 600 700 560
Let the three terminals (A, B, C) be the origins and none of the shipping companies (SC)
have any knowledge of the others regarding scheduling and routings. We set one
coordinator that collects and updates the information from the shipping companies for the
coordinated MDFLB based on the origins-and-destinations of freight as shown in Figure
4.8. The total demand requirement is 3060 containers which are equally distributed
among the origins (e.g. each origin has 1020 containers).
56
Figure 4.8: Locations of terminals, shipping companies, and coordinator
Links 𝑥 1
and 𝑥 2
correspond to the I-710 and I-110 freeways respectively. A set of links
connects a pair of origin-destination in the constructed network. Two transportation
modes are taken into account for the experiments in this chapter e.g. trucks and trains.
We model the double-track railway system as the constructed railway network as shown
in Figure 4.9 Containers are transferred to the destinations by two different ways. Either
they are delivered directly by trucks from the origins to the destinations or they are
transferred partially by trains as illustrated in Figure 4.9 The solid line indicates the
aggregated roads and the dotted line represents a railway in the constructed network.
A
Coordinator
SC
X
1
X
2
Train Station
Train Station
B
C
SC
SC
57
Figure 4.9: Possible links between a pair of origin-destination
Five scenarios are defined for different situations within a day to have time dependent
traffic conditions in the region. Rush hour, noon, and night time correspond to the first
three scenarios respectively. Accidents are introduced in the region of study for the fourth
scenario, and real time traffic volume for an individual link changes significantly during
the optimization procedure in the fifth scenario. Each scenario contains a 5-hour horizon
which is divided into five time periods with different traffic conditions. The system
updates solutions every hour to capture new conditions.
Initially, traffic volumes for all the links in the transportation network are assigned in
VISUM as the baseline for the scenarios which includes rush hour, noon and night time
based on available historical traffic data from the region. The historical traffic data are
obtained from Southern California Association of Governments (SCAG). Traffic data is
only available for a small portion of the links in the selected region; hence, the dynamic
traffic assignment is used to estimate volumes for the other links. Then, trucks/trains are
loaded into network dynamically using the optimization algorithm and the integrated
simulators. First, we assume that no interaction exists among the shipping companies,
and each of these companies operates independently (uncoordinated). Then, the
coordinator collects the origins-to-destinations information of the shipping companies
and determines the MDFLB (coordinated). Table 4.2 illustrates the role of the coordinator
that coordinates the shipping companies’ data. Traffic condition is set to be rush hour.
A
D1
Intermediary node
58
Table 4.2: Comparison between uncoordinated and coordinated MDFLB
Destination
Final solution
Travel time (min),
uncoordinated
Number of routes
chosen to each
destination,
no opt.
Final solution
Travel time (min),
coordinated
Number of routes
chosen to each
destination,
opt.
Simulated
system
Travel time (min),
coordinated
1 30 14 23 19 23
2 28 12 22 17 22
3 28 13 21 16 22
4 41 18 31 24 31
5 42 18 29 26 30
6 41 17 30 27 30
The first column represents destination number. The second column gives the final
solution in terms of average travel time of transferring containers from the origins to the
assigned destination using all the transportation modes (e.g trucks and trains) without
incorporating the coordinator. The third and fifth columns are the number of routes
chosen to each destination. The fourth column represents average travel time by taking
into account information sharing within the shipping companies using the coordinator. To
verify the results from the optimization models, we use a simulated system which
consists of the microscopic traffic flow simulator (VISSIM) and the railway simulation
system as representation of the real system shown in Figure 4.2 [82]. The microscopic
traffic flow simulator (VISSIM) simulates single vehicle-driver unit which takes into
account the dynamics of the individual vehicles. In VISSIM, each entity e.g. car, truck,
bus, person is simulated individually. The sixth column indicates the average travel time
of the coordinated MDFLB using the simulated system. In the optimization models, the
travel times are estimated using the macroscopic traffic flow simulator and the railway
simulation system. Our methodology uses the macroscopic traffic flow simulator since
the microscopic traffic flow simulator would be too computationally intense to use in our
iteration approach. To test the accuracy of these estimates, we take the final volumes
59
from the optimization solution and feed them to the simulated system to estimate the
travel times of the actual system. As Table 4.2 shows, the results from 4 and 6 closely
model each other suggesting that using a macroscopic traffic simulator is sufficient to
estimate the parameters of the optimization model. Figure 4.10 demonstrates the
convergence of the iterating approach for the coordinated MDFLB in the rush hour
condition. The total cost (transportation cost) is reduced by more than 49% from the
initial solution using the proposed methodology.
Figure 4.10: Iterating approach convergence (coordinated rush hour)
As mentioned earlier, the links’ volume in the network are time-dependent and change
due to various reasons such as load deployment and incidents. To demonstrate the time
varying characteristic of the links’ volume, two links are selected as shown in Figure 4.8
( 𝑥 1
, 𝑥 2
). Links 𝑥 1
and 𝑥 2
correspond to the I-710 and I-110 freeways respectively.
Containers are deployed within time interval 𝑘 = [0, 5]. Figure 4.11 illustrates the links’
volumes with respect to time for the uncoordinated and coordinated MDFLB in the rush
hour scenario. Note that in the coordinated MDFLB, the link volumes are more balanced.
60
(a)
(b)
Figure 4.11: (a) Uncoordinated MDFLB (b) Coordinated MDFLB
The following tables demonstrate the average travel times of transferring containers from
the origins to the assigned destination using trucks and trains for each scenario which
corresponds to different time-periods within a day (rush hour, noon, night time). The
coordinator is taken into account for these scenarios to collect the information from the
shipping companies (coordinated).
Uncoordinated
Coordinated
61
Table 4.3: Rush hour
Destination
Initial solution
Travel time
(min), no opt.
Number of routes
chosen to each
destination,
no opt.
Final solution
Travel time
(min), opt.
Number of routes
chosen to each
destination,
opt.
Percentage
improved
1 49 3 23 19 53.06
2 41 3 22 17 48.78
3 38 3 21 16 44.74
4 64 3 31 24 51.56
5 58 3 29 26 50.85
6 57 3 30 27 47.37
Table 4.4: Noon
Destination
Initial solution
Travel time
(min), no opt.
Number of routes
chosen to each
destination,
no opt.
Final solution
Travel time
(min), opt.
Number of routes
chosen to each
destination,
opt.
Percentage improved
1 24 3 13 15 45.83
2 23 3 13 14 43.48
3 20 3 12 12 40.00
4 26 3 18 20 30.77
5 27 3 17 20 37.04
6 25 3 16 18 36.00
Table 4.5: Night
Destination
Initial solution
Travel time
(min), no opt.
Number of routes
chosen to each
destination,
no opt.
Final solution
Travel time
(min), opt.
Number of routes
chosen to each
destination,
opt.
Percentage improved
1 16 3 12 11 25.00
2 17 3 11 9 35.29
3 15 3 11 10 26.67
4 23 3 15 14 34.78
5 25 3 16 14 36.00
6 22 3 16 15 27.27
The first column represents destination number which includes 6 main destinations in the
region as shown in Figure 4.7 The second column indicates initial solution (average
travel time) in terms of minutes when no optimization is applied for the containers’
transfer decision. For the initial solution, the fastest route to each destination is found for
a single container. Then, all the containers are transferred to destinations using those
62
routes. The third column represents the number of routes chosen to each destination with
no optimization taken into account. As stated earlier, only one route (the fastest route) is
chosen to deliver demands from each origin to each destination. Two transportation
modes (train and truck) are taken into consideration for transferring containers, and three
types of nodes (e.g. origins, destinations, intermediary) are defined in the network where
train stations are in the intermediary node category. The fourth column indicates the final
solution (average travel time) in terms of minutes from the optimization models to
transfer all the demand from the origins to the assigned destinations using both
transportation modes. The fifth column is the number of routes chosen to each destination
using the optimization models and the integrated simulators. Expectedly, more routes are
used to deliver containers to each destination. For instance, 19 different routes are chosen
to deliver containers from the port complex to destination 1 in the rush hour scenario. The
number of routes and volumes which are assigned to each route are obtained from the
final solution of the optimization model. Finally, the last column indicates the percentage
improvement of the initial solution in the final solution.
In the fourth scenario, it is assumed that accidents occur during the rush hour. In this
case, the coordinated MDFLB finds solutions to cope with the new situation. Accidents
are introduced on two main freeways (I-110 and I710) and a main street causing the
capacity of these links to reduce by a half during the rush hour period. The locations of
the accidents are shown in Figure 4.12 by circles.
63
Figure 4.12: Location of the accidents
Table 4.6 demonstrates the results for the fourth scenario (accidents during the rush hour
period). The results demonstrate the effectiveness of the proposed methodology in case of
accidents. Expectedly, the improvement of percentage between the initial and final
solutions increases in the situations where there exist more traffic congestion in the
region such as rush hour and accidents. The same approach can be applied for other
events such as road closure.
Table 4.6: Accidents
Destination
Initial solution
Travel time
(min), no opt.
Number of routes
chosen to each
destination,
no opt.
Final solution
Travel time
(min), opt.
Number of routes
chosen to each
destination,
Opt.
Percentage
improved
1 69 3 30 28 56.52
2 64 3 30 27 53.13
3 59 3 29 26 50.85
4 85 3 46 34 45.88
5 92 3 53 32 42.39
6 90 3 51 34 43.33
Accident
Accident
Accident
64
In the fifth scenario, the assumption is that the traffic volume for link 𝑥 1
corresponding
to the I-710 freeway in Figure 4.8 increases significantly during the optimization
procedure due to an unknown reason. In this case, the coordinated MDFLB captures the
new situation and provides new transfer decisions. The following figure illustrates the
optimization result in terms of number of containers assigned to link 𝑥 1
.
Figure 4.13: Number of containers assigned
During the optimization, the link’s volume increases significantly given the real time
traffic data. The simulation model framework captures the new condition; therefore,
fewer numbers of trucks are assigned to link 𝑥 1
. The proposed model yields promising
results specifically in the event of rush hour and accidents. Traffic conditions are time
dependent and variable during a day. The first three scenarios correspond to different
hours of a day e.g. rush hour, noon, and night time. The accidents are introduced for the
fourth scenario creating more congestion in the region of study. The real time traffic
volume changes significantly for a specific link in the fifth scenario. The coordinator is
responsible to collect the shipping companies’ information for the coordinated MDFLB.
65
4.5 Conclusions
We used the optimization model and the integrated simulators to effectively address the
uncoordinated and coordinated MDFLB. The coordinator was introduced to collect and
update the information from the shipping companies to improve the MDFLB solutions.
Also, the proposed model takes into account the nonlinear nature of the link travel time
with respect to the volume in the objective function of the optimization model. The
optimal routing in road network model and the railway simulation model were used to
estimate the volumes and corresponding travel times of the links.
Furthermore, we focused on the transportation network in the Port of LA/LB
complex in the case simulation study. The computational results demonstrated the
effectiveness of the proposed model in reducing traffic congestion and average travel
times. The total cost was reduced by more than 56% in the coordinated MDFLB with
rush hour condition. The microscopic traffic flow simulator was used to evaluate the final
solution obtained from the integrated simulators and the optimization models to calibrate
the macroscopic traffic flow simulator’s parameters.
66
Chapter 5
Multimodal Routing Under Disruptions
5.1 Introduction
Multimodal transportation is a critical infrastructure in the global economy and its
utilization has grown significantly in recent years. The United State Department of
Transportation announced that the multimodal shipments (ship-train-truck) have
increased more than 40% in value from 1993 to 2002, and Twenty-foot Equivalent Unit
(TEU) transportation has shown more than 20% growth in 10 years since 1993. Trucks
handled 82.9% of the U.S. ground transportation in 2002 carrying 9.197 billion tons
while trains transported 1.895 billion tons [97].
Sea-ports, rail roads, and roads are the main components of the multimodal
transportation. The quality of performance of each individual component and
interconnection along the network affects the quality of service provided to the
customers. In today’s economy, more than 80% of the world’s trade travels by water.
Sea-ports play a major role in the international cargo trade. More than 95% of
international trade with the United States involves some form of maritime transportations
[3]. Sea-ports consist of cranes, storage yards and transportation services to deliver
cargos to the customers.
The freight transportation industry has major impact on the nation’s Gross Domestic
Product (GDP). The industry needs to maintain high quality of service and efficiency for
the stakeholders. The quality of service is one of the main issues of service providers due
67
to competition in the private sectors. Transportation systems are very complex since there
are many factors involved in forming reliable delivery of goods from origins to
destinations. Therefore, it becomes necessary that managers and policymakers determine
general rules to maintain high quality of services. The planning levels can be classified
into three different categories as illustrated in Figure 5.1.
Figure 5.1: Multimodal transportation planning levels
As shown in Figure 5.1, the strategic planning level represents the long-term planning. It
involves large capital investments to improve infrastructures such as constructing
terminals, railroads, and highways. At this level, managers provide general policies for
the tactical planning level. The tactical planning is the medium-term planning. The
objective of the tactical level is to optimize resource allocation to improve the efficiency
of transportation network. Duties at this level include assigning services and determining
frequency of routes as well as developing specific rules and goals for the operational
level. Finally, the operational planning level is concerned with daily activities such as
assigning vehicles and crews, and dispatching vehicles. The freight transportation
network can be modeled as the SNDP [30] which contains sets of nodes and arcs for
Strategic Planning
Long term planning, determining general policies and large
capital investments
Tactical Planning
Medium term planning, optimize resource allocation
Operational Planning
Short term planning, daily operation
68
different planning levels shown in Figure 5.1. Nodes represent terminals, hubs, and
intermediate terminals. Rails, roads and sea links form set of arcs with different
characteristics. Both nodes and arcs have limited capacities that impose some constraints
on the SNDP formulation.
Any changes to business operations such as disruptions or facility maintenance may
have a significant impact on the freight transportation. The planning and scheduling of
goods in the multimodal freight transportation is a very difficult task due to the complex
nature of large-scale transportation network. The Freight Chain Disruption Problem
(FCDP) is such an example which can be modeled using the framework of the SNDP. In
this research, we focus on the dynamic service selection and traffic distribution by taking
into account alternative routes and resources. In the dynamic service selection, the
resources are assigned to which services are offered while in the dynamic traffic
distribution, routes are determined to fulfill demands for customers over a multi-period
time horizon. Figure 5.2 illustrates the multimodal transportation structure for the freight
transportation including road and rail networks.
Figure 5.2: Multimodal physical network
The main contribution of this chapter is to address the Freight Chain Disruption Problem
(FCDP) that is modeled using the framework of the SNDP. The optimization models are
used to formulate and solve the FCDP which helps stakeholders to borrow other
resources or look for alternative routes to satisfy customers’ demand. Moreover, the
Sea Carriers Sea Terminals
Road Network
Rail Network
Inland Regions
69
proposed modeling approach will make the freight distribution process become more
resilient in the presence of disruptions. It is significant useful when a large-scale
disruption takes place at a freight chain and as a result the freight chain becomes partially
non-functional. The computational results demonstrate the effectiveness of the proposed
methodology.
5.2 Multimodal Routing under Road Network Disruptions due to
Earthquake
The baseline traffic condition for the links of the microscopic traffic simulator (VISSIM)
is set using data (e.g. bridge damages, recoveries) provided by the REDARS
methodology [98] to evaluate vehicles’ delay in the ports of LA/LB region due to
earthquake. The REDARS methodology evaluates the first level estimates of system-
wide bridge damage, costs and times to restore system-wide traffic flows, and economic
losses from disruption of passenger and freight traffic. The following figures illustrate
post-earthquake system states at times of 3 days, 4-12 days, and 13-49 days after the
Newport-Inglewood earthquake scenario in the designated region provided by REDARS.
Figure 5.3: Bridge damage estimation in the Newport-Inglewood earthquake
70
Figure 5.4: Link closure 3 days after the Newport-Inglewood earthquake
Figure 5.5: Link closure 12 days after the Newport-Inglewood earthquake
Figure 5.6: Link closure 49 days after the Newport-Inglewood earthquake
71
We use the integrated model including the microscopic traffic simulator (VISSIM) and
the optimization model in (4.18) to reduce the vehicles’ delay by rerouting the vehicles.
The vehicles’ delay (in terms of PCU-hour delay) is evaluated given the bridge damages
and link closures information due to the earthquake scenario by REDARS. Then, the
optimization model reroute vehicles to minimize the total travel times in the
transportation network. Table 5.1 illustrates the passenger and freight PCU-hour delays
due to the earthquake scenario.
Table 5.1: Vehicles’ delay in the Newport-Inglewood earthquake scenario
Days
Passenger
(PCU-hour Delay)
Passenger
(PCU-hour Delay) Using Optimization
PCU-
hour
Delay
Reduced
for
Passenger
(%)
Freight
(PCU-
hour
Delay)
Freight
(PCU-hour
Delay) Using
Optimization
PCU-
hour
Delay
Reduced
for
Freight
(%)
0-3 301,227 215,680 28.40 234,241 174,306 25.59
3-12 261,653 205,047 21.63 210,136 169,443 19.37
12-49 59,180 48,596 17.88 27,456 23,788 13.36
The first column indicates the time-intervals after the earthquake scenario. The PCU-hour
delay for the passenger cars is presented in the second column using VISSIM and bridge
damages and recoveries data provided by the REDARS methodology. In the third
column, the optimization algorithm is used to reroute vehicles to minimize the total travel
time. The fourth column is the percentage reduction in the PCU-hour delay for the
passenger cars due to the vehicles’ rerouting. The same description is applied to the 5
th
to
7
th
columns for freight in Table 5.1.
72
5.3 Multimodal Routing under Port Disruptions
In a large magnitude disruption occurring at a sea port, distributers need to reroute freight
to other terminals or ports to meet customers’ demands. The framework of the SNDP is
used to model the Port Disruption Problem (PDP) in order to mitigate the impact of
disruptions. Disruptions are classified based on their level of impact. In the large
magnitude of disruption, the assigned port is able to process goods partially; therefore,
the service network needs to be reconfigured in order to transport part of goods to the
alternative ports in the region.
The PDP formulation contains a temporal aspect that determines multimodal routing
in the transportation network. In the time-space network, the original dynamic network is
expanded in the time dimension by making a separate copy of every link and node (static
network) at every time 𝑘 ∈ 𝐾 [63]. Figure 5.7 illustrates the time-space network with
different types of nodes, links, and specified time space {1,…..,w}. The ground
transportation is shown as a solid line where blue and black solid lines indicate railroad
and road respectively. On the other hand, the sea link is represented by dotted lines. Port
A is connected to Port B by dotted line indicating the sea link. Train is used to transfer
goods from Port A to the terminal, whereas goods can be carried by trucks from Port B to
the terminal. The optimal solution to the PDP formulation indicates the best goods’
transferring decision with regard to the transportation network constraints.
73
Figure 5.7: Time-space transportation network
In the event of a large level of disruption at a particular port, the service network needs to
be reconfigured. The port is not able to operate at full capacity; therefore, the other ports
in the region will cooperate with the damaged port to meet demands of customers. The
service network is reconfigured to use sea links originating from the disrupted port to the
other ports in the region allowing containerized cargos to reach their destinations in the
best feasible way. Let us demonstrate the reconfiguration of the service network by an
example. The original service network is shown in Figure 5.8. Port 1 is operating in the
normal condition. The IT, and D represent intermediary terminals and destination
respectively in Figure 5.8.
t=1
t=2
t=3
t=w
Port A Port B Terminal
74
Figure 5.8: Original service network
As mentioned earlier, in the event of a large level of disruption at a particular port, the
service network needs to be reconfigured. The fictitious nodes adjacent to the ports are
defined in the network representing the connection of ports by sea links. The solutions to
the PDP determine whether or not sea links need to be created in the network. Figure 5.9
illustrates the reconfigured transportation network by opening sea links. The fictitious
nodes are represented by circles. In the original service network, Port 1 is the origin but
the fictitious node (red circle) is the origin in the reconfigured service network. It is
assumed that a large level of disruption occurs at Port 1; therefore, the port itself is
unable to process all the assigned goods. As a result, part of goods is transferred to the
other nearby ports by opening the sea links. The availability and capacity of ships are two
critical variables characterizing the sea links connecting the fictitious nodes.
Goods are transferred by large container vessels with the capacity of more than
12,000 TEUs which is shown as a red solid line in Figure 5.9, whereas feeder vessels
with the smaller capacity e.g. 3,000 TEUs are responsible for carrying goods between
sea-ports in the local region. The dotted black line indicates the sea link between two
IT
Port 1
IT
IT
D
Sea
Land
75
ports in Figure 5.9. The optimal solution of the PDP identifies the amounts of goods
which are assigned to the purple lines with regard to the transportation network
constraints.
Figure 5.9: Reconfiguration of service network
Consider a graph G = (N, L) which represents a physical network, where N is a set of
nodes and L is a set of links. Nodes (N) are classified into three categories which are
origins ( 𝑛𝑛 0
), intermediary ( 𝑛𝑛 𝐼 ), and destinations ( 𝑛𝑛 𝐷𝐷 ). A set of arcs in the network is
characterized by the set of transportation links L (roads/railways/ships) offered between
the origins and destinations. Let 𝑐 𝑖𝑖 𝑗 ( 𝑘 ) indicate the cost per unit flow for pair ( 𝑖𝑖 , 𝑗 ) at
time 𝑘 ∈ 𝐾 . The cost per unit flow ( 𝑐 𝑖𝑖 𝑗 ( 𝑘 )) at time 𝑘 ∈ 𝐾 is defined as the sum of the
transportation cost per unit flow ( ℎ
𝑖𝑖 𝑗 ( 𝑘 )) of link ( 𝑖𝑖 , 𝑗 ) ∈ 𝐿 and the terminal cost per unit
flow ( 𝑔 𝑖𝑖 𝑗 ( 𝑘 )) of node 𝑗 . The terminal cost per unit flow is only considered for the
intermediary terminals which are the connections between origin and destination.
𝑐 𝑖𝑖 𝑗 ( 𝑘 ) = ℎ
𝑖𝑖 𝑗 ( 𝑘 ) + 𝑔 𝑖𝑖 𝑗 ( 𝑘 ) ∀( 𝑖𝑖 , 𝑗 ) ∈ 𝐿 , ∀ 𝑘 ∈ 𝐾 (5.1)
IT
Port 1
Port 2
Port 3
IT
IT
IT
IT
D
Sea
Land
76
𝑤𝑤 ℎ 𝑒𝑓𝑓𝑒 �
𝑔 𝑖𝑖 𝑗 ( 𝑘 ) > 0 𝑖𝑖𝑓𝑓 𝑗 ∈ 𝑛𝑛 𝐼 𝑔 𝑖𝑖 𝑗 ( 𝑘 ) = 0 𝑂 . 𝑊𝑊 .
(5.2)
The terminal cost per unit flow consists of the terminal operational cost per unit flow
(loading/unloading) and the storage cost per unit flow. Obviously, the terminal cost per
unit flow increases in the presence of disruption. The amount of flow between pair
( 𝑖𝑖 , 𝑗 ) at time 𝑘 ∈ 𝐾 is represented by 𝑥 𝑖𝑖 𝑗 ( 𝑘 ). Let 𝑢 𝑖𝑖 𝑗 ( 𝑘 ) be the capacity of the link
connecting pair ( 𝑖𝑖 , 𝑗 ), and 𝐷𝐷 𝑖𝑖 represents the amounts of supply or demand of node 𝑖𝑖
which can be defined as follows:
�
𝐷𝐷 𝑖𝑖 = 𝑠 𝑢 𝑝𝑝𝑙 𝑦 𝑖𝑖𝑓𝑓 𝑖𝑖 ∈ 𝑛𝑛 0
𝐷𝐷 𝑖𝑖 = − 𝑑𝑒 𝑚𝑎 𝑛𝑛𝑑 𝑖𝑖𝑓𝑓 𝑖𝑖 ∈ 𝑛𝑛 𝐷𝐷 𝐷𝐷 𝑖𝑖 = 0 𝑖𝑖𝑓𝑓 𝑖𝑖 ∈ 𝑛𝑛 𝐼 (5.3)
Note that ∑ 𝐷𝐷 𝑖𝑖 = 0
𝑖𝑖 ∈ 𝑁 and 𝑥 𝑖𝑖 𝑖𝑖 ( 𝑘 ) = 0 for all nodes in the network. Therefore, the
optimization formulation for the PDP can expressed as follows:
minimize
� � 𝑐 𝑖𝑖 𝑗 ( 𝑘 )
( 𝑖𝑖 , 𝑗 ) ∈ 𝐿𝐿 𝑂𝑂 ∈ 𝐾 𝑥 𝑖𝑖 𝑗 ( 𝑘 )
(5.4)
subject to
� � 𝑥 𝑖𝑖 𝑗 ( 𝑘 )
𝑗 :( 𝑖𝑖 , 𝑗 ) ∈ 𝐿𝐿 − � � 𝑥 𝑗𝑖𝑖
( 𝑘 )
𝑗 :( 𝑗 , 𝑖𝑖 ) ∈ 𝐿𝐿 =
𝑂𝑂 ∈ 𝐾 𝑂𝑂 ∈ 𝐾 𝐷𝐷 𝑖𝑖
∀ 𝑖𝑖 ∈ 𝑁𝑁 (5.5)
0 ≤ 𝑥 𝑖𝑖 𝑗 ( 𝑘 ) ≤ 𝑢 𝑖𝑖 𝑗 ( 𝑘 ) ∀( 𝑖𝑖 , 𝑗 ) ∈ 𝐿 , ∀ 𝑘 ∈ 𝐾 (5.6)
𝑥 𝑖𝑖 𝑗 ( 𝑘 ) ∈ {0,1,2, … } ∀( 𝑖𝑖 , 𝑗 ) ∈ 𝐿 , ∀ 𝑘 ∈ 𝐾 (5.7)
Objective function (5.4) aims to minimize the total cost of the flow. Constraint (5.5)
provides demand satisfaction requirements, and constraint (5.6) is the capacity restriction
of the flow on each link. The formulation is similar to the MDFLB which was described
77
in Chapter 4. The only difference is that the links’ costs are not variable due to amount of
flow assigned on the links in the PDP problem. Therefore, the non-basic link with the
most negative reduced cost ( 𝑐 𝑖𝑖 𝑗 ( 𝑘 ) − 𝜆 𝑖𝑖 ( 𝑘 ) + 𝜆 𝑗 ( 𝑘 )) by taking into account the capacity
constraint is added to the basic links. The variable 𝜆 is a dual variable corresponding to
constraint (5.6). The process continues until all the non-basic links with negative reduced
cost are covered within the network (stopping condition).
Case study
The freight transportation industry has grown rapidly in the last decade. Ports, hubs, and
terminals maintain reliable transportation of goods. Any disruption to the flow of
transportation may yield negative economic impact on the region. The ports of Los
Angeles (LA) and Long beach (LB) handle more cargo than any other ports in the United
States. Table 5.2 describes the important role of the ports of LA and LB in the U.S
economy. Four ports out of the first seven ports (in terms of annual TEUs processed) are
located on the nation’s west coast [3].
Table 5.2: Ranking of North American ports by TEUs processed annually
Rank Port 2011 (TEUs) 2010 (TEUs)
Absolute Change
(TEUs)
Percentage Change
1 Los Angeles 7,940,511 7,831,902 108,609 1.4%
2 Long Beach 6,061,091 6,263,499 -202,408 -3.2%
3 New York 5,503,455 5,292,025 211,460 4%
4 Savannah 2,944,678 2,825,179 119,499 4.2%
5 Vancouver 2,507,032 2,514,309 -7,277 -0.3%
6 Oakland 2,342,504 2,330,214 12,290 0.5%
7 Seattle 2,033,535 2,133,548 -100,013 -4.7%
78
The disruptions and distribution of containers at the LA and LB ports which we will
consider as one “port complex” are taken into account for the scenarios. Table 5.3
illustrates cargo growth in the Port of LA in the 100-year time span from 1907 to 2007.
Cargo values have increased more than 2400% in 25 years since 1982 [79].
Table 5.3: Cargo growth in value (Port of Los Angeles)
1907 1932 1957 1982 2007
Cargo value($M) 45.4 790.6 1000 9400 238400
Four major ports are located in the west coast region: LA/LB, Oakland, Portland, and
Seattle. It is assumed that a disruption occurs in the port complex and the optimal
solution of the PDP generates distribution of freight based on the constraints and carrier
availability. In the case of a large disruption, the sea links are included in the optimal
solution, and a fraction of goods is transported to the other three ports in the west coast
region to satisfy customers’ demands.
The structure of graph network is defined based on the 2010 Origin-Destination
statistics and map from the Freight Analysis Framework (FAF) of the U.S. Department of
Transportation [43]. The FAF data covers 123 domestic regions and 8 foreign regions
with statistics regarding to imports and exports. The FAF data are categorized by the
mode of transportation, the commodity type, and the volume of commodities
imported/exported in terms of tons.
The experiments are restricted to the west coast region with 16 domestic regions.
Furthermore, one region is assigned to the remainder of the United States as shown in
79
Figure 5.10. Four major ports on the nation’s west coast and four fictitious nodes
corresponding to the ports form 8 nodes in the structured graph network. Therefore, the
structure of graph network contains 25 nodes. Obviously, several customers are located in
each domestic region. The selected west coast region is illustrated in Figure 5.10 where
green regions are tied to cities, and white regions represent remaining land not near to a
major city.
Figure 5.10: The selected domestic regions in the west coast
The selected domestic region consists of 16 regions in 7 states including California,
Oregon, Washington, Arizona, Nevada, Utah, and Idaho. There exist three transportation
modes (ship, train, truck) among the coastal states and two transportation modes in the
other four states. The network graph can be constructed by drawing aggregated links
connecting the regions in Figure 5.11. The constructed graph network that covers all the
domestic regions in the west coast, four major ports (LA/LB, Oakland, Portland, and
Seattle), and four fictitious nodes as shown in Figure 5.11.
80
Figure 5.11: Aggregated graph network for west coast
The constructed graph network shown in Figure 5.11 consists of 16 domestic regions and
one region for the rest of the United States ( 𝑅𝑅 𝑟𝑟 ). The fictitious nodes are represented
by 𝑛𝑛 𝑓𝑓𝑖𝑖
, and four major ports in the west coast are characterized by 𝑛𝑛 𝐿𝐿𝐿𝐿 / 𝐿𝐿𝐿𝐿
, 𝑛𝑛 𝑂𝑂 𝑂𝑂𝑂𝑂
, 𝑛𝑛 𝑃𝑃 𝑃𝑃 , and
𝑛𝑛 𝑆𝑆 𝑆𝑆 which correspond to the ports of LA/LB, Oakland, Portland and Seattle respectively.
Table 5.4 demonstrates corresponding regions for the domestic regions which are defined
in the FAF map. Note that each region consists of multiple customers with the specific
demand.
Table 5.4: Corresponding regions of FAF map
FAF Region Corresponding Region
𝑅𝑅 5
Tucson, AZ
𝑅𝑅 6
Remainder of Arizona
𝑅𝑅 8
LA, LB, Riverside
𝑅𝑅 9
San Diego
𝑅𝑅 1 0
Sacramento
𝑅𝑅 1 1
San Jose, San Francisco
𝑅𝑅 1 2
Remainder of California
𝑅𝑅 28
Idaho
𝑅𝑅 59
Las Vegas
𝑅𝑅 60
Remainder of Nevada
𝑛𝑛 𝑓𝑓 1
𝑛𝑛 𝑓𝑓 2
𝑛𝑛 𝑓𝑓 3
𝑛𝑛 𝑓𝑓 4
𝑛𝑛 𝐿𝐿𝐿𝐿 / 𝐿𝐿𝐿𝐿
𝑛𝑛 𝑂𝑂𝑂𝑂 𝑂𝑂
𝑛𝑛 𝑃𝑃𝑃𝑃
𝑛𝑛 𝑆𝑆𝑆𝑆
𝑅𝑅 8
𝑅𝑅 1 2
𝑅𝑅 9
𝑅𝑅 5
𝑅𝑅 6
𝑅𝑅 5 9
𝑅𝑅 6 0
𝑅𝑅 1 1
𝑅𝑅 1 0
𝑅𝑅 8 5
𝑅𝑅 8 4
𝑅𝑅 1 0 9
𝑅𝑅 1 1 0
𝑅𝑅 2 8
𝑅𝑅 1 0 2
𝑅𝑅 1 0 3
𝑅𝑅 𝑟𝑟
81
𝑅𝑅 84
Portland, OR
𝑅𝑅 85
Remainder of Oregon
𝑅𝑅 1 0 2
Salt Lake City, UT
𝑅𝑅 1 0 3
Remainder of Utah
𝑅𝑅 1 0 9
Seattle, WA
𝑅𝑅 1 1 0
Remainder of Washington
The information regarding to the cargo imports from the selected ports in the west coast
region to the specified regions is required in order to set the baseline of the transportation
network. The FAF provides data for imports, exports, and the total flow from various
foreign regions or countries to the domestic regions. The import data originating from the
four major ports in the west coast to 16 domestic regions and one region for the rest of
the United States are illustrated in Table 5.5.
Table 5.5: Volume (ton/day) of imported goods to the domestic regions
𝒏 𝑳𝑨 / 𝑳𝑩
𝒏 𝑶𝒂 𝒌 𝒏 𝑷𝒐
𝒏 𝑺𝒆
𝑹 𝟓 1,433 56 22 47
𝑹 𝟔 12,128 328 48 72
𝑹 𝟖 131,783 512 478 910
𝑹 𝟗 9,723 49 73 25
𝑹 𝟏𝟎
579 610 76 44
𝑹 𝟏𝟏
2,745 12,415 66 101
𝑹 𝟏𝟐
5,981 1,393 211 114
𝑹 𝟐𝟖
455 1,102 1,503 746
𝑹 𝟓𝟗
1,788 71 25 24
𝑹 𝟔𝟎
112 102 43 29
𝑹 𝟖𝟒
446 65 7,239 1,127
𝑹 𝟖𝟓
232 51 980 901
𝑹 𝟏 𝟎𝟐
873 1,512 983 374
𝑹 𝟏 𝟎𝟑
720 1,629 824 89
𝑹 𝟏 𝟎𝟗
1,004 105 890 25,834
𝑹 𝟏 𝟏𝟎
450 191 1,555 2,719
𝑹 𝒓 14,328 984 1,259 3,254
𝑻𝒐 𝒕𝒂𝒍 184,780 21,175 16,275 36,410
Cargos are transported to the destinations via multimodal transportation modes including
truck, train, and ship. Each individual transportation mode has specific characteristics
such as capacity and maximum speed. The transportation cost per unit (price per
ton.mile) for truck transportation is estimated to be 0.37 cents; whereas, train
82
transportation is much cheaper to be about 0.03 cents. The ship transportation cost unit is
about 0.01 cents which is cheaper than the other two transportation modes [21].
Tsunami, flood and many other natural disasters may cause disruption to sea-ports
that have significant impact on the distribution of freight. The scenarios are defined based
on the capacity reduction due to disruption. Let us define 15 scenarios where the capacity
of the port complex is reduced in 2% decrements. Table 5.6 illustrates the effectiveness
of the proposed methodology.
Table 5.6: Computational results
Capacity
Reduction (%)
Total Cost
($M) [using
no other
ports]
Total Cost
($M) [using
other ports]
Absolute
Change ($M)
Percentage
Change (%)
𝒏 𝑶𝒂 𝒌 𝒏 𝑷𝒐
𝒏 𝑺𝒆
0 9.11 9.11 0 0 0 0 0
2 9.14 9.14 0 0 0 0 0
4 9.19 9.19 0 0 0 0 0
6 9.28 9.26 -0.02 -0.22 1 0 0
8 9.51 9.42 -0.09 -0.95 1 0 0
10 9.90 9.71 -0.19 -1.92 1 0 0
12 10.71 10.20 -0.51 -4.76 1 1 0
14 12.12 11.33 -0.79 -6.52 1 1 0
16 14.19 13.06 -1.13 -7.96 1 1 0
18 18.01 16.13 -1.88 -10.44 1 1 0
20 26.65 20.44 -6.21 -23.30 1 1 1
22 41.29 29.68 -11.61 -28.12 1 1 1
24 65.14 44.09 -21.05 -32.32 1 1 1
26 101.02 66.08 -34.94 -34.59 1 1 1
28 154.42 98.19 -56.23 -36.41 1 1 1
30 232.10 144.15 -87.95 -37.89 1 1 1
83
The first column is the percentage reduction of the LA/LB port complex capacity due to
disruption. The capacity is reduced in 2% decrements up to 30% of the capacity of the
port complex. The second column represents the total cost of freight distribution when
goods are required to distribute through the LA/LB port complex without using the other
ports in the region. The third column indicates freight distribution with no restriction on
using the other ports and sea links; therefore, the other major west coast ports in the west
coast region are taken into account in order to use their resources to handle part of the
freight transportation process. The fourth and fifth columns represent the differences
between the total costs obtained in no using the other ports column (second column) and
the proposed methodology column (third column). Columns 6-8 are binary indicating
whether the other ports in the west coast region are engaged in the freight transportation
process.
The computational results in Table 5.6 demonstrate the effectiveness of the proposed
methodology. In 0% capacity reduction scenario (normal condition), there is no need to
use the other ports. In 2% and 4% capacity reduction scenarios, the PDP solutions
indicate that transferring some cargos to the other port within the region will not reduce
the total cost. Expectedly, the total cost of distribution increases as the magnitude of
disruption rises. The fifth column illustrates the significant reduction in the total cost
when the other ports are taken into account in the presence of large magnitude of
disruptions. The proposed methodology yields promising results and a huge advantage
compared to the regular freight distribution methods specifically in the event of large-
scale disruption.
84
5.4 Conclusions
The multimodal routing under disruptions methodology is proposed to effectively use the
regional resources in the freight transportation network in order to satisfy customers’
demand with the minimum total cost. It is significant useful when a large-scale disruption
takes place at a freight chain and as a result the freight chain becomes partially non-
functional. The optimization model and reconfiguration of service network aim to find
alternative routes and resources to handle part of customers’ demands. The proposed
approach yields the freight distribution process become more robust and resilient in the
presence of disruptions. Moreover, the methodology can be used in the tactical planning
of freight distribution.
Furthermore, the transportation network in the US west coast region is considered for
the case study. The computational result demonstrates the effectiveness of the proposed
methodology in reducing the impact of port disruption. The relative total cost is
significantly reduced in the large-scale disruptions at the sea-port. The proposed
methodology is a constructive, valuable system of routing of containers in the tactical or
operational planning.
85
Chapter 6
Concluding Remarks and Proposed Future Topics
6.1 Final Conclusions
Today’s transportation network is complex, and any disruptions to part of the network
could result delays to drivers. Delays due to these disruptions yield economic impact
depending on the magnitude of disruption. We developed an optimal routing
methodology in road network to guide an individual driver through a complex urban
network. At its core, the challenge behind the optimal routing in road network is an
effective real time estimation of traffic flows for all links in a transportation network. If
we can effectively estimate the travel from one zone to another within a network, we can
assign traffic to the links (roads) in the network and generate traffic flows given roadway
capacities. Then, we minimize the travel time for a driver travelling from a node in the
network (his origin) to his destination node. To accurately estimate traffic flows, we first
made an initial traffic flow estimation based on general information from the region of
interest such as number of commuters to specific businesses. We used the four step
model to generate, distribute, and assign trips by travel mode in order to estimate the
initial traffic flows for all links in the transportation network. We took into account the
historical loop conductor traffic flow data, and data from monitoring events in the offline
estimation model. Finally, we added the real-time traffic data and tried to match the
simulated flows with the existing traffic conditions. The traffic flow estimation model
needs continuous and dynamic adjustment so that the routing system would always be
86
directing the driver along the path that requires the least time from the driver’s ever-
changing origin (as he progresses forward) to his static destination.
In order to accurately estimate traffic flows, we needed to analyze the impact of on-
road and off-road events on links across the network. Detour routes adjacent to certain
events may not necessarily be the time-minimizing routes because drivers avoiding the
event may clog the adjacent links as well. We classified events as on-road and off-road
events. We further divided these events into planned and unplanned; ideally, we were
able to input planned events into the traffic flow estimation model before they occur. It is
fairly straightforward to monitor the effects of the planned on-road events such as road
and lane closures, since we can change the capacities of links in the traffic flow simulator
according to the planned on-road events. Accidents are unplanned on-road events, and it
is difficult to accurately estimate the accident’s effects. This is because accidents vary in
size, intensity, and visibility. Ideally, we were able to preemptively identify
characteristic changes in traffic flow signifying an accident. Finally, off-road cultural
events such as concerts and sporting events can create huge car attraction rates. These
events would also release thousands of automobiles on to the network at their
destinations. We estimated the initial and historical traffic flow pattern for these off-road
events by monitoring major events like USC football games.
The model for the optimal routing in road network can be used in the optimum
multimodal routing methodology. We used an optimization model and the integrated
simulators to effectively address the uncoordinated and coordinated MDFLB. The
proposed model takes into account the nonlinear nature of the link travel time with
respect to the volume in the objective function of the optimization model. The optimum
87
routing in road network and the railway simulation model were used to estimate the
volumes and corresponding travel times of the links. In the computational experiment, we
considered freight movements from the Port of Los Angeles to 6 main destinations within
the local region. The computational results demonstrated the effectiveness of the
proposed model in reducing traffic congestion and average travel times. The microscopic
traffic flow simulator was used to evaluate the final solution obtained from the integrated
simulators in order to calibrate the macroscopic traffic flow simulator’s parameters.
Furthermore, we addressed the multimodal routing under disruptions. First, the road
and bridge damages due to an earthquake scenario were given to evaluate vehicles’ delay
using a microscopic traffic flow simulation. Then, the integrated model consists of a
microscopic traffic flow simulator and an optimization model was used to reduce
vehicles’ delay by rerouting vehicles in the network. Furthermore, we assumed that
disruption took place at terminals inside a port; therefore, other ports in the local region
are used to handle part of freight transferring process. The problem is solved by
reconfiguring the service network to use other available routes and resources in the
region.
The computational experiment was defined to test the effectiveness of the
methodology to minimize cost given a disruption at a port. We used the U.S. Department
of Transportation 2010 Origin-Destination statistics and map from the Fright Analysis
Framework (FAF) to simulate a disruption at a major port on the west coast. For the case
study, we limited the scenario to the western United States and to four major west coast
ports. With a major disruption occurring at one of the ports, we allowed the
reconfiguration of the service network by using sea links in addition to roads and
88
railways. We defined 15 port disruption scenarios by reducing port capacity in 2%
decrements. The reduction in cost showed the success of the proposed methodology to
address the multimodal routing under disruptions. We could easily expand the solution to
the entire US intermodal transportation network. The solution could aid in the tactical
and strategic planning at ports in terms of handling disruptions.
6.2 Proposed Future Topics
The optimum routing methodology results can be improved by incorporating more traffic
data into the model. More traffic data can be obtained using smart phone and probe
vehicle data. In the optimum multimodal routing, the simulators were used to evaluate the
links’ cost according to the assigned flows to the links. The links’ capacities were the
only constraints taken into account in the optimization formulation. Other constraints
such as time windows and capacity of intermediary terminals can be implemented in the
optimization formulation.
Furthermore, in the multimodal routing under disruptions, various disruption
scenarios such as terrorist attack or tsunami can be evaluated by the proposed
methodology. Furthermore, air transportation is another mode choice in the multimodal
transportation especially for the international freight transportation. The multimodal
routing formulation can be modified to include air transportation as the transportation
mode choice. It is much more expensive than the other transportation mode choices but it
will be an appropriate choice if the good value is incorporated in the multimodal routing
formulation. High value goods are mostly transferred by the air transportation mode
choice.
89
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Abstract (if available)
Abstract
The road and rail networks, ports and terminals, and sea carriers are part of the multimodal freight transportation chain that handles most imports and exports of goods in the United States. Most of the ports and terminals are located adjacent to urban regions. Urban traffic conditions are extremely time-dependent, and change drastically throughout a day. A major problem in getting traffic flow information in real time is that the vast majority of links are not equipped with traffic sensors. Another problem is that factors affecting traffic flows such as accidents, public events, and road closures are often unforeseen, suggesting that traffic flow forecast is a challenging task. On the other hand, any disruption to the freight transportation chain may have devastating effects on the economy and society. Reconfiguration and routing strategies are needed to mitigate the impact of potential disruptions along the freight transportation chain. ❧ In this research, we use an optimization modeling approach to address the multimodal routing under normal condition and disruptions. The optimization modeling approach consists of traffic flow data completion and optimum routing in road network, optimum multimodal routing, and multimodal routing under disruptions. First, we use a dynamic traffic flow simulator to generate flows in all links in real time using available traffic information, estimated demand, and historical traffic data from the links equipped with sensors. Using the traffic flow data completion methodology and an optimization model, the optimum routing in road network is developed for an individual driver to reach destination in a complex urban network. Then, we use the developed optimum routing in road network and a railway simulation system to address the optimum multimodal routing. The proposed model evaluates volume and travel time of each route based on current traffic conditions and number of trucks or trains that have been assigned dynamically on each route by the optimization models using the optimum routing in road network and a railway simulation system. Finally, the proposed optimization modeling approach will make freight distribution process become more resilient in the presence of disruptions. Our approach is based on reconfiguration to include available resources and alternative routes which in the absence of disruption were not cost effective, and therefore under normal conditions were not feasible choices. Several case studies are presented to demonstrate the effectiveness of the proposed optimization modeling approach.
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Asset Metadata
Creator
Abadi, Afshin
(author)
Core Title
Optimum multimodal routing under normal condition and disruptions
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
09/10/2014
Defense Date
09/04/2014
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
disruption,freight,multimodal,OAI-PMH Harvest,optimization
Format
application/pdf
(imt)
Language
English
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Electronically uploaded by the author
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Advisor
Ioannou, Petros (
committee chair
), Dessouky, Maged M. (
committee member
), Prasanna, Viktor K. (
committee member
)
Creator Email
abadi@usc.edu,afshinabadi@gmail.com
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https://doi.org/10.25549/usctheses-c3-473801
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etd-AbadiAfshi-2918.pdf
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473801
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Abadi, Afshin
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Tags
disruption
multimodal
optimization