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In this paper, an ordinal optimization based approach is proposed to solve for a good enough schedule that minimizes expected sum of storage expenses and tardiness penalties of stochastic classical job shop scheduling problem using limited computation time. The proposed approach consists of exploration and exploitation stage. The exploration stage uses a genetic algorithm to select a good candidate solution set, where the objective function is evaluated with an artificial neural network that is trained beforehand. The exploitation stage composes of multiple substages, which allocate the computing resource and budget by iteratively and adaptively selecting the candidate solutions. At each substage, remaining solutions are simulated and some of them are eliminated, and the solution obtained in the last substage is the good enough schedule that we seek. The proposed approach is applied to a SCJSSP with random processing time in truncated normal, uniform, and exponential distributions. The test results demonstrated that the obtaining good enough schedule is successful in the aspects of solution quality and computational efficiency.