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Models and algorithms for the freight movement problem in drayage operations

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Models and algorithms for the freight movement problem in drayage operations

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Content MODELS AND ALGORITHMS FOR THE FREIGHT MOVEMENT PROBLEM IN
DRAYAGE OPERATIONS
by
SIYUAN YAO
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA In
Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(INDUSTRIAL AND SYSTEMS ENGINEERING)
May 2023
Copyright 2023 Siyuan Yao

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Acknowledgments
Words cannot express my gratitude toward my advisor and chair of my committee: Dr.
Maged Dessouky. He is the best advisor, a brilliant scholar and educator, and my role model in
work and life, guiding me to be a rigorous scholar and a strong individual. I could not complete
the journey without the best committee I could ever wish for: Dr. Petros Ioannou and Dr. John
Carlsson. Their knowledge and expertise have motivated and guided me in the pursuit of my
intellectual passion. I also would like to thank Dr. Genevieve Giuliano, who can always provide
invaluable suggestions from a social scientist's perspective and inspire my research. In addition, I
want to thank Metrans for funding my research.
My sincere gratitude goes to my professors at Daniel J. Epstein Department of Industrial
& Systems Engineering at the University of Southern California: Dr. Andres Gomez, Dr. Qiang
Huang, Dr. Joing-Shi Pang, Dr. Meisam Razaviyayn, Dr. Sheldon Ross, Dr. Phebe Vayanos, and
Dr. Sima Parisay. I also wish to express my thanks for all the administrative support provided by
Shelly Lewis, Roxanna Carter, and Grace Owh. You have all been my solid support and
inspiration. Thank you for making my life here at Viterbi warm and cheerful.
I am also grateful for having a group of brilliant lab- and classmates: Dr. Shichun Hu, Dr.
Santiago Carvajal, Dr. Haochen Jia, Dr. Yuanxiang Wang, Wei Gu, Zheyu Wang, Pengfei Chen,
Wentao Zhao, Xiaocheng Liu, Zuhayer Mahtab, Tanvir Kaisar, Mingdong Lyu, Ying Peng,
Tianjian Huang, Han Xu, Anguo Hu. I feel very lucky to have known you. Thank you for
making my Ph.D. life at USC so colorful and delightful! Keep on with your excellent work!
Lastly, I am deeply grateful to my family and friends. I want to express special thanks to
my mother for always encouraging me to pursue my goals and providing unconditional love and
support. Your spirit will always be with me. I also appreciate my father and parents-in-law for

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their encouragement, love, and unwavering support. I also want to express my appreciation to
my wife, Dr. Yiqi Li, for her encouragement, support, patience, and love. In addition, I want to
thank my six-month-old daughter--Ellie Yao--for bringing joy and happiness to our family and
motivating me to be a better person. I am lucky to have amazing friends--Dr. Yu Hou, Ketian
Xu, Kejia Pu, Weibo Qiu, Dr. Junda Xiong, Abner Li, Jier Dong, Mingyang Li. I am also
grateful to my friends in the Pasadena Basketball Group. In the end, thanks, Offer, my
Chinchilla, for your company.

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Table of Contents
Acknowledgments........................................................................................................................... ii
List of Tables ................................................................................................................................. vi
List of Figures ............................................................................................................................... vii
Abstract ........................................................................................................................................ viii
Chapter 1: Introduction ............................................................................................................... 1
1.1 Background ........................................................................................................................... 1
1.2 Motivation ............................................................................................................................. 5
1.3 Research Gap and Contribution ............................................................................................ 8
1.4 Structure of the Dissertation .................................................................................................. 9
Chapter 2: Literature Review .................................................................................................... 11
2.1 Empty Container Repositioning Problem ........................................................................... 11
2.2 Freight Movement Problem with Dynamic Transportation Network ................................. 15
2.3 Drayage Routing Problem with Alternative Fuel Vehicles ................................................. 18
Chapter 3: Empty Container Repositioning Problems with Double Container Trucks ...... 22
3.1 Problem Description ............................................................................................................ 22
3.2 Double Container Flexible pickup and drop-off Model ...................................................... 24
3.3 Stochastic Double Container Flexible pickup and drop-off Model .................................... 29
3.4 Computational Results ........................................................................................................ 34
Chapter 4: Freight Routing and Touring with Network Dynamics ....................................... 41
4.1 Problem Description ............................................................................................................ 41
4.2 Mixed Integer Programming Model .................................................................................... 42
4.3 Solution Methodology ......................................................................................................... 46
4.4 Subproblem Formulations ................................................................................................... 51
4.5 Load-Balancing with Touring Algorithm ........................................................................... 54
4.6 Computational Results ........................................................................................................ 56
Chapter 5: Mixed Fleet Drayage Routing Problem ................................................................. 63
5.1 Problem Description ............................................................................................................ 63
5.2 Mixed Fleet Drayage Routing Problem .............................................................................. 65
5.3 Solving framework for MFDRP .......................................................................................... 70
5.4 Computational Results ........................................................................................................ 77

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Chapter 6: Conclusion and Future Work ................................................................................. 83
6.1 Conclusion ........................................................................................................................... 83
6.2 Future Work ........................................................................................................................ 85
References ..................................................................................................................................... 86
Appendices .................................................................................................................................... 97

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List of Tables
Table 1. Parameter Settings for DCAM and DCFM Comparison ................................................ 34
Table 2. Average Ratio of Model Cost ......................................................................................... 35
Table 3. Parameter Settings for DCFM and SDCFM Comparison .............................................. 37
Table 4. Average of the Ratios for DCFM and SDCFM Comparison.......................................... 38
Table 5. Truck Information for DCFM and SDCFM ................................................................... 38
Table 6. Average of the Ratios v.s. Perfect Information Condition ............................................. 39
Table 7. Standard Deviation of the Ratios v.s. Perfect Information Condition ............................ 39
Table 8. Multi-way ANOVA Analysis Results ............................................................................ 40
Table 9. Experiment Parameters ................................................................................................... 58
Table 10. Evaluation Results for Different Approaches ............................................................... 60
Table 11. Small Instances Results ................................................................................................ 79
Table 12. Net Daily Emissions Savings, Relative to Max-DHDTs .............................................. 82

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List of Figures
Figure 1. Global Container Throughput by Year ............................................................................ 1
Figure 2. Present Container Movements ......................................................................................... 3
Figure 3. Container Movements with Street Turns ......................................................................... 3
Figure 4. Static Network v.s. Dynamic Network in Truck Flow Balancing ................................... 7
Figure 5. Two Restricted Routing Patterns ..................................................................................... 7
Figure 6. A More Realistic BEHDT Route ..................................................................................... 8
Figure 7. Ideal Container Flow ..................................................................................................... 23
Figure 8. Example of a Small-size Problem ................................................................................. 25
Figure 9. Framework for Solving Two-day ECRP with Stochastic Demand ............................... 30
Figure 10. Convergence Plot for DCFM and SDCFM ................................................................. 37
Figure 11. The Master Optimization Framework ......................................................................... 48
Figure 12. Example of Connecting Delivery Trips with Pickup Trips ......................................... 50
Figure 13. Study Area ................................................................................................................... 57
Figure 14. Data Flow in Solution Procedure ................................................................................ 59
Figure 15. Model Costs Using Different Approaches .................................................................. 61
Figure 16. A Sample Network ...................................................................................................... 64
Figure 17. Battery Consumption Rates under Different States ..................................................... 65
Figure 18. Solution Framework for Modified ALNS ................................................................... 73
Figure 19. A Truck Route in the Initial Solution .......................................................................... 73
Figure 20. Two Types of Charging Insertions .............................................................................. 74
Figure 21. Charging Curve ........................................................................................................... 78
Figure 22. Avg. Number of HDTs Required for Each Target Year ............................................. 81
Figure 23. Avg. Truck Miles Required for Each Target Year ...................................................... 82

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Abstract
The increasing demand for freight transportation has caused social issues such as traffic
congestion and air pollution, making optimizing drayage operations essential in the logistics
industry. To address these problems, we investigate drayage operations problems from three
perspectives. First, we propose two container repositioning models using double container
trucks. Both models allow a flexible pickup and drop-off policy for double container trucks. A
rolling horizon based solving framework is developed to solve the models under deterministic
and stochastic demand. Second, we study a freight routing and touring problem on a load-
dependent dynamic transportation network. Instead of using explicit mathematical expressions to
model network dynamics, traffic simulation models are introduced into our optimization loop,
which allows more efficient and accurate network approximations. The problem is decomposed
into two subproblems. The first subproblem can be solved by a modified load-balancing
algorithm, which provides a partial integer solution for the second subproblem. We prove that
solving the second subproblem’s linear programming relaxation can provide an integer solution
with the partial solution input. A co-simulation optimization based load-balancing with touring
solution framework is proposed to tackle the freight routing and touring problem. Third, we
explore the potential of substituting diesel heavy-duty trucks with battery-electric heavy-duty
trucks in daily drayage operations to reduce emissions. A mixed integer programming model is
formulated. A modified adaptive large neighborhood search algorithm is developed to efficiently
solve the proposed mixed fleet drayage routing problem.
Keywords: Drayage Operations, Freight Movement, Mixed Integer Programming, Truck

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Models and Algorithms for the Freight Movement Problem in Drayage Operations
Chapter 1: Introduction
1.1 Background
Due to increasing international trade, the world has witnessed rapid growth in the
transportation of merchandise. Seventy percent of merchandise is shipped by sea, among which
52 percent is transported in containers (WSC, n.d.; UNCTAD, 2019). Figure 1 provides an
overview of global container throughput at ports from 2015 to 2021 with a forecast for 2022
through 2025 (Statista, 2019). Although global containerized trade decreased in 2020 because of
COVID-19, we see a significant rebound and sustained growth in the following years, rendering
the research effort that aims at increasing the efficiency of freight movements significant and
relevant. This dissertation focuses on drayage service, a short-haul pickup and delivery service
for transporting freights among ports, warehouses, and other facilities, and three drayage
operations models will be proposed and validated using data from San Pedro Bay area.

Figure 1. Global Container Throughput by Year

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As the second-largest metropolitan region in the United States, the Los Angeles area has
two nationally leading ports– Port of Los Angeles (POLA) and Port of Long Beach (POLB). In
2021, POLA/LB handled 20.1 million TEUs, including 10.1 million imports and 10.0 million
exports ( Port of Los Angeles, 2023; Port of Long Beach, n.d.). Of the 10 million exports, 7.3
million were empty, caused by the imbalance in international trade. Before being shipped out,
containers need to be transloaded because of the differences in container size between the United
States convention and cargo ships. Whereas cargo ships generally use standard-height 40-foot
containers, 53-foot containers are used by trains and trucks in the United States because of their
larger capacity.
A benefit of greater efficiency would be less traffic congestion and, thus, less air
pollution between international ports and transloading stations, also referred to as warehouses,
where cargos are consolidated into their corresponding sized containers. Most of these containers
are transported by heavy-duty trucks (HDTs). Taking POLA/LB area as an example, under
current efficiency, a conservative estimate of congestion suggests that container movements will
double by 2035 if annual growth is a mere six percent, affecting the three main transportation
corridors in Southern California, I-110, I-710, and I-405. According to the Federal Highway
Administration’s study of regional freight transportation (FHWA, 2023), the increasing freight
movements in the Southern California region cause social issues such as air pollution, traffic
congestion, and land-use problems. The effort in this dissertation aims at optimizing drayage
operations, which helps relieve traffic congestion and air pollution in urban centers.
Better freight movements can be achieved by managing the large number of empty
containers in an efficient way. Since the 1950s, the problem has been studied as the Empty
Container Repositioning Problem (ECRP). To reduce truck costs, logistic providers in some

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international markets are adopting double container trucks. Because DCTs can carry two
containers simultaneously, employing DCTs can significantly improve drayage efficiency by
reducing fleet size, truck miles, and management costs.
In addition, we consider different strategies for empty container movements. Figure 2
shows current container movements with empty and full container loading states in the
POLA/LB. Although this type of container movement is easy to execute, there is room to
improve the container movement plan by introducing additional routes between these locations.
Street turn and depot-direct (Jula et al., 2006) are two empty container reuse strategies that have
been investigated extensively. Street turn allows an empty container to go from an importer to an
exporter directly instead of via the port, as shown in Figure 3. This strategy reduces costs by 15
to 17 percent over current container movements (Hjortnaes et al., 2017). Depot-direct introduces
depots into the container movement network, thus allowing temporary storage of containers and
providing more truck routing choices. This dissertation applies these two strategies to solve
ECRP with DCTs.

Figure 2. Present Container Movements

Figure 3. Container Movements with Street Turns

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Another direction for drayage operations research is balancing truck flows over space and
time, especially when truck flows cause significant impact on transportation networks. When
freight demand is not large, a small number of trucks will not significantly impact the existing
transportation network. Therefore, assuming a static transportation network is appropriate.
However, the assumption becomes unrealistic when truck flows increase significantly. In this
case, capturing network dynamics and variations caused by truck flow changes becomes
essential when optimizing freight movements.
It is hard to model transportation systems with closed-form mathematical expressions
explicitly because of the complexity. These networks have heterogeneous road sections with
disparate physical characterizations (e.g., road width, slope, pavement condition) and traffic
regulations. In addition, to realistically capturing the network dynamics, we also need to consider
the interaction of passenger and truck flows. To better estimate network conditions, we take
advantage of the availability of fast computers and software tools and employ complex traffic
simulation models. Introducing traffic simulation models into the optimization loops is known as
a CO-SiMulation Optimization (COSMO) approach. This dissertation investigates the
application of COSMO based approach in solving the freight routing and touring problems
(FRTP) with large-scale demand for drayage problems.
Besides optimizing drayage operations by minimizing fleet sizes, travel time, and travel
miles, reducing emissions caused by HDTs has attracted increased research attention in recent
years. In the United States, the transport sector accounts for 27 percent of greenhouse gases
(GHG) and 56 percent of NOx, among which trucks generated 22 percent of GHGs and 32
percent of NOx (U.S. EPA, 2022). Much of this is caused by the freight shipping industry.

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California established GHG reduction goals in 2006, being the only state to have done so
by that time. It enacted AB 32 and set a goal of reducing GHG emissions below the 1990 level
by 2020, cutting GHG emissions by 30 percent. Later, California enacted SB 32 to further reduce
GHG emissions by another 10 percent by 2030. The law does not state specific approaches or
requirements for reaching the GHG reduction goal, but a more efficient freight movement system
is an avenue for reducing emissions, given its disproportionate GHG share in the transport sector.
One of the most promising directions for reducing GHG emissions in the trucking
industry is adopting zero-emission trucks in freight operations. Burke and Sinha (2020)
investigated the zero-emission HDT markets. Their report indicated that although both hydrogen
fuel cells and batteries can be used as alternative fueling options, battery-electric heavy-duty
trucks (BEHDTs) are the only zero-emission HDTs currently available on the market, while
hydrogen cells are still in the testing stage. Thus, this dissertation considers the possibility of
reducing emissions by substituting BEHDTs for diesel heavy-duty trucks (DHDTs) in drayage
operations.
1.2 Motivation
This dissertation explores the potential of employing DCTs as carriers in the ECRP with
stochastic demand. First, DCTs allow more flexible routing decisions than single-container
trucks. Second, employing DCTs has the potential to alleviate traffic congestion and air pollution
by reducing the number of deployed trucks and the total miles of trucks traveled. Third, the cost
per container (i.e., fuel consumption, operational cost, and management cost) is lower for DCTs
than for traditional single container trucks. Although DCTs are allowed on public roads
(USDOT, U.S., 2004), single container trucks still dominate both on the roads and in the research
(Dejax and Crainic, 1987; Dong and Song, 2009; Funke and Kopfer, 2016; Li and Zheng, 2014).

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To the best of our knowledge, Minvielle and Dessouky’s (2021) work was the first to propose a
double container assignment model (DCAM) to study the ECRP with DCTs. However, their
model assumes a DCT can only pick up containers when empty. We relax the assumption by
modeling the ECRP with DCTs with different decision variables, which achieves a more flexible
routing ability for DCTs than DCAM. Another natural extension of the model is to incorporate
the future stochastic demand into the ECRP with DCTs. In practice, as each day unfolds, the
availability of containers for pickup is known. Transloading stations around the port area make
daily demand requests to a control center. The control center helps the trucking companies plan
their daily truck routing schedule to fulfill the demand requested from all the locations. However,
operations could improve if they consider tomorrow’s demand which is most likely uncertain
today. This unexplored demand data could be used to coordinate containers better to meet
tomorrow’s demand since the initial container allocation for the second day depends on the end
state of the first day.
Then, we consider large-scale demand in daily drayage operations. To satisfy large-scale
daily demand, truck flows can create significant impact on the transportation network. Figure 4
provides a simple case to show the benefits of considering network dynamics when balancing
truck flows over different routes when coping with large-scale demand. Suppose we have three
thousand units of demand that need to be transported from location A to B, and three candidate
routes between these two locations exist. The free flow travel time on Route 1, 2, and 3, is 45
minutes, 30 minutes, and 40 minutes, respectively. All the demand will be transported using
route 2 (the fastest route) if the network is static, which takes 30 minutes. In reality, the actual
travel time on route 2 may increase due to the overwhelming truck flows. For example, it can
become 60 minutes after applying 3000 units of truck flows, which means the route is no longer

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the fastest and the solution is not optimal. After dynamically balancing the truck flows to other
routes, we may find a better solution.

Figure 4. Static Network v.s. Dynamic Network in Truck Flow Balancing

Finally, we consider emission reduction in drayage operations by substituting DHDTs
with BEHDTs. This research is inspired by an earlier work by Giuliano et al. (2020). They
developed an optimization model and tested it using real-world data collected from the
POLA/LB. However, they simplified truck trips by assuming trips must start and end at the port
and may only have one or two stops outside the port. In addition, they assumed that the port was
the only charging location available. Figure 5 shows the two types of trips in their work.

Figure 5. Two Restricted Routing Patterns

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However, in reality, trucks can visit multiple locations, and charging stations can also be
considered. By relaxing these assumptions, the BEHDTs’ routing potential can be improved.
Figure 6 illustrates a possible one-day route for a BEHDT. The BEDHT starts at the depot. It
first conducts a few pickups and deliveries (demand satisfaction trips). As the battery level goes
down, the BEDHT visits charging station C (charging trip) and then continues its pickup and
delivery. By the end of the day, the truck goes back to the depot.

Figure 6. A More Realistic BEHDT Route

1.3 Research Gap and Contribution
In this dissertation, we contribute to the routing of trucks for drayage operations by
proposing mixed integer programming (MIP) models, developing solution frameworks, and
examining the potential of adopting DCTs and BEHDTs in drayage operations.
First, we study ECRPs with DCTs considering future stochastic demand. We propose two
ECRP models—the double container flexible pickup and drop-off model (DCFM) and stochastic
double container flexible pickup and drop-off model (SDCFM). DCFM allows a DCT to pick up
and drop off containers as long as its capacity permits. SDCFM enhances DCFM by considering
the second-day stochastic demand. We solve these models by using a modified rolling horizon
based framework.

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Second, we consider the freight routing and touring problem on a dynamic transportation
network. The problem is first formulated as a MIP model and then decomposed into two
subproblems—a dynamic traffic assignment (DTA) problem and a touring decision problem. A
modified load-balancing algorithm can solve the first subproblem and generate a partial solution.
The solution is treated as the input for solving the second subproblem. We prove that the second
subproblem can generate an integer solution by solving its linear programming relaxation (LPR).
Traffic simulation models are introduced to the solving framework to capture the network
changes caused by flow variations. By iteratively solving the two subproblems with updated
network states generated by the traffic simulation model, the integrated solution approach
provides routing and touring decisions for freight movements and reduces the fleet size and total
travel miles for daily drayage operations.
The last problem, a mixed fleet drayage routing problem (MFDRP), investigates the
potential of substituting DHDTs with BEHDTs in drayage operations. We consider charging
locations outside the depot with non-linear charging times for BEHDTs. Furthermore, load-
dependent energy consumptions are used for BEHDTs to simulate more realistic scenarios. An
adaptive large neighborhood search (ALNS) algorithm is proposed to solve practical-size
problems.
1.4 Structure of the Dissertation
The rest of the dissertation is structured as follows. Chapter 2 reviews the related
literature on the topics of empty container repositioning problems, freight movement problems
with dynamic transportation networks, and drayage routing problems with alternative fuel trucks.
We study the ECRP with DCTs in Chapter 3. Then we apply a COSMO based approach to solve
FRTP with large-scale demand in Chapter 4. In Chapter 5, BEHDTs are considered substitutes

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for DHDTs in drayage operations to reduce GHG and NOx emissions. Chapter 6 presents
conclusions and future research directions.

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Chapter 2: Literature Review
Researchers in drayage operations is mainly interested in studying how to optimize
freight movements. In this section, we review the literature regarding (1) empty container
repositioning problems, which focuses on how to efficiently allocate empty containers to better
assist the transloading process in the port area; (2) freight movement problems with dynamic
transportation network, which considers the network dynamics according to flow assignments;
and (3) drayage routing problems with alternative fuel vehicles, which investigates potential
benefits of employing alternative fuel vehicles into drayage operations.
2.1 Empty Container Repositioning Problem
The empty container repositioning problem involves finding the most efficient empty
container movements that help the container transloading process in port areas. Dejax and
Crainic (1987) performed one of the earliest studies that revealed the value of managing empty
container flows and characterized the empty vehicle flow problem. Crainic et al. (1993) were the
first to introduce the ECRP to the literature and provide a modeling framework. It included
stochasticity as part of the single- and multi-commodity problem and presented a mathematical
formulation. Shen and Khoong (1995) studied the problem as a decision-support system with
communications between different operation levels for a shipping company. Constraint
relaxations were performed to make the model sensitive to the changes in container demands and
supplies. Then Choong et al. (2002) built a time-extended optimization model to study the ECRP
by investigating the influence of the planning horizon. Tan et al. (2006) constructed a hybrid
multi-objective model to solve the ECRP. They developed a heuristic method by setting Pareto
improvement conditions so that the results would move towards a Pareto efficiency. Similarly,
Bandeira et al. (2009) proposed a two-step method to solve the ECRP. Their first step was to find

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a feasible solution that satisfies customers’ demand, and the second was to minimize the total
cost.
More recent papers addressed the ECRP through advanced models and solution
procedures. Li et al. (2014) built a model that maximized profits by reusing empty containers.
They tested their model under a deterministic scenario based on real-world data from four ports
on the Eastern Chinese coast. The solution not only achieved more profits but also benefited the
environment. Song and Dong (2015) took a more in-depth approach to ECRP, which discussed
the importance of studying ECRP and investigated different solution approaches to solve ECRP.
In their study, they subdivided the problems into two parts. The first part was inland empty
container reuse, which can be solved by inventory control-based models. The second part, the
focus of the current study, was a pickup and delivery problem between ports and nearby
locations.
Historically, ECRP has also been studied as a Traveling Salesman Problem (TSP) with
pickup and delivery. Zhang et al. (2009) proposed a multiple TSP model to tackle the multi-
depot container transportation problem. They solved the problem with a clustering method and a
tabu search heuristic. Their follow-up research (2010) enhanced their methodology by proposing
a window partition based approach, which addressed the problem and lowered the computation
time. Because of the asymmetricity in the transportation network, Braekers et al. (2013)
formulated the ECRP as an asymmetric multiple TSP. They proposed two solution approaches to
solve the TSP. The first approach, the sequential approach, requires predetermined empty
container allocation while the second one, the integrated approach, allows flexibility in container
allocation.

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Sterzik and Kopfer (2013) proposed a model for the ECRP which can optimize vehicle
routing and container repositioning at the same time. A tabu search heuristic was designed to
solve the model in both small- and large-scale instances. The research was augmented by Sterzik
et al. (2015) by considering multiple trucking companies. They modified Clarke and Wright’s
(1964) savings algorithm to generate an initial solution for their model. The study indicated that
if truck companies could share containers, total operational costs would fall, especially when the
demand had a tight time window.
As the trucking industry develops, multi-container trucks have been employed around the
globe. However, limited research attention has been paid to solving the ECRP using multi-
container trucks. Minvielle and Dessouky (2021) proposed the first ECRP model using DCTs
under deterministic demand, which optimizes truck routing and container repositioning
simultaneously. They formulated a container assignment problem and then developed a heuristic
to solve the problem. Their use of container movements as their decision variables imposed the
pickup and drop-off requirement. They defined delivery sequences as consisting of a first and
second leg, which led to the assumption that DCTs pick up loads only after delivering all the
containers. This limitation can be overcome by using DCTs with loading states as the decision
variables. Furthermore, incorporating stochastic demand can further improve freight movement
efficiency with DCTs.
Considering stochasticity when modeling the ECRP has become a trend in the literature.
One way of considering stochasticity is by incorporating future demand into the analysis. Di
Francesco et al. (2009) studied the ECRP with uncertainty. They considered opinions from
shipping companies and built a multi-scenario optimization model. They claimed that historical
data might be deficient, and future demand might be probabilistically independent of the past.

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However, with the development of forecast algorithms, demand predictions play a more critical
role in modeling container movements. Although it may be hard to predict demand in the long
run, the next day demand is easier to forecast. In our SDCFM, the next day stochastic demand is
considered for repositioning containers at the end of the day for the following day's freight
operations. Meng et al. (2014) indicated the research potential of incorporating port operations
into the ECRP. Following this path, we incorporate the uncertainty of port operations into our
ECRP models. Lu et al. (2020) studied a two-depot ECRP with stochastic demand. They
formulated the problem as a stochastic dynamic programming model and proposed an
approximation model to improve solving efficiency.
Two-stage stochastic models are widely used in modeling stochastic ECRPs. Long et al.
(2015) developed a two-stage stochastic programming model and solved it with a rolling
horizon-based framework. The solving framework is applicable to our ECRP models. Like Long
et al. (2015), we use a modified rolling horizon in solving ECRPs over a continuous study
horizon because it can connect the final container allocation state of one day with the next day's
initial state, allowing us to incorporate future demand into our model. Zhao et al. (2018)
developed a chance-constrained model for stochastic ECRP, incorporating a CO2 emission cost
into the objective. Hosseini and Sahlin (2019) modeled ECRP with uncertainty using a chance-
constrained method. The model required a predetermined study horizon, which means the
solution quality might depend on the initial conditions. Sarmadi et al. (2020) enhanced Bender’s
decomposition method (Benders, 1962) to solve the problem. Instead of using a sample average
approximation, Han et al. (2018) proposed a dynamic programming approach to solve stochastic
ECRP. In their study, a myopic model was tested to compare the solution quality against a static
model and the dynamic programming approach.

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Studies have also focused on the policy that defines rules for balancing empty containers
and the environmental impact caused by container movements. Ng et al. (2012) investigated
ECRP from an inventory control-based perspective. They proposed a policy for transferring
empty containers between two ports under stochastic scenarios while allowing backlog for unmet
demand. Dong et al. (2014) evaluated the effectiveness of several control policies for moving
empty containers between ports. Legros et al. (2019) proposed a detention-time-based policy for
ECRP that encourages street turns.
Besides proposing mathematical models and solution approaches, the Tioga Group
(2002) investigated the container movements in the POLA/LB. They indicated several research
potentials, such as empty container reuse and off-dock return depots. Their extensive research
has served as a foundation for later container movement models that use San Pedro Bay as their
test scenarios. For example, Chang et al. (2008) studied the container substitution problem to
reduce empty container interchange costs. They applied a branch-and-bound method to solve
their model. We also adopt the scenarios provided by the Tioga Group (2002) to validate our
ECRP models.
2.2 Freight Movement Problem with Dynamic Transportation Network
In a real-world transportation network, the capacity and travel time on an arc are load-
dependent, varying over time and traffic conditions. This dissertation includes network dynamics
in the freight routing and touring model caused by the change in truck flows. This part relates to
the dynamic traffic assignment (DTA), where dynamic models are considered to achieve traffic
balancing. In our FRTP, we combine the DTA with a truck routing optimization in order to
achieve load balancing in response to truck routes and achieve system optimality. The system
optimal dynamic traffic assignment problem was introduced in the literature by Merchant and

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Nemhauser (1978) and can be modeled with path-based formulations with equilibrium
constraints (Peeta & Mahmassani, 1995; Shen et al., 2006). Shen et al. (2006) proposed a
solution procedure for solving path-based SO-DTA problems. Instead of evaluating the travel
cost on each path, they introduced a path marginal cost (PMC) estimation to transform the SO-
DTA model into a variational inequality problem. However, for real-world size problems,
candidate routes for all pairs of ODs grow exponentially with respect to the size of the
transportation network, which makes PMC estimation computationally expensive. Zhang and
Qian (2020) proposed another approach to solve the SO-DTA problem by using a subgradient
method. Both PMC and subgradient approaches depend on the approximation of network states.
However, the transportation network has heterogeneous road sections with disparate physical
characterizations (e.g., road width, slope, pavement condition) and traffic regulations. As a
result, network states are hard to be approximated with closed-form mathematical expressions.
Although mathematical formulations are still used in recent studies (Das & Rama
Chilukuri, 2020; Gore et al., 2022) to model traffic-dependent networks, traffic simulation
models have also been employed in studying DTA systems by Mahmassani (2001) who stated
two properties for combining a simulation model into a DTA problem. One is that the
transportation network has to be descriptive, which means the network can be evaluated at a
given point and the latter network states partially depend on the current state. The second is
normative, which requires the system to provide drivers with acceptable routing decisions while
considering systematic objectives. The proposed method was applied to successive studies
(Mahmassani et al., 2007; Zhou et al., 2008) on multimodal transportation systems with
predetermined demand.

17

Another advantage of using traffic simulation models is that the non-homogeneity
between passenger and truck flows can be considered. Such non-homogeneity is caused by the
physical sizes (Dong et al., 2014), driving behaviors (Kong & Guo, 2016), and dynamics
between passenger vehicles and trucks (Ahn et al., 2002; Barth et al., 2001; Brodrick et al.,
2002). As a result, truck flows can affect passenger flows. Studying passenger and truck flows in
the same network increases the computational difficulty of solving freight operation problems.
Abadi et al. (2016) proposed a load-balancing algorithm, which iteratively updates network costs
for the DTA problem using real-time traffic simulation models that capture the traffic dynamics
and the impact of truck flows on passenger flows. In each iteration, for each pair of ODs, the
algorithm searches for lower cost routes in the transportation network to balance the traffic
assignment across different routes. Later, Zhao et al. (2018) improved the load-balancing
algorithm by deriving the marginal cost on network arcs to reduce the computational time. Both
studies showed the effectiveness of the load-balancing algorithm in providing acceptable routing
decisions to truck drivers. Another extension was proposed by Chen et al. (2021) that considered
a mixed fleet of diesel and electric trucks.
However, the load-balancing algorithm (Abadi et al., 2016; Chen et al., 2021; Zhao et al.,
2018) focuses on the container assignment part, which means truck drivers only get routing
information for the container delivery process. Once a truck finishes a delivery task, the model
stops considering the truck in the network. In actual practice, trucks can be reused to initiate
another delivery task, which reduces the total number of trucks for daily freight movements. We
extend the earlier work of Abadi et al. (2016) and Zhao et al. (2018) to construct truck tours
beyond just the initial delivery and propose a co-simulation optimization based load-balancing
with touring (LBT) approach to solve the FRTP. Considering only delivery flows (i.e., truck

18

flows that deliver containers from suppliers to warehouses) is not enough for freight
management, it is also essential to determine where a truck should travel next after delivering a
container. To develop complete tours for the trucks, we introduce a pickup flow optimization
procedure based on the load-balancing algorithm solution, which completes truck touring
decisions.
The idea of touring in this case is similar to the vehicle routing problems in the literature
(Azad et al., 2022; Wang et al., 2016; Yao et al., 2019). However, most studies on this topic
share common assumptions that costs between ODs are homogeneous and travel times are not
load-dependent (i.e., travel time on a link does not depend on the number of vehicles on the
link). In practice, especially in urban areas, it is likely to have multiple candidate routes between
a pair of locations with different costs and travel times depending on traffic flow assignments.
2.3 Drayage Routing Problem with Alternative Fuel Vehicles
The drayage routing problem is a variation of the vehicle routing problem (VRP) with all
vehicles having containerized capacity. The goal is to minimize the cost of regional freight
movements. Prior research on drayage routing problems has typically used the multi-traveling
salesperson problem (m-TSP) formulations (Jula et al., 2005; Shiri & Huynh, 2018; Wang &
Regan, 2002; Zhang et al., 2009, 2010). Most earlier work focused on minimizing the travel
distance for freight pickup and delivery. Jula et al. (2005) proposed an exact method for small-
size problems with dynamic programming and a genetic algorithm for large-size problems.
Zhang et al. (2009) formulated the m-TSP with multi-depots. They developed a reactive tabu
search to solve the model and solved small-scale problems optimally. Instead of purely
minimizing travel distance for daily drayage operations, Zhang et al. (2010) extended their

19

model by considering trucks as a limited resource. The model was solved by a window partition
method inspired by Wang and Regan (2002).
Due to globalization and economic prosperity, freight movements have been increasing
during the past decades, leading to social issues associated with air pollution, especially in GHG
emissions, NOx, and particulates. Erdoğan and Miller-Hooks (2012) introduced the green vehicle
routing problem (GVRP) to the literature to address the need to consider emissions while
optimizing vehicle routes. They formulated GVRP as a MIP and solved the problem with a
modified saving algorithm and a heuristic to improve the initial solution.
Lin et al. (2014) conducted a comprehensive survey of GVRPs. They indicated that
GVRP should consider heterogeneous fleets instead of identical vehicles. Behnke and
Kirschstein (2017) formulated the GVRP with heterogeneous fleets. Instead of searching for the
shortest path, they calculate the emission-minimal path for each vehicle class. Koyuncu and
Yavuz (2019) proposed two GVRP formulations with node- and arc-duplicating forms. They
compared both models’ computational performance under various refueling policies, mixed
fleets, and charging locations. Another variation of the mixed fleet vehicle routing problem is
considering a heterogeneous driving range which was introduced by Juan et al. (2014). A more
recent study on the multi-range VRP by Eskandarpour et al. (2019) separated the objective
function into monetary costs and environmental impacts. They applied a large neighborhood
search to solve their optimization model. Bruglieri et al. (2019) investigated the GVRP with
capacitated fuel stations with constant refueling time.
Asghari and Mirzapour Al-e-hashem (2021) summarized the most recent studies of
GVRP. One of the future directions was to consider various energy consumption rates for
electric battery vehicles. As a greener substitute for diesel trucks, researchers extended the

20

GVRP by considering battery-powered trucks. Elangovan et al. (2021) analyzed the GHG
emissions for diesel and battery-electric light-duty trucks. The results showed that electric light-
duty trucks emitted about 30 percent less GHGs and consumed 66 percent less energy than diesel
trucks.
Unlike commercial pickup and delivery services with small packages and commodities,
drayage operations are containerized. The HDTs’ driving range depends on the carrying loads,
and BEHDTs are particularly sensitive, ranging from 70 miles to 100 miles. Based on payload,
Giuliano et al. (2020) interviewed BEDHT drivers and found that range limitations significantly
curtail the tasks they can undertake. This suggests the need for charge planning to employ
BEHDTs in drayage operations.
The electric vehicle routing problem (EVRP), an extension of the VRP that considers the
limited driving range for electric vehicles and charging stations, is extensively studied. Lin et al.
(2016) presented a general EVRP formulation that minimized fleet size, travel time, and energy
costs. Pelletier et al. (2019) extended the EVRP into the stochastic world with uncertain energy
consumption. They formulated the problem as a robust optimization problem and developed a
large neighborhood search heuristic to solve it. Because electric vehicles usually have strict
limits on working time with a relatively long recharging time, electric vehicle drivers tend to
visit charging stations as few times as possible. As a result, full recharging policies are
commonly accepted in the literature (Wen et al., 2016; Yu et al., 2019; Zhang et al., 2018; Zhao
& Lu, 2019).
Felipe et al. (2014) considered heterogeneous charging facilities for electric vehicles.
Recent EVRP studies have considered the possibility of battery swapping (Ban et al., 2021; Hof
et al., 2017; Masmoudi et al., 2018), but the high cost and large capacity of the batteries for

21

BEHDTs make it challenging to perform in practice. We use a concave piecewise-linear
charging function to calculate charging times for BEHDTs in our model, since researchers have
widely used piecewise linear approximations as the charging function (Keskin et al., 2019;
Pelletier et al., 2017; Szablowski & Bralewski, 2019).
Most EVRPs cannot be solved optimally since VRP itself has been known to be NP-hard.
Exact methods such as branch-and-cut (Koç & Karaoglan, 2016), branch-and-price (Yu et al.,
2019), and the MIP-based reduction procedure (Leggieri & Haouari, 2017) can solve small-scale
problems optimally. However, for large instances, an optimal solution cannot be obtained within
a reasonable amount of computational time. Therefore, most of the studies in the literature have
used heuristics to solve their models. Tabu search is a class of local search methods that have
been used in different variations of VRPs. The most successful variation is ALNS proposed by
Ropke and Pisinger (2006), which studies show is one of the most effective metaheuristics for
solving vehicle routing problems (Demir et al., 2012; Dessouky et al., 2020; Lahyani et al., 2019;
Shao & Dessouky, 2020). In this dissertation, we propose a modified ALNS algorithm to solve
our mixed fleet drayage routing problem efficiently.

22

Chapter 3: Empty Container Repositioning Problems with Double Container Trucks
This chapter presents empty container repositioning models with double container
trucks— the double container flexible pickup and drop-off model and the stochastic double
container flexible pickup and drop-off model. We develop a solving scheme based on a rolling
horizon-based framework to solve these two models. Instead of solving a one-day problem, the
proposed solving scheme can solve ECRPs with a continuous study horizon. At the end of this
chapter, we show the computational results of the proposed models and draw conclusions.
3.1 Problem Description
This chapter considers ECRP using DCTs. In daily drayage operations, empty containers
are transported between ports and warehouses to fulfill transloading demand. We enhance the
double container assignment model (DCAM) proposed by Minvielle and Dessouky (2021) by (1)
relaxing the pickup and drop-off requirements for trucks, incorporating port operations, and (2)
considering stochastic future demand into the container movement model. The first improvement
increases the model flexibility and reduces truck miles in the truck routing decisions by allowing
a double-container truck to pick up a new container after dropping off a single container. Instead
of using DCTs as the main decision variables, our model further defines truck states to represent
the loading state of a double container truck. The second improvement focuses on modeling a
transportation system that provides a more realistic scenario in the container shipping industry.
That is, the daily demand in the port region depends on the port states. For example, if there is a
ship leaving the port, the demand for export containers goes up; and if there is a ship arriving, the
demand for import containers will increase. Therefore, modeling the port states and the road
network as a transportation system also helps to estimate the second-day stochastic demand more

23

accurately so that strategic container movements can be made on the first day cconsidering the
second-day delivery.
We define a directed graph network consisting of a set of location nodes (L) with a set of
directed links (D) connecting these nodes. A node in the network can be one of the following
four types of locations: ports, importers, exporters, and depots. Let SI, SE, SD, and SP be the sets
of importers, exporters, depots, and ports nodes, respectively. All of them are subsets of L. There
are three types of containers—import containers, export containers, and empty containers. A set
of container types (Z) with index z equals 1, 2, and 3 represents empty containers, export
containers, and import containers, respectively. An importer has the demand for import
containers, and then it has the option to turn a container with import commodities to an empty
container. An exporter can load an empty container with export commodities and then send the
export containers to the port. A depot is a location that stores containers temporarily without
transloading ability. In each location, the transloading process takes a fixed period of time (𝜏 𝑖 ,
where 𝑖 ∈𝐿 ), which is called a turnover time. For each transloading station, the capacity is
limited by its space and location. All demand needs to be satisfied at all locations by the end of
the day. An ideal container transloading process using DCTs is shown below in Figure 7.

Figure 7. Ideal Container Flow

24

However, because of the United States trade imbalance, the total number of import
containers is significantly greater than the export ones, which generates numerous empty
containers in the system. If these empty containers are not managed in an efficient way, the
transloading process can be delayed due to capacity limitations. The analysis time horizon for
each day is discretized into |T| time units.
3.2 Double Container Flexible pickup and drop-off Model
In this section, we propose a DCFM to overcome the assumption that a truck must drop
off all of its loads before another pickup, as in the earlier DCAM model. Allowing a DCT to pick
up containers whenever capacity permits can potentially reduce the total truck miles. Figure 8
provides a small-size problem that helps illustrate the benefits of relaxing this assumption. In the
network, there are four nodes connected by directed edges, including one port (node 1), two
importers (nodes 2 and 3), and one exporter (node 4). For the empty containers, export
containers, and import containers of each node, the demand or supply is described as a triplet
(𝑖,𝑗 ,𝑘 ) , where a positive number represents demand and a negative number indicates supply. For
example, (0, 2, -2) means the port (node 1) has no empty container demand, two export container
demand, and two import container supplies. A DCT starts its task from the port. With the pickup
and drop-off assumption, the truck has to return to node 2 after delivering both import containers
at nodes 2 and 3. As a result, the truck has to follow route (a), which results in a total travel
distance of 20 miles. However, if the truck can pick up loads whenever capacity permits, the
total travel distance can be reduced to 14 miles, as shown in route (b). The truck can pick up the
empty container at the same time as dropping off an import container at node 2 to reduce the
total travel distance.

25

Figure 8. Example of a Small-size Problem

The DCTs can carry combinations of two of the three types of containers. We introduce a
set of truck loading states (R) containing all ten possible carrying combinations with set index r.
Let vector 𝑘 𝑟 be a three-element vector where 𝑘 𝑟 =(𝑘 𝑟 ,1
,𝑘 𝑟 ,2
,𝑘 𝑟 ,3
)
𝑇 represents a truck in state r
carries 𝑘 𝑟 ,1
empty containers, 𝑘 𝑟 ,2
export containers, and 𝑘 𝑟 ,3
import containers. There are ten
possible truck states and they are (0,0,0), (0,0,1), (0,0,2), (0,1,0), (0,1,1), (0,2,0), (1,0,0), (1,0,1),
(1,1,0), and (2,0,0). For example, state (0,0,1) represents the truck state of carrying zero empty
containers, zero import containers, and one export container. The notations employed throughout
the ECRP section are defined as follows:
𝑡 The index of a time interval, 𝑡 ∈𝑇 ;
𝑐 𝑖 The capacity at location i;
𝑝 𝑖 ,𝑧 The initial number of type 𝑧 containers at location 𝑖 ;
𝛿 𝑖 ,𝑗 The distance from node 𝑖 to node 𝑗 in miles;
𝑜 𝑖 ,𝑗 ,𝑡 The travel time of going from location 𝑖 to 𝑗 arriving at time 𝑡 ;
𝑑 𝑖 ,𝑧 ,𝑡 The demand for type 𝑧 containers at location 𝑖 at time 𝑡 ;
𝜆 The weighting factor for the total truck miles and total number of trucks in use;
𝑣 𝑖 The number of vehicles at location 𝑖 at the beginning of today;

26

𝑥 𝑖 ,𝑗 ,𝑡 𝑟 The number of trucks in state 𝑟 leaving from location 𝑖 to 𝑗 with a departure time 𝑡 ;
𝑎 𝑖 ,𝑧 ,𝑡 The number of type 𝑧 containers that have been supplied by location 𝑖 at time 𝑡 ;
𝑏 𝑖 ,𝑧 ,𝑡 The number of type 𝑧 containers that have been received by location 𝑖 at time 𝑡 ;
𝑙 𝑖 ,𝑡 The number of vehicles leaving location 𝑖 at time 𝑡 ;
𝑒 𝑖 ,𝑡 The number of vehicles arriving at location 𝑖 at time 𝑡 ;
The DCFM can be formulated in the following way:
𝑚𝑖𝑛 𝑇𝐶 (𝑥 𝑖 ,𝑗 ,𝑡 𝑟 ,𝑉 𝑖 )= 𝜆 (∑∑∑∑𝛿 𝑖 ,𝑗 ∗𝑥 𝑖 ,𝑗 ,𝑡 𝑟 𝑟 ∈𝑅 𝑡 ∈𝑇 𝑗 ∈𝐿 𝑖 ∈𝐿 )+(1−𝜆 )(∑𝑣 𝑖 𝑖 ∈𝐿 ) (1)
Subject to the following constraints:
∑∑∑𝑘 𝑟 ,𝑧 ∙𝑥 𝑖 ,𝑗 ,𝜎 𝑟 𝜎 ≤𝑡 𝑟 ∈𝑅 𝑗 ∈𝐿 =𝑎 𝑖 ,𝑧 ,𝑡 ∀𝑖 ∈𝐿 ,∀𝑡 ∈𝑇 ,∀𝑧 ∈𝑍 (2)

∑∑∑𝑘 𝑟 ,𝑧 ∙𝑥 𝑖 ,𝑗 ,𝜎 𝑟 𝜎 ≤𝛾 𝑟 ∈𝑅 𝑖 ∈𝐿 =𝑏 𝑗 ,𝑧 ,𝑡 ∀𝑗 ∈𝐿 ,∀𝑡 ∈𝑇 ,∀𝑧 ∈𝑍 (3)

0≤𝑓 𝑖 (𝑡 )≤𝑏 𝑖 ,3,𝜋 ∀𝑖 ∈𝑆𝐼 ,∀𝑡 ∈𝑇 (4)

0≤𝑓 𝑖 (𝑡 )≤𝑏 𝑖 ,1,𝜋 ∀𝑖 ∈𝑆𝐸 ,∀𝑡 ∈𝑇 (5)

𝑔 𝑖 ,𝑧 (𝑡 )≥0 ∀𝑖 ∈𝐿 ,∀𝑡 ∈𝑇 ,∀𝑧 ∈𝑍 (6)

𝑏 𝑖 ,3,𝑡 −𝑎 𝑖 ,3,𝑡 ≥𝑑 𝑖 ,3,𝑡 ∀𝑖 ∈𝑆𝐼 ,∀𝑡 ∈𝑇 (7)

𝑎 𝑖 ,2,𝑡 −𝑏 𝑖 ,2,𝑡 ≥𝑑 𝑖 ,2,𝑡 ∀𝑖 ∈𝑆𝐸 ,∀𝑡 ∈𝑇 (8)

𝑏 𝑖 ,1,𝑡 −𝑎 𝑖 ,1,𝑡 ≥𝑑 𝑖 ,1,𝑡 ∀𝑖 ∈𝑆𝑃 ,∀𝑡 ∈𝑇 (9)

𝑏 𝑖 ,2,𝑡 −𝑎 𝑖 ,2,𝑡 ≥𝑑 𝑖 ,2,𝑡 ∀𝑖 ∈𝑆𝑃 ,∀𝑡 ∈𝑇 (10)

𝑎 𝑖 ,3,𝑡 −𝑏 𝑖 ,3,𝑡 ≥𝑑 𝑖 ,3,𝑡 ∀𝑖 ∈𝑆𝑃 ,∀𝑡 ∈𝑇 (11)

∑(𝑝 𝑖 ,𝑧 +𝑎 𝑖 ,𝑧 ,𝑡 −1
−𝑏 𝑖 ,𝑧 ,𝑡 )
𝑧 ∈𝑍
≤𝑐 𝑖 ∀𝑖 ∈𝐿 ,∀𝑡 ∈{𝑇 |𝑡 ≥2} (12)

27

∑∑𝑥 𝑖 ,𝑗 ,𝑡 𝑟 𝑟 ∈𝑅 𝑗 ∈𝐿 =𝑙 𝑖 ,𝑡 ∀𝑖 ∈𝐿 ,∀𝑡 ∈𝑇 (13)

∑∑𝑥 𝑖 ,𝑗 ,𝛾 𝑟 𝑟 ∈𝑅 𝑖 ∈𝐿 =𝑒 𝑗 ,𝑡 ∀𝑗 ∈𝐿 ,∀𝑡 ∈𝑇 (14)

𝑣 𝑖 +∑(𝑒 𝑖 ,𝜎 −𝑙 𝑖 ,𝜎 )
𝜎 ≤𝑡 ≥0 ∀𝑖 ∈𝐿 ,∀𝑡 ∈𝑇 (15)

𝑥 𝑖 ,𝑗 ,𝑡 𝑟 ,𝑣 𝑖 ∈𝑍 0+
∀𝑖 ∈𝐿 ,∀𝑗 ∈𝐿 ,∀𝑟 ∈𝑅 ,∀𝑡 ∈𝑇 (16)

The objective function (1) includes the total truck miles to deliver the demand for one
day and the total vehicle cost. Constraint (2) represents that, for each type of container, the
number of containers that have been provided by a location at time 𝑡 equals the sum of
containers left before time 𝑡 . Constraint (3) computes the containers of each type that have been
received by location 𝑗 at time 𝑡 which is equal to the sum of the containers that left location 𝑖 at
time 𝑡 −𝑜 𝑖 ,𝑗 ,𝑡
. Since the quantity 𝑡 −𝑜 𝑖 ,𝑗 ,𝑡
can be negative, we define 𝛾 =𝑚𝑎𝑥 {0,𝑡 −𝑜 𝑖 ,𝑗 ,𝑡 }
and set 𝑥 𝑖 ,𝑗 ,0
𝑟 =0 for all locations and loading states 𝑟 .
Let 𝜋 be the latest time point when the containers can arrive while still completing the
transloading process at time 𝑡 . That is, 𝜋 =𝑚𝑎𝑥 {1,𝑡 −𝜏 𝑖 }. Since the transloading process takes
a fixed time δ
𝑖 at each location 𝑖 , containers arriving at time 𝑡 −𝜏 𝑖 can be transformed into other
types of containers according to their location. At the beginning of the day, where 𝑡 −𝜏 𝑖 <1, we
can assume there is no completed transloading process. In constraints (4) and (5), 𝑓 𝑖 (𝑡 ) is a
function that calculates the total number of containers that have completed the transloading
process at location 𝑖 ∈𝐿 at time 𝑡 . Therefore,
𝑓 𝑖 (𝑡 )={
max{𝑓 𝑖 (𝑡 −1),𝑎 𝑖 ,1,𝑡 −𝑏 𝑖 ,1,𝑡 −𝑝 𝑖 ,1
} ∀𝑖 ∈𝑆𝐼 ,∀𝑡 ∈{𝑇 |𝑡 ≥2}
max{𝑓 𝑖 (𝑡 −1),𝑎 𝑖 ,2,𝑡 −𝑏 𝑖 ,2,𝑡 −𝑝 𝑖 ,2
} ∀𝑖 ∈𝑆𝐸 ,∀𝑡 ∈{𝑇 |𝑡 ≥2}
0 ∀𝑖 ∈𝑆𝐼 ⋃𝑆𝐸 ,𝑡 =1
(17)

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Constraint (4) is the transloading feasibility constraint for importers, which means the
number of unloaded import containers cannot exceed the number of import containers it has
received. For exporters, we have a similar constraint (5) to represent the transloading feasibility.
Both constraints (4) and (5) can be linearized when solving the problem.
In (6), 𝑔 𝑖 ,𝑧 (𝑡 ) is a function that calculates the remaining number of type 𝑧 containers at
location 𝑖 at time 𝑡 . Taking an importer 𝑖 ∈𝑆𝐼 as an example, constraints (6) can be understood
in the following sense. The importer containers that have been received by time 𝜋 can be
unloaded to empty containers. Thus, the total number of empty containers remaining at time 𝑡 is
the summation of the unloaded import containers, the empty containers that have been received,
and the initial empty containers at the location minus the total number of empty containers that
have been supplied by location 𝑖 , i.e.,
𝑔 𝑖 ,1
(𝑡 )=𝑓 𝑖 (𝑡 )+𝑏 𝑖 ,1,𝑡 +𝑝 𝑖 ,1
−𝑎 𝑖 ,1,𝑡 ∀𝑖 ∈𝑆𝐼 ,∀𝑡 ∈𝑇 (18)

The number of export containers that remain is the total number of export containers the
location has received and the initial number of export containers at the location minus the
number that has been supplied by the location, i.e.,
𝑔 𝑖 ,2
(𝑡 )=𝑏 𝑖 ,2,𝑡 +𝑝 𝑖 ,2
−𝑎 𝑖 ,2,𝑡 ∀𝑖 ∈𝑆𝐼 ,∀𝑡 ∈𝑇 (19)

For the import containers, the number is the sum of the number of import containers at a
location initially and the number the location has received minus the total number of import
containers that were unloaded and import containers that were supplied, i.e.,
𝑔 𝑖 ,3
(𝑡 )=𝑏 𝑖 ,3,𝑡 +𝑝 𝑖 ,3
−(𝑓 𝑖 (𝑡 )+𝑎 𝑖 ,3,𝑡 ) ∀𝑖 ∈𝑆𝐼 ,∀𝑡 ∈𝑇 (20)

For each type of location (SI, SE, SD, SP), the corresponding function can be constructed
accordingly (please see Appendix A for all 𝑔 𝑖 ,𝑧 (𝑡 ) functions).

29

Constraints (7)-(11) ensure that the demand at all the locations is met. Constraint (7)
shows that the total number of import containers unloaded at the location has to be greater than
the demand. Constraint (8) means each exporter has to send out enough export containers to
satisfy the demand. Unlike an importer or exporter which has only one type of container demand,
ports have demand for sending import containers to transloading stations, constraint (9), and
shipping export containers and empty containers out of the country, constraints (10) and (11).
Constraint (12) is the capacity constraint at each location. At time 1, the initial container
allocation is known, which satisfies the capacity constraint by definition. Constraints (13) and
(14) model the vehicle leaving and arriving at a specific location at time 𝑡 . The parameter 𝜏 is
used in the same way as the definition above in constraint (3). Constraint (15) states that the total
number of trucks leaving location 𝑖 at time 𝑡 cannot exceed the number of trucks that location 𝑖
has. The last set of constraints states that all the decision variables are non-negative integers.
3.3 Stochastic Double Container Flexible pickup and drop-off Model
As aforementioned, long-term demand prediction is challenging and often inaccurate
compared to predicting short time demand. In this study, we only include tomorrow’s demand in
the model. In sea transportation, the demand at all the locations depends on the port. In practice,
the port can have a set of scenarios (Θ) , which can be described by the ships’ arrival and
departure information. The nearby warehouses can request supply and demand according to the
current port scenario at the beginning of the day. For example, if ships are leaving the port today,
exporters will send more export containers to the port. Therefore, instead of focusing only on the
road network, the port state also plays an important role in the transportation system. We develop
a Markov process to simulate the port scenarios, and the transitional probabilities can be
estimated from historical data.

30

In practice, at the beginning of the day, the demand at each location can be determined by
the state of the port. The demand at each location follows a distribution based on the scenarios of
the ports. Since the final container allocation condition today will be the starting condition the
next day, the benefit of including tomorrow’s demand is that trucks can make strategic
movements for tomorrow’s demand. At the beginning of tomorrow, the state of the port and the
demand at each location are updated, and the problem-solving cycle repeats itself, as shown in
Figure 9.

Figure 9. Framework for Solving Two-day ECRP with Stochastic Demand

The SDCFM is proposed to advance the solution obtained by DCFM. SDCFM models
the double container movements for two consecutive days. It requires strictly meeting all of
today’s demand. However, since tomorrow’s demand is stochastic, the model allows for a certain
amount of unmet demand for tomorrow with a penalty cost. Additional notations are introduced
as follows:
𝑡 ̅
The index of the second-day time interval, 𝑡 ̅
∈𝑇̅
;
𝑠 The index of the port scenarios, 𝑠 ∈Θ;
𝜃 𝑠 The probability for scenario 𝑠 ;

31

𝜑 The penalty cost for not fulfilling one unit of tomorrow’s demand;
𝜇 𝑖 The expected number of import containers that arrive at port 𝑖 , 𝑖 ∈𝑆𝑃 ;
𝑑 ̅
𝑖 ,𝑧 ,𝑡 ̅
𝑠 The demand for type 𝑧 containers at location 𝑖 at time 𝑡 ̅
in scenario s;
𝑥 ̅
𝑖 ,𝑗 ,𝑡 ̅
𝑟 The number of trucks in state 𝑟 going from location 𝑖 to 𝑗 with a departure time 𝑡 ̅
;
𝑎̅
𝑖 ,𝑧 ,𝑡 ̅
The number of type 𝑧 containers that have been supplied by location 𝑖 at time 𝑡 ̅
;
𝑏̅
𝑖 ,𝑧 ,𝑡 ̅
The number of type 𝑧 containers that have been received by location 𝑖 at time 𝑡 ̅
;
𝜔 𝑖 ,𝑧 ,𝑡 ̅
𝑠 The unmet demand vector of type 𝑧 containers at location 𝑖 at time 𝑡 ̅
in scenario 𝑠 ;
𝑝 ̅
𝑖 ,𝑧 The number of type 𝑧 containers at location 𝑖 at the beginning of tomorrow.
The SDCFM can be formulated in the following way:
𝑚𝑖𝑛 𝑇𝐶 (𝑥 𝑖 ,𝑗 ,𝑡 𝑟 ,𝑣 𝑖 )+𝜆 ∑∑∑∑𝛿 𝑖 ,𝑗 ∙𝑥 ̅
𝑖 ,𝑗 ,𝑡 ̅
𝑟 𝑟 ∈𝑅 𝑡 ̅
∈𝑇̅ 𝑗 ∈𝐿 𝑖 ∈𝐿 +∑∑∑∑𝜑 ∗𝜃 𝑠 ∗
𝑧 ∈𝑍 𝑡 ̅
∈𝑇̅ 𝑖 ∈𝐿 𝜔 𝑖 ,𝑧 ,𝑡 ̅
𝑠
𝑠 ∈Θ
(21)

The problem constraints consist of (2)-(16) and
∑∑∑𝑘 𝑟 ,𝑧 ∙𝑥 ̅
𝑖 ,𝑗 ,𝜎 𝑟 𝜎 ≤𝑡 ̅ 𝑟 ∈𝑅 𝑗 ∈𝐿 =𝑎̅
𝑖 ,𝑧 ,𝑡 ̅
∀𝑖 ∈𝐿 ,∀𝑡 ̅
∈𝑇̅
,∀𝑧 ∈𝑍 (22)

∑∑∑𝑘 𝑟 ,𝑧 ∙𝑥 ̅
𝑖 ,𝑗 ,𝜎 𝑟 𝜎 ≤𝛾̅ 𝑟 ∈𝑅 𝑖 ∈𝐿 =𝑏̅
𝑗 ,𝑧 ,𝑡 ̅
∀𝑗 ∈𝐿 ,∀𝑡 ̅
∈𝑇̅
,∀𝑧 ∈𝑍 (23)

0≤𝑓 ̅
𝑖 (𝑡 ̅ )≤𝑏̅
𝑖 ,3,𝜋̅
∀𝑖 ∈𝑆𝐼 ,∀𝑡 ̅
∈𝑇̅
(24)
0≤𝑓 ̅
𝑖 (𝑡 ̅ )≤𝑏̅
𝑖 ,1,𝜋̅
∀𝑖 ∈𝑆𝐸 ,∀𝑡 ̅
∈𝑇̅
(25)
𝑔 ̅
𝑖𝑧
(𝑡 ̅ )≥0 ∀𝑖 ∈𝐿 ,∀𝑡 ̅
∈𝑇̅
,∀𝑧 ∈𝑍 (26)
𝑏̅
𝑖 ,3,𝑡 ̅
−𝑎̅
𝑖 ,3,𝑡 ̅
+𝜔 𝑖 ,3,𝑡 ̅
𝑠 ≥𝑑 ̅
𝑖 ,3,𝑡 ̅
𝑠 ∀𝑖 ∈𝑆𝐼 ,∀𝑡 ̅
∈𝑇̅
,∀𝑠 ∈Θ (27)
𝑎̅
𝑖 ,2,𝑡 ̅
−𝑏̅
𝑖 ,2,𝑡 ̅
+𝜔 𝑖 ,2,𝑡 ̅
𝑠 ≥𝑑 ̅
𝑖 ,2,𝑡 ̅
𝑠 ∀𝑖 ∈𝑆𝐸 ,∀𝑡 ̅
∈𝑇̅
,∀𝑠 ∈Θ (28)
𝑏̅
𝑖 ,1,𝑡 ̅
−𝑎̅
𝑖 ,1,𝑡 ̅
+𝜔 𝑖 ,1,𝑡 ̅
𝑠 ≥𝑑 ̅
𝑖 ,1,𝑡 ̅
𝑠 ∀𝑖 ∈𝑆𝑃 ,∀𝑡 ̅
∈𝑇̅
,∀𝑠 ∈Θ (29)
𝑏̅
𝑖 ,2,𝑡 ̅
−𝑎̅
𝑖 ,2,𝑡 ̅
+𝜔 𝑖 ,2,𝑡 ̅
𝑠 ≥𝑑 ̅
𝑖 ,2,𝑡 ̅
𝑠 ∀𝑖 ∈𝑆𝑃 ,∀𝑡 ̅
∈𝑇̅
,∀𝑠 ∈Θ (30)

32

𝑎̅
𝑖 ,3,𝑡 ̅
−𝑏̅
𝑖 ,3,𝑡 ̅
+𝜔 𝑖 ,3,𝑡 ̅
𝑠 ≥𝑑 ̅
𝑖 ,3,𝑡 ̅
𝑠 ∀𝑖 ∈𝑆𝑃 ,∀𝑡 ̅
∈𝑇̅
,∀𝑠 ∈Θ (31)
𝜔 𝑖 ,𝑧 ,𝑡 ̅
𝑠 ≥0 ∀𝑖 ∈𝐿 ,∀𝑡 ̅
∈𝑇̅
,∀𝑠 ∈Θ,∀𝑧 ∈𝑍 (32)
𝑝 ̅
𝑖 ,𝑧 =ℎ
𝑖 ,𝑧 (𝑡 ∗
)≥0 ∀𝑖 ∈𝐿 ,∀𝑧 ∈𝑍 (33)

∑(𝑝 ̅
𝑖 ,𝑧 +𝑎̅
𝑖 ,𝑧 ,𝑡 ̅
−1
−𝑏̅
𝑖 ,𝑧 ,𝑡 ̅
)
𝑧 ∈𝑍
≤𝑐 𝑖 ∀𝑖 ∈𝐿 ,∀𝑡 ̅
∈{𝑇̅
|𝑡 ̅
≥2} (34)

𝑥 ̅
𝑖 ,𝑗 ,𝑡 ̅
𝑟 ∈ℤ
+
∀𝑖 ∈𝐿 ,∀𝑗 ∈𝐿 ,∀𝑟 ∈𝑅 ,∀𝑡 ̅
∈𝑇̅
(35)

The objective function (21) includes the total truck miles for today and tomorrow’s
container movement, the total vehicle cost, and the penalty cost for not fulfilling tomorrow’s
demand. Constraints (22)-(26) are similar to that of constraints (2)-(6) in DCFM, respectively.
Constraints (27)-(31) are the demand constraints for tomorrow. Unlike the first day which
requires the demand to be satisfied at the end of the day, shortages are allowed for the next-day
demand with a penalty cost applied in the objective function. Constraint (32) requires that the
unmet demand cannot be negative to ensure the nonnegativity of the penalty cost.
Constraint (33) is the initial container allocation for tomorrow, and it depends on the final
state at the end of the first day, where 𝑡 ∗
is the last time interval index of set T and ℎ
𝑖 ,𝑧 is the type
𝑧 container initial allocation function for the second day at location 𝑖 . Taking an importer 𝑖 ∈𝑆𝐼
as an example, import containers remaining overnight will be unloaded and converted into empty
containers while all the other containers will remain the same, i.e.,
ℎ
𝑖 ,𝑧 (𝑡 ∗
)={
𝑏 𝑖 ,1,𝑡 ∗+𝑏 𝑖 ,3,𝑡 ∗+𝑝 𝑖 ,3
+𝑝 𝑖 ,1
−(𝑎 𝑖 ,1,𝑡 ∗+𝑎 𝑖 ,3,𝑡 ∗) 𝑖𝑓 𝑧 =1
𝑏 𝑖 ,2,𝑡 ∗+𝑝 𝑖 ,2
−𝑎 𝑖 ,2,𝑡 ∗ 𝑖𝑓 𝑧 =2
0 𝑖𝑓 𝑧 =3
, ∀𝑖 ∈𝑆𝐼 (36)
The same idea can be applied to construct corresponding ℎ
𝑖 ,𝑧 functions for exporters and
depots (please see Appendix A for all ℎ
𝑖 ,𝑧 (𝑡 ∗
) functions ). However, the container allocations at
the beginning of the next day in the ports are different from the non-port locations because

33

export and empty containers will be shipped out from the port, and the import containers will be
ready at the port overnight. Therefore,
ℎ
𝑖 ,𝑧 (𝑡 ∗
)={
𝑏 𝑖 ,1,𝑡 ∗+𝑝 𝑖 ,1
−𝑎 𝑖 ,1,𝑡 ∗−𝑑 𝑖 ,1,𝑡 ∗ 𝑖𝑓 𝑧 =1
𝑏 𝑖 ,2,𝑡 ∗+𝑝 𝑖 ,2
−𝑎 𝑖 ,2,𝑡 ∗−𝑑 𝑖 ,2,𝑡 ∗ 𝑖𝑓 𝑧 =2
𝑏 𝑖 ,3,𝑡 ∗+𝑝 𝑖 ,3
−𝑎 𝑖 ,3,𝑡 ∗+𝜇 𝑖 𝑖𝑓 𝑧 =3
, ∀𝑖 ∈𝑆𝑃 (37)
Constraint (34) states that the capacity of a location cannot be exceeded within every time
interval for tomorrow.
SDCFM is a simplified two-stage stochastic programming model. The first stage models
the container movements for the first day with deterministic demand, and the second stage
models the container movements for the next day for different port scenarios. Compared to a
traditional two-stage stochastic model, the first stage of the two approaches is the same. In the
second stage, however, SDCFM requires about 50 percent fewer variables and constraints than a
traditional two-stage model when considering the three port scenarios. In SDCFM, the
complexity increases at a sub-linear speed with respect to the increasing number of port
scenarios, meaning only the number of the unmet demand vector (𝜔 𝑖 ,𝑡 ,𝑠 ) and their related
constraints (27)-(32) will increase. Instead, using a traditional two-stage stochastic programming
method, the model complexity grows at linear speed, because there must be a set of variables and
constraints to model each port scenario. For example, we need to subdivide the decision variable
𝑥 ̅
𝑖 ,𝑗 ,𝑡 ̅
𝑟 into 𝑥 ̅
𝑖 ,𝑗 ,𝑡 ̅
𝑟 ,𝑠 to model the second stage container movements under port scenario 𝑠 , which is
also the case for the other decision variables, such as 𝑎̅
𝑖 ,𝑧 ,𝑡 ̅
, 𝑏̅
𝑖 ,𝑧 ,𝑡 ̅
, and so on. Also, the number of
corresponding constraints will increase as well. Since the model is solved as a mixed-integer
programming problem, fewer variables and constraints can reduce the computational time for
finding the best solution.

34

3.4 Computational Results
We test these models on real-world data from the POLA/LB. We assume ten transloading
stations, including five importers, three exporters, and two depots. The distance matrix used in
this study can be found in Minvielle and Dessouky (2021). All the locations are within 15 miles
of the port because the street turn is more likely to occur at a short range. The travel time
between non-port locations is less than one hour. Furthermore, the travel time between the port
and non-port locations is two hours because it takes time for a truck to go through several
checkpoints when entering or leaving the port. The model is developed using Julia 1.1 and
solved by Gurobi 8.1 on an i9-8950HK CPU. As mentioned, DCFM and SDCFM are mixed
integer programming models. The stopping criteria for solving these models were either reaching
20 minutes of CPU time or finding the optimal solution. If the optimal solution is not found
within 20 minutes of CPU time, we record the best solution found.
3.4.1 Impact of Incorporating Flexibility in Pickups
We first test the benefit of allowing a double-container truck to pick up a new container
after dropping off a single container. To make a fair comparison with DCAM, we let 𝜆 =1,
meaning that both models only minimize the total truck miles with the same demand in the
network. The base case parameters used in these experiments are listed in Table 1.
Table 1. Parameter Settings for DCAM and DCFM Comparison
Parameter name Parameter value
Daily horizon 12 hours
Time interval 1 hour
Transloading station capacity 20 units of containers
Port Capacity 1500 units of containers
Importer demand 65 units of containers
Exporter demand 55 units of containers

35

In the base case experiment, DCFM outperforms DCAM by reducing the total number of
truck miles from 2252.5 miles to 1913.7 miles, a 14.7 percent reduction. From this base case
test, we conduct three groups of experiments with one of the following parameters altered while
fixing the two other parameters: the capacity of the transloading stations, the demand of the
importers, and the demand of the exporters. To test the effect of changing the capacity, ten runs
are conducted with the capacity increasing from 16 to 25 in increments of one. Then, in the
second series, the importer demand is increased from 55 to 100 in increments of five. In the
third series, the exporter demand is increased from 45 to 80 also in increments of five. The
model costs are compared using the cost ratio of DCAM over DCFM. The average ratios for
each series of experiments are given in Table 2. The result shows that DCFM performs around
16 percent better than DCAM, which shows that allowing pickup before dropping off both
containers can reduce the total truck miles.
Table 2. Average Ratio of Model Cost

Transloading Station Capacity Importer Demand Exporter Demand
Avg. Ratio 1.165 1.175 1.142

3.4.2 Impact of Incorporating Flexibility in Pickups
The second series of experiments investigates the benefits of including second-day
stochastic demand into DCFM. One of the benefits of using a rolling horizon-based framework is
that it can incorporate different scenario-based demand forecasting approaches. In the
experiments, we use a Markov model to estimate future demand. The port has three typical
scenarios: (1) the number of import containers arriving is equal to the sum of export containers
and empty containers leaving the port (referred to as balanced state), (2) the number of import
containers arriving is larger than the sum of export containers and empty containers leaving the
port (referred to as import dominant state), and (3) the number of import containers arriving is

36

less than the sum of export containers and empty containers leaving the port (referred to as
export dominant state). We assume the three states follow a Markov chain with specific
transitional probabilities. The demand at each location follows a distribution based on the port
state. Based on yesterday and current port states, the port states of tomorrow follow a certain
probability distribution.
The transitional probabilities and demand distributions can be found in Dessouky et al.
(2020). The data contains three different transitional probability and demand probability
distributions where distribution 1 has the smallest variance, and distribution 3 has the highest
variance. Other parameters used in the experiments can be found in Table 3. We assume that
using one additional truck is much more costly than detours with existing trucks (𝜆 =0.01 in
these experiments). A 10-day problem is simulated with the first two days of the study horizon
assumed to be state one (i.e., balanced state). Since SDCFM is a two-day formulation covering
the first-day deterministic demand and second-day stochastic demand, we solve the 10-day
problem as a rolling horizon problem. The first problem assumes that today is day one and
tomorrow is day two. The solution being recorded is only today’s container movements. Then,
we repeat SDCFM with day two as today and day three as tomorrow, and then this continues
until we have found a solution for all ten days. Figure 10 is the average model convergence plot
from 20 randomly sampled numerical experiments. It shows that both DCFM and SDCFM can
provide a near-optimal solution within 20 CPU minutes.

37

Table 3. Parameter Settings for DCFM and SDCFM Comparison
Parameter name Parameter value
Penalty cost (𝜑 ) 200 per unit
Daily horizon 12 hours
Time interval 15 minutes
Importer/Exporter capacity 20 units of containers
Depot capacity 25 units of containers
Port Capacity 1500 units of containers

Figure 10. Convergence Plot for DCFM and SDCFM

We compare the SDCFM solution with solving the deterministic version of the problem,
where each day is solved without considering tomorrow’s uncertain demand. We refer to this
solution as the DCFM solution. For each setting, we run five experiments. It is worth mentioning
that on the first day, all the containers are located at the port. Thus, it takes several days for the
model to warm up. As previously mentioned, SDCFM optimizes two days’ container movements
while only the first day’s solution is recorded. Thus, we compare the two solutions over the last
five days of the simulation. The results are compared by using the ratio of the DCFM solution

38

over the SDCFM solution. Table 4 shows the average ratio of the last five days' cost (C5) and the
total number of trucks used in the last five days (T5).
Table 4. Average of the Ratios for DCFM and SDCFM Comparison
Transitional Probability 1 Transitional Probability 2 Transitional Probability 3
C5 T5 C5 T5 C5 T5
Demand Distribution 1 1.053 1.131 1.053 1.134 1.095 1.129
Demand Distribution 2 1.102 1.147 1.099 1.144 1.091 1.128
Demand Distribution 3 1.047 1.166 1.121 1.163 1.109 1.152

As shown in Table 4, SDCFM performs better than DCFM in terms of both C5 and T5
under all scenarios. In particular, by incorporating the future demand, the model reduces the
average five-day cost by 8.3 percent on average and reduces the total number of trucks used in
the last five days was 14.3 percent on average. The results show that a larger variance in the
demand distribution will reduce the benefits of incorporating future demand. Table 5 shows the
mean and variance of the daily used trucks in each model during the last five days. In Table 5,
the variance of the daily used truck numbers increases as the variabilities increase in the demand
and transitional probability distributions. From the results, when there is a high variance in
demand, both models perform less effectively.
Table 5. Truck Information for DCFM and SDCFM
Transitional Probability 1 Transitional Probability 2 Transitional Probability 3
DCFM SDCFM DCFM SDCFM DCFM SDCFM
Mean Var Mean Var Mean Var Mean Var Mean Var Mean Var
Demand Distribution 1 49.9 2.1 44.1 2.2 49.8 2.2 43.9 2.2 50.4 2.0 44.7 2.9
Demand Distribution 2 49.6 2.1 43.2 2.3 49.6 2.2 43.3 2.2 50.1 2.5 44.4 3.4
Demand Distribution 3 50.5 3.3 43.3 2.6 50.4 3.3 43.3 2.5 50.9 2.9 44.2 4.6

We next compare both DCFM and SDCFM against a lower-bound solution. The lower
bound solution assumes that there is perfect information about the demand over the ten days. In

39

this case, we assume the sampled demand over the ten days is known a priori, and model DCFM
is run over a 10-day horizon instead of a single day. Since, in this model, we have a significantly
large number of integer variables, it is difficult to find an optimal solution. However, we use the
linear programming relaxation of this formulation as the lower bound. In this case, the total cost
for the ten days is compared using the ratio of either SDCFM or the DCFM results over the
linear program relaxation of the perfect information model result. Table 6 and Table 7 show the
results of the average ratios and the standard deviation of the ratios, respectively.
Table 6. Average of the Ratios v.s. Perfect Information Condition
Transitional Probability 1 Transitional Probability 2 Transitional Probability 3
DCFM SDCFM DCFM SDCFM DCFM SDCFM
Demand Distribution 1 1.063 1.010 1.064 1.011 1.064 1.015
Demand Distribution 2 1.060 1.009 1.061 1.008 1.057 1.010
Demand Distribution 3 1.068 1.004 1.069 1.006 1.061 1.010

Table 7. Standard Deviation of the Ratios v.s. Perfect Information Condition
Transitional Probability 1 Transitional Probability 2 Transitional Probability 3
DCFM SDCFM DCFM SDCFM DCFM SDCFM
Demand Distribution 1 0.006 0.009 0.004 0.006 0.005 0.012
Demand Distribution 2 0.009 0.016 0.008 0.013 0.006 0.015
Demand Distribution 3 0.010 0.024 0.010 0.022 0.015 0.023

As seen in Table 6, the SDCFM performs close to the linear programming relaxation. In
Table 7, the results show that SDCFM has a slightly higher standard deviation than DCFM
because DCFM focuses on minimizing today’s cost with all demand satisfied. Therefore, the
model will not consider the container movements for future pickup and delivery. However,
strategic placements are made today by SDCFM to minimize tomorrow’s container movement in
advance, which increases the standard deviations of the solution from SDCFM slightly. Such a
minor increase is caused by SDCFM’s possible redundant container movements when future

40

demand is considered. In the short term, SDCFM may perform worse than DCFM, especially in
the warm-up phase. Nevertheless, in the long run, SDCFM outperforms DCFM in these
experiments.
In Table 7, the results also show that when the demand distributions have high standard
deviations, the ratios of both models increase. As the demand distribution varies, both DCFM
and SDCFM have higher standard deviations. Since DCFM only focuses on one-day container
movements, demand variations have less impact on DCFM than SDCFM, the performance of
which depends on the prediction accuracy of the next day’s demand. A demand distribution with
high variance reduces the accuracy of prediction. In this case, the strategic placement of
containers may be incorrect which worsens the model’s performance in this case. However, in
general, solutions from SDCFM have a smaller average total cost but slightly larger standard
deviation than those from DCFM. The DCFM generates the same solution regardless of
tomorrow’s demand, while the SDCFM considers the distribution of tomorrow’s demand in
generating today’s solution. We also conduct a multi-way ANOVA analysis considering effects
on models, demand distributions, transitional probabilities, and their interactions. The results are
given in Table 8 and only the model effect is significant (F(1) = 398.2, p < .001), which supports
our conclusion that SDCFM can improve DCFM by incorporating the future demand.
Table 8. Multi-way ANOVA Analysis Results
Df SS MS F Value Pr(>F)

Model 1 0.06479 0.06479 398.231 <2e-16
***
Demand Distribution 2 0.00021 0.00010 0.637 0.532

Transitional Probability 2 0.00000 0.00000 0.007 0.993

Model x Demand Distribution 2 0.00034 0.00017 1.057 0.352

Model x Transitional Probability 2 0.00026 0.00013 0.794 0.456

Residuals 80 0.01302 0.00016

41

Chapter 4: Freight Routing and Touring with Network Dynamics
This chapter presents a freight routing and routing problem on a dynamic transportation
network. We first present a MIP model for the problem. A co-simulation optimization based
load-balancing with touring approach is proposed to solve the problem. At the end of this
chapter, we show the computational results of the LBT approach and explore insights from the
results.
4.1 Problem Description
This chapter focuses on FRTP with predetermined supply and demand on a dynamic
transportation network. Each supplier has a given number of containers to be transported to
warehouses at the start of each day. Regional traffic congestion is likely to arise if container
movements are not appropriately managed by considering traffic conditions and the impact such
movements have on traffic. Therefore, a central coordinator is proposed to provide truck
scheduling and routing information to drivers to avoid the possibility of traffic congestion as a
result of routing trucks on overlapping routes that appear to be of lower travel time initially. The
goal of the central coordinator is to minimize the total number of trucks to satisfy the demand
and reduce the truck traveling cost via balancing the truck flows on the transportation network
over a day by taking into account the possible impact of the routed trucks on traffic flow states.
Here are some terminologies used in this study: (1) trip is defined as a truck going from
one location to another; (2) truck routing represents the routing decision (which road segments to
travel on) for a trip; (3) truck touring represents the sequence of trips for trucks in the study
horizon; (4) delivery flows are the aggregated truck trips going from suppliers to warehouses
carrying containers; (5) pickup flows are the aggregated truck trips without carrying containers.

42

4.2 Mixed Integer Programming Model
Traditionally, the pickup and delivery problem is modeled with vehicles as the main
decision variables (Savelsbergh & Sol, 1995). However, we use a flow-based formulation instead
because, in practice, there are thousands of trucks on the road for daily container movements,
and depending on the demand and transportation network conditions, the total number of trucks
for daily operations can be significant. The problem is defined on a transportation network 𝐺 =
(𝑁 ,𝐴 ) where the set 𝑁 and 𝐴 include all the nodes and directed arcs in the network, respectively.
The nodes include the suppliers, the warehouses, the truck depot, and the intersections in the
transportation network. The arcs represent the road sections in the transportation network. The
demand (𝑑 𝑖 ,𝑗 ) from location 𝑖 to 𝑗 is predetermined, which needs to be satisfied by the end of the
day. We assume that all the trucks start from the depot (with the location index 0) and return to
the depot by the end of the day. In addition, we discretize the study horizon into |𝐾 | intervals.
The rest of the notation used in the rest of the chapter is as follows:
𝑖 ,𝑗 The index of the node set, 𝑖 ,𝑗 ∈𝑁 and the index of the truck depot is 0;
𝑎 The index of the arcs set, 𝑎 ∈𝐴 ;
𝑘 The index of the time interval, 𝑘 ∈𝐾 ;
𝑑 𝑖 ,𝑗 The demand from location 𝑖 to location 𝑗 in the number of containers, 𝑖 ,𝑗 ∈𝑁 ;
𝑅 𝑖 ,𝑗 The candidate route set for location 𝑖 to 𝑗 with index 𝑟 ;
𝑥 𝑖 ,𝑗 ,𝑘 𝑟 The delivery flow from location 𝑖 to location 𝑗 using route 𝑟 leaving at time 𝑘 ;
𝑦 𝑖 ,𝑗 ,𝑘 𝑟 The pickup flow from location 𝑖 to location 𝑗 using route 𝑟 leaving at time 𝑘 ;
𝑐 𝑖 ,𝑗 ,𝑘 𝑟 The travel cost from location 𝑖 to 𝑗 using route 𝑟 leaving at time 𝑘 , 𝑟 ∈𝑅 𝑖 ,𝑗 ;
𝜆 The weighting factor for the travel cost and the truck cost;

43

𝑝 𝑖 ,𝑘 The delivery flow leaving location 𝑖 at time 𝑘 ;
𝑞 𝑗 ,𝑘 The cumulative delivery flow that has arrived at location 𝑗 by time 𝑘 ;
The container pickup and delivery problem (MP) can be formulated as,
minimize
𝑥 𝑖 ,𝑗 ,𝑘 𝑟 ,𝑦 𝑗 ,𝑖 ,𝑘 𝑟 𝑇𝑟𝑢𝑐𝑘𝐶𝑜𝑠𝑡 +𝑇𝑟𝑎𝑣𝑒𝑙𝐶𝑜𝑠𝑡
=(𝜆 ∑∑ ∑ 𝑦 0,𝑖 ,𝑘 𝑟 𝑟 ∈𝑅 0,𝑖 𝑘 ∈𝐾 𝑖 ∈𝑁 )+(∑∑∑ ∑ 𝑥 𝑖 ,𝑗 ,𝑘 𝑟 𝑟 ∈𝑅 𝑖 ,𝑗 𝑘 ∈𝐾 𝑗 ∈𝑁 𝑖 ∈𝑁 ⋅𝑐 𝑖 ,𝑗 ,𝑘 𝑟 +∑∑∑ ∑ 𝑦 𝑗 ,𝑖 ,𝑘 𝑟 𝑟 ∈𝑅 𝑗 ,𝑖 𝑘 ∈𝐾 𝑖 ∈𝑁 𝑗 ∈𝑁 ⋅𝑐 𝑗 ,𝑖 ,𝑘 𝑟 ) (38)

Subject to
𝑑 𝑖 ,𝑗 = ∑ ∑ 𝑥 𝑖 ,𝑗 ,𝑘 𝑟 𝑟 ∈𝑅 𝑖 ,𝑗 𝑘 ∈𝐾 ∀𝑖 ∈𝑁 ,𝑗 ∈𝑁 (39)

𝑝 𝑖 ,𝑘 =∑ ∑ 𝑥 𝑖 ,𝑗 ,𝑘 𝑟 𝑟 ∈𝑅 𝑖 ,𝑗 𝑗 ∈𝑁 ∀𝑖 ∈𝑁 \{0},𝑘 ∈𝐾 (40)

𝑞 𝑗 ,𝑘 =∑ ∑ ∑𝑥 𝑖 ,𝑗 ,𝜏 𝑟 ⋅𝜙 𝑖 ,𝑗 ,𝜏 ,𝑘 𝑟 𝜏 ≤𝑘 𝑟 ∈𝑅 𝑖 ,𝑗 𝑖 ∈𝑁
∀𝑗 ∈𝑁 \{0},𝑘 ∈𝐾 (41)

𝑞 𝑗 ,𝑘 ≥∑∑ ∑ y
𝑗 ,𝑖 ,𝜏 𝑟 𝑟 ∈𝑅 𝑗 ,𝑖 𝑖 ∈𝑁 𝜏 ≤𝑘 ∀𝑗 ∈𝑁 \{0},𝑘 ∈𝐾 (42)

∑𝑝 𝑖 ,𝜏 𝜏 ≤𝑘 ≤∑ ∑ ∑y
𝑗 ,𝑖 ,𝜏 𝑟 𝜏 ≤𝑘 𝑟 ∈𝑅 𝑗 ,𝑖 ⋅𝜙̅

𝑗 ,𝑖 ,𝜏 ,𝑘 𝑟 𝑗 ∈𝑁 ∀𝑖 ∈𝑁 \{0},𝑘 ∈𝐾 (43)

∑ ∑ ∑ 𝑦 𝑗 ,0,𝑘 𝑟 𝑟 ∈𝑅 𝑗 ,0
𝑘 ∈𝐾 𝑗 ∈𝑁 \{0}
= ∑ ∑ ∑ 𝑦 0,𝑖 ,𝑘 𝑟 𝑟 ∈𝑅 0,𝑖 𝑘 ∈𝐾 𝑖 ∈𝑁 \{0}
(44)

𝑥 𝑖 ,𝑗 ,𝑘 𝑟 ∈𝑍 0+
∀𝑖 ∈𝑁 ,𝑗 ∈𝑁 ,𝑘 ∈𝐾 ,𝑟 ∈𝑅 𝑖 ,𝑗 (45)

𝑦 𝑖 ,𝑗 ,𝑘 𝑟 ∈𝑍 0+
∀𝑖 ∈𝑁 ,𝑗 ∈𝑁 ,𝑘 ∈𝐾 ,𝑟 ∈𝑅 𝑖 ,𝑗 (46)

The objective function (38) minimizes the total cost for the pickup and delivery process.
The first term is the cost of employing a new truck to the system, and the second term is the
truck movement cost in the transportation network. Constraint (39) indicates that the demand for

44

each OD pair needs to be satisfied in the study horizon. Constraint (40) calculates the total
delivery flows leaving location 𝑖 at time 𝑘 . Constraint (41) captures the relationship between the
cumulative arrival of trucks at location 𝑗 until time 𝑘 , where 𝜙 𝑖 ,𝑗 ,𝜏 ,𝑘 𝑟 is a binary indicator. Since
the truck flow going from location 𝑖 to 𝑗 leaving at time 𝜏 has dynamic travel time based on the
transportation network states, we use the binary indicator to capture the flow arrival and service
condition: 𝜙 𝑖 ,𝑗 ,𝜏 ,𝑘 𝑟 =1 if and only if the truck from 𝑖 to 𝑗 leaving at time 𝜏 with route 𝑟 is
available for another delivery task at time 𝑘 . Constraint (42) states that the number of pickup
trips originating from location 𝑗 cannot exceed the number of delivery trips to their destination.
Constraint (43) means the total number of delivery trips starting from location 𝑖 has to be less or
equal to the total number of pickup trips to their destination, where 𝜙̅

𝑗 ,𝑖 ,𝜏 ,𝑘 𝑟 is a binary indicator
for flow y
𝑗 ,𝑖 ,𝑘 𝑟 similar to 𝜙 𝑖 ,𝑗 ,𝜏 ,𝑘 𝑟 . If the pickup flows from non-depot locations cannot provide
enough trucks, additional trucks will be employed into the system starting from the truck depot.
Constraint (44) is the flow conservation constraint at the depot. Constraints (45−46) are
domain constraints for the problem, which sets all the decision variables to be non-negative
integers.
In constraints (41) and (43) , the binary indicators 𝜙 𝑖 ,𝑗 ,𝜏 ,𝑘 𝑟 and 𝜙̅

𝑗 ,𝑖 ,𝜏 ,𝑘 𝑟 are hard to describe
due to the network dynamics. For example, if there is only one truck going from location 𝑖 to 𝑗
leaving at time 𝜏 using route 𝑟 , then it can finish the delivery at time 𝜏 +1. However, if there are
hundreds of trucks using the same route, then it is likely to take a much longer time for the
delivery process, and that is why we need to consider the dynamic network states to be generated

45

by the simulator. In addition, the travel cost 𝑐 𝑖 ,𝑗 ,𝑘 𝑟 is associated with travel time on route 𝑟 from 𝑖
to 𝑗 , which is load-dependent. Therefore, we need to estimate the travel time in the network.
We introduce additional notation to describe the transportation network states: (1) 𝑣 𝑎 ,𝑘 is
the traffic volume on arc 𝑎 at time 𝑘 , (2) 𝑡 𝑎 ,𝑘 is the travel time on arc 𝑎 at time 𝑘 , and (3) 𝑋 𝑘 ,𝑌 𝑘 ,
and 𝑍 𝑘 are the delivery flow information, the pickup flow information, and the passenger traffic
information in the network by time 𝑘 , respectively (i.e., 𝑋 𝑘 =[𝑥 𝑖 ,𝑗 ,𝜏 𝑟 :∀𝑖 ∈𝑁 ,𝑗 ∈𝑁 ,𝑟 ∈𝑅 𝑖 ,𝑗 ,𝜏 ≤
𝑘 ]). The travel volume on an arc 𝑎 at time 𝑘 +1 can be expressed as a nonlinear function of the
traffic volume at time 𝑘 , the traffic information in the transportation network by time 𝑘 +1, and
the flow on arc 𝑎 at time 𝑘 +1 can be estimated as follows,
𝑣 𝑎 ,𝑘 +1
=𝑓 𝑎 (𝑣 𝑎 ,𝑘 ,𝑋 𝑘 +1
,𝑌 𝑘 +1
,𝑍 𝑘 +1
) ∀𝑎 ∈𝐴 ,𝑘 ∈𝐾 \{|𝐾 |} (47)
Let 𝑉 𝑘 =(𝑣 1,𝑘 ,𝑣 2,𝑘 ,…𝑣 |𝐴 |,𝑘 ) be the traffic volume information in the transportation
network at time 𝑘 . Because the arc travel time is determined by the flows on interdependent arcs
in the network, once the traffic volume in the transportation network is known, the travel time on
arc 𝑎 at time 𝑘 can be expressed as the following function:
𝑡 𝑎 ,𝑘 =𝑔 𝑎 (𝑉 𝑘 ) ∀𝑎 ∈𝐴 ,𝑘 ∈𝐾 (48)

With (48) , the travel time from location 𝑖 to location 𝑗 using route 𝑟 can be expressed as
the summation of the travel time on each arc. Suppose a route 𝑟 from location 𝑖 to 𝑗 consists of
arcs 𝑎 1
,𝑎 2
,…,𝑎 𝑛 , the travel time on the route leaving at time 𝑘 can be written as,
𝛿 𝑖 ,𝑗 ,𝑘 𝑟 =∑𝑡 𝑎 𝑢 ,𝜃 𝑢 n
𝑢 =1
(49)
where 𝜃 𝑢 is the entry time of the flow on arc 𝑎 𝑢 , and therefore,
𝜃 1
=𝑘 (50)
𝜃 𝑢 +1
=𝜃 𝑢 +𝑡 𝑎 𝑢 ,𝜃 𝑢 ∀𝑢 ∈{1,2,…,𝑛 −1} (51)

46

If we can estimate the travel time in the transportation network for every pair of ODs on
the different routes, we can calculate the binary indicators 𝜙 𝑖 ,𝑗 ,𝜏 ,𝑘 𝑟 and 𝜙̅

𝑗 ,𝑖 ,𝜏 ,𝑘 𝑟 in the following
way,
𝜙 𝑖 ,𝑗 ,𝜏 ,𝑘 𝑟 ={
1 𝑖𝑓 𝛿 𝑖 ,𝑗 ,τ
𝑟 ≤𝑘 −𝜏 −𝛽 𝑗 0 𝑂𝑡 ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(52)
𝜙̅

𝑗 ,𝑖 ,𝜏 ,𝑘 𝑟 ={
1 𝑖𝑓 𝛿 𝑗 ,𝑖 ,τ
𝑟 ≤𝑘 −𝜏 −𝛼 𝑖 0 𝑂𝑡 ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(53)
where 𝛼 𝑖 is the loading service time at location 𝑖 and 𝛽 𝑗 is the unloading service time at location
𝑗 .
We assume that the travel cost 𝑐 𝑖 ,𝑗 ,𝑘 𝑟 is a strictly monotonic function that can be expressed
in the following way,
𝑐 𝑖 ,𝑗 ,𝑘 𝑟 =ℎ(𝛿 𝑖 ,𝑗 ,𝑘 𝑟 ) ∀𝑖 ∈𝑁 ,𝑗 ∈𝑁 ,𝑟 ∈𝑅 𝑖 ,𝑗 ,𝑘 ∈𝐾 (54)
However, it is challenging to explicitly express the functions in constraints (47−51)
due to the complexity of the transportation network and the nonlinear relationship between the
traffic flows and the network conditions. Therefore, instead of using analytical expressions for
these functions, we use simulation models to approximate the transportation network states by
running them simultaneously with the optimization approach.
4.3 Solution Methodology
The FRTP can be decomposed into two subproblems: (1) Finding the least-cost flows for
a DTA problem and (2) determining the touring sequence for trucks during the study horizon.
We refer to the first subproblem as SP1 and the second subproblem as SP2. In the first
subproblem, we only solve the delivery flows (𝑥 𝑖 ,𝑗 ,𝑘 𝑟 ) in the FRTP. Then, based on the delivery
flow solution, we find the best pickup flow solution (𝑦 𝑖 ,𝑗 ,𝑘 𝑟 ) that can support the delivery flow
solution. In this case, the pickup flow solution can be understood as the best response to the

47

delivery flow solution. By combining delivery flows with pickup flows, we can construct truck
touring decisions and provide an efficient solution for the regional freight management problem.
The detailed formulations of the two subproblems are discussed in the following section.
To solve the FRTP, we introduce the following solution framework consisting of initial
solution generation, load balancing process, traffic simulation, pickup flow optimization, and
network cost evaluation, which is shown in Figure 11. In the solution framework, we start with
an initial delivery flow solution to the problem. For example, the initial solution can be
constructed by a greedy approach. Each truck can carry one unit of demand between an OD pair
using the shortest route at the beginning of the day. With the initial solution, the load-balancing
process can start distributing demand across the network over the study horizon. By solving SP1,
the delivery flow solution and network state approximation can be generated by the load
balancing process. Taking the load-balancing process results as data inputs, the pickup flow
optimization model is used to compute the best pickup flow solution given the delivery flows.
However, these pickup flows will affect the transportation network states. As a result, we have to
check the pickup flow solution feasibility after updating the network states with both delivery
and pickup flows. Finally, the solution framework terminates with some heuristic stopping
criteria. We record the best solution found by the iterative process as the final solution.

48

Figure 11. The Master Optimization Framework

The framework provides an overall procedure for our LBT approach, and each
component is explained as follows:
Initial Solution Generation. The initial solution is generated by assigning one container
to one truck. For example, if a truck is assigned to deliver one container from location 𝑖 to
location 𝑗 , the touring sequence for the truck is 0
𝑟 1
→𝑖 𝑟 2
→𝑗 𝑟 3
→0, where 𝑟 1
, 𝑟 2
, and 𝑟 3
can be any
route from the truck depot to location 𝑖 , from location 𝑖 to location 𝑗 , and from location 𝑗 to the
depot, respectively. The initial solution is generated by aggregating the truck trips on every route
for each OD pair.
Load-Balancing Process. The load-balancing process has two components: the traffic
simulator and the modified load-balancing algorithm. The traffic simulator is used to generate
estimates of the transportation network states. The load-balancing algorithm proposed by Zhao et
al. (2018) iteratively optimizes the delivery flow in the network by minimizing the marginal cost
for delivery flows between each pair of ODs. Instead of investigating every possible route in the
transportation network, the load balancing algorithm deals with a higher level network named the
service network, which only includes important nodes in the transportation network such as
major intersections, interchanges, highway entrances and exists. In each iteration, the load-

49

balancing algorithm searches for routes with lower costs for all pairs of ODs in the service
network. If cheaper cost routes exist, these routes are considered as candidate routes in the next
iteration. As a result, the algorithm reduces the traffic flows on expensive routes and increases
the flows on cheaper routes. In this case, the algorithm terminates when all the routes for each
OD pair have the same cost. Taking the initial solution as an example, because the traffic flows
enter the transportation network simultaneously, it is likely that the network will be congested,
which generates a high system cost. The load-balancing algorithm balances the truck flows over
the study horizon repeatedly to reduce traffic congestion. The original load-balancing algorithm
does not provide integer solutions since the solution can also be understood as the proportion of
the demand for the OD pairs. However, when considering pickup and delivery problems, integer
solutions are required to find the best pickup solution. Therefore, the load-balancing algorithm of
Zhao et al. (2018) is modified by rounding the step size to integers to generate integer solutions.
The previously developed load-balancing procedure does not consider truck tours. In other
words, it is limited to the delivery flows for the DTA problem calculated without reusing trucks
to make additional deliveries. Therefore, we further optimize pickup flows that can support the
delivery flows to reuse the trucks in the network to generate complete truck tours.
Pickup Flow Optimization. In this step, we solve the touring decision problem, which
aims to construct minimum cost pickup flows for the delivery flow solution generated by the
load balancing process. The intuition of the procedure is to connect delivery flows with feasible
pickup flows. Suppose the load balancing process gives two delivery flows 𝑥 ̅
𝑖 1
,𝑗 1
𝑘 1
𝑟 1
and 𝑥 ̅
𝑖 2
,𝑗 2
𝑘 2
𝑟 2

and there exists a route 𝑟 ̅∈𝑅 𝑗 1
,𝑖 2
such that,
𝑘 1
+𝛿 𝑖 1
,𝑗 1
,𝑘 1
𝑟 1
+𝛿 𝑗 1
,𝑖 2
,𝑘 1
+𝛿 𝑖 1
,𝑗 1
,𝑘 1
𝑟 1
𝑟 ̅
≤𝑘 2
(55)

50

We can introduce a pickup flow 𝑦 𝑗 1
,𝑖 2
,𝑘 1
+𝛿 𝑖 1
,𝑗 1
,𝑘 1
𝑟 1
𝑟 ̅
to connect delivery flows 𝑥 ̅
𝑖 1
,𝑗 1
𝑘 1
𝑟 1
and
𝑥 ̅
𝑖 2
,𝑗 2
𝑘 2
𝑟 2
with fewer trucks employed in the system. Figure 12 provides a simple example of
connecting two delivery trips with a pickup trip (trip 3) in this process. The touring sequence 𝑖 1
𝑟 1
→𝑗 1
𝑟 ̅
→𝑖 2
𝑟 2
→𝑗 2
could be a part of the tour for a truck during the study horizon.

Figure 12. Example of Connecting Delivery Trips with Pickup Trips

In the pickup flow optimization, we not only minimize the travel time for pickup flows
but also minimize the total number of trucks for the transportation system. Details of the model
will be discussed in the next section. The pickup flows will change the transportation network
states; therefore, we need to update the network states using the traffic simulation model.
Traffic Simulation. The purpose of the simulator is to estimate the traffic network states
by taking into account the complex dynamics of traffic flow that the traditional simple
mathematical models often used cannot accurately capture. In addition, it takes into account the
impact of the routing decisions on traffic flow generated by the pickup flow optimization and

51

modified load balancing algorithm and updates the traffic network states to be subsequently used
for the next iteration of optimization and load balancing.
Feasibility Check. Because changing the pickup flows will affect the travel time in the
transportation network, the current pickup flow solution may violate the updated travel times.
Therefore, we have to validate the solution feasibility before checking the model convergence. If
the current pickup flow solution is not feasible, we re-optimize pickup flows with the updated
network states.
Stopping Criteria. The solution procedure stops when the difference in the system cost
between two iterations is less than a threshold or when the maximum running time Τ
𝑐𝑎𝑝 is
reached.
4.4 Subproblem Formulations
4.4.1 Delivery Flow Optimization
In the first subproblem, we solve a DTA problem (SP1),
minimize
𝑥 𝑖 ,𝑗 ,𝑘 𝑟 ∑∑∑ ∑ 𝑥 𝑖 ,𝑗 ,𝑘 𝑟 𝑟 ∈𝑅 𝑖 ,𝑗 𝑘 ∈𝐾 𝑗 ∈𝐽 𝑖 ∈𝐼 ⋅𝑐 𝑖 ,𝑗 ,𝑘 𝑟 (56)

Subject to constraints (47−51) , (54) and
∑ ∑ 𝑥 𝑖 ,𝑗 ,𝑘 𝑟 𝑟 ∈𝑅 𝑖 ,𝑗 𝑘 ∈𝐾 =𝑑 𝑖 ,𝑗 ∀𝑖 ∈𝐼 ,𝑗 ∈𝐽 (57)

∑∑ ∑ ∑𝑥 𝑖 ,𝑗 ,𝑘 𝑟 𝑘 ≤𝜗 𝑟 ∈𝑅 𝑖 ,𝑗 𝑗 ∈𝐽 𝑖 ∈𝐼 =0 (58)

∑∑ ∑ ∑ 𝑥 𝑖 ,𝑗 ,𝑘 𝑟 𝑘 ≥|𝐾 |−2𝜗 𝑟 ∈𝑅 𝑖 ,𝑗 𝑗 ∈𝐽 𝑖 ∈𝐼 =0 (59)

𝑥 𝑖 ,𝑗 ,𝑘 𝑟 ∈𝑍 0+
∀𝑖 ∈𝐼 ,𝑗 ∈𝐽 ,𝑘 ∈𝐾 ,𝑟 ∈𝑅 𝑖 ,𝑗 (60)

52

The objective function minimizes the total travel cost for delivery flows in the
transportation network. Constraint (57) means that at the end of the day, all the demand needs to
be satisfied by the delivery flows. Constraint (58) prevents any delivery flow until time 𝜗 , where
𝜗 is the maximum travel time to go from the truck depot to any location since it takes time for a
truck to travel from the depot to the supplier and pick up the first container. Because the problem
is regional, we assume that 𝜗 ≪|𝐾 |. In this case, any truck starting from the depot at the
beginning of the day will be ready for its first delivery trip at time 𝜗 from a supplier to a
warehouse. Constraint (59) is similar to constraint (58) which ensures every truck has enough
time to go back to the depot after its final delivery task. Constraint (60) is the domain constraint
for the problem.
Note that in SP1, we only optimize the delivery flows in the transportation network. The
pickup flows are considered fixed and are only included in the traffic simulator to estimate the
network state. The least-cost delivery flows can be solved by the load-balancing process (Zhao et
al., 2018). Because the process minimizes the marginal cost between each OD pair, we can
represent travel costs (𝑐 𝑖 ,𝑗 ,𝑘 𝑟 ) on any route 𝑟 using the corresponding marginal cost 𝑐 ̅
𝑖 ,𝑗 ,𝑘 without
the route index. As a result, the travel time will be the same on all the routes for each pair of
ODs.
4.4.2 Pickup Flow Optimization
The delivery flow solution generated by the load balancing process has one important
property. Between a pair of OD, the travel times on the different selected routes are the same due
to two facts: (1) the load balancing process minimizes marginal costs for all OD pairs, and (2)
the traffic simulator provides equilibrium solutions. Therefore, we aggregate truck flows on
different routes for all the OD pairs. Unlike the delivery flows that only go from supplier to

53

warehouses, there are three types of aggregated pickup flows in the network: (1) 𝑦̃
0,𝑖 ,𝑘 : the flow
that starts from the depot at time 𝑘 and goes to supplier 𝑖 ; (2) 𝑦̃
𝑗 ,𝑖 ,𝑘 : the flow that goes from
warehouse 𝑗 to supplier 𝑖 leaving at time 𝑘 ; (3) 𝑦̃
𝑗 ,0,𝑘 : the flow that returns to the depot from
location 𝑗 at time 𝑘 .
Additional notation is introduced as follows,
𝑐 ̅
𝑖 ,𝑗 ,𝑘 The travel cost from location 𝑖 to 𝑗 leaving at time 𝑘 ;
𝛿 ̅
𝑖 ,𝑗 ,𝑘 The travel time from location 𝑖 to 𝑗 leaving at time 𝑘 ;
𝑥 ̅
𝑖 ,𝑗 ,𝑘 𝑟 The current delivery flow from location 𝑖 to 𝑗 leaving at time 𝑘 .
Given the delivery flow solution (𝑥 ̅
𝑖 ,𝑗 ,𝑘 𝑟 ) from the load-balancing process and the network
states (𝑐 ̅
𝑖 ,𝑗 ,𝑘 𝑎𝑛𝑑 𝛿 ̅
𝑖 ,𝑗 ,𝑘 ) approximated by the traffic simulation model, we formulate SP2 as
follows:
minimize
𝑦̃
0,𝑖 ,𝑘 ,𝑦̃
𝑖 ,𝑗 ,𝑘 ,𝑦̃
𝑗 ,0,𝑘 ∑∑(𝜆 +𝑐 ̅
0,𝑖 ,𝑘 )𝑦̃
0,𝑖 ,𝑘 𝑘 ∈𝐾 𝑖 ∈𝐼 +∑∑∑𝑦̃
𝑗 ,𝑖 ,𝑘 𝑘 ∈𝐾 𝑖 ∈𝐼 𝑗 ∈𝐽 ⋅𝑐 ̅
𝑗 ,𝑖 ,𝑘 + ∑∑𝑦̃
𝑗 ,0,𝑘 ⋅𝑐 ̅
𝑗 ,0,𝑘 𝑘 ∈𝐾 𝑗 ∈𝐽 (61)

Subject to
∑ ∑ ∑ 𝑥 ̅
𝑖 ,𝑗 ,𝜏 𝑟 𝜏 ≤𝑘 −𝛿̅
𝑖 ,𝑗 ,𝜏 −𝛽 𝑗 𝑟 ∈𝑅 𝑖 ,𝑗 𝑖 ∈𝐼 ≥∑∑𝑦̃
𝑗 ,𝑖 ,𝑘 𝑖 ∈𝐼 𝜏 ≤𝑘 ∀𝑗 ∈𝐽 ,𝑘 ∈𝐾 (62)

∑∑ ∑ 𝑥 ̅
𝑖 ,𝑗 ,𝑘 𝑟 𝑟 ∈𝑅 𝑖 ,𝑗 𝑗 ∈𝐽 𝜏 ≤𝑘 ≤ ∑ 𝑦̃
0,𝑖 ,𝜏 τ≤𝑘 −𝛿̅
0,𝑖 ,𝜏 +∑ ∑ 𝑦̃
𝑗 ,𝑖 ,𝑘 𝜏 ≤𝑘 −𝛿̅
𝑗 ,𝑖 ,𝑘 −𝛼 𝑖

𝑗 ∈𝐽 ∀𝑖 ∈𝐼 ,𝑘 ∈𝐾 (63)

∑∑𝑦̃
𝑗 ,0,𝑘 𝑘 ∈𝐾 𝑗 ∈𝐽 =∑∑𝑦̃
0,𝑖 ,𝑘 𝑘 ∈𝐾 𝑖 ∈𝐼 (64)

𝑦̃
𝑗 ,𝑖 ,𝑘 ∈𝑍 0+
∀𝑖 ∈𝐼 ,𝑗 ∈𝐽 ,𝑘 ∈𝐾 (65)

𝑦̃
0,𝑖 ,𝑘 ∈𝑍 0+
∀𝑖 ∈𝐼 ,𝑘 ∈𝐾 (66)
𝑦̃
𝑗 ,0,𝑘 ∈𝑍 0+
∀𝑗 ∈𝐽 ,𝑘 ∈𝐾 (67)

The objective function (61) minimizes the number of used trucks and the pickup flow
cost simultaneously. Constraints (62−64) are the flow conservation constraints for

54

warehouses, suppliers, and the depot. Constraints (65−67) are the domain constraints for the
pickup flows. We next show that the relaxation of the above formulation provides an integer
solution.
Proposition: Given the network states (𝑐 ̅
𝑖 ,𝑗 ,𝑘 𝑎𝑛𝑑 𝛿 ̅
𝑖 ,𝑗 ,𝑘 ) and delivery flows (𝑥 ̅
𝑖 ,𝑗 ,𝑘 𝑟 ) , there
exists a solution to the Linear Programming Relaxation (LPR) of SP2. In addition, with integer
delivery flows, solving the LPR of SP2 yields an integer solution.
Proof: See Appendix B ∎
By solving the LPR of the SP2, we have the aggregated pickup flows with minimum
costs for the current delivery flow solution. The traffic simulation model can be used to distribute
the aggregated flows onto different routes. However, due to the network dynamics caused by the
shifting pickup flows, the travel time and cost parameters also need to be updated. The current
pickup flow solution may not be feasible after we update the network states. Therefore, we need
to iteratively solve the pickup flow solutions until it meets the feasibility condition. In order to
make the model converge faster, we can use a larger time interval or extend the service time at
every location to have a wider time window.
4.5 Load-Balancing with Touring Algorithm
The optimization procedure for the MP is as follows:

Algorithm 1: LBT algorithm
Step 1: Set master iteration counter 𝜅 =0.
Step 2: Set 𝑦̃
0,𝑖 ,0
=∑ 𝑑 𝑖 ,𝑗 𝑗 ∈𝐽 ,∀𝑖 ∈𝐼 .
Step 3: Set 𝑦̃
𝑗 ,0,|𝐾 −𝜗 |
=∑ 𝑑 𝑖 ,𝑗 𝑖 ∈𝐼 ,∀𝑗 ∈𝐽 .
Step 4: Set the rest of the aggregated pickup flows to zero.

55

Step 5: Set 𝑥 𝑖 ,𝑗 ,𝑘 𝑟 = {
𝑑 𝑖 ,𝑗 𝑖𝑓 𝑟 =1 𝑎𝑛𝑑 𝑘 =ϑ
0 𝑜𝑡 ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ,
where 𝜗 is the maximum travel time from the truck depot to any location.
Step 6: Optimize the least-cost delivery flow 𝑥 𝑖 ,𝑗 ,𝑘 𝑟 and network states with the load-balancing
process.
Step 7: Solve the LPR of SP2 and update the aggregated pickup flows.
Step 8: Update the network cost using the traffic simulation model with the current aggregated
pickup flows and check pickup flow solution feasibility. If it is not feasible, go back to
step 7.
Step 9: 𝜅 =𝜅 +1.
Step 10: Check for the stopping criteria. Denote 𝑇 𝐶 𝜅 as the system cost for master iteration 𝜅 and
𝜖 as a predefined cost threshold. If at least one of the following criteria is satisfied, the
optimization procedure will terminate. Otherwise, go to step 6.
1. 𝜅 =Κ
2. |𝑇 𝐶 𝜅 −1
−𝑇 𝐶 𝜅 |≤𝜖

The LBT algorithm above uses the best-response strategy. The idea is that the delivery
flows and the pickup flows are optimized based on each other’s solutions. Steps 2-5 construct the
initial solution for the optimization procedure. Step 6 is the load balancing process, which solves
SP1. In this step, we treat the aggregated pickup flows as fixed parameters and use the traffic
simulation model to distribute the aggregated flows onto the transportation network. Step 7
solves the SP2, which generates the aggregated pickup flows in the demand network. In
comparison to Step 6, solving the LPR of SP2 takes less computational power. Therefore, before
reusing the load balancing algorithm, we check the pickup flow solution feasibility in Step 8.

56

Since the problem involves a large number of nonconvex dynamic functions, it is difficult to find
a global optimal solution. We use two heuristic rules to stop the procedure at an approximate
local minimum. The first is that the maximum running time (Τ
𝑐𝑎𝑝 ) is reached and the second is
that the differences of the system costs in two consecutive iterations are less than a threshold (𝜖 ) .
4.6 Computational Results
4.6.1 Experiment Setting
In this section, we show the effectiveness of the proposed approach to solve a freight
movement problem on a regional transportation network in the San Pedro Bay area (Figure 13).
In the figure, 𝑁𝑜𝑑𝑒 0 represents the truck depot; 𝑁𝑜𝑑𝑒𝑠 1−9 are the warehouse nodes; and
𝑁𝑜𝑑𝑒 10 and 𝑁𝑜𝑑𝑒 11 represent the Port of Los Angeles and Port of Long Beach respectively.
These nodes are connected by interstate highways (I-105, I-110, I-405, and I-710), state routes
(Route-1, Route-19, Route-47, and Route-91), and other local roads. We also cluster the demand
in the study area for the nine warehouses to approximate the demand distribution in the region.
The demand data are obtained from a survey used in the study by Giuliano et al. (2021), which
investigated the container movements in the study area. We use the data as the basis for the
spatial distribution of the demand in our study. Based on the port statistics, the demand in
Giuliano et al. (2021) is scaled up to match the entire demand
1
. We test scenarios with the total
demand from 13000 to 18000 units of containers with a step size of 1000.

1
The demand data and service network can be found in this link: https://bit.ly/39HR7Jo

57

Figure 13. Study Area

At the beginning of the day, all the demand at each location is known. All the trucks start
from the depot and return to the depot by the end of the day. We assume that the travel cost for a
trip is proportional to the travel time,
𝑐 𝑖 ,𝑗 ,𝑘 𝑟 =𝜌 ∗𝛿 𝑖 ,𝑗 ,𝑘 𝑟 ∀𝑖 ∈𝑁 ,𝑗 ∈𝑁 ,𝑟 ∈𝑅 𝑖 ,𝑗 ,𝑘 ∈𝐾 (68)

In the experiments, we use heuristic stopping criteria to terminate the solution procedure:
(1) the maximum running time (Τ
𝑐𝑎𝑝 ) is reached, and (2) the difference of the system costs
between two iterations is smaller than a threshold (𝜖 ) . Since the previously developed load-
balancing algorithm (Y. Zhao, Ioannou, et al., 2018) takes around eight hours to stop, we set the
maximum running time to eight hours in this study. Most of the parameters are taken from
(Giuliano et al., 2021; Y. Zhao, Ioannou, et al., 2018) and are adjusted based on our dataset. The
other parameters are given in Table 9.

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Table 9. Experiment Parameters
Parameter name Parameter value
Daily horizon 10 hours
Time interval 15 minutes
Ports' service time 1 hour
Warehouses' service time 30 minutes
Weighting factor $50/truck
Weighting factor $25/hour
Maximum travel time 1 hour
Stopping threshold $100
Maximum running time 8 hours

4.6.2 Testing Platform
A macroscopic traffic simulation software VISUM is used to model the traffic flow and
predict the transportation network states. The simulation model captures transportation network
parameters such as intersections, road sections, road types, lane numbers, and speed limits. The
data input to the traffic simulator includes the passenger traffic and truck flows represented by
the OD matrices. The traffic simulator generates the network states, such as travel time and
traffic volume, on every route between every pair of ODs. In our experiments, the passenger
traffic data used by the simulator are tuned based on Google Maps travel times.
After building the simulation environment, we develop the LBT algorithm using Python.
Gurobi 9.1.2 is used to solve the LPR of the pickup flow optimization problem. Using the
VISUM COM API, the data flows are transmitted between the traffic simulator and Python
scripts. Both the simulation model and Python scripts are run on a virtual machine with an 8-core
3.70 GHz CPU and 16 GB of memory. In this study, the transportation network contains 27686
nodes and 74414 road sections. The service network used in the load-balancing process contains
39 nodes. Based on the data input, the computational time for the network state estimates in the

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simulation is about 45-60 CPU seconds. The data flow in the solution procedure is given in
Figure 14.

Figure 14. Data Flow in Solution Procedure

4.6.3 Comparison of System Costs
We test three solution approaches in this study. In Approach 1, we only run the modified
load balancing algorithm without the pickup flow optimization. In this case, there is no truck
reuse in the network. Every truck starts from the depot, delivers a container from one location to
another, and then returns to the depot after the delivery. This is the approach introduced by Zhao
et al. (2018). Approach 2 considers truck reuse and the pickup flow optimization procedure with
one iteration. After solving the delivery flows, we iteratively solve the pickup flow optimization
problem until finding a feasible pickup flow for the current delivery flow. In Approach 3, we
execute the complete solution procedure with the stopping criteria. We evaluate the model

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performance by computing the total number of trips leaving the truck depot, the total truck travel
time, the total travel distance, and the system cost at different demand levels.
Table 10. Evaluation Results for Different Approaches
Demand
Approach 1 Approach 2 Approach 3
Number
of Trips
Leaving
the Depot
Truck
Travel
Distance
(mi)
Truck
Travel
Time
(hr)
Number
of Trips
Leaving
the Depot
Truck
Travel
Distance
(mi)
Truck
Travel
Time
(hr)
Number
of Trips
Leaving
the Depot
Truck
Travel
Distance
(mi)
Truck
Travel
Time
(hr)
13000 13000 294739 6464 6022 282380 6664 4022 265217 5853
14000 14000 307711 6796 6447 284632 6914 4446 284037 6671
15000 15000 322203 7278 7062 308658 7576 4324 306701 7307
16000 16000 338802 7666 7493 311892 7815 4572 311125 7516
17000 17000 356554 8028 7408 322741 7966 5308 321333 7645
18000 18000 371415 8389 7804 333261 8553 5858 332797 8121

As shown in Table 10, the number of trucks used and the total travel time for the
container movements generally increase with the demand levels. However, note that special
cases may arise: under a higher demand level, it is possible to observe fewer trucks. For
example, in Approach 3, 4446 trucks are needed under the demand level of 14000 with the total
travel time of 6671 hours; and in comparison, under the demand level of 15000, the truck
number drops to 4324 and the travel time increases to 7307 hours. The reason this occurs is that
the optimization procedure is designed to minimize the system cost instead of merely minimizing
the total number of trucks or the total travel time. In addition, under the same demand level, as
the solution approaches shift from 1 to 3, the total number of trucks used in the transportation
system and the total travel distance reduce. This means that by iteratively updating the delivery
flows and pickup flows, the truck reuse rate can be improved. Moreover, constructing truck trips
can also reduce the total travel distance in the pickup and delivery procedure, since once a truck
finishes its delivery task, the pickup flow optimization procedure motivates the truck to find

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nearby tasks and start its next delivery task instead of returning to the truck depot. The model
costs using the different approaches are given in Figure 15.

Figure 15. Model Costs Using Different Approaches

Comparing Approaches 1 and 2, a 41 percent to 46 percent reduction in system cost can
be observed by performing one iteration of the pickup flow optimization procedure. Similarly,
comparing Approaches 2 and 3, the execution of the complete solution procedure can further
reduce the system cost by about 20 percent. The computational time for running experiments
under these three approaches is worth mentioning. With the current settings, the experiments in
Approach 1 take about eight hours to meet the stopping criteria for the load balancing procedure
due to the large-scale demand. In Approach 2, we take the load balancing solution from the
experiments in Approach 1, and then run the pickup flow optimization procedure for one
iteration. Based on our experiments, the pickup flow optimization procedure takes a few minutes.
Specifically, solving the LPR of the optimization model only takes less than a second, and the
rest of the time is spent on running the simulation model iteratively. Therefore, the

62

computational time for Approach 2 is similar to Approach 1 (i.e., approximately eight hours).
For Approach 3, we limit the computational time for running the load balancing procedure in
each iteration to 20 minutes. Since the maximum number of iterations in our setting is 20, the
total computational time for running experiments under Approach 3 is also about eight hours. In
summary, the results of the numerical experiments show that the proposed solution procedure
provides the best quality solution for regional freight movements.

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Chapter 5: Mixed Fleet Drayage Routing Problem
This chapter presents a mixed fleet drayage routing problem with DHDTs and BEHDTs.
We first present a MIP formulation for the problem with model linearization and variable
elimination. A modified ALNS is proposed to efficiently solve the problem. At the end of the
chapter, results from numerical experiments are presented.
5.1 Problem Description
In this chapter, we explore the potential of substituting BEHDTs for DHDTs to reduce
emissions in the freight logistics industry. Unlike a DHDT, which has a range of hundreds of
miles per refill, under the current battery technology, a BEHDT has a more limited range
between charging and the range depending on the loading state of the truck. Therefore, we
assume only BEHDTs need to be charged during working hours. In addition, all the trucks start
and end at the truck depot.
The problem is defined on a directed graph 𝐺 =(𝑉 ,𝐴 ) with a vertex set 𝑉 =
{0,1,…,𝑛 ,𝑛 +1,…,𝑛 +𝑚 ,𝑛 +𝑚 +1} and an arc set 𝐴 ={(𝑖,𝑗 )|𝑖 ≠𝑗 𝑎𝑛𝑑 𝑖 ,𝑗 ∈𝑉 }. In
addition, both the vertices 0 and 𝑛 +𝑚 +1 represent the truck departure depot and arrival depot,
respectively, although in reality both can represent the same physical location. 𝐶 ={1,…,𝑛 } is
the customer location set and 𝐹 ={𝑛 +1,…,𝑛 +𝑚 } is the location set of charging stations. A
sample network is given in Figure 16.

64

Figure 16. A Sample Network
We use an arc-based formulation for the MFDRP using two types of trucks. A set 𝐷
represents available DHDTs, and a set 𝐸 represents all the BEHDTs. Each location 𝑖 (𝑖 ∈𝑉 ) is
associated with a quadruple {𝑞 𝑖 ,𝑤 𝑖 ,𝑠 𝑖 ,𝑟 ̅
𝑖 } where 𝑞 𝑖 ∈{−1,0,1} represents the loading state and
𝑞 𝑖 =1(−1) means a pickup (drop-off) of a container at location 𝑖 , and 𝑞 𝑖 =0 means no pickup
or drop-off at location 𝑖 (e.g., only charging occurs at location 𝑖 ); 𝑤 𝑖 ∈{0,𝛾 𝑒 ,𝛾 𝑙 } represents the
load weight where 𝛾 𝑒 is the weight for an empty container and 𝛾 𝑙 is the weight for a loaded
container; 𝑠 𝑖 represents the service time; and binary indicator 𝑟 ̅
𝑖 represents whether a BEHDT
can be charged at location 𝑖 . To simplify the model, we assume that a truck can only conduct one
of the following three tasks at each non-depot location: pickup, drop-off, or charging. However,
the modeling framework allows for multiple tasks at the same location by copying the node. In
this way, only a single task is applied at each node. Binary variable 𝑥 𝑖 ,𝑗 𝑑 =1 if and only if arc
(𝑖,𝑗 ) is traveled by a DHDT 𝑑 ∈𝐷 ; binary variable 𝑥 𝑖 ,𝑗 𝑒 =1 if and only if arc (𝑖,𝑗 ) is traveled by
a BEHDT 𝑒 ∈𝐸 . Due to the range limitation on BEHDTs, we introduce binary variable 𝑟 𝑖 𝑒 to
determine if truck 𝑒 recharges at location 𝑖 and non-negative variable 𝑏 𝑖 𝑒 to keep track of truck
𝑒 ’s battery level upon arrival at location 𝑖 . Non-negative variable 𝑧 𝑖 𝑘 represents the arrival time
of truck 𝑘 (𝑘 ∈𝐷 ∪𝐸 ) at location 𝑖 . Since BEHDTs have disparate power consumption rates

65

under different loading states, we introduce a default battery consumption rate 𝑝 , a load-
dependent consumption rate 𝑝 ′
, and a variable 𝑙 𝑖 𝑘 which represents the container loading states at
location 𝑖 for truck 𝑘 . Figure 17 provides battery consumption rates for BEHDT under different
load states.

Figure 17. Battery Consumption Rates under Different States

5.2 Mixed Fleet Drayage Routing Problem
5.2.1 Mixed Integer Programming Model
The notation used in this chapter is as follows:
Sets
𝐾 The set of all the trucks;
𝐷 The set of DHDTs, 𝐷 ∈𝐾 ;
𝐸 The set of BEHDTs, 𝐸 ∈𝐾 ;
𝑉 The set of locations, 𝑉 ={0,1,…,𝑛 +𝑚 +1};
𝐶 The set of customers, 𝐶 ={1,…,𝑛 };
𝐹 The set of charging stations, 𝐹 ={𝑛 +1,…,𝑛 +𝑚 };
𝐴 The set of arcs in the network;
Ω The set of emissions, Ω={CO
2
,NO
x
}, with set index 𝜔 .
Parameters
𝑝 The default hourly battery consumption rate;

66

𝑝 ′ The load dependent hourly battery consumption rate;
𝑡 𝑖 ,𝑗 Travel time on arc (𝑖,𝑗 ) ;
𝑎 𝑖 ,𝑗 Distance of arc (𝑖 ,𝑗 ) ;
𝑞 𝑖 Loading state at location 𝑖 ;
𝑤 𝑖 Load weight at location 𝑖 ;
𝑠 𝑖 Service time at location 𝑖 ;
𝑟 ̅
𝑖 Charging station indicator for location 𝑖 ;
𝑐 0
𝑑 The daily DHDT employment cost;
𝑐 0
𝑒 The daily BEHDT employment cost;
𝑐 1
𝑑 The DHDT fuel cost per mile;
𝑐 1
𝑒 The BEHDT charging cost per mile;
𝑐 𝜔 𝑑 The Emission cost of DHDT for pollutant 𝜔 ;
𝑐 𝜔 𝑒 The Emission cost of BEHDT for pollutant 𝜔 ;
𝑧 ̅ The maximum working time for truck drivers.
Variables
𝑥 𝑖 ,𝑗 𝑘 Binary variable for a truck 𝑘 traveling on arc (𝑖 ,𝑗 ) , 𝑘 ∈𝐾 ;
𝑧 𝑖 𝑘 Arrival time at location 𝑖 for truck 𝑘 , 𝑘 ∈𝐾 ;
𝑙 𝑖 𝑘 Truck 𝑘 ’s container loading state when leaving location 𝑖 , 𝑘 ∈𝐾 ;
𝑟 𝑖 𝑒 Binary indicator for BEHDT 𝑒 recharging at location 𝑖 , 𝑒 ∈𝐸 ;
𝑏 𝑖 𝑒 Battery level for BEHDT 𝑒 upon arriving at location 𝑗 , 𝑒 ∈𝐸 .
The MFDRP can be formulated as,
min
𝑥 𝑖 ,𝑗 𝑑 ,𝑥 𝑖 ,𝑗 𝑒 (𝑐 0
𝑑 ∑∑𝑥 0,𝑗 𝑑 𝑗 ∈𝐶 𝑑 ∈𝐷 +𝑐 0
𝑒 ∑∑𝑥 0,𝑗 𝑒 𝑗 ∈𝐶 𝑒 ∈𝐸 )+( ∑ 𝑎 𝑖 ,𝑗 ((𝑐 1
𝑑 +∑𝑐 𝜔 𝑑 𝜔 ∈Ω
)∑𝑥 𝑖 ,𝑗 𝑑 𝑑 ∈𝐷 +(𝑐 1
𝑒 +∑𝑐 𝜔 𝑒 𝜔 ∈Ω
)∑𝑥 𝑖 ,𝑗 𝑒 𝑒 ∈𝐸 )
(𝑖 ,𝑗 )∈𝐴 ) (69)
Subject to
∑∑𝑥 𝑖 ,𝑗 𝑘 𝑘 ∈𝐾 𝑗 ∈𝑉 =1 ∀𝑖 ∈𝐶 (70)

∑𝑥 𝑖 ,𝑗 𝑘 𝑖 ∈𝑉 −∑𝑥 𝑗 ,𝑖 𝑘 𝑖 ∈𝑉 =0 ∀𝑗 ∈𝐶 ∪𝐹 ,𝑘 ∈𝐾 (71)

67

𝑥 𝑖 ,0
𝑘 =𝑥 𝑛 +𝑚 +1,𝑖 𝑘 =0 ∀𝑖 ∈𝐶 ∪𝐹 ,𝑘 ∈𝐾 (72)
∑ 𝑥 0,𝑖 𝑘 𝑖 ∈𝐶 ∪𝐹 = ∑ 𝑥 𝑖 ,𝑛 +𝑚 +1
𝑘 𝑖 ∈𝐶 ∪𝐹 ∀𝑘 ∈𝐾 (73)

𝑟 𝑖 𝑒 ≤𝑟 ̅
𝑖 ∀𝑖 ∈𝑉 ,𝑒 ∈𝐸 (74)
𝑥 𝑖 ,𝑗 𝑒 ((1−𝑟 𝑖 𝑒 )𝑏 𝑖 𝑒 +𝑟 𝑖 𝑒 −𝑡 𝑖 ,𝑗 (𝑝 +𝑝 ′
𝑙 𝑖 𝑒 ))≤𝑏 𝑗 𝑒 ∀𝑗 ∈𝑉 \{0},𝑖 ∈𝑉 ,𝑒 ∈𝐸 (75)
𝑏 0
𝑒 =1 ∀𝑒 ∈𝐸 (76)
𝑏 𝑖 𝑒 ≥0 ∀𝑖 ∈𝑉 ,𝑒 ∈𝐸 (77)
𝑥 𝑖 ,𝑗 𝑑 (𝑧 𝑖 𝑑 +𝑠 𝑖 +𝑡 𝑖 ,𝑗 )≤𝑧 𝑗 𝑑 ∀𝑖,𝑗 ∈𝑉 ,𝑑 ∈𝐷 (78)
𝑥 𝑖 ,𝑗 𝑒 (𝑧 𝑖 𝑒 +𝑟 𝑖 𝑒 ⋅𝑓 (𝑏 𝑖 𝑒 )+𝑠 𝑖 +𝑡 𝑖 ,𝑗 )≤𝑧 𝑗 𝑒 ∀𝑖 ,𝑗 ∈𝑉 ,𝑒 ∈𝐸 (79)
𝑧 𝑛 +1
𝑘 −𝑧 0
𝑘 ≤𝑧 ̅ ∀𝑘 ∈𝐾 (80)
𝑧 0
𝑘 =0 ∀𝑘 ∈𝐾 (81)
𝑥 𝑖 ,𝑗 𝑘 (𝑙 𝑖 𝑘 +𝑞 𝑗 ⋅w
j
)≤𝑙 𝑗 𝑘 ∀𝑖 ∈𝑉 ,𝑗 ∈𝑉 \{0},𝑘 ∈𝐾 (82)
𝑥 𝑖 ,𝑗 𝑘 𝑙 𝑖 𝑘 =0 ∀𝑘 ∈𝐾 ,𝑖 ∈𝑉 ,𝑞 𝑗 >0 (83)
𝑙 0
𝑘 =0 ∀𝑘 ∈𝐾 (84)
𝑙 𝑖 𝑘 ≥0 ∀𝑖 ∈𝑉 ,𝑘 ∈𝐾 (85)
𝑥 𝑖 ,𝑗 𝑘 ∈{0,1} ∀(𝑖,𝑗 )∈𝐴 ,𝑘 ∈𝐾 (86)
𝑟 𝑖 𝑒 ∈{0,1} ∀𝑖 ∈𝑉 ,𝑒 ∈𝐸 (87)
𝑧 𝑖 𝑘 ∈𝑅 ∀𝑖 ∈𝑉 ,𝑘 ∈𝐾 (88)

The objective function (69) is to minimize the cost for one day’s drayage operation,
including daily truck employment costs, fuel costs, and emission costs. Constraints (70−73)
are the traditional vehicle routing constraints, such as demand satisfaction and flow conservation
constraints. Constraint (74) indicates that charging only occurs at locations with charging
stations. Constraint (75) defines the remaining battery level of a BEHDT upon arrival at location

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𝑗 . The constraint only becomes effective only when truck 𝑒 travels on arc (𝑖,𝑗 ) . When 𝑥 𝑖 ,𝑗 𝑒 =1,
the remaining battery for truck 𝑒 after traveling arc (𝑖,𝑗 ) can be represented as follows:
𝑏 𝑗 𝑒 ={
𝑏 𝑖 𝑒 −𝑡 𝑖 ,𝑗 (𝑝 +𝑝 ′
𝑙 𝑖 𝑒 ) 𝑖𝑓 𝑟 𝑖 𝑒 =0
1−𝑡 𝑖 ,𝑗 (𝑝 +𝑝 ′
𝑙 𝑖 𝑒 ) 𝑖𝑓 𝑟 𝑖 𝑒 =1
(89)
The battery consumption rate on arc (𝑖,𝑗 ) is determined by the container load state of
truck 𝑒 when leaving location 𝑖 . Constraint (76) states that all the BEHDTs are fully charged in
the truck depot at the beginning of the day. Constraint (77) ensures every BEHDT can arrive at
the next destination with a non-negative battery level. Constraints (78−79) keep track of the
arrival time for all the trucks and eliminate sub-tours. In constraint (79) , 𝑓 is the charging time
function, which depends on the battery level of BEHDT 𝑒 upon arrival at location 𝑖 . In this
report, we assume a piece-wise linear charging function in our experiments. Constraints
(80−81) bound the working time for all the trucks. Constraints (82−85) are the truck
capacity constraints. The rest of the constraints are domain constraints.
5.2.2 Model Pre-processing
To improve the formulation of the proposed model, we first linearize the MIP
formulation. Then, we eliminate variables based on feasibility rules. In the end, we strengthen
the formulation by eliminating symmetric solutions.
Model Linearization. The proposed MFDRP has a few non-linear constraints (75) ,
(78) , (79) , (82) and (83). All these constraints share one property: the non-linear terms have at
most one non-negative continuous variable multiplied by multiple binary variables. We linearize
our model using the following trick.

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The expression 𝑦 =𝑢 ∑ 𝑣 𝑖 𝑁 𝑖 =1
, where 𝑢 is a non-negative variable with a maximum value
of 𝑈 and 𝑣 1
−𝑣 𝑁 are binary variables, can be linearized by introducing the following
constraints:
𝑦 ≤𝑈𝑣
𝑖 ∀𝑖 ∈{1,2,…,𝑁 } (90)
𝑦 ≤𝑢 (91)
𝑦 ≥𝑢 +𝑈 (−𝑁 +∑𝑣 𝑖 𝑁 𝑖 =1
) (92)
𝑦 ≥0 (93)
Variable Elimination. The first elimination rule is based on the truck capacity. A
location 𝑖 can have three types of freight loads: loaded container, empty container, and no
container. By constraints (82−85) , once a truck picks up a container, it cannot pick up other
containers before dropping off the current load. It is worth noting that BEHDTs can visit
charging stations at any time. Therefore, variable 𝑥 𝑖 ,𝑗 𝑘 and arc (𝑖 ,𝑗 ) are eliminated if the
following condition holds:
{
𝑞 𝑗 >0 𝑖𝑓 𝑞 𝑖 >0
𝑞 𝑗 <0 𝑖𝑓 𝑞 𝑖 <0
∀𝑖 ∈𝐶 (94)
The second elimination rule is based on BEHDT range. Variable 𝑥 𝑖 ,𝑗 𝑒 is eliminated if arc
(𝑖,𝑗 ) satisfies the following conditions.
{

𝑚𝑖𝑛 𝑖 ′∈𝑉 𝑡 𝑖 ′
,𝑖 𝑝 +𝑡 𝑖 ,𝑗 ∗(𝑝 +𝑝 ′
)+ min
𝑗 ′
∈𝐹 ∪{𝑛 +1}
𝑡 𝑗 ,𝑗 ′
𝑝 >1 𝑖𝑓 𝑞 𝑖 =1
𝑚𝑖𝑛 𝑖 ′∈𝑉 𝑡 𝑖 ′
,𝑖 𝑝 +𝑡 𝑖 ,𝑗 (𝑝 +𝑝 ′
𝛾 ) + min
𝑗 ′
∈𝐹 ∪{𝑛 +1}
𝑡 𝑗 ,𝑗 ′
𝑝 >1 𝑖𝑓 𝑞 𝑖 =𝛾 𝑚𝑖𝑛 𝑖 ′∈𝑉 𝑡 𝑖 ′
,𝑖 (𝑝 +𝑝 ′
)+𝑡 𝑖 ,𝑗 𝑝 + min
𝑗 ′
∈𝐹 ∪{𝑛 +1}
𝑡 𝑗 ,𝑗 ′
𝑝 >1 𝑖𝑓 𝑞 𝑖 = −1
𝑚𝑖𝑛 𝑖 ′∈𝑉 𝑡 𝑖 ′
,𝑖 (𝑝 +𝑝 ′
𝛾 )+𝑡 𝑖 ,𝑗 𝑝 + min
𝑗 ′
∈𝐹 ∪{𝑛 +1}
𝑡 𝑗 ,𝑗 ′
𝑝 >1 𝑖𝑓 𝑞 𝑖 = −𝛾 ∀𝑖 ∈𝐶 ,𝑗 ∈𝐶 (95)
These four conditions are the cases that a BEHDT traveling on arc (𝑖 ,𝑗 ) connecting two
customer locations. Taking the first condition as an example, the first term represents the

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minimum battery consumption on prior arc (𝑖 ′
,𝑖) . The second term is the battery consumption on
arc (𝑖,𝑗 ) . The last term is the minimum battery consumption that allows the BEHDT to reach the
nearest charging station or depot. These three trips cannot exceed the battery limit. Otherwise,
arc (𝑖,𝑗 ) is not a valid arc for a BEHDT in the model.
Symmetry Elimination. Although two different types of trucks are employed in the
model, trucks in the same truck set are homogeneous, meaning all the DHDTs are the same and
all the BEHDTs are the same. In this case, multiple solutions can be made identical by switching
the numbering of the trucks. To avoid the symmetry of the solutions, we label DHDTs and
BEHDTs using numerical values. DHDTs are numbered from one to |𝐷 |, followed by BEHDTs
from |𝐷 |+1 to |𝐷 |+|𝐸 |. Additional constraints (96−97) are introduced to force trucks with
smaller labels to be employed first.
∑𝑥 0,𝑖 𝑑 𝑖 ∈𝑉 ≤∑𝑥 0,𝑖 𝑑 −1
𝑖 ∈𝑉 ∀𝑑 ∈{2,3,…,|𝐷 |} (96)

∑𝑥 0,𝑖 𝑒 𝑖 ∈𝑉 ≤∑𝑥 0,𝑖 𝑒 −1
𝑖 ∈𝑉 ∀𝑒 ∈{|𝐷 |+2,|𝐷 |+3,…,|𝐷 |+|𝐸 |} (97)

5.3 Solving framework for MFDRP
Although the proposed MFDRP can be solved optimally by commercial solvers, it takes
hours to even find feasible solutions for problems of practical size. Therefore, we propose a
modified ALNS heuristic to solve the problem. There are two main steps to solve the MFDRP
with ALNS, (1) find an initial feasible solution, and (2) explore neighborhoods of the current
solution and move towards a better solution.

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5.3.1 Initial Solution Construction
In the proposed model, containers may be empty or loaded. Before constructing truck
routes, we first match supply and demand based on container types. The formal definition of a
matched supply and demand (also referred to as task) is the following:
τ=(𝑜 ,𝑑 ,𝜉 )
where 𝑜 is a supply node, 𝑑 is a demand node, and 𝜉 is a binary indicator for the demand type
(i.e., 𝜉 =0 for empty container demand and 𝜉 =1 for loaded container demand).
There are many ways to match supply and demand. For example, the distance between a
supply node 𝑖 and a demand node 𝑗 can be considered as the preferences for pair (𝑖,𝑗 ) , which
turns the matching problem into a stable marriage problem and can be solved by the Gale-
Shapley algorithm (Gale & Shapley, 1962). Once all the customer nodes have been matched, we
store every matched task 𝜏 into a task list (Τ) and use the following algorithm for initial solution
construction.

Algorithm 2: Initial Solution Construction
Step 0. 𝐷 ={}, 𝑘 =0.
Step 1. 𝑘 =𝑘 +1
Step 2. Introduce a new truck 𝑘 with its schedule 𝜋 𝑘 =[] and initialize the truck location ι=0
Step 3. Search for the nearest 𝑡𝑎𝑠𝑘 s.t.
𝜏 ∗
=(𝑜 ∗
,𝑑 ∗
,𝜉 ∗
)=arg min
(𝑜 ,𝑑 ,𝜉 )∈Τ
𝑎 ι,o

Step 4. If working time for truck 𝑘 with 𝜏 ∗
≤𝑧 ̅ :
Append 𝜏 ∗
to 𝜋 𝑘
Remove 𝜏 ∗
from Τ
Update truck location ι=𝑑 ∗

Else:

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Append truck 𝑘 to 𝐷
Go to Step 1
Step 5. If Τ is not empty, go to Step 3, else STOP.

For initial solution construction, only DHDTs are used in the system. Algorithm 1 is a
greedy algorithm for solving a bin-packing problem. In this algorithm, trucks are considered as
capacitated bins with limited working time. Each ordered list 𝜋 𝑘 holds the truck schedule for
truck 𝑘 . Every time a new truck is introduced into the system, it will search for the nearest
remaining 𝜏 ∗
in Τ based on the truck’s last location. If the current truck can finish 𝜏 ∗
and return
to the depot within 𝑧 ̅ , we append 𝜏 ∗
to the ordered list and remove 𝜏 ∗
from the task list Τ.
Otherwise, the truck is added to the diesel truck set 𝐷 and a new truck 𝑘 +1 is introduced to the
system to take the remaining tasks. The algorithm stops when all tasks are assigned.
5.3.2 Modified ALNS
The ALNS is adapted from Dessouky et al. (2020). The modified ALNS in our study
considers truck substituting between DHDTs and BEHDTs. In addition, we introduce charging
station insertion and task re-match operations in our ALNS. The framework of our ALNS is
given in Figure 18.

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Figure 18. Solution Framework for Modified ALNS

Substitute Truck Types. After running Algorithm 2, only DHDT routes are
generated. Figure 19 shows a truck route in the initial solution.

Figure 19. A Truck Route in the Initial Solution

When we substitute a BEHDT for a DHDT, charging station insertions are necessary to
ensure non-negative battery levels for the BEHDT. Figure 20 gives two types of charging station
insertions, within and between task insertions. In addition, because the charging time for
BEHDTs is non-negligible, we cut off tasks that exceed the working time limit and add them
back into the task list Τ. On the other hand, when substituting a BEHDT for a DHDT, we only
need to eliminate all the charging stations along the route.

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Figure 20. Two Types of Charging Insertions

Re-match Tasks. Initially, all the tasks are generated in a greedy manner only
considering the distance between supply and demand nodes. It is easy to see such a matching
strategy might lead to a sub-optimal solution and trap the model solution in a local minima.
Therefore, it is necessary to re-match supply and demand nodes along the procedure. In each
iteration, trucks that have less than 𝛿 tasks will be removed from the truck sets. These tasks will
be added back to the task list Τ. Before inserting these unassigned tasks to trucks, in each
iteration with a certain probability 𝜌 , we randomly remove 𝜂 tasks from all the trucks and add
them back to the task list Τ and rematch these tasks.
Insert Tasks into Trucks. At this point, if the task list Τ is not empty, we need to insert
tasks into existing trucks or employ additional trucks to ensure demand satisfaction.
Optimize Routes. For each DHDT, we minimize its travel distance with working time
limit constraints. For each BEHDT, additional battery constraints are considered when
determining its routes. Although finding the best route for every truck is a TSP which is NP-
hard, the best route can be found fast when each truck can have at most
𝑧 ̅
𝑠 𝑖 tasks. In practice, a
truck usually serves less than twenty customers in daily drayage operations.

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Stopping Criteria. The heuristic terminates when the following two conditions hold: (1)
The maximum number of iterations (Ψ) is reached, and (2) the improvement between successive
iterations is not greater than Δ.
The solution to the problem is a collection of truck routes for DHDTs and BEHDTs
which are stored in 𝐷 and 𝐸 , respectively. Two functions are used in the ALNS (1) 𝐶 𝑠𝑦𝑠 is a
function that calculates the system costs based on the truck routes in 𝐷 and 𝐸 , and (2) 𝑔 is a
function that calculates travel distances for any truck 𝑘 . The complete ALNS heuristic is given in
Algorithm 3.

Algorithm 3: Modified ALNS
Step 0. Set 𝜓 =0, 𝐸 ={}, 𝑑 ̅
=
𝑐 0
𝑒 −𝑐 0
𝑑 𝑐 1
𝑑 −𝑐 1
𝑒 +∑ (𝑐 𝜔 𝑑 −𝑐 𝜔 𝑒 )
𝜔 ∈Ω
,
Step 1. 𝜒 ∗
=(𝐷 ,𝐸 ) , Κ
∗
=𝐶 𝑠𝑦𝑠 (𝐷 ,𝐸 ) , 𝜓 ∗
=1,
Step 2. For truck 𝑘 in 𝐷 :
If 𝑔 (𝑘 )>𝑑 ̅
:
Remove truck 𝑘 from 𝐷
Insert charging stations into 𝜋 𝑘 with greedy insertion
If working time for truck 𝑘 >𝑧 ̅ :
Cut off the tasks that exceed 𝑧 ̅ and add these tasks to Τ
Add 𝑘 to 𝐸
Else if truck 𝑘 has less than 𝛿 tasks:
Add all the tasks in truck 𝑘 to Τ
Remove truck 𝑘 from 𝐷
Step 3. For truck 𝑘 in 𝐸 :
If 𝑔 (𝑘 )≤𝑑 ̅
:
Remove truck 𝑘 from 𝐸

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Remove charging stations from 𝜋 𝑘
Add 𝑘 to 𝐷
Step 4. With probability 𝜌 , randomly remove 𝜂 tasks from all trucks to Τ and re-match OD pairs
for all 𝜏 ∈Τ based on the demand type
Step 5. While Τ is not empty:
Select a 𝜏 ′ from Τ
If 𝜏 ′ can be inserted into any active truck 𝑘 ∈𝐷 ∪𝐸 :
Insert 𝜏 ′ into 𝜋 𝑘
Else:
Assign 𝜏 ′ to a new truck
Step 6. Optimize routes for all active trucks
Step 7. If 𝐶 𝑠𝑦𝑠 (𝐷 ,𝐸 )≤ Κ
∗
:
𝜒 ∗
=(𝐷 ,𝐸 )
Κ
∗
=𝐶 𝑠𝑦𝑠 (𝐷 ,𝐸 )
𝜓 ∗
= 𝜓
Step 8. 𝜓 = 𝜓 +1
Step 9. If 𝜓 −𝜓 ∗
≥Δ or 𝜓 = Ψ, STOP; otherwise go to Step 2.

The algorithm starts with the solution generated by Algorithm 2. In Step 0, we initialize a
few parameters, where 𝜓 is the iteration counter, 𝐸 is the set for BEHDTs, and 𝑑 ̅
is the travel
distance threshold for substituting a DHDT to a BEDHT. In Step 1, we introduce 𝜒 ∗
to store the
best solution, Κ
∗
to store the minimum system cost, and 𝜓 ∗
to store the iteration that finds the
best solution. Step 2 and Step 3 are the truck type substituting process. If a DHDT travels more
than the threshold 𝑑 ̅
, we substitute the DHDT to a BEHDT, perform charging station insertions,
and cut off tasks that exceed the working time limit 𝑧 ̅ . If a DHDT travels less than 𝑑 ̅
and has
fewer than 𝛿 tasks, it is removed from the system and its tasks are added back to the task list Τ.

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On the other hand, if a BEHDT travels less than 𝑑 ̅
, it is substituted back to a DHDT by
removing all charging stations on its route. Step 4 is the task re-matching process. In each
iteration, there is a probability 𝜌 of performing a re-match for the remaining tasks in Τ. Every
time a re-match happens, additional 𝛿 tasks are added back to Τ from the truck routes to better
explore the solution space and escape from a local minima. In Step 5, we try to insert the
remaining tasks in Τ into existing trucks. Every time a task cannot be inserted into existing
trucks, it is assigned to a new truck. Step 6 is the route optimization procedure, which finds the
best route for each individual truck. In Step 7, the best solution is updated if the system costs in
the current iteration is lower than the minimum system costs. We then update the iteration
number and check the stopping criteria.
5.4 Computational Results
In this section, we present the numerical results of small-size and practical-size instances.
Most of the parameters are taken from Giuliano et al. (2021). We first run the modified ALNS on
small instances, which can also be solved by a standard commercial optimization solver Gurobi
using the MFDRP formulation. The optimal solutions obtained from the Gurobi solver are also
considered as benchmark cases to show the effectiveness of the proposed ALNS heuristic. Then,
we use the modified ALNS to solve practical size problems to explore the potential of employing
BEHDTs in drayage operations in years 2022, 2025, and 2030.
All the numerical experiments are conducted on a square grid map with randomly
generated customer and charging station nodes on the map. The depot is located at the center of
the map. All the trucks start and end at the depot. DHDTs are assumed to be able to finish one-
day operations without refueling and BEHDTs are fully charged at the beginning of the day.
Once a BEHDT arrives at a charging station, it cannot leave the station until it is fully charged.

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We use a two-stage charging model in this study. The battery has a faster (slow) charging speed
𝑣 1
(𝑣 2
) before (after) the battery level threshold, as shown in Figure 21. We assume all the trucks
are driving on well-maintained roads with an average speed of 40 mph. The service time for all
customer nodes is half an hour.

Figure 21. Charging Curve

All simulated experiments are solved on a computer with an Intel i7-12800H CPU of 4.8
GHz and a RAM of 64 GB. All the scripts are programmed in Python 3.8. The MFDRP model is
solved by Gurobi 9.5.2 with Python API.
5.4.1 Experiments with Small-size Instances
As a non-linear MIP model, the MFDRP cannot be efficiently solved by state-of-the-art
commercial optimization solvers when the problem size is large. However, for very small
instances, the model can be solved optimally within 4 CPU minutes. A 50 miles by 50 miles grid
map is generated with 8-12 customer nodes and 2 charging station nodes. The location of the
customer and charging station nodes are randomly generated on the map. All the small cases are
solved with Gurobi 9.5.2 as well as the modified ALNS. We record the best solution found

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within 2 CPU hours for all the experiments. The parameters that are used in this section can be
found in Appendix C.
Table 11. Small Instances Results
Name
Gurobi Solver ALNS
Obj LB Gap Time (s) Obj Gap Imp Time (s)
4F4E2C1 836.74 836.74 0.00% 209.0 836.74 0.00% 0.00% 3.2
4F4E2C2 822.81 822.81 0.00% 74.9 828.93 0.74% -0.74% 2.9
4F4E2C3 818.16 818.16 0.00% 148.0 818.16 0.00% 0.00% 3.3
4F4E2C4 860.54 860.47 0.01% 88.8 867.18 0.77% -0.77% 2.8
4F4E2C5 770.68 770.68 0.00% 216.3 779.63 1.15% -1.16% 1.8
Avg 821.79 821.77 0.00% 147.4 826.13 0.53% -0.54% 2.8
6F4E2C1 770.71 243.43 68.41% 7200.0 770.71 68.41% 0.00% 8.3
6F4E2C2 835.80 270.85 67.59% 7200.0 873.01 68.98% -4.45% 4.3
6F4E2C3 831.95 303.65 63.50% 7200.0 872.57 65.20% -4.88% 3.8
6F4E2C4 816.84 238.85 70.76% 7200.0 835.44 71.41% -2.28% 3.2
6F4E2C5 776.94 230.31 70.36% 7200.0 770.30 70.10% 0.85% 4.6
Avg 806.45 257.42 68.13% 7200.0 824.40 68.82% -2.15% 4.84
8F4E2C1 1115.21 223.09 80.00% 7200.0 830.39 73.13% 25.54% 7.8
8F4E2C2 1206.20 305.24 74.69% 7200.0 1251.97 75.62% -3.79% 10.2
8F4E2C3 1168.02 318.66 72.72% 7200.0 867.79 63.28% 25.70% 4.6
8F4E2C4 1248.14 332.73 73.34% 7200.0 1299.55 74.40% -4.12% 9.2
8F4E2C5 1266.20 339.48 73.19% 7200.0 1283.10 73.54% -1.33% 5.7
Avg 1200.76 303.84 74.79% 7200.0 1106.56 71.99% 8.40% 7.5

Table 11 shows the numerical results for the small instances. In total, we conducted 15
instances with three different settings and each setting has five randomly generated instances.
The instance name shows the types of nodes on the map. For example, instance 8F4E2C1 means
it is the first instance with eight loaded container demand nodes, four empty container demand
nodes, and two charging station nodes. The first group of columns is the numerical results
retrieved directly from the Gurobi solver. Within the group, the first column (Obj) shows the
objective function value of the best solution found; the second column (LB) is the best lower
bound; the third column (Gap) is the optimality gap between the best objective function value
and the best lower bound; the fourth column (Time) is the solving time in seconds for the Gurobi

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solver. The second group of columns is the results from our ALNS. Since ALNS does not
provide any lower bounds in the solving procedure, we compare the results with the solver
solution and calculate improvements (Imp) compared to the best solution from the Gurobi solver.
As shown in Table 1, all the cases under 4F4E2C can be solved optimally within a few hundred
seconds by the Gurobi solver, while ALNS can also solve these cases nearly optimally within a
few seconds. However, when the problem size increases, the solver performance drops down
significantly. None of them can be solved optimally within two CPU hours. ALNS consistently
solves the problem in a matter of seconds and provides good quality solutions. In some instances,
such as 6F4E2C5, 8F4E2C1, and 8F4E2C3, ALNS even outperforms the best solution from the
solver.
5.4.2 Experiments with Practical-size Instances
We conduct simulations to explore the potential of substituting DHDTs with BEHDTs
from different aspects including the fleet size, travel distances, and emissions. In addition, three
years (2022, 2025, and 2030) are considered in our experiments. For each target year, we
considered three situations including (1) all trucks are DHDTs (Max-DHDTs), (2) nearly half of
the trucks are BEHDTs (Mid-Point), and (3) all trucks are BEHDTs (Max-BEHDTs). In total, we
have nine groups of experiments. The year 2022 with all DHDTs is considered as our base case.
We modify parameters from Giuliano et al. (2021), which provides comprehensive information
including daily employment costs, emission rates, and battery capacity improvements for both
DHDTs and BEHDTs for years 2022, 2025 and 2030. Other parameters are listed in Appendix C.
Similar to Giuliano et al. (2021), we study scenarios with daily demand for empty and
loaded containers as 135 and 176 respectively. We randomly generate demand nodes on the map
in each experiment and run five replications under each setting.

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As seen in Figure 22, in 2022, to satisfy the daily demand, the Max-DHDTs scenario
requires 71.6 trucks on average. As the share of BEHDTs increases, the fleet size increases
significantly. To reach the maximum BEHDT share in the truck fleet, there is a 47.2 percent
increase in the fleet size, and the average fleet size becomes 105.4. Additional trucks are needed
due to the range limitations and extra charging time. The fleet size for reaching maximum
BEHDT share reduces sharply over the target years as battery technology improves. In 2025 and
2030, the fleet sizes both increased by 3.4 percent to reach the maximum BEHDT share in the
fleet.

Figure 22. Avg. Number of HDTs Required for Each Target Year

Figure 23 provides the truck miles for daily demand satisfaction. In 2022, the truck miles
increase as the BEHDT share increases. These extra truck miles are caused by detours for
charging. In 2025, we still can see a slightly increase in the truck miles under different scenarios.
However, in 2030, the truck miles are very similar under different scenarios. As battery

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technology significantly improves in future years, there is less need to charge during the
workday. Thus, there is less need to take detours for charging during working time.

Figure 23. Avg. Truck Miles Required for Each Target Year

Table 12 shows the emissions reductions relative to the Max-DHDT scenario for each
target year. In all target years, the Max-BEHDT scenario has the most emissions savings. In
2022, over 50 percent reductions in CO2 emissions and 93 percent reductions in NOx can be
achieved by employing BEHDTs in the fleet. In 2025 and 2030, with fewer BEHDT charging
detours, CO2 emissions can be reduced by an additional 10 percent. In addition, NOx emissions
can be reduced to zero when only using BEHDTs in drayage operations.
Table 12. Net Daily Emissions Savings, Relative to Max-DHDTs
Pollutant Year Mid-point Max-BEHDTs
Net saving (kg) % of saving Net saving (kg) % of saving
CO 2 2022 3531.63 14.5% 12375.98 50.9%

2025 8655.44 40.2% 13249.23 61.5%

2030 7254.03 37.6% 11864.52 61.4%
NO X 2022 4.90 31.1% 14.78 93.6%

2025 5.07 64.0% 7.94 100.0%

2030 4.92 62.0% 7.94 100.0%

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Chapter 6: Conclusion and Future Work
6.1 Conclusion
This dissertation focuses on drayage operations from three perspectives: (1) employing
double container trucks to optimize empty container repositioning, (2) balancing truck flows on a
load-dependent dynamic transportation network under large-scale demand, and (3) substituting
BEHDTs for DHDTs in drayage operations to reduce emissions.
In Chapter 3, We present two empty container repositioning models using double
container trucks. Both DCFM and SDCFM allow DCTs to flexibly pick up and drop off
containers if capacity permits. Future stochastic demand is considered in the second model to
improve container allocations for second-day pickup and delivery. In order to solve real-world
ECRPs, a rolling horizon based solution framework is designed, which connects day-one final
container allocations with day-two initial states. Experimental results show that DCTs can save
16 percent of travel distances by allowing flexible pickup and drop-off compared to a state-of-
the-art DCT model. In addition, the total system cost can be further reduced by 8 percent,
considering the future stochastic demand.
In Chapter 4, we propose a load balancing with routing approach to solve the freight
routing and touring problem on a dynamic transportation network with large-scale demand. The
approach involves using a traffic simulator that captures the complexity of traffic flow and how
it is affected by truck routing decisions, especially in the case of high demand. The use of the
simulation model in real-time replaces the use of simplified mathematical models traditionally
used to solve similar problems. In complex environments, simplified mathematical models
cannot capture the complexity of nonlinearities and dynamics and may lead to highly inaccurate
results. On the other hand, simulation traffic flow models can be as complex as the computer

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speed and memory allows leading to more accurate decisions. We use a macroscopic simulator
for the load balancing with touring approach on real-world data to evaluate the performance of
the proposed methodology. The proposed approach is tested on a realistic transportation network
in the San Pedro Bay area which includes two ports. Based on the current container throughput,
we can achieve over 40 percent reduction in total system cost by reusing trucks for freight
movements. In addition, by iteratively optimizing delivery and pickup flows, the total system
cost can be reduced by another 20 percent.
In Chapter 5, we examine the potential of employing BEHDTs in daily drayage
operations as a substitute for DHDTs. We consider sufficient charging locations on the map with
non-linear charging time for BEHDTs. We first formulate the mixed fleet drayage routing
problem as a non-linear MIP and then improve the formulation with linearization and variable
elimination. Then, a modified ALNS is proposed to solve the problem more efficiently. We
conduct randomized experiments based on data from the Ports of Los Angeles and Long Beach.
The results indicate that employing BEHDTs as substitutes for DHDTs will increase the fleet
size under the same level of demand, especially given today’s battery technology. Additional
truck miles caused by employing BEHDTs decrease as battery technology improves. These
additional truck miles are caused by BEHDT charging detours. With improved battery capacity,
BEHDTs are able to travel longer per charge, requiring fewer detours for charging. From an
emission reduction perspective, significant reductions can be achieved by employing BEHDTs as
substitutes for DHDTs.

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6.2 Future Work
The following steps of the dissertation are (1) studying a mixed fleet drayage routing
problem with dynamic load-dependent transportation network and (2) integrating empty
container repositioning into drayage routing and touring operations with dynamic transportation
network.
First, the mixed fleet drayage routing problem in this work assumes that the travel times
between ODs depend only on the physical distance. This assumption simplifies the complexity of
the transportation network. More realistic transportation network states can be approximated by
traffic simulation models, which improves the model accuracy.
Second, the freight routing and touring problem only involves container pickup and
delivery demand in the network, where the container transloading process is not considered. In
order to model a more realistic drayage system, the empty container repositioning process can be
considered to further assist freight transloading and transshipment. This extension also improves
trucks’ routing options on the network by adding empty container movement trips. Additional
transloading related constraints (e.g., container conservation and transloading constraints) are
needed to formulate the problem, which can significantly increase the problem's complexity.

86

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97

Appendices
Appendix A.
Function 𝒈 𝒊 ,𝒛
𝑔 𝑖 ,𝑧 (𝑡 )={
𝑓 𝑖 (𝑡 )+𝑏 𝑖 ,1,𝑡 +𝑝 𝑖 ,1
−𝑎 𝑖 ,1,𝑡 𝑖𝑓 𝑧 =1
𝑏 𝑖 ,2,𝑡 +𝑝 𝑖 ,2
−𝑎 𝑖 ,2,𝑡 𝑖𝑓 𝑧 =2
𝑏 𝑖 ,3,𝑡 +𝑝 𝑖 ,3
−(𝑓 𝑖 (𝑡 )+𝑎 𝑖 ,3,𝑡 ) 𝑖𝑓 𝑧 =3
, ∀𝑖 ∈𝑆𝐼 ,∀𝑡 ∈𝑇
𝑔 𝑖 ,𝑧 (𝑡 )={
𝑏 𝑖 ,1,𝑡 +𝑝 𝑖 ,1
−(𝑓 𝑖 (𝑡 )+𝑎 𝑖 ,1,𝑡 ) 𝑖𝑓 𝑧 =1
𝑓 𝑖 (𝑡 )+𝑏 𝑖 ,2,𝑡 +𝑝 𝑖 ,2
−𝑎 𝑖 ,2,𝑡 𝑖𝑓 𝑧 =2
𝑏 𝑖 ,3,𝑡 +𝑝 𝑖 ,3
−𝑎 𝑖 ,3,𝑡 𝑖𝑓 𝑧 =3
, ∀𝑖 ∈𝑆𝐸 ,∀𝑡 ∈𝑇
𝑔 𝑖 ,𝑧 (𝑡 )={
𝑏 𝑖 ,1,𝑡 +𝑝 𝑖 ,1
−𝑎 𝑖 ,1,𝑡 𝑖𝑓 𝑧 =1
𝑏 𝑖 ,2,𝑡 +𝑝 𝑖 ,2
−𝑎 𝑖 ,2,𝑡 𝑖𝑓 𝑧 =2
𝑏 𝑖 ,3,𝑡 +𝑝 𝑖 ,3
−𝑎 𝑖 ,3,𝑡 𝑖𝑓 𝑧 =3
, ∀𝑖 ∈𝑆𝐷 ,∀𝑡 ∈𝑇
𝑔 𝑖 ,𝑧 (𝑡 )={
𝑏 𝑖 ,1,𝑡 +𝑝 𝑖 ,1
−𝑎 𝑖 ,1,𝑡 𝑖𝑓 𝑧 =1
𝑏 𝑖 ,2,𝑡 +𝑝 𝑖 ,2
−𝑎 𝑖 ,2,𝑡 𝑖𝑓 𝑧 =2
𝑏 𝑖 ,3,𝑡 +𝑝 𝑖 ,3
−𝑎 𝑖 ,3,𝑡 𝑖𝑓 𝑧 =3
, ∀𝑖 ∈𝑆𝑃 ,∀𝑡 ∈𝑇
Function 𝒉 𝒊 ,𝒛
ℎ
𝑖 ,𝑧 (𝑡 ∗
)={
𝑏 𝑖 ,1,𝑡 ∗+𝑏 𝑖 ,3,𝑡 ∗+𝑝 𝑖 ,3
+𝑝 𝑖 ,1
−𝑎 𝑖 ,1,𝑡 ∗+𝑎 𝑖 ,3,𝑡 ∗ 𝑖𝑓 𝑧 =1
𝑏 𝑖 ,2,𝑡 ∗+𝑝 𝑖 ,2
−𝑎 𝑖 ,2,𝑡 ∗ 𝑖𝑓 𝑧 =2
0 𝑖𝑓 𝑧 =3
, ∀𝑖 ∈𝑆𝐼
ℎ
𝑖 ,𝑧 (𝑡 ∗
)={
0 𝑖𝑓 𝑧 =1
𝑏 𝑖 ,2,𝑡 ∗+𝑏 𝑖 ,1,𝑡 ∗+𝑝 𝑖 ,2
+𝑝 𝑖 ,1
−𝑎 𝑖 ,2,𝑡 ∗+𝑎 𝑖 ,1,𝑡 ∗ 𝑖𝑓 𝑧 =2
𝑏 𝑖 ,3,𝑡 ∗+𝑝 𝑖 ,3
−𝑎 𝑖 ,3,𝑡 ∗ 𝑖𝑓 𝑧 =3
, ∀𝑖 ∈𝑆𝐸
ℎ
𝑖 ,𝑧 (𝑡 ∗
)={
𝑏 𝑖 ,1,𝑡 ∗+𝑝 𝑖 ,1
−𝑎 𝑖 ,1,𝑡 ∗ 𝑖𝑓 𝑧 =1
𝑏 𝑖 ,2,𝑡 ∗+𝑝 𝑖 ,2
−𝑎 𝑖 ,2,𝑡 ∗ 𝑖𝑓 𝑧 =2
𝑏 𝑖 ,3,𝑡 ∗+𝑝 𝑖 ,3
−𝑎 𝑖 ,3,𝑡 ∗ 𝑖𝑓 𝑧 =3
, ∀𝑖 ∈𝑆𝐷
ℎ
𝑖 ,𝑧 (𝑡 ∗
)={
𝑏 𝑖 ,1,𝑡 ∗+𝑝 𝑖 ,1
−𝑎 𝑖 ,1,𝑡 ∗−𝑑 𝑖 ,1,𝑡 ∗ 𝑖𝑓 𝑧 =1
𝑏 𝑖 ,2,𝑡 ∗+𝑝 𝑖 ,2
−𝑎 𝑖 ,2,𝑡 ∗−𝑑 𝑖 ,2,𝑡 ∗ 𝑖𝑓 𝑧 =2
𝑏 𝑖 ,3,𝑡 ∗+𝑝 𝑖 ,3
−𝑎 𝑖 ,3,𝑡 ∗+𝜇 𝑖 𝑖𝑓 𝑧 =3
, ∀𝑖 ∈𝑆𝑃

98

Appendix B.
Proof for Proposition:
To show the existence of the solution, we let
𝑦̃
0,𝑖 ,0
=∑∑ ∑ 𝑥 ̅
𝑖 ,𝑗 ,𝑘 𝑟 𝑟 ∈𝑅 𝑖 ,𝑗 𝑗 ∈𝐽 𝑘 ∈𝐾 ∀𝑖 ∈𝐼
𝑦̃
𝑗 ,0,|𝐾 |
=∑∑ ∑ 𝑥 ̅
𝑖 ,𝑗 ,𝑘 𝑟 𝑟 ∈𝑅 𝑖 ,𝑗 𝑖 ∈𝐼 𝑘 ∈𝐾 ∀𝑗 ∈𝐽
𝑦̃
𝑗 ,𝑖 ,𝑘 =0 ∀𝑖 ∈𝐼 ,𝑗 ∈𝐽 ,𝑘 ∈𝐾
The solution satisfies the constraints of the LPR of SP2. Therefore, the feasible region for
the LRP is not empty, meaning the LPR must have at least one solution.
The remaining is to show that the LPR of SP2 yields an integer solution when delivery
flows are integer numbers. The problem can be converted into a minimum-cost flow problem by
constructing the set of nodes (𝑉 ) , demand (𝐷 ) , edges (𝐸 ) , and weights (𝑊 ) in a directed
weighted graph 𝐺 𝑝 =(𝑉 ,𝐷 ,𝐸 ,𝑊 ) .
1. Node construction
There are three types of nodes in the graph: a source node 𝑠 , a sink node 𝑡 , and a
collection of space-time nodes 𝑣 𝑘 , where 𝑣 ∈𝐼 ∪𝐽 and 𝑘 ∈𝐾 .
2. Demand construction
In the problem, we know that the maximum number of trucks is bounded by the overall
demand. Therefore, we have the demand at the source node 𝑠 ,
𝐷 𝑠 =−∑∑∑ ∑ 𝑥 ̅
𝑖 ,𝑗 ,𝑘 𝑟 𝑟 ∈𝑅 𝑖 ,𝑗 𝑘 ∈𝐾 𝑗 ∈𝐽 𝑖 ∈𝐼
The demand at all space-time nodes 𝑣 𝑘 follows,
𝐷 𝑣 𝑘 =
{

∑ ∑ 𝑥 ̅
𝑖 ,𝑗 ,𝑘 𝑟 𝑟 ∈𝑅 𝑖 ,𝑗 𝑗 ∈𝐽 𝑖𝑓 𝑣 ∈𝐼 ,𝑘 ∈𝐾 ∑ ∑ ( ∑ 𝑥 ̅
𝑖 ,𝑗 ,𝜏 𝑟 𝜏 ≤𝑘 −1−𝛿̅
𝑖 ,𝑗 ,𝜏 −𝛽 𝑗 − ∑ 𝑥 ̅
𝑖 ,𝑗 ,𝜏 𝑟 𝜏 ≤𝑘 −𝛿̅
𝑖 ,𝑗 ,𝜏 −𝛽 𝑗 )
𝑟 ∈𝑅 𝑖 ,𝑗 𝑖 ∈𝐼 𝑖𝑓 𝑣 ∈𝐽 ,𝑘 ≥1+𝛿 ̅
𝑖 ,𝑗 ,𝜏 +𝛽 𝑗 0 𝑜𝑡 ℎ𝑒𝑟𝑤𝑖𝑠𝑒
The demand at the sink node balances the network demand.
3. Edge construction
There are five types of edges in the graph that include:

99

I. Directed edge (𝑠 ,𝑖 𝑘 ) with weight 𝜆 +𝑐 ̅
0,𝑖 ,𝜏 can be constructed for all 𝑖 ∈𝐼 ,𝑘 ∈𝐾 , where
the truck leaving at time 𝜏 can arrive at supplier 𝑖 at time 𝑘 . The flow on edge (𝑠 ,𝑖 𝑘 ) is
𝑦̃
0,𝑖 ,𝜏 ;
II. Directed edge (𝑗 𝑘 ,𝑖 𝑘 ′
) with weight 𝑐 ̅
𝑗 ,𝑖 ,𝑘 can be constructed if and only if a truck leaving
warehouse 𝑗 at time 𝑘 and is ready to go to supplier 𝑖 at time 𝑘 ′ , the flow on edge (𝑗 𝑘 ,𝑖 𝑘 ′
)
is 𝑦̃
𝑗 ,𝑖 ,𝑘 ;
III. Directed edge (𝑣 𝑘 ,𝑣 𝑘 +1
) with weight 0 can be constructed for all 𝑣 ∈𝐼 ∪𝐽 ,𝑘 ∈𝐾 \{|𝐾 |};
IV. Directed edge (𝑗 k
,𝑡 ) with weight 𝑐 ̅
𝑗 ,0,𝑘 can be constructed for all 𝑗 ∈𝐽 . The flow on edge
(𝑗 k
,𝑡 ) is 𝑦̃
𝑗 ,0,𝑘 ;
V. Directed edge (𝑠 ,𝑡 ) with weight 0 is a source-sink edge used to balance the demand.
With this construction, the minimum-cost flow problem defined on 𝐺 𝑝 is equivalent to SP2.
In addition, the demand in graph 𝐺 𝑝 is integer-valued. Since the incidence matrix of 𝐺 𝑝 is totally
unimodular, the minimum-cost flow problem yields an integer solution with integer demand on
the right-hand side. ∎

100

Appendix C.
Table A1. Parameters for small-size instances
𝑐 0
𝑑 300
𝑐 0
𝑒 360
𝑐 1
𝑑 0.58
𝑐 1
𝑒 0.38
𝑐 𝑐 𝑜 2
𝑑 0.1
𝑐 𝑁 𝑂 𝑥 𝑑 0.5
𝑐 𝑐 𝑜 2
𝑒 0
𝑐 𝑁 𝑂 𝑥 𝑑 0
𝑧 ̅ 8
Ψ 400
Δ 200
𝛽 0.3
𝜂 4
𝛿 2
𝑣 1
0.8
𝑣 2
0.2
𝛼 80

101

Table A2. Parameters for practical-size instances
Year 2022 2025 2030
𝑐 0
𝑑 150 150 150
𝑐 0
𝑒 380 250 180
𝑐 1
𝑑 1.36 1.26 1.16
𝑐 1
𝑒 0.49 0.50 0.47
𝑐 𝑐 𝑜 2
𝑑 0.1501 0.1329 0.1191
𝑐 𝑁 𝑂 𝑥 𝑑 0.0010 0.0005 0.0005
𝑐 𝑐 𝑜 2
𝑒 0.0506 0.0475 0.0444
𝑐 𝑁 𝑂 𝑥 𝑑 0 0 0
Ψ 1000
Δ 400
𝛽 0.3
𝜂 4
𝛿 2
𝑧 ̅ 8
𝑣 1
0.8
𝑣 2
0.2
𝛼 80

Abstract (if available)

Abstract The increasing demand for freight transportation has caused social issues such as traffic congestion and air pollution, making optimizing drayage operations essential in the logistics industry. To address these problems, we investigate drayage operations problems from three perspectives. First, we propose two container repositioning models using double container trucks. Both models allow a flexible pickup and drop-off policy for double container trucks. A rolling horizon based solving framework is developed to solve the models under deterministic and stochastic demand. Second, we study a freight routing and touring problem on a load-dependent dynamic transportation network. Instead of using explicit mathematical expressions to model network dynamics, traffic simulation models are introduced into our optimization loop, which allows more efficient and accurate network approximations. The problem is decomposed into two subproblems. The first subproblem can be solved by a modified load-balancing algorithm, which provides a partial integer solution for the second subproblem. We prove that solving the second subproblem’s linear programming relaxation can provide an integer solution with the partial solution input. A co-simulation optimization based load-balancing with touring solution framework is proposed to tackle the freight routing and touring problem. Third, we explore the potential of substituting diesel heavy-duty trucks with battery-electric heavy-duty trucks in daily drayage operations to reduce emissions. A mixed integer programming model is formulated. A modified adaptive large neighborhood search algorithm is developed to efficiently solve the proposed mixed fleet drayage routing problem.

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Asset Metadata

Creator Yao, Siyuan (author)

Core Title Models and algorithms for the freight movement problem in drayage operations

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Degree Conferral Date 2023-05

Publication Date 03/29/2023

Defense Date 03/01/2023

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag drayage operations,freight movement,mixed integer programming,OAI-PMH Harvest,Truck

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Dessouky, Maged (committee chair), Carlsson, John (committee member), Ioannou, Petros (committee member)

Creator Email siyuanya@usc.edu,syyao0908@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC112850146

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