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Global intermodal transportation involves the movement of shipments between inland terminals located in different continents by 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 shipment requests with transport services in an integrated global network. The characteristics of the global intermodal shipment matching problem include acceptance and matching decisions, soft time windows, capacitated services, and transshipments between multimodal services. The objective of the problem is to maximize the total profits which consist of revenues, travel costs, transfer costs, storage costs, delay costs, and carbon tax. Travel time uncertainty has significant effects on the feasibility and profitability of matching plans. However, travel time uncertainty has not been considered in global intermodal transport yet leading to significant delays and infeasible transshipments. To fill in this gap, this paper proposes a chance-constrained programming model in which travel times are assumed stochastic. We conduct numerical experiments to validate the performance of the stochastic model in comparison to a deterministic model and a robust model. The experiment results show that the stochastic model outperforms the benchmarks in total profits.
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A global intermodal shipment matching problem
under travel time uncertainty
W. Guo, B. Atasoy, W. Beelaerts van Blokland, and R. R. Negenborn
Dept. of Maritime & Transport Technology, Delft University of Technology, Delft,
The Netherlands
W.Guo-2@tudelft.nl B.Atasoy@tudelft.nl
W.W.A.BeelaertsvanBlokland@tudelft.nl R.R.Negenborn@tudelft.nl
Abstract. Global intermodal transportation involves the movement of
shipments between inland terminals located in different continents by us-
ing 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 shipment requests with transport ser-
vices in an integrated global network. The characteristics of the global
intermodal shipment matching problem include acceptance and match-
ing decisions, soft time windows, capacitated services, and transship-
ments between multimodal services. The objective of the problem is
to maximize the total profits which consist of revenues, travel costs,
transfer costs, storage costs, delay costs, and carbon tax. Travel time
uncertainty has significant effects on the feasibility and profitability of
matching plans. However, travel time uncertainty has not been con-
sidered in global intermodal transport yet leading to significant delays
and infeasible transshipments. To fill in this gap, this paper proposes
a chance-constrained programming model in which travel times are as-
sumed stochastic. We conduct numerical experiments to validate the per-
formance of the stochastic model in comparison to a deterministic model
and a robust model. The experiment results show that the stochastic
model outperforms the benchmarks in total profits.
Keywords: Global intermodal transportation Shipment matching prob-
lem Travel time uncertainty Chance-constrained programming
1 Introduction
With the increasing volumes of global trade and the trend towards time-sensitive
shipments, efficient global transportation becomes increasingly important in
global supply chains [18]. Intermodal transportation is the provision of efficient,
effective, and sustainable transport services thanks to the horizontal and vertical
collaboration among players [15]. However, implementing intermodality in global
transport is still challenging from several aspects, including: the design of collab-
oration contracts and pricing strategies that ensure fairness and attractiveness
among players at the strategic level [8]; integrated service network design that
determines service frequencies and time schedules at the tactical level [12]; and
2 W. Guo et al.
1 5
840 hours
2 1
Jan 1, 15:00
2 6
Jan 2, 15:00
56
5 hours
2 5
Jan 1, 10:00
1 6
Jan 1, 11:00
Jan 1, 9:00
720 hours
960 hours
r1
r2
r3
Shipment list
s2
s3
s4
Service list
Global Intermodal Matching Platform
Shipment
requests
Transport
services
Acceptance and matching decisions
Global intermodal transportation
Shippers Carriers
15 hours
360 hours
2 6
Jan 1, 12:00
960 hours
r4
1 5
Jan 2, 11:00
s1 680 hours
30 TEU
40 TEU
10 TEU
20 TEU
200 TEU
50 TEU
50 TEU
50 TEU
Fig. 1. A global intermodal matching platform.
integrated transport plan that assigns specific shipments with transport services
under a dynamic or stochastic environment at the operational level [15]. This pa-
per investigates a global intermodal shipment matching (GISM) problem under
travel time uncertainty at the operational level.
With the development of digitization in the logistics industry, increasing
matching/booking platforms have appeared in freight transportation [13], such
as Uber Freight and Quicargo. We consider a platform owned by a global opera-
tor that receives shipment requests from shippers and receives transport services
from carriers, as shown in Fig. 1. The global operator could be a logistics service
provider or an alliance formed by multiple carriers, such as Maersk and COSCO
Shipping lines. A shipment request is defined as a batch of containers that must
be transported from its origin to its destination within a specific time window.
For example, shipment r1 consists of 30 containers which require to be trans-
ported from origin terminal 1 to destination terminal 5 with a release time of
Jan 1, 9:00, and a lead time of 840 hours. A transport service is characterized
by its mode, origin, destination, time schedule, and free capacity. For example,
ship service s1 with capacity 200 TEU (twenty-foot equivalent unit) will depart
from terminal 1 on Jan 2, 11:00, and arrive to terminal 5 with an estimated
travel time of 680 hours. The platform aims to provide optimal acceptance and
matching decisions in a global intermodal network. A match between a shipment
request and a transport service represents that the shipment will be transported
by the service from the service’s origin to the service’s destination. The platform
combines the matched services into itineraries to provide integrated transport
for global shipments. For instance, shipment r2 will be transported by barge ser-
vice s2 from origin terminal 2 to transshipment terminal 1 and by ship service
s1 from transshipment terminal 1 to destination terminal 5. The objective of the
platform is to maximize the total profits which consists of revenues and costs.
A GISM problem with travel time uncertainty 3
Due to travel time uncertainty and the utilization of multimodal services, the
matches made for accepted requests might become suboptimal or even infeasi-
ble at transshipment terminals. Thanks to the development in data analytics,
probability distributions of uncertainties are often available to transport systems
[3]. However, while stochastic approaches that incorporate stochastic informa-
tion of travel times in decision-making processes have been well investigated in
vehicle routing problems [2, 9] and inland intermodal transport planning [1, 6],
the stochastic approach for the GISM problem under travel time uncertainty is
still missing. This paper contributes to the literature by developing a chance-
constrained programming model to set confidence levels of chance constraints
regarding infeasible transshipments in a global intermodal network.
In the literature, most similar to our work are the work of Demir et al. [1]
and Guo et al. [4]. Demir et al. [1] investigated an inland intermodal service
network design problem with travel time uncertainty. In comparison to [1], this
paper considers fixed time schedules of multimodal services in a global network,
and develops a model that integrates acceptance and matching decisions. Guo
et al. [4] studied an inland intermodal shipment matching problem with request
uncertainty. In comparison to [4], this paper considers travel time uncertainty in
a global intermodal network.
The remainder of this paper is structured as follows. In Section 2, we provide
a detailed problem description, followed by a mathematical formulation in Sec-
tion 3. In Section 4, we develop the Chance-constrained programming model. In
Section 5, we present the experimental results. Finally, in Section 6, we provide
concluding remarks and directions for future research.
2 Problem description
Let Nbe the set of terminals. Each terminal iNis characterized by its
loading/unloading cost lcm
i, loading/unloading time ltm
iwith mode mM=
{ship,barge,train,truck}, and storage cost per container per hour cstorage
i.
We assume terminal operators provide unlimited loading/unloading and storage
capacity to the global operator.
Let Rbe the set of shipment requests. Each request rRis characterized
by its container type CTr(i.e., dry or reefer), origin terminal or, destination
terminal dr, container volume ur, release time Trelease
r(i.e., the time when the
shipment is available for transport process), lead time LDr, freight rate pr, and
delay cost cdelay
r. The due time of request ris Tdue
r=Trelease
r+LDr.
Let Sbe the set of services. Each service sSis characterized by its mode
MTsM, origin terminal os, destination terminal ds, total free capacity Us,
free capacity Uk
sin terms of container type kK={dry,reefer}, estimated
travel time ts, travel cost cs, and generation of carbon emissions ek
sfor container
type k. We consider ship, barge and train services as time scheduled services with
scheduled departure time T Dsand scheduled arrival time T Asfor sSship
Sbarge Strain. Each truck service consists of a fleet of trucks that have flexible
departure times. We define T Drs as a variable that indicates the departure time
4 W. Guo et al.
of service sStruck with shipment rR. Moreover, different services with the
same mode might be operated by the same vehicle. For two successive services
operated by the same vehicle, transshipment is unnecessary at the intermediate
terminal. Let lsq be equal to 0 if services sand qare operated by the same
vehicle, and service sis the preceding service of service q, 1 otherwise.
In practice, travel time uncertainties are quite common resulting from weather
conditions and traffic congestion [1]. In this paper, we use common assumption
that the travel times ˜
tssSare continuous random variables following nor-
mal distributions, and are statistically independent [2]. Let ˜
tsN(µs, σ2
s), in
which µsis the mean travel time between terminal osand terminal ds, and σs
is the corresponding standard deviation. Due to the travel time uncertainties,
the actual departure and arrival time of service sSare also uncertain. The
distribution of the departure time of service sis based on the distribution of the
arrival time of its preceding service; the distribution of the arrival time of service
sis based on the distributions of the departure and travel time of service s. For
vehicle vV, we define the itinerary of vehicle vas the sequence of services that
the vehicle operated, and define In
vas the nth service of vehicle v. Therefore, the
departure time of service s=In
vfollows normal distribution given by:
˜
T DsN(T DI1
v+X
j∈{1...n1}
µIj
v+X
j∈{1...n1}
2ltM Tv
dIj
v
,X
j∈{1...n1}
σ2
Ij
v),
where MTvis the mode of vehicle v. We denote ˜
T DsN(µ+
s, σ+
s
2). Similarly,
the arrival time of service s=In
vfollows the normal distribution given by:
˜
T AsN(T DI1
v+X
j∈{1...n}
µIj
v+X
j∈{1...n1}
2ltM Tv
dIj
v
,X
j∈{1...n}
σ2
Ij
v).
We denote ˜
T AsN(µ
s, σ
s
2).
Travel time uncertainty of services in a global intermodal network may lead to
infeasible transshipments in addition to the commonly studied outcome of late or
early delivery at destinations [9, 14]. An illustrative example is shown in Fig. 2. A
shipment is planned to be transported by a train service from its origin terminal
to port A, by a ship service from port A to port B, and by two barge services
from port B to its destination terminal according to fixed time schedules. The
outcomes of travel time uncertainty in global intermodal transportation include
late delivery at destination terminal under realization 1 which causes delayed
costs, early delivery at destination terminal under realization 2 which causes
storage costs, and infeasible transshipment at port B under realization 3 which
requires re-planning from port B to destination terminal.
The objective of the platform is to maximize the total profits by optimizing
acceptance and matching decisions over a given planning horizon T. The total
profits consist of revenues received from shippers, travel costs paid to carriers,
transfer costs and storage costs paid to terminal operators, delay costs paid to
shippers, and carbon tax charged by institutional authorities.
The notation used in this paper is shown in Table 1.
A GISM problem with travel time uncertainty 5
Port B
Port A
Destination
terminal
Origin
terminal
Dec 2 15:00
Dec 1 21:00
Dec 3 15:00
Jan 2 09:00
Jan 1 07:00
Jan 2 21:00
Jan 3 05:00
Jan 3 10:00
Time schedules
Departure:
Arrival:
Inland
terminal
Realization 1
Transportation Transshipment
Dec 2 16:00
Dec 1 21:00
Dec 3 15:00
Jan 2 09:00
Jan 1 16:00
Jan 2 22:00
Jan 3 06:00
Jan 3 12:00
Realization 2
Dec 2 16:00
Dec 1 21:00
Dec 3 15:00
Jan 2 09:00
Jan 1 16:00
Jan 2 20:00
Jan 3 04:00
Jan 3 08:00
Realization 3
Dec 2 16:00
Dec 1 21:00
Dec 3 15:00
Jan 2 09:00
Jan 2 10:00
Departure:
Arrival:
Departure:
Arrival:
Departure:
Arrival:
Delay
Delay
Delay
Delay Delay
Delay Delay
Late delivery
Early arrival
Early delivery
Infeasible
transshipment
Fig. 2. Possible outcomes of travel time uncertainty in global transport.
Table 1: Notation.
Sets:
NTerminals N
KContainer types, K={dry,reefer}
RShipment requests
RkRequests with container type kK
MModes, M={ship,barge,train,truck}
VSet of vehicles V=Vship Vbarge Vtrain Vtruck
SServices, S=Sship Sbarge Strain Struck
S+
iServices departing at terminal i,S+
i=S+ship
iS+barge
iS+train
iS+truck
i
S
iServices arriving at terminal i,S
i=Sship
iSbarge
iStrain
iStruck
i
Deterministic parameters
TLength of the planning horizon
αConfidence level
CTrContainer type of request rR, C TrK
orOrigin terminal of request rR,orN
drDestination terminal of request rR, drN
urContainer volume of request rR
Trelease
rRelease time of request rR
Tdue
rDue time of request rR
prFreight rate of request rR
LDrLead time of request rR,LDr=Tdue
rTrelease
r
cdelay
rDelay cost of request rRper container per hour overdue
MTsMode of service sS, M TsM
osOrigin terminal of service sS, osN
dsDestination terminal of service sS, dsN
UsFree capacity of service sS
6 W. Guo et al.
Uk
sFree capacity of service sSregarding container type kK
csTravel cost of service sSper container
ek
sCarbon emissions of service sSper container with type kK
M T vMode of vehicle vV
In
vThe nth service of vehicle vV\Vtruck, In
vS\Struck
T DsScheduled departure time of service sS\Struck
T AsScheduled arrival time of service sS\Struck
tsEstimated travel time of service sS
lsq Binary variable; 0 if services sand qare operated by the same vehicle, and
service sis the preceding service of service q, 1 otherwise
lcm
iLoading/unloading cost per container at terminal iNwith mode mM
ltm
iLoading/unloading time at terminal iNwith mode mM
cstorage
iStorage cost at terminal iper container per hour
cemission Activity-based carbon tax charged by institutional authorities
MA large number used for binary constraints
Random variables
˜
tsTravel time of service sS,˜
tsN(µs, σ2
s)
˜
T DsDeparture time of service sS\Struck,˜
T DsN(µ+
s, σ+
s
2)
˜
T AsArrival time of service sS\Struck,˜
T AsN(µ
s, σ
s
2)
Variables
yrBinary variable; 1 if request rRis accepted
xrs Binary variable; 1 if request rRis matched with service sS, 0 otherwise
zrsq Binary variable; 1 if request rRis matched with service sS,xrs = 1
and service qS,xrq = 1, 0 otherwise
T Drs Departure time of truck service sStruck with request rR
fri Transshipment cost of request rRat terminal iNper container
˜wri Storage time of request rRat terminal iN
˜
Tdelay
rDelay of request rRat destination terminal drN
3 Mathematical formulation
Let yrbe the binary variable which is 1 if request rRis accepted, otherwise
0. We use the binary variable xrs to represent the match between request rR
and service sS. A match between request rand service smeans shipment
rwill be transported by service sfrom terminal osto terminal ds. Due to the
travel time uncertainty, the transport plan might become infeasible and requires
re-planning. Therefore, the costs generated by accepted requests are uncertain
and hard to estimate. We use ˜
Cr(x) to denote the random cost generated for
request rRwhich consists of travel costs, transfer costs, storage costs, delay
costs, and carbon tax. The mathematical formulation of the GISM problem is:
P0 max
y,xX
rR
pruryrX
rR
˜
Cr(x)(1)
subject to
yrX
sS+
or
xrs,rR, (2)
A GISM problem with travel time uncertainty 7
yrX
sS
dr
xrs,rR, (3)
X
sS+
i
xrs 1,rR, i N\{dr},(4)
X
sS
i
xrs 1,rR, i N\{or},(5)
X
sS
or
xrs 0,rR, (6)
X
sS+
dr
xrs 0,rR, (7)
X
sS+
i
xrs =X
sS
i
xrs,rR, i N\{or, dr},(8)
X
rR
xrsurUs,sS, (9)
X
rRk
xrsurUk
s,sS, k = reefer,(10)
Trelease
r+ltM Ts
orT Drs +M(1 xrs ),rR, s S+truck
or,(11)
Trelease
r+ltM Ts
or˜
T Ds+M(1 xrs),rR, s S+
or\S+truck
or,(12)
˜
T As+ltM Ts
i+ltM Tq
i˜
T Dq+M(1 xrs) + M(1 xr q ),rR,
iN\{or, dr}, s S
i\Struck
i, q S+
i\S+truck
i, lsq = 1,(13)
T Drs +˜
ts+ltM Ts
i+ltM Tq
i˜
T Dq+M(1 xrs) + M(1 xr q ),rR,
iN\{or, dr}, s Struck
i, q S+
i\S+truck
i,(14)
˜
T As+ltM Ts
i+ltM Tq
iT Drq +M(1 xrs ) + M(1 xrq ),rR,
iN\{or, dr}, s S
i\Struck
i, q S+truck
i,(15)
T Drs +˜
ts+ltM Ts
i+ltM Tq
iT Drq +M(1 xrs ) + M(1 xrq ),rR,
iN\{or, dr}, s Struck
i, q S+truck
i.(16)
Constraints (2-3) ensure that request rRwill not be accepted by the
platform if there is no matching possibilities. Constraints (4-5) ensure that at
most one service transports request rdeparting from or arriving to a terminal.
Constraints (6-7) are designed to eliminate subtours. Subtours might be formed
since in one OD pair, there exist services in both directions. Constraints (8) en-
sure flow conservation. Constraints (9) ensure that the total container volumes of
requests matched with service sdo not exceed its total free capacity. Constraints
(10) ensure that the total volumes of reefer containers matched with service s
cannot exceed its free capacity on reefer slots. Constraints (11-12) ensure that
the departure time of service sminus loading time must be earlier than the
release time of request r, if request rwill be transported by service sdepart
its origin terminal. Here, Mis a large enough number which ensures the time
compatibility between shipment rand service swhen binary variable xrs equals
8 W. Guo et al.
to 1, but leaves the constraints “open” if xrs is 0. Constraints (13-16) ensure
that the arrival time of service sS
iplus loading and unloading time must be
earlier than the departure time of service qS+
iif request rwill be transported
by service sentering terminal iand by service qleaving terminal i.
4 Chance-constrained programming model
In the literature, different techniques have been developed to deal with travel
time uncertainty: deterministic, stochastic, and robust programming [14]. While
deterministic programming considers average travel times and robust program-
ming considers minimum and maximum travel times, stochastic programming
considers the probability distributions of travel times. Chance-constrained pro-
gramming (CCP) is one of the major stochastic approaches to solve optimization
problems under travel time uncertainty [9]. In this section, we develop a CCP
model to approximate stochastic constraints (12-16) and random cost ˜
Cr(x) for
request rin model P0. The CCP model does not take into account the correction
costs caused by the re-planning of requests.
Under the CCP, each stochastic constraint will hold at least with probability
α, where αis referred to as the confidence level provided by the platform. A high
αmeans the matches have a low probability causing infeasible transshipments.
When α= 0.5, the CCP model becomes a deterministic model; when α= 1,
the CCP model becomes a robust model. The objective is to maximize expected
total profits while ensuring that the probability of infeasible transshipments does
not exceed α. The formulation of the CCP model is:
P1 max
y,xX
rR
pruryr X
rRX
sS
csxrsur+X
rRX
iN
friur
+X
rRX
iN
cstorage
iE( ˜wri)ur+X
rR
cdelay
rE(˜
Tdelay
r)ur
+X
kKX
rRkX
sS
cemissionek
sxrsur
(17)
subject to constraints (2-11),
P{Trelease
r+ltM Ts
or˜
T Ds+M(1 xrs)} ≥ α, rR, s S+
or\S+truck
or,(18)
P{˜
T As+ltM Ts
i+ltM Tq
i˜
T Dq+M(1 xrs) + M(1 xr q )} ≥ α,
rR, i N\{or, dr}, s S
i\Struck
i, q S+
i\S+truck
i, lsq = 1,(19)
P{T Drs +˜
ts+ltM Ts
i+ltM Tq
i˜
T Dq+M(1 xrs) + M(1 xr q )} ≥ α,
rR, i N\{or, dr}, s Struck
i, q S+
i\S+truck
i,(20)
P{˜
T As+ltM Ts
i+ltM Tq
iT Drq +M(1 xrs ) + M(1 xrq )} ≥ α,
rR, i N\{or, dr}, s S
i\Struck
i, q S+truck
i,(21)
P{T Drs +˜
ts+ltM Ts
i+ltM Tq
iT Drq +M(1 xrs ) + M(1 xrq )} ≥ α,
rR, i N\{or, dr}, s Struck
i, q S+truck
i,(22)
A GISM problem with travel time uncertainty 9
fri =X
sS+
i
xrslcMTs
i,rR, i =or,(23)
fri =X
sS
i
xrslcMTs
i,rR, i =dr,(24)
fri =X
sS+
iX
qS
ilcM Ts
i+lcM Tq
izrsq lsq ,rR, i N\{or, dr},(25)
zrsq xrs ,rR, s S, q S, (26)
zrsq xrq ,rR, s S, q S, (27)
zrsq xrs +xrq 1,rR, s S, q S, (28)
E( ˜wror)E(˜
T Ds)ltM Ts
orTrelease
r+M(xrs 1),rR, s S+
or\S+truck
or,(29)
E( ˜wror)T Drs ltM Ts
orTrelease
r+M(xrs 1),rR, s S+truck
or,(30)
E( ˜wri)E(˜
T Dq)E(˜
T As)ltM Ts
iltM Tq
i+M(xrs 1) + M(xrq 1),
rR, i N\{or, dr}, s S
i\Struck
i, q S+
i\S+truck
i,(31)
E( ˜wri)E(˜
T Dq)T Drs E(˜
ts)ltM Ts
iltM Tq
i+M(xrs 1) + M(xrq 1),
rR, i N\{or, dr}, s Struck
i, q S+
i\S+truck
i,(32)
E( ˜wri)T Dr q E(˜
T As)ltM Ts
iltM Tq
i+M(xrs 1) + M(xrq 1),
rR, i N\{or, dr}, s S
i\Struck
i, q S+truck
i,(33)
E( ˜wri)T Dr q T Drs E(˜
ts)ltM Ts
iltM Tq
i+M(xrs 1) + M(xrq 1),
rR, i N\{or, dr}, s Struck
i, q S+truck
i,(34)
E( ˜wrdr)Tdue
rE(˜
T As)ltM Ts
dr+M(xrs 1),rR, s S
dr\Struck
dr,(35)
E( ˜wrdr)Tdue
rT Drs E(˜
ts)ltM Ts
dr+M(xrs 1),rR, s Struck
dr,(36)
E(˜
Tdelay
r)E(˜
T As) + ltM Ts
drTdue
r+M(xrs 1),rR, s S
dr\Struck
dr,(37)
E(˜
Tdelay
r)T Drs +E(˜
ts) + ltM Ts
drTdue
r+M(xrs 1),rR, s Struck
dr,(38)
where fri is the planned loading and unloading cost of request rat terminal
i;E( ˜wri) is the expected storage time of request rat terminal i;E(˜
Tdelay
r) is
the expected delay in delivery of request rat destination terminal dr; P is the
probability measure; zrsq is a binary variable which equals to 1 if request rhas
to transfer between service sand q, 0 otherwise; E(˜
T Ds) = µ+
s,E(˜
T As) = µ
s,
E(˜
ts) = µs.
The objective function P1 is to maximize the expected total profits which
consist of total revenues, travel costs, transfer costs, storage costs, delay costs and
carbon tax. Constraints (18-22) ensure that the possibility of feasible transship-
ment at terminals will be higher than the confidence level α. Constraints (23-25)
calculate the loading costs at origin terminals, the unloading costs at destina-
tion terminals, and the loading and unloading costs at transshipment terminals.
Constraints (26-28) ensure that binary variable zrsq equals to 1 if xrs = 1 and
xrq = 1, 0 otherwise. Constraints (29-36) calculate the storage time at origin,
transshipment, and destination terminals. Constraints (37-38) calculate delayed
time at destination terminals.
10 W. Guo et al.
To solve the CCP model, the traditional method is to convert the chance
constraints into their corresponding deterministic equations. Based on the prop-
erties of normal distributions, chance constraints (18-22) can be linearized as:
Trelease
r+ltM Ts
or+M(xrs 1) µ+
s
σ+
s
φ1(1 α),rR, s S+
or\S+truck
or,(39)
ltM Ts
i+ltM Tq
i+M(xrs 1) + M(xrq 1) (µ+
qµ
s)
q(σ+
q)2+ (σ
s)2
φ1(1 α),
rR, i N\{or, dr}, s S
i\Struck
i, q S+
i\S+truck
i, lsq = 1,
(40)
T Drs +ltM Ts
i+ltM Tq
i+M(xrs 1) + M(xrq 1) (µ+
qµs)
q(σ+
q)2+ (σs)2
φ1(1 α),
rR, i N\{or, dr}, s Struck
i, q S+
i\S+truck
i,
(41)
T Drq ltM Ts
iltM Tq
i+M(1 xrs) + M(1 xr q )µ
s
σ
s
φ1(α),
rR, i N\{or, dr}, s S
i\Struck
i, q S+truck
i,
(42)
T Drq T Drs ltM Ts
iltM Tq
i+M(1 xrs) + M(1 xr q )µs
σs
φ1(α),
rR, i N\{or, dr}, s Struck
i, q S+truck
i,
(43)
where φ1is the inverse function of standardized normal distribution.
5 Numerical experiments
We evaluate the performance of the CCP on the GISM problem in comparison
to a deterministic approach (DA) which uses average travel times (i.e., α= 0.5)
and a robust approach (RA) which considers the maximum and minimum travel
times (i.e., α= 1). Compared with the CCP, the DA is a risk neutral approach in
which decision makers are indifferent to uncertainties, and the RA is a risk averse
approach that seeks guarantee. 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.
Unless otherwise stated, the benchmark values of coefficients are set as fol-
lows: loading cost (unit: /TEU) lcship
i= 18, lcbarge
i= 18, lctrain
i= 12, lctruck
i=
12 for iN; loading time (unit: hours) ltship
i= 12, ltbarge
i= 4, lttrain
i= 2,
lttruck
i= 1 for iN; storage cost (unit: /TEU-h) cstorage
i= 1 for iN;
carbon tax (unit: /kg) cemission = 0.07.
We consider a global intermodal network that consists of two terminals in
Europe and three terminals in Asia that are connected by Suez Canal Route
(SCR), Northern Sea Route (NSR), and Eurasia Land Bridge (ELB), as shown
in Fig. 3. Compared with the SCR, the NSR has a shorter travel time but a
higher travel cost caused by ice-breaking fees [11]. With the implementation of
IMO 2020 regulations, shipping liner companies are required to use low-sulfur
A GISM problem with travel time uncertainty 11
Rotterdam
Shanghai
Wuhan
Chongqing
Duisburg
17
16
14
1
2
13
3
4
5
6
8
7
15
9
10
11
12
SeaTerminal Inland waterway Railway Roadway 1Service number
with direction
18
Northern Sea Route
Fig. 3. The topology of a global intermodal network.
Table 2. Service data.
Service.
ID
Mode Origin Destination Total
capacity
(TEU)
Reefer
slots
(TEU)
Departure
time
Arrival
time
Travel
time (h)
Travel
cost
( )
Carbon
emissions-
dry (kg)
Carbon
emissions-
reefer (kg)
Preceding
service
Succeeding
service
1 barge Chongqing Wuhan 160 50 144 235 91 192 313 940 2
2 barge Wuhan Shanghai 160 50 243 328 85 178 291 874 1
3 barge Shanghai Wuhan 160 50 144 229 85 178 291 874 4
4 barge Wuhan Chongqing 160 50 237 328 91 192 313 940 3
5 train Chongqing Shanghai 90 30 144 181 37 269 526 1578
6 train Shanghai Chongqing 90 30 144 181 37 269 526 1578
7 truck Shanghai Chongqing 200 60 22 1823 1489 4466
8 truck Chongqing Shanghai 200 60 22 1823 1489 4466
9 barge Rotterdam Duisburg 160 30 1010 1027 17 35 57 170
10 barge Duisburg Rotterdam 160 30 750 767 17 35 57 170
11 train Rotterdam Duisburg 90 30 910 917 7 48 92 276
12 train Duisburg Rotterdam 90 30 750 757 7 48 92 276
13 truck Rotterdam Duisburg 200 60 3 334 219 658
14 truck Duisburg Rotterdam 200 60 3 334 219 658
15 ship Shanghai Rotterdam 200 50 350 988 638 1441 2161 6483
16 ship Shanghai Rotterdam 200 50 350 900 550 2240 1631 4894
17 train Chongqing Duisburg 90 30 350 723 373 2007 3517 10551
18 ship Shanghai Rotterdam 200 50 518 1156 638 1441 2161 6483
fuels on the sea, which in turn increases travel costs in the SCR and the NSR
[10]. As an alternative, the ELB becomes more and more competitive thanks to
its shortest travel time. However, without subsidies from governments, the ELB
is still the most expensive route.
We consider 18 services operating on the network: 8 in Asia, 6 in Europe, and
4 connecting Asia and Europe as presented in Table 2. The hinterland-related
data is adapted from the work of [5]; the intercontinental-related data is adapted
from the works of [7], [16], [17]. We consider 6 shipment requests received by the
platform at time 0. The detailed request data is shown in Table 3. Compared
with reefer shipments (requests 1, 3, 5), dry shipments (requests 2, 4, 6) have
longer lead times, lower freight rates, and lower delay costs.
5.1 Impact of different objective functions
The effects of objective functions are tested under a deterministic environment
without travel time uncertainties, i.e., mean of travel times µs=ts, standard
deviation σs= 0,sS. We set the confidence level α= 0.5 for the CCP
model, and therefore φ1(α) = φ1(1 α) = 0.
12 W. Guo et al.
Table 3. Request data.
Requests Container
type
Origin Destination Container volume
(TEU)
Release
time
Lead time
(h)
Freight rate
( /TEU)
Delay cost
( /TEU-h)
1 reefer Shanghai Rotterdam 5 100 720 4000 20
2 dry Shanghai Rotterdam 5 100 840 3500 17.5
3 reefer Wuhan Rotterdam 5 100 600 4500 22.5
4 dry Wuhan Rotterdam 5 100 960 3000 15
5 reefer Chongqing Duisburg 5 100 480 5000 25
6 dry Chongqing Duisburg 5 100 1080 2500 12.5
Table 4. Impact of different objective functions.
Cases Objective
function
Total
profits
Revenue Travel
costs
Transfer
costs
Storage
costs
Delay
costs
Carbon
tax
Rejections Delay
(TEU-h)
Emission
(kg)
1 Travel
costs
-67978 112500 48061 2040 6914 113163 10300 0 4945 147146
2 Transfer
costs
-34695 112500 50677 1320 8890 74925 11383 0 3416 162611
3 Storage
costs
-47333 112500 59413 2400 4814 81063 12144 0 3482 173483
4 Delay
costs
1590 112500 63648 1560 9317 21439 14947 0 873 213529
5 Carbon
tax
-67375 112500 72030 2040 8367 89363 8076 0 3773 115366
6 Total
costs
4946 112500 63282 2100 5983 21439 14750 0873 210711
7 Total
profits
13107 87500 53249 1980 4743 3364 11057 1150 157957
The results generated under different objective functions are shown in Ta-
ble 4. Under cases 1 to 6, all the requests are accepted. Comparing case 6 with
cases 1 to 5, the total profit is the highest. It means that considering the trade-
off among logistics costs, delays, and emissions is very important. While cases 1
to 6 are designed to minimize different costs, case 7 aims to maximize the total
profit that consists of revenue and total costs. Compared with cases 1 to 6, the
total profit is significantly higher under case 7. Comparing case 6 and case 7
shows that it may be necessary to reject the requests that are not profitable.
5.2 Comparing deterministic, stochastic, and robust approaches
To investigate the differences between solutions generated by the CCP, DA, and
RA under travel time uncertainty, we set the mean of travel times µs=tsfor
sS, standard deviation of travel times σs= 0.1tsfor sS\Struck,σs= 0.5ts
for sStruck. Besides, we let 0.9tsbe the fixed lower bound for travel times of
service sS. Under the realization of travel times as shown in Table 5, barge
service 2 is delayed, the transfers between barge service 2 and ship service 15 and
16 are therefore becoming infeasible. Regarding the CCP, we set the confidence
level α= 0.7, and therefore φ1(α) = 0.524, φ1(1 α) = 0.524.
Due to travel time uncertainty, the planned profits are different from the
actual profits. Table 6 shows the results received before the realization of travel
times. We note that the DA generates the highest planned profits with the lowest
number of rejections and the highest delay in deliveries. In comparison, the CCP
takes into account the trade-off between feasibility and profitability. It rejects
A GISM problem with travel time uncertainty 13
Table 5. The realization of travel times.
Service. ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Actual travel time 98 99 89 101 40 36 23 21 18 15 7 7 3 4 631 537 384 657
Actual departure time 144 250 144 241 144 144 1010 750 910 750 350 350 350 518
Actual arrival time 242 349 233 342 184 180 1028 765 917 757 981 887 734 1175
Table 6. Results received before the realization of travel times.
Approaches Planned Rejection Delay Planned itinerary of requests
profits (TEU-h) 1 2 3 4 5 6
DA 13107 1 150 3,4,17,10 16 4,17,14 2,15 1,2,15,9
CCP 6553 4 0 6,17,10 16
RA 4217 5 0 16
Table 7. Results received after the realization of travel times.
Approaches Actual Infeasible Rejection Delay Actual itinerary of requests
profits transshipments (TEU-h) 1 2 3 4 5 6
DA -438 2 1 911 3,4,17,10 16 4,17,14 2,18 1,2,18,13
CCP 6533 0 4 0 6,17,10 16
RA 4151 0 5 0 16
requests 3 to 6 which might be non-profitable under travel time uncertainties
and chooses rail service 6 instead of barge services 3 and 4 for request 2. The
RA is the most conservative approach which has the lowest planned profits
and the highest number of rejections. Regarding the results received after the
realization of travel times, Table 7 shows that the DA generates the lowest actual
profits due to infeasible transshipments at Shanghai Port for requests 4 and 6.
In comparison, the CCP has the highest actual profits thanks to the rejection of
non-profitable requests 4 and 6. Compared with the DA and the CCP, the RA
is the safest approach which avoids the possibility of infeasible transshipments
but loses the opportunity to get higher profits.
The difference among the deterministic, stochastic, and robust solutions is
graphically represented in Fig. 4. Under the DA, request 5 is rejected; requests
1 and 3 with reefer shipments are assigned to the ELB; requests 2, 4, 6 with
dry shipments are assigned to the SCR and NSR. Due to travel time varia-
tions, requests 4 and 6 switch from service 15 to 18 at Shanghai Port. Under
the CCP, request 1 arrives Chongqing terminal by using rail service 6 which is
faster than barges services 3 and 4. Under the RA, all the requests that require
transshipments at terminals are rejected.
6 Conclusions and future research
In this paper, we investigated a stochastic shipment matching problem in global
intermodal transport. The problem is stochastic since the uncertainties in travel
times are incorporated. We developed a chance-constrained programming (CCP)
model to address travel time uncertainties. We conducted experiments to val-
idate the performance of the CCP in comparison to a deterministic approach
(DA) in which decisions are made based on estimated travel times and a robust
14 W. Guo et al.
(b) Stochastic solutions
Shanghai
Wuhan
Chongqing
Duisburg
4,6
6
Rotterdam
2
15
Shanghai
Wuhan
Chongqing
Duisburg
4,6
6
Rotterdam
2
18
Shanghai
Wuhan
Chongqing
Duisburg 1
Rotterdam
2
Wuhan
Chongqing
Duisburg
2
Rotterdam
Shanghai
(a) Deterministic solutions
(c) Robust solutions
SeaTerminal Inland waterway Railway Roadway Request number
with transport route
1
NSR NSR
NSR NSR
Fig. 4. Comparison of deterministic, stochastic and robust solutions.
approach (RA) in which decisions are made based on maximum and minimum
travel times. The experimental results indicate that the CCP increases total
profits by 1591.55% in comparison to the DA and by 57.38% in comparison to
the RA under the designed case.
This research can be extended in several promising directions. First, due to
the computational complexity, we only conducted small experiments in this pa-
per, future research can be extended to large-scale instances by designing efficient
algorithms that benefit from parallelization and distributed structure. Second,
this paper used fixed settings of parameters, conducting sensitivity analysis of
parameters is a promising future research direction. Third, due to the fluctuation
of freight rates in spot markets, future requests are quite uncertain. Combining
travel time uncertainty with spot request uncertainty in global intermodal trans-
port planning deserves further research.
Acknowledgments
This research is financially supported by the China Scholarship Council (Grant
201606950003) and “Complexity Methods for Predictive Synchromodality” (project
439.16.120) of the Netherlands Organisation for Scientific Research (NWO).
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Stochastic vehicle routing, which deals with routing problems in which some of the key problem parameters are not known with certainty, has been an active, but fairly small research area for almost 50 years. However, over the past 15 years we have witnessed a steady increase in the number of papers targeting stochastic versions of the vehicle routing problem (VRP). This increase may be explained by the larger amount of data available to better analyze and understand various stochastic phenomena at hand, coupled with methodological advances that have yielded solution tools capable of handling some of the computational challenges involved in such problems. In this paper, we first briefly sketch the state-of-The-Art in stochastic vehicle routing by examining the main classes of stochastic VRPs (problems with stochastic demands, with stochastic customers, and with stochastic travel or service times), the modeling paradigms that have been used to formulate them, and existing exact and approximate solution methods that have been proposed to tackle them. We then identify and discuss two groups of critical issues and challenges that need to be addressed to advance research in this area. These revolve around the expression of stochastic phenomena and the development of new recourse strategies. Based on this discussion, we conclude the paper by proposing a number of promising research directions.