A Stochastic Drone-Scheduling Problem with Uncertain Energy Consumption

Abstract

In this paper, we present a stochastic drone-scheduling problem where the energy consumption of drones between any two nodes is uncertain. Considering uncertain energy consumption as opposed to deterministic energy consumption can effectively enhance the safety of drone flights. To address this issue, we developed a two-stage stochastic programming model with recourse cost, and we employed a fixed-sample sampling strategy based on Monte Carlo simulation to characterize uncertain variables, followed by the design of an efficient variable neighborhood search algorithm to solve the model. Case study results indicate the superiority of our algorithm over genetic algorithms. Additionally, a comparison between deterministic and stochastic models suggests that considering the uncertainty in energy consumption can significantly enhance the average returns of unmanned aerial vehicle scheduling systems.

1. Introduction

The traditional logistics industry has been consistently grappling with three major challenges: high logistic costs, particularly labor costs; increased traffic congestion; and the exacerbation of environmental pollution due to emissions from transportation vehicles [1]. These challenges have persisted despite continuous efforts from both the industrial and academic sectors to explore new service models and technologies aimed at addressing these issues. Moreover, the first two challenges also lead to a decline in the quality of logistics services, which severely impacts the customer service experience. Issues such as delivery delays caused by traffic congestion significantly affect customer satisfaction. These factors further hinder the sustainable development of logistics services.

Drones, favored by both the industrial and academic sectors for their advantages, such as being unaffected by traffic congestion and low cost, have found wide applications in fields including logistics [2,3], transportation [4], healthcare [5], and power distribution [6]. The utilization of drones for last-mile delivery in logistics has emerged as a prominent research focus in recent years, with a steady increase in publications in renowned international journals. Presently, the applications of drones in logistics distribution can be broadly categorized into two types: strategic and tactical analysis involving drone station facility location and assignment [7], and operational-level research focusing on drone-scheduling issues [8]. From our communication with Meituan, the largest food delivery platform in China, and SF-EXPRESS, an express company, companies that have already launched drone applications for food delivery and express delivery, respectively, the biggest challenge they are currently facing in drone applications is the operational level of drone scheduling, rather than strategic and tactical drone facility location and allocation. The biggest issue encountered in drone applications is safety, and crashes caused by inaccurate energy consumption calculations are a major factor affecting the safe flight of drones. Furthermore, regarding the application of drones, the government only allows drone services in certain areas, which strictly limits research on the facility location of drones. Considering the above reasons, this article studies the optimization of drone scheduling considering stochastic energy consumption from an operational perspective.

When investigating drone-scheduling issues at the operational level, safety emerges as a paramount challenge that cannot be overlooked. Among these safety considerations, energy consumption plays a crucial role in drone flight safety [3,9]. As far as our knowledge extends, the majority of existing literature adopts fixed flight times or distance to represent drone energy consumption [10]. However, such a representation does not accurately capture the energy consumption of drones. This is because energy consumption is influenced not only by factors such as payload but also by variables like speed, weather conditions, and wind speed, and the formulation of an energy consumption function using a linear or nonlinear model is exceedingly challenging, and in most cases, there are insufficient data to support such modeling efforts [11]. Therefore, investigating drone energy consumption under uncertainty is highly imperative for addressing drone scheduling problems effectively.

Motivated by the above challenges, we investigate a drone-scheduling problem considering the uncertainty of energy consumption. The main contributions of this paper are as follows:

• We study a drone-scheduling problem with the uncertainty of energy consumption. In order to promote energy efficiency and drone safety services, we considered the uncertainty of energy consumption during drone flight in order to provide efficient logistic services to customers while also ensuring drone safety.
• We have defined the above problem and constructed it as a two-stage stochastic programming model. To our knowledge, this is the first stochastic programming model to study this problem. Furthermore, we designed a fixed sample size method based on Monte Carlo simulation to characterize uncertain variables and constructed a variable neighborhood search algorithm for solving this problem.
• We have generated a series of case studies to demonstrate the superiority of our model and algorithm, as well as to evaluate the benefits of considering uncertain energy consumption. From an algorithmic perspective, the algorithm we designed is effective and superior. From a model perspective, considering uncertain energy consumption can indeed bring benefits, improve the service efficiency of drone-scheduling systems, and serve more customers.

The rest of this paper is organized as follows: Section 2 gives a literature review related to our study. The problem is described and the model is formulated in Section 3. Section 4 designs the solution approach. The numerical experiments are carried out in Section 5, and Section 6 concludes the paper.

2. Literature Review

Currently, the drone-scheduling problem falls mainly into two categories: one is the joint scheduling problem of drones and trucks, and the other is the drone-scheduling problem exclusively [12]. This paper primarily focuses on the second category of problems; hence, we will introduce the drone-scheduling problem exclusively (see Table 1). In meal delivery, online meal delivery platforms initially hired fixed delivery personnel to provide last-mile delivery services to customers [13]. However, with the drastic increase in order volume, they had to resort to hiring temporary idle labor from the community to supplement the manpower shortage [14]. Nevertheless, the influx of a large number of delivery personnel has brought about a series of issues, such as exacerbating traffic congestion and environmental pollution, among others. The maturation of drone technology has provided new solutions for online meal delivery platforms.

Kim et al. presented a facility location problem for drones considering uncertain flight distance. This article is one of the few that address the issue of uncertain flight distances, sharing similarities with our study [9]. However, the focus of this article lies in the facility location and configuration of drone bases rather than real-time drone-scheduling problems. In addition, Cheng et al. characterized the energy consumption of drones as a nonlinear function of payload and drone weight, demonstrating that this nonlinear function more accurately captures drone energy consumption compared to linear functions, ensuring the safety of drone last-mile services [11]. Ioannidis et al. proposed a data-driven approach to investigate the efficiency of the drone-scheduling system in Greece, taking into account the uncertainty associated with data such as energy consumption and flight distance. Empirical analysis results indicate that, compared to traditional transportation methods, the drone service model reduces operational costs by 89.44% and can significantly minimize carbon emissions by up to 77.42% [15].

Other researchers focused on other stochastic variants of the drone-scheduling problem. Liu considered the dynamic order information and established a mixed integer model to characterize meal-delivery problems with drones, presenting an optimization-driven, rolling-horizon algorithm to solve this issue [2]. Huang et al. proposed a dynamic task-scheduling problem with stochastic task arrival times and due dates. To address this issue, they designed a simulated annealing-based local search method to solve this problem [10]. Yu et al. focused on the drone route problem considering stochastic demand. They designed an approximation algorithm for solving this problem [16]. Glick et al. investigated the drone-scheduling problem in the field of medical delivery under random demands. They utilized actual data to simulate the probability distribution of random variables and then constructed a stochastic optimization model. The research focused on balancing the reliability, failure rate, and scale of drone scheduling [17]. Hamdi et al. considered the uncertain weather conditions and proposed a drone-as-a-service model, with a focus on drone scheduling, path planning, and combinatorial issues within this service model. Case studies validate the superiority of their constructed models and methods [18]. In addition to demand uncertainty, there is also uncertainty in supply. Wang et al. addressed the uncertainty in vaccine supply by developing a two-stage robust model to optimize vaccination facility location and scheduling. A column-and-constraint generation algorithm was designed to solve the model [19].

In summary, the majority of scholars focus on drone scheduling under deterministic conditions or consider other stochastic variables. However, the literature addressing the uncertainty of energy consumption in drone-scheduling problems remains scarce. Nevertheless, characterizing the uncertainty of drone energy consumption is crucial as it directly relates to the safety of drone flights, presenting an opportunity for the research in this paper.

Table 1. Summary of the literature on drone-only scheduling problem.

References	Problem	Objective	Energy Consumption	Capacity	Algorithms
Liu [2]	Scheduling	Minimize total delivery lateness	Payload size	Multiple	Rolling-horizon algorithm
Kim et al. [9]	Location and scheduling	Minimize costs	Random endurance	Single	Linear programming
Huang et al. [10]	Scheduling	Minimize total delivery lateness	Payload weight	Multiple	K-mean simulated annealing
Cheng et al. [11]	Scheduling	Minimize costs	Payload weight	Multiple	Branch-and-cut
Ioannidis et al. [15]	Location and scheduling	Minimize distances and energy consumption	Payload capacity	Single	Data-driven methodology
Yu et al. [16]	Scheduling	Minimize costs	Fix endurance	Multiple	Approximation algorithm
Hamdi et al. [18]	Scheduling	Minimize delivery time	Fix endurance	Single	Heuristics
Wang et al. [19]	Location and scheduling	Maximize the vaccination profit	Fix endurance	Single	Column-and-constraint generation
this paper	Scheduling	Maximize on-time deliveries	Payload weight	Single	HVNS and FSS

Table 1. Summary of the literature on drone-only scheduling problem.

References	Problem	Objective	Energy Consumption	Capacity	Algorithms
Liu [2]	Scheduling	Minimize total delivery lateness	Payload size	Multiple	Rolling-horizon algorithm
Kim et al. [9]	Location and scheduling	Minimize costs	Random endurance	Single	Linear programming
Huang et al. [10]	Scheduling	Minimize total delivery lateness	Payload weight	Multiple	K-mean simulated annealing
Cheng et al. [11]	Scheduling	Minimize costs	Payload weight	Multiple	Branch-and-cut
Ioannidis et al. [15]	Location and scheduling	Minimize distances and energy consumption	Payload capacity	Single	Data-driven methodology
Yu et al. [16]	Scheduling	Minimize costs	Fix endurance	Multiple	Approximation algorithm
Hamdi et al. [18]	Scheduling	Minimize delivery time	Fix endurance	Single	Heuristics
Wang et al. [19]	Location and scheduling	Maximize the vaccination profit	Fix endurance	Single	Column-and-constraint generation
this paper	Scheduling	Maximize on-time deliveries	Payload weight	Single	HVNS and FSS

3. Problem Formulation

In this section, we first introduce the drone-scheduling problem. Then, a two-stage stochastic optimization model with recourse is formulated. In the first stage, we established a deterministic model under uncertain energy consumption. In the second stage, considering the uncertainty of energy consumption, we established a stochastic optimization model with recourse cost after the realization of energy consumption. The architectural diagram illustrating the overall scheme can be seen in Figure 1.

3.1. Problem Description

Consider a drone-scheduling system (see Figure 2) where there is a drone station 0, a set of demand locations $M=\{1,2,\dots ,m\}$, and homogeneous drones K. Drone $k\in K$ can take off and land at the drone station. Each demand has a deadline time $l_i$ and travel time $t_i$. An income $h_i$ is obtained after drones serve demand i. Drones can perform multiple deliveries until the remaining energy is insufficient to complete the next delivery, each drone needs to return to the station after completing a delivery trip. Other notation descriptions can be seen in

Table 2. Notation description.

Notation	Description
M	$\{1,2,\dots ,m\}$: Set of all demand locations
K	$\{1,2,\dots ,k\}$: Set of all drones
R	$\{1,2,\dots ,r\}$: Set of trips per drone
Parameter	Description
$h_i$	The income of customer $i\in M$
$t_i$	The travel time between demand location $i\in M$ and drone station
$l_i$	The latest delivery time of demand $i\in M$
$w_i^{kr}$	The time the demand $i\in M$ is served by drone $k\in K$ in trip r
$E_i^{kr}$	The accumulated energy consumption of drone $k\in K$ after serving customer i at trip r
$e_i$	The energy consumed by customer $i\in M$
E	The energy with fully charged battery
Decision Variables	Description
$x_i^{kr}$	1, if demand $i\in M$ is served by trip $r\in R$ of drone $k\in K$,
	and 0, otherwise

.

3.2. Formulation

The first-stage optimization model is as follows:

$$\begin{array}{r}Max{\displaystyle \sum _{i\in M}}{\displaystyle \sum _{k\in K}}{\displaystyle \sum _{r\in R}}{x}_i^{kr}{h}_i-E\left[f(x,ϵ)\right],\end{array}$$

(1)

$$\begin{array}{r}s.t.{\displaystyle \sum _{k\in K}}{\displaystyle \sum _{r\in R}}{x}_i^{kr}\le 1,\forall i\in M,\end{array}$$

(2)

$$\begin{array}{r}{\displaystyle \sum _{i\in M}}{x}_i^{kr}\le 1,\forall k\in K,r\in R,\end{array}$$

(3)

$$\begin{array}{r}2{\displaystyle \sum _{i\in M}}{\displaystyle \sum _{r\in R}}{x}_i^{kr}{e}_i\le E,\forall j\in N,k\in K,\end{array}$$

(4)

$$\begin{array}{r}{\displaystyle \sum _{i\in M}}{\displaystyle \sum _{k\in K}}{\displaystyle \sum _{r\in R}}{x}_i^{kr}\le |K|,\end{array}$$

(5)

$$\begin{array}{r}{w}_i^{kr}+x_i^{kr}{t}_i\le l_i,\forall i\in M,k\in K,r\in R,\end{array}$$

(6)

$$\begin{array}{r}{w}_i^{kr}+2\ast x_i^{kr}{t}_i\le w_u^{k(r+1)},\forall i,u\in M,i\ne u,k\in K,r\in R,\end{array}$$

(7)

$$\begin{array}{r}{x}_i^{k(r+1)}\le {\displaystyle \sum _{u\in M}}{x}_u^{kr},\forall i,u\in M,u\ne i,k\in K,r\in R,\end{array}$$

(8)

$$\begin{array}{r}{E}_i^{kr}\le E,\forall i\in M,k\in K,r\in R,\end{array}$$

(9)

$$\begin{array}{r}{E}_i^{kr}+e_j{x}_j^{k,r+1}\le E_j^{k,r+1}+E(1-x_j^{k,r+1}),\forall i\in M,k\in K,r,r+1\in R,\end{array}$$

(10)

$$\begin{array}{r}{x}_i^{kr}\in \{0,1\},\forall i\in M,k\in K,r\in R,\end{array}$$

(11)

The objective (1) is to maximize the total incomes, and $E\left[f\right(x,ϵ\left)\right]$ means expected recourse cost in the second stage because of uncertainty. Constraint (2) ensures that each demand is served at most once. Constraint (3) ensures that each trip of drones only serves once. Constraint (4) enforces drone capacity constraints. The total number of drones assigned is forced by Constraint (5). Constraint (6) ensures that demand has to be served before the latest delivery time. The consecutive deliveries need to start afterward and are supposed to be non-overlapping in Constraints (7) and (8). Constraints (9) and (10) ensure the energy constraints. The variables are defined in Constraint (11).

In the above model, the maximal number |R| of trips per drone can be computed as follows: for the drone station, we sort these customers in ascending order of energy consumed to serve customers, then accumulate the energy of the first c customers till the sum of their energy consumed is greater than E; as a result, the $\left|R\right|=c-1$.

3.3. Two-Stage Stochastic Optimization Model

The second-stage stochastic optimization with recourse cost can be formulated as follows:

$$\begin{array}{r}f(x,ϵ)={\displaystyle \sum _{i\in M}}{\displaystyle \sum _{k\in K}}{\displaystyle \sum _{r\in R}}{\alpha }_i^{kr}{p}_i+\beta {\displaystyle \sum _{k\in K}}max\{(r_k-E),0\},\end{array}$$

(12)

$$\begin{array}{r}\mathit{\text{s.t.}}{\alpha }_i^{kr}=\left\{\begin{array}{cc}1\hfill & E_i^{kr}>E,\forall i\in M,k\in K,r\in R\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.\end{array}$$

(13)

$$\begin{array}{r}{p}_i=0.2h_i,\forall i\in M\end{array}$$

(14)

$$\begin{array}{r}{r}_k={\displaystyle \sum _{i\in M}}{\displaystyle \sum _{r\in R}}{x}_i^{kr}{e}_i,\forall i\in M,r\in R\end{array}$$

(15)

$$\begin{array}{r}{\alpha }_i^{kr}\in \{0,1\},\forall i\in M,k\in K,r\in R\end{array}$$

(16)

where ${\alpha }_i^{kr}$ is the 0–1 variable judging whether the energy constraint is violated, as described in Constraint (13). $p_i$ is the income loss once the energy constraint is violated. $r_k$ is the accumulated energy consumption of drone k. $\beta$ means the penalty cost caused by drone energy consumption exceeding the threshold.

In the case of the stochastic environment, stochastic variables have a great effect on the decisions in the first-stage model. One of the notable impacts is that, in a stochastic environment, the solution obtained by the first-stage model is infeasible, necessitating managers to expend some costs for rectification [20]. Subsequently, we will elucidate the calculation of the penalty cost incurred due to the infeasibility of the solution, referred to herein as the expected recourse cost (ERC). First, we record the number of drones, denoted as |K|, and the number of serviced orders, denoted as s, within the solution, with an initial cost denoted as c set to zero. Consider a sample of size |N|. Second, we iterate through each drone in the solution, calculate the penalty cost incurred by this drone’s service to orders within this sample, and accumulate it into the initial cost c, repeating the computation until all drones have been traversed. Finally, we obtain the total penalty cost. Then, we divide the total cost by the sample size to obtain the expected recourse cost. This can be seen in Algorithm 1. The two-stage framework and method for handling uncertain variables proposed by Lyu et al. [21] and Zhou et al. [22], respectively, have inspired this article. We will use them to design the algorithm framework and handle uncertain energy consumption.

Algorithm 1: The procedure of calculating ERC

Input: The initial value of ERC $Su{m}_0$, the number of drones |K|, the number of            orders $O_k$, served by drone k, the sample size |N| Output: The value of ERC. Initialize $Su{m}_0\leftarrow 0$ For $k\leftarrow 1,2,\dots ,\left|K\right|$, do     For $i\leftarrow 1,2,\dots ,O_k$, do        For $j\leftarrow 1,2,\dots ,\left|N\right|$, do           Calculate penalty cost $f(x,ϵ)$           let $Su{m}_0+=f(x,ϵ)$        End For     End For End For $E\left[f(x,\xi )\right]=Su{m}_0/\left|N\right|$.

4. Solution Approach

In this section, we devised a hybrid variable neighborhood search algorithm (HVNS) based on the Monte Carlo simulation to solve the stochastic drone-scheduling problem [23,24,25,26]. In this algorithm, we first generate an initial solution under deterministic conditions. Subsequently, we iteratively update this initial solution through a neighborhood structure, considering stochastic energy consumption during the iteration process, while also calculating the expected recourse cost. This iterative process continues until the algorithm’s termination condition is met. Finally, the current optimal solution is output. The framework of HVNS is described in Algorithm 2.

Algorithm 2: The framework of HVNS

Input: The sample of size |N|, initial solution $S_0$, the number of shaking            operations c, the max iterations of non-improvement $N_t$ Output: Current best solution $S_b$. Initialize $S_b\leftarrow S_0,\delta \leftarrow 0$ For $i\leftarrow 1,2,\dots ,c$, do     While $\delta \le N_t$        Select f-i shaking operations to improve current solution and obtain a        new solution $S_{im}$.        Calculate the expected profit on $S_{im}$ using Equation (1) and Algorithm 1.        If $S_{im}$ has a higher expected profit than $S_b$ then           $S_b\leftarrow S_{im},\delta \leftarrow 0$        Else           $\delta \leftarrow \delta +1$        end If     end While end For

4.1. Construction Heuristic

In this section, we will design a Construction heuristic to construct an initial feasible solution, incorporating randomness into a greedy algorithm. Specifically, we iterate through each unallocated order and assign it to a drone that satisfies the constraints. We retain $N_o$ schemes with the maximum profit for order allocation. From these, we randomly select one allocation scheme to complete the delivery service for the order until all orders have been traversed. The detailed steps of the Construction heuristic are as follows:

(1)

For unserved orders, iterate through all drones to find allocation schemes that satisfy the model constraints.

(2)

Calculate the profits of all schemes and retain the top $N_0$ ranked schemes.

(3)

Randomly select one scheme from the $N_0$ retained schemes to execute.

(4)

Repeat steps (1)–(3) until all orders have been traversed.

4.2. Shaking Operations

In this section, we will design efficient Shaking Operations to improve the initial feasible solution. This is the most crucial and important stage of the HVNS algorithm. When designing Shaking Operations, we will not only refer to existing efficient operators but also combine the characteristics of drone-scheduling problems to design new efficient operators targeted at improving the solution. Each Shaking Operation will consist of a pair of operators, namely the destruction operator and the reconstruction operator. The destruction operator will transform the current solution into an incomplete solution, while the reconstruction operator will transform this incomplete solution back into a complete solution. The aim is to enhance the total expected revenue of the solution under uncertain conditions. Specifically, the destruction and reconstruction operators are as follows.

Destruction operators:

(1)

Random-α-orders: Randomly delete $\alpha$ orders that have been assigned to drones.

(2)

Greedy-α-orders: Delete $\alpha$ orders with the least profit that have been served by drones.

(3)

Random-β-drones: Randomly select $\beta$ drones and delete these orders delivered by selected drones.

(4)

Greedy-β-drones: Select $\beta$ drones with the least profit and delete these orders delivered by selected drones.

Reconstruction operators:

After implementing the destruction operator, we obtain a disrupted solution $S_d$. Subsequently, we utilize the reconstruction operator to reassign unserved orders to drones in descending order of profit. For each order, there may be multiple allocation schemes or no drones available to serve the order. If there are multiple allocation schemes, we select the one with the minimum travel time. If it is the latter case, the order is placed into the unserved order set.

After each Shaking Operation is implemented, we can obtain a new solution $S_{im}$. If the total expected revenue of this new solution is greater than the current best solution, we accept the new solution and replace the current best solution with it. Otherwise, we do not accept the new solution, and the current best solution remains unchanged. Repeat the above process until the algorithm termination condition is met. The detailed algorithm flow is shown in Algorithm 2.

5. Numerical Experiments

In this section, we show the results of a set of numerical experiments to examine the performance of this algorithm. To avoid bias related to the data generation, the demand locations are generated randomly within a radius of 10 km. The demands to be delivered at customer locations are randomly generated. Initially, we first verify the computational efficiency and effectiveness of the algorithm. Subsequently, we demonstrate the superiority of models considering randomness. Finally, we analyze the computational efficiency and effectiveness of the algorithm under different sample sizes. The algorithm is coded in Python language. All above experiments are executed on an Intel(R) Core(TM) i5-10210U CPU @ 1.60 GHz 2.11 GHz processor with 16 GB of memory running under a Windows 10 operating system.

5.1. Instance Description

To ensure the practicality of our model and algorithm, we construct scenarios based on real-world data. Firstly, concerning drone-related parameters [11], the initial effective flight time of the drone ranges from 0 to 10 min, with a flight speed of 50 kg/h, a fully charged load of 0.3 kWh, an order loading and unloading time of 1 min, a drone’s own weight of 2 kg, and a battery weight of 1 kg. Additionally, the fluid density of air is considered as $\rho$ = 1.204 kg/m², and the area of the spinning blade disc is denoted as $\zeta$ = 0.0064 m². Secondly, concerning order parameters, orders are generated within the time range of [0, 10], with weights ranging from 0.5 kg to 2 kg. All orders are promised to be delivered to customers within 20 min. Failure to meet this deadline results in disqualification for drone allocation. The revenue for each order randomly ranges from 0.1 to 0.9.

For the parameters involved in the algorithm, we set $N_t=100$, $N=100$, and $c=4$. When the number of orders is not greater than 20, $\alpha =3$ and $\beta =1$; otherwise, $\alpha =5$ and $\beta =2$.

5.2. Computational Efficiency

In this section, we will first verify the superiority of the HVNS algorithm designed in this paper. To achieve this, we compare HVNS with a commonly used metaheuristic algorithm, the Genetic Algorithm (GA). For the Genetic Algorithm, we adopt a real-coded representation for the solution, with a crossover probability of 0.8 and a mutation probability of 0.05, and the roulette wheel selection method to choose individuals for the next generation.

Initially, we utilize the HVNS to solve the model and obtain solutions (including average profit and CPU time). Subsequently, we use the GA algorithm with the same time expenditure and obtain solutions (also including average profit and CPU time). The results are presented in

Table 3. The computational performance of the GA and the HVNS algorithm.

K	M	GA		HVNS		Improve (%)
		Obj_val	CPU_t (s)	Obj_val	CPU_t (s)
		avg.	avg.	avg.	avg.	avg.
5	10	4.93	0.02	4.95	0.01	0.4
	20	7.26	0.85	8.09	0.82	11.4
	30	8.47	5.24	9.53	5.19	12.5
	40	9.08	7.75	9.95	7.66	9.5
10	30	13.37	6.40	14.22	6.27	6.3
	40	14.74	7.19	16.28	7.00	10.5
	50	15.65	21.15	17.55	20.87	12.2
	60	16.75	23.25	18.65	22.99	11.4

.

In Table 3, the first two columns represent the number of drones and the number of orders, respectively, the third and fourth columns represent the average revenue and CPU time of the solution obtained by using the GA algorithm, and the fifth and sixth columns represent the average value of the solution obtained by using HVNS. Revenue and CPU time, the last column represents the average revenue increase percentage of the solution obtained by the HVNS algorithm compared to GA. From the table, we can see that the average benefits of the solutions obtained by the HVNS designed in this article have been improved, and as the scale of the problem expands, the benefits increase more obviously.

To better visualize the advantages of our designed HVNS algorithm, we plotted an error graph comparing results with GA, as shown in Figure 3. From Figure 3, it is evident that our algorithm yields significantly higher solution benefits compared to GA, with a maximum increase of up to 12.5%. Furthermore, Figure 3 illustrates that as the scale of the instances increases, the superiority of our algorithm becomes more pronounced, with the magnitude of benefits showing an increasing trend.

5.3. Advantage of Considering Stochasticity

In this section, we will analyze the advantages of considering stochastic energy consumption in the model. Specifically, we will first obtain the deterministic solution and then randomly generate a large sample of 1000 instances to evaluate this solution, thereby obtaining the expected incomes and CPU time of this solution under large samples. Subsequently, we will consider stochastic energy consumption and utilize the algorithm designed in this paper to obtain the solution under stochastic conditions. We will then evaluate the generated solution using the same large sample to obtain the expected incomes and CPU time of this solution under large samples. Finally, we will compare the expected incomes and CPU time of the solutions under the two scenarios to demonstrate the advantages of considering stochasticity.

In

Table 4. The comparison analysis under deterministic and stochastic environment.

K	M	Deterministic		Stochastic		Improve (%)
		Obj_val	CPU_t (s)	Obj_val	CPU_t (s)
		avg.	avg.	avg.	avg.	avg.	max.
5	10	4.91	0.01	4.95	0.01	0.8	10.1
	20	7.58	0.42	8.09	0.82	6.7	14.1
	30	9.07	1.96	9.53	5.19	5.4	10.1
	40	9.60	2.55	9.95	7.66	3.6	8.1
10	30	13.42	2.97	14.22	6.27	5.9	11.9
	40	15.27	4.07	16.28	7.00	6.6	11.6
	50	16.66	5.32	17.55	20.87	5.3	9.6
	60	17.88	11.9	18.65	22.99	4.3	7.1

, we present the results of cases (each category comprising 10 specific instances) for deterministic and stochastic scenarios along with a comparative analysis. The first two columns denote the number of drones and orders, respectively. The third and fourth columns represent the average revenue and average time consumption for each category in deterministic scenarios, while the fifth and sixth columns depict the average revenue and average time consumption for each category in stochastic scenarios. The last three columns, respectively, illustrate the average improvement percentage, maximum improvement percentage, and minimum improvement percentage.

From Table 4, it is evident that, with improvement under stochastic conditions compared to deterministic scenarios, the revenue of the drone-scheduling system has undergone varying degrees of improvement under stochastic conditions. The maximum improvement is observed in the second category of instances (with five drones, 20 orders), with a revenue increase of 14.1%. In other categories, the average revenue increased by 4% to 7%. This indicates that considering stochastic energy consumption can significantly enhance the overall expected revenue of drone-scheduling systems.

Similarly, we employed error plots, as depicted in Figure 4, to illustrate the disparity in returns between stochastic and deterministic environments. From Figure 4, it is evident that solutions incorporating random variables yield significantly higher returns compared to those derived under deterministic conditions. On average, across all cases, the returns increased by 5.1% (12.4 vs. 11.8).

5.4. Comparison Analysis under Different Sample Sizes

Table 5. The comparison analysis under different sample sizes.

K	M	Size = 20		Size = 50		Size = 100		Size = 200
		Obj_val	CPU_t (s)	Obj_val	CPU_t (s)	Obj_val	CPU_t (s)	Obj_val	CPU_t (s)
5	10	4.95	0.001	4.97	0.001	4.95	0.01	4.95	0.00
	20	8.01	0.53	8.06	0.63	8.09	0.82	8.03	1.22
	30	9.40	1.93	9.48	2.31	9.53	5.19	9.47	5.54
	40	9.97	3.14	9.98	3.38	9.99	7.66	9.98	8.65
10	30	14.23	2.86	14.26	3.00	14.22	6.27	14.29	9.86
	40	16.13	4.87	16.22	6.87	16.28	7.00	16.29	14.37
	50	17.47	6.97	17.47	9.789	17.55	20.87	17.65	23.73
	60	18.59	10.30	18.54	9.697	18.65	22.99	18.65	26.30

illustrates the variations in objective values and CPU time across different sample sizes for various cases. It is evident from the table that as the sample size increases from 20 to 100, the solution exhibits a slight augmentation. However, with the sample size escalating from 100 to 200, the quality of the solution remains nearly unchanged (99.27 → 99.29). Concerning CPU time, it consistently escalates with an increase in the sample size. From the tabulated data, it can be inferred that a sample size of 100 is deemed adequate for the problem formulated within this model design.

6. Conclusions

This paper addresses the uncertainty of energy consumption in drone-scheduling problems by constructing a two-stage stochastic optimization model considering recourse costs, with the objective of maximizing revenue. We employ a fixed-sample-size method based on Monte Carlo simulation to characterize the uncertain energy consumption and design an efficient hybrid variable neighborhood search algorithm to solve the problem. After conducting a series of case studies, we conclude the key findings of this paper as follows: First, the stochastic nature of energy consumption significantly affects profitability, with an average profit increase of 5.1% across all cases when uncertainty is considered. Second, our proposed hybrid variable neighborhood search algorithm outperforms commonly used genetic algorithms. For eight types of cases, the increase in revenue ranges from 0.4% to 12.5% when running the algorithm for similar times. Last, by incorporating stochastic optimization, the model enhances the overall profitability of drone-scheduling systems, demonstrating the importance of accounting for energy consumption variability. These results validate the effectiveness and superiority of our approach in addressing the challenges posed by uncertain energy consumption in drone scheduling.

In future work, we will incorporate drone battery replacement decisions into the model to investigate drone-scheduling problems under battery replacement modes while also designing efficient exact algorithms to solve the model.