An Energy-Efficient Logistic Drone Routing Method Considering Dynamic Drone Speed and Payload

Abstract

Unmanned aerial vehicles (UAVs), or drones, are recognized for their potential to improve efficiency in last-mile delivery. Unlike the vehicle routing problem, drone route design is challenging due to several operational signatures, such as speed optimization, multi-trip operation, and energy consumption estimation. Drone energy consumption is a nonlinear function of both speed and payload. Moreover, the high speed of drones can significantly curtail the drone range, thereby limiting the efficiency of drone delivery systems. This paper addresses the trade-off between speed and flight range in a multi-trip drone routing problem with variable flight speeds (DRP–VFS). We propose a new model to specifically consider energy constraints using a nonlinear energy consumption model and treat drone speeds as decision variables. The DRP–VFS is initially formulated using mixed-integer linear programming (MILP) to minimize energy consumption. To solve large-scale instances, we propose a three-phase adaptive large neighborhood search (ALNS) algorithm and compare its performance with a commercial MIP solver. The experimental results demonstrate that the proposed method is capable of effectively identifying suboptimal solutions in practical scenarios. Furthermore, results indicate that operating drones at variable speeds leads to about 21% energy savings compared to fixed speeds, with advantages in cost savings and range extension.

1. Introduction

Over the past few decades, developments in technology, particularly carbon fiber and lithium polymer batteries, have facilitated the use of drones for wireless communications, plant protection, and last-mile delivery. For example, Zipline has completed over 14,000 deliveries of medical supplies in Rwanda since 2016 [1]. In 2019, Google’s Wing and Amazon received Federal Aviation Administration (FAA) approval to begin commercial deliveries by drone [2,3].

In recent years, researchers have shown a growing interest in last-mile delivery that incorporates drones [4,5]. Drone delivery problems can be categorized as drone-only or truck–drone tandem problems. The first uses only drones for parcel delivery, while the second uses both drones and trucks to deliver. The alternative objectives of drone delivery problems are to minimize delivery time or to minimize operational cost [6]. Numerous variants of these problems have been suggested, such as heterogeneous drones [7] or multiple trucks [8,9]. Furthermore, the efficacy of drone delivery system is significantly dependent on an available and robust communication network [10]. There are potential risks in drone delivery systems, such as jamming, information leakage, and spoofing, which must be seriously considered. Ensuring communications security is also paramount to the implementation of drone delivery [11,12]. The growth in drone usage for delivery may also increase the risk to consumer privacy. As the privacy risk increases, the consumers’ intention to adopt drone delivery service decreases [13]. Consequently, customer privacy protection is required for the implementation of drone delivery. Additionally, the operation of drones in low-altitude airspace can result in significant noise pollution, potentially leading to health impairment, which has a continuous influence on urban residents [14]. Therefore, optimizing drone routing strategies is imperative to mitigate emissions and prevent privacy infringements during delivery.

Compared with traditional trucks, drone delivery is faster and less affected by road systems and can additionally save a lot of labor and time costs [15]. These advantages allow logistics companies to dispatch drones for parcel delivery. On the other hand, a drone’s endurance is limited by its battery capacity, which is further impacted by parcel weight, speed, and weather conditions [16,17,18]. In the literature involving drone delivery, one typical assumption is that drones fly at constant or maximum speed, regardless of their payload or speed. Thus, speed is not taken into consideration during the decision-making procedure. In fact, operating drones at a fixed speed can lead to increased energy consumption, thus resulting in an energy-infeasible route or an inability to serve long-distance customers, as drone energy consumption is severely impacted by parcel weight and speed. Therefore, it is crucial to consider the impact of payload and speed on endurance in drone routing problems.

The motivation for considering flight speeds as decision variables is the influence of drone energy consumption on flight endurance and range. Figure 1a shows the power rate of drones under different payloads, using the model suggested by [17]. The circular marks indicate optimal drone speeds that maximize the drone endurance under any onboard battery capacity, which can be seen more clearly from Figure 1c. Figure 1b shows the drone energy consumption per unit traveling distance with varying payloads. The triangular symbols indicate optimal drone speeds that maximize the total traveling distance with fixed battery capacity, which can be observed more distinctly in Figure 1d. It can be seen from Figure 1b that the drone energy consumption per unit distance initially decreases and subsequently rises again as the speed increases. This is because there is a balance between the flight duration and the energy consumption. Flying at lower speeds requires less power, which leads to a longer flight duration. In contrast, higher speeds result in more power being consumed, but the makespan is reduced. Therefore, the flight range notably depends on its speed, and faster flight speeds are not always optimal. Consequently, higher speeds can cause faster deliveries but with reduced range and increased energy consumption. This is essential within the scope of drone delivery systems because, in some cases, the customer is sufficiently remote from the warehouse, and flying at maximum speed can lead to a shorter range that falls short of the distance necessary to reach the customer. It may be beneficial to fly at lower speeds, which can improve range and serve more customers. Moreover, speed adjustments can enhance endurance, which enables drones to serve more customers. As such, it is imperative to factor in drone speeds during deliveries when considering energy consumption.

This paper significantly contributes to the field of sustainability by focusing on the use of drones for delivery, a method that can reduce reliance on traditional road-based delivery vehicles, leading to decreased carbon emissions and traffic congestion—a key goal in sustainable urban planning. Moreover, the paper emphasizes energy-efficient routing methods, which are crucial for sustainable resource use. By optimizing drone speed and payload, the proposed model ensures that energy is used as efficiently as possible, reducing the environmental impact of drone operations and increasing their economic viability, thereby promoting the sustainability of drone-based delivery systems. Furthermore, by developing drone routing methods that consider the dynamic drone speed and payload, this paper contributes to the advancement of sustainable logistics and transportation systems. This is especially crucial as the demand for e-commerce continues to grow, necessitating delivery systems that are efficient, cost-effective, and environmentally friendly. Additionally, drone delivery can also contribute to social sustainability by improving access to goods and services in remote or underserved areas. In summary, this paper plays a significant role in advancing sustainable logistics practices.

This paper introduces the multi-trip drone routing problem with variable flight speeds (DRP–VFS), an extension of the DRP proposed in [19]. In this problem, a fleet of homogeneous drones delivers multiple parcels to customers while minimizing energy consumption. In this case, drones can perform multiple trips and can only be dispatched from and returned to the depot once per trip. While the existing literature on the DRP assumes that drones fly at constant speeds in advance, the DRP–VFS considers drone speeds as decision variables when drones make deliveries.

This paper makes the following contributions:

• We introduce and formally define a multi-trip drone routing problem, aiming to achieve operational cost minimization where drone speeds are considered as decision variables rather than constants;
• We clearly consider drone energy consumption as a nonlinear function of both flight speed and payload rather than supposing that flight endurance and range are constants;
• We propose a three-phased method integrating variable drone speed to solve DRP–VFS, which can improve the solution in terms of computational time to a greater extent than MILP.

The rest of this paper is structured as follows. In Section 2, we present a review of the literature. Section 3 presents the formal description of DRP–VFS and its MILP formulation. In Section 4, an efficient heuristic method proposed to solve large-scale instances is introduced. Section 5 demonstrates the case study and verifies the effectiveness of the proposed method. In the end, conclusions are drawn and future research work is discussed in Section 6.

2. Literature Review

The growth in online shopping has increased the demand for faster, more reliable and environmentally friendly parcel delivery. One key challenge for package delivery is the last-mile delivery. The increasing deployment of traditional vehicles for delivery may exacerbate traffic congestion and air pollution [20]. Drones, with their ability of fly in the air, can reduce operation costs, delivery time, and emissions, making them a promising alternative to conventional vehicles [21]. Over the past years, there have been increasing studies on the civil logistics using drones [22,23,24]. These studies reported that operational planning for drone delivery is a well-studied problem. There already exists many papers on the vehicle routing problem (VRP) involving conventional vehicles and their variants [25,26]. However, adapting the VRP to drones faces additional challenges since delivery by drones must account for extra constraints such as lower battery capacity and higher sensitivity of energy cost to payload and flight speed [6].

Most drone studies have assumed either a fixed endurance or a constant speed ([15,27,28,29,30,31]). Obviously, under these assumptions, drones flying at higher speed have longer ranges. Other studies consider a constant range of UAVs. These two assumptions are equivalent when drone speeds are constant. However, both of them fail to consider the influence of payload and speed on the drone’s energy consumption, thus limiting the flexibility of drone delivery. For example, drones can extend their flight endurance by adjusting speed to serve customers further away from the depot.

Some studies have improved the fixed-endurance/-range model by considering drone power consumption as dependent on payload while remaining unrelated to drone speed. Ref. [32] applies the UAV power rate function proposed by [33] and assumes that the UAV maintains the maximum power during flight to calculate the load-dependent UAV speed during delivery. Ref. [34] proposes a package delivery schedule utilizing drones and considers the drone’s power rate varies linearly with its payload. Ref. [6] proposes two drone routing problems where customers are served by drones only. The authors consider the power rate function as linearly dependent on payload weight and develop a simulated annealing (SA) algorithm to address the problems. The power rate function adopted in [6] pertains solely to hovering without encompassing that of actual flight. Ref. [19] adopts the same energy consumption as in [6], but, instead of approximating the power rate function, they develop an exact algorithm by adding logical and subgradient cuts to solve a DRP with time windows. Ref. [35] uses the UAV power rate proposed by [36] as a piecewise linear function of package weight in their DRP. Ref. [37] also considers the influence of package weight on energy consumption using the same model as [6]. However, different from the proposed model based on varying speeds, the studies in the above papers only consider constant drone speeds. Moreover, the proposed energy-consumption models are mainly about hovering and do not consider the energy cost of actual flight status. Other studies apply an energy consumption model related to the payload and the drone speed during forward flight. Ref. [38] adopts an energy model from [33], which is linearly dependent on payload and speed. Ref. [39] also consider the energy-consumption function introduced in [16] in the multiple flying sidekicks traveling salesman problem (mFSTSP). Ref. [40] investigates a truck–drone tandem routing challenge aimed at reducing the overall delivery duration, which characterizes drone energy consumption as a function of the package weight and the flight time. However, the UAV speeds are fixed in these two papers, limiting the potential time and energy savings from flying at varying speeds.

Some recent studies have considered variable flight speeds in drone routing problems. For instance, Ref. [41] considers the reciprocal of the flight speed to be approximately linear with the weight of loads. However, they consider drone speed as a function of the load, which can be regarded as a fixed parameter related to given loads rather than as a decision variable to be optimized. Ref. [42] considers drone speeds as decision variables and uses the energy model introduced by [17]. Furthermore, Ref. [43] employs the multi-rotor UAV energy-consumption model developed by [16] and considers power rate as a function of a UAV’s payload and speed. Ref. [44] also uses the energy model in [42] and [17] to solve a two-echelon vehicle routing problem with drones. These two studies treat drone speeds as decision variables and demonstrate that operating drones at varying speeds contribute to delivery cost and time reductions. Moreover, Ref. [42] emphasizes that using fixed speeds may result in infeasibility in some instances. These two papers consider drone and truck tandem delivery, whereas our focus is solely on drone-only deliveries. As far as we know, this paper presents for the first time a DRP–VFS that considers UAV energy consumption and variable flight speeds.

Table 1. Summary of studies on the drone routing problem in the literature.

Reference	Delivery Type		Energy Model	Drone Speed	Objective	Solution Method
	Truck & Drone	Drone
[27]	✓			Constant	Delivery time	Exact & Heuristic
[28]	✓			Constant	Delivery time	−
[29]	✓			Constant	Operational cost	Heuristic
[30]	✓			Constant	Operational cost	Heuristic
[15]	✓			Constant	Delivery time	Exact & Heuristic
[31]	✓			Constant	Delivery time	Exact
[32]	✓		[33]	Constant	Vehicle waiting time	Heuristic
[34]		✓	Hovering	Constant	Drone fleet size	Exact & Heuristic
[6]		✓	Hovering	Constant	Operational cost	Exact & Heuristic
[19]		✓	Hovering	Constant	Operational cost	Exact
[35]	✓		[36]	Constant	Delivery time	Heuristic
[37]	✓		Hovering	Constant	Delivery time	Exact & Heuristic
[38]		✓	[33]	Constant	Operational cost	Exact
[39]	✓		[16]	Constant	Delivery time	Exact & Heuristic
[41]	✓		Linear	Variable	Delivery time	Exact
[42]	✓		[17]	Variable	Operational cost	Exact & Heuristic
[43]	✓		[16]	Variable	Delivery time	Heuristic
[44]	✓		[17]	Constant	Delivery time	Exact & Heuristic
[45]	✓		[36]	Variable	Operational cost	Exact
This paper		✓	[17]	Variable	Operational cost	Exact & Heuristic

demonstrates a summary of relevant studies on the drone delivery problem reviewed in this paper. The studies are compared using the following factors: (i) the type of delivery (truck–drone tandem or drone-only), (ii) whether the endurance of drones is fixed or evaluated by energy consumption model, (iii) the drone speed (constant or variable), (iv) the objective function used, and (v) whether the solution methods are exact or heuristic. In the literature mentioned above, it is evident that only a few papers consider the energy cost model and even fewer take drone speeds into account as decision variables. Because the finite drone range is susceptible to speed and payload weight, it is essential to consider variable drone speed and payload in drone delivery systems. In this study, a DRP–VFS is proposed to consider the energy constraints and variable speed and payload of the drone for last-mile delivery systems. Also, a three-phase heuristic method is proposed to solve large-scale scenarios. The performance of the proposed heuristic is evaluated by comparing it to the exact method.

3. Model Formulation

This section presents a formulation of the DRP–VFS and introduces piecewise linearization to approximate the nonlinear energy-consumption function.

3.1. Problem Definition

The multi-trip drone routing problem with variable flight speeds (DRP–VFS) is defined on a complete directed graph $G=(N,A)$, where $N=\left\{0,1,\dots ,n+1\right\}$ represents the set of all nodes, with the nodes 0 and $n+1$ corresponding to the depot (distribution center) from which all drones must originate and to which they must return. The set of customers is denoted by $N^{\prime }=\left\{1,\dots ,n\right\}$, associated with the distinct parcels. To simplify notation, we introduce $N^{+}=\left\{0,\dots ,n\right\}$ to indicate the set of nodes from which a drone can depart and $N^{-}=\left\{1,\dots ,n+1\right\}$ to describe the set of nodes that a drone may visit. Let $A=\left\{\left(i,j\right):i\in \left\{0\right\},j\in N^{\prime }\mathrm{and}i\in N^{\prime },j\in N^{-},i\ne j\right\}$ denote the set of arcs. Additionally, we define sets ${\delta }^{-}\left(i\right)$ and ${\delta }^{+}\left(i\right)$ to denote the forward and backward nodes, respectively, relative to node i.

Each customer has a positive demand $q_i$ and a hard time window $\left[a_i,b_i\right]$. The drones are allowed to arrive early but need to wait until $a_i$ to begin serving the customer. A fleet of U homogeneous multi-rotor drones departs from the depot, each with a fully charged battery. Q represents the drone’s maximum payload, and we suppose that $q_i\le Q,\forall i\in N^{\prime }$, while E is the maximum energy provided by a fully charged battery. The service time for each customer is $s_i$. Every drone can implement multiple trips and serve several customers within one delivery. $d_{ij}$ represents flight distance on arc $\left(i,j\right)$, which is assumed to be symmetric. Since the drone speeds are decision variables in this paper, the flight time on arc $\left(i,j\right)$ is non-constant, as discussed later. The maximum speed limit is represented by $V_{max}$.

We make the following assumptions:

• Parcels are delivered by drones only;
• There is only one depot from which drones can depart;
• We neglect the time of loading parcels and swapping batteries;
• Without loss of generality, we ignore the influence of weather, i.e., wind impact is not considered;
• Drones can fly at a constant speed between two locations. The speed of each flight can vary.

3.2. Mathematical Model

DRP–VFS involves two sets of binary variables: $x_{ij}=1$ if a drone flies across arc $\left(i,j\right)$ and $x_{ij}=0$ otherwise. $z_{ij}=1$ if a trip ends at node i and is immediately succeeded by another trip with node j as the first customer. Additionally, there are six sets of continuous variables: $m_{ij}$ is the parcel’s weight transported across the arc $\left(i,j\right)$ (kg); $v_{ij}$ denotes the drone speed from node $i\in N^{+}$ to node $j\in N^{-}$ (m/s); ${\tau }_i$ is the service start time of customer $i\in N^{-}$ (second); $t_{ij}$ represents the time required to fly through arc $\left(i,j\right)$; $f_i$ denotes the total energy that a drone consumes when it arrives at customer i (kWh); $e_{ij}$ indicates the energy consumed through arc $\left(i,j\right)$, measured in kWh.

The constraints of route feasibility can be expressed as follows:

$$\begin{array}{cc}\hfill \sum _{j\in {\delta }^{+}\left(i\right)}{x}_{ij}& =1\forall i\in N^{\prime },\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \sum _{j\in {\delta }^{-}\left(i\right)}{x}_{ji}& =1\forall i\in N^{\prime },\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \sum _{j\in N^{\prime }}{x}_{0j}& =\sum _{j\in N^{\prime }}{x}_{j,n+1}.\hfill \end{array}$$

(3)

Constraints (1) and (2) guarantee that each customer is served only once. Constraint (3) guarantees that the number of trips leaving the depot equals the number returning to the depot.

We apply demand constraints through

$$\begin{array}{cc}\hfill & \sum _{j\in {\delta }^{-}\left(i\right)}{m}_{ji}-\sum _{j\in {\delta }^{+}\left(i\right)}{m}_{ij}=q_i\forall i\in N^{\prime },\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & m_{i,n+1}=0\forall i\in N^{\prime },\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & m_{ij}\le Q{x}_{ij}\forall \left(i,j\right)\in A.\hfill \end{array}$$

(6)

Constraint (4) ensures that the demand of each customer can be satisfied and also eliminates sub-tours. Consequently, the payload decreases to zero when the drone returns to the depot as specified by Constraint (5). With Constraint (6), the total weight of parcels carried by a drone in a trip must not exceed its payload capacity.

The following equations are formulated to enforce time-related constraints:

$$\begin{array}{cc}\hfill & {\tau }_i+s_i+d_{ij}/v_{ij}\le M\left(1-x_{ij}\right)+{\tau }_j\forall i\in N^{\prime },j\in N^{-},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & a_i\le {\tau }_i\le b_i\forall i\in N^{-},\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & {\tau }_i+s_i+\left(t_{i,n+1}+t_{0j}\right)\le M\left(1-z_{ij}\right)+{\tau }_j\forall i,j\in N^{\prime },i\ne j.\hfill \end{array}$$

(9)

Constraint (7) requires that the total of the arrival time and service time in i and travel time across the arc $\left(i,j\right)$ is not greater than the arrival time at the customer j (the immediate successor to i). Constraint (8) indicates that the arrival time to customer i falls within its designated time window $\left[a_i,b_i\right]$. Constraint (9) indicates the temporal relationship between successive trips executed by one drone, including the return time. Note that Constraint (7) is nonlinear due to $\frac{1}{v_{ij}}$; we introduce a piecewise linear method to convert it to a linear one in Section 3.3.

In this paper, we allow drones to change batteries and place parcels for new trips when they return to the depot. We adopt a 2-index formulation inspired by [19] to formulate the multi-trip characteristics of drone delivery. Therefore, we can achieve the reusability of drones by incorporating the following constraints:

$$\begin{array}{cc}\hfill \sum _{\underset{i\ne j}{i\in N^{\prime }}}{z}_{ij}& \le x_{0j}j\in N^{\prime },\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \sum _{\underset{j\ne i}{j\in N^{\prime }}}{z}_{ij}& \le x_{i,n+1}i\in N^{\prime },\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \sum _{j\in N^{\prime }}{x}_{0j}& -\sum _{i\in N^{\prime }}\sum _{\underset{j\ne i}{j\in N^{\prime }}}{z}_{ij}\le U.\hfill \end{array}$$

(12)

Constraints (10) and (11) establish a relationship between variables $\mathbf{x}$ and $\mathbf{z}$. Constraint (12) limits the number of drones that can be used.

To evaluate the energy usage during delivery, we adopt the power rate model, according to [17]:

$$P\left(v,w\right)=\frac{\sigma }{8}\rho sD{\Omega }^2{R}^3(1+\frac{3v^2}{U_{tip}^2})+\frac{\kappa w^{\frac{3}{2}}}{\sqrt{2\rho D}}{\left(\sqrt{\frac{v^4}{4v_0^4}+1}-\frac{v^2}{2v_0^2}\right)}^{\frac{1}{2}}+\frac{1}{2}\rho S_{FP}{v}^3,$$

(13)

where $\sigma$ is the average profile drag coefficient, $\rho$ is the air density, s is the rotor solidity, $\Omega$ is the angular velocity of the rotor, $U_{tip}$ is the blade tip speed, R is the rotor radius, D is the rotor disc area, $\kappa$ is the correction factor to ideal induced power, $v_0$ is the induced velocity in hovering, and $S_{FP}$ is the equivalent flat plate area of the drone. w is the total weight of the drone, including the unloaded weight $m_v$ of the drone and the weight $m_p$ of parcels. Thus, $w=\left(m_v+m_p\right)g$ on the forward flight to customers, where g is the gravitational acceleration. To simplify the above formula, define $P_0=\frac{\sigma }{8}\rho sD{\Omega }^2{R}^3$ and $P_i=\frac{\kappa w^{\frac{3}{2}}}{\sqrt{2\rho D}}$ as the blade profile power and induced power when hovering.

By transforming (13), we derive an energy-consumption formula denoted as $E_0\left(v,m\right)$ ($\mathrm{J}$/$\mathrm{m}$) for a drone’s travel per unit distance at a constant speed v, calculated as

$$E_0\left(v,w\right)=\frac{P\left(v,w\right)}{v}=P_0\left(\frac{1}{v}+\frac{3v}{U_{tip}^2}\right)+P_i{\left(\sqrt{v^{-4}+\frac{1}{4v_0^4}}-\frac{1}{2v_0^2}\right)}^{\frac{1}{2}}+\frac{1}{2}\rho S_{FP}{v}^2.$$

(14)

Thus, the energy consumption for drone on arc $\left(i,j\right)$, denoted as $e_{ij}$, can be expressed as

$$e_{ij}=\left\{\begin{array}{cc}{E}_0\left(v_{ij},w_{ij}\right)·d_{ij}\hfill & x_{ij}=1,\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(15)

where $d_{ij}$ and $w_{ij}=\left(m_v+m_{ij}\right)g$ are the distance and payload on arc $\left(i,j\right)$. It is worth noting that (15) is nonlinear when $x_{ij}=1$ and a piecewise linear method is applied to deal with them in the next subsection.

Hence, drones’ energy constraints are written as

$$\begin{array}{cc}\hfill & f_0=0,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & f_i+e_{ij}\le M\left(1-x_{ij}\right)+f_j\forall \left(i,j\right)\in A,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & f_{n+1}\le E.\hfill \end{array}$$

(18)

Constraint (16) signifies that the initial energy consumption of each trip is set to zero. Constraint (17) indicates the energy consumed across the arc $\left(i,j\right)$. Constraint (18) guarantees that the battery’s energy capacity constraint is imposed.

The domain of the variables is shown below:

$$\begin{array}{cc}\hfill & x_{ij}\in \left\{0,1\right\}\forall \left(i,j\right)\in A,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & z_{ij}\in \left\{0,1\right\}\forall i,j\in N^{\prime },\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & m_{ij},e_{ij},t_{ij}\ge 0\forall \left(i,j\right)\in A,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & 0\le v_{ij}\le V_{max}\forall \left(i,j\right)\in A,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & f_i\ge 0\forall i\in N,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & {\tau }_i\ge 0\forall i\in N^{-}.\hfill \end{array}$$

(24)

3.3. Piecewise Linearization

Piecewise linearization can approximate a nonlinear univariate function using a set of non-negative variables of special ordered sets type 2 (SOS2), where at most two adjacent variables can be positive, while the sum of all variables equals one ([46]). For univariate functions like $1/v$, the piecewise linearization approximation is realized by sampling K points $\left[V_1,V_2,\dots ,V_K\right]$ as breakpoints on the v-axis. After that, the linear segments $\left[\left(V_1,1/V_1\right),\dots ,\left(V_K,1/V_K\right)\right]$ are used to approximate $1/v$.

In this case, we define a set of continuous variable ${\alpha }_{ij}^k$ for each breakpoint. The value of $v_{ij}$ on arc $\left(i,j\right)$ is represented by

$$v_{ij}=\sum _{k=1}^K{V}_k{\alpha }_{ij}^k.$$

(25)

The approximation value of the reciprocal of speed $1/v_{ij}$ is $1/{\tilde{v}}_{ij}$, which can be calculated by (26):

$$\frac{1}{{\tilde{v}}_{ij}}=\sum _{k=1}^K\frac{{\alpha }_{ij}^k}{V_k}.$$

(26)

The SOS2 variables ${\alpha }_{ij}^k$ need to satisfy the following constraints:

$$\begin{array}{cc}\hfill & \sum _{k=1}^K{\alpha }_{ij}^k=1,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & 0\le {\alpha }_{ij}^k\le 1k=1,2,\dots ,K.\hfill \end{array}$$

(28)

Most modern MIP solvers, such as Gurobi and CPLEX, can address specially ordered sets of types 1 and 2 automatically. Consequently, Constraints (25)–(28) are sufficient to produce the correct computation. Consequently, the nonlinear constraints in (7) are approximated by replacing $1/v_{ij}$ with $1/{\tilde{v}}_{ij}$.

Similarly, we introduce a set of SOS3 variables ${\beta }_{ij}^{st}$ to linearly approximate the two-variable energy-consumption function dependent on drone speed and parcel weight in (14). Similar to the SOS2, the SOS3 variables require that at most three adjacent components be greater than zero. The difference is that the set of SOS2 variables is one-dimensional, while the set of SOS3 variables is two-dimensional. Consider again S sampling coordinates $\left[V_1,V_2,\dots ,V_S\right]$ on the v-axis and T sampling coordinates $\left[M_1,M_2,\dots ,M_T\right]$ on the m-axis. The function $E_0\left(v,m\right)$ is determined by each breakpoint $\left(V_s,M_t\right)\left(s=1,\dots ,S;t=1,\dots ,T\right)$.

In our problem, the values of $v_{ij}$ and $m_{ij}$ on arc $\left(i,j\right)$ are represented by

$$\begin{array}{cc}\hfill & v_{ij}=\sum _{s=1}^S{V}_s{\beta }_{ij}^{st},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & m_{ij}=\sum _{t=1}^T{M}_t{\beta }_{ij}^{st}.\hfill \end{array}$$

(30)

The approximation value of the energy consumption is ${\tilde{e}}_{ij}$, which can be calculated by using (31):

$${\tilde{e}}_{ij}=\sum _{s=1}^S\sum _{t=1}^T{E}_0\left(V_s,M_t\right){\beta }_{ij}^{st}.$$

(31)

The SOS3 variables ${\beta }_{ij}^{st}$ should satisfy (32) and (33):

$$\begin{array}{cc}& \sum _{s=1}^S\sum _{t=1}^T{\beta }_{ij}^{st}=1,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & 0\le {\beta }_{ij}^{st}\le 1s=1,2,\dots ,S,t=1,2,\dots ,T.\hfill \end{array}$$

(33)

However, unlike SOS1 and SOS2, modern MIP solvers do not have an automatic syntax for imposing SOS3. For this reason, we utilize the triangle method proposed by [47] to impose corresponding constraints. Consider the rectangle corresponding to intervals $\left[V_s,V_{s+1}\right)$ and $\left[M_t,M_{t+1}\right)$ and the two triangles generated by its diagonal $\left[\left(V_s,M_t\right)\left(V_{s+1},M_{t+1}\right)\right]$. We introduce two sets of binary variables $u_{ij}^{st}$ and $l_{ij}^{st}$ to represent the upper and lower triangles in the rectangle, respectively, with dummy values $u_{ij}^{0*}=u_{ij}^{*0}=u_{ij}^{S*}=u_{ij}^{*T}=0$ and $l_{ij}^{0*}=l_{ij}^{*0}=l_{ij}^{S*}=l_{ij}^{*T}=0$ at the boundaries. The additional constraints are shown as follows:

$$\begin{array}{cc}\hfill & \sum _{s=1}^{S-1}\sum _{t=1}^{T-1}\left(u_{ij}^{st}+l_{ij}^{st}\right)=1,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & {\beta }_{ij}^{st}\le u_{ij}^{st}+l_{ij}^{st}+u_{ij}^{s,t-1}+l_{ij}^{s-1,t-1}+u_{ij}^{s-1,t-1}+l_{ij}^{s-1,t}.\hfill \end{array}$$

(35)

Constraint (34) requires that just one triangle be chosen from all possible triangles. Constraint (35) guarantees that the only ${\beta }_{ij}^{st}$ values greater than zero can be those corresponding to the three vertices of such a triangle. Accordingly, the nonlinear energy-related constraints in (17) are approximated by replacing $e_{ij}$ with ${\tilde{e}}_{ij}$.

The objective function is to minimize the total energy consumption during delivery:

$$min\sum _{\left(i,j\right)\in A}{e}_{ij}.$$

(36)

The model described above is an MILP model, which can be solved efficiently by commercial MIP solvers such as Gurobi. While MILP solvers find optimal solutions, the computational time required is substantial, especially for scenarios with several customers. In Section 4, we propose a three-phased heuristic algorithm aimed at finding suboptimal solutions within computational time constraints.

4. Solution Method

The DRP–VFS extends the classical VRP and is an NP-hard problem. Owing to the NP-hardness of the DRP–VFS, a three-phased heuristic is proposed to effectively tackle practical large-scale scenarios.

The solution of DRP–VFS can be represented by $\mathcal{R}=\left\{{\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_{\mathcal{K}}\right\}$, where $\mathcal{K}$ is the number of trips. The ${\mathbf{r}}_k$ contains a vector of customers that a drone will visit, which can be expressed as ${\mathbf{r}}_k=\left\{r_1,r_2,\dots ,r_{C_k}\right\}$, where $C_k$ is the number of customers assigned to the kth trip.

4.1. Initialization

There are numerous heuristic approaches that can efficiently search for a feasible solution for the VRP, such as the CW savings algorithm and sweep algorithm. In our method, we initially apply an improved k-means algorithm that incorporates capacity and Euclidean distance constraints to partition customers into several subsets (Algorithm 1). Subsequently, a traveling salesman problem with time windows (TSPTW) is solved within each subset to determine the customer sequence that a drone serves. This algorithm is executed while satisfying the capacity, energy, and time window constraints. The number of clusters is calculated by the total demands and capacity of the drone as

$$\mathcal{K}=\lceil \sum _{i=1}^n{q}_i/Q\rceil$$

(37)

The $\mathcal{K}$ centroids are randomly initialized in the customers’ coordinate range. The Euclidean distances between each customer and all $\mathcal{K}$ centroids are calculated. We sort customers by demand from largest to smallest and assign each customer to the nearest centroids. The total demands for each subset of clusters cannot exceed the drone’s capacity. The selected customer will be assigned to the next nearest centroid if the capacity constraint is not satisfied. The centroid of the kth cluster $\left(x_k,y_k\right)$ is calculated based on its members using (38):

$$\begin{array}{c}\hfill x_k=\sum _{i=1}^{C_k}{x}_i/C_k,\\ \hfill y_k=\sum _{i=1}^{C_k}{y}_i/C_k.\end{array}$$

(38)

The iterative procedure is repeated until the cluster is unchanged or the maximum iteration is reached.

Algorithm 1 K-means Clustering.

Require: a set of customers $R=\left\{r_1,r_2,\dots ,r_n\right\}$.  1: Initialization: calculate $\mathcal{K}$ using (37) and initialize the centroid randomly. Sort customers by demand.  2: while not converged do  3:     for each customer $r_i\in R$ do  4:       Calculate the distance from $r_i$ to each of the $\mathcal{K}$ clusters and sort it in descending order.  5:       while $r_i$ is not assigned do  6:           Assign $r_i$ to the nearest centroid without violating the capacity constraint.  7:           if $r_i$ is not assigned then  8:              Choose the next nearest centroid.  9:           end if 10:       end while 11:     end for 12:     Calculate the new centroid of the clusters using (38). 13: end while Ensure: $\mathcal{K}$ clusters

The TSPTW is a special case of DRP–VFS since only one drone is used. Thus, we just need to remove the reusability Constraints (10)–(12) and modify Constraint (3) to obtain the following:

$$\sum _{r\in {\mathbf{r}}_k}{x}_{0r}=\sum _{r\in {\mathbf{r}}_k}{x}_{r,n+1}=1k=1,2,\dots ,\mathcal{K}.$$

(39)

Therefore, an initial feasible solution can be obtained quickly.

4.2. Local Search

In this section, we propose a speed-optimization model to adjust the flight speed in a trip and calculate the cost of a DRP–VFS solution, and the ALNS algorithm is used to enhance the solution quality.

4.2.1. Speed Optimization

The speed-optimization problem (SOP) involves adjusting the flight speed in order to minimize energy consumption. The SOP is defined on a route $\mathbf{r}=\left\{r_0,r_1,\dots ,r_{C+1}\right\}$, consisting of customers served by a single drone, where $r_0$ and $r_{C+1}$ represent the depots corresponding to 0 and $n+1$, respectively. $v_i$ and $m_i$ denote the flight speed and payload carried by a drone between node $r_i$ and its successor. The SOP is formulated as follows:

$$min\sum _{i=0}^C{e}_i,$$

(40)

subject to

$$\begin{array}{cc}\hfill & m_i=\sum _{j=i+1}^C{q}_{j}i=0,\dots ,C,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & m_0\le Q,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & {\tau }_i+s_i+d_i/v_i\le {\tau }_{i+1}i=0,\dots ,C,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & a_i\le {\tau }_i\le b_{i}i=1,\dots ,C,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & e_i=E_0\left(v_i,m_i\right)d_{i}i=0,\dots ,C,\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & f_0=0,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & f_i+e_i\le f_{i+1}i=0,\dots ,C,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill & f_{C+1}\le E.\hfill \end{array}$$

(48)

The nonlinear energy function is linearized using the SOS2 variables introduced in Section 3.3. Since $m_i$ is calculated in advance using (42), the $E_0$ between node $r_i$ and its successor depends only on $v_i$. We introduce ${\gamma }_i^s$ to linearly approximate the energy-consumption function. Hence, the value of $v_i$ and the approximate value of ${\tilde{e}}_i$ can be represented by

$$\begin{array}{cc}\hfill & v_i=\sum _{s=1}^S{V}_s{\gamma }_i^s,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & {\tilde{e}}_i=\sum _{s=1}^S{E}_0\left(V_s,m_i\right){\gamma }_i^s.\hfill \end{array}$$

(50)

To deal with $1/v_i$, the same method can be used. However, to speed up the solving process, we introduce a convexification method, i.e., replace $1/v_i$ with an auxiliary variable ${\lambda }_i$ to avoid reintroducing integer variables:

$${\tau }_i+s_i+d_i{\lambda }_i\le {\tau }_{i+1}.$$

(51)

Meanwhile, the following inequality should be imposed to ensure that ${\lambda }_i$ converges to $1/v_i$:

$${\lambda }_i\ge 1/v_i.$$

(52)

It can be seen that Constraint (52) is a quadratic and convex constraint. Therefore, the SOP is modeled using mixed-integer quadratically constrained programming (MIQCP) and can be solved by commercial solvers.

4.2.2. ALNS Heuristics

In this subsection, an adaptive large neighborhood search algorithm is adopted to improve the initial solution obtained by Section 4.1. The ALNS framework has demonstrated successful results in solving multiple variants of vehicle routing problems [48]. By removing selected nodes from the solution and reinserting them, we obtain a neighborhood of the given solution. The operators are dynamically chosen based on their previous performance. For this reason, each operator is given a score that increases if it enhances the current solution. A new solution is accepted if it meets the criteria specified by the local search scheme, such as simulated annealing.

In this paper, we apply two removal operators and two insertion operators to effectively destroy and repair a DRP–VFS solution. In addition, for every $\Gamma$ iteration, the TSPTW model is solved once within each trip to adjust the customer sequence. The framework of the ALNS algorithm is presented in Algorithm 2. Figure 2a–c depict the initial solution, the solution after applying two destroy operators, and the solution after executing two repair operators.

The destroy operators involve primarily removing several nodes from the current solution and placing them in a removal list $\mathcal{L}$. We present the removal operators involved in our approach as follows:

(1)

Random removal: This operator removes several nodes at random from the current solution;

(2)

Worst removal: This operator removes the highest-cost node from the solution, where the cost is determined by solving SOP.

The insertion operators are applied to repair a partly destroyed solution by reinserting the nodes from $\mathcal{L}$ back into the solution. The insertion operators employed in our algorithm include:

(1)

Greedy insertion: This operator repeatedly removes a node from $\mathcal{L}$ and inserts it into the lowest-cost position of a route;

(2)

Regret insertion: An obvious disadvantage of greedy insertion is that it defers node insertion to later iterations, where few feasible moves are available. The regret operator in our algorithm uses a two-regret criterion. We define $\Delta c_i^j$ as the cost change incurred by inserting node i into the route where the cost is the jth-cheapest. The two-regret criterion inserts the node i based on $i^{*}=\underset{i\in \mathcal{L}}{\mathrm{argmax}}\left(\Delta c_i^2-\Delta c_i^1\right)$, where $\Delta c_i^1$ and $\Delta c_i^2$ are the best and second-best insertions into node i. We iterate the procedure until no more nodes in $\mathcal{L}$ can be inserted.

We describe the adaptive weight adjustment process below. The operators are chosen by the roulette-wheel method. In the beginning, all operators are equally likely. They are updated as $P_o^{t+1}=P_o^t\left(1-\Theta \right)+\Theta {\pi }_o/{\omega }_o$ during program execution, where $P_o^t$ is the probability of operator o at the tth iteration, $\Theta$ is the parameter of the roulette wheel, ${\pi }_o$ denotes the score of operator o, and ${\omega }_o$ indicates the number of times that it was selected in the previous $\Gamma$ iterations. The score reflects the performance of each operator. When a new best solution is found, the related operators’ scores are increased by ${\sigma }_1$. If the new solution outperforms the current one, the score is increased by ${\sigma }_2$. If the new solution falls short of the current one but meets the acceptance criteria, the score is increased by ${\sigma }_3$.

Our ALNS algorithm applies the simulated annealing as an acceptance criterion. We calculate the probability to determine whether to update the current solution and to decrease the initial temperature T according to $T=\Phi T$, where $\Phi$ is a constant that indicates the cooling rate. The algorithm produces the best solution that has been found after $\Lambda$ iterations.

Algorithm 2 ALNS with simulated annealing.

Require: Removal operators $\mathcal{D}$, insertion operators $\mathcal{I}$, cooling rate $\Phi$.  1: Produce an initial solution using the K-means algorithm in Section 4.1.  2: Initialize temperature T and counter $t\leftarrow 1$.  3: Initialize probability $P_d^t$ for each destroy operator and probability $P_i^t$ for each repair operator.  4: Let ${\mathcal{R}}_{current}\leftarrow {\mathcal{R}}_{best}\leftarrow {\mathcal{R}}_{init}$.  5: while  $t\le \Lambda$  do  6:     Select a removal operator $d^{*}\in \mathcal{D}$ and a insertion operator $i^{*}\in \mathcal{I}$.  7:     Apply operator $d^{*}$ to ${\mathcal{R}}_{current}$ to generate ${\mathcal{R}}_{new}^{\prime }$.  8:     Apply operator $i^{*}$ to ${\mathcal{R}}_{new}^{\prime }$ to generate ${\mathcal{R}}_{new}$.  9:     if $c\left({\mathcal{R}}_{new}\right)<c\left({\mathcal{R}}_{current}\right)$ then 10:       ${\mathcal{R}}_{current}\leftarrow {\mathcal{R}}_{new}$ 11:     else 12:       Let $\nu \leftarrow e^{-\left(c\left({\mathcal{R}}_{new}\right)-c\left({\mathcal{R}}_{current}\right)/T\right)}$ 13:     end if 14:     Get a random number $\xi \in \left[0,1\right]$ 15:     if $\xi \le \nu$ then 16:       ${\mathcal{R}}_{current}\leftarrow {\mathcal{R}}_{new}$ 17:     end if 18:     if $c\left({\mathcal{R}}_{current}\right)<c\left({\mathcal{R}}_{best}\right)$ then 19:       ${\mathcal{R}}_{best}\leftarrow {\mathcal{R}}_{current}$ 20:     end if 21:     $T\leftarrow \Phi T$ 22:     if $tmod\Gamma =0$ then 23:       Apply the adaptive weight adjustment procedure to update probabilities. 24:       Solve the TSPTW model within each route to adjust the customer sequence. 25:     end if 26:     $t\leftarrow t+1$ 27: end while Ensure:${\mathcal{R}}_{best}$

4.3. Assignment

In this section, we describe a list schedule algorithm to minimize the number of drones. The delivery time, denoted by h, is the latest allowable departure time from the depot, whereas the arrival time, denoted by l, indicates when the drone reaches the depot upon finishing the delivery. The two times are calculated by using (53):

$$\begin{array}{cc}\hfill h& ={\tau }_1-d_0/v_0,\hfill \\ \hfill l& ={\tau }_C+s_C+d_C/v_C,\hfill \end{array}$$

(53)

where ${\tau }_1$ and ${\tau }_C$ are the starts of service time at the first and last nodes in a route; $d_0,v_0$ and $d_C,v_C$ are the distance and speed during the first and last flights in a route; and $s_C$ is the service time of the last node.

We allocate $\mathcal{K}$ routes to U drones using the list scheduling algorithm (Algorithm 3). First, we sort routes by delivery time in ascending order. A binary heap is used to obtain the route with minimum arrival time quickly. Next, it traverses the $\mathcal{K}$ routes; if the delivery time of ${\mathbf{r}}_k$ is greater or equal to the arrival time of the heap top route, then ${\mathbf{r}}_k$ is appended to this route. Otherwise, a new drone should be added. The time complexity is $\mathcal{O}\left(\mathcal{K}log\mathcal{K}\right)$. In conclusion, our heuristic method finds a near-optimal solution ${\mathcal{R}}^{*}$ for DRP–VFS.

Algorithm 3 List Schedule.

Require: a solution $\mathcal{R}=\left\{{\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_{\mathcal{K}}\right\}$, the number of drones U.  1: Initialize near-optimal solution ${\mathcal{R}}^{*}\leftarrow Ø$  2: Initialize binary heap $\mathcal{Q}\leftarrow Ø$  3: for  $u\in \left\{1,\dots ,U\right\}$  do  4:     $\mathcal{Q}\leftarrow \mathcal{Q}\cup (Ø,0)$  5: end for  6: Sort $\mathcal{R}$ by delivery time in ascending order.  7: for  $k\in \left\{1,\dots ,\mathcal{K}\right\}$  do  8:     $\left(\mathbf{r},l\right)\leftarrow \mathrm{POPMINIMUMARRIVALTIMEELEMENT}\left(\mathcal{Q}\right)$  9:     if $l\le h_k$ then 10:       $\mathbf{r}\leftarrow \mathbf{r}\cup {\mathbf{r}}_k$ 11:       $l\leftarrow l_k$ 12:     else 13:       $\mathrm{PUSHELEMENT}\left(\mathcal{Q},({\mathbf{r}}_k,l_k)\right)$ 14:     end if 15:     $\mathrm{PUSHELEMENT}\left(\mathcal{Q},(\mathbf{r},l)\right)$ 16: end for 17: while $\mathcal{Q}$ is not empty do 18:     $\mathbf{r}\leftarrow \mathrm{POPELEMENT}\left(\mathcal{Q}\right)$ 19:     ${\mathcal{R}}^{*}\leftarrow {\mathcal{R}}^{*}\cup \mathbf{r}$ 20: end while Ensure: ${\mathcal{R}}^{*}$.

5. Results and Discussion

In this section, numerical experiments have been carried out to assess the computational efficiency of the proposed heuristic method for the DRP–VFS. Firstly, the computational performance of the ALNS algorithm is revealed by comparing its solution quality and CPU time to those of the exact method with small-scale scenarios in Section 5.2. Secondly, the performance of ALNS implementation in large-scale instances is demonstrated in Section 5.3. Finally, we compare the solutions generated by fixed-speed and variable-speed in Section 5.5. The results demonstrate that the energy consumption of variable-speed solutions decreases notably compared to that of fixed-speed solutions. All computational work was conducted on a computer configuration with an Intel i7 CPU and 16 GB RAM. The heuristic algorithm was coded in Python 3.7 and Gurobi 9.5.0 was used to solve the mathematical models.

5.1. Parameter Settings

We use two sets of benchmark instances introduced by [19], named $SetA$ and $SetB$. $SetA$ is created based on [49] and [6] and contains 10–50 customers. $SetB$ is an extension of Solomon’s instance, which contains 10–100 customers. We use $SetA$ to validate the effectiveness of MILP and ALNS implementation and energy savings with variable flight speeds. We use $SetB$ to validate the efficiency of ALNS implementation on large-scale instances. We execute the MILP implementation once for each instance. We execute the ALNS algorithm 10 times for each instance and calculate the average objective value and computational time. The parameter settings of drones and the ALNS algorithm are shown in

Table 2. Parameters of typical values for drones.

Notation	Description	Values
$m_v$	Mass of drone $\left(\mathrm{k}\mathrm{g}\right)$	2
s	Rotor solidity	0.05
$v_0$	Induced velocity in hover	4.03
$\kappa$	Correction factor	1.1
$\rho$	Air density $\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)$	1.225
$\sigma$	Profile drag coefficient	0.012
$\Omega$	Angular velocity of the rotor $(\mathrm{rad}/\mathrm{s})$	300
D	Rotor disc area in ${\mathrm{m}}^2$, $D=\pi R^2$	0.503
R	Rotor radius $\left(\mathrm{m}\right)$	0.4
$S_{FP}$	Equivalent flat plate area $\left({\mathrm{m}}^2\right)$	0.0151
$U_{tip}$	Blade tip speed $(\mathrm{m}/\mathrm{s})$	120

and

Table 3. Parameter settings of ALNS algorithm.

Parameter	$\Lambda$	$\Gamma$	$\Theta$	$\Phi$	T	${\sigma }_1$	${\sigma }_2$	${\sigma }_3$
Setting	1000	50	0.4	0.9	30	30	20	10

. For $SetA$ instances, the battery energy capacity is set to $E=0.27k\mathrm{W}\mathrm{h}$; for $SetB$ instances, we set $E=0.027k\mathrm{W}\mathrm{h}$.

5.2. Performance Comparison between MILP and ALNS Implementations

In this section, we compare the performance of the MILP implementation with the ALNS implementation to demonstrate that the ALNS algorithm is effective in finding near-optimal solutions to small DRP–VFS instances. The results of MILP and ALNS are presented in

Table 4. Comparison of MILP and ALNS for Set A instances with size 10–20.

Cust	Inst	MILP				ALNS				Gap (%)
		Energy	Distance	CPU	UAVs *	Energy	Distance	CPU	UAVs *	Energy	Distance	CPU
10	1	0.0133	4.475	19.5	2	0.0133	4.475	83.9	2	0.00	0.00	330.26
	2	0.0163	5.447	19.9	2	0.0163	5.447	80.4	2	0.00	0.00	304.02
	3	0.0137	4.582	18.4	2	0.0146	4.981	99.6	2.5	6.57	8.71	441.30
	4	0.0156	5.281	77.9	2	0.0157	5.320	110.9	2.1	0.64	0.74	42.36
	5	0.0144	4.845	52.5	2	0.0144	4.845	84.7	3	0.00	0.00	61.33
15	1	0.018	6.025	124.8	3	0.0184	6.121	127.7	4.6	2.22	1.59	2.32
	2	0.0221	7.382	228.9	3	0.0225	7.513	134.6	3.8	1.81	1.77	−41.20
	3	0.0183	6.042	93.4	4	0.0186	6.140	102.4	4	1.64	1.62	9.64
	4	0.0213	7.195	112.2	3	0.0227	7.515	137.3	3.8	6.57	4.45	22.37
	5	0.0231	7.74	161.6	3	0.0232	7.760	115.4	3.3	0.43	0.26	−28.59
20	1	0.0306	10.208	658.7	4	0.0307	10.211	130.2	4	0.33	0.03	−80.23
	2	0.0262	8.638	586.1	4	0.0270	9.026	158.4	4	3.05	4.49	−72.97
	3	0.0281	9.345	1039.2	4	0.0282	9.453	186.1	4	0.36	1.16	−82.09
	4	0.0266	8.819	495.1	4	0.0268	8.949	109.4	5	0.75	1.47	−77.90
	5	0.0222	7.331	555.4	5	0.0229	7.585	164.9	6	3.15	3.46	−70.31

* The UAVs used within each ALNS instance are the average value of 10 runs. Gap (%) = $\frac{\mathrm{ALNS}-\mathrm{MILP}}{\mathrm{MILP}}$ × 100%.

. CPU is the computational time taken to solve the problem. The last three columns in Table 4 are the percentage differences of ALNS versus MILP in terms of energy consumption, total flight distance, and computational time. The reason why ALNS needs more drones than MILP is that the time window is not fully utilized when calculating the delivery time and arrival time using (53). For example, when appending ${\mathbf{r}}_1$ to ${\mathbf{r}}_2$, the delivery time of ${\mathbf{r}}_2$ is possible to postpone backwards while satisfying the time window constraint. Nevertheless, ALNS does not take this scenario into account due to the challenges associated with its implementation.

It can be observed that the CPU time of MILP grows exponentially with the number of customers. For instance, the average CPU time increases from $144.2$ $\mathrm{s}$ to $666.9$ $\mathrm{s}$, up by $362.5$% when solving the 20-customer problem compared to the 15-customer problem. The average CPU time of the ALNS increases from $123.5$ $\mathrm{s}$ to $149.4$ $\mathrm{s}$, a growth of $21.3$%. Moreover, for the first 2 instances with 10 customers, ALNS finds the optimal solution. It can be obviously found from Table 4 that the differences in energy consumption and travel distance between ALNS and MILP are less than 10%, which reveals that ALNS is capable of finding near-optimal solutions in small scenarios.

5.3. Performance of ALNS in Large-Scale Instances

Furthermore, the ALNS algorithm is employed to solve large-scale instances.

Table 5. ALNS results for Set B instances with size 100.

Instance	Energy (Wh)	Distance (km)	CPU (s)	Trips *	UAVs *
c201	8.035	2.639	500.0	30	12.3
c202	8.020	2.639	633.6	37.2	14.2
c203	8.002	2.641	754.2	37	16.8
c204	8.007	2.644	870.5	37.1	21.1
Average	8.016	2.641	689.6	37.8	16.1

* The trips and UAVs used within each ALNS instance are the average value of 10 runs.

indicates the results of $SetB$ instances with 100 customers. Trips is the number of trips that drones execute. The ALNS algorithm can find feasible solutions for large instances within a limited time. However, due to its high spatial complexity, the MILP model cannot be solved. As shown in Table 5, the average computational time is $689.6$ $\mathrm{s}$, indicating the high computational efficiency of the proposed heuristic algorithm.

5.4. Comparison between Variable-Speed and Fixed-Speed

This section provides the energy savings achievable by enabling drones to fly at variable speeds. The MILP implementation is run with varying speeds and three fixed speeds (10/20/30 $\mathrm{m}$/$\mathrm{s}$) to generate solutions for the DRP–VFS.

Table 6. Comparison of energy consumption for the variable-speed case versus the fixed-speed cases (10, 20, and 30 m/s).

Instance	Energy Consumption (kWh)
	10 m/s	20 m/s	30 m/s	Variable-Speed (% desc. over 10/20/30 m/s)
1	0.0571	0.0309	0.0361	0.0306 (−46.39/−0.94/−15.29)
2	0.0495	0.0265	0.0308	0.0262 (−47.14/−1.15/−14.99)
3	0.0521	0.0283	0.0331	0.0281 (−46.09/−0.82/−15.23)
4	0.0496	0.0269	0.0314	0.0266 (−46.34/−1.18/−15.22)
5	0.0419	0.0224	0.0261	0.0222 (−47.06/−1.06/−14.93)

illustrates that the average total energy consumption can be reduced by up to 46.61% compared to the 10 $\mathrm{m}$/$\mathrm{s}$ case, 1.03% vs. the 20 $\mathrm{m}$/$\mathrm{s}$ case, and 15.13% vs. the 30 $\mathrm{m}$/$\mathrm{s}$ case. The drone speeds in variable speed cases are all located at maximum-range speed since this is the most energy-efficient speed for drones. The results in Table 6 demonstrate that variable flight speeds result in lower energy consumption than fixed speeds in most instances. This reduction in energy consumption has the potential to facilitate the use of smaller-capacity batteries, resulting in weight savings and possibly extension of the flight range. Conversely, employing an inappropriate flight speed across different routes may result in insufficient energy to serve customers situated farther away from the depot.

5.5. Sensitivity Analysis

This section provides a sensitivity analysis of our exact algorithms. We changed the capacity of a drone from 1.5 kg to 2 kg; the results of 15 customers in set A are shown in

Table 7. Result with different payload capacities.

Instance	Q = 1.5 kg			Q = 2 kg			Gap (%)
	Energy (kWh)	Distance (km)	UAVs	Energy (kWh)	Distance (km)	UAVs	Energy	Distance
1	0.0159	6.03	4	0.0147	5.47	2	−7.24	−9.29
2	0.0195	7.38	5	0.0169	6.28	3	−13.28	−14.91
3	0.0161	6.04	4	0.0139	5.08	4	−13.79	−15.89
4	0.0188	7.20	4	0.0166	6.19	3	−11.55	−14.03
5	0.0203	7.74	4	0.0172	6.38	4	−15.52	−17.57

. As can be seen from the table, increasing the UAV payload reduces energy consumption, flight distance, and the number of UAVs used since each UAV can carry more packages and serve more customers in one trip.

The UAV routes generated by the different capacity in instance 1 are shown in

Table 8. The schedules generated by different capacities for Set A with size 15.

Q = 1.5 kg		Q = 2 kg
Trips	Energy (kWh)	Trips	Energy (kWh)
$\left[\right[0,9,16],[0,10,3,5,16\left]\right]$	0.0046	$\left[\right[0,9,10,3,16],[0,8,6,16],[0,13,5,16\left]\right]$	0.0080
$\left[\right[0,15,16],[0,13,16],[0,8,16\left]\right]$	0.0039	$\left[\right[0,11,16],[0,15,16],[0,12,7,14,2,4,1,16\left]\right]$	0.0067
$\left[\right[0,12,7,14,16],[0,6,1,4,16\left]\right]$	0.0056
$\left[\right[0,11,2,16\left]\right]$	0.0017

. Node 0 represents the depot. The connected trips indicate that they are executed by the same UAV. As the drone capacity is increased from 1.5 kg to 2 kg, the number of drones required for delivery is reduced from 4 to 2. Taking trip $[0,9,10,3,16]$ as an example, the package weights for customers 9, 10, and 3 are 1.4 kg, 0.5 kg, and 0.1 kg, respectively. With a 1.5 kg drone capacity, the drone can only visit nine customers in a single trip. However, with a 2 kg capacity, the drone is capable of serving all three customers.

6. Conclusions and Future Work

This paper extends the multi-trip DRP with time windows to incorporate variable flight speeds. We proposed an MILP formulation for DRP–VFS that minimizes total cost while considering the energy consumption model, flight speed, payload weight, and drone reuse. To address practical instances, we proposed a three-phased ALNS heuristic algorithm. Numerical results indicate that optimizing drone speeds is an important consideration for drone delivery. Variable speed resulted in about 46% and 15% improvements in total energy consumption compared to fixed-speed 10 $\mathrm{m}$/$\mathrm{s}$ and 30 $\mathrm{m}$/$\mathrm{s}$ cases, respectively. Numerical experiments also revealed that the three-phased heuristic performed well in both solution quality and CPU time in small and large DRP–VFS instances. Logistics industry managers can utilize our findings for drone fleet management, route planning, and resource allocation. Furthermore, this research enhances the comprehension of optimizing energy efficiency in drone delivery systems, thereby fostering advancements in sustainable aerial transportation.

The limitation of our work is that we use a greedy algorithm to assign routes to drones, which does not fully leverage the time window to minimize the drone fleet size. A promising direction for enhancement could involve adopting a hierarchical optimization framework that simultaneously addresses route planning and drone allocation. Additionally, more destruction and repair operations can be applied to improve solution quality.

There are multiple opportunities for further research in this scope. For instance, exact solution methods can be explored to address the nonlinear energy function. To extend the flight range of drones, multiple depots or recharging stations might be investigated. The development of drone delivery strategies often faces the challenge of dealing with uncertainty, e.g., uncertain customer demand and travel time [50]. Robust optimization can be explored to address these uncertainty-related constraints. It is of great potential to take weather conditions as well as ground and low-altitude traffic congestion into account, which may affect the delivery times in drone-only or truck–drone tandem delivery. Additionally, ethical considerations in drone delivery systems, such as user privacy protection and noise pollution reduction, are of paramount importance and warrant further investigation.