Related work

In recent years, the rapid development of MEC and UAV technologies has led to a growing body of research focused on optimizing the collaborative operation of these two technologies. In UAV-assisted MEC networks, much of the research has concentrated on optimizing individual performance metrics, such as computation resource allocation, task offloading strategies, energy efficiency, and latency optimization. Current studies primarily focus on improving specific objectives, such as reducing computation delay, minimizing energy consumption, or enhancing task offloading efficiency, under predefined constraints. However, there remains a lack of research that comprehensively considers multiple performance metrics, particularly in the context of optimizing overall system utility.

Many studies focus on optimizing a single performance metric, such as computation delay, energy efficiency, or task offloading efficiency. For instance, [5] proposed a trajectory prediction-based edge offloading strategy to enhance the processing efficiency of computational tasks by optimizing the task offloading path. This method aims to reduce transmission latency through task offloading path optimization but does not consider the broader system utility, such as the balance between the utility of service providers (UAVs, base stations) and the costs incurred by users. Similarly, [2,6–10] applied game theory to examine how UAV-assisted MEC can improve system performance through computation offloading optimization, as well as optimize user scheduling, UAV management, and power allocation. While these game-theoretic approaches effectively optimize task offloading decisions, they focus mainly on optimizing individual objectives, such as energy efficiency or task offloading efficiency, without fully considering the global system utility. [11] proposed an efficient energy minimization scheme that optimizes UAV trajectory, resource allocation, and task offloading strategies, significantly reducing the network’s energy consumption. This approach transformed a non-convex optimization problem with multi-variable coupling into a more manageable form by introducing auxiliary variables, successfully achieving energy consumption optimization. However, the scheme primarily focuses on energy consumption. [6] optimized UAV trajectories and area division through a task offloading framework, reducing energy consumption during the task offloading process. The optimal transportation theory introduced in this study provided services to multiple users, successfully optimizing task offloading paths and energy efficiency. While it excels in terms of energy efficiency and offloading efficiency, it does not address the balance of benefits and costs between service providers and users, remaining limited to optimizing a single objective.

In UAV-assisted MEC systems, Deep Reinforcement Learning (DRL) [12–14] has been widely applied to optimize task offloading and trajectory design. [15] proposed a framework based on Double Deep Q-Learning (D-DQN) to optimize the 3D trajectory design and task offloading strategy of UAVs through online learning in dynamic environments. This method addresses dynamic optimization issues to some extent, enhancing task offloading and energy efficiency. However, it does not consider the trade-off between user costs and system revenues. [16] introduced a DRL-based approach aimed at optimizing UAV flight trajectories to maximize average secrecy rates in wireless security. While this study effectively improved system security, it focused solely on secrecy metrics and did not account for the balance among multiple performance metrics such as task offloading, energy efficiency, and latency. [17] proposed an iterative algorithm that jointly optimized UAV beamforming, CPU frequency, trajectory design, and user transmission power, successfully reducing the system’s energy consumption. Although this study provided valuable insights into resource allocation in UAV-assisted MEC systems, it remained focused on optimizing a single objective (energy efficiency).

In UAV-assisted MEC systems and Industrial Internet of Things (IIoT) environments [18,19], many studies have applied heuristic algorithms to optimize task offloading, resource allocation, and energy efficiency. [20] proposed a user satisfaction-oriented task offloading and UAV scheduling strategy to optimize task offloading and user satisfaction. This study employed a genetic algorithm (GA) and simulated annealing (SA) to combine task offloading decisions with UAV scheduling strategies, improving overall user satisfaction. However, it remains based on single-objective optimization and does not comprehensively balance multiple performance metrics. [21] focused on resource allocation in IIoT environments and proposed a learning-based resource allocation scheme using a Cooperative Particle Swarm Optimization (LCPSO) algorithm, successfully optimizing response time in forest fire monitoring. Although this approach improved response time, it did not sufficiently address the weight balancing problem across multiple performance metrics [22]. Furthermore, many existing studies, when faced with complex constraints, tend to rely on heuristic algorithms and convex optimization methods to solve problems. For example, an energy consumption minimization scheme based on a low-complexity resource allocation algorithm was proposed, which optimized user delay and energy efficiency, particularly in Heterogeneous Non-Orthogonal Multiple Access (HNOMA) networks. While this scheme significantly improved energy efficiency, it remained focused on energy efficiency alone, without a comprehensive improvement in system utility.

Previous research [23,24] applied the BSUM algorithm to optimize overall energy consumption and latency in task offloading within MEC systems. The novelty of this study lies in proposing a global system utility optimization scheme based on the BSUM framework. By comprehensively considering the differences between user costs and service provider (UAV and base station) revenues, a new multidimensional optimization framework is introduced. However, despite these advancements, most existing studies focus on optimizing individual performance metrics, such as computation delay, energy efficiency, or task offloading efficiency. These approaches typically overlook the balance between the revenues of service providers (UAVs and base stations) and the costs incurred by users. For instance, as shown in Table 1, studies like those of [23] primarily focus on optimizing specific metrics, such as task offloading and energy efficiency, but often fail to address the balance between provider revenues and user costs.

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Table 1. Summary of key contributions and methodologies of selected studies on UAV-assisted MEC.

https://doi.org/10.1371/journal.pone.0342583.t001

Furthermore, while some studies utilize the BSUM algorithm or other optimization techniques, they generally focus on single-objective optimization, such as minimizing energy consumption or enhancing task offloading efficiency, without considering the trade-off between service provider revenue and user costs. The current literature also lacks a unified, comprehensive framework capable of simultaneously addressing task offloading, resource allocation, and pricing strategies, while ensuring an optimal balance between service providers’ revenues and user costs.

Base station layer

In the MEC network, ground base stations possess powerful computing capabilities and a continuous power supply. The set of users requiring task offloading is denoted as , where represents the maximum number of users in the set. Similarly, the sets of base stations and UAVs serving these users are represented as and , where and denote the maximum number of base stations and UAVs in the respective sets.

For each base station b, user i can either directly offload tasks to the base station for computation or offload tasks to the base station via UAV j as a relay. The base station sets a price for the computational resources used by user i, and a price for the relay UAV j. The base station’s computational resources F^b are limited, and the resources allocated to user i are denoted by . Let represent the user’s task offloading decision variable, where indicates that the task is offloaded to the base station via a wireless link, and otherwise. Similarly, indicates whether the task of user i is offloaded to the base station through UAV j as a relay.

The offloading decisions can be mathematically expressed as follows:

(1)

(2)

The base station’s utility function is defined as , where P_b represents the profit of the base station from providing computation services to users, and C_b is the cost of using UAVs. These can be expressed as:

(3)

(4)

UAV layer design

In the MEC network, UAVs can serve either as base stations to receive and compute tasks offloaded by users, or as relay nodes to forward users’ tasks to base stations for computation. When a UAV functions as a base station, it sets a pricing for the computing resources allocated to user i. Similarly, the UAV’s computing resources F_j are limited. Let denote the user task offloading decision variable, indicating whether user i offloads the task to UAV j for computation.

(5)

When the UAV acts as a base station, it sets the pricing for the computing resources allocated to users, and similarly, the UAV’s computing resources are also constrained. The utility function of the UAV layer can be expressed as , where and represent the UAV’s revenue when acting as a base station and a relay, respectively, while and represent the computational cost when the UAV acts as a base station and the communication cost for transmitting tasks as a relay, as shown below:

(6)

(7)

where represents the computing resources allocated to user i by UAV j, and

(8)

(9)

where P_j is the transmission power of UAV j, and is the transmission rate from UAV j to base station b. Let represent the number of CPU cycles required to complete the computation for user i.

User layer design

In the MEC network, the utility of the user is determined by both the task offloading cost and the processing cost , which includes computation delay and communication cost during task transmission. Let represent the user’s decision variable indicating whether user i performs local computation for the task.

(10)

The task offloading cost is defined as:

(11)

The computation cost is expressed as:

(12)

where P_i is the transmission power of user i, and represent the user’s offloading decisions. Based on these, the utility of the user layer can be expressed as (Table 2).

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Table 2. Summary of assumptions, variables, and system parameters.

https://doi.org/10.1371/journal.pone.0342583.t002

System utility and optimization problem

In the user offloading system designed for our MEC network, the goal is to maximize the overall system utility while ensuring that constraints, such as UAV and base station computing resources and offloading decisions, are satisfied. In the proposed system model, the interactions between users, UAVs, and base stations are critical for optimizing overall system utility. Users offload their computational tasks to the nearest UAV or base station, depending on resource availability and pricing strategies set by the UAV and base station. UAVs, acting as either relays or computation units, interact with both users and base stations to maximize the overall system utility by optimizing the offloading decisions, resource allocation, and pricing schemes. Base stations, with powerful computing resources, set prices and allocate resources to users either directly or via UAVs. Thus, the system-level utility depends on the joint decisions made by users, UAVs, and base stations, making their interactions a key part of the optimization problem.

The overall system utility is defined as the difference between the total revenue generated by the service providers (UAVs and base stations) and the costs incurred by the users. Therefore, the system utility is influenced by task offloading decisions, resource allocation (computation and communication), and pricing strategies. The utility function for the overall system is defined as U = U_b  +  U_j − U_i, where U_b represents the utility of the base station, U_j represents the utility of the UAV, and U_i represents the utility of the user.

To facilitate the optimization process and make the problem compatible with standard optimization techniques, we transform the objective from maximizing U to minimizing −U. This transformation is commonly used in optimization, as most solvers are designed to minimize objective functions. Maximizing U is equivalent to minimizing −U, as the optimal solution remains unchanged, and the problem is computationally more convenient when expressed as a minimization problem.

Thus, the objective becomes minimizing −U, which is mathematically equivalent to maximizing U. Our optimization problem can be reformulated as:

(13)

This transformation ensures that the optimization problem remains consistent with the goal of maximizing system utility, while leveraging optimization techniques that are optimized for minimization problems.

Finally, we can express the formula for the overall system utility as:

(14)

The bounds for the pricing and offloading parameters, such as , , and , are determined based on practical considerations and typical network operations. The minimum values are constrained by the operational costs of maintaining the UAVs and base stations, ensuring that these services remain economically viable. On the other hand, the maximum values are constrained by market conditions and competitive pricing, ensuring that pricing does not become prohibitively high for users, which could lead to inefficiencies in offloading decisions. The offloading parameters are similarly bounded by the available computational and communication resources of the network, ensuring that offloading decisions remain feasible within the system’s capabilities.

(15)

These bounds ensure that the system operates within realistic parameters while maintaining both efficiency and feasibility.

Eq (17) represents the constraint on the computing resources allocated to all users by the base station and UAV. Eq (18) defines a range constraint on the pricing set by the base station and UAV for all users. Eq (19) indicates that each user can only choose one offloading method, and the decision is binary. Due to the coupling, non-linearity, binary constraints, and the non-convex structure of the decision variables in the objective function, convex optimization techniques cannot be directly applied to solve the problem.

To effectively solve the non-convex offloading optimization problem, we adopt the Block Successive Upper Bound Minimization (BSUM) method. BSUM decomposes the problem into several subproblems and constructs a local convex approximation for each subproblem, effectively avoiding the convergence issues that traditional methods, such as the Alternating Direction Method of Multipliers (ADMM), may encounter when dealing with non-convex problems. Additionally, BSUM offers lower computational complexity, making it suitable for solving large-scale, multi-task decision-variable optimization problems. In our case, BSUM demonstrates superior performance, particularly when handling complex non-convex optimization problems that involve both discrete and continuous decision variables.

In the BSUM framework, the upper bound of the objective function is minimized iteratively over the variables , , and . To apply the BSUM framework, the binary decision variable constraints in Eq (19) are relaxed into continuous constraints, i.e., , , , and . This relaxation is a standard approach in non-convex optimization that simplifies the problem, enabling the use of convex optimization techniques more effectively. By treating the binary variables as continuous, we can solve the problem using standard optimization methods. However, this relaxation introduces potential errors, as the continuous solutions may not directly satisfy the original binary constraints. To mitigate this, a post-processing step is applied in which the continuous values are rounded to the nearest binary values (0 or 1), ensuring that the final solution adheres to the original binary constraints.

These methods guarantee that the solution complies with the original problem’s binary constraints. Although the relaxation introduces some error, it does not significantly affect the optimality of the solution, as the resulting binary solution is close to the continuous optimal solution. This relaxation technique provides a more tractable approach to solving complex non-convex problems, while still maintaining feasibility and optimality with respect to the original binary constraints. We can now introduce the feasible set of , , and .

(16)

(17)

(18)

(19)

Finally, we define the closest upper bound function T_n for the objective function in Eq (13). For each iteration t, , where N is the index set. To ensure that the upper bound function is more beneficial for our optimization, we apply a quadratic penalty to the objective function in Eq (13), as follows:

(20)

where is the penalty parameter, which can also be applied to other variables. Additionally, in each iteration t, the upper bound function is minimized with respect to using the values , which are the solutions from the previous iteration. The solution at iteration t + 1 can be updated by solving the following optimization problem:

(21)

(22)

(23)

The iterative procedure in the BSUM framework updates the decision variables (M, J, and ) in each iteration by minimizing a surrogate objective function for each respective variable. Each subproblem is solved independently, and the solution is updated iteratively until convergence. The convergence of this iterative process is guaranteed under the assumptions of the BSUM framework, which ensures convergence to a stationary point due to the block structure and the use of upper-bound minimization. However, since the problem is non-convex, convergence to the global optimum is not guaranteed. Each iteration involves solving three subproblems corresponding to M, J, and , with the complexity of each subproblem depending on the number of users, UAVs, and base stations. Specifically, solving each subproblem requires linear or quadratic optimization, so the overall complexity per iteration is ⋅ , where n represents the number of decision variables and m is the number of iterations required for convergence.

In the BSUM framework, we must also solve subproblems for three sets of variables. Since task offloading, pricing, and resource allocation involve multiple decision variables and complex constraints, and the optimization objectives are often nonlinear, traditional exact optimization methods (such as linear programming and integer programming) are frequently unable to find the global optimal solution within a reasonable time for large-scale problems, due to the vast solution space. Heuristic algorithms, which do not rely on the continuity or differentiability of the problem, are more suitable for handling complex combinatorial optimization problems. For the optimization of pricing variables, resource allocation, and offloading strategies, the interdependencies between variables and the complex constraints make exact solutions difficult. Heuristic algorithms can flexibly adapt to these changes, handle discrete decision spaces and complex constraints, and thus provide effective solutions. Therefore, the use of heuristic algorithms, such as GAs and SA, offers an effective solution.

Algorithm 1 Joint optimization of system utility based on the BSUM framework.

1: Initialization: Set t = 0, , and find the initial feasible solution.

2: repeat 3:   Select the index set N.

4:   Set .

5:   Set .

6:   Interface with GA: Pass the continuous decision variables (, , ) to the GA to optimize the discrete decision variables.

7:   Update with GA output: Receive the optimized discrete variables from GA and update , , .

8:   Interface with SA: Pass the updated continuous variables to the SA algorithm for further refinement of the continuous decision variables.

9:   Update with SA output: Receive the refined continuous variables from SA and update , , .

10:   Set t = t + 1.

11: unit

12: Final Solution: Set , , and as the required solution.

Optimization of Pricing Variables: GA. The optimization of pricing variables involves the pricing set by the base station for users, the pricing set by the base station for UAVs, and the pricing set by the UAV for users. These pricing variables need to be optimized under multiple constraints, with the goal of maximizing system benefits and reducing costs.

(24)

The Genetic Algorithm (GA) is used to optimize the pricing variables, including those set by base stations and UAVs. The population size is set to 200, striking a balance between solution diversity and computational efficiency. Tournament selection is employed with a tournament size of 4, which helps maintain diversity and avoid premature convergence. The GA terminates either after 50 generations or when the fitness improvement is smaller than a threshold ().

The crossover rate is set to 0.9, and the mutation rate is set to 0.25, based on preliminary testing. A sensitivity analysis showed that the algorithm remains stable across a range of parameters, demonstrating robustness. The GA evaluates and selects solutions, iterating through crossover and mutation steps to improve the pricing strategy and maximize overall system utility.

Algorithm 2 Optimizing pricing variables using GA.

1: Randomly initialize the population .

2: Define the fitness function based on the system objective.

3: Select the index set N.

4: Select individuals based on their fitness.

5: Perform crossover and mutation operations to generate new individuals.

6: Update the population with the newly generated individuals.

7: Repeat the steps until convergence (maximum generations or fitness threshold is reached).

8: repeat

9:  

10: until

11: Output the optimized pricing variables .

Optimization of Resource Allocation Variables: SA Algorithm. The optimization of resource allocation variables involves the distribution of computing resources and by the base station and UAV. The optimization objective is to maximize the utilization of computing resources while ensuring that the resource allocation meets the system’s demands and constraints.

(25)

The SA algorithm is used to optimize the resource allocation variables. The SA algorithm perturbs the current solution randomly and uses the Metropolis criterion to decide whether to accept the new solution, thereby avoiding getting stuck in local optima.

Algorithm 3 Optimizing resource allocation variables using SA.

1: Initialize the resource allocation variables .

2: Define the objective function for resource allocation.

3: Set the initial temperature and cooling coefficient.

4: while the temperature is greater than the final temperature do

5:   Generate a new solution by perturbing the current solution.

6:   Calculate the objective function value of the new solution.

7:   Accept the new solution based on the Metropolis criterion, with the acceptance probability depending on the temperature.

8:   Lower the temperature.

9: end while

10: Output the optimized resource allocation variables .

The optimization of offloading strategy variables involves the user’s task offloading decisions . These decisions need to be optimized based on the user’s computational requirements, network conditions, and resource availability to maximize the system’s utility.

(26)

The offloading strategy variables are also optimized using the SA algorithm. The algorithm steps are outlined in Algorithm 3, where the offloading strategy is randomly adjusted, and the new offloading scheme is evaluated based on the objective function. This process gradually converges to the optimal solution.

The solution methods for these three optimization problems involve the use of GAs and SA algorithms to optimize the pricing variables, resource allocation variables, and offloading strategy variables, respectively. The GA optimizes pricing and resource allocation through a global search, while the SA algorithm optimizes the offloading strategy by gradually cooling down, ensuring optimal performance of the entire system under complex constraints.