Two-echelon Multi-trip Vehicle Routing Problem with Synchronization for An Integrated Water-and Land-based Transportation System

Abstract

This study considers an integrated water-and land-based transportation (IWLT) system for waste collection. Research on the issue is motivated by increased heavy street movements that damage quay walls as well as congestion. We present a novel two-echelon vehicle routing problem with satellite synchronization based on a two-index formulation and evaluate it on small-sized instances for 10 waste points and 4 hubs. We compare the proposed synchronized IWLT approach with three benchmarks that can reduce issues associated with heavy loads. It is shown that the proposed system can provide better solutions with less collection cost, reduced street movements and lightweight garbage vehicles.

Full text

Two-echelon Multi-trip Vehicle Routing Problem with Synchronization for An
Integrated Water- and Land-based Transportation System
Çi˘
gdem Karademir*1, Breno A. Beirigo1, Rudy R. Negenborn1, and Bilge Atasoy1
1Dept. of Maritime & Transport Technology, Faculty 3mE, Delft University of Technology, Delft
University of Technology, The Netherlands
SHORT SUMMARY
This study considers an integrated water- and land-based transportation (IWLT) system for waste
collection. Research on the issue is motivated by increased heavy street movements that dam-
age quay walls as well as congestion. We present a novel two-echelon vehicle routing problem
with satellite synchronization based on a two-index formulation and evaluate it on small-sized in-
stances for 10 waste points and 4 hubs. We compare the proposed synchronized IWLT approach
with three benchmarks that can reduce issues associated with heavy loads. It is shown that the
proposed system can provide better solutions with less collection cost, reduced street movements
and lightweight garbage vehicles.
Keywords: City logistics; Integrated water- and land-based transportation; Multi-trip; Satellite
synchronization; Two-echelon vehicle routing problem
1. INTRODUCTION
The interest towards logistics over inland waterways has been increasing recently as cities plan to
reduce on-street congestion and emissions (Amsterdam.nl,2019). Amsterdam, the Netherlands,
for example, aims at further harnessing its extensive waterway network, which covers 25% of the
city’s central area, to improve its waste collection system. This system is based on heavy garbage
trucks, which, besides worsening congestions, contribute to damaging the city’s historical and
fragile quay walls, resulting in billions of euros in maintenance costs (TheMayor.EU,2021).
In order to reduce heavy vehicle movements and prevent quay wall damage, this study proposes
an integrated water- and land-based transportation (IWLT) system that eliminates heavy garbage
trucks entirely. As pointed out by Anderluh, Nolz, Hemmelmayr, and Crainic (2021), two-echelon
systems may help alleviate the impact of growing freight movements on society, the economy, and
the environment caused by the development of e-commerce and same-day delivery services. At the
first echelon, small garbage cars collect waste. Next, at the second echelon, waste is consolidated
onto larger vessels that meet the cars at the hubs. Finally, full vessels sail to a central waste facility.
We model this hybrid waste collection system as a variant of the two-echelon capacitated vehicle
routing problem (2ECVRP) introduced by Gonzalez-Feliu (2008). Typically, 2ECVRPs aim at
routing and consolidating freight through intermediate satellites connecting echelons before trans-
ferring it to a final destination (Perboli, Tadei, & Vigo,2011). Unlike most 2ECVRP models
in the literature, which consider a delivery scenario where items are first consolidated and then
dispatched, we model a reverse logistics problem (see Figure 1).
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Figure 1: Two-echelon waste collection network.
The first echelon problem consists of finding routes for cars and selecting the best hubs for the
transfer tasks, while the second echelon problem consists of finding routes for the vessels to serve
these transfer tasks. Due to their low capacity, cars may execute multiple transfer tasks, leading
to multiple trips to hubs. Therefore, vessels and cars should be in sync to be present at hubs to
perform the transfer tasks. Synchronizing vessels and cars is further constrained by time windows
(collection hours vary per neighborhood) and physical space (only a single vessel can access a hub
at a time and perform a single transfer task at once).
Most studies have focused on the basic variant of 2ECVRP where the synchronization is required
only for cargo flow (Cattaruzza, Absi, Feillet, & González-Feliu,2017). Since satellites may lack
storage capabilities or feature temporary storage capacity, temporal synchronization may also be
necessary: first echelon vehicles should not arrive after the departure of second echelon vehicles
(Li, Chen, Wang, & Bai,2021). Crainic, Ricciardi, and Storchi (2009) first introduce the 2ECVRPs
with time windows and temporal satellite synchronization (2E-VRPTW-SS) and propose a general
model for a multi-depot multi-trip variant considering heterogeneous vehicles.
Although many studies have focused on multi-trip 2E-VRPTW-SS variants (see, e.g., Grangier,
Gendreau, Lehuédé, & Rousseau,2016;He & Li,2019;Anderluh et al.,2021), to the best of our
knowledge, none has considered vehicle transfer constraints. They assume that multiple transfer
operations can occur simultaneously at a hub, which is not feasible for the Amsterdam waste
collection use case where space for maneuvering cars and vessels is limited. Hence, in this study,
we model a multi-trip 2E-VRPTW-SS with one-to-one transfers (2E-MVRPTW-SS). Additionally,
from a practical view, we analyze different logistic systems that integrate water and land for waste
collection problems in cities.
The remainder of the paper is organized as follows. Section 2formulates the problem as a mixed-
integer linear programming problem (MILP). Section 3lays out an experimental study where we
compare the solution quality of the proposed model with three benchmarks on modified Solomon’s
(1987) test instances. Finally, Section 4is devoted to the conclusions and further directions.
2. PROBLEM FORMULATION
The 2E-MVRPTW-SS is formulated as a MILP. The first echelon is the street level, where we
have K1identical electric cars with a capacity of Q1units. These cars start their journey at the
main depot (d), visit a set of waste points (C), and return to the main depot without exceeding
their capacity at any point. Each waste point irequires qiunits of waste to be collected within a
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time window of (ai,bi) associated with a service time of si.ti j and ci j denote the shortest travel
time and travel cost between points iand j, respectively. Cars can unload the waste onto vessels
at a set of transshipment hubs Sover multiple trips. Transferring waste requires Utime units. The
second echelon is the water level where K2identical vessels with a capacity of Q2units start at the
waste centre (w), visit hub(s) if cars require transfer task(s), and return to the waste centre. The xi j
Table 1: Notation for the 2E-MVRPTW-SS.
Sets
dThe depot for electric cars
wThe waste centre for vessels
CWaste points, {1,...,n}
CdWaste points and the depot d,C∪ {d}
SHubs
Parameters
ti j Shortest travel time from waste point ito j
ci j Cost of travelling from node ito j
Q1Capacity of an electric car
Q2Capacity of a vessel
K1The number of available electric cars
K2The number of available vessels
siService duration of node
qiWaste amount at node
aiEarliest collection time of node i
biLatest collection time of node i
UConstant duration for a transfer task
β1The fixed cost of an electric car
β2The fixed cost of vessel
MSufficiently large number for constraint linearization
Variables
xi j (Binary) 1 if node jis visited immediately after node i, 0 otherwise
vip (Binary) 1 if the hub pis visited immediately after node iis served, 0 otherwise
miTotal load on the car after node iis visited
hiService start time at node i
εiExtra travel cost to visit a hub after node i
yip,jr (Binary) 1 if the transfer task i p is served by a vessel immediately after the transfer task jr
yw,ip (Binary) 1 if the transfer task ip is served as the first task by a vessel
yip,w(Binary) 1 if the transfer task i p is served as the last task by a vessel
uiService start time for the transfer task requested immediately after visiting node i
liTotal load on the vessel at the departure from the transfer task requested immediately after visiting i
variable determines whether a car serves point jimmediately after serving point iwhile vis decides
whether the car visits hub simmediately after serving point i.xi j gives the order in which waste
points are assigned to a car while vis provides the selected hub and the last point in a trip. The hub
selection decisions by vis enable us to correctly calculate the total load on the car and the earliest
arrival time of a car at the hub saccording to the service decisions of the last point i. If vis is 1,
then there is a transfer task at hub swith a demand equal to the total waste on the car after serving
point idenoted by miand earliest service start time equal to the service end time at point idenoted
by hi+siplus travel time to hub s. In this way, the model jointly decides the first echelon routes
and transfer tasks for the second level routing problem. The second echelon sub-problem (vessel
routing) is a basic VRP where a fleet of vessels serves all the transfer tasks required by the first
echelon decisions respecting capacity of the vessels and the maximum time duration, operational
times represented by awand bwfor the waste centre. The vessel routing decisions are taken by y
variables. The synchronization is achieved by the earliest service start time for a transfer task at a
hub and delayed arrival time to the next point, accordingly to the service end time of the transfer
task plus travel time to next point. All sets, parameters and decision variables are presented in
Table 1.
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min(∑
i∈C
β1xdi +∑
i∈C
∑
p∈S
β2yw,ip) +( ∑
i,j∈Cd
ci jxi j +∑
i∈C
εi)+( ∑
i,j∈C
∑
r,p∈S
cpryi p jr +∑
i∈C
∑
p∈S
cwp yw,ip +cpwyi p,w)
(1)
subject to
∑
j∈Cd
xi j =1i∈C(2)
∑
j∈Cd
xji =1i∈C(3)
∑
i∈C
xdi =∑
i∈C
xid ≤K1(4)
∑
p∈S
vip ≥xid i∈C(5)
εi≥(cip +cp j −ci j )(xi j +vip −1)i∈C,j∈Cd,i=j,p∈S(6)
mj−mi≥qj−Q1(1−xi j +∑
p∈S
vip )i,j∈C,i=j(7)
ai≤hi≤bii∈C(8)
ad+td j ≤hjj∈C(9)
hi+si+ti j ≤hj+M(1−xi j)i∈C,j∈Cd,i=j(10)
ui+ (U+tpj )vip ≤hj+M(1−xi j )i∈C,j∈Cd,i=j,p∈S(11)
ui≥hi+si+tip vip i∈C,p∈S(12)
∑
p∈S
vip ≤1i∈C(13)
yw,ip +∑
j∈C
∑
r∈S
yjr,ip =vip i∈C,p∈S(14)
∑
j∈C
∑
r∈S
yip,jr +yip,w=vi p i∈,p∈S(15)
∑
i∈C
∑
p∈S
yw,ip =∑
i∈C
∑
p∈S
yip,w≤K2(16)
li≥mii∈C(17)
lj−li≥mj−Q2(1−∑
p∈S
∑
r∈S
yip,jr)i,j∈C,i=j(18)
aw+twp vip ≤uii∈C,p∈S(19)
ui+U+tpr ≤uj+M(1−yip,jr)i,j∈C,p,r∈S,i=j(20)
ui+U+tpwvi p ≤bwi∈C,p∈S(21)
xi j ∈ {0,1}i,j∈Cd,i=j(22)
vip ,yw,ip,yi p,w∈ {0,1}i∈C,p∈S(23)
yip,jr ∈ {0,1}i,j∈C,i=j,p,r∈S(24)
mi≥qii∈C(25)
εi≥0i∈C(26)
Objective function (1) minimizes the total number of used vehicles (vessels or cars) and the trans-
portation costs for cars and vessels. The first part is the number of total vehicles, the second is
the transportation cost for street level and the last part is the transportation cost for water level.
Constraints (2) and (3) ensure that each waste point is served exactly once by a car while con-
straints (4) indicate that the number of leaving and returning cars must be equal and should not
exceed available fleet size. Constraints (5) guarantee that a car must visit a hub before returning
to the depot in order to transfer the collected waste in its last trip. Constraints (6) calculate the
additional travel cost to visit a hub sbetween points iand j, assuming triangle inequality holds for
all i,jpairs. Constraints (7) are capacity constraints. Constraints (8)– (10) sequentially calculate
4

service start times at the points with respect to their time windows and operational time horizon
of the cars. Constraints (11) delay the arrival time to next point jif there exist any transfer task
just before point jwhile constraints (12) ensure the transfer task must be performed after the car
arrives at the selected hub. Constraints (14) and (15) assign a single vessel to hub ponly if there
exists a transfer task decision at that hub. If no transfer task is assigned to a hub pimmediately
after i, then all second echelon constraints regarding this task become redundant. Constraints (16)
indicate that the number of leaving and returning vessels to the waste centre must be equal and not
larger than the fleet size. Constraints (17) ensure that the waste load for a transfer task must be at
least the amount of collected waste on the car after serving last point ijust before the transfer task
occurs while (18) indicate the waste load on the vessel while performing transfer tasks. (19)-(21)
state temporal limitations for transfer tasks. Finally, (22) - (26) define ranges for each decision
variable.
Modelling one-to-one transfers
A hub can only perform one transfer task at a time, meaning that any two operations cannot
overlap. Let fi j be the time difference between the service start time of the transfer tasks requested
immediately after collecting node iand j. If they are assigned to the same hub p, then we need to
ensure that they need to be at least Uunits of time distant away from each other in order to finish
one before starting the other. The temporal distance between two operations assigned to the same
hub is equal to:
|ui−uj| ≥ U(vi p +vj p −1)i,j∈C,i=j,p∈S(27)
It can be linearized such that:
ui−uj≤fi j i,j∈C,i<j(28)
uj−ui≤fi j i,j∈C,i<j(29)
fi j =fji i,j∈C,i=j(30)
fi j ≥U(vip +vj p −1)i,j∈C,i<j,p∈S(31)
3. COMPUTATIONAL EXPERIMENTS
The proposed IWLT system, where cars and vessels operate in synchronization, is referred to as
a two-echelon VRP with flexible vessels system (2-echelon-F) and evaluated with respect to three
benchmarks: single echelon VRP with large trucks (1-echelon-T), single echelon VRP with small
cars (1-echelon-C), and two-echelon VRP with stationary barges (2-echelon-S) system.
The models are implemented in a computer with Intel Core(TM) i7-3820 3.60 GHz and 32 GB
RAM. They are solved by a commercial solver, CPLEX 12.10. The computation time limit is set
to an hour for every instance.
Only large trucks: 1-echelon-T
To assess the proposed IWLT collection system, the traditional collection system is modeled as-
suming that large garbage trucks start from a depot in the city, collect waste, deliver it to the waste
centre and return to the depot. It is formulated as a capacitated VRP with time windows consider-
ing the collection hours of the neighbourhoods. The proposed model by Bard, Kontoravdis and Yu
(2002) is modified to include the trip to the waste centre at the end of each route before returning
to the depot.
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Only small electric cars: 1-echelon-C
Since large trucks will be removed from the streets by 2025 (Amsterdam.nl,2019), another option
is to use smaller electric cars instead of large trucks. They are allowed to perform multiple trip to
the waste centre due to their relatively smaller capacities with respect to total waste. The model
used in 1-echelon-T system is modified to include multi-trip for the vehicles such that:
mj≥mi+qj−Q1(1−xi j +vi)i,j∈C,i=j(32)
hi+si+ti j +εi+Uvi≤hj+M(1−xi j)i,j∈C,i=j(33)
εi≥(ciw +cw j −ci j)(xi j +vi−1)i∈C,j∈Cd,i=j(34)
Let mibe the load on the car after collecting the waste point i,videcide whether there is a waste
centre visit immediately after iand εibe the additional travel time to the waste centre. The cost of
additional travel time (εi) is also added to the objective as a part of travel cost.
Stationary barges: 2-echelon-S
In this case, large barges are placed along the canals as temporary dump sites for the cars. Instead
of delivering the collected waste to the waste centre as in the traditional setting or to the vessels as
in the proposed setting, the cars dump the waste into these barges that are placed at a hub during
the collection hours. The barges are taken to the waste centre by tugboats when they are full or
no longer needed. The number of barges is assigned such that the waste generated in the city can
be contained, bn=⌈∑i∈Cqi
Qb⌉where Qbis the capacity of a barge. The objective still includes the
travel cost of the tugboats to place the barges to the selected hubs and take them back to the waste
centre in order to account for the water level logistic costs in the IWLT setting. The proposed
2E-MTVRPTW-SS formulation is modified as follows:
vip ≤zpi∈C,p∈S(35)
∑
p∈S
zp=bn=K2(36)
yip,jr =0i,j∈C,p,r∈S,p=r(37)
Constraints (35) allow the cars to use selected hubs as dump sites while constraints (36) ensure
that at most bnhubs are selected. Constraints (37) prevent inter-movements between hubs meaning
that they can only be placed at a single hub. Note that these constraints are added to the model ex-
plained in Section 2and subject to the hub capacity, multi-trip, synchronization and time window
constraints.
Test Instances
We use modified Solomon’s VRPTW instances (Solomon,1987) as proposed by Grangier et al.
(2016) for geographical configuration. The only difference is in locating the hubs, where we
choose hubs outside of the city while Grangier et al. (2016) locate them at the center of the
network. Keeping the transfer operations away from the public is primarily motivated by hygienic
concerns, noise and the lack of space in the city. Let xmin,xmax ,ymin, and ymax be the minimum and
maximum values of the coordinates of the nodes to collect. Four hubs are located at (xmin,ymin),
(xmin,ymax ), (xmax,ymin ), and (xmax,ymax ). The earliest and latest operational times of the hubs, and
the waste centre are equal to the ones of the depot as given in the instances.
To better observe multiple trips and transfer tasks, the capacity of electric cars is set to 50 units
6

Table 2: Results on the instances with 10 waste points and four hubs derived from
Solomon’s VRPTW problems (Solomon,1987)
Street Level Water Level
NV Travel Time Weighted Avg.
Load NV Travel Time Weighted Avg.
Load
C
1-echelon-T 1 227,99 (base) 92,11 (base) - - -
1-echelon-C 1 392,28 (+72%) 26,52 (-71%) - - -
2-echelon-S 1 208,65 (-8%) 28,63 (-69%) 1 119,61 79,38
2-echelon-F 1 181,16 (-21%) 28,19 (-69%) 1 148,24 106,47
R
1-echelon-T 1 277,71 (base) 84,24 (base) - - -
1-echelon-C 1,3 413,59 (+49%) 27,61 (-67%) - - -
2-echelon-S 1 263,82 (-5%) 31,02 (-63%) 1 120,44 86,45
2-echelon-F 1 205,66 (-26%) 26,65 (-68%) 1 191,28 151,46
RC
1-echelon-T 1 209,88 (base) 136,57 (base) - - -
1-echelon-C 2 408,04 (+94%) 38,71 (-72%) - - -
2-echelon-Sf2 250,48 (+19%) 42,02 (-69%) 1 62,43 120,63
2-echelon-Ff2 197,56 (-6%) 35,91 (-74%) 1 163,58 146,23
in 1-echelon-C, 2-echelon-S, and 2-echelon-F systems. The capacity of the trucks, barges and
vessels are set to 250 units.
The objective is minimizing the number of trucks or cars first, and then minimizing the travel cost
for 1-echelon-T and 1-echelon-C systems. Similarly for 2-echelon-S and 2-echelon-F systems,
the first priority is to minimize the number of cars, then the travel cost. The cost of water-level
logistics is also minimized in the same order but relatively less important than the street level cost
such that the fixed cost of the cars (β1) is set to 1000, the travel cost on the streets is equal to 1
while the fixed cost for the vessels (β2) 100 and the travel cost is 0.1 of the travel times. The main
motivation is to reduce the heavy movements and congestion on the streets. Lastly, Uis assumed
to be 150 time units.
For testing different approaches, we assume that waste points are the first ten nodes of Solomon
class "2" instances with wide time windows and long scheduling horizons, which is more similar
to the structure of the collection hours for neighbourhoods.
Results and Discussion
Table 2summarizes the average results of the instances in each type for the problems with ten
waste points for all approaches and four hubs for 2-echelon-S and 2-echelon-F systems. Based
on the geographical distribution of the waste points, cases are divided into three categories: C
type for clustered locations, R type for random locations, and RC type for randomly clustered
locations. For both levels, NV is the number of the vehicles (cars, barges, or vessels), Travel Time
is the total travel time of the vehicles on their own network, while Weighted Avg.Load is the
weighted average of the load on the vehicles per travel time considering non-empty movements.
All instances are solved to optimality except the ones labeled with superscript f, where the best
feasible solutions are presented.
1-echelon-T system has typically been preferred by decision-makers for the advantage of easy and
direct access to service areas, as well as larger capacity of trucks, which leads to fewer trucks
needed for waste collection. Therefore, 1-echelon-T system cannot be beaten in terms of the
number of vehicles for all instances as expected. 1-echelon-C system requires frequent visits to
the waste centre due to the smaller capacity of electric cars, resulting in under-utilization of the
working hours. It causes more cars to use in R type problems on average compared to 2-echelon-S
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and 2-echelon-F systems, where small cars also operate. 2-echelon IWLT systems can reduce the
number of cars down to the number of trucks for R and C type problems, but failed to use less cars
for RC type problems.
Travel Time for the street level represents the vehicle movements on the streets. The models, also,
minimize the travel cost which is mostly proportional to the travel time. The results show that the
proposed 2-echelon-F system reduces the burden on the road infrastructure by partially shifting
the movements to inland waterways. It reduces the total travel time on the streets for all type of
problems more than 2-echelon-S system which shows the added value of the flexibility.
The cost of waste collection in Amsterdam is not only about the logistic cost but also the damage
on quay walls. 2-echelon-S system produces the largest values for Weighted Avg.Load among
three systems, where the cars are used, compared to 1-echelon-T. 2-echelon-F system can achieve
the lowest Weighted Avg.Load for almost all scenarios. It shows us 2-echelon-F system has the
potential to reduce heavy street movements by providing cheaper solutions in terms of the fleet
size, total street travel time and lightweight operating garbage cars.
4. CONCLUSIONS
In this study, we consider an integrated water- and land-based waste collection system that aims
to remove heavy large garbage trucks from the streets to reduce the damage on the quay walls
as well as the congestion. We provide a new formulation for the 2E-MTVRPTW-SS considering
one-to-one transfers, and compare the proposed approach with three different benchmarks in terms
of the fleet size, average travel time on the streets, and weighted average load of the vehicles per
non-empty movements. The proposed system with synchronized mobile vessels and electric cars
is shown to be a promising solution for the issues with the current system. It can reduce the total
travel time of the garbage cars on the street by 18% and the weighted average loads of the cars by
70% on average across all scenarios without increasing the fleet size of the cars significantly even
if they have way less capacities than the traditional garbage trucks.
The proposed model is a computationally heavy problem. Decomposition-based exact methods or
heuristics can be developed for the problem to solve larger instances in order to gain more insights
into the gains of such a system. The gains observed in small instances gives indications on the
potential improvements that can be obtained for larger instances.
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ACKNOWLEDGMENT
This research is supported by the project “Sustainable Transportation and Logistics over Water:
Electrification, Automation and Optimization (TRiLOGy)” of the Netherlands Organization for
Scientific Research (NWO), domain Science (ENW), and by the Researchlab Autonomous Ship-
ping (RAS) at TUDelft. We also thank Marcel Stiphout from the Municipality of Amsterdam for
his assistance in designing the two-echelon systems.
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