Chance-constrained selection of the best: An indifference-zone-free procedure

This paper considers a variant of the selection of the best problem, namely the chance-constrained selection of the best (CCSB) problem, where the goal is to select the best alternative among a finite set of alternatives based on a primary performance measure, subject to probabilistic constraints on one or more secondary performance measures. We first focus on the case with a single secondary performance measure and develop an indifference-zone-free procedure that guarantees the probability of correct selection. This is achieved by investigating the marginal probability of falsely eliminating the true best alternative at each time point where the procedure conducts optimality checks between different alternatives based on the primary performance measure and feasibility checks for each individual alternative based on the secondary performance measure. We show that this procedure is efficient in terms of the growth rates of the decision boundaries for the continuation regions, which are used to conduct optimality and feasibility checks. We further demonstrate that the procedure can be easily extended to handle the general CCSB problem with multiple secondary performance measures. Moreover, with minor adjustments, we improve the growth rates of the decision boundaries to reach their theoretical lower bounds, significantly enhancing the finite-sample performance of the procedure. Comprehensive numerical experiments confirm the competitiveness and efficiency of our procedures.