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Two types of simulations (forward and backward simulations) and four types of bisimulations (forward, backward, forward–backward, and backward–forward bisimulations) between fuzzy automata have been introduced by Ćirić et al. (2012). If there is at least one simulation/bisimulation of some of these types between the given fuzzy automata, it has been proved that there is the greatest simulation/bisimulation of this kind. In the present work, for any of the above-mentioned types of simulations/bisimulations we provide an efficient algorithm for deciding whether there is a simulation/bisimulation of this type between the given fuzzy automata, and for computing the greatest one, whenever it exists. The algorithms are based on the method developed by Ignjatović  et al. (2010), which comes down to the computing of the greatest post-fixed point, contained in a given fuzzy relation, of an isotone function on the lattice of fuzzy relations. The corresponding algorithms are also provided for nondeterministic automata.