This paper addresses job shop scheduling problems with fuzzy processing time and fuzzy trapezoid or doublet due date. An efficient random key genetic algorithm (RKGA) is suggested to maximize the minimum agreement index and to minimize the maximum fuzzy completion time. In RKGA, a random key representation and a new decoding strategy are proposed and two-point crossover (TPX) and discrete crossover (DX) are considered. RKGA is applied to some fuzzy scheduling instances and performance analyses on random key representation, and the comparison between RKGA and other algorithms are done. Computation results validate the effectiveness of random key representation and the promising advantage of RKGA on fuzzy scheduling.