Managing Redundancy in a Multiple-Robot Orienteering Problem for Equilibrium and Robustness

Abstract

In our work, the Robust Multiple-robot Orienteering Problem with Workload Balancing is constructed for the first time. Our primary contribution lies in the rigorous formulation of this problem as a three-stage optimization task. It leverages the Robust Multiple-robot Orienteering Problem (RMOP) as the initial stage. The Path Replanning stage and the workload balancing stage are introduced to minimize walk redundancy and achieve workload equilibrium. The resultant solution upholds the optimality inherent to the original RMOP. Additionally, we craft a suite of heuristic strategies to mitigate redundancy and employ Monte Carlo sampling to tackle the problem. Our algorithm analysis indicates that the method has asymptotic convergence properties and a feasible time complexity under certain conditions. Local parallelization of the algorithm can further improve its performance. Our simulation studies demonstrate that our approach can efficaciously attain a balance between robustness and workload without compromising performance in the presence of adversarial challenges.

1. Introduction

In various missions, whether drone is conducting post-disaster search and rescue operations [1,2,3,4] or a ground mobile robot is exploring and perceiving the surrounding environment [5,6,7,8,9], the initial goal usually involves exploring and mapping the environment. The task of an agent exploring its surroundings can be modeled as the Orienteering Problem (OP) in many scenarios [7,8,9]. OP refers to determining a walk with a given budget from a given starting point, maximizing the total reward collected along the walk [7]. The reward depends on the vertices visited along the walk. The focus of robotic exploring tasks is gradually shifting to unmanned systems composed of multiple robots. Such autonomous systems can help solve new problems or improve existing solutions. In exploring unknown environments, using multiple robots can shorten the overall exploration time, improve accuracy, and be more robust to failures [10,11,12,13]. In [14], a Multiple-robot Orienteering Problem (MOP) aims to design paths for n robots, maximizing the sum of rewards collected by all robots. Recently, Shi et al. proposed the Robust Multiple-robot Orienteering Problem (RMOP) [15], which further considers robots operating in adversarial or fail-prone environments.

The applications of this methodology extend far beyond military domains, holding significant potential in numerous civilian scenarios as well. For instance, in a large-scale post-disaster search and rescue operation, as illustrated in Figure 1, the disaster area can be abstracted as a network where nodes and links represent the critical elements of the rescue mission. The nodes are categorized into two types: target points for searching for survivors (mission points) and waypoints that serve as relays. In the complex and hazardous post-disaster environment, direct communication and passage between mission points are often obstructed, necessitating that the drone fleet first establishes a reliable communication relay network. These relay points can also be assigned additional functions, such as serving as temporary drone maintenance, recharging, or data-uplink stations. By assigning differentiated reward values to different types of nodes (both mission and relay points), they can be handled uniformly. Considering the urgency of disaster rescue missions and the dynamic nature of the environment (e.g., aftershocks, building collapses), the planning of rescue routes must possess a high degree of robustness. Therefore, the RMOP methodology can provide powerful decision support for such tasks, ensuring that in a worst-case scenario where some drones fail or lose contact, the overall mission reward (such as search and rescue coverage and communication network integrity) is still maximized. Beyond disaster response [4,16,17], this robust multi-robot path planning method shows broad application prospects in various other fields, including environmental monitoring [15,18], precision agriculture [18], logistics and transportation [7,19], and tourist route planning [7,19].

RMOP focuses on selecting collections that can withstand the worst-case scenario of removing certain subsets of projects. Under certain circumstances, RMOP can effectively balance the trade-off between excessive overlap (resulting in insufficient reward) and insufficient overlap (resulting in insufficient redundancy). However, its effectiveness diminishes for sparse graphs—a common characteristic of networks formed by drones operating deep within enemy territory with minimal connections. To overcome these challenges, we propose the Robust Multiple-robot Orienteering Problem with Workload Balancing (RMOPWB). It can realize the redundancy balance of the system well.

1.1. Related Works

Although the OP is NP-hard [14], there are some approximation algorithms for many variants of OP. The review by Vansteenwegen et al. [7] provides a summary of various algorithms for OP and its variants. Blum et al. [14] have presented a polynomial-time 4–approximation for OP when the objective function is modular, which is extended to yield a 5–approximation for the MOP assuming all robots start at different vertices. If the reward function is submodular, Chekuri and Pal [20] present a recursive greedy algorithm for a single robot. In [21], Singh et al. demonstrated the utilization of OP and MOP for active information gathering. Their algorithms sequentially determine paths for each robot using the single-robot algorithms by Blum et al. [14] and Chekuri and Pal [20] as subroutines. Atanasov et al. [22] recently presented a decentralized version for multi-robot information gathering following similar principles to [21]. Recently, Jorgensen et al. introduced the Matroids Team-Surviving Orienteers Problem (MTSO) [8], which takes into account individual robot failures by assuming a given probability of failure associated with every edge in the environment. Robust submodular maximization [23,24,25,26,27] focuses on selecting sets that can withstand the worst-case removal of some subset of items. Using robust submodular maximization, RMOP considers worst-case failures and is better suited for operation in adversarial or stochastic conditions when seeking worst-case guarantees.

The assignment of multiple targets to each robot is often treated as a partitioning problem in multi-robot exploration research. A simplified variant of this problem is to assign a single task to each robot, which has been widely studied in other fields. The Hungarian method can optimally solve this problem in polynomial time, and the authors of [28] have applied this method to multi-robot exploration. As described in [29], the issue for the effective coordination of multi-robot systems is the allocation of target points so that each robot can explore different regions of the environment. In [30], the authors use the MiniMax criterion to minimize overall exploration time and use minimum spanning trees (MSTs) to approximate the path costs of robots.

1.2. Contributions

This paper, for the first time, simultaneously considers robustness and load balancing in orienteering tasks in multi-robot systems, presenting a comprehensive approach that enhances the resilience and efficiency of these systems. Our method is validated in a high-fidelity simulation environment. A sample of task allocation results is shown in Figure 2, wherein the environment is explored by six UAVs with the proposed method.

Our principal contributions are highlighted as follows:

• Path Replanning for Workload Management. We introduce a Path Replanning stage, specifically designed to optimize the total workload of multi-robot systems, which we mathematically formalize as Problem 2.
• Load Balancing for Equitable Distribution. We introduce a workload balancing stage, which we formulate as Problem 3. The complete framework, comprising three sequential stages (Problem 1, Problem 2, and Problem 3), is termed RMOPWB. This framework has been empirically validated through numerical simulations, demonstrating its effectiveness in achieving balanced workload distribution.
• Efficient Algorithm for Complex Optimization. To address the complexity of this optimization problem, we propose an efficient algorithm that combines heuristic methods with Monte Carlo sampling. This approach dramatically reduces redundant operational paths while preserving essential redundancy to enhance system robustness.

The remainder of this paper is organized as follows. Section 2 establishes the theoretical foundation for our work. We begin in Section 2.1 with a formal review of the existing RMOP. In Section 2.2, we introduce our novel problem formulation, the RMOPWB, which explicitly addresses load distribution among agents. Section 2.3 is then dedicated to detailing our proposed algorithm for solving the RMOPWB. Subsequently, Section 3 presents the results of extensive numerical simulations designed to validate the performance and efficiency of our proposed method. Finally, Section 4 concludes the paper with a summary of our key findings and a discussion of potential avenues for future research.

2. Materials and Methods

2.1. Review of RMOP

We use calligraphic letters to denote sets (e.g., $\mathcal{A}$). Given a set $\mathcal{B}$, $\mathcal{A}\setminus \mathcal{B}$ denotes the set of elements in $\mathcal{A}$ but not in $\mathcal{B}$. For elements u and v, $\{u,v\}$ denotes an unordered pair and $\langle u,v\rangle$ denotes an ordered pair.

Definition 1

(Submodularity). For a set $\mathcal{V}$, a function $f:2^{\mathcal{V}}\to \mathbb{R}$ is submodular if and only if for any sets $\mathcal{A}\subseteq \mathcal{V}$ and $\mathcal{B}\subseteq \mathcal{V}$, we have $f\left(\mathcal{A}\right)+f\left(\mathcal{B}\right)\ge f(\mathcal{A}\cup \mathcal{B})+f(\mathcal{A}\cap \mathcal{B})$.

Definition 2

(Normalized Monotonicity). For a set $\mathcal{V}$, a function $f:2^{\mathcal{V}}\to \mathbb{R}$ is called normalized, monotonically non-decreasing if and only if for any $\mathcal{A}\subseteq \mathcal{B}\subseteq \mathcal{V}$, we have $f\left(\mathcal{A}\right)\subseteq f\left(\mathcal{B}\right)$ and $\mathcal{A}=\varnothing$ if and only if $f\left(\mathcal{A}\right)=0$.

As a shorthand, we refer to a normalized, monotonically non-decreasing function as a monotone function. Let $G(\mathcal{V},\mathcal{E})$ be a graph where $\mathcal{V}$ is a nonempty finite set and $\mathcal{E}$ is a set of unordered pairs of distinct elements of $\mathcal{V}$. The vertex u and the edge e are incident if $u\in e$. A walk of a graph G is an alternating sequence of vertices and edges $v_0,e_1,v_1,e_2,v_2,\dots ,e_m,v_m$ beginning and ending with vertices such that each edge $e_i$ is incident to $v_{i-1}$ and $v_i$. The walk is also denoted by $\mathcal{W}=\langle v_0,v_1,\dots ,v_m\rangle$. Let $\mathcal{T}$ denote the set of all walks. The set of all connected induced subgraphs of $\mathcal{W}$ is denoted by $I\left(\mathcal{W}\right)$. We use ${\mathcal{V}}_{\mathcal{W}}$ to represent the vertices of $\mathcal{W}$ and ${\mathcal{E}}_{\mathcal{W}}:=\{\{v_0,v_1\},\dots ,\{v_{m-1},v_m\}\}$ to denote its edges. Thus, $\mathcal{W}$ corresponds to the subgraph $G({\mathcal{V}}_{\mathcal{W}},{\mathcal{E}}_{\mathcal{W}})$, and for convenience, we may also refer to $\mathcal{W}$ itself as this subgraph. The cost of a walk $\mathcal{W}$, denoted by $C\left(\mathcal{W}\right)$, is the sum of the edge weights along the walk. The edge weights are assumed to be metric. The walk of robot i, denoted by ${\mathcal{W}}_i$, must begin at a specific vertex $v_{s_i}$. The reward function $g\left({\mathcal{V}}^{\prime }\right):2^{\mathcal{V}}\to {\mathbb{R}}_{+},{\mathcal{V}}^{\prime }\subseteq \mathcal{V}$ is assumed to be a monotone submodular function. Let $\mathcal{S}$ be the union of all n walks of the robots in the team, i.e., $\mathcal{S}=\{{\mathcal{W}}_1,{\mathcal{W}}_2,\dots ,{\mathcal{W}}_n\}$. The team reward $f\left({\mathcal{S}}^{\prime }\right):2^{\mathcal{S}}\to {\mathbb{R}}_{+},{\mathcal{S}}^{\prime }\subseteq \mathcal{S}$ is given by

$$\begin{array}{c}\hfill f\left({\mathcal{S}}^{\prime }\right)=g\left(\bigcup _{{\mathcal{W}}_i\in {\mathcal{S}}^{\prime }}{V}_{{\mathcal{W}}_i}\right).\end{array}$$

(1)

Multiple robots can visit the same vertex, but only one visit is accounted for when computing the reward of the team. The reward of a vertex cannot be accumulated twice for the reward of the team. In [15], there is the mathematical representation of the RMOP:

Problem 1

(Stage #1). Given a graph $G(\mathcal{V},\mathcal{E})$, n robots with starting positions $\{v_{s_1},v_{s_2},\dots ,v_{s_n}\}$, a budget constraint B, a robot reward function $g\left({\mathcal{V}}^{\prime }\right):2^{\mathcal{V}}\to {\mathbb{R}}_{+}$ , and a team reward function f as defined in (1), the Robust Multiple-robot Orienteering Problem seeks to find a collection of n walks ${\mathcal{S}}_1=\{{\mathcal{W}}_1,{\mathcal{W}}_2,\dots ,{\mathcal{W}}_n\}$ that are robust to the worst-case failure of α robots:

$$\begin{array}{cc}\hfill \underset{\mathcal{S}\subseteq \mathcal{T}}{max}\underset{\mathcal{A}\subseteq \mathcal{S}}{min}& f(\mathcal{S}\setminus \mathcal{A})\hfill \\ \hfill s.t.& \left|\mathcal{S}\right|=n,\hfill \\ \hfill & \left|\mathcal{A}\right|\le \alpha ,0<\alpha <n\hfill \\ \hfill & C\left({\mathcal{W}}_j\right)\le B,\forall j\hfill \end{array}$$

(2)

where $v_{s_j}$ must additionally be the starting vertex when constructing a walk ${\mathcal{W}}_j$ for robot j.

2.2. Problem Formulation

The concept of system workload is pivotal in ensuring the longevity and sustainability of a system, particularly within the context of multi-robot operations. This consideration encompasses two principal facets: the aggregate workload, referred to as total workload, and the distribution of this workload, known as Workload Balance. While robustness in system design often necessitates redundancy to enhance resilience, the efficient management of the system load demands a reduction in such redundancy. Striking an equilibrium between these two objectives is the key to achieving redundancy balance. Moreover, the resilience of the system in the face of adversarial attacks is quantified by a function denoted as

$$Rd(\mathcal{S},\alpha )=\underset{\mathcal{A}\subseteq \mathcal{S}}{min}f(\mathcal{S}\setminus \mathcal{A}),\left|\mathcal{A}\right|\le \alpha$$

which represents the cumulative rewards garnered by the robots after an attack by $\alpha$ adversaries.

2.2.1. Minimizing Total Workload

To minimize the total workload, the initial step is termed Path Replanning (stage #2 in Figure 3). This stage must satisfy certain criteria to be formally represented. Following stage #1, there emerges a set of n walks, denoted as $\{{\mathcal{W}}_1,{\mathcal{W}}_2,\dots ,{\mathcal{W}}_n\}$. For each walk ${\mathcal{W}}_i$, the objective is to eliminate unnecessary vertices while ensuring that the resulting condensed connected graph remains a subgraph of ${\mathcal{W}}_i$. Drawing on the concept of an induced subgraph, the post-replanning walk ${\mathcal{W}}_i^{\prime }$ should correspond to a variant of the Traveling Salesman Problem (TSP) walk on some induced subgraph. Let $T\left(I_i\right)$ denote a variant TSP walk on a connected undirected graph $I_i$. The walk $T\left(I_i\right)$ starts at vertex $v_{s_i}$ and traverses all vertices in $I_i$ without returning to the starting vertex, optimizing the budget expenditure for robot i. The set of reconfigured walks $\{{\mathcal{W}}_1^{\prime },{\mathcal{W}}_2^{\prime },\dots ,{\mathcal{W}}_n^{\prime }\}$ must satisfy the following four conditions:

(1)

Budget constraint: $C\left({\mathcal{W}}_i^{\prime }\right)\le B,\forall i\in \{1,\dots ,n\}$.

(2)

TSP-variant walk: ${\mathcal{W}}_i^{\prime }=T\left(I_i\right),\exists I_i\in I\left({\mathcal{W}}_i\right),\forall i$.

(3)

Reward preservation: $Rd({\mathcal{W}}_i,\alpha )=Rd({\mathcal{W}}_i^{\prime },\alpha ),\forall i$.

It can be shown that the walks $\left\{{\mathcal{W}}_i^{\prime }\right\}$ must be optimal for stage #1. Through the above discussion, we formalize the Path Replanning stage:

Problem 2

(Stage #2). After stage #1, we obtain a collection of n walks, denoted as ${\mathcal{S}}_1=\{{\mathcal{W}}_1,{\mathcal{W}}_2,\dots ,{\mathcal{W}}_n\}$. The objective function for stage #1 reaches its optimal value R. The Path Replanning problem is formulated as follows:

$$\begin{array}{cc}\hfill \underset{I_i\in I\left({\mathcal{W}}_i\right)}{min}& C\left(T\left(I_1\right)\right)+C\left(T\left(I_2\right)\right)+\cdots +C\left(T\left(I_n\right)\right)\hfill \\ \hfill s.t.& \underset{\mathcal{A}}{min}f(\mathcal{D}\setminus \mathcal{A})=R,\left|\mathcal{A}\right|\le \alpha ,\mathcal{A}\subseteq \mathcal{D}\hfill \\ \hfill & \mathcal{D}=\{T\left(I_1\right),\dots ,T\left(I_n\right)\},C\left(T\left(I_i\right)\right)\le B.\forall i\hfill \end{array}$$

(3)

The optimal solution $\{I_1,I_2,\dots ,I_n\}$ yields a new set of n walks denoted as ${\mathcal{S}}_2=\{T\left(I_1\right),\dots ,T\left(I_n\right)\}$. For notational convenience, we may refer to $\{I_1,I_2,\dots ,I_n\}$ simply as ${\mathcal{S}}_2$.

Example 1.

As illustrated in Figure 3, the red robot in subfigure (1-1) traverses a path in 5 steps. However, the final step proves redundant as it duplicates the path of the yellow robot, creating unnecessary overlap. By eliminating this superfluous movement, the solution maintains its optimality for Problem 1. This demonstrates that removing such redundancies preserves the solution’s optimal characteristics.

Remark 1.

To mitigate excessive redundancy, one could impose budget constraints B on individual robots. Although this approach effectively prevents path overpopulation, it requires a priori budget allocation for each robot. Furthermore, in multi-robot systems with heterogeneous budget requirements, this method demonstrates limited adaptability to dynamic environments with frequent budget variations. In contrast, Problem 2’s solution achieves superior redundancy balancing.

2.2.2. Achieve Workload Balancing

Now considering workload balancing (represented as stage #3 in Figure 3), we proceed with the formalization. Let $\mathcal{II}\left({\mathcal{S}}_1\right)$ denote the set of optimal solutions from stage #2 for a given ${\mathcal{S}}_1\in {\mathcal{S}}_1^{*}$, where ${\mathcal{S}}_1^{*}$ represents the optimal solution set of stage #1. The workload balancing stage is then formalized using the MiniMax criterion:

Problem 3

(Stage #3). Following stages #1 and #2, we obtain a set of n walks defined as ${\mathcal{S}}^{*}:={\bigcup }_{{\mathcal{S}}_1\in {\mathcal{S}}_1^{*}}\mathcal{II}\left({\mathcal{S}}_1\right)$. Employing the MiniMaxoptimization strategy coupled with a load function $h:\mathcal{T}\to {\mathbb{R}}_{+}$, we identify solutions that achieve optimal load balancing:

$$\underset{{\mathcal{S}}_2\in {\mathcal{S}}^{*}}{min}\underset{{\mathcal{W}}_i\in {\mathcal{S}}_2}{max}h\left({\mathcal{W}}_i\right).$$

(4)

We may define the load function as the total budget expenditure of a single robot; i.e., let $h\left(\mathcal{W}\right)=C\left(\mathcal{W}\right)$. The integrated problem comprising stages #1–#3 is designated as the Robust Multiple Orienteering Problem with Workload Balancing (RMOPWB).

Example 2.

The budget variations of robotic walks from stage #1 to stage #2 are obtained by solving Problem 2, as shown in Figure 3:

$$\mathrm{subfigure}\left(2\text{-}1\right)\text{:}(5,5)\to (4,3),\mathrm{subfigure}\left(2\text{-}2\right)\text{:}(5,5)\to (5,2).$$

Subsequent solution of Problem 3 yields the configuration presented in subfigure (3-1).

2.3. Algorithm for RMOPWB

This section proposes an efficient method for solving RMOPWB. Figure 4 illustrates the entire process of the algorithm. In Algorithm 1, the main procedure of the algorithm for solving RMOPWB is shown. To achieve workload balancing, perform Monte Carlo sampling on the solution set ${\mathcal{S}}^{*}$. After setting the Monte Carlo sampling times K and the load function h, the algorithm executes K loops, and each loop obtains a sample (i.e., a solution to the first two stages of RMOPWB) through two functions (RMOP and PR). After sampling, calculate the load of each solution according to the load function h and select the most balanced solution.

The RMOP function serves as the solver for the RMOP (Stage #1). This stochastic solver typically generates multiple optimal RMOP solutions. The PR function, designed for stage #2, reconstructs input walks to minimize path redundancy through an optimization process. For a formal algorithm description, let L denote the workload balancing metric, derived from the load function h. The metric is formally expressed as $L\left(\mathcal{S}\right)={max}_{{\mathcal{W}}_i\in \mathcal{S}}h\left({\mathcal{W}}_i\right)$.

Algorithm 1: Solve RMOPWB

Algorithm 2: Function PR (Path Replanning)

Algorithm 3:$\mathit{CutCircle}$ and $\mathit{BackWalks}$

2.3.1. Path Replan

Two heuristic methodologies are explored: one for solitary robotic operations and the other for collaborative robotic endeavors. These approaches are implemented through the $\mathit{CutCircle}$ and $\mathit{BackWalks}$ functions, respectively. Algorithm 2 formalizes the Path Replanning process using a while-loop structure, where each iteration processes a walk ${\mathcal{S}}_1\left(i\right)$ via the $\mathit{CutCircle}$ function. Following loop termination, the complete walk set undergoes optimization through the $\mathit{BackWalks}$ function. Implementation details for both functions are provided in Algorithm 3.

• Implementation of the CutCircle function. When a robot exhausts its traversal budget, it may return to a previously visited vertex. This revisit does not necessarily constitute redundancy—in some cases (e.g., example 3 in Figure 5), the robot may restart exploration in a new direction to access high-reward vertices. A cycle is identified when the current vertex connects back to a previously visited counterpart. Such a cycle is deemed redundant and removable only when all constituent vertices are revisits (as demonstrated in example 2 of Figure 5).
• Implementation of the BackWalks function. In crowded environments with multiple robots, walk redundancy primarily manifests as inter-robot path overlaps. The solution involves backtracking walks from terminal vertices to starting points (see examples 1, 4, and 5 in Figure 5). The algorithm sequentially checks each walk’s terminal vertex for presence in other robots’ walks (example 4). For implementation, we must verify whether removing overlapping terminal vertices preserves solution optimality relative to Problem 1.

2.3.2. Algorithm Analysis

The guarantee of optimality for our proposed algorithm is conditional upon the characteristics of the employed RMOP Solver. As illustrated in the workflow of Figure 4 and the procedure in Algorithm 1, if the RMOP Solver is optimal, each execution will yield a random optimal solution ${\mathcal{S}}_1$, where ${\mathcal{S}}_1\in {\mathcal{S}}_1^{*}$, the set of all optimal solutions for stage #1. Under this condition, our Monte Carlo sampling framework is theoretically guaranteed to identify the globally optimal RMOPWB solution. Specifically, for each random optimal solution drawn from the RMOP solver, there is a probability of at least $1/|{\mathcal{S}}_1^{*}|$ of obtaining the solution that corresponds to the global optimum of the RMOPWB problem. While this provides a theoretical guarantee, achieving the optimal trade-off between the sample size and the associated computational cost remains an essential practical consideration, particularly when the cardinality of ${\mathcal{S}}_1^{*}$ is large.

In Algorithm 1, the main computational expenses come from the execution of the functions denoted as RMOP and PR. The total computation time is $K\times (R_1+R_2)$, where K symbolizes the quantity of Monte Carlo samples. $R_1$ and $R_2$ correspond to the respective computational time required for RMOP and PR. Upon the implementation of parallel sampling, the computation time is significantly reduced, culminating in an expression of $R_1+R_2$. $R_2$ consists of two processes: the CutCircle function and the BackWalks function. Suppose that basic operations like sorting and comparison take one unit of time. CutCircle determines for each vertex whether there is a repeated cycle. If there is, remove the cycle. So there is

$$CC\left({\mathcal{W}}_i\right)=\sum _{l=1}^{M_i}(l+l)=M_i(M_i+1).$$

(5)

In the above formula, $M_i$ represents the length of ${\mathcal{W}}_i$. $CC\left({\mathcal{W}}_i\right)$ represents the time taken by CutCircle to process the ${\mathcal{W}}_i$. Then consider an upper bound on the computation time of the BackWalks function:

$$BW=\sum _{j=1}^n{M}_j\left[\sum _{k_1=1}^{j-1}{M}_{k_1}+(M_j-1)+\sum _{k_2=j+1}^n{M}_{k_2}\right]\le \sum _{j=1}^n{M}_j\sum _{k=1}^n{M}_k={\left(\sum _{j=1}^n{M}_j\right)}^2.$$

(6)

Utilizing (5) and (6), deduce the upper bound of the time complexity when employing parallelization techniques:

$$O(R_1+R_2)=O(R_1+\sum _{i=1}^{n}CC\left({\mathcal{W}}_i\right)+BW)=O\left(R_1+2{\left[\sum _{j=1}^n{M}_j\right]}^2\right).$$

3. Results

This section validates our proposed algorithm through a series of numerical simulations. All simulations are conducted on a workstation running Ubuntu 22.04, equipped with a 24-core Intel i9-13900HX CPU (2.20 GHz) and 32 GB RAM. The implementation is developed in Python 3.10.13 under the ROS2 Humble framework.

3.1. Simulation Setup

Simulation Environment: Our foundational simulations are conducted using randomly generated graphs. For more realistic validation, we also implement a higher-fidelity simulation environment in ROS2 Humble on Ubuntu 22.04. In these ROS2 scenarios, the mission topology graphs are predefined to model specific operational environments. It is important to note that these are not merely abstract graphs with uniform edge weights. For instance, the ’City’ environment (Figure 2) models a dense urban area where travel costs between tasks are non-uniform and are estimated via trajectory planning that accounts for obstacles and no-fly zones. Similarly, the ’Maze’ environment (Figure 6) introduces highly constrained navigation challenges, where direct paths are often impossible. This underlying complexity ensures our validation is performed on non-trivial, realistic problem instances.

Baseline algorithms: Given that the RMOPWB (Algorithm 1) is introduced in this paper, there exists no pre-existing efficient algorithm for comparative analysis. Consequently, the RMOPWB algorithm, which is an evolution of the established RMOP algorithm, is juxtaposed against its predecessor as a benchmark. This comparative framework serves to demonstrate the effectiveness of our proposed method. The Sequential Greedy Assignment (SGA) proposed in [15] is used as our RMOP Solver. The GCB algorithm [31] is employed as a subroutine to address the OP due to its efficiency and ease of implementation.

3.2. Simulation Instances

Figure 7a shows the first example. Due to the walks overlapping, we also show the walks in

Table 1. The detailed path planning information of the robot system in Figure 7a.

Figure 7a		Figure 7(a2)	Figure 7(a3)	Figure 7(a4)
Paths	Robot 0	6,19,6,17,0,16	6,19,6,17	6,19,6,17
	Robot 1	20,8,13,1,13,1	20,8,13,1	20,8,13,1
	Robot 2	0,16,4,15,14,11	0,16	0,16,4,15
	Robot 3	13,8,20,5,20,3	13,8,20,5,20,3	13,8
	Robot 4	4,15,14,11,14,4	4,15,14,11	4,15,14,11
	Robot 5	16,4,15,14,11,14	16,4,15	16,4,15,14
Indicators	Reward	124	124	124
	Budget	131.47	73.54	69.51
	WBV	23.93	21.69	13.73

in detail and calculate the total reward (Reward), total budget (Budget), and system Workload Balance value (WBV). In Table 1 and Figure 7(a2), the walks obtained by planning have enough redundancy to allow the robot system to deal with adversarial attacks. In Table 1, it can be observed that, while ensuring the total reward remains unchanged, our algorithm is capable of finding better walks. Figure 7(a3) presents a Path Replanning process that significantly reduces the redundancy of the original walks. The walks for robots 2 and 5 have been substantially shortened. These changes have directly led to a reduction in the total budget from $131.47$ to $73.54$, while the workload is not well balanced. Figure 7(a4) demonstrates the outcome after workload balancing. At this stage, our algorithm shortens the walks for robot 3 from $[13,8,20,5,20,3]$ to $[13,8]$ by adding some walks to robot 2 and 5. Consequently, all walks are shortened to some extent, greatly balancing the workload of the system (the WB value drops from $21.69$ to $13.73$). Overall, from Figure 7a and Table 1, it can be seen that our algorithm is highly effective in achieving a balance between system robustness and system workload.

A second example is illustrated in Figure 7b and detailed in

Table 2. The detailed path planning information of the robot system in Figure 7b.

Figure 7b		Figure 7(b2)	Figure 7(b3)	Figure 7(b4)
Paths	Robot 0	7,20,11,3,9	7,20,11	7,20,11,3,9
	Robot 1	4,16,14,0,13,15	4,16,14,0,13,15	4,16,14
	Robot 2	0,14,16,6,12,6	0,14,16,6	0,13,12
	Robot 3	5,21,8,15,12,6	5,21,8	5,21,8
	Robot 4	14,16,6,12,15,8	14,16,6,12,15,8	14,16,6,12,15
	Robot 5	3,9,3,11,20	3,9,3,11	3,20,17,5
Indicators	Reward	189	189	189
	Budget	135.86	93.78	82.38
	WBV	23.82	22.87	20.85

. Our algorithm effectively diminishes the total workload and achieves workload balancing, all while maintaining the original reward. A key observation is the variation in the set of attacked robots, as depicted in Figure 7b. In the scenario shown in Figure 7(b2), robots 0 and 5 are targeted. In contrast, Figure 7(b3,b4) show attacks on robot sets $\{3,4\}$ and $\{1,4\}$, respectively. This indicates that the attacker’s actions in stages #2 and #3 are responsive to our path planning strategy. The attackers within our model are assumed to be rational, meaning they possess prior knowledge of our RMOPWB strategy. Conversely, in a situation involving irrational attackers, the optimization for stages #2 and #3 would disregard their presence, operating under the assumption that these attackers are only aware of the RMOP strategy.

We further validate our approach through simulations conducted in ROS2 for both the Maze environment shown in Figure 6 and the City environment from Figure 2. In these scenarios, a ground station assigns mission targets to the drone fleet, and the resulting assignments are visualized by the color of the target points. A more detailed comparison of the methods’ performance is presented in

Table 3. Performance comparison of different methods in terms of total reward, total budget spent, and workload balancing value in the City (Figure 2) and Maze (Figure 6) scenarios.

Scene	UAV Num	Method	Reward	Budget	WBV	Runtime (ms)
City	3	RMOP	121.8	72.4	24.3	51
$\left|\mathcal{V}\right|=18$		RMOPWB1	121.8	66.5	23.4	192
$\left|\mathcal{E}\right|=22$		RMOPWB2	121.8	64.8	23.6	76
City	7	RMOP	228.5	125.3	26.1	95
$\left|\mathcal{V}\right|=18$		RMOPWB1	228.5	95.9	22.9	403
$\left|\mathcal{E}\right|=22$		RMOPWB2	228.5	91.4	23.0	151
City	11	RMOP	298.7	187.6	25.6	176
$\left|\mathcal{V}\right|=18$		RMOPWB1	298.7	122.5	21.1	783
$\left|\mathcal{E}\right|=22$		RMOPWB2	298.7	128.0	21.9	212
Maze	3	RMOP	82.1	56.7	28.3	42
$\left|\mathcal{V}\right|=15$		RMOPWB1	82.1	50.1	27.9	141
$\left|\mathcal{E}\right|=20$		RMOPWB2	82.1	51.8	27.1	59
Maze	7	RMOP	155.6	79.6	30.8	78
$\left|\mathcal{V}\right|=15$		RMOPWB1	155.6	64.2	26.8	286
$\left|\mathcal{E}\right|=20$		RMOPWB2	155.6	62.5	25.6	109
Maze	11	RMOP	218.9	116.8	32.7	116
$\left|\mathcal{V}\right|=15$		RMOPWB1	218.9	72.2	25.8	529
$\left|\mathcal{E}\right|=20$		RMOPWB2	218.9	70.9	27.3	161

The distinction between RMOPWB¹ and RMOPWB² is whether the K Monte Carlo sampling steps in Algorithm 1 are performed with parallel processing. RMOPWB² utilizes parallelization, while RMOPWB¹ does not.

. The results in Table 3 clearly demonstrate that our RMOPWB algorithm consistently enables the system to collect greater rewards while simultaneously maintaining a lower total workload and achieving balanced load distribution. More importantly, the parallelized implementation of RMOPWB significantly enhances computational efficiency, though this typically requires a computational platform with greater parallel processing capabilities.

3.3. Simulation Results

Our multi-robot planning scenario is characterized by five key parameters, which fall into three distinct categories: system parameters (the number of robots and the individual robot budget), topological parameters (the number of vertices and edges), and an adversarial parameter (the number of attackers).

Figure 8 illustrates the influence of two critical parameters on the performance of the robotic system. In Figure 8a, the number of robots is varied from 2 to 6. The rewards remain consistent without a decrease. Notably, the RMOPWB algorithm demonstrates a substantial reduction in both the total budget and the load balancing value. Specifically, in the second subfigure of Figure 8a, as the number of robots increases, the budget required by the RMOP algorithm surges significantly, while the budget for the RMOPWB algorithm exhibits only a modest increment. This trend is attributed to the fact that, with other variables held constant, an increased number of robots leads to scene congestion and spatial limitations. A similar phenomenon can be observed in the workload balancing value. In Figure 8b, there are analogous results. As the budget expands, robots are permitted longer walks, suggesting an increasingly congested operational environment. Our RMOPWB reduces system redundancy and becomes increasingly more effective as the budget grows.

Figure 9 illustrates the influence of two key parameters of the task topology: the number of vertices and the number of edges. First, we examine the impact of the number of vertices in Figure 9a. In this setup, the environment’s topology is constructed as a minimum spanning tree (MST), which is a planar graph. Thus, the number of edges is determined by the number of vertices. In contrast, Figure 9b shows the results as we directly increase the number of edges from 10 to 18. Both Figure 9a,b demonstrate that with our proposed algorithm, the total rewards collected by the robot team do not decrease. In both scenarios, increasing either the vertices or the edges leads to a sparser spatial distribution of the robots. This contrasts with the results shown in Figure 8 and causes the performance of the RMOPWB and RMOP algorithms to converge. Despite this convergence, our proposed RMOPWB algorithm still demonstrates a consistent, albeit smaller, improvement over RMOP. Lastly, Figure 10 presents the analysis of the impact of the number of attackers. As the number of attackers increases, the number of available robots decreases. This creates an experimental condition analogous to that in Figure 8a, and thus, the results show a similar trend.

Taken together, these results lead to a clear conclusion: our RMOPWB algorithm is most impactful in dense, confined, or resource-limited scenarios where it can significantly reduce system redundancy and improve workload balancing while ensuring mission rewards are not compromised. However, it is important to critically acknowledge the associated trade-offs. The two additional optimization stages in RMOPWB mean it is computationally more intensive than the simpler RMOP algorithm. While this makes RMOP inherently faster, the results presented in Table 3 demonstrate that our parallelized implementation of RMOPWB yields acceptable computation times for practical applications. This makes RMOPWB a highly effective solution, balancing superior planning quality with manageable computational demands.

4. Discussion

The paper simultaneously considers robustness and workload equilibrium in orienteering tasks for the first time. It is strictly expressed as a three-stage optimization problem called RMOPWB. The approach extends the traditional MOP framework by integrating robust optimization principles and workload balancing. Our contributions include the development of heuristic strategies and the utilization of Monte Carlo sampling techniques to mitigate redundancy. The numerical simulations show that the algorithm can maintain the total rewards, while effectively reducing the excessive redundancy. Building on the comprehensive validation presented here, our immediate future work will focus on deploying and evaluating the RMOPWB algorithm on a physical fleet of UAVs. The reliability and performance demonstrated in our high-fidelity ROS2 environment establish the necessary foundation for this important next step.