## 1 Introduction
A self-adaptive Evolutionary Algorithm is proposed in [PERSON] et al. (2018) for the berth scheduling problem. [PERSON] et al. [PERSON] et al. (2018) provided a full comparison among Variable Neighborhood Search, Tabu Search, Simulated Annealing and Evolutionary Algorithm to solve a model for large-scale problem instances of the Vehicle Routing Problem. They demonstrated that the Evolutionary Algorithm outperforms the other metaheuristic algorithms developed for the model. In recent years there has been a flourishing of articles that try leveraging Machine Learning (ML) to solve combinatorial optimization problems. In this context, ML may help substitute heavy computations by a fast approximation. This is the case of learning variable and node selection in branch-based algorithms for Mixed-Integer Linear Programming (MIP). See [PERSON] and [PERSON] (2018) for a recent review. A wider perspective is taken by [PERSON] et al. [PERSON] et al. (2018) that identify three broad approaches to leveraging machine learning for combinatorial optimization problems: learning alongside optimization algorithms, learning to configure optimization algorithms, and end-to-end learning to approximately solve optimization problems. Some results of this line of research have been recently used in industry: in November 2019, following the work of [PERSON] et al. [PERSON] et al. (2018), IBM announced that version 12.10 of its well-known commercial optimization solver CPLEX implements, for the first time, a ML-based classifier to make automatic decisions over some algorithmic settings. In particular, an ML-based classifier is invoked by default to decide if the binary component of a Mixed-Integer Quadratic Optimization problem should benefit from the application of a linearization procedure, which transforms the problem into a MIP problem. In this paper, we make use of supervised and unsupervised techniques to improve the quality of the solutions generated by a heuristic for the Time-Dependent Travelling Salesman Problem with no increased computing time. This can be useful in a real-time setting where a speed update (or the arrival of a new customer request) may lead to the reoptimization of the planned route. As far as we know, this is the first attempt to use ML to solve a time-dependent routing problem. For a comparative analysis of machine learning heuristics for solving the classical (time-invariant) Travelling Salesman Problem, see [PERSON] and [PERSON] [12]. In Time-Dependent Vehicle Routing Problems, the aim is to design routes for a fleet of vehicles on a graph whose arc traversal times vary over time, in order to optimize a given criterion, possibly subject to side constraints. See [PERSON] et al. [13] for a review of the field. In this paper, we study the Time-Dependent Travelling Salesman Problem (TDTSP) which amounts to find a Hamiltonian tour of least total duration on a given time-dependent graph. The TDTSP was first addressed by [PERSON] and [PERSON] [14]; they proposed a _Mixed Integer Programming_ (MIP) formulation for the problem. Subsequently, [PERSON] and [PERSON] [15] developed an approximate dynamic programming algorithm, whilst [PERSON] et al. [16] presented two heuristics. [PERSON] [17] and [PERSON] et al. [18] presented some meta-heuristic approaches to solve the TDTSP. [PERSON] et al. [19] showed some properties of the TDTSP and derived lower and upper bounding procedures; they also devised a set of valid inequalities used in a branch-and-cut algorithm. [PERSON] et al. [20] derived some properties of the problem that were used in a branch-and-bound algorithm that outperformed the [PERSON] et al. [19] branch-and-cut procedure. In [PERSON] et al. [21] a parameterized family of lower bounds was developed to enhance this branch-and-bound approach. Moreover, [PERSON] et al. [22] presented a new global constraint useful in a _Constraint Programming_ approach.