# The Vehicle Routing Problem: State-of-the-Art Classification and Review
## 3 VRP and Its Variants
### VRP Variants
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Authors** & \multicolumn{3}{c}{**Model Features**} \\ \cline{3-5}  & **Customer-Related Aspects** & **Vehicle-Related Aspects** & **Deptot-Related Aspects** \\ \hline \multirow{4}{*}{\(1\)} & \multirow{4}{*}{([PERSON], [PERSON], [PERSON], [PERSON], [PERSON], & & two vehicles are expected to serve one customer each, while one vehicle is expected to serve two customers after visiting the required supplier and manufacturer nodes} & **Deptot-Related Aspects** \\ \cline{3-5}  & & & two vehicles are expected to serve one customer each, while one vehicle is expected to serve two customers after visiting the required supplier and manufacturer nodes & **Deptot-Related Aspects** \\ \cline{2-5}  &  
The objectives of VRPs can also be diversified according to different stakeholder requirements. The traditional objective of the standard VRP is to minimize a cost function, which is considered to be the total distance traveled by all vehicles. However, recent studies have focused on various negative externalities of transportation, including carbon emissions and duration. For an objective discussion, we classified single and multiple objectives according to the diversity of objectives and then listed the objectives used in different studies. The papers with the same numbers as those in Tables 2-4 are listed in Tables 8-10. Additionally, we discussed the test instances used in different studies.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Authors**} & \multicolumn{2}{c}{**Model Features**} \\ \cline{3-4}  & & **Customer-Related Aspects** & **Vehicle-Related Aspects** & **Depot-Related Aspects** \\ \hline
1 & ([PERSON] \& [PERSON] [82]) & arc-based & with drones & single depot \\ \hline
2 & ([PERSON], [PERSON], \& [PERSON] [83]) & classical & electric freight vehicles & single depot \\ \hline
3 & ([PERSON], [PERSON], \& [PERSON] [84]) & classical & homogenous vehicles & single depot \\ \hline
4 & ([PERSON], [PERSON], [PERSON], \& [PERSON] [85]) & classical & green vehicle & single depot \\ \hline
5 & ([PERSON], [PERSON], \& [PERSON] [86]) & classical & green vehicle & multiple depots \\ \hline
6 & ([PERSON], [PERSON], [PERSON], [PERSON], \& [PERSON] [87]) & classical & electric commercial vehicles & single depot \\ \hline
7 & ([PERSON], [PERSON], [PERSON], \& [PERSON] [88]) & classical & electric two-cehelon vehicle & single depot \\ \hline
8 & ([PERSON], [PERSON], [PERSON], \& [PERSON] [89]) & classical & unmanned aerial vehicles & single depot \\ \hline
9 & ([PERSON], [PERSON], \& [PERSON] [90]) & classical & consistent vehicle & single depot \\ \hline
10 & ([PERSON], [PERSON], \& [PERSON] [91]) & time windows & electric vehicle & single depot \\ \hline
11 & ([PERSON], [PERSON], [PERSON], [PERSON], \& [PERSON] [92]) & classical & feed vehicle & single depot \\ \hline
12 & ([PERSON] \& [PERSON] [93]) & classical & homogenous vehicles & single depot \\ \hline
13 & ([PERSON] et al. [94]) & classical & prize-collecting vehicle & single depot \\ \hline
14 & ([PERSON], [PERSON], \& [PERSON] [95]) & classical & unmanned aerial vehicles & single depot \\ \hline
15 & ([PERSON], [PERSON], \& [PERSON] [96]) & classical & homogenous vehicles & single depot \\ \hline
16 & ([PERSON], [PERSON], \& [PERSON] [97]) & time window & homogenous vehicles & single depot \\ \hline
17 & ([PERSON], [PERSON], [PERSON], \& [PERSON] [98]) & classical & electric vehicle & single depot \\ \hline
18 & ([PERSON], [PERSON], \& [PERSON] [99]) & time window & homogenous vehicles & single depot \\ \hline
19 & ([PERSON], [PERSON], \& [PERSON] [100]) & time window & homogenous vehicles & single depot \\ \hline
20 & ([PERSON], [PERSON], \& [PERSON] [101]) & classical & homogenous vehicles & single depot \\ \hline \hline \end{tabular}
\end{table}
Table 7: Model categories of VRPs published in 2019.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Multi-Object**} & \multirow{2}{*}{**Single-Object**} & \multirow{2}{*}{**Dataset**} & \multirow{2}{*}{**Max Nodes**} & \multirow{2}{*}{**Other Settings**} \\ \hline
1 & cost, green emissions & & self-generation & 72 & \\ \hline
2 & minimize the & & & driving and flight times \\  & operational cost & self-generation & 400 & of trucks and drones are assumed to be deterministic \\ \hline
3 & energy consumption & real data & map & \\ \hline
4 & minimize the total & based on the TDVRPTW instances & 100 & multiple trips per vehicle, time-dependent travel times \\  & travel distance & proposed in [102] & 100 & time-dependent travel times \\ \hline
5 & tune constant waiting & 100 customer EVRPTW-SP instances & 108 & stochastic waiting times \\  & times & from [103] & 108 & at recharging stations \\ \hline
6 & minimize costs, service & & & \\  & waiting times, and & & & VRPTW-SP instances from [103] & 41 \\  & number of vehicles in & & & \\  & multiple service periods & & & \\ \hline \hline \end{tabular}
\end{table}
Table 8: Model objectives of VRPs published in 2021.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Multi-Object** & **Single-Object** & **Dataset** & **Max Nodes** & **Other Settings** \\ \hline
7 & & reduce greenhouse gas (GHG) emissions & instances for emission-oriented vehicle routing on a multigraph (uni-halle.de) & 100 & different vehicle–load combinations \\ \hline
8 & minimize cost consisting of total GHG emissions & & adapted Solomon instances introduced in [104] & 100 & \\ \hline
9 & time-related and distance-related variable costs & & adapted Solomon instances introduced in [104] & 100 & \\ \hline
10 & & minimize the total duration of delivery routes (cost) & test instances proposed in [105] & 150 & \\ \hline
11 & & minimize the total duration of routes & exact branch-and-price (BP) method reported in [106] & 200 & \\ \hline
12 & & minimize the distribution cost & from the well-known Solomon \(\backslash\) WRTW instances presented in [107] & 200 & \\ \hline
13 & & minimize transportation cost & self-generation & 100 & \\ \hline
14 & & stochastic and correlated customer demands & instance A-n32-k5 (available from 31 August 2020) & 31 & \\ \hline
15 & & minimize number of vehicles, total travel distance, makespan, total waiting time, and total delay time arrivals & & same testing datasets used in [109,110] & 250 \\ \hline
16 & & minimize the total distance & 02 instances from seven standard benchmarks in [111, 112, 113, 114] & 200 & \\ \hline
17 & & minimize the total distance & GVRP instances generated in [115] & 483 & \\ \hline
18 & & minimize the total distance & benchmarks instances proposed in [111] & 199 & \\ \hline
19 & & minimize the total travel time of vehicles and drones & benchmarks instances from [95] & 200 & \\ \hline
20 &
\begin{tabular}{c} VRPSD-PDC (reduce traveling costs and the number of required vehicles) \\ \end{tabular} & & self-generation & 60 & optimal restocking \\ \hline
21 & & minimize the total cost & self-generation & 20 & \\ \hline
22 & & minimize the total distance & self-generation & 20 & \\ \hline
23 & & minimize the sum of route completion times & \(/\)data.mendeley.com/datasets/ kxfckwdb9/draft7a=eb5 cc79-b4 & 200 & delivery robots \\ \hline
24 & & & & \\ \hline \hline \end{tabular}
\begin{tabular}{c} minimize distance, economic dimension cost, and environmental dimension cost \\ \end{tabular} & & five instances from [116] & 43 & \\ \hline
25 & duration minimizing & test instances adopted from [102] and newly generated instances & 100 & \\ \hline
26 & total travel and charging time & adapted Solomon instances & 36 & \\ \hline \hline \end{tabular}