Synchronized Drone and Truck Routing Problem: A Multi-Stakeholder

Perspective

Maria Elena Bruni

1,3 a

, Sara Khodaparasti

2 b

, Giuseppe Muratore

and Vincenzo Gentile

Department of Mechanical, Energy and Management Engineering, University of Calabria, Rende, Italy

Department of Management and Production Engineering, Polytechnic University of Turin, Turin, Italy

CIRRELT, Montreal, Canada

Keywords:

Last-Mile Delivery, Synchronized Truck and Drone Delivery, Vehicle Routing Problem, Multi-Objective

Optimization, Meta-Heuristic.

Abstract:

This paper introduces a synchronized truck and drone routing problem consisting a ﬂeet of capacitated trucks

and drones dispatching last-mile deliveries. Each truck, while traveling on its route to serve customers, could

stop and launch drones which perform multiple back and forth trips between the truck and one or more de-

livery destination, serving one customer at a time. The synchronization between trucks and drones moves

is mandatory since the drone is launched and retrieved by the truck, that should wait for delivery comple-

tion and drone retrieval. Following a multi-stakeholder perspective, different customer- and business-oriented

objectives are included to account for the presence of different (public an private) actors and end-users with

conﬂicting interests and preferences. We formulate the problem as a mixed-integer program to determine the

optimal truck routes and sequence of drone trips under the synchronization assumption. The proposed model

is then solved by a meta-heuristic solution approach.

1 INTRODUCTION

The last few years have seen a sharp increase of the

paradigms of on-demand economy and e-commerce.

New emerging business models are no longer driven

by suppliers, but are more and more inﬂuenced by

customers’ preferences and expectations in terms of

time and cost, with a consequent disruptive impact

on the delivery process (Perboli and Rosano, 2019).

To survive in such a competitive market, e-commerce

companies should face the upward trend for fast and

cheap deliveries, optimizing the last-mile process. It

is well-recognized that routing inefﬁciencies, in the

last leg of of the distribution, may contribute to de-

layed deliveries, loss of revenues and proﬁts and

above all, unsatisﬁed customers. To catch up with

these trends, in the last decade, new delivery sys-

tems, exploiting the complementary nature of trucks

and drones, have been investigated with the aim of

assessing the value of using drones within logistics

chain. Drones are considered as a potential solu-

tion to the last-mile challenges, because of their high

https://orcid.org/0000-0002-3152-5294

https://orcid.org/0000-0003-3858-2571

travel speed and ability to access areas regardless of

road infrastructure, to deliver small packages in a fast

and sustainable way, signiﬁcantly accelerating deliv-

ery times and reducing human intervention. Even

though the speed of drones can offer signiﬁcant time

savings with respect to the traditional vehicles trav-

eling congested routes, drone-aided delivery systems

are challenged by intrinsic characteristics of aerial ve-

hicles, such as limited payload capacities, battery en-

durance and sensitivity to weather conditions. More-

over, the debate on the regulatory and safety issues

of air control administrations and long-term viabil-

ity, is far to be resolved. Despite all these issues,

the idea of drone-aided delivery is gaining ground

from both practitioners, persuaded to apply sustain-

able, cost- and time-efﬁcient transport alternatives,

and researchers, promoting multi-modal and collab-

orative delivery systems combining terrestrial (such

as trucks) and aerial vehicles (Unmanned Aerial Ve-

hicles (UAVs) or drones).

The design of an efﬁcient multi-modal joint deliv-

ery system is not straightforward for the main chal-

lenges of coordinating and synchronizing trucks and

drones, especially when a set of truck routes needs

to be synchronized with the drones movements. In

328

Bruni, M. E., Khodaparasti, S., Muratore, G. and Gentile, V.

Synchronized Drone and Truck Routing Problem: A Multi-Stakeholder Perspective.

DOI: 10.5220/0013217600003893

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 14th International Conference on Operations Research and Enterprise Systems (ICORES 2025), pages 328-339

ISBN: 978-989-758-732-0; ISSN: 2184-4372

a synchronized multi-truck and multi-drone model, a

mixed ﬂeet of homogeneous capacitated trucks, that

deliver heavy packages, and multiple sidekick drones

that carry out lightweight deliveries are employed.

Each truck dispatched from the depot travels a route

starting from a central depot, visits a subset of cus-

tomers and returns back to the depot after the last de-

livery completion. The truck also serves as a mobile

depot and launch and retrieve platform for the side-

kick drones and moves them to the proximity of the

drone delivery area, where the truck stops, launches

the drone to deliver a single customer order and re-

trieves it back once the delivery mission is completed.

The drone is allowed to perform multiple one-to-one

delivery trips between the stationary truck and cus-

tomer locations as long as its battery is not depleted.

Due to safety regulations and to better monitor the

drone moves, the truck driver awaits at the location

until all scheduled multi-trip deliveries are performed

and the drone is retrieved. This fundamental assump-

tion makes the relative movements non-simultaneous,

however, the synchronization is still relevant since the

truck departure can be scheduled only after the drone

retrieval. Obviously, this joint scheme requires more

complex evaluation and planning models and proce-

dures, compared to those existing in the literature as

not only the operations of each transport mode should

be optimized, but also the interactions between dif-

ferent vehicle types, such as the synchronization and

coordination of the traditional vehicle with the drones

are involved. That is the focus of the present study.

We extend the synchronized drone and truck delivery

with non-simultaneous relative movements between

one truck and drones to the case with multiple trucks.

On the other hand, the complexity of drone-

aided delivery applications goes far beyond the syn-

chronization problem since the social and environ-

mental concerns of the public stakeholder (in terms

of pollution, trafﬁc congestion, and other externali-

ties), the economic interests of the delivery system

owner, and the expectations of customers call for

a multi-stakeholder optimization approach. To ful-

ﬁll this goal, we adopt a multi-objective perspective

and model the synchronized drone and truck rout-

ing problem considering the following four objective

functions of i) total proﬁt, (assuming that each de-

livery brings a proﬁt to the stakeholder that should

decide which customers to serve), ii) total traveling

cost for trucks (it may also represent environmen-

tal costs), iii) sum of arrival times to each customer

visited by a truck or a drone (also called latency)

reﬂecting the customers’ satisfaction and ﬁnally iv)

number of trucks dispatched from the depot (cost-

efﬁciency of the transport system). Since the ﬂeet

cost –proportional to the ﬂeet size– is a consider-

able contribution to the operating costs, the decision

maker could optimize the number of trucks to be em-

ployed. Except the latency goal, that is a customer-

oriented objective, the other three goals are business-

oriented. In general, not all the delivery requests are

proﬁtable, especially those prolonging the arrival time

of other requests, and therefore, some requests might

be skipped. This imposes a selective structure to the

problem exacerbating its complexity.

The complication of drone-aided deliveries is also

linked to intrinsic drone-related features –such as lim-

ited drone payload, battery capacity, and variable

energy consumption in drone battery due to wind

and weather condition– restricting the drone delivery

range and number of allowed back and forth trips be-

tween the truck and customer locations. The litera-

ture is abundant in contributions that treat the drone-

related issues as those in the traditional terrestrial ve-

hicle routing problems, either by setting a maximum

drone endurance in terms of travel distance/time or

approximating the energy consumption as a linear

function of drone payload and travel time that later

brings the validity and applicability of the obtained

solutions into question. To ﬁll this gap, we model

the energy consumption in drone battery as a non-

linear function of drone payload and travel time and

adopt a distributionally robust optimization approach

to capture the uncertainty of energy consumption due

to variations in drone speed and travel time.

The contributions of this paper are manifold. We

address a novel multi-trip synchronized truck and

drone routing problem, under a multi-criteria set-

ting addressing both business- and customer-oriented

goals where the realistic drone-related features in

terms of load-dependent energy consumption rates

and the ﬂuctuations in weather condition causing

speed and travel time variations are taken into ac-

count. To solve the problem, we apply an evolution-

ary meta-heuristic algorithm that efﬁciently handles

instances of reasonable size, as shown in the compu-

tational experiments.

The remainder of this paper is organized as fol-

lows. Section 2 presents a detailed review on the

relevant literature. Section 3 describes the prob-

lem setting and the mathematical formulation. Sec-

tion 4 proposes the Non-Dominated Sorting Ge-

netic Algorithm-II (NSGA-II), as a well-known meta-

heuristic for multi-objective problems. Section 5

presents computational results evaluating the validity

of the proposed model and the efﬁciency of the solu-

tion approach. Finally, Section 6 concludes the paper

and summarizes some directions for future research.

Synchronized Drone and Truck Routing Problem: A Multi-Stakeholder Perspective

329

2 LITERATURE REVIEW

In the last decade, a large body of the operations re-

search literature has focused on routing problems for

drone delivery. Several new challenges, derived from

the multi-modal nature of delivery system combin-

ing the terrestrial and aerial vehicle types with ex-

tremely different characteristics, has stimulated a ris-

ing interest in the optimization community unveiling

different drone integration schemes in the last-mile

delivery context. The ﬂourishing literature on drone

delivery involves different problem variants catego-

rized based on problem features, such as number of

trucks and drones, the number of trips allowed for the

drones and/or trucks, the interrelation between trucks

and drones as independent or sidekick working units

that brings the parallel (unsynchronized) or synchro-

nized movements into question, different launch and

retrieve policies (either at the depot or while the truck

is en-route) simultaneous and non-simultaneous rela-

tive movements.

When the routing problem combines one truck

and one or more drones, it can be view as a variant

of the traditional Traveling Salesman Problem (TSP).

Depending on the synchronization level between the

vehicles, it is referred to as the Flying Sidekick Trav-

eling Salesman Problem (FSTSP) or Parallel Drone

Scheduling TSP (PDSTSP). The fundamental differ-

ence between these two problems is the synchro-

nization issue. While in the FSTSP problem trucks

and drones operations must be coordinated, in the

PDSTSP the synchronisation issue is not relevant as

truck and drones work independently. In the FSTSP,

the truck carries the drone and it can decide to launch

the drone to deliver the orders of a few customers in

the nearby while the truck could continue its route to

serve other customers and ﬁnally meet the drone at a

rendezvous location, with the objective of minimizing

the makespan, that is the time required by the truck to

return to the depot. In the PDSTSP, the drones are di-

rectly deployed from the depot, serving one customer

at a time but possibly performing multiple trips.

In this Section, we focus on the literature on

PDSTSP since our contribution essentially belongs to

the class of PDSTSP in which the synchronization is-

sues are also incorporated. Following the research

stream on PDSTSP, Murray and Chu (2015) pro-

posed an MILP formulation and a heuristic method

for PDSTSP, tested on small instances with 10 and 20

customers. The heuristic builds a TSP tour for cus-

tomers visited by the truck and after assigns the re-

maining customers to drones. Mbiadou Saleu et al.

(2018), proposed the ﬁrst iterative two-step heuristic

for the problem. First, the customer sequence is de-

termined and then, it is decomposed into a tour for

the truck and a set of trips for the drones. DellAm-

ico (2020) presented a simpliﬁed mixed integer lin-

ear programming model for the PDSTSP and a set

of matheuristic algorithms tested on instances with

48–229 customers. Several PDSTSP variants have

been also introduced and studied in the literature.

Kim and Moon (2019) considered a variant in which

drones can be deployed from the depot but also from

dedicated drone stations to address the limitations of

the drone ﬂight range. Ham (2018) studied a multi-

truck multi-depot pickup and delivery variant of the

PDSTSP where a drone, after the delivery task, can

either ﬂy back to the depot or reach another pickup

customer. Several publications in the literature have

investigated the objective function of minimizing the

total transportation cost in routing problems using

the truck-drone combination for delivery. Moshref-

Javadi et al. (2020a) extended the PDSTSP, imple-

menting a different operating condition, to better re-

ﬂect the realities associated with the complex nature

of last-mile distribution logistic problems. Speciﬁ-

cally, this variant of the problem considers that drones

can either take off from the depot or to a node reached

by the by the truck (this node could be a customer or

a parking area or a locker point). The drone delivers a

set of packages, servicing one customer at a time per-

forming back and forth trips, and returns to the truck,

waiting for the return of all the drones launched. An

efﬁcient ALNS meta-heuristic was designed to solve

real-world-size instances and tested on several small

instances in the benchmark that found near-optimal

solutions within low computational time. The model

and solution approach were also tested on a real-world

case study of e-commerce deliveries in Sao Paulo,

Brazil. They compare their model with the classi-

cal traveling repairman problem, showing the effec-

tiveness of using a multi-modal system delivery com-

pared to the traditional delivery system with terres-

trial vehicle. Moshref-Javadi et al. (2020b) propose a

hybrid tabu search-simulated annealing algorithm for

solving their variant on real-world-size problem in-

stances.

The present paper extends the PDTSP problem

along several dimensions. First of all, more than

one truck is considered. Secondly, this variant of the

problem considers synchronized shipping operations,

where the drones can be launched at any truck stop.

Different constraints for the trucks and the drones are

also considered. First, the total weight of parcels

loaded into the truck should not exceed the truck ca-

pacity and drones have limited payload. Second, we

impose a restriction on the energy consumed by each

drone during the whole service. We should notice

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

330

that, since drones are faster than trucks, any solu-

tion which ignores the battery endurance, will exten-

sively employ more drones against trucks. We remove

the restrictive (and often unrealistic) assumption that

the drone battery is fully charged/swapped while the

truck is moving, rather considering that the drone bat-

teries will be charged at the end of the service, when

the truck comes back into the depot.

Previous studies do not consider neither truck

capacity constraints nor intrinsic characteristic of

drones.

3 SYNCHRONIZED MULTI-TRIP

DRONE AND TRUCK ROUTING

PROBLEM

3.1 Problem Description

In the Synchronized Multi-Trip Drone and Truck

Routing Problem (SMTDTRP), a ﬂeet of capacitated

trucks, each hosting at least one drone, departs from a

single depot to service a subset of potential customers.

Some customer orders are required to be delivered

only by truck (cases where the order volume and mass

is beyond the drone payload capacity, the travel time

to reach the delivery location by a drone violates the

drone maximum endurance, or the local meteorolog-

ical at the delivery site prevents the use of drone)

while other customers, ordering light packages, can

be served by drones. A proﬁt is associated with each

customer and the vehicles can skip serving some cus-

tomers, depending on the proﬁt rate, thus framing the

problem as a selective one. Clearly, adding the deci-

sion on the selection of customers to be served exac-

erbates the problem complexity. The trucks are able

to dispatch drones from the vehicle roof and, start-

ing from the depot, visit every customer exactly once.

Drones can be dispatched from the truck delivering

light packages to nearby customers while the truck is

servicing a truck-only customer and then waits for the

drone to retrieve it. The drone is allowed to perform

multiple back and forth trips (but can serve only one

customer per operation), and then returns to the truck

from which the drone started its mission. The truck

should wait until the dispatched drones have com-

pleted the delivery operations. The drones, with lim-

ited payload and battery capacity, are fully charged

at the beginning of the service and the battery cannot

be recharged nor swapped until the truck returns to

the depot. The non-simultaneous relative movements

between truck and drone holds in many applications

especially those in which the drone track should be

Figure 1: A nominal routing plan.

frequently monitored due to safety regulations. Fig-

ure 1 represents a possible routing plan for an exam-

ple with two trucks. One truck serves customers 4,

15, 9 and 10 and hosts one drone (Drone 1) perform-

ing deliveries to customers 18, 13, 11, and 12. The

other truck (Truck 2) visits customers 5, 6, 7, and 8

and hosts two drones that deliver the orders of cus-

tomers 1, 2, and 19 (by Drone 2) and customers 3, 17,

and 20 (by Drone 3).

The energy consumption in drone battery depends

(linearly) on travel time, and (non-linearly) on drone

frame mass (W ), battery mass (m), and payload (p).

Assuming that the drone battery capacity is denoted

by C , the total drone energy consumption within a trip

r, with travel time d

, to deliver a parcel of mass p

can be evaluated as follows.

= k

′

(W + m + p

)

3/2

+ k

′

(W + m)

3/2

(1)

where k

′

is a constant depending on the air density, the

area of the spinning blade disc and the gravity. The

ﬁrst term in (1) represents the energy consumption

during the trip towards the customer’s location while

the drone is carrying the order and the second term

denotes the energy consumption during the drone re-

turn trip to the truck when the order is delivered and

thus drone payload is zero.

We assume that drones perform one-to-one de-

livery trips r ∈ R until the battery is depleted, i.e.,

∑

r∈R

≤ C. Uncertain weather conditions, i.e., wind

speed and direction, could drastically affect the en-

ergy consumption. To guarantee that under different

weather scenarios, the drones operate in a safe and ef-

ﬁcient fashion, we adopt a distributionally robust opti-

mization framework in which we suppose the battery

consumption is uncertain and although its probabil-

ity distribution is not completely speciﬁed, we have

Synchronized Drone and Truck Routing Problem: A Multi-Stakeholder Perspective

331

some partial information on its distribution function

knowing that it belongs to a class of distributions

with known mean and covariance. This assumption

of course broadens the applicability of the approach,

that can be applied to different real-life scenarios. It

is worthwhile mentioning that the selection of the ap-

propriate distribution functions for energy consump-

tion is not straightforward (for more information on

energy consumption functions in drone battery see

Zhang et al. (2021)).

Knowing this, we may apply the probabilistic

chance constraint (2) to account for both the uncer-

tainty of energy consumption and limited battery ca-

pacity.

∑

r∈R

≤ C) ≥ 1 − α (2)

where α ∈ [0,1] is the risk level and thus 1−α denotes

the decision maker’s risk-aversion tendency. Under

this assumption, the distributionally robust counter-

part for constraint (2) can be written as

µ +

1 − α

σ ≤ C (3)

where µ =

∑

r∈R

µ(E

) represents the expected value

corresponding to the total energy consumption and

σ =

∑

r∈R

σ(E

) is the total variance (Lam, 2019;

Calaﬁore and Ghaoui, 2006).

The problem consists in deciding the subset and

the sequence of potential customers that should be

serviced by trucks and drones, as well as possible

assignments of customers to drones performing back

and forth trips. In particular, four objectives are of

our interest: maximizing the total proﬁt assigned to

delivered orders, minimizing the total latency (arrival

time to customers), minimizing the total travel cost,

and minimizing the number of deployed trucks.

3.2 Mathematical Model

The formulation makes use of the notation in Table 1

and the mathematical model is cast as (4)-(27).

max

∑

i∈V

∑

j∈V

′

∑

k∈K

i j

∑

i∈V

∑

j∈V

′

∑

k∈K

∑

u∈U

∑

r∈R

kur

i j

(4)

min

∑

i∈V

′

∑

k∈K

) (5)

min

∑

i∈V

∑

j∈V

′

∑

k∈K

i j

(6)

min

∑

j∈V

′

∑

k∈K

0 j

(7)

∑

j∈V

i j

∑

j∈V

i ∈ V, k ∈ K (8)

∑

j∈V

′

i j

≤ 1 i ∈ V, k ∈ K (9)

∑

j∈V

′

∑

u∈U

∑

r∈R

kur

i j

≤ M y

i ∈ V, k ∈ K (10)

∑

i∈V

i j

≥ y

j ∈ V

′

, k ∈ K (11)

+ z

+ d

i j

− M (1 − x

i j

) ≤ t

i ∈ V, j ∈ V

′

i ̸= j, k ∈ K (12)

∑

j∈V

′

∑

r∈R

(

i j

) f

kur

i j

≤ z

i ∈ V, k ∈ K, u ∈ U

(13)

∑

j∈V

′

kur

i j

≥

∑

j∈V

′

ku(r+1)

i j

i ∈ V, k ∈ K, u ∈ U

r ∈ R (14)

∑

i∈V

∑

u∈U

∑

r∈R

∑

k∈K

kur

i j

∑

i∈V

∑

k∈K

i j

≤ 1 j ∈ V

′

(15)

∑

j∈V

′

kur

i j

≤ 1 i ∈ V, k ∈ K, u ∈ U, r ∈ R (16)

i j

− M (1 − f

kur

i j

) +

∑

j∈V

′

∑

′

∈R

′

(

i j

′

) f

kur

′

i j

′

≤

i ∈ V, j ∈ V

′

, i ̸= j, u ∈ U, r ∈ R, k ∈ K (17)

∑

i∈V

∑

j∈V

′

∑

k∈K

∑

r∈R

i j

+ e

) f

kur

i j

≤ C u ∈ U (18)

∑

i∈V

∑

j∈V

′

i j

∑

i∈V

∑

j∈V

′

∑

u∈U

∑

r∈R

kur

i j

≤ Q

k ∈ K (19)

∑

k∈K

kur

i j

≤ 1 i ∈ V, j ∈ V

′

, r ∈ R, u ∈ U (20)

kur

i j

≥ M

∑

′

∈K

′

̸=k

′

, i, i

′

∈ V, j, j

′

∈ V

′

r, r

′

∈ R, u ∈ U, k ∈ K (21)

kur

i j

= 0 i ∈ V, j ∈ V

, k ∈ K, u ∈ U, r ∈ R (22)

i j

= 0 i ∈ V, j ∈ V

, k ∈ K (23)

i j

∈ {0, 1} i ∈ V, j ∈ V

′

, k ∈ K (24)

kur

i j

∈ {0, 1} i ∈ V, j ∈ V

′

, k ∈ K, u ∈ U,

r ∈ R (25)

∈ {0, 1} i ∈ V, k ∈ K (26)

, t

≥ 0 i ∈ V, k ∈ K (27)

Objective (4) maximizes the total proﬁt of visited cus-

tomers and encompasses two terms corresponding to

the proﬁt of customers visited by trucks and drones;

objective (5) minimizes the total latency, deﬁned as

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

332

Table 1: Notation for the mathematical model.

Sets

′

Set of customers

V = V

′

∪ {0} Set of nodes including the depot (0) indexed by i, j, j

′

⊂ V

′

Set of customers to be visited by trucks

⊂ V

′

Set of customers to be visited by drones

U Set of drones indexed by u

K Set of trucks indexed by k

R Set of drone trips indexed by r, r

′

Parameters

Demand of customer i

Proﬁt associated to customer i

Capacity of truck k

i j

Time-dependent travel cost (such as environmental cost)

i j

Travel time for truck between node i and j

i j

Travel time for drone between node i and j

i j

= k

′

(W + m + q

)

3/2

i j

Energy consumption of a drone launched from node i to deliver the order of customer j

= k

′

(W + m)

3/2

Energy consumption of a drone on its return trip after delivering order of customer j and coming back to the launch node i

C Drone battery capacity

M A sufﬁciently large number

Decision variables

i j

Binary variable that takes 1 if truck k travels from node i to reach node j

kur

i j

Binary variable that takes 1 if truck k stops at node i and launches drone u to visit customer j in its rth trip from node i; otherwise, 0

Binary variable takes 1 if truck k stops at node i to launch at least one drone; otherwise, 0.

Waiting time of truck k at node i

Waiting time of customer i served by truck k

Waiting time of customer i served by a drone launched by truck k

the sum of customers’ arrival times visited by either

trucks or drones; Objective (6) minimizes the total

truck travel cost, and objective (7) minimizes the total

number of dispatched trucks. Constraints (8) ensure

the continuity of ﬂow for each truck; Constraints (9)

ensure that the each truck can exit each customer at

most once. Constraints (10) relates the variables y

and f

kur

i j

and ensures that if a customer is supposed

to be visited by a drone, then it is mandatory that the

truck stops at a customer to launch the drone. Con-

straints (11) ensure that a truck can only stop at a

customer location to launch a drone, only if the truck

serves that customer. Constraints (12) deﬁne the ar-

rival time of a customer visited by a truck. Constraints

(13) deﬁne the waiting time of each truck as the max-

imum delivery time of all drones launched from that

truck. Constraints (14) sets the logical relation be-

tween the order of drone trips; so a drone can oper-

ate its r + 1 tour only if it has already performed r

trips. Constraints (15) show that each customer can

be served at most once either by a truck or a drone.

Constraints (16) indicate that the drones can serve at

most one customer within each trip. Constraints (17)

deﬁnes the arrival time of customer j visited by a

drone in terms of arrival time of the customer i that

the drone was launched from, and the time that the

drone spent to serve customers served before j. Con-

straints (18) deﬁne the limited drone battery over all

missions. Constraints (19) require that the total pay-

load of each truck over its tour is below the truck

capacity. Constraints (20) ensure that each drone is

assigned to at most one truck. Constraints (21) en-

sure that if a drone is deployed by a truck it cannot be

used by any other truck. Constraints (22) requires that

those customers allowed to be served only by trucks

cannot be visited by drones. In a similar way, Con-

straints (23) ensure that the those customers allowed

to be served by drones cannot be visited by trucks.

Finally, constraints (24)-(27) deﬁne the nature of vari-

ables.

4 SOLUTION METHODOLOGY

In this section, we present the sketch of the NSGA-

II solution algorithm that we apply to solve the pro-

posed model. Known as the multi-objective variant of

the genetic algorithm, the NSGA-II is one of the most

popular and promising evolutionary algorithms. The

algorithm efﬁciency is supported by the low compu-

tational complexity, limited to O(MN

), where M and

N are, respectively, the number of objective functions

and the population size. The algorithm is based on

the main idea of ﬁnding multiple non-dominated so-

lutions in a single simulation run as common in multi-

objective evolutionary algorithms. The NSGA-II ben-

eﬁts from the elitism concept and does not require

any sharing parameter that signiﬁcantly contributes to

its credibility (Deb et al., 2002). In a nutshell, the

algorithm main elements are: a fast non-dominated

sorting mechanism and a fast crowded distance es-

timation mechanism acting as intensiﬁcation and di-

versiﬁcation tools. The fast non-dominated sorting

mechanism sorts the population individuals into dif-

ferent fronts, where the front index represents the

non-domination level (rank) of the solutions in the

front. All the solutions in the ﬁrst front are not dom-

inated by any other solution in the population and

Synchronized Drone and Truck Routing Problem: A Multi-Stakeholder Perspective

333

5 6

8 0

(6,1,2, 3) (8, 17,19,20) (15,18) (9,13) (10, 11,12)

Figure 2: Chromosome representation for solution in Figure

dominate all elements in the other fronts; the elements

in the second front are dominated only by the solu-

tions in the ﬁrst front and dominate all solutions in the

third fronts; in a similar way, the elements in the third

front are dominated only by those in the ﬁrst and the

second front, and so on. Instead, the crowding dis-

tance estimation mechanism is used to maintain di-

versity in NSGA-II. The crowding distance provides

the density of solutions surrounding a particular solu-

tion in a nondominated set. The solutions in a non-

dominated front are ﬁrst sorted in an ascending order

of magnitude, according to each normalized objective

function value. Then, for each objective, to the so-

lutions with smallest and largest objective function

values is associated a distance equal to inﬁnity. For

the remaining pairs of adjacent solutions, the distance

value is evaluated as the absolute normalized differ-

ence in the corresponding objective function values.

The overall crowding-distance value is the sum of

distance values corresponding to each objective. In

what follows, we present the details on applying the

NSGA-II algorithm for the SMTDTRP.

4.1 Solution Representation

The population P includes N individuals correspond-

ing to solutions of the problem. Each solution con-

tains a set of tours, one tour for each truck, and each

tour is an ordered list of visited customers. In the so-

lution representation, to each drone is associated an

ordered list of nodes allowed to be visited. Each ele-

ment in the list contains the launch point of the drone

(necessarily a node in set V

, a truck-eligible cus-

tomer) followed by the set of customers visited in the

ordered trips.

Figure 2 displays the encoded chromosome corre-

sponding to the solution in Figure 1.

4.2 Initial Population Generation

To generate a population the following steps are fol-

lowed.

Each solution in the initial population is generated

following steps 1-6:

1. Select a random integer number N

∈ [1,|K|] rep-

resenting the number of deployed trucks (tours) in

the current solution.

2. To each truck k, assign a random integer value u

that speciﬁes the number of drones assigned to the

truck.

3. For each customer in V

, repeat i to iii that builds

the truck tours:

• i. select randomly a customer i ∈ V

, to be re-

moved from V

and update V

• ii. select randomly a truck with free capacity to

add the order of customer i to.

• iii. Add customer i to the end of tour of truck k.

4. For each customer in V

, repeat i’ to vi:

• i’. select randomly a customer i ∈ V

, remove

it from set V

and update V

• ii’. select randomly a truck k

with residual ca-

pacity greater than payload of the order of cus-

tomer i.

• iii’. Randomly select a customer i

within the

tour of truck k

as the drone launch point

• v. Select randomly an available drone u

as-

signed to truck t

• vi. If the drone energy consumption is not vio-

lated, add customer i to the path of drone u

5. Update the objective function values and add the

solution to the population.

4.3 Fitness Evaluation

The ﬁtness of the individuals is evaluated based on the

non-dominance rank and the crowding distance mea-

sure.

The crowding distance of a solution sol

that be-

longs to front F

with rank l is deﬁned as:

cd(sol

) =

∑

m=1

i+1

− f

i−1

max

− f

min

(28)

where f

is the objective value of sol

with respect

to objective m and f

max

= max

sol

∈F

and f

min

sol

∈F

, respectively, represent the highest and

the lowest objective function among all individuals in

4.4 Parent Selection

As common in the classical genetic algorithm, a tour-

nament selection procedure is applied that chooses a

pair of individuals (parents) from the population to

be recombined using the crossover operators and later

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

334

modiﬁed by the mutation operations. The N new indi-

viduals are then merged into the population: next, this

extended population with size of 2N is sorted thanks

to the non-dominated sorting procedure. To shrink

the size of the population to N individuals, starting

from the ﬁrst rank, all the solutions in the front are

added in the new population. If the resulting number

of solutions in the front is still less than N, the so-

lutions of the subsequent fronts in the order of their

ranks are inserted to complete the population. In case

the cardinality of a front is above the number of in-

dividuals needed to be added to the population set,

the crowded-comparison feature breaks the tie in fa-

vor of solutions with higher crowding distance. This

procedure respects the elitism as the non-dominance

sorting is applied on both the parents and offsprings.

4.5 Crossover Operators

The Crossover is one of the main operators in the

genetic algorithm that uses the information of two

individuals (current solutions) to produce offsprings,

representing new solutions. Given a crossover prob-

ability p

crossover

, the single truck and multi-truck

crossover operators could be applied. To this end,

we ﬁrst need to explain the notion of Complementary

truck and complementary map:

Two trucks are complementary iff the truck (indecies)

and the set of visited customer are different (they do

not have any visited node in common).

Let I and I

′

be two individuals, the complementary

map of I is a list that includes all set of trucks from I

′

which are complementary to the trucks in I.

Crossover single truck

Let I and I

′

be two individuals, their offspring is gen-

erated by starting from the tour of one of the two indi-

viduals followed by adding the tour of the second one

skipping the customers already present.

Crossover multi-truck

In this case, one of the individuals is randomly se-

lected as the ﬁrst parent (p

) and the other would be

the second one (p

). We create a map M in which

for each truck in the ﬁrst individual, we store all the

complementary trucks belonging to the second indi-

vidual. We choose a random truck v and we add it to

the solution. From the map M we take the comple-

mentary list relating to vehicle v. If this list is empty,

the crossover ends. Otherwise we choose a random

number nc of complementary vehicles to add to the

solution. This number can range from 1 to NC, where

NC is the size of complementaries. We randomly take

nc vehicles from complementaries and add them to

the solution.

4.6 Mutation

With probability p

mutation

the following mutation op-

erators are applied.

• Intratour swap

This operator swaps the order of two randomly se-

lected customers within the truck tour. The swap

is done at least once and up to L times where L

denotes the length of truck tour.

• Delete nodes

This operator randomly deletes a prespeciﬁed

number of customers, let say D, from the tour of

a given truck where 0 ≤ D ≤ L − 2

5 COMPUTATIONAL RESULTS

In this Section, we report the the computational ex-

periments carried out on a set of benchmark instances

adapted from the literature in order to investigate the

efﬁciency of NSGA-II algorithm for the proposed

mathematical model.

All the experiments were executed on an Intel Core

i7-4700HQ, with 2.40 GHz CPU, 16 GB RAM and

the NSGA-II algorithm was coded in Java language

Arnold et al. (2005). In the following, we describe

the adopted test beds and present the computational

results.

5.1 Data Sets and Parameter Settings

As the benchmark, we used the set of instances from

Dewilde et al. (2013) originally designed for the trav-

eling repairman problem with proﬁts. In particular,

we considered four classes of instances with 50, 100,

200, and 500 customers and selected the ﬁrst two

instances of each class. We modiﬁed the instances

appropriately to account for the presence of drones

and variations in travel speed. In our setting, it is

assumed that up to 75% of customers can be vis-

ited by trucks and the remaining ones by the drones

(|V

| = ⌊0.75V

′

⌋, |V

| = |V

′

| − |V

|). Regarding the

uncertainty of drone energy consumption, we ran-

domly generated the variance of the energy consump-

tion in each drone trip as σ

= χ µ where µ is the ex-

pected energy consumption and χ is a randomly gen-

erated value where χ ∈ [0.1, 0.3]. To reﬂect the highly

risk-averse attitude of the decision-maker, parameter

α was set to 0.1. We considered a homogeneous ﬂeet

of drones with a fully charged battery of 0.27 KW /h,

payload capacity of 1 kg, where the drone speed is set

to 43.2 km/h (12 m/s). Other drone-related features

are as follows:

Synchronized Drone and Truck Routing Problem: A Multi-Stakeholder Perspective

335

Table 2: Parameter settings.

Parameter Value

Population size N 200

Number of iterations 250

crossover

0.8

mutation

0.5

Table 3: Computational results.

Instance TP TL TTD TNDT CPU

50-1 28931.9 80008.2 184046.9 3.5 5

50-2 32039.7 96638.0 209477.7 3.6 3

100-1 89579.3 165380.7 325997.0 5.2 7

100-2 74263.8 159259.4 299150.4 4.4 7

200-1 316691.3 363168.8 634370.7 8.5 24

200-2 330863.0 429824.7 701401.6 8.2 28

500-1 1484141.8 2525309.0 3915138.7 11.2 42

500-2 1573664.9 2388172.2 4185311.2 12.9 41

TP: Total Proﬁt, TL: Total Latency, TTD: Total Traveled

Distance, TNDT: Total Number of Dispatched Trucks.

The drone frame and battery mass are both equal to

1.5 kg, constant k

′

was calculated as a function of

′

2ρξ h

where the gravity g is set to 9.81 (N/kg),

the ﬂuid density of air ρ is set to 1.204kg/m

; the area

of spinning blade disc ξ is 0.0064 m

and each drone

has eight rotors h = 8. For all instances, we set a ﬂeet

of ﬁve trucks and two drones. The truck speed was

set to 29.88 km/h and the capacity of trucks depen-

dent on the instance were set as the eighty percent of

total demands divided by the size of terrestrial ﬂeet.

Also, the travel distance between nodes i and j across

arc (i, j) is calculated following the Euclidean norm.

Other algorithm-related parameter settings are as re-

ported in Table 2.

Table 3 reports the average of best results within

three random executions for each objective (Columns

2-5). Column 6 presents the algorithm CPU time

in seconds. Figure 3 displays the algorithm compu-

tational time in seconds conﬁrming the high perfor-

mance of NSGA-II. Also, as expected, with the in-

crease in problem size, the computational time also

increases. Figure 4 illustrates the number of Pareto-

optimal solutions found by the algorithm for each

instance. We observe that the size of approximate

Pareto-optimal set increases with the increase in the

number of customers. This also justiﬁes the increase

in the computational time since provides the decision-

maker with more choices.

In order to present some insights on the impact

Table 4: Sensitivity analysis with respect to the problem

size.

Instance ∆|V

′

| ∆ Proﬁt ∆ Latency ∆ Distance ∆ Fleet Size

100 2.0 2.7 -1.9 -1.6 -1.4

200 4.0 10.6 -4.5 -3.4 -2.4

500 10.0 50.2 -28.1 -20.6 -3.4

Figure 3: NSGA-II: Computational time.

Figure 4: NSGA-II: Number of Pareto-optimal solutions.

of problem size (in terms of number of customers)

on the problem objectives, we consider the two cases

with 50 customers as the baseline and report the ratio

of change (depicted by ∆. in Table 4) with respect to

each criterion for cases with more than 50 customers.

For example, the value of 50.2 in the last row indi-

cates that the total proﬁt of the logistics manager is

multiplied 50 times as the size of customers increase

from 50 to 500. The ratio of changes in the ﬂeet size

in also low while the increase in total traveled dis-

tance and latency criteria remains acceptable. This

shows that it is beneﬁcial for the logistics stakeholder

to enlarge the business size and service area.

Figure 5 illustrates the percentage of customers

served by drones and trucks for each instance, verify-

ing that no matter what the problem size is, on average

around 70% of customers are served by the trucks.

In order to better understand the relation among

Figure 5: Percentage of customers served by trucks and

drones.

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

336

different criteria (in terms of harmony or conﬂict),

Figure 6 visualizes the trade-off among the Pareto-

optimal solutions for all cases, thanks to the chord

chord diagram Koochaksaraei et al. (2017). We

should clarify that ob j0, ob j1,ob j2, ob j3, respec-

tively, correspond to the proﬁt, total traveled distance,

latency, and the ﬂeet size objectives. Our ﬁrst obser-

vation is on the conﬂict of the proﬁt criterion with the

latency and travel distance objectives supported by

links in the end (ﬁrst) point of the Ob j0 connected to

the ﬁrst (end) points of Ob j1, Ob j2, and ob j3 and that

makes sense since higher proﬁt gains require, serv-

ing more customers, and this requires traveling more

distance, increasing the waiting time of customers,

and of course more vehicles. In addition, the pres-

ence of many links between the end (ﬁrst) points of

the Ob j0 and Ob j2 shows that there is a strong con-

ﬂict with the proﬁt and traveled distance. In general,

we observe a harmony among the all objectives, but

the proﬁt. Fortunately, there are links connecting the

middle middle bound of each objective and such area

is dense enough conﬁrming the possibility to ﬁnd a

trade-off among all objectives. Last but not least, we

observe that the chord diagrams gets denser with the

increase of the number of customers that is quite rea-

sonable since in general we expected with the increase

of problem size, the number of Pareto-optimal solu-

tions increase. This means that the decision-maker

has more choices to select among. We should em-

phasize that in all cases, almost all the points on red

arches are fully covered and connected by some links

to other points in other arches. This also conﬁrms the

diversity of solutions and the fact that the objective

range (feasible region) is covered appropriately.

In Figure 7, we illustrate the Pareto-optimal fron-

tier as a three-dimensional scatter plot, where the

color of each point, mapped into the color bar, refers

to the forth objective. We can see that there are plenty

of solutions in the middle of the three-dimensional

plot which are colored in Yellow and Orange that cor-

respond to those solutions in which all objectives are

more or less balanced. The density of the solutions

and its wide coverage are promising results, conﬁrm-

ing the efﬁciency and validity of the NSGA-II for the

proposed model.

6 CONCLUSIONS

In this paper, we studied the synchronized last-mile

delivery problem under a joint and collaborative sys-

tem consisting of terrestrial and sidekick aerial vehi-

cles. This novel variant, requiring the timing synchro-

nization between drone and truck, can be classiﬁed as

(a) 50-1 (b) 50-2

(e) 200-1 (f) 200-2

(g) 500-1 (h) 500-2

Figure 6: Visualization of the relation among objectives:

chord diagram.

Synchronized Drone and Truck Routing Problem: A Multi-Stakeholder Perspective

337

·10

Ob j0

Ob j 1

Ob j 2

Ob j 3

1 2 3 4

(a) Instance 50-1

·10

Ob j0

Ob j 1

Ob j 2

Ob j3

1 2 3 4

(b) Instance 50-2

0.5

1.5

·10

Ob j0

Ob j 1

Ob j 2

Ob j3

2 4

0.5

1.5

·10

Ob j0

Ob j 1

Ob j 2

Ob j3

2 4

(d) Instance 100-2

·10

Ob j0

Ob j 1

Ob j 2

Ob j3

2 4

8 10 12 14

(e) Instance 200-1

·10

0.5

·10

Ob j0

Ob j 1

Ob j 2

Ob j3

2 4

8 10 12 14

(f) Instance 200-2

·10

0.5

·10

Ob j0

Ob j 1

Ob j 2

Ob j3

(g) Instance 500-1

·10

0.5

·10

Ob j0

Ob j 1

Ob j 2

Ob j3

(h) Instance 500-2

Figure 7: Scatter plot representing the Pareto-optimal frontier in 4D.

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

338

a synchronized version of parallel drone scheduling

vehicle routing problem. We formulated the prob-

lem as a multi-objective model involving both busi-

ness and customer-oriented goals and developed an

evolutionary-based meta-heuristic approach to solve

the model. The efﬁciency of the solution approach

was investigated through computational experiments

carried out on different instances adopted from the

benchmark. Several interesting research avenues are

open for further investigation. One relates to the de-

sign of exact solution methods or matheuristics and

compares the solutions of such methods with those

presented in this study. Another interesting research

topic could be focused on the extension of the present

model to the case where drones can serve more than

one customer per trip or being retrieved at arbitrary

locations besides customer sites.

ACKNOWLEDGEMENTS

The second author is supported by PNRR project

NEST “Network4EnergySustainableTransition”-

PE00000021.

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