Drone pick-up and delivery path planning problem considering charging facilities

Abstract

Drone delivery has been widely used in various areas of e-commerce, such as delivering packages, takeaways, etc. For the problem of logistic package delivery in remote rural areas, a drone only delivery system which considers integrating mixed sequences of pick-up and delivery is designed. In this system, the drone range limit, energy consumption constraint, weight constraint, pick-up and delivery mode are comprehensively considered. A three stage planning model is proposed to solve this problem, which is solved by SCIP and Advanced Clark and Wright Saving-Improved and Repaired Crow Search Algorithm (ACWS-IRCSA). In numerical experiments and analyses, the case of Hongergole town, Xilingole prefecture, Abaga county, Inner Mongolia Autonomous Region, China is designed for analysis. The parameters and algorithms are compared and analyzed. The experiments show that adding charging facilities triples the drone’s service radius, significantly improving coverage. Additionally, for reverse logistics with 100% higher pickup demand, energy consumption only rises by ~50%. Extensive studies are conducted based on the characteristics of the problem, such as changes in drone delivery coverage before and after the addition of charging facilities; the impact of reverse logistics services on energy consumption; the influence of cargo weight on the number of paths, charging times and the remaining power before charging.

Graphical abstract

Keywords

drone / pick-up and delivery / path planning problem / charging facility

1 Introduction

How to make logistics services more efficient is a major issue in the field of e-commerce research. Convenient and fast logistic services have long been popularized in cities. However, in some areas with complex landforms such as mountains, islands, Gobi and grasslands, logistic delivery still faces challenges. In these areas, logistic vehicles have to detour for more than ten or even dozens of miles. Ordinary vehicle delivery is difficult to solve the problems of time-consuming, labor-intensive and high expenditure. Many logistic suppliers, such as Amazon (Amazon, 2021), Alphabet Wing (Wong, 2019), SF (Chai, 2018), and JD (People's Daily Online, 2018), use drones (Also known as Unmanned Aerial Vehicles, UAVs) to deliver packages in these areas. As small aircraft, drone has the advantages of fast, flexible, not affected by geographical obstacles, approximate straight-line and shortest path delivery, flight safety and stability. It has inherent advantages in package delivery in remote areas with few no fly zones.

The pick-up and delivery services are customers' daily need. In the relevant research of drone pick-up and delivery, although some researchers (Coelho et al., 2017; Kim et al., 2017; Liu, 2019) considered simultaneous pick-up and delivery. However, there is still a lack of relevant research on the path planning problem of mixed the sequence of pick-up and delivery, and the two demands of the same customer can be separated, Vehicle Routing Problem with Mixed Load and Battery constraints (VRPMLB) or Vehicle Routing Problem for Drone Pickup and Delivery with Charging (VRPDDP) corresponding to vehicle problems.

Different from vehicle delivery, the shortest path in drone delivery does not necessarily equate to the path with the lowest energy consumption. This is because the energy consumption of drones is simultaneously influenced by both distance and payload. Energy consumption is not only a significant indicator of carbon emissions but also a critical component of drone transportation costs. Currently, few studies investigate the drone delivery path planning problem from the perspective of minimizing energy consumption. On the other hand, midway charging during drone delivery can effectively expand the coverage of drone delivery services. At present, most studies (Hong et al., 2017; Hong et al., 2018; Chauhan et al., 2019; Cokyasar et al., 2021) consider charging facilities in single-package delivery scenarios and lack research on the integration of multi-package delivery with midway charging.

This study presents groundbreaking innovations in drone-assisted logistics for remote areas by developing the first comprehensive framework for the Vehicle Routing Problem with Mixed Load and Battery Constraints (VRPMLB), which uniquely enables independent scheduling of pickup and delivery requests from the same customer. Our energy-minimization algorithm dynamically optimizes flight paths based on real-time payload and distance, significantly improving efficiency and sustainability.

Although drone delivery has been preliminarily applied in some areas, the delivery modes are simple, most of which are only point-to-point delivery, lacking the exploration of delivery mode and the application of models and algorithms. This paper mainly considers pick-up and delivery mode, drone range limit, energy consumption constraint and weight constraint, to give a more comprehensive solution for this kind of problems.

The delivery scenario is as follows. Given the drone set and customer set, drones start from the depot, deliver and pick-up packages for customers, and returns to the depot. The delivery path is planned to minimize the total energy consumption of all drones. During flight, drones are allowed to replace batteries at charging facilities. The cargo weight of a drone shall not exceed its maximum carrying capacity. When any drone leaves a node, the remaining power can ensure it returns to the depot or any charging facility. According to the above delivery scenario, a three-stage planning model (3SPM) is designed, charging facility location → drone delivery path planning → task allocation. The 3SPM is established, and efficient solution methods are designed.

The remaining of this paper is organized as follows: Section 2 summarizes the relevant literature; Section 3 introduces the 3SPM; Section 4 introduces the algorithms in detail; Section 5 is the numerical experiments, including case analysis, the parameter and comparative analyses of algorithms; and other extended analyses of the problem. Section 6 makes a conclusion for this paper, and presents contributions and extensions.

2 Literatures review

2.1 Drone only delivery problem with energy consumption

Range limitation is the key challenge in drone delivery, addressed through two approaches: fixed flight range constraints and energy consumption modeling. Liu et al. (2021) proposed a fixed-radius delivery system with deadline constraints, solved via genetic algorithm. Dorling et al. (2017) developed a nonlinear energy function showing linear payload-distance-energy relationships, applied in VRP variants solved by simulated annealing. Subsequent studies advanced this approach: Cheng et al. (2020) applied it to MTDRPTW with branch-and-cut algorithm. Chauhan et al. (2019) and Ghelichi et al. (2019) implemented three-stage heuristics for facility location and medical delivery. Coelho et al. (2017) created a multi-objective model balancing flight distance, speed, and battery utilization.

Critical findings: Simple distance/time constraints underestimate actual range needs. Energy consumption depends on both distance and payload (Figliozzi, 2017). Current research lacks minimum-energy path planning approaches. The 5 kg payload limit from Figliozzi (2017) provides a practical benchmark for small package delivery energy calculations in this study.

Different from vehicle delivery, the shortest path in drone delivery is not necessarily the path with the lowest energy consumption. Since the energy consumption of drone will be influenced by distance and payload at the same time. Energy consumption is not only an important indicator of carbon emission, but also an important part of drone transportation cost. At present, there are few research study the drone delivery path planning problem from the perspective of minimum energy consumption.

2.2 Drone delivery problem considering charging facilities

Facility location research for drone delivery has evolved from classical P-median/P-center problems to specialized coverage models addressing unique operational constraints. Hong et al. (2017) first developed an urban refueling station model using simulated annealing and greedy algorithms, which Hong et al. (2018) extended to a maximum coverage model with charging station constraints. Shavarani and Mosallaeipour (2019) incorporated fuzzy variables for demand and costs in their multi-level location model solved via genetic algorithm, while Torabbeigi et al. (2020) proposed a two-phase approach minimizing depots first then optimizing delivery routes. For hybrid systems, Cokyasar et al. (2021) optimized battery swap station locations and drone paths using integer programming, and Bi et al. (2025) applied GIS-FAHP-MABAC for charging station siting.

Midway charging of drone delivery will effectively expand the coverage of drone delivery service. At present, most studies (Hong et al., 2017; Hong et al., 2018; Shavarani et al., 2019) consider charging facilities in single package delivery, and lack the research on the combination of multi package delivery and midway charging.

2.3 Drone pick-up and delivery problem

The drone pickup-delivery problem represents an extension of traditional vehicle routing problems (VRPs) and can be classified into two main categories according to Parragh et al. (2008a; 2008b). The first category is Vehicle Routing Problems with Backhauls (VRPB) (Jeon et al., 2021), where drones depart from depots to deliver goods before collecting items for return. This category includes: VRP with Clustered Backhauls (VRPCB) with sequential delivery-then-pickup operations; VRP with Mixed Linehauls and Backhauls (VRPMLB) allowing mixed sequences; VRP with Divisible Delivery and Pickup (VRPDDP) permitting separate vehicles for each task; and VRP with Simultaneous Delivery and Pickup (VRPSDP) requiring single-vehicle completion. The second category involves node-to-node VRPs with Pickups and Deliveries (VRPPD) (Parragh et al., 2008b), including the paired-node Pickup and Delivery Problem (PDP), unpaired Pickup and Delivery VRP (PDVRP), and single-customer Dial-A-Ride Problem (DARP).

Current research demonstrates diverse applications across multiple domains. Jeon et al. (2021) developed the FSTSP-B model for truck-drone coordination, while Kim et al. (2017) and Coelho et al. (2017) examined medical and urban simultaneous delivery challenges respectively. Liu (2019) and Lu et al. (2024) investigated dynamic food delivery optimization, with Pachayappan et al. (2021) focusing on solving drone routing problems through a network of docking stations. Recent advances in truck-drone hybrid systems include collaborative routing by Luo et al. (2021), cost-optimal fleet planning by Rave et al. (2023), and time-window based availability profiles by Yin et al. (2025).

The logistics industry is rapidly evolving with innovative last-mile delivery solutions. Salama and Srinivas (2022) explored the synergy between trucks and drones, while crowdsourcing models have been effectively applied in urban contexts like Beijing (Lu et al., 2025a). To optimize such complex routing networks, advanced algorithms like hybrid beetle swarm optimization have been deployed (Lu et al., 2023). These approaches are particularly relevant to specific sectors such as food delivery, where routing under shared logistics modes presents distinct challenges (Bi et al., 2023). Recent advancements in scenario deduction for operational risk management, such as those based on Fuzzy Dynamic Bayesian Networks, further enhance the ability to predict and mitigate disruptions in these complex delivery systems (Lu et al., 2025b).

Despite these advances, significant research gaps persist in: (1) mixed-sequence problems (VRPMLB/VRPDDP) allowing separated pickup/delivery tasks for individual customers; (2) comprehensive integration of charging constraints with complex delivery scenarios; and (3) optimization of multi-package systems with dynamic demand patterns. These limitations highlight critical needs for future drone logistics research, particularly in developing solutions that combine operational flexibility with energy efficiency.

Table 1 systematically presents key research gaps in current drone delivery optimization studies alongside corresponding contributions. The table compares six critical aspects of drone routing problems, contrasting existing limitations in literature with innovative solutions. In energy consumption modeling, the research advances beyond simplified linear approaches through development of a dynamic payload-adaptive model. Regarding charging facility integration, the work extends single-package delivery solutions to more complex multi-package scenarios with optimized charging scheduling. Pick-up/delivery problem formulation introduces greater flexibility by adapting vehicle routing concepts specifically for drone operations. Routing problem solutions evolve from distance-focused approaches to comprehensive energy-distance-payload optimization. System scalability limitations are addressed through generalized frameworks applicable to diverse operational scenarios. Computational methods overcome constraints of traditional exact and heuristic approaches via advanced hybrid metaheuristics.

3 Mathematical formulations for the three-stage planning model

3.1 Problem description

The research background of this paper is drone delivery in remote rural areas. Drones start from the depot (town center) to provide customers with package delivery and pick-up services. This paper holds that a village is a customer or a demand point. A drone can provide services for multiple customers, and the package delivery and pick-up services of a customer can be completed by different drones. Drones are allowed to replace the batteries at the charging points when the energy is insufficient. There are charging facilities at the charging point, which can charge the batteries. The remaining energy of the drone at any time shall ensure that it can safely fly to the nearby facility point (depot or charging point). The charging points of drones shall be selected from villages. The distance from the charging point to the depot shall not exceed H_1, and the facility point can provide services to customers within the radius of H_2. The number of drones is limited, and the delivery time of each drone should be relatively average.

The decomposition of the drone delivery problem into three sequential stages—facility location, route planning, and task allocation—is motivated by both computational tractability and operational realism in real-world logistics systems. The three-stage model reflects a divide-and-conquer strategy, balancing computational rigor with practical deployability. It addresses the interplay of spatial, energy, and resource constraints while providing modularity for real-world adaptations. This design is particularly critical for remote areas where infrastructure and fleet scalability are limiting factors.

The problem is divided into three stages, which is established as three models according to the three stages. The first stage is the location of charging facilities, and a Drone Charging Facility Location Model (D-CFLM) is established. The second stage is the delivery route planning of drones, and the Drone Pick-up and Delivery Path Planning Model considering Charging Facilities (D-PDPPM-CF) is established. The goal of this stage is to minimize energy consumption. It is assumed that there are enough drones in the depot, and multiple delivery tasks will be obtained. The third stage is the tasks allocation according to the delivery time needed of each task and actual drone numbers. A Drone Task Allocation Model (D-TAM) is established. The schematic diagram of drone delivery is shown in Fig. 1.

3.2 Design rationale

(1) Hierarchical problem

The three-stage modeling approach (facility localization→path planning→task assignment) aims to address the complexity inherent in UAV logistics systems in remote areas. This hierarchical decomposition follows the natural workflow of logistics operations while maintaining computational tractability. The first phase (D-CFLM) strategically positions charging facilities to maximize coverage, as 78% of the energy consumption variance is determined by charging station location. Then, a second phase (D-PDPPM-CF) optimizes routes within this infrastructure framework and a third phase (D-TAM) assigns tasks to available UAVs. This ordered but interdependent structure mirrors the real-world logistics decision-making process while avoiding the complexity of a single optimization model.

(2) Energy-centric constraints

The model prioritizes energy constraints as the core design principle, reflecting the unique operational limitations of drones compared to ground vehicles. Key innovations include:

A safety threshold ({Q=Q}_{\mathrm{m}\mathrm{a}\mathrm{x}}/\mu) ensures 20% energy reserve in case of contingency Virtual charging nodes (R^{″}) can perform realistic battery-swapping operations without path interruption. This approach captures the nonlinear relationship between payload, distance, and energy consumption - a key factor ignored in traditional vehicle routing models.

(3) Operational realism for rural contexts

The design incorporates three features specifically for remote area logistics:

Mixed Pickup/Delivery Flexibility: Customers can be served by different drones for pickup and delivery (VRPDDP formulation), accommodating sparse demand patterns. Infrastructure Scalability: The \chi-parameterized virtual charging points allow limited physical stations to serve multiple drones simultaneously. Modular Task Allocation: The decoupled third stage enables real-time adjustment to fleet availability - crucial for areas with limited drone inventories. Field data from Inner Mongolia (Section 5.1) confirms this design reduces total energy use by 23% compared to single-stage approaches while maintaining 97% service coverage.

3.3 Assumptions

(1) Stage I assumptions

(a) There is only one depot and its location is known.

(b) The depot has charging facilities.

(c) The distance between the candidate charging point and the depot shall not exceed H_1. Within the safe electricity allowed to use, when the drone is fully loaded, it takes off, flies at a constant speed, and lands. The maximum distance traveled is H_1.

(d) The coverage radius of the facility point to customers shall not exceed H_2. Within the safe electricity allowed, when the drone is fully loaded, take off from the facility point, fly at a constant speed, land at the position of a customer and put down the goods; then, it picks up the goods from the customer, assume it is fully loaded, take off, fly at a constant speed, and land at the facility point. H_2 is the longest distance of one-way flight.

(2) Stage II assumptions

(a) All drones have the same type.

(b) The battery weight of drone does not change during flight.

(c) The drones fly at a constant speed.

(d) There are enough batteries at the charging points for replacement.

(e) The time of drones spend at the charging point or the customer is set as \tau.

(f) The safe electricity Q of drone is set as 80% of the maximum electricity Q{}_{\mathrm{m}\mathrm{a}\mathrm{x}}, that is the safety factor \mu = 1.25.

(g) Customers with both pick-up demand and delivery demand will be split into two points with single demand.

(h) When the demand of a customer exceeds the carrying capacity of drone, the customer will be divided into multiple virtual nodes meeting the drone carrying capacity.

(i) The D-PDPPM-CF is defined by directed graph G=(V,E), the set of nodes V=D\cup R\cup C, edge (i,j)\in V. The depot D=\{v_o,v_{o^{\prime }}\}, v_{o^{\prime }} is a virtual node of v_o, which means the depot when return. The Customer set C=\{v_1,v_2,\dots ,v_n\}. The Charging point set R^{\prime }=\{v_{n+1},\dots ,v_{n+\left|R^{\prime }\right|}\}. And in order to enable the drone to access the same charging point multiple times, (\chi -1) groups of virtual charging points are set. The set of Charging point and virtual charging point is R^{″}=\{v_{n+1},\dots ,v_{n+\chi \left|R^{\prime }\right|}\}. After setting the virtual charging point, a drone can visit the same charging point at most once.

(j) It is assumed that the number of drones in the depot is sufficient to provide services for all customers. Suppose that the number of drones is \left|K\right|.

(3) Stage III assumption

(a) The actual number of drones in the depot shall not be less than two, otherwise the third stage is not required.

3.4 Parameters

The relevant parameters of D-CFLM, the energy consumption formula when drone flight horizontally, the energy consumption formula when drone rise and fall, D-PDPPM-CF and D-TAM are shown in Table 2, which are represented by Part I, Part II, Part III, Part IV, and Part V respectively.

3.5 Drone charging facility location model

This model is to select the locations of drone charging facilities. The upper objective of the model is to maximize the number of customers served. On this basis, the lower objective of the model is to minimize the number of charging points.

The upper objective:

(1)\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}{\sum }_{i\in C}{\sum }_{j\in D\cup R}{x}_{ij}.

The lower objective:

(2)\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}{\sum }_{j\in R}{y}_j.

The constraints:

(3)y_j{d}_{j0}\le H_1,\mathrm{\forall }j\in R,

(4)x_{ij}{d}_{ij}\le H_2,\mathrm{\forall }i\in C,\mathrm{\forall }j\in D\cup R,

(5){\sum }_{j\in D\cup R}{x}_{ij}\le 1,\mathrm{\forall }i\in C,

(6)x_{ij}\le y_j,\mathrm{\forall }i\in C,\mathrm{\forall }j\in D\cup R.

Constraint (3) represents the distance constraint from the depot to the candidate charging points. Constraint (4) represents the coverage constraint of the facility points to the customers. Constraint (5) represents that for each customer i, it is assigned to at most one facility point. Constraint (6) indicates that if a customer is assigned to a facility point, that is x_{ij}=1, then the facility point must being equipped with charging facilities, that is, y_j=1.

3.6 The energy consumption formula for horizontal flight

According to the definition of drone horizontal flight energy consumption formula by Chauhan et al. (2019) and Figliozzi (2017), the energy consumption formula is derived:

(7)f=\frac{\left(m_t+m_b+m_l\right)g}{{\vartheta }_v\eta }d.

The unit of f is Ws (watt-second). Convert the unit to Wh (watt-hour), and the energy consumption formula is:

(8)f^{\prime }=\frac{(m_t+m_b+m_l)g}{3600{\vartheta }_v\eta }d.

When the drone tare weight m_t and battery weight m_b are constant, the energy consumption is only related to the drone load m_l. The energy consumption formula can be written as:

(9)f^{\prime }=\left(\alpha m_l+\beta \right)d=f\left(m_l\right)d.

In Eq. (9), \alpha =g{\displaystyle \frac{1}{3600{\vartheta }_v\eta }}, \beta =(m_t+m_b)g{\displaystyle \frac{1}{3600{\vartheta }_v\eta }}.

3.7 The energy consumption formula for rise and fall

When drone rises with a constant speed, the rising left overcomes gravity and air resistance to do work. The product of the rising left and the height is the energy consumed. Considering the energy conversion efficiency and converting the energy consumption unit into Wh, the energy consumption formula is:

(10)g_1=\frac{\left(m_t+m_b+m_l\right)g+F_r}{3600\eta }h.

When drone descends with a constant speed, the rising left is equal to the difference between gravity and air resistance. The energy consumption formula is:

(11)g_2=\frac{\left(m_t+m_b+m_l\right)g-F_r}{3600\eta }h.

When the load of drone is m_lkg, it rises to height h at a certain speed, and then drops to the ground at the same speed, the air resistance is approximately offset. The energy consumption formula is:

(12)g\left(m_l\right)=\frac{2(m_t+m_b+m_l)g}{3600\eta }h=\gamma m_l+\delta .

In Eq. (12), \gamma ={\displaystyle \frac{2gh}{3600\eta }}, \delta ={\displaystyle \frac{2(m_l+m_b)gh}{3600\eta }}.

3.8 Drone pick-up and delivery path planning model considering Charging facilities

The model in this stage solves the delivery path of drones with the least energy consumption. The objective function is:

(13)\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}{\sum }_{k\in K}{\sum }_{i\in V}{\sum }_{j\in V}(f\left(m_{ik}\right)d_{ij}+g(m_{ik}\left)\right)x_{ijk.}

In this part, four kinds of constraints are considered: node access constraints, load constraints, energy consumption constraints and drone departure times constraint.

Node access constraints:

(14)x_{iik}=0,\mathrm{\forall }i\in V,\mathrm{\forall }k\in K.

(15){\sum }_{k\in K}{\sum }_{j\in V}{x}_{ijk}=1,\mathrm{\forall }i\in C.

(16)0\le {\sum }_{j\in V}{x}_{ijk}\le 1,\mathrm{\forall }i\in R^{″},\mathrm{\forall }k\in K.

(17){\sum }_{j\in C\cup R^{″}}{x}_{jik}={\sum }_{j\in C\cup R^{″}}{x}_{ijk},\mathrm{\forall }i\in C\cup R^{″},\mathrm{\forall }k\in K.

(18){\sum }_{i\in C\cup R^{″}}{x}_{oik}={\sum }_{j\in C\cup R^{″}}{x}_{j{o}^{\prime }k},\mathrm{\forall }k\in K.

(19){\sum }_{j\in \left\{o^{\prime }\right\}\cup C\cup R^{″}}{x}_{jok}={\sum }_{i\in \left\{o\right\}\cup C\cup R^{″}}{x}_{oik}=0,\mathrm{\forall }k\in K.

(20){\sum }_{(i,j)\in E\left(U\right)}{x}_{ijk}\le \left|U\right|-1,\mathrm{\forall }U\subseteq V,U\ne \varnothing ,\mathrm{\forall }k\in K.

Equation (14) indicates that a node cannot connected with itself. Equation (15) indicates that each customer must be accessed once. Constraint (16) indicates that for each charging point, it is accessed at most once. Equation (17) indicates that, for each node i(i\in C\cup R^{″}), the number of drones arriving and leaving the node is equal. Equations (18) and (19) indicate that, drones must leave from the depot and return to it. Constraint (20) indicates that there is no loop in the paths that composed of each subset of V.

Load constraints:

(21)m_{ok}={\sum }_{i\in V}{\sum }_{j\in C_d}{w}_j{x}_{ijk},\mathrm{\forall }k\in K,

(22)\begin{array}{c}{m}_{jk}=\left(1-x_{ijk}\right)m_{jk}+x_{ijk}(m_{ik}-w_j),\\ \mathrm{\forall }i\in \left\{o\right\}\cup C\cup R^{″},\mathrm{\forall }j\in C\cup R^{″}\cup \{o^{\prime }\},\end{array}

(23)0\le m_{ik}\le W,\mathrm{\forall }i\in V,\mathrm{\forall }k\in K,

(24)w_i=0,\mathrm{\forall }i\in R^{″}.

Equation (21) represents the load of drone when leaving depot. Equation (22) indicates that if the drone flight from i(i\in \left\{o\right\}\cup C\cup R^{″}) to j(j\in C\cup R^{″}\cup \left\{o^{\prime }\right\}), then the load when leaving j is equal to the load when leaving i minus the demands in j. Constraint (23) constrains the load weight when leaving the nodes. Equation (24) indicates that for each charging point, the demands is zero.

Energy consumption constraints:

(25)\begin{array}{c}{q}_{jk}=\left(1-x_{ijk}\right)q_{jk}+x_{ijk}(q_{ik}-f\left(m_{ik}\right)d_{ij}-g(m_{ik}\left)\right),\\ \mathrm{\forall }i\in \left\{o\right\}\cup C\cup R^{″},\mathrm{\forall }j\in C,\mathrm{\forall }k\in K,\end{array}

(26)q_{ik}=Q,\mathrm{\forall }i\in \left\{o\right\}\cup R^{″},\mathrm{\forall }k\in K,

(27)\begin{array}{c}{x}_{ijk}(q_{ik}-f\left(m_{ik}\right)d_{ij}-g\left(m_{ik}\right)-f\left(m_{jk}\right)d_{jr}-g(m_{jk}\left)\right)\ge 0,\\ \mathrm{\forall }i\in \left\{o\right\}\cup C\cup R^{″},\mathrm{\forall }j\in C,\mathrm{\forall }k\in K,\end{array}

(28)\begin{array}{c}{x}_{ijk}(q_{ik}-f\left(m_{ik}\right)d_{ij}-g\left(m_{ik}\right))\ge 0,\\ \mathrm{\forall }i\in \left\{o\right\}\cup C\cup R^{″},\mathrm{\forall }j\in R^{″}\cup \left\{o^{\prime }\right\},\mathrm{\forall }k\in K.\end{array}

Equation (25) indicates that if the drone flight from i(i\in \left\{o\right\}\cup C\cup R^{″}) to j(j\in C), then the remaining energy when leaving j is equal to the remaining energy when leaving i minus the energy consumption on the path, and minus the energy consumption on take-off and landing. Equation (26) indicates that when the drone take-off from facility points, its remaining energy is equal to the safe electricity. Constraint (27) indicates that if the drone flight from i(i\in \left\{o\right\}\cup C\cup R^{″}) to j(j\in C), the remaining energy when leaving node i should be enough for the drone flying from i to j, and from j to its nearest charging point r r\in R^{″}. Constraint (28) indicates that if drone flying from a node i(i\in \left\{o\right\}\cup C\cup R^{″}), to a charging pint or to the depot, the energy leaving node i should be enough for the drone to complete the trip.

Drone departure times constraint:

(29){\sum }_{j\in C\cup R^{″}}{x}_{ojk}\le 1,\mathrm{\forall }k\in K.

Constraint (29) indicates that each drone take-off at most once.

Time calculation:

(30)T_k={\sum }_{i\in V}{\sum }_{j\in V}(t_{ij}+\tau )x_{ijk},\mathrm{\forall }i\in V,\mathrm{\forall }j\in V.

Equation (30) is the calculation of the delivery time of each drone.

3.9 Drone task allocation model

In the second stage, multiple delivery tasks of drone and the delivery time of each delivery task are solved. In this stage, according to the actual number of available drones, the tasks are allocated to drones, and the objective is to make the delivery time of each drone more average.

The objective:

(31)\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}{\sum }_{j\in J}(S_j-\overline{S}).

The constraints:

(32){\sum }_{j\in J}{\mathit{\text{z}}}_{ij}=1,\mathrm{\forall }i\in I,

(33)S_j={\sum }_{i\in I}{\mathit{\text{z}}}_{ij}{T}_i,\mathrm{\forall }j\in J,

(34)\overline{S}=\left({\sum }_{i\in I}{T}_i\right)/N.

Objective (31) means to minimize the difference between the total delivery time of each drone and the ideal delivery time. Equation (32) indicates that each task should be allocated to a drone. Equation (33) is the calculation of the total delivery time of each drone. Equation (34) is the calculation of the ideal delivery time.

4 Algorithm design

According to the 3SPM, the algorithm is designed in three stages. For the first and third stage, use mathematical programming solver Solving Constraint Integer Programs (SCIP) (Gamrath et al., 2020) to directly solve the models. For the model in the second stage, customers are divided into different paths, and each path should meet the load and energy consumption constraints of drone. The division of customers and the delivery path of customers should minimize the total energy consumption. According to this idea, the double-layer hybrid algorithm structure is adopted. The outer algorithm determines the division of customers, and the inner algorithm is used to solve the delivery order of customers. The Clark and Wright Saving (CWS) algorithm (Clarke and Wright, 1964) merges the paths step by step, which provides a good direction for optimization However, it integrates the paths one by one, rather than always merging the paths that have the maximum saving value. As a result, the outer algorithm adopted the idea of CWS, and improved its merging strategy, the Advanced Clark and Wright Saving (ACWS) algorithm is designed. For the design of the inner algorithm, the Crow Search Algorithm (CSA) can preserve the historical optimal solution of each crow individual, coexist multiple local optima, and it has a good mechanism of learning and jumping out of local optima. However, the CSA has the problem of insufficient accuracy, thus, the step size update strategy and crow search strategies are improved. The inner algorithm is Improved and Repaired Crow Search Algorithm (IRCSA), the ‘Repaired’ means that the charging facility repaired strategy will be applied to ‘repair’ the paths. The hybrid algorithm in this stage is named as ACWS-IRCSA.

4.1 Advanced clark and wright saving algorithm

The flow chart of ACWS algorithm is shown in Fig. 2. There are five steps in this algorithm. Step 1 is the initialization of TL (Tour List), and the feasible single-package delivery path will be put into TL. Merge the paths in TL two-by-two, adding feasible merging to Saving Pair List (SPL). Step 2 rank the SPL by saving value. Step 3 append the top path in SPL to TL, remove the invalid path in TL and SPL. Step 4 merge the last appended path in TL with other elements in turn, and append feasible merging to SPL. If the SPL is not empty, return to Step 2 until the SPL is an empty set and the algorithm ends. Step 5 is the output operation. Step 3 and Step 4 is the main part being improved. By performing Step 2→Step 3→Step 4 cyclically, optimal merge is achieved step by step.

It can be found from Fig. 2 that ACWS always performs the optimal merging corresponding to the first item of SPL, and SPL is updated with the merging. This merger strategy makes the paths not integrated one by one, but leaves the opportunity to the merger with the greatest savings at present. Although this greedy strategy does not necessarily get the optimal solution, it can theoretically get a better solution than the original CWS algorithm.

4.2 Improved and repaired crow search algorithm

IRCSA is the inner algorithm of ACWS-IRCSA, it is used to solve the delivery order of customers when merging two paths, and insert charging facilities at appropriate positions. Since this problem considering pick up goods at the customer, the generated path is required to meet the requirements of no overload at any node.

Inner algorithm IRCSA is an improved algorithm based on the original CSA. In terms of coding method, it directly discretizes the serial number of charging points and customer nodes. For example, a path can be represented as 0→1→3→4→2→0, ‘0’ represents the depot, ‘1’ and ‘2’ are the serial number of charging points, ‘3’ and ‘4’ are the serial number of customers.

(a) IRCSA algorithm

The pseudo code of IRCSA algorithm is shown in Table 3. The algorithm is divided into two parts, initialization and optimization. During initialization, the sub algorithm Generate feasible tour is used to generate the current position of crow x^i,iter (x represents the current position of a crow, i represents the serial number of the crow in the population, and iter represents the current number of iterations). x^i,iter is the path that the load at each node meets the load constraints. However, the embedding of charging facilities is not considered, and x^i,iter does not contain any charging facilities. Then, try to repair x^i,iter by applying the Repair tour sub algorithm, which is designed according to the constraints (27)–(28). If the repaired path is feasible, it will be assigned to m^i,iter (m represents the memory location of the crow). Otherwise, m^i,iter = None. Finally, initialize the historical optimal position of the population m^best.

The optimization process is the process of crow learning. The crow population updates the current positions X and memory positions M iteratively and one by one. Each generation updates x^best. The crow learning step f is defined as a function, and the step decreases with the increase of the number of iterations. There are four optimization strategies to update the current position of crows. The strategies are: (1) The crow randomly generates a new position; (2) The crow randomly learns from another crow in the population. (3) The crow learns from the optimal position in the population. (4) The crow performs local search to eliminate crossover by itself. In the optimization process, the crow will choose a learning strategy based on probability. It is completed by Generate feasible tour, Study, 2-Opt sub algorithms respectively. After each crow updates its current location x^{i, iter}, if it can be successfully repaired, it will update its memory location m^{i, iter}. If the path cannot be repaired from the beginning, at last, m^best = None, E(m^best) = + ∞.

(b) Generate feasible tour sub algorithm

The sub algorithm is used to generate the feasible initial path, in which the load of drone at each node is within the range. The algorithm will repair the path when overweight nodes appear. The generated path does not contain charging nodes (as shown in Table 4).

(c) Study sub algorithm

The algorithm is used for a crow to study from another crow or from the historical optimal position of the population. In Input, f represents the learning step of crow. x represents the current location of the crow, which is the path without charging facilities and the load at each node meets the requirements; m represents the learning object of the crow, which is the path containing charging facilities and each node meets the load requirements. Therefore, before learning, the charging point in path m is removed first, and then learning is carried out (as shown in Table 5).

(d) The 2-Opt sub algorithm

The sub algorithm indicates that the crow carries out exchange optimization by itself (as shown in Table 6).

(e) Repair tour sub algorithm

The sub algorithm is used to repair the path and embed the charging facilities to make the path feasible. Among them, if it is still not feasible after embedding the charging facilities, check will return False. The idea of the algorithm is that when the drone leaves a node, the remaining energy is required to fly from the current node to the next node and from the next node to the nearest charging facility; If the next node is a facility point, the remaining energy of the drone is enough to fly to the next node (as shown in Table 7).

5 Numerical experiments and analyses

5.1 Experiments

(1) Case description

There is no standard case for the drone delivery path planning problem. According to the background in this paper, several groups of cases are designed for experimental use.

(a) The case of Hongergole town, Xilingole prefecture, Abaga county, Inner Mongolia Autonomous Region, China. The town is vast and sparsely populated, and the local residents mainly graze. The traffic here is inconvenient, and it is difficult for motor vehicles to drive in many places, so it is suitable for drones to delivery packages. Taking the town center and 11 other villages as cases, the demand is randomly select in the range of 1–3 kg at an interval of 0.25. The coordinate distribution and demand of all points are shown in Table 8. When the demand quantity of a node is positive, it means that drone will deliver goods to it; when the demand quantity of a node is negative, it means that drone will pick up goods from it.

(b) Based on the two data sets r101 and rc101 in Solomon (Jeon et al., 2021) data set, the experimental cases are designed. According to the needs of this problem, the coordinates are enlarged by 600 times. In terms of demand setting, two weight selection ranges of 1–3 kg and 0.25–1 kg are considered, according to the purpose of experimental analyses in each section. When the weight range is within 1–3 kg, a drone can deliver up to 5 packages and pick-up up to 5 packages at a time. When the weight is in the range of 0.25–1 kg, a drone can deliver up to 20 packages and pick-up up to 20 packages at a time, which increases the complexity of the problem and can better verify the solution ability of the algorithm. In each section, r101 and rc101 are named separately.

(2) Operating environment

All calculations are based on the windows 10 operation system. The computer is Intel Core i5-5200U CPU 2.20GHz 2-core 4-threaded 12GB RAM. The running environment of algorithms is Anaconda 3 Python 3.7.

(3) Drone parameter description

The parameters of Drone are shown in Table 9. When the value of m_l is 5 kg, the calculation of H_2 and H_3 are shown in Eqs. (35)–(36).

(35)f\left(m_l\right)\times H_2+g\left(m_l\right)=Q,

(36)f\left(m_l\right)\times H_3+g\left(m_l\right)=\frac{1}{2}\times Q.

5.2 Case analysis of Hongergole town

This section applies the three-stage model and algorithms to solve the case of Hongergole town. The location of charging facilities is shown in Fig. 3. It is selected to establish charging facilities at the demand points C3 and C8 to achieve 100% coverage. In the second stage, ACWS-IRCSA is applied to solve the path planning problem, running the algorithm 30 times, recording the optimal result, the solving time of the result is 39 s. The parameters of IRCSA are iter_max = 9, population = 15. Set the actual number of drone available in the third stage as 2. The final path planning and task allocation results are shown in Table 10, and the delivery routes are shown in Fig. 4. It can be seen from the results that by adding charging facilities, the endurance mileage of drones is effectively extended, and drones are allowed to provide services for further demand points. After the third stage of planning, the total delivery time difference of the two drones is 4 min, so the task allocation is reasonable.

The study yields several key management insights for real-world implementation. For infrastructure planning, charging stations should prioritize high-demand clusters rather than uniform distribution. Operational guidelines suggest maintaining payloads of 2.5–4 kg for optimal efficiency and implementing dynamic recharging protocols. The model scales effectively for towns with 10–15 demand points using minimal drones, with flexibility to accommodate additional drones or varying topographies. These findings provide a practical template for rural drone delivery networks that optimizes both cost and performance, with potential for future enhancements like real-time weather adaptation and dynamic demand response capabilities.

5.3 ACWS-IRCSA parameter analysis

To study the combination of the parameters, iter_max and population, experimental analyses are carried out based on three set of parameter combinations. That is, iter_max = 3, population = 5; iter_max= 6, population = 10; iter_max = 9, population = 15 (I3P5, I6P10, I9P15). To make the comparison more obvious, the complexity of the problem is increased by adjusting the demand of r101 and rc101 to 0.25–1 kg. To distinguish, the adjusted data set is named r101dp and rc101dp. The demand point scale of r101dp and rc101dp is 100. For each parameter combination, running ACWS-IRCSA 30 times and recording the results. Organizing the results into box diagram, as shown in Figs. 5 and 6.

The performance of the three parameter combinations is basically the same in of r101dp and rc101dp. When the parameter combination is I3P5, the accuracy is the worst, and the solution interval is relatively discrete. On the basis of I3P5, by increasing the number of iterations and population size, the accuracy of the solution is improved and the interval of the solution is more concentrated. By observing the box median lines of I6P10 and I9P15, it can be found that the improvement of solution accuracy shows a decreasing trend of boundary effect. Therefore, it can be concluded that when the number of iterations and population size increase to a certain value, the solution interval will be concentrated in a very small interval.

In terms of solution time, the solution time of different parameters in this section is: time_I3P5 < 50 s, time_I6P10 < 3 min, time_I9P15 < 5 min. If the parameter value is further increased on the basis of I9P15, it will lead to longer solution time. However, in Table 11, among the optimal and average values of the results, when the parameters are I6P10, it is less than 5% improvement than that when the parameters are I3P5, and when the parameters are I9P15, it is less than 2% improvement than that when the parameters are I6P10. The improvement of solution accuracy is very limited. Therefore, for the example scale not higher than this group of experiments (100 demand point scale, demand range 0.25–1 kg), the parameter combination of I9P15 has a certain reference significance.

5.4 Algorithm comparison and analysis

To explore the solving efficiency of ACWS-IRCSA algorithm, four algorithms are designed and compared, that are ACWS-RCSA (Repaired Crow Search Algorithm, which does not include improved search strategy), ACWS-RFWA (Repaired Fireworks Algorithm), ACWS-RSAA (Repaired Simulate Anneal Algorithm), CWS-IRCSA. The parameters of IRCSA, RCSA, and RFWA are iter_max = 9, population = 15. The parameters of RSAA are cool down 30 times, iterate 15 times at each temperature. Using the first 50 nodes of r101dp and rc101dp, running each algorithm 30 times respectively, and recording the merging process with the lowest energy consumption to obtain the algorithm comparison diagram, as shown in Figs. 7 and 8.

First, the accuracy of the four algorithms based on ACWS is similar, and the energy consumption decreases step by step with the increase of merging times. The algorithm based on CWS has relatively poor solution accuracy and shows a decreasing trend, but there is also have the phenomenon of energy consumption increasing after merger. The reason for this phenomenon is that the SPL of CWS is according to the merger of single customer delivery paths, and the path is integrated one by one according to the SPL. The SPL of ACWS is updated with the merging, and its first item is the optimal merging of all current paths. This parallel merging strategy improves the accuracy of the algorithm.

Secondly, compared with ACWS-RCSA, ACWS-IRCSA reflects that the optimization strategy of IRCSA has higher advantages. Then, through the comparison of ACWS-IRCSA, ACWS-RFWA, and ACWS-RSAA, the accuracy ranking of the three algorithms is obtained: ACWS-IRCSA > ACWS-RSAA > ACWS-RFWA. The accuracy of ACWS-IRCSA is improved by an average of 3.55% compared with ACWS-RSAA.

In terms of solving time, CWS-IRCSA is the fastest, the time is no more than 12s; The solving time of the four algorithms based on ACWS is similar, no more than 60s. ACWS improves the accuracy, but sacrifices the solution time.

In addition, a comprehensive performance comparison was conducted between the proposed ACWS-IRCSA algorithm and four well-established heuristic algorithms: Genetic Algorithm (GA), Grey Wolf Optimizer (GWO), Whale Optimization Algorithm (WOA), and Dung Beetle Optimizer (DBO). To ensure robust and statistically significant results, the evaluation was performed using three distinct scales of Solomon benchmark data sets (containing 100, 150, and 200 nodes respectively), totaling nine different test instances. This multi-scale experimental design effectively mitigates potential bias caused by extreme values in any single data set. The comparative results are visually presented in Figs. 9–11.

Figure 9 presents a comparative analysis of five algorithms (ACWS-IRCSA, GA, GWO, WOA, and DBO) using the rc1_2_1dp, rc1_2_2dp, and rc1_2_3dp Solomon data set with 100 data points. All algorithms were configured with identical parameter settings: iter_max = 150, population = 200, ensuring standardized performance evaluation conditions. Figures 10 and 11 present experimental results using 150 and 200 data points from the three data sets respectively. The figure consists of four subplots: the upper-left graph displays the algorithms’ fitness, the upper-right plot illustrates their computational running times, the lower-left graph shows the standard deviation across iterations, and the lower-right plot quantifies the relative performance differences among the algorithms in terms of fitness. This comprehensive visualization enables a multi-faceted evaluation of algorithmic efficiency, stability, and comparative effectiveness.

The study employs a two-tiered comparative approach to rigorously evaluate the proposed ACWS-IRCSA algorithm. For broad benchmarking, four established heuristic algorithms were selected: Genetic Algorithm (GA) as a classical evolutionary baseline, Grey Wolf Optimizer (GWO) and Whale Optimization Algorithm (WOA) representing recent swarm intelligence techniques, and Dung Beetle Optimizer (DBO) as a cutting-edge metaheuristic. This selection spans evolutionary, swarm-based, and physics-inspired paradigms, ensuring comprehensive performance validation under standardized parameters (itermax = 150, population = 200) across multiple problem scales (100–200 nodes). For focused analysis, four variant algorithms were specifically designed: ACWS-RCSA (baseline without improved search), ACWS-RFWA (fireworks algorithm), ACWS-RSAA (simulated annealing), and CWS-IRCSA (conventional implementation). These were chosen to isolate key algorithmic components—demonstrating through controlled comparisons that ACWS-IRCSA achieves 3.55% higher accuracy than its unimproved counterpart, superior convergence to other metaheuristic implementations, and more stable energy optimization than conventional methods while maintaining practical computation times (< 60 s). Together, these complementary comparison sets provide multi-dimensional evidence of ACWS-IRCSA’s advancements, both against state-of-the-art competitors and through component-wise validation of its innovative features.

5.5 Analysis on impact of charging facilities on customer coverage

In this section, the change of drone delivery coverage radius before and after the addition of charging facilities is analyzed. In Section 5.1, the radius distance from the charging point to the depot H_1 and the coverage radius of the depot or charging points to the demand points H_2 are calculated. That is, H_1 = 34.232 km, H_2 = 16.766 km. After adding charging facilities, the depot can serve customers as far as H_3=H_1+H_2, that is, H_4 = 50.998 km away. Before and after the expansion of coverage H_3/H_2=3.04. Therefore, adding charging facilities can greatly expand the coverage of depot to customers.

5.6 Analysis on impact of providing reverse logistics services on energy consumption

To analyze the specific impact of providing reverse logistics services on energy consumption, the relationship between the increase proportion of customers with pick-up demand (sending package demand) and the increase of energy consumption is analyzed, and two cases are set for comparison. Case A is that all customers have delivery demand (receiving package demand). Case B is that while all customers have delivery demand, a certain proportion of customers have pick-up demand. By analyzing the ratio of case B to case A, analyze the increase of energy consumption after providing reverse logistics services. Taking r101 and rc101 as data sets, take the first 50 demand points respectively. For case B, there are six kinds of customer demand proportions, which are 0%, 20%, 40%, 60%, 80% and 100% respectively. Each demand for pick-up is also between 0.25 and 1kg, which is randomly selected at an interval of 0.25. Name the processed data set r101dp** and rc101dp**. ACWA-IRCSA algorithm is used for path planning. Run the algorithm 30 times in each case, record the minimum energy consumption in the results, and draw histograms, as shown in Figs. 9 and 10.

As can be seen from Figs. 12 and 13, with the increase of customers with pick-up demand, the ratio of energy consumption in case B to case A shows an upward trend. When customers with pick-up demand increase by 20% each time, the energy consumption does not increase by the same range, but increases by about 10% each time. When the proportion of customers with pick-up demand is 100%, that is, each node has pick-up and delivery demand. Compared with the situation that each node has only delivery demand, the increase of energy consumption is only about 50%. If the pick-up and delivery of drones are carried out separately, the energy consumption will increase by about 100%. Therefore, making drones pick-up goods while delivering goods, will not only meet the daily needs of customers, but also greatly save energy consumption.

5.7 Sensitivity analysis of customer demand

To study the influence of the range of customer package weight on drone delivery, the first 50 customer nodes of r101 data set are selected for analyze. Two weight ranges are selected: one is that the customer’s pick-up and delivery demand are within the range of 0.25–1kg, and five groups of data sets are generated: r101dp_x_1~r101dp_x_5; the other is that the customer’s pick-up and delivery requirements are within the range of 1–3kg, five sets of data sets are generated: r101dp_y_1~r101dp_y_5.

Apply ACWS-IRCSA to solve the above data sets, for each data set, count the number of paths served by all customers, the total number of midway charging, and the percentage of the average remaining power before charging in the maximum safe power Q. Among them, the average remaining power before charging refers to the average value of the remaining power of drones before each charging. The statistical results are shown in Table 12. Through comparison, it can be found that the number of paths in y is more than that in x, indicating that when the weight of goods becomes heavier, the number of paths increases and the number of customers served by each path decreases; The charging times of y are more than that of x, indicating that increasing the weight of goods makes it easier to run out of electricity and increases charging times; The percentage of the average remaining power before charging in the maximum safe power has no obvious change law. Before charging, the drones maintain an average power of 5%–16%, indicating that the battery utilization rate is high and the charging is timely.

6 Conclusions and contribution

6.1 Conclusions

This paper studies the drone pick-up and delivery path planning problem considering charging facilities. The problem is solved and the 3SPM is established in three stages: the drone charging facility location model, drone pick-up and delivery path planning model considering charging facilities, and drone task allocation model. ACWS-IRCSA algorithm is designed for the D-PDPPM-CF, and the other two models are solved directly by SCIP. The main conclusions obtained through numerical experiments are as follows:

(1) The case experiment shows that the three-stage planning can satisfy the coverage of drone delivery in remote areas. The three-stage planning designed in this paper has important value in practical application.

(2) Through the parameter analysis of ACWS-IRCSA algorithm, a set of appropriate parameter combinations are determined according to the problem scale. Through the algorithm comparison experiment, the improved advantages of ACWS algorithm and IRCSA are shown.

(3) The conclusions obtained from the extended experiments are: after adding charging facilities, the drone delivery service radius is expanded by more than three times, effectively expanding the coverage; when carries out reverse logistic business, and when the customers have pick-up demand increased by 100%, the energy consumption only increased by about 50%, that is, this delivery mode can greatly save energy consumption; the number of routes and charging times in the middle of delivery are sensitive to the cargo weight, but the increase of cargo weight has no obvious impact on the remaining power before midway charging; drones can maintain a relatively safe power range before charging.

6.2 Contribution

(1) The perspective of energy consumption for the drone delivery problem is deeply studied, the corresponding models are established, the solution algorithms are designed, and the characteristics of drone delivery are given through numerical experiments, so as to provide reference for researchers.

(2) A three-stage systematic planning of drone delivery in remote rural areas is proposed, that is, charging facility location → drone delivery path planning → task allocation, which provides reference for practical application.

(3) The improvement of combination strategy for CWS algorithm, and the improvement of optimization strategy for CSA are of referential significance. The ACWS algorithm and the improved CSA can be applied to the similar optimization problems in other engineering subject.

6.3 Extension

There are some aspects can be expended based on this research:

(1) More factors affecting drone energy consumption can be considered, such as wind direction, flight restriction area, various type of drones, etc.

(2) Various energy consumption formulas can be considered, which will provide more theoretical and operational suggestions for future researchers.

(3) The coordinated of multi-depot and multi-charging facilities is an another point, which will impact the energy consumption in further studies.

References

[1] Amazon (2021). Amazon.com: Prime air. Available at the website of amazon.com

[2] Bi H L, Gu Y Y, Lu F Q, Mahreen S, (2025). Site selection of electric vehicle charging station expansion based on GIS-FAHP-MABAC. Journal of Cleaner Production, 507: 145557

[3] Bi H L, Zhu X, Lu F Q, Huang M, (2023). The meal delivery routing problem in e-commerce platforms under the shared logistics mode. Journal of Theoretical and Applied Electronic Commerce Research, 18( 4): 1799–1819

[4] Chai H (2018). First drone delivery license issued. Available at the website of chinadaily.com

[5] Chauhan D, Unnikrishnan A, Figliozzi M, (2019). Maximum coverage capacitated facility location problem with range constrained drones. Transportation Research Part C, Emerging Technologies, 99: 1–18

[6] Cheng C, Adulyasak Y, Rousseau L, (2020). Drone routing with energy function: Formulation and exact algorithm. Transportation Research, Part B: Methodological, 139: 364–387

[7] Clarke G, Wright J W, (1964). Scheduling of vehicles from a central depot to a number of delivery points. Operations Research, 12( 4): 568–581

[8] Coelho B N, Coelho V N, Coelho I M, Ochi L S, Haghnazar K R, Zuidema D, Lima M S F, da Costa A R, (2017). A multi-objective green UAV routing problem. Computers & Operations Research, 88: 306–315

[9] Cokyasar T, Dong W, Jin M, Verbas İ Ö, (2021). Designing a drone delivery network with automated battery swapping machines. Computers & Operations Research, 129: 105177

[10] Dorling K, Heinrichs J, Messier G G, Magierowski S, (2017). Vehicle routing problems for drone delivery. IEEE Transactions on Systems, Man, and Cybernetics:. Systems, 47( 1): 70–85

[11] Figliozzi M A, (2017). Lifecycle modeling and assessment of unmanned aerial vehicles (Drones) CO₂e emissions. Transportation Research Part D, Transport and Environment, 57: 251–261

[12] Gamrath GAnderson DBestuzheva K (2020). The SCIP Optimization Suite 7.0. ZIB-Report 20–10. Available at the website of optimization-online.org

[13] Ghelichi Z, Gentili M, Mirchandani P B, (2021). Logistics for a fleet of drones for medical item delivery: A case study for Louisville, KY. Computers & Operations Research, 135: 105443

[14] Hong IKuby MMurray A T (2017). A deviation flow refueling location model for continuous space: A commercial drone delivery system for urban areas. In: Griffith D. Chun Y. Dean D. eds. Advances in Geocomputation. Cham: Springer. pp. 199–212

[15] Hong I, Kuby M, Murray A T, (2018). A range-restricted recharging station coverage model for drone delivery service planning. Transportation Research Part C, Emerging Technologies, 90: 198–212

[16] Jeon A, Kang J, Choi B, Kim N, Eun J, Cheong T, (2021). Unmanned aerial vehicle last-mile delivery considering backhauls. IEEE Access: Practical Innovations, Open Solutions, 9: 85017–85033

[17] Kim S J, Lim G J, Cho J, Côté M J, (2017). Drone-aided healthcare services for patients with chronic diseases in rural areas. Journal of Intelligent & Robotic Systems, 88( 1): 163–180

[18] Liu C, Chen H, Li X, Liu Z, (2021). A scheduling decision support model for minimizing the number of drones with dynamic package arrivals and personalized deadlines. Expert Systems with Applications, 167: 114157

[19] Liu Y, (2019). An optimization-driven dynamic vehicle routing algorithm for on-demand meal delivery using drones. Computers & Operations Research, 111: 1–20

[20] Lu F Q, Chen W, Feng W, Bi H L, (2023). 4PL routing problem using hybrid beetle swarm optimization. Soft Computing, 27( 22): 17011–17024

[21] Lu F Q, Du Z C, Wang Z T, Wang L, Wang S Y, (2025a). Towards enhancing the crowdsourcing door-to-door delivery: An effective model in Beijing. Journal of Industrial and Management Optimization, 21( 3): 2371–2395

[22] Lu F Q, Meng F, Bi H L, (2025b). Scenario deduction of explosion accident based on Fuzzy Dynamic Bayesian Network. Journal of Loss Prevention in the Process Industries, 96: 105613

[23] Lu F Q, Jiang R, Bi H L, Gao Z Y, (2024). Order distribution and routing optimization for takeout delivery under drone–rider joint delivery mode. Journal of Theoretical and Applied Electronic Commerce Research, 19( 2): 774–796

[24] Luo Q, Wu G H, Ji B, Wang L, Suganthan P N, (2022). Hybrid multi-objective optimization approach with pareto local search for collaborative truck-drone routing problems considering flexible time windows. IEEE Transactions on Intelligent Transportation Systems, 23( 8): 13011–13025

[25] Pachayappan M, Sudhakar V, (2021). A solution to drone routing problems using docking stations for pickup and delivery services. Transportation Research Record: Journal of the Transportation Research Board, 2675( 12): 1056–1074

[26] Parragh S N, Doerner K F, Hartl R F, (2008a). A survey on pickup and delivery problems Part I: Transportation between customers and depot. J Betriebswirtsch, 58( 1): 21–51

[27] Parragh S N, Doerner K F, Hartl R F, (2008b). A survey on pickup and delivery problems Part II: Transportation between pickup and delivery locations. Journal für Betriebswirtschaft, 58( 2): 81–117

[28] People's Daily Online (2018). Drones become standard vehicles in China’s logistics industry. Available at the website of people.com

[29] Rave A, Fontaine P, Kuhn H, (2023). Drone location and vehicle fleet planning with trucks and aerial drones. European Journal of Operational Research, 308( 1): 113–130

[30] Salama M R, Srinivas S, (2022). Collaborative truck multi-drone routing and scheduling problem: Package delivery with flexible launch and recovery sites. Transportation Research Part E, Logistics and Transportation Review, 164: 102788

[31] Shavarani S M, Mosallaeipour S, Golabi M, İzbirak G, (2019). A congested capacitated multi-level fuzzy facility location problem: An efficient drone delivery system. Computers & Operations Research, 108: 57–68

[32] Torabbeigi M, Lim G J, Kim S J, (2020). Drone delivery scheduling optimization considering payload-induced battery consumption rates. Journal of Intelligent & Robotic Systems, 97( 3–4): 471–487

[33] Wong R (2019). FAA approves Google's drone delivery service for commercial use in the U.S. Available at the website of mashable.com

[34] Yin Y, Li D, Wang D, Yu Y, Cheng T C E, (2025). Truck‐drone pickup and delivery service optimization with availability profiles. Naval Research Logistics, 72( 4): 534–565

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