# Dynamic Response Threshold Model for Self-Organized Task Allocation in a Swarm of Foraging Robots
## 4 Discussion
### Different Sizes of Foraging Arena  
We designed a set of experiments to investigate the effect of the size of the foraging arena on the foraging performance. We ran experiments for a swarm with \(L=3\), \(d=1\), and keeping the other parameters the same as in the comparison experiments.  
As shown in Figure 16, the DRTM (\(V_{f}=2.35\)) maintains a number of food tokens that is closer to the desired level compared to the ARTM (\(V_{f}=4.61\)). The total number of food tokens collected by the DRTM is 165.32 and the net food is 17.75; the corresponding values for the ARTM are 170.90 and 15.22, respectively. The DRTM (\(E_{e}=10.74\%\)) consumes less energy than the ARTM does (\(E_{e}=8.91\%\)), meaning that the DRTM gives rise to more efficient foraging.  
Figure 16: Net number of food tokens in nest (\(N_{f}=20\), \(E_{a}=10\), \(N_{r}=20\), \(L=3\)).  
Figure 15: Number of OAEs (\(N_{f}=20\), \(E_{a}=10\), \(N_{r}=30\), \(L=4\)).  
As shown in Figure 17, the ARTM swarm (\(M_{T}\) = 1882.32) clearly produces many more OAEs than the DRTM one does (\(M_{T}\) = 1423.08). By taking the traffic information into account, the DRTM (\(M_{V}\) = 71.15) allocates an appropriate number of robots to go foraging and, therefore, responds more effectively to traffic congestion compared to the ARTM (\(M_{V}\) = 94.12).  
Foraging robots are more likely to find food in a smaller arena, thereby decreasing the ADF. However, swarm foraging in a smaller arena produces many more OAEs, thereby reducing the energy efficiency. Compared with the comparison experiments, the ARTM swarm produces too many OAEs, and so we conclude that the DRTM deals better with traffic congestion than the ARTM does. Therefore, using too small an arena reduces the foraging efficiency.  
Table 4 lists the results of \(t\)-tests between the DRTM and ARTM for each of the preceding subsections (listed are the t value, the two-tailed \(p\) value, and the significance of the results). The sample size and number of degrees of freedom were set as 50 and 98, respectively, for all cases in Table 3. Based on the results of these statistical tests, in Table 4, it is clear that the results of DRTM are significantly better than the results of ARTM.  
In summary, the foraging efficiency was affected by many factors, such as the number of food tokens in the foraging arena (\(N_{f}\)), the energy consumption per OAE (\(E_{a}\)), the total number of robots (\(N_{r}\)), and the size of the foraging arena (\(L\)\(\times\)\(L\)). We performed the simulation experiments many times under different conditions, and the experimental results showed that the DRTM gives rise to more efficient foraging compared to the ARTM.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Section** & **Compare** & \(t\)**-Value** & **Two-Tailed \(P\)** & **Significance** \\ \hline
5.1 & \(F_{t}\) & 4.9663 & 2.89 \(\times\) 10\({}^{-6}\) & YES \\
5.2 & \(M_{T}\) & 2.7761 & 6.59 \(\times\) 10\({}^{-3}\) & YES \\
5.3 & \(F_{t}\) & 3.2744 & 1.46 \(\times\) 10\({}^{-3}\) & YES \\
5.4 & \(M_{T}\) & 2.3803 & 0.0192 & YES \\
5.5 & \(F_{t}\) & 4.7869 & 6.00 \(\times\) 10\({}^{-6}\) & YES \\
5.6 & \(M_{T}\) & 3.9267 & 1.60 \(\times\) 10\({}^{-4}\) & YES \\
5.7 & \(F_{t}\) & 2.1899 & 0.0310 & YES \\
5.8 & \(M_{T}\) & 2.6984 & 8.21 \(\times\) 10\({}^{-3}\) & YES \\ \hline \hline \end{tabular}
\end{table}
Table 4: Results of \(t\)-tests between DRTM and ARTM regarding total amount of food (\(F_{t}\)) and total amount of obstacle avoidance (\(M_{T}\)) for all cases.  
Figure 17: Number of OAEs (\(N_{f}\) = 20, \(E_{a}\) = 10, \(N_{r}\) = 20, \(L\) = 3).