## 4 ITLCO Algorithm
### Improved Scheme
#### 4.2.2 Incremental Pheromone Update Model
\begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Detect Environmental Change and Update Pheromone** \\ \hline Output & - Updated pheromone evaporation rate rho \\  & - Updated pheromone matrix tau \\  & 1. Function detect\_change(f\_history, H\_history, gamma): \\  & // Calculate the absolute difference in entropy between the last two time steps \\  & delta\_H \(\leftarrow\) 1H\_history[last] - H\_history[second\_last] \(|\) \\  & // Check if the entropy change exceeds the threshold \\  & return delta\_H \(>\) gamma * H\_history[second\_last] \\  & 2. Main process: \\  & // Call the detect\_change function to check for environmental \\  & change \\ Steps & if detect\_change(f\_history, H\_history, gamma) then \\  & // Double the pheromone evaporation rate, but limit it to a \\  & maximum of 0.99 \\  & rho \(\leftarrow\) min(0.99, 2 * rho) \\  & // Partially reset the pheromone matrix by initializing it \\  & and multiplying by 0.5 \\  & tau \(\leftarrow\) initialize\_pheromone(\(\ldots\)) * 0.5 \\  & end if \\  & 3. Return the updated parameters: \\  & return rho, tau \\ \hline \hline \end{tabular}  
The pseudocode for sliding window maintenance is shown in Table 3.  
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Calculate Window—Size and Update Pheromone** \\ \hline Input & - M\_base: Base value for window size \\  & - t: Current iteration number \\  & - T: Total number of iterations \\  & - solution\_buffer: Buffer storing solution history \\  & - update\_with\_window: Function to update pheromone using a \\  & window of recent solutions \\ Output & - Updated pheromone matrix tau \\  & 1. Calculate the window size: \\  & window\_size \(\leftarrow\) M\_base * (1 + \(\sin(2*\pi*\) t/T)) \\  & // Convert window\_size to an integer (assuming it should be \\  & an integer in the context) \\  & window\_size \(\leftarrow\) floor(window\_size) \\ Steps & 2. Get the recent solutions: \\  & recent\_solutions \(\leftarrow\) solution\_buffer[-window\_size:] \\  & 3. Update the pheromone: \\  & tau \(\leftarrow\) update\_with\_window(recent\_solutions) \\  & 4. Return the updated pheromone matrix: \\  & return tau \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Sliding window maintenance. (3) Dynamic environment adaptation strategy  
In the dynamic environment adaptation strategy for the TLCO algorithm, different approaches are employed to handle various environmental change scenarios. For gradually changing environments, a method of retaining partial historical experience through exponential decay is adopted. Specifically, the pheromone update formula is as follows:  
\[E_{h\ell}(\iota+1)=0.8E_{h\ell}(\iota)+0.2\Delta E_{h\ell}^{\rm new} \tag{48}\]  
where \(\Delta E_{h\ell}^{\rm new}\) is a new pheromone. This formula ensures that the algorithm can leverage past knowledge while also incorporating new information, enabling it to adapt smoothly to slow-paced environmental changes. On the other hand, when a drastic environmental change is detected, a more aggressive approach is taken. In this case, 10% of the elite pheromone is preserved, while the pheromone in the remaining non-elite areas is reset to a uniform distribution. The reset pheromone \(E_{\rm rest}\) is calculated as follows:  
\[E_{\rm rest}=E_{\rm elite}+\frac{1-E_{\rm elite}}{N_{\rm non-elite}} \tag{49}\]  
where \(E_{\rm elite}\) is the elite pheromone and \(N_{\rm non-elite}\) represents the number of non-elite areas. This strategy allows the algorithm to quickly discard obsolete information and explore new search spaces, enhancing its ability to respond effectively to sudden environmental shifts. By combining these two methods, the TLCO algorithm can better adapt to dynamic environments, improving its performance and efficiency in optimization tasks.