The periodic rural postman problem with irregular services on mixed graphs

In this paper, we deal with an extension of the rural postman problem in which some links of a mixed graph must be traversed a given number of times over a time horizon. These links represent entities that must be serviced a specified number of times in some subsets of days (or periods) of the time horizon. The aim is to design a set of minimum-cost tours, one for each day/period of the time horizon, that satisfy the service requirements. We refer to this problem as the periodic rural postman problem with irregular services (PRPP–IS). Some practical applications of the problem can be found in road maintenance operations and road network surveillance, for example. In order to solve the PRPP–IS, we propose a mathematical model and a branch-and-cut algorithm. As far as we know, this is the first exact method devised for a periodic arc routing problem. In the solution framework, constraints ensuring connectivity and other valid inequalities are identified by using specific separation procedures. Most valid inequalities consider the particular nature of the PRPP–IS. We show the effectiveness of the solution approach through an extensive experimental phase.