A flow-based model for the multivehicle covering tour problem with route balancing

This paper introduces a new flow-based mathematical formulation for the multivehicle covering tour problem. This problem determines a set of vehicle routes for the available fleet; therefore, the total distance traveled is minimal. In the proposed mathematical formulation, each vertex represents a location of interest, and these vertices belong to three sets: vertices that can be visited; vertices that must be visited; and vertices that must be covered, in the sense that each vertex must be close to a visited vertex in a route. The proposed formulation aims to find solutions with balanced routes (all the routes with an equal or similar number of visited nodes) and extends the previous formulations proposed in the literature by considering a two-index vehicle flow formulation with a decision variable to find the balanced routes. It constitutes an NP-hard integer programming problem based on a two-commodity flow model, which is solved by applying the branch-and-cut and biased random-key genetic algorithms. This paper reports computational results for some adapted instances from TSPLIB to evaluate the efficiency of the proposed model. The computational results validate the branch-and-cut and genetic algorithms, whereby the latter performs better in large-size instances with reduced computational runtime. Finally, the used approaches obtained competitive solutions compared to the literature.