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We consider the problem of finding a system with the best primary performance measure among a finite number of simulated systems in the presence of subjective stochastic constraints on secondary performance measures. When no feasible system exists, the decision maker may be willing to relax some constraint thresholds. We take multiple threshold values for each constraint as a user’s input and propose indifference-zone procedures that perform the phases of feasibility check and selection-of-the-best sequentially or simultaneously. Given that there is no change in the underlying simulated systems, our procedures recycle simulation observations to conduct feasibility checks across all potential thresholds. We prove that the proposed procedures yield the best system in the most desirable feasible region possible with at least a pre-specified probability. Our experimental results show that our procedures perform well with respect to the number of observations required to make a decision, as compared with straight-forward procedures that repeatedly solve the problem for each set of constraint thresholds, and that our simultaneously-running procedure provides the best overall performance.