In group decision-making (GDM) model, experts often evaluate their opinion by using triangular fuzzy numbers. The preference relations with triangular fuzzy numbers are in consistent in nature, so we turned to neutrosophic in this paper. It is very important to take into account consistency of expert opinion and consensus degree in GDM. In order to distinguish the typical consistency, the concept of additive approximation consistency is proposed for triangular neutrosophic additive reciprocal matrices. The properties of triangular neutrosophic additive reciprocal matrices with additive approximation consistency are studied in detail. Second, by using (n- 1) restricted preference values, a triangular neutrosophic additive reciprocal preference relation with additive approximation consistency is constructed. The differences among expert’s opinions are measured using consensus degree. For generating a collective triangular neutrosophic additive reciprocal matrix with additive approximation consistency, the neutrosophic triangular weighted aggregation operator is used. Finally, a novel algorithm for the group decision-making problem with triangular neutrosophic additive reciprocal preference relations is presented. A numerical example is carried out to illustrate the proposed definitions and algorithm. © 2017, Springer-Verlag GmbH Germany.