# The Vehicle Routing Problem: State-of-the-Art Classification and Review
## 4 Solutions for VRPs
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{2}{c}{**Solution**} & \multirow{2}{*}{**Operating Environment**} \\ \cline{2-2} \cline{4-4}  & \multicolumn{1}{c}{**Exact Methods**} & \multicolumn{1}{c}{**Heuristic Algorithms**} & \multicolumn{1}{c}{**Meta-Heuristic Algorithms**} & \multicolumn{1}{c}{**Operating Environment**} \\ \hline
43 &
\begin{tabular}{c} new two-index-based \\ mathematical formulation \\ \end{tabular} & & / \\ \hline
44 & & \
\begin{tabular}{c} NSGA-II as a static \\ optimizer when the \\ environment does not \\ change \\ \end{tabular} &
\begin{tabular}{c} \(16\) GB RAM, Intel Core i7-10700 \\ @2.9 GHz. \\ \end{tabular} \\ \hline
45 &
\begin{tabular}{c} First is the difficulty of \\ obtaining exact moment \\ measures for the ambiguity set \\ _Pi_ and second is when the \\ distribution function is \\ continuous \\ \end{tabular} & & / \\ \hline
46 & &
\begin{tabular}{c} improved differential \\ evolution (DE) algorithm \\ \end{tabular} & MATLAB R2014a, Windows 7 (x32) \\ \hline
47 & \
\begin{tabular}{c} heuristic dynamic \\ programming \\ \end{tabular} &
\begin{tabular}{c} Coded in C and C++, IntelXeon \\ E3-1226 v3 @3.30 GHz and \\ 16 GB RAM \\ \end{tabular} \\ \hline
48 & & \
\begin{tabular}{c} MCWS-LS heuristic, \\ S-ALNS algorithm with \\ SA \\ \end{tabular} &
\begin{tabular}{c} Coded in JAVA, computations \\ executed on a Dell XPS PC with a \\ 2.80 GHz Intel Xeon CPU \\ (E5-2680-V2) and 32 GB RAM \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Cont. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{2}{c}{**Solution**} & \multirow{2}{*}{**Operating Environment**} \\ \cline{2-2} \cline{4-4}  & \multicolumn{1}{c}{**Exact Methods**} & \multicolumn{1}{c}{**Heuristic Algorithms**} & \multicolumn{1}{c}{**Meta-Heuristic Algorithms**} & \multicolumn{1}{c}{**Operating Environment**} \\ \hline
1 & \
\begin{tabular}{c} Evolutionary algorithm \\ \end{tabular} & &
\begin{tabular}{c} MATLAB (2016a), Intel(R) CoreTMi7-7700K CPU \\ and 32 GB RAM, Windows 10 OS \\ \end{tabular} \\ \hline
2 & & \
\begin{tabular}{c} GA \\ \end{tabular} &
\begin{tabular}{c} Intel (R) Core i5-7600 @3.50 GHz with 8 GB RAM \\ equipped with an NVIDIA GeForce GTX 1060 \\ graphics card, This GPU has 1280 cores and its base \\ and boost clocks are 1506 MHz and 1708 MHz, \\ respectively, C++ CUDA 8.0 (V8.0.61) \\ \end{tabular} \\ \hline
3 &
\begin{tabular}{c} DTRC, LNS \\ \end{tabular} & & \\ \hline
4 & \
\begin{tabular}{c} Dynamic programming \\ algorithm and integer \\ program, ILNS algorithm \\ \end{tabular} & &
\begin{tabular}{c} Intel CoreTM i5 @3.40 GHz CPU with 8 GB RAM. \\ The branch-and-bound solver of CPLEXM 12.9.0 \\ was used as the exact solver and all other \\ algorithms were coded in MATLAB. When \\ necessary, CPLEXM is called from MATLAB. \\ \end{tabular} \\ \hline
5 & \
\begin{tabular}{c} ALNS algorithm, VND \\ algorithm \\ \end{tabular} &
\begin{tabular}{c} 3.60 GHz AMD Ryzen 7 3700X CPU with 32 GB \\ RAM, Windows 10 OS \\ \end{tabular} \\ \hline
6 &
\begin{tabular}{c} Improved artificial fish swarm \\ algorithm, push forward \\ insertion heuristic (PFIH) \\ \end{tabular} & / \\ \hline
7 &
\begin{tabular}{c} NSGA-II \\ \end{tabular} & / \\ \hline
8 &
\begin{tabular}{c} Proposed EVNS algorithm \\ \end{tabular} & / \\ \hline
9 & \
\begin{tabular}{c} Nearest-neighbor heuristic, \\ insertion heuristic \\ \end{tabular} &
\begin{tabular}{c} C language, desktop PC with an Intel Core i7-3820 \\ CPU @3.6 GHz and 32 GB RAM \\ \end{tabular} \\ \hline
10 &
\begin{tabular}{c} ALNS algorithm \\ \end{tabular} & & \\ \hline
11 & \
\begin{tabular}{c} Particle swarm \\ optimization (PSO), GA \\ \end{tabular} &
\begin{tabular}{c} Intel Core i7 @2.80 GHz, 8 GB RAM, model \\ implemented in CPLEX (version 12.6.2) with \\ C\# (VS2015) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Solutions to VRPs published in 2020.