An Optimal Selection Problem for a Sequence with a Random Number of Applicants Per Period

In the classical secretary problem, an employer interviews and rates applicants one at a time, and wants to stop at an optimal applicant. Such optimal stopping problems also arise in the selection of projects, military targets, and resource alternatives. We consider a generalization of this optimal stopping problem in which the employer wishes to hire a certain number of applicants, the total number of applicants is known, and the number of applicants per day is random. We also treat a variation in which a random number of applicants arrive randomly over time. We derive several properties of the optimal policy and the total expected reward obtainable from this policy. For an optimal policy, in order for an applicant to be selected by the decision maker, the applicant's value must be large in absolute magnitude and be high from among all the applicants.