Distributionally Robust Chance-Constrained Task Assignment for Heterogeneous UAVs with Time Windows Under Uncertain Fuel Consumption

Abstract

This paper addresses the cooperative task assignment problem for heterogeneous unmanned aerial vehicles with time windows considering uncertain fuel consumption. In the scenario where probabilistic fuel consumption exists and its distribution needs to be estimated from historical data samples, we first formulate the problem as a chance-constrained combinatorial optimization problem and utilize the sample average approximation method to solve it. Further, to address the issue of ambiguous distribution, we introduce distributionally robust chance constraints, which consider a set of probability distributions that are contained within a 1-Wasserstein ball centered around the empirical distribution of field data. We approximate the distributionally robust chance-constrained cooperative task assignment problem by applying a CVaR-based tractable approximation such that the problem can be transformed into a deterministic mixed-integer linear programming problem, which can be efficiently solved by state-of-the-art optimization solvers. Finally, we conduct a series of numerical experiments, which not only verify the computational efficiency of the distributionally robust chance-constrainted models but also reduce the degree of constraint violation in out-of-sample tests compared with a sample average approximation method.

1. Introduction

1.1. Motivation

Unmanned aerial vehicles (UAVs) play an important role in missions such as the suppression of enemy air defenses because of its flexibility and low-cost. The multi-UAVs cooperative task assignment problem (CTAP) [1] has been proposed in the existing literature to optimize the optimal path of UAVs. Subsequently, to meet the time requirements for the task, scholars have introduced time windows to CTAP, resulting in the cooperative task assignment problem with time windows (CTAPTW). It describes a heterogeneous UAV fleet comprising reconnaissance UAVs, attack UAVs, and integrated UAVs assigned to perform a sequence of reconnaissance, attack, and verification tasks on specified targets. Specially, these three tasks for each target must be executed in a strict chronological order and completed within a predefined time window.

In the existing literature of CTAP as well as the CTAPTW, there are many related research works on the deterministic optimization model. For example, Chen et al. [2] modeled the reconnaissance task assignment as a multiple time-window-based Dubins traveling salesmen problem and employed a modified multi-objective symbiotic organisms search algorithm to optimize UAVs task sequences. Wu et al. [3] addressed the task assignment profit maximization problem with dynamic reconnaissance and confirmation tasks under critical time and multi-UAV coupling constraints by extending the consensus-based bundle algorithm and shortening operation time using a distributed genetic algorithm. Compared with the single types of tasks considered in [2], literature [3] studies the task coordination of heterogeneous UAVs in performing three types of tasks, including reconnaissance, attack, and verification.

In addition to multiple constraints like time windows and task coordination, the existence of uncertain parameters is also a major challenge for CTAP [4]. To address parameter uncertainties in CTAP, various uncertain optimization methods have been developed based on different theoretical frameworks. For instance, stochastic optimization (SO), which models uncertain parameters as random variables with known probability distribution has been applied to CTAP. Yang et al. [5] adopted this approach to solve a chance-constrained task profit maximization problem under resource constraints, where resource consumption was modeled as a random variable. Meanwhile, robust optimization (RO), which focuses on optimization under the worst-case scenarios of parameter variations within a predefined uncertainty set [6], has also been explored.

However, existing methods face a critical limitation: they cannot handle scenarios in CTAP where parameters exhibit randomness with ambiguous probability distributions (i.e., where distributional information is incomplete or imprecise). This limitation can lead to suboptimal or even infeasible solutions when confronted with real-world uncertainties. To address this issue, distributionally robust optimization (DRO) [7] has been proposed and widely applied in finance [8,9], energy systems [10], and supply chain management [11,12]. In finance, DRO has been used in portfolio selection [13], taking into account the uncertainty in asset returns. In energy systems, DRO helps make robust decisions under uncertain renewable energy production. In supply chain management, DRO can be applied to handle uncertainties in demand [14] and transportation costs [15].

Motivated by the above research works, this paper proposes a distributionally robust chance-constrained (DRCC) framework to characterize the ambiguity of random variables in CTAPTW within a defined distributional ambiguity set (DAS) based on Wasserstein distance [16]. By considering worst-case distribution, DRCC yields a more conservative and reliable solution. This approach ensures that the proposed solution remains effective when the true probability distribution cannot be precisely known. Furthermore, a key advantage of using the Wasserstein distance lies in its ability to regulate the level of conservatism by adjusting the radius of the Wasserstein ball. Specifically, selecting an appropriate radius can guarantee that the true probability distribution is contained in the DAS with a specified confidence level. Therefore, the DRCC method can ensure the reliability of the task assignment solutions.

1.2. Literature Review

The CTAP shares structural similarities with the vehicle routing problem (VRP) in that both involve optimizing resource allocation and path planning. However, CTAP introduces additional complexity by incorporating two critical issues: the heterogeneity of UAVs [17], which differ in flight performance, payload capacity, and energy consumption rates; and task cooperative constraints [18] among multiple task types, such as reconnaissance, attack, and verification tasks, which require cooperative execution to achieve task objectives. These distinct characteristics of CTAP distinguish it as a unique class of combinatorial optimization problems. Currently, the problem of parameter uncertainty in VRP [19] has been extensively explored, leading to the development of diverse solution methods. In contrast, the research on parameter uncertainty in the CTAP remains in its infancy, which highlights the urgent need for more in-depth research to develop effective strategies for deriving optimal task assignment.

In VRP and CTAP, the RO method has been widely used due to its computational tractability. As a powerful method in addressing optimization problems under uncertainty, RO does not require that the probability distribution of uncertain parameters is known exactly. Building on the budgeted uncertainty sets proposed by Bertsimas and Sim [20], Chen et al. [21] modeled failure rates as uncertain variables, formulating CTAP via an RO framework and solving it with a modified two-part wolf pack search algorithm. In the meantime, Lu et al. [22] proposed a robust formulation where uncertain demand is supported by cardinality-constrained sets. This approach specifies the demand range without assuming any probability distribution to characterize uncertainty. In contrast to [21,22], which introduces additional constraints and variables to encapsulate uncertainties, Munari et al. [23] directly derived the robust counterpart from the deterministic model using a novel compact RO formulation, effectively addressing demand and travel time uncertainties. Jeong et al. [24] proposed a two-stage adaptive RO framework to minimize worst-case energy consumption while ensuring services are delivered within appointed time windows, solving the model via a column-and-constraint generation heuristic algorithm integrated with variable neighborhood search and an alternating direction algorithm. However, RO fails to incorporate historical data and its distribution, which may lead to excessively conservative solutions that neglect task profit.

Compared with RO, SO has the advantage of leveraging historical data to construct its distribution. In the context of SO, research diverges based on the type of random variables considered. For problems involving random travel times, Evers et al. [25] tackled real-time operational uncertainties in UAV task planning with time windows by a dynamic re-planning strategy to maximize expected profit. Complementing this, Bergsma et al. [26] applied the sample average approximation (SAA) method with Monte Carlo simulation to routing optimization, though Jaillet et al. [27] highlighted that SAA and its reliance on accurate distribution assumptions may yield suboptimal solutions if these assumptions deviate from reality. Shifting focus to random demand, Yu et al. [28] minimized expected travel costs when demand fluctuations could be bounded within known probabilistic parameters. These studies collectively demonstrate how different random variables, including travel times, real-time operational durations, and demand fluctuations, promote distinct methodological approaches in SO. However, in practice, identifying the precise probability distribution of a random variable is often challenging due to insufficient historical. Furthermore, any deviation between the assumed and true distributions can lead to infeasible decisions, which is an unmanageable issue in combat scenarios.

Meanwhile, as a specific form of constraint in SO, chance-constrained programming (CCP) has been proposed as a viable alternative for addressing uncertainty in CTAP. This approach enables controlled tolerance of risk by formulating constraints to hold with a specified probability, thus balancing solution robustness against excessive conservatism in uncertain environments. CCP allows for a controlled tolerance of risky outcomes, balancing solution conservatism by ensuring constraints are satisfied with a specified probability rather than for all possible scenarios. For example, Yang et al. [29] formulated the problem as a chance-constrained framework for simultaneous task assignment and path planning. By leveraging the mean and variance of each path’s travel cost, their approach ensures that the total path cost remains below a minimum threshold with high probability for any realization of stochastic travel costs. In addition, a risk-averse task allocation problem was formulated as a deterministic integer optimization model to address the chance-constrained optimization problem under payoff uncertainty in [30]. The approach assumes payoffs follow Gaussian distributions with known means and variances, transforming probabilistic constraints into deterministic equivalents for tractable computation.

However, while chance constraints can control risks by adjusting confidence levels, they suffer from limitations that restrict their applicability. First, this approach requires full knowledge of the probability distribution, which is particularly difficult to obtain with limited historical data, especially given the high uncertainty inherent in combat environments. Second, chance constraints are often non-convex [31], posing significant challenges for solving such optimization problems.

Rather than assuming a perfectly known probability distribution for uncertain parameters, DRO constructs a DAS encompassing a family of probability distributions. By identifying decisions that perform optimally under the worst-case distribution within this set, DRO takes into account parameter uncertainties, effectively bridging the gap between deterministic and fully stochastic approaches. Notably, this method has been scarcely applied in CTAP thus far. As a novel data-driven paradigm, DRO has emerged to address the limitations of traditional SO and RO by leveraging partial distributional knowledge within a specified ambiguity set. Its core objective is to optimize the expected value under the worst-case distribution within this ambiguity set, thereby explicitly addressing probabilistic ambiguity. By constructing the DAS from statistical insights derived from available uncertain data, DRO yields solutions that are both data driven and less conservative than conventional approaches.

Although DRO has seen growing applications in various process optimization domains, no prior studies have applied DRO to optimize CTAP under uncertainty. Inspired by Ordoudis et al. [32], this paper addresses CTAPTW under uncertainty by employing a novel data-driven DRCC optimization method to replace deterministic total fuel consumption constraints. The DRCC ensures that the total flight fuel consumption of each UAV meets constraints with a specified probability, valid within a subset of probability distributions encapsulated in a Wasserstein DAS. This DAS is structured as a 1-Wasserstein metric ball centered on an empirical distribution derived from limited training datasets, enabling RO under scarce data conditions. The DRCC is approximated using the CVaR constraint, which allows the CTAP involving uncertain fuel consumption to be approximately reformulated as a MILP. This reformulation enables the problem to be solved exactly and relatively easily by off-the-shelf solvers.

1.3. Proposed Approaches

Based on the aforementioned research gaps, this paper constructs a DRCC optimization model to study the multi-UAV CTAPTW in an uncertain environment. The main contributions are as follows:

• A multi-UAV cooperative task assignment model with time window:different from traditional CTAP models that often ignore time constraints, the model proposed in this paper explicitly incorporates time window constraints to ensure that tasks are executed in a specified sequence within specific time intervals. This design is more in line with the strict requirements for time coordination in actual task scenarios.
• Construction of a DRCC optimization model to handle uncertainty: this paper pioneers the application of DRCC to the CTAPTW in uncertain environments. To tackle the uncertainty in fuel consumption, a DAS is constructed based on partial distribution information of fuel consumption. By leveraging CVaR approximation to handle chance constraints, the DRCC model is transformed into a deterministic MILP model, which can be efficiently solved using standard optimization solvers. Furthermore, the size parameters of the uncertainty set, such as the radius of the Wasserstein ball, are rigorously derived. This ensures that the true probability distribution is contained within the uncertainty set at a specified confidence level, thereby striking a balanced trade-off between the conservatism and optimization efficiency of the model.
• Experimental verification and performance advantages: through numerical experiments, this paper verifies the computational efficiency and robustness of the DRCC model. The experimental results show that compared with the RO method and the SAA method, the DRCC model not only ensures higher task returns while significantly reducing the total flight distance and saving fuel consumption but also significantly reduces the constraint violation rates within and outside the samples. Moreover, this paper studies the influence of Wasserstein radius and risk tolerance parameters on the model, and investigates the adjustment strategies of the two to enable the DRCC model to flexibly adapt to the requirements of different scenarios and achieve a better balance between optimality and conservatism.

1.4. Organization of the Paper

The structure of this paper is as follows. In Section 2, we present the nominal deterministic MILP formulation of the CTAPTW for heterogeneous multi-UAVs. Section 3 focuses on modeling fuel consumption CCP model, which is transformed using both SAA and DRCC models, transforming the CCP model with risk tolerance into a MILP model using CVaR-based tractable approximation and duality theory. Section 4 presents a case study that includes performance evaluation, sample size sensitivity analysis, parameter analysis of distributional robustness, and impact analysis of the risk tolerance, illustrating the feasibility and effectiveness of our DRCC models. Finally, Section 5 offers the conclusions of the study.

2. The Nominal Cooperative Task Assignment Problem with Time Windows

This section describes the cooperative task assignment problem for heterogeneous UAVs with time windows and presents its mathematical formulation. The analysis herein assumes that the fuel consumption between targets for each UAV is known with certainty. This case is referred to hereinafter as the nominal problem. A MILP formulation for the CTAPTW is put forward.

2.1. Description of the UAV Cooperative Task Assignment Problem with Time Windows

The problem considered in this paper is to assign $N_v$ heterogeneous UAVs to complete different types of tasks on $N_t$ targets within the given time window.

There are $N_t$ stationary ground targets considered in the problem, along with an aircraft base denoted as $T_0$. The target set T is composed of $T_1,T_2,\cdots ,T_{N_t}$. For each of these $N_t$ targets, there are three sequential tasks to be carried out: reconnaissance, attack, and verification. The task set $M_T$ consists of three elements: “C” representing reconnaissance, “A” representing attack, and “V” representing verification. Each target requires the execution of three tasks, so the total number of tasks is $3N_t$.

The $N_v$ heterogeneous UAVs are made up of $N_{v_1}$ reconnaissance UAVs, $N_{v_2}$ attack UAVs, and $N_{v_3}$ integrated UAVs. Different types of UAVs have different task execution capabilities. Integrated UAVs can perform all three tasks of reconnaissance, attack, and verification. Reconnaissance UAVs can only carry out reconnaissance and verification tasks, not attack tasks. Attack UAVs are only capable of performing attack tasks.

2.2. Modeling Assumptions and Notation

Prior to formulating the problem, the following assumptions are made.

Table 1. System of notation.

$N_t$	Number of targets
$N_v$	Number of UAVs
$i,j$	Index of targets, $i,j=\{1,\dots ,N_t\}$
$D_{i,j}$	Euclidean geometric distance between targets i and j
k	Index of UAVs, $k=\{1,\dots ,N_v\}$
$V_k$	Flight speed of UAV k
m	Index of task type, $m=\{1,2,3\}$
$l_k$	Index of training samples for UAV k, $l_k=\{1,\dots ,L_k\}$
${\rho }_k$	Wasserstein radius of distributionally robust chance constraint for UAV k
$1-{ϵ}_k$	Individual chance constraint’s confidence level for UAV k
${\xi }_{i,j}^{k,m}$	Nominal fuel consumption from target i to j for UAV k performing task m
${\tilde{\xi }}_{i,j}^{k,m}$	Uncertain fuel consumption from target i to j for UAV k performing task m
${\widehat{\mathsf{\xi }}}_{i,j}^{k,m,l}$	Fuel consumption from target i to j for UAV k performing task m according to training sample l
${\overline{\xi }}_{i,j}^{k,m}$	Uncertain fuel consumption upper bound from target i to j for UAV k performing task m
${\underline{\xi }}_{i,j}^{k,m}$	Uncertain fuel consumption lower bound from target i to j for UAV k performing task m
C	Reconnaissance task corresponding $m=1$
A	Attack task corresponding $m=2$
V	Verification task corresponding $m=3$
${\omega }_{j,k,m}$	Task profit gained by UAV k performing the task m of target j
$p_j^m$	Duration of the task m of the target j
${\Delta }_{t_m}$	Time interval after executing the task m
$s_{j,k}^m$	Start time of the task m of target j if it is allocated to UAV k
$f_{j,k}^m$	End time of the task m of target j if it is allocated to UAV k

outlines the notation adopted in this paper.

• Known target locations: the geographical locations of all the targets are precisely known, which allows for the straightforward calculation of distances between targets, crucial for task assignment and path optimization for the UAVs.
• Limited UAV capabilities and safe flight: UAVs have inherent limitations in onboard resources, such as finite fuel. To ensure safe and efficient operations, UAVs are assigned to fly at different altitudes. This altitude separation strategy guarantees that the UAVs have collision-free flight paths, minimizing the risk of mid-air collisions and enhancing overall task safety.

2.3. Formulation of the UAV Cooperative Task Assignment Problem with Time Windows

2.3.1. Objective and Decision Variables

The problem aims to maximize the total task profit including reconnaissance task profit, attack task profit and verification task profit achieved by a heterogeneous UAV fleet. We have the following:

• Let $x_{i,j}^{k,m}\in \{0,1\}$ be the binary decision variable, which represents UAV k flying from target i to j to perform task m (reconnaissance: $m=1$, attack: $m=2$, verification: $m=3$).
• Let $t_{j,k}^m\ge 0$ be the non-negative real decision variable, which represents the time when UAV k executes task m of target j.

The objective is to maximize the the total task profit of the completed tasks:

$$max\sum _{j=1}^{N_t}\sum _{k=1}^{N_v}\sum _{m=1}^3{\omega }_{j,k,m}{x}_{i,j}^{k,m},$$

(1)

where ${\omega }_{j,k,m}$ represents the task profit that UAV k performs the task m of target j to gain. In particular, the task profit quantifies the strategic or operational value of UAV k completing task m of target j. It is derived from factors such as target importance, task urgency, and UAV capability matching (e.g., attack UAVs gain more profit from attack tasks due to specialization).

2.3.2. UAV Heterogeneity Constraints

• Reconnaissance UAVs cannot perform attack ($m=2$) task:

  $$x_{i,j}^{k,2}=0,k=1,2,\cdots ,N_{v_1}.$$

  (2)
• Attack UAVs cannot perform reconnaissance ($m=1$) and verification ($m=3$) task:

  $$x_{i,j}^{k,1}=0,x_{i,j}^{k,3}=0k=N_{v_1}+1,\cdots ,N_{v_1}+N_{v_2}.$$

  (3)

2.3.3. Task Execution Requirements

• All three types of tasks must be completed exactly once per target j:

  $$\sum _{i=0}^{N_t}\sum _{k=1}^{N_v}{x}_{i,j}^{k,m}=1,m=1,2,3,j=1,2,\cdots ,N_t.$$

  (4)
• The same UAVs entering a target must also exit it:

  $$\sum _{i=0}^{N_t}{x}_{i,j}^{k,m_1}-\sum _{l=1}^{N_t}{x}_{j,l}^{k,m_2}=0,m_1,m_2=1,2,3,k=1,2,\cdots ,N_v,j=1,2,\cdots ,N_t,$$

  (5)

  which is a typical constraint in the traditional VRP.

2.3.4. Temporal Coordination

• Time windows: task m of target j when assigned to UAV k is completed within the specified time window $[s_{j,k}^m,f_{j,k}^m]$:

  $$s_{j,k}^m{x}_{i,j}^{k,m}\le t_{j,k}^m,(f_{j,k}^m-p_j^m)x_{i,j}^{k,m}\le t_{j,k}^m,i,j=1,2,\cdots ,N_t,k=1,2,\cdots ,N_v,m=1,2,3,$$

  (6)

  where $s_{j,k}^m$ and $f_{j,k}^m$ represent, respectively, the start time and end time of task m for target j when assigned to UAV k, and $p_j^m$ denotes the duration of the task m.
• Task sequencing: reconnaissance ($m=1$) must precede attack ($m=2$), which precedes verification ($m=3$):

  $$\sum _{k=1}^{N_v}{t}_{j,k}^1+p_j^1+{\Delta }_{t_1}\le \sum _{k=1}^{N_v}{t}_{j,k}^2,\sum _{k=1}^{N_v}{t}_{j,k}^2+p_j^2+{\Delta }_{t_2}\le \sum _{k=1}^{N_v}{t}_j^3,j=1,2,\cdots ,N_t,$$

  (7)

  where ${\Delta }_{t_m}$ denotes the time interval after executing the task m.
• Travel time consistency: UAV k must be guaranteed to reach target j to execute subsequent tasks after completing tasks at target i:

  $$t_{i,k}^{m_1}+p_i^{m_1}+D_{i,j}/V_k\le t_{j,k}^{m_2}+K(1-x_{i,j}^{k,m_2}),i,j=1,2,\cdots ,N_t,k=1,2,\cdots ,N_v,$$

  (8)

  where $D_{i,j}$ represents the distance between targets i and j; $V_k$ denotes the flight speed of UAV k; and K is defined as a big-M parameter, set to $max\left\{f_{j,k}^m+p_j^m\right\}$.

2.3.5. Modeling Fuel Consumption Constraints

In CTAPTW, for a certain UAV, fuel consumption is not only related to the flight distance but also to the type of task being performed. The fuel consumption of reconnaissance, attack, and verification tasks carried out against the same target varies among different tasks. When carrying out the reconnaissance task, UAVs usually maintain low-speed flight, with less fuel consumption. The attack task requires UAVs to have high maneuverability, resulting in a significant increase in fuel consumption. The flight requirements of the verification task lie between the two. Although it does not require as high maneuverability as the attack task, it has a wider coverage area, and its fuel consumption is generally higher than that of the reconnaissance task. It implies that the fuel consumption ${\xi }_{i,j}^{k,m}$ from target i to j to execute task m for UAV k satisfies the following specific relationships:

$${\xi }_{i,j}^{k,1}<{\xi }_{i,j}^{k,3}<{\xi }_{i,j}^{k,2},$$

(9)

where ${\xi }_{i,j}^{k,1}$ corresponds to the fuel consumption of the reconnaissance task, ${\xi }_{i,j}^{k,2}$ represents the fuel consumption of the attack task, and ${\xi }_{i,j}^{k,3}$ denotes the fuel consumption of the verification task. Thus, we have

$${\xi }_{i,j}^{k,m}=c_m{D}_{i,j},i,j=1,2,\cdots ,N_t,k=1,2,\cdots ,N_v,m=1,2,3,$$

where $c_m$ denote the fuel consumption rate under the condition of performing task m. Because the total fuel consumption of UAV k cannot exceed the maximum fuel load $F^k$ of the UAV k, we have

$$\sum _{i=0}^{N_t}\sum _{j=1}^{N_t}\sum _{m=1}^3{\xi }_{i,j}^{k,m}{x}_{i,j}^{k,m}\le F^k,k=1,2,\cdots ,N_v.$$

(10)

2.3.6. The Nominal Model

The nominal cooperative task assignment problem with time windows for heterogeneous UAVs considering fuel consumption can be formulated as a combinatorial optimization problem:

$$\begin{array}{cc}max\hfill & {\displaystyle \sum _{j=1}^{N_t}}{\displaystyle \sum _{k=1}^{N_v}}{\displaystyle \sum _{m=1}^3}{\omega }_{j,k,m}{x}_{i,j}^{k,m}\\ \mathrm{s}.\mathrm{t}.\hfill & \left(2\right)–\left(8\right),\left(10\right),\end{array}$$

which includes $3(N_v\times N_t+N_v\times N_t^2)$ decision variables and $N_v+(2N_v\times L_k)+(3N_v\times L_k\times N_t^2)$ constraints.

3. Modeling Fuel Consumption Chance-Constrained Programming Model

Aircraft maneuvering actions and meteorological conditions, among others, can significantly affect the uncertainty of fuel consumption. The violent maneuvers such as climbing and diving increase resistance and engine power requirements and make fuel consumption difficult to estimate. Meteorological factors change fuel consumption by influencing ground speed, flight duration and aerodynamic characteristics.

The presence of uncertain fuel consumption poses a challenge under deterministic conditions. Utilizing mean fuel consumption values or other measures of central tendency may lead to frequent violations of the constraint, potentially causing incomplete or unexecuted task actions. Conversely, enforcing the constraint for all possible realizations of uncertain fuel consumption would yield an overly conservative task scheme. To address this parameter uncertainty, a common strategy is to employ probabilistic or chance constraints. As such, the original deterministic constraint (10) can be reformulated as the following chance constraint:

$$\mathbb{P}\left[\sum _{i=1}^{N_t}\sum _{j=1}^{N_t}\sum _{m=1}^3{\tilde{\xi }}_{i,j}^{k,m}{x}_{i,j}^{k,m}\le F^k\right]\ge 1-{ϵ}_k,k=1,2,\cdots ,N_v,$$

(11)

which stipulates that the original constraint is satisfied with a probability of at least $1-{ϵ}_k$, where ${ϵ}_k\in (0,1)$ represents a risk tolerance parameter that prescribes the maximum acceptable violation probability of the break fuel consumption constraint for UAV k. Note that, in the above equation, ${\tilde{\xi }}_{i,j}^{k,m}$ is the random variable of the fuel consumption from target i to j to execute task m of target j for UAV k, which follows a known distribution G.

Within the deterministic model framework, the fuel consumption values adhere to the relationship specified in (9). As a result, the relationships regarding both the expectation and variance among the three task types are retained in the stochastic context:

$$\mathbb{E}[{\tilde{\xi }}_{i,j}^{k,1}]<\mathbb{E}[{\tilde{\xi }}_{i,j}^{k,3}]<\mathbb{E}[{\tilde{\xi }}_{i,j}^{k,2}],$$

(12)

$$\mathrm{Var}[{\tilde{\xi }}_{i,j}^{k,1}]<\mathrm{Var}[{\tilde{\xi }}_{i,j}^{k,3}]<\mathrm{Var}[{\tilde{\xi }}_{i,j}^{k,2}].$$

(13)

3.1. Sample Average Approximation

Inspired by the existing literature [33] and leveraging the given sample data, we employ the SAA method to tackle chance-constrained optimization problems. This approach involves approximating the underlying probabilistic constraints by replacing expected values with empirical averages computed from the sample set, transforming the original stochastic model into a deterministic optimization problem that can be solved using conventional algorithms. Specifically, we first generate a finite sample of random variables based on historical data, then reformulate the chance constraints to ensure they are satisfied with a specified confidence level by enforcing constraints on the sample averages. This strategy not only inherits the computational tractability of deterministic models but also leverages statistical theory to guarantee that the solution remains valid in the presence of uncertainty.

Firstly, for the convenience of expression, the following random variable $X_k={\sum }_{i=1}^{N_t}{\sum }_{j=1}^{N_t}{\sum }_{m=1}^3{\tilde{\xi }}_{i,j}^{k,m}{x}_{i,j}^{k,m}$ is introduced to represent the total uncertain fuel consumption of UAV k. Then, the VaR equivalent form of chance constraint (11) is written as follows:

$${\mathrm{VaR}}_{1-{ϵ}_k}\left(X_k\right)\le F^k,k=1,2,\cdots ,N_v,$$

(14)

where VaR represents the maximum possible loss at confidence $(1-{ϵ}_k)$ level, and its definition is given as follows:

$${\mathrm{VaR}}_{1-{ϵ}_k}\left(X_k\right)=min\{{\eta }_k:F_{X_k}\left({\eta }_k\right)\ge 1-{ϵ}_k\},k=1,2,\cdots ,N_v,$$

(15)

where $F_{X_k}$ represents cumulative distribution function of UAV k fuel consumption.

However, because VaR is typically non-convex and discontinuous, CVaR is used to approximate it, enabling related problems to be solved more efficiently. And further, the CVaR approximation constraint of (11) is given by

$${\mathrm{CVaR}}_{1-{ϵ}_k}\left(X_k\right)\le F^k,k=1,2,\cdots ,N_v,$$

(16)

where the CVaR at confidence level $(1-{ϵ}_k)\in (0,1]$ is given by

$${\mathrm{CVaR}}_{1-{ϵ}_k}\left(X_k\right)=min\left\{{\eta }_k+{\displaystyle \frac{1}{{ϵ}_k}}\mathbb{E}\left({[X_k-{\eta }_k]}_{+}\right):\eta \in \mathbb{R}\right\},k=1,2,\cdots ,N_v,$$

(17)

where ${\left(a\right)}_{+}:=max\{0,a\}$.

It is well-established that the minimum value in the definition of CVaR (17) is achieved at the VaR with a confidence level of $(1-{ϵ}_k)$ [33], which forms the theoretical basis for leveraging CVaR in optimization problems. One of the most appealing features of CVaR is its tractability. For random variables with finite distributions, CVaR can be reformulated as a linear programming (LP) problem, which can be integrated into optimization models as demonstrated by Munari et al. [23]. The LP-based formulation not only simplifies the computational complexity but also enables an efficient solution using standard optimization solvers.

More precisely, consider a discrete random variable $X_k$ with realizations $X_k^1,\cdots ,X_k^N$ and associated probabilities $p_1,\cdots ,p_N$. By leveraging the finite distribution structure, we approximate the expectation $\mathbb{E}\left[{\left(X_k-{\eta }_k\right)}_{+}\right]$ using the sample average:

$$\mathbb{E}\left[{\left(X_k-{\eta }_k\right)}_{+}\right]\approx {\displaystyle \frac{1}{L_k}}\sum _{l=1}^{L_k}max\left\{0,X_k^l-{\eta }_k\right\},$$

(18)

where $X_k^l={\sum }_{i=1}^{N_t}{\sum }_{j=1}^{N_t}{\sum }_{m=1}^3{\widehat{\xi }}_{i,j}^{k,m,l}{x}_{i,j}^{k,m}$ is the total fuel consumption of UAV k under the l-th sample.

To linearize the non-linear $max\{0,X_k^l-{\eta }_k\}$ term, we introduce non-negative auxiliary variables $w_{k,l}$ with two constraints. Then, the CVaR optimization problem defined in (17) can be equivalently reformulated as the following LP for given UAV k:

$$\begin{array}{cc}\hfill \underset{{\eta }_k,w_{k,l}}{\min }& {\eta }_k+{\displaystyle \frac{1}{{ϵ}_k}}\sum _{l=1}^L{p}_{k,l}{w}_{k,l}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& w_{k,l}\ge X_k^l-{\eta }_k,l=1,\cdots ,L,\hfill \\ \hfill & w_{k,l}\ge 0,l=1,\cdots ,L,\hfill \end{array}$$

(19)

where ${\eta }_k\in \mathbb{R}$ represents the VaR threshold at confidence level $(1-{ϵ}_k)$.

Given the maximum fuel load $F^k$ of UAV k in (10), the optimization problem in (19) can equivalently be formulated as the following constraints:

$${\eta }_k+{\displaystyle \frac{1}{{ϵ}_k}}\sum _{l=1}^{L_k}{p}_{k,l}{w}_{k,l}\le F^k,k=1,2,\cdots ,N_v,$$

(20)

$$w_{k,l}\ge \sum _{i=1}^{N_t}\sum _{j=1}^{N_t}\sum _{m=1}^3{\widehat{\xi }}_{i,j}^{k,m,l}{x}_{i,j}^{k,m}-{\eta }_k,k=1,2,\cdots ,N_v,l=1,2,\cdots ,L_k,$$

(21)

$$w\in {\mathbb{R}}_{+}^N,$$

(22)

which introduces $N_v+(N_v\times L_k)$ auxiliary variables including $w_{k,l}$ and ${\eta }_k$, and $N_v+(N_v\times L_k)$ constraints.

Finally, based on the objective function (1) and constraints (2)–(8), (20)–(22), the SAA model of the multi-UAV CTAPTW is established.

3.2. Distributionally Robust Chance-Constrained Programming Model

Given the lack of the exact distribution of the empirical distribution ${\mathbb{P}}_N$ and the potential overfitting issue arising from the SAA-based method, this section employs distributionally robust programming to address the fuel consumption uncertainty:

$$\underset{P\in \mathcal{P}}{min}\mathbb{P}\left[\sum _{i=1}^{N_t}\sum _{j=1}^{N_t}\sum _{m=1}^3{\tilde{\xi }}_{i,j}^{k,m}{x}_{i,j}^{k,m}\le F^k\right]\ge 1-{ϵ}_k,k=1,2,\cdots ,N_v,$$

(23)

where $\mathcal{P}$ represents ambiguity set of distribution P. Constraints (23) indicate that, considering the uncertainty of the distribution, the original chance constraints (11) still hold under the worst-case scenario.

3.2.1. Illustrative Example

To justify the application of a DRCC in addressing the distributional ambiguity of the random fuel consumption ${\tilde{\xi }}_{i,j}^k$, we consider a single-path scenario from target i to j for an integrated UAV. At the onset of the task, three distinct task types are available for selection: reconnaissance ($m=1$), attack ($m=2$), and verification ($m=3$). Each task type is associated with a specific task profit, denoted as ${\omega }_{j,k,1}$, ${\omega }_{j,k,2}$, and ${\omega }_{j,k,3}$, respectively. These profits follow the order

$${\omega }_{j,k,1}<{\omega }_{j,k,3}<{\omega }_{j,k,2},$$

(24)

At this moment, the planner aims to devise a task scheme that ensures the successful completion of the task with a probability of at least 95% within a 30 kg (with adaptability to other energy metrics) fuel consumption. In the context of chance constraints, this requirement corresponds to setting the risk tolerance parameter ${ϵ}_k=0.05$.

For simplicity, assume that the task planner has a priori knowledge that fuel consumption follows a normal distribution. Consider an attack task from target i to j, which has a mean fuel consumption of 25 kg and a standard deviation of 5 kg. In contrast, a reconnaissance task has a mean fuel consumption of 20 kg and a standard deviation of 3 kg.

However, in reality, the task planner does not know the true probability distribution parameters. Instead, the planner only has access to a limited sample of three historical fuel-consumption realizations for each task. For the attack task, these realizations are 15 kg, 20 kg, and 25 kg; for the reconnaissance task, they are 17 kg, 20 kg, and 23 kg. Based on the true distribution of the reconnaissance task, option $m=2$ (the attack task) is infeasible. This is because, with a fuel consumption limit of 30 kg, the probability of completing the attack task is only $84.13\%$, which is lower than the required confidence level. Thus, the best feasible option is the reconnaissance task, which yields a task profit of ${\omega }_{j,k,1}$. Nevertheless, the planner will base its decision on the available fuel consumption sample. Calculating the sample average and standard deviation for the attack-task fuel consumption sample, which are 20 kg and 5 kg respectively, the estimated probability of completing the attack task is $97.72\%$. From the perspective of the fuel consumption sample, reconnaissance tasks seem more feasible. But since the attack task offers a higher profit, the task planner will be inclined to choose the attack task.

Therefore, due to sampling error, the planner may mistakenly select an infeasible option (the attack task) instead of the best feasible one (the reconnaissance task). When implementing the chosen action, the planner is faced with two choices: either violate the chance constraint, rendering the scheme infeasible, or abandon the task action entirely. It should be emphasized that abandoning this operation is not permitted.

Employing a DRCC with an appropriately sized DAS can effectively preclude the selection of infeasible options. To illustrate, consider a scenario where the DAS includes all normal distributions whose means lie within 5 units of the sample average. In such a case, the true distribution of the fuel consumption is guaranteed to be included within the set. This approach allows the planner to adopt a more prudent strategy, hedging against the inherent distributional ambiguity associated with uncertain fuel consumption. Instead of over-relying on limited fuel consumption samples, which frequently result in suboptimal out-of-sample performance, the DRCC with a well-defined DAS provides a more reliable framework for decision making. This ensures that the selected options are more likely to be feasible under a wider range of potential distributions, enhancing the robustness and effectiveness of the overall planning process.

3.2.2. Structure of the Distributional Ambiguity Set

To construct the DAS, it is assumed that the planner has access to a finite number $L_k$ of independent training samples ${\widehat{\xi }}_{i,j}^{k,m,l}$ with $l=1,2,\cdots ,L_k$ for UAV k. These samples can be used to construct the empirical distribution ${\widehat{P}}_k^{L_k}$ for the training samples. Furthermore, we require that all distributions in ${\mathcal{P}}_k$ are supported on the bounded polyhedral set ${\Xi }_k=\{{\tilde{\xi }}_k\in {\mathbb{R}}^{\Gamma }:H_k{\tilde{\xi }}_k\le h_k\}$, where $h_k\in {\mathbb{R}}^{O_k}$, $H_k\in {\mathbb{R}}^{O_k\times \Gamma }$, $O_r$ is the number of inequalities defining the polyhedral support set ${\Xi }_k$, and $\Gamma =N_t\times N_t!$ is the total number of flight paths available for each UAV. For example, ${\Xi }_k$ can be the Cartesian product of the closed interval sets $\left[{\underline{\xi }}_{i,j}^{k,m},{\overline{\xi }}_{i,j}^{k,m}\right]$, where the lower bound ${\underline{\xi }}_{i,j}^{k,m}$ and upper bound ${\overline{\xi }}_{i,j}^{k,m}$ can be taken as multiples of the nominal fuel consumption values. In this case, $h_r={\left[\begin{array}{cc}{\overline{\xi }}^{k\top }& -{\underline{\xi }}^{k\top }\end{array}\right]}^{\top }$ and $H_k={\left[\begin{array}{cc}{\mathbb{I}}_{\Gamma }& -{\mathbb{I}}_{\Gamma }\end{array}\right]}^{\top }$, where “⊤” is the matrix transpose operator, ${\mathbb{I}}_{\Gamma }$ is the identity matrix of size $\Gamma$, and ${\overline{\xi }}^k(-{\underline{\xi }}^k)\in {\mathbb{R}}^{\Gamma }$ is a vector that represents the upper (lower) bound of the fuel consumption required to implement the task by UAV k.

Due to its desirable statistical properties, the so-called Wasserstein ambiguity set has witnessed an explosion of interest. The Wasserstein ambiguity set, defined as the ${\rho }_k$-radius Wasserstein ball of distributions around the empirical distribution ${\widehat{P}}_k^{L_k}$, is given by

$${\mathcal{P}}_k^{\mathbb{W}}:=\left\{P_k\in {\mathcal{M}}_k\left({\Xi }_k\right):\mathbb{W}(P_k,{\widehat{P}}_k^{L_k})\le {\rho }_k\right\},k=1,2,\cdots ,N_v,$$

(25)

where ${\mathcal{M}}_k\left({\Xi }_k\right)$ is defined as the set of all distributions $P_k$ supported on ${\Xi }_k$, and $\mathbb{W}(P_k,{\widehat{P}}_k^{L_k})$ represents the Wasserstein distance between the $P_k$ and ${\widehat{P}}_k^{L_k}$, which is defined in [34] as

$$\mathbb{W}(P_{k1},P_{k2}):=inf\left\{{\int }_{{\Xi }^2}∥{\tilde{\xi }}_{k1}-{\tilde{\xi }}_{k2}∥\Pi (d{\tilde{\xi }}_{k1},d{\tilde{\xi }}_{k2})\right\},k=1,2,\cdots ,N_v.$$

(26)

The Wasserstein distance involves an optimization problem on computing the optimal transportation cost from the distribution $P_k$ to the empirical distribution ${\widehat{P}}_k^{L_k}$. Here we adopt the 1-norm to maintain linearity. Choosing an appropriate radius ${\rho }_k$ such that the ambiguity set ${\mathcal{P}}_k^{\mathbb{W}}$ can encompass the unknown true distribution with a high degree of confidence is of vital importance. In this study, the Wasserstein radius is selected according to the relationship between the radius and sample size defined in Theorem 3.4 of [35]:

$${\rho }_k=\left\{\begin{array}{cc}{\left({\displaystyle \frac{log\left(q_1{\beta }^{-1}\right)}{q_2{L}_k}}\right)}^{1/max\{n,2\}}\hfill & \mathrm{if}{L}_k\ge {\displaystyle \frac{log\left(q_1{\beta }^{-1}\right)}{q_2}},\hfill \\ {\left({\displaystyle \frac{log\left(q_1{\beta }^{-1}\right)}{q_2{L}_k}}\right)}^{1/a}\hfill & \mathrm{if}{L}_k<{\displaystyle \frac{log\left(q_1{\beta }^{-1}\right)}{q_2}},\hfill \end{array}\right.$$

(27)

where $L_k$ is the fuel consumption sample size for UAV k, n is the dimension of the random vector, i.e., $n=1$, $a>1$ is the tail exponent of the distribution ($a=2$ for normal distributions), and $q_1$ and $q_2$ are positive constants dependent on a, the distribution’s moments, and m, which are derived from measure concentration results. This formula ensures that the true distribution $P_k$ lies within the Wasserstein ball centered at the empirical distribution ${\widehat{P}}_k^{L_k}$ with probability of at least $1-\beta$.

3.2.3. Tractable Approximation

The DRCC (23) remains challenging to solve, even in the special scenario where the DAS is unitary, reducing it to a classical chance constraint. To tackle this, we adopt the tractable approximation put forward by Ordoudis et al. [32]. This approximation is carried out in two steps.

First, as demonstrated by Nemirovski and Shapiro [36], we conservatively approximate the chance constraint (23) with a CVaR constraint:

$$\underset{P\in \mathcal{P}}{max}\mathbb{P}{\text{-CVaR}}_{{ϵ}_k}\left[\sum _{i=1}^{N_t}\sum _{j=1}^{N_t}\sum _{m=1}^3{\tilde{\xi }}_{i,j}^{k,m}{x}_{i,j}^{k,m}-F^k\right]\le 0,k=1,2,\cdots ,N_v.$$

(28)

Second, we use standard duality techniques to reformulate the worst-case CVaR constraints in a tractable manner. For more in-depth details, refer to the proof of Proposition 1 in [32]. Consequently, (23) can be approximated as follows:

$${\lambda }_k{\rho }_k+{\displaystyle \frac{1}{L_k}}\sum _{l=1}^{L_k}{s}_k^l\le 0,k=1,2,\cdots ,N_v,$$

(29)

$${\tau }_k\le s_k^l,k=1,2,\cdots ,N_v,l=1,2,\cdots ,L_k,$$

(30)

$$\left|{ϵ}_k({\gamma }_{i,j}^{k,m,l}-{\sigma }_{i,j}^{k,m,l})-x_{i,j}^{k,m}\right|\le {ϵ}_k{\lambda }_k,i,j=1,2,\cdots ,N_t,k=1,2,\cdots ,N_v,m=1,2,3,l=1,2,\cdots ,L_k,$$

(31)

$$\begin{array}{cc}\hfill & \sum _{i=1}^{N_t}\sum _{j=1}^{N_t}\sum _{m=1}^3{\widehat{\mathsf{\xi }}}_{i,j}^{k,m,l}{x}_{i,j}^{k,m}-F^k+\sum _{i=1}^{N_t}\sum _{j=1}^{N_t}\sum _{m=1}^3({\overline{\xi }}_{i,j}^{k,m}-{\widehat{\mathsf{\xi }}}_{i,j}^{k,m,l}){\gamma }_{i,j}^{k,m,l}+\sum _{i=1}^{N_t}\sum _{j=1}^{N_t}\sum _{m=1}^3({\widehat{\mathsf{\xi }}}_{i,j}^{k,m,l}-{\underline{\xi }}_{i,j}^{k,m}){\sigma }_{i,j}^{k,m,l}+({ϵ}_k-1){\tau }_k\le {ϵ}_k{s}_k^l,\hfill \\ \hfill & k=1,2,\cdots ,N_v,l=1,2,\cdots ,L_k,\hfill \end{array}$$

(32)

$${\gamma }_{i,j}^{k,m,l},{\sigma }_{i,j}^{k,m,l}\ge 0,i,j=1,2,\cdots ,N_t,k=1,2,\cdots ,N_v,m=1,2,3,l=1,2,\cdots ,L_k,$$

(33)

$$s_k^l,{\lambda }_k,{\tau }_k\in \mathbb{R},k=1,2,\cdots ,N_v,l=1,2,\cdots ,L_k,$$

(34)

where ${\lambda }_k$, $s_k^l$, ${\tau }_k$, ${\gamma }_{i,j}^{k,m,l}$, and ${\sigma }_{i,j}^{k,m,l}$ are the resulting dual variables. Constraints (29)–(32) represent the dual transformation of the original constraints (28), and both have the same physical significance with (23). Constraints (33) and (34) define the domains of the dual variables. Specifically, constraint (31) can be further simplified as follows:

$${ϵ}_k({\gamma }_{i,j}^{k,m,l}-{\sigma }_{i,j}^{k,m,l})-x_{i,j}^{k,m}\le {ϵ}_k{\lambda }_k,i,j=1,2,\cdots ,N_t,k=1,2,\cdots ,N_v,m=1,2,3,l=1,2,\cdots ,L_k,$$

(35)

$${ϵ}_k({\gamma }_{i,j}^{k,m,l}-{\sigma }_{i,j}^{k,m,l})-x_{i,j}^{k,m}\ge -{ϵ}_k{\lambda }_k,i,j=1,2,\cdots ,N_t,k=1,2,\cdots ,N_v,m=1,2,3,l=1,2,\cdots ,L_k.$$

(36)

Based on the objective function and constraints analysed above, the DRCC model of the multi-UAV CTAPTW is established:

$$\begin{array}{cc}max\hfill & {\displaystyle \sum _{j=1}^{N_t}}{\displaystyle \sum _{k=1}^{N_v}}{\displaystyle \sum _{m=1}^3}{\omega }_{j,k,m}{x}_{i,j}^{k,m}\\ \mathrm{s}.\mathrm{t}.\hfill & \left(2\right)–\left(8\right),\left(29\right),\left(30\right),\left(32\right)–\left(36\right),\end{array}$$

which is an MILP that introduces $2N_v+(N_v\times L_k)+(6N_v\times L_k\times N_t^2)$ dual variables including ${\lambda }_k$, $s_k^l$, ${\tau }_k$, ${\gamma }_{i,j}^{k,m,l}$ and ${\sigma }_{i,j}^{k,m,l}$, along with $N_v+(2N_v\times L_k)+(3N_v\times L_k\times N_t^2)$ constraints. Despite its increased complexity, this problem can be efficiently solved using CPLEX solvers.

4. Numerical Experiments

In this section, numerical experiments are conducted using MATLAB 2022b with CPLEX 12.10.0 on an Intel Core i7-13620H CPU to evaluate the performance of the proposed SAA and DRCC models and compare their performance against each other. Moreover, the sensitivity analysis of significant parameters in DRCC model is revealed. Since all subsequent experiments aim to verify the performance of DRCC, and the setting of the time window has no impact on the simulation conclusion regarding the robustness of fuel consumption, the time window parameters are thus set as follows: $s_{j,k}^m=0$ and $f_{j,k}^m=200,j=1,2,\cdots ,N_t,k=1,2,\cdots ,N_v,m=1,2,3$.

4.1. Performance Evaluation

In the simulation experiment, two instances are used to verify the universality of the experimental conclusion: 4UAVs vs. 10Targets and 6UAVs vs. 20Targets. The fuel capacity, coordinates of the UAV base $T_0$, and coordinates of targets $T_1$–$T_{20}$ (randomly generated near the base $T_0$) are detailed in

Table 2. Parameters of heterogeneous UAVs and coordinates of base $T_0$ and targets $T_1$–$T_{20}$.

Instance	UAV Type	Fuel (kg)	Target	x (km)	y (km)	Target	x (km)	y (km)
			$T_1$	39	33	$T_2$	47	49
	2 × Reconnaissance UAVs	(480, 490)	$T_3$	34	12	$T_4$	55	50
4UAVs vs. 10Targets	1 × Attack UAV	(500)	$T_5$	62	42	$T_6$	54	18
	1 × Integrated UAV	(510)	$T_7$	38	62	$T_8$	25	25
			$T_9$	25	50	$T_{10}$	16	30
			$T_{11}$	42	28	$T_{12}$	58	35
	2 × Reconnaissance UAVs	(950, 960)	$T_{13}$	30	45	$T_{14}$	20	15
6UAVs vs. 20Targets	2 × Attack UAVs	(970, 980)	$T_{15}$	60	10	$T_{16}$	18	55
	2 × Integrated UAVs	(990, 1000)	$T_{17}$	45	60	$T_{18}$	33	38
			$T_{19}$	50	22	$T_{20}$	28	18
Base location: $T_0$ ($x=40$ km, $y=40$ km)

.

In the simulation experiment, the fuel consumption rates in different task types are modeled as random variables following normal distributions. Specifically, for UAV k from target i to j, the random variables ${\tilde{\xi }}_{i,j}^{k,1}$, ${\tilde{\xi }}_{i,j}^{k,2}$, and ${\tilde{\xi }}_{i,j}^{k,3}$ representing the fuel consumption for reconnaissance, attack, and verification tasks respectively, can be expressed as

$${\tilde{\xi }}_{i,j}^{k,m}={\tilde{c}}_m{D}_{i,j},i,j=1,2,\cdots ,N_t,k=1,2,\cdots ,N_v,m=1,2,3,$$

(37)

where the fuel consumption rate ${\tilde{c}}_m$, as a coefficient related to the task type, follows the same distribution as the fuel consumption ${\tilde{\xi }}_{i,j}^{k,m}$ given that $D_{i,j}$ can be regarded as a constant. Specifically, ${\tilde{c}}_1$, ${\tilde{c}}_2$, and ${\tilde{c}}_3$ for reconnaissance, attack, and verification tasks respectively satisfy

$${\tilde{c}}_m\sim \mathcal{N}({\mu }^m,{\left({\sigma }^m\right)}^2),m=1,2,3,$$

(38)

then the mean and standard deviation settings of ${\tilde{c}}_m$ for different task types are summarized in

Table 3. The mean-standard deviation setting of fuel consumption rate under different task types.

Task Types	Fuel Consumption Rate	Mean Value	Standard Deviation
		(${\mu }^m$)	(${\sigma }^m$)
Reconnaissance ($m=1$)	${\tilde{c}}_1$	1.8	0.5
Attack ($m=2$)	${\tilde{c}}_2$	3.2	1.6
Verification ($m=3$)	${\tilde{c}}_3$	2.4	1.0

. However, given that extreme fuel consumption values are rarely observed in real UAV operations, we refined the model by adopting a truncated normal distribution. Specifically, we imposed lower and upper bounds (${\underline{\xi }}_{i,j}^{k,m}$ and ${\overline{\xi }}_{i,j}^{k,m}$) on fuel consumption samples as defined in Section 3.2.2. This approach retains the modeling convenience of the normal distribution while enhancing the practical relevance of the results.

To verify the effectiveness of the DRCC model in the CTAPTW problem, we first obtained a sample set of $L_k=30$ based on the fuel consumption distribution defined above to conduct a comparative experiment between the SAA model and the DRCC model. Then, with reference to measure concentration theory, we set $q_1=1$ and $q_2=0.1$. According to Formula (27), the estimated value of the radius is obtained as ${\rho }_k=0.67$. Therefore, the radii ranging from 0 to 2 are adopted in the subsequent simulation experiments. To verify the generalization ability, we conducted out-of-sample tests using a sample set of 1000. Set the fuel consumption chance constraint risk value of each UAV uniformly at ${ϵ}_k=0.2,k=1,2,\cdots ,N_v$, and focus on quantitative analysis of three types of indicators: the objective of task profit, total distance, and constraint violation degree (within/outside the sample). The result comparison is presented in

Table 4. Performance comparison of the NM, RO, SAA, and DRCC models.

Instance	Model	Task Profit	Total Distance (km)	In-Sample Violation (%)	Out-Sample Violation (%)	Gap	Time (s)
4UAVs vs. 10Targets	NM	988	607.47	30.0	52.50	0	52.64
	RO	977	574.49	35.0	53.10	0	130.23
	SAA (${\rho }_k=0$)	984	671.67	5.0	13.6	0	8.58
	DRCC (${\rho }_k=1$)	984	638.78	0.0	6.8	0	26.16
6UAVs vs. 20Targets	NM	2032	1116.9	50.0	55.0	0	3404.92
	RO	1988	1117.8	80.0	85.0	0	2611.33
	SAA (${\rho }_k=0$)	1996	1563.1	0.0	0.0	0	1678.05
	DRCC (${\rho }_k=1$)	1996	1491.1	0.0	0.0	0	4800.13

, and the comparisons of task assignment between the SAA and DRCC models in two scenarios (4UAVs vs. 10Targets and 6UAVs vs. 20Targets) are shown respectively in Figure 1 and Figure 2.

As shown in Table 4, the performance of the DRCC model is comprehensively superior to NM, RO, and SAA models, which can be analyzed from the following aspects:

In terms of task profit, the DRCC model achieves the same profit as the SAA model under two experimental scenario, both of which are higher than the RO model. This indicates that although DRCC considers the worst-case scenario within the distribution ambiguity set and adopts a relatively conservative strategy, it does not sacrifice profit due to differences in optimization logic, avoiding the excessive conservatism of RO that leads to profit loss. In contrast, the NM model, which ignores uncertainty, achieves a slightly higher profit in theory, but its practical reliability is severely compromised by high constraint violation rates, making it unsuitable for actual combat scenarios.

Regarding total distance, the DRCC model shows obvious advantages in resource utilization efficiency. Whether in the 4UAVs vs. 10Targets scenario or the 6UAVs vs. 20Targets scenario, the total distance of the DRCC model is shorter than that of the SAA model as shown in Figure 1 and Figure 2. Although accounting for the uncertainty of fuel consumption distribution, the DRCC not only optimizes path planning to reduce unnecessary flight mileage but also improves the utilization efficiency of the UAV flight range, which is crucial for enhancing the ability to cope with fuel consumption uncertainty.

The constraint violation rate directly reflects the reliability of the model in meeting fuel consumption constraints, which is a core indicator to verify the practical applicability of the model. In the in-sample test, the violation rate of the DRCC model is 0.0%, and that of the SAA model is 5.0% in the 4UAVs and 10Targets scenario. Both are far lower than the violation rates of the NM and RO models. This indicates that within the training sample distribution, DRCC has stricter control over fuel consumption constraints. In the out-of-sample test, the advantage of DRCC in robustness is more prominent: the violation rate of SAA rises to 13.6%, while that of DRCC is controlled at 6.8%, and the violation rates of NM and RO are as high as 52.50% and 53.10% respectively. This fully demonstrates that DRCC has stronger adaptability to fuel consumption fluctuations, effectively reducing the risk of constraint failure in actual implementation.

In addition, as can be observed from the computational time data, all models exhibit a significant “robustness-computational efficiency” trade-off, with inherent scalability bottlenecks becoming evident as the scenario scale expands. In small-scale scenarios, the SAA model achieves the optimal efficiency due to the absence of iterative optimization for uncertainty. Although the computational time of the DRCC model is higher than that of the SAA model, it remains lower than those of the NM and RO models, thus achieving a favorable balance between robustness and efficiency. However, when the scenario is scaled up to the “6UAVs vs. 20Targets” case, the computational time of all models increases significantly, with the DRCC model even reaching 4800.13 s. This indicates that while the DRCC model can enhance robustness through complex constraints, the dimensionality of these constraints grows non-linearly with the number of UAVs/targets, leading to a sharp surge in computational costs when the scenario scale expands.

In summary, compared with NM, RO, and SAA, the DRCC model achieves a better balance among task profit, resource utilization efficiency, and constraint reliability. It not only avoids the excessive conservatism of RO and the unreliability of NM but also overcomes the defect that SAA is sensitive to distribution assumptions and prone to out-of-sample performance degradation. Therefore, the DRCC model is more suitable for complex actual combat environments and can provide a more robust optimization solution for the CTAPTW under uncertain fuel consumption.

4.2. Sample Size Sensitivity Analysis

4.2.1. Impact of Sample Size on Objective Values

This study leverages the DRCC method to examine how task profit objectives evolve with increasing sample sizes. The following experiments were all carried out under the instance of 4UAVs vs. 10Targets. The experimental design is structured as follows: the fuel consumption samples for different task types follow a normal distribution with distinct means and standard deviations, and their set values are presented in Table 3. Four sample size groups are set, specifically 10, 30, 50, and 100. The risk value for the fuel consumption chance constraint of each UAV is uniformly set at ${ϵ}_k=0.2$, and the Wasserstein radius is set as ${\rho }_k=0.1$ for all $k=1,2,\cdots ,N_v$. For each group, 30 repeated experiments are conducted to mitigate random errors. Subsequently, a box plot is generated using the 30 sets of objective values from these trials, facilitating quantitative analysis of the relationship between sample sizes and task profit objectives.

The results of 30 repeated experiments with normally distributed samples, as illustrated in Figure 3, demonstrate a clear pattern in the behavior of task profit objectives with varying sample sizes. When the sample size is 10, the median of the task profit objectives stands at approximately 968, defining the central tendency for this sample size. As the sample size increases to 100, the median drops and stabilizes at 949. This progression from 10 to 100 samples clearly shows that the median initially declines steadily and then approaches a stable state, indicating that larger samples in the DRCC method systematically lower the central level of task profit objectives. Regarding the dispersion of the data, Figure 3 also offers critical insights. With a sample size of 10, the box plot exhibits a substantial span of 15, reflecting high variability and strong influence from sample randomness. As the sample size increases to 100, the box span is narrower and minimized to just 2, signifying that the data points cluster closely together. This consistent reduction in dispersion with increasing sample size underscores the enhanced stability and repeatability of DRCC results as the impact of random fluctuations diminishes.

In summary, the findings yield two key conclusions about the DRCC method. First, the central tendency of task profit objectives, as measured by the median, decreases monotonically with increasing sample size. Second, larger sample sizes effectively mitigate random interference, leading to a significant reduction in data dispersion and a marked improvement in result stability. These results suggest that increasing the sample size in DRCC not only lowers the central value of the objectives but also enhances the overall reliability of the method.

4.2.2. Impact of Sample Size on the Degree of Out-of-Sample Constraint Violation

This study focuses on exploring the influence of sample sizes on out-of-sample constraint violation under the DRCC method. The risk value for the fuel consumption chance constraint of each UAV is uniformly set at ${ϵ}_k=0.2$, and the Wasserstein radius is set as ${\rho }_k=0.1$ for all $k=1,2,\cdots ,N_v$. Four sample sizes are established-specifically 10, 30, 50, and 100-with nested samples adopted. For each sample size, relevant experiments are conducted to collect data on out-of-sample constraint violation values and task profit objective values, which are then visualized using a combined box plot and line plot in Figure 4.

As depicted in Figure 4, the task profit objective exhibits a consistent downward trajectory as the sample size rises from 10 to 100. This trend underscores that an increase in sample size, which reflects the tightening of constraints in the DRCC model, drives more conservative decision-making and ultimately leads to a reduction in the task profit objective. In addition, the out-of-sample constraint violation behavior evolves distinctly with increasing sample sizes. In this progression from 10 to 100 samples, the box plot shows an increasingly narrow range, with all violation values eventually reaching 0. This indicates that sufficiently large samples strengthen the constraints of the DRCC model to such an extent that out-of-sample violations are eliminated, thereby achieving optimal stability in constraint satisfaction.

Notably, for the UAV fleet, the total flight distance shows a trend of first decreasing and then stabilizing as the sample size increases. When the sample size rises from 10 to 50, the total distance decreases from 948.04 km to 902.17 km, which is mainly due to the model optimizing path planning more cautiously under the constraint of increasing sample information. This avoids redundant routes to reduce fuel consumption risks. When the sample size further increases to 100, the total distance remains stable at 902.17 km, indicating that with sufficient sample data, the model tends to a more optimal and stable path scheme, balancing the trade-off between profit reduction and distance optimization under the compressed feasible domain. This phenomenon demonstrates that increasing sample size enhances the model’s ability to control fuel consumption uncertainty but also leads to a certain loss of task profit. At the same time, it promotes more efficient path planning of the UAV fleet, which is of great significance for improving the operational reliability and resource utilization of UAV missions.

Table 5. The influence of sample size on the DRCC models.

Sample Sizes	Task Profit	Total Distance (km)	Out-Sample Constraint Violation (%)	Gap	Time (s)
10	963	948.04	14.8	0	10.77
30	956	931.86	8.1	0	48.36
50	953	902.17	4.0	0	65.58
100	947	902.17	0.0	0	329.30

further quantifies these trends and reveals additional critical insights: as the sample size increases, while both the task profit and the out-of-sample constraint violation rate drop markedly, the model’s runtime increases significantly, soaring from 10.77 s to 329.30 s. This substantial growth in runtime is attributed to the increased complexity introduced by more constraints as the sample size expands, highlighting a trade-off between solution robustness and computational efficiency in the DRCC model.

4.3. Parameter Analysis of Distributional Robustness

To explore the influence law of the Wasserstein radius on the objective of task profit in the DRCC model, we first fixed the risk tolerance ${ϵ}_k$ of each UAV at 0.01, 0.05, and 0.1 respectively. Then, experiments were carried out using 30 sets of samples. By varying the Wasserstein radius (within the range of 0 to 2.0), we observed the changes in the objective of task profit and the feasibility of the model solution. Finally, the variation of task profit with the Wasserstein radius is presented in

Table 6. The relationship between the objective of task profit and the Wasserstein radius.

Wasserstein Radius		Task Profit (Gap)
	${ϵ}_{\mathit{k}}=\mathbf{0}.\mathbf{01}$	${ϵ}_{\mathit{k}}=\mathbf{0}.\mathbf{05}$	${ϵ}_{\mathit{k}}=\mathbf{0}.\mathbf{1}$
0	953 (0)	953 (0)	953 (0)
0.1	949 (0)	953 (0)	953 (0)
0.2	923 (0)	953 (0)	953 (0)
0.3	897 (0)	953 (0)	953 (0)
0.4	−	949 (0)	953 (0)
0.5	−	949 (0)	953 (0)
0.6	−	938 (0)	953 (0)
0.7	−	938 (0)	953 (0)
0.8	−	934 (0)	949 (0)
0.9	−	934 (0)	949 (0)
1.0	−	923 (0)	949 (0)
1.5	−	897 (0)	938 (0)
2.0	−	−	934 (0)
Infeasible Ratio (%)	69.23	7.69	0

, and the changing trend of the feasible results is clearly shown in Figure 5.

As indicated by the data in Figure 5 and Table 6, an increase in the Wasserstein radius exhibits a significant negative correlation with task profit, and the magnitude of this effect is strongly dependent on the risk tolerance ${ϵ}_k$. When the risk tolerance value is relatively low, the model becomes unsolvable even at a very small radius. This indicates that the low-risk constraint makes the model highly sensitive to even minor increases in the Wasserstein radius. Even a very small uncertainty can break through the feasible region, leading to a sharp decline in profits or solution failure. Thus, when the risk tolerance value is lower (${ϵ}_k=0.01$), a Wasserstein radius in the range of 0 to 0.1 yields higher task profit. As the risk value increases, the model allows for a larger range of solvable radius. Therefore, the high-risk setting significantly expands the feasible domain and shows superior adaptability to the increase in the Wasserstein radius. Specifically, when the risk tolerance is higher (${ϵ}_k=0.1$), a Wasserstein radius in the range of 0 to 1 constitutes a better choice.

As shown in

Table 7. The influence of Wasserstein radius on the DRCC models (${ϵ}_k=0.05$).

Wasserstein Radius	Task Profit	Total Distance (km)	Gap	Time (s)
0.1	953	658.63	0	21.86
0.5	949	653.97	0	31.95
1.0	923	636.07	0	49.66

, the total flight distance of the fleet decreases from 658.63 km to 636.07 km. This reduction is a strategic adaptation to greater fuel consumption uncertainty. The shorter paths inherently reduce the risk of exceeding fuel limits when faced with unpredictable consumption fluctuations. By shortening distances, the model mitigates the impact of distributional ambiguity, ensuring mission feasibility even under the most adverse fuel consumption scenarios. Therefore, as a regulator of model robustness, Wasserstein radius can adjust the degree of distribution ambiguity, thereby controlling the conservativeness of the model. Additionally, the solving time increases with the radius, which is attributable to the expanded constraint space caused by a larger ambiguity set, requiring more computational effort to explore feasible solutions.

4.4. Impact Analysis of the Risk Tolerance

To clarify the influence law of risk tolerance on task profits in the DRCC model, we first fix the Wasserstein radius ${\rho }_k$ of the ambiguity set for the fuel consumption distribution of each UAV at 0, 0.5, and 1, representing three levels of distribution uncertainty. Experiments are then carried out using 30 sets of fuel consumption samples. By gradually increasing the risk tolerance (within the range of 0.1 to 1.0), we observe the dynamic changes in task profits and explore the adaptive relationship between risk preference and profits. The relationship between the risk tolerance obtained from the experiments and the changes in task profits is presented in

Table 8. The relationship between the objective of task profit and the risk tolerance.

Risk Tolerance	Task Profit (Gap)
	${\mathit{\rho }}_{\mathit{k}}=\mathbf{0}$	${\mathit{\rho }}_{\mathit{k}}=\mathbf{0.5}$	${\mathit{\rho }}_{\mathit{k}}=\mathbf{1}$
0.1	956 (0)	954 (0)	953 (0)
0.2	958 (0)	958 (0)	956 (0)
0.3	958 (0)	958 (0)	958 (0)
0.4	961 (0)	961 (0)	958 (0)
0.5	961 (0)	961 (0)	961 (0)
0.6	961 (0)	961 (0)	961 (0)
0.7	963 (0)	963 (0)	963 (0)
0.8	963 (0)	963 (0)	963 (0)
0.9	963 (0)	963 (0)	963 (0)
1.0	963 (0)	963 (0)	963 (0)
Infeasible Ratio (%)	0	0	0

and clearly illustrated in Figure 6.

Combining the data in Figure 6 and the Table 8, enhancing risk tolerance exerts a significant positive impact on task profits, with the task profit growth rate strongly correlated to the Wasserstein radius ${\rho }_k$. As risk tolerance increases, task profits rise continuously before stabilizing at 963. This suggests that under conditions of low distributional uncertainty, moderately elevating risk tolerance can rapidly unleash task profit potential, and once a critical threshold is crossed, task profits enter a steady state. When distributional uncertainty is lower ($\rho =0.5$), selecting a lower risk tolerance (e.g., ${ϵ}_k=0.4$) yields a task profit of 961. In contrast to low-uncertainty scenarios, higher uncertainty settings involve greater distributional fluctuations, resulting in a slower trajectory of task profit growth. Here, a higher level of risk tolerance is required for sustained growth. For instance, when distributional uncertainty is higher ($\rho =1$), choosing a higher risk tolerance (e.g., ${ϵ}_k=0.5$) also achieves a task profit of 961. Howevre, under high distributional uncertainty, while the initial task profit base is lower, high risk tolerance can offset the impacts of uncertainty, allowing task profits to match those in other scenarios.

As shown in

Table 9. The influence of risk tolerance on the DRCC models (${\rho }_k=0.5$).

Risk Tolerance	Task Profit	Total Distance (km)	Gap	Time (s)
0.1	954	885.02	0	1.02
0.5	961	899.05	0	1.19
1.0	963	938.53	0	1.16

, the total flight distance of the UAV fleet increases from 885.02 km to 938.53 km. The expansion of distance suggests that to achieve higher profits, the model tends to include longer paths that were originally avoided to reduce fuel consumption risks. A higher risk tolerance means the model allows a greater probability of exceeding fuel limits, which aligns with the longer flight distances.

In summary, adjusting risk tolerance enables the model to balance profit maximization and risk control: lower risk tolerance prioritizes operational stability with shorter distances and lower profits, while higher risk tolerance pursues higher returns at the cost of longer flights and increased fuel shortage risks. This flexibility allows the model to adapt to diverse mission requirements.

5. Conclusions

This study proposes a data-driven DRCC model to address the multi-UAVs CTAPTW. The model is capable of handling complex scenarios involving heterogeneous UAVs, multiple consecutive tasks, and time windows.

Unlike prior studies, this approach incorporates time windows for real-world task assignment. A DAS from partial fuel consumption data is constructed, then DRCC is used to find optimal decisions under worst-case distributions. Transformed via CVaR into a solvable MILP, it balances conservatism and efficiency with derived Wasserstein radius and risk tolerance. Experiments show that it outperforms RO and SAA. Not only can it achieve higher profits, but it also greatly reduces fuel consumption and constraint violations to achieve a better balance between optimality and robustness.

However, the proposed model has limitations in real-world UAV operations: it is currently designed for pre-task assignments, while dynamic real-world contexts may require real-time adjustments to handle unforeseen changes; additionally, for large-scale problems, real-time model adaptation requires advancements in computational speed and decision-making frameworks for timely responses. Furthermore, the dependence on sample quality and the correlation between the selection of the Wasserstein radius and sample size are also issues that cannot be ignored. Insufficient data may lead to overly conservative or non-robust decisions. Future research in CTAP could focus on multi-objective optimization to balance the benefits of multiple objectives.