The Multidimensional Assignment Problem (MAP) (abbreviated
-AP in the case of
dimensions) is an extension of the well-known assignment problem. The most studied case of MAP is 3-AP, though the problems with larger values of
have also a number of applications. We consider several known and new MAP local search heuristics for MAP as well as their combinations. Computational experiments with three instance families are provided and discussed. As a result, we select dominating local search heuristics. One of the most interesting conclusions is that combination of two heuristics may yield a superior heuristic with respect to both solution quality and the running time.