Job shop scheduling with multi-objective has been extensively investigated; however, multi-objective stochastic job shop scheduling problem is seldom considered. In this paper, a simplified multi-objective genetic algorithm (SMGA) is proposed for the problem with exponential processing time. The objective is to minimize makespan and total tardiness ratio simultaneously. In SMGA, the chromosome of the problem is ordered operations list, an effective schedule building procedure is proposed, a novel crossover is used, and a simplified binary tournament selection and a simple external archive updating strategy are adopted. SMGA is finally tested on some benchmark problems and compared with some methods from literature. Computational results demonstrate that the good performance of SMGA on the problem.