## 4 A General Solving Methodology
The constructive heuristics for solving the VRP and TOP are based on the savings concept [76]. Despite being structurally similar for both problems, their particularities are introduced to adequately cope with each respective case, as follows:  
* Firstly, a dummy solution is constructed. This hypothetical solution is composed of a set of routes, each of them being designed to serve one customer. The vehicle departs from the origin depot, visits the customer and continues the trip towards the destination depot. In the case of the TOP, this stage takes into account the maximum tour length when designing these dummy routes. That is, those dummy routes whose total travel time is greater than this limit are automatically discarded. Similarly, in the case of the VRP, this stage takes into account the maximum loading capacity of each vehicle (i.e., if the demand of a customer is higher than this capacity, this customer is discard). * Secondly, a savings list (\(SL\)) is created, which includes all the edges connecting two different locations. For each edge \((i,j)\in SL\), a savings value is computed according to Equation (1), for the VRP, and (2), for the TOP. In both cases, \(t_{ij}\) represents the time- or distance-based cost associated with traveling from node \(i\) to node \(j\), \(0\) is the origin depot. In the case of the TOP, \(n\) represents the destination  
Figure 3: High-level flowchart of the proposed solution method. depot, while \(u_{i}\) and \(u_{j}\) represent the rewards obtained when customers \(i\) and \(j\) are visited for the first time. In the case of the TOP, considering a linear combination of both travel time and reward allows us to define an 'enriched savings' concept that reflects not only the desire of maximizing the total collected reward, but also takes into account how far a customer is from the rest of nodes on the emerging route [29]. Once computed, the \(SL\) list is sorted in descending order of savings value, which implies that edges with the highest savings are placed at the top of the list. \[s_{ij}=t_{0i}+t_{j0}-t_{ij}\] (1) \[s_{ij}=\alpha(t_{in}+t_{0j}-t_{ij})+(1-\alpha)(u_{i}+u_{j})\] (2)
* The sorted \(SL\) reflects the most promising movements to reduce the corresponding costs. In this way, the savings edge at the top of the \(SL\) is selected to perform the merging of the associated routes. This procedure is performed only if a feasible combined route is generated. For the VRP, two routes can only be merged if the vehicle capacity is not exceeded. Alternatively, for the TOP, two routes can only be merged if the total travel distance does not exceed the maximum tour length. Once the selected savings edge is checked, it is deleted from the \(SL\). Then, the new edge at the top is selected to continue this procedure, which is repeated until the \(SL\) is empty. At the end of this process, a feasible solution is generated. 2. The described heuristics are deterministic, which implies that the same decisions are made whenever they start from the same configuration. To change this behavior, these deterministic heuristics are transformed into a probabilistic algorithms by'smoothing' the selection of candidates from the \(SL\) using a probability distribution. This concept is called biased-randomization (BR), and is described in [PERSON] et al. [77]. In our case, the geometric probability distribution was adopted, as suggested in [PERSON] et al. [78]. Introducing BR decisions in our heuristics requires dealing with additional parameters, such as the \(\beta\in(0,1)\), which defines the geometric distribution. The value of this parameter was set after a quick tuning process over a random sample, establishing a good performance for both algorithms whenever \(\beta\) falls in the interval \((0.3,0.4)\). Algorithm 1 describes the heuristic operational structure. Note that the difference between the two algorithms, designed to solve their respective problems, consists in how the \(SL\) is constructed (line 2).