# Bio-Inspired Traffic Pattern Generation for Multi-AMR Systems
## 3 Methodology
### Illustrative Example  
To illustrate the key mechanisms of the optimization process, we present a simplified 4 \(\times\) 4 test scenario, shown in Figure 2. The environment contains four AMRs with pickup and dropoff stations positioned in the corners. For each generated task \(T\), the due time \(t_{due}\) is set to \(d+3\), where \(d\) is the minimum path length for the AMR to complete the task. This buffer of 3 time units provides some flexibility in conflict resolution while maintaining pressure on the system to optimize routes. The collision penalty parameter \(t_{p}\) is set to 6 units, which creates a significant incentive for the optimization process to develop conflict-avoiding movement patterns. The optimization process results in two symmetrical solutions, shown in Figure 2a,b, demonstrating how different movement patterns can evolve under the same optimization parameters due to the stochastic nature of the process. Both solutions generate inner and outer rings, preferring movements in opposite directions. The temporal evolution of movement patterns during the optimization process can be observed through successive snapshots taken every 20,000 iterations, as shown in Figure 3. For clarity, only costs lower than the neutral value of 0.5 are displayed. During the first 100,000 iterations, an inner roundabout pattern emerges through the combined effect of the movement reward mechanism and the collision penalty. The latter is particularly important in penalizing the initial shortest routes along the outer edges of the environment, where head-on collisions are most likely to occur. This naturally leads to the formation of a central circular pattern that provides conflict-free paths between stations. Between iterations 100,000 and 150,000, the delay penalty becomes the dominant factor, driving the propagation of costs outward from the established inner ring. In this phase, the organized movement pattern is effectively extended to the outer regions of the environment. In the final phase of optimization (iterations 150,000 to 200,000), the established movement patterns are reinforced through continued application of the movement reward mechanism, while the evaporation mechanism evaporates the rest. Figure 2: Solutions to the test problem: (**a**) first solution and (**b**) second solution. Further experiments with modified parameters reveal how different conditions influence the emerging movement patterns. When only a single AMR operates in the environment, the optimization generates bidirectional paths that directly connect pickup and dropoff stations, as shown in Figure 4a. Without the need to avoid conflicts, the system naturally converges to these shortest paths between stations. The importance of the collision penalty mechanism becomes evident when it is disabled, as illustrated in Figure 4b, which shows the optimization results using 4 AMRs. In this case, the movement reward mechanism alone leads to the formation of a single-direction outer ring, where the direction (clockwise or counterclockwise) is randomly determined in the early stages of optimization and subsequently reinforced through positive feedback. If, instead, additional tasks are introduced between the top two stations, creating higher traffic demand along the top row, a distinct hybrid pattern emerges (Figure 4c). A bidirectional path is formed between these frequently connected stations, demonstrating the algorithm's ability to adapt to non-uniform task distribution patterns. A sensitivity analysis was performed by varying key parameters, specifically the collision penalty \(p_{c}\) in Phase 1 and the delay factor \(\alpha_{d}\) in Phase 2. As illustrated in Figure 5, alternative parameter values can sometimes achieve similar performance levels. However, the default parameter configuration converges more reliably to the optimal solution.