A multi-objective shortest path problem

We study a multi-objective shortest path problem. We formulate a multi-objective Markov decision process model for the problem and solve it by the policy improvement procedure. We first show by counter-examples the differences between the multi-objective shortest path problem and the single-objective one. We next introduce "locally efficient policies" and present an algorithm for finding all efficient paths from all nodes to the destination. Moreover, as an application, we consider a shortest path problem with a constraint. Last, the algorithm is illustrated by a numerical example.