Mining method selection (MMS) is one of the core contents in mining design. As the influencing factors of MMS are in general of ambiguity or uncertainty, MMS can therefore be deemed as a complex fuzzy multi-criteria decision making problem. To resolve this problem, an extended multi-objective optimization by ratio analysis plus the full multiplicative form (MULTIMOORA) approach is studied in this paper. Three major contributions of this study are included. At first, comparative importance is crisp number in most stepwise weight assessment ratio analysis (SWARA) method, while linguistic terms are utilized to determine the criteria weights in the improved SWARA. Then, several Heronian mean (HM) operators related to linguistic neutrosophic numbers (LNNs) are defined to cope with evaluating indicators’ interrelations. Special cases are discussed, and primary properties are proved as well. At last, in combination with improved SWARA and HM operators, a novel decision making method, extended MULTIMOORA approach based on LNNs, is proposed. Applying this new approach to selecting the optimal mining method through an example, the feasibility is indicated. In addition, analyses of the strengths of the presented method are performed in comparison with other existing approaches.