# Effect of Formation Size on Flocking Formation Performance for the Goal Reach Problem
## 3 Results  
### Simulation Environment  
The flock controller's experimental setups consisted of different combinations between swarm sizes and arena conditions. According to [14], the used arena size should be 50 \(\times\) 50 m for flock sizes between 5 and 20 agents. The initial zone was 10 \(\times\) 10 m, and the goal zone had a radius of 10 m. However, these specifications were used for a maximum of 20 agents in the flock and may not be suitable when scaling up the flock size. Therefore, we considered the ratio between these specifications to scale the arena, based on the flock size. As such, the arena size could be represented as 100% of the space, the initiating zone may be represented as 4%, and the goal zone about 12.6% of the arena size (according to the circle area \(A=\pi r^{2}\), as shown in Figure 3). These ratios were used in the experiments, regardless of the position of the initial and goal zones. In contrast to [14], we assumed that the initial zone had a circular shape, but still used the same ratio in the initial zone specifications. Scaling up the arena requires more specification, in order to determine its size. Suppose that each agent occupies a 1 m\({}^{2}\) size of the arena; this allows the maximum required space for an agent to be \(\frac{50\times 50}{20}=125\) m\({}^{2}\) of the arena's space. Therefore, using the proper ratio to calculate the suitable space of the arena is achieved by multiplying the maximum flock size by 125. In this work, we conducted experiments with flock sizes in the range of 10-100 agents in steps of 10. To ensure the containment of agents and obstacles, and to have a standard environment for comparison, the arena specifications for the maximum flock size (100 agents) will be considered as follows: The arena size was \(112\times 112\) m, the radius of the initial zone was 13 m, and the goal zone's radius was 22 m. The experimental parameters are summarized in Table 2.  
Tools known as agent-based modeling and simulation (ABMS) tools help researchers to understand the desired model through the simulation of real-world scenarios [33; 34]. These tools use a shared environment, in which agents communicate and interact dynamically with each other [34]. Different ABMS tools have been developed to serve different problem fields. NetLogo [35] has an easy-to-use interface and is capable of simulating complex phenomena [33]. We used NetLogo version 6.2.0 as a simulation tool, on a macOS machine with 2 GHz and 16 GB RAM. NetLogo is based on two concepts representing the simulation world: patch and turtle. The patch represents the world's grid, as shown in Figure 4. On the other hand, turtles are agents who move around the world. The world is split into a \begin{table}
\begin{tabular}{c l} \hline \hline
**Parameter** & **Value** \\ \hline \(n\) & 10–100 agents in steps of 10 \\ Arena size & \(112\times 112\) m \\ Initiate zone radius & 13 m \\ Goal zone radius & 22 m \\ Patch size & \(5.5\times 5.5\) pixels, corresponding to 1 m\({}^{2}\) \\ \(d_{c}\) & 0.55 patch \\ step & 1 s \\ Agent speed & Random number \(\in\)[0–0.5] Patch \\ \(\Delta r\) & 1 patch \\ \(\Delta o\) & 13 patch \\ \(\Delta a\) & 1 patch \\ Priority weights & \(w_{1}=0.4\), \(w_{2}=0.075\), \(w_{3}=0.025\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameter values. Figure 3: Arena specifications, including ratios. grid of patches, and is two-dimensional. Turtles may move across patches, described as square pieces of \"ground\". The patch size was set as \(5.5\times 5.5\) pixels. Agents were set to fit a single patch size. On the other hand, the size of obstacles varied between one to two patches [23]. Obstacles were added manually, as adding obstacles randomly to the arena may lead to scenarios where the flock never applies obstacle avoidance. Collisions were considered to occur when the distance between two agents (or an agent and an obstacle) was less than \(d_{c}=0.55\). A time step does not directly map to real-world time (seconds).