# Managing Redundancy in a Multiple-Robot Orienteering Problem for Equilibrium and Robustness
## 3 Results
### Simulation Instances  
Figure 6(a) shows the first example. Due to the walks overlapping, we also show the walks in Table 1 in detail and calculate the total reward (Reward), total budget (Budget), and system Workload Balance value (WBV). In Table 1 and Figure 6(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 6(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 6(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 6(a) 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 6(b) and detailed in Table 2. 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 6(b). In the scenario shown in Figure 6(b2), robots 0 and 5 are targeted. In contrast, Figures 6(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. 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.  
Figure 7: Two mission topology graphs. Six robots are set and numbered in order as \(0,1,2,3,4,5\). The budget of each robot is 24. The initial positions of the robots are randomly distributed and \(|\mathcal{A}|=2\). For all vertices in the two graphs, different colors are used to represent rewards and the reward values are labeled in black font on the upper right corner of each vertex. The vertex number is labeled in gray font. For the two scenarios, the first subgraph, (a1) and (b1), comprehensively displays the task topology information and the initial state of the robots. The subsequent subgraphs illustrate the three stages of the RMOPWB algorithm. Specifically, (a2) and (b2) demonstrate the task allocation results for the six robots in Stage #1. Following this, (a3) and (b3) depict the task allocation results for each robot in Stage #2. Finally, (a4) and (b4) show the final paths and the rewards obtained by each robot after completing Stage #3.