## 4 Experimental Results and Discussion  
### Comparisons with Best Known Results  
This section describes computational experiments carried out to evaluate the performance of the proposed algorithm. The algorithm was coded in Python and ran on a personal computer with the following specifications: Intel Core i7 (Intel Corporation, California, USA)--8550U 1.8 GHz with 16 GB memory. The proposed algorithm was tested in [PERSON]'s VRPTW benchmark instances with 100 customers (see [18]), as well as in [PERSON] and [PERSON]'s instances with 1000 customers (see [30]). In the specific datasets, C1 and C2 problems have geographically clustered customers; in R1 and R2, geographical data are randomly generated; and in RC1 and RC2, a mix of random and clustered structures is included. R1, C1 and RC1 have a short scheduling horizon, narrow time windows and low vehicle capacities, respectively, while R2, C2 and RC2 have a long scheduling horizon, wide time windows and high vehicle capacities, defining the number of customers serviced by the same vehicle. Tables 1 and 2 present a summary of the results obtained from our proposed MOLNS algorithm in [PERSON]'s instances, as well as the best-published solutions in the literature. The column labeled BP gives the best-published results, while column MOLNS the results of our Large Neighborhood Search algorithm. Each solution is characterized by the number of vehicles (NV) needed and the total distance traveled (TD). The published work in which the optimum solution is proposed is presented in column \"Reference\", the computational time needed to obtain each solution is given in column \"Time\" in seconds and the percentage deviation in the total distance traveled between the BP and the LNS is presented in column \"Deviation\". Each instance was run three times, while the computational time limit was set to be 5 min, which is a reasonable time for a logistics company to obtain the routing of their vehicles and the scheduling of their deliveries. In most cases, the best-obtained results needed less time than the time limit. Since we face the multiobjective VRPTW in some instances, more than one solution was obtained. In order for the BP solution and the solution obtained from the MOLNS to be comparable, the number of vehicles must be the same. Therefore, the deviation is measured in the total distance traveled between the BP solution and the solution of the MOLNS. It can be easily observed that instances with geographically clustered customers (C1 and C2) have more efficient results than problems with random geographical data, as the mean deviation in that case is 0.30%. In randomly generated geographical data (R), the mean deviation is higher and reaches 3.37%, while in mixed geographical customers' data (RC) it is 3.47%. Finally, the mean deviation for all instances is less than 2.85%, which can be considered very efficient, since the optimal results have emerged from multiple researchers over time. In total,  
Figure 4: The general framework of the MOLNS. 32.47% of the comparable solutions have a deviation of less than 1%, most of them in C instances. Simultaneously, 98.70% of the results deviate less than 10% from the optimum. Finally, three new Pareto-optimal solutions are proposed in [PERSON]'s benchmark instances, in RC202, RC207 and R201. In RC202 problem, a decrease in the total distance traveled is accomplished, but at the expense of an increased number of vehicles. In R201 and RC202 instances, we managed to improve the solutions by decreasing the total distance traveled, while the number of vehicles remains steady. The exact routes and the schedule of deliveries in each of the three new optimal solutions that are proposed in the current research are given in Appendix A.  
Additionally, when comparing results obtained from algorithms, it is important to take into consideration the computational time needed.