In this paper we present new Jahn-Graef-Younes type algorithms for solving discrete vector optimization problems. In order to determine all minimal elements of a finite set with respect to an ordering cone, the original approach proposed by Jahn in 2006 (known as the Jahn-Graef-Younes method) consists of a forward iteration (Graef-Younes method), followed by a backward iteration. Our methods involve additional sorting procedures based on scalar cone-monotone functions. In particular, we analyze the case where the ordering cone is polyhedral. Computational results, obtained in MATLAB, allow us to compare our new algorithms with the original Jahn-Graef-Younes method.