# A GRASP Approach for the Energy-Minimizing Electric Vehicle Routing Problem with Drones

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Electric Vehicle Routing Problems

#### 2.2. Routing Problems with Drones

## 3. The Electric Vehicle Routing Problem with Drones

#### 3.1. EVRPD Route Example

#### 3.2. Mathematical Formulation of the EVRPD

- Only drones may visit the customers;
- Each customer requires a single package;
- Each stop may be visited only once by an EV;
- The deployment and retrieval locations are identical;
- EVs remain stationary at the deployment/retrieval locations;
- Given sufficient range, drones may carry multiple deliveries;
- There is no limit to the concurrent number of drones operating from an EV;
- There are sufficient vehicles of both types to meet the demand;
- The handling time of drones is considered to be negligible;
- Ideal environmental conditions are assumed.

## 4. The proposed GRASP Approach

#### 4.1. Greedy Randomized Adaptive Search Procedure

Algorithm 1 Overall GRASP Algorithm. |

#### 4.2. GRASP-VL RCL Construction

Algorithm 2 Value-based RCL construction. |

#### 4.3. GRASP-CRD RCL Construction

Algorithm 3 Cardinality-based RCL construction. |

#### 4.4. Local Search

- Intra-EV-Intra-Drone-Intra-route Swap 1-1: Swaps positions of two customers belonging to the same drone route;
- Intra-EV-Intra-Drone-Inter-route Exchange 1-1: Exchanges positions of two customers belonging to two different routes of the same drone;
- Intra-EV-Intra-Drone-Inter-route Relocation 1-0: Relocates a customer to another route of the same drone;
- Intra-EV-Inter-Drone-Inter-route Exchange 1-1: Exchanges positions of two customers belonging to routes of two different drones;
- Intra-EV-Inter-Drone-Inter-route Relocation 1-0: Relocates a customer to a route of a different drone;
- Inter-EV-Inter-Drone-Inter-route Exchange 1-1: Exchange 1-1: Exchanges positions of two customers belonging to two drones’ routes, and those drones belong to different EVs;
- Inter-EV-Inter-Drone-Inter-route Relocation 1-0: Relocates a customer to a route of a drone which belongs to a different EV;
- EV-route-Intra-route 2-Opt: Performs the 2-opt operator in the EV’s route.

Algorithm 4 Local search. |

## 5. Computational Results

^{®}Core i7-4770 CPU (3.40 GHz) and 7.7 GB RAM, running the Fedora Workstation 35 OS. For each algorithm, each instance is solved 15 times during the experiments.

#### 5.1. Parameter Sensitivity

#### 5.2. Experimental Results

#### 5.3. Comparison with Other Approaches

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Kyriakakis, N.A.; Stamadianos, T.; Marinaki, M.; Marinakis, Y. The electric vehicle routing problem with drones: An energy minimization approach for aerial deliveries. Clean. Logist. Supply Chain.
**2022**, 4, 100041. [Google Scholar] [CrossRef] - Kara, İ.; Kara, B.Y.; Yetis, M.K. Energy Minimizing Vehicle Routing Problem. In Proceedings of the Combinatorial Optimization and Applications, Xi’an, China, 14–16 August 2007; Springer: Berlin/Heidelberg, Germany, 2007; pp. 62–71. [Google Scholar]
- Conrad, R.G.; Figliozzi, M.A. The recharging vehicle routing problem. In Proceedings of the 2011 Industrial Engineering Research Conference (IISE), Reno, NV, USA, 21–25 May 2011; p. 8. [Google Scholar]
- Erdoğan, S.; Miller-Hooks, E. A green vehicle routing problem. Transp. Res. Part E Logist. Transp. Rev.
**2012**, 48, 100–114. [Google Scholar] [CrossRef] - Schneider, M.; Stenger, A.; Goeke, D. The Electric Vehicle-Routing Problem with Time Windows and Recharging Stations. Transp. Sci.
**2014**, 48, 500–520. [Google Scholar] [CrossRef] - Desaulniers, G.; Errico, F.; Irnich, S.; Schneider, M. Exact algorithms for electric vehicle-routing problems with time windows. Oper. Res.
**2016**, 64, 1388–1405. [Google Scholar] [CrossRef] - Keskin, M.; Çatay, B. A matheuristic method for the electric vehicle routing problem with time windows and fast chargers. Comput. Oper. Res.
**2018**, 100, 172–188. [Google Scholar] [CrossRef] - Montoya, A.; Guéret, C.; Mendoza, J.E.; Villegas, J.G. The electric vehicle routing problem with nonlinear charging function. Transp. Res. Part B Methodol.
**2017**, 103, 87–110. [Google Scholar] [CrossRef] - Froger, A.; Mendoza, J.E.; Jabali, O.; Laporte, G. A Matheuristic for the Electric Vehicle Routing Problem with Capacitated Charging Stations; Research Report; Centre Interuniversitaire de Recherche sur les Reseaux D’entreprise, la Logistique et le Transport (CIRRELT): Montreal, QC, Canada, 2017. [Google Scholar]
- Zuo, X.; Xiao, Y.; You, M.; Kaku, I.; Xu, Y. A new formulation of the electric vehicle routing problem with time windows considering concave nonlinear charging function. J. Clean. Prod.
**2019**, 236, 117687. [Google Scholar] [CrossRef] - Keskin, M.; Laporte, G.; Çatay, B. Electric Vehicle Routing Problem with Time-Dependent Waiting Times at Recharging Stations. Comput. Oper. Res.
**2019**, 107, 77–94. [Google Scholar] [CrossRef] - Çatay, B.; Keskin, M. The impact of quick charging stations on the route planning of electric vehicles. In Proceedings of the 2017 IEEE Symposium on Computers and Communications (ISCC), Heraklion, Greece, 3–6 July 2017; pp. 152–157. [Google Scholar]
- Ferro, G.; Paolucci, M.; Robba, M. An Optimization Model For Electrical Vehicles Routing with time of use energy pricing and partial Recharging. IFAC-PapersOnLine
**2018**, 51, 212–217. [Google Scholar] [CrossRef] - Ding, N.; Batta, R.; Kwon, C. Conflict-Free Electric Vehicle Routing Problem with Capacitated Charging Stations and Partial Recharge; Technical Report; Department of Industrial and Systems Engineering, University at Buffalo: Buffalo, NY, USA, 2015. [Google Scholar]
- Chakraborty, N.; Mondal, A.; Mondal, S. Intelligent charge scheduling and eco-routing mechanism for electric vehicles: A multi-objective heuristic approach. Sustain. Cities Soc.
**2021**, 69, 102820. [Google Scholar] [CrossRef] - Keskin, M.; Çatay, B.; Laporte, G. A simulation-based heuristic for the electric vehicle routing problem with time windows and stochastic waiting times at recharging stations. Comput. Oper. Res.
**2021**, 125, 105060. [Google Scholar] [CrossRef] - Basso, R.; Kulcsár, B.; Sanchez-Diaz, I. Electric vehicle routing problem with machine learning for energy prediction. Transp. Res. Part B Methodol.
**2021**, 145, 24–55. [Google Scholar] [CrossRef] - Lin, B.; Ghaddar, B.; Nathwani, J. Electric vehicle routing with charging/discharging under time-variant electricity prices. Transp. Res. Part C Emerg. Technol.
**2021**, 130, 103285. [Google Scholar] [CrossRef] - Xiao, Y.; Zuo, X.; Kaku, I.; Zhou, S.; Pan, X. Development of energy consumption optimization model for the electric vehicle routing problem with time windows. J. Clean. Prod.
**2019**, 225, 647–663. [Google Scholar] [CrossRef] - Shao, S.; Guan, W.; Bi, J. Electric vehicle-routing problem with charging demands and energy consumption. IET Intell. Transp. Syst.
**2017**, 12, 202–212. [Google Scholar] [CrossRef] - Rastani, S.; Yuksel, T.; Çatay, B. Effects of ambient temperature on the route planning of electric freight vehicles. Transp. Res. Part D Transp. Environ.
**2019**, 74, 124–141. [Google Scholar] [CrossRef] - Lin, J.; Zhou, W.; Wolfson, O. Electric Vehicle Routing Problem. Transp. Res. Procedia
**2016**, 12, 508–521. [Google Scholar] [CrossRef] - Zhang, S.; Gajpal, Y.; Appadoo, S.; Abdulkader, M. Electric vehicle routing problem with recharging stations for minimizing energy consumption. Int. J. Prod. Econ.
**2018**, 203, 404–413. [Google Scholar] [CrossRef] - Basso, R.; Kulcsár, B.; Egardt, B.; Lindroth, P.; Sanchez-Diaz, I. Energy consumption estimation integrated into the Electric Vehicle Routing Problem. Transp. Res. Part D Transp. Environ.
**2019**, 69, 141–167. [Google Scholar] [CrossRef] - Omidvar, A.; Tavakkoli-Moghaddam, R. Sustainable vehicle routing: Strategies for congestion management and refueling scheduling. In Proceedings of the 2012 IEEE International Energy Conference and Exhibition (ENERGYCON), Florence, Italy, 9–12 September 2012; pp. 1089–1094. [Google Scholar]
- Kullman, N.D.; Goodson, J.; Mendoza, J.E. Dynamic Electric Vehicle Routing: Heuristics and Dual Bounds; Hal Science: Lyon, France, 2018; working paper or preprint. [Google Scholar]
- Zhang, S.; Chen, M.; Zhang, W.; Zhuang, X. Fuzzy optimization model for electric vehicle routing problem with time windows and recharging stations. Expert Syst. Appl.
**2020**, 145, 113123. [Google Scholar] [CrossRef] - Breunig, U.; Baldacci, R.; Hartl, R.F.; Vidal, T. The electric two-echelon vehicle routing problem. Comput. Oper. Res.
**2019**, 103, 198–210. [Google Scholar] [CrossRef] - Jie, W.; Yang, J.; Zhang, M.; Huang, Y. The two-echelon capacitated electric vehicle routing problem with battery swapping stations: Formulation and efficient methodology. Eur. J. Oper. Res.
**2019**, 272, 879–904. [Google Scholar] [CrossRef] - Murray, C.C.; Chu, A.G. The flying sidekick traveling salesman problem: Optimization of drone-assisted parcel delivery. Transp. Res. Part C Emerg. Technol.
**2015**, 54, 86–109. [Google Scholar] [CrossRef] - Kitjacharoenchai, P.; Ventresca, M.; Moshref-Javadi, M.; Lee, S.; Tanchoco, J.M.; Brunese, P.A. Multiple traveling salesman problem with drones: Mathematical model and heuristic approach. Comput. Ind. Eng.
**2019**, 129, 14–30. [Google Scholar] [CrossRef] - Murray, C.C.; Raj, R. The multiple flying sidekicks traveling salesman problem: Parcel delivery with multiple drones. Transp. Res. Part C Emerg. Technol.
**2020**, 110, 368–398. [Google Scholar] [CrossRef] - Raj, R.; Murray, C. The multiple flying sidekicks traveling salesman problem with variable drone speeds. Transp. Res. Part C Emerg. Technol.
**2020**, 120, 102813. [Google Scholar] [CrossRef] - Luo, Z.; Poon, M.; Zhang, Z.; Liu, Z.; Lim, A. The Multi-visit Traveling Salesman Problem with Multi-Drones. Transp. Res. Part C Emerg. Technol.
**2021**, 128, 103172. [Google Scholar] [CrossRef] - Pina-Pardo, J.C.; Silva, D.F.; Smith, A.E. The traveling salesman problem with release dates and drone resupply. Comput. Oper. Res.
**2021**, 129, 105170. [Google Scholar] [CrossRef] - Wang, Z.; Sheu, J.B. Vehicle routing problem with drones. Transp. Res. Part B Methodol.
**2019**, 122, 350–364. [Google Scholar] [CrossRef] - Schermer, D.; Moeini, M.; Wendt, O. A matheuristic for the vehicle routing problem with drones and its variants. Transp. Res. Part C Emerg. Technol.
**2019**, 106, 166–204. [Google Scholar] [CrossRef] - Hu, M.; Liu, W.; Lu, J.; Fu, R.; Peng, K.; Ma, X.; Liu, J. On the joint design of routing and scheduling for vehicle-assisted multi-UAV inspection. Future Gener. Comput. Syst.
**2019**, 94, 214–223. [Google Scholar] [CrossRef] - Sacramento, D.; Pisinger, D.; Ropke, S. An adaptive large neighborhood search metaheuristic for the vehicle routing problem with drones. Transp. Res. Part C Emerg. Technol.
**2019**, 102, 289–315. [Google Scholar] [CrossRef] - Rossello, N.B.; Garone, E. Carrier-vehicle system for delivery in city environments. IFAC-PapersOnLine
**2020**, 53, 15253–15258. [Google Scholar] [CrossRef] - Karak, A.; Abdelghany, K. The hybrid vehicle-drone routing problem for pick-up and delivery services. Transp. Res. Part C Emerg. Technol.
**2019**, 102, 427–449. [Google Scholar] [CrossRef] - Nguyen, M.A.; Dang, G.T.H.; Hà, M.H.; Pham, M.T. The min-cost parallel drone scheduling vehicle routing problem. Eur. J. Oper. Res.
**2021**. [Google Scholar] [CrossRef] - Kitjacharoenchai, P.; Min, B.C.; Lee, S. Two echelon vehicle routing problem with drones in last mile delivery. Int. J. Prod. Econ.
**2020**, 225, 107598. [Google Scholar] [CrossRef] - Li, H.; Wang, H.; Chen, J.; Bai, M. Two-echelon vehicle routing problem with time windows and mobile satellites. Transp. Res. Part B Methodol.
**2020**, 138, 179–201. [Google Scholar] [CrossRef] - Liu, C.; Chen, H.; Li, X.; Liu, Z. A scheduling decision support model for minimizing the number of drones with dynamic package arrivals and personalized deadlines. Expert Syst. Appl.
**2021**, 167, 114157. [Google Scholar] [CrossRef] - Kyriakakis, N.A.; Stamadianos, T.; Marinaki, M.; Matsatsinis, N.; Marinakis, Y. A Bee Colony Optimization Approach for the Electric Vehicle Routing Problem with Drones. In Proceedings of the Machine Learning, Optimization, and Data Science: 8th International Workshop (LOD 2022), Certosa di Pontignano, Italy, 19–22 September 2022; Revised Selected Papers, Part II. Springer: Berlin/Heidelberg, Germany, 2023; pp. 219–233. [Google Scholar]
- Mara, S.T.W.; Sarker, R.; Essam, D.; Elsayed, S. Solving electric vehicle–drone routing problem using memetic algorithm. Swarm Evol. Comput.
**2023**, 79, 101295. [Google Scholar] [CrossRef] - Macrina, G.; Pugliese, L.D.P.; Guerriero, F.; Laporte, G. Drone-aided routing: A literature review. Transp. Res. Part C Emerg. Technol.
**2020**, 120, 102762. [Google Scholar] [CrossRef] - Chung, S.H.; Sah, B.; Lee, J. Optimization for drone and drone-truck combined operations: A review of the state of the art and future directions. Comput. Oper. Res.
**2020**, 123, 105004. [Google Scholar] [CrossRef] - Moshref-Javadi, M.; Winkenbach, M. Applications and Research avenues for drone-based models in logistics: A classification and review. Expert Syst. Appl.
**2021**, 177, 114854. [Google Scholar] [CrossRef] - Stamadianos, T.; Kyriakakis, N.A.; Marinaki, M.; Marinakis, Y. Routing Problems with Electric and Autonomous Vehicles: Review and Potential for Future Research. Oper. Res. Forum
**2023**, 4, 46. [Google Scholar] [CrossRef] - Kyriakakis, N.A.; Aronis, S.; Marinaki, M.; Marinakis, Y. A GRASP/VND algorithm for the energy minimizing drone routing problem with pickups and deliveries. Comput. Ind. Eng.
**2023**, 182, 109340. [Google Scholar] [CrossRef] - Perboli, G.; Tadei, R.; Vigo, D. The two-echelon capacitated vehicle routing problem: Models and math-based heuristics. Transp. Sci.
**2011**, 45, 364–380. [Google Scholar] [CrossRef] - Kancharla, S.R.; Ramadurai, G. Electric vehicle routing problem with non-linear charging and load-dependent discharging. Expert Syst. Appl.
**2020**, 160, 113714. [Google Scholar] [CrossRef] - Feo, T.; Resende, M. Greedy Randomized Adaptive Search Procedures. J. Glob. Optim.
**1995**, 6, 109–133. [Google Scholar] [CrossRef] - Laguna, M.; Marti, R. GRASP and Path Relinking for 2-Layer Straight Line Crossing Minimization. INFORMS J. Comput.
**1999**, 11, 44–52. [Google Scholar] [CrossRef] - Aiex, R.; Resende, M.; Pardalos, P.; Toraldo, G. GRASP with Path Relinking for Three-Index Assignment. INFORMS J. Comput.
**2005**, 17, 224–247. [Google Scholar] [CrossRef] - Marinakis, Y. Multiple Phase Neighborhood Search-GRASP for the Capacitated Vehicle Routing Problem. Expert Syst. Appl.
**2012**, 39, 6807–6815. [Google Scholar] [CrossRef] - Resende, M.G.C.; Ribeiro, C.C. Parallel GRASP heuristics. In Optimization by GRASP: Greedy Randomized Adaptive Search Procedures; Springer: New York, NY, USA, 2016; pp. 205–227. [Google Scholar] [CrossRef]
- Kyriakakis, N.A.; Marinaki, M.; Matsatsinis, N.; Marinakis, Y. A cumulative unmanned aerial vehicle routing problem approach for humanitarian coverage path planning. Eur. J. Oper. Res.
**2022**, 300, 992–1004. [Google Scholar] [CrossRef] - Mladenović, N.; Hansen, P. Variable neighborhood search. Comput. Oper. Res.
**1997**, 24, 1097–1100. [Google Scholar] [CrossRef] - Mjirda, A.; Todosijević, R.; Hanafi, S.; Hansen, P.; Mladenović, N. Sequential variable neighborhood descent variants: An empirical study on the traveling salesman problem. Int. Trans. Oper. Res.
**2017**, 24, 615–633. [Google Scholar] [CrossRef]

**Figure 1.**Route example of the EVRPD (d is the arc distance, f is the arc payload weight, p is the package’s weight class, s, ${s}^{\prime}$ are the starting and ending nodes, respectively and 1, 2, 3 are the intermediate nodes).

**Figure 3.**Gap% of solutions to the best solution found for each instance with different parameter settings.

**Figure 5.**Gap% of average solution obtained to the best previously known solution for each instance.

Problem | Description |
---|---|

Traveling Salesman Problem with a Flying sidekick (FSTSP) | The FSTSP involves optimizing the route of a traveling salesman who is accompanied by a flying sidekick, also known as the Traveling Salesman Problem with Drones (TSPD). The most common objectives are the minimization of cost and time. |

Vehicle Routing Problem with Drones (VRPD) | In VRPD, the objective is to optimize the routes of a fleet of vehicles (e.g., trucks) along with the use of drones. Various extensions of VRP have been adapted for VRPD so far. |

Drone Routing Problem (DRP) | The DRP focuses on optimizing the routes of drones for various applications. The primary objective is to find efficient paths for the drones to visit a set of locations, possibly taking into account additional constraints. |

Electric Vehicle Routing Problem (EVRP) | EVRP is a variant of VRP that specifically addresses the unique characteristics and constraints associated with EVs. Charging time and energy consumption minimization are among the most common objectives. |

Electric Vehicle Routing Problem with Drones (EVRPD) | EVRPD combines EVs and drones in routing applications, aiming to minimize the overall energy consumption and share the travel distance between two means of transportation. |

Package Class | Weight Range (Weight Units) | Weight Accounted (Weight Units) |
---|---|---|

PC1 | (0.0, 1.0] | 1 |

PC2 | (1.0, 2.0] | 2 |

PC3 | (2.0, 3.0] | 3 |

**Table 3.**List of all possible loading cases, with maximum payload weight of 4 units and maximum quantity of 3 items (PC: Package Class).

Case | Compartment 1 | Compartment 2 | Compartment 3 | Packages (Quantity) | Payload (Weight Units) |
---|---|---|---|---|---|

1 | PC 1 | - | - | 1 | 1.0 |

2 | PC 1 | PC 1 | - | 2 | 2.0 |

3 | PC 1 | PC 1 | PC 1 | 3 | 3.0 |

4 | PC 1 | PC 2 | - | 2 | 3.0 |

5 | PC 1 | PC 1 | PC 2 | 3 | 4.0 |

6 | PC 2 | - | - | 1 | 2.0 |

7 | PC 2 | PC 2 | - | 2 | 4.0 |

8 | PC 3 | - | - | 1 | 3.0 |

9 | PC 3 | PC 1 | - | 2 | 4.0 |

Instance | Number of Customers | Satellite Positions | Drones per EV | Number of EVs |
---|---|---|---|---|

EVRPD-n22-k4-s10-14 | 21 | 2 | 3 | 2 |

EVRPD-n22-k4-s11-12 | 21 | 2 | 3 | 2 |

EVRPD-n22-k4-s12-16 | 21 | 2 | 3 | 2 |

EVRPD-n22-k4-s6-17 | 21 | 2 | 3 | 2 |

EVRPD-n22-k4-s8-14 | 21 | 2 | 3 | 2 |

EVRPD-n22-k4-s9-19 | 21 | 2 | 3 | 2 |

EVRPD-n33-k4-s1-9 | 32 | 2 | 3 | 2 |

EVRPD-n33-k4-s14-22 | 32 | 2 | 3 | 2 |

EVRPD-n33-k4-s2-13 | 32 | 2 | 3 | 2 |

EVRPD-n33-k4-s3-17 | 32 | 2 | 3 | 2 |

EVRPD-n33-k4-s4-5 | 32 | 2 | 3 | 2 |

EVRPD-n33-k4-s7-25 | 32 | 2 | 3 | 2 |

EVRPD-n51-k5-s11-19 | 50 | 2 | 3 | 3 |

EVRPD-n51-k5-s11-19-27-47 | 50 | 4 | 3 | 3 |

EVRPD-n51-k5-s2-17 | 50 | 2 | 3 | 3 |

EVRPD-n51-k5-s2-4-17-46 | 50 | 4 | 3 | 3 |

EVRPD-n51-k5-s27-47 | 50 | 2 | 3 | 3 |

EVRPD-n51-k5-s32-37 | 50 | 2 | 4 | 3 |

EVRPD-n51-k5-s4-46 | 50 | 2 | 3 | 3 |

EVRPD-n51-k5-s6-12 | 50 | 2 | 3 | 3 |

EVRPD-n51-k5-s6-12-32-37 | 50 | 4 | 3 | 3 |

Instance | BKV | GRASP-CRD | GRASP-VL | ||||
---|---|---|---|---|---|---|---|

${\mathit{C}\mathit{o}\mathit{s}\mathit{t}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}$ | ${\mathit{C}\mathit{o}\mathit{s}\mathit{t}}_{\mathit{a}\mathit{v}\mathit{g}}$ | ${\mathit{T}}_{\mathit{a}\mathit{v}\mathit{g}}$(s) | ${\mathit{C}\mathit{o}\mathit{s}\mathit{t}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}$ | ${\mathit{C}\mathit{o}\mathit{s}\mathit{t}}_{\mathit{a}\mathit{v}\mathit{g}}$ | ${\mathit{T}}_{\mathit{a}\mathit{v}\mathit{g}}$(s) | ||

EVRPD-n22-k4-s10-14 | 1144.28 | 1136.29 | 1142.48 | 23.51 | 1293.90 | 1295.43 | 22.10 |

EVRPD-n22-k4-s11-12 | 1403.94 | 1405.41 | 1410.21 | 23.14 | 1463.78 | 1464.55 | 21.18 |

EVRPD-n22-k4-s12-16 | 1240.95 | 1239.74 | 1247.42 | 24.56 | 1421.66 | 1423.80 | 22.34 |

EVRPD-n22-k4-s6-17 | 1610.70 | 1602.32 | 1617.77 | 23.06 | 2053.43 | 2054.62 | 22.95 |

EVRPD-n22-k4-s8-14 | 1191.20 | 1189.32 | 1192.77 | 24.19 | 1293.83 | 1295.47 | 23.19 |

EVRPD-n22-k4-s9-19 | 1873.95 | 1874.20 | 1879.94 | 22.87 | 2278.81 | 2279.25 | 20.49 |

EVRPD-n33-k4-s1-9 | 3599.16 | 3615.09 | 3622.40 | 40.14 | 3793.41 | 3798.25 | 36.91 |

EVRPD-n33-k4-s14-22 | 4033.19 | 4038.24 | 4049.83 | 40.44 | 4038.50 | 4046.52 | 37.28 |

EVRPD-n33-k4-s2-13 | 3428.85 | 3434.82 | 3442.55 | 38.95 | 3461.27 | 3505.41 | 34.31 |

EVRPD-n33-k4-s3-17 | 3307.26 | 3331.42 | 3375.77 | 40.01 | 3328.80 | 3366.99 | 36.02 |

EVRPD-n33-k4-s4-5 | 3795.61 | 3810.39 | 3878.80 | 42.30 | 4591.91 | 4613.57 | 38.16 |

EVRPD-n33-k4-s7-25 | 3819.62 | 3821.37 | 3830.03 | 39.41 | 3909.77 | 3922.74 | 34.20 |

EVRPD-n51-k5-s11-19 | 3061.89 | 3104.68 | 3144.38 | 59.59 | 3300.97 | 3389.15 | 51.03 |

EVRPD-n51-k5-s11-19-27-47 | 1916.57 | 1968.93 | 2089.22 | 56.66 | 1958.55 | 1992.22 | 49.86 |

EVRPD-n51-k5-s2-17 | 2891.04 | 2953.84 | 3000.79 | 56.59 | 3038.33 | 3122.75 | 51.22 |

EVRPD-n51-k5-s2-4-17-46 | 2895.94 | 2960.97 | 3028.70 | 55.39 | 3014.60 | 3089.14 | 49.29 |

EVRPD-n51-k5-s27-47 | 1917.50 | 1975.07 | 2070.98 | 59.00 | 1950.62 | 1983.73 | 49.16 |

EVRPD-n51-k5-s32-37 | 4918.59 | 4998.20 | 5074.82 | 62.82 | 4973.86 | 5030.61 | 54.48 |

EVRPD-n51-k5-s4-46 | 4170.25 | 4205.47 | 4233.21 | 58.56 | 5066.18 | 5148.52 | 51.53 |

EVRPD-n51-k5-s6-12 | 2540.91 | 2634.67 | 2674.66 | 54.15 | 2739.78 | 2769.80 | 49.82 |

EVRPD-n51-k5-s6-12-32-37 | 2543.73 | 2582.77 | 2723.38 | 56.93 | 2609.31 | 2669.19 | 49.00 |

Average | 2756.34 | 2796.67 | 42.96 | 2932.44 | 2964.84 | 38.31 |

Algorithm | Avg. ${\mathit{C}\mathit{o}\mathit{s}\mathit{t}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}$ | Avg. ${\mathit{C}\mathit{o}\mathit{s}\mathit{t}}_{\mathit{a}\mathit{v}\mathit{g}}$ | Avg. ${\mathit{T}\mathit{i}\mathit{m}\mathit{e}}_{\mathit{a}\mathit{v}\mathit{g}}$ |
---|---|---|---|

ACS | 2736.84 | 2749.76 | 167.99 |

HACS | 2738.77 | 2752.68 | 73.72 |

MMAS | 2730.26 | 2745.36 | 41.75 |

HMMAS | 2734.24 | 2752.78 | 43.39 |

BCO | 2737.87 | 2763.89 | 43.81 |

GRASP-CRD | 2756.34 | 2796.67 | 42.96 |

GRASP-VL | 2932.44 | 2964.84 | 38.31 |

Other Algorithm | ACS | ACSVND | MMAS | MMASVND | BCO |
---|---|---|---|---|---|

Number of instances | 21 | 21 | 21 | 21 | 21 |

W-value | 54.0 | 48.0 | 21.0 | 30.0 | 24.0 |

Significance level ${\alpha}_{s}$ | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 |

p-value | 0.0319 | 0.0333 | 0.0017 | 0.0018 | 0.0007 |

${H}_{0}$ rejected | Yes | Yes | Yes | Yes | Yes |

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**MDPI and ACS Style**

Kyriakakis, N.A.; Stamadianos, T.; Marinaki, M.; Marinakis, Y.
A GRASP Approach for the Energy-Minimizing Electric Vehicle Routing Problem with Drones. *World Electr. Veh. J.* **2023**, *14*, 354.
https://doi.org/10.3390/wevj14120354

**AMA Style**

Kyriakakis NA, Stamadianos T, Marinaki M, Marinakis Y.
A GRASP Approach for the Energy-Minimizing Electric Vehicle Routing Problem with Drones. *World Electric Vehicle Journal*. 2023; 14(12):354.
https://doi.org/10.3390/wevj14120354

**Chicago/Turabian Style**

Kyriakakis, Nikolaos A., Themistoklis Stamadianos, Magdalene Marinaki, and Yannis Marinakis.
2023. "A GRASP Approach for the Energy-Minimizing Electric Vehicle Routing Problem with Drones" *World Electric Vehicle Journal* 14, no. 12: 354.
https://doi.org/10.3390/wevj14120354