The Assignment problem is a fundamental and well-studied problem in the intersection of Social Choice, Computational Economics and Discrete Allocation. In the Assignment problem, a group of agents expresses preferences over a set of items, and the task is to find a pareto optimal allocation of items to agents. We introduce a generalized version of this problem, where each agent is equipped with multiple incomplete preference lists: each list (called a layer) is a ranking of items in a possibly different way according to a different criterion. We introduce the concept of global optimality, which extends the notion of pareto optimality to the multi-layered setting, and we focus on the problem of deciding whether a globally optimal assignment exists. We study this problem from the perspective of Parameterized Complexity: we consider several natural parameters such as the number of layers, the number of agents, the number of items, and the maximum length of a preference list. We present a comprehensive picture of the parameterized complexity of the problem with respect to these parameters.