## 2 Mathematical Formulation
\] \[y_{ij} = \text{number of remaining nodes after the node $i$ on a route if $x_{ij}=1;\;0$, otherwise.}\] \[v^{k}_{0j} = \text{the sum of demands of the nodes after node 0 on the route k if $w^{k}_{0j}=1;\;0$, otherwise.}\] \[v_{ij} = \text{sum of demands of the nodes after node $i$ on a route if $x_{ij}=1;\;0$, otherwise.}\] \[t^{k}_{0j} = \text{Arrival time of node $j$ from node 0 on a route $k$ if $w^{k}_{0j}=1;\;0$, otherwise.}\] \[t_{ij} = \text{Cumulative time at node $j$ on a route if $x_{ij}=1;\;0$, otherwise.}\] \[I_{ij} = \text{Tardiness in the arrival time to node $i$ if node $j$ is served first($p_{ij}=1$);\;0$, otherwise.}\]  
The corresponding mathematical model is stated as follows:  
\[min\;F_{1}=L=\sum_{i\in V^{\prime}}c_{0i}y_{0i}+\sum_{i\in V^{\prime}}\sum_{ \begin{subarray}{c}j\in V^{\prime}\\ j\
eq i\end{subarray} }c_{ij}y_{ij}, \tag{2}\]  
\[min\;F_{2}=T=\sum_{i\in V^{\prime}}\sum_{ \begin{subarray}{c}j\in V^{\prime} \\ j\
eq i\end{subarray} }I_{ij}, \tag{3}\]  
Figure 2: Solution routes minimizing the total tardiness: total latency = 2972.84; total tardiness = 0.  
subject to:  
\[\sum_{j\in V^{\prime}}\sum_{k\in K}w^{k}_{0j}=K, \tag{4}\] \[\sum_{i\in V^{\prime}}x_{i0}=K,\] (5) \[\sum_{j\in V^{\prime}}\sum_{k\in K}w^{k}_{0j}=1, \forall k\in K,\] (6) \[\sum_{k\in K}w^{k}_{0j}+\sum_{i\in V^{\prime}}x_{ij}=1, \forall j\in V^{\prime},\] (7) \[\sum_{j\in V}x_{ij}=1, \forall i\in V^{\prime},\] (8) \[y_{0j}+\sum_{ \begin{subarray}{c}i\in V^{\prime}\\ i\
eq j\end{subarray} }y_{ij}-\sum_{ \begin{subarray}{c}i\in V^{\prime}\\ i\
eq j\end{subarray} }y_{ji}=1, \forall j\in V^{\prime},\] (9) \[y_{0j}\geq\sum_{k\in K}w^{k}_{0j}, \forall j\in V^{\prime},\] (10) \[y_{ij}\geq x_{ij}, \forall i\in V^{\prime};\forall j\in V^{\prime};i\
eq j,\] (11) \[y_{0j}\leq(N-K+1)\sum_{k\in K}w^{k}_{0j}, \forall j\in V^{\prime},\] (12) \[y_{ij}\leq(N-K)\sum_{k\in K}x_{ij}, \forall i\in V^{\prime};\forall j\in V^{\prime},\] (13) \[v^{k}_{0j}\geq d_{j}w^{k}_{0j}, \forall j\in V^{\prime};\forall k\in K,\] (14) \[v^{k}_{0j}\leq Q^{k}w^{k}_{0j}, \forall j\in V^{\prime};k\in K,\] (15) \[v_{ij}\geq d_{j}x_{ij}, \forall i\in V^{\prime};j\in V^{\prime};i\
eq j,\] (16) \[v_{ij}\leq(Q_{max}-d_{i})x_{ij}, \forall i\in V^{\prime};j\in V;i\
eq j,\] (17) \[\sum_{k\in K}v^{k}_{0j}+\sum_{ \begin{subarray}{c}i\in V^{\prime} \\ i\
eq j\end{subarray} }v_{ij}-\sum_{ \begin{subarray}{c}i\in V\\ i\
eq j\end{subarray} }v_{ji}=d_{j}, \forall j\in V^{\prime},\] (18) \[t^{k}_{0j}=c_{0j}w^{k}_{0j}, \forall j\in V^{\prime};j\in V^{\prime};k\in K,\] (19) \[t^{k}_{0j}\leq Mw^{k}_{0j}, \forall i\in V^{\prime};j\in V^{\prime};i\
eq j;k\in K,\] (20) \[t_{ij}\geq c_{ij}x_{ij}, \forall i\in V^{\prime};j\in V^{\prime};i\
eq j,\] (21) \[t_{ij}\leq(M-c_{ij})x_{ij}, \forall i\in V^{\prime};j\in V;i\
eq j,\] (22) \[\sum_{ \begin{subarray}{c}h\in V\\ h\
eq i\end{subarray} }t_{ih}-\sum_{ \begin{subarray}{c}h\in V^{\prime}\\ h\
eq i\end{subarray} }t_{hi}-\sum_{ \begin{subarray}{c}k\in K\\ h\
eq j\end{subarray} }t^{k}_{0i}=\sum_{ \begin{subarray}{c}j\in V\\ j\
eq i\end{subarray} }c_{ij}x_{ij}, \forall i\in V^{\prime},\] (23) \[I_{ij}\geq p_{ij}\Bigg{(}\sum_{k\in K}t^{k}_{0i}+\sum_{ \begin{subarray}{c}h\in V^{\prime}\\ h\
eq i\end{subarray} }t_{hi}-\sum_{ \begin{subarray}{c}h\in V^{\prime}\\ h\
eq j\end{subarray} }t_{hj}-\sum_{k\in K}t^{k}_{0j}\Bigg{)}, \forall i\in V^{\prime};j\in V^{\prime};i\
eq j, \tag{24}\]\[w^{k}_{0j}\in\{0,1\}, \forall j\in V^{\prime};\forall k\in K, \tag{25}\] \[x_{ij}\in\{0,1\}, \forall i\in V^{\prime};\forall j\in V,\] (26) \[y_{ij}\geq 0, \forall i\in V;\forall j\in V,\] (27) \[v^{k}_{0j}\geq 0, \forall j\in V^{\prime};k\in K,\] (28) \[v_{ij}\geq 0, \forall i\in V^{\prime};j\in V,\] (29) \[t^{k}_{0j}\geq 0, \forall j\in V^{\prime};k\in K,\] (30) \[t_{ij}\geq 0, \forall i\in V^{\prime};j\in V,\] (31) \[I_{ij}\geq 0, \forall i\in V^{\prime};j\in V^{\prime}. \tag{32}\]  
In this formulation, the objective functions in Equations (2) and (3) seek a trade-off between the objective of the total latency and the total tardiness of the system.