Branch-Price-and-Cut algorithms for the team orienteering problem with interval-varying profits

This paper studies the team orienteering problem (TOP), wherein each vertex requires two visits, and the service profit of each vertex depends on the time interval between these two visits. Motivated by scenarios like perishable product delivery, home health care scheduling, and earth observation satellite scheduling, the paper addresses two variants: the periodic orienteering problem with a focus on the number of days (NoDs) and the team orienteering problem with attention to a combination of time windows (CoTWs). It develops mixed-integer linear programming (MILP) models for both variants and devises exact Branch-Price-and-Cut (BPC) algorithms tailored to their block diagonal structures. Furthermore, this paper proposes two strategies to improve algorithm performance. A simplification strategy streamlines the directed graph network by removing redundant vertices and arcs without compromising optimality, thereby accelerating the solution of pricing problems. Additionally, a matheuristic algorithm is proposed to obtain integer solutions quickly. Computational results demonstrate the effectiveness of these algorithms, showcasing their superiority over CPLEX. The BPC algorithm, coupled with the simplification strategy, exhibits an average computational time reduction of 70 % compared to the basic strategy. The effectiveness of the matheuristic algorithm is also confirmed. Finally, the findings presented herein offer valuable insights for refining and applying BPC algorithms.