## 2 Methods
### Formulation of the Two-Objective ILP Model  
In this section, a two-objective integer linear programming (ILP) model of OP-MATSP is established. We first propose the sets, parameters, decision variables, constraints, and objective function of the two-objective integer linear programming (ILP) model for the multi-depot asymmetric traveling salesman problem (MATSP). Then, we can give the two-objective ILP model for OP-MATSP by changing the objective function. 1. Sets \(V_{n}\): the city node set. \(V_{m}\): the depot node set. Since each traveling salesman starts from a unique depot node, we assume that the set of traveling salesmen is equal to the depot node set \(V_{m}\). \(V\): the city and depot node set. 2. Parameters \(c_{ij}\): the cost from node \(i\) to node \(j\). \(q_{i}\): the own cost of city node \(i\). 3. Decision variables \(x_{ij}^{m}\): if \(m\)th traveling salesman visits the node \(i\) and moves to the next node \(j\), then \(x_{ij}^{m}=1\); otherwise, \(x_{ij}^{m}=0\), \(i,j\in V\), \(m\in V_{m}\). Figure 4: Task sequences for the multi-robot task assignment. \(y_{i}^{m}\): if \(m\)th traveling salesman visits the node \(i\), then \(y_{i}^{m}=1\); otherwise, \(y_{i}^{m}=0\), \(i\in V\), \(m\in V_{m}\). 4. Constraints 1. The \(m\)th traveling salesman can only visit the city node from the \(m\)th depot node, but not other depot nodes. To guarantee the above assumptions, we set the following constraints: \[x_{ij}^{m}=x_{ji}^{m}=0,\forall i,j\in V_{m},i\
eq j,\forall m\in V_{m}\] 5. To prevent the traveling salesman from repeatedly visiting the same city node or depot node, the following constraint is established: \[x_{ii}^{m}=0,\forall i\in V,\forall m\in V_{m}\] 6. The relationship between decision variables \(x_{ij}^{m}\) and \(y_{i}^{m}\) can be expressed as: \[\sum_{j\in V}x_{ji}^{m}=y_{i}^{m},\forall i\in V_{n},\forall m\in V_{m}\] 7 \[\sum_{j\in V}x_{ij}^{m}=y_{i}^{m},\forall i\in V_{n},\forall m\in V_{m}\] 8. The number of node sequences is the same as the number of traveling salesmen. To ensure each traveling salesman visits only one node sequence, we add the following constraints: \[\sum_{j\in V_{n}}x_{ij}^{m}=1,\forall i,m\in V_{m}\] 9 \[\sum_{j\in V_{n}}x_{ji}^{m}=1,\forall i,m\in V_{m}\] 10. The city node can only be visited by one traveling salesman. To prevent one city node from being repeatedly visited by different traveling salesmen, the following constraints are established: \[\sum_{j\in V}\sum_{m\in V_{m}}x_{ij}^{m}=1,\forall i\in V_{n},i\
eq j\] 11 \[\sum_{i\in V}\sum_{m\in V_{m}}x_{ij}^{m}=1,\forall j\in V_{n},i\
eq j\] 12. To prevent the traveling salesman from visiting the previous node after visiting the current node, we establish the following constraints: \[\sum_{m\in V_{m}}x_{ij}^{m}+\sum_{m\in V_{m}}x_{ji}^{m}<2,\forall i,j\in V,i\
eq j\] 13. To ensure the continuity of node sequence and avoid the fracture of node sequence, the following flow balance constraint is established: \[x_{ij}^{m}=x_{ji}^{m}=0,\forall i,j\in V_{m},i\
eq j,\forall m\in V_{m}\]0 14. We need to ensure that every node is visited and each traveling salesman's route forms a Hamiltonian cycle. To prevent sub-tours in the traveling salesman's routing, we have to add sub-tour elimination constraints (SECs). The SECs for the MATSP in this paper are Danzig-Fulkerson-Johnson (DFJ) [24]: \[x_{ij}^{m}=x_{ji}^{m}=0,\forall i,j\in V_{m},i\
eq j,\forall m\in V_{m}\]1 (15)where \(|\mathcal{S}|\) represents the cardinality of subset \(\mathcal{S}\), and \(|V|\) represents the cardinality of set \(V\). In each subset \(\mathcal{S}\), sub-tours are prevented. To ensure that each traveling salesman starts from the depot node and visits at least one city node, we add the following constraints:  
\[x_{ij}^{m}=x_{ji}^{m}=0,\forall i,j\in V_{m},i\
eq j,\forall m\in V_{m}\]2
5. Objective optimization functions We will first propose two objective optimization functions based on MATSP.