In risk assessment, the decision is generally made according to the most probable situation. However, the fact that there are rarely realized outcomes and that the degree of effectiveness of the precaution applied against any risk is not measured are the indetermanicies encountered in risk assessment. In the literature, there are various risk assesment approaches based on fuzzy logic to handle the uncertainties in specialist evaluation. Therefore, in this study, a neutrophic set based Failure Mode and Effect Analysis (FMEA) method, includes probability, severity and detectability parameters, is proposed for the first time in the literature to handle inconsistencies, subjectivities and indecisions in the risk assessment process. In the proposed method, membership degrees of truth (T), indetarmancy (I) and falsity (F) used in neutrosophic set definition are preferred in the probability, severity and detectability parameters of FMEA method, while different dimensions of expert assessment of a risk and uncertanities regarding risk assessment is taken into account. Mamdani fuzzy inference system is used instead of multiplication in determining risk depending on FMEA parameters. The risk assessment method developed in the study was applied in the chemical industry. The results obtained from the application were compared with classical and fuzzy FMEA. The results showed the success of the proposed method in considering the instabilities and subjectivity-related uncertainties in risk assessment compared to existing methods.