The Allocated Maximal Backup Covering Tour Problem with Minimum Building Cost

This study addresses the Allocated Maximal Backup Covering Tour Problem (AMBCTP) that is a generalization of the Covering Tour Problem (CTP). This problem is defined on an undirected graph , where W is a set of vertices that must be collectively covered by a vehicle. The AMBCTP consist of determining a minimum length vehicle route on a subset of V, subject to side constraints, such that every vertex of W is within a pre-specified distance from the route. Maximizing number of vertices of W set which are covered for second or more times is another objective in this problem. Moreover, allocation cost of the every vertex of W to one vertex of V which is belonging to the tour is minimized. Transmission vehicle from each city, in health care teams example that provided by Current and Schilling (1994) for this problem, requires to build a clinic in it, therefore we considered a fixed and variable cost (i.e. building cost) for visited cities, that must be minimized. Mathematical formulation of the AMBCTP, that is a multi-objective problem, is proposed. We used a powerful Multi-objective Decision Making (MODM) method for optimizing it. Finally a numerical example is provided to demonstrate the validity of the model.