Two-Stage Robust Optimization for Coupled Multi-Agent Task Allocation in Disaster Response Under Demand Uncertainty

Abstract

Multi-agent systems (MASs), with unmanned aerial vehicles (UAVs) as a representative embodiment, have become increasingly vital in time-sensitive disaster response scenarios, where multiple agents must collaborate to execute “observe-and-intervene” emergency tasks and jointly cope with dynamic environmental uncertainties. Existing research on task allocation mostly eliminates uncertainty through deterministic models; the few studies that directly consider uncertainty focus primarily on time uncertainty, overlooking the critical importance of demand uncertainty. To this end, this study accounts for the impact of harsh environmental conditions and incident complexity factors on intervention resource demands. We establish an uncertainty set for these demands and construct a two-stage robust optimization model to solve the coupled multi-agent task allocation problem. Compared with deterministic models, this framework enhances risk resistance while simultaneously reducing the conservatism of decisions. Furthermore, to overcome the computational challenges of large-scale instances, a Learning-Enhanced Column and Constraint Generation (LE-C&CG) algorithm is proposed. Experimental results demonstrate that LE-C&CG converges over an order of magnitude faster than standard Benders and C&CG algorithms, consistently achieving a 0% optimality gap within fractions of a second, making it highly suitable for time-critical emergency applications.

1. Introduction

With the advancement of intelligent technologies, multi-agent systems (MASs), particularly unmanned aerial vehicles (UAVs), are increasingly playing a vital role in dynamic and hazardous disaster response scenarios [1]. However, cooperating to execute coupled “observe-and-intervene” tasks in these environments exposes MASs to various unpredictable factors. Therefore, incorporating uncertainty into task allocation models is essential for generating robust and practical decision-making plans.

The existing literature on MAS task allocation has largely relied on deterministic models to simplify these challenges [2,3]. Studies in universe environments frequently focus on large-scale coverage with strict time windows [4,5,6,7,8,9], occasionally utilizing predictive models like LSTM or multi-agent reinforcement learning (MARL) to adapt to dynamic changes [10,11]. Conversely, research in high-risk environments emphasizes strict sequential task chains [12,13,14,15,16,17,18], agent survivability [19,20], and target heterogeneity [21,22]. Furthermore, optimization often requires runtime recalculation to resolve unexpected failures as tasks unfold [23].

Despite these advancements, the literature explicitly addressing uncertainty remains comparatively scarce. While some studies model position, trajectory, or time uncertainties [24,25,26,27,28,29,30], critical demand uncertainty is frequently overlooked. Additionally, traditional solution methods like fuzzy and stochastic programming rely on subjective priors or exact probability distributions, which contradict the unpredictable nature of real-world emergencies [31,32,33,34,35]. Robust optimization effectively avoids these restrictive assumptions, often drawing inspiration from worst-case game-theoretic frameworks [36,37]. To mitigate the over-conservatism inherent in single-stage static models [38,39,40], two-stage robust optimization offers a more flexible framework that balances risk mitigation with decision efficiency [41,42].

Addressing the severe demand uncertainty in hazardous disaster environments, this paper first establishes a deterministic baseline model minimizing total operational cost for coupled MAS task allocation. We then construct a two-stage robust optimization model to maximize the satisfaction of uncertain resource demands while balancing costs and emergency benefits. Finally, to overcome the computational bottlenecks of large-scale instances, we propose the Learning Enhanced C&CG (LE-C&CG) algorithm, leveraging the recourse function’s monotonicity to replace the MIP-based subproblem solver with a specialized heuristic separation oracle.

The main contributions of this paper are as follows:

(1)

Systematic Modeling of Coupled Tasks under Demand Uncertainty: Our framework mathematically captures the strict sequential dependencies of “observe-and-intervene” operations. Furthermore, we explicitly formulate a budgeted uncertainty set for physical intervention resource demands, directly addressing the unpredictable payload surges caused by hazardous disaster environments.

(2)

Balanced Risk Mitigation via Two-Stage Robustness: By formulating the problem within a two-stage robust optimization paradigm, the model effectively balances upfront task allocation costs against the worst-case recourse penalties of payload shortages. This strategic structure significantly mitigates the over-conservatism typically associated with single-stage static robust approaches, providing decision-makers with a flexible risk-hedging tool.

(3)

Structural Acceleration via the LE-C&CG Algorithm: We propose the Learning Enhanced C&CG (LE-C&CG) algorithm, which leverages a rigorously proven monotonicity property of the recourse function in our model. By replacing the generic NP-hard subproblem with an $O(TlogT)$ greedy heuristic separation oracle, LE-C&CG achieves orders-of-magnitude faster convergence, rendering the robust framework practically viable for large-scale, real-time emergency applications.

The structure of this paper is arranged as follows. Section 2 introduces the deterministic model for the coupled MAS task allocation problem, defines the uncertainty set, and develops the two-stage robust optimization framework. In Section 3, we present the enhanced solution algorithm designed to mitigate computational complexity. Section 4 provides a thorough experimental assessment based on emergency response scenarios to validate both the model’s robustness and the algorithm’s efficiency. To conclude, Section 5 summarizes our findings and proposes potential directions for subsequent investigations.

2. Model

This section considers the coupled multi-agent task allocation problem utilizing integrated observation-and-intervention UAVs to perform sequential tasks at various incident sites. Initially, we formulate a mixed-integer linear programming baseline to formally define the physical and operational constraints of the multi-UAV system. Subsequently, we construct a two-stage robust optimization model accounting for uncertain intervention resource demands to hedge against extreme environmental perturbations.

2.1. Deterministic Model

2.1.1. Problem Description

Consider a fleet of M observation and intervention integrated UAVs available to perform either observation or intervention tasks at various incident sites. The attributes of these UAVs are represented by the set U, with each UAV defined by the following tuple:

$$U=\left\{U_1,\dots ,U_M\right\},U_i=<va{l}_i^U,p_i^U,S_i^U,v_i,c_i,F_i>$$

(1)

Specifically, $va{l}_i^U$ represents the replacement cost of the UAV, incurring corresponding systemic losses if the UAV fails due to severe environmental hazards. The parameter $p_i^U$ denotes the probability of the UAV failing under such extreme conditions. $S_i^U$ signifies the upper capacity limit of intervention resources carried by the UAV, representing the collection of emergency payloads required to mitigate the hazard at the incident site. Additionally, $v_i$ describes the flight speed of the UAV, $c_i$ indicates its unit fuel consumption, and $F_i$ limits the maximum fuel capacity.

Furthermore, there are N distributed incident sites requiring response, characterized by the set T, with each incident defined by the following tuple:

$$T=\left\{T_1,\dots ,T_N\right\},T_j=<po{s}_j^T,va{l}_j^T,p_j^T,S_j^T,t_j^0,t_j^1,t_j^L>$$

(2)

Here, $po{s}_j^T$ represents the location of the incident site and $va{l}_j^T$ denotes the benefits generated upon a successful response to the incident. $p_j^T$ represents the probability of successful intervention by the assigned UAVs. $S_j^T$ defines the baseline intervention resource demands. $t_j^0$ denotes the time of the observation task and $t_j^1$ denotes the time of the intervention task. $t_j^L$ dictates the time limit of the incident, representing the latest permissible start time for the intervention task. In addition, the observation task must be completed prior to executing the intervention task for each incident, ultimately completing a total of $2N$ tasks.

2.1.2. Constraints

The decision variable is 0-1 variable $x_{ij}^h$, which indicates whether UAV i performs an observation or intervention task at incident site j, where $h=0$ indicates an observation task and $h=1$ indicates an intervention task.

$$x_{ij}^h\in \left\{0,1\right\},\forall i\in U,j\in T,h\in \left\{0,1\right\}$$

(3)

The first constraint is regarding intervention resource demand constraints. When UAV i performs an intervention task at site j, its current remaining intervention resource should be greater than or equal to the influence resource demands of site j.

$$\left(S_i^U-\sum _{k=1}^{j-1}{S}_k^T\right)x_{ij}^h>S_j^T,\forall i\in U,\forall j\in T$$

(4)

The second and third constraints ensure the temporal feasibility of the coupled “observe-and-intervene” tasks within the macro-level allocation framework. Since the explicit flight sequences are deferred to the post-processing heuristic, the MILP model guarantees temporal feasibility by evaluating the direct-flight response times.

Given the strict logical dependency that an observation must precede an intervention, the completion time of the intervention task depends on both the observation’s completion time and the intervention UAV’s arrival time. To ensure that the entire coupled operation finishes before the incident’s time limit $t_j^L$, we introduce two coupled linear constraints.

First, the observation task must be completed early enough to leave sufficient time for the subsequent intervention execution. Thus, the arrival and execution time of the assigned observation UAV, plus the required intervention duration $t_j^1$, must not exceed the deadline:

$$\sum _{i=1}^M\left({\displaystyle \frac{D_{i,j}}{v_i}}+t_j^0\right)x_{ij}^0+t_j^1\le t_j^L,\forall j\in T$$

(5)

Second, the UAV assigned to the intervention task must independently be capable of arriving at the site and completing its operation within the deadline:

$$\sum _{i=1}^M\left({\displaystyle \frac{D_{i,j}}{v_i}}\right)x_{ij}^1+t_j^1\le t_j^L,\forall j\in T$$

(6)

Together, Equations (5) and (6) provide a strictly necessary feasibility boundary for the temporal coupling without relying on non-linear routing variables. They effectively restrict the maximum allowable assignment distances for both observation and intervention agents.

The fourth constraint is the fuel capacity constraint, restricting the total flight distance accumulated by UAV i for both observation and intervention tasks from exceeding its maximum range limit $F_i$. Let $(X_i^U,Y_i^U)$ and $(X_j^T,Y_j^T)$ denote the initial spatial coordinates of UAV i and target j, respectively. The distance is defined as $D_{i,j}=\sqrt{{(X_i^U-X_j^T)}^2+{(Y_i^U-Y_j^T)}^2}$:

$$c_i\sum _{j=1}^N\sum _{h\in \{0,1\}}{D}_{i,j}{x}_{ij}^h\le F_i,\forall i\in U$$

(7)

The fifth constraint ensures that every required task (both observation $h=0$ and intervention $h=1$) at each incident site must be executed exactly once by one of the available UAVs.

$$\sum _{i=1}^M{x}_{ij}^h=1,\forall j\in T,\forall h\in \{0,1\}$$

(8)

2.1.3. Objective Function

The objective function of this model is divided into costs and benefits, expressed in the form of minimizing the total cost.

The first component $f_1$ is the hazard risk cost, combining the UAV replacement cost $va{l}_i^U$ and the failure probability $p_i^U$ under severe environmental conditions.

$$f_1=va{l}_i^U\left(1-\prod _{j=1}^N\left(1-p_i^U\right)\right)$$

(9)

The second cost is the UAV’s operational range cost, $f_2$, which is minimized to reduce total flight distance.

$$f_2=D_{i,j}$$

(10)

The reward component, $f_3$, represents the value gained from successfully mitigating an incident site. We assume a single-intervention protocol where each site is dedicated by exactly one UAV.

$$f_3=va{l}_j^T{p}_j^T$$

(11)

Normalizing and aggregating these three distinct components yields the comprehensive objective function f.

$$minf=\sum _{i=1}^M\sum _{j=1}^N\sum _{h=0}^1\left({\displaystyle \frac{va{l}_i^U}{{max}_i\left\{va{l}_i^U\right\}}}\left(1-\prod _{j=1}^N\left(1-p_i^U\right)\right)+{\displaystyle \frac{D_{i,j}}{{max}_i\left\{D_{i,j}\right\}}}-{\displaystyle \frac{va{l}_j^T}{{max}_j\left\{va{l}_j^T\right\}}}{p}_j^T\right)x_{ij}^h$$

(12)

Consequently, the complete deterministic model is formulated as follows:

$$\begin{array}{cc}\hfill & minf=\sum _{i=1}^M\sum _{j=1}^N\sum _{h=0}^1\left({\displaystyle \frac{va{l}_i^U}{{max}_i\left\{va{l}_i^U\right\}}}\left(1-\prod _{j=1}^N\left(1-p_i^U\right)\right)+{\displaystyle \frac{D_{i,j}}{{max}_i\left\{D_{i,j}\right\}}}-{\displaystyle \frac{va{l}_j^T}{{max}_j\left\{va{l}_j^T\right\}}}{p}_j^T\right)x_{ij}^h\hfill \\ \hfill & s.t.\left(S_i^U-\sum _{k=1}^{j-1}{S}_k^T\right)x_{ij}^h>S_j^T,\forall i\in U,\forall j\in T\hfill \\ \hfill & \phantom{s.t.}\sum _{i=1}^M\left({\displaystyle \frac{D_{i,j}}{v_i}}+t_j^0\right)x_{ij}^0+t_j^1\le t_j^L,\forall j\in T\hfill \\ \hfill & \phantom{s.t.}\sum _{i=1}^M\left({\displaystyle \frac{D_{i,j}}{v_i}}\right)x_{ij}^1+t_j^1\le t_j^L,\forall j\in T\hfill \\ \hfill & \phantom{s.t.}{c}_i\sum _{j=1}^N\sum _{h\in \{0,1\}}{D}_{i,j}{x}_{ij}^h\le F_i,\forall i\in U\hfill \\ \hfill & \phantom{s.t.}\sum _{i=1}^M{x}_{ij}^h=1,\forall j\in T,\forall h\in \{0,1\}\hfill \\ \hfill & \phantom{s.t.}{x}_{ij}^h\in \left\{0,1\right\},\forall i\in U,j\in T,h\in \left\{0,1\right\}\hfill \end{array}$$

(13)

2.2. Two-StageRobust Optimization Model

Building upon the deterministic model, this section develops a two-stage robust optimization formulation accounting for the uncertainty of intervention resource demands. Recognizing the over-conservatism of single-stage robust models allocating highly redundant resources to prevent extreme risks, we construct a two-stage paradigm. The first stage formulates a baseline task plan minimizing the total operational cost based solely on baseline demands, operating prior to the realization of any uncertainty. Subsequently, following the revelation of the actual demands, the second stage minimizes the additional penalty cost incurred under the worst-case scenario. The two-stage model can find the optimal balance between prevention investment and failure risk and develop a plan that can effectively resist disturbances without being overly conservative and causing costs to get out of control.

2.2.1. Demand Uncertainty Set

The actual intervention resource demand heavily depends on unpredictable environmental hazards and incident complexity factors. Severe weather conditions, such as strong winds or limited visibility, degrade intervention efficiency, necessitating higher resource consumption. Simultaneously, the inherent complexity of the incident site, evolving rapidly during a crisis, further escalates the resource requirements.

Acknowledging the incomplete prior knowledge regarding these escalating factors, we construct a budgeted uncertainty set $\mathsf{\Xi }$ bounding the actual intervention resource demands. In real-world disaster response scenarios, while an entire region may experience hazardous conditions, extreme localized demand surges (e.g., a sudden secondary explosion or a localized structural collapse) rarely occur simultaneously at every single site. The independent binary activation bounded by a budget $\mathsf{\Gamma }$ is highly appropriate here because it effectively models this spatial sparsity of extreme events. It allows decision-makers to hedge against a realistic number of concurrent worst-case escalations without falling into the trap of absolute worst-case over-conservatism.

$$\mathsf{\Xi }:\left\{{\tilde{S}}_j^T\in {\mathbb{R}}^{\left|T\right|}|\begin{array}{cc}\hfill & {\tilde{S}}_j^T={\widehat{S}}_j^T+{\xi }_j·({\mathsf{\Delta }}_j^{env}+{\mathsf{\Delta }}_j^{prot}),\forall j\in T\hfill \\ \hfill & \sum _{j\in T}{\xi }_j\le \mathsf{\Gamma }\hfill \\ \hfill & {\xi }_j\in \left\{0,1\right\},\forall j\in T\hfill \end{array}\right\}$$

(14)

In this formulation, ${\tilde{S}}_j^T$ represents the actual intervention resource demand, aggregating the nominal baseline requirement ${\widehat{S}}_j^T$ and the potential demand increments. The binary variable ${\xi }_j$ dictates the activation of the uncertainty bound for a specific site, triggering the environmental demand increment ${\mathsf{\Delta }}_j^{env}$ and the incident complexity demand increment ${\mathsf{\Delta }}_j^{prot}$. The uncertainty budget $\mathsf{\Gamma }$ restricts the total number of simultaneous severe disruptions.

2.2.2. Development of Two-Stage Robust Optimization Model

Formulating the first stage as the master problem, the objective function integrates the deterministic operational costs and the anticipated worst-case penalty function derived from the second stage. The constraints mirror the deterministic model excluding the rigid resource capacity limitations, transitioning the strict capacity compliance into flexible constraints evaluated within the recourse problem.

$$\begin{array}{cc}\hfill & minf=\left(\sum _{i=1}^M\sum _{j=1}^N\sum _{h=0}^1{C}_{ij}^h{x}_{ij}^h+\underset{{\tilde{S}}_j^T\in \mathsf{\Xi }}{max}Q\left(x,{\tilde{S}}_j^T\right)\right)\hfill \\ \hfill & C_{ij}^h={\displaystyle \frac{va{l}_i^U}{{max}_i\left\{va{l}_i^U\right\}}}\left(1-\prod _{j=1}^N\left(1-p_i^U\right)\right)+{\displaystyle \frac{D_{i,j}}{{max}_i\left\{D_{i,j}\right\}}}-{\displaystyle \frac{va{l}_j^T}{{max}_j\left\{va{l}_j^T\right\}}}{p}_j^T\hfill \\ \hfill & s.t.\sum _{i=1}^M\left({\displaystyle \frac{D_{i,j}}{v_i}}+t_j^0\right)x_{ij}^0+t_j^1\le t_j^L,\forall j\in T\hfill \\ \hfill & \phantom{s.t.}\sum _{i=1}^M\left({\displaystyle \frac{D_{i,j}}{v_i}}\right)x_{ij}^1+t_j^1\le t_j^L,\forall j\in T\hfill \\ \hfill & \phantom{s.t.}{c}_i\sum _{j=1}^N\sum _{h\in \{0,1\}}{D}_{i,j}{x}_{ij}^h\le F_i,\forall i\in U\hfill \\ \hfill & \phantom{s.t.}\sum _{i=1}^M{x}_{ij}^h=1,\forall j\in T,\forall h\in \{0,1\}\hfill \\ \hfill & \phantom{s.t.}{x}_{ij}^h\in \left\{0,1\right\},\forall i\in U,j\in T,h\in \left\{0,1\right\}\hfill \end{array}$$

(15)

Functioning as the recourse sub-problem, the second stage translates the deficit between the available intervention payloads and the realized actual demands into a quantifiable penalty cost. To clarify the physical interpretation, the resource gap is evaluated per UAV rather than per site. Let $P_i$ denote the unit penalty cost for UAV i and $y_i$ represent the continuous recourse variable capturing the payload deficit of UAV i. The second-stage problem is formulated as follows:

$$\begin{array}{cc}\hfill & Q\left(\mathit{x},\mathit{\xi }\right)=\underset{\mathit{y}\ge \mathbf{0}}{min}\sum _{i=1}^M{P}_i{y}_i\hfill \\ \hfill & s.t.y_i\ge \sum _{j\in T}\left({\widehat{S}}_j^T+{\xi }_j{\mathsf{\Delta }}_j\right)x_{ij}^1-S_i^U,\forall i\in U\hfill \end{array}$$

(16)

In Equation (16), the constraint explicitly calculates the capacity shortage: it sums the realized demand (${\widehat{S}}_j^T+{\xi }_j{\mathsf{\Delta }}_j$, where ${\mathsf{\Delta }}_j={\mathsf{\Delta }}_j^{env}+{\mathsf{\Delta }}_j^{prot}$) of all target sites j assigned to UAV i (where $x_{ij}^1=1$), and subtracts the UAV’s physical payload capacity $S_i^U$. If the assigned demand exceeds the capacity, $y_i$ takes the positive difference; otherwise, $y_i=0$. The objective seeks the worst-case uncertainty realization $\mathit{\xi }\in \mathsf{\Xi }$ that maximizes this total system penalty.

3. Algorithm

The proposed two-stage robust model features a “min-max-min” structure bounded by an uncertainty budget $\mathsf{\Gamma }$. While the standard Column-and-Constraint Generation (C&CG) algorithm [43] solves this exactly, its reliance on dualization and Big-M constraints inherently ignores structural properties like monotonicity, leading to exponential computational growth. To overcome this, we propose the Learning-Enhanced C&CG (LE-C&CG) algorithm. By leveraging the monotonic nature of the recourse function—where penalty costs non-decreasingly scale with perturbation intensity—LE-C&CG replaces the computationally expensive MIP-based subproblem with a fast $O(TlogT)$ greedy heuristic separation oracle. For any fixed first-stage allocation, this oracle rapidly identifies worst-case scenarios by simply ranking and selecting the top $\mathsf{\Gamma }$ perturbed targets that maximize marginal penalty costs. This avoids auxiliary dual variables and generates high-quality cuts, significantly accelerating convergence to meet the stringent real-time demands of large-scale UAV task allocation.

To rigorously justify the heuristic separation oracle within the LE-C&CG framework, we establish the following property regarding the recourse function.

Proposition 1

(Monotonicity of the Recourse Function). For any fixed first-stage task allocation matrix $\mathit{x}$, the recourse penalty function $Q(\mathit{x},\mathit{\xi })$ is monotonically non-decreasing with respect to the uncertainty realization vector ξ.

Proof. 

Let ${\mathit{\xi }}^A,{\mathit{\xi }}^B\in \mathsf{\Xi }$ be two distinct uncertainty realizations such that ${\mathit{\xi }}^A\le {\mathit{\xi }}^B$ element-wise. Because the demand perturbation is strictly non-negative (${\mathsf{\Delta }}_j\ge 0$), the total demand assigned to any UAV $i\in \mathcal{U}$ satisfies

$$\sum _{j\in \mathcal{T}}\left({\overline{S}}_j^T+{\xi }_j^A{\mathsf{\Delta }}_j\right)x_{ij}^1\le \sum _{j\in \mathcal{T}}\left({\overline{S}}_j^T+{\xi }_j^B{\mathsf{\Delta }}_j\right)x_{ij}^1$$

Subtracting the constant payload capacity $S_i^U$ from both sides preserves the inequality. Since the recourse gap is defined by a standard maximum function $y_i=max(0,{\mathrm{load}}_i-S_i^U)$, which is monotonically non-decreasing, and the penalty coefficient satisfies $P_i>0$, we have

$$\sum _{i\in \mathcal{U}}{P}_{i}max\left(0,{\mathrm{load}}_i^A-S_i^U\right)\le \sum _{i\in \mathcal{U}}{P}_{i}max\left(0,{\mathrm{load}}_i^B-S_i^U\right)$$

Hence, $Q(\mathit{x},{\mathit{\xi }}^A)\le Q(\mathit{x},{\mathit{\xi }}^B)$. This guarantees that the worst-case scenario inherently lies at the upper boundary of the uncertainty budget $\mathsf{\Gamma }$, fundamentally validating the greedy selection strategy in the LE-C&CG framework.    □

Building upon this monotonicity, the identification of worst-case scenarios can be reinterpreted as a combinatorial ranking and selection process. The specific procedural logic is detailed in Algorithm 1.

Algorithm 1: Learning-Enhanced C&CG (LE-C&CG)

To strictly control the computational complexity within the two-stage robust optimization framework, our primary mathematical model focuses on the macro-level task allocation. By utilizing binary assignment variables $x_{ij}^h$, the model guarantees resource reliability against demand uncertainty without suffering from the exponential complexity explosion typically caused by explicit routing variables (e.g., tracking sequences from node to node). Consequently, the exact flight sequences are not implicitly ordered within the main MILP formulation.

To bridge the gap between static robust allocation and dynamic operational execution (as visualized in the subsequent experimental evaluations), we employ a decoupled Post-Processing Nearest-Neighbor Heuristic. Once the optimal robust allocation ${\mathit{x}}^{*}$ is determined by the LE-C&CG algorithm, this heuristic constructs the micro-level flight trajectories and temporal schedules.

It simulates real-time execution by greedily dispatching available UAVs to their assigned, physically closest, and logically valid targets. Specifically, it strictly enforces the “observe-before-intervene” timing prerequisite dynamically: an intervention task at a specific site can only be initiated if its corresponding observation task has been fully completed by the system. The detailed state-updating logic and scheduling procedure are outlined in Algorithm 2.

Algorithm 2: Post-Processing Dynamic Routing and Scheduling

4. Experiments

4.1. Experimental Settings

To evaluate the practical utility of the proposed two-stage robust optimization model, we construct realistic disaster response simulations within a $100\mathrm{km}\times 100\mathrm{km}$ region. All UAVs are dispatched from a centralized base to randomly distributed emergency sites, each possessing a baseline intervention requirement that may unpredictably escalate due to environmental hazards and incident complexity.

The primary parameter settings are detailed in

Table 1. Simulation parameter settings.

Parameter Name	Symbol	Value
Number of Incident Sites	$\left|T\right|$	≤$1.5\left|U\right|$
Task Area Range (km)	$Area$	$[20,100]\times [20,100]$
UAV Payload Capacity (kg)	$S_i^U$	$\mathrm{rand}(15,50)$
UAV Maximum Range (km)	$F_i$	$\mathrm{rand}(3000,8000)$
Unit Distance Consumption	$c_i$	$\mathrm{rand}(0.8,2.0)$
UAV Flight Speed (km/min)	$v_i$	$\mathrm{rand}(2.5,4.0)$
Intervention Success Probability	$p_i^U$	$\mathrm{rand}(0.5,0.8)$
UAV Replacement Cost	$va{l}_i^U$	$\mathrm{rand}(50,150)$
Basic Intervention Demand (kg)	$S_j^T$	10
Incident Urgency Value	$va{l}_j^T$	$\mathrm{rand}(80,200)$
Failure Probability	$p_j^T$	$\mathrm{rand}(0.02,0.08)$
Observation Task Time (min)	$t^0$	$\mathrm{rand}(5,15)$
Penalty Cost for Unmet Demand	$P_j$	5000
Demand Perturbation Magnitude	$\mathsf{\Delta }$	$8.0$
Uncertainty Budget	$\mathsf{\Gamma }$	$max(1,\lfloor 0.4·|T|\rfloor )$

. While exact coordinates and MAS specifications are synthesized to avoid disclosing sensitive data, the parameter ranges are strictly benchmarked against real-world industrial and emergency-response hybrid VTOL (Vertical Take-Off and Landing) UAVs. For example, payload capacities ($S_i^U\in [15,50]$ kg) reflect standard medium-to-heavy lift multi-rotors used in firefighting, and flight speeds ($v_i\in [2.5,4.0]$ km/min) align with rapid-response fixed-wing hybrid drones. Crucially, the unmet demand penalty $P_j$ is deliberately set to 5000, acting as a “Big-M” weight to ensure that the optimization prioritizes disaster mitigation over marginal operational savings.

To comprehensively validate the model and algorithms, we designed 4 experimental groups with varying fleet sizes and proportionally scaled incident sites, yielding 12 distinct instances. All mathematical models were formulated and solved using the Gurobi Optimizer via the gurobipy interface. Computations were executed in Python 3.8 (PyCharm 2024) on a machine equipped with an AMD 7945HX CPU (AMD, Santa Clara, CA, USA) and an NVIDIA RTX 4060 GPU (NVIDIA, Santa Clara, CA, USA).

This section describes 4 groups of experiments with varying fleet sizes, each scaling up the number of incident sites, yielding 12 distinct experimental instances to thoroughly evaluate the proposed model and algorithms.

4.2. Experimental Results and Discussion

4.2.1. Effectiveness Analysis

Table 2. UAV task allocation schedule.

UAVs	Incident Sites	Allocation Summary	Cost	Iter
5	4	U1: {2, 3, 4}; U2: {1}	1.97	3
	5	U1: {3}; U2: {4, 5}; U3: {1, 2}	2.24	3
	6	U1: {3, 4, 5, 6}; U2: {1, 2}	3.09	3
Detailed Plan: U1: [T3(0) → T3(1) → T5(0) → T5(1) → T4(0) → T4(1) → T6(0) → T6(1)]; U2: [T2(0) → T2(1) → T1(0) → T1(1)]
10	8	U1: {3, 8}; U2: {4, 7}; U3: {5, 6}; U4: {1, 2}	2.69	3
	10	U1: {3, 8, 10}; U2: {4, 9}; U3: {6, 7}; …	3.18	3
	12	U1: {3, 8, 12}; U2: {9, 10, 11}; U3: {4, 6, 7}…	4.92	3
Detailed Plan: U1: [T3(0) → T3(1) → T8(0) → T8(1) → T12(0) → T12(1)]; U2: [T10(0) → T10(1) → T11(0) → T11(1) → T9(0) → T9(1)]; U3: [T4(0) → T4(1) → T6(0) → T6(1) → T7(0) → T7(1)]; U4: [T5(0) → T5(1) → T2(0) → T2(1) → T1(0) → T1(1)]
15	12	U1: {3, 8, 12}; U2: {9, 10, 11}; …	4.26	3
	15	U1: {3, 8, 12}; U2: {10, 11, 14, 15}; …	4.88	3
	18	U1: {3, 8}; U2: {10, 12, 14}; U3: {15}…	3.75	3
Detailed Plan: U1: [T3(0) → T3(1) → T8(0) → T8(1)]; U2: [T12(0) → T12(1) → T10(0) → T10(1) → T14(0) → T14(1)]; U3: [T15(0) → T15(1)]; U4: [T11(0) → T11(1) → T16(0) → T16(1) → T9(0) → T9(1)]; U5: [T4(0) → T4(1) → T17(0) → T17(1) → T7(0) → T7(1)]; U6: [T6(0) → T6(1) → T13(0) → T13(1) → T5(0) → T5(1) → T2(0) → T2(1)]; U7: [T1(0) → T1(1) → T18(0) → T18(1)]
20	16	U1: {3}; U2: {8, 10, 12, 14}; …	5.79	3
	20	U1: {3}; U2: {8, 10, 12, 19}; …	4.51	3
	24	U1: {3, 8, 19}; U2: {10, 12, 14, 21}; …	5.33	3
Detailed Plan: U1: [T3(0) → T3(1) → T8(0) → T8(1) → T19(0) → T19(1)]; U2: [T12(0) → T12(1) → T10(0) → T10(1) → T14(0) → T14(1) → T21(0) → T21(1)]; U3: [T15(0) → T15(1) → T11(0) → T11(1) → T9(0) → T9(1)]; U4: [T24(0) → T24(1) → T16(0) → T16(1)]; U5: [T23(0) → T23(1) → T4(0) → T4(1)]; U6: [T20(0) → T20(1) → T7(0) → T7(1)]; U7: [T22(0) → T22(1)]; U8: [T17(0) → T17(1) → T6(0) → T6(1) → T13(0) → T13(1) → T5(0) → T5(1)]; U9: [T2(0) → T2(1)]; U10: [T1(0) → T1(1) → T18(0) → T18(1)]

shows the results of the task allocation for different numbers of UAVs and incident sites and lists the detailed allocation plan for the maximum number of incident sites in each group. The data in Table 2 show that when the number of UAVs is fixed, expanding the target set results in higher total allocation costs. This is because a larger number of targets introduces greater cumulative demand variability, intensifying the allocation constraints and driving up resource expenditure. Table 2 also shows that increasing the number of UAVs can reduce the total cost of task allocation. More UAVs expand the choice space, help find better solutions, and further reduce overall cost.

Figure 1 shows part of the task allocation scheme for 20 UAVs and 24 incident sites. UAVs with different numbers are responsible for performing different tasks for different sites. UAVs 0, 1, and 2 chose sites that are closer to perform tasks. UAVs 3 and 4 chose sites that are moderately distant to perform tasks. UAVs 5, 6, 7, and 8 chose sites that are far away to perform tasks. The task allocation plan shows that UAV task selection comprehensively considers the shortest range and maximum demand satisfaction, further verifying the effectiveness and feasibility of the model.

4.2.2. Robust Analysis

This section describes the use of Monte Carlo simulation to conduct 500 independent evaluation runs for each experimental group. To rigorously test the out-of-sample performance of the optimization models, the uncertain demands in these simulations were not simply drawn from the extreme boundaries of the predefined discrete uncertainty set $\mathsf{\Xi }$ (which would inherently and trivially favor the robust model). Instead, we simulate realistic, continuous environmental noise.

Specifically, the sampling procedure is defined as follows: In each simulation run, every incident site $j\in T$ has an independent probability $p=\mathsf{\Gamma }/\left|T\right|$ of encountering a severe environmental hazard or complexity escalation. Furthermore, if an escalation is triggered, the actual additional resource demand is not fixed at the theoretical worst-case maximum ${\mathsf{\Delta }}_j$ but is rather sampled from a continuous uniform distribution $\mathcal{U}(0,{\mathsf{\Delta }}_j)$. Thus, the realized demand for site j is calculated as ${\tilde{S}}_j^T={\widehat{S}}_j^T+{\delta }_j$, where ${\delta }_j\sim \mathcal{U}(0,{\mathsf{\Delta }}_j)$ with probability p, and ${\delta }_j=0$ otherwise.

Under this probabilistic and continuous evaluation framework, we assess the robustness between the two-stage robust optimization model and the deterministic model through four key indicators: Average Real Cost, Violation Probability, Average Violation Magnitude, and Price of Robustness.

Among them, Average Real Cost is the total cost including penalty costs divided by the number of simulations; Violation Probability is the probability that the UAV will have insufficient influence resource requirements in the simulation; Average Violation Magnitude represents the average demand gap when insufficient influence resource requirements occur; Price of Robustness represents the additional cost paid by the two-stage robust optimization model for higher robustness.

Figure 2 shows the changes in Average Real Cost for the two models. Overall, the total costs of the deterministic models are all higher than those of the two-stage robust optimization models, indicating that the two-stage robust optimization models can avoid greater cost losses by paying a small additional cost up front. With a fixed fleet size, total costs rise for both model types; however, the deterministic model exhibits a sharper cost escalation, leading to a progressively larger performance disparity compared to the robust optimization model.

Figure 3 show the changes in Violation Probability and Average Violation Magnitude of the two models respectively. Overall, the deterministic model has a higher probability of insufficient demand and a larger demand gap. Even when the scale is larger, the probability of insufficient demand is high for both models, but the demand gap is smaller for the two-stage robust optimization model.

Figure 4 shows the changes in Price of Robustness of the two-stage robust optimization model. Overall, in most cases, the additional cost paid was less than 1%, with only one group raising it to 2.7%. However, based on the previous three figures, we can see that by paying a small amount of additional costs, both the demand gap and the total cost can be significantly reduced. Collectively, these four metrics demonstrate that the two-stage robust optimization model effectively incorporates risk considerations, offering superior robustness compared to the deterministic approach.

4.2.3. Algorithm Analysis

This section evaluates the performance of LE-C&CG through three sub-experiments: solution time analysis, sub-problem complexity analysis, and convergence trajectory comparison. The remaining two comparison objects are the Benders algorithm and the standard C&CG algorithm. Among them, the CPU calculation time analysis records the changes in the CPU solution time of the three algorithms as the scale increases; the sub-problem complexity analysis records the solution time changes of LE-C&CG and standard C&CG under the condition of a fixed scale and as the uncertainty budget increases; the convergence trajectory analysis tracks the iterative progress of all three algorithms as they approach the optimal solution.

Figure 5 shows the results of the solution time analysis. Overall, the solution times of LE-C&CG are all one order of magnitude lower than the other two algorithms, indicating that LE-C&CG is optimal regardless of scale. As the scale increases, the solution time of all three algorithms will become longer, but LE-C&CG tends to increase by a smaller amount.

Figure 6 shows the results of the subproblem complexity analysis. Overall, the solution time of LE-C&CG is always lower than 0.05 s, while the solution time of standard C&CG is always higher than 0.25 s. In addition, the uncertain budget growth has not limited the performance of LE-C&CG.

Figure 7 shows the results of algorithm convergence. The curve of LE-C&CG shows a “vertical drop” shape, which can reduce the gap to 0% in a very short time. In contrast, Benders and C&CG need to go through a long iterative process to slowly converge. This evidence confirms the algorithm’s rapid convergence capabilities, satisfying the stringent real-time decision-making requirements inherent in dynamic disaster response operations.

4.2.4. Sensitivity Analysis

This section conducts sensitivity analysis on the three factors of uncertainty budget $\mathsf{\Gamma }$, additional demand $\mathsf{\Delta }$ and penalty cost coefficient $P_j$ and analyzes the impact of the relevant parameters of the uncertainty set on the model. Among them, the parameter $\mathsf{\Gamma }$ reflects the decision-maker’s estimate of the number of incident sites that will occur simultaneously in emergencies in the environment. The parameter $Delta$ represents the upper limit of the worst-case demand surge for a single site; the penalty coefficient $P_j$ reflects the acceptance of using excess demand and partial task failure.

Figure 8 shows the total cost and robustness penalty as a function of the ratio of the uncertainty budget to the number of objectives. Subgraph (a) shows that the total robust cost shows a weak upward trend, but is generally controlled at the same level. This shows that the impact of the uncertain budget on the two-stage robust optimization model is limited and the model’s risk resistance is strong enough. Subgraph (b) shows that the upward trend of robust cost is more obvious. When the uncertainty budget is high, the robustness cost increases even more, but is generally controlled at around 6%, which shows that when the uncertainty level is high, the model can still achieve robustness at a lower cost.

Figure 9 shows reliability and average violation proportion as a function of additional demand. Subgraph (a) shows that reliability shows a downward trend with the increase in additional demand, and the degree of decrease increases with the increase in scale. Subgraph (b) shows that the average violation ratio shows an upward trend with the increase in additional demand. Notably, the average violation ratio exhibits higher sensitivity to $\mathsf{\Delta }$ in smaller-scale scenarios, suggesting that additional demand fluctuations exert a more improved influence on the model when fewer resources are available.

Figure 10 shows the variation of reliability and average violation proportion with penalty cost coefficient $P_j$. Subgraph (a) shows that reliability shows a stable trend with the increase in penalty cost coefficient, but the decline rate increases with the increase in scale. Subgraph (b) shows that the average violation ratio shows a stable trend as the penalty cost coefficient increases, but the decline increases with the increase in scale.

5. Conclusions

5.1. Summary of Contributions

This study addressed the coupled multi-agent task allocation problem for time-sensitive emergency response by incorporating uncertain intervention resource demands caused by environmental hazards and incident complexities. We constructed a two-stage robust optimization model that effectively balances the trade-off between upfront resource investment and potential risk penalties. To overcome the computational bottlenecks of large-scale instances, we proposed the Learning-Enhanced Column and Constraint Generation (LE-C&CG) algorithm, which substitutes the traditional Mixed-Integer Programming subproblem with a specialized greedy heuristic oracle. Extensive simulations verified that the proposed framework significantly enhances task reliability compared to deterministic approaches, reducing violation probabilities while avoiding excessive conservatism, with a marginal Price of Robustness generally under 1%. Moreover, the LE-C&CG algorithm demonstrated superior scalability, achieving convergence an order of magnitude faster than standard Benders and C&CG benchmarks, consistently reaching a 0% optimality gap in under 0.05 s.

5.2. Discussion and Limitations

While existing two-stage robust optimization applications often focus on static resource sizing or independent task scheduling without complex spatial–temporal coupling, our framework provides a rigorous algorithmic foundation that explicitly encodes the strict sequential dependencies of “observe-and-intervene” operations and achieves sub-second decision-making efficiency. Despite these operational and computational advantages over traditional exact dualization methods, several challenges remain for practical deployment in real-world disaster response systems. A critical hurdle is the integration of the proposed model into live command-and-control infrastructures, which requires addressing communication delays and ensuring seamless dynamic updates as ground-truth data shifts during operations.

Furthermore, some theoretical limitations persist. The current uncertainty set relies on estimated intervals and budgets, which may not fully capture the complex, non-linear correlations found in real-world disaster data. Additionally, the efficiency of the heuristic oracle depends on the monotonicity of the penalty function. Future research will focus on: (1) incorporating more realistic, data-driven uncertainty factors; (2) investigating AI-assisted column generation techniques to handle non-monotonicity; and (3) developing robust compensation mechanisms to maintain coordination under intermittent communication failure.