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This paper investigates a dynamic and stochastic shipment matching problem faced by network operators in hinterland synchromodal transportation. We consider a platform that receives contractual and spot shipment requests from shippers, and receives multimodal services from carriers. The platform aims to provide optimal matches between shipment requests and multimodal services within a finite horizon under spot request uncertainty. Due to the capacity limitation of multimodal services, the matching decisions made for current requests will affect the ability to make good matches for future requests. To solve the problem, this paper proposes an anticipatory approach which consists of a rolling horizon framework that handles dynamic events, a sample average approximation method that addresses uncertainties, and a progressive hedging algorithm that generates solutions at each decision epoch. Compared with the greedy approach which is commonly used in practice, the anticipatory approach has total cost savings up to 8.18% under realistic instances. The experimental results highlight the benefits of incorporating stochastic information in dynamic decision making processes of the synchromodal matching system.
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Vol.:(0123456789)
Flexible Services and Manufacturing Journal
https://doi.org/10.1007/s10696-021-09428-5
1 3
Anticipatory approach fordynamic andstochastic
shipment matching inhinterland synchromodal
transportation
WenjingGuo1,2 · BilgeAtasoy1· WouterBeelaertsvanBlokland1·
RudyR.Negenborn1
Accepted: 30 June 2021
© The Author(s) 2021
Abstract
This paper investigates a dynamic and stochastic shipment matching problem faced
by network operators in hinterland synchromodal transportation. We consider a
platform that receives contractual and spot shipment requests from shippers, and
receives multimodal services from carriers. The platform aims to provide optimal
matches between shipment requests and multimodal services within a finite hori-
zon under spot request uncertainty. Due to the capacity limitation of multimodal
services, the matching decisions made for current requests will affect the ability to
make good matches for future requests. To solve the problem, this paper proposes
an anticipatory approach which consists of a rolling horizon framework that handles
dynamic events, a sample average approximation method that addresses uncertain-
ties, and a progressive hedging algorithm that generates solutions at each decision
epoch. Compared with the greedy approach which is commonly used in practice, the
anticipatory approach has total cost savings up to 8.18% under realistic instances.
The experimental results highlight the benefits of incorporating stochastic informa-
tion in dynamic decision making processes of the synchromodal matching system.
Keywords Synchromodal transportation· Dynamic shipment matching· Stochastic
spot requests· Anticipatory approach
* Wenjing Guo
guo.wenjing@courrier.uqam.ca
1 Department ofMaritime andTransport Technology, Delft University ofTechnology, Delft,
TheNetherlands
2 Department ofAnalytics, Operations andInformation Technologies, University ofQuebec
atMontreal andCIRRELT, Montreal, Canada
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W.Guo et al.
1 3
1 Introduction
Hinterland transportation is the movement of shipments between deep-sea ports
and inland terminals by trucks, trains, barges, or any combination of them (Stead-
ieSeifi etal. 2014). Typically, a hinterland transport system is made up of mul-
tiple stakeholders that interact with each other, including network operators,
shippers, carriers, terminal operators, and institutional authorities (Crainic etal.
2018). Network operators (e.g., logistics service providers and alliances formed
by multiple carriers) control the transport system. Shippers (e.g., manufacturers,
ocean carriers, and freight forwarders) generate freight transport demand and out-
source transport activities to network operators. Carriers (e.g., truck, train, and
barge companies) provide transport services and supply timely transport capac-
ity to network operators. Terminal operators handle transshipment operations at
terminals. Institutional authorities (e.g., governments and public administrations)
charge tax, give incentives, and regulate transport activities to network operators,
such as the charging of carbon emissions.
As shippers become more time-sensitive that require shipments to be deliv-
ered within tight time windows, trucks are used more often which contributes
to road traffic congestion, transport costs, and carbon emissions (Demir et al.
2016). However, due to the increasing environmental issues and the enforced
regulations, companies in the transport industry are required to control carbon
emissions (Demir etal. 2016). Synchromodal transportation, as an emerging and
attractive concept, aims to manage different types of shipments considering the
trade-off among costs, delays, and emissions through integrated real-time plan-
ning and synchronization of activities (Giusti etal. 2019). Under synchromodal-
ity, shippers only specify shipments’ origin, destination, volume, release time,
and due time, and leave the choice of modes, routes, and departure and arrival
times to network operators. For example, for time-sensitive shipments, network
operators can assign trucks for transportation; but if time available, barges, trains
or barge-truck can be assigned taking into account their impact on costs, time,
and emissions.
With the development of digitization in the logistics industry, increasing online
booking platforms have appeared in freight transportation, such as Uber Freight,
Quicargo, and Maersk Spot. In this paper, we consider a synchromodal match-
ing platform owned by a network operator (e.g., European Gateway Services or
Contargo) that receives contractual and spot shipment requests from shippers and
receives time-scheduled services (e.g., trains) and departure time-flexible ser-
vices (e.g., trucks) from carriers. The platform aims to provide optimal matches
between shipment requests and transport services over a given planning horizon.
Having a match between a shipment and a service means that the shipment will
be transported by the service from the service’s origin terminal to the service’s
destination terminal. The platform combines the matched services into ship-
ments’ itineraries.
In practice, container transport companies receive shipment requests from
both long-term contracts and spot markets (Meng et al. 2019). Different from
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Anticipatory approach fordynamic andstochastic shipment…
the contractual requests received from large shippers whose information is
known before the operational planning horizon, the information of spot requests
is unknown and revealed dynamically (Guo etal. 2020). The demand from the
spot market is influenced by many factors, such as global economy, seasonality,
fluctuations of freight rate, and competitions from other companies (Wang and
Meng 2021). Due to the capacity limitation of multimodal services, the capacity
assigned to current requests will be unavailable for future requests which might
be more profitable. Thanks to the advancements in information technologies,
such as increased use of sensors in transport infrastructures, communication tech-
nologies, open data sources, and data analytics, exploiting stochastic information
of spot requests is increasingly achievable (Gendreau etal. 2016). With the sto-
chastic information, network operators might hold some barge and train capaci-
ties available for spot requests which are predicted to be more profitable.
In this paper, we define the matching of shipments and services under spot request
uncertainty with the aim to minimize total costs over a given planning horizon as the
dynamic and stochastic shipment matching (DSSM) problem. The complexity of the
DSSM problem lies in three aspects. First, spot requests arrive in the platform in
real-time which calls for a dynamic approach that handles dynamic events. Second,
the stochastic information of spot requests is available which calls for a stochastic
approach that addresses uncertainties. Third, the computation complexity of the
optimization problem calls for an efficient algorithm that generates timely solutions
at each decision epoch.
In the literature, Guo et al. (2020) developed a myopic approach to solve the
DSSM problem which does not consider the stochasticity of spot requests. The
myopic approach involves a rolling horizon framework that handles dynamic events
and a preprocessing-based heuristic algorithm that generates timely solutions at
each decision epoch. As an extension of Guo etal. (2020), this paper proposes an
anticipatory approach to incorporate the stochastic information of spot requests in
the dynamic shipment matching processes. The anticipatory approach involves a
sample average approximation method that addresses spot request uncertainties and
a progressive hedging algorithm that solves the deterministic formulations at each
decision epoch of a rolling horizon framework.
The remainder of this paper is structured as follows. We briefly review the rel-
evant literature and specify our contributions in Sect.2. In Sect. 3, we describe the
DSSM problem. In Sect. 4, we design the rolling horizon framework, the sample
average approximation method, and the progressive hedging algorithm. In Sect.5,
we describe the experimental setup, and present the experimental results. Finally, in
Sect.6, we provide concluding remarks and directions for future research.
2 Literature review
In the past decades, because of economic factors and environmental concerns, differ-
ent management concepts have appeared in the literature and in the logistics indus-
try: multimodal, intermodal, co-modal and synchromodal transportation. While
multimodality refers to the utilization of multiple modes, intermodality emphasizes
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W.Guo et al.
1 3
the utilization of standardized loading units (i.e., containers), namely the vertical
integration of different modes (SteadieSeifi etal. 2014); co-modality focuses on the
optimal and sustainable utilization of different modes on their own or in combina-
tion, namely the horizontal integration of different modes. As an extension of inter-
modality and co-modality, synchromodality adds the (real-time) flexibility in plan-
ning when disturbances happen (Giusti etal. 2019).
The implementation of synchromodal transportation relies on collaboration
among stakeholders, information technologies, and integrated planning at differ-
ent decision levels. Typically, synchromodal transport planning can be divided into
three levels: strategic, tactical, and operational level. While strategic and tactical
planning focus on physical network design (e.g., hub location) and service network
design (e.g., service selection, service frequency) in long and medium time hori-
zons, operational planning deals with the routing of shipments under dynamic and
stochastic environments (Giusti etal. 2019).
In the literature, the majority of the studies (e.g., Ayar and Yaman 2011; Chang
2008; Moccia etal. 2010; van Riessen etal. 2014) related to synchromodal transport
planning are conducted in a static and deterministic environment, namely, all the
inputs are known beforehand and decisions do not change once they are set. How-
ever, in practice, there are many sources of uncertainties in synchromodal trans-
portation, such as demand uncertainty. With the growing amount of historical data,
the stochastic information about uncertainties is available. Incorporating stochastic
information in decision-making processes has been proven to have better perfor-
mance than the corresponding myopic approaches in many fields, such as vehicle
routing problems (Albareda-Sambola etal. 2014) and dial-a-ride problems (Schilde
etal. 2011).
In the field of stochastic synchromodal transport planning, Demir etal. (2016)
studied a green intermodal service network design problem with demand and travel
time uncertainties. In this study, the origins, destinations, time windows of ship-
ments are known in advance, but the actual demand (i.e, the number of contain-
ers) is uncertain. A sample average approximation method was proposed to gener-
ate robust plans. Hrušovský etal. (2016) proposed a hybrid approach combining a
deterministic model with a simulation model to investigate an intermodal transport
planning problem with travel time uncertainty. Sun etal. (2018) established a fuzzy
chance-constrained mixed integer nonlinear programming model to describe rail
service capacity uncertainty and road traffic congestion. Generally, stochastic trans-
port planning problems have the probability distributions of random variables and
the optimization process is performed before their realization. The transport plan
will not be updated after the realization, thus, it is often referred to as a-priori opti-
mization (Ritzinger etal. 2015).
The trend towards digitalization in transportation allows gathering real-time
information and thus dynamic decision making. In synchromodal transportation,
some input data are revealed during the execution of the plan. The most common
dynamic events are the arrival of new shipment requests, but demands and travel
times are possible dynamics as well. In the literature, Li etal. (2015) presented a
receding horizon intermodal container flow control approach to deal with the
dynamic transport demands and dynamic traffic conditions. Mes and Iacob (2015)
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Anticipatory approach fordynamic andstochastic shipment…
considered the real-time planning of shipment requests under a synchromodal net-
work with the objective to minimize costs, delays, and emissions. van Heeswijk
etal. (2016) proposed an online planning algorithm to schedule the transport of less
than truckload freight via intermodal networks. Guo etal. (2020) developed a rolling
horizon approach to handle shipment requests that arrive dynamically in a synchro-
modal matching platform.
The advances in information and communication technologies as well as the
computing power allow the incorporation of stochastic information of future events
in dynamic decision-making processes. Approaches for dynamic and stochastic
transport planning problems can be divided into two categories: methods based on
preprocessed decisions and methods based on online decisions. Solution approaches
in the first group (preprocessed decisions) determine the values and policies of
decision making before the execution of the transport plan (Ritzinger etal. 2015).
Therefore, possible states need to be constructed in advance and evaluated based
on possible dynamic events and stochastic information over a planning horizon.
For example, van Riessen etal. (2016) designed a decision tree to derive real-time
decision rules for suitable allocation of shipment requests to services. Rivera and
Mes (2017) proposed an algorithm based on approximate dynamic programming to
tackle the curse of dimensionality of a Markov decision process model. The second
group (online decisions) focuses on the computation when a dynamic event occurs.
Specifically, decisions are made online with respect to the current system state and
the available stochastic information. SteadieSeifi (2017) proposed a rolling hori-
zon approach to handle dynamic demands. At each iteration of the rolling horizon
framework, the author proposed a scenario-based two-stage stochastic programming
model to incorporate the stochastic information of future demands.
In this paper, we investigate the dynamic and stochastic shipment matching
(DSSM) problem in synchromodal transportation at the operational level. The for-
mulation characteristics of the DSSM problem include: (1) contractual and spot
shipment requests; (2) stochastic information of spot requests; (3) unsplittable ship-
ments, i.e., a shipment should be delivered as a whole; (4) soft time windows, i.e.,
delay in delivery is available but with a penalty; (5) capacitated and time-sched-
uled barge and train services; (6) departure time-flexible truck services with time-
dependent travel times; (7) transshipment operations at terminals; (8) minimizing
generalized costs which consist of transport costs, delay costs, and carbon tax over a
planning horizon. The formulation characteristics, solution approaches, and experi-
mental size of related articles are summarized in Table1.
Our work has three main contributions to the literature. First, we introduce the
stochasticity of spot requests in the dynamic shipment matching processes. Second,
we propose an anticipatory approach to solve the problem under realistic instances
in a reasonable time. The anticipatory approach uses a sample average approxima-
tion method to address spot request uncertainty and applies a progressive hedging
algorithm to get solutions at each decision epoch of a rolling horizon framework.
This approach enables to consider a large set of scenarios (within 1min of computa-
tion time) to more accurately represent the stochasticity and this in turn increases the
benefits of incorporating stochastic information in dynamic decision-making pro-
cesses. Third, thanks to the above developed methodologies we propose a platform
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W.Guo et al.
1 3
Table 1 Formulation characteristics, solution approaches and experiment size of related articles
a
Information of shipment requests consists of shipments’ origin, destination, container volume (i.e., demand), announce time, release time, and due time
b
All the articles consider time-scheduled barge or train services
c
C Costs, D Delays, E Emissions
d
SAA Sample average approximation method, SO Simulation-optimization, HA Hybrid algorithm, MILP Mixed integer linear programming, RHA Rolling horizon
approach, GA Greedy approach, CA Consolidation algorithm, DT Decision trees, ADP Approximate dynamic programming, STSP Scenario-based two-stage stochastic
programming, PHA Progressive hedging algorithm
e
Instances follow naming convention of I-a-b-c where a represents the number of terminals, b is the number of services, and c is the number of shipment requests
Articles Dynamic
information
a
Stochastic
information
a
Integrity Time windows Barge/train
services
b
Truck services Transshipment Objectives
c
Methods
d
Maximum
instance size
e
Demir etal.
(2016)
Demand,
travel times
Splittable Soft Capacitated Flexible
C, D, E SAA I-20-100-20;
I-20-250-5;
I-20-500-1
Hrušovský
etal. (2016)
Travel times Splittable Soft Capacitated Flexible
C, D, E SO I-20-250-20
Sun etal.
(2018)
Service capac-
ity
Unsplittable Soft Capacitated Flexible, time-
dependent
C, D, E MILP I-12-25-10
Li etal. (2015) Demand,
travel times
Splittable Capacitated Flexible
C RHA I-6-54-1
Mes and Iacob
(2015)
Shipment
requests
Unsplittable Soft Capacitated Flexible
C, D, E GA I-6-110-1728
van Heeswijk
etal. (2016)
Shipment
requests
- Unsplittable Hard Uncapacitated Flexible
C, D, E CA I-37-110-1006
Guo etal.
(2020)
Shipment
requests
Unsplittable Soft Capacitated Flexible, time-
dependent
C, D, E RHA I-10-116-1600
van Riessen
etal. (2016)
Shipment
requests
Demand Splittable Soft Capacitated Flexible C, D DT I-2-4-20
Rivera and
Mes (2017)
Shipment
requests
Shipment
requests
Splittable Hard Capacitated Flexible C ADP I-12-29-40
SteadieSeifi
(2017)
Demand Demand Splittable Hard Capacitated Scheduled C RHA, STSP I-20-400-200
This paper Shipment
requests
Shipment
requests
Unsplittable Soft Capacitated Flexible, time-
dependent
C, D, E RHA, SAA,
PHA
I-10-116-1600
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Anticipatory approach fordynamic andstochastic shipment…
in which companies can manage different types of shipments (e.g., time-sensitive
shipments) under a synchromodal network considering the trade-off among costs,
delays, and emissions. Such a platform provides the means for a more efficient,
effective and sustainable decision-making framework for transportation systems.
3 Problem description andpreprocessing procedures
In this section, we first describe the DSSM problem in detail, and then briefly pre-
sent the preprocessing procedures designed to reduce the computational complexity.
3.1 Problem description
We consider an online matching platform that receives contractual and spot ship-
ment requests from shippers, receives time-scheduled and departure-time flexible
services from carriers, and receives unlimited handling services (i.e., loading and
unloading) from terminal operators. Let N be the set of terminals. Let
lcbarge
i
,
lctrain
i
,
be the loading/unloading cost coefficient of barge, train, and truck services at
terminal
iN
, respectively. Let
ltbarge
i
,
lttrain
i
,
lttruck
i
be the loading/unloading time of
barge, train, and truck services at terminal
iN
, respectively. Let
cstorage
i
be the stor-
age cost coefficient at terminal
iN
. The
CO2
emissions-related cost coefficient is
set as
cemission
.
Let R be the set of shipment requests. Each shipment request
rR
is charac-
terized by its announce time
tannounce
r
(i.e., the time when the platform receives the
request), release time
trelease
r
(i.e., the time when the shipment is available for hinter-
land transportation) at origin terminal
or
, due time
tdue
r
(i.e., the time that the ship-
ment needs to be delivered) at destination terminal
dr
, expiry date
texpire
r
(i.e., the
time that the matching decisions for request r cannot be further postponed), and con-
tainer volume
ur
. Delay in delivery is available but with a penalty cost per container
per hour overdue
cdelay
r
.
Requests R can be divided into two groups: contractual requests
Rcontract
and spot requests
Rspot
. While
Rcontract
are known beforehand,
Rspot
are
unknown and revealed dynamically. However, the probability distributions
{𝜋o,𝜋d,𝜋u,𝜋t
announce
,𝜋t
release
,𝜋t
due
,𝜋t
expire
}
of spot requests’ origin, destination, volume,
announce time, release time, due time, and expiry date are assumed available from
historic data. In addition, shippers require their shipments to be transported as a
whole, and ask to receive the transport plan before shipments’ release time, namely
the expiry date is equal to the release time,
t
release
r
=t
expire
r
.
Let V be the set of transport services, all the services are received before the plan-
ning horizon. According to the time schedules, services can be divided into two
groups:
Time-scheduled barge and train services. Each barge or train service
vVbarge Vtrain
is characterized by its departure time
tdepature
v
at origin terminal
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W.Guo et al.
1 3
ov
, arrival time
tarrival
v
at destination terminal
dv
, free capacity
Uv
, transport cost
cv
and carbon emissions
ev
.
Departure time-flexible truck services. We view each truck service as a fleet
of trucks which has flexible departure times and an unlimited capacity. Thus, a
truck service might have multiple departure times for different shipments. Due
to traffic congestion at several time periods throughout a day, the travel time of
truck services is time-dependent (Ichoua etal. 2003). Therefore, each truck ser-
vice
vVtruck
is characterized by its origin terminal
ov
, destination terminal
dv
,
time-dependent travel time function
ttruck
v
(𝜏
)
, transport cost
cv
, and carbon emis-
sions
ev
.
The objective of the platform is to provide optimal online matches in total costs
between shipment requests and transport services over a planning horizon T. The
total costs consist of transit costs generated by using services, transfer costs and
storage costs generated at transshipment terminals, penalty costs caused by delay in
delivery, and carbon tax charged for services’ carbon emissions.
3.2 Preprocessing procedures
In this section, we briefly present the preprocessing procedures that aim to reduce
the computational complexity of the DSSM problem by identifying infeasible
matches between shipments and services. It consists of two steps: the preprocessing
of feasible path and the preprocessing of feasible matches.
Preprocessing of feasible path. We define a path p as a combination of services
in sequence. A path p is feasible if the services inside a combination satisfy time-
spatial compatibility. Specifically, for two consecutive services
vi,vi+1
within
path p, the destination of service
vi
must be the same as the origin of service
vi+1
;
the arrival time of service
vi
must be earlier than the departure time of service
vi+1
minus loading and unloading time at transshipment terminal
d
v
i
. The set P
denotes the collection of feasible paths.
Preprocessing of feasible matches. A match (r, p) means shipment r will be
transported by path p from its origin to its destination. A match between request
rR
and path p=
[
v
1
, ..., v
l]
P
is feasible if it satisfies time-spatial compat-
ibility:
Spatial compatibility. The origin terminal of shipment request r should be the
same as the origin of service
v1
; the destination of request r should be the
same as the destination of service
vl
.
Time compatibility. The release time of request r should be earlier than the
departure time of service
v1
minus loading time at origin terminal
or
.
Let
Pr
be the set of feasible paths for request r, and let
crp
denote the costs
of matching request r with path p including transport costs, delay costs and car-
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Anticipatory approach fordynamic andstochastic shipment…
bon tax. The details of the preprocessing procedures are presented in Guo etal.
(2020).
An illustrative example of shipment matching with feasible paths is shown in Fig.1.
Here, a shipment needs to be transported from origin terminal 1 to destination termi-
nal 8 after release time 00:00, and the due time of the shipment is 24:00. The ship-
ment can be matched with different paths (i.e., service combinations). Using feasible
path 1, the shipment will be loaded at origin terminal 1 and transported by a barge
service to transshipment terminal 5, and then the shipment is transferred to a train
service which delivers the shipment to its destination terminal.
The notation used in this paper is presented in Table2.
4 Solution approaches
In this section, we propose an anticipatory approach (AA) to solve the DSSM prob-
lem and use the myopic approach (MA) proposed by Guo etal. (2020) as a bench-
mark. Both the AA and the MA are implemented under a rolling horizon frame-
work. However, the MA is based on deterministic information only while the AA
incorporates stochastic information of future requests at each decision epoch, as
shown in Fig.2.
4.1 Myopic approach
The MA presented in Guo et al. (2020) utilizes a rolling horizon framework to
handle dynamic events, which is known as an efficient periodic re-optimization
Fig. 1 An illustrative example of shipment matching with different paths
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Table 2 Notation
Sets
RShipment requests received within a planning horizon,
R=
̂
R0
̂
R1...
̂
RT
̂
R0
Contractual requests that are received before the planning horizon
̂
Rt
Spot requests that are received during time interval
(t1, t]
,
t>0
Rt
Shipment requests that are received before stage t and will expire before stage
t+1
̄
Rt
Shipment requests that are active at stage t
𝜔rk
Set of sampled requests received at stage
kK={t+1, ..., min{t+H,T}}
under
scenario
𝛾∈{1, ..., 𝛤}
VTransport services within a planning horizon,
V=Vbarge Vtrain Vtruck
PFeasible paths
Pr
Feasible paths for shipment r
Prv
Feasible paths for shipment r including service v
NTerminals
Parameters
or
Origin terminal of shipment request
rR
dr
Destination terminal of shipment request
rR
ur
Container volume of shipment request
rR
tannounce
r
Announce time of shipment request
rR
trelease
r
Release time of shipment request
rR
tdue
r
Due time of shipment request
rR
texpire
r
Expiry date of shipment request
rR
ov
Origin terminal of service
vV,ovN
dv
Destination terminal of service
vV,dvN
tdepature
v
Departure time of time-scheduled service
vVbarge Vtrain
tarrival
v
Arrival time of time-scheduled service
vVbarge Vtrain
ttruck
v
(𝜏
)
Time-dependent travel time of truck service
vVtruck
cv
Transport cost of service
vV
per container
ev
Carbon emissions of service
vV
per container
Ut
v
Free capacity of service
vVbarge Vtrain
at stage
t∈{0, 1, ..., T}
crp
The cost of matching request
rR
with path
pP
TThe planning horizon,
t∈{0, 1, ..., T}
𝛤
Number of scenarios
HLength of prediction horizon
Niteration
Maximum iteration number
̄xt
rp
The ‘overall design vector’ for request
rRt
matching with path
pP
̄yt
rp
The ‘overall design vector’ for request
r
̄
R
t
R
t
matching with path
pP
𝜆t𝛾
rp
Lagrangian multipliers for request
rRt
matching with path
pP
̃
𝜆t𝛾
rp
Lagrangian multipliers for request
r
̄
RtRt
matching with path
pP
𝜌t𝛾
rp
Penalty factors for request
rRt
matching with path
pP
̃𝜌 t𝛾
rp
Penalty factors for request
r
̄
RtRt
matching with path
pP
𝜂
A small positive number designed to control the termination of simulations
𝛼
A constant designed to control the updating rate of penalty factors
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Anticipatory approach fordynamic andstochastic shipment…
approach for dynamic problems (e.g., Arslan etal. 2019; Najmi etal. 2017; Wang
and Kopfer 2015; Yang et al. 2004). The planning horizon is rolled forward to
incorporate the dynamically released information, and the process continues
until the end of the horizon. Under the MA, the system is optimized periodically
at pre-specified points in time called optimization times (i.e., decision epochs).
Let
̂
R
t={rR
|
t1<tannounce
r
t
}
be the set of spot requests received during
time interval
(t1, t]
,
t>0
. At decision epoch t, decisions for all active shipment
requests
̄
Rt
are made. Request r is active if it is already announced but not expired
yet, formally ̄
R
t={rR
|
tannounce
r
t,t
expire
r
>t
}
. However, the decision for request
r
̄
Rt
is fixed only if
r
Rt={rR
|
tannounce
r
t,t<t
expire
r
t+1
}
, namely the
request will expire before the next decision epoch. The platform will inform ship-
pers the decisions only if a match is fixed for them. Thus, only the matches fixed at
stage t have effects on the free capacity of service
vVbarge Vtrain
at stage
t+1
.
We define
xt
rp
as the binary variable which is 1 if request in
rRt
is matched with
path
pP
and define
yt
rp
as the binary variable which is 1 if request in
r
̄
RtRt
is
matched with path
pP
. Let
Prv
be the set of feasible paths for shipment request r
Table 2 (continued)
Random variables
Rspot
Spot requests received over the planning horizon. The probability distributions
{𝜋o,𝜋d,𝜋u,𝜋t
announce
,𝜋t
release
,𝜋t
due
,𝜋t
expire
}
are assumed known
Decision variables
xt
rp
A binary variable equal to 1 if request
rRt
is matched with path
pP
, 0 other-
wise
yt
rp
A binary variable equal to 1 if request
r
̄
RtRt
is matched with path
pP
, 0
otherwise
z𝛾k
rp
A binary variable equal to 1 if request
r𝜔𝛾k
is matched with path
pP
under
scenario
𝛾∈{1, ..., 𝛤}
at stage
kK
, 0 otherwise
xt𝛾
rp
Binary variable; 1 if request
rRt
is matched with path
pP
under scenario
𝛾
yt𝛾
rp
Binary variable; 1 if request
r
̄
R
t
R
t
is matched with path
pP
under scenario
𝛾
(a) (b)
Fig. 2 Illustration of the myopic approach and the anticipatory approach
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W.Guo et al.
1 3
including service v,
Prv ={pPr|vp}
. Under the MA, the objective function is
to minimize the total costs of the current-stage decisions made for active requests
̄
Rt
. The formulation of the DSSM problem at stage
t∈{0, 1, ..., T}
under the MA is:
subject to
Constraints (23) ensure that each request will be matched with one feasible path
only. Constraints (4) ensure that the total container volumes of shipments assigned
to service
vVbarge Vtrain
does not exceed its free capacity at stage t. Constraints
(5) represent that the free capacity of service
vVbarge Vtrain
at the next stage is
only influenced by the free capacity of service v at the current stage and the match-
ing decisions made for requests
Rt
which will expire before the next stage.
4.2 Anticipatory approach
In this section, we propose the AA to incorporate the stochastic information of
future requests at each decision epoch of the rolling horizon framework, in contrast
to the MA in which dynamic decisions are made based on deterministic information
only. The implementation of the AA for a synchromodal matching system is shown
in Algorithm1. Before the planning horizon, the system applies the preprocessing
of feasible path to get the set of feasible paths. At each decision epoch of the roll-
ing horizon framework, the system generates scenarios of future requests by ran-
domly sampling from their probability distributions, applies the preprocessing pro-
cedure to obtain feasible matches for active requests and sampled requests, utilizes
a sample average approximation method presented in Sect.4.2.1 to get deterministic
(1)
𝐏𝟏
min
xt,yt
rRt
pP
r
crpxt
rp +
r
̄
R
t
R
t
pP
r
crpyt
rp
(2)
pPr
xt
rp =1, rRt
,
(3)
p
P
r
yt
rp =1, r
̄
RtRt
,
(4)
r
Rt
pP
rv
urxt
rp +
r
̄
R
t
R
t
pP
rv
uryt
rp Ut
v,vVbarge Vtrain
,
(5)
U
t+1
v=Ut
v
r
Rt
p
P
rv
urxt
rp,vVbarge Vtrain
,
(6)
xt
rp ∈{
0, 1
}
,
r
R
t
,p
P
,
(7)
yt
rp
∈{0, 1},r
̄
R
t
R
t
,pP
.
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1 3
Anticipatory approach fordynamic andstochastic shipment…
formulations, and utilizes a progressive hedging algorithm presented in Sect.4.2.2
to generate solutions. The state of the system is updated based on the decisions
made for requests
Rt
. Then the system is rolled forward to obtain the decisions for
the next stage.
4.2.1 Sample average approximation method
The sample average approximation method is an approach to solve stochastic
optimization problems by generating scenarios. In this technique, the expected
objective function is approximated by a sample average estimate derived
from a random sample (Verweij et al. 2003). At decision epoch t, a sample
{𝜔1,𝜔2, ..., 𝜔𝛾, ..., 𝜔𝛤}
of
𝛤
scenarios is generated by randomly sampling from the
probability distributions of spot requests
{
𝜋
o,
𝜋
d,
𝜋
u,
𝜋
t
announce
,
𝜋
t
release
,
𝜋
t
due
,
𝜋
t
expire
}
.
For companies that do not have accurate probability distributions, scenar-
ios can also be sampled randomly from their historical operational data.
Each scenario includes a realization of shipment requests from stage
t+1
to
stage
t+H
,
𝜔𝛾={𝜔𝛾(t+1),𝜔𝛾(t+2), ..., 𝜔𝛾(t+H)}
. Here, H is the prediction hori-
zon that is just long enough to obtain good decisions at stage t. The expected
cost over the prediction horizon is approximated by the sample average func-
tion
1
𝛤𝛤
𝛾=1t+H
k=t+1r𝜔
𝛾k
pPr
crpz𝛾
k
rp
, which is an unbiased estimator of future
costs as the sample size
𝛤
goes to infinity and the prediction horizon
t+H=T
(Ruszczyński and Shapiro 2003). We define K as the set of predicted time stages
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W.Guo et al.
1 3
at stage t,
K={t+1, ..., min{t+H,T}},t∈{0, 1, ..., T1}
;
K=�when t=T
.
Let
z𝛾k
rp
be the binary variable which equals to 1 if request
r𝜔𝛾k
is matched with
path
pP
under scenario
𝛾∈{1, .., 𝛤}
at stage
kK
. The formulation of the
DSSM problem at stage t changes to:
subject to Constraints (23, 57),
In formulation
𝐏𝟐
,
xt
and
yt
are first-stage decisions which do not depend on the
scenarios,
zt
is the second-stage decision which depends on the corresponding sce-
narios. However, only
xt
will be implemented at each decision epoch,
yt
and
zt
will
be released after the optimization.
4.2.2 Progressive hedging algorithm
Formulation
𝐏𝟐
is a large-scale deterministic binary integer program which is
non-convex and highly complex to solve. In this section, we apply the progressive
hedging algorithm (PHA) to solve the formulation. The PHA is first proposed
by Rockafellar and Wets (1991) and has been implemented in many applications,
such as stochastic network design problems (Crainic et al. 2014) and stochas-
tic resource allocation problems (Watson and Woodruff 2010). It is a horizontal
decomposition method which decomposes
𝐏𝟐
by scenarios rather than by time
stages, and iteratively solves penalized version of the scenario-based subprob-
lems to gradually enforce implementability (also called non-anticipativity) (Gade
etal. 2016).
In
𝐏𝟐
, the condition that the first-stage decisions
xt
,
yt
must not depend on the
realization of random variables is implicit. In the PHA scheme, we write the non-
anticipativity constraints explicitly. We define
xt𝛾
rp
as the binary variable which
equals to 1 if request
rRt
is matched with path
pP
under scenario
𝛾
,
yt𝛾
rp
as
the binary variable which equals to 1 if request
r
̄
RtRt
is matched with path
pP
under scenario
𝛾
. Let
̄xt
and
̄yt
be the ‘overall design vector. The DSSM
problem is then reformulated as:
(8)
𝐏𝟐
min
xt,yt,zt
rRt
pP
r
crpxt
rp +
r
̄
R
t
R
t
pP
r
crpyt
rp +1
𝛤
𝛤
𝛾=1
kK
r𝜔𝛾k
pP
r
crpz𝛾
k
rp
(9)
pP
r
z𝛾k
rp =1, 𝛾∈{1, ..., 𝛤},kK,r𝜔𝛾k
,
(10)
r
Rt
pPrv
urxt
rp +
r̄
RtRt
pPrv
uryt
rp +
kK
r𝜔𝛾k
pPrv
urz𝛾k
rp Ut
v
,
𝛾∈{1, ..., 𝛤},vV
barge
V
train
,
(11)
z𝛾k
rp
∈{0, 1},𝛾∈{1, ..., 𝛤},kK,r𝜔
𝛾k
,pP
.
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Anticipatory approach fordynamic andstochastic shipment…
subject to
Constraints (1718) are the non-anticipatory constraints which stipulate that in all
feasible solutions, the first-stage decisions are not allowed to depend on scenarios.
Therefore, the newly added variables do not affect the optimal solution, and thus P3
is equivalent to P2.
Following the PHA scheme, we drop off the constant coefficient
𝛤1
, and move the
non-anticipativity constraints (1718) into the objective function based on augmented
Lagrangian strategy, which yields the objective function as follows:
(12)
𝐏𝟑
min
xt,yt,zt
1
𝛤
𝛤
𝛾=1
rRt
pPr
crpxt𝛾
rp +
r̄
RtRt
pPr
crpyt𝛾
rp +
kK
r𝜔𝛾k
pPr
crpz𝛾k
rp
(13)
pPr
xt𝛾
rp =1, 𝛾∈{1, ..., 𝛤},rRt
,
(14)
pPr
yt𝛾
rp =1, 𝛾∈{1, ..., 𝛤},r̄
RtRt
,
(15)
pPr
z𝛾k
rp =1, 𝛾∈{1, ..., 𝛤},kK,r𝜔𝛾k
,
(16)
r
Rt
pPrv
urxt𝛾
rp +
r̄
RtRt
pPrv
uryt𝛾
rp +
kK
r𝜔𝛾k
pPrv
urz𝛾k
rp Ut
v
,
𝛾∈{1, ..., 𝛤},vV
barge
V
train
,
(17)
xt𝛾
rp
=̄x
t
rp
,𝛾∈{1, ..., 𝛤},rR
t
,pPr
,
(18)
yt𝛾
rp
=̄y
t
rp
,𝛾∈{1, ..., 𝛤},r
̄
R
t
R
t
,pPr
,
(19)
U
t+1
v=Ut
v
r
Rt
p
P
rv
ur
̄xt
rp,vVbarge Vtrain
,
(20)
xt𝛾
rp ∈{
0, 1
}
,
𝛾
∈{
1, ..., 𝛤
}
,r
R
t
,p
P
,
(21)
yt𝛾
rp
∈{0, 1},𝛾∈{1, ..., 𝛤},r
̄
R
t
R
t
,pP
,
(22)
z𝛾k
rp
∈{0, 1},𝛾∈{1, ..., 𝛤},kK,r𝜔
𝛾k
,pP
.
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W.Guo et al.
1 3
subject to Constraints (1316, 1922).
In formulation P4,
𝜆t𝛾
rp
and
̃
𝜆t𝛾
rp
are Lagrangian multipliers,
𝜌t𝛾
rp
and
̃𝜌 t𝛾
rp
are penalty fac-
tors. Given the binary requirements for variables
xt
,yt
,̄xt
,̄yt
, the objective function can
be further formulated as:
subject to Constraints (1316, 1922).
For a given overall design
̄xt
,
̄yt
, the relaxed formulation P5 is separable on a sce-
nario basis. As it contains
𝛤
scenarios, it can be broken down into
𝛤
individual sub-
problems. An arbitrary subproblem indexed by
𝛾∈{1, ..., 𝛤}
by dropping constant
terms has the following form:
subject to
(23)
𝐏𝟒
min
xt,yt,zt
𝛤
𝛾=1
rRt
pPr
crpxt𝛾
rp +
r̄
RtRt
pPr
crpyt𝛾
rp +
kK
r𝜔𝛾k
pPr
crpz𝛾
k
rp
+
rRt
pPr
𝜆t𝛾
rpxt𝛾
rp ̄xt
rp+1
2
rRt
pPr
𝜌t𝛾
rpxt𝛾
rp ̄xt
rp2
+
r̄
RtRt
pPr
̃
𝜆t𝛾
rp
yt𝛾
rp ̄yt
rp
+1
2
r̄
RtRt
pPr
̃𝜌 t𝛾
rp
yt𝛾
rp ̄yt
rp
2
(24)
𝐏𝟓
min
xt,yt,zt
𝛤
𝛾=1
rRt
pPr
crp +𝜆t𝛾
rp +1
2𝜌t𝛾
rp 𝜌t𝛾
rp ̄xt
rp
xt𝛾
rp 𝜆t𝛾
rp ̄xt𝛾
rp +1
2𝜌t𝛾
rp ̄xt
rp
+
r̄
RtRt
pPrcrp +̃
𝜆t𝛾
rp +1
2̃𝜌 t𝛾
rp ̃𝜌 t𝛾
rp ̄yt
rpyt𝛾
rp ̃
𝜆t𝛾
rp ̄yt𝛾
rp +1
2̃𝜌 t𝛾
rp ̄yt
rp
+
kK
r𝜔
𝛾k
pP
r
crpz𝛾k
rp
(25)
𝐏𝟔
min
xt𝛾,yt𝛾,zt𝛾
rRt
pPr
(
crp +𝜆t𝛾
rp +
1
2𝜌t𝛾
rp 𝜌t𝛾
rp ̄xt
rp
)
xt𝛾
rp
+
r̄
RtRt
pPr
(crp +̃
𝜆t𝛾
rp +1
2̃𝜌 t𝛾
rp ̃𝜌 t𝛾
rp ̄yt
rp)yt𝛾
rp
+
k
K
r𝜔
𝛾k
p
P
r
crpz𝛾k
rp
(26)
pPr
xt𝛾
rp =1, rRt
,
(27)
pPr
yt𝛾
rp =1, r̄
RtRt
,
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Anticipatory approach fordynamic andstochastic shipment…
Formulation
𝐏𝟔
is a scenario-based binary integer program which can be solved by
using commercial solvers within an acceptable computational time, such as CPLEX.
For a given scenario subproblem
𝛾
, the Lagrangian multiplier
𝜆t𝛾
rp
(
̃
𝜆
t𝛾
rp
) and the
penalty parameter
𝜌t𝛾
rp
(
̃𝜌 t𝛾
rp
) contribute to penalize the difference in terms of values
between the local variable
xt𝛾
rp
(
yt𝛾
rp
) and the current overall design
̄xt
rp
(
̄yt
rp
).
The pseudocode of the PHA at each decision epoch is shown in Algorithm2.
Each iteration of the PHA involves an optimization (Step 2) for scenario-based sub-
problems, an aggregation (Step 3) which corresponds to a projection of the individ-
ual scenario solutions onto the subspace of non-anticipative policies, a termination
criteria (Step 4) to make sure the algorithm converges to within a tolerance, and a
modification (Step 5) to update multipliers.
The key to success in implementing the PHA under a rolling horizon framework
is to choose a proper
𝜌
-value to avoid slow convergence. However, in the literature,
there are no conclusive results on the selection of
𝜌
-value. In this paper, we choose
the
𝜌
in proportion to the matching cost of the associated request and path, namely
𝜌t
rp
=𝛼crp
for
rRt
,
pP
. This method will be evaluated in the experiments in
comparison to a commonly used method in container transportation
𝜌n+1=𝛼𝜌n
(Crainic etal. 2011; Dong etal. 2015).
(28)
p
P
r
z𝛾k
rp =1, kK,r𝜔𝛾k
,
(29)
r
Rt
pPrv
urxt𝛾
rp +
r̄
RtRt
pPrv
uryt𝛾
rp +
kK
r𝜔𝛾k
pPrv
urz𝛾k
rp Ut
v
,
vV
barge
V
train
,
(30)
xt𝛾
rp
∈{0, 1},rR
t
,pP
,
(31)
yt𝛾
rp
∈{0, 1},r
̄
R
t
R
t
,pP
,
(32)
z𝛾k
rp
∈{0, 1},kK,r𝜔
𝛾k
,pP
.
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W.Guo et al.
1 3
5 Numerical experiments
In this section, we evaluate the performance of the anticipatory approach (AA) on
the DSSM problem in comparison to the myopic approach (MA) proposed by Guo
etal. (2020) and the commonly used greedy approach (GA) in the container trans-
port industry (van Riessen et al. 2016). The GA is sometimes also referred to as
a first come first served approach (Meng etal. 2019). Under the GA, a shipment
request is assigned to the cheapest feasible path at the time of request arrival. To
provide a theoretical lower bound of the AA, we also report the optimal solutions
obtained when all the input information is known beforehand. The approaches are
implemented in MATLAB, and all experiments are executed on 3.70 GHz Intel
Xeon processors with 32 GB of RAM. The optimization problems are solved with
CPLEX 12.6.3.
5.1 Experimental setup
In this paper, we use the hinterland synchromodal network designed by Guo et al.
(2020) for the numerical experiments, which includes 3 deep-sea terminals in the
port of Rotterdam (i.e., node 1, 2, and 3) and 7 inland terminals in the Netherlands,
Belgium, and Germany (i.e., node 4, 5, 6, 7, 8, 9, and 10), as shown in Fig.3. The
network consists of 116 services, including 49 barge services, 33 train services,
and 34 truck services. The detailed information of the services is presented in the
Appendix.
We generate several instances to represent different characteristics of shipment
requests within a given planning horizon. Each shipment request is characterized by
its origin, destination, container volume, announce time, release time, expiry date,
and due time. We assume that:
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1 3
Anticipatory approach fordynamic andstochastic shipment…
the origins of shipments are independent and identically distributed among
{1, 2, 3}
with probabilities
{0.66, 0.2, 0.14}
; the destinations of shipments are
independent and identically distributed among
{4, 5, 6, 7, 8, 9, 10}
with probabili-
ties
{0.306, 0.317, 0.153, 0.076, 0.071, 0.034, 0.043}
;
the container volumes of shipment requests which arrive before the planning
horizon (i.e., contractual requests) are drawn independently from a uniform
distribution with range [10, 30], the average container volume of contractual
requests
UAVE
1
=
20
; the container volumes of spot requests are drawn indepen-
dently from uniform distributions with range [1,9], the average container vol-
ume of spot requests
UAVE
2
=
5
;
the announce time of contractual requests is 0, while the frequency of spot
requests arriving in the system belongs to Poisson distributions with mean
AT AVE
;
the release time of contractual requests is drawn independently from a uniform
distribution with range [1, 120]; the release time of spot requests is generated
based on its announce time,
trelease
r
=
t
announce
r
+𝛥T ,
𝛥T
belongs to a uniform
distribution with range [1,6]; the expiry date is equal to the release time;
the due time of shipment requests is generated based on its release time and
lead time,
tdue
r
=t
release
r
+LD
r
, the lead time of shipments is independent and
identically distributed among
{24, 48, 72}
(unit: hours) with probabilities
{0.15, 0.6, 0.25}
. The delay cost coefficients of shipments with lead time 24, 48,
and 72h are 100, 70, and 50 €/h-TEU, respectively.
We use
EU n1n2
to represent an instance with
n1
contractual requests and
n2
spot requests. We set
AT AVE
to 20, 10, 6, 5, and 4min (i.e., about 0.33, 0.17, 0.1,
0.08, and 0.07h per request) for instances EU-300-400, EU-200-800, EU-100-1200,
EU-50-1400, and EU-0-1600, respectively, as shown in Fig.4. We define the degree
of dynamism as the ratio between the number of containers from spot requests and
Fig. 3 The topology of the hinterland synchromodal network derived from Guo etal. (2020)
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W.Guo et al.
1 3
the total number of containers, namely, degree of dynamism= n2U
AVE
2
n
1UAVE
1
+n2UAVE
2
. There-
fore, the degrees of dynamism for instances EU-300-400, EU-200-800, EU-100-
1200, EU-50-1400, and EU-0-1600 are 25%, 50%, 75%, 87.5%, and 100%,
respectively.
The length of the planning horizon is set to 168h for all the instances. The length
of the optimization interval is set to 1h in the MA and the AA. At each decision
epoch of the AA, a sample is generated randomly based on the probability distribu-
tions presented above. In case of sample instability, for each instance, we replicate
the optimization process 10 times under the AA.
5.2 Impact ofthedegree ofdynamism
To test the influence of the degree of dynamism, we set the number of scenarios
to 10, and the length of prediction horizon to 12h. We use ‘gaps in total costs’ as
the performance indicator which is given by (benchmark value - objective value)/
benchmark value. Here, the total cost generated by the MA is the benchmark value,
while the total cost generated by the AA is the objective value. Therefore, the higher
the ‘gaps in total costs’, the better the performance of the AA in reducing total costs.
Fig.5 shows that the AA has better performance than the MA in all the instances
in reducing total costs, and the gap between the AA and the MA grows with the
increasing of the degree of dynamism from 25% to 87.5%. Nevertheless, further
increasing the degree of dynamism to 100%, the gap in total costs stays around 4%.
Fig. 4 Arrival frequency of instances
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Anticipatory approach fordynamic andstochastic shipment…
5.3 Impact ofthenumber ofscenarios andthelength ofprediction horizon
With regards to the number of scenarios, we set the degree of dynamism to
87.5% (i.e., instance EU-50-1400), and the length of prediction horizon to 12h.
The number of scenarios is varied from 1 to 30. Figure6a shows that increas-
ing the number of scenarios, the gap in total costs between the AA and the MA
becomes larger. The reason is that the larger the number of scenarios, the more
accurate the representation of the future. On the other hand, we set the number
of scenarios to 10, and vary the length of prediction horizon from 1 to 24h for
instance EU-50-1400. Figure6b shows that the length of prediction horizon has
high influences on the performance of the AA in reducing total costs. The longer
the prediction horizon, the more the stochastic information of future requests
Fig. 5 Impact of the degree of dynamism
Fig. 6 Impact of the number of scenarios and the length of prediction horizon
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W.Guo et al.
1 3
will be considered. The system thus reserves capacities for predicted future
requests which are more ‘valuable. In turn, the performance of the system over
the planning horizon becomes better.
5.4 Impact oftheselection of
‑value
To test the impact of the selection of
𝜌
-value, we design 10 instances with dif-
ferent number of requests, different number of scenarios and different length of
prediction horizon. The proposed cost proportional method (i.e.,
𝜌rp
=𝛼crp
) is
evaluated in comparison to the typical iterative method (i.e.,
𝜌n+1=̂𝛼 𝜌 n
). We set
𝛼=1, ̂𝛼 =1.1, 𝜌0=1
. Table3 shows that the costs generated by these two methods
are almost the same in all the instances. However, the number of iterations (i.e., N.
Iteration) and the computation time (i.e., CPU) under the typical iterative method
are way much higher than the cost proportional method. The larger the number of
scenarios and the length of prediction horizon, the higher the gaps between these
two methods. We also notice that the CPU increases dramatically with the increase
of shipment requests under the typical iterative method. In comparison, all these
instances can be solved by the cost proportional method within 20s. With the cost
proportional method, the PHA can be implemented under a rolling horizon frame-
work to provide timely solutions at each decision epoch.
5.5 Comparison betweentheGA, theMA, andtheAA
In this section, we test the performance of the AA in comparison to the MA and the
GA. While the result obtained from the GA provides an upper bound of the AA, we
Table 3 Impact of the selection of
𝜌
-value
Instances
𝛤
H
𝜌rp
=𝛼crp
𝜌
n
+1
=
̂𝛼 𝜌 n
Costs (€) N. iteration CPU (s) Costs (€) N. iteration CPU (s)
EU-50-0 5 6 144553 2 1.25 144549 35 29.43
EU-50-0 10 12 195831 3 2.66 195810 51 44.39
EU-50-0 10 24 283651 2 4.10 283654 50 99.60
EU-50-0 10 48 434789 3 17.95 434763 50 275.49
EU-50-0 30 12 189193 2 1.84 189194 48 137.82
EU-50-0 30 24 286633 2 4.10 286631 52 111.17
EU-50-0 30 48 438920 3 17.39 442021 94 589.13
EU-100-0 10 12 292268 2 2.20 292274 27 31.36
EU-200-0 10 12 422272 3 3.65 422273 69 87.17
EU-300-0 10 12 634021 5 13.06 632923 60 182.77
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Anticipatory approach fordynamic andstochastic shipment…
Table 4 Comparison between the GA, the MA, and the AA
Instances GA MA AA Theoretical lower bounds
Costs CPU Costs CPU Improv (%) Costs CPU Improv (%) Costs CPU Improv (%)
EU-300-400 975784 0.02 965934 0.76 1.01 945127 64.42 3.14 937292 8.22 3.94
EU-200-800 971170 0.02 933842 0.59 3.84 912194 13.70 6.07 900644 16.98 7.26
EU-100-1200 1004494 0.02 980743 0.42 2.36 922322 29.49 8.18 914585 29.93 8.95
EU-50-1400 994406 0.02 972295 0.41 2.22 924220 44.12 7.06 892168 42.33 10.28
EU-0-1600 971799 0.01 948457 0.32 2.40 912294 62.04 6.12 841668 56.16 13.39
Average 2.37 6.12 8.77
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W.Guo et al.
1 3
use the solutions obtained when all the input information is known in advance as
the theoretical lower bounds. Specifically, we assume all the contractual and spot
requests are received before the planning horizon, which gives rise to an optimi-
zation problem that includes all the shipments and services. Due to the computa-
tional complexity, the problem is solved by the heuristic algorithm designed in Guo
etal. (2020). We set
𝛤=100, H=48, Niteration =100, 𝛼=1, 𝜂=0.001
for the AA.
The comparison between the GA, the MA, and the AA is shown in Table4. We
consider three performance indicators: the total costs (€), the ave.CPU (s), and the
improvements. The ave.CPU of the GA, the MA, and the AA is the average com-
putation time per stage over the planning horizon (i.e., 168 time stages). Although
the AA needs to solve a large number of subproblems at each decision epoch due to
the iteration of Lagrangian multipliers, applying the parallel computing techniques
enables to use multiple CPUs to solve the subproblems in a single iteration of the
AA simultaneously. We use the results obtained from the GA as the benchmark, the
improvements between the MA/AA and the GA are given by (benchmark value -
objective value)/benchmark value. Table4 shows that the AA outperforms the GA
and the MA in all the instances. While the MA has average improvements of about
2.37% in comparison to the GA, the AA has average improvements of about 6.12%.
Impressively, we notice that with the designed AA, the gap between the AA and the
theoretical lower bounds is no more than 2.65% on average.
6 Conclusions andfuture research
In this paper, we introduced a dynamic and stochastic shipment matching (DSSM)
problem in hinterland synchromodal transportation. The problem is considered
dynamic since spot requests arrive in the system in real-time. The problem is con-
sidered stochastic since the information of spot requests is not known with certainty.
To solve the problem, we developed an anticipatory approach (AA) which uses a
sample average approximation method to address spot request uncertainties and a
progressive hedging algorithm to generate solutions at each decision epoch of a roll-
ing horizon framework.
We validated the performance of the AA on the DSSM problem in comparison
with the myopic approach (MA) proposed by Guo etal. (2020) in which dynamic
decisions are made based on deterministic information only and the greedy approach
(GA) which is commonly used in practice. The experimental results indicate that the
AA outperforms the GA and the MA in all the instances of the synchromodal match-
ing system. Compared with the GA, the AA has total cost savings up to 8.18%.
From a managerial viewpoint, with the proposed AA, the utilization of barges,
trains, and trucks can be managed more efficiently by taking into account the time-
sensitivity of current received requests and the predicted future requests. Besides,
the proposed approach enables the decision makers to dynamically update the deci-
sions of the previously received shipments when the newly received ones can be
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Anticipatory approach fordynamic andstochastic shipment…
better served with the previously matched services. This increases the adaptive
nature of transport systems to meet today’s environment. Furthermore, the experi-
mental results show that the more the stochastic information is incorporated, the bet-
ter the performance of the AA. However, the computational complexity increases
with the increase of stochastic information. To implement such an approach in prac-
tice, the trade-off between solution quality and computational complexity must be
considered.
Future research can be conducted under three directions. First, due to the capac-
ity limitation of road infrastructures, the number of trucks is limited in a synchro-
modal network. Therefore, the rejection of shipment requests can be considered in
the online matching processes to avoid infeasible solutions. Another research direc-
tion is to investigate the benefits of incorporating ad hoc services (i.e., dynamic ser-
vices). Considering the excess capacity of services from carriers, the online match-
ing of contractual requests, spot requests, dedicated services, and ad hoc services
gives rise to a new variant of the dynamic shipment matching problem in synchro-
modal transportation. Third, due to the existence of traffic congestion and terminal
congestion in synchromodal transportation, travel time of services and transfer time
at terminals are usually uncertain. Combining multiple uncertainties in dynamic
shipment matching is a promising research direction.
Appendix
The detailed information of barge, train and truck services is presented in Tables5,
6 and 7. The barge and train connections are derived from European Gateway Ser-
vices (http:// www. europ eanga teway servi ces. com/ en/). We assume there exists truck
connections between all the terminals. The distance of services used in this paper is
obtained from European Gateway Services, InlandLinks (https:// www. inlan dlinks.
eu/ en), and Google maps.
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W.Guo et al.
1 3
Table 5 Barge services in the numerical experiments
Barge services Origin Destination Capacity (TEU) Departure time Arrival time Transit time (h) Transit cost
(€/TEU)
Distance (km) Carbon emis-
sions (kg/
TEU)
1 Delta Euromax 160 53 54 1 2.10 15 3.43
2 Delta HOME 160 53 55.5 2.5 5.25 37.5 8.58
3 Delta Moerdijk 160 3 8 5 10.50 75 17.16
4 Delta Moerdijk 160 15 20 5 10.50 75 17.16
5 Delta Moerdijk 160 27 32 5 10.50 75 17.16
6 Delta Moerdijk 160 39 44 5 10.50 75 17.16
7 Delta Moerdijk 160 51 56 5 10.50 75 17.16
8 Delta Moerdijk 160 63 68 5 10.50 75 17.16
9 Delta Moerdijk 160 75 80 5 10.50 75 17.16
10 Delta Moerdijk 160 87 92 5 10.50 75 17.16
11 Delta Moerdijk 160 99 104 5 10.50 75 17.16
12 Delta Moerdijk 160 111 116 5 10.50 75 17.16
13 Delta Moerdijk 160 123 128 5 10.50 75 17.16
14 Delta Moerdijk 160 135 140 5 10.50 75 17.16
15 Delta Moerdijk 160 147 152 5 10.50 75 17.16
16 Delta Moerdijk 160 159 164 5 10.50 75 17.16
17 Delta Venlo 160 12 25 13 27.30 195 44.62
18 Delta Venlo 160 18 31 13 27.30 195 44.62
19 Delta Venlo 160 36 49 13 27.30 195 44.62
20 Delta Venlo 160 42 55 13 27.30 195 44.62
21 Delta Venlo 160 60 73 13 27.30 195 44.62
22 Delta Venlo 160 66 79 13 27.30 195 44.62
23 Delta Venlo 160 90 103 13 27.30 195 44.62
24 Delta Venlo 160 96 109 13 27.30 195 44.62
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Anticipatory approach fordynamic andstochastic shipment…
Table 5 (continued)
Barge services Origin Destination Capacity (TEU) Departure time Arrival time Transit time (h) Transit cost
(€/TEU)
Distance (km) Carbon emis-
sions (kg/
TEU)
25 Delta Venlo 160 120 133 13 27.30 195 44.62
26 Delta Duisburg 160 82 98 16 33.60 240 54.91
27 Delta Duisburg 160 102 118 16 33.60 240 54.91
28 Delta Willebroek 160 68 79 11 23.10 165 37.75
29 Delta Willebroek 160 98 109 11 23.10 165 37.75
30 Delta Willebroek 160 146 157 11 23.10 165 37.75
31 Delta Neuss 160 80 97 17 35.70 255 58.34
32 Euromax Moerdijk 160 3 8.5 5.5 11.55 82.5 18.88
33 Euromax Moerdijk 160 51 56.5 5.5 11.55 82.5 18.88
34 Euromax Moerdijk 160 99 104.5 5.5 11.55 82.5 18.88
35 Euromax Venlo 160 27 40.5 13.5 28.35 202.5 46.33
36 Euromax Venlo 160 75 88.5 13.5 28.35 202.5 46.33
37 Euromax Duisburg 160 103 119.5 16.5 34.65 247.5 56.63
38 Euromax Willebroek 160 112 123.5 11.5 24.15 172.5 39.47
39 Euromax Neuss 160 66 83.5 17.5 36.75 262.5 60.06
40 HOME Moerdijk 160 5 8 3 6.30 45 10.30
41 HOME Moerdijk 160 53 56 3 6.30 45 10.30
42 HOME Moerdijk 160 101 104 3 6.30 45 10.30
43 HOME Venlo 160 99 110 11 23.10 165 37.75
44 HOME Venlo 160 126 137 11 23.10 165 37.75
45 HOME Duisburg 160 51 66.5 15.5 32.55 232.5 53.20
46 HOME Willebroek 160 20 30.5 10.5 22.05 157.5 36.04
47 Moerdijk Venlo 160 95 105 10 21.00 150 34.32
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W.Guo et al.
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Table 5 (continued)
Barge services Origin Destination Capacity (TEU) Departure time Arrival time Transit time (h) Transit cost
(€/TEU)
Distance (km) Carbon emis-
sions (kg/
TEU)
48 Moerdijk Duisburg 160 71 83 12 25.20 180 41.18
49 Duisburg Neuss 160 120 122.5 2.5 5.25 37.5 8.58
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Anticipatory approach fordynamic andstochastic shipment…
Table 6 Train services in the numerical experiments
Train services Origin Destination Capacity
(TEU)
Departure time Arrival time Transit time (h) Transit cost
(€/TEU)
Distance (km) Carbon emis-
sions (kg/
TEU)
1 Delta Venlo 90 16 20 4 30.33 180 56.63
2 Delta Venlo 90 40 44 4 30.33 180 56.63
3 Delta Venlo 90 9 13 4 30.33 180 56.63
4 Delta Venlo 90 33 37 4 30.33 180 56.63
5 Delta Venlo 90 57 61 4 30.33 180 56.63
6 Delta Venlo 90 81 85 4 30.33 180 56.63
7 Delta Venlo 90 105 109 4 30.33 180 56.63
8 Delta Venlo 90 129 133 4 30.33 180 56.63
9 Delta Duisburg 90 41 47 6 44.73 270 84.94
10 Delta Duisburg 90 75 81 6 44.73 270 84.94
11 Delta Duisburg 90 99 105 6 44.73 270 84.94
12 Delta Duisburg 90 113 119 6 44.73 270 84.94
13 Delta Neuss 90 110 115 5 37.53 225 70.79
14 Delta Dortmund 90 88 95 7 51.93 315 99.10
15 Delta Nuremberg 90 51 66 15 109.53 675 212.36
16 Delta Nuremberg 90 99 114 15 109.53 675 212.36
17 Euromax Venlo 90 78 82.5 4.5 33.93 202.5 63.71
18 Euromax Venlo 90 102 106.5 4.5 33.93 202.5 63.71
19 Euromax Duisburg 90 75 81.5 6.5 48.33 292.5 92.02
20 Euromax Duisburg 90 99 105.5 6.5 48.33 292.5 92.02
21 Euromax Neuss 90 77 82.5 5.5 41.13 247.5 77.86
22 Euromax Dortmund 90 78 85.5 7.5 55.53 337.5 106.18
23 Euromax Nuremberg 90 79 94.5 15.5 113.13 697.5 219.43
24 HOME Venlo 90 86 89.5 3.5 26.73 157.5 49.55
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W.Guo et al.
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Table 6 (continued)
Train services Origin Destination Capacity
(TEU)
Departure time Arrival time Transit time (h) Transit cost
(€/TEU)
Distance (km) Carbon emis-
sions (kg/
TEU)
25 HOME Duisburg 90 27 32.5 5.5 41.13 247.5 77.86
26 HOME Duisburg 90 75 80.5 5.5 41.13 247.5 77.86
27 Moerdijk Venlo 90 75 78 3 23.13 135 42.47
28 Moerdijk Duisburg 90 77 81 4 30.33 180 56.63
29 Venlo Neuss 90 112 113.5 1.5 12.33 67.5 21.24
30 Venlo Dortmund 90 113 115.5 2.5 19.53 112.5 35.39
31 Venlo Nuremberg 90 114 125 11 80.73 495 155.73
32 Duisburg Dortmund 90 121 122.5 1.5 12.33 67.5 21.24
33 Duisburg Nuremberg 90 122 132 10 73.53 450 141.57
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Anticipatory approach fordynamic andstochastic shipment…
Table 7 Truck services in the numerical experiments
Truck services Origin Destination Transit time (h) Transit
cost (€/
TEU)
Distance (km) Carbon emis-
sions (kg/
TEU)
1 Delta Euromax 0.2 92.00 15 13.30
2 Delta HOME 0.5 115.40 37.5 33.25
3 Delta Moerdijk 1.0 154.40 75 66.50
4 Delta Venlo 2.6 279.20 195 172.89
5 Delta Duisburg 3.2 326.00 240 212.78
6 Delta Willebroek 2.0 232.40 150 132.99
7 Delta Neuss 3.5 349.40 262.5 232.73
8 Delta Dortmund 4.0 388.40 300 265.98
9 Delta Nuremberg 9.0 778.40 675 598.46
10 Euromax HOME 0.6 123.20 45 39.90
11 Euromax Moerdijk 1.2 170.00 90 79.79
12 Euromax Venlo 2.8 294.80 210 186.19
13 Euromax Duisburg 3.3 333.80 247.5 219.43
14 Euromax Willebroek 2.2 248.00 165 146.29
15 Euromax Neuss 3.6 357.20 270 239.38
16 Euromax Dortmund 4.2 404.00 315 279.28
17 Euromax Nuremberg 9.5 817.40 712.5 631.70
18 HOME Moerdijk 0.6 123.20 45 39.90
19 HOME Venlo 2.3 255.80 172.5 152.94
20 HOME Duisburg 2.7 287.00 202.5 179.54
21 HOME Willebroek 1.5 193.40 112.5 99.74
22 HOME Neuss 3.0 310.40 225 199.49
23 HOME Dortmund 3.4 341.60 255 226.08
24 HOME Nuremberg 8.8 762.80 660 585.16
25 Moerdijk Venlo 1.8 216.80 135 119.69
26 Moerdijk Duisburg 2.4 263.60 180 159.59
27 Moerdijk Willebroek 1.4 175.20 95 84.23
28 Venlo Duisburg 0.8 138.80 60 53.20
29 Venlo Neuss 0.9 146.60 67.5 59.85
30 Venlo Dortmund 1.5 193.40 112.5 99.74
31 Venlo Nuremberg 6.6 591.20 495 438.87
32 Duisburg Neuss 0.5 115.40 37.5 33.25
33 Duisburg Dortmund 0.9 146.60 67.5 59.85
34 Duisburg Nuremberg 6 544.40 450 398.97
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W.Guo et al.
1 3
Acknowledgements This research is supported by the China Scholarship Council under Grant
201606950003, the project “Complexity Methods for Predictive Synchromodality” (Project 439.16.120)
of the Netherlands Organisation for Scientific Research (NWO), and the Natural Sciences and Engineer-
ing Council of Canada (NSERC) through its Cooperative Research and Development Grants Program.
Declarations
Conflict of interest The authors declare that they have no conflict of interest.
Ethical approval This article does not contain any studies with human participants or animals performed
by any of the authors.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as
you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-
mons licence, and indicate if changes were made. The images or other third party material in this article
are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the
material. If material is not included in the article’s Creative Commons licence and your intended use is
not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission
directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen
ses/ by/4. 0/.
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maps and institutional affiliations.
Wenjing Guo is a postdoctoral researcher at CIRRELT and UQAM. Her research interests include opera-
tions research, intermodal transportation, dynamic optimization, stochastic optimization, and distributed
optimization. Her main ambition is to combine advanced approaches with practical applications consider-
ing the trend towards sustainability, ecommerce, and digitalization in freight transportation. She received
her PhD degree from Delft University of Technology in 2020 in the area of digital platform-based syn-
chromodal transport planning.
Bilge Atasoy is an assistant professor at TU Delft within the Department of Maritime and Transport
Technology. Her main research interests lie at the intersection of optimization and behavioral models.
She applies the scientific methodologies in the field of various transport and logistics problems in order
to increase sustainability and efficiency. One of the application areas is transport and logistics over water
where fleet management models are developed to optimize several decisions (e.g., fleet size, needed
capacity, routes and schedules for vessels). Prior to joining TU Delft, Bilge was a research scientist at
MIT at the Intelligent Transportation Systems (ITS) Lab where she is now a research affiliate. At MIT
she led several research projects in the areas of real-time optimization, travel behavior and choice-based
optimization. Bilge obtained her PhD from EPFL in November 2013 in the area of integrated supply and
demand models in transportation problems which received the best PhD Thesis Award from the Swiss
Operations Research Society. She received her MSc and BSc degrees in Industrial Engineering from
Bogazici University, Istanbul, in 2009 and 2007, respectively.
Wouter Beelaerts van Blokland researches theories supporting Advanced Operations and Production
Management such as Lean manufacturing, value chain and system, (maintenance) supply chains and
value creation by innovation. Central theme is performance measurement with KPI’s regarding the flow
of components or sub systems through processes, to support the process performance regarding the coor-
dination of assets and resources. Currently he is assistant professor at Delft University of Technology
within the section Multi-Machine Engineering, part of the department Marine & Transport Technology
of the faculty of Mechanical, Marine and Materials engineering. He achieved his PhD at the Delft Uni-
versity of Technology in 2010. After working in the drive and control and aerospace industry he started
this PhD in 2004 with the faculty of Aerospace Engineering to research the effect of leveraging value on
suppliers by aircraft manufacturers for the co-development and co-production of aircraft. In that time
he started to lecture on Lean Operations Performance Assessment and Value Engineering from which
several start-up companies were initiated such as “Type22” and Fly Aeolus. He was nominated for the
Delft Entrepreneurial Scientist Award 2010, category Entrepreneurial Motivator and principal lecturer
to the team on the project “Formation Flyer”, which won the National Prize on aeronautics (2011) in the
Netherlands.
Rudy R. Negenborn is a full professor in Multi-Machine Operations & Logistics. He is head of the Sec-
tion Transport Engineering & Logistics of Department Maritime & Transport Technology. His research
interests include intelligent infrastructures & logistics, decision making and coordination for transport
technology (including smart vessels) in general, whereby he proposes multi-agent system and model pre-
dictive control approaches that benefit from real-time information availability and the potential of commu-
nication. As such, his research anticipates the massive introduction of sensing, computation, and commu-
nication technologies. This is materialized into innovative solutions for smart equipment, transport hubs,
ports and (synchromodal) transport networks. He has over 200 peer reviewed academic publications. He
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
1 3
Anticipatory approach fordynamic andstochastic shipment…
leads NWO, EU and industry funded research, and is on the editorial board of the series on “Intelligent
Systems, Control and Automation: Science and Engineering”. He was moreover general chair of the 6th
International Conference on Computational Logistics, has acted as member of the organizing committee
of several other international conferences (including IEEE control conferences and maritime systems &
logistics conferences) and was guest editor of special journal issues on autonomous vessels and computa-
tional logistics. In addition, he is the editor of the books “Intelligent Infrastructures”, “Distributed Model
Predictive Control Made Easy”, and “Transport of Water versus Transport over Water”.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
1.
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... Rivera and Mes [24] developed an approximate dynamic programming algorithm to assign the newly arrived ship-ments to a barge or trucks by incorporating the probability distributions of future shipments. Guo et al. [10] proposed a stochastic programming-based rolling horizon approach to create online matches between newly received shipments and services in an inland synchromodal transportation network by integrating sampled shipments appearing in the near future. However, none of the above studies considered dynamic and stochastic travel times. ...
... The itinerary of shipment r at stage t + 1 is decided by the current itinerary and the matching decisions, as shown in (8). The position of shipment r at stage t + 1 is decided by the arrival time of the matched service s ∈ S −(t+1) , as shown in (9)(10). Set R t+1 consists of the accepted shipments that arrive at new terminals at stage t + 1, as shown in (11). ...
... Methods based on approximation strategies to solve SDP models have attracted increasing interest in the literature [21]. These methods can be divided into two groups: methods based on online decisions which focus on the computation when a dynamic event occurs with respect to the current system state and the available stochastic information, such as stochastic programming-based rolling horizon approaches [10]; and methods based on preprocessed decisions which estimate the value functions and determine the behavior policies before the execution of transport plans, such as reinforcement learning approaches [29]. A policy is defined as a mapping from perceived states of the environment to decisions to be taken when in those states [29]. ...
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Global synchromodal transportation involves the movement of container shipments between inland terminals located in different continents using ships, barges, trains, trucks, or any combination among them through integrated planning at a network level. One of the challenges faced by global operators is the matching of accepted shipments with services in an integrated global synchromodal transport network with dynamic and stochastic travel times. The travel times of services are unknown and revealed dynamically during the execution of transport plans, but the stochastic information of travel times are assumed available. Matching decisions can be updated before shipments arrive at their destination terminals. The objective of the problem is to maximize the total profits that are expressed in terms of a combination of revenues, travel costs, transfer costs, storage costs, delay costs, and carbon tax over a given planning horizon. We propose a sequential decision process model to describe the problem. In order to address the curse of dimensionality, we develop a reinforcement learning approach to learn the value of matching a shipment with a service through simulations. Specifically, we adopt the Q-learning algorithm to update value function estimations and use the epsilon-greedy strategy to balance exploitation and exploration. Online decisions are created based on the estimated value functions. The performance of the reinforcement learning approach is evaluated in comparison to a myopic approach that does not consider uncertainties and a stochastic approach that sets chance constraints on feasible transshipment under a rolling horizon framework.
... Three main types were identified: Binary, flow and state space. With binary models, we denounce models where decisions are described with binary variables, e.g., in [58] batches of containers are as a group matched with a service if the corresponding variable is 1. Next to the binary variables there are often continuous variables describing time limitations. ...
... The work on co-planning takes outset in the restricted information which can be communicated between the agents and a clear limits to their responsibilities. It is discussed further in Chapter 6. [5] x x Unclear [9] x x Unclear [10] x x [16] x [30] x [31] x x [36] x x [56] x [58] x [57] reject x [70] x [85] x x [88] x [92] x [105] x [112] x 5) x [110] x x [111] x [121] x [137] x Barge [126] x x [136] Truck [139] x [130] x x [134] x [165] x [164] x [175] x [178] x [184] x Terminal ...
Thesis
Full-text available
Under the synchromodal transport paradigm, transport providers decide how freight is transported. Thereby, real-time information on the transport system can be used to integrate the routing decisions of both freight and vehicles to utilize the transport capacity well between multiple stakeholders. This dissertation proposes co-planning, where consciously chosen information is exchanged between cooperating partners that plan individually towards shared goals. In the dissertation multiple routing methods based on model predictive control are presented. The conclusions illustrate that co-planning can contribute to make freight transport more efficient and thereby alleviate the environmental impacts.
... • A Markov decision process model is proposed to describe the SSM problem in hinterland transportation. Due to the curse of dimensionality, a stochastic approach is proposed to solve the problem under realistic instances in [33] (see also Chapter 5). ...
Thesis
With the increasing volumes of containers in global trade, efficient global container transport planning becomes more and more important. To improve the competitiveness in global supply chains, stakeholders turn to collaborate with each other at vertical as well as horizontal level, namely synchromodal transportation. Synchromodality is the provision of efficient, effective, and sustainable transport plans for all the shipments involved in an integrated network driven by advanced information technologies. However, the decision-making processes of a global synchromodal transport system is very complex. First, time-dependent travel times caused by traffic congestion need to be considered. Second, a dynamic approach that handles real-time shipment requests in a synchromodal network is required. Third, spot requests received from spot markets are unknown in advance. Fourth, travel time uncertainty is not handled yet for global synchromodal transport networks. Fifth, distributed approaches that stimulate cooperation among multiple stakeholders involved in global container transportation are still missing. This thesis addresses the above-mentioned challenges with dynamic, stochastic, and coordinated models.
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Supply chain management is becoming demand driven and logistics service providers need to use real-time information efficiently and integrate new technologies into their business. Synchromodal logistics has emerged recently to improve flexibility in supply chain, cooperation among stakeholders, and optimal utilization of resources. We survey the existing scientific literature and real-life developments on synchromodality. We focus on the critical success factors of synchromodality and six categories of existing enabling technologies. We identify some open research issues and propose the introduction of a new stakeholder, which takes on the role of an orchestrator, coordinating and providing services through a technology-based platform.
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