Hierarchical classes models are quasi-order retaining Boolean decomposition models for N-way N-mode binary data. To fit these models to data, rationally started alternating least squares (or, equivalently, alternating least absolute deviations) algorithms have been proposed. Extensive simulation studies showed that these algorithms succeed quite well in recovering the underlying truth but frequently end in a local minimum. In this paper we evaluate whether or not this local minimum problem can be mitigated by means of two common strategies for avoiding local minima in combinatorial data analysis: simulated annealing (SA) and use of a multistart procedure. In particular, we propose a generic SA algorithm for hierarchical classes analysis and three different types of random starts. The effectiveness of the SA algorithm and the random starts is evaluated by reanalyzing data sets of previous simulation studies. The reported results support the use of the proposed SA algorithm in combination with a random multistart procedure, regardless of the properties of the data set under study.