The Multi-Visit Vehicle Routing Problem with Drones under Carbon Trading Mechanism

Abstract

In the context of the carbon trading mechanism, this study investigated a multi-visit vehicle routing problem with a truck-drone collaborative delivery model. This issue involves the route of a truck fleet and drones, each truck equipped with a drone, allowing drones to provide services to multiple customers. Considering the carbon emissions during both the truck’s travel and the drone’s flight, this study established a mixed integer programming model to minimize the sum of fixed costs, transportation costs, and carbon trading costs. A two-stage heuristic algorithm was proposed to solve the problem. The first stage employed a “Scanning and Heuristic Insertion” algorithm to generate an initial feasible solution. In the second stage, an enhanced variable neighborhood search algorithm was designed with problem-specific neighborhood structures and customized search strategies. The effectiveness of the proposed algorithm was validated with numerical experiments. Additionally, this study analyzed the impact of various factors on carbon trading costs, revealing that there exists an optimal combination of drones and trucks. It was also observed that changes in carbon quotas do not affect carbon emissions but do alter the total delivery costs. These results provide insights for logistics enterprise operations management and government policy-making.

1. Introduction

With the rapid development of e-commerce and the global economy, the demand for commercial deliveries has surged, leading to a significant increase in carbon emissions [1]. Traditional truck deliveries heavily rely on fossil fuels, producing substantial amounts of carbon dioxide and toxic pollutants, which severely impact the environment [2]. As environmental awareness grows and sustainable development goals become more widespread [3], delivery companies face increasing pressure to address sustainability issues [4]. Consequently, many governments have mandated environmental protection measures within supply chains and have implemented corresponding carbon emission policies.

The carbon trading mechanism is recognized as an essential policy tool to address environmental issues. It is widely regarded as an effective, flexible, and low-cost method for reducing carbon emissions [5]. Carbon trading typically refers to carbon emissions trading, a market-based mechanism for controlling carbon emissions. The government sets an overall carbon emission cap and allocates it to participating enterprises. These enterprises can then buy and sell carbon emission allowances in the market, adjusting their own carbon emissions as needed [6]. In this context, companies need to explore new models for low-carbon commercial delivery, and the truck–drone collaborative delivery model has emerged as a significant innovation for addressing carbon emissions and enhancing delivery efficiency.

Drone delivery, as an emerging low-carbon delivery method, has gained attention due to its high efficiency and significant reduction in energy consumption and carbon emissions [7]. However, the limited payload capacity and flight distance of drones restrict their application range, necessitating traditional vehicles for many customers [8]. Truck deliveries in complex road conditions or traffic congestion often increase time, fuel consumption, and carbon emissions, whereas drones can significantly reduce these costs through straight-line flights. Therefore, the truck–drone collaborative delivery model has emerged, which not only reduces the energy consumption and carbon emissions of trucks but also uses trucks as charging stations for drones, extending flight times and improving delivery efficiency.

In summary, the interaction between carbon trading mechanisms and truck–drone collaborative delivery manifests in several ways. Firstly, carbon emissions trading motivates companies to adopt low-carbon delivery models like truck–drone collaboration to reduce emissions. Secondly, this delivery model optimizes resource utilization and improves delivery efficiency, thereby reducing carbon emissions and achieving both environmental and economic benefits. Lastly, in the carbon trading market, companies can gain carbon credits by reducing emissions and providing economic benefits. Therefore, this paper proposes the multi-visit vehicle routing problem with drones under carbon trading mechanisms (MVRPD-CRs), which incorporates carbon emissions into route optimization based on carbon trading mechanisms. By considering the impact of carbon trading costs on total costs, this study aimed to balance economic and environmental benefits. A mixed-integer programming model was established with the objective of minimizing the overall costs of the truck–drone collaborative delivery network, seeking to determine the most cost-effective delivery routes for trucks and drones.

The contributions of this paper are summarized as follows: (1) To our knowledge, this is the first study to introduce the carbon trading mechanism into the multi-visit drone delivery problem, naming it the multi-visit vehicle routing problem with drones under the carbon trading mechanism. A mixed-integer programming model was proposed to minimize the fixed costs associated with drones and trucks, as well as the combined travel distance costs and carbon trading costs. (2) We designed a two-stage heuristic algorithm to solve the problem, initially generating solutions with a scanning and heuristic insertion method and then optimizing them using an enhanced variable neighborhood search algorithm. (3) Extensive computational experiments demonstrate the model’s and algorithm’s effectiveness, providing insights into the impact of various factors on carbon trading costs and total delivery costs.

The remainder of this paper is organized as follows. Section 2 reviews the related literature on the collaborative truck–drone operation. Section 3 describes and formulates the MFS-VRPD, and Section 4 discusses the ALNS metaheuristic. Detailed results of the numerical experiments and sensitivity analysis are presented in Section 5. Finally, Section 6 provides some concluding remarks.

2. Literature Review

This study aimed to develop a truck–drone collaborative delivery route optimization model that considers the carbon trading mechanism and analyzes the impact of the carbon trading mechanism on the total delivery cost. The research problem features characteristics of the green vehicle routing problem and the truck–drone collaborative delivery problem. Therefore, we first review the literature on GVRP from the perspective of carbon emissions and then discuss the truck–drone collaborative delivery problem.

2.1. Green Vehicle Routing Problem

The vehicle routing problem (VRP) was first proposed by Dantzig and Ramser [9]. In recent years, with increasing attention to environmental issues and the implementation of policies to reduce energy consumption and carbon emissions, scholars have recognized the importance of incorporating these factors into the VRP.

Chen et al. [10] studied the multi-compartment vehicle routing problem (MCVRPTW) with time windows considering carbon emissions and designed a variable neighborhood search (VNS) method with local search and shaking operations as the main framework to solve the MCVRPTW problem. Li et al. [11] investigated the vehicle routing problem considering carbon emissions, established two mathematical models, and solved them using a simulated annealing method. Numerical results indicated that distribution distance is a key factor affecting carbon emissions, and optimizing the allocation sequence can further reduce carbon emissions. Pan [12] considered the shortest path problem, incorporating carbon emissions, and designed an improved algorithm based on the problem’s characteristics, demonstrating good performance. Addressing carbon emissions caused by traffic congestion, Figliozzi proposed a path optimization problem with time windows [13]. Building on this, Xiao studied the impact of different vehicle sizes on carbon emissions [14].

Given the low carbon emission advantages of electric vehicles, they are widely used. Wang et al. established a multi-objective optimization model for electric vehicle routing, aiming to minimize travel time, energy consumption, and charging costs [15]. Liao, Liu, and Fu compared the cost and environmental impact on the logistics companies of electric vehicles and traditional fuel vehicles under a carbon trading mechanism, proposing a hybrid genetic algorithm. This study showed that using electric vehicles can effectively reduce carbon emissions, but customer satisfaction may be relatively low [16].

Goodchild et al. [17] found that using drones in logistics delivery can reduce carbon emissions. Figliozzi et al. [18] discovered that in sparsely populated areas with low transport density, the unit distance energy consumption and emissions of drones are lower than those of trucks. Figliozzi evaluated the potential of drones and ground autonomous delivery robots in reducing carbon dioxide emissions in the delivery industry, showing that these new vehicles can reduce energy consumption [19]. Chiang et al. [20] studied the green vehicle routing problem in the truck–drone collaborative delivery model. They assessed the environmental and economic sustainability of drone-assisted truck transport, showing that drone-assisted truck transport significantly reduces emissions.

Most existing studies consider carbon emissions a constraint or convert them into carbon trading costs to simultaneously consider economic and social benefits. For example, Wang et al. considered fuel consumption and refrigeration equipment’s carbon emissions in cold chain logistics, converting them into carbon trading costs in the objective function [21]. Additionally, research by Chen, Liao, and Yu indicated that worsening traffic conditions would increase carbon emissions, while carbon trading could reduce emissions to some extent [22]. Kwon et al. considered the case where companies have fixed carbon quotas, incorporating carbon trading costs into transportation costs, finding that companies could gain revenue rather than costs when selling carbon emission quotas [23]. It is evident that carbon emission quotas under the carbon trading mechanism can affect carbon trading costs, thereby impacting the total cost of enterprises. However, few studies on the green vehicle routing problem have considered the impact of carbon quotas on vehicle routes under the carbon trading mechanism. Moreover, systematic research on multi-modal transportation collaborative delivery under the carbon trading mechanism is lacking.

2.2. Truck–Drone Collaborative Delivery Problem

Truck–drone collaborative delivery, leveraging complementary advantages, has gradually become a research hotspot in the logistics delivery field. Murray and Chu [24] introduced the flying sidekick traveling salesman problem (FSTSP) in truck–drone collaborative delivery. Dell’Amico et al. [25] proposed new formulations for the FSTSP problem and a set of effective inequalities, solving them using an exact branch-and-cut algorithm. The results showed that the new formulations outperformed the original FSTSP formulations. Freitas et al. [26] proposed a hybrid heuristic algorithm to solve the FSTSP problem, where the initial solution was obtained by solving the optimal TSP problem using a solver, followed by a general variable neighborhood descent algorithm (RVND) for obtaining the optimal delivery routes for trucks and drones. Ha et al. [27] constructed a Greedy Random Adaptive Search Procedure (GRASP) and traveling salesman problem-local search (TSP-LS) heuristic to solve the FSTSP.

Agatz et al. [28] proposed a similar model, the traveling salesman problem with a drone (TSP-D), assuming that a truck carries a drone, which delivers one package per flight and returns to the truck to change the battery for subsequent flights. Bouman et al. [29] proposed a dynamic programming algorithm for the TSP-D, capable of solving instances with up to 20 customer nodes. Poikonen et al. [30] proposed different heuristic algorithms based on a branch-and-bound approach, considering only a subset of potential parcel delivery orders at each node. By solving the dynamic programming, they provided approximate lower bounds for each node and other heuristic variants. Phan et al. [31] further studied a new variant based on the TSP-D, called the mTSP problem, showing that the ALNS algorithm was more effective than the extended GRASA algorithm. Agardi et al. [32] proposed a variant of the TSP-D problem, where the launch and recovery points of the drone must be at the same customer point, allowing multiple visits by the drone to minimize total flying and driving distances. Marinelli et al. [33] proposed that trucks could launch and recover drones at customer nodes or along the delivery route (in transit), analyzing the advantages of launching and recovering drones during transit. Kim et al. [34] developed a TSP-D problem with drone stations to overcome drone flight range limitations, assuming that drone stations were facilities storing drones and charging equipment, providing sufficient drones. Cavani et al. [35] studied the multiple-drone traveling salesman problem (TSP-mD), establishing a mixed-integer linear programming (MILP) model and solving it using a branch-and-cut algorithm capable of solving instances with up to 24 customers. Based on the TSP-D variant, Meng et al. [36] considered practical factors such as simultaneous pickup and delivery, truck capacity, and drone load-dependent energy consumption. Jeong considered the impact of geographical constraints such as no-fly zones on drone applications [37]. Existing research has not fully addressed the impact of complex traffic restrictions, no-fly zones, and other practical constraints on green delivery.

Compared with FSTP and TSP-D, research on the collaboration of multiple drones with a fleet of trucks is more complex. Wang et al. [38] introduced the drone vehicle routing problem (VRP-D), considering a fleet of trucks, each equipped with one or more drones to directly deliver packages or launch drones to deliver packages. Drones could be launched from warehouses or any customer location and delivered by trucks, with customer types including those that could only be delivered by truck or only by drone. Poikonen et al. [39] studied the VRP-D problem of trucks carrying drones to deliver packages, with the objective of minimizing the completion time for delivering all packages and returning all vehicles to the distribution center. Wang et al. [40] considered the possibility of multiple visits to customer points by drones, with the objective of minimizing total cost. The results showed that effective drone flight time could reduce delivery costs by about 10%. Schermer et al. [41] proposed two heuristic methods for solving the VRP-D, divided into two main stages: initialization and improvement. The initialization stage used a path-first, cluster-second heuristic algorithm, followed by multiple local search actions to improve the solution. Sacramento et al. [42] considered truck capacity and maximum drone flight duration, establishing a mixed-integer programming model to study the VRP-D with the objective of minimizing cost under time constraints, and proposed an adaptive large neighborhood search metaheuristic for solving large-scale instances. Liu et al. [43] considered the problem of drones delivering multiple packages and proposed a hybrid heuristic algorithm combining nearest-neighbor and cost-saving strategies.

In summary, existing research on truck–drone collaborative delivery has not fully addressed the impact of complex traffic restrictions, no-fly zones, and other practical constraints on green delivery in urban environments. Moreover, truck–drone collaborative delivery research has only considered carbon emissions, lacking studies on the carbon trading mechanism. Additionally, there is a lack of in-depth comparative analysis on the impact of the carbon trading mechanism on logistics delivery. Therefore, this study considered the carbon trading mechanism, incorporating carbon emission factors into route optimization to balance economic and environmental benefits, proposing a truck–drone collaborative delivery route optimization problem considering the carbon trading mechanism.

3. Problem Formulation

3.1. Problem Description

This study considered a combined logistics system with a fleet of homogeneous trucks and multiple drones to provide delivery service to customers. Trucks and drones originate from and return to a central depot. Each truck, equipped with a maximum carrying capacity, transports a drone while serving customers along its predetermined route. At both the central depot and any customer node traversed by the trucks, the drone may be deployed to execute independent sub-routes. Within each sub-route, the drone has the capability to service one or several customers, constrained by its maximum payload capacity. Notably, customer points within restricted traffic zones are serviced by drones to avoid detours by trucks. Conversely, customer points within no-fly zones can only be serviced by trucks, which complete their delivery tasks along the designated route. This concurrent operation enables efficient utilization of resources and maximizes the delivery throughput. As illustrated in Figure 1, this study develops a collaborative logistics network consisting of trucks and unmanned aerial vehicles (UAVs) to fulfill the delivery requirements of 14 customers. In this distribution scheme, trucks are responsible for the delivery tasks within no-fly zones, while UAVs undertake the delivery services in areas with traffic restrictions. The pertinent assumptions of the problem are outlined as follows:

• Each customer must be serviced once by either a truck or a drone;
• Drones may launch from or return to the central depot or any customer node visited by trucks, but they can only be launched and retrieved by the same truck. Therefore, when truck k is deployed, the corresponding drone k is also enabled;
• Each truck has a limited load capacity and a constant speed;
• Each drone can serve multiple customers per flight as long as its load capacity and flight distance are within limits;
• Drones can be launched and retrieved multiple times. However, the drone cannot be re-launched before being retracted, and the situation where the retraction point coincides with the launch point is not considered;
• If a drone arrives at a rendezvous node before the truck, it may land to conserve energy while awaiting the truck’s arrival for synchronization. Conversely, if the truck arrives first, it will wait for the drone to synchronize operations;
• When delivering by truck or drone, time windows are temporarily not considered.

3.2. Notations

All the notations used in the mathematical model are listed in

Table 1. Notation in the mathematical model.

Notation	Description
Sets:
C	$\mathrm{Set} \mathrm{of} \mathrm{customers}, C=\left\{i\right|i=1,2,\cdots ,n\}$
N	$\mathrm{Set} \mathrm{of} \mathrm{all} \mathrm{nodes}, \mathrm{N}=\{0_s$$\}\cup \mathrm{C}\cup \{0_r$$\}, \mathrm{where} 0_s$$\mathrm{and} 0_r$ correspond to the same depot location, denoting the origin depot and destination depot, respectively.
$C_0/C_{+}$	$\mathrm{Set} \mathrm{of} \mathrm{all} \mathrm{customers} \mathrm{plus} \mathrm{the} \mathrm{starting}/\mathrm{ending} \mathrm{depot}, C_0=\left\{0_s\right\}\cup C,C_{+}=\left\{0_r\right\}\cup C$
A	$\mathrm{Set} \mathrm{of} \mathrm{all} \mathrm{arcs}, A=\{(i,j)|i\in C_O,j\in C_{+},i\ne j\}$
K	$\mathrm{Set} \mathrm{of} \mathrm{trucks}/\mathrm{drones}, K=\left\{k\right|k=1,2,\cdots ,a\}$
Parameters:
$Q_T/Q_D$	Maximum load capacity of each truck/drone
$c_T/c_D$	The fixed cost of trucks/drones
$C_T/C_D$	The per-unit cost associated with the utilization of trucks/drones
$w_T/w_D$	The weight of the truck/drone itself
E	Maximum flight distance from drone launch to retrieve
$q_i$	The package weight of customer i
$d_{ij}$	The distance from node i to node j
M	A sufficiently large positive integer
Decision variables:
$x_{ij}^k$	$0-1 \mathrm{variables}, \mathrm{if} \mathrm{truck} \mathrm{k} \mathrm{travels} \mathrm{from} \mathrm{node} i \mathrm{to} \mathrm{node} j, x_{ij}^k$$\mathrm{is} 1; \mathrm{otherwise}, x_{ij}^k$ is 0
$y_{ij}^k$	$0-1 \mathrm{variables}, \mathrm{if} \mathrm{drone} k \mathrm{fly} \mathrm{from} \mathrm{node} i \mathrm{to} \mathrm{node} j, y_{ij}^k$$\mathrm{is} 1; \mathrm{otherwise}, y_{ij}^k$ is 0
$S{t}_i^k$	$0-1 \mathrm{variables}, \mathrm{if} \mathrm{truck} k \mathrm{server} \mathrm{customer} i, S{t}_i^k$$\mathrm{is} 1; \mathrm{otherwise}, S{t}_i^k$ is 0
$S{d}_i^k$	$0-1 \mathrm{variables}, \mathrm{if} \mathrm{drone} k \mathrm{server} \mathrm{customer} i, S{d}_i^k$$\mathrm{is} 1; \mathrm{otherwise}, S{d}_i^k$ is 0
$z^k$	$0-1 \mathrm{variables}, \mathrm{if} \mathrm{truck}/\mathrm{drone} k \mathrm{is} \mathrm{deployed}, z^k$$\mathrm{is} 1; \mathrm{otherwise}, z^k$ is 0
$r_i^k$	$0-1 \mathrm{variables}, \mathrm{if} \mathrm{truck} k \mathrm{launches}/\mathrm{retrieves} \mathrm{drone} \mathrm{k} \mathrm{at} \mathrm{node} i, r_i^k$$\mathrm{is} 1; \mathrm{otherwise}, r_i^k$ is 0
$G{t}_i$	$0-1 \mathrm{variables}, \mathrm{if} \mathrm{node} i \mathrm{is} \mathrm{located} \mathrm{in} \mathrm{the} \mathrm{truck} \mathrm{restricted} \mathrm{area}, G{t}_i$$\mathrm{is} 1; \mathrm{otherwise}, G{t}_i$ is 0
$G{d}_i$	$0-1 \mathrm{variables}, \mathrm{if} \mathrm{node} i \mathrm{is} \mathrm{in} \mathrm{the} \mathrm{drone} \mathrm{no}-\mathrm{fly} \mathrm{zone}, G{d}_i$$\mathrm{is} 1; \mathrm{otherwise}, G{d}_i$ is 0
$L{t}_i^k\in [0,Q_T]$	Load carried on truck k when it leaves node i
$A{d}_i^k\in [0,Q_D]$	Total parcel weight carried on drone k when it arrives node i
$L{d}_i^k\in [0,Q_D]$	Total parcel weight carried on drone k when it leaves node i
$E_i^k\in [0,E]$	The remaining flight distance when drone k arrives at node i

.

3.3. Carbon Trading Cost

McKinnon’s research suggests that carbon emissions from trucks can be estimated based on the weight (or volume) they transport and the distance traveled [44]. In truck–drone cooperative delivery systems, the total weight carried by both trucks and drones is influenced by the selected delivery routes. During the delivery process, a drone is mounted on the truck or flies independently to complete package deliveries. When the drone is unloaded, the truck’s total weight decreases, enhancing fuel efficiency, reducing delivery path costs, and lowering carbon emissions. Assuming that $w_{ij}^k$ represents the load capacity of the truck k in arc(i, j), and $u_{ij}^k$ denotes the payload of drone k through arc(i, j), then they are calculated as follows:

$$w_{ij}^k=w_T\cdot x_{ij}^k+L{t}_i^k$$

(1)

$$u_{ij}^k=w_D\cdot y_{ij}^k+L{d}_j^k$$

(2)

Then, using the Goodchild method [17], the fuel consumption of trucks per unit distance is related to both the transportation distance and the load. The weighted average emission of trucks is denoted by $WAER$, and the variable carbon emissions of trucks over arc (i, j) are calculated as follows:

$$E{C}^T=WAER\cdot {\displaystyle \sum _{i\in N}{\displaystyle \sum _{j\in N}{d}_{ij}}}\cdot w_{ij}^k$$

(3)

The carbon emissions of drones are related to the load they carry and the distance they fly. The calculation of drone carbon emissions references the method by Chiang et al. [20]. Here, $PGFER$ represents the carbon emissions per unit watt of electricity consumed during drone flight, and $AER$ represents the average power consumption per kilometer per kilogram. The formula for calculating drone carbon emissions is given by the following:

$${\mathrm{EC}}^U=PGPER\cdot AER\cdot {\displaystyle \sum _{i\in N}{\displaystyle \sum _{j\in N}{d}_{ij}\cdot u_{ij}^k}}$$

(4)

Under a government-implemented carbon trading mechanism, firms are allocated a certain carbon emission quota. If their actual emissions are below this quota, they can sell the surplus emissions allowances for profit; otherwise, they must purchase additional allowances from the carbon trading market, incurring costs. Let $\stackrel{-}{CE}$ represent the total carbon emissions and $\epsilon$ denote the unit carbon trading price. The carbon trading cost in a truck–drone collaborative delivery model is given by the following:

$$\mathrm{CC}=(E{C}^U+E{C}^T-\stackrel{-}{CE})\epsilon$$

(5)

3.4. The Mathematical Model

The problem described in the previous section can be formulated using a MILP model. The objective is to minimize the total cost of the entire delivery operation, as outlined below:

$$\min F=(c_T+c_D){\displaystyle \sum _{k\in K}{z}^k}+C_T{\displaystyle \sum _{k\in K}{\displaystyle \sum _{i\in C_0}{\displaystyle \sum _{j\in C_{+}}{d}_{ij}{x}_{ij}^k}}}+C_D{\displaystyle \sum _{i\in C_0}{\displaystyle \sum _{j\in C_{+}}{\displaystyle \sum _{k\in K}{d}_{ij}{y}_{ij}^k}}}+CC$$

(6)

The first term represents the fixed costs incurred from deploying trucks and drones, the following two terms account for the transportation costs associated with trucks and drones, and the final term reflects the carbon trading cost.

The constraints fall into several categories, as follows.

• The routing constraints are the following:

$$s.t.{\displaystyle \sum _{i\in C}{x}_{0_{s}i}^k}={\displaystyle \sum _{i\in C}{y}_{0_{s}i}^k}=1;\forall k\in K$$

(7)

$${\displaystyle \sum _{i\in C}{x}_{i{0}_r}^k={\displaystyle \sum _{i=C}{y}_{i{0}_r}^k}=1,\forall k\in K}$$

(8)

$${\displaystyle \sum _{k\in K}(S{t}_i^k}+S{d}_i^k)=1,\forall i\in C$$

(9)

$${\displaystyle \sum _{i\in C_0}{x}_{im}^k}={\displaystyle \sum _{j\in C_{+}}{x}_{mj}^k}=S{t}_m^k,\forall k\in K,\forall m\in C$$

(10)

$${\displaystyle \sum _{i\in C_0}{y}_{im}^k}={\displaystyle \sum _{j\in C_{+}}{y}_{mj}^k=}S{d}_m^k+S{t}_m^k,\forall k\in K,\forall m\in C$$

(11)

$$u_i-u_j+n(x_{ij}^k)\le n-1;\forall i\in C_0,\forall jC\in C_{+},\forall k\in K$$

(12)

$$u_i-u_j+n(y_{ij}^k)\le n-1;\forall i\in C_0,\forall jC\in C_{+},\forall k\in K$$

(13)

$$G{t}_i\le {\displaystyle \sum _{k\in K}S{d}_i^k};\forall i\in C$$

(14)

$$G{d}_i\le {\displaystyle \sum _{k\in K}S{t}_i^k};\forall i\in C$$

(15)

$${\displaystyle \sum _{i\in C_0}{x}_{ij}^k}+{\displaystyle \sum _{i\in C_0}{y}_{ij}^k}\ge 1-r_j^k;\forall k\in K,\forall j\in C_{+}$$

(16)

$${\displaystyle \sum _{i\in C_0}{x}_{ij}^k}+{\displaystyle \sum _{i\in C_0}{y}_{ij}^k}\le 1+r_j^k;\forall k\in K,\forall j\in C_{+}$$

(17)

$${\displaystyle \sum _{i\in C_0}{x}_{im}^k}+{\displaystyle \sum _{i\in C_0}{y}_{im}^k}+{\displaystyle \sum _{j\in C_{+}}{x}_{mj}^k}+{\displaystyle \sum _{j\in C_{+}}{y}_{mj}^k}\ge 3r_m^k;\forall k\in K,\forall m\in C$$

(18)

$${\displaystyle \sum _{i\in C_0}{x}_{im}^k}+{\displaystyle \sum _{i\in C_0}{y}_{im}^k}+{\displaystyle \sum _{j\in C_{+}}{x}_{mj}^k}+{\displaystyle \sum _{j\in C_{+}}{y}_{mj}^k}\le 2+2r_m^k;\forall k\in K,\forall m\in C$$

(19)

Constraints (7) and (8) indicate that each truck and each drone both depart from the starting depot and return to the ending depot. Constraint (9) ensures that each customer is served once by a truck or drone. Constraint (10) ensures the entry and exit balance for the nodes in the truck distribution path, and all customer nodes traversed by trucks receive deliveries directly from the trucks, while constraint (11) ensures that the customer nodes not visited by trucks are serviced by drones deployed from the trucks. Constraints (12) and (13) ensure that there are no sub-tours in the truck routes and drone flight paths, respectively [45]. Constraint (14) ensures that the truck routes avoid restricted traffic zones, while constraint (15) ensures that the drone delivery paths avoid no-fly zones. Constraints (16) to (19) establish the rules for truck and drone arrivals and departures. Constraints (16) and (17) ensure that each customer is visited only once by either a truck or a drone, independently. However, when a truck synchronizes with a drone at a specific customer node, constraints (16) and (17) become nonbinding. In such cases, constraints (18) and (19) describe three scenarios: the drone can be retrieved by the truck, launched by the truck, or first collected and then launched at the node.

• The loading constraints are the following:

$$L{t}_j^k\ge L{t}_i^k-q_j-L{d}_i^k-M(1-x_{ij}^k);\forall k\in K,\forall i\in C_0,\forall j\in C_{+}$$

(20)

$$L{t}_j^k\le L{t}_i^k-q_j-L{d}_i^k+M(1-x_{ij}^k);\forall k\in K,\forall i\in C_0,\forall j\in C_{+}$$

(21)

$${\displaystyle \sum _{i\in C}{q}_i\cdot (S{t}_i^k+S{d}_i^k)+w_D\le }{Q}_T;\forall k\in K$$

(22)

$$A{d}_j^k\le L{d}_i^k+M(1-y_{ij}^k);\forall k\in K,\forall i\in C_o,\forall j\in C_{+}$$

(23)

$$A{d}_j^k\ge L{d}_i^k-M(1-y_{ij}^k);\forall k\in K,\forall i\in C_o,\forall j\in C_{+}$$

(24)

$$L{d}_i^k\le A{d}_i^k-q_i(1-r_i^k)+M(1-y_{ij}^k);\forall k\in K,\forall i\in C_0,\forall j\in C_{+}$$

(25)

$$A{d}_i^k\le (1-r_i^k)M;\forall k\in K,\forall i\in C_{+}$$

(26)

Constraints (20) to (22) ensure that truck capacities are not exceeded. Constraints (20) and (21) update the delivery load of truck k as it moves from node i to node j. Constraint (22) ensures that the truck’s capacity is respected at all times. Constraints (23) to (26) guarantee the feasibility of drone payloads. Constraints (23) and (24) update the drone load when it travels from node i to node j after serving node i. Constraint (25) specifies that the drone’s load is updated when it leaves node i after serving customer i by itself. Constraint (26) ensures that all collected packages are unloaded before a drone begins a new sub-route.

• The drone flight distance constraints are the following:

$$E_i^k\le E+M(1-r_i^k);\forall k\in K,\forall i\in N$$

(27)

$$E_i^k\ge 0-M(1-r_i^k);\forall k\in K,\forall i\in N$$

(28)

$$E_j^k\ge E_i^k-d_{ij}·(2-x_{ij}^k-y_{ij}^k)-M(1-y_{ij}^k);\forall k\in K,\forall i\in C_0,\forall j\in C_{+}$$

(29)

$$E_j^k\ge E_i^k-M·(2-x_{ij}^k-y_{ij}^k);\forall k\in K,\forall i\in C_0,\forall j\in C_{+}$$

(30)

$$E_j^k\le E_i^k+M·(2-x_{ij}^k-y_{ij}^k);\forall k\in K,\forall i\in C_0,\forall j\in C_{+}$$

(31)

Constraints (27)–(31) manage the drone flight distance feasibility. Constraints (27) and (28) indicate that the remaining flight distance at the drone’s launch node must be greater than the maximum flight distance and that the remaining flight distance at the drone’s recovery node must be greater than zero. Constraint (29) updates the remaining flight distance when drone k flies alone from node i to node j. Constraints (30) and (31) ensure that the remaining flight distance of the drone remains unchanged when the truck carrying the drone moves from node i to node j.

4. Solution Method

Due to the NP-hard nature of the multi-visit vehicle routing problem with drones under the carbon trading mechanism (MVRPD-CR), exact algorithms are computationally intensive and time-consuming. Given the characteristics of the problem under a carbon trading mechanism, which involves optimizing routes for both trucks and drones while considering constraints such as drone payload capacity, flight endurance, no-fly zones, truck payload limits, restricted areas, and carbon emission allowances, the solution space becomes highly constrained, and the complexity significantly increases. To address these challenges, a two-phase algorithm was proposed to approximate the optimal solution (SHIA-EVNS). In the first phase, a “Sweep and Heuristic Insertion” approach was employed to construct an initial feasible solution, which serves to expand the search space for the subsequent optimization phase. During the second phase, an enhanced variable neighborhood search (EVNS) algorithm was utilized to refine and optimize the routes for both trucks and drones, aiming to minimize the overall cost. The pseudocode for SHIA-EVNS is presented in Algorithm 1.

Algorithm 1: SHIA-EVNS

$\mathbf{Inputs}: c_T, c_D, C_T, C_D, d_{ij}, Q_T, Q_D, E, WAER, PGFER, AER, \stackrel{-}{CE}, \epsilon$

$\mathbf{Outputs}: \mathrm{Best} \mathrm{drone}-\mathrm{truck} \mathrm{solution} S_{B\mathrm{e}st}$

1: Truck-only routes $\leftarrow$ scanning method

$2: \mathrm{Initial} \mathrm{drone}-\mathrm{truck} \mathrm{solution} S_{Initial}\leftarrow$ heuristic insertion method

$3: \mathrm{Best} \mathrm{drone}-\mathrm{truck} \mathrm{solution} S_{B\mathrm{e}st}\leftarrow$ EVNS algorithm

$4: \mathbf{Return} S_{B\mathrm{e}st}$

4.1. Initial Solution Construction

Given the dynamic nature of truck payload capacity across different customer locations, traditional greedy algorithms and random initialization methods require frequent checks for payload constraints, thus increasing computational time. Consequently, a “Scan and Heuristic Insertion” method (SAHI) was proposed to construct an initial solution. This approach ensures that the generated initial solution adheres to the payload feasibility constraints of truck routes while also expanding the solution space for the subsequent phase by integrating drone deliveries, thereby enhancing both the efficiency and quality of the search process.

The method initially applies the scan algorithm to build complete routes for truck-only deliveries, determining the number of trucks required, followed by a heuristic insertion phase to integrate drone deliveries into the truck routes, converting some customer deliveries to drone services to form the initial solution. The detailed steps are outlined below:

Step 1: Coordinate System Establishment

Establish a coordinate system with the central depot as the pole and the line connecting any customer point to the central depot as the polar axis, transforming the remaining customer locations into polar coordinates.

Step 2: Customer Point Scanning

Initiate scanning with the customer point at the minimum angle, proceeding counterclockwise, sequentially adding customer points to the truck route while calculating current cumulative demand $Q_i$. Once $Q_i+w_D$ exceeds the truck’s payload capacity $Q_T$, designate the customer node i as the starting point for the next truck route and continue scanning the remaining customers.

Step 3: Truck Route Formation

Connect the first and last customer points in each truck route to the central depot; the initial truck routes are fully determined, as depicted in Figure 2a.

Step 4: Drone Delivery Integration via Heuristic Insertion

For each truck path, use heuristic insertion to convert some customers into drone delivery, meeting their flight distance and load constraints. The specific operation is as follows:

(1)

Select the node near the central depot as a drone launch point within each truck route, and the launch point cannot be located in a restricted area;

(2)

For each potential drone delivery point, calculate the drone delivery ratio $z=\sigma /Q_i$, where $\sigma$ represents the distance between the drone’s launched node and the truck’s delivery node, and $Q_i$ is the demand for the distribution node. Sort the calculated ratios $z$ and select the delivery node corresponding to the smallest $z$ for drone delivery. Once the delivery point is located in a restricted area, then it will be prioritized for delivery by drones;

(3)

Remove the selected drone delivery point from the truck route, add it to the drone route, and update the drone’s remaining flight distance and payload, as illustrated in Figure 2b;

(4)

Reiterate Operations (2) and (3) until the drone’s payload and flight distance constraints are exceeded, marking the last point as a retrieved point. If a delivery point falls within a no-fly zone, exclude it from the drone route and add it back to the truck route;

(5)

Adjust the truck delivery route according to the updated drone delivery points;

(6)

Select the node near the retrieved point as the newly launched point;

(7)

Repeat Operations (2) to (6) until all customer points on each truck route are selected, as shown in Figure 2c;

4.2. Enhanced Variable Neighborhood Search (EVNS)

To enhance both the efficiency and quality of the solution search, the variable neighborhood search (VNS) algorithm was enhanced, with the design of neighborhood structures serving as a pivotal element in determining the effectiveness of VNS. For each initial route of individual trucks, eleven distinct neighborhood structures were devised to refine the initial solution. Subsequently, a Randomized Variable Neighborhood Descent (RVND) strategy was employed for local search to further optimize the route configurations. To circumvent entrapment in local optima, a simulated annealing acceptance criterion was integrated to facilitate escape from such suboptimal states.

4.2.1. Neighborhood Construction

Three neighborhood operation methods were designed: intra-route mixed insertion, inter-route mixed insertion, and two-point exchange.

Intra-Route Mixed Insertion: Within a single truck delivery route, without altering the mode of customer service, a random customer point is selected for insertion at the position that minimizes greedy insertion costs. This includes three types of insertions: truck point insertion, drone insertion, and launch/recovery point insertion, as illustrated in Figure 3.

• Truck Point Insertion: Under the condition of not violating restricted area constraints, two points are randomly chosen within the truck route. Point (A) is inserted before point (B). Should the insertion result in an increase in the number of restricted arcs traversed, nodes are randomly replaced until the restricted area constraints are satisfied;
• Drone Point Insertion: A random drone service point is selected and inserted into one of the drone sub-routes associated with the same truck. The insertion location was chosen to minimize cost and could potentially be in another sub-route;
• Launch/Recovery Point Insertion: The launch or recovery point is inserted into a new position within its respective truck route. If the insertion causes the launch point to follow the recovery point or vice versa, the affected drone sub-route must be reversed. If the operation results in an increased number of restricted arcs, the operation is abandoned.

Inter-Route Mixed Insertion: For a single truck, under conditions where the mode of customer service is changed, a feasible position with minimal greedy insertion cost is randomly selected. This encompasses three approaches, as depicted in Figure 4.

• Truck Point Insertion: A truck point is inserted into a drone route of the same truck, shifting responsibility for delivery to the drone, or a new drone sub-route is established for this truck point, choosing the less costly insertion location. If the truck point is within a no-fly zone, a new truck point is selected;
• Drone Point Insertion: A drone point is inserted into a truck route, enabling the truck to complete the delivery. The insertion is executed if it does not lead to an increase in restricted area arcs. Otherwise, a new selection is made;
• Launch/Recovery Point Insertion: The original launch point transforms into a drone service point, with the selection of a truck route point with the least cost as the new launch point. The original launch point is vacated in the truck route, while the new launch point remains unchanged. The handling of recovery points follows the same procedure.

Two-Point Exchange: For a single truck, two points are randomly selected from the serviced customers for exchange, encompassing five forms, as shown in Figure 5.

• Truck Point Exchange: Two customer points are randomly picked from the truck route for exchange. The exchange is executed if it does not lead to an increase in restricted area arcs. Otherwise, one of the nodes is replaced until the exchange can be performed;
• Drone Point Exchange: Within the truck–drone route, two drone visit points are randomly selected, and their corresponding launch points are swapped in totality;
• Truck–Drone Inter-Route Exchange: A truck point is exchanged into a drone route of the same truck, transitioning to drone delivery, or a new drone sub-route is established for the truck point, selecting the location with minimal cost. If the truck point lies in a no-fly zone, a new selection is initiated;
• Launch/Recovery Point Same-Route Exchange: The launch (or landing) point of a drone is converted into a new launch (or landing) point, with a change in the delivery sequence;
• Launch/Recovery Point Inter-Route Exchange: The launch (or landing) point of a drone is removed from its existing route and inserted into another drone route, forming a novel drone delivery route. The original route’s launch (or landing) point is correspondingly adjusted.

4.2.2. Local Search

The VNS algorithm expands the search space of the solution by calling local search in each iteration, drawing on the random VND strategy to obtain the local optimal solution by continuously changing the order of neighborhood movement [26]. The pseudocode for random VND is presented in Algorithm 2.

Algorithm 2: random VND

$1. \mathbf{Input}: S^{\prime }\leftarrow$$\mathrm{Generate} \mathrm{neighborhood} \mathrm{solution}, \mathrm{F}(S^{\prime })\leftarrow$$\mathrm{Calculate} \mathrm{the} \mathrm{objective} \mathrm{function} \mathrm{of} S^{\prime }$$, N_k\leftarrow$$\mathrm{Neighborhood} \mathrm{structure} \mathrm{k}, k=1,\dots ,11$

$2. \mathrm{Set} k\leftarrow 1$

$3. \mathbf{While} k\le 11$$\mathbf{Do}$

$4. S^{″}\leftarrow$$\mathrm{Find} \mathrm{the} \mathrm{best} \mathrm{neighbor} \mathrm{of} S^{\prime }$$\mathrm{in} N_k$

$5. \mathbf{If} F(S^{″})<F(S^{\prime })$ then

$6. S^{\prime }\leftarrow S^{″},F(S^{\prime })\leftarrow F(S^{″}),k\leftarrow 1$

$7. \mathbf{Else}: k\leftarrow k+1$

8. End While

$9.\mathbf{Output} S^{\prime }, F(S^{\prime })$

4.2.3. Solution Stopping Criteria

We adopted a maximum iteration criterion to terminate the search in the proposed algorithm, concluding the loop after a predefined number of iterations. During the iterations, we used a simulated annealing criterion to accept new solutions. If the new solution was better than the current solution, it was accepted. Otherwise, the probability of accepting the new solution is given by the following formula:

$$P(S_{temp},S_{current})=\min \{\exp {\displaystyle \frac{S_{temp}-S_{current}}{T_n}},1\}$$

(32)

In the formula, $T_n=T_o{c}^n$ is the current temperature gradually decreasing during iteration, $T_o$ is the initial temperature, and $c^n$ is the cooling rate at the nth iteration. Assuming $T_o=-\mathsf{\Phi }\left|S_{current}\right|/\ln 0.5$, $\mathsf{\Phi }$ is the initial temperature control parameter, $S_{current}$ is the current solution, and $S_{temp}$ is the new solution. If $S_{temp}$ is worse than $S_{current}$ under the initial temperature condition, there is still a 50% probability that it will be accepted.

5. Numerical Experiments and Analysis

5.1. Case Study and Relevant Parameters

These experiments utilized cases A, B, and P from the Solomon dataset as test cases. For each experiment, several customers were randomly designated as restricted areas and no-fly zones. The customer demand data in the cases are adjusted to meet the standard that 80% of packages weigh less than 2.3 kg. Relevant parameters include the following:

• Drone maximum flight distance: 20 km [28];
• Fixed cost per truck: 200 CNY per vehicle, delivery cost: 1.5 CNY per km;
• Fixed cost per drone: 45 CNY per unit, delivery cost: 0.3 CNY per km;
• Maximum truck payload: 150 kg, self-weight: 30 kg;
• Maximum drone payload: 5 kg, self-weight: 1 kg;
• Carbon emission coefficient for trucks: 1.2603 kg/mi, CO₂ emission from drone charging: 0.0003773 kg/Wh;
• Carbon quota: 150 kg, carbon price: 0.5 CNY/kg;
• Drone energy consumption per km: 3.333 Wh/km [46];
• Average carbon emission price: 0.1332 CNY/kg.

All algorithms were implemented using MATLAB R2022b and executed on a computer with AMD R5 3550H @ 3.7 GHz, 16 GB RAM (America), running Windows 11. The maximum iteration count for the algorithm was 1000, the initial temperature was 0.01, and the cooling factor was 0.98. The experimental results for each case were averaged over 10 runs.

5.2. Model and Algorithm Validation

5.2.1. Experiment with Small-Scale Instances

To evaluate the performance of SHIA-EVNS, we compared the solutions of the proposed algorithm across 13 instances from three sets against the optimal or near-optimal solutions obtained from CPLEX. The results are summarized in

Table 2. Results of small-scale examples.

Instance	CPLEX		SHIA-EVNS		Gap (%)
	$\mathit{O}\mathit{b}{\mathit{j}}_{\mathit{C}\mathit{P}\mathit{L}\mathit{E}\mathit{X}}$	${\mathit{T}}_{\mathit{C}\mathit{P}\mathit{L}\mathit{E}\mathit{X}}$	$\mathit{O}\mathit{b}{\mathit{j}}_{\mathit{S}\mathit{H}\mathit{I}\mathit{A}-\mathit{E}\mathit{V}\mathit{N}\mathit{S}}$	${\mathit{T}}_{\mathit{S}\mathit{H}\mathit{I}\mathit{A}-\mathit{E}\mathit{V}\mathit{N}\mathit{S}}$
A1-n8	333.4	890.6	336.7	27.42	0.68
A2-n8	335.6	834.2	338.1	26.22	0.74
A3-n8	368.9	783.5	370.5	26.82	0.43
B1-n8	407.7	913.6	409.4	27.04	0.41
B2-n8	412.1	858.3	414.9	27.83	0.67
P-n8	465.4	927.4	469.7	28.15	0.92
P-n8	438.9	912.7	438.9	29.20	0.00
P-n16	493.2	1234.6	495.5	29.86	0.46
P-n19	536.3	1385.3	508.9	30.37	0.51
P-n20	638.2 *	2643.6	522.5	31.22	−18.1
P-n21	694.3 *	3600	544.3	31.95	−21.6
P-n22	720.1 *	3600	561.8	32.36	−21.9
p-n23	785.3 *	3600	587.5	33.05	−25.1
Avg.	421.27	1706.4	461.36	29.34	−6.29

* The optimal solution cannot be obtained within a time limit of 3600 s.

. The first column lists the instances, followed by four columns detailing the best objective values and running times for both CPLEX and SHIA-EVNS. The last column, $Gap=(Ob{j}_{SHIA-EVNS}-Ob{j}_{CPLEX})/Ob{j}_{CPLEX}\times 100\%$, presents the relative gap between the best solutions from SHIA-EVNS and CPLEX. CPLEX terminates once an optimal solution is found or the time limit of 3600 s is reached.

From

Table 3. Results of large-scale examples.

Set	Instances	GA			TPSCA			SHIA-EVNS
		Obj1	T1(s)	Gap1	Obj2	T2(s)	Gap2	Obj3	T3(s)
1	A-n32	755.3	25.82	13.45%	675.0	34.25	1.39%	665.7	30.05
1	A-n33	706.2	26.08	15.54%	638.5	35.07	4.46%	611.2	31.18
1	A-n34	808.4	26.34	17.85%	746.4	35.23	8.82%	685.9	31.45
1	A-n36	826.1	27.26	14.18%	786.3	36.21	8.68%	723.5	32.56
1	A-n37	858.3	27.75	13.29%	808.6	36.84	5.79%	764.3	33.12
1	A-n38	892.9	28.19	11.87%	845.4	37.22	5.92%	798.1	33.64
1	A-n39	927.4	28.64	12.67%	878.3	38.35	6.70%	823.1	34.14
2	A-n44	962.6	29.19	9.39%	921.3	39.08	4.71%	879.9	34.85
2	A-n45	991.4	30.11	9.26%	946.4	39.83	4.30%	907.3	35.31
2	A-n48	1068.5	30.84	14.69%	971.3	40.28	4.26%	931.6	36.33
3	A-n53	1122.6	31.76	12.74%	1084.5	41.35	8.91%	995.7	37.18
3	A-n54	1148.5	32.32	11.99%	1107.6	42.64	8.00%	1025.5	38.27
3	A-n55	1186.8	33.08	13.41%	1175.3	43.18	12.32%	1046.4	38.83
4	A-n62	1308.7	34.51	13.68%	1246.8	44.27	8.30%	1151.2	39.24
4	A-n63	1347.5	35.36	13.62%	1289.4	45.16	8.72%	1185.9	39.87
4	A-n65	1399.4	35.89	13.01%	1308.6	46.27	5.66%	1238.4	40.24
4	A-n69	1477.8	36.25	13.11%	1398.8	47.05	7.06%	1306.5	41.32
	Avg.	1046.37	30.55	13.07%	989.36	40.13	6.47%	926.44	35.74

, we observe that for smaller instances, the quality of CPLEX solutions surpasses that of SHIA-EVNS. However, as the problem size increases, SHIA-EVNS outperforms CPLEX in solution quality. Regarding solving time, SHIA-EVNS has an average solving time of 29.34 s, which is significantly shorter than that of CPLEX. In terms of solution quality, the average Gap value between SHIA-EVNS and CPLEX is −6.29%, indicating that SHIA-EVNS solutions are, on average, 6.29% better than those of CPLEX.

In summary, SHIA-EVNS demonstrates superior solution quality and efficiency compared with CPLEX. As problem size increases, SHIA-EVNS can handle larger scale problems in a shorter time, providing optimal approximate solutions for highly practical optimization problems.

5.2.2. Experiment with Large-Scale Instances

Due to the increasing size of the instances, obtaining an accurate solution with CPLEX becomes challenging. Therefore, GA [20] and TPSCA [37] were chosen for comparison. To adapt these two algorithms to MVRPD-CR, the number of no-fly zones and restricted traffic zones in the comparative algorithms, as well as the handling of these zones, were consistent with SHIA-EVNS. The weight value for TPCSA was set at 0.2, with a maximum of 1000 iterations and a population size of 500. For GA, the maximum number of iterations was 1000, the population size was 150, the crossover probability was 0.7, and the mutation probability was 0.3. The comparison results are shown in Table 3 and Figure 6, where $Gap1=(Obj1-Obj3)/Obj3\times 100\%$ and $Gap2=(Obj2-Obj3)/Obj3\times 100\%$ represent the differences in the best objective values between the comparison algorithms and SHIA-EVNS.

From Table 3, it can be seen that in terms of solution quality, SHIA-EVNS outperforms the other two methods in all cases. In more than 40 large-scale instances, the SHIA-EVNS algorithm shows significant advantages. Compared with GA, the solution quality of SHIA-EVNS improves by 9.26% to 14.69%. Compared with TPSCA, SHIA-EVNS improves solution quality by 1.39% to 12.32%. SHIA-EVNS effectively balances local and global search capabilities.

Regarding solving time, as the size of the instances increases, the solving time generally increases. However, SHIA-EVNS can solve problems in a relatively short time. The running time of SHIA-EVNS was slightly longer than that of GA, with an average difference of 5.19 s, due to the diverse neighborhood searches of SHIA-EVNS. Despite this, it can still obtain better solutions than GA within 42 s. Compared with TPSCA, the running time of SHIA-EVNS was reduced by 4.39 s. Especially when $n\ge 60$, SHIA-EVNS obtained a feasible solution within 42 s, whereas TPSCA required more than 44 s.

The test instances were divided into four sets based on the customer node intervals. Figure 6 shows a box plot visualization of the distribution of objective values obtained from running each of the three algorithms 10 times for each test set. The middle line of each box represents the median of the 10 results, the upper and lower edges represent the upper and lower quartiles, and the vertical lines at the ends represent the maximum and minimum values. As shown in Figure 6, the distribution of approximate optimal solutions obtained by the SHIA-EVNS algorithm was generally more concentrated compared with other algorithms. Moreover, the maximum value of SHIA-EVNS was still lower than the minimum value of other algorithms in Test Set 3, verifying the stability and superiority of SHIA-EVNS.

5.3. Comparison of Different Distribution Methods

To demonstrate the effectiveness of the truck–drone collaborative distribution model in reducing carbon trading costs, it was compared with the conventional truck-only distribution model. For customer points in restricted traffic zones, trucks detour to complete deliveries. Both distribution models were implemented using the SHIA-EVNS algorithm, and the results are presented in

Table 4. Results of different delivery methods.

Instances	Truck–Drone Collaborative					Truck-Only				N	TD	EC
	${\mathit{N}}^{\mathit{T}}$	${\mathit{E}}^{\mathit{T}}$	${\mathit{E}}^{\mathit{U}}$	$\mathit{T}\mathit{D}$	$\mathit{C}\mathit{C}$	${\mathit{N}}^{\mathit{T}}$	${\mathit{E}}^{\mathit{T}}$	TD	CC
A-n32	3	49.21	0.98	675.0	−49.9	4	73.21	874.6	−38.4	1	29.5%	46.77%
A-n33	3	40.15	0.67	611.2	−54.9	4	65.42	838.1	−42.9	1	37.1%	60.26%
A-n34	3	51.39	0.89	685.9	−48.8	4	75.83	887.3	−37.1	1	29.4%	43.56%
A-n36	3	55.27	0.78	723.5	−46.9	4	79.36	918.4	−35.3	1	26.9%	41.58%
A-n37	3	58.39	1.36	764.3	−45.1	4	86.32	939.5	−31.8	1	22.9%	44.46%
A-n38	4	60.82	1.45	798.1	−43.8	5	92.45	986.2	−28.8	1	23.8%	48.46%
A-n39	4	65.93	2.07	823.1	−41.2	5	98.67	1046.5	−25.7	1	27.1%	46.26%
A-n44	4	71.78	1.49	879.9	−38.4	5	109.7	1128.3	−20.2	1	28.2%	49.72%
A-n45	4	79.98	1.97	907.3	−34.0	5	119.8	1209.4	−15.1	1	33.2%	46.19%
A-n48	5	86.35	2.32	931.6	−30.6	6	128.4	1287.1	−10.8	1	37.5%	44.81%
A-n53	5	94.24	2.67	995.7	−26.5	6	139.5	1369.5	−5.25	1	37.5%	43.95%
A-n54	5	105.9	2.98	1025.5	−20.5	6	147.6	1457.3	−1.2	1	42.1%	35.56%
A-n55	5	109.4	3.02	1046.4	−18.8	7	158.3	1568.2	4.15	2	49.8%	40.81%
A-n62	6	119.3	3.67	1151.2	−13.5	7	165.5	1633.4	7.75	1	41.8%	34.58%
A-n63	6	127.5	3.44	1185.9	−9.5	7	178.6	1748.1	14.3	1	47.4%	36.86%
A-n65	6	136.4	3.65	1238.4	−4.9	7	189.4	1869.3	19.7	1	50.9%	35.23%
A-n69	6	146.1	1.76	1306.5	−1.1	7	200.3	2003.7	25.2	1	51.2%	35.46%
Avg.		85.77	2.07	926.44	−31.08		124.0	1280.3	−13.02		36.3%	43.20%

. In the table, $N^T$ is the number of trucks, $E^T$ denotes carbon emissions from trucks, $E^U$ is carbon emissions from drones, $TD$ is the total costs, and $CC$ represents the carbon trading cost when $CC<0$ indicates that enterprises can earn profits by selling carbon emission rights. Additionally, $N$ denotes the reduction in the number of trucks, $TD$ represents the percentage improvement in total cost, and $\mathrm{EC}$ indicates the percentage improvement in carbon emissions.

From Table 4, it is evident that compared with the traditional single-truck distribution model, the truck–drone collaborative distribution model offers substantial advantages in reducing carbon emissions, saving total costs, and decreasing carbon trading expenditures. Specifically, the collaborative distribution method reduces total costs by 36.3% and decreases average carbon emissions by 43.2%. As carbon emissions decline, companies gain additional carbon quotas and can sell surplus quotas for extra revenue, averaging an additional income of 18.06 yuan from carbon trading.

With the expansion of case scale, the benefits of collaborative distribution become more pronounced due to the drones’ ability to serve customers in restricted zones, reducing truck travel distances and lowering delivery costs. Drones, with their lower carbon footprint and increased route flexibility for multi-point deliveries, contribute to a decrease in the number of required trucks.

In summary, the truck–drone collaborative delivery model has demonstrated significant economic and environmental benefits, especially in large-scale delivery tasks.

5.4. Analysis of the Carbon Trading Mechanism

Carbon trading prices and quotas are not static but dynamically adjusted based on market supply and demand. To explore the impact of variations in carbon trading prices and quotas on corporate economic benefits and environmental effects, the A-n65 case from Section 4.2 is analyzed. The total cost, carbon trading cost, and carbon emissions are calculated for carbon quota ranges from 0 kg to 300 kg (increasing by 50 kg increments) and carbon trading prices from 0.5 yuan/kg to 1 yuan/kg (increasing by 0.25 yuan/kg increments). The results are summarized in

Table 5. Results of changes in carbon trading prices and quotas.

Carbon Trading Price	Carbon Quota	Carbon Emissions	Carbon Trading Cost	The Total Cost
0.5	0	140.05	70.03	1313.31
	50	140.05	45.02	1288.3
	100	140.05	20.02	1263.3
	150	140.05	−4.9	1238.4
	200	140.05	−29.9	1213.4
	250	140.05	−54.9	1188.4
	300	140.05	−79.9	1163.4
0.75	0	134.67	101.0	1389.5
	50	134.67	63.5	1352.0
	100	134.67	26.0	1324.0
	150	134.67	−11.5	1286.5
	200	134.67	−48.9	1249.1
	250	134.67	−86.4	1211.6
	300	134.67	−123.9	1174.1
1	0	128.53	128.53	1497.2
	50	128.53	78.53	1447.2
	100	128.53	25.53	1394.2
	150	128.53	−21.4	1347.2
	200	128.53	−71.4	1297.2
	250	128.53	−121.4	1247.2
	300	128.53	−171.4	1197.2

.

From Table 5, it is observed that with a fixed unit carbon price, increasing the carbon quota from 150 kg to 300 kg allows companies to profit from selling excess carbon emission rights, thereby reducing total costs. However, the carbon emissions of the obtained solutions remain nearly constant across different carbon quotas, as there is no direct correlation between the carbon quota and the decision variables in the model. When the carbon quota is fixed, an increase in the unit carbon price leads to higher carbon trading and total costs if the company’s quota is less than its operational requirements. Conversely, if the quota exceeds the company’s needs, the profits from selling excess quotas increase with rising unit carbon prices, thus decreasing total costs.

Carbon emissions decrease step-wise with an increase in the unit carbon price, indicating that beyond a certain threshold, the cost savings from substituting truck deliveries with drones outweigh the increase in fixed and electricity costs. This prompts companies to favor drone deliveries, resulting in a significant reduction in carbon emissions. Therefore, governments can promote the concurrent development of corporate economic and environmental benefits by reasonably setting carbon trading policies and controlling carbon trading prices and quotas within certain ranges.

5.5. Sensitivity Analysis

To validate the influence of critical parameters, a sensitivity analysis was conducted based on four parameters: traffic restriction ratio ($\alpha$), no-fly zone ratio ($\beta$), maximum drone payload ($Q_D$), and maximum drone flight distance (E). Each parameter was set at five levels, generating 40 scenarios for the cases A-n32 and A-n65. The average results from 10 runs for each scenario are depicted in Figure 7.

Figure 7a illustrates the impact of the no-fly zone ratio on costs. As the proportion of no-fly zones increases, the number of packages deliverable by drones decreases, pushing costs closer to those of truck-only distribution. The effect of the traffic restriction ratio on costs is shown in Figure 7b. As the truck restriction ratio rises, the number of drone-delivered packages increases, initially reducing costs. However, due to the drones’ limited payload, multiple launches are required to complete multiple sub-routes for delivery, causing costs to rise after a 15% restriction ratio, suggesting an optimal balance exists between drone and truck usage. Figure 7c depicts the impact of drone payload on costs. Higher drone payload capacities lead to cost reductions, with the most significant decrease occurring when $Q_D\le 20$. Figure 7d exhibits the effect of maximum drone flight distance on costs. Costs decrease rapidly at first and then more gradually as the maximum flight distance increases.

In summary, optimal performance is achieved when there are fewer drone no-fly zones, greater drone payload and flight distance, and fewer traffic restrictions, allowing drones to leverage their advantages more effectively and minimize total costs.

6. Conclusions

Under the carbon trading framework, this study addressed the collaborative truck–drone delivery vehicle routing problem, considering constraints such as no-fly zones for drones and restricted areas for trucks. By introducing carbon trading factors, we constructed a mixed-integer programming model and developed a two-stage solution method. First, the “Scan and Heuristic Insertion” technique constructs an initial solution, allocating delivery tasks to trucks and determining the optimal number of trucks and drones. Subsequently, a variable neighborhood search algorithm with multiple neighborhood strategies optimizes the delivery routes for both trucks and drones, aiming to minimize total costs. The simulation results lead to the following conclusions:

(i)

The effectiveness of the model and algorithm are validated through different scale comparisons. For small-scale instances, the SHIA-EVNS algorithm generally outperforms CPLEX in solution time and results. For large-scale instances, SHIA-EVNS outperforms GA and TPSCA in both aspects;

(ii)

Compared with truck-only delivery, the truck–drone collaboration reduces carbon trading costs, lowering total costs by 36.3%;

(iii)

Carbon emissions decrease step-wise with higher unit carbon prices. If the carbon quota is below the operational need, both carbon trading and total costs increase with the unit price. If the quota exceeds the need, both costs decrease with the unit price;

(iv)

Sensitivity analysis of parameters indicates that there is an optimal combination of drones and trucks; more drones do not necessarily lead to better outcomes.

Achieving the strategic goals of “peak carbon dioxide emissions” and “carbon neutrality” requires joint efforts from both the government and enterprises. Enterprises must enhance their awareness of carbon reduction and incorporate environmental considerations into production and transportation decisions. At the same time, the government should strengthen the supervision of carbon emissions, prudently regulate carbon quotas and trading prices, and create an environment conducive to the simultaneous development of corporate economic viability and environmental protection. The experience and data accumulated by enterprises in reducing carbon emissions can, in turn, assist government departments in formulating more scientific and reasonable carbon quota management and carbon trading mechanisms. This two-way interaction not only helps optimize resource allocation but also enhances the overall level of environmental governance in society.

Future research directions should incorporate real-world complexities such as traffic congestion and dynamic customer demands into the model to further optimize total delivery costs. This comprehensive approach will not only refine the theoretical foundations of collaborative delivery systems but also enhance their practical applicability in real-world logistics operations, thereby promoting sustainable and efficient supply chain management practices. By continuously improving and optimizing carbon trading mechanisms and collaborative delivery models, enterprises and governments can jointly advance the development of green logistics, achieving a win-win situation for both economic benefits and environmental protection.