# The Vehicle Routing Problem: State-of-the-Art Classification and Review
## 3 VRP and Its Variants
### VRP Variants
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Multi-Object** & **Single-Object** & **Dataset** & **Max Nodes** & **Other Settings** \\ \hline
1 & & minimize total cost & self-generation & 13 & 1 \\ \hline
2 & & minimize total cost & test instances adopted from [135] & 320 & 2 \\ \hline
3 & & minimize the makespan through constraints & test instances adopted from [143] & 100 & 3 \\ \hline
4 & & minimize the total & test instances adopted from [145, 115] & 100 & 4 \\ \hline
5 & revenue maximization, and travel time, emission, and cost minimization & & self-generation & 35 & 5 \\ \hline
6 & stage one: path minimization; stage two: route minimization & & self-generation & 20 & 6 \\ for total energy consumption & & & & \\ \hline
7 & & & sets two and three from [145], set five from [146], and set six from [147]; charging stations follow the guidelines from [129] & 200 & 7 \\  & & (instances of the electric VRP with time windows and recharging stations) and [103] & & & \\ \hline
8 & & minimize the total time required to complete monitoring tasks & self-generation & 724 & 8 \\ \hline
9 & & maximize the net acquired profit & from the traditional VRP instances in [112] & 199 & 9 \\ \hline
10 & & minimize total cost & VRPTW instances presented by [PERSON] et al. [129] & 100 & 10 \\ \hline
11 & & minimize total cost & self-generation & 200 & 11 \\ \hline
12 & & minimize the number of routes & instance set from [148], MDVRP experiments using the benchmark set from [134] & 1000 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Model objectives of VRPs published in 2019. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Multi-Object** & **Single-Object** & **Dataset** & **Max Nodes** & **Other Settings** \\ \hline
13 & & maximize the total & & test instances adopted & based on dataset & limited number of resources \\ \hline
13 & & minimize the total cost & & test instances adopted from [129] & 100 & \\ \hline
13 & & minimize the total cost & & self-generation & 200 & \\ \hline
13 & & minimize fixed and variable vehicle costs & test instances adopted from [137, 138, 107] & 100 & \\ \hline
14 & & minimize the total cost & & & \\ \hline
15 & & minimize the total cost & & & \\ \hline
16 & & minimize fixed and variable vehicle costs & test instances adopted from [138, 107, 139, 140] & 100 & \\ \hline
17 & & minimize the total cost & test instances adopted from [141] & based on dataset & \\ \hline
18 & & minimize the total cost & test instances adopted from [107] & 200 & \\ \hline
19 & & minimize the total cost & test instances adopted from [107] & 100 & \\ \hline
20 & & minimize the total cost & test instances adopted from [138, 142] & 100 & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: _Cont._As shown in the tables above, there are different main model objectives for different years. The results are summarized in Figure 1.  
One can see that the time window still occupies a large proportion of model objectives and is the mainstream of current research on the VRP and its variants. This trend is closely related to the concept of the \"to C\" distribution, where customers focus on service satisfaction. There have been various extensions of the VRP, including the VRPTW and time-dependent problems such as those discussed in [17; 24; 41; 75; 89]. Additional research has focused on heterogeneous vehicle problems that are closely related to real-life vehicle applications. With the increasing focus on environmental protection, electric vehicle distribution has also gradually become a mainstream research topic. Relevant research can be found in [49; 72; 81; 89; 116]. Single-objective models still occupy a certain research space, where the objective value setting is still largely based on cost metrics (e.g., cost, distance, and CO\({}_{2}\) emission). However, unlike cost metrics in past research, the costs in the  
Figure 1: Different model objectives in different years.