The clustered prize-collecting arc routing problem

Prize-collecting arc routing problems are arc routing problems where, in addition to the cost function, there is a profit function on the edges that must only be taken into account the first time that an edge is traversed. This work presents the clustered prize-collecting arc routing problem where there are clusters of arcs and it is required that all or none of the edges of a cluster be serviced. The paper studies properties and dominance conditions used for formulating the problem as a linear integer program. An exact algorithm for finding an optimal solution to the problem is also proposed. At the root node of the enumeration tree, the algorithm generates upper and lower bounds obtained from solving an iterative linear programming-based algorithm in which violated cuts are generated when possible. A simple heuristic that generates feasible solutions provides lower bounds at each iteration. The numerical results from a series of computational experiments with various types of instances illustrate the good behavior of the algorithm. Over 75% of the instances were solved at the root node, and the remaining instances were solved with a small additional computational effort.