README.md

Capacitated Vehicle routing problem (CVRP) Branch-and-cut algorithm using the CVRPSEP package

A C++ implementation of the CVRP undirected 3-index model [1] branch-and-cut algorithm using GUROBI's API and CVRPSEP package [2].

Note: in this code it is used this fork of the CVRPSEP package which fixes some minor issues. Please refer to [3] for further details.

CVRP undirected 3-index formulation

Let $G = (V, E)$ be an undirected graph, $K$ a set of vehicles with capacity of $Q$, and let vertex 0 be the depot and vertices $V' = V \setminus {0}$ be the customers. Consider that there is a demand $d_i$ for each $i \in V'$. The undirected three-index CVRP model can be defined as below [1]:

$$ \begin{align} \min & \sum\limits_{(i, j) \in E} c_{ij} \sum\limits_{k \in K} x_{ij}^k \\ \text{subject to} & \\ & \sum\limits_{k \in K} y_{i}^k = 1, & \forall i \in V', \\ & \sum\limits_{k \in K} y_{0}^k = K, \\ & \sum\limits_{(i, j) \in E} x_{ij}^k = 2y_i^k, & \forall i \in V, \forall k \in K, \\ & \sum\limits_{i \in V} d_i y_{i}^k \leq Q, & \forall k \in K,\\ & \sum\limits_{(i, j) \in S} x_{ij}^k \leq |S| - 1, & \forall S \subseteq V', |S| \geq 2, \forall k \in K,\\ & y_{i}^{k} \in {0, 1}, & \forall i \in V, \forall k \in K, \\ & x_{ij}^{k} \in {0, 1}, & \forall i, j \in V', \forall k \in K,\\ & x_{0j}^{k} \in {0, 1 , 2}, & \forall j \in V', \forall k \in K. \end{align} $$

Prerequisites

• CMake.

• C++17 compiler or an early version.

• Boost library (program_options)

• GUROBI solver (9 or an early version). Academics can obtain it via this link.

Compile and run instructions

Go to the source code folder and to compile type:

cmake -H. -Bbuild -DCMAKE_BUILD_TYPE=Debug
cmake --build build

for the debug version or

cmake -H. -Bbuild
cmake --build build

for the release version.

To run with a configuration file:

$ ./build/bc_cvrp -f [configuration file path]

See the "example.cfg" file at the "input" folder for an example of the input configuration file and

$ ./build/bc_cvrp --help

to see usage.

References

[1] P. Toth and D. Vigo. The Vehicle Routing Problem, Discrete Mathematics and Applications, SIAM, 2002

[2] J. Lysgaard, A.N. Letchford and R.W. Eglese. A New Branch-and-Cut Algorithm for the Capacitated Vehicle Routing Problem, Mathematical Programming, vol. 100 (2), pp. 423-445

[3] CVRPSEP package SAS software repository.