# Logistics Distribution Vehicle Routing Problem with Time Window under Pallet 3D Loading Constraint
## 3 Modeling
### Model Construction
\begin{table}
\begin{tabular}{c c} \hline
**Variable Name** & **Meaning of Variables** \\ \hline \(AT^{t}_{kj}\) & The time at which the vehicle arrives at the customer point \(j\) in the service cycle \(t\). \(k\in K,j\in J,t\in T\) \\ \(V^{t}_{km}\) & The volume of the pallet \(m\) of the vehicle \(k\) in service cycle \(t\). \(k\in K,m\in M,t\in T\) \\ \((\overline{L}^{t}_{k^{\prime}},\overline{W}^{t}_{k^{\prime}},\overline{H}^{t} _{k})\) & The length, width, and height of the carriage \(k\) in service cycle \(t\). \(k\in K,t\in T\) \\ \(V^{p}\) & The volume of the pallet base itself before loading. \\ \((l^{t}_{km^{\prime}},w^{t}_{km^{\prime}},h^{t}_{km^{\prime}})\) & Length, width, and height of the goods \(n\) on the pallet \(m\) of the vehicle \(k\) in service cycle \(t\). \(k\in K,t\in T,m\in M,n\in N\) \\ \((L^{t}_{km^{\prime}},W^{t}_{km^{\prime}},H^{t}_{km})\) & Length, width, and height of the pallet \(m\) of the vehicle \(k\) in service cycle \(t\). \(k\in K,m\in M,t\in T\) \\ \(|S^{t}_{k}|\) & The number of customer nodes served vehicle \(k\) during service cycle \(t\). \(S^{t}_{k}\subseteq J,t\in T\) \\ \(g^{t}_{k}\) & Weight of the cargo carried by the vehicle \(k\) during service cycle \(t\). \(k\in K,t\in T\) \\ \(G\) & The maximum load capacity of the vehicle \\ \(DT^{t}_{ik}\) & The time of departure that the distribution vehicle \(k\) from the distribution center during the service cycle \(t\). \(i\in J,k\in K,t\in T\) \\ \(RT^{t}_{ik}\) & The time that distribution vehicle to return to the distribution center during the service cycle \(t\). \(i\in J,k\in K,t\in T\) \\ \(TT^{t}_{ijk}\) & The time that the distribution vehicle takes to travel from the distribution center to customer point \(j\) during service cycle \(t\). \(i,j\in J,k\in K,t\in T\) \\ \(AT^{t}_{jk}\) & Time of arrival that distribution vehicle \(k\) at customer point \(j\) during service cycle \(t\). \(j\in J,k\in K,t\in T\) \\ \((\overline{x}^{t}_{j_{km}^{\prime}},\overline{y}^{t}_{j_{km}^{\prime}}, \overline{z}^{t}_{j_{km}^{\prime}})\) & Coordinates of the pallets delivered to the unloading point in the vehicle \(k\) during service cycle \(t\). \(j\in J,k\in K,m\in M,t\in T\) \\ \(h^{t}_{p_{km}}\) & Height of the pallet \(m\) delivered to the unloading point in the vehicle \(k\) during service cycle \(t\). \(j\in J,k\in K,m\in M,t\in T\) \\ \(l^{t}_{p_{km}}\) & Length of the pallet \(m\) in the vehicle \(k\) to the unloading point \(j\) in service cycle \(t\). \(j\in J,k\in K,m\in M,t\in T\) \\ \(n^{t}_{kij}\) & In service cycle \(t\), if the vehicle \(k\) is transported from point \(i\) to point \(j\), then the decision variable is \(u^{t}_{kij}=1\); \\ \(u^{t}_{kij}=0\), and \(i,j\in J,k\in K,t\in T\). \\ \(u^{t}_{k}\) & During service cycle \(t\), if the vehicle \(k\) is used, then the decision variable \\ \(u^{t}_{k}=1\); otherwise, \(u^{t}_{k}=0\), and \(k\in K,t\in T\). \\ \(u^{t}_{k_{km}}\) & If the nth load of the pallet \(m\) of the vehicle \(k\) is loaded onto the pallet in service cycle \(t\), then \(u^{t}_{k_{km}}=1\); otherwise, \(u^{t}_{k_{km}}=0\). \\ \((x^{t}_{kmm^{\prime}},y^{t}_{kmm^{\prime}},z^{t}_{kmm})\) & The coordinates of the load \(n\) in the pallet \(m\) in the vehicle \(k\) to the discharge point \(j\) in service cycle \(t\). \(k\in K,t\in T,m\in M,n\in N\) \\ \hline \end{tabular}
\end{table}