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One of the most popular tools used for risk assessment of marine machinery systems is Failure Mode and Effect Analysis (FMEA). With this analysis tool, risk is represented in the form of a risk priority number (RPN) which is computed by multiplying the severity rating (S) by the occurrence probability (O) and the detection rating (D) for all failure modes of the system. This conventional FMEA has been criticised as having several limitations such as inability to aggregate imprecise ratings of multiple experts and inability to incorporate more than three risk criteria. These challenges have been addressed in this paper by developing two novel methodologies for prioritising the risk of failure modes for marine machinery systems. The first methodology integrates an averaging technique with VIKOR (Vlsekriterijumska Optimizacija Ikompromisno Resenje, meaning: Multicriteria Optimisation and Compromise Solution). The second methodology integrates an averaging technique with the Compromise Programming (CP) technique. While the averaging technique is applied as a means of aggregating imprecise risk criteria ratings from multiple experts, VIKOR and CP are used in the ranking of risk of failure modes. The applicability and suitability of these methodologies for risk prioritisation is demonstrated using two case studies. [ABSTRACT FROM AUTHOR]