An optimal path of a moving vehicle on a sphere

We consider the problem of finding an optimal path of a moving vehicle, which during its journey provides a certain type of service to a group of existing facilities on a sphere. Service requests from each facility are assumed to follow a Poisson process and the conditional expected cost incurred by a service call is assumed to be proportional to the shortest arc distance between the facility and the vehicle's location at the time of the service request. This problem is formulated as a variational problem and solved by repeatedly applying a dynamic programming procedure to a sequence of staged networks that are constructed via perturbation.