## 4 The Algorithms
### Transportation Role
#### 4.3.1 Transport of an Agent
If the agent \(a_{\{f,1,v\}}\) operating in \(s_{\{1,f\}}\) finds that the value of the pheromone mark requesting the addition of an agent \(Pena_{\{2,m,d\}}(t)\), at the entrance \(e_{2_{m}}\), exceeds the threshold value \(TPea_{\{d,2,x\}}\), it takes the request; the value of \(Pena_{\{2,m,d\}}(t)\) is set to zero and transfers the request to the entrance it has previously used to enter the actual subspace  
was calculated as the average value of five repetitions of the experiment. Every repetition had 10,000 iterations. The value of the pheromone \(Pa_{\{f,k,v\}}\) was set to \(Pmax_{\{f,k,v\}}\) (therefore the parameter \(TRA_{\{f,k,v\}}\leq Pa_{\{f,k,v\}}\) has to be set by this rule). Figure 3 shows the number of agents added into the group in the 10,000 th iteration based on two different initial pheromone mark settings. The parameter \(TRA_{\{f,k,v\}}\), expressed as a percentage of \(Pmax_{\{f,k,v\}}\), is on the \(X\) axis, and the \(Y\) axis shows the number of agents added into the group in the 10,000 th iteration. Axis \(Z\) shows the value of \(Pmax_{\{f,k,v\}}\). \(TPea_{\{f,k,v\}}\) is set to \(TPea_{\{f,k,v\}}=3*Pmax_{\{f,k,v\}}\) in Figure 3a, and \(TPea_{\{f,k,v\}}=5*Pmax_{\{f,k,v\}}\) in Figure 3b. Figure 4 shows another view of the results of the same experiments; however, the \(Y\) axis shows how many times the exploration space was explored (as the average from five repetitions of the experiments measured after 10,000 iterations). Different colors of the points represent the values of \(TCa_{\{f,k,v\}}\). Figure 3: The number of agents added in the group at the 10,000 th iteration of the exploration process: \(Pa_{\{f,k,v\}}=Pmax_{\{f,k,v\}}\), the exploration space size represented \(10\times 8\times 5\) cells, \(TPea_{\{f,k,v\}}=3*Pmax_{\{f,k,v\}}\), and \(TPea_{\{f,k,v\}}=5*Pmax_{\{f,k,v\}}\). The results demonstrated that, with the current parameter settings, adding more agents into the space led to a situation where the space was explored more times. We can also assume that decreasing \(Pmax_{\{f,k,p\}}\) (resp. \(Pa_{\{f,k,p\}}\)) leads to an increasing number of agents in the space. Decreasing \(TRa_{\{f,k,p\}}\) had a similar effect. On the contrary, increasing \(TCa_{\{f,k,p\}}\) increased the number of the agents in the space. The average number of conflicts between the agents increased with the number of the agents in the group. However, the graph is similar to the graph in Figure 4. The agents' performance as a group is shown in the graph in Figure 5.  
Our previous study of this algorithm proved that a group achieved the best results when \(Pmax_{\{f,k,p\}}\) was roughly equal to the number of the cells of the explored space. This can be seen in Figure 5 (\(Pmax_{\{f,k,p\}}=400\sim 450\), \(TRa_{\{f,k,p\}}=90\%\sim 100\%\) of \(Pmax_{\{f,k,p\}}\)). A high value of \(Pmax_{\{f,k,p\}}\) caused the agents to explore the entire space and then to decide whether or not they need help (therefore, we focus on this range of \(Pmax_{\{f,k,p\}}\) in our investigation). The next part of Figure 5 shows the situation when \(Pmax_{\{f,k,p\}}\) was set incorrectly with respect to \(TRa_{\{f,k,p\}}\) (the number of explorations per agent in the group is less than three). Agents with this setting were unable to create a highly efficient group in the exploration space. Reducing \(TRa_{\{f,k,p\}}\) led to a higher efficiency of the group. Figure 6 shows how the agents were added during the iterations. The curves represent (except  
Figure 4: The number of explorations of the entire space in the 10,000 th iteration: \(TPea_{\{f,k,p\}}=3*Pmax_{\{f,k,p\}}\). Figure 5: The average number of explorations per agent at the 10,000 th iteration: \(TPea_{\{f,k,p\}}=3*Pmax_{\{f,k,p\}}\), \(Pa_{\{f,k,p\}}=Pmax_{\{f,k,p\}}\), and the exploration space was \(10\times 8\times 5\) cells. for the graph in Figure 6) the averages of the population sizes measured in five experiments that lasted for 10,000 iterations.