# 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**} & \\ \cline{2-3}  & **Exact Methods** & **Heuristic Algorithms** & **Meta-Heuristic Algorithms** & **Operating Environment** \\ \hline \multirow{2}{*}{1} & & & C\# on a computer with an Intel I7 CPU @2.69 GHz & C\# on a computer with an Intel I7 CPU @2.69 GHz \\  & & & and 16 GB RAM. Gurobi 8.0.0 chosen as an MIP solver & MIP solver \\ \hline \multirow{2}{*}{2} & & C++, executed on a cluster of 27 machines, each with two Intel(R) Xeon(R) X5675 CPUs @3.07 GHz & \\  & & method & with 96 GB RAM running on Linux. Each machine had 12 cores and each instance was executed using a single thread & \\ \hline \multirow{2}{*}{3} & & & Intel\({}^{\text{0}}\) Xeon® Gold 6126 CPUs and 16 GB RAM, algorithms in Java SE 8 and Gurobi Optimizer 8.1.0 for solving MILPs \\ \hline \multirow{2}{*}{4} & Path-based exact solution & & Intel i7-5500U @2.4 GHz with 16 GB RAM, coded in Java \\ \hline \multirow{2}{*}{5} & & improved ACO algorithm & / \\ \hline \multirow{2}{*}{6} & Bellman–Ford algorithm & / \\ \hline \multirow{6}{*}{7} & & Coded in Fortran 77 and run on a single thread of a 3.6 GHz Intel i7-4790 CPU with 32 GB RAM. Relies on CPLEX 12.5.1 for the resolution of linear programs and some integer sub-problems. \\ \cline{1-1}  & & LNS & Metaheuristic was coded in Java (IRE 1.8.0-151), and run on a single thread of a 3.4 GHz Intel i7-3770 CPU & \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Solutions to VRPs published in 2019. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{2}{c}{**Solution**} & \\ \cline{2-3}  & **Exact Methods** & **Heuristic Algorithms** & **Meta-Heuristic Algorithms** & **Operating Environment** \\ \hline \multirow{2}{*}{12} & & & Coded in GAMS 23.9 and solved using the Gurobi 7.5 solver hosted on the NEOS server. The server runs on an Intel XeonE5-2430 @2.2 GHz with 3 GB RAM. ALNS algorithm coded in Python and tested on a PC running a 3.6 GHz Intel Core-i7-7700 CPU with 16 GB RAM & \\ \hline \multirow{2}{*}{13} & VND tabu search algorithm & & Algorithm coded in C++ and run on a 3.30 GHz \\  & with holding list & & Intel\({}^{\text{0}}\) Core(TM) i5-2400 CPU \\ \hline \multirow{2}{*}{14} & & improved ant colony optimization (ACO) & Coded in Visual C++ and implemented on an Intel Core i5 CPU @3 GHz with 8 GB RAM \\ \hline \multirow{2}{*}{15} & IVNS algorithm, VND procedure & & Coded in C++ and run on a Linux cluster system with an AMD Optron 4184 CPU (2.8 GHz and 2 GB RAM) running Ubuntu 12.04. For the general CPLEX solver, the latest version of 12.6 was used \\ \hline \multirow{2}{*}{16} & & constructive heuristic based on LNS & Programmed in C\# and run on an Intel® CoreTM \\  & & meta-heuristic & i7-6500U CPU @2.5 GHz with 16 GB RAM \\ \hline \multirow{2}{*}{17} & PFHH method, neighborhood search, tabu search & & \\ \hline \multirow{2}{*}{18} & & FA based on coordinates & Gurobi 4.5.1 with Python 3.0 on an Intel(R) Core(TM) i7-7700 HQ CPU @2.80 GHz with 8 GB RAM. \\ \hline \multirow{2}{*}{19} & & & CPLEX 12.9, coded in C++. The code for the heuristic and exact method was executed on a Windows 8 computer configured with an Intel(R) Core(TM) 3.2 GHz CPU with 8 GB RAM \\ \hline \multirow{2}{*}{20} & & GA & MatLab on a 2.10 GHz PC, with 8 GBytes of RAM \\ \hline \hline \end{tabular}
\end{table} 
Table 12: \(Cont.\)As shown in the tables above. Heuristic algorithms and meta-heuristic algorithms are still the mainstream solution methods, although branch and X methods will continue to increase in popularity in 2021. As mentioned previously, with the rapid growth in the processing speed and memory capacity of computers (i.e., operating environments), more complex instances of the VRP can be solved.