The scheduling problem with fuzzy processing times and fuzzy due dates are concerned in this paper. The concept of tardiness is much more flexibly interpreted based on the concepts of possibility and necessity measures in the fuzzy sets theory. Certain types of the objective function are taken into account through the different combinations of possibility and necessity measures. We turn these objective functions into fuzzy-valued functions. To obtain an optimal schedule which minimizes these fuzzy-valued objective functions, the ant colony algorithm is invoked to solve the optimal problem. One numerical example is also provided to clarify the discussion by using the commercial software Matlab.