Drone-Truck Combined Operations Problem: A Comparative Study on
Operational Models and Solution Algorithms
Yida Dinga , Jiaqi Wanb , Wei Linc , Kai Wanga , Xiaoqian Sund,∗
a
School of Vehicle and Mobility, Tsinghua University, 100084 Beijing, China
Department of Precision Instrument, Tsinghua University, 100084 Beijing, China
c
Department of Computer and Engineering, The Chinese University of Hong Kong, Hong Kong SAR, China
d
School of Electronic and Information Engineering, Beihang University, 100191 Beijing, China
b
Abstract
Throughout the recent decade, various researchers have proposed to leverage the concept of
urban air mobility for an effective construction of package delivery service to address problems such
as last-mile delivery and recurrent traffic jams in metropolitan regions. To resolve the relatively
short travel range and limited capacity issues of drones, researchers have proposed to further enhance
the utility of drones by deploying them with ground vehicles (e.g., trucks) in tandem. This new
concept leads to a new transportation planning problem, i.e., drone-truck combined operations
problem (DTCO). Both assignment decisions (i.e., which vehicle will serve which customer) and
routing decisions (i.e., the sequence of visits for each customer assigned by the vehicle) are involved
in DTCO. In this paper, a systematic evaluation and experimental benchmark are conducted for
the algorithms of three typical DTCO models: the flying sidekick traveling salesman problem
(FSTSP), the traveling salesman problem with drone (TSP-D), and the parallel drone scheduling
traveling salesman problem (PDSTSP). We evaluate the algorithms by various metrics including
result quality, runtime and scalability. We conduct sensitivity analysis to assess the performance
of a drone delivery system under various system parameters, such as drone speed, drone range and
customer geographic distribution. Finally, a case study is conducted to demonstrate the superior
performance of DTCO model as compared to classical TSP model based on the city road network
data of Beijing. Our research provides insights for developing new methodologies for large-scale
real-world drone logistics problems.
1. Introduction
In recent years, companies have been exploring new technologies to enhance the delivery of goods
ordered online. One such technology that has recently gained attention is the use of unmanned aerial
vehicles (i.e., drones) for parcel delivery. A key advantage of using drones for delivery operations
is that they can operate without a costly human pilot. On the other hand, drones can fly over
congested roads without adhering to the road network, making them faster than regular delivery
vehicles.
∗
Corresponding author
Email addresses: dyd23@mails.tsinghua.edu.cn (Yida Ding), wjq23@mails.tsinghua.edu.cn (Jiaqi Wan),
louislin@link.cuhk.edu.hk (Wei Lin), cwangkai@tsinghua.edu.cn (Kai Wang), sunxq@buaa.edu.cn (Xiaoqian
Sun)
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1 INTRODUCTION
Figure 1: Two on-site experiments of drone-truck combined operation by UPS (left) and Meituan (right)
On the industry side, Amazon, Alibaba, and Google are currently conducting practical tests on
the feasibility of using drones for parcel delivery (Agatz et al., 2018). These trials involve multipropeller drones that can transport parcels weighing around two kilograms across a distance of 20
kilometers. Although there are examples of drones that are currently used for deliveries, they are
limited to non-urban areas. For instance, DHL Parcel has started using drones to deliver essential
items, such as medication, to one of Germany’s North Sea islands. Despite being automated, these
drones still require continuous monitoring. However, aeronautics experts anticipate that drones will
become capable of autonomously and safely operating in urban environments within the next ten
years due to significant advancements in obstacle detection and avoidance technology.
However, there are several limitations of using drones for parcel delivery. One of the most significant constraints is the weight and size restrictions, as most commercial drones have a weight limit
of around two kilograms, which limits their ability to deliver larger and heavier items. Additionally,
drones have limited flight range, making it challenging to deliver parcels to areas that are far away
or not easily accessible. Weather conditions, such as rain, wind, and fog, can also affect drone
operations, potentially delaying deliveries. Furthermore, there are strict regulations governing the
use of airspace, which can limit the ability of drones to deliver parcels to certain areas, such as
urban environments or near airports.
As drones continue to gain attention in the delivery industry, researchers and companies are
exploring ways to extend their effective range and capacity. One promising solution is to allow
drones to collaborate with delivery vehicles, such as vans or trucks, to complete the last mile of
the delivery journey. This approach takes advantage of the strengths of both drones and vehicles.
Drones can quickly and easily navigate through urban environments and deliver small, lightweight
packages, while vehicles can carry larger and heavier items over longer distances. By combining
these capabilities, companies can offer faster and more efficient deliveries to customers. In this
scenario, the delivery vehicle acts as a base station for the drone, allowing it to recharge, swap
batteries, or transfer packages, which would enable the drone to cover greater distances and deliver
more parcels. This approach has already been tested by several companies, such as Mercedes-Benz,
UPS, and Amazon, and it shows great potential to revolutionize the delivery industry by providing
more cost-effective and sustainable delivery solutions. Figure 1 shows two on-site experiments of
drone-truck combined operation by UPS (left) and Meituan (right).
The widespread interest of drone-truck combined operations (DTCO) problem has led to a
plethora of research papers. For an overview of the literature of DTCO problems, the reader is
referred to several excellent surveys and reviews in this area (Chung et al., 2020; Pasha et al.,
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2 PROBLEM FORMULATIONS
3
2022; Otto et al., 2018). Most DTCO problems are NP-hard, leaving room for devising and tuning
solution techniques, be it exact methods or heuristics. Solution techniques have appeared in a wide
range of venues, given the practical application of the problem to disrupt the logistic industry.
Moreover, although some larger problems can be solved quite well, it is not well understood which
solution approaches work best for various problems and why.
In this paper, a systematic evaluation and experimental benchmark are conducted for the algorithms of three typical DTCO models: the flying sidekick traveling salesman problem (FSTSP),
the traveling salesman problem with drone (TSP-D), and the parallel drone scheduling traveling
salesman problem (PDSTSP). These three models provide the basic structures for DTCO problems,
and the majority of DTCO papers incorporate extensions on top of them. The comparative study
includes various metrics such as runtime, scalability, and result quality. Through evaluating the
pros and cons of different models and algorithms, this paper aims to summarize previous works.
Patterns and similarities among different solution techniques would be identified and extracted, providing insights for developing new methodologies for large-scale real-world drone logistics problems.
The major contribution of this study is summarized as follows:
• We reimplement typical exact/heuristic algorithms for the three typical DTCO models and
compare the performance of solution algorithms regarding runtime, scalability, and result
quality.
• We conduct sensitivity analysis to assess the performance of a drone delivery system under various system parameters, such as drone speed, drone range and customer geographic
distribution.
• We conduct a case study to demonstrate the superior performance of DTCO model as compared to classical TSP model based on the city road network data of Beijing.
The remainder of this paper is as follows: Section 2 reviews models for the FSTSP, TSP-D, and
PDSTSP problems with assumptions and formulations. Section 3 summarizes multiple exact and
heuristic solutions for each problem. Section 4 conducts various experimental evaluations to each
model, sensitivity analysis to assess the performance of the system under varying parameters, and
a case study based on the city road network data of Beijing. Section 5 concludes the paper and
discusses future research directions.
2. Problem Formulations
This section starts with a brief review of three typical DTCO problems and their formulations;
the goal of this section is to set the notation and terminology for the remainder of this study. The
following problems are formalized as defined in the literature: FSTSP in Section 2.1, TSP-D in
Section 2.2 and PDSTSP in Section 2.3.
2.1. Flying Sidekick Traveling Salesman Problem (FSTSP)
2.1.1. Basic Model
The flying sidekick traveling salesman problem (FSTSP) is a variant of the classical Traveling
Salesman Problem where a delivery system consists of a truck and a drone. The drone can launch
from the truck with a single package to deliver to a customer and then rendezvous with the truck
at another location. The truck can also deliver packages to other customers while the drone is
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Table 1: Sets, parameters and decision variables of the FSTSP model
Sets
C
C′
N
set of all customers, C = {1, 2, ..., c}
set of drone-eligible customers, C ′ ⊆ C
set of all nodes, where the first node and the last node are the depot node, N =
{0, 1, ..., c + 1}
set of nodes that the truck may depart from, N0 = {0, 1, ..., c}
set of nodes that the truck may arrive at, N+ = {1, 2, ..., c + 1}
set of all feasible drone sorties
N0
N+
P
Parameters
τi,j
′
τi,j
sL , sR , e
Dec. var.
xij
yijk
tj
t′j
Pij
ui
the travel time from node i ∈ N to node j ∈ N for the truck
the travel time from node i ∈ N to node j ∈ N for the drone
the launch time, rendezvous time and flight endurance for the drone
1 if
1 if
the
the
1 if
the
the truck travels from i ∈ N0 to j ∈ N+ with i ̸= j, and 0 otherwise
the drone travels from i ∈ N0 to j ∈ C ′ and back to k ∈ C+
time at which the truck arrives at node j ∈ N
time at which the drone arrives at node j ∈ N+
customer i ∈ C is visited before customer j ∈ {C : j ̸= i} in the truck route
position of node i ∈ N+ in the truck’s path
flying. The objective is to minimize the total time spent by the truck and the drone to visit all the
customers. Over the course of a delivery cycle the drone may make multiple sorties, each consisting
of three locations. A drone sortie may begin at either the depot or from a customer location where
it is loaded by the truck driver with a parcel. The second location in a sortie must be a customer
node served by the drone. The final node of a sortie may be either the depot or the location of the
truck. The sets, parameters and decision variables of the FSTSP model are tabulated in Table 1.
Considering the realistic scenario and modeling complexity, some assumptions are proposed as
follows. Since drones have restricted capacity, they are capable of serving only one customer during
each sortie. Given the space needed to land a drone and the potential hazards involved, the drone
cannot land temporarily to wait for a truck. Thus, the time that drone spends waiting for the truck
needs to be counted toward the endurance of the drone. After the drone is collected by the truck,
it can depart from the same location again to start a new sortie. If the drone is collected by the
truck, this must take place at the location of a customer served by the truck. Neither truck nor
drone may revisit any customer nodes which is served before. If the drone is collected at the depot,
it is taken out of service (i.e., it cannot be relaunched from the depot). The time it takes to load
the drone and replace the battery needs to be taken into account.
The mathematical model of the FSTSP proposed by Murray and Chu (2015) is as follows:
min tc+1
X
X
s.t.
xij +
i∈N0
i̸=j
(1)
X
yijk = 1 ∀j ∈ C
(2)
i∈N0 k∈N+
i̸=j ⟨i,j,k⟩∈P
X
x0j = 1
(3)
xi,c+1 = 1
(4)
j∈N+
X
i∈N0
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2 PROBLEM FORMULATIONS
ui − uj + 1 ⩽ (c + 2) (1 − xij )
X
X
xjk ∀j ∈ C
xij =
i∈N0
i̸=j
X
(5)
∀i ∈ C, j ∈ {N+ : j ̸= i}
(6)
k∈N+
k̸=j
X
(7)
yijk ⩽ 1 ∀i ∈ N0
j∈C k∈N+
j̸=i ⟨i,j,k⟩∈P
X
X
(8)
yijk ⩽ 1 ∀k ∈ N+
i∈N0 j∈C
i̸=k ⟨i,j,k⟩∈P
2yijk ⩽
X
xhi +
y0jk ⩽
xlk
∀i ∈ C, j ∈ {C : j ̸= i}, k ∈ {N+ : ⟨i, j, k⟩ ∈ P }
(9)
l∈C
l̸=k
h∈N0
h̸=i
X
X
xhk
(10)
∀j ∈ C, k ∈ {N+ : ⟨0, j, k⟩ ∈ P }
h∈N0
h̸=k



uk − ui ⩾ 1 − (c + 2) 
1 −
X
j∈C
⟨i,j,k⟩∈P


yijk 

∀i ∈ C, k ∈ {N+ : k ̸= i}
(11)

X

t′i ⩾ ti − M 
1
−

X
j∈C k∈N+
j̸=i ⟨i,j,k⟩∈P
(12)

yijk 
 ∀i ∈ C


X

t′i ⩽ ti + M 
1 −
X
j∈C k∈N+
j̸=i ⟨i,j,k⟩∈P

yijk 


(13)
∀i ∈ C

X

t′k ⩾ tk − M 
1
−

X
i∈N0 j∈C,C
i̸=k ⟨i,j,k⟩∈P


yijk 

∀k ∈ N+
(14)
∀k ∈ N+
(15)

X

t′k ⩽ tk + M 
1 −
X
X
X
i∈N0 j∈C
i̸=k ⟨i,j,k⟩∈P
tk ≥ th + τhk + sL (
l∈C m∈N+
l̸=k ⟨k,l,m⟩∈P

yijk 

yklm ) + sR (
X
X
yijk ) − M (1 − xhk )
(16)
i∈N0 j∈C
i̸=k ⟨i,j,k⟩∈P
∀h ∈ N0 , k ∈ {N+ : k =
̸ h}
X
t′j ≥ t′i + τij′ − M (1 −
yijk ) ∀j ∈ C ′ , i ∈ {N0 : i ̸= j}
(17)
k∈N+
⟨i,j,k⟩∈P
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′
t′k ≥ t′j + τjk
+ sR − M (1 −
X
yijk ) ∀j ∈ C ′ , k ∈ {N+ : k ̸= j}
(18)
i∈N0
⟨i,j,k⟩∈P
t′k − (t′j − τij′ ) ≤ e + M (1 − yijk )
(19)
∀k ∈ N+ , j ∈ {C : j ̸= k}, i ∈ {N0 : ⟨i, j, k⟩ ∈ P }
ui − uj ≥ 1 − (c + 2)pij
(20)
∀i ∈ C, j ∈ {C : j ̸= i}
ui − uj ≤ −1 + (c + 2)(1 − pij ) ∀i ∈ C, j ∈ {C : j ̸= i}
(21)
pij + pji = 1 ∀i ∈ C, j ∈ {C : i ̸= j}
X
X
yijk −
t′l ≥ t′k − M (3 −
(22)
j∈C
⟨i,j,k⟩∈P
j̸=l
X
ylmn − pil )
(23)
m∈C n∈{N+ :n̸=i,n̸=j}
m∈{i,j,k}
/
⟨l,m,n⟩∈P
∀i ∈ N0 , k ∈ {N+ : k ̸= i}, l ∈ {C : l ̸= i, l ̸= k}
t0 = 0
(24)
t′0
(25)
=0
p0j = 1 ∀j ∈ C
(26)
xij ∈ {0, 1} ∀i ∈ N0 , j ∈ {N+ : j ̸= i}
(27)
yijk ∈ {0, 1} ∀i ∈ N0 , j ∈ {C : j ̸= i}, k ∈ {N+ : (i, j, k) ∈ P }
(28)
1 ≤ ui ≤ c + 2 ∀i ∈ N+
(29)
ti ≥ 0 ∀i ∈ N
(30)
t′i
(31)
≥ 0 ∀i ∈ N
pij ∈ {0, 1} ∀i ∈ N0 , j ∈ {C : j ̸= i}
(32)
The objective function (1) aims to minimize the latest time at which either the truck or the
drone returns to the depot. Constraint (2) ensures that each customer is served exactly once, either
by a drone or a truck. Constraint (3) and Constraint (4) require that the truck will only depart from
and arrive at the depot exactly once, respectively. Subtour elimination constraints for the truck are
provided by Constraint (5). Constraint (6) indicates that a truck visiting node j must also depart
from j, while Constraint (7) states that the drone may launch from any particular node, including
the depot, at most once. Similarly, Constraint (8) indicates that the drone may rendezvous at any
particular node (including customers and the ending depot) at most once. Constraint (9) specifies
that if the drone takes off from customer i and the truck picks it up at node k, then the truck must
be assigned to both customer i and node k. Similarly, Constraint (10) ensures that if the drone
launches from the starting depot 0 and is collected at node k, then the truck must be assigned to
node k. Constraint (11) requires that if the drone is launched from customer i and collected at
node k, the truck must visit customer i before node k. Constraints (12) and (13) make sure that
the truck and the drone are synchronized in time when the drone is launched from a customer node
i. Similarly, Constraints (14) and (15) ensure synchronization between the truck and the drone
when the drone returns to the truck at node k. These constraints ensure that the truck and the
drone arrive at node k at the same time. Constraint (16) states that when the truck travels from
node h to node k, the time at which it arrives at node k must take into account the time it took
for the truck to travel from node h to k. Constraint (17) specifies that if the drone is launched
from node i, then the time when it arrives at node j should include the travel time from i to j.
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Constraint (18) ensures that the arrival time at node k takes into account the travel time from node
j to k as well as the recovery service time at node k denoted as sR . Constraint (19) deals with the
maximum time that the drone can stay airborne. Constraints (20) to (22) determine the correct
values of pij . Constraint (23) ensures that if the drone takes off from node i, lands back at node k,
and subsequently takes off from node l, the launch time t′l must not be earlier than the return time
to k, t′k . Constraints (27) to (32) provide definitions for the decision variables used in the model.
2.1.2. Extended Models
After Murray and Chu introduced the MILP model for the FSTSP, numerous subsequent studies
building upon this model have suggested various extensions aimed at enhancing the modeling of
FSTSP to align it more closely with real-world scenarios. This section will present a selection of
notable extensions.
D. Schermer et al. suggested the relaxation of the assumption that the launch and retrieval of
the drone are restricted to the locations of customers and the depot (Schermer et al., 2019). They
promoted the idea of launching and retrieving drones at arbitrary locations. The proposed new
problem is called the Vehicle Routing Problem with Drones and En Route Operations (VRPDERO)
and is formulated as a mixed integer linear programming problem.
In contrast to minimizing the time at which the truck and drone finish the service, Ha et al.
(2018) consider a new variant of the FSTSP in which the objective is to minimize operational costs
including total transportation cost and the waiting cost for either vehicle. For the transportation
cost, the truck and the drone have their transportation cost per unit of distance. For the waiting
cost, the idle cost of the truck and the drone are estimated in their model, respectively.
Jeong et al. (2019) incorporated more real-world factors into their modeling. They asserted that
the flight range of the drone is influenced by the payload weight due to its battery-powered nature.
Additionally, there are restrictions on where the drone can be operated, including no-fly zones like
airports, heliports, and areas near military facilities. According to the physical properties of the
drones and the experimental results, they established a linear model to depict the relationship
between drone payload and battery capacity and derived the relationship between payload and
endurance. For no-fly zones, they assumed that they were all circular and deduced the distance
needed to get around them.
Ham (2018) considered the situation of multiple trucks and multiple drones coordinating in
asynchronous mode. The problem is subject to several constraints such as drone capacity, flight
endurance, time windows, drop and pickup synchronization, and multi-visit. The authors proposed
two constraint programming (CP) models to solve the problem exactly. The first model (CP1) used
state-dependent transition distances to capture the different types of drone tasks (drop and pickup).
The second model (CP2) used virtual pickup and drop tasks to represent the drone capacity and
synchronization.
In another work,Murray and Raj (2020) extended the basic FSTSP model to consider multiple
heterogeneous drones. These drones may have different flight speeds, maximum loads, and ranges.
Besides, the truck cannot retrieve multiple drones at the same time due to its capacity limitation.
As a result, they specifically introduced a queuing system for the drone during both the launch and
retrieval stages, leading to an additional scheduling challenge.
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Table 2: Sets, parameters and decision variables of the TSP-D model
Sets
O
O− (i)
O+ (i)
O(i)
O− (S)
O+ (S)
Parameters
to
Dec. var.
αo
βi
set of operations, o ∈ O
subset of operations with start node i ∈ N
subset of operations with end node i ∈ N
subset of operations that contain node i ∈ N
subset of all operations with the start node in S and the end node in N \S, for each
set of nodes S ⊂ N
subset of all operations with the end node in S and the start node in N \S, for each
set of nodes S ⊂ N
the travelling time for the truck and drone to take operation o ∈ O in a synchronized
manner
1 if the operation o ∈ O is chosen in the solution, and 0 otherwise
1 if node i ∈ N is chosen as a start node in at least one operation, and 0 otherwise
2.2. Traveling Salesman Problem with Drone (TSP-D)
2.2.1. Basic Model
The goal of the TSP-D problem is to determine the most time-efficient route that visits all
customer locations, utilizing either the truck or the drone. The problem aims to find a route for
both the truck and the drone that minimizes the total joint time to serve all delivery tasks. Due
to its limited capacity, the drone has to return to the vehicle to pick up the parcel before each new
delivery. Accordingly, the route of the drone needs to be synchronized with that of the truck. The
routing of drone and truck should satisfy customer-eligibility issue, drone’s flying range issue and
etc.
The TSP-D is very similar to the FSTSP, and the primary difference lies in their assumptions
and formulation. The TSP-D uses the concept of operations, which contains at most one drone node
(i.e., a node that is only visited by a drone), two combined nodes (i.e., a node that is visited by
both truck and drone), and a nonnegative number of truck nodes (i.e., a node that is only visited by
a truck). A drone is allowed to meet a truck at the same node from which it launches in operation,
while it is not allowed in the concept of sortie. The operation can also take into account other
practical restrictions on the path of the drone and/or truck, for example, maximum flight distance
and time, nodes that cannot be visited by the drone, maximum waiting times, etc. Readers may
refer to Agatz et al. (2018) for more explanation on the concept of operations. Besides, the major
advantage of the TSP-D lies in its simple formulation compared to the FSTSP. The sets, parameters
and decision variables of the TSP-D are tabulated in Table 2. The integer programming model of
the TSP-D is formulated as follows:
min
X
(33)
to αo
o∈O
s.t.
X
αo ≥ 1 ∀i ∈ N
(34)
o∈O(i)
X
αo ≤ c · βi
∀i ∈ N
(35)
o∈O+ (i)
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X
X
αo =
o∈O+ (i)
X
∀i ∈ N
(36)
∀S ⊂ C, i ∈ S
(37)
αo
o∈O− (i)
αo ≥ βi
o∈O+ (S)
X
αo ≥ 1
(38)
o∈O+ (0)
β0 = 1
(39)
αo ∈ {0, 1} ∀o ∈ O
(40)
βi ∈ {0, 1} ∀i ∈ N
(41)
The objective function (33) aims to minimize the tour cost, which is the sum of the costs of all
the operations chosen. Constraint (34) ensures that all nodes are covered by at least one operation.
Constraint (35) sets yv to 1 if at least one operation that starts at node v is selected. The left-hand
side of Constraint (36) is less than or equal to c since each operation must contain at least one
previously unvisited node in any optimal solution.
Considering operations as arcs from their start node to their end node, Constraints (36) and (37)
ensure that the chosen operations span a Eulerian graph such that a Eulerian cycle in this graph
represents a feasible truck-and-drone tour. Besides, Constraints (37) and (38) ensure that the graph
is connected. Constraint (39) ensures that this tour starts (and ends) at the depot. Constraints (40)
and (41) restrict the variables xo and yv to be binary.
The extended models of the TSP-D is similar to those of the FSTSP since both of the basic
model considers a single drone operating in tandem with a single truck. Readers may refer to
Section 2.1.2 for the extended models of the FSTSP.
2.3. Parallel Drone Scheduling Traveling Salesman Problem (PDSTSP)
2.3.1. Basic Model
The parallel drone scheduling traveling salesman problem (PDSTSP) involves the optimal routing of a delivery truck and multiple delivery drones to serve a set of customers in an independent
manner (Murray and Chu, 2015). Regarding the application scenario, the FSTSP and TSP-D
mainly consider the case in which the depot is relatively far from the customer locations and the
drone is operating in tandem with the truck. In contrast, the PDSTSP primarily deals with the
case in which a significant proportion of customers are located within the flight range of drone from
the depot. In the PDSTSP, a single truck and a fleet of identical drones depart and return to a
single depot. The drone serves customers directly from the depot in a point-to-point manner, while
the truck serves the customers in a TSP route manner, so that the synchronization effect between
both types of vehicles is not considered. The objective is to minimize the latest time that a vehicle
returns to the depot, such that each customer is served exactly once. The sets, parameters and
decision variables of the PDSTSP are tabulated in Table 3. The mixed integer linear programming
formulation of the PDSTSP is formulated as follows:
(42)
min z
s.t. z ≥
X X
τi,j x̂i,j
(43)
i∈N0 j∈N+
j̸=i
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Table 3: Sets, parameters and decision variables of the PDSTSP model
Sets
D
C′
C ′′
Dec. var.
x̂i,j
ŷi,d
ûi
set of identical drones, d ∈ D
subset of customers that may receive delivery of their parcel via a drone, i.e., these
customers are drone-eligible, C ′ ⊆ C
subset of drone-eligible customers that are also within the drone’s range from the
′
′
depot, i.e., customer i ∈ C ′ is in set C ′′ if τ0,i
+ τi,c+1
≤e
1 if the truck travels from node i ∈ N0 to node j ∈ {N+ : j ̸= i}, and 0 otherwise
1 if customer i ∈ C ′′ is served by drone d ∈ D, and 0 otherwise
the position of node i ∈ N+ in the truck’s path, 1 ≤ ûi ≤ c + 2
X
z≥
′
′
τ0,i
+ τi,c+1
ŷi,d
∀d ∈ D
(44)
i∈C ′′
X
x̂i,j +
i∈N0
i̸=j
X
(45)
ŷj,d = 1 ∀j ∈ C
d∈D
j∈C ′′
X
x̂0,j = 1
(46)
x̂i,c+1 = 1
(47)
j∈N+
X
i∈N0
X
i∈N0
i̸=j
x̂i,j =
X
(48)
∀j ∈ C
x̂j,k
k∈N+
k̸=j
ûi − ûj + 1 ≤ (c + 2) (1 − x̂i,j )
∀i ∈ C, j ∈ {N+ : j ̸= i}
(49)
1 ≤ ûi ≤ c + 2 ∀i ∈ N+
(50)
x̂i,j ∈ {0, 1} ∀i ∈ N0 , j ∈ {N+ : j ̸= i}
(51)
′′
ŷi,d ∈ {0, 1} ∀i ∈ C , d ∈ D
(52)
The objective function (42) aims to minimize the latest return time for both type of vehicles
to return to the depot. Constraints (43) and (44) incorporate the routing decisions of both type of
vehicles to provide lower bounds on z. Constraint (45) ensures that each customer is served exactly
once by either a truck or a drone. Constraints (46) and (47) ensure the truck to leave and return to
the depot exactly once. Constraint (48) is the flow balance constraint for the truck at any customer
node, while Constraint (49) is a standard subtour elimination constraint. Constraints (50) to (52)
specify the decision variable definitions.
2.3.2. Extended Models
Many studies in the literature extended the PDSTSP to incorporate more realistic factors into
the model. Mbiadou Saleu et al. (2018) presented a new heuristic algorithm for solving the PDSTSP,
in which multiple drones and multiple delivery trucks work in coordination to deliver small packages
to geographically distributed customers. This heuristic algorithm is an iterative two-step algorithm.
The first step of the algorithm is to assign all drones to delivery trucks in order to form a subproblem
on each delivery truck. The second step is to use a greedy algorithm in each subproblem to determine
the flight path of the drones. The iterative process of this algorithm is performed in the second
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step until the paths of all drones are determined.
Lei and Chen (2022) presented an improved variable neighborhood search (VNS) algorithm
for solving the PDSTSP. The VNS algorithm uses three methods to generate the initial solution
and applies a shaking process to diversify the search space. The algorithm improves the solution
iteratively by applying a variable neighborhood descent (VND) process. The VNS algorithm is
tested on several datasets and compared with other algorithms.
Ham (2018) presented a new constraint programming (CP) application for multi-truck, multidrone, and multi-depot scheduling problems subject to time windows, drop-pickup, and multiple
visits, with the goal of minimizing the total cost. The algorithm uses a CP model to describe the
problem and solves it using a CP solver. It is worth noting that the model, unlike other algorithms,
considers the multi-truck case.
Dell’Amico et al. (2020) proposed a mathematical heuristic algorithm for the PDSTSP. In this
paper, the authors consider the PDSTSP, where a set of customers to be delivered is assigned to
a truck and a fleet of drones with the objective of minimizing the total time required to serve all
customers. The authors propose a set of mathematical heuristics to solve this problem. Among
the algorithms proposed by the authors are greedy-based algorithm, simulated annealing-based
algorithm, genetic algorithm-based algorithm, ant colony algorithm-based algorithm, etc. All these
heuristic algorithms are designed to minimize the total time required to serve all customers.
3. Solution Techniques
3.1. Solution Techniques for FSTSP
The MILP model for the FSTSP can be directly solved by standard optimization solvers such
as CPLEX and Gurobi. Although the optimal solution is guaranteed, they take extensive amount
of runtime which is unaffordable in practical application. Therefore, various heuristic methods
are proposed in the literature to obtain a near-optimal feasible solution in reasonable time, which
presents a trade-off between runtime and result quality. In this section, three heuristic methods
adopted in this benchmark are introduced.
3.1.1. Greedy Randomized Adaptive Search Procedure
The greedy randomized adaptive search procedure (GRASP) is an effective metaheuristic proposed by Ha et al. (2018). The main idea of the metaheuristic is to constantly generate candidate
solutions and improve them through local search, finally reaching to a close-to-optimal solution.
It starts by initializing the best objective value to infinity and setting the iteration counter to
zero. The algorithm then enters a loop that will repeat until the maximum number of iterations is
reached. In each iteration, the algorithm generates a candidate solution by solving the TSP problem for the given set of customers using any TSP algorithm (e.g., local search-based algorithms,
2-opt heuristic, etc.). Then, the metaheuristic constructs an auxiliary graph using the candidate
solution, and the solution for FSTSP (i.e., the tour for truck and drone) can be extracted based on
the auxiliary graph. If the new tour has a better objective value than the current best solution, the
best solution and best objective value will be updated. The algorithm repeats this process until
the maximum number of iterations is reached or a stopping criterion is met.
3.1.2. Dynamic Customer Relocation Algorithm
The dynamic customer relocation algorithm (DCRA), proposed by Murray and Chu (2015),
is based on a combination of TSP and relocation techniques. The main idea of the algorithm is
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Figure 2: Three operators in the greedy partitioning heuristic
that, in each iteration, it calculates the potential cost savings by redistributing customers that each
drone can deliver to and selects the redistribution scheme with the largest cost savings to improve
the optimization solution. Specifically, the main functions of this algorithm include calcSavings,
relocateAsTruck, and relocateAsUAV. The calcSavings function calculates the potential cost savings
when a customer is removed from the truck’s route. The relocateAsTruck function attempts to reassign customers to the truck’s route, while the relocateAsUAV function tries to reassign customers
to the drone’s route.
3.1.3. Greedy Partitioning Heuristic
The greedy partitioning (GP) heuristic is a heuristic algorithm proposed by Agatz et al. (2018)
for solving the FSTSP problem. This algorithm follows a strategy of first generating TSP routes
and then classifying customer attributes (i.e., drone-visited or truck-visited). As is visualized in
Figure 2, the algorithm consists of three operators: MakeFly, PushLeft, and PushRight. Starting
from the initial TSP solution, at each step, this heuristic attempts to modify the current solution
by performing one of the three different operators. Among all feasible operators, the one that saves
the most delivery time is selected, and labels are assigned to nodes to indicate whether they are
served by the drone, the truck, or both during the task deliveries. Once all nodes are assigned
with the appropriate labels, it is straightforward to obtain the FSTSP solution, i.e., the drone tour
and the truck tour. The algorithm terminates after a maximum number of iterations, and its time
complexity is O(N logN ) with the application of appropriate data structure. This algorithm heavily
relies on the initial TSP solution, so it considers the use of multiple local search heuristics to modify
the initial TSP solution.
3.2. Solution Techniques for TSP-D
The mathematical model for the TSP-D problem is essentially a linear integer programming
problem, which can be solved using commercial optimization software such as CPLEX and Gurobi.
However, such mathematical programming solvers often have relatively low computational efficiency
and may not meet the requirements for solving large-scale problems. Therefore, it is necessary to
design efficient exact solution algorithms to improve solving efficiency. Compared to large-scale
heuristic algorithms for solving the FSTSP, exact solution algorithms can ensure that the returned
solution is globally optimal when the algorithm terminates and can be rigorously proven theoretically. In this section, two exact solution approaches adopted in this benchmark are introduced.
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3.2.1. Three-pass Dynamic Programming
The three-pass dynamic programming method (3PDP) is proposed by Bouman et al. (2018a)
for solving the TSP-D, inspired by the Bellman-Held-Karp dynamic programming algorithm for the
TSP (Bellman, 1962). In the first pass of the hybrid approach, the dynamic programming algorithm
for the regular TSP is adapted to find the shortest path for the truck for each start node, end node,
and set of truck nodes covered by the path. In the second pass, the truck paths generated in the
first pass of the approach are combined with drone nodes to generate operations, which represents
the least costly way to cover a set of nodes for given start node and end node. In the third pass,
the optimal sequence of operations is computed, which covers all locations and starts and ends at
the depot. To achieve this, the authors iteratively solve the subproblem of finding the best sequence
of operations starting at the depot, covering a specific set of nodes, and ending at a specific node.
3.2.2. A* Search with MST-based Heuristic
The algorithm outlined in Section 3.2.1 features a disadvantage that even if an instance has
clearly many subproblems that are clearly not relevant to the optimal solution of the TSP-D, they
are still solved by the DP and the results are tabulated into a table of exponential size. Based
on this potential drawback, Bouman et al. (2018a) propose an A*-based searching algorithm that
attempts to skip irrelevant subproblems and searches in an informed manner. The optimal drone
tour and truck tour are obtained by executing A* algorithm to find a shortest path from the source
state to the destination state on the state-graph. Regarding the heuristic function incorporated in
A* algorithm to estimate the distance from a given state to the sink state, the authors consider the
length function of a minimum spanning tree (MST) connecting the remaining unvisited nodes and
the depot, and theoretically prove that the MST-based heuristic never overestimates the distance
to the goal state, which ensures the approach to find an optimal solution.
3.3. Solution Techniques for PDSTSP
Similar to the FSTSP and the TSP-D, the MILP formulation of the PDSTSP can be directly
solved by commericial optimization solvers but the runtime is unaffordable in practical application.
Accordingly, existing literature mainly focus on developing heuristic algorithms to obtain a nearoptimal feasible solution in reasonable time. In this section, two heuristic methods adopted in
this benchmark are introduced, including the two-step iterative heuristic method and the random
restart local search.
3.3.1. Two-step Iterative Heuristic Method
Murray and Chu (2015) propose a two-step iterative heuristic (TSIH) to solve the PDSTSP. The
core of this algorithm is to divide the PDSTSP into two independent optimization problems, i.e.,
the Parallel Machine Scheduling (PMS) problem for the customers to be served by the drones, and
the TSP for the customers to be served by the truck. The premise of solving these two problems is
to select the partitioning of customers to be served either by the truck or by a drone.
The algorithm begins by assuming that the drones will serve all eligible customers and the
truck will serve the remaining customers. A PMS is solved to determine the assignment of drones
to customers and the corresponding makespan, while a TSP is solved to determine the truck tour
to visit the customers in the truck’s partition. The algorithm iteratively improves the objective
function value (i.e., the makespan) by reassigning individual customers to either the drone or truck
partitions, in an attempt to balance the working duration of the drones and the truck. Figure 3
visualizes the procedure to update the customer assignment of drones and truck.
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4 COMPUTATIONAL RESULTS
Figure 3: The procedure to update the customer assignment of drones and truck.
3.3.2. Random Restart Local Search (RRLS)
Dell’Amico et al. (2020) propose a random restart local search approach to solve the PDSTSP.
This specific local search approach is finely tuned to accommodate the unique attributes of the
PDSTSP. This algorithm combines the advantages of both model-based and random search methods, which allows it to handle complex optimization problems effectively. The algorithm first
optimizes the truck tour with TSP heuristics, and then delegate the MILP model to adjust the
truck tour by inserting specific drone nodes. The truck tour is iteratively re-optimized once it
has been adapted by the MILP solver. When the algorithm reaches a local minimum, it disrupts a
portion of the truck’s tour, restarting the entire process in an attempt to escape the local minimum.
4. Computational Results
In this section, we report the computational results of the three DTCO models. Section 4.1
provides a general description of the experimental dataset. Section 4.2 evaluates the optimality,
efficiency and scalablity of different algorithms for DTCO models. Section 4.3 then analyzes the
impact of different parameters on system performance. Finally, Section 4.4 conducts a case study
to demonstrate the superior performance of DTCO model as compared to classical TSP model.
4.1. Dataset Description
For the purpose of obtaining a benchmark comparsion, we conduct experiments based on the
TSP-D instances generated by Bouman et al. (2018b). The node locations are created on a twodimensional plane and it is assumed that the time taken by the truck to travel between any two
locations is proportional to their Euclidean distance. This dataset contains three categories of
instances. Among them, the Basic dataset consists of a total of 462 instances, with the number
of nodes (i.e., customer locations and the depot) ranging from 5 to 375. The Maxradius dataset,
which restricts the drone from flying beyond a certain distance in a single operation, comprises a
total of 440 instances, with the number of nodes ranging from 5 to 200. The Novisit dataset, which
imposes restrictions on the drone from visiting certain prohibited nodes, includes a total of 2400
instances, with the number of nodes ranging from 10 to 80.
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Regarding the spatial distribution of nodes, three types of instances are distinguished in each
dataset, briefly described as belows:
• uniform instances: The x and y coordinates for every node are drawn independently from
{0, 1, ..., 100}
• single-center instances: For each location of the node, the angle a is uniformly drawn from
[0, 2π] and the distance r is drawn from a normal distribution with mean 0 and standard
deviation 50. The x and y coordinates for each node is r cos a and r sin a, respectively. In
general, this type of instance mimics a circular city.
• double-center instances: They are generated in the same way as the single-center instances,
but every node is translated in the x direction by 200 units with a probability of 50%. In
general, this type of instance mimics a city with two centers.
In all cases, the first node generated is the depot node. By default, the drone is assumed to be
′ = 2, but experiments are also conducted
twice as fast as the truck on all edges, i.e., α = τi,j /τi,j
with the case where the truck and drone have equal speed (α = 1) and the case where the drone
is three times as fast as the truck (α = 3). All the experiments in this paper were performed
by a laptop with 16GB RAM and an Intel Core i7-12700 CPU. The code used in this study was
implemented in Python 3.7, and the source code is available online.1
4.2. Benchmark Comparison of Models and Algorithms
To evaluate the performance of different algorithms for the FSTSP, we first conduct benchmark
comparison on the Basic dataset. The instances used in this base comparison are small-scale singlecentered instances with α = 2. More specifically, we select single-centered instances of less than ten
nodes since the Gurobi solver cannot solve larger scale instance in reasonable amount of runtime,
and the speed ratio of the drone and truck is equal to two. Regarding the computational time
of solving these instances, all heuristic methods can generate the results within 0.1 seconds, while
the Gurobi solver takes runtime between 398 to 3601 seconds, which demonstrates the efficiency of
heuristic methods in solving the three typical DTCO models.
To compare the solution algorithms for three types of DTCO model, we unify the settings of
the three models in the following manner. The FSTSP and the TSP-D both consider a single
drone operating in tandem with a single truck. Despite slightly different assumptions regarding
the modelling side, the performance of algorithms for the FSTSP can be directly compared with
that of the TSP-D. However, the PDSTSP is distinct from the other two models since it considers
the drones operating independently with the truck. To unify the problem setting, we set the
number of drones in PDSTSP to one, thus all models consider the operation of a single drone and
a single truck. Then, the comparison of results for the FSTSP and the PDSTSP can present the
difference between two operating mode, i.e., operating in a synchronized manner and operating in
an independent manner.
Figure 4 visualizes the objective function values of various algorithms in solving the DTCO
models. Different colors of boxes are presented to distinguish the algorithms for different type of
DTCO model: green boxes for the TSP-D results, blue boxes for the FSTSP results, red boxes for the
PDSTSP results and purple box for the standard TSP result. We first compare the Gurobi optimal
1
https://github.com/LouisLINWEI/DTCO
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Figure 4: Comparison of the objective function values of various algorithms in solving the DTCO models
solutions for all four models, which represents how different operating mode affects the system
performance (i.e., the total delivery time). The optimal solution of the TSP-D model is 261.9 on
average, being the lowest amount of delivery time among all four models. The optimal solution of
the FSTSP model is 264.9 on average, being slightly higher than that of the TSP-D. In contrast, the
PDSTSP yields an optimal solution of 282.9 on average, which is of 8.0% higher than that of the
FSTSP. This difference of result demonstrates the superiority of operating the truck and drone in a
synchronized manner over an independent manner, since the synchronization can effectively reduce
the waiting time for either type of vehicle. The optimal solution of the TSP model is 431.0 on
average, which is of 64.6% higher than that of the FSTSP. This significant difference demonstrates
the superiority of deploying truck-drone coordinated system over the truck-only delivery system.
Regarding the results of algorithms in solving the TSP-D model, both A*-MST and 3PDP
algorithms achieve the same optimal solution as the Gurobi optimal solution, since both algorithms
search for the optimal sequence of operations in an exact manner (i.e., the algorithms will not
terminate until the optimal solution is found). Regarding the results of algorithms in solving the
FSTSP model, none of the three heuristic methods (i.e., DCRA, GRASP and GP) can achieve
the optimal solution as Gurobi solver. The DCRA generally performs better than the other two
heuristics, primarily due to its incorporation of cost saving mechanism and customer redistribution
scheme. Regarding the results of algorithms in solving the PDSTSP model, both of the heuristics
TSIH and RRLS achieve close objective value to the Gurobi optimal solution, which demonstrates
the reliability of both heuristics in solving the PDSTSP.
In the real-world application, the number of customers to serve is much larger than 10, leading
to combinatorial challenge for commercial optimization software and exact solution approach. In
our experiment, the Gurobi solver and the exact algorithms for TSP-D cannot obtain the optimal
solution for instances with 20 nodes within three hours. Accordingly, heuristic algorithms are
necessary to obtain solutions of high quality in an efficient manner. In the following, we conduct
scalability analysis regarding five heuristics applied to solve the FSTSP and the PDSTSP. Figure 5
visualizes the comparison of the five heuristics on medium/large-scale instances regarding solution
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Figure 5: Comparison of the five heuristics on medium/large-scale instances regarding solution quality and solving
efficiency.
quality and solving efficiency. The number of nodes for the instances ranges from 10 to 175. In
Figure 5(a), the ordinate saving over TSP is defined by the relative difference between the objective
value of the heuristic and the optimal objective value of standard TSP. As the size of instances
increases, the saving statistics decreases. A possible explanation is that the solution space increases
exponentially with the number of nodes, while the search space of the heuristics only increases
linearly. The objective value of the heuristics for the FSTSP is generally similar to that of the
PDSTSP. The GRASP and RRLS perform better than the other heuristics regarding the FSTSP
and the PDSTSP, respectively. In Figure 5(b), the runtime of heuristics grows with respect to the
size of instances. The GP and TSIH have significantly lower runtime than the other heuristics on
large-scale instances.
4.3. Impact of Different Parameters on System Performance
In this section, we analyze how various parameters affect the performance of the drone-truck
combined delivery system. The results of this study can provide valuable insights for companies experimenting with this innovative mode of transportation. To evaluate the performance, we compare
the total delivery/completion time of the FSTSP model with that of the truck-only TSP model. In
all experiments, the Dynamic Customer Relocation Algorithm is used, since it performs the best in
the previous experiments.
4.3.1. Distribution of Customer Locations
Figure 6 visualizes box plots of savings of the solutions of the FSTSP over the truck-only TSP.
Two types of instances are used in this study, i.e., the single-center instances and the uniform
instances, with number of nodes ranging from 10 to 250. It can be observed that the combination
of a drone and a truck leads to substantial time savings in single-center instances, on average,
between 20% to 30%. The time savings in uniform instances range from 13% to 23%. The results
suggest that the FSTSP is especially beneficial if the demand locations are clustered around a
center, which is intuitive since the time needed to serve customers far from the center is long.
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Figure 6: Comparison of savings of solutions for instances with different customer distribution
4.3.2. Drone Speed
The speed of the drone may be a major factor affecting the performance of the system. Drones
are thought to travel much faster than trucks, since they can conduct point-to-point travel rather
than adhering to the road network. Table 4 presents the characteristics of FSTSP solutions for
different drone speeds. We set up three different scenarios in this experiment, in which the speed
ratio of the drone and truck α is equal to one, two and three, respectively. The uniform instances
are used in this experiment, and we average the objective values over 10 instances for each group
of instances with c number of nodes. In addition to the savings over TSP statistics, we also record
the ratio of truck nodes, drone nodes and combined nodes (i.e., nodes that are visited by both
vehicles), as well as the utilization rate of both vehicles. The utilization rate is measured by the
actual travelling time of the vehicle divided by the total makespan.
The savings over TSP increase as the speed of the drone increases, since the drone is able to
complete the delivery task in less time for the same distance. In line with this observation, the
number of drone nodes also increases, as faster drones could visit more customer nodes while the
truck is delivering goods. Besides, we observe that the utilization rate of the drone generally reduces
as it becomes faster, since it has to wait for the truck to rendezvous more.
4.3.3. Drone Range
In this section, we investigate the effect of drone range on the system performance. We conduct
a set of experiments for different instance sizes, i.e., c = 20, 50, 75, 100, and under different drone
ranges. Figure 7 visualizes the savings of the FSTSP system over the truck-only TSP, and the
number of customers served by the drone for different ranges. The x-coordinates represent increasing
the range of drone by various percentage based on the default value of range.
As expected, we observe that the savings of the system increase as the flight range of the
drone increases. Figure 7(a) also shows that the savings of the medium/large-scale instances (c =
50, 75, 100) are more sensitive to the increase of drone range than small-scale instances (i.e., c = 20).
A possible explanation for this result is that the average distance between customer locations are
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Table 4: Characteristics of the FSTSP solutions for different drone speeds (α)
Savings (%)
α=1
α=2
α=3
c = 10
c = 50
c = 100
c = 10
c = 50
c = 100
c = 10
c = 50
c = 100
19.4
19.6
18.3
31.0
29.4
22.1
36.6
37.2
29.3
Number of nodes (%)
Truck
24.0
53.3
63.5
16.0
42.8
50.1
4.0
29.0
40.9
Drone
6.6
7.4
6.4
16.2
15.2
14.2
23.0
19.4
18.7
Combined
69.4
39.3
30.1
67.8
42.0
35.7
73.0
51.6
40.4
Utilization (%)
Truck
100.0
97.2
96.7
93.3
88.0
82.5
93.7
75.5
73.5
Drone
33.8
69.0
73.0
37.1
60.1
62.2
31.3
45.4
52.3
Figure 7: Characteristics of the FSTSP solutions for different flight ranges
smaller in large-scale instances, so the drone can cover more customers under a specific range.
Accordingly, the increase in drone range can cover much more customers in large-scale instances.
Figure 7(b) shows the change in the number of drone nodes with respect to the change of drone
range. As the drone travels larger amount of distance, it is assigned to more delivery tasks. For
the instances of c = 20, 50, 100, the number of drone nodes increases monotonically with respect
to the increase of flight range, which is intuitive since the drone with larger range can serve more
customers. However, for the instances of c = 75, we observe a decline of the percentage of drone
nodes. This observation may be associated with the fact that the longer-range drone may be
scheduled to fewer but longer drone flights.
4.4. Case Study
In this section, we conduct a real-world case study, in which we schedule the parcel delivery
operations among universities in Haidian District, Beijing. As is shown in Figure 8, we consider a
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Figure 8: A case study to demonstrate the TSP-D operations among nine universities in Beijing
region of size approximately 8 km×8 km on Amap Web API (Amap, 2023). The delivery network
behind the case study is composed of a set of locations and a set of arcs including both road arcs
and flying arcs. The nodes include a JD Haidian depot (indexed by zero) and nine universities
(Beihang University, Tsinghua University and etc) in Haidian District, Beijing. The longitude and
latitude of each location are obtained by reverse geocoding module in Amap Web API, and the road
paths between every two locations are obtained by path planning module in Amap Web API. As
for the drone, the distance between every two locations is obtained based on the haversine distance,
since the movement of the drone does not adhere to the road network. The distance matrix for
truck and that for drone is accordingly different.
The model and related specifications of the drone and the truck is shown in Figure 8. The “H4
quadcopter”, operated by SF Express for delivering medical supplies, has a cruise speed of 43.2
km/h and a range of 15 km, with a takeoff weight of 25 kg and maximum payload of 10 kg. The
“ISUZU N series” truck operated by SF Express has a cruise speed of 40 km/h and a maximum
range of 800 km. The maximum payload of the truck is 20 t and we assume that it can carry a
drone.
Given the road-network distance matrix for truck and the haversine distance matrix for drone,
we then solve the underling TSP and TSP-D problem respectively. We utilize the dynamic programming approach to obtain the optimal solution of the truck-only TSP, and the three-pass dynamic
programming approach in Section 3.2.1 to obtain the optimal solution of the TSP-D. The routing
decisions of TSP and TSP-D are visualized in Figure 9, with the corresponding objective value and
optimal solution of paths.
As for the routing of the truck, detours are common which lead to relatively long travelling
distance and accordingly, the travelling time. The truck-only TSP mode takes 2743 s (i.e., 46 min)
to serve all the universities and return to the depot. When effectively combining the truck with
the drone, the total travelling distance of the truck decreases from 30480 m to 21076 m, which is
an over 30% decrease. Four nodes which are scheduled to be served by the truck in TSP is now
assigned to the drone in TSP-D. The final objective value of TSP-D is 1905 s (i.e., 32 min). This
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5 CONCLUSION
Figure 9: Routing decisions of truck-only TSP (left) and TSP-D (right) in the case study
leads to a large saving of delivery time with respect to the truck-only TSP mode,
Saving = (2743.20 − 1904.79)/2743.20 × 100% = 30.56%
This amount of delivery time saving is beneficial for logistics companies to enhance the efficiency of
delivery service. The result also demonstrates the effectiveness of the parallelization of two types
of vehicles, which can make up for the weakness of each other, i.e., the range concern of the drone
and the disadvantage of detour for the truck.
We conduct a sensitivity analysis regarding the variation of drone speed on the objective value
of TSP-D, visualized in Figure 10. The baseline scenario (left) considers a drone speed of 43.2
km/h, and the second scenario (middle) considers an increase of 20% while the third scenario
(right) considers an increase of 50%. As is shown in Figure 10, the routing structures of the three
scenarios are significantly different. The truck experiences smaller driving distance while the drone
travels more as the speed of the drone increases. The drone even visits the same location repeatedly
to ensure a valid connection to the truck. The objective values of the three scenarios are 32, 31
and 29 minutes, respectively. This indicates that the increase of drone speed leads to only a small
magnitude of decrease of objective value.
5. Conclusion
In this paper, we conduct a systematic evaluation and experimental benchmark for the algorithms of three typical DTCO models: the flying sidekick traveling salesman problem (FSTSP), the
traveling salesman problem with drone (TSP-D), and the parallel drone scheduling traveling salesman problem (PDSTSP). We evaluate the algorithms by various metrics including result quality,
runtime and scalability, and we observe the superiority of heuristics GRASP and RRLS in solving
the FSTSP and PDSTSP. Through the sensitivity analysis regarding the impact of different parameters on system performance, we observe that a higher drone speed, larger drone range or a more
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5 CONCLUSION
Figure 10: A sensitivity analysis regarding the variation of drone speed on the objective value of TSP-D
clustered customer distribution can improve the delivery efficiency of the DTCO system. Finally,
we conduct a case study to demonstrate the superior performance of DTCO model as compared
to classical TSP model based on the city road network data of Beijing. The outcome of our research provides insights for developing new methodologies for large-scale real-world drone logistics
problems. Future study may incorporate more realistic objective functions and constraints into the
models as well as the algorithms, for instance, including the fixed setup cost and operating cost
into the objective function and introducing drone’s energy consumption model into the constraints.
Furthermore, large-scale real-world transportation data can be used for the purpose of verification.
September 19, 2023
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