## 4 Problem Definition and Mathematical Model
The edge is connecting vertices \(i\) and \(j\) is denoted by \((i,j)\) and has an associated routing cost \(c_{ij}(i,j\in V)\) shown in Figure 2. The distance and times matrices are symmetric. Figure 2: Example for routing cost between dealership \(i\) and \(j\). * Fleet: Given a heterogeneous fleet of auto-carriers, composed by a set \(T\) of auto-carrier types. Each auto-carrier type \(t(t\in T)\) has a maximum vehicles capacity \(W_{t}\) and is formed \(A_{k}^{1,2}\) by loading platforms (levels, shown in Figure 3). There are \(K_{t}\) auto-carriers available for each type \(t\). * Demand: The demand of dealership \(i\) consists of a set \(M\) of vehicles \((i\in V\backslash\{0\})\). Each vehicle \(m_{i}\in M\) demanded by dealership \(i\) belongs to a vehicle type (or vehicle model) shown in Figure 4, which is defined by a height \(h_{m}\) and a vehicle identification number (VIN). In this VRPTW, each dealership \(i\in V\backslash\{0\}\) has an associated time window \([e_{i},l_{i}]\), with a time allowed service for arriving auto-carriers to it and service time or delay \(d_{i}\). If \((i,j)\) is an arc of the solution and \(a_{i}\) and \(a_{j}\) are the arrival times to the dealerships \(i\) and \(j\), time window imply that necessarily must be fulfilled \(a_{i}\leq l_{i}\) and \(a_{j}\leq l_{j}\). On the other hand, if \(a_{i}\leq e_{i}\), then the auto-carrier must wait until the dealership \"opens\" so necessarily \(a_{j}=e_{i}+d_{i}+c_{ij}\). Using the nodes \(0\) and \(n+1\) to represent the NCSY and the set \(K\) to represent the auto-carriers, the problem is formulated for a heterogeneous fleet of auto-carriers, according to [28]:  
\[\min\sum_{k\in K}\sum_{(i,j)\in E}c_{ij}^{k}x_{ij}^{k} \tag{1}\]  
subject to  
\[\sum_{k\in K}\sum_{j\in\Delta-(i)}x_{ij}^{k}=1\qquad\forall i\in V\backslash \{0,n+1\} \tag{2}\]  
\[\sum_{j\in\Delta+(0)}x_{0j}^{k}=1\qquad\forall k\in K \tag{3}\]  
\[\sum_{j\in\Delta+(i)}x_{ij}^{k}-\sum_{j\in\Delta-(i)}x_{ji}^{k}=0\qquad \forall k\in K,i\in V\backslash\{0,n+1\} \tag{4}\]  
\[\sum_{i\in V\backslash\{0,n+1\}}x_{ij}^{k}-\sum_{j\in\Delta+(i)}x_{ji}^{k} \leq W^{k}\qquad\forall k\in K \tag{5}\]  
\[y_{j}^{k}-y_{i}^{k}\geq d_{i}+a_{ij}^{k}-H(1-x_{ij}^{k})\qquad\forall i,j\in V \backslash\{0,n+1\},k\in K \tag{6}\]  
\[e_{i}\leq y_{i}^{k}\leq l_{i}\qquad\forall i\in V\backslash\{0,n+1\},k\in K \tag{7}\]  
\[x_{ij}^{k}\in 0,1\qquad\forall(i,j)\in E,k\in K\]  
Figure 2:0 Vehicle types (vehicle, vehicle, van). Figure 2:1 Auto-carrier levels. Figure 2:2  
The \(x_{ij}^{k}\) variables indicate if the arc \((i,j)\) is used by the auto-carrier \(k\). The \(y_{i}^{k}\) variables indicate the arrival time at the dealership \(i\) when it is visited by the \(k\) auto-carrier (if the dealership is not visited by the auto-carrier, the variable has no meaning). The objective function (1) minimizes the total routing cost. Constraint (2) state that each dealership is visited exactly once, while constraints (3) and (4) determine that each auto-carrier \(k\in K\) goes through a path of 0 to \(n+1\). The capacity of each auto-carrier is imposed in (5). Since \(H\) is a sufficiently large constant, restriction (6) ensures that if an \(k\) auto-carrier travels from \(i\) to \(j\), it cannot reach \(j\) before \(y_{i}+d_{i}+a_{ij}^{k}\). These constraints also eliminate subtours and constraints (7) enforce time windows restriction. The use of a heterogeneous fleet and the nature of the demand (vehicles and vans) impose the allocation constraints:  
Figure 2:3  
Figure 2:4  
Figure 2:5  
The constraint (8) considers the assignment of a vehicle \(m\) with a height \(h_{m}\) greater than 2.5 m (meters) that corresponds to a van, which occupies an allocated space in level \(A_{k}^{1}\) and two spaces on level \(A_{k}^{2}\), using three spaces of the capacity \(W\) of the auto-carrier \(k\). A vehicle \(m\) with height \(h_{m}\) greater than 1.8 m and less than 2.5 m, its assignment corresponds to a space \(W_{k}^{1}\) and can only be accommodated at level \(A_{k}^{1}\), i.e., the constraint (9).