Drone-Based Instant Delivery Hub-and-Spoke Network Optimization

: Drone-based transportation is emerging as a novel mode in city logistics, featuring first-mile pickup and last-mile instant delivery using drones and truck transshipment. A fundamental challenge involves coordinating merchants, drones, transshipment hubs, trucks, and consumer communities through the hub-and-spoke network (HSN). This study formulated the optimization problem for HSN to minimize logistics costs and loss of orders constrained by service time limits. The ε -constraint model, two evolutionary algorithms based on Non-dominated Sorting Genetic Algorithm II (NSGA-II) using permutation (EAp) and rand key-based (EAr) encoding/decoding schemes were devised to solve the bi-objective mathematical program. Three groups of twelve experiments were conducted using ideal datasets and datasets generated from Shenzhen city to validate the models and algorithms. Relaxing the logistics objective by 10% and subsequently minimizing the loss of orders can significantly reduce average unmet orders by 24.61%; when spokes were beyond 20, the ε -constraint model failed to achieve solutions within an acceptable time. While EAp and EAr demonstrated competence, EAr proved to be more competitive in computation time, hypervolume, spacing metric, and the number of non-dominated solutions on the Pareto fronts. Key parameters influencing the HSN solutions include drone and truck speeds, acceptable delivery times, and the processing and waiting time at hubs.


Introduction
Applying drones to urban logistics, particularly in the domain of instant delivery, has emerged as a technological trend, drawing extensive global attention and discourse [1].Various countries and companies have made significant strides in advancing drone technology and its applications.China continues to develop drone technologies, particularly achieving noteworthy milestones in urban logistics and instant delivery [2].Companies like Meituan.com,JD.com, and SF Express (sf-express.com)have undertaken substantial efforts in pilot projects for drone-based instant deliveries [3].Amazon and Google have invested in developing instant-delivery drone technology [4].European nations actively apply drone technologies in city logistics; e.g., France and the UK are conducting numerous pilot projects [5].Singapore is a global logistics and technology hub dedicated to advancing drone technologies in city logistics [6].
Large-scale applications of drones in city logistics meet challenges.First, nations must establish pertinent regulations and policies to ensure safe and compliant drone operations within city areas [7].Second, drone technologies must be mature enough to operate safely and reliably in urban environments while handling complex meteorological and environmental conditions [8].Implementing an effective air traffic management system for drones is crucial to coordinate and prevent collisions [9].Given that using drones raises privacy and security concerns, it is essential to take appropriate measures to safeguard personal information and urban infrastructure.Public acceptance also requires extensive societal outreach and education [10].Constraints and challenges encountered in the application of Instant delivery aims to significantly enhance the efficiency of cargo transportation and promptly respond to consumer demands by applying advanced technologies.It integrates the Internet of Things (IoT), artificial intelligence and machine learning (ML), automation and robotics, and big data analytics.It enables the real-time monitoring of the location and status of packages, demand forecasting, routing optimization, and the utilization of autonomous vehicles or robots for delivery tasks.Chinese e-commerce giants such as Alibaba and JD.com have conducted trials of autonomous delivery vehicles [3], and Amazon offers the Prime Now service for two-hour delivery while exploring the possibilities of drone delivery [4].In instant delivery, drones are considered a robust solution for addressing the last-mile delivery challenge due to their rapidness and flexibility.Drone applications extend beyond last-mile delivery to support emergency supplies transportation and delivery to remote areas.
Instant delivery is limited by the short-order engagement time ranges, e.g., one hour, so advanced technologies and decision-making methods are essential for saving time and speeding the delivery processes.Table 1 lists studies on the operations research of instant delivery, which differs from general delivery in the order engagement time limits [16].Most studies reviewed apply UAVs (Unmanned Aerial Vehicles) or drones to cooperate with logistics vehicles.The column "TD" (truck and drone) summarizes the transportation devices used.Three studies coordinate trucks and drones for efficient instant delivery.Although even complicated instant delivery problems can be formulated or transferred into mixed-integer linear programs (MILPs), they are challenging to solve by exact algorithms within acceptable times.Various heuristics can be applied to solve the problems.As shown at the end of Table 1, this study investigates the solution of using trucks and drones simultaneously and devises bi-objective MILPs that are solved by algorithms based on ε-constraint and NSGA-II.

Study
Problem Features TD Model Algorithm [16] Heterogeneous instant delivery order scheduling and routing problems.T MILP CG [17] Instant delivery by vehicles and drones.TD MILP ACO [18] An auction-based trading platform to enable procurement for instant delivery.T Game ML [19] A facility location and demand allocation problem for drone-based instant delivery.TD -HA [20] Online instant delivery with dynamic orders.T MILP GA [21] Time-dependent instant delivery considering cost, customer satisfaction, and traffic.T MILP GA, VNS [22] A VRP in an instant delivery peak period.

T MILP ACO
This study An HSN coordinating trucks for transshipments among hubs and drones for first-mile pickup and last-mile distribution.

Hub-and-Spoke Network Optimization
The HSN is a classic model for designing logistics and transportation systems.The network features a set of hubs connecting peripheral spokes, which are interconnected to utilize the economy of scale of logistics.HSN is widely applied in the logistics industry, especially in building efficient supply chain systems for retail and e-commerce [23].The aviation industry also adopts this concept by connecting different destinations through hub airports to enhance flight operational efficiency [24].The design of public transportation routes can similarly draw inspiration from the HSN concept to achieve efficient connections [12].The theoretical and technical challenges in designing and optimizing HSNs include addressing distinct features, developing faster solutions, and considering uncertain and dynamic demands, traffic, and network structures.
Many studies have investigated HSN optimization, as presented in Table 2.The HSNs have been applied in waterways [25], maritime [26], roads [27], railroad [28], and aviation transportation and logistics systems.Generally, the hubs are not fully connected to waterways and maritime transportation systems.In this study, the hub connections are constrained by order engagement time ranges.In the column "PF" (partial or complete connections among hubs), some studies consider the partial connections practically [25,[28][29][30].The HSN involves logistics costs, efficiency, and emissions, so it should be formulated as a multi-objective model.Although it can be formulated as MILPs, due to its complexities, various math-heuristics and metaheuristics have been developed in the literature.As elucidated in the last row in Table 2, we considered minimizing the logistics cost and order loss simultaneously in a bi-objective model, which is solved by ε-constraint and NSGA-II.

Drone-Based Facility Location Problems
The application of drone technology to logistics is gradually becoming widespread, bringing forth new features and challenges in facility location problems.These features consider the rational spatial utilization and safety requirements of drone takeoff and landing facilities, charging stations, and maintenance to ensure sustained and stable drone operations.Various new technologies and methods with drone-based logistics systems are applied to facility locations.Virtual reality and simulation technologies are used to simulate location selection scenarios virtually, evaluating the operational effectiveness of drones in different locations [35].Data analytics and predictive technologies help assess the demands and potential traffic at various locations, while autonomous navigation and environmental perception technologies ensure safe and effective drone flights in urban environments [12].However, limitations include short endurance [36], susceptibility to weather conditions, regulation challenges [7], and privacy protection.
Table 3 reviews pioneering studies on drone-based routing and location optimization in three aspects: problem features, model, and algorithm.Three classification criteria are further developed to study the problem features: FL (first-mile pickup or last-mile delivery), RN (routing or network optimization), and FD (fixed or dynamic drone launching and returning locations).In all reviewed studies, drones undertake the last-mile delivery tasks.Most studies concern the truck/vehicle and drone routing problem, and some consider the locations of drone launching and returning, while these locations depend on the truck routes.As for models, although some studies formulate the problems as MILPs, they also analyze the problems and devise models by scenario-based sensitivities.The above studies correlate well with methodology, while the large-scale drone application in urban distribution may challenge management, the environment, and safety.Due to the complexities of the truck-drone routing problem (TDRP) and location optimization models, exact algorithms (e.g., B&PC and BD) and various heuristic algorithms (e.g., ALNS, GA, and TS) contribute to obtaining optima or approximate solutions.As indicated in the last row of Table 3, this study was activated by drone-based urban instant delivery and aimed to optimize the HSN coordinating drones and trucks.In particular, the drones undertake the first-mile pickup from merchants to hubs and last-mile delivery from hubs to consumers, while the trucks serve the transshipment among hubs.Therefore, the drone launching and returning hubs incur a strategic decision.We formulated the problem as a bi-objective MILP, which was solved using ε-constraints and NSGA-II.

Problem Statement
As a Chinese technology retail company, Meituan (www.meituan.com,accessed on 4 June 2024) has launched an advanced drone-based instant delivery service in Shenzhen, primarily applied in scenarios such as tourist attractions, parks, community office buildings, and residential areas, providing consumers with marvelous, efficient, safe, and convenient takeaway ordering experiences.Shenzhen City is critical in developing drone technologies and applications in China and worldwide.The local governments set up positive technology, economy, and regulation environments to promote drone experiments and marketable applications.Meituan's drone-based instant delivery service began active exploration in 2017 and successfully initiated the first domestic industrial park drone-based instant delivery route in China in Shenzhen in 2021.By 2023, the service operated 17 routes in cities, including Shenzhen and Shanghai, completing over 184,000 orders and delivering efficient consumer services [42].
As a notable feature, drone-based instant delivery utilizes cutting-edge technologies, including navigation, obstacle avoidance, three-dimensional visual mapping, and routing control intelligent devices and technologies.Advanced technologies enable drones to use autonomous flight and airborne traffic management, enhancing flight safety and stability.Drone-based instant delivery effectively mitigates road congestion and environmental pollution [43,44], aligning with green and low-carbon development targets.
Despite the significant achievements, drone-based instant delivery still faces challenges and difficulties, including issues related to the city logistics system with drones, operating costs, technologies, regulations, and consumer behaviors.Addressing these challenges requires further in-depth exploration and refinement to adapt to market demands and enhance the quality and sustainability of the service.
To improve logistics operations management performances, we investigated the solutions based on HSN for cooperating with drone-based first-mile pickup and last-mile delivery.
Typically, an HSN consists of a set of spokes (S) and a minor set of hubs (H).The logistics flow among two given spokes, i, j, is denoted by W ij , and the distance is C ij .Indeed, the terrain will impact the distance between any two points and even the climate.Thus, we can further compute C ij considering real-world 3D trajectories, even considering constraints by airspace regulations and discretization.The HSN operator should select a set of spokes to be fully connected hubs to achieve the economy of scale of logistics operations among the hubs.Thus, the unit consolidation cost is defined from spokes to hubs, the transshipment cost among hubs, and the distribution cost from hubs to spokes as P C , P T , P D , generally, P C > P D > P T .Without a loss of generality and for simplification purposes, we created an ideal example, as presented in Figure 1, to explain the related concepts and entities.In Figure 1, the HSN has 16 spokes and four hubs.Here, each spoke is assigned to only one hub entitled single-allocation HSN.
lutions based on HSN for cooperating with drone-based first-mile pickup and last-mile delivery.
Typically, an HSN consists of a set of spokes () and a minor set of hubs ().The logistics flow among two given spokes, , , is denoted by  , and the distance is  .Indeed, the terrain will impact the distance between any two points and even the climate.Thus, we can further compute  considering real-world 3D trajectories, even considering constraints by airspace regulations and discretization.The HSN operator should select a set of spokes to be fully connected hubs to achieve the economy of scale of logistics operations among the hubs.Thus, the unit consolidation cost is defined from spokes to hubs, the transshipment cost among hubs, and the distribution cost from hubs to spokes as  ,  ,  , generally,    .Without a loss of generality and for simplification purposes, we created an ideal example, as presented in Figure 1, to explain the related concepts and entities.In Figure 1, the HSN has 16 spokes and four hubs.Here, each spoke is assigned to only one hub entitled single-allocation HSN.
As illustrated in Figure 1, drones pick up packages from spokes and send them to hubs.Then, the packages are gathered and transshipped among the hubs by trucks.Finally, some drones load the packages from the hubs and deliver them to the target spokes.In the above processes, the consumers will only raise orders within an instant order engagement time range (denoted by ), generally one hour in the urban catering and takeaway delivery scenarios.The drones can only take tasks within flying duration limits.The trucks and drones can only travel at given speeds (denoted by  , ).Under the above premises, the HSN hubs are not always connected entirely under the constraint of  .Thus, the hubs incur transshipment ranges.Considering this feature, the HSN should be sparse because the long connections among hubs will be eliminated.

Baseline Hub-and-Spoke Optimization Model
The parameters and decision variables are defined in the baseline HSN optimization model. is a set of spokes,  1,2,3, … . is the distance between two spokes, ,  ∈ As illustrated in Figure 1, drones pick up packages from spokes and send them to hubs.Then, the packages are gathered and transshipped among the hubs by trucks.Finally, some drones load the packages from the hubs and deliver them to the target spokes.In the above processes, the consumers will only raise orders within an instant order engagement time range (denoted by T order ), generally one hour in the urban catering and takeaway delivery scenarios.The drones can only take tasks within flying duration limits.The trucks and drones can only travel at given speeds (denoted by S truck , S drone ).Under the above premises, the HSN hubs are not always connected entirely under the constraint of T order .Thus, the hubs incur transshipment ranges.Considering this feature, the HSN should be sparse because the long connections among hubs will be eliminated.

Baseline Hub-and-Spoke Optimization Model
The parameters and decision variables are defined in the baseline HSN optimization model.S is a set of spokes, S = {1, 2, 3, . ..}. C ij is the distance between two spokes, i, j ∈ S, C ii = 0. P C , P D , P T are the costs of transporting a unit of packages for a unit of distance from spokes to hubs, among hubs, and from hubs to spokes.The cargo flow from a spoke i ∈ S to another j ∈ S is W ij .O i , D i represent the in-and out-flows of the spoke i ∈ S, namely, Two decision variables are defined: x ik and y ikh .When x ik = 1, the spoke i ∈ S is assigned to the hub k ∈ S and x kk = 1; then, k is a hub.y ikh is the cargo flow from the spoke i through the hub k and then the hub h, i, k, h ∈ S. The hub set is denoted by H, H ⊆ S. In literature, the cost of i ∈ S to be a hub is denoted by F i .Hub investment is a strategic problem.We introduced the number of hubs to be chosen, H, namely, H = |H|.
The above notations are summarized as follows.

Sets
S, H, H S is a set of nodes, namely, spokes and potential hubs; H is a set of hubs, and H is the number of hubs.

C ij
Distance from the node i ∈ S to j ∈ S, C ii = 0 for all i ∈ S; assume that the triangle inequality is satisfied, The consolidation (from spokes to hubs), transshipment (among hubs), and distribution (from hubs to spokes) cost discounts.
Cargo flow destinated to the node i ∈ S H Number of hubs.

Variables
x ik ∈ {0, 1} x ik = 1 if the spoke i is assigned to the hub k; otherwise, 0. x kk = 1 represents that k is chosen as a hub.
y ikh ≥ 0 The flow originated from the spoke i ∈ S, through the hubs k and h sequentially to the target spoke (i, k, h ∈ S).

[M1]minz
where subject to Drones 2024, 8, 247 ∑ h̸ =k x ik ∈ {0, 1}, ∀i, k The constraint (5) constrains that a spoke can be assigned to one and only one hub.In (6), a spoke can accept flows from spokes only when it is chosen as a hub.The number of hubs is given in (7).The flow originating from a spoke is determined by (8) and (9).The variable domains are given in (10) and (11).

Extended Bi-Objective Optimization Models
Unlike the general HSN solutions defined in [M1], the investigated instant logistics solution uses drones for first-mile pickup and last-mile delivery [45].The consumers raise takeaways orders under the consideration of the acceptable order engagement time range (T order ) between their locations and the merchants.Therefore, some orders may be unacceptable because the total logistics time exceeds T order .
For a given pair of spokes i, j, the total logistics time accomplishing the pickup, transshipment, and delivery is denoted by , which consists of five parts: drone-based pickup time, processing and waiting time at the consolidation hub, truck transshipment time from the consolidation hub to the distribution hub, the processing and waiting time at the distribution hub, and the drone-based delivery time.To simplify the formulation, we use A(i) to represent the hub assigned to the spoke i.If ∼ t order ij > T order , the order is likely to be lost because the consumer may not accept a later delivery.Indeed, the consumer raises the order to consider (13), and the takeaway ordering platforms must consider such criteria.
The drone-based logistics solution changes the components of time in the order engagement.Therefore, the decision-makers must improve the logistics network and operations to minimize the loss of orders.
To formulate the minimization of the loss of orders, we introduced additional variables besides ∼ t order ij .First, t tr ijkh represents the transshipment time by a truck from the hub k to the hub h in the engagement of the order from the spoke i to the spoke j. t tr ijkh can be determined by (14) when k and h are hubs of i and j.Second, t tr ij is the transshipment time by a truck, which is computed based on t tr ijkh , as calculated in (15).Third, a binary variable z ij is defined, indicating whether the order (i, j) will be lost or not, as denoted in (16).Finally, the loss of orders can be expressed in (17).
A revised model was then formulated as [M2].
The known data of the model can be represented by a vector (I), and the variables can be represented by X, as denoted by ( 18) and (19).I = S, W, C, P C , P D , P T , D, O, S drone , S truck , T tr , T order (18)

Solving Bi-Objective Models by ε-Constraints
As formulated in Section 4.2, [M2] is a bi-objective model, which can be solved based on ε-constraints First, we can optimize [M2] by minimizing the single objective z hsn , which can be denoted by [M2a], as defined in (20).We can use f [M2a] (•) to represent the solving process (21), where z hsn * , X * are the solution values of the variables.
[M2a]min z hsn ((1) − ( 17)) (20) Second, we can formulate a new model to consider the relaxation of z hsn by considering a new constraint with a relaxation coefficient ε ≥ 0, as denoted in (22).
[M2(ε)]min z loss ((1) − (17)); Third, by iterating the acceptable values of ε, we can obtain the pairs of z hsn , z loss to construct Pareto fronts between the two objectives, z hsn and z loss .The ε-constraint method, as formulated in (22), converts the bi-objective model [M2] into a computable model.For each value of ε, [M2] is solved to obtain the two objectives through the constraint (z hsn ≤ (1 + ε)z hsn * ) and objective (z loss ) in (22).When [M2] can be solved mathematically, it can provide a tight Pareto front.However, solving [M2] is time-consuming and does not apply to practical instances with more than 20 nodes (see Section 6.4.4).

Evaluating Solutions with Determined Hubs
Algorithm 1 evaluates solutions with given hubs, assigns hubs to spokes, and calculates the logistics cost z hsn and the order loss z loss .Step 2 Calculate logistics cost Calculate loss of orders z loss Step 4 Return z hsn , z loss Algorithm 1 can be denoted by f eval as follows: z hsn , z sum , x * ← f eval (I; hubs) 5.3.Evolutionary Algorithms Based on NSGA-II 5.3.1.Introducing NSGA-II-Based EAs NSGA-II addresses the multi-objective optimization problems (MOP).The algorithm has been proven effective in solving real-world MOPs in various domains, promoting diversity and convergence of Pareto-optimal solutions.Many MOP solution algorithms have been developed, while NSGA-II is widely used in literature and practice and is a typical base of most other MOP algorithms.NSGA-II ranks the population through nondominated sorting, calculates the crowding distance of individuals in the population to maintain diversity, and obtains an approximate solution when the termination condition is met [46].The basic flowchart of the NSGA-II algorithm is shown in Figure 2. The algorithm employs a fast, non-dominated sorting method, reducing the algorithm's computational complexity from O mN 3 to O mN 2 (m is the number of objective functions, and N is the population size), significantly decreasing the computation time [47].The introduced method for calculating crowding distance avoids the issue of manually specifying the sharing radius in fitness-sharing strategies [48].The elite strategy is adopted, merging parent and offspring individuals for non-dominated sorting, expanding the search space [46].Individuals with higher priority are selected to generate the next generation of parent populations.Within the same priority level, the binary tournament strategy is employed based on crowding distance to select high-quality individuals in the population [49], which ensures that excellent individuals have a greater chance of being retained.

Inputs
The number of hubs, H; A vector of indexed spokes, S; A vector of permutated spokes, chroms.

Outputs
Values of the two objectives, z hsn , z sum Because H, S are parts of the known data I of the HSN, we can use f p (chroms) for simplicity.

2.
Random key-based encoding/decoding scheme The random key-based encoding scheme uses a vector of 2H genes with values in [0, 1].These values are mapped to coordinates in the space determined by the coordinates of all spokes.Then, the spoke closest to each coordinate is set to a hub.Algorithm 4 gives the steps where X − , X + , Y − , Y + are the minimum and maximum of the coordinates of the spokes, calculated as Equation (23).

Inputs
The number of hubs, H; A vector of spokes, and its coordinates, S, X, Y; The ranges of spokes' coordinates, X − , X + , Y − , Y + ; A vector of random keys, chroms.

Outputs
Values of the two objectives, z hsn , z sum .

Steps Step 1
Initialize the set of hubs, hubs = ∅.
Step 2 Get the random keys x rk i , y rk i for each potential hub i.
Map the random keys into coordinates for each hub i.
For each i ∈ 1, 2, • • • , H : Find the nearest spoke to the coordinates x co i , y co i for the hub i.
Update the hub set: hubs ← hubs ∪ {s i } .
Step 5 Obtain the objective values by Algorithm 1.
Algorithm 4 can be denoted by f r (•): z hsn , z loss ← f r H, S; chroms .
Because H, S are parts of the known data I of the HSN, we can use f r (chroms) for simplicity.
The notated function f decode (•) is introduced to represent the above two algorithms, f p and f r , namely, decode ∈ {p, r}.

Evolutionary Operators
In NSGA-II, parent individuals are selected using the binary tournament method.The process can be described as four steps.First, two individuals from the current population will be randomly selected as competitors.Second, compare competitors based on nondominated relationships: if one individual dominates the other, choose the more dominant one.If two individuals have a non-dominated relationship, further compare their crowding distances.Third, select the winner based on the comparison results and add it to the breeding pool.Finally, repeat the above steps until the size of the breeding pool reaches the desired quantity.
The permutation-based encoding/coding scheme employs the partially matched crossover (PMX) operator [50] and the Inversion Mutation (IM) operator.PMX is a crossover operation used in genetic algorithms, typically applied to solve permutation-based problems.PMX exchanges partial genes between two parent individuals to generate offspring individuals.IM selects and reverses a substring to introduce mutation and develop a new individual.
The random key-based encoding/decoding scheme uses the simulated binary crossover (SBX) and polynomial mutation operators [51].SBX is a real-value crossover operator.It blends information from two parents to create offspring solutions by simulating the binary crossover process for real-value variables.SBX uses a B (beta) distribution to balance exploration and exploitation, and ensures that the offspring solutions are created within the feasible ranges of the variable spaces.The polynomial mutation operator generates parameters by polynomial distribution to control the magnitude of perturbations.

Datasets
To demonstrate and verify the devised models and solution algorithms, we developed an ideal dataset (as presented in Figure 1, entitled [IDEAL]) with 12 spokes and four hubs and a dataset generated using the geography map of Baoan District, Shenzhen City, China, considering that Baoan should be a leading district for developing drone-based urban logistics system (see Figure 3).The Baoan District provides the environment for applying drones to instant delivery.The dataset is entitled SaHbic.For example, S16H4i3 is the ith dataset with 16 spokes and four hubs.
The Baoan dataset was constructed using the following three steps.First, we obtained the geographical map from OpenStreetMap (www.openstreetmap.org,accessed on 4 June 2024).Second, we used the Baoan political polygon to extract the road network and catering positions from the map.Third, the spokes were randomly chosen from the merchant positions, and thus the datasets were generated.
We constructed the datasets based on actual data from Baoan District, Shenzhen City.The datasets are scalable; we can derive new ones from them by choosing nodes and connections.In the models and algorithms devised in Sections 4 and 5, we focus on the distinct features of coupling HSN and drone technologies in the formulations and computational strategies while they do not impose particular assumptions on the datasets.China, considering that Baoan should be a leading district for developing drone-based urban logistics system (see Figure 3).The Baoan District provides the environment for applying drones to instant delivery.The dataset is entitled .For example, 1643 is the ith dataset with 16 spokes and four hubs.
In [IDEAL], the coordinates of the nodes were regularly set to 0,1,2,3,4 .The distances among the nodes were calculated using Euclidean distance.Additional parameters were set to  4,    5,   1,  2.1, and  0.1.The Baoan dataset was constructed using the following three steps.First, we obtained the geographical map from OpenStreetMap (www.openstreetmap.org,accessed on 4 June 2024).Second, we used the Baoan political polygon to extract the road network and catering positions from the map.Third, the spokes were randomly chosen from the merchant positions, and thus the datasets were generated.

Shenzhen City
We constructed the datasets based on actual data from Baoan District, Shenzhen City.The datasets are scalable; we can derive new ones from them by choosing nodes and connections.In the models and algorithms devised in Sections 4 and 5, we focus on the distinct features of coupling HSN and drone technologies in the formulations and computational strategies while they do not impose particular assumptions on the datasets.

Assessment Metrics for Multi-Objective Optimization Solutions
We used three metrics, namely the number of non-dominated solutions (NS), hypervolume (HV), and Spacing, to assess the performance of multi-objective optimization algorithms.Each metric offers unique insights into different aspects of Pareto fronts, contributing to a comprehensive assessment of algorithmic effectiveness.
(1) The NS measures diversity, emphasizing the quantity and diversity of solutions on Pareto fronts [52].The higher the NS, the broader the exploration of the Pareto fronts, facilitating the search for global optimal solutions.
(2) The HV measures the volume occupied by the non-dominated set, depicting the spatial coverage of non-dominated solutions, underscoring the convergence of solutions on Pareto fronts [53].A larger HV is preferable in optimization problems with a minimization objective.The formula for calculating HV is formulated by Equation (24).The Lebesgue measure, denoted as   , is employed to quantify the volume of the non-dominated solution set, representing its extent in the objective space; || signifies the number of solutions in the non-dominated set;  represents the HV formed by the reference point and the ith solution in the set.

Assessment Metrics for Multi-Objective Optimization Solutions
We used three metrics, namely the number of non-dominated solutions (NS), hypervolume (HV), and Spacing, to assess the performance of multi-objective optimization algorithms.Each metric offers unique insights into different aspects of Pareto fronts, contributing to a comprehensive assessment of algorithmic effectiveness.
(1) The NS measures diversity, emphasizing the quantity and diversity of solutions on Pareto fronts [52].The higher the NS, the broader the exploration of the Pareto fronts, facilitating the search for global optimal solutions.
(2) The HV measures the volume occupied by the non-dominated set, depicting the spatial coverage of non-dominated solutions, underscoring the convergence of solutions on Pareto fronts [53].A larger HV is preferable in optimization problems with a minimization objective.The formula for calculating HV is formulated by Equation (24).The Lebesgue measure, denoted as δ(S), is employed to quantify the volume of the non-dominated solution set, representing its extent in the objective space; |S| signifies the number of solutions in the non-dominated set; v i represents the HV formed by the reference point and the ith solution in the set.
(3) The Spacing metric quantifies the dispersion of solutions, capturing the relative distances between solutions [54].Higher Spacing values imply a more favorable distribution of solutions, while lower values may indicate a concentration of the Pareto front solutions in specific local regions.Consequently, higher Spacing values are generally considered indicative of superior performance.The formula for calculating Spacing is as formulated by Equation (25), where d i represents the minimum distance from the ith solution to other solutions in the set P, |P| denotes the number of solutions in set P, and d signifies the mean of all d i .

Experimental Settings
In Table 4, we developed three groups of experiments to reveal the distinct features of the devised models and algorithms.In the first group, we compared the models and algorithms' computing performances and optimality capabilities and determined the recommended values of the critical parameters.In the second group, we investigated the parameter sensitivities and impacts on computing performances.We examined four typical managerial strategies or scenarios in the third group, considering overdue delivery, rush hours, drone delays, and hub busyness and exploration.

Results
As presented in Table 4 for the 12 experiments, the results with figures, tables, and explanations are given in the following sub-section.

Demonstration
The devised models and algorithms solve the dataset [IDEAL] and get the same results as Figure 4.This dataset is typically ideal, and its Pareto front has only one solution.In the solution, z hsn = 874 and z loss = 216; namely, there are 216 orders whose engagement times are beyond T order .For example, because T order = 2.1, the order (i = 13, j = 5) with direct distance 2 will not be met because the drone-based instant delivery distance contains five parts, 13 → 10 → 6 → 5 .The distance is beyond 2 because there is additional processing time at hubs 10 and 6.

Compare [M1] and [M2]
As studied in Section 4, [M1] is a baseline HSN optimization model, while the extended [M2] optimizes the logistics cost and order loss in the bi-objective program.As revealed in Table 5, [M2a] minimizes z hsn and so gets the same results as [M1].The model [M2(ε = 0.1)] relaxes the objective z hsn by 10% and then minimizes z loss .We can find that the unmet orders are significantly reduced by 24.61% on average.

Analyze the Hyperparameters of [EAp] and [EAr]
Table 6 presents the results of crossing experiments on the mutation probability (P m ) and the crossover probability (P c ).Here, P m and P c iterate five values, [0, 0.25, 0.5, 0.5, 1], and we calculate four metrics, namely computing time (CT), hypervolume (HV), Spacing, and the number of non-dominated solutions (NS).We cannot recommend concrete values based on the experimental results because four metrics are involved.However, we can judge that bigger values are more beneficial than minor ones.We set P m to 0.25 and P c to 1.00 in the following experiments.[EAp] uses individuals with genes of the same length as spokes and encodes them as permutations of spoke indices.[EAr] maps 2H and keys as coordinates to hubs.So, [EAr] uses fewer genes for solutions.In Table 7, the computing time of [EAr] is less than [EAp]; as for HV and Space, [EAr] is also competitive, while the NS values are not stable.In Figure 6, [EAr] can generate a better Pareto front than [EAp], although the competitiveness is minor.Comprehensively, [EAr] is chosen as the algorithm in the following experiments.

Sensitivities of Four Parameters
As studied in Section 4, the drone-based HSN will be affected much by the key parameters, including S truck , S drone , T order , and T tr .As shown in Table 8, the computing time (CT) is unstable for all the parameters and variations; the number of non-denominated solutions (NS) is stable and can obtain enough.The Spacing metrics also incur uncertainties while the trends are apparent.The impacts on HV are clear.Figure 7 gives five Pareto fronts (−10%, −5%, 0%, +5%, 10%) for each parameter's sensitivity analysis results.

Numbers of Hubs Impacting on the Performances
[EAr] uses 2H individuals in the encoding and decoding schemes so that the hubs will impact the solution performances.In Table 9, when increasing the number of hubs (NS), the computing time (CT) decreases, the HV and NS values also lower with a tendency, while the Space metrics are not stable.In Figure 8, from five to 10 hubs, the Pareto front moves to the corner fast.

Numbers of Spokes Impacting on the Performances
As studied above, the number of hubs can affect the HSN solution difficulty, while Table 10 indicates that the number of spokes is the most affected.The computing time (CT) drastically increases when the number of spokes (NS) increases from 10 to 100.In Spacing metrics, more spokes help find diversities of solutions on the Pareto fronts.

Study the Overdue Delivery Strategy
In drone-based instant delivery, the merchants should have a strong attitude to extend the acceptable order engagement time range (T order ) because it will enlarge the coverage of the transshipment hubs and the consumer group.However, they must consider the costs to extend the overdue delivery.Consumers may accept comparatively delayed delivery for three reasons: first, drone-based delivery is fantastic; second, drone-based delivery is much cleaner and contactless; third, it is at least a new choice.Table 11 presents the results of increasing T order , and Figure 9 depicts five Pareto fronts.From 0% to 5% and 5% to 10%, the Pareto front moves fast.6.4.10.Impacts of Urban Rush Hours on the Solutions Big cities generally have several rush hours, which will affect the truck speeds in the transshipment stage.

Impacts of Drone Delays on the Solutions
The drones may delay transportation because of air and various emergency conditions.Although the reasons may be diverse, the impacts on the logistics are mainly reflected by reduced drone speeds.Table 13 shows the results of gradually reducing drone speeds.Figure 11 further depicts the Pareto fronts.In the illustrated case, a slight delay (5%) may be acceptable, and a 10% delay is severe.

Impacts of Busy Hubs and Even Explosion Periods on the Solutions
In [M2], the busy hubs and explosion degrees are formulated by the parameter T tr , which depends on the efficiency of transshipment at the hubs.Table 14 presents the impacts of the four performance metrics by increasing T tr .When its value is tight, CT increases, HV drops, and NS drops, making it difficult to find solutions.Figure 12 indicates that bigger T tr values will drastically make more unmet orders.

Managerial Implications
Based on the results analyzed above for the devised models and algorithms, we can conclude that the following generations have managerial implications for drone-based instant delivery service providers and merchants.
(1) Drone-based pickup and delivery can utilize the economy of scale of transportation while additional interaction time is incurred, including the processing times at hubs, transshipment operations between drones and trucks, and drones and loading/unloading cabinets (Section 6.4).
(2) Introducing drones for urban instant pickup and delivery may prolong the order engagement time.Optimizing the logistics networks and tuning the parameters can help reduce costs and mitigate order losses (Section 6.4.9).
(3) HSN and extended HSN optimization, particularly when considering urban dronebased instant delivery, pose challenges for on-the-shelf mathematical program solvers.As examined in Section 6, mathematical programs can produce good Pareto fronts for smallscale instances, while [EAr] and [EAp] are competitive in solving large-scale instances and cannot ensure optimality.It is practical to develop heuristics and intelligent algorithms (Sections 6.4.4 and 6.4.5).
(4) More hubs will demand more computing resources and challenge the solution methods.More hubs could be practical in drone-based instant delivery, given that transshipment hubs should be more flexible than traditional city logistics (Sections 6.4.7 and 6.4.8).
(5) Practical applications of the devised models and algorithms must address largescale instances, e.g., with more than 100 spokes.So, fast solution algorithms with promising computing performance are essential (Sections 6.4.4 and 6.4.5).
(6) Multi-objective Pareto front analysis assesses the impacts of extending the order engagement time ranges on the solutions, which must reflect the costs.So, drone-based instant delivery service providers and merchants should prioritize balancing expenses and revenues (from Sections 6.4.7-6.4.12).(7) The Pareto front analysis of the drone delays can contribute to investing and backing up additional drones.Drone-based instant delivery service providers should carefully weigh the Pareto front improvement and drone investment costs (from Sections 6.4.7-6.4.12).
(8) Among all parameters and strategies, mitigating drone delays at hubs may impact the solution most, and addressing it is achievable through effective management, advanced technologies, and innovations (Section 6.4).

Conclusions
Drone-based instant delivery with simultaneous first and last-mile pickup is a challenging decision-making and optimization problem.Inspired by the Shenzhen City drone-based instant delivery case, this study investigated the hub-and-spoke network (HSN) optimization problem, considering logistics costs and order loss minimization as a bi-objective optimization problem.Extending the baseline HSN model, we devised a bi-objective mixed-integer linear program (MILP), solved by the ε-constraint method and two evolutionary algorithms using permutation (EAp) and rand keys (EAr) as encoding and decoding schemes.Using an ideal dataset and the datasets generated in the Shenzhen City scenarios, the solution methods are compared by experiments, the parameter sensitivities are analyzed numerically, and three groups and twelve experiments are conducted to demonstrate the problem features and validate the devised models and algorithms.
The examined drone-based logistics networks and operations optimization problems are exciting and challenging in formulation and algorithm development.Based on the problem and the methods devised, the following research directions should help enable the development of drone-based instant delivery.First, extending the solution algorithms by mathematic heuristics and advanced intelligent algorithms for large-scale practical instances is beneficial.For example, the Baoan District of Shenzhen City contains more than 20,000 takeaway merchants and 200,000 buildings.The existing algorithms based on NSGA-II can be adapted to other multi-objective optimization algorithms that may optimize computation performance and optimality.Further, we will try to find solutions to the multi-objective model by using methods that solve mathematical programs exactly.Second, the instant delivery scenarios are full of uncertainties and dynamics.We should extend the deterministic solutions to stochastic programs and robust optimization.We can even incorporate the potential economic and environmental impacts of widespread drone use in urban logistics into the models.Third, some cities are trying different logistics service modes.The layouts and traffic conditions of other cities may also impact network optimization and scheduling results, and thus, it is beneficial to develop dataset generation schemes.We can create the mode featured by drone-based pickup/delivery and truckbased transshipment to other creative modes, e.g., crowdsourcing and cloud logistics.

Figure 1 .
Figure 1.A demonstrative diagram of drone-based instant delivery.

Figure 1 .
Figure 1.A demonstrative diagram of drone-based instant delivery.

Algorithm 1 .
Evaluate an HSN solution determined by given hubs (eval) Inputs Data : I.A set of hubs, hubs.Outputs Logistics cost and loss of orders, z hsn , z loss .Steps Step 1 Assign spokes to hubs A(s) = arg min k∈hubs distance(s, k), ∀s ∈ S.

Figure 2 .Algorithm 3 .
Figure 2. A diagram of NSGA-II.5.3.2.An Evolutionary Algorithm Framework Based on NSGA-IIWe devised an EA framework (Algorithm 2) based on NSGA-II to obtain Pareto fronts as defined in [M2].The population size is set to P n , the maximum number of generations is P gen , the mutation probability is p m , and the crossover probability is p c .Based on the characteristics of the model, we encoded the decision variables and thus devised decoding schemes correspondingly denoted by f decode (•), as defined by Algorithms 3 and 4 in Section 5.3.3.

Figure 3 .
Figure 3. Shenzhen City and its Baoan District.

Figure 3 .
Figure 3. Shenzhen City and its Baoan District.

Figure 7 .
Figure 7. Pareto fronts of sensitivities of four parameters.(a) Truck speed; (b) Drone speeds; (c) Order time range; (d) Process time at hubs.

Figure 9 .
Figure 9. Pareto fronts of extending order time overdue.

Figure 11 .
Figure 11.Pareto fronts impacted by drone delays in transportation.

Table 1 .
Studies on instant delivery.

Table 2 .
Pioneering studies on hub-and-spoke network optimization.

Table 3 .
Related studies on drone-based routing and location optimization.
. Steps Step 1 Slice the first H genes in chroms in to s. s = chroms 1, . . ., H . Step 2 Map each index in s into the spoke as a hub, denoted by hubs.hubs = {S[i]|∀i ∈ s}.Step 3 Obtain the objective values by Algorithm 1. z hsn , z loss , x * ← f [A1] (I; hubs) .

Table 6 .
Tuning the mutation and crossover probabilities of [EAp] and [EAr].

Table 8 .
Sensitivity analysis of S truck , S drone , T order , and T tr .

Table 9 .
Impacts of hubs on the solutions.

Table 10 .
Results of datasets with different spokes on the solution performances.

Table 11 .
Impacts of overdue delivery on the solutions.

Table 12 .
Impacts of truck speeds on the solutions.

Table 13 .
Impacts of drone speeds on the solutions.

Table 14 .
Impacts of busy hubs on the solutions.