Motivated by the practical needs of simulation-based optimization, this paper considers a problem for selecting the best feasible design from a finite set of alternatives, subject to stochastic constraints given in several secondary objectives. We propose an efficient ranking and selection procedure that maximizes the accuracy of the selection under a limited simulation budget. The proposed procedure sequentially updates the simulation data of designs with a heuristic policy that allocates further simulation replications according to the evaluation results of data based on a statistical hypothesis test. Compared to recent studies, such as OCBA-CO and SCORE, this procedure can be more efficient when the simulation model involves large stochastic noise because its heuristic policy considers the precision of the sample mean ignored by the previous studies. Several experimental results of benchmarks demonstrate its improved efficiency, and a case study on the design of military network system shows its effectiveness for practical problems.