## 2 Mathematical Formulation  
In this section, we present the formal definition of the BCCVRP-Pr as well as its corresponding mathematical formulation. First, the following parameters and variables are considered:  
**Parameters**  
* \(N\): Number of customers
* \(K\): Number of vehicles available  
The BCCVRP-Pr can be formally defined as an undirected graph \(G=(V,A)\) where \(V=\{0,1,2,...,N\}\) denote the node set and \(A\) is the set containing all arcs. The node \(0\) corresponds to the depot and the rest of the nodes form the set of customers \(V^{\prime}=\{1,2,...,N\}\). Each arc \((i,j)\in A\) has an associated travel time \(c_{ij}\) and each node \(j\in V^{\prime}\) has an associated demand \(d_{j}\). A heterogeneous fleet of \(K\) vehicles is available, each with capacity \(Q^{k}\), \(k\in\{1,2,...,K\}\). It is assumed that the vehicle's overall capacity is enough to satisfy the total demand of all the clients. In addition, to consider the priority, a precedence matrix \(P\) is defined in which \(p_{ij}=1\) represents that customer \(i\) should be serviced before customer \(j\) and \(p_{ij}=0\) means that customer \(i\) can be served after the customer \(j\). The following additional parameters are considered:  
* \(Q_{max}\): Maximum capacity of any of the vehicles
* \(M\): Maximum travel distance allowed (the same for all vehicles)  
The goal is to find \(k\) tours that have only node 0 in the first position, and to cover all the nodes, while minimizing the sum of the latencies of the trips. The demand of all customers must be satisfied without exceeding the capacity of each vehicle. Customers should be served (preferably) according to their priority level, minimizing the total tardiness of the system. For each path, the tardiness arises when a customer with lower priority is served before a customer with higher priority (even if both customers belong to different routes). In other words, the arrival time (\(t_{i}\)) for a customer with a lower priority index is less than the arrival time of a customer with a higher priority index (\(t_{j}\)). Qualitatively, tardiness is associated with customer dissatisfaction. However, because latency is estimated as a function of distance, then tardiness is obtained as the difference between their arrival times (when \(p_{ij}=1\)), \(I_{ij}=t_{j}-t_{i}\). Hence, the total tardiness of the system is computed, as shown in Equation (1):  
\[Total\ tardiness=\sum_{i\in\mathcal{V}^{\prime}}\sum_{j\in\mathcal{V}^{\prime}} I_{ij} \tag{1}\]  
Figures 1 and 2 graphically exhibit the conflict among the values of latency and tardiness for an instance that contains 12 nodes. The number above each node denotes the customer priority index (higher values indicate higher priorities). In Figure 1, the minimum total tardiness was obtained subject to the minimum total latency. Correspondingly, Figure 2 indicates the optimal total latency while assuring the minimum total tardiness. Figure 1: Solution routes minimizing the total latency: total latency = 1595.31; total tardiness = 4304.55. The formulation is based on the model presented in [10] for the classical CCVRP and following the single flow formulation proposed for the multiple traveling salesman problem [27]. **Variables**  
\[w^{k}_{0j} = \begin{cases}1,&\text{if the vehicle $k$ is assigned to a path from node 0 to customer $j$,}\\ 0,&\text{otherwise.}\\ \end{cases} \] \[x_{ij} = \begin{cases}1,&\text{if the arc $(i,j)$ is in the path of a vehicle,}\\ 0,&\text{otherwise.}\\ \end{cases}