## 4 Proposed Methods
### Ant Colony Optimization
3:\(\Psi_{\text{best}}^{k}\leftarrow\{\ |\,\text{for all }k\in\{1,2,\ldots,N_{\text{u}}\}\). 4:\(L_{\text{unv}}^{k}\leftarrow\{\ |\ \}_{i=1}^{N}\), for all \(k\in\{1,2,\ldots,N_{\text{u}}\}\). 5:For\(r=1\) to \(N_{\text{m}}\)
6:\(v_{\text{curr}}^{k}=v_{\text{start}}^{k,r}\)
7:Initialize(\(v_{\text{curr}}^{k}\), \(v_{\text{start}}^{k,r}\));
8:\(L_{\text{unv}}^{k}\leftarrow\{\ |\ \}_{i=1}^{N}\), \(L_{\text{e}}^{k}\leftarrow\{\ |\,\}_{\psi}^{k}\leftarrow\{\ |\,L_{\xi}^{k}\leftarrow\{\ |\,L_{\xi,\text{sel}}^{k}\leftarrow\{\ |\,L_{\text{vis}}^{k}\leftarrow\{\ |\ \}}\). 9:For each \(k\)-th ant species
10:While\(L_{\text{unv}}^{k}\
eq\{\ |\ \}\)
11:\(L_{\text{e}}^{k}\leftarrow\)CreateEdgeList(\(v_{\text{curr}}^{k}\)). 12:For each edge in \(L_{\text{e}}^{k}\), i.e., \({}^{q}e\),
13:If\(\text{end}_{\text{e}}(^{q}e)\in L_{\text{unv}}^{k}\). 14:If\(\text{IsCollidated}(L_{\text{vis}}^{l},\text{end}_{\text{e}}(^{q}e))\) is TRUE
15:continue;
16:End
17:\(\psi=\)CreateSubTrajectoriesList(\(L_{\psi}^{k}\),\(\theta\)). 18:\(L_{\psi}^{k}\leftarrow{}^{p}\psi\). 19:For each \(p\)-th trajectory in \(L_{\psi}^{k}\) i.e., \({}^{p}\psi\),
20:\(L_{\xi}^{k}\leftarrow\)CreateAugmentedEdgesList(\(L_{\psi}^{k}\), \({}^{p}\psi\)). 21:End
22:\(L_{\text{c,sel}}^{k}\leftarrow\)SelectAugmentedEdge(\(L_{\xi}^{k}\)). 23:\(\{\text{sk}\}_{\text{m,max}}^{k}\leftarrow\)CalculateMaxArrivalTime(\(L_{\xi,\text{sel}}^{k}\)). 24:Remove(\(L_{\text{unv}}^{k}\), \(\text{end}_{\text{e}}(^{q}e)\)). 25:Else
26:continue. 27:End
28:End
29:If\(L_{\psi}^{k}==\{\ \}\)
30:\({}^{h}\tau^{k}=\)ReducePheromoneTrailsAmount(\(L_{\xi,\text{sel}}^{k}\), \({}^{h}\tau^{k}\))
31:go to Line 5. 32:End
33:\(L_{\psi}^{k}\leftarrow\{\ |\ \}\). 34:End//end while
35:End//end for
36:\({}^{h}\tau^{k}=\)CalculatePheromoneTrailsAmount(\(L_{\xi,\text{sel}}^{k}\), \({}^{h}\tau^{k}\)), for all \(k\in\{1,2,\ldots,N_{\text{u}}\}\). 37:\(P\left(L_{\xi,\text{sel}}^{k}\right)=\)CalculateProbability(\(L_{\xi,\text{sel}}^{k}\),\({}^{h}\tau^{k}\)), for all \(k\in\{1,2,\ldots,N_{\text{u}}\}\). 38:\(\text{sk}_{\text{best}}^{k}=\)min(\(\text{sk}_{\text{i,max}}^{k}\))
39:\(L_{\text{unv}}^{k}\leftarrow\{\ |\ \}_{i=1}^{N_{\text{u}}^{k},\text{max}}\)
40:End
41:Outputs: \(\psi_{\text{best}}^{k}\) with \(\text{sk}_{\text{i,max}}^{k}=\text{sk}_{\text{best}}^{k}\), for all \(k\in\{1,2,\ldots,N_{\text{u}}\}\). ```  
**Algorithm 3** Main Algorithm  
In Lines 19-21, the function _CreateAugmentedEdgesList_ (\(L_{\psi^{\prime}}^{k}\), \({}^{p}\psi\)) is used to extract the augmented edges that is spanned by the \(p\)-th trajectory in \(L_{\psi}^{k}\). This function uses augmented-edge adjacency matrix \({}^{\xi}B\) as a reference to find the correct augmented edge that is spanned by the \(p\)-th trajectory in \(L_{\psi}^{k}\). The result is \({}^{h}\xi\) and is pushed to \(L^{k}_{\xi}\), and we select the augmented edge, whose probability, calculated using Equation (11), is the largest. The selected augmented edge is then pushed into \(L^{k}_{\xi,\text{sel}}\) (Line 22). In addition, the maximum traveling time, \(t^{k}_{i,max}\), is calculated in Line 23. Here, the final value of \(t^{k}_{i,max}\) is the traveling time of the slowest ant species. Moreover, the member of \(L^{k}_{\text{unv}}\) that is the same to \(\mathit{end}_{\text{e}}({}^{\text{\small 4}e})\) is removed (Line 24), since the \(\mathit{end}_{\text{e}}({}^{\text{\small 4}e})\) is selected to be visited. After all ants from all species complete their routes, the pheromone trails on all augmented edges are updated in Line 31 by using Equation (11). By using the current pheromone trails, the probability of all augmented edges in \(L^{k}_{\xi}\), \(k\in\{1,2,\ldots,N_{\text{u}}\}\) is calculated by using \(\mathit{CalculateProbability}(L^{k}_{\xi,\text{sel}}{}^{h}\tau k)\) in Line 37. The global minimum, \(t^{k}_{\text{best}}\), is then calculated in Line 38. Finally, the output is \(\psi^{k}_{\text{best}}\) with \(t^{k}_{\text{i,max}}=t^{k}_{\text{best}}\), for all \(k\in\{1,2,\ldots,N_{\text{u}}\}\).