Summary: The Algorithm in this paper is designed to find the shortest path in a network given time-dependent cost functions. It has the following features: it is recursive; it takes place both in a backward dynamic programming phase and in a forward evaluation phase; it does not need a time-grid such as in \textit{L. Cooke} and \textit{E. Halsey}, J. Math. Anal. Appl., 1966, 14, 493-498, and \textit{M. M. Kostreva} and \textit{M. M. Wiecek}, J. Math. Anal. Appl., 173, 289-308 (1993) `` Algorithm One''; it requires only boundedness (above and below) of the cost functions; it reduces to backward multi-objective dynamic programming if there are constant costs. This algorithm has been successfully applied to multi-stage decision problems where the costs are a function of the time when the decision is made. There are examples of further applications to tactical delay in production scheduling and to production control.