## 4 ITLCO Algorithm
### TLCO Algorithm  
In the design philosophy of the TLCO algorithm, the distinct roles of worker termites, soldier termites, and reproductive termites play a crucial role. Within the colony, worker termites are typically the most numerous caste, responsible for foraging for new food sources and constructing the nest. In the TLCO algorithm, worker termites account for 70% of the total population, represented by particles \(\xi_{\rm worker}\), whose capability is to explore new search spaces to find potential solutions. Soldier termites, representing 30% of the population and denoted by particles \(\xi_{\rm soldier}\), are responsible for defending the colony in reality and improving and optimizing solutions by exploiting the vicinity of the current best solution. Reproductive termites, whose primary task in reality is to reproduce new individuals, are represented by particles \(\xi_{\rm reproductive}\) and emerge to explore new areas when worker and soldier termites fail to find better solutions over extended periods. The movement strategy in the TLCO algorithm is based on a combination of random walks and Levy flights, allowing for both long-distance and short-distance movements. The step length of movement is adjusted by varying the parameter \(\beta_{3}\) from 1.5 to 2 during iterations, ensuring that the algorithm starts from an initial population of termites and updates their positions according to the movement strategy for each caste. The best solution found by each caste is used to update the global best solution. This process continues until the termination condition is met, ensuring the algorithm converges to the optimal solution. The movement equation is as follows:  
\[\overrightarrow{\xi_{\rm new}}=\varpi_{1}\overrightarrow{\xi_{\rm end}}+ \varpi_{2}\Big{|}\varpi_{3}\overrightarrow{\xi^{*}}-\varpi_{4}\overrightarrow {\xi_{\rm end}}\Big{|} \tag{39}\]  
where \(\overrightarrow{\xi_{\rm new}}\) is the new position of the termite; \(\overrightarrow{\xi_{\rm end}}\) is the old position of the termite; \(\varpi_{1},\varpi_{2}\), \(\varpi_{3}\), and \(\varpi_{4}\) are parameters controlling the direction and distance of movement. The actual execution process of the TLCO algorithm is as follows: Suppose the initial number of individuals in the termite colony is \(\wp\), then \(\overrightarrow{\xi_{\rm c,worker}},1\leq\varsigma\leq 0.7\wp\) represents the position of the worker termites, and \(\overrightarrow{\xi_{\rm co,collider}},0.7\wp\leq\varsigma\leq\wp\) represents the position of the soldier termites. Worker termites use long step lengths to cover extensive areas, thereby enhancing the algorithm's exploration capability. Soldier termites employ short step lengths to focus on the vicinity of the best solution, thereby enhancing the algorithm's exploitation capability. When \(\overrightarrow{\xi_{\rm c,worker}}\) and \(\overrightarrow{\xi_{\rm co,collider}}\) frequently conduct inefficient suboptimal searches, the reproductive termites \(\overrightarrow{\xi_{\rm c, reproductive}}\) can move to new positions far from local optima and discover new optimal search spaces to replace the suboptimal ones. (40) represents the frequency of appearance of the reproductive termites \(\overrightarrow{\xi_{\varsigma,\text{ reproductive}}}\) that replace \(\overrightarrow{\xi_{\varsigma,\text{worker}}}\) and \(\overrightarrow{\xi_{\varsigma,\text{soddier}}}\) at the \(t\)-th iteration. \[\left\{ \begin{array}{l}\lambda_{\text{worker}}(t)=1-\dfrac{1}{1+e^{- \sigma(t-0.5t_{\text{max}})}}\\ \lambda_{\text{soddier}}(t)=1-\lambda_{\text{worker}}(t)\end{array} \right. \tag{40}\]  
where \(\lambda_{\text{worker}}(t)\) and \(\lambda_{\text{soddier}}(t)\in\) [0,1] are the distribution values of the worker and soldier termites at the \(t\)-th iteration, respectively; \(\sigma=\frac{1}{0.1t_{\text{max}}}\) is a constant; and \(t_{\text{max}}\) is the maximum number of iterations. According to (40), \(\overrightarrow{\xi_{\varsigma,\text{ reproductive}}}\) will appear in the following two scenarios:  
\[\text{If }\left(