# Optimization of Truck-Drone Parcel Delivery Using Metaheuristics
## 6 Conclusions  
In this study, we addressed the TSP-D problem and its variants. Various algorithms, including population-based metaheuristic algorithms and solution-based metaheuristic algorithms, have been proposed for TSP-D and its variants, such as the FSTSP. We used GRASP as a single-solution metaheuristic algorithm. The proposed algorithm starts by constructing the entire truck route as a classical TSP. The obtained route is then partitioned into truck nodes and drone nodes, yielding an initial TSP-D solution. Following that, the proposed GRASP algorithm is applied to minimize the total delivery time for the initial TSP-D solution. Two versions of the LS procedure in GRASP were used: HCLS and LS with SA. Both versions use a self-adaptive neighborhood. The proposed approach was tested  
Figure 11: Comparison of the average solution cost (in _secs_) obtained with the HCLS variant of the proposed GRASP, HGVNS [23], and LS [10] for instances of size \(10,20,50,75, and 100\): (**a**) Uniform distribution, \(\alpha=1\). (**b**) Uniform distribution, \(\alpha=2\). (**c**) Single-centered distribution. \(\alpha=1\) (**d**) Single-centered distribution, \(\alpha=2\). on the \"Instances of TSP with Drone\" benchmark dataset. Generally, the HCLS variant outperformed the SA variant of GRASP. Subsequently, the performance of the proposed GRASP HCLS variant was compared with two state-of-the-art algorithms: LS [10] and HGVNS [23]. Overall, the proposed approach was found to have a performance comparable to those of the two rival algorithms. Nonetheless, under some (type, size, \(\alpha\)) combinations, our approach outperformed the rival algorithms. The main contribution of this study is the self-adaptive selection of the search neighborhood, which was implemented for two neighborhoods (associated with \(Swap\) and \(TwoOpt\) moves), but which could be easily extended to a larger number of neighborhoods. The self-adaptive selection of the search neighborhood for the TSP-D problem may also inspire other methods, which in turn may significantly outperform existing ones. We recommend that the proposed GRASP approach be further improved by increasing the number of iterations in the LS procedure and tuning the \(\delta\) value. Moreover, it would be interesting to investigate different parameters for the SA variant and to add more neighborhood alternatives into the search. Conceptualization, S.A. ([PERSON]); methodology, S.A. ([PERSON]) and T.A. ; software, T.A., S.A. ([PERSON]). Y.A., L.A. and H.A. ; validation, T.A., S.A. ([PERSON]). Y.A., L.A. and H.A. ; writing--original draft preparation, S.A. ([PERSON]), T.A., S.A. ([PERSON]) and Y.A. ; writing--review and editing, S.A. ([PERSON]); visualization, L.A. and H.A. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The data used in this study are openly available in Instances for the TSP with Drone at doi:10.5281/zenodo.1204676, reference number [43]. The authors would like to thank the anonymous reviewers for their constructive comments. Special thanks to [PERSON], Department of Computer Science, King Saud University, for her valuable advice that helped improve the quality of this work. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: